Chemisorption effects on the thin-film conductivity

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Surface Science 277 (1992) 429-441 North-Holland

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science

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Chemisorption

effects on the thin-film conductivity

H. Geistlinger Technische Hochschule Leipzig, Fachbereich Automatisierung, Postfach 66, D(O)-7030 Leipzig, Germany

Received 31 January 1992; accepted for publication 2 June 1992

An investigation of chemisorption effects on the electrical conductivity of thin, wide bandgap semiconductors is presented in the framework of the Volkenstein model. The controversially discussed problem of a neutral, weak-chemisorbed state is reconsidered on the basis of the one-electron theory of chemisorption (considered by Einstein and Schrieffer). An analytically tractable variational method is developed for the ground state in lattice space. It turns out that in case of atop-anion binding a stable, weak-chemisorbed state can arise. For the case of acceptor-like chemisorption the pressure dependence of the thin-film conductivity is derived, solving selfconsistently the one-dimensional Poisson equation. In a wide pressure region the conductivity shows a power-law behaviour. It is found that the power is determined by different, fundamental parameters and cannot be expressed by only one parameter, e.g., the constant, stochiometric exponent in the mass-action law. Furthermore, the theoretical values of the power vary between 0 and 1 in agreement with the variety of experimental results.

1. Introduction

The influence of chemisorption on the electrical conductivity of thin-film semiconductors, especially compound metal-oxide semiconductors (MO-SC) with wide bandgap (> 2 eV>, is the subject of numerous experimental and theoretical studies. The investigation of catalytic and electronic properties of MO-SC surfaces is a basic problem in the field of heterogeneous catalysis and of semiconductor surface physics. The microscopic understanding of charge transfer processes at the phase boundary is of central importance in order to support a systematic development of reversible and selective thin-film gas sensors [1,2]. Thin MO-SC films (e.g., SnO,, ZnO, TiO,, Ga,O,), which have been produced by PVDtechniques (electron-beam, reactive sputtering and thermal evaporation) [3-111 and CVD-techniques [ll] exhibit in general a polycrystalline structure. Unfortunately, all the specific features of the electronic, catalytic and geometric properties of polycrystalline semiconductors (grain boundary potential, activated conductivity, field effect at the grain-grain junctions, overlapping 0039-6028/92/$05.00

between the surface and grain boundary potentials, two-dimensional geometry, etc.) are neglected in the majority of theoretical work dealing with electronic and adsorptive phenomena on semiconductor surfaces (see ref. [12], p. 81). The complexity of the problem is illustrated by a schematic representation of the most important electronic features of the conductivity, which can be affected by chemisorption (fig. 1): electronic transfer of delocalized conduction band electrons to localized surface states and vice versa cl), changing of grain boundary potentials (2) and/or contact potentials (31, changing of surface-defect (4) or bulk-defect concentration in the high-temperature region (T > 800 K). Most of the empirical work on chemisorption effects on semiconductors, which focuses on application for thin-film gas sensors or MO catalysts, chose theoretical interpretations of the experimental data starting from the hypothesis that chemical interactions between the ambient gas phase and the semiconductor surface are accompanied by electronic exchange reactions. The presence of chemisorbed atoms or molecules is represented by a localized term within the energy

0 1992 - Elsevier Science Publishers B.V. All rights reserved

H. Geistlinger

430

o-o

0

/ Chemisorption

free particles

adsorbed

effects on the thin-film conductivity

the partial pressure px of the gas phase with the concentration of physisorbed species [Xphys]; (2) Applying the mass-action law to the catalytic-surface reaction

particles a ’

porous metalsata

surface

bulk

Fig. 1. Schematic presentation of the gas/solid interactions of a semiconducting, polycrystalline MO film. Chemisorption changes the surface potential (0, the grain boundary (2) and contact potential (3) by electron transfer. At higher temperatures (T > 800 K) bulk-defect diffusion will change the surface (4) and bulk-defect concentrations.

gap of the semiconductor, whose occupation probability is given by the Fermi-Dirac distribution. Therefore, chemisorption leads to a charge displacement in the phase-boundary region and according to the solution of the Poisson equation to a band bending in the surface region. The connection between partial pressure of the gas phase and the concentration of chemisorbed species is interposed by physical adsorption, considering the “kinetic” interaction between the gas phase and the physisorbed state, where the physisorbed state acts as the precursor state of the chemisorbed state. This picture implicates that physisorbed and chemisorbed states are states to one and the same electron system of a stable lattice ions-adion configuration (physisorbed state - unoccupied state, chemisorbed state - occupied state); hence their occupation probability is determined by the Fermi-Dirac distribution [13]. (In this paper, we will call this heuristic picture the charge-transfer model (CTM).) Clifford [14] has consequently transformed this “philosophy” of CTM into a theoretical model: (1) Proposing a kinetic equation, which connects

phys +

e-H

X&,.

