KINETICS OF OXIDE GROWTH ON METAL SURFACES

A. Vlad

Faculty of Science, University of Oradea, RO-410087 Oradea, Romania

Max-Planck-Institut für Metallforschung, D-70569, Stuttgart, Germany

Abstract:

A short review of the principles governing the oxidation of metals is presented.

The initial stage of the oxidation process involves the chemisorption of oxygen, frequently

followed by dissociation and at least partial ionization. The growth of continuous oxide

films or scales is considered in terms of various rate-limiting processes such as anion or

cation diffusion through the bulk oxide, mass or electron transport across one of the

interfaces, or electron transfer processes associated with the chemisorption step.

INTRODUCTION

Metals are generally unstable in the climatic conditions of Earth.

Thermodynamically, only a metal as noble as gold should survive as a

native metal, resisting conversion into a oxide, halide, sulfide or other

compound. The destruction of metals by corrosion is the costly aspect of the

low temperature oxidation: the important metals used in engineering and

construction, like iron, copper, aluminium all corrode to different degrees.

However, the oxides are not just harmful and under specific conditions a

passivating oxide may form which acts as a protective layer against further

oxidation. The discovery of Atalla et. al. [1] in 1959 that the thermal

oxidation of silicon passivates its surface was a crucial step in the

semiconductor device technology. There are also other fields were oxide

layers play an important role, like in heterogeneous catalysis or in the high

temperature resistant coatings. In order to control the oxidation and use it as

a tool, understanding the mechanism at an atomic scale is mandatory.

INITIAL STAGES OF GAS-METAL INTERACTION

The initial stages of the gas-metal interaction may be very complex

and imply a number of different physical or chemical processes. Oxygen

molecules from the gas phase must first come into contact with the metallic

surface. The rate of impingement of the molecules on the unit surface area is

42

given by the kinetic theory as being directly proportional to the gas pressure,

p and inversely proportional to the square root of the mass of the gas

molecule, m, and of the temperature, T:

p/(2πmkBT)1/2

(1.1)

where kB is the Boltzmann constant.

A fundamental property of the gas-surface interaction is the sticking

probability, that is the probability that an gas atom or molecule which hits

the surface ends up is an adsorbed state on the surface. The sticking

probability is influenced by several external variables, like the surface

structure, cleanliness of the surface, the temperature and the gas pressure.

Two main adsorption mechanisms can be distinguished: physisorption and

chemisorption.

Physisorption

In the process of physisorption of a molecule on a surface, the

electronic structure the adsorbate and of the substrate is hardly perturbed.

The bonding takes place via weak van der Waals interactions, where an

attractive force appears due to the mutually induced dipole moments.

The interaction of a diatomic molecule, like O2, with a metallic surface may

be described by the potential energy diagrams such as that shown in Fig. 1.1

[2]. The curve I (blue) represents the potential energy of the molecule as it is

attracted to the surface by the long-range forces and becomes physisorbed in

a non-activated process. ∆Hp denotes the heat of physisorption. The

potential has a shallow minimum at a few Ǻ from the surface. At closer

distances, the electron wave function of the adsorbate and the surface atoms

start to overlap leading to a strong repulsion. Characteristic for the

physisorption is a low bonding energy (5-100 meV). Thus, it is assumed that

it becomes important at relatively high gas pressures and low temperatures

(RT: kBT » 25 meV) and primarily as a precursor to chemisorption [2].

Figure 1.1: Schematic potential energy diagram for the interaction of a

diatomic gas with a metallic substrate.

43

Chemisorption

The strongest adsorption mechanism is the chemisorption which

involves a rearrangement of the valence electrons of the metal and adsorbate

in order for a chemical bond to form. The bonding energies of

chemisorption are relatively high (> 1 eV) and involve short bond distances

of a few Ǻ. The curve II (red) in Fig. 1.1 represents the potential energy of a

molecule which has been dissociated prior to chemisorption. Ea and Ed

represent the activation energies for chemisorption and desorption,

respectively, whereas ∆Hc denotes the heat of chemisorption. Depending on

the point of intersection between the two curves, chemisorption can be

either an activated or a non-activated process: it the point of intersection P is

above the zero potential energy an activation energy, Ea, will be required for

the chemisorption to occur. On the other hand, if P lies below zero, no

activation energy is necessary for chemisorption.

