THE DIFFUSION OF ALUMINIUM AND GALLIUM I N ZINC OXIDE
[Nanuscript received September 12, 19681
Summary
Solubilities and rates of substitutional diffusion of aluminium and gallium in
polycrystalline zinc oxide have been determined in the temperature range 750-1000"
by a chemical method previously described. The solubility of aluminium in zinc oxide,
expressed in ions
is given by the expression n = 1 .Ox 10aSexp(- 1.08k-lT-l),
and the solubility of gallium by .n = 2.7 x 1021exp(-0.59k-1T-1), where the activation energies are expressed in eV. The rates of diffusion of both aluminium and
gallium were found to be dependent on surface concentration up to a limiting value,
a t which the diffusion coefficient of aluminium is D = 6 3 x
exp(- 2 . 74k-IT-1)
cmZsec-l, and that of gallium is D = 3.6 x lo4exp(- 3.75k-lT-l) om2 sac-l.
It is shown that supersaturation of both aluminium and gallium in the zinc
oxide crystals occurs above 800'.
+
The conductivity of zinc oxide is increased by the substitutional addition of
trivalent element~,l-~
and decreased by the incorporation of monovalent cation^.^,^*^
Solubilities and rates of substitutional diffusion of monovalent and trivalent elements
in zinc oxide are therefore of considerable interest. The diffusion and precipitation
of substitutional indium in zinc oxide single crystals has been investigated by
Th0mas.l Lander4 has reported the solubility and diffusion of lithium in single
crystals of zinc oxide. The diffusion coefficients reported by both Thomas1 and
Lander4 were obtained by following the change of conductivity induced in the zinc
oxide single crystal by the diffusing element as a function of time.
I n a previous paper6 i t was shown that chemical methods778 could be extended
to determine the concentrations of trivalent and monovalent cations occupying
zinc lattice sites in zinc oxide that had been doped by heating in air a t temperatures
above 550". Because of their simplicity and high sensitivity these methods were
suggested as a convenient and accurate means of determining rates of substitutional
diffusion of monovalent and trivalent elements in zinc oxide.
This paper describes the determination of the solubilities and rates of substitutional diffusion of aluminium and gallium in polycrystalline zinc oxide in the
* Australian Defence Scientific Service, Department of Supply, Defence Standards Laboratories, P.O. Box 1935P, Adelaide, S.A. 5001.
Thomas, D. G., Physics Chem. Solids, 1959, 9, 31.
2 Bogner, G., and Mollwo, E., Physics Chem. Solids, 1958, 6 , 1936.
a Heiland, G., 2.Phys., 1957, 15, 148.
4 Lander, J. J., Physics Chem. Solids, 1960, 15, 324.
Kasai, P. H., Phys. Rev., 1963, 130, 989.
6 Norman, V. J., Aust. J. Chem., 1968, 21, 299.
7 Norman, V. J., Analyst, 1964, 89, 261.
Norman, V. J., Aust. J. Chem., 1966, 19, 1133.
Aust. J . Chem., 1969, 22, 325-9
V. J. PU'ORMAN
326
temperature range 750-1000" by the chemical method.' The method also furnishes
information on the supersaturation and precipitation of aluminium and gallium a t
these temperatures.
( a )Chemical Method for the Determination of Substitutional Trivalent Cations
The chemioal method,' which was originally developed for the determination
of excess zino in zino oxide, is a photometric one based on the reduction of dichromate,
and essentially counts the electrons necessary to convert the non-stoicheiometric
species into Zn2+ and 02-ions.
The introduction of a trivalent element into substitutional positions in the
zinc oxide lattice may be expressed
Heating zinc oxide in air a t temperatures above 550" results in the removal
of interstitial ~ i n c l and
, ~ reactive chemisorbed o ~ y g e n .The
~ method, applied to
samples that have been doped with a trivalent element by heating in air a t temperatures above 550°, then becomes a direct measure of excess electrons, which are
equivalent to the number of trivalent cations occupying zinc lattice sites as represented in equation (1).
(b)Solubility and Diflusion of Aluminium and Gallium
A high-purity "guaranteed reagent" grade zinc oxide was used throughout
the investigation. Electron-microscopic examination showed that the zinc oxide
consisted essentially of equiaxed crystals with a small proportion only of acicular
crystals. The average particle radius was estimated to be 0.16 pm. For the calculation of diffusion coefficients the particles were assumed to be spherical.
The zinc oxide was doped with a soluble decomposable salt of the trivalent
~
of the doped zinc oxide
metal according to the procedure used p r e v i ~ u s l y .Samples
were heated in air for appropriate times a t the temperatures indicated, then allowed
to cool in air. The fractional saturations of these samples were determined a t each
temperature; both the saturation concentration and the mean concentration after
time t were directly measured by the chemioal method.' The diffusion coe&cients
were calculated from the fractional saturations. The diffusion coefficient for diffusion into a sphere is given by the equationlo
where r is the radius of the sphere, and ti is the time required to reach 50% saturation.