For the first time Clifford succeeded in describing from a uniform point of view the numerous power laws for the conductivity, u apxm, which were experimentally observed, without involving a priori a multitude of catalytic reactions for the same adsorbate-adsorbent system. As already shown by Volkenstein [15], this CTM is inconsistent from the quantum-chemical point of view, since physisorbed and chemisorbed states are states of different electron systems, which belong to different stable lattice ions-adion configurations; this inconsistency will be elucidated in detail in section 2 of this paper. Based on the phenomenological electron theory of chemisorption developed by Volkenstein (noted as Volkenstein model (VM)), which is at present widely accepted in the theory of heterogeneous catalysis [ 121 and semiconductor surface physics [16], we study in this paper for the first time the exactly soluble case of a homogeneous, crystalline thin film (wide bandgap > 2 eV> and calculate the pressure dependence of the conductivity, a(p). In a forthcoming paper we will consider the influence of an inner grain boundary potential/ contact potential on chemisorption (see fig. 11, which is from the mathematical point of view rather complicated, since one has to solve selfconsistently the Poisson equation for a two-dimensional geometry [34]. Our paper is organized as follows: In section 2.1 we discuss qualitatively the problem of physical and chemical adsorption from a quantumchemical point of view. In section 2.2 we show by a variational treatment of the ground state of the total Hamiltonian of the one-electron problem that the neutral, weak-chemisorbed state is a stable state. Then we derive the Volkenstein isotherm (section 3) and solve numerically the one-dimensional Poisson equation with respect to the fundamental parameter 5 = D/L, (D - film thickness, L, - Debye length) (section 4). For numeri-

H. Geistlinger / Chemisorption effects on the thin-film conducticity

cal demonstration of the proposed theory we compare the theoretical curves with experimental data of oxygen chemisorption on thin films of ZnO and CdS which both are wide band gap semiconductors (section 5).

2. Qualitative discussion total Hamiltonian

of the spectrum

adsorbent-lattice

surface

431

adatom

of the

2.1. Physical and chemical adrorption

It seems to be worthwhile to point out the notions “physisorbed” and “chemisorbed” species, since in the literature these notions are sometimes attributed to the strength of the binding and sometimes to particular types of adsorptive interactions. In general one distinguishes between two types of adsorption with respect to the underlying adsorptive forces - physical and chemical adsorption. Physisorption is caused by dispersion forces, forces of electrostatic nature and electrical image forces, while chemisorption is based on the stronger covalent forces (overlapping between the adsorbateand adsorbent-wavefunctions) and hence is connected with a partial electron transfer between adsorbent and adsorbate. Starting from the physical picture that physical adsorption arises at wide distances, R, from the adsorbent (R z=-lattice constant), the overlapping between the Bloch function, @&r), of the conduction-band electron and the wavefunction, xa(rZ), of the adelectron tends to zero and therefore it is sufficient to restrict the perturbation theory to the second-order approximation of the dipole-dipole interaction [17]. This leads to the attractive van der Waals interaction (London dispersion interaction) A@‘=

_-.

const R6

’

hence a particle from the gas phase is attracted to the semiconductor surface. Whether the adparticle further approaches to the surface, depends on the “quantum-mechanical nature” (symmetry and spin direction of the Bloch electron or of the localized electron of a biographic surface state).

Fig. 2. Schematic presentation: for two different adsorption sites - Xchem and Xphys - the adsorption curves, W(R), and the overlapping between the Bloch function, &(r,), and the adelectron wavefunction, xa(rZ), of an univalent adatom are shown. The strong covalent interaction between the chemisorption site Xchem and the adatom leads to a hybrid wavefunction, Nr,, r2).

In that case of repulsive forces (for instance: parallel spin direction or a short-range, repulsive potential of a surface defect) acting on the adpartitle, a weak minimum in the curve of the potential energy, W(R), can arise; the adparticle will be physisorbed at Rphys (see fig. 2). On the other hand, if the “nature” of the adsorption site yields stronger attractive forces by a further approach (for instance: antiparallel spin direction or short-range, attractive potential of a surface defect), it will come to a strong overlapping and hence to a covalent, chemical binding; hence the adparticle will be chemisorbed at Rehem (see fig. 2). The different binding conditions (minimum in the potential curves) produced by the different extent of overlapping of the wavefunctions is represented schematically in fig. 2. From fig. 2 it is evident, that physical and chemical adsorption belong to different adsorption curves, W(R), and therefore represent electron states to different stable (rigid) lattice ionsadion configurations; it turns out, that there is no correlation between physisorbed and chemisorbed states, given by the same Fermi-Dirac distribution [19]. Of course, there is a “kinetic” interaction between physisorbed and chemisorbed states, similar to that between the gas phase and the

H. Geistlinger / Chemisorption effects on the thin-film conductivi@

432

0

-Z

Fig. 3. Energy diagram of the chemisorbed, univalent adatom proposed by Volkenstein: The neutral, weak-chemisorbed state Ei and the strong-chemisorbed state EL; a free conduction-band electron (so-called “free valence” of the surface) will be trapped by the weak-chemisorbed adatom.

chemisorbed species. Throughout the subsequent considerations we suppose that physisorption sites and chemisorption sites are quasi-isolated in the quantum-statistical sense and we take into account only the “kinetic” interaction between the gas phase and the chemisorption sites. (x - electron affinity, t,, - centre of the conduction band.) 2.2. Hamiltonian and energy functional of a single ada tom