In a simplified picture, whether a molecule will be physisorbed or

chemisorbed on a surface depends primarily upon the energetic position of

the molecular orbitals of oxygen molecule with respect to the Fermi level of

the metal. In the frame of the Molecular Orbital Theory, the bonding within

a molecule is described in terms of bonding (σ, π, ..) and anti-bonding

(σ∗, π∗, ..) molecular orbitals which are constructed by combining the

available atomic orbitals. The molecular orbital energy level diagram for an

oxygen molecule in the ground state is presented in Fig. 1.2.

Figure 1.2: Molecular orbital energy level diagram for an oxygen molecule

in the ground state.

44

In principle, three situations can be distinguished: if the anti-bonding

molecular orbital of the oxygen molecule lies higher in energy than the

Fermi level of the metallic substrate, no charge transfer will take place,

therefore the oxygen molecule will be physisorbed. An incomplete filling of

the O2 anti-bonding molecular orbitals leads to the adsorption of the charged

chemisorbed oxygen molecule on the surface. This is the situation when the

anti-bonding orbital is situated partially below the substrate Fermi level. A

favorable situation for the dissociative chemisorption occurs whenever the

anti-bonding orbital is completely filled with electrons; the O2 anti-bonding

orbital lies below the Fermi level of the metal.

With increasing number of oxygen adatoms, repulsive lateral

interactions between the adsorbates combined with the adsorbate-substrate

interaction may lead to the formation of ordered chemisorbed layers. For

instance, a quarter of a monolayer of oxygen on Rh(111) forms a (2 x 2)

reconstruction (a hexagonal lattice similar to the substrate structure, but with

twice the distance between the O atoms as compared to the Rh atoms) [3].

Oxide nucleation and growth

Provided that the mobility is high enough, the adsorbates may mix

with the substrate atoms and a two-dimensional so-called surface oxide may

form. For instance, depending on the partial pressure, temperature and

orientation, the oxidation of transition metals (Rh, Pd, Ag) can lead to the

formation of surface oxides which may or may not bear a resemblance to the

corresponding bulk oxides [4] and ref. herein.

The oxide nucleation is an activated process and it was reported that

the activation energy should decrease with increasing the oxygen pressure

[2]. At low temperatures the thermal activation energy for atomic motion is

small. Therefore, at relatively low temperatures and oxygen partial pressures

the oxide nucleation is expected to take place mainly at defect sites (e.g.

kinks or step edges). The possibility that the oxide would nucleate at any

surface site increases with increasing the oxygen pressure.

After the formation of oxide islands or clusters, oxidation mostly

continues laterally until the first oxide monolayer is closed. Alternatively, a

three-dimensional growth may also be observed. A recent study on Cu

oxidation [5] has shown the dependence of the oxidation behavior on the

crystal orientation and temperature. The kinetic data on the nucleation and

growth of oxide islands showed a highly enhanced initial oxidation rate on

the Cu(110) surface as compared with Cu(100). It was reported that the

45

dominant mechanism for the nucleation and growth on the (100) and (110)

oriented crystals is the surface diffusion of oxygen. A markedly different

oxidation behavior was observed of Cu(111) shows a dramatically different:

the initial stages of oxidation are dominated by the nucleation of three-

dimensional oxide islands at T < 550oC and by two-dimensional oxide

growth at T > 550oC.

KINETICS OF OXIDE FILM FORMATION

As soon as a thin continuous film of oxide has formed on a metal

surface, the metal and gaseous reactants are spatially separated by a barrier

and the reaction can continue only if cations, anions, or both and electrons

diffuse through the oxide layer. The rate-determining step in the oxidation

reaction depends on the system in question, the thickness and nature of the

oxide film, as well as on the pressure and temperature of the system.

As a function of the rate determining processes, two limiting cases

may be discussed. On one hand, if the transport through the existing oxide

layer is considered to be faster than the processes taking place at the surface

(the rate of impingement of the oxygen molecules, oxygen dissociation,

chemical reactions), the surface reactions are regarded to be rate limiting.

This limiting case is therefore designated as being surface reaction

controlled. On the other hand, one can consider the possibility that the

transport of defect species through the oxide can be so slow relative to the

surface/interfacial reactions that it becomes rate-limiting. This is the case

denoted as transport current controlled or diffusion controlled, if the

relevant currents are due to diffusion. Based on these assumptions, a number

of models have been proposed to explain the kinetics of oxide film growth

for different thickness regimes. An arbitrary film-thickness terminology has

been proposed by Fromhold [6] which is listed in Table 1.1.

