A discussion of epitaxial growth is presented for those situations (OMVPE,
CBE, ALE, MOMBE, GSMBE, etc.) when the kinetics of surface processes associated
with molecular precursors may be rate limiting. Emphasis is placed on the
identification of various {\it characteristic length scales} associated with
the surface processes. Study of the relative magnitudes of these lengths
permits one to identify regimes of qualitatively different growth kinetics as a
function of temperature and deposition flux. The approach is illustrated with a
simple model which takes account of deposition, diffusion, desorption,
dissociation, and step incorporation of a single precursor species, as well as
the usual processes of atomic diffusion and step incorporation. Experimental
implications are discussed in some detail.Comment: 10 pages, 2 figure

Vvedensky, D. D.

Zangwill, A.

English

arXiv

A discussion of epitaxial growth is presented for those situations (OMVPE, CBE, ALE, MOMBE, GSMBE, etc.) when the kinetics of surface processes associated with molecular precursors may be rate limiting. Emphasis is placed on the identification of various {\it characteristic length scales} associated with the surface processes. Study of the relative magnitudes of these lengths permits one to identify regimes of qualitatively different growth kinetics as a function of temperature and deposition flux. The approach is illustrated with a simple model which takes account of deposition, diffusion, desorption, dissociation, and step incorporation of a single precursor species, as well as the usual processes of atomic diffusion and step incorporation. Experimental implications are discussed in some detail

