Sept., 1960
INTERRUPTED OXIDATION OF
IXSB
1113
OXIDA4TIOKOF IXTERRIETALLIC CORIPOUKDS. 11. IKTERRUPTED OXID-4TIOK OF InSb' RrA
A
~ J. ROSENBERG ~ ~ ~ - ~
Liricnln Laboratory, 3fnssnchzisett.s Institzife of TeChnOlOgy, Lexingfon 73, JInssachztsetts Recezued .l!aich 7 , 1960
The later stages of the oaidation of InSb above 300' are limited by the migration of atomic defects through the grox-ing film of In.03. By utilizing a new kinetic procedure it has been shown that the concentration of defects is sufficient t o confine any space charge to negligibly thin regions a t the boundaries of the oxide film. The motion of defects is, accordingly, a field-free diffusion process. The experimental method is based on rate measurements following interruptions of the reaetion, and permits the simultaneous determination of D,the diffusion coefficient, and Co, the solubility, of the mobile defect in In20a. The necessary mathematical formulation has a simple analytical form and contains a rigorous internal check. In the temperature range 308-405", the quantities D and Co are given by D = 0.0078 esp( -31200/RT) cm.2/sec., and Co = 9.4 X loazexp( -5400/RT) defects/cm.3. The results find a consistent statistical interpretation when it is assumed that the defects are interstitial indium cations, and that a heavy mass approximation applies to conduction electrons in In& The values of D and Co decrease gradually with increasing film thickness in the range 600-1500 the effect is not related, however, to thtmnnl aging of the o d e . Evidence is also prcsmted that antimony can penetrate In-03during the interruptions.
w.;
I. Introduction I n the preceding article of this seriesj2 the oxidation of InSb in the temperature range 212494' was described. After a brief initial reaction during which Sb203 vaporizes from the surface, a protective oxide film consisting largely of InzOa forms and restricts further reaction to the simultaneous formation of In203 and of elemental antimony which accumulates a t the In203-InSb interface. 'The present article is concerned with the mechanism of the latter phase of the oxidation. The driving force of the reaction is the free energy of oxidation of InSb which resolves itself kinetically into a gradient of electrochemical potential under which a diffusible species is driven through the growing oxide film, whence3 D
fined to layers whose thicknesses are similar at each interface and are given to an order of magnitude by
x
-(
(3)
where K is the dielectric constant of the oxide; n is the average mobile charge density, and the other quantities have their usual meaning. As long as X is much larger than A, the more mobile charge carriers are constrained to m o w with the slower ones in the bulk of the film, n-hich then remains neutral. The reaction is then under thermal diffusion control, and eq. 1 yield.: dt
=
h.?X-l;
I:?
=
QD ICo - CI1
(1)
where Q is the volume increase associated with the oxidation of one substrate atom, and Co and C1 are the diffusing ion concentrations at the iiiner where jl(x) is the flus of the ith species across a and outer interfaces, re~pectively.~ boundary a t a distance, x, measured from the inner When X is comparable to A, however, fieldsurface of the oxide, D,is its diffusion coefficient, induced migration in the boundary layers can inn,(x) is its concentration. pl(x) is its chemical fluence the rate. h general analytical solution of potential, and z, is its charge. NL is the Lo- eq. 1 is not possible, but for the case where X is Schmidt number, e is the electronic charge, E is much smaller than the A's, sereral special ~ o l u the electrostatic field and R and T have their tions have been d e ~ e l o p e d . ~ -Each ~ aqsumes usual meaning. For uncharged species, z , = electronic equilibrium a t the oxide-O? interface. 0 and eq. 1 can be solved to give the rate of increase While this assumption has been of the film thickness, ,Y since it requires that the electrons I1ioT.e easily over a considerable energy barrier, it does lead to practical solutions which give a qualitative picture where IC, is :t constant for given experimental condi- of thin film oxidation kinetics. For this ciondition tions. Generally, however, the diffusing species Grinileyja writes eq. 1 RS are ionized and can move independently of t,heir count,ercharges. The ionization of adsorbed oxygen atoms and the differing tendency of ions and electrons to enter the oxide creates local charge where E depends on X and t,he defect character of the oxide film. For p-type oxides, in which the inibalaiices a t the boundaries of the ~ x i d e .These ~ predominant defects are cation vacancies and eleccharge imb:ilances, called space charges, are contron holes, E = 1 a t very low X and increases to *Materials Rvsearcli Lab.. TTCO, Inc., Kalthani, Mass. 2, 3, or even higher values depending upon the (1) T h e w o r c reported was performed a t Lincoln Laboratory, valence on the vacancy. For n-type oxides in a center for rewerch operated b y Massachusetts Institute of Techjdz) =
- & n,(z) {graddz) + N,z,eE(z))
(1)
nology with t h e joint support of t h e Ti. 8. Army, Navy and Air Force. ( 2 ) A. J. RoPenbrra and AI. (:. Lavine, THISJOURTAL,64, 11333 (1960).
