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Morphological instability of a vapour-grown crystal: in situ study of (0001) cadmium face
Journal of Crystal Growth 118 (1992) 243—258 North-Holland
J oor~o or
CRYSTAL
GROWTH
Morphological instability of a vapour-grown crystal: in situ study of (0001) cadmium face M. Gauch and G. Quentel Centre de Recherches cur lea Mécanismes de la Croissance Cristalline F- 13288 Marseille Cedex 9, France
~‘,
Campus de Luminy, Case 913,
Received 25 June 1991; manuscript received in final form 11 November 1991
From in-Situ observation of a vapour grown cadmium single crystal, the morphological evolution of the (00(11) face is described in a large domain of supersaturation and crystal temperature. At low supersaturation, a great morphological instability results from the competition between composite pyramids which leads to a single dominant-growth pyramid as supersaturation is increased. The mechanism of growth is explained through the characteristic starlike sha~emorphology of the pyramid (twenty four vicinal facets) of elementary step height (5.6 A) Beyond a- 30%, bunching occurs in some sectors, as a result of the fluctuating interstcp distance and the pyramid slope increases very slowly with the supersaturation. The spiral-growth mechanism is predominant because of the too small interstep distance which inhibits the two dimensionnal nucleation. The morphological evolution is reproductihie and reversible with the supersaturation. The mean free surface diffusion path and the edge free energy are estimated.
1. Introduction The growth of the F faces of a real crystal proceeds, at low supersaturation, through the spreading of spiral layers originating at the outcroping of screw dislocations; this mechanism predicted by Frank [1] has been intensively observed experimentally [2,31 and the classical theories of BCF [41and Chernov [5] settle the basic principles of growth of the face in the simplest situations of spirals cooperation. Dislocations are often arranged in groups of various size which are then growth centres for pyramids of different activities whose competition is sharply responsible for growth rate fluctuations [6,7](together sometimes with impurity adsorption). The rate of growth is essentially determined by the dominant growth pyramid which is supersaturation dependent. This explains the increasing in(crest in in-situ experiments allowing a better —
—
*
Laboratoire propre du CNRS associé aux Universités ii et III.
0022-0248/92/$05.0() © 1992
—
insight in the mechanism of growth of individual hillocks versus the supersaturation [8—131.Most of these experiments, mainly relative to solution growth, show, among other interesting features, a non-linear dependence of the slope of individual hillocks on the supersaturation [10,11,141. In a previous paper [15], we reported briefly about the mechanism of competition between growth pyramids on the basal face of a large vapour grown cadmium crystal, in the case of small deviations around the thermodynamical equilibrium, but the main part was devoted to the ellipsometric detection of step-train fluctuations. This study has been extended to the high supersaturation range and to the influence of the crystal temperature with the same two in-situ optical techniques in order to get a better comprehension of the mechanisms involved [161. The present work presents the morphological evolution of the face derived from the observation through an intcrferential differential binocular with a high angular resolution. This information about the topography of the face is absolutely necessary for the interpreta-
Elsevier Science Publishers B.V. All rights reserved
24—I
31 (lain/i. (L Qiu’nn’I / Morphological uscia/ui/us’ of ‘apousr—I,’rosu’n crystal: (000/) ( ~/
lion of the ellipsometric results described in a second part which will he devoted to the kinetical aspects of these growth instabilities analysed by real—time ellipsonietry [17].
2. Experimental set-up The experimental set-up is about the same as the one described previously [15], but with a new ellipsometer configuration. A special metal glass growth cell, enclosed in a large aluminium furnace (fig. I) with associated heaters, allows the birth and subsequent evolution (growth or sublimation) of a large single crystal from the vapour;
the process is in-situ followed by two optical complementary techniques, the only ones possible in this case of high vapour pressure: Observalion of the (0001) face is possible through specially built differential-interferential binocular with a high angular resolution, while the ellipsometric signals give a dynamic analogical picture both at a macroscopic (pyramidal hillocks conlpc—
tition) and
which is more Important scopic scale (steps movement). 2. 1.
—
—
micro-
-
(jiiwt/i appurallLs
The growth cell (fig. I ) is outgassed and evacuated down to 10 n Torr and then supplied with high purity cadmium (99.9999~2~) after several high vacuum distillations. The crystal grows on a polished glass substrate at the point K; its temperature i~.is controlled by the outside cooling finger CF connected with a heat pipe and two heaters, one for the crystal and the other to prevent condensation around it. The furnace imposes the temperature T5 of the cadmium vapour. All teniperatures are measured with thermocouples and regulated with ±0.2°C accuracy. Moreover, J7~ can he slaved on ‘I~ such that the difference il — 1~. remains constant. The ellipsometric beam runs through optical windows Al and A2 placed on two bellows (B I and B2). Orientation adjustment, as well as temperature regulation (370°C±0.5°C). prevents fluctuations of the residual birefringence of the windows. =
r~. Hi
Fig. I Experimental set—up: growth apparatus (cell and its turnace ). I ntcrferent ial differential binocular and eli ipsoillet rue arrangement. (I) He—Ne laser: (2) lamp: (3) polarizer: (4) camera: (5) binocular (with anal ‘ser inside): (6) Savart polariscope (SF I: (7) rotating analyzer: (8) photom ci) Ii t1 icr: (9) cornpu te r.
M. Gauch, G. Quentel
/ Morphological instability of u’apour-grown crystal: (0001) Cd
245
2.2. Interference differential binocular binocular (20 cm) used to observe the crystal face, To improve the resolution of the long focal it has been modified following the method developed by Françon [18] for the microscope: The sample is illuminated by reflection of a parallel and polarized white ligth beam (fig. 1), The reflected beam is split into two waves (ordinary and extraordinary) by the Savart polariscope (SP), which then interfere. The resulting optical path difference S is converted into the associated colour of the Newton tint scale. The higher sensitivity is reached when the polariscope is adjusted to get the flat tint as a reference. Two kind of defects can be detected from the estimation of the different colours of the picture: angular dcviations a of the surface such as the pyramid slope (fig. 2a) and bunches of steps (fig. 2b).
.
1780mm
‘CRYSTAL
SIZE
3
“un
© • ~ 14.8°io
19.2%
.~ .
Sal
.‘.
236%
28.3%
33%
]O%
Fig. 3. (a) Determination of the pyramid slope from the laser reflection pattern of the crystal. (b) Example of evolution of the reflection pattern with the supersaturation.
2.3. Measurement of pyramid slope The differential interferential binocular is a very sensitive method for detecting a slight variation of a pyramid slope but the quantitative estimation lies on a colour appreciation. Another
very simple method (fig. 3) has been used in our case when only one pyramid spreads all over the face. The laser beam of the ellipsometer is tern-
1
2
‘(SP)
o
(SP)
h
_______
~2
37~
cx
Fig. 2. Principle of the interferential differential binocular for the detection of two kinds of defects. (a) Angular deviation a through the relation a =[(n~ + n~)/(n~ — n~)][(h, — 9 1)/2e] with n0 and n,u respectively the extraordinary and ordinary refractive indexes. (b) The height Ii of macrosteps is derived from h = I tan a. The optical paths in the three zones 1, 2 and 3 are, respectively. b~,~ = — b~ and h~.
31. ( rails/i, (I. Qiu’uoeI
,.‘
Morpbioloi
5’iccu/ iii sushi/its’ of ‘apouur—s,’roui’ui rri’stal: 1000/) ( ‘3
porary deviated to get the reflected picture of the face. The angle t is derived from the mean distance between the spots by the relation tan (5 2 sr = (I/f). 2.4. Real ti/tic t’llip.s’otttt’trv It is now well established that ellipsonietry is very sensitive to the sample surface roughness [15,19]. which may be taken into account in the measurements of optical characteristics of thin films, particularly in the submonolayer region. We showed previously [15.20] that this feature can he used to follow the roughness evolution of a crystal face during the growth (or sublimation) from the vapour. bor such kinetical studies, the ellipsonietric signals can he recorded at a fixed ~vavelengh. hut require automatic data aequisition. The home-built computer-driven ellipsometer is based on the method of the rotating anal— yser. The ellipsometric results are specially interesting for the analysis of the behaviour of ~ step-train versus the supersaturation and are the subject of a following paper, in order to concen— trate the present paper on the whole description of the process of growth and evaporation as seen through the optical system. 2.5. E.vpcriincnral procedure The driving force for growth is the supersaturation /3 defined ~
/3
=
the thermodynamic equilibrium (r’r 0). Then. steady states of growth are successively settled by small supersaturation steps to avoid ~t drastic modification of the pyramid distribution. Regeneration of a smooth face from too perturbed ones can be achieved by an evaporation—regrowth process. which is also a good procedure for the
experimental determination of the thermody— narnic equilibrium (see section 3.fi).
3. Experimental results 3’ 1.Morpliolog.v oJ t’tithtiiii,it
sing/c crystal
‘l’he cadmium single crystal is obtained in four successive steps [21 a]: (I) deposition of a polycrystalline film ( i~K 1~): (2) fusion in a droplet (J~> 321°C): (3) controlled freezing by lowering both ‘I~ and T, (at this stage, a small platelet appears at the bottom of the droplet and nitgrates by capillarity forces to the top): (4) crystal— lization of the floating raft from the melt [21 b] results in a rounded single crystal with a (0001) basal face on the top. The raft grows first very slowly by nucleation of successive layers, then. when the radius exceeds a critical value r equal to 0.53 droplet radius, the crystallization in— creases sharply owing to a spiral growth mechanism from a centred dislocation [22]. The subsequent growth conditions lead to the polygonization of the single crystal in the hexagonal system with a large basal (0001) face on top.
