886
Langmuir 1987,3, 886-896
Metal Oxidation Kinetics from the Viewpoint of a Physicist: The Microscopic Motion of Charged Defects through Oxidest Albert T.Fromhold, Jr. Department of Physics, Auburn University, Auburn, Alabama 36849 Received November 22, 1986 A unified survey is given of the kinetics of metal oxidation from the viewpoint that nearly all situations can be simulated by the microscopic migration of charged point defects through the oxide. The difference equations which characterize the hopping model can be used to show that the currents and local stresses in the oxide in the intermediate and lower field limitsare linearly related to the macroscopic electrochemical potential differences of the defect species across the oxide. The thrust of the presentation is to show the common relationship between theoretical developments in the diverse areas of plasma oxidation, anodic oxidation, and thermal oxidation. Promising areas for experimental measurements are delineated.
Introduction Prologue. Physicists have an innate penchant for the generalization and unification of physical phenomena. Einstein's lifelong preoccupation with developing a generalized field theory can be cited as one well-known example of this tendency. Such is the motivation underlying the present effort. My intent is to overview the phenomenon of metal oxidation with an emphasis on the microscopic migration of charged point defects through the oxide. The principal aim is to show the common basis for, and the relationship between, the diverse areas of plasma oxidation, anodic oxidation, and thermal oxidation. References are made, as appropriate, to specific comprehensive theoretical treatments of the various aspects in order to mitigate the burden of working through mathematical and computational details. This enables a better focus on the primary goal. To keep the amount of background knowledge required to a minimum, let us begin with some basic definitions and terminology. Definitions. It is useful to distinguish between thermal oxidation, anodic oxidation, and plasma oxidation, though a common microscopic basis can be employed to treat all three. Thermal oxidation1 usually refers to cases in which all charged species must migrate through the oxide film in order for reaction to occur, whatever the specific transport processes which may be operative for each of the various species. The adjective "thermal" probably has its origin in the fact that the parent metal is usually heated above room temperature to promote the reaction and also from the fact that the primary transport mechanism for species transport is a thermally activated diffusion. However, the first condition is not required: Thermal oxidation has been carried out at temperatures even below room t e m p e r a t ~ r e . ~Moreover, ,~ the second condition is not limited to thermal oxidation, but is also generally characteristic4 of the motion of the ionic species in both anodic and plasma oxidation. Both anodic oxidation5v6and plasma o~idation,~ in contrast to thermal oxidation, involve external circuitry in series with a carrier medium. The external circuitry provides a path for the electronic species external to the oxide film, so that the electronic species need not migrate through the oxide film itself in order for reaction to occur. In anodic oxidation, the parent metal is placed in a solu'Based on invited paper presented a t the symposium on "Corrosion", 191st National Meeting of the American Chemical Society, New York, NY, April 13-18, 1986.
tion and made the anode by means of an externally applied potential source such as a battery. The parent metal thus becomes positively charged, and the positive charge a(0) leads to a surface-chargeelectric field E(0) (designated Eo) which drives the ionic species through the oxide to enable more oxide to grow. Plasma oxidation generally takes place because of a gas discharge plasma established in the ambient oxygen between the parent metal sample and a secondary electrode. Various techniques, both ac and dc, including voltages ranging from very high to low, are used. Whatever the character of the voltage causing the plasma discharge, the external circuitry in series with the ionized gas and the oxide film provides an outside path for movement of the electronic species. The parent metal can thus become charged, leading to field-driven ionic motion through the oxide layer. Since in thermal oxidation there is no external circuitry, there is no outside path available for the transfer of electrons. In this situation the electrons, like the ions, must travel through the oxide layer itself. For example, the electrons can tunnel through the thin oxide layer or be thermally emitted from the parent metal into the oxide over a metal-oxide work function barrier.8 Oxidation kinetics is the measurement or evaluation of the oxide film thickness L ( t )as a function of time t. Simultaneously, the electric field driving the motion of the charged species, or the voltage across the oxide film, can be obtained as a function of time. Terminology. The metal atoms come into contact with the oxygen atoms only after one or the other have migrated through the existing oxide layer. The continued chemical reaction of oxide formation thus hinges on the occurrence of this migrati~n.~ We specifically limit our consideration to the situation where the growth rate of the oxide is limited by the rate of migration through the oxide layer. What is the nature of the species that migrates? It may be extra atoms (interstitials) in the oxide matrix, these being either metal or oxygen, which move from interstitial position to interstitial position due to the energy of thermal excitation. Interstitials are extra atoms, in excess of the (1)Pilling, N. B.; Bedworth, R. E. J . Inst. Met. 1923, 29, 529. (2) Rhodin, T. N., Jr. J.Am. Chem. SOC.1950, 72, 5102. (3) Cathcart, J. V.; Smith, G. P. J . Electrochem. SOC.1960, 107,141. (4) Verwey, E. J. W. Physica 1935, 2, 1059. (5) Young, L. Anodic Oxide Films; Academic: New York, 1961. (6) Vermilyea, D. A. J. Electrochem. SOC.1964, I l l , 883. (7) O'Hanlon, J. F. In Oxides and Oxide Films; Vijh, A. K., Ed.; Marcel Dekker: New York, 1977; Vol. 5. (8) Cabrera, N.; Mott, N. F. Rep. Prog. Phys. 1949, 12, 163. (9) Wagner, C., 2. Phys. Chem. 1933, B21,25.
0743-7~63/87/2403-0886$01.50/0 0 1987 American Chemical Society
Physicists' Viewpoint of Metal Oxidation
ELI,
n
0-
0- 0-
M+ Mt Mt
Metal
Mf M+ M+ M+
0-
IZl- M+ Oxvaen 00- 000-
m- M+ I
00- 0-
t
0-
Figure 1. Concentrations of diffusing ionic species. Cation interstitials (ci) and anion vacancies (av) initiate at the metal-oxide interface, whereas anion interstitials (ai) and cation vacancies (cv)
initiate at the oxide-oxygen interface.