For an univalent adatom Volkenstein proposed in analogy to the Hl and H, binding the following physical picture of the chemisorption process at the semiconductor surface: (i) One-electron problem. The single adelectron can promote a weak binding of the adatom to the surface atom or surface impurity (see fig. 3, E,O). (ii) Two-electron problem. The singlet-pairing of a conduction-band electron and the adelectron leads to a strong binding and a charge transfer from the semiconductor to the adatom occurs, if one considers acceptor-like chemisorption (see fig. 3, El). There is a controversy in literature about the existence of a neutral, weak-chemisorbed state [15,20]. Because the basic idea in the VM is introducing a neutral form of chemisorption, we will show by a quantum-mechanical, variational treatment of the one-electron problem that in the spectrum of the total Hamiltonian (adsorbent + adatom) a stable, weak-chemisorbed state can arise.

Since we are only interested in demonstrating the existence of the weak-chemisorbed state and the qualitative dependence on the semiconductor-bulk parameters, we have allowed ourselves a number of simplyfying assumptions. We deal with the (100) surface of a simple cubic, semi-infinite lattice and calculate the ground state energy of the atop binding (fig. 4 (1 and 2)) by a variational treatment in tight-binding approximation [21]. This one-electron problem of chemisorption is described by the Hamiltonian:

where V,attice(r) is the periodic potential of the lattice ions, Wr -d) the Coulomb potential of the adion and U(d) the electrostatic potential of the adion and the lattice ions. In order to calculate U(d), one has to carry out a Madelung summation over the semi-infinite lattice assuming that the valence-band electrons are localized at the anions. We approximate this Madelung sum by an effective-continuum approximation using a screened Coulomb interaction between the adsorption centre and the adion. Since we are interested in heteropolar semiconductor surfaces (for instance the ZnO(lOi0) surface with the ionicity fi = 0.69, we have to consider the atop-cation and the atop-anion binding (see fig. 4) (1 and 2)). In order to calculate the energy function of the total Hamiltonian (1) we make for the trial function the ansatz:

I$>=xla)+

Cci(y2)IiY,

(2)

Fig. 4. Diagram illustrating a possible system of surface-binding types for an ionic surface of a simple cubic lattice: atopcation binding Cl), atop-anion binding (2), bridge binding (3) and centered binding (4).

433

H. Geistlinger / Chemisorption effects on the thin-film conducticity

where I a> are the adelectron-wavefunctions, I i) the Wannier functions of the conduction band to the lattice vector Rj (isum over the semi-infinite lattice) and x and y2 are the variational parameters. Using the tight-binding approximation of nearest neighbours (TB) and taking into account the transfer amplitudes (a I W IO)” and (a I H, IO>“,we obtain for the energy functional in atomic units (length unit a, = 0.0506 nm, energy unit 2 Ry = 27.211 eV; details of the calculation see Appendix 1) 8(x,

Y2> 4

(The terms T, f< y2) and cOu( y2, d) are calculated in Appendix 1). Inserting eq. (4a) into eq. (4b), it is easy to solve the nonlinear equation for y2, then the identity in eq. (4a) determines x. Depending on the distance d one gets the pair (X, T2) satisfying the variational equations.

2.3. Discussion A binding of the adatom to the surface will only arise, if A8=8(x,

2

+(1-x’)

COU(Y27

-x+-J&

Er

eff

i

4 I

where x is the electron affinity of the semiconductor, m,, the effective mass of the conduction band and I E&co) I the ionisation energy of the isolated adatom. We note that the transfer amplitudes do not depend on the variational parameters, but on the surface-adion distance d and vanish for d + 03. In this limit case one gets x(d + m) = 1 and the energy functional (3) gives E&w). The variation of the energy functional (3) leads to: &= X

=p+/T+f,

(da)

(4b) where P(Y~) =

C,/2T,

I-L = 2

C,=E,(m)+X-*+ eff

T=(uIWIO)~+(~IH~IO)~.

(5)

meff/mo7 COU(Y,, Er

4 ’

(6)

(7)

y2, d) -E,(m)

~0,

(8)

(binding condition). Physically, one would expect that for p = ,y/ I E&w) I = 1 the delocalization of the adelectron will decrease its kinetic energy and a stable binding can arise; this delocahzation effect is the reason for the Hi binding. Now, we discuss the numerical solutions of the variational equations for the two interesting cases: (i) atop-cation binding (fig. 4 (1)) and (ii) atop-anion binding (fig. 4 (2)); using typical semiconductor-bulk parameters: electron affinity x = 4 eV, the centre of the conduction band t&, = - 3 eV, lattice constant a = 0.5 nm, meff = 0.2m, and E, = 10. (i) Atop-cation binding. Only in the interval 1.2 5 p 5 1.5 there exist solutions satisfying the binding condition (8). One obtains the absolute energy minimum at d + m, AZ( d + 03) = I E,(m) I -x x2(d +@J) =o

(p=

1.2).