1’s. /1’,.. .3.2. Growth of (0001) basal face i ‘cents sIlpersatli -
where I’, = k exp( — iHS/RTV) is the pressure of the cadmium vapour and P1. the equilibrium vapour pressure of the crystal. The experimentally useful expression is then /3
=
exp
[
( i//5/R
) ( IT/T
~. ~ -
with if!5 = 21 kcal mol ‘ for cadmium. In the text, we used the relative supersaturation ci = /3 I (u > 0 for growth). In order to study the (0001) face morphology evolution versus the supersaturation, the first step consists of obtaining a face as smooth as possible, in the limit of resolution of the observation, near
Most of the growth studies have been carried out at a high temperature of the crystal (i~.about 300°C)for a reasonable time of observation. but a careful study of the low temperature case has also been made (not reported here). Whatever the parameters (ii.. u). the growth is always a non-stationnarv process which takes place by the birth and spreading of growth spiral layers originating at screw dislocations (or group of screw dislocations). These layers build growth pyramids of dif’f’erents activities, depending on the number
M. Gauch, C. Quentel / Morphological instability of i’apour-grown crystal: (0001) C’cl
of cooperating screw dislocations [4]. So, fluctuations can be observed in the topography of the basal face resulting from a competition between the growth pyramids, even at a constant supersaturation. Starting with a smooth (0001) face, where no hillocks appear at the resolution of the differential interferential binocular and near the thermodynamic equilibrium, the topography is very unstable. A slight supersaturation step results in a rather large distribution of hexagonal pyramids of different activities and hence of different slopes. Because the pyramid activity increases with the slope P, it is the steepest which remains the
t~O
t=42r —.
dominant growth centre for identical step height, and gradually ousts the more shallow ones. Surprisingly, the morphological stability increases as u increases: At constant supersaturation (10%
—
t=14
t=26’
= 38%; only one pyramid (C) invades the whole face. (b). (c) A step in 45%) reveals two other more active pyramids (A. B). (d) Here. A becomes the leader growth.
Fig. 4. Growth pyramids competition (a) At a-
supersaturation (38%
U=38%
247
24s
ti. (iOta/i. (I. ljsui’uuli’/
tlorp/iologh’sl uuista/ui/ili’ of’ s’a/uoisr-t,’i’oui’ui risial: 100011 (4
fluctuating sizes and hence activities due to the rvthmical bunching of the steps emanating from the dislocation source (see section 3.7): after some time, the topography becomes stable with only one dominant growth pyramid (C ~tt 38~’. A at 45~) spreading all over the face. ‘[his growthleader is supersaturation dependent until a sit— persaturation of about 4(Y’~is reached. .3.3. Morphology of g,i~tnt/i pyramids The polygonization of the hillocks is a general feature observed at low supersaturation. 3,7. 1. Tvinc’al morphologic’s Three typical morphologies arc observed in most experiments where supersaturatioti is slowly increased; they represent the typical stages of a reversible evolution with the supersaturation. ‘l’he first growth pyramids which appear oti a very smooth face at low supersaturation have the starlike aspect M3 shown in fig. 5 with 24 sectors. It is the most general morphology of the hexago— nal system. Raising the supersaturation leads to ~t reduction of the number of facets: the narrow sectors B around the [10.0] directions first disap— pear, giving F~itwelve-facet pyramid M2 and then the only six vicinal faces M I remain at higher supersaturation. The next step of the evolution is the deformation and roughening of the pyramid
described below. Notice the re-entering angle of the sectors around the bissectrices of the vicinal faces in the M2 and M3 morphologies. ‘I’he straight edge between two adjacent sectors mdicatesaconstant ratio of the growth rates and hence a stationary growth from the centre to (lie edges of the hillock. .7. .7.2. I)c’i ‘Ic/lions front ideal case Other kind of morphologies are also often observed on the (0001) face of cadmium, else— where described in the literature on some mmeral substances [2b]. ‘I’hese are conical hillocks. fl at tops and hollow core pyritm ids: -( ‘onica/ lu/lock’s (fig. (a) are obtained more easily at low temperature of the crystal or in the first stage of the growth from a rough, previously evaporated. (0001) face. A high rate of growth favours the appearance of this kind of hillock. It means an isotropic lateral rate of growth. but we never observe a conical hillock spreading all over the face. — F/cit top.s are often associated with a hollow co/c’ and are not stable. Tsukamoto and Van der I-lock [23] suggest that it was the result of the neutralization of the activities of pair of opposite sign dislocations. Therefore, the pyramid slope decreases suddendy and cannot he resolved by the optical system. Another explanation may be the gliding of the dislocation its shown in figs.
itt F
trig © (A) Morphology ol the ohscrsed pyranids: l’or s~ells.ontrcilled condition’, ol growth. M3 step height is 5.0
‘\:
The evolution M3
—~
M2
—‘
is
often uubsersed: In this case, the
MI ss0h the supersaturatioul nieans in increasing step height. (I)) t)irection iidcxiie (or M3 niorphologv.
M. Gauch, G. Quentel
d
/ Morphological instability of
i’apour-groe’n crystal: (0001)
Cd
249
5201w
Fig. 6. Some other kinds of typical morphologies. (a)—(c) Flat top resulting from the gliding of a dislocations group. (d) Schematic reconstitution: the direction 7 D) of the gliding is (II (1). (e) Conical hillocks at low crystal temperature. (f) Flat hole under evaporation conditions at low temperature (7’~ 200°C).The size of the hole mci eases with the supersaturation. Markers in (e) and (f) represent 830 and 76 j.sm, respectively.
11.
25))
(jail/i, (j. Qia’nrs’f ,i Morpliolory’al i,istahilitv of i a/iour—y’,’on’,i ri’~al: (0001 ) (‘ii
ba—bd. In the case of evaporation, the corre— sponding picture of the flat tcp isa flat hole (fig
3.4. Moip/iologic’ci/ c’i’olution ot’ the domincintgrowl/i pyramid at lug/i supersaturation
(‘if).
Above a critical supersaturation (T( of about 30%, two main new morphological features are observed (1) The slope of the pyramid increases very slowly with increasing supersaturation (fig. 7). (2) A characteristic roughness of almost paralId grooves appears in the sectors A on both sides of the [21.0] directions (fig. 8). The narrow’ sectors B around the [11.0] directions remain smooth and stable even at high supersaturation (a-> 100% at 310°C). The grooves propagate from the centre towards the edges of the face in the [11.0] direction and increase in size with the distance from the centre of the pyramid. They are an expression of the morphological instability of the initial step-train due to the overlap of the diffusion fields and perhaps the influence of itiipurities [25]. In any case, they arc interpreted as kinematic waves [5.25] (steps whose the density varies during their motion) which show a typical hehaviour with the supersaturation: a supersaturatioti step leads first to an increasing density of macrosteps. which further decreases while reaching the new stationary growth state (figs. 8a—8c). Higher macrosteps are observed near the edges during this transient state. lit the steady state, the number of permanent waves observed is an increasing function of the
3.3.7. Dependence of pyramid slope I’ on supersaturation a-
Because the initial topography of the face in a given experiment depends on its history (number and nature of evaporation—growth sequences) the slope !‘ of the unique pyramid versus supersaturation a- isa reversible and repriductihle law characteristic of the conditions (Ii.. a-) only for a given (000l)face of a crystal. For example, a high growth rate leads to steeper hillocks. The slope measurements are not possible when several pyramids coexist on the face because the map changes with the supersaturation. Fig. 7 shows the 1’(o-) curves for three beautiful pyramids spreading over the whole face (three different crystals and experiments). The slope values lie in a rather narrow range (5 x l0~—3x l0~ rad) even at high supersaturation. Unlike curve A which can he explained only by the increase of the number of cooperating spirals, curves B and C show a critical supersaturation (Ti of about 30~ above which the slope increases very slowly with a-. The break in the P(a-) curve is more pronounced at higher temperature. An explanation, based on the back-stress effect [24], is discussed below, 0.003
p(rd)
-
0.000
I
0
20
40
I
60
60
iOU
Fig. 7. Pyramid slope I’ versus the supersaturation a-: (A) 3l5A’: (Bl 305’ (‘: )(‘) 195A’.
20
M. Couch, G. Qut nrel / Morphological instability of i’apour-grown crystal: (000/) Ccl
-
()
251
.- .‘~
64%
44%
60%
95%
Fig. 8. Pyramid roughnening at high supersaturation (T
5. 2n) C). (a) (e) Transient response to a supersaturation step. (d)—(g) Schematic evolution of the morphology of the grooves versus the supersaturation. The number of steady-state grooves increases with the supersaturation. The sectors B remain smooth.
supersaturation (figs. 8d—8g). At the lowest supersaturation (a- 30%), they appear first near the edges of the faces where they are stopped by the lateral faces; then, the roughness increases and spreads towards the centre as supersatura-
tion is increased. At the same time the relative orientation of the A sectors changes and inverses. This is shown by the bending and fading of the edge along the [21.01directions. At this point, the two sectors lie in the same plane and the waves of
252
31. (lane/i,
(7.