ordinary atoms in the matrix of the oxide, and hence constitute "defects" in the oxide. Thus the "defect concentrations" referred to in our presentation constitute departures from the normal concentrations of the same species as found in the "perfect" oxide lattice. Defect concentrations thus do not include any of the concentrations of atoms of the species in question which are normally contained in the oxide lattice. In addition to excess atoms of either species (namely, either metal or oxygen), there can also be missing atoms (vacancies) of either species. The ordinary atoms of the oxide can then move by thermally activated motion into these vacant sites. The position of each vacant site changes when this occurs, so we speak of "vacancy motion". It usually is the case that these point defects carry a charge, so we often refer to them as "charged particles". Concentrations of diffusing ionic species thus mean to us the local concentrations of excess metal atoms (cation interstitials),deficit metal atoms (cation vacancies),excess oxygen atoms (anion interstitials), and deficit oxygen atoms (anion vacancies) (see Figure 1). In the case of cation interstitials or anion interstitials, the charge is positive or negative, respectively. Vacancies of cations or anions represent missing positive or negative charges, respectively, so locally the oxide carries a net excess of negative or positive charge, respectively. Thus we speak of cation vacancies as being negatively charged and anion vacancies as being positively charged. The parent metal represents a source of cation interstitials and a sink for cation vacancies. Similarly, the oxygen interface of the oxide film represents a source of anion interstitials and a sink for anion vacancies. The gradient of cation interstitials and the gradient of anion vacancies are thus in the direction to yield particle currents of these species from metal to the opposite interface. On the other hand, the gradient of cation vacancies and the gradient of anion interstitials are such as to yield particle
Langmuir, Vol. 3, No. 6, 1987 887 currents of these species toward the metal interface. It is also clear that in the case of thermal oxidation the parent metal constitutes a source for electrons and a sink for electron holes. At the opposite interface, namely, the oxideoxygen interface, the conversion of neutral oxygen into chemisorbed oxygen requires electrons, so it constitutes a sink for electrons and a source for electron holes. The gradient of electrons is therefore such as to yield a current of electrons from metal to gas (parallel to the positive x-direction),whereas the gradient of electron holes yields a current of electron holes from gas to metal (namely, in the negative x-direction). It is convenient to express the charge q of any point defect in terms of the magnitude e of the charge carried C. Thus we write by an electron, viz., e = le1 =. 1.6 X the defect charge for any given species labeled s as QB
= zse
(1)
where 2, carries information on both sign of the charge as well as size (in units of e). Usually 2 can be associated with the valence of the atom. While we speak of defect concentrations within a normal oxide, it is to be recognized that the distinction between defects and ordinary atoms of the oxide matrix is only statistical. Any specific normal oxide atom may become an interstitial if thermally activated into an adjacent interstitial position, thus simultaneously creating a vacancy in the oxide. The motion of any vacancy through the oxide will involve the sequential motion of a large number of normal atoms of the oxide matrix. The normal atoms of the background oxide matrix are, therefore, involved when we speak of motion of a defect species through the lattice. However, the mere fact that there is a statistical participation by many atoms of the oxide lattice in no way invalidates the quantitative aspects of a model based upon the consideration of mobile defect species moving through a seemingly inert and immobile diffusing medium.
Basic Equations for Linear Transport Phenomenological Approach. The normal oxide (viz., a hypothetically inert perfect matrix through which diffusion occurs) is locally charge neutral, so there would be no macroscopic electric field due to space charge. Defects are usually charged, so defect concentrations can represent sources for macroscopic electric fields in the oxide. An electric field E exerts a force
Fs(m = qJ3
(2)
in a given direction on a charged particle, the force being proportional to the charge qs = Zse and being parallel to the field direction. A force can produce motion of the defects, the proportionality factor 3,being characteristic of the defect species s and the oxide. In an isotropic crystal the net direction of motion of the defects will be parallel to the direction of the force. The linear relation
Js(m= 3 p s ( m C , = ZSe3,C$
(3)
often describes sufficiently well the particle current Js(E) (in particles per unit area per second) of species s produced by the action of an electric field E on a concentration C, (in number of defects per unit volume) of that defect species. The product of particle current JSCmand the charge Zse carried per particle of species s is referred to as the particle electrical conduction current d,",
6," = ZSeJs(@ (4) When the relationship between 6," and E is linear, as given by eq 3 and 4, then conduction is said to be ohmic.
Fromhold
888 Langmuir, Vol. 3, No. 6, 1987
When the relation is nonlinear, then conduction is said to be nonohmic. A concentration gradient of any defect species likewise can lead to a net migration of that species due to random thermal motion of the species. This is of course the phenomenon of diffusion. The linear relation J,(O)
= D, (-dC,/dx)
(5)
often describes sufficiently well the particle current J S ( O ) (in particles per unit area per second) of species s produced by the action of a concentration gradient (dC,/dx) at position x of the species in question. The diffusion coefficient D,is characteristic of both the species and the oxide. The concentration gradient can be viewed as a force F8(D)producing motion of the species. Similar to the case of electric fields, we write
F,(D)C, Comparing the relations, we see that
(6)
F,(O)C, a -(dC,/dx)
(7)
J,(O)
0~
field. The defect concentration C, is sometimes referred to as the carrier Concentration associated with defect species s. The Linear Diffusion Equation. The linear relation given by eq 16 can be found in many other forms in the literature, one popular form being the so-called linear diffusion equation dC, J , = -D, - FJC, dx where p , is termed the mobility of species s, which can be shown’O to be related to the diffusion coefficient D, by the Einstein relation Ps = ZseDs/kBT (18)
+
A combination of electric-field-produced and concentration-gradient-produced current is sometimes referred to as field-modified diffusion,although this term is sometimes used more frequently in the limit where current varies nonlinearly with the field, as in anodic oxidation cases. Since expression 16 can be written as
so
Because the chemical potential u, of defect species s can be written as U, = U> kBT In c, (9)
+
the constant u,O being a reference value, we see that 0:
-du, / dx
(10)
That is, the diffusion force is proportional to the gradient of the chemical potential. The electric force F,m = 2,eE can be expressed in terms of the gradient of a potential, namely, the electric potential V. Since E = -dV/dx (11) we write dV d F,(E) = -Z,e - = -- (2,eV) dx dx The quantity 2,eV is the electrostatic potential energy of a charged point defect labeled s. Since the chemical potential u, is similarly an energy per particle, in this case the “chemical” energy as contrasted to the “electrostatic” energy, the proportionality factor in eq 10 is unity, so F,@) = -du,/dx (13) The total force on particle s is the sum of the two forces F,(D)and F,O, so we write
--d
dx
+ Z,eV)
(us
dii, (14) dx
= --
where E, = u,
+ z,eV = u? + kBT In c, + 2,eV
(15)
is known as the electrochemical potential of defect species s. The linear relation
then gives an expression for the total current produced by the combined effects of concentration gradient and electric
dC, -kBTBs - + Z,eB,CJ dx (19) we see by comparison with eq 17 that kBTB, = D, (20)
Our considerations for ionic defects can be extended to electronic defects, which by definition are either excess electrons or a deficiency of electrons (electron holes) relative to the electron concentration in the normal oxide. Whereas an excess electron constitutes an excess negative charge, an electron hole constitutes a missing negative charge and so constitutes in an oxide a local excess of positive charge at the position of the missing electron. Electron holes thus constitute species which are positively charged. The migration of an electron hole takes place by sequential motion of normal oxide electrons into the hole, quite analogous to the motion of an atom vacancy already described. The electrons and electron holes have chemical and electrochemical potentials in the same way as ionic defects, and motion of these electronic defects takes place in accordance with the linear diffusion equation (eq 17) in exactly the same way as described for ions. The linear dependence of current on force is an approximation to the true state of affairs, albeit often an excellent approximation. It often serves well in the domain of electric fields below lo5 V/cm (lo7 V/m), which as a practical matter is usually the case in thermal oxidation when the oxide film thickness exceeds 100 A. The linear approximation must be replaced by an exponential dependence when the fields are larger, which sometimes is the case in thermal oxidation when film thicknesses are less than 50 A and in the case of anodic oxidation where (10) Fromhold, A. T., Jr. Theory of Metal Oxidation; North-Holland: Amsterdam, 1976; Vol. I, p 488.
