kinetics of the mbe growth of si(001)2 × 1

Surface Science Letters 279 (1992) L207-L212 North-Holland

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letters

Surface Science Letters

Kinetics of the MBE growth of Si(0Ol)2 x 1

Ivan Markov 1

Department of Materials Science and Engineering, National Tsing Hua University, Hsinchu, Taiwan, ROC

Received 20 August 1992; accepted for publication 1 October 1992

The time evolution of growth pyramids consisting of 2D islands is followed. The islands are elongated along the dimer rows and rotated by 90* in every next monolayer. The S A steps propagate by formation and growth of 1D nuclei whereas the S B steps advance by direct incorporation of adatoms from the terraces. It is shown that at high enough temperatures ( > 1000 K) both kinds of steps grow in a diffusion regime which leads to comparable rates fo step advance and fast damping of the RHEED intensity oscillations. At lower temperatures (400-500°C) the S B steps still grow in a diffusion regime whereas the S A steps propagate in a kinetic regime. The latter leads to considerable elongation of the consecutive 2D islands and slower damping of the oscillations.

1. Introduction

Understanding the elementary processes involved in epitaxial growth is essential for control- ling the growth on an atomic scale and thus for fabrication of novel optical and electronic de- vices. The MBE growth of Si(001)2 × 1 is of particular theoretical interest as the steps on its surface are inequivalent and propagate with different rates, on one hand, and the surface diffusion is highly anisotropic, on the other. Since Sakamoto et al. [1] reported the first experimental observations of RHEED intensity oscillations during Si(001) MBE many studies both theoretical [2-7] and experimental [8-12] have been de- voted to the problem. In particular Sakamoto et al. [1,8] reported that high temperature annealing leads to the formation of a single domain Si(001)2 x 1 surface. RHEED intensity oscillations with a bilayer period have been observed in the [110] and [110] azimuth and with a monolayer period in the [100] azimuth on such an annealed surface. The R H E E D intensity oscillations

i Permanent address: Institute of Physical Chemistry, Bulgar- ian Academy of Sciences, 1040 Sofia, Bulgaria.

showed remarkable behaviour with temperature. Deposition at high (1000°C) and extremely low (25°C) temperatures gave rise to a small number of fast damping oscillations whereas at intermediate temperatures (400-500°C) high intensity slowly damping oscillations have been recorded. 2200 oscillations have been thus recorded at a substrate temperature of 500°C after annealing for 20 min at 1000°C [8]. The amplitude de- creased approximately 25 times after 1000 oscillations. On the contrary on a double domain surface without (or with insufficient) annealing to produce single domain surface no azimuthal dependence of the oscillations has been observed and the latter displayed a monolayer period and a decaying envelope [10]. Heun et al. [12] reported oscillations with a monolayer period of the LEED intensity of the 00-beam during growth of a well oriented double domain Si(001)2 x 1 surface (tilt angle less than 0.1 °) in an azimuth aligned 15 ° off the [001] azimuth. The spot profile analysis (SPA-LEED) during growth showed that the oscillations recorded at 640 K are best explained if a simultaneous growth of four monolayers was assumed. The variation of the step density with the number of monolayers deposited showed a visible asymmetry. Besides, some time after the

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1.208 L Marko~ / Kinetics of the MBE growth o f Si(O01)2 × I

beginning of the deposition a constant length-to- width ratio 3:1 at 640 K of the growing monolayer islands has been established.

This Letter describes a kinetic 2D nucleation model which explains qualitatively some of the observations outlined above.