= -0.061

eV, (9)

The physical meaning of eq. (9) is that the adatom loses its electron to the semiconductor (x2 is the probability to find the electron at the adion), since the macro-potential x > I E,(m) I; then the adion leaves the surface in order to minimalize the electrostatic repulsion term U(d). The reason why the delocalization effect lowering the kinetic energy of the electron cannot promote a stable binding for the limit case x = I Es(m) I, is the increase of the kinetic energy by the smaller effective mass of the conduction band.

H. Geistlinger / Chemisorption effects on the thin-film conducticit):

434

(ii) Atop-anion

binding. At the distance z= = lattice constant a the energy has an absolla, lute minimum: A8(d) x*(J)

= -0.065 =0.32

chemisorption site (see Appendix 2) we calculate the occupation probabilities of the weak- and strong-chemisorbed states, respectively:

eV,

(/!I = 1.2).

(IO) In case of the atop-anion binding there exists a stable one-electron surface state, where the electron is still localized to the adatom and partially localized to the semiconductor.

3. The Volkenstein

where E, = E; - Ez. Using eqs. (ll)-(14) obtains the Volkenstein isotherm

isotherm

As we have noted in the previous section, we consider only the interaction between the acceptor-like species of the gas phase (e.g., oxygen) and the chemisorption sites Xchem; the “kinetic” interaction between physisorbed species and chemisorbed sites will be neglected. We assume according to VM that the spectrum of chemisorbed states exists of two discrete levels [23] - the weak-chemisorbed state Ei and the strong chemisorbed state E; (see fig. 3). For simplicity we shall treat the case of nonactivated and non-dissociative adsorption; then we obtain in the adsorptive-electronic equilibrium (for details we refer to the monograph by Volkenstein [ 151): cxp( 1 - 0) = ~‘8’ exp( -Q’/kT) + v-B_ exp( -Q-//CT),

(11)

where cy= soA/ dm is the usual kinetic coefficient of the Langmuir isotherm, u”, Y- are typical phonon frequencies of the order lo-i3 l/s and the total coverage 0 is the ratio of the total number of chemisorbed species N = No + N- and the number of chemisorption sites Nchem: @- - N

Nthem

f”

and f-

=f”e

+f-e

= e” + e-.

are the probabilities

Q0 = E,(w) - E,O,

(12)

and

Q-=Ec+E,(w)

-E, (13)

are the adsorption heats of the weak- and strong-chemisorbed state, respectively. With the help of the grand partition function of the

one

O(P) = & with @=b(~“[I+&exp(~)]}‘.

(I5b)

We note that in contradiction to physical adsorption with b in the Langmuir isotherm being a function of temperature only, here the coefficient p, in accordance with eqs. (14b) and (15a), depends on the difference of the bulk Fermi level and surface state energy E,. That means that in chemical adsorption the adsorptivity of the surface depends not only on the external parameters p and T, but also on the electronic state of the adsorptive system as a whole, which is determined by the location of the Fermi level.

4. Pressure dependence conductivity

of the thin-film

The aim of this section is to derive the connection between the gas pressure and the thin-film conductivity. This problem will be solved under the following assumptions [24] (for the purpose of clarity the notations of important surface and bulk parameters are listed in table 1): (i) Acceptor-like chemisorption on n-type, nondegenerate, wide band gap (> 2 eV) MO semiconductor, where the 0 vacancy acts as a double donor (e.g., n-type ZnO: E - -0.05 eV, E,, = -0.4 eV [26] or -2 eV [2$.(ii) The thin, semiconducting film is ideally iso-

435

H. Geistiinger / Chemisorption effects on the thin-film conducticity Table 1 List of symbols Surface A E!’ EP 4 NO NN chcm

of important

surface

where nd = [V,] is the concentration of oxygen vacancies, 6(z) is the Dirac delta function and the chemisorbed surface charge density is given by

and bulk parameters

parameters Area of the surface Energy of the weak-chemisorbed adelectron Energy of the strong-chemisorbed adelectron = E,- - E: Number of weak-chemisorbed species Number of strong-chemisorbed species Total number of chemisorption sites = (N” f N- )/Nchemr total coverage Surface charge density

Q, = -ee-(E,,

E,,, E,, n” n(z) p(z) V(z) 4

V-E, 4= - kT

Bulk concentration of neutral oxygen vacancies, where the single vacancy is occupied by two donor electrons Energies of the first and second donor electron Nondegenerate electron concentration at flat bands Electron concentration Hole concentration Potential energy (sometimes also shortly denoted as “potential”) Fermi level

lated from the substrate, no electron transfer between the thin film and substrate can arise; (iii) Acceptor-like and donor-like biographic surface states are compensated. The general expression of the thin-film conductivity according to space-charge region (SCR) theory of chemisorption [12] is given by a( P) = +ZcR<4 2, P)>, (16) where psCR(T) is the electron mobility in the SCR and’the averaged value of free conductionband electrons is

(n(z, P)> = ;/;pd

.rrrO exp( - ‘(Lip)).