Zangwill, A

Vvedensky, DD

Spiral - Imperial College Digital Repository

arXiv:0712.1289v1  [cond-mat.mtrl-sci]  8 Dec 2007Regimes of Precursor-Mediated Epitaxial GrowthA. ZangwillSchool of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332D.D. VvedenskyThe Blackett Laboratory, Imperial College, London SW7 2BZ, United KingdomAbstractA discussion of epitaxial growth is presented for those situations (OMVPE, CBE, ALE,MOMBE, GSMBE, etc.) when the kinetics of surface processes associated with molecularprecursors may be rate limiting. Emphasis is placed on the identification of various charac-teristic length scales associated with the surface processes. Study of the relative magnitudesof these lengths permits one to identify regimes of qualitatively different growth kinetics asa function of temperature and deposition flux. The approach is illustrated with a simplemodel which takes account of deposition, diffusion, desorption, dissociation, and step incor-poration of a single precursor species, as well as the usual processes of atomic diffusion andstep incorporation. Experimental implications are discussed in some detail.A well-founded conceptual and computational framework now exists for the theoreticaldescription of molecular beam epitaxy (MBE). Surprisingly, it has proved sufficient to focusalmost exclusively on the diffusion and incorporation kinetics of single adatoms both inatomistic Monte Carlo simulations1,2 and in studies based on reaction-diffusion equations.3−5Within this context, not only is there an analytic theory of the transition between stepflow and two-dimensional (2D) island nucleation and coalescence4−6, as monitored, e.g.,by the disappearance of reflection high-energy electron diffraction (RHEED) oscillations asa function of growth conditions but, for GaAs(001) under sufficiently As-rich conditions,quantitative agreement with RHEED oscillation data2 can be achieved without any explicitreference to the As source (whether As2 or As4) in the theoretical model. Unfortunately,there is increasing evidence that these simple adatom models are not adequate to describethe growth kinetics for the general case when the atomic constituents of the growing filmare delivered to the substrate in the form of heteroatomic molecules. For example, the verydifferent kinetics observed for surface reactions7,8 and growth9,10 when trimethylgallium(TMG) or triethylgallium (TEG) is deposited onto GaAs makes clear that no single universaltheoretical model is likely to be proposed for this case.The purpose of the present Letter is to demonstrate the existence and nature of variousqualitatively distinct kinetic regimes of epitaxial growth which can occur whenever surfacechemical processes are important. We are motivated by detailed surface diffraction studiesof GSMBE (using electrons)11−13 and OMVPE (using x-rays)14,15 which yield considerableinformation about the morphological evolution of the surface as growth proceeds. Since theexistence and evolution of characteristic length scales emerge quite naturally from such stud-ies we organize our discussion around the identification of these quantities. For this purpose,it is essential to take explicit account of surface diffusion, a feature missing from essentiallyall existing theoretical models of such growth. The latter16,17 typically take the form of spa-tially uniform coupled rate equations and focus attention on the prediction of quantities suchas the net growth rate and the average concentration of various surface species. Nonetheless,as will become clear below, these earlier studies readily can be generalized so that the ratherdifferent point of view we advocate here can be adopted.–2–To illustrate our method, we first describe a simple growth scenario and then work outthe qualitative consequences which might be observed in a diffraction experiment of the sortnoted earlier. We regard the model as a physically reasonable minimal generalization of theadatom models used to study MBE. It is not intended to describe any particular materialsystem in detail.Consider deposition onto a vicinal surface of a molecule that contains the atomic con-stituent of the growing substrate. We include (see Fig. 1) the processes of (i) desorptionof the molecule back into the gas phase, (ii) surface diffusion of the molecule in a weaklybound precursor state, (iii) decomposition of the molecule on a terrace to release the atomicconstituent, and (iv) decomposition of the molecule at a step edge and incorporation thereof the atomic species. Included as well are (v) surface diffusion of the atomic species and (vi)incorporation of these atoms at a step edge. For simplicity only, we ignore all site-blockingeffects and the presence (and fate) of all molecular decomposition fragments. Desorption ofthe atomic species is also neglected.A quantitative theory results if we generalize the analysis of Burton, Cabrera andFrank18 (BCF) to include the processes described above. The one-dimensional continuummodel of BCF often is used to analyze MBE on vicinal surfaces.3 In the present case, theevolution in time and space of the surface concentration of precursor molecules n(x, t) andadatoms c(x, t) is determined by the following coupled reaction-diffusion equations:∂n∂t= DM∂2n∂x2−nτ− κn + J (1)∂c∂t= DA∂2c∂x2+ κn (2)Here, DM andDA respectively denote the surface diffusion constant of the precursor moleculeand the atom, κ and τ−1 are the rate constants for decomposition and desorption of theprecursor molecule on a terrace, and J is the molecular deposition flux. Equations (1) and(2) are supplemented by the boundary conditionsDAcx(0, t) = βA[c(0, t)− c0], −DAcx(ℓ, t) = βA[c(ℓ, t)− c0] (3)DMnx(0, t) = βMn(0, t), −DMnx(ℓ, t) = βMn(ℓ, t) (4)–3–where ℓ is the terrace length and c0 is the equilibrium concentration of atoms at the stepedge. As discussed in detail elsewhere,3 the choice (3) indicates that atoms incident on a stepincorporate into the solid at a rate proportional to βA. The