Qitetitel / MOt’p/u)lOgu’al instability of’ i apout’—gro ss’,i it’s’s/al: (0001) ( ~l
one encroach upon the other. A further increase of the supersaturation leads to a curved profile of the facet and hence of the whole pyramid. The roughness increases again and becomes irregular. This morphological coarsening is reversible. Lowering the supersaturation lead to the inverse evolution of the morphology; the edge between the two A sectors becomes again visible. indicating the inversion of the corresponding angle. The roughness covers a regular pattern of grooves which progressively disappear. The original pyramid with M3 morphology is restored and the slope again decreases if the supersaturation de-
creases. Some qualitative interpretation will he mentioned in the discussion. 3.5. Ei’aporation process’ The competition between pyramids, previously described for growth, takes place also in the first stages of the evaporation at low undersaturation leading to a reduction of the number of pyramids: we observed that the pyramids dominated during growth appear again with the reverse mechanism of their disparition. hut this is a transitory stage towards a more and more complex topography. A
Edge position 2mm
~tt
7
50
°D3 “
St K U
5
~
15
V
2t
34~m/~n 3t
25
Fig. 9. Regeneration of a smooth (0001) face from the growth of a pre—evaporated crystal. ‘l’he iiearly isotropic edge velocity is 34 Cn’i/rnin,
M. Gauch, G. Quencel
/ Morphological instability of i’apour-grown
long time evaporation at a low rate leads to a randomly rough face showing the emerging points of all the dislocations active in evaporation. The case of high undersaturation is quite different, as shown in fig 9a: the higher rate of evaporation at
Fig. 9 shows the regrowth wave origInating from the highest point of the previously strongly evaporated face; the mean lateral rate of growth in the [12.0] direction is 0.6 mm/s and the height of the front about 280 A. The polygonization of this (0001) very smooth facet begins when it reaches a sufficient size. Previous leader-growth hillocks will then appear when the wave spreads over the corresponding outcropping dislocation and we can again observe the first stages of the growth pyramids. The centring of the tip of. the face in the picture results from a faster evaporation rate of the edges of the face as previously reported. 3.7. Distribution of growth steps
We said that the pyramid competition was in fact that of a group of spirals originating from screw dislocations spacially distributed with interdistances d such as d < 19r*, where r* is the critical radius of the nuclei in the conditions of growth. The scheme of fig. 10 shows how the slope of any facet of the resulting pyramid is thus periodically modulated. This permanent so-called “rythmical bunching” [26] is revealed in our case by the colour modulation of the pyramid vicinal facets. The step-trains beneath the resolution of the binocular are also detected by a slight supersaturation step which leads to a perturbation in the spatially distributed steps; this instability disappears after some time depending on the growth
crystal: (0001) Cd
253
6
/
/
-
/
/ /
3
‘
-
—
—
\
4 -
Profiie XX’
. . . , , Fig. 10. Schematic constructn)n ol . a pyramid tram the cooperation of six elementary spirals when the distance d between dislocations groups follows: ii < r °/2, where r is the critical nucleus radius. The radial section XX’ shows a modulated profile.
rate. Notice also that the growth front step is revealed by the colour variation induced by the mean pyramid slope modification. When the single crystal grows at low temperature (about ‘J~< 250°C), the step-trains are always observed because any perturbation is very slowly removed. Step shape etolution: The growth step shape is the periphery of a cross section of the hillock. So, following the shape of the step from the centre to the edges gives a tridimensionnal picture of the corresponding hillock. Fig. 11 schematizes this evolution: The round initial shape becomes progressively hexagonal and then reaches a typical starlike shape far from the centre. The discussion below explains this evolution. It means that the initial growth step winds itself in an archimedian spiral, as stated in the BCF theory, before it polygonizes progressively. Moreover, the interstep distance increases from the centre to the edges and the lateral rate of growth becomes
tI. (lain/i. (I, Qieiioi’/
254
.
K
~“
~
OOO1~-
“
3 2~
,/
.%‘Ior~t/io/oy’ii’alinstability
(000/) ( ‘il
that along the (II .0K directions. the two layers of atoms hunch in a 5.6 A layer growing laterally at ,
the lowest velocity (fig. I 2c): along the K 10.0K directions, the dissociation of the 56 A layer
‘,‘
/
of i ‘apoier-io’ois’ti it .sial:
/
occurs and the sub-layers travel independently ,
l:ig. II. Scliem:ttic csolution of the step shape Iron the centre to the edges ol’ a pyramid with a M3 morphology: (I) round: (2) lic\agollal: (3). (4) 24 edges polygonc (starlike shape).
(fig l2h) giving icgul ir intcrl tcing structurc which builds thc n trrow scctors B This mccli t nism still rcm tins in thc c tsc of composttc p~rmud without bunching of thc stcps Whcn thc pyr tmid step is high (scvcr ml coop crating growth centres), the dissociation is less probable and the narrow sectors B around the K II .0~directions do not appear. So, we can con— elude that the most pols’gomzc’d pyramids’ (M3
:ttuisotropic. ‘[he nuorphology of the spiral indeed changes in the radial directions,
morphology) hate an cleiiic’ntar~’stc’p height ot 5.~ .1 . even if they result froni a multi—spiral coopera— tion. These are the less steep hillocks observed it low supersaturation.
\
i
--
nO.11
“.:
4. 1.2. Dislocation glide
4. i)iscussion
The modification of the morphology atter several evaporation—regrowth sequences is cxplained by the growth centre displacemetut clearly visible in fig. bd in the (11.0K direction which is the principal direction of gliding in hcp metals when c/a > 1.633. In the regrowth process, the high wave front can bent the dislocation lines met [30] and sweep them away from the periphery of the face. Asa consequence, the niore centred sources are less exposed to this sweeping, and this explains the well centred unique pyramid generally observed after several treatments. ‘[he observation of high—slope pyramids of weak activ— ity at the periphery’ of the (0001) face indicates a concentration of dislocations which induce a high step spiral layer of slow’ lateral rate of growth. At temperature of the crystal lower than (T~ is the melting point temperature. which is 321°C for cadmium), the motion of the dislocations is no more possible: this explains the great stability of the pyramid map at low temperature.
4, 1. (/~~ntb, ,net’ha,’ij,rnis
ln the hexagonal compact structure of cadmium, the periodicity in the direction of the c axis consists of successive layers AB AB ABAB . . . of atoms with a i~- rotated trigonal symmetry (fig. 12). The Burgers vector is then hit = 2mnCii. with c~= c/2. where c’ is the cell parameter (c’ = 5.6 A is the AB layer thickness). ‘l’he nuost probable step height is c’ = 5.6 A (iii = I) in order to have a minimization of the elastic energy of the dislocation. 4.1. 1. Shape ci’olution of .s’lep.s’ and pyramids’ The starlike shape of the pyramid is an expression of the anisotropy in the lateral velocity of the steps, enhanced by the dissociation of the 5.6 A high spiral layer in two layers of 2.8 A height along the six (10.0K directions (fig. 12), itS wits proposed by Frank [27] for the interpretation of SiC experiments [281. The mechanism is summitrized as follows: (21.0K is the fastest direction of growth for the A layer hut the slowest for the B layer; it is the inverse situation for K 11.0K; the velocities’ ~trc the same in the K 10.0K direction. From these anisotropic growth rates, it results
4.2. Kinetic fratures
•
4.2.1. Ei’olution of th,c’ pyramid slope P nil/i I/u’ supersaturation ain the classical theory of BCF [4]. the slope of an in dislocation cooperating spiral with a step
/ Morphological instability of i’apoor-grosvn crystal:
M. Gauch, G. Quentel
heigth h is given by
speed of the steps
2kT~ ln( 1 P
mh =
mh =
+
‘
where I is the interstep distance, 1~the substrate temperature, a- the relative supersaturation, ‘y the edge-free energy, £1 the atom volume in the crystal and k the Boltzmann constant. The normal rate of growth R depends on the lateral [21
0]
// /
®
/
,,
\
/
=
/
/ /
[21.01
/
/ /
Lever
A
.L~erB
/
,/
[lit]
‘
//
/
///////
A
~
/
/
/ I
/
/7
/
/ ‘ ‘
=
[lot]
/
/
\\\
=
=
/
~ ~/
\
i’
[lot]
/
/1 /
/
/ /
/
255
1 as R Pi’~ h(w/2~r) Some qualitative and quantitative results can i’i11/l(a-, T). be derived from the P(a-) curves of fig. 7, which correspond in all cases to a single M3 morphology pyramid on the whole face. P, h, and sometimes R are derived from experiments. (a) A type curve: The lateral growth rate of the steps is deduced from the measurements of R and P from the relation ‘It R/P. As shown
a-)
l9yfl
(0001) Ccl
/
[ITO]
A
Atom
layer A
////// Atom
layer B
~t,O1
~t.0I
Fig. 12. Growth mechanism of cadmium layers. (A) Trigonal symmetry of the atoms_layers A and B, which explains the anisotropy of the lateral velocity L’. P is the resulting profile of a spiral layer. 1(2.) = 2v/(i’(ii ~ . (b) Interlacing structure of the A and B layers. (C) Resulting profiles in three directions; [21.0]: IA> CB; [11.0]: ~ 1A [110.0]: o~=
250
(bun/i,
:ti,
I :ehlc ,l , Quantitative characteri,ation
(I, (jui’nti’/
/
Morp/io/oi,’ii’al oi,stahilui’ of i ‘aftoiir-~’t’ois’,iirs’stal: (000/) ( il
Interstep distance: 0) a pyramid derived from the
experimental slope versus a- curve
ii
in
a- <
Slope!’
R
i~/
/
(rad)
(A/n)
(pm/n)
)nm)
10)) -
4.2 .
152
2~
l257 407 2)(S
12.7 2
(.2 l.%
((.44/ 1.2 ‘/ 0) I))
(5)
3.1
27
“
x II)
‘
in table I. we find that it decreases in inverse ratio to the supersaturation; the interstep distance 1 li/I’ is evaluated assuming the element;try step height to he Ii 5.6 The decrease of t’ii suppsesa drastic decrease of the interstep distance, so that overlapping of the diffusion fields occurs. The experinuental law’ is verified to he P(a-) (1.1 76a-”, with n 2.1. In the BCF regime, we expect I’ Ka-” with a ~ I: moreover. itssuming that the nuean distances d,. between cooperating growth centres remain constant. (lie condition il. < ~l does not hold when I decreases as the supersaturation increases and the number of cooperating centres decreases. As a consequence, we expect more ~tnd nuore individual pyramids on the face. It is just the opposite we observe: Curve A means an iiucrease of the iuumher in of cooperating growth centres. The estimation of in (see table 1) is derived from the above expression of P with the edge free eiucrgy y deduced from our experimcnts. (h) Curve B is the pyramid slope variation for tile experiment given in fig. 7. A critical supersatur~mtion a-~( 38%) is evidenced by two kinetic regimes: for a- < a-i. P(a-) is roughly linear with the supersaturation; above a-i, P(a-) Ka-” with 0.3
=
~.
