yt /2d.)
$
+ [1 -
= [(y - 1)2
+ 4Ay]- 1/2de ly
(27)
1.0,------------------,
+ 1-
y
+ 2yx(0)
(n)
+ 2yx(0) ]tanh($yt /2d.) 1)2 + 4Ay}1/2. We shaH first study Y
where (P== [(y the t -+ 00 limit of this solution, and then the rate of convergence to the asymptotic steady state. On physical grounds one would expect the degree of orientational order far from the substrate in a thick film to be spatially unvarying and independent of the type of substrate. s This is indeed the case: Eq. (23) shows that xU) converges to the t-+ 00 limit (23) for aU x(O) in the interval. [0,1]. Two limits are of interest here. For smally, 4162
(26)
J. Appl. Phys., Vol. 60. No. 12, 15 December 1986
8
"
o
2
3
y-
FIG. 1. Plot of the asymptotic degree of orientational order
x~
vs
y = (d,ldb)r for several values of fl. Bradley, Harper, and Smith
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is a measure of how long the film must be grown before the asymptotic ordering is approached. Like x"" , 7 is independent of x(O). The coefficient A does depend on x(O), however. As shown in Fig. 2, the relaxation time 7 is a decreasing function of b. for fixed r. This is to be expected since as b. is increased, the ion beam becomes more effective in inducing orientational order. For y< I, (28) rr/de = 1 + (1 - 2b.)y + O(y), and in particular, 7 = d.ly for r = O. Thus, defy is simply the relaxation time when the ion beam is turned off. In the large y limit, on the other hand, yT/de =y-] + (1- 2b.)y-2 + O(y-3). (29)
Therefore, Y7 (1 - 2b.)db/de. V. DISCUSSION
Weare now in a position to discuss the significance of the parameters b., r*, de' and db and to suggest means to measure them. This task is simple for the first two of these parameters. By definition, b. is the fraction of orientations ifJ which channel the ion beam. It can be determined simply by extrapolating Xoo to the limit r-+O. The parameter r* is the ion/atom arrival rate ratio where deposition and resputtering balance to give no net deposition or removal; it can be
found using the fact that the fraction of material resputtered is r/r* in the steady state. The length de is the thickness of film which must be deposited for the steady state to be approached in the absence of ion bombardment, since y7 = de when r = O. Thus, by studying the approach of the film surface to its final state with the ion beam turned off, de can be determined. Since convergence can only be observed if the initial and final states differ, care must be taken to prepare an initial state which is not too close to the randomly oriented final state. One way this could be achieved is by first depositing a layer with the ion beam turned on, and then depositing on top of this layer with the ion beam turned off. The length db was defined by the relation
p = 1 - f2/f]
= (dJd b )r.
Therefore, it measures the extent to which the resputtering yields from aligned and misaligned material differ. Since y7 = db/rwhen de = 00, the time needed for the steady state to be approached in the limit of perfect epitaxy is db/y when r = 1. This does not provide a practicable means of measuring db' because de and db cannot be varied independently. However, db can be determined by performing a two-parameter fit to data for x as a function of r. This fit would yield values for b. and dJd b. Combining this with the value of de found by convergence measurements would then yield db' The observations of Yu et at. 2-4 are in qualitative agreement with our theory. In particular, their data for x"" rise slowly as r is increased from zero until, as a critical value of r is approached, x begins to grow more rapidly. (See Fig. 3.) Although there is considerable scatter in the data, we can see that b.....,
a.J
0.21-
0
I
0.0 0.0
• 0.1
• -:• ... •• 0.2
0.3
0.4
0.5
CURRENT DENSITY AT SUBSTRATE (mAlcm 2 )
a
2
3
FIG. 2. Plot of the relaxation time 1'vsy = (d,ld b )rforseveral values of fl.. 4163
J. Appl. Phys., Vol. 60, No. 12, 15 December 1986
0.0 0.5 1.0 1.5 ION/ATOM ARRIVAL RATE RATIO
FIG. 3. The degree of orientational order in Nb films plotted vs the ionl atom arrival rate ratio (from Ref. 3). Bradley, Harper, and Smith