choice (4) guarantees that everyprecursor molecule incident on a step decomposes (and its atomic constituent incorporatedinto the step) at a rate proportional to βM . The steady-state (time-independent) solutionsof these equations have simple analytic solutions3 and one obtains an exact expression forthe growth rate. We do not exhibit the explicit formula here since our aim in this paper isto identify the qualitative regimes of growth implied by (1)–(4). Instead, we proceed to theidentification of the relevant length scales.We begin with xs =√DMτ and ℓκ =√DM/κ. The quantity xs is the averagedistance a molecule diffuses before desorbing while ℓκ is the average distance a moleculediffuses before decomposing to release an adatom. Thus, the first two regimes of importanceare distinguished by whether xs/ℓκ ≫ 1 or xs/ℓκ ≪ 1, i.e., whether the diffusing speciesare predominantly adatoms or molecules. In the first case, we must consider the pertinentlength scales associated with the kinetics of the adatoms. The quantity Jeff (x) = κn(x) isthe effective “flux” of adatoms due to the decomposition of molecules. It then is natural todefine the length ℓA =4√DA/Jeff , where Jeff is the constant value obtained by averagingJeff(x) over a terrace. Clearly, ℓA corresponds to the distance an adatom diffuses beforeanother adatom is released by a molecular decomposition reaction.There now are two new regimes to consider: ℓA/ℓ ≪ 1 and ℓA/ℓ ≫ 1. In the first case,the typical migration distance of a free adatom is much smaller than the terrace length, theencounter probability of adatoms is high, and growth proceeds by the nucleation, growthand coalescence of 2D islands on the terraces. In the second case, adatoms diffuse to thestep edges before another atom is released, so growth proceeds by the advancement of steps,i.e., step flow. We note that the usefulness of the length scale ℓA has been demonstratedpreviously5 for the case of MBE where Jeff is replaced by the vapor phase deposition flux ofatoms.Within the atomic step flow regime, there are two additional possibilities to consider.To see this, we define a length dA = DA/βA which is the additional distance an atom–4–diffuses (after its first arrival at a step) before incorporation into the solid occurs. If dA ≪ ℓ,incorporation occurs very soon after the first encounter with a step. We refer to this as “fastatomic step flow.” Clearly, ℓA ≫ ℓ ≫ dA in this regime. On the other hand, if dA ≫ ℓ,step edge incorporation requires several attempts. If dA ≪ ℓA, growth is reaction-limitedat the step edges of the original surface and occurs by “slow atomic step flow”. However,if dA ≫ ℓA, attractive interactions between atoms on the terraces become important andgrowth occurs by a combination of slow step flow and 2D island formation. For growth undertypical conditions of MBE, this latter mode appears to be appropriate.19,20 In this regimewe have dA ≫ ℓA ≫ ℓ.If xs/ℓκ ≪ 1, the migrating species are predominantly the molecular precursors. Todetermine the growth regimes in this case, we first construct the molecular analogue to ℓA,i.e., ℓM =4√DM/J . The quantity ℓM is the average distance a molecule travels beforeencountering another molecule deposited by the incoming flux. We thus are lead to considerthe regimes ℓM/ℓ ≪ 1 and ℓM/ℓ ≫ 1. In the first case the molecules diffuse to the step edgebefore encountering another molecule, while in the second case, the molecules collide on theterraces before arriving at the step edge. Both cases require further consideration.If ℓM/ℓ ≪ 1, the encounter probability of the molecules on the terraces is high andseveral scenarios can occur depending upon the precursor mean density. Possibilities includecollisional decomposition with concomitant island nucleation, the formation of a molecularfilm, and the formation of a liquid-like state of adsorbed molecules. Since our model doesnot include interactions among the molecules we refer to this simply as the “molecularinteraction” regime.If ℓM/ℓ ≫ 1, it is useful to determine whether one or several encounters with a stepare required before the molecule decomposes. This issue is addressed by introducing thelength dM = DM/βM which may be regarded as the distance a precursor diffuses beforea successful step- catalyzed decomposition and incorporation reaction occurs. Again, tworegimes are distinguished: dM/xs ≪ 1 or dM/xs ≫ 1. If dM/xs ≪ 1, most of the moleculesimmediately decompose upon arrival at a step edge. We call this the regime of “fast molecularstep flow”. Note that ℓκ ≫ xs ≫ dM in this case.–5–When dM/xs ≫ 1, we must specify the relative magnitudes of dM and ℓκ since bothare much greater than the mean desorption length xs. If ℓκ is the shorter of the two, themolecule decomposes on the terrace and the growth mode is determined by atomic kinetics asdiscussed earlier. But if dM is the shorter length, the molecules decompose predominantly atthe step edge, albeit with some difficulty. We refer to this latter regime as “slow molecularstep flow” to indicate that even though molecular diffusion to the step edge is fast, thesubsequent attachment of the atom is slow.The preceding discussion can be summarized conveniently in graphical form. To do so,note that the model parameters DA, DM , κ, τ , βA and βM typically exhibit an Arrheniusform. Thus, the molecular deposition flux J , the mean terrace width ℓ and the substratetemperature T are the true control parameters of the problem. We choose to display thevarious regimes discussed above as a “kinetic phase diagram” in the T -J plane (Figure2). This diagram was generated by making several physically reasonable choices for theenergy barriers and pre-exponential factors in the foregoing rate constants and computingthe relative magnitudes of the various characteristic lengths. Only the gross topology of thephase fields should be noted since the precise placement of the phase boundaries depends onthe details of the system in question.What are the experimental