Langmuir, Vol. 3, No. 6, 1987 889
Physicists’ Viewpoint of Metal Oxidation the applied field is generally larger than the built-in field in thermal oxidation. The linear approximation was used by Wagner in his pioneering treatment? of thick oxide film growth. Space-Charge and Electric Fields. The defect concentrations C,, whether ionic or electronic in nature, vary with position x in the oxide, so in general we write C, = C,(x). The chemical potential u, thus is position dependent, u, = u,(x). The electric potential V likewise varies with position, so V = V(x). The electrochemical potential E, is thus also position dependent, so in general, 6,= E,(x). Because these quantities vary with the macroscopic position x in the oxide, we say that they are “local functions of position”. The local space-charge density p ( x ) is the sum of the individual local charge densities Z,eC,(x), so we write r
P(X)
= Cz,eC,(x) s=l
(22)
The parameter r in the summation denotes the total number of defect species in the oxide, with 1 Is I1 denoting the various ionic defect species and l + l Is 5 r denoting the electronic species. Whenever p ( x ) = 0, we say that we have local charge neutrality. Although a neutral metal reacting with neutral oxygen does not create any net charge, it does result in a separation of charge. This separation of charge occurs because the mobilities of the various charged defects differ greatly from defect to defect. A mobile electronic species migrating from the metal through the oxide, for example, leaves behind an equal quantity of positive charge. Charge densities u,(O) and u,(L)are thus established a t interfaces of the oxide. The interface at x = 0 is designated the metal-oxide interface and that at x = L is designated the oxide-oxygen interface, the distance L representing the thickness of a planar oxide. If we add all charges in the system, we must obtain zero, since no net charge is created during the oxidation. This condition of overall charge neutrality can be expressed” for a planar oxide as
Local charge densities p ( x ) constitute sources for the macroscopic electrostatic field E ( x ) . Poisson’s equation of electrostatics, namely, d2V = _P-
(24) dx2 e (in SI units), together with the relation E(x) = -dV(x)/dx between electric field E(x) and electrostatic potential V(x), leads directly to the relation dE(x) -P(X) --
(25) dx e for a planar oxide. The static dielectric constant e, which constitutes a measure of the screening effect of the oxide for electric fields, is associated with the electric polarization of the oxide by applied electric fields. Integration of the differential relation given by eq 25 yields
E ( x ) = E,
+ fSo’p(x)
dx = Eo
+
(11) Fromhold, A. T., Jr. Theory of Metal Olcidation. North-Holland Amsterdam, 1980; Vol. 11, p 58.
where the so-called “surface-chargefield” Eo can be associated with the charge density u(0) by means of Gauss’s law,
Eo
=
do)/€
(27)
The second term in the electric field in eq 26, namely, that involving the space charge p ( x ) , is referred to as the space-charge field. (To convert eq 26 in SI units to Gaussian cgs units, the e should be replaced by E / ~ A . ) Stresses in Growing Oxides. In analogy with the local charge density p ( x ) , we can define a local stress density S(x). Physically we see that the forces F,producing the currents J, must be countered by reverse forces exerted by the oxide medium on the charged migrating defects, since otherwise the forces F, would lead to ever accelerating charged particles and thus runaway currents J,. The scattering of the accelerated charged defects in the oxide medium is the physical origin of the reverse force.12 This scattering produces the local stress in the oxide medium. Since the total local force on species s is F,C,, namely, the product of force F, per defect with the concentration of such defects, the local stress S ( x ) is given by S ( x ) = kF,C, 8’
1
(28)
The local stress S(x) is the total force per unit volume sustained by the growing oxide at position x . Substituting for F,C, from eq 16 into eq 28 yields r
S(x) =
CJ,/B,
s=l
(29)
which is interesting since each term has the sign of the direction of the particle current. Thus, for example, cation interstitials and electrons migrating from the metal through the oxide to the oxide-oxygen interface where reaction occurs yield stresses which are locally additive. The Steady-State Approximation. For a planar oxide, any position dependence of J , leads to a time-dependent change in the concentration of species s. The so-called “equation of continuity” V.J = -ac,/at
(30)
is merely a mathematical statement of particle conservation in the absence of sinks and sources for defects within the oxide. It reduces for a planar oxide to the form
a c , / a t = -eJ,/ax
(31)
It is often a good approximation to neglect any time dependence of the defect concentration C, ec,/at = o
(32)
This is termed the “steady-state approximation”. When this holds, eq 31 reduces to =o
(33)
J A x ) = JAO)
(34)
aJ,/ax so that
The particle currenta in a planar oxide are thus independent of position for the steady state. This is called steady-state transport, which leads to the concept of “steady-state growth” of oxides. It can be noted from eq (12) (a) Fromhold, A. T., Jr. Surf. Sci. 1972,29, 396. (b) Fromhold, A. T., Jr.; Coriell, 5. R.; Kruger, J. J. Phys. SOC.Jpn. 1973,34, 1452. ( c ) Fromhold, A. T., Jr. In Stress Effects and the Olcidation of Metals; Cathcmt, 3. V., Ed.;The Metallurgical Society of AIME: New York, 1975.