It was found, from the plot of mean concentration against time, a t any temperature, that the diffusion of both aluminium and gallium into the spherical particles
of zino oxide obeyed simple diffusion theory up to the time required to reach 70 or
80% saturation. The diffusion, however, departed from ideal diffusion as solubility
Heilend, G., Mollwo, E., and Stockmann, F., "Solid State Physics." Vol. 8, p. 215.
(Academic Press: New York 1959.)
10 Darken, L. S., and Gurry, R. W., "Physical Chemistry of Metals." p. 447. (McGrawHill: New York 1953.)
9
DIFFUSION I N ZINC OXIDE
327
equilibrium was approached. This has been attributed to the particle size distribution in the sample of zinc oxide used, and the departure of a significant proportion
of particles from the average size accepted. For this reason, diffusion coefficients a t
each temperature were calculated from fractional saturations of between 50 and 60%.
Fig. 1.-Saturation solubilities
of aluminium and gallium
3
8
%
3
B
in zinc oxide.
-
10l8
The saturation solubility curves of aluminium and gallium in zinc oxide in the
temperature range 750-1000", determined on samples which had been allowed to
cool in air after heating, are shown in Figure 1. The straight lines are given by the
equations :
Aluminium n
=
1 .O x loz3exp( -1-08k-IT-l)
ions ~ m - ~
The activation energy is expressed in electron-volts.
The concentration of the dopant, up to a limiting value, was found to have a
marked effect on the rate of diffusion of both aluminium and gallium. I n initial
experiments, a concentration of 1 mg of doping metal per gram of zinc oxide was
used. It was found, however, that the rate of diffusion of aluminium in zinc oxide
rose sharply with increasing aluminium concentration up to 3 mg aluminium per
gram of zinc oxide ; increasing the doping concentration beyond this figure had no
further effect. With gallium, a doping concentration of 25 mg gallium per gram of
zinc oxide was necessary to effect the maximum rate of diffusion.
The diffusion of aluminium and gallium in polycrystalline zinc oxide in the
temperature range 750-1000", and the dependence of the diffusion rate upon concentration, are shown in Figure 2 .
The equations for the straight lines are:
AIuminium (3 mg A1 g-1 ZnO) D
Gallium (25 mg Ga g-l ZnO) D
= 5 a3 x 10-2exp(-2
= 3.6 x
q74k-ITF1) crn2 seo-l
lo4exp(-3.75k-lT-l)
cm2see-I
All points plotted in Figure 2 represent the average of several determinations
performed a t different times using different fractional saturations within the 50-60%
range. The results thus include variations due to temperature fluctuations during
V. J. NORMAN
328
heating. The agreement between these replicate determinations was good, in most
cases being within &5% of the average value. The maximum variation encountered
was 110%.
Temperature
950
I
900
850
I
I
Temperature ("c)
(OC)
800
,
750
slope 2.74 eV
Fig. 2.-Effect
of concentration on the diffusion of ( a ) aluminium, (b) gallium, in zinc oxide.
A, 1 mg A1 g-I ZnO; B, 1.5 mg A1 g-I ZnO; C , 3, 5, 10 mg A1 g-I ZnO; D, 4 mg Ga g-l ZnO;
E, 8 mg G a g-I ZnO; F, 25 mg Ga g-I ZnO.
It is not suggested that the diffusion mechanism itself is concentrationdependent. To permit diffusion to proceed a t the maximum rate, it is necessary to
have a sufficiently high concentration of dopant to provide adequate coverage of the
zinc oxide crystals to fulfil the diffusion requirement that the zinc oxide surfaces
are brought instantaneously to, and maintained at, a constant surface concentration.
The actual concentration of dopant necessary to provide adequate coverage will be
dependent both on the particle size of the zinc oxide sample, and the particle size
of the doping salt as crystallized from solution.
Neither Thomas1 nor Lander4 have reported the concentration of the doping
solutions used in their experiments.
(c)Supersaturation and Precipitation
Information on supersaturation and precipitation of aluminium and gallium
was obtained by heating duplicate samples of the doped zinc oxide simultaneously
a t each temperature for the same time. One of the samples was quenched from
the furnace temperature in the water of the analytical solution as rapidly as possible.
The remaining sample was allowed to cool in air prior to analysis.
The precipitated phase (presumably1 a mixed ZnO-Alto, or ZnO-Ga,O,
compound) does not register in the analysis,' which records only trivalent ions
occupying zinc lattice sites. The difference between the concentration of the dif-
DIFFUSION I N ZINC OXIDE
329
fusing element in the quenched and air-cooled samples therefore represents the
extent of supersaturation and subsequent precipitation of the element a t that
temperature. No significant supersaturation of aluminium or gallium was observed
below 800".
TABLE1
SUPERSATURATION O F ALUMINIUM AND GALLIUM I N ZINC O X I D E
A1 (ions ~ m - ~ )
Temp.
A
Air-cooled
Ga (ions cm-*)
1
Water-quenched
r
Air-cooled
A
>
Water-quenched
Diffusion coefficients calculated from the results of analyses of water-quenched
samples were in good agreement with those obtained from the analyses of air-cooled
samples.
The results of analyses of water-quenched and air-cooled samples of aluminium
doped and gallium doped zinc oxide a t solubility equilibrium are shown in Table 1.