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Yu et af.2-4 also found that aligned Nb films several thousand angstroms thick could be grown without bombardment on thin layers of niobium which had been deposited with ion bombardment. This shows that de was at least several thousand A. This was true for db as well, because de and db were comparable in these experiments. Since de and db were large compared to the interatomic spacing, we are indeed justified in using our macroscopic theory to interpret these results. Several interesting avenues for experiment remain to be explored. By studying the approach of a growing film to its steady state, the prediction that x (t) converges exponentially to x'" could be checked. Moreover, it would be exciting to verify that the convergence time 1" has a peak as r is varied. Finally, a basic assumption of the theory is that the dominant ordering mechanism is the difference in resputtering yields from aligned and misaligned crystallites. The validity of this assumption could be tested by measuring the azimuthal dependence of the sputtering yield from the (lID) surface of a niobium crystal. To our knowledge, all work done to date on sputtering from crystals has probed the dependence of the sputtering yield on the polar angle alone. 6 Our theory has a number of implications for the efficacy of thin-film orientation by ion bombardment. As we commented above, if db/d. is greater than r* and a is small, ion bombardment cannot induce appreciable orientational order. To ensure that db/d. is much smaJjer than r*, we should look for circumstances in which the resputtering yields from channeling and nonchanneling orientations differ markedly, and in which epitaxy is good. If db/d. «r*, the increase in x '" obtained by an increase in r becomes smaller and smaller once r exceeds db/db' Moreover, this increase in order is bought only at the expense of slower and slower film growth. A value of r several times larger than db / de will yield a wellordered film and still give a reasonable deposition rate. Finally, values of r in the immediate vicinity of db/d. should be avoided since the convergence time 1" may be excessively long. Several variables can be adjusted in an experiment to optimize orientational order. The epitaxial coherence length de can be modified by changing the substrate temperature. The parameters a, db' and r* are functions of the angle e between the beam and the surface, the ion species, and the ion energy. As an example, consider the effect of changing the ion species. 9 An ion which is too large will not channel well, so a will be small. There also will be little difrerence between the resputtering yields from al.igned and misaligned materi.al, so db win be large. To maximize x'" ' therefore, the
4164
J. Appl. Phys., Vol. 60, NO.12,15 December 1986
smallest ions which do not react chemically with the substrate should be employed.
VI. CONCLUSIONS In this paper a macroscopic theory of thin-film orientation by ion bombardment has been developed. We found that if the total acceptance angle of the channeling directions is small, the asymptotic degree of orientational order x'" grows very slowly as a function of the ion/atom flux ratio r until r reaches the critical value db / d., The steady-state degree of order x", then rises more rapidly and tends asymptotically to its maximum value. In some systems this critical value of r will not be attainable, however, since the material may be resputtered more rapidly than it is deposited. when r = db/de' The orientational order at the surface was shown to converge exponentially toward the asymptotic steady state. Interestingly, the associated convergence time has a peak in the vicinity of the critical value of r. From a practical standpoint, our results indicate that values of r several times larger than db/d. must be employed if good orientational order is to be obtained. Ion/atom arrival rate ratios close to db/d. are to be avoided, moreover, because the time needed to reach the steady state may be excessively long. Finally, experiments to test the theory and determine the values of its parameters were suggested. ACKNOWLEDGMENT
We would like to thank P. N. Strenski for helpful discussions.
's. Mader, in Handbook of Thin Film Technology, edited by R. Giang and L. I. Maissel (McGraw-Hill, New York, 1970). 2L. S. Yu, J. M. E. Harper, J. J. Cuomo, and D. A. Smith, Appl. Phys. Lett. 47, 932 (1985). 3L. S. Yu, J. M. E. Harper, J. J. Cuomo, and D. A. Smith, 1. Vac. Sci. Technol. A 4, 443 (1986). 4L. S. Yu, M. S. thesis, Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA (1985). ~1.
1. Cuomo, C. R. Guarnieri, R. H. Hammond, 1. M. E. Harper, S. R. Herd, and D. S. Yee, IBM Tech. DiscI. Bull. 25, 3331 (1982). "For a review, see H. E. Roosendaal, in Sputtering by Particle Bombardment I, Vol. 47 of Topics in Applied Physics, edited by R. Behrisch (Springer, Berlin, 1981). 7Since the layer thickness after resputtering d, varies with I, it is simplest to use the time rather than the film thickness as the continuous independent variable. 80f course x ~ will be the same only for substrates which pin the same crystal axis normal to the substrate surface. 9The effect of varying (J on the growth of Nb films is discussed in Ref. 2 and
3.
Bradley. Harper, and Smith
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