(17)

r

1 + 2 exp

-en(z)

+ep(z),

[Ln=/T],

and inserting the non-degenerate electron- and hole concentrations, the Poisson equation can be rewritten as

d24

1

-=l+ dt2

(2/K,)exp(-4)+1

+‘exp(-‘)

+P exp(-E,/kT) exp(4+

(21)

where l = Nc/nd, p = N,/n, and K, = exp(E,,/kT). With respect to the two appropriate boundary conditions (i) global charge neutrality d4 dt

r=D/L,

=

0,

(224

and (ii) Gauss law

d4

27

Qs<-%

*=+o= en,L,

P>.

(22b)

’

we can integrate once [28], and neglecting hole contribution we obtain

2(4-4,)

P) 6(z) 1

r=e

7

(20)

For a planar geometry (z = 0: surface, z > 0: bulk) the potential V(z, p) is determined by the one-dimensional Poisson equation with the charge density P(Z) = Q,(%

(19)

In eq. (18) we have assumed that all oxygen vacancies are singly ionized (E, < Ed, ), therefore p(z) takes only into account the statistics of the second donor electron. Using reduced variables

Bulk parameters

Wol

P)%,,,/~.

+ln

+S[ew(-4)

2 exp(-4) +K2 2 exp( -4,)

the

+K2 II

‘P

’

(23)

where 4n = 4(t = D/L,). We note that the pressure dependence of the potential is involved by the boundary condition (22b), where the surface

436

H. Geistlinger / Chemisorption effects on the thin-film conductivity

charge density is determined by the coverage of the charged, strong-chemisorbed species 9-=f-0. Since the occupation probabilities are dependent on the difference E, - (Ef + 1/(z = 0, p) and hence on the solution +(z, p), therefore one has to solve the problem selfconsistently: Inserting 8- (using eqs. (12)-X15)) in eq. (19) the boundary condition (22b) and eq. (23) lead to a nonlinear equation, F(&, +o, p) = 0, for the reduced potential at the surface &, = +(t = 0). Since one does not know the function c#@) one has to iterate the numerical integration of eq. (23) successively with a variable value $o and hence with a variable boundary value C#Q,and check, if the boundary condition (22a) is satisfied at the film thickness D of interest.

1

lg(/no)

\

\ -911, I, I, I, I ,-7 -5 -3 -1 1 3 lg(Po&‘a) Fig. 5. Oxygen pressure dependence of the normalized, averaged electron density in the low-temperature region (T = 480 K, D = 55 nm): (i) highly compensated case [Vo] = lo’* cmm3 (solid line), (ii) uncompensated case [Vo] = lOI cmm3 (dashed line). According to D a ~0,” the corresponding m values are indicated in the power-law region.

5. Results Ld, = 15.1 nm) is chosen 5.1. po, dependence of the thin-film conductivity

Due to additional scattering by chemisorbed species, the electron mobility pzcR (eq. (16)) depends both on temperature and on partial oxygen pressure. The problem of dominant scattering mechanisms in thin films is not considered here; we shall discuss only the po, dependence of the averaged density of free electrons of the conduction band (n)(p,J assuming that the electron mobility is weakly affected by adatom scattering. Usually, in this case ptcR(T> is replaced by ptlk CT), then a/a0 = (n>/n’. For numerical demonstration we consider thin ZnO films and use the following standard values, if no others are indicated explicitly: E, = 3.2 eV, m eff = 0.25mo, E,, = -0.05 eV, E,, = -0.4 eV, E&O;) = - 1.1 eV, Q” = 0.1 eV, the sticking co= 4 x 1o12 efficient s 0 = lo-” cmm2 and N&J/4 cm-*. In fig. 5 (n)(p,,)/n’ is shown over the whole pO, region in the low-temperature region (T = 480 K) for two oxygen-vacancy concentrations: (i) highly compensated case (as considered in ref. [lo]) [V,,] = 1012 crne3 (solid line, index “i”); (ii) uncompensated case [V,] = 1017 cm-’ (dashed line, index “d”); the essential parameter 5 = D/L,, (Lb = 4.8 pm,

equal to two. Fig. 5 indicates that uncompensated thin films are more sensitive to po, chemisorption and both curves exhibit a nearly linear po, dependence for lop4 spO, I lo2 kPa. Usually, this linear dependence in the low-temperature region is attributed to the 0; chemisorption, applying the mass-action law one obtains (n) ape: 1301. Furthermore, it can be seen from fig. 5 that the averaged electron density cannot be described by a power-law behaviour in the whole po, region.

5.2. Thickness dependence of the averaged electron density

We are now interested in the dependence of the averaged electron density on the technologically important parameter - the film thickness D. We note that in many cases the measured thickness is of the order of pm, however, the physical thickness is much smaller due to reduced thickness of grain-grain junctions and/or to grain boundary penetration of chemisorbed species. In fig. 6 the normalized, averaged electron density ( n)(po,)/nO (uncompensated case, T = 480 K) is shown in dependence on the parameter ,$ = 2; 3; 4; 6; the values correspond to D = 30; 40; 60; 90 nm. In the pO, region, where (n) can be described by a power law (henceforth called

H. Geistlinger

/ Chemisorption

effects on the thin-film

437

conducticity

I lg (/no) F E 1.1145

k(poJkPa) Fig. 6. Parametric 5 dependence of the electron density (6 = D/L,). In the power-law region the corresponding m values are indicated.