L (Ǻ) Terminology

< 5 ultra-thin

5-50 very thin

50-500 thin

500-5000 intermediate

5000-50000 thick

> 50000 very thick

Table 1.1: Terminology of oxide films as a function of thickness.

46

The thick and very thick film regime is best described by the theory

proposed by Wagner (1933) [7], whereas for the kinetics of thinner oxide

the most relevant is the Cabrera-Mott theory (1949) [8] which was further

developed by Fromhold and Cook [9, 10]. The above mentioned theories

will be briefly presented in the following sections

Thin film growth

In 1939, Mott [11] proposed a model to explain the limiting-

thickness behavior of thin oxide films in the low temperature regime. The

thermal diffusion of ions was found to limit the oxide growth in the first

stage, since the ionic current Ji is lower than the current due to tunneling of

electrons through the oxide, Je. The growth law in this regime is parabolic.

However, as the film thickens the electron current Je drops down

considerably and becomes the rate-limiting step. The growth rate in this

stage was found to be direct logarithmic.

A less restrictive model in terms of temperature and oxide thickness

range was proposed in 1949 by Cabrera and Mott [8]. It was based on the

electron transport either by tunneling or by thermionic emission from the

metal into the oxide conduction band and ionic diffusion. The electrons

were considered to be faster than the ions and an equilibrium contact

potential is established between the metal and adsorbed oxygen. This

contact potential was termed as Mott potential, VM, and is defined as:

(1.2)

where e is the magnitude of the electronic charge, χ0 is the metal-oxide

work function and χL is the energy difference between the conduction band

in the oxide and O- level of adsorbed oxygen. Figure 1.3 gives a schematic

potential energy diagram for the metal/oxide/oxygen system, for which VM

is negative in sign. In the absence of space charge, a positive electric field

E0 = VM/L(t) is build-up in the oxide which speeds up the ions and slows

down the electrons. The lowering of the energy barrier for ionic motion by

large electric fields is termed as non-linear diffusion.

47

Figure 1.3: Schematic band structure of the metal-oxide-oxygen system.

Further developments have been made by Fromhold and Cook

(1967) in a theory that constitutes a synthesis of the Mott and Cabrera-Mott

models [9, 10]. The central concept of this theory is that of coupled charge

currents. The assumption that the steady state ionic and electronic currents,

qiJi and qeJe, respectively, are equal in magnitude, but have opposite signs is

made. This is referred to as the coupled-currents condition (or kinetic

condition) and mathematically can be written as:

(1.3)

where qi and qe are the electric charges of single defects of the diffusing

ionic and electronic species, respectively, Ji is the ion current density and Je

is the electron current density. This assumption is valid whenever the space-

charge contribution to the electrostatic potential gradient is negligible

relative to the surface-charge contribution. For oxide films thinner than 50

nm, this condition is usually satisfied and a uniform electric field may be

considered. The surface-charge field E0 as a function of oxide thickness is

thus determined from the coupled-currents condition (Eq. 1.3) and

substituted into one of the two currents Ji and Je to obtain the growth rate:

(1.4)

48

Jc denotes the rate-limiting particle current and Rc represents the volume of

oxide formed per particle of the flux Jc which is transported across the

oxide. The oxide film thickness then is numerically evaluated as a function

of time.

The main theoretical task is to find suitable expressions for the ion and

electron currents. Depending on the oxidation conditions and the properties

of the involved materials, either electron or ion transport processes may

become the rate-limiting step in the oxidation process.

Electron-tunnel current limited growth

At low temperatures the thermal energy kBT is insufficient for

thermionic emission to occur and the electron transport through the oxide

takes place by quantum mechanical tunnel effect. Considering the zero

temperature case, if χL > χ0, tunneling of electrons from the filled part of

the metal conduction band which lies above the O- level takes place (Fig.

1.3). If the metal Fermi level EF and the O- level of adsorbed oxygen are

equalized, the current drops to zero and the system is in equilibrium with the

potential VM existing across the film. The tunnel electron current during the

establishment of this potential is given by:

(1.5)

where VK = -E0L is the kinetic potential. Whenever the kinetic potential

equals the Mott potential, the electron tunnel current is zero, corresponding

to a current equilibrium of the electronic species.