=
:
//2 X
r>a-~:
=
a-~/2 a-
=
(a-1/a-)i
//2X
5
[24aj,
2a-) [24h]. //2 X~ (a-i / The experimental value of the exponent (0.3— 0.4) for the high supersaturation case is in good agreement with the theory if we keep in mind the pyramid morphological imperfections: the conical shape around the centre becomes hexagonal ;tnd decomposes into interlacing layers as going toor
=
wards the edges. Moreover, instabilities of the growth steps modulate the slope because of the rythmical bunching; therefore, the reflection pattern itS well as the colour analysis gives its the mean value of the slope. 4.2.2. iwo-chmnensional nucleation
It is obvious that the spiral growth is the predominant mechanism in our experiments, even at the highest supersaturations applied. hut what about the two-dimensional nucleation? Following Van Erk et al. [10]. 2D nucleation will occur at sufficiently high supersaturation if the interstep distance I is larger than the total capture area 2X., around the step. In this case. the step velocity is increased hut the pyramid slope decre~tses. We can explain in this wity the apparent decrease ot (lie slope itu fig. 7B near a- 35%: the two-dinuensional nucleation mechanism occurs between 3tY~and 38% and it is inhibited beyond a- 38~ by the overlapping of the diffusion fields. This is consistent with the estimation of the nucleation rate which becomes significant around 40~. Nevertheless, 2D growth will also occur during the transient state following the application of a supersaturation step because this perturbation gives rise to waves of high density of steps and hence the fluctuating interstep distance becomes periodically larger than the capture area 2 X~. 4.2.3. Estimatiomi of I/ic’ macan free surface diffusion pat/i X, and the edge ene,gv y for the c’adtnium At a- a1, the slope P~ h/2X5 assuming an elementary step height of 5.6 ~&.we find X 2000 ±20() ~ an edge energy y 10 ~ erg/cm =
=
utive steps. Moreover, introducing the hack-stress effect [24] heads to the following relations for the
1
M. Gauch, C. Quentel / Morphological instability oJ i’apour-grown crystal: (000/) Ccl
257
is deduced from the slope of curve B in the low supersaturation case (a- < 30%) using the above theoretical expression of P(a-). Unfortunately, no experimental data were found in the literature for comparison and only a theoretical estimation is possible using the formulae: X5(T~)
bunches cover the whole hillock. However, keeping in mind that the roughness of grooves is observed only in those sectors around the (21.0K directions, it means that impurities adsorb on these vicinal facets selectively. The reason for this
a)) exp(3CD/kT) and ‘y 2CD/2a11. For a hcp metal, CD L%H5~0/6with JiHSUh the sublimation energy. Using the cadmium values (a)) 2.98 A, 26.6 kcal), we find X5(305°C) 950 A and y i0~ erg/cm.
steps into kinematic waves seems more realistic at high temperature, but the impurity adsorption is probably more active in low-temperature experiments (T~<240°C). The kinetic behaviour of the steps will be discussed with the aid of the ellipsometric results described in the forthcoming paper.
=
=
=
=
=
4.3. Roughening of the face at high supersaturation The theory of kinematic waves [5] can explain the observed formation of reversible piling up of steps into macroscopic grooves in the case of an interstep distance perturbation which can be a supersaturation step or the action of impurities. The direction of propagation is that of the maximum rate of growth (21.0K and so the bunching of steps is the higher. The density of steps and the macroscopic apparent step height increase in this direction with the distance from the centre of the pyramid. The decay of the number of grooves observed with time is proof of a shock-wave but macroscopic steps exist also. Assuming an oscillating collective behaviour of the steps, this perturbation increases with the distance from the centre. The first steps reaching the edges of the face are stopped and the following steps pile up behind and form a macroscopic groove; this instahihity increases with the supersaturation. In spite of the high purity of the cadmium material and the careful multi-distillation process of the cell filling and closing, a possible influence of some impurities may be taken into account; Van der Eerden and Müller-Krumbhaar [30,31] recently stated a new impurity adsorption model to explain the dynamical process of formation of macrosteps. According to this theory, macrosteps are formed far away from the centre by first a doubling and then a quadrupling of steps and so on; moreover, the bunching of steps is possible only beyond a given supersaturation and extends logarithmically with time. Although quantitative data were not yet available, this last point only holds at very high supersaturation where the
is not understood. In conclusion, the coupling of
~•
Conclusions
through a number of in-situ experiments on a large cadmium single crystal, the growth and evaporation processes of the (0001) face have been carefully investigated for a large spectrum of crystal temperature T~ and super(under) saturation a-, with an optical differential— interferential technique. The following main conclusions can be put forward: (a) Growth proceeds by a spiral mechanism whatever the conditions (T~,a-), and the morphology of the face is determined by a competition between pyramids of different activities; Unlike the classical theory of BCF, the number of’ pyramids decreases when the supersaturation increases: the number of pyramids, maximum near the thermodynamic equilibrium (a- < 10%), decreases steeply as the supersaturation increases; beyond about 20%, only one pyramid spreads all over the face and controls the growth. (b) The rythmical bunching of the growth (or evaporation) steps is a frequently observed feature indicating the composite nature of the pyramids. (c) Analysis of the pyramid morphology leads to identifying elementary step-height pyramids showing a starlike shape composed of 24 vicinal facets; these are the most commonly observed in our well-defined conditions of growth. Moreover, all kind of pyramids have also been found such as hollow cores and flat-top pyramids.
tI.
258
(biiic/i, ( i. (lnc,iti’l ,/ ,‘sI,,rp/is>/oç’ii’a/ iii stability of i a~ioiir-c’roii’ni’rs:s’ial: (0001) ( il
(ci) ‘l’lue morphological evolution of an isolated pyramid, with elementary steps, with the supersaturation is a reversible process which shows two interesting features: (i) The pyramid slope in— creases very slowly above a supersaturation /38% when the overlapping of the diffusion fields of the steps sets on. This is interpreted itsa consequence of the hack-stress effect. (ii) Beyond a-i, the bunching of steps is increasingly marked froi’n the edges to the top of the pyrat’nid facets adjacent to the (21.0K directions as the supersat— uration iiucreases. This morphological instability heads td) a characteristic and reproducible dynamical roughening which is also a reversible process. As a consequeiuce. the vicinal facets take a curved profile. At the highest supersaturation, the roughness appears to be randomly distributed. . (e) 7 I he evaporation process ,is also governed in the first stages by the pyramid competition. Sue.
cessive evaporatioiu—regrowth cycles sweep the dislocations towards the edges; this explains the centring of the unique pyramid. -, (I) Iwo—dimensional nucleation is inhibited by the overlapping of the step diffusion fields. The general consequence of this great morphuological instability is that the normal rate of growth of the (0001 )face is not a simple function of the thuermodynamical condittons (It. a-). hut depends widely on the morphology of the leadergrowth pyramids which are supersaturation dependent, and on the step behaviour (bunching and orientation changes). .
.
.
Acknowledgments
ertien oI Solid State Materials, Vol. 7. Ed. W.R. Wilcox (Dekkcr, New York. l982).
[4] W.K. l3urton. N. Cabrera and CF. [“rank. Roy Soc London A 243 (1951) 299
Phil, Trans.
[5] A.A Chernon. Soviet Phvs -Usp 4 (1961) 116 [6] S.D. Luhetkin and W.J. Dunning, J. (‘rystal Growth 43 (197$) 77. [7] Y. Zundelevich :tnd K. Glaser. .1. (‘r~stal Giowth ‘1990)) 9 ,
.
,
102
.
[8] AL. Morizan and W.J. Dunning. J, ( rvstal ( rowttn (1970) 75) 9] W.P..1. nan Fnckevori,
R ,.Janssen-vari
Rosrnalen and
\W.l I. van cler Linden. .1. Crystal Growth 49 (I 981)) 5)12. 11(11 W. van Eik. I l.J. van Flock-Martens and G. Baitels, .1. [II]
Crystal Growth 4$ (l9$0) 62l , ‘‘ K. I sukamoto. II. Ohha and I. Sunasiawa, .1. ;i’ow’th 63 (l9$3) IS
(.
i’vstsl
12] A.A. Chernov. L.N. Rashkovich and A.A. Mkrtchan, .1.
(‘ryst:tl Growth 74 1(986) (II. [13] K. Maiwa, K. ‘Fsukanioto and I. Sunagawa. J. (‘rvst~il Growth 11)2 (1990) 43. . [141 ((957) \ A ( hcrnon oid I N R nhkovich I ( rynt il ( iowth 84 389.
[IS] M. Gauch and C. Quentel. Surt’scc Sc). 108 (19$)) 617. [16] M. Gauch, Thesis. University of Aix—Marseille III (1989). [17] M. Gauch and U. Quentel, to he published. [IS] M. l”rancon. Rev. Opt. 34 (2) (1952) 65. II
9j
C A. E~enstermaker and FL. Me( rackin, Surface Sui. In
(li69) $5. 120] G. Quentel and M. Gauch, Act:i Electron, 2-I (1982) l3~). [21] (a) R. Kainchew and (‘6. Nancy, Phys. Status Solidi I)) (1965) 779. (6) B. Mutat’tschiev and J. Sell. Surface Se), 12 1 196$) 7 [221 R. Buckle, K.C Pointon and P. T’sakiropoulos. .1. Mater, Sei. IS (1980) 2921.
[23] K. I’sukarnoto and B. van der I loek, J. Cr stal Growth 57 ) 1982) 131. [24] (a) N. Cabrera and R.V. (,‘olenian. in: ‘l’he Art and Science of Growing Crystals. Ed .J..I. Gilman (Wiley. New York, 1963) p. 3: (6) P. Beunema and (it-I, Gilmer. in: Cr stal Growth: An
Introduction. Ed. P. 1 lartman (North—I lolland .Amster—
‘l’he authors wish to thank Professor R. Kern and Professor A. Baronnet for very helpful suggestions and encouragement during this work and A. Ranguis for his technical assistance. References [I] F.C. Frank, Disc’. Faraday Soc. 3 (1949) 48, [2] A. Baronnet. Progr. (‘rystal Growth Characterization I (1978) 212. [3] I. Sunagawa and P. Bennema. in: Preparation and Prop-
dam, 1973) p. 282. [25] N. (‘abrera and D.A. Vermilyea. in: (irowth and Pcrtec-
tion of C rystals, Fds, RH. Doremus. B.W. Roberts and D. Turnhull (Wiley. New York. 195$). [26] I. Sunagawa and P. Bennema .J.C rystal Growth
46
11979) 451. [27] F.C. Frank, Phil. Mag. 42 (1951) 0)14. [2$] AR. Verma. Phil, Mag. 42 (1951)1384.