2. Model We consider growth of pyramids consisting of

consecutive 2D islands formed on top of each other as shown in fig. 1. Islands of the new monolayer nucleate instantaneously at the mo- ment the lowest level is completely filled. Thus a constant thickness of the growth front is main- tained. The islands are elongated along the dimer rows and are surrounded by two long S A and two short S B' edges (in the notation of Chadi [13]). The S A steps are smooth [14] and propagate by consecutive formation and growth of 1D nuclei (atomic rows with a critical length determined by kinetic factors) [15,16]. In the case of steps with finite length l A, as in the case under study, the steps propagate by the row-by-row mode in com- plete analogy with the layer-by-layer growth of defectless crystal faces. The advance rate of the S A steps is given by V A = 2Jal A [15] where J is the rate of 1D nucleation and 2a (a = 3.84 ,~) is the width of a dimer row. Surface diffusion is allowed only in the direction of the dimer rows in accordance with recent findings that this is the direction of the fast diffusion [17,18]. Then SA steps grow at the expense of adatoms diffusing on the lower terraces whereas S B steps are fed by

i B ~ ~ a Ira1

IAI Fig. 1. Schemat icrepresenta t ion of a growth pyramid consisting of three monolayer islands. The islands are elongated along the dimer rows which rotate by 90 ° on each next terrace.

adatoms diffusing to them on their upper terraces. This is also in agreement with recent find- ing by Roland and Gilmer [19] that attachment of adatoms to the S A steps from the above terrace is less probable than from the lower terrace. The reverse is valid for rebonded S B steps. Thus the S A steps of the first monolayer islands grow at the expense of atoms diffusing to them on the crystal surface between the pyramids whereas the S B steps advance at the expense of the atoms adsorbed on the terrace formed between them and the S A edges of the second monolayer islands (fig. 1). Finally, the atoms adsorbed on top of the uppermost island feed only its S B edges. This simplification reduces the diffusion problem to a one-dimensional one. Exchange of adatoms between neighbouring terraces is not allowed al- though recent work [7] showed that it is one of the probable mechanisms.

The diffusion of adatoms to the island edges is governed by the equation (re-evaporation is ex- cluded)

d2n( x ) / d x 2 + R/DS = 0, (1)

where n(x) is the adatom concentration, R (cm- 2 s -1) is the atom arrival rate and D~ = a2u ex p [ - Esd/kT] is the surface diffusion coeffi- cient, u and Esd being the vibrational frequency and the activation energy for surface diffusion, respectively.

Solving eq. (1) for each terrace subject to the boundary conditions [6]

-a2Os dn( x ) / d x = ~no'B

X = ( I A , n - - / B , n + l ) / 2

a2Os dn( x ) / d x =/3Atr A

x = - (IA,. - - / . . . + , ) / 2

allows us to write a system of equations for the growth rates of the separate island edges. Here trA, B = nA,B/n e -- 1 are the supersaturations in the step vicinities, n A and n B are the respective adatom concentrations and ne = n o e x p ( - A W / kT) is the equilibrium adatom concentration, n o = 1/a 2 and AW being the atom density in the crystal plane and the energy to transfer an atom from a kink position on the correspondLng terrace

I. Markov / Kinetics of the MBE growth of Si(O01)2 x 1 L209

[20]. flA = [~rAlA, n and /3 s are the kinetic coefficients [21] of the respective steps, or in other words, the rates with which the atoms are incor- porated into the kink sites. Note that the kinetic coefficients of the S A steps are proportional to the step lengths whereas thos6 of the S B steps are independent of them [15]. The kinetic coefficients depend exponentially on a kinetic barrier for the incorporation of atoms into the kink sites [21]. They are also proportional to the equilibrium kink densities being therefore a function of the work for thermally activated formation of kinks [15,211.

3. Results and discussion

The surface coverage of any monolayer is 0. = IA, nls,nN s where N~ is the density of the growth pyramids. Then the time evolution of the surface coverage On(t) of each monolayer is easily found from dOn(t) /d t = (lA,n d l s , n /d t + lB,ndlA, / d t ) N s. The general expressions for dOn(t)/dt are a bit cumbersome, so we do not give them, in- stead, we write the system for a bilayer pyramid

dO1 MA MB ~rr = 1 - 01--~--(1 + + M s ) - 02-~--(1 + 1MA),

d02 MA MB = 01--~-(1 + ½Ms) + 02--~- (1 + ½MA)

d r

(2)