890 Langmuir, Vol. 3, No. 6, 1987
Fromhold
29 that the stress S ( x ) is independent of position x in the limit of steady-state growth. Although the steady state often constitutes an excellent approximation, it can be violated quite seriously if the ambient conditions are suddenly perturbed. This could occur due to a sudden change in oxygen pressure or temperature or, in addition in the case of anodic oxidation, to a sudden change in the applied voltage or current. Non-steady-state transients would then be produced, with characteristic time constants determined by the migration parameters of the system.13 Oxide Growth Rate. Each ionic defect of species s which traverses the oxide yields formation of new oxide when chemical reaction occurs. In the steady state, we can write the oxide growth rate as (35)
where 1 migrating ionic species contribute to growth. The parameter R, is the oxide volume formed per transported defect of species s, having a sign determined by the direction of motion of the defect. When conditions prevail such that only a single migrating species need be considered, we speak of single-carrier transport. Generally we must consider more than one defect species, and we then use the term multispecies transport. Multispecies diffusion generally involves charged particles and electric fields, though it could also include cases in which two or more uncharged atomic species moved through the oxide. Likewise, single carrier transport could refer to the diffusion of a single uncharged atom species, though the term “carrier” usually means “charge carrier”. Approximation Valid for Certain Limiting Cases. The mathematical formalism outlined above can be simplified in various limits. Two distinctly different limits for the electric field can be delineated, these being termed the “homogeneous-field approximation” and the “local space-charge neutrality approximation”. The homogeneous-field approximation consists of stating that
E ( x ) = Eo (36) for the transport equations (e.g., eq 17), whereas the local space-charge neutrality approximation consists of stating that r
s=l
If p ( x ) # 0, then the electric field is said to be nonhomogeneous. In the homogeneous-field approximation the field Eo is still considered to vary as the oxide thickness increases. If the field is considered to be independent of thickness, we designate this condition the “constant-field approximation”. Note that the constant-field approximation is far more restrictive than the homogeneous-field approximation. The actual variation of the surface-charge field Eo with oxide thickness L is determined by the socalled “coupled-currents condition”, which states that to a good approximation the net charge transported by all species through the oxide layer is zero. Mathematically this is given by s=l
It is consistent with the fact that the newly formed oxide (13) (a) Butler, W. H.; Kinzer, E. T., Jr.; Fromhold, A. T., Jr. Phys. Lett. A 1972, 40A, 57. (b) Butler, W. H.; Fromhold, A. T., Jr. J. Phys. Chem. Solids 1974, 35, 1099. (c) Reference 10,p 94.
molecules are charge neutral. The coupled-currents condition (eq 38) has in practice yielded Eo values which are without exception L dependent, so the constant-field approximation does not seem to be a viable approximation for oxidation kinetics. In contrast, the homogeneous-field approximation turns out to be excellent so long as the experimental conditions remain unchanged. If surface erosion of metal atoms from the parent metal constitutes the rate-limiting step, as contrasted to migration through the oxide layer, then an almost constant field may be required to yield the required flux of atoms. If the voltage established across the layer produces this flux, then the voltage would be required to increase linearly with thickness if homogeneous-field conditions prevailed. (Such would not be required if space charge were large.) Thus we maintain as a possibility the constant-field approximation. It in fact has been found to provide a good empirical fit to the kinetics of oxidation of some metals at lower temperature^.'^
Basic Equations for Nonlinear Transport Verwey Model. The above description of currents obtained as a linear response to the force is to a certain extent empirical and phenomenological, based as it is on a liited clans of observations. However, the linear relation is found to be quite inadequate in extreme cases, as, for example, when the electric fields are very large, as is generally the case in anodic oxidation. Empirically it is found in that case that the current increases exponentially with the electric field. The exponential dependence of current on electric field can generally be understood in accordance with the model of Verwey; in which the electric field lowers the barrier for forward hopping over microscopic distances between adjacent potential energy minima. If the distance between adjacent potential energy maxima and minima is a , then the field lowering of the energy is ZeEa. An activation energy barrier of height Win zero field is reduced to W - ZeEa. On the other hand, reverse hopping against the field then must overcome a higher barrier given by W + ZeEa. By “hopping” we mean random jumps of defects promoted by the thermal vibrational motion of the oxide at temperature T. A defect thus is viewed as a particle in incessant motion in a “cage” of microscopic dimensions, somewhat analogous to the picture of atoms of a gas rapidly moving about in a container of macroscopic dimensions in accordance with the kinetic theory of gases. The collision frequency of the individual defects with the “walls” of the “potential energy cage” is called the attempt frequency Y . In physics the terminology of “phonons” and “phonon frequencies” is often used for thermal vibrations of a crystalline lattice. For media with localized disorder, there are localized phonon modes. By associating the attempt frequency Y with a typical phonon frequency, we come to the conclusion that v should have a value in the range 1013 to 10l6cycles per second. The attempt frequency v must be multiplied by the classical Boltzmann probability ptot to obtain the hopping frequency, where ptot is the probability that the particle will have an energy exceeding the barrier height Q,, after any given thermal excitation. In analogy with the above picture, the hopping frequency pbtv denotes the escape frequency of each defect from its partly transparent microscopic cage. (14) Reference 10, Appendix A; also Figure 1.12.