(3) See I> l O I 7 cm -3
(6)
If eq. G is satisfied, there exists a practical kiiietic procedure for measuring the diffusion coefficient and the concentration of the particles whose diffusion controls the growth of the oxide layer. This procedure is based 011 the controlled interruption of the oxidation. 11. Theory of Interrupted Oxidation Kinetics Consider the generalized oxidation of a metal, taking place a t a flat surface y v
+ 2 02 = M,O,
(7)
The reaction is assumed to he controlled hy the field-free diffusion of ions or M atoms through interstitial positions in the growing monocrystalline oxide lattice. 31 atoms enter the oxide a t x = 0, migrate through the film under the influence of their concentration gradient, and complete the reaction with oxygen a t the oxide/oxygen interface (x = 9).The concentratioii of interstitials a t J = 0 is C0while that a t x = X i s C,. These values correspond to those which would be observed if imlnted M,O, were at equilibrium with oxygen a t n pressure (Po2)equalling, respectively, the dissocintion pressure of the oxide (PCliss) and the ambient pressure of the evperiment ( P J . A i tany rnomcnt ( 7 ) G Rupp echt Z. Phyazk. 139, 504 (1954). (8)n r a n he determined in lirinciple by niemuring the elertrical conductivitv 0 of the film and ZBS iniing carrier niobilities for use in t h e expression. r = n r e p . B y partially immersing a n oxidized sample in mercury, u can h e measitmi, h u t the method IS insensitive, and, in t h e present case led only t o a lower limit of lO’5/cc. for n.
Vol. 64
( a ) DURING FILM GROWTH CO I I
x=o
X’X X
(b) DURING INTERRUPTION I
I
I
x=O
X=X 1
Fig. 1.-Concentration of interstitial cations in a film of M,O, growing upon a metal substrate: ( a ) during growth;
(b) during interruption.
D ( d 2 C / b x 2 )= dC/dt = 0 throughout the film so that C varies linearly with x as shown in Fig. la. The diffusion coefficient, D,is taken to be independent of C. Assume nom that the ambient O2 pressure is dropped from P I to J)dlss. Experimentally, this is easily accomplished by removing excess gaseous 0 2 from the apparatus and sealing off the sample enclosure. The sample then acts as a source or a sink, and the oxygen pressure in the enclosure automatically adjusts itself t o P d l s s . 31 atoms continue to enter the film a t x = 0, but are no longer removed a t x = X. The M concentration approaches the equilibrium value, Ca, throughout the film, and reaction ceases (Fig. lb). If the pressure is then raised to P I , C(X) is again reduced to C1. The cations resume their outward flow and the concentration distribution of Fig. l a is re-established If COX is large enough, this “restoration” of the steady-state concentration gradient should resolve itself as a relatively fast initial uptake of oxygen followed by a gradual return to thc rate given by eq. 4. lllathematically, the problem may be approximated :is follons. It is assumed that cations may be created or destroyed only a t x = 0 and s = X, and that their diffusion coefficient in the oxide is independent of C. Then, a t a givcn temperature As bouiidary conditions, one takes at t = 0: at t > 0 :
(‘(z) = C, for nll
(’(0)
=
C(X) =
Co
c,
z (0)
where t is measured from re-admission of 0 2 . The equation is readily solved by standard techn i q u e ~giving ,~ (0) J. Crank, “The Mathematics of Diffusion.” Oxford, 1956, p . 11
ff
INTERRUPTED OXIDATIONOF IxSn
Sept., 1900
1143 TABLE I
NUMERICAL SOLUTION FOR
TEE
DIFFUSIONEQUATION OF
INTERRUPTED KINETICS,EQ. 12 '
e-D'n2t
+ P
sin (2m f 1) T
+
,,2m+l
>
e-D'(2n+1)2L
(10)
where D' = D r 2 / X 2 . By taking the derivative of C(s) with respect to x , evaluating it a t x = X , and integrating from t = 0 to t = t, one obtains an expression for A, the number of h'f atoms crossing unit area of the outer interface in time t.