(2)

where r = R t / n o is the number of the deposited monolayers,

flA,Bn0 MA'S 2Dsn¢ (lAa -- lS,2), (3)

and M = M A + M B + MAMB. As seen M A and M s represent ratios of the

kinetic coefficients and the rates of surface diffusion 2Ds / ( IA , 1 - - lB,2)" They depend on time through the time dependence of the edge lengths LA, 1 and IB, 2 and are exponential functions of temperature. Depending on the temperature they can be either greater or smaller than unity. Thus at T--1200 K, with the typical values of the constants involved; R = 1 × 1013 e m - 2 s -1, v = 1

X 1013 S -1, Esd = 0.6 eV [17,18], to A = 0.5 eV and o~ B = 0.05 eV (work required for thermally activated kink formation [22]), i* = 1 (i* is the number of atoms in the critical nuclei) the values 5 X 10 4 and 20 are obtained for M s and M A, respectively. At T = 800 K M B and M A have the values 5 × 102 and 0.4, and at T = 600 K - 50 and 0.01, respectively. In the above estimations /B,2 has been neglected in comparison with /Aa and lA,2 has been approximated by lBa. In the case when both crystallization rates flA and /3 B are much greater than the surface diffusion rate the latter is rate limiting and the corresponding steps propagate in a diffusion regime. In the reverse case the steps grow in a kinetic regime and the adatom concentration is uniform all over the terraces. In the classical paper of Burton et al. [20] the successive steps are equivalent and the transition from diffusion to kinetic regime of growth takes place at one and the same temperature. In the growth of Si(100) the steps are inequivalent and the transition from one growth regime to another for both kinds of steps takes place at different temperatures. Thus there are temperature intervals in which both steps grow in one and the same regime, either a diffusion or a kinetic one, and an interval in which the steps grow in different regimes. Three cases could be clearly distinguished:

(i) High temperatures ( T > 1000 K) 1 <<M A << M B and M - - M A M s. Both steps propagate in a diffusion regime and their propagation rates are comparable. The system for the surface coverages reads

d 0 1 / d r 1 + 0 2 ) , = - + ( o l

dO. /dr = 1 + 0~) - 1 "~(On_ 1 ~(On"~On+l) , (4)

dON~dr = 1 ~( ON_ 1 "[- ON) ,

where the index N denotes the uppermost monolayer. As seen, the kinetic coefficients do not enter the equations. In this sense they resemble the equations derived previously under the as- sumption of growth of isotropic crystal surfaces (growth pyramids consisting of circular islands) and rapid exchange of atoms between the edges and the adatom population on the terraces (diffusion regime of growth) [23-25].

L2Itl l. Markot, / Kinetics c~f the MBE growth of Si(O01)2 x l

1.20 Z O

F.., .¢

~ 0 . 8 0

Z

0.40

ra,,1

0.00 0.00 1.00 2.00 3.00

NUMBER OF MONOLAYERS

Fig. 2. Overall step density variation as a function of the number of deposited monolaycrs in the high temperature region (cq. (4)). The figures denote the numbcr of the simul-

taneously growing monolaycrs.

(ii) Low temperatures (600 < T < 800 K) M A << 1 << M B and M -- M B. The S B steps propagate in a diffusion regime whereas the S A steps advance in a kinetic regime due to the difficulties connected with the formation of 1D nuclei. The

propagation rate of the S B steps is much greater than that of the S A steps and the islands grow strongly elongated. The system for the surface coverages now reads

dO, /dr= 1 - ½(QO~ + 02) ,

d 0 , / d ~ " = I 2 q_ On ) I 2 ~(QO,_, + 5(QO, +O, ,), (5)

d O N / d , r = 1 2 ON), g ( Q O N _ 1 +

where Q = 13'Ano/2DsneN s is a constant originating from M a. The analogous quantity for the S B steps does not enter the equations as in case (i). Typical values of Q with R = 1 X 1013 cm -2 s -~, ~ ,=1×1013 s l, Es a = 0 . 6 e V [ 1 7 , 1 8 ] , N s = 2 . 0 × 10 ~l cm- 2 (critical nucleus consisting of one atom is assumed), to a = 0.5 eV [22], range from 0.01 to 0.4 for T varying from 600 and 800 K, respectively.