Physicists’ Viewpoint of Metal Oxidation
Langmuir, Vol. 3, No. 6,1987 891
To deduce pbt, recall that the Boltzmann probability p that a particle have an energy G after any thermal excitation is given by P
a
exP(-G /kBT)
(39)
where k B is Boltzmann’s constant and T is the absolute temperature. With the proportionality factor in this relation denoted by pot P = Po exP(-Q/kBT)
(40)
Let us evaluate pofrom the requirement that the particle have some energy value in the range (0 IG I m). Integrating p from zero to infinity yields pokBT,and setting this equal to unity gives po = (l/kBT). Then the total classical probability ptotfor crossing the barrier in any one attempt is simply the total probability that the particle will have an energy exceeding 6”. This is given15aby integrating p from G,, to infinity, Ptot
=
lm
POexP(-g/kBT) dQ = exP(-G,ax/kBT)
(41)
mu
The hopping frequency vpbt is thus [v exp(-E,,/kBT)] for each individual defect. A density n of carriers (number per unit area perpendicular to the hopping direction) will lead to a foward particle current density $0 given by $0 =
nu exp(-Gmax(fl/kBT)
(42)
and will lead to a reverse particle current density Jcr)given by j(r) = nv eXP(-G,,,(d/kgT)
(43)
where in our situation the maximum barriers Gmax(O and €mm(r)for forward and reverse motion are given respectively by
(45)
= $0 - J(r)
(46)
Discrete Barrier Hopping Equations. For a sequential series of N barriers labeled by the index k in a direction perpendicular to the planar oxide fii, we obtain a sequence of net currents J k , J k = nk-1V eXp[-(W - ZeEka)/kBT] - nkv eXp[-(W + zf?Eka)/kBn = V eXp(-W/kBT)[nk-l eXp(ZeE@/kBT) - nk eXp(-zeEkU/k~T)] k = 1, 2, 3, ..., N (47) These are the so-called discrete barrier hopping equations for the particle currents J k and the areal densities nk. The areal density nk-l precedes the barrier labeled k, while the areal density nk follows the barrier labeled k. The position Xk” of the maximum in the barrier labeled k is given by = (2k - 1)a
k = 1, 2, 3,
..., N
(48)
The film thickness is given by L = 2Na. The potential minima Xkmin are located at Xkmin
=
2ka
k = 0, 1, 2,
k = 0 , 1 , 2 ,...,N
p(Xkmh)
(50)
= ZeC(xkmin)= Zenk/2a
(51)
This species alone would then give rise to a position dependence of the electric field in accordance with eq 26. If there are several hopping species, then we must introduce a species index s. This is most readily accomplished by a superscript for the discrete variables, thus and wkcS).For simplicity, howleading to Z(”, nk(,), Jk(’), ever, in the case of the continuum variable C the species index s will continue to be affixed as a subscript to give C,, as is conventional to the linear diffusion equation (eq 17). In the case of several charged species, the space-charge density given by eq 51 must be replaced by I
p(Xkmin)
= CZ(s)eCs(xkmin) = CZ(s)enk(s)//2a(52) s=1
s=l
The integral of the charge density of species s can be written as a sum, k Q(Xkmax)
= x X b - p ( x ) dx = Cp(xjmin)2a= j-1
r
r
k
s=l
j=l
Z(s)enj(s)=
nj(s) (53)
The electric field given by eq 26 then takes on values at the barrier maxima Xk” given by
Ek = E(Xk-)
= Eo
+ Q(Xkmax)/t
(54)
With this factor of space charge included, the series of difference equations generally requires elaborate computer calculations. On the other hand, in the homogeneous-field limit, the equations can be solved by algebraic methods.15b The situation parallels that of the linear differential equations already presented (eq 17), in that the solution is simple for the homogeneous-field limit but more involved when local space charge contributes significantly to the electric field. Approximations Valid in Various Limits of the Electric Field. In the intermediate and lower field limits, we should expect the hopping formulation to yield the same predictions as the phenomenological linear approach previously considered. Equation 47 reduces for zero field to the form = (nk-1 - nk)V eXp(-W/kBT)
(55) However, the concentration gradient evaluated locally at position XkmaX is J k
(49)
The defect concentrations corresponding to planar areal Using this expression in eq 55 gives (15)Fromhold, A. T.,Jr.; Fromhold, R. G. In Reactions of Solids with Gases; Bamford, C. H.; Tipper, C. F. H.; Compton, R. G., Ede.; Elsevier: Oxford, 1984;Vol. 21, (a) p 38, (b) pp 48-51.
..., N
The difference equations (eq 47) are interlocked in the sense that each nk is involved in two equations, namely, in the specific equations for barriers k and k + 1. The solution of the set must therefore be carried out self-consistently. Hopping transport, as described by eq 47, is simply a microscopic world view of what we know as macroscopic diffusion and conduction processes. Space-Charge and Electric Fields in Discrete Variable Notation. The problem of solving the difference equations is even more involved than otherwise if the nk are large enough to constitute an appreciable space charge,
j=ls=l
The net current density Pet) over the given barrier is then
Xkmm
= nk/2a
(44)
= W + ZeEa
P e t )
C(Xkmin)
k
= W - ZeEa
€,(fl
densities separated by 2a are given by
892 Langmuir, Vol. 3, No. 6, 1987 Comparison with eq 5 yields the diffusion coefficient D in terms of the microscopic parameter values,
D = 4a2v eXp(-W/kBT)
(58)
Thus we see one important relation between the microscopic hopping approach and the phenomenological approach. If the electric field is relatively small but nonzero, the exponential functions in eq 47 can be expanded to give
The first term is the zero-field gradient term already noted in eq 55. The second term can be interpreted in terms of an average defect concentration at the location of barrier k,
In terms of this quantity, the second term in eq 59 becomes 4a2V eXp(-W/kBT)(Ze/kgT)C(xk"")E(xk) = (zeD/kgT)E(xk)C(xk"") = pE(Xk)C(Xkm") (61) The total particle current of this species at position Xkm" as given by eq 59 then takes the form
This result is identical with the linear diffusion equation (eq 17). Our correlation of the microscopic physical approach with the phenomenological approach is essentially complete. Steady-StateCondition in Discrete Variable Notation. The steady-state condition as given by eq 33 has in discrete notation the form k = 1, 2, 3, ..., N-1 (63) Jk+l = Jk for a planar oxide, quite analogous to eq 34. Using this condition as a reasonable approximation, the series of difference equations can be used to relate the areal densities from potential minimum to potential minimum, in much the same way that the differential equation for C(x) relates the defect density from one position to another position within the oxide. Boundary Conditions. Evaluation in the limit of the steady-state of either the differential equation for C(x) or the set of difference equations for nk hinges on the specificiation of appropriate boundary values of C(0) = 2ano and C(L) = 2anN at the metal-oxide and oxide-oxygen interfaces. These are called the boundary concentrations. If transport through the oxide is rate determining for all species, then the boundary values of the various defect species are established by interfacial equilibrium conditions and hence are independent of the oxide film thickness. The free energy of formation of the oxide represents one important constraint on the boundary concentrations of the various defect species. Essentially, the sum of the chemical potential differences across the oxide for the elemental number of charged particles required to form a new oxide molecule must equal the free energy of formation of the molecule.
Figure 2. Sequential place-exchange events: (a) before place exchange; (b) after first event; (c) after second event; (d) after
third event.
In the other extreme in which the rate is limited at an interface, a linear rate of oxide film formation can often be expected. It is possible to combine these limiting behaviors into a single growth expression, as was done in one especially simple way by Deal and Grovel6 in the case of silicon oxidation.