0.2

400

600

800

1000

T/K “power-law region”), it is obvious, that the power m strongly depends on the film thickness.

Fig. 8. The temperature dependence of the exponent m of the conductivity (a a ~0,“). The theoretical curves for different surface-state energies E, (in eV) and film thicknesses D (in nm) (thick, solid lines, denoted by I E, I/D) are compared with experimental data (+) for thin ZnO films from ref. [lo].

5.3. Power-law behaviour: (n) ap0,” In fig. 7 the pO, dependence of the averaged electron density (uncompensated case, D = 55 nm, E,(O;) = - 0.8 eV) is shown for different temperatures; it can be seen that the power m decreases with increasing temperature. Restricting to the power-law region, we have calculated the temperature dependence m(T) for two different surface-energy values (EJO;) = -0.8 and - 1.1 eV) chasing the film thicknesses D = 45 and 55 nm, respectively. The m(T) curves are represented in fig. 8 (thick solid lines denoted

1 lg(a>/no) -1

855 K 720 K

-2

620 K

-3

480 K

-41 -4

-2

0

2

lg(P%/kPa)

Fig. 7. Parametric temperature dependence of the normalized, averaged electron density.

by: 0.8/45; 0.8/55; 1.1/45; 1.1/55) and compared with experimental data (+ ) [10,31]. The theoretical fit to the experimental data given by Hirschwald et al. (thin solid line) exhibits a transition from low-temperature behaviour due to chemisorption (m = 0.5 is derived by the mass-action law of O- chemisorption) to hightemperature behaviour due to bulk diffusion of 0 vacancies (m = 0.25). As fig. 8 shows, the drastic drop of m,,,(T) in the low-temperature region cannot be described only by chemisorption effects, because all theochange less than the exretical curves m theor perimental one. In our oppinion this drastic drop cannot be understood by chemisorption theory, only. A comprehensive theory for the whole temperature region (400-1200 K) gives a low-temperature limit the chemisorption theory, because the vacancy concentration is frozen in, and as hightemperature limit the bulk-diffusion theory, because all chemisorbed species are desorbed. In order to describe the mean temperature region one has to solve a difficult, selfconsistent problem. Obviously, there is a critical temperature T,, where a,,, has reached the same order as vCchem, hence with increasing temperature the power m

H. Geistlinger / Chemisorption effects on the thin-film conducticity

438

0.07 eV/ Decade

0.1

’ -8

’

’ -6

-4

-2

0

k(PO2Ma) Fig. 9. Comparison of the calculated pressure dependence of the activation energy (upper, thick solid line - uncompensated case; lower, thick solid line - highly compensated case; .$ = 2; T = 400 K, so = lo-’ cm21 with the experimental data ( +) for thin CdS films from ref. 1291.

is determined by the power law of adi,. If one considers the curve 1.1/45 in fig. 8 and assumed T, = 800 K, then one gets a drastic drop for m(T) (dashed line). dependence of the EA(pOr) for thin CdS films

5.4. PO,

activation

energy

Suggesting an activated behaviour of the conductivity a(pO,) a exp[ -EA(poZ)/kTl, we have plotted in fig. 9 the activation energy versus 0, pressure in the low-temperature region for the uncompensated and highly compensated case (5 = 2, T = 400 K). The curves exhibit a wide linear region, where the activation energy increases at the rate 0.07/poz decade; the good agreement of the curve for the uncompensated case with the experimental values (+) of thin CdS films [29] is remarkable.

charged, strong-chemisorbed form, and have derived in the second part the pressure dependence of the thin film conductivity. The main results of our theory are: the lowtemperature behaviour of the thin-film conductivity due to chemisorption is determined by different, fundamental parameters - 5 = D/L,,, D, T, E,. It was shown that the power-law behaviour cannot be expressed by only one parameter - the constant, stochiometric exponent in the mass-action law, which is applied in the majority of experimental and theoretical papers, which deal with thin-film gas sensors. Based on the proposed microscopic model it should be possible to interpret the experimental data of chemisorption effects on thin films; e.g., the observed m(T) dependence [5,101. It is worthwile to remark that the theoretical values of m vary between 0 and 1; this is no trivial result in the VM and agrees with the large experimental experience that 0 I m,,,(T) I 1. Furthermore, if one can neglect the grain boundary potentials, the thin-film theory should be applicable in a first approximation to thin, polycrystalline films, thick films and sintered species of MO-SCs, if one assumes in the both latter cases, that the current pathes cross the grain-grain junctions diameters of which are of the order of the Debye length. Theoretical analysis on this subject are under consideration. Acknowledgements

The author is grateful to Dr. T. Braunschweig and Dipl.-Phys. U. Roland for stimulating discussions; and Dr. J. Lagois for critical reading of the manuscript.