(iii) Extremely tow temperatures (300 < T < 500 K) M A<<M B<<I and M ~ M B. Both steps propagate in a kinetic regime. We will not consider this case in detail as the rate of 2D nucleation is very high and the surface becomes kineti- cally rough. The model is no longer valid.

The systems of equations are easily solved numerically, subjected to the boundary conditions 0,(1) = 0,+1(0) and 0N(0) = 0. Once we have solu-

0.80 Z o

0.60

>

>,., [,.,, ~,,-q 0.4.0 O0 Z r.z.l

0.20 r~ [- o~

0.00 0.00

(a)

3,4

1.0o 2.00 3.00 N U ~ E R OF MONOLAYERS

Z 0.50

o [-,, ~w~ 0.40

0.30

Z o.2o

.i 0.10

0.00 o.oo 1 .oo 2.oo .oo

NUMBER OF M O N O ~ Y E R S Fig. 3. Overall step density variation as a function of the number of deposited monolayers m the low temperature region (eq. (5)): (a) T = 800 K (Q = 0.4), the figures denote the number of monolayers which grow simultaneously, (b) T - -600 K (Q = 0!01), curve 1

- layer-by-layer growth, curve 2 - s imultaneous growth of two, three and four monolayers.

I. Markov / Kinetics of the MBE growth of Si(O01)2 × 1 L211

tions for On(z) one can calculate the total step density according to [25] L(¢) = (4wNs)l/2En{1 - 0n(¢)}{-In[1 -0n(~')]} 1/2. Fig. 2 shows the variation of the total step density in terms of (4¢rNs) 1/2 in the high temperature case (eq. (4)). As seen the amplitude decreases with increasing thickness of the growth front. The curves are visibly asym- metric and the maximum being shifted to the left of ~-= 0.5 is in good qualitative agreement with the experimental observations during MBE growth of Si(001)2 x 1 [12]. It is worth noting that the high temperature case does not differ signifi- cantly from the case of circular island growth [25]. The reason is obviously due to the nearly equal velocities of step advance of the both kinds of steps in a diffusion regime. The situation changes dramatically in the low temperature interval. Fig. 3a shows the total step density variation at T = 800 K (eq. (5) with Q = 0.4). Whereas the step density originated from the growth of bilayer pyramids shows nearly the same behaviour as in the high temperature case, the step densities from three and four-layer pyramids are indistinguishable. It follows that the fourth monolayer cannot nucleate due to the very small size of the third monolayer islands and in turn the very low adatom concentration on top of them. This effect becomes much more pronounced with decreasing temperature (fig. 3b). At T--600 K (Q = 0.01) the total step densities originating from the growth of two, three and four-layer pyramids are indistinguishable from each other which means that even the third monolayer cannot nucleate because of the very small size of the second monolayer islands. What is more important, how- ever, is that the curves nearly coincide with the one for layer-by-layer growth which is an addi- tional proof that the second layer islands are very small and give negligible contribution to the overall step density. The advance rate of the S A steps are much lower than the advance rates of the S B steps and the islands grow strongly elongated. The growth shape ratio IA,JIB, ~ increases with decreasing temperature. The uppermost islands become smaller with increasing growth shape ratio. Nuclei on top of them can be formed when the islands become long enough. This takes place at a much later stage (if at all) and the growth

proceeds by simultaneous propagation of a few number of monolayers or in a layer-by-layer mode in the limiting case. Slowly damping RHEED intensity oscillations are to be expected in this case as found by Sakamoto et al. [1,8]. At still lower temperatures the number of nuclei increases drastically and the surface becomes kinet- ically rough which leads to fast damping of the oscillations. At very high temperatures the nucleation is suppressed and the growth proceeds by propagation of the steps. We thus conclude that the large number of oscillations recorded by Sakamoto et al. [8] at intermediate temperatures is due to the anisotropy of growth which results in a considerable elongation of the monolayer islands.

Acknowledgement

This research was supported by the National Science Council of the Republic of China.

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kinetics of the mbe growth of si(001)2 × 1

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