Extending the Limits of Validity of the Hopping Model Vitreous Substances. A more extensive discussion of the hopping model can be found in several revie~s.l'-~~ In particular, it can be argued that the model applies to vitreous substances and amorphous substances, which have short-range order at best, in addition to crystalline substances, which have long-range order. Nonsimultaneous Place Exchange. The hopping model can also be used even if the mobile defects are more complex entities than single vacancies or interstitials. To give an example, consider a "place exchange" event in which a chemisorbed oxygen ion at the oxide-gas interface exchanges position with an adjacent normal metal ion in the oxide medium. If row a in Figure 2 represents the normal array of metal-oxygen-metal atoms in the oxide, with the left-most positive charge representing a metal ion located just within the parent metal surface and the right-most negative charge representing a chemisorbed oxygen ion at the oxygen phase, then row b illustrates the new order after the place-exchange event. Note that the event is equivalent to the motion of one metal ion one nearest-neighbor distance to the right, with the simultaneous motion of one oxygen ion one nearest-neighbor distance to the left. The positive charge has moved to the (16) Deal, B. E.; Grove, A. S. J. Appl. Phys. 1965, 36,3770. (17) Fromhold, A. T., Jr. In Oxides and Oxide Films; Diggle, J. W.; Vijh, A. K., Eds.; Marcel Dekker: New York, 1976; Vol. 3. (18) Fromhold, A. T., Jr. In Passivity of Metals; Frankenthal, R. P.; Kruger, J., Eds.; The Electrochemical Society: Princeton, NJ, 1978. (19) Dignam, M. J. In Oxides and Oxide Films; Diggle, J. W., Ed.; Marcel Dekker: New York, 1972; Vol. 1.
Physicists’ Viewpoint of Metal Oxidation
Langmuir, Vol. 3, No. 6, 1987 893
At Oxide- Oxygen Interface
Chain I
Surface Layer
Bulk Oxide
0
I
+ - + - + - - I
I
I
+
-
+ -I+l
4
-
N
t - + - + - +
2
3
0
+
-
+
-
+
-
+I-I
5
+ + - + - -?. +-+-!--I+-+ +-[3+-+-+ - + - + - + __-_ - + - + - + - +
N
- + - + - + - +
I
2 3 4
7-7 I - ! +
ll
Figure 3. Successivechains of hopon events leading to new oxide growth at the oxide-oxygen interface. Rows show ion configuration near the oxide-oxygen interface following sequential place-exchange events.
right and the negative charge has moved to the left, though at this point the distances are only of the order of one nearest-neighbor distance. Row c in Figure 2 illustrates a second place-exchange event which occws some time after the first. The adjacent negative charges side-by-sideafter the f i t place-exchange event repel one another, thus constituting an unstable configuration which promotes the second place exchange in which the adjacent cation-anion pair flip positions. The new order after the second place exchange illustrated by row c sets the stage for a third place exchange, which is illustrated in row d of the figure. The sequence of place exchange events can continue until finally the metal ion in the parent metal has been incorporated as new oxide. At that point in time the sequence of place-exchange events has led to an overall result which (if the ions of a given species are indistinguishable) is physically identical with the movement of one metal ion through the oxide from metal to gas phase and the movement of one oxygen ion through the oxide from gas to metal. Chemical reactants as well as charge have been transported. If we call the sequence of events just described a “chainn,then successive chains lead to continuous buildup of new oxide at the oxygen interface (Figure 3), with a simultaneous continuous buildup of new oxide at the metal interface (Figure 4). Direct application of the equations of the hopping model to treat quantitatively the place-exchange events just described can be justified.20 The essential point is that the position of the excess charge (see Figure 2) continuously changes during the course of a chain. The moving local (20) Fromhold, A.
T.,Jr. J. Electrochem. Soc. 1980, 127, 411.
Figure 4. Successive chains of hopon events leading to new oxide growth at the metal-oxide interface. Rows show ion configuration near the metal-oxide interface following sequential place-exchange events.
excess charge is a “virtual” defect which has been referred to20 as a “hoponn. Since the hopon can in principle move in either direction, ik3 motion under thermal activation will be statistical. A net motion of hopons in one direction will result if there is a hopon concentration gradient. Because the event is envisioned as being thermally activated, there is an activation energy barrier for each place-exchange event. The barrier can be modified by an existing macroscopic electric field which aids the motion of the positive charge along the field while simultaneously aiding the motion of the negative charge in the opposite direction. Thus we have all essential ingredients for direct applicability of the hopping model. Summary of Manifold Aspects of Hopon Transport. The most important deductionsz0 for transport by the hopon process are the following: 1. New oxide forms simultaneously at both interfaces. (The new metal ions are incorporated at the metal-oxide interface, and the new anions are incorporated at the opposite interface, with the initial step occurring in principle at either interface.) 2. The microscopic order of the cations and anions within the already formed layer is preserved during oxide growth. 3. The transport numbers for the oppositely charged ionic species are of the same order of magnitude. (A cation and an anion carrying equal magnitude but opposite charges advance equal distances in opposite directions in each hopon event, so equal amounts of charge are carried by each of the two species and equal quantities of oxide will then form at the two interfaces.) 4. The transport rate will be markedly dependent upon built-in and applied electric fields. (Charge is transported in each direction, but the sign of the charge transported
894 Langmuir, Vol. 3, No. 6, 1987
Fromhold other under low-field conditions.) Thus when transport numbers are found to be nearly equal for the two different ions making up an oxide, there is an indication that transport is occurring by a correlated process rather than by a normal point defect hopping mechanism. Coupled with the transport number data is the powerful evidence from radioactive tracer measurements indicating that radioactively "tagged" oxygen is incorporated into the oxide at the oxygen-bearing phase. This contrasts markedly with the results to be expected for an oxygen point defect transport mechanism where the migrating oxygen would move entirely through the oxide layer before chemically reacting with the metal. Thus, cogent arguments based on the independent data provided by markers and tracers can be made for the hopon process in certain real systems. Whether one microscopic process or another dominates during oxide growth is of course a function of the microscopic physical characteristics of the oxide system in question. The inclusion of the hopon process within the framework of the point defect hopping model broadens our perspective and generalizes the model numerical calculations carried out to date. That is to say, the hopping model which we use to evaluate the kinetics of oxide growth can simulate a variety of distinctly different microscopic diffusion mechanisms.