For most practical purposes C1 1). The model predicts, furthermore, that B should be independent of X ; the following results establish, howeTTer, that B diminishes steadily with increasing X. An oxidation was interrupted at several stages of film growth. The data obtained upon resumption of reaction a t each stage are shown in I;ig. 5 . The formal predictions of the thermal diffusion model are separately satisfied in all but the fir.;t stage, the solid lines being drawn to eq. 12. The interrupted rate parameters are summarized in Table 11. 0.267B2/QR is close to unity in each case. B and RA- diminish systematically with increasing X. The decline cannot be attributed t o D or Coalone, since both decline. D. The Effect of the Interruption Conditions upon the Rate Parameters-Prorided that the time of interruption, t,, is long enough t o permit equilibrium of the interstitial atomq throughout the film, i e . , L1 > X 2 / D , the basic diffusion parameters should be unaffected bj7 further increases in t,. d n experiment was performed in which t , n as increased successively from 4.5 hours to 16 hour.: for a sample in which X 2 D was estimated to be 3.0 hours. The results, shown in Fig. 6, confirm the invariance of the diffusion parameters. The p:trameter 2, however, shows a definite time-dependence. Another experiment was designed to examine the effect of the temperature of iiiterruptioii, T,, upoil the kinetics. During the interruption of a reaction being conducted a t 342", the temptmture was
ARTHURJ . ROSEKBERG
1148
vARlATION O F INTERRUPTED
Q R RN 0.267B2/QR ( X C O )eq. 15 { D / X 2 ]eq. 17
co
TABLE I1 RATEPARAMETERS WITH FILMTHICKNESS, 367"
X 1017 oxygen atoms/cm.2 X 1015 oxygen atoms/cm.2 x 1015 oxygen atoms/cm.2/min.'/z X 10'6 oxygen atoms/cm.2 X 10i4 oxygen atoms/cm.z/min. x 1 0 3 1 oxygen at,orns2/cm.'/niin
A'
Z B
X X X X X
1.70 2.62 1 .53 3.40 3.18 5.39 0.59
10'6 oxygen atoms/cm. 10-smin.-l 10-6cm. cm.2/sec. 1020 indium d e f e c t ~ / c m . ~
raised to 494" for two hours before equilibrating for 16 hours a t 342". The results are shown in Fig. 7 . Ideally the diffusion kinetics should have been independent of this treatment, but its severity apparently introduces irreversible changes in the oxide structure, and small changes in B and R were noted. The principal effect is again associated, however, with the parameter, 2. These results demonstrate that the apparent decreases in D and Cowhich are observed with increasing extent of reaction, are not related to aging of the oxide, but are truly dependent upon the oxide thickness. E. The Parameter 2.-It was assumed a t the outset that the parameter Z given by the intercept of the AN 21s. t'/z curves, reflects a reaction of the oxide which proceeds parallel with, but independent of, the slower diffusion reaction described by eq. 12. A striking confirmation of this view lies in the results of the preceding section: substantial increases in 2 were observed when the time and temperature of the interruptions were increased, while the parameters of eq. 12 were virtually unaffected. The noteworthy features of Z which are evident from all of the foregoing results are: (i) the observed values of 2 lie in the range 1-5 X 10'j oxygen atoms/cm.2, corresponding to 1.3-8 oxygen atoms/surface atom. It is thus evident that the reaction which underlies 2 cannot be a simple chemisorption of oxygen on the surface atoms of the oxide; (ii) the reaction occurring during the interruption which sets the stage for 2 is time- and temperature-dependent ; (iii) the reaction in which 2 is actually measured is complete within one minute a t the higher temperatures and on the thinner films, while it is measurably slower a t the lower temperatures and on the thicker films.
V. Discussion
Yol. 64
2.80
3 73 2.46 1.34 3.38 1.43 5.33 0.99 1.01 14.2 6.15 8.90 11.1
5.02 2.73 1. I 4 4.10 0.87 4.36 0.96 1.20 7.25 8.28 8.25 9.65
9.43 2.46 0.89 0.77 0.318 2.99 0.99 1.99 1.59 15.56 6.41 8.50
7.11 3.18 0.97 5.13 0.502 3.57 0.97 1.49
3.35 11.73 6.56 8.45
AN ,/PARAMETERS = N x 10-1'OXYGEN A T O M S / ~ ~ ~ ( curves disploced vert c o l l y )
,
2
0
I
I
I
1
4
6
B
IO
. A 2
14
6
8
V'M INu T ES .
Fig. 5.-Variation of interrupted rate parameters with Y X lo-" oxygen a t o n ~ s / c m . ~ . film thickness. Key = i For all but N = 1.8 Xo 10'7, the solid lines are calculated from eq. 12. 2' = 367 . The oxide thickness is obtained by multiplying N by Q / F = 1.64 X 10-23 (section IVA). Thus the measurements cover the range 300-1500 A.
