Homoepitaxial growth kinetics in the presence of a Schwoebel barrier

Computational Materials Science 17 (2000) 510±514

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Homoepitaxial growth kinetics in the presence of a Schwoebel barrier Vladimir I. Tro®mov *, Vladimir G. Mokerov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Mokhovaya Str. 11, 103 907 Moscow, Russian Federation

Abstract Nowadays it is well-recognized that the additional barrier to downhill adatom di€usion at the step edge plays an important role in the epitaxial growth. Very recently we have developed a simple model for homoepitaxial layer growth kinetics which allows to take into account the Schwoebel barrier impact on adatoms interlayer di€usion by using the concept of a feeding zone, as we have proposed earlier. This paper is devoted to further re®nement and extension of the model to the cases of an arbitrary nucleus size and coalescence behaviour of growing islands. The model consists of an in®nite set of coupled non-linear rate equations for adatom and 2D island surface densities and coverage in each successive growing layer. These equations in combination with an integral condition determining the new layer formation onset fully describe homoepitaxial growth kinetics at predetermined ®ve model parameters, characterizing adatoms di€usion rate, critical nucleus size and stability, Schwoebel barrier e€ect, and coalescence. The growth mechanisms and kinetics in a wide range of parameter values are studied and growth mechanism phase diagrams in various parameter spaces are constructed and discussed. Ó 2000 Elsevier Science B.V. All rights reserved.

1. Introduction In the past decade, there has been a renewal of interest in molecular beam epitaxial growth (MBE) due to its abundant applications in modern materials science, ranging from superconducting thin ®lms and quantum wells, to more esoteric applications in nanotechnology and biology. The ability to grow various layered structures with high crystalline quality and atomically sharp interfaces is necessary in these applications. The detailed understanding of the basic mechanisms of MBE growth at successive deposition stages as a function of the controlling deposition parameters is

* Corresponding author. Tel.: +7-95-203-3689; fax: +7-95-2038414. E-mail address: [email protected] (V.I. Tro®mov).

therefore of both fundamental and technological importance. Nowadays it is well-accepted that one of the most important growth parameters is the socalled Schwoebel (S) barrier, i.e., the additional potential barrier at descending step edge [1] that inhibits an interlayer adatom di€usion thus giving rise to growth instabilities producing ``wedding cake'' structures on a growing surface. The growth behaviour in the presence of S barrier is a subject of great current activity [2±8]. Recently we have developed a simple kinetic model for MBE growth on a singular surface and we have shown that in the absence of S barrier with decreasing of the ratio of the di€usion and deposition rates the growth mechanism crosses over from smooth layer-by-layer growth to rough multilayer one and eventually to very rough Poisson random deposition growth process [9]. In this paper, we extend the model by including S barrier and investigate

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V.I. Tro®mov, V.G. Mokerov / Computational Materials Science 17 (2000) 510±514

the growth kinetics as a function of the S barrier height. 2. Model At ®rst we describe brie¯y our model [9] which combines a familiar rate equations approach [10] with the concept of a feeding zone [11] allowing to take into account an interlayer di€usion. Let the growth proceed in the so-called complete condensation regime and only single adatoms migrate on a substrate surface with di€usion coecient D (cm2 sÿ1 ). Then, the rate equations for surface densities of adatoms (n1 ) and 2D islands (N1 ) in the ®rst monolayer [10,11] in dimensionless densities n1 ˆ n1 =N0 ; N1 ˆ N 1 =N0 and time s ˆ Jt=N0 can be written in the form: dn1 =ds ˆ …1 ÿ n1 † ÿ 2r11 ln21 ÿ hriln1 N1 ; 

dN1 =ds ˆ …1 ÿ n1 †gi lni1 ‡1 ÿ 2dN1 dn1 =ds;

…1† …2†

where l and gi ± the model parameters l ˆ N02 D=J ;

gi ˆ r1i Ci exp …Ei =kT †:

the island perimeter, di€use to a lower level whereas the atoms deposited onto the central part of the island remain there thus forming a feeding zone (FZ) for the next (second) layer. The speci®c area n1a of an FZ is given by an expression [9,11] n1a …t† ˆ n1 …R…t† ÿ k†:

Here J (cmÿ2 sÿ1 ) is the incoming ¯ux, n1 the substrate coverage, P r11 the adatom±adatom capture number, hri ˆ r1i N i =N the average capture number, r1i the adatom-i-size island capture number, Ni the density of i-size islands, Ci the con®guration constant, i the critical nucleus size, Ei the bonding energy of atoms in a critical nucleus, N0 (cmÿ2 ) the surface lattice sites density and coecient d accounts for a coalescence behaviour. If impinging islands simply cease to grow at their contact boundary but continue to grow in all other available directions, d ˆ 0 and if these islands rapidly (in a liquid-like manner) merge into a whole larger island, d ˆ 1. The atoms incident upon the tops of growing ®rst layer islands migrate over their surface with the same di€usion coecient D and until an average island radius R is less than a some critical value Rc all these atoms migrate across the island surface and attach to the island edge. We will assume that when R > Rc only the atoms landing onto a ring band of some width k adjoining to