[29] H. Klapper, Phys. Status Solidi A 14 (I 97) 99, [30] J.P. van der Eerden and H. Miiller.Krumhhaar. [leetrochim, Acts 31(1986) 1)107. [31] J P van der Eerden and H. MdlIer-Krumhhaar, Phvs. Rev, Letters 57 (1986) 2431.
J oor~o or
CRYSTAL
GROWTH
Morphological instability of a vapour-grown crystal: in situ study of (0001) cadmium face M. Gauch and G. Quentel Centre de Recherches cur lea Mécanismes de la Croissance Cristalline F- 13288 Marseille Cedex 9, France
~‘,
Campus de Luminy, Case 913,
Received 25 June 1991; manuscript received in final form 11 November 1991
From in-Situ observation of a vapour grown cadmium single crystal, the morphological evolution of the (00(11) face is described in a large domain of supersaturation and crystal temperature. At low supersaturation, a great morphological instability results from the competition between composite pyramids which leads to a single dominant-growth pyramid as supersaturation is increased. The mechanism of growth is explained through the characteristic starlike sha~emorphology of the pyramid (twenty four vicinal facets) of elementary step height (5.6 A) Beyond a- 30%, bunching occurs in some sectors, as a result of the fluctuating interstcp distance and the pyramid slope increases very slowly with the supersaturation. The spiral-growth mechanism is predominant because of the too small interstep distance which inhibits the two dimensionnal nucleation. The morphological evolution is reproductihie and reversible with the supersaturation. The mean free surface diffusion path and the edge free energy are estimated.
1. Introduction The growth of the F faces of a real crystal proceeds, at low supersaturation, through the spreading of spiral layers originating at the outcroping of screw dislocations; this mechanism predicted by Frank [1] has been intensively observed experimentally [2,31 and the classical theories of BCF [41and Chernov [5] settle the basic principles of growth of the face in the simplest situations of spirals cooperation. Dislocations are often arranged in groups of various size which are then growth centres for pyramids of different activities whose competition is sharply responsible for growth rate fluctuations [6,7](together sometimes with impurity adsorption). The rate of growth is essentially determined by the dominant growth pyramid which is supersaturation dependent. This explains the increasing in(crest in in-situ experiments allowing a better —
—
*
Laboratoire propre du CNRS associé aux Universités ii et III.
0022-0248/92/$05.0() © 1992
—
insight in the mechanism of growth of individual hillocks versus the supersaturation [8—131.Most of these experiments, mainly relative to solution growth, show, among other interesting features, a non-linear dependence of the slope of individual hillocks on the supersaturation [10,11,141. In a previous paper [15], we reported briefly about the mechanism of competition between growth pyramids on the basal face of a large vapour grown cadmium crystal, in the case of small deviations around the thermodynamical equilibrium, but the main part was devoted to the ellipsometric detection of step-train fluctuations. This study has been extended to the high supersaturation range and to the influence of the crystal temperature with the same two in-situ optical techniques in order to get a better comprehension of the mechanisms involved [161. The present work presents the morphological evolution of the face derived from the observation through an intcrferential differential binocular with a high angular resolution. This information about the topography of the face is absolutely necessary for the interpreta-
Elsevier Science Publishers B.V. All rights reserved
24—I
31 (lain/i. (L Qiu’nn’I / Morphological uscia/ui/us’ of ‘apousr—I,’rosu’n crystal: (000/) ( ~/
lion of the ellipsometric results described in a second part which will he devoted to the kinetical aspects of these growth instabilities analysed by real—time ellipsonietry [17].
2. Experimental set-up The experimental set-up is about the same as the one described previously [15], but with a new ellipsometer configuration. A special metal glass growth cell, enclosed in a large aluminium furnace (fig. I) with associated heaters, allows the birth and subsequent evolution (growth or sublimation) of a large single crystal from the vapour;
the process is in-situ followed by two optical complementary techniques, the only ones possible in this case of high vapour pressure: Observalion of the (0001) face is possible through specially built differential-interferential binocular with a high angular resolution, while the ellipsometric signals give a dynamic analogical picture both at a macroscopic (pyramidal hillocks conlpc—
tition) and
which is more Important scopic scale (steps movement). 2. 1.
—
—
micro-
-
(jiiwt/i appurallLs
The growth cell (fig. I ) is outgassed and evacuated down to 10 n Torr and then supplied with high purity cadmium (99.9999~2~) after several high vacuum distillations. The crystal grows on a polished glass substrate at the point K; its temperature i~.is controlled by the outside cooling finger CF connected with a heat pipe and two heaters, one for the crystal and the other to prevent condensation around it. The furnace imposes the temperature T5 of the cadmium vapour. All teniperatures are measured with thermocouples and regulated with ±0.2°C accuracy. Moreover, J7~ can he slaved on ‘I~ such that the difference il — 1~. remains constant. The ellipsometric beam runs through optical windows Al and A2 placed on two bellows (B I and B2). Orientation adjustment, as well as temperature regulation (370°C±0.5°C). prevents fluctuations of the residual birefringence of the windows. =
r~. Hi
Fig. I Experimental set—up: growth apparatus (cell and its turnace ). I ntcrferent ial differential binocular and eli ipsoillet rue arrangement. (I) He—Ne laser: (2) lamp: (3) polarizer: (4) camera: (5) binocular (with anal ‘ser inside): (6) Savart polariscope (SF I: (7) rotating analyzer: (8) photom ci) Ii t1 icr: (9) cornpu te r.
M. Gauch, G. Quentel
/ Morphological instability of u’apour-grown crystal: (0001) Cd
245
2.2. Interference differential binocular binocular (20 cm) used to observe the crystal face, To improve the resolution of the long focal it has been modified following the method developed by Françon [18] for the microscope: The sample is illuminated by reflection of a parallel and polarized white ligth beam (fig. 1), The reflected beam is split into two waves (ordinary and extraordinary) by the Savart polariscope (SP), which then interfere. The resulting optical path difference S is converted into the associated colour of the Newton tint scale. The higher sensitivity is reached when the polariscope is adjusted to get the flat tint as a reference. Two kind of defects can be detected from the estimation of the different colours of the picture: angular dcviations a of the surface such as the pyramid slope (fig. 2a) and bunches of steps (fig. 2b).
.
1780mm
‘CRYSTAL
SIZE
3
“un
© • ~ 14.8°io
19.2%
.~ .
Sal
.‘.
236%
28.3%
33%
]O%
Fig. 3. (a) Determination of the pyramid slope from the laser reflection pattern of the crystal. (b) Example of evolution of the reflection pattern with the supersaturation.
2.3. Measurement of pyramid slope The differential interferential binocular is a very sensitive method for detecting a slight variation of a pyramid slope but the quantitative estimation lies on a colour appreciation. Another
very simple method (fig. 3) has been used in our case when only one pyramid spreads all over the face. The laser beam of the ellipsometer is tern-
1
2
‘(SP)
o
(SP)
h
_______
~2
37~
cx
Fig. 2. Principle of the interferential differential binocular for the detection of two kinds of defects. (a) Angular deviation a through the relation a =[(n~ + n~)/(n~ — n~)][(h, — 9 1)/2e] with n0 and n,u respectively the extraordinary and ordinary refractive indexes. (b) The height Ii of macrosteps is derived from h = I tan a. The optical paths in the three zones 1, 2 and 3 are, respectively. b~,~ = — b~ and h~.
31. ( rails/i, (I. Qiu’uoeI
,.‘
Morpbioloi
5’iccu/ iii sushi/its’ of ‘apouur—s,’roui’ui rri’stal: 1000/) ( ‘3
porary deviated to get the reflected picture of the face. The angle t is derived from the mean distance between the spots by the relation tan (5 2 sr = (I/f). 2.4. Real ti/tic t’llip.s’otttt’trv It is now well established that ellipsonietry is very sensitive to the sample surface roughness [15,19]. which may be taken into account in the measurements of optical characteristics of thin films, particularly in the submonolayer region. We showed previously [15.20] that this feature can he used to follow the roughness evolution of a crystal face during the growth (or sublimation) from the vapour. bor such kinetical studies, the ellipsonietric signals can he recorded at a fixed ~vavelengh. hut require automatic data aequisition. The home-built computer-driven ellipsometer is based on the method of the rotating anal— yser. The ellipsometric results are specially interesting for the analysis of the behaviour of ~ step-train versus the supersaturation and are the subject of a following paper, in order to concen— trate the present paper on the whole description of the process of growth and evaporation as seen through the optical system. 2.5. E.vpcriincnral procedure The driving force for growth is the supersaturation /3 defined ~
/3
=
the thermodynamic equilibrium (r’r 0). Then. steady states of growth are successively settled by small supersaturation steps to avoid ~t drastic modification of the pyramid distribution. Regeneration of a smooth face from too perturbed ones can be achieved by an evaporation—regrowth process. which is also a good procedure for the
experimental determination of the thermody— narnic equilibrium (see section 3.fi).
3. Experimental results 3’ 1.Morpliolog.v oJ t’tithtiiii,it
sing/c crystal
‘l’he cadmium single crystal is obtained in four successive steps [21 a]: (I) deposition of a polycrystalline film ( i~K 1~): (2) fusion in a droplet (J~> 321°C): (3) controlled freezing by lowering both ‘I~ and T, (at this stage, a small platelet appears at the bottom of the droplet and nitgrates by capillarity forces to the top): (4) crystal— lization of the floating raft from the melt [21 b] results in a rounded single crystal with a (0001) basal face on the top. The raft grows first very slowly by nucleation of successive layers, then. when the radius exceeds a critical value r equal to 0.53 droplet radius, the crystallization in— creases sharply owing to a spiral growth mechanism from a centred dislocation [22]. The subsequent growth conditions lead to the polygonization of the single crystal in the hexagonal system with a large basal (0001) face on top.