0
I
The excellent description of the experimental __ 0 5 0 data given by eq. 12 over a broad range of temperaJM I N U T E s ture and film thickness leaves little doubt that the Fig. 6.-The effect of the interruption time: lower curve, oxidation of InSb above 300" is controlled by the oxidation a t 342" resumed after 4.5 hours interruption; diffusion of defects through the growing oxide film. upper curve, after 16 hours interruption. The diffusion equation itself cannot be used to identify the defect, since the equation applies sure, and increases upon heating a sample under equally to metal or oxygen ions, atoms, or vacan- vacuum. This can only be true if In203is a metalexcess semiconductor, and the mobile defect must, cies. It was shown by Rupprecht' that the conductiv- accordingly, be either oxygen ion (or atom) vacani:y of In203decreases with increasing oxygen pres- cies or interstitial indium ions (or atoms). For
INTERRUPTED OXIDATION OF INSB
Sept., 1960
11-19
-
which gives Co lo2' in the temperature range of this study. This high value, which corresponds to 5% occupation of the available interstitial sites, complicates the interpretation of eq. 22. By examining certain limiting models, however, it can be shown that the preexponential factor in eq. 22 is of the proper magnitude. Some light is also shed on the interpretation of the exponential term, which is the key to understanding the relative rates of oxidation of In and InSb presented in the preceding article. Consider the transfer of an indium atom from InSb to an interstitial position in In203by the reaction
-1
InSb(s) = In(In203)
I'
p '
In(In?03) = In+" (In&)
/
-
2 5
Jrn
.
reasons which will become clear, it has been assumed that the latter are the predominant defects in Inz03. The Diffusion Coefficient D.-The fundamental parameters of the diffusion equation-the diffusion coefficient D and the equilibrium concentration Cyoof the diffusing species-are given by eq. 22 and 23. The physical plausibility of these equations provides additional support for the model. Thus, the expression for D is D = 0.0078 exp( -31,20O/RT)~m.~/sec. ( 2 3 ) This may be interpreted through the use of the absolute rate theory'O which gives
? !h
P
&P/R e-AH*/RT
(2.1)
nhere d is the jump distance of an interstitial particle from one position into one of the p equivalent positions surrounding it, while AS* and AH* are the entropy and enthalpy of activation for the jump, and the other symbols have their usual meaning. It has been shown theoretically that AS* must have a omall positive value.'l This requirement is met since, by t,iking d = 2-3 A. arid p = 2-6, one obtains 0 < i\S* < 4 e.u. upon comparing ecl. 23 and 24. The a,ctivation enthalpy is difficult to access theoretically but the d u e , AH* = 31,000 cal., is certainly not unreasonable. The Concentration Co-The experiment a1 exprcsbioi1 for Co is
x
1022
no = foN exp(-wo/kt)
10
Fig. 7 -'l'tie eilect of tiit: lilterruptun temperature: (a) oxidation a t 342" was resumed after interrupting the reaction for 16 hours a t 342": ( b ) oxidation a t 342" was resumed after interrupting t+ reaction for 2 hours a t 494" followed by 16 hours a t 342
Pa = 9.4
+ z electrons
(26)
of In(Iiizo3),In+" (In203) and conduction electrons, respectively, and N is the total coilcentration of interstitial positions in In&, one has
1
I) =
(25)
If no,a+, and ne are the equilibrium concentration I
0
+ Sb(s)
This can be followed by the ionization of In(InzOg)
ne =
(27)
(29)
2123
where f o and j+ are factors which relate to the change of the vibrational entropy in the intermediate vicinity of an interstitial. y e is the activity coefficient and J? is the partition function of conduction electrons in In203. ye increases with n e and can be written12
-
ye =
ney
-
(30)
When n,/r < 1, y 0 and y 1; but when ne//r ? 1, y > 0 and y e > 1. Substituting (a?),(29) and (30) into (28) one obtains (31)
The quantities w0and w+ are the energies of reactions 25 and 26, respectively. As a general rule wo is positive, but w+ will have either a much smaller positive value or can be negative, i.e. wo
> w+
(32)
The relative magnitudes of no and n+ depend upoii r, which in turn depends on the band structure of In203 and the energy levels associated with the interstitial atoms. These are not known, but the consequences of three limiting cases are of interest. In each case a simple band model is assumed, i.e., the energy bands are spherical and the (kinetic) energy of electrons is proportional to the square of the wave vector.I3 Case 1.-The interstitial atonis depress z eigenstates from the top of the conduction barid to a discrete level just tielow the conduction band. In this caseL3
e\p( -540O/RT) indium tlcfects ~ m . - ~ (22)
( I O ) S. Glasstone. I