in opposite directions is opposite, leading to a net charge transport; the virtual charge acted upon by the field to promote a hopon event is thus the sum of the anion and cation charge magnitudes, so the potential energy lowers the zero-field activation barrier for the event.) 5. The local region surrounding a hopon carries a net charge which is equal to the virtual charge of the hopon, so space charge contributes to the macroscopic electric field and thus modifies the transport rate accordingly. 6. A concentration gradient in the density of hopons leads to a net transport away from the high-density region even in zero field, so the gradient represents a driving force which acts in addition to the driving force of the electric field. 7. Stress effects follow from the local driving forces of the concentration gradient and the electric field, analogous to the stresses given by eq 28 for individual hopping defects. 8. There being a one-to-one correspondence between every aspect of the rate of hopon transport and the rate of transport of individual point defects, the kinetics of oxide formation are thus predicted to be the same as the kinetics deduced by means of the point defect hopping model. Correlation with Experimental Observations. The higher the electric field, the more likely it is that hopon motion will be induced, since the activation energy for the place-exchange event is apt to be large. High electric fields are common in anodic oxidation, so this is a promising field to search for evidence. Experiments by Amsel and Samuelz1show that oxygen atoms are fixed according to their order of arrival during the formation of amorphous compact layers on aluminum and tantalum. That is, the oxygen atoms belonging to a new layer of oxide are located a t the oxygen-solution interface. Experiments by PringleZzainvolving radioactive noble gases show that both Ta and 0 migrate during the anodic oxidation of tantalum, and these migrations occur simultaneously with the transport numbers being of the same order of magnitude (t = 0.243 for Ta and t = 0.757 for 0). Furthermore, the new oxide was formed a t the metal-oxide and oxide-electrolyte interfaces only. In other experiments,22btantalum was anodized first in l60electrolyte and then in l80electrolyte, and afterward the oxide was sectioned by slow dissolution in concentrated HF almost saturated with NH4F. The l80present was detected by bombarding the oxide with 3.042-MeV protons and counting the neutrons emitted in the 180(p,n)18Freaction. I t was found that the l80incorporated last was outside the l60layer incorporated first, in agreement with the conclusions of Amsel and Samuel.21 In addition, there was found a small mixing of the populations of the two oxygen isotopes at the boundary between them, and the degree of mixing was found to be proportional to the square root of the thickness added in the l80electrolyte. It is quite significant that such experiments indicate that oxide is formed at both interfaces of the oxide film during some anodic processes. This would happen for ordinary independent motion of point defects only if the ionic conductivity is the same order of magnitude for metal ions and oxygen ions, which is considered to be unlikely for point defect motion in most systems. (Transport numbers t, giving relative measures of the conductivity contributions of the various carriers usually differ widely from one an-
Applications of the Mathematical Formalism Anodic Oxidation. Our first illustration of the application of the discrete model formalism described above is to the case of anodic oxide film formation. This is a single-current situation in which the nonlinear dependence of current on the electric field must be considered. It is common practice to consider the limit of the hopping equations in which the applied field is so large that the forward current $0 exceeds $) sufficiently to neglect the latter. This is equivalent to choosing the concentration gradient to be larger than it could be expected to be for equilibrium conditions at the interfaces. In addition, it is common practice to neglect space charge effects. To check the importance of the effects usually neglected, namely, the back current associated with the true concentration gradient and the effects of the space charge, numerical computationsB%were carried out for a sequence of constant voltages including both of these effects. The results of one set of computations are shown as solid curves in Figure 5. To separate out the specific contribution of the space charge, the computations were repeated without including the effects of space charge. The zero-spacecharge results are shown as dashed curves in the figure. For a detailed interpretation, reference can be made to the literature.23 In brief, the common curvature in both solid and dashed curves illustrated on the log-log plot in the figure in the small thickness domain represents the nonlinear effect of the very high electric field. (The accumulated space charge is relatively small in the limit of small thicknesses.) On the other hand, the marked difference in slopes of the solid and dashed curves (cf. curves in figure) corresponding to a given voltage reflects the effects of space charge. (Note that reverse polarity potentials can yield anodic oxide growth if the concentration gradient is sufficiently large.) Two important points are to be noted: First, the current at a given voltage and oxide thickness is predicted to be much less whenever space-charge effects are appreciable, as can be concluded from current ratios exceeding 100 for
(21) Amsel, G.; Samuel, D. J. Phys. Chem. SoEids 1962, 23, 1707. (22) (a) Pringle, J. P. S. J. Electrochem. SOC.1973, 120, 398. (b) Pringle, J. P. S. J.Electrochem. SOC.1973, 120, 1391.
(23) Fromhold, A. T.,Jr.; Kruger, J. J.Electrochem. SOC.1973, 120, 722. (24) Fromhold, A. T., Jr. J.Electrochem. SOC.1977, 124, 538.
Langmuir, Vol. 3, No. 6, 1987 895
Physicists' Viewpoint of Metal Oxidation
I10 millitorr
40
20 I 20
0
I
l
/
I
I
I
I
I
I 40
I
I 60
I
I
I
I
80
I
/
100
t (minutes)
Figure 6. Plasma-produced oxide film thickness vs. time of oxidation in different oxygen pressures: (solid curves) theoretical computations; (points) experimental data. I
IO0
101
I
I02
\
I
t
I03
N (MONOLAYERS)
Figure 5. Defect particle current vs. anodic oxide thickness under constanbvoltage conditions including concentration gradient due to fixed unequal boundary valuea of defect densities, (solid curves) including space-charge effects and (dashed curves) ignoring space-chargeeffects.
a given voltage and f i i thickness, which are evident from the figure. Second, the usual interpretation in which space-charge effects are omitted yields erroneous values for the microscopic parameters whenever space-charge effects are truly appreciable. Plasma Oxidation. Our next application of the discrete model formalism is to the case of plasma oxidation. This likewise involves a single current in which the nonlinear dependence of current on the electric field must be considered, so the model is quite similar to the problem of anodic oxidation. An additional effect which must often be considered in the plasma oxidation case is the loss of oxide due to sputtering by impinging high-energy ions from the gaseous plasma. (However, an analogous dissolution of oxide into the anodic solution sometimes occurs in anodic oxidation.) By neglecting explicitly the back current, the hopping equations can be expressed in a form which can be integrated without difficulty. The fundamental equations25 include the net particle current Jnet = Jo - Jsput (64) where J, ut is the equivalent current due to sputtering by ion bomgardment and Jo = [\kDC(O)/2a] exp[-V(L)/E*L] (65) where V(L), the electrostatic potential a t the outer interface of the oxide, is a measure of the voltage developed across the oxide layer. The parameters \k and E* are definedz6by \k = e(1 + (L/x*))-[l+(~*/L)l (66)
E* = kBT/Zea
(67)
where X* = ek~T/4?r(Ze)~Uc(O) (68) The parameter e 2.718 in eq 66 is the base of the Naperian logarithm system. The parameter \k is a measure
(25) Fromhold, A. T.,Jr.; Baker, J. M. J. Appl. Phys. 1980,51,6377.