implications of such a diagram? For definiteness, consider adiffraction experiment where the time evolution of the intensities of both Bragg reflectionsand diffuse scattering are recorded. By tuning the control parameters so that oscillations inthe former disappear, one straddles one of the curved phase boundaries in Figure 2 (whereeither ℓ = ℓM or ℓ = ℓA) and the surface diffusion constant of the primary migratingspecies can be extracted. By comparison with corresponding MBE results, one likely candetermine if one is to the left or right of the long vertical line in Figure 2. At elevateddeposition rates, the diffuse scattering from such an experiment reveals the time evolutionof the mean size of 2D islands and their average separation. An abrupt change in thesequantities with increasing temperature could be interpreted consistently (from Figure 2) asa kinetic transition from a regime of molecular interaction to a regime of 2D atomic islandformation. Recent extensive surface x-ray scattering measurements of OMVPE growth of–6–GaAs(001) onto a slightly misoriented substrate14,15 provide an attractive data base for ananalysis of the sort we propose.As a more concrete example, consider a recent systematic study of the disappearance ofRHEED oscillations as a function of temperature for GaAs(001) grown by MOMBE usingTMG as the source of gallium.10 A transition to step flow is observed to occur for both vicinalA-type surfaces (Ga-terminated step edges) and vicinal B-type surfaces (As-terminated stepedges). But, when compared to the result found in MBE, the transition temperature TC isincreased on the A surface and decreased on the B surface, both by approximately 30 ◦C.We speculate that this behavior reflects the presence of slow molecular step flow whereAs-terminated step sites are required to catalyze precursor21 decomposition. In that way,the B surface would be in slow step flow automatically. A reduced TC is expected due tothe presence of Ga-terminated kink sites exposed by thermal fluctuations of the step edge.Conversely, more temperature is needed to effect step flow on the A surface since a significantdensity of As- terminated kink sites (again exposed due to thermal fluctuations) is requiredin that case.Evidently, the simple model described in this paper must be generalized if additionalimportant kinetic steps are demonstrated to exist for any particular growth problem. Thiswould complicate both the length scale analysis and the construction of the kinetic phasediagram. Nonetheless, such an exercise fosters a mode of thinking which, we believe, ulti-mately will help create a conceptual framework for the chemically-based epitaxial growthtechniques comparable to that already achieved for MBE.The authors thank Ahmet Erbil, Tom Foxon, Bruce Joyce, Tadaaki Kaneko and To-moya Shitara for valuable discussions. Work performed at Georgia Tech is supported bythe U.S. Department of Energy under grant No. DE-FG05-88ER45369. Work performed atImperial College is supported by Imperial College and the Research Development Corpora-tion of Japan under the auspices of the “Atomic Arrangement: Design and Control for NewMaterials” Joint Research Program.–7–References1. A. Madhukar and S.V. Ghaisas, CRC Crit. Rev. Solid State Mater. Sci. 13, 1434 (1987).2. T. Shitara, D.D. Vvedensky, M.R. Wilby, J. Zhang, J.H. Neave and B.A. Joyce, Appl.Phys. Lett. 60, 1504 (1992).3. R. Ghez and S.S. Iyer, IBM J. Res. Develop. 32, 804 (1988).4. A.K. Myers-Beaghton and D.D. Vvedensky, Phys. Rev. A 44, 2457 (1991).5. A.K. Myers-Beaghton and D.D. Vvedensky, Phys. Rev. B 42, 5544 (1990).6. V. Fuenzalida, Phys. Rev. B 44, 10835 (1991).7. V.M. Donnelly, J.A. McCaulley and R.J. Shul, in Chemical Perspectives of Microelec-tronic Materials II, edited by L.V.Interrante, K.F.Jensen, L.H. Dubois and M.E. Gross(Materials Research Society, Pittsburgh, 1991), pp. 15-23.8. M.L. Yu, U. Memmert, N.I. Buchan and T.F. Kuech, in Chemical Perspectives of Mi-croelectronic Materials II, edited by L.V.Interrante, K.F.Jensen, L.H. Dubois and M.E.Gross (Materials Research Society, Pittsburgh, 1991), pp. 37-46.9. Y. Okuno, H. Asahi, T. Kaneko, T.W. Kang, and S. Gonda, J. Crystal Growth 105,185 (1990).10. T. Kaneko, T. Jones, and B.A. Joyce (unpublished).11. W.K. Liu, S.M. Mokler, N. Ohtani, J. Zhang, and B.A. Joyce, Appl. Phys. Lett. 60, 56(1992).12. M. Hiroi, K. Koyama, T. Tatsumi, and H. Hirayama, Appl. Phys. Lett. 60, 1723 (1992).13. S.M. Mokler, W.K. Liu, N. Ohtani, and B.A. Joyce, Appl. Phys. Lett. 60, 2255 (1992).14. F.J. Lamelas, P.H. Fuoss, P. Imperatori, D.W. Kisker, G.B. Stephenson, and S. Brennan,Appl. Phys. Lett. 60, 2610 (1992).15. D.W. Kisker, G.B. Stephenson, P.H. Fuoss, F.J. Lamelas, S. Brennan, and P. Imperatori,J. Crystal Growth (in press).16. A. Robertson, Jr., T.H. Chiu, W.T. Tsang, and J.E. Cunningham, J. Appl. Phys. 64,877 (1988).17. S.M. Gates and S.K. Kulkarni, Appl. Phys. Lett. 60, 53 (1992).–8–18. W.K. Burton, N. Cabrera and F.C. Frank, Philos. Trans. R. Soc. London, Sect. A 243,299 (1951).19. H.J.W. Zandvliet, H.B. Elswijk, D. D.Dijkkamp, E.J. van Loenen, and J. Dieleman, J.Appl. Phys. 70, 2614 (1991).20. T. Shitara, J. Zhang, J.H. Neave, and B.A. Joyce, J. Appl. Phys. 71, 4299 (1992).21. Presumably, the mobile precursor in this case is not TMG, but dimethylgallium ormonomethylgallium. See, e.g., Refs. 7 and 8.–9–Figure CaptionsFigure 1. Schematic illustration of the dominant kinetic processes included in the modeldefined by equations (1)–(4).Figure 2. Kinetic phase diagram of the different growth regimes exhibited by the modeldefined by equations (1)–(4) as a function of molecular deposition flux (J) and substratetemperature (T). Each phase boundary is the locus of points where the relevant ratio oflength scales is unity.–10–

Regimes of precursor-mediated epitaxial growth

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Regimes of Precursor-Mediated Epitaxial Growth

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