1’s. /1’,.. .3.2. Growth of (0001) basal face i ‘cents sIlpersatli -
where I’, = k exp( — iHS/RTV) is the pressure of the cadmium vapour and P1. the equilibrium vapour pressure of the crystal. The experimentally useful expression is then /3
=
exp
[
( i//5/R
) ( IT/T
~. ~ -
with if!5 = 21 kcal mol ‘ for cadmium. In the text, we used the relative supersaturation ci = /3 I (u > 0 for growth). In order to study the (0001) face morphology evolution versus the supersaturation, the first step consists of obtaining a face as smooth as possible, in the limit of resolution of the observation, near
Most of the growth studies have been carried out at a high temperature of the crystal (i~.about 300°C)for a reasonable time of observation. but a careful study of the low temperature case has also been made (not reported here). Whatever the parameters (ii.. u). the growth is always a non-stationnarv process which takes place by the birth and spreading of growth spiral layers originating at screw dislocations (or group of screw dislocations). These layers build growth pyramids of dif’f’erents activities, depending on the number
M. Gauch, C. Quentel / Morphological instability of i’apour-grown crystal: (0001) C’cl
of cooperating screw dislocations [4]. So, fluctuations can be observed in the topography of the basal face resulting from a competition between the growth pyramids, even at a constant supersaturation. Starting with a smooth (0001) face, where no hillocks appear at the resolution of the differential interferential binocular and near the thermodynamic equilibrium, the topography is very unstable. A slight supersaturation step results in a rather large distribution of hexagonal pyramids of different activities and hence of different slopes. Because the pyramid activity increases with the slope P, it is the steepest which remains the
t~O
t=42r —.
dominant growth centre for identical step height, and gradually ousts the more shallow ones. Surprisingly, the morphological stability increases as u increases: At constant supersaturation (10%
—
t=14
t=26’
= 38%; only one pyramid (C) invades the whole face. (b). (c) A step in 45%) reveals two other more active pyramids (A. B). (d) Here. A becomes the leader growth.
Fig. 4. Growth pyramids competition (a) At a-
supersaturation (38%
U=38%
247
24s
ti. (iOta/i. (I. ljsui’uuli’/
tlorp/iologh’sl uuista/ui/ili’ of’ s’a/uoisr-t,’i’oui’ui risial: 100011 (4
fluctuating sizes and hence activities due to the rvthmical bunching of the steps emanating from the dislocation source (see section 3.7): after some time, the topography becomes stable with only one dominant growth pyramid (C ~tt 38~’. A at 45~) spreading all over the face. ‘[his growthleader is supersaturation dependent until a sit— persaturation of about 4(Y’~is reached. .3.3. Morphology of g,i~tnt/i pyramids The polygonization of the hillocks is a general feature observed at low supersaturation. 3,7. 1. Tvinc’al morphologic’s Three typical morphologies arc observed in most experiments where supersaturatioti is slowly increased; they represent the typical stages of a reversible evolution with the supersaturation. ‘l’he first growth pyramids which appear oti a very smooth face at low supersaturation have the starlike aspect M3 shown in fig. 5 with 24 sectors. It is the most general morphology of the hexago— nal system. Raising the supersaturation leads to ~t reduction of the number of facets: the narrow sectors B around the [10.0] directions first disap— pear, giving F~itwelve-facet pyramid M2 and then the only six vicinal faces M I remain at higher supersaturation. The next step of the evolution is the deformation and roughening of the pyramid
described below. Notice the re-entering angle of the sectors around the bissectrices of the vicinal faces in the M2 and M3 morphologies. ‘I’he straight edge between two adjacent sectors mdicatesaconstant ratio of the growth rates and hence a stationary growth from the centre to (lie edges of the hillock. .7. .7.2. I)c’i ‘Ic/lions front ideal case Other kind of morphologies are also often observed on the (0001) face of cadmium, else— where described in the literature on some mmeral substances [2b]. ‘I’hese are conical hillocks. fl at tops and hollow core pyritm ids: -( ‘onica/ lu/lock’s (fig. (a) are obtained more easily at low temperature of the crystal or in the first stage of the growth from a rough, previously evaporated. (0001) face. A high rate of growth favours the appearance of this kind of hillock. It means an isotropic lateral rate of growth. but we never observe a conical hillock spreading all over the face. — F/cit top.s are often associated with a hollow co/c’ and are not stable. Tsukamoto and Van der I-lock [23] suggest that it was the result of the neutralization of the activities of pair of opposite sign dislocations. Therefore, the pyramid slope decreases suddendy and cannot he resolved by the optical system. Another explanation may be the gliding of the dislocation its shown in figs.
itt F
trig © (A) Morphology ol the ohscrsed pyranids: l’or s~ells.ontrcilled condition’, ol growth. M3 step height is 5.0
‘\:
The evolution M3
—~
M2
—‘
is
often uubsersed: In this case, the
MI ss0h the supersaturatioul nieans in increasing step height. (I)) t)irection iidcxiie (or M3 niorphologv.
M. Gauch, G. Quentel
d
/ Morphological instability of
i’apour-groe’n crystal: (0001)
Cd
249
5201w
Fig. 6. Some other kinds of typical morphologies. (a)—(c) Flat top resulting from the gliding of a dislocations group. (d) Schematic reconstitution: the direction 7 D) of the gliding is (II (1). (e) Conical hillocks at low crystal temperature. (f) Flat hole under evaporation conditions at low temperature (7’~ 200°C).The size of the hole mci eases with the supersaturation. Markers in (e) and (f) represent 830 and 76 j.sm, respectively.
11.
25))
(jail/i, (j. Qia’nrs’f ,i Morpliolory’al i,istahilitv of i a/iour—y’,’on’,i ri’~al: (0001 ) (‘ii
ba—bd. In the case of evaporation, the corre— sponding picture of the flat tcp isa flat hole (fig
3.4. Moip/iologic’ci/ c’i’olution ot’ the domincintgrowl/i pyramid at lug/i supersaturation
(‘if).
Above a critical supersaturation (T( of about 30%, two main new morphological features are observed (1) The slope of the pyramid increases very slowly with increasing supersaturation (fig. 7). (2) A characteristic roughness of almost paralId grooves appears in the sectors A on both sides of the [21.0] directions (fig. 8). The narrow’ sectors B around the [11.0] directions remain smooth and stable even at high supersaturation (a-> 100% at 310°C). The grooves propagate from the centre towards the edges of the face in the [11.0] direction and increase in size with the distance from the centre of the pyramid. They are an expression of the morphological instability of the initial step-train due to the overlap of the diffusion fields and perhaps the influence of itiipurities [25]. In any case, they arc interpreted as kinematic waves [5.25] (steps whose the density varies during their motion) which show a typical hehaviour with the supersaturation: a supersaturatioti step leads first to an increasing density of macrosteps. which further decreases while reaching the new stationary growth state (figs. 8a—8c). Higher macrosteps are observed near the edges during this transient state. lit the steady state, the number of permanent waves observed is an increasing function of the
3.3.7. Dependence of pyramid slope I’ on supersaturation a-
Because the initial topography of the face in a given experiment depends on its history (number and nature of evaporation—growth sequences) the slope !‘ of the unique pyramid versus supersaturation a- isa reversible and repriductihle law characteristic of the conditions (Ii.. a-) only for a given (000l)face of a crystal. For example, a high growth rate leads to steeper hillocks. The slope measurements are not possible when several pyramids coexist on the face because the map changes with the supersaturation. Fig. 7 shows the 1’(o-) curves for three beautiful pyramids spreading over the whole face (three different crystals and experiments). The slope values lie in a rather narrow range (5 x l0~—3x l0~ rad) even at high supersaturation. Unlike curve A which can he explained only by the increase of the number of cooperating spirals, curves B and C show a critical supersaturation (Ti of about 30~ above which the slope increases very slowly with a-. The break in the P(a-) curve is more pronounced at higher temperature. An explanation, based on the back-stress effect [24], is discussed below, 0.003
p(rd)
-
0.000
I
0
20
40
I
60
60
iOU
Fig. 7. Pyramid slope I’ versus the supersaturation a-: (A) 3l5A’: (Bl 305’ (‘: )(‘) 195A’.
20
M. Couch, G. Qut nrel / Morphological instability of i’apour-grown crystal: (000/) Ccl
-
()
251
.- .‘~
64%
44%
60%
95%
Fig. 8. Pyramid roughnening at high supersaturation (T
5. 2n) C). (a) (e) Transient response to a supersaturation step. (d)—(g) Schematic evolution of the morphology of the grooves versus the supersaturation. The number of steady-state grooves increases with the supersaturation. The sectors B remain smooth.
supersaturation (figs. 8d—8g). At the lowest supersaturation (a- 30%), they appear first near the edges of the faces where they are stopped by the lateral faces; then, the roughness increases and spreads towards the centre as supersatura-
tion is increased. At the same time the relative orientation of the A sectors changes and inverses. This is shown by the bending and fading of the edge along the [21.01directions. At this point, the two sectors lie in the same plane and the waves of
252
31. (lane/i,
(7.