…4†

It is clear that k  Rc . For simplicity we will suppose that k ˆ Rc and evaluate Rc from the analysis of the nucleation process on top of the island with an account of the presence of S barrier at the island edge. In a quasi steadystate approximation, the adatom density distribution on top of an island obeys the di€usion equation Dr2 n ‡ J ˆ 0

…5†

with the boundary condition at the island edge ± D…dn=dr†R ˆ n…R†ms a. Solution of Eq. (5) gives n…r† ˆ B ÿ Jr2 =…4D†; B ˆ JR2 ‰1 ‡ 2a=…xR†Š=…4D†:

…3†

511

…6†

ÿ1=2

Here a ˆ N0 ; x ˆ ms =md ; ms  exp …ÿEs =kT † and md  exp …ÿEd =kT † ± di€usional frequencies for adatom hopping over descending step at the island edge and on ¯at terrace with activation energy Es and Ed , respectively, so that the third model parameter x ˆ exp…ÿEB =kT †;

…7†

where EB ˆ Es ÿ Ed is the Schwoebel barrier at an island edge which might in principle be positive, zero, or even negative, here we will suppose that EB P 0, and hence x 6 1. Now we can calculate local nucleation rate on top of an island I…r† ˆ l gi J …n=N0 †

i ‡1

and then R R a total nucleation rate of a new layer [11] U ˆ 0 2prI…r† dr. Thus for a critical island size qc ˆ Rc =a given by the relation Z qc ‰U …q†=…dq=ds†Š dq ˆ 1; q ˆ R=a; 0

512

V.I. Tro®mov, V.G. Mokerov / Computational Materials Science 17 (2000) 510±514

after integration we obtain Z sc   ÿ1  i ‡2 q2…i ‡2† f‰1 ‡ 2=…xq†Š pgi ‰…i ‡ 2†4i ‡1 li Š i ‡2

ÿ ‰2=…xq†Š

0

gds ˆ 1;

qc ˆ q…sc †:

…8† 1=2

Setting now q…s† ˆ ‰ÿ ln…1 ÿ n1 †=…pN1 †Š , from (4) to (8) we ®nd ®nally the speci®c area of an FZ for the second layer n1a ˆ 1 ÿ exp fÿ‰…ÿ ln…1 ÿ n1 ††1=2 ÿ …ÿ ln…1 ÿ n1c ††1=2 Š2 g;

…9†

s > sc ;

n1c ˆ n1 …sc † is a critical substrate coverage, i.e. coverage at which second layer nucleates. Accordingly, the third rate equation for substrate coverage is  1 s < sc ; …10† dn1 =ds ˆ 1 ÿ n1a s > sc : Eqs. (1), (2) and (10) with formulas (8) and (9) fully determine the growth kinetics of the ®rst layer which serves as a substrate for the second one, the second layer serves as a substrate for the third one, etc. Thus, proceeding in this way we obtain eventually an in®nite set of coupled rate equations describing the layer growth kinetics in the presence of a Schwoebel barrier dnk =ds ˆ …nkÿ1;a ÿ nk † ÿ 2r11 ln2k ÿ hrilnk Nk ; 

dNk =ds ˆ …nkÿ1;a ÿ nk †gi lnik ‡1 ÿ 2dNk dnk =ds; dnk ˆ ds



nkÿ1;a ; nkÿ1;a ÿ nka ;

s < sck ; s > sck ;

…11†

where n0a  1 and nka with k P 1 are given by 8 s < sck ; < 0; 1=2 nka ˆ 1 ÿ exp fÿ‰…ÿ ln…1 ÿ nk †† : ÿ…ÿ ln…1 ÿ nck ††1=2 Š2 g; s > sck ; …12† and sck are calculated by formula (8) with substi1=2 tution q ˆ ‰ÿ ln …1 ÿ nk †=…pNk †Š . 3. Results and discussion As an example we consider the typical case i ˆ 1 for epitaxy of semiconductors in complete

condensation regime, then mi ˆ r11 ˆ 2 and d ˆ 0. In the numerical integration of Eq. (11) the capture number hri was calculated in the lattice approximation [10] hri ˆ ÿ4p…1 ÿ n†=‰ ln n ‡ …1 ÿ n†…3 ÿ n†=2Š:

…13†

By using obtained coverage kinetics ni …s† in successivePgrowing layers the average ®lm thickness hhi ˆ ni …s† was calculated. The latter is linear with time in the complete condensation regime, hhi ˆ s. For more detailed quantitative characterization of the growth morphology the r.m.s. roughness r was calculated. 2 1=2

r  ‰hh2 i ÿ hhi Š

2 1=2

ˆ ‰R…2i ÿ 1†ni ÿ …Rni † Š

; …14†

where h and r are given in monolayers. It should be noted that for the 1st layer growth Eq. (11) are solved exactly and in the case of perfect S barrier (EB ) 1), x ) 0 Eq. (11) give naturally Poisson growth kinetics nn ˆ 1 ÿ

nÿ1 k X s kˆ0

K!

eÿs ;

n ˆ 1; 2; . . .