1,
40 0
I 40
,
I 80
,
I
,
120
I 160
,
, 200
t (minutes)
Figure 7. Oxide film thickness vs. time under increasing and decreasing growth conditions: (initial growth phase, 5 mtorr of oxygen pressure; next growth phase, 20 mtorr of oxygen pressure; decomposition phase: 5 mtorr of oxygen pressure. of the effects of space charge, reducing to unity in the limit of negligible space-charge effects. The growth rate &/dt of the oxide is given by RJ,,,,. This model has been applied to experimental data26on the oxidation of lead. These results are shown in Figures 6 and 7. The experimental data points can be noted to fall directly on the computed solid curves. The oxide film thicknesses, which are below 100 A, were measured by the technique of ellipsometry by Greiner.26 The second of these figures shows how the sputtering of the oxide by impinging oxygen ions from the plasma can reduce the thickness of the oxide. Further details of the comparison of theoretical and experimental plasma oxidation kinetics curves can be found in the l i t e r a t ~ r e . ~ ~ Thermal Oxidation. Let us now examine applications of the theoretical formalism for charged particle hopping to the problem of thermal oxidation. In thermal oxidation there is no external circuitry; thus there is no outside path available for the transfer of electrons. In this situation the electrons, like the ions, must migrate through the oxide layer itself. Because both electrons and ions must be transported through the oxide in thermal oxidation, any application must involve at least two mobile defect species (one ionic and one electronic). The electric field is jointly produced by all mobile charged species. It can be deduced in the limit of steady-state growth by means of the cou(26) Greiner, J. H. J. Appl. Phys. 1971, 42, 5151.
896 Langmuir, Vol. 3, No. 6, 1987
pled-currents condition given by eq 38. For example, the electrons can tunnel through the thin oxide layer simultaneously with field-modified diffusion of an ionic species, and coupling of these two currents in accordance with the coupled-currents condition (defined by eq 38) leads to the “electron tunnel model” of oxidation kinetimn Similarly, there is an “electron emission model” for thermal oxidation* in which the electrons are thermally emitted over a metal-oxide work function barrier simultaneously with diffusion of ionic species, the coupling of the two currents in accordance with the coupled currents condition again yielding the surface-charge field Eo as a function of oxide film thickness. Space-charge-modified thermal electron emission also leads to an important thermal oxidation Apart from the nondiffusion transport of electrons described above, there is of course the transport of electrons by field-modified diffusion. In that case the hopping transport equations are applicable to the electronic species in the same way as utilized for the quantitative description of ionic transport. This leads to the space-charge-modified ‘electron diffusion model” of thermal oxidation kinetics.s0 Numerical computations carried out into the thick-film limit have shown that this model exhibits the phenomenon of local space charge n e ~ t r a l i t y . ~ ~ Because the built-in voltages for thermal oxidation are often not very large, compared to the values which may be attained in anodic oxidation where the voltages are applied, the linear dependence of the current on electric field proves often to be an adequate approximation for oxides with thicknesses exceeding 50 A. For oxide film thicknesses below 50 A or so, however, where electron tunneling can be effective, the voltages can be larger and the electric fields much higher. In that case the nonlinear relationship between current and field (i.e., nonlinear diffusion) must be utilized. Numerical evaluation^^'^^^^^ carried out for the three different oxidation models, each considering the nonlinear diffusion of ions but with the different electron transport mechanisms of electron tunneling, thermal electron emission, and nonlinear electron diffusion, delineate those regions of thermal oxidation where nonlinear diffusion effech do occur. In almost all cases, nonlinear effects due (27) Fromhold, A. T., Jr.; Cook, E. L. Phys. Reo. 1967,158,600. (28) Fromhold, A. T., dr.; Cook, E. L. Phys. Reu. 1967,163,650. (29) Mosley, R.B.; Fromhold, A. T., Jr. Oxid. Met. 1974,8, 19,47. (30)Fromhold, A. T., Jr.; Cook, E. L. Phys. Reo. 1968,175,877.
(31)Fromhold, A. T.,Jr. Phys. Lett. A 1976,58A,118.
Fromhold to very large electric fields are found only in the region of thicknesses below 50 A. The most general evaluations of coupled-currentsmodels outlined above involve extensive numerical calculations. In certain limits, however, approximations lead to analytical evaluations of the kinetics which are consistent with the corresponding numerical evaluations in those limits. The most important of these analytical approaches lead usually written in the form to parabolic laws ( t cc L2),
L cc t ’ / 2 (69) for the growth of thin oxide films as well as for the growth of thick oxide films. The thin-film parabolic growth is baaed on the neglect of space charge and the consideration that both ions and electrons are diffusing through the existing oxide film in conformity with the linear diffusion equation (eq 17). The thick-film parabolic growth law,= which holds in the limit where the oxide film is thick enough for local space-charge neutrality to be a good approximation, again considers both ions and electrons to be diffusing through the existing oxide film in conformity with the linear diffusion equation. It is somewhat incredible that the thin-film and the thick-film parabolic laws have vastly different rate constants, a point which is still too little appreciated by many researchers in the field. Some information on the differences between these rate constants is given in the literat ~ r e The . ~ ~ experimental measurement of rate constants together with the voltages developed during the oxide growth represents a very important experimental area of future research. At present there are insufficient data to carry out a definitive correlation with the theoretical predictions. The concentration gradients are also very important in the thermal oxidation problem, precisely because the built-in voltages are not inordinately large. Since the concentration gradients must therefore be treated so carefully, it is important for quantitative estimates of the rate constant to have good estimates of the defect densities near each of the two interfaces of the growing oxide layer. Such measurements have still not been carried out, despite the multitude of new experimental techniques which have become available over the past two decades for probing surfaces and thin films. This is another challenging and important area for the experimentalist. (32) Fromhold, A. T.,Jr. J. Phys. Chem. Solids 1972,33,95. (33) Fromhold, A. T.,Jr. J. Phys. SOC.Jpn. 1980,48, 2022.