Qitetitel / MOt’p/u)lOgu’al instability of’ i apout’—gro ss’,i it’s’s/al: (0001) ( ~l
one encroach upon the other. A further increase of the supersaturation leads to a curved profile of the facet and hence of the whole pyramid. The roughness increases again and becomes irregular. This morphological coarsening is reversible. Lowering the supersaturation lead to the inverse evolution of the morphology; the edge between the two A sectors becomes again visible. indicating the inversion of the corresponding angle. The roughness covers a regular pattern of grooves which progressively disappear. The original pyramid with M3 morphology is restored and the slope again decreases if the supersaturation de-
creases. Some qualitative interpretation will he mentioned in the discussion. 3.5. Ei’aporation process’ The competition between pyramids, previously described for growth, takes place also in the first stages of the evaporation at low undersaturation leading to a reduction of the number of pyramids: we observed that the pyramids dominated during growth appear again with the reverse mechanism of their disparition. hut this is a transitory stage towards a more and more complex topography. A
Edge position 2mm
~tt
7
50
°D3 “
St K U
5
~
15
V
2t
34~m/~n 3t
25
Fig. 9. Regeneration of a smooth (0001) face from the growth of a pre—evaporated crystal. ‘l’he iiearly isotropic edge velocity is 34 Cn’i/rnin,
M. Gauch, G. Quencel
/ Morphological instability of i’apour-grown
long time evaporation at a low rate leads to a randomly rough face showing the emerging points of all the dislocations active in evaporation. The case of high undersaturation is quite different, as shown in fig 9a: the higher rate of evaporation at
Fig. 9 shows the regrowth wave origInating from the highest point of the previously strongly evaporated face; the mean lateral rate of growth in the [12.0] direction is 0.6 mm/s and the height of the front about 280 A. The polygonization of this (0001) very smooth facet begins when it reaches a sufficient size. Previous leader-growth hillocks will then appear when the wave spreads over the corresponding outcropping dislocation and we can again observe the first stages of the growth pyramids. The centring of the tip of. the face in the picture results from a faster evaporation rate of the edges of the face as previously reported. 3.7. Distribution of growth steps
We said that the pyramid competition was in fact that of a group of spirals originating from screw dislocations spacially distributed with interdistances d such as d < 19r*, where r* is the critical radius of the nuclei in the conditions of growth. The scheme of fig. 10 shows how the slope of any facet of the resulting pyramid is thus periodically modulated. This permanent so-called “rythmical bunching” [26] is revealed in our case by the colour modulation of the pyramid vicinal facets. The step-trains beneath the resolution of the binocular are also detected by a slight supersaturation step which leads to a perturbation in the spatially distributed steps; this instability disappears after some time depending on the growth
crystal: (0001) Cd
253
6
/
/
-
/
/ /
3
‘
-
—
—
\
4 -
Profiie XX’
. . . , , Fig. 10. Schematic constructn)n ol . a pyramid tram the cooperation of six elementary spirals when the distance d between dislocations groups follows: ii < r °/2, where r is the critical nucleus radius. The radial section XX’ shows a modulated profile.
rate. Notice also that the growth front step is revealed by the colour variation induced by the mean pyramid slope modification. When the single crystal grows at low temperature (about ‘J~< 250°C), the step-trains are always observed because any perturbation is very slowly removed. Step shape etolution: The growth step shape is the periphery of a cross section of the hillock. So, following the shape of the step from the centre to the edges gives a tridimensionnal picture of the corresponding hillock. Fig. 11 schematizes this evolution: The round initial shape becomes progressively hexagonal and then reaches a typical starlike shape far from the centre. The discussion below explains this evolution. It means that the initial growth step winds itself in an archimedian spiral, as stated in the BCF theory, before it polygonizes progressively. Moreover, the interstep distance increases from the centre to the edges and the lateral rate of growth becomes
tI. (lain/i. (I, Qieiioi’/
254
.
K
~“
~
OOO1~-
“
3 2~
,/
.%‘Ior~t/io/oy’ii’alinstability
(000/) ( ‘il
that along the (II .0K directions. the two layers of atoms hunch in a 5.6 A layer growing laterally at ,
the lowest velocity (fig. I 2c): along the K 10.0K directions, the dissociation of the 56 A layer
‘,‘
/
of i ‘apoier-io’ois’ti it .sial:
/
occurs and the sub-layers travel independently ,
l:ig. II. Scliem:ttic csolution of the step shape Iron the centre to the edges ol’ a pyramid with a M3 morphology: (I) round: (2) lic\agollal: (3). (4) 24 edges polygonc (starlike shape).
(fig l2h) giving icgul ir intcrl tcing structurc which builds thc n trrow scctors B This mccli t nism still rcm tins in thc c tsc of composttc p~rmud without bunching of thc stcps Whcn thc pyr tmid step is high (scvcr ml coop crating growth centres), the dissociation is less probable and the narrow sectors B around the K II .0~directions do not appear. So, we can con— elude that the most pols’gomzc’d pyramids’ (M3
:ttuisotropic. ‘[he nuorphology of the spiral indeed changes in the radial directions,
morphology) hate an cleiiic’ntar~’stc’p height ot 5.~ .1 . even if they result froni a multi—spiral coopera— tion. These are the less steep hillocks observed it low supersaturation.
\
i
--
nO.11
“.:
4. 1.2. Dislocation glide
4. i)iscussion
The modification of the morphology atter several evaporation—regrowth sequences is cxplained by the growth centre displacemetut clearly visible in fig. bd in the (11.0K direction which is the principal direction of gliding in hcp metals when c/a > 1.633. In the regrowth process, the high wave front can bent the dislocation lines met [30] and sweep them away from the periphery of the face. Asa consequence, the niore centred sources are less exposed to this sweeping, and this explains the well centred unique pyramid generally observed after several treatments. ‘[he observation of high—slope pyramids of weak activ— ity at the periphery’ of the (0001) face indicates a concentration of dislocations which induce a high step spiral layer of slow’ lateral rate of growth. At temperature of the crystal lower than (T~ is the melting point temperature. which is 321°C for cadmium), the motion of the dislocations is no more possible: this explains the great stability of the pyramid map at low temperature.
4, 1. (/~~ntb, ,net’ha,’ij,rnis
ln the hexagonal compact structure of cadmium, the periodicity in the direction of the c axis consists of successive layers AB AB ABAB . . . of atoms with a i~- rotated trigonal symmetry (fig. 12). The Burgers vector is then hit = 2mnCii. with c~= c/2. where c’ is the cell parameter (c’ = 5.6 A is the AB layer thickness). ‘l’he nuost probable step height is c’ = 5.6 A (iii = I) in order to have a minimization of the elastic energy of the dislocation. 4.1. 1. Shape ci’olution of .s’lep.s’ and pyramids’ The starlike shape of the pyramid is an expression of the anisotropy in the lateral velocity of the steps, enhanced by the dissociation of the 5.6 A high spiral layer in two layers of 2.8 A height along the six (10.0K directions (fig. 12), itS wits proposed by Frank [27] for the interpretation of SiC experiments [281. The mechanism is summitrized as follows: (21.0K is the fastest direction of growth for the A layer hut the slowest for the B layer; it is the inverse situation for K 11.0K; the velocities’ ~trc the same in the K 10.0K direction. From these anisotropic growth rates, it results
4.2. Kinetic fratures
•
4.2.1. Ei’olution of th,c’ pyramid slope P nil/i I/u’ supersaturation ain the classical theory of BCF [4]. the slope of an in dislocation cooperating spiral with a step
/ Morphological instability of i’apoor-grosvn crystal:
M. Gauch, G. Quentel
heigth h is given by
speed of the steps
2kT~ ln( 1 P
mh =
mh =
+
‘
where I is the interstep distance, 1~the substrate temperature, a- the relative supersaturation, ‘y the edge-free energy, £1 the atom volume in the crystal and k the Boltzmann constant. The normal rate of growth R depends on the lateral [21
0]
// /
®
/
,,
\
/
=
/
/ /
[21.01
/
/ /
Lever
A
.L~erB
/
,/
[lit]
‘
//
/
///////
A
~
/
/
/ I
/
/7
/
/ ‘ ‘
=
[lot]
/
/
\\\
=
=
/
~ ~/
\
i’
[lot]
/
/1 /
/
/ /
/
255
1 as R Pi’~ h(w/2~r) Some qualitative and quantitative results can i’i11/l(a-, T). be derived from the P(a-) curves of fig. 7, which correspond in all cases to a single M3 morphology pyramid on the whole face. P, h, and sometimes R are derived from experiments. (a) A type curve: The lateral growth rate of the steps is deduced from the measurements of R and P from the relation ‘It R/P. As shown
a-)
l9yfl
(0001) Ccl
/
[ITO]
A
Atom
layer A
////// Atom
layer B
~t,O1
~t.0I
Fig. 12. Growth mechanism of cadmium layers. (A) Trigonal symmetry of the atoms_layers A and B, which explains the anisotropy of the lateral velocity L’. P is the resulting profile of a spiral layer. 1(2.) = 2v/(i’(ii ~ . (b) Interlacing structure of the A and B layers. (C) Resulting profiles in three directions; [21.0]: IA> CB; [11.0]: ~ 1A [110.0]: o~=
250
(bun/i,
:ti,
I :ehlc ,l , Quantitative characteri,ation
(I, (jui’nti’/
/
Morp/io/oi,’ii’al oi,stahilui’ of i ‘aftoiir-~’t’ois’,iirs’stal: (000/) ( il
Interstep distance: 0) a pyramid derived from the
experimental slope versus a- curve
ii
in
a- <
Slope!’
R
i~/
/
(rad)
(A/n)
(pm/n)
)nm)
10)) -
4.2 .
152
2~
l257 407 2)(S
12.7 2
(.2 l.%
((.44/ 1.2 ‘/ 0) I))
(5)
3.1
27
“
x II)
‘
in table I. we find that it decreases in inverse ratio to the supersaturation; the interstep distance 1 li/I’ is evaluated assuming the element;try step height to he Ii 5.6 The decrease of t’ii suppsesa drastic decrease of the interstep distance, so that overlapping of the diffusion fields occurs. The experinuental law’ is verified to he P(a-) (1.1 76a-”, with n 2.1. In the BCF regime, we expect I’ Ka-” with a ~ I: moreover. itssuming that the nuean distances d,. between cooperating growth centres remain constant. (lie condition il. < ~l does not hold when I decreases as the supersaturation increases and the number of cooperating centres decreases. As a consequence, we expect more ~tnd nuore individual pyramids on the face. It is just the opposite we observe: Curve A means an iiucrease of the iuumher in of cooperating growth centres. The estimation of in (see table 1) is derived from the above expression of P with the edge free eiucrgy y deduced from our experimcnts. (h) Curve B is the pyramid slope variation for tile experiment given in fig. 7. A critical supersatur~mtion a-~( 38%) is evidenced by two kinetic regimes: for a- < a-i. P(a-) is roughly linear with the supersaturation; above a-i, P(a-) Ka-” with 0.3
=
~.