The non-zero temperature situation is more complex, since tunneling

in both directions occurs simultaneously. However, it has been shown that

the there is a relatively small temperature dependence for tunneling, which

implies that the saturation oxide thickness is almost independent of

temperature.

Numerical calculations shown that in the first stages the oxide

growth rate is limited by ionic diffusion and the growth law is inverse

logarithmic, as obtained by Cabrera and Mott. In Fig. 1.4 the film thickness

versus the logarithm of time for different values of the Mott potential is

shown. A sharp transition to a second growth stage occurs at a thickness

49

around 20-30 Ǻ. In this latter growth stage ionic current equilibrium exists.

The rate is limited by electron tunnelling and a direct logarithmic growth

law results.

Figure 1.4: Film thickness versus the logarithm of time for different values

of Mott potential. Curves 1-4: VM = 0, VM = -0.25, VM = -0.5 and VM = -0.75

V, respectively. Dashed curve, VM = +0.1 V [9].

Thermal electron emission current limited growth

At sufficiently high temperatures (300 - 600oC) thermionic emission

of electrons over the metal-oxide work function χ0 becomes a likely

transport mechanism. The region in film thickness where electron tunneling

and electron thermal emission occur simultaneously to an appreciable extent

is very limited, since the electron tunneling becomes increasingly difficult

as the oxide film thickness exceeds 20-30 Ǻ. The ion diffusion current Ji is

produced by the concentration gradient of ionic defects in the oxide which is

due to the differing chemical reactions at the metal-oxide x = 0 and the

oxide-oxygen x = L interface and can be expressed mathematically by

equation 1.8. A negative electric field E0 is established which opposes the

ionic diffusion, but has the proper polarity to facilitate the emission of

electrons from the metal into the oxide conduction band (Schottky

emission):

(1.6)

The equation for the electron current due to thermionic emission Je

is given by [2]:

(1.7)

50

where A is a constant equal to 4πmkB2 =h

3 = 7.5 x 10

20 electron/cm

2 sec K

2.

It is important to notice that the electron current Je is independent on the

oxide thickness, as opposed to the previously discussed case of electron

tunneling. The ionic diffusion potential VD = -E0L(t) is also independent of

the film thickness. Therefore, at low oxide thicknesses a high electric field is

established which leads to a rapid initial growth. As the film thickens, the

magnitude of the electric field decreases and a linear growth rate is

observed. The slope of the curves is found to increase with temperature.

This can be observed also in Fig. 1.5 [10] which illustrates the temperature

dependence of aluminium oxide thickness versus time during the oxidation

of molten aluminium.

Figure 1.5: Calculated thickness evolution as a function of time at different

temperatures. The symbols represent experimental data measured for liquid

Al oxidation [10].

The overall agreement between the experimental data and theoretical

predictions is good. However, the data taken at 973 K are more scattered

around the theoretical values, due to the fact that in this thickness regime (<

50 Ǻ) the electron tunneling is modifying the growth kinetics.

Ion diffusion current limited growth

Thermally activated ionic motion in the presence of an electric field

is considered to be the primary mechanism for ion transport in coherent

oxides. The ionic current is expected to be the rate limiting process for

51

semiconducting oxides or for oxides which manifest large defect electron

conductivity.

In the steady-state approximation and in the absence of space-charge effects,

the ionic diffusion current density is given by:

(1.8)

where 2a is the ionic jump distance, νi is the frequency at which each ion

attempts to surmount the energy barrier Wi for diffusion, kB is the

Boltzmann constant, T is the temperature, Zie is represents effective charge

per particle of ionic species, E0 is the surface-charge field, L is the thickness

of the oxide layer, Ci(0) and Ci(L) are the defect concentrations of the

diffusing ionic species at the metal-oxide interface (x = 0) and the oxide-

oxygen interface (x = L), respectively. A logarithmic growth rate with a

limiting thickness behavior is predicted and the limiting thickness is

expected to increase considerably upon increasing the oxidation

temperature. Kinetics of Cr2O3 growth during Cr(110) oxidation are

following the above mentioned behavior [12]. In Fig. 1.6 the thickness of

Cr2O3 on Cr(110) is plotted as a function of the oxidation time and for

different oxidation temperatures.

Figure 1.6: The thickness of Cr2O3 on Cr(110) plotted as a function of the

oxidation time and for different oxidation temperatures [12].