6. Conclusions

In the first part of this paper the ground state energy of the one-electron problem of chemisorption (weak-chemisorbed state) was calculated by the Ritz variational method. It turns out that there exists a stable solution for the case of atop-anion binding. On this bases we have introduced two forms of chemisorption according to the VM: the neutral, weak-chemisorbed and the

Appendix 1. Calculation 8(x, y2, d) 1321 (i) Normalization (2)). With

of the energy functional

of the trial function

1(lr) (eq.

la)=(:)3’2exp(-y,Ir--dl), ci(y2) =N exp( -YZ&),

(Al.1)

439

H. Geistlinger / Chemisorption effects on the thin-film conducticity

we calculate the normalization constant N replacing the sum over i by an integral (continuum approximation (CA)), and obtain N2 = 2(1

-x2)& ?r’

(alH,,lO)=

(ILIH”W -h2

AJ2m,

Cci(Y2)Cj(Y2)fij i,i

X (a I Ho

~~~l~>~il~oIO~-~alO~t,, (Al .7)

where

n tcKl=

=x2(al

(2Tj3

/ szd3k

E(k)*

Assuming for the strong-localized tion at R = 0

I a)

i

(Al .8)

Ii> +x2(a I Vjattice I a>,

(A1.3) where a is the lattice constant, we obtain an approximate, d-dependent expression for the amplitude (a IO)(d). If one integrates over the whole space in elliptic coordinates, one yields (u

I())(d)

=

( ~)3’2d3expy)

X

cosh( k,) + k,2

E(k=O)-*

h2AR. (A1.4)

(A1.5)

we substitute for cg in accordance malization condition (A1.2): c2= 0

-

n

=(l

/ 0

d -

2

+

= (1

-x’)f(YZ)Y

with R = $r(~,)~.

+

CCi(Yt)Cj(Y2)(il W I .i) i,i

+2xCci(Y2)(aIWIi),

(Al-lo)

where for the both last terms a screened Coulomb potential is used:

-x2)(1--ew(-y2~,) + 2(Y2Pd2

(f+~,),k=f(f+y,).

with the nor-

“d3Rj exp( - y2Ri)

[2Y,P,

(A1.9)

~~I~IJI)=x2h4Wla)

W(r-d) x

1’

(iii) Coulomb interaction.

Using (TB) for the third term:

I H” IO>,

where c=

eff

N2

Wannier func-

+ 2xCci(Y2)

where we have supposed that x and ci(y2) are real functions and we have omitted the upper index “c” for the conduction band. Assuming there is a small overlapping between the adelectron wavefunction I a) and the lattice potential, we neglect the last term in (A1.3). The first term is easy to calculate. For the second term we obtain in CA and effective-mass approximation @MA)

2xc,(a

amplitude

(A1.2)

where 0 denotes the volume of the primitive cell. (ii) Kinetic energy of the electron.

+

In order to calculate the transfer (a 1Ho IO) we use consequently TB:

11)

(Al .6)

= -

e2 47x&

I r -d I *

The first term is easy to calculate and for the second term we use CA. A complete, analytical integral was not achieved, because one has to

440

H. Geistlinger/ Chemisorptioneffects on the thin-filmconductivity

integrate only over the semi-infinite last Coulomb integral

lattice; the

COU(Y,Y d) = 4y,jjd

XX 1+xYzd i

This yields for the occupation probabilities of the single-occupied state (the weak-chemisorbed state)

[1+ (5ii4”i

xexp( -2x)

1 =

+1

(Al.ll)

(A2.3)

was integrated numerically. For the third term we use TB; c,, is given by eq. (A1.6) and if one approximates the Wannier function by eq. (A1.8), then one obtains the following analytical expression for the matrix element

and of the double-occupied chemisorbed state -

state - the strong-

1

(alWlO> (A2.4)

x((c+l)

sinh(k.)

+ckf[k,

where E, = E, t 3 -El; if we denote the energies in accordance with VM then E, = E; - Ez.

cosh(k,) (A1.12)

-sinh(k.)]},

References

where c and k, are defined in eq. (A1.9).

Appendix 2. chemisorption

Grand centre

partition

function

of

a

The general expression of the grand partition function is given by Z them

=

(A2.1)

where j is the number of electrons which occupy the chemisorption centre, Ejm the energy of the mth quantum state with j electrons and p the chemical potential [16]. If one considers only quantum states to stable lattice ion-ad&z configurations, then in case of VM: j = 1, 2, and if one assumes that the single-occupied state is spin-degenerated (E, f = E, 3.= E,), one obtains Z

chem=2exp(~)+exp( 2p-jF2Ti). (A2.2)