=
:
//2 X
r>a-~:
=
a-~/2 a-
=
(a-1/a-)i
//2X
5
[24aj,
2a-) [24h]. //2 X~ (a-i / The experimental value of the exponent (0.3— 0.4) for the high supersaturation case is in good agreement with the theory if we keep in mind the pyramid morphological imperfections: the conical shape around the centre becomes hexagonal ;tnd decomposes into interlacing layers as going toor
=
wards the edges. Moreover, instabilities of the growth steps modulate the slope because of the rythmical bunching; therefore, the reflection pattern itS well as the colour analysis gives its the mean value of the slope. 4.2.2. iwo-chmnensional nucleation
It is obvious that the spiral growth is the predominant mechanism in our experiments, even at the highest supersaturations applied. hut what about the two-dimensional nucleation? Following Van Erk et al. [10]. 2D nucleation will occur at sufficiently high supersaturation if the interstep distance I is larger than the total capture area 2X., around the step. In this case. the step velocity is increased hut the pyramid slope decre~tses. We can explain in this wity the apparent decrease ot (lie slope itu fig. 7B near a- 35%: the two-dinuensional nucleation mechanism occurs between 3tY~and 38% and it is inhibited beyond a- 38~ by the overlapping of the diffusion fields. This is consistent with the estimation of the nucleation rate which becomes significant around 40~. Nevertheless, 2D growth will also occur during the transient state following the application of a supersaturation step because this perturbation gives rise to waves of high density of steps and hence the fluctuating interstep distance becomes periodically larger than the capture area 2 X~. 4.2.3. Estimatiomi of I/ic’ macan free surface diffusion pat/i X, and the edge ene,gv y for the c’adtnium At a- a1, the slope P~ h/2X5 assuming an elementary step height of 5.6 ~&.we find X 2000 ±20() ~ an edge energy y 10 ~ erg/cm =
=
utive steps. Moreover, introducing the hack-stress effect [24] heads to the following relations for the
1
M. Gauch, C. Quentel / Morphological instability oJ i’apour-grown crystal: (000/) Ccl
257
is deduced from the slope of curve B in the low supersaturation case (a- < 30%) using the above theoretical expression of P(a-). Unfortunately, no experimental data were found in the literature for comparison and only a theoretical estimation is possible using the formulae: X5(T~)
bunches cover the whole hillock. However, keeping in mind that the roughness of grooves is observed only in those sectors around the (21.0K directions, it means that impurities adsorb on these vicinal facets selectively. The reason for this
a)) exp(3CD/kT) and ‘y 2CD/2a11. For a hcp metal, CD L%H5~0/6with JiHSUh the sublimation energy. Using the cadmium values (a)) 2.98 A, 26.6 kcal), we find X5(305°C) 950 A and y i0~ erg/cm.
steps into kinematic waves seems more realistic at high temperature, but the impurity adsorption is probably more active in low-temperature experiments (T~<240°C). The kinetic behaviour of the steps will be discussed with the aid of the ellipsometric results described in the forthcoming paper.
=
=
=
=
=
4.3. Roughening of the face at high supersaturation The theory of kinematic waves [5] can explain the observed formation of reversible piling up of steps into macroscopic grooves in the case of an interstep distance perturbation which can be a supersaturation step or the action of impurities. The direction of propagation is that of the maximum rate of growth (21.0K and so the bunching of steps is the higher. The density of steps and the macroscopic apparent step height increase in this direction with the distance from the centre of the pyramid. The decay of the number of grooves observed with time is proof of a shock-wave but macroscopic steps exist also. Assuming an oscillating collective behaviour of the steps, this perturbation increases with the distance from the centre. The first steps reaching the edges of the face are stopped and the following steps pile up behind and form a macroscopic groove; this instahihity increases with the supersaturation. In spite of the high purity of the cadmium material and the careful multi-distillation process of the cell filling and closing, a possible influence of some impurities may be taken into account; Van der Eerden and Müller-Krumbhaar [30,31] recently stated a new impurity adsorption model to explain the dynamical process of formation of macrosteps. According to this theory, macrosteps are formed far away from the centre by first a doubling and then a quadrupling of steps and so on; moreover, the bunching of steps is possible only beyond a given supersaturation and extends logarithmically with time. Although quantitative data were not yet available, this last point only holds at very high supersaturation where the
is not understood. In conclusion, the coupling of
~•
Conclusions
through a number of in-situ experiments on a large cadmium single crystal, the growth and evaporation processes of the (0001) face have been carefully investigated for a large spectrum of crystal temperature T~ and super(under) saturation a-, with an optical differential— interferential technique. The following main conclusions can be put forward: (a) Growth proceeds by a spiral mechanism whatever the conditions (T~,a-), and the morphology of the face is determined by a competition between pyramids of different activities; Unlike the classical theory of BCF, the number of’ pyramids decreases when the supersaturation increases: the number of pyramids, maximum near the thermodynamic equilibrium (a- < 10%), decreases steeply as the supersaturation increases; beyond about 20%, only one pyramid spreads all over the face and controls the growth. (b) The rythmical bunching of the growth (or evaporation) steps is a frequently observed feature indicating the composite nature of the pyramids. (c) Analysis of the pyramid morphology leads to identifying elementary step-height pyramids showing a starlike shape composed of 24 vicinal facets; these are the most commonly observed in our well-defined conditions of growth. Moreover, all kind of pyramids have also been found such as hollow cores and flat-top pyramids.
tI.
258
(biiic/i, ( i. (lnc,iti’l ,/ ,‘sI,,rp/is>/oç’ii’a/ iii stability of i a~ioiir-c’roii’ni’rs:s’ial: (0001) ( il
(ci) ‘l’lue morphological evolution of an isolated pyramid, with elementary steps, with the supersaturation is a reversible process which shows two interesting features: (i) The pyramid slope in— creases very slowly above a supersaturation /38% when the overlapping of the diffusion fields of the steps sets on. This is interpreted itsa consequence of the hack-stress effect. (ii) Beyond a-i, the bunching of steps is increasingly marked froi’n the edges to the top of the pyrat’nid facets adjacent to the (21.0K directions as the supersat— uration iiucreases. This morphological instability heads td) a characteristic and reproducible dynamical roughening which is also a reversible process. As a consequeiuce. the vicinal facets take a curved profile. At the highest supersaturation, the roughness appears to be randomly distributed. . (e) 7 I he evaporation process ,is also governed in the first stages by the pyramid competition. Sue.
cessive evaporatioiu—regrowth cycles sweep the dislocations towards the edges; this explains the centring of the unique pyramid. -, (I) Iwo—dimensional nucleation is inhibited by the overlapping of the step diffusion fields. The general consequence of this great morphuological instability is that the normal rate of growth of the (0001 )face is not a simple function of the thuermodynamical condittons (It. a-). hut depends widely on the morphology of the leadergrowth pyramids which are supersaturation dependent, and on the step behaviour (bunching and orientation changes). .
.
.
Acknowledgments
ertien oI Solid State Materials, Vol. 7. Ed. W.R. Wilcox (Dekkcr, New York. l982).
[4] W.K. l3urton. N. Cabrera and CF. [“rank. Roy Soc London A 243 (1951) 299
Phil, Trans.
[5] A.A Chernon. Soviet Phvs -Usp 4 (1961) 116 [6] S.D. Luhetkin and W.J. Dunning, J. (‘rystal Growth 43 (197$) 77. [7] Y. Zundelevich :tnd K. Glaser. .1. (‘r~stal Giowth ‘1990)) 9 ,
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,
102
.
[8] AL. Morizan and W.J. Dunning. J, ( rvstal ( rowttn (1970) 75) 9] W.P..1. nan Fnckevori,
R ,.Janssen-vari
Rosrnalen and
\W.l I. van cler Linden. .1. Crystal Growth 49 (I 981)) 5)12. 11(11 W. van Eik. I l.J. van Flock-Martens and G. Baitels, .1. [II]
Crystal Growth 4$ (l9$0) 62l , ‘‘ K. I sukamoto. II. Ohha and I. Sunasiawa, .1. ;i’ow’th 63 (l9$3) IS
(.
i’vstsl
12] A.A. Chernov. L.N. Rashkovich and A.A. Mkrtchan, .1.
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[IS] M. Gauch and C. Quentel. Surt’scc Sc). 108 (19$)) 617. [16] M. Gauch, Thesis. University of Aix—Marseille III (1989). [17] M. Gauch and U. Quentel, to he published. [IS] M. l”rancon. Rev. Opt. 34 (2) (1952) 65. II
9j
C A. E~enstermaker and FL. Me( rackin, Surface Sui. In
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Introduction. Ed. P. 1 lartman (North—I lolland .Amster—
‘l’he authors wish to thank Professor R. Kern and Professor A. Baronnet for very helpful suggestions and encouragement during this work and A. Ranguis for his technical assistance. References [I] F.C. Frank, Disc’. Faraday Soc. 3 (1949) 48, [2] A. Baronnet. Progr. (‘rystal Growth Characterization I (1978) 212. [3] I. Sunagawa and P. Bennema. in: Preparation and Prop-
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46
11979) 451. [27] F.C. Frank, Phil. Mag. 42 (1951) 0)14. [2$] AR. Verma. Phil, Mag. 42 (1951)1384.
[29] H. Klapper, Phys. Status Solidi A 14 (I 97) 99, [30] J.P. van der Eerden and H. Miiller.Krumhhaar. [leetrochim, Acts 31(1986) 1)107. [31] J P van der Eerden and H. MdlIer-Krumhhaar, Phvs. Rev, Letters 57 (1986) 2431.