…15†

with in®nitely growing roughness r ˆ s1=2 . In the absence of S barrier, as we have shown [9], with decrease of l the growth mode crosses over from the layer-by-layer growth to rough multilayer growth and ®nally to random Poisson deposition process. Since the incorporation of S barrier can only destroy the epitaxial growth we investigate here at more carefully the growth kinetics as a function of x, i.e. the Schwoebel barrier height, in the most interesting range of l…> 104 † corresponding to the smooth layer-by-layer growth in the absence of S barrier. Calculations showed as it might be expected that the impact of x is qualitatively the same as that of l. In Fig. 1 a typical example for the case of l ˆ 109 is shown corresponding to the absence of S barrier to nearly ideal layer-by-layer growth when rms roughness r oscillates with time between rmax ˆ 0.5 and 1=2 rmin ˆ 0 as r ˆ …n ÿ n2 † . As is seen in Fig. 1, at x ˆ 1 roughness oscillates in the same manner with period exactly equal to the time of deposition of one monolayer (ML). With decrease of x

V.I. Tro®mov, V.G. Mokerov / Computational Materials Science 17 (2000) 510±514

513

Fig. 1. Successive layer coverage kinetics curves (on the left) and corresponding r.m.s. roughness kinetics curves (on the right) at l ˆ 109 and x (1, 0.02, 0.015, 0.01, 0.006) from top to bottom.

oscillations fade out and at x ˆ 0.02 when the growth front consists of 2±3 simultaneously evolving MLs r saturates at rs ˆ 0.5. With further decrease of x a stable growth persists up to x ˆ 0.01 when the growth front consists of 4±5 MLs and rs ˆ 1. Between values of x ˆ 0.01 and 0.006 the growth front and hence the rms roughness begins to diverse and at x 6 0:006 r diverses

more and more and eventually runs into the limiting Poisson law r ˆ s1=2 . The same behaviour of r with x was established for other values of l. The results are summarized in the form of a ``phase diagram'' of the growth modes in parametric space l±x (Fig. 2). Here, layer-by-layer growth includes both an ideal layerby-layer growth and layer-by-layer growth with r

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V.I. Tro®mov, V.G. Mokerov / Computational Materials Science 17 (2000) 510±514

values of EB ˆ 0.1 and 0.3 eV correspond (at 4000°C) to values of x ˆ 0.18 and 0.006, respectively, which at l ˆ 106 ±108 (for typical Ed ˆ 0.5± 0.8 eV) as is seen in Fig. 2, just correspond to smooth layer-by-layer and rough multilayer growth, respectively. Thus, the diagram proposed may be used for prediction of the MBE growth modes depending on deposition conditions.

References

Fig. 2. Phase diagram of the growth modes in the parameter space l±x.

oscillating between rmax ˆ 0.5 and rmin , increasing with decrease of x; the smooth multilayer growth denotes the growth with stable with time roughness which increases with decrease of x, and the rough multilayer growth designates the growth with r diverging with time. Diagram predictions agree with analysis [7] showing the smooth growth on Ag(1 1 1) and Pt(1 1 1) at EB ˆ 0.150 and 0.165 eV, respectively, and MC simulations [12] revealing the oscillating behaviour of r(s) at EB ˆ 0.1 eV and absence of oscillations at EB ˆ 0.3 eV. Really,

[1] L. Schwoebel, E.J. Shipsey, J. Appl. Phys. 37 (1966) 3682. [2] J. Villain, J. Phys. (France) I 1 (1991) 19. [3] J. Krug, M. Pliscke, M. Siegert, Phys. Rev.Lett. 70 (1993) 3271. [4] M.D. Johnson, C. Orme, A.W. Hunt, et al., Phys. Rev. Lett. 72 (1994) 116. [5] J. Terso€, A.W. Denier van der Gon, R.M. Tromp, Phys. Rev. Lett. 72 (1994) 266. [6] J.G. Amar, F. Family, Thin Solid Films 272 (1996) 208. [7] J.A. Meyer, J. Vrijmoeth, et al., Phys. Rev. B. 51 (1995) 14790. [8] P. Smilauer, S. Harris, Phys. Rev. B. 51 (1995) 14798. [9] V.I. Tro®mov, V.G. Mokerov, A.G. Shumyankov, Thin Solid Films 306 (1997) 105. [10] J.A. Venables, G.D.T. Spiller, M. Hanbuken, Rept. Progr. Phys. 47 (1984) 399. [11] V.I. Tro®mov, V.A. Osadchenko, Growth and Morphology of Thin Films (in Russian), Energoatomizdat, Moscow, 1993. [12] M. Breeman, Th. Michely, G. Comsa, Surf. Sci. 370 (1997) L193.