[l] W. Gopel, Prog. Surf. Sci. 20 (1985) 9. [2] J. Zemel and P. Bergfeld, Eds., Chemically Sensitive Electronic Devices (Elsevier, Lausanne, 1980). [3] K.D. Schierbaum, U. Weimar and W. Gopel, in: Proc. “Sensor 91”, Vol. 4, Niirnberg, 1991, p. 123, and references therein. [4] R. Sanjines, V. Demarne and F. Levy, Thin Solid Films 193/194 (1990) 935; V. Demarne, R. Sanjines, D. Rosenfeld, F. Levy and A. Grisel, Proc. EUROSENSORS V, Rome, 1991, Sensors Actuators B 7 (1992) 704. [5] G. Sbergveglieri, S. Groppelli, P. Nelli, F. Quaranta, A. Valentini and L. Vasanelli, Proc. EUROSENSORS V, Rome, 1991, Sensors Actuators B 7 (1992) 747. G. Sbergveglieri, A. Camanzi, G. Faglia, S. Groppelli and P. Nelli, Semicond. Sci. Technol. 5 (1990) 1231. [6] M. Fleischer and H. Meixner, Proc. EUROSENSORS V, Rome, 1991, Sensors Actuators B 7 (1992) 257. [7] T. Seiyama, Ed., Chemical Sensor Technology (Elsevier, Tokyo, 1988). [8] U. Lampe and J. Miiller, Sensors Actuators B 18 (1989) 269; K.-S. Weissenrieder and J. Miiller, in: Proc. “Sensor 91”, Niirnberg, 1991. [9] J. Zemel, Thin Solid Films 163 (1988) 189. [lo] P. Bonasewicz, W. Hirschwald and G. Neumann, Thin Solid Films 142 (1986) 77; Phys. Status Solidi (a) 97 (1986) 593; Appl. Surf. Sci. 28 (1987) 135.

H. Geistlinger [ll] [12] [13] [14]

[15] [16] [17]

[18] [19]

[20] [21]

1221 [23]

[24]

/ Chemisorption

R. Lalauze, P. Breuil and C. Pijolat, Sensors Actuators B 3 (1991) 175. V.F. Kiselev and O.V. Krylov, Electronic Phenomena in Adsorption and Catalysis (Springer, New York, 1986). S.R. Morrison, The Chemical Physics of Surfaces (Plenum, New York, 1977). P.K. Clifford, in: Proc. 1st Int. Conf. Chemical Sensors, Fukuoka, Japan (Elsevier, Kodanska, 1983) p. 135; P.K Clifford and D.T. Tuma, Sensors Actuators B 3 (1983) 233, 255. Th. Volkenstein, The Electron Theory of Catalysis on Semiconductors (McMillan, New York, 1963). A. Many, Y. Goldstein and N.B. Grover, Semiconductor Surfaces (North-Holland, Amsterdam, 1965). This argument is strictly valid only for s states, for which the first-order approximation vanishes because of symmetry arguments [IS]. L.D. Landau and E.M. Lifshitz, Quantenmechanik (Akademie-Verlag, Berlin, 1978). In the framework of the Born-Oppenheimer approximation one seperates the motions of ions and electrons and calculates then the grand partition function of electron states of an electron system to one rigid ion configuration. K. Hauffe, Adv. Catal. 7 (1955) 213. This one-electron theory of chemisorption was developed by Einstein and Schrieffer for transition metals using the Green function technique [22]. The authors do not take into account the d dependence (surface-adion distance) of the nearest-neighbour matrix element (0 I W I a> of the interaction potential W. T.L. Einstein and J.R. Schrieffer, Phys. Rev. B 7 (1973) 3629. A generalization to a continuous spectrum, e.g., to a uniform, Gaussian or exponential density of states, is possible. Kogan and Sandomarski [25] derived a generalized analytical expression for the Volkenstein isotherm, including the integration of the Poisson equation, in p and B/p

effects on the thin-film conducticity

441

coordinates for a non-degenerate, n-type semiconductor for the limit case of small occupation of the different types of chemisorbed species - f ‘, f- < 1 justifying Boltzmann statistics - and consider only thick films D >z [25] t.t&. Kogan and W.B. Sandomarski, J. Fiz. Chim. 33 (1959) 1709 [in Russian]. [26] A. Piippl, Thesis, University of Leipzig, 1990. [27] U. Schwing and B. Hoffmann, J. Appl. Phys. 57 (1985) 5372. [28] For standard arguments of the first analytical integration of the one-dimensional Poisson equation in the case of constant donor concentration and D za L, we refer to the monograph by Many, Goldstein and Grover [16]. The case of constant donor concentration and D - L, is considered by Baidyaroy and Mark [29]. Both cases are based on the CTM. [29] S. Baidyaroy and P. Mark, Surf. Sci. 30 (1972) 53. [30] S. Strlssler, A. Reis and D. Wiesner, in: Polycrystalline Semiconductors, Ed. G. Harbeke (Springer, New York, 1985) p. 209. [31] Hirschwald et al. [lo] suggest that the ZnO films are highly compensated, therefore one has to compare with the theoretical curves of the highly compensated case. Since the fundamental dependence of m is given by the .$ dependence, the comparison is approximately justified. [32] The calculation scheme, which we apply to the one-electron problem of chemisorption, was developed in order to describe the electron binding to a short-range bulk-impurity [33]. The difficulties arises from the calculation of the kinetic energy, both the ,va term in the chemisorption problem and the $(O) term in the problem of a shortrange bulk-impurity potential have to be excluded from the effective-mass approximation. [33] H. Geistlinger and W. Weller, Phys. Status Solidi (b) 128 (1985) 709. [34] H. Geistlinger, Proc. EUROSENSORS V, Rome, 1991, Sensors Actuators B 7 (1992) 619.