52

After a very rapid increase during the first 10 s, the oxide thickness

changes very slowly on a linear time scale, indicating a limiting thickness

behavior. The solid lines are model calculations for the time dependence

assuming that Cr-ion diffusion is rate limiting. This assumption is well

justified by the large electron defect conductivity existing in Cr2O3.

Thick film growth

Thick oxide films are usually formed at elevated temperatures and as

the oxide grows out of the thin film regime previously discussed the growth

law most commonly observed is parabolic. Summaries of much of this work

are available in the book by Kubaschewski and Hopkins [13] and in the

review articles by Lawless [2] and Atkinson [14].

The mechanism of oxidation at elevated temperatures must depend

primarily on the detailed nature of the oxide formed. A solid oxide will

normally contain a varied array of defects. These defects may take the form

of point defects (vacancies or interstitials), line defects (dislocations) and

planar defects (stacking faults or grain-boundaries). These defects are

responsible for material transport through the oxide and thus play a critical

role in the oxidation process. Macroscopic defects in the form of pores or

cracks are frequently found in oxide scales and material transport is no

longer rate limiting.

The best known and most thoroughly tested theory of parabolic

growth of oxide films at elevated temperatures was developed by Wagner

[7], based on the idea of ambipolar diffusion of the reactants through the

volume of compact oxide as the rate controlling process. It was assumed

that cations, anions and electrons are the diffusing species with the ions

moving through the oxide via lattice defects under the influence of an

electrochemical potential gradient. Thermodynamic equilibrium is assumed

to exist between metal and oxide at the metal-oxide interface, and between

oxide and oxygen gas at the oxide-oxygen interface. The ions and electrons

are presumed to migrate independently of one another, and the effects of

electric field transport are considered to be negligible. The assumptions of

charge-neutrality for each volume element of the oxide and a zero net

charge transport through the oxide are also made.Wagner's theory shows

that the growth of the oxide films obeys parabolic time dependence:

(1.9)

53

where x is the film thickness and kp is the parabolic rate constant. Wagner's

theory is based upon diffusion across the film being the slowest, and

therefore, the rate-limiting step in the overall sequence of reactions. Phase

boundary reactions are considered to be rapid with respect to the rate-

determining diffusion processes.

Oxidation of metals vs. alloys oxidation

It should be noted that the although the previously discussed theories

have been proven to successfully describe the kinetics of oxide growth on

different systems, they are still rather restrictive as concerning the

conditions of validity. It was suggested that in practice these conditions are

often not observed. The case of alloy oxidation is far more complex, since

additional factors come into play. For instance, concurrent processes of

oxidation-induced

chemical segregation and selective oxidation induce compositional changes

in the alloy subsurface during oxidation must be accounted for. Depletion of

the active species in the near-surface region of the metallic alloy may yield

departure from the rate law otherwise expected. A review of the oxidation of

alloys was written by Wallwork [15].

REFERENCES:

[1] E. T. M. Atalla and E. J. Scheibner, Bell Syst. Tech. J. 38, 749 (1959).

[2] K. R. Lawless, Rep. Prog. Phys. 37, 231 (1974).

[3] J. Gustafson, Ph.D. thesis, Lund University, 2006.

[4] E. Lundgren, A. Mikkelsen and P. Varga, J. Phys.: Condens. Matter. 18,

481 (2006).

[5] G. Zhou and J. C. Yang, J. Mater. Res. 20, 1684 (2005).

[6] J. A. T. Fromhold, Theory of Metal Oxidation, Vol. I - Fundamentals of

Defects in Crystalline Solids (North-Holland Publishing Company,

Amsterdam - New York - Oxford, 1976).

[7] C. Wagner, Z. Physik. Chem. B 21, 25 (1933).

[8] N. Cabrera and N. F. Mott, Rep. Prog. Phys. 12, 163 (1949).

[9] A. T. Fromhold and E. L. Cook, Phys. Rev. Lett. 158, 600 (1967).

[10] A. T. Fromhold and E. L. Cook, Phys. Rev. Lett. 17, 1212 (1966)

[11] N. F. Mott, Rep. Prog. Phys. 35, 1175 (1939).

[12] A. Stierle and H. Zabel, Europhys. Lett. 37, 365 (1997).

54

[13] O. Kubaschewski and B. E. Hopkins, Oxidation of Metals and Alloys

(Butterworths, London, 1962).

[14] A. Atkinson, Rev. Mod. Phys. 57, 437 (1985).

[15] G. R. Wallwork, Rep. Prog. Phys. 39, 401 (1976).