bending of the covalent bonds. In all our simulations, each atom is allowed to occupy either a substitutional or interstitial position, and transi- tions between these ...
MATERIALS MIENCE & ENGlWEERiNG ELSEVIER
B
Materials Science and Engineering B37 (1996) 25-29
Surface morphology due to enhanced migration in heteroepitaxial growth of compound semiconductors M. Djafari ‘Laboratoire bLaboratoire
Rouhaniaab, A.M. d’Analyse et d’iirchitecture de Physique des Solides,
Gu&“, R. Malek”,
G. Bouyssoua, D. Estitvea
des Systimes, CIVRS, 7 Ave. Colonel Roche, 31077 Toulouse, France Univ. Paul Sabatier, 118 Route de hlarbonne, 31062 Toulouse, France
Abstract
Monte Carlo technique and Valence force field approximation are used to investigate the heteroepitaxial growth with high lattice mismatch. It is shown that the growing surface becomes rough very rapidly. Its morphology is stabilized and shows V-grooves
with
(111) facets. The grooves are filled with
highly mobile
liquid-like
atoms that can be incorporated
in stable
solid-like positions through collective motions and reactions. Keywords:
Semiconductors; Heteroepitaxial growth; Monte Carlo technique; Valence force field approximation
1. Introduction
The heteroe&taxial growth of lattice mismatched materials has been extensively investigated during recent years, both on experimental and theoretical grounds. The initial goal was to use cheap substrates for the growth of thin layers, as in the case of CdTe-GaAs [l] and CdTe-Si [2]. The present effort is, however, directed towards the improvement of device performances, for example in Ge,Si, -,--Si thin film alloys, or the design of new devices, such as InAs-GaAs quantum dots [3,4]. On the theoretical side, different types of growth simulation have been carried out using the Monte Carlo technique or Molecular Dynamics [5,6]. Molecular Dynamics allows the description of atomic movements and is therefore useful to simulate highly disordered systems [7] or to investigate elementary mechanisms, such as defect creation [S], in detail. As, a result of this detailed description in Molecular Dynamics treatments, the total simulation time is always very short compared with experimental growth times. In contrast, in the Monte Carlo technique, where only atomic hoppings are treated, simulations on longer periods become possible. The main experimental feature is the roughening of the surface which may lead to island formation [3,4] or to the creation of V-shaped defects [9-l 11, depending 0921-5107/96/915.00 0 1996 - Elsevier Science S.A. All rights reserved
on the kinetic behavior of the surface atoms. This kinetic is obviously governed by the strain which enhances the atomic motion on the surface. In this paper we will analyze the surface roughening as a result of strain enhanced migrations and examine how this roughening can affect the subsequent growth mode. The simulation procedure is first described, leading to the need for an energy model to determine the strain and the stress in the deposited layer. This energy model is then presented and some preliminary results are discussed. 2. Simulation procedure
The simulation procedure is based on the Monte Carlo technique. In this scheme, starting from a given configuration, all possible events which include the adsorption (and respectively desorption) of molecules from (and into) the gas phase and the intralayer and interlayer migrations are listed and a time is attributed to each of them according to the Poisson probability law. The probabilities are themselves calculated using an Arrhenius-type expression in which the activation energies are assumed to be stress dependent [12]. This means that the more stressed atoms tend to move more rapidly than the less stressed ones. An energy model is
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44. Djafasi Roullarzi et al. / Materials Science and Engineering B37 (1996) 25-29
therefore necessary to determine the stress and the strain in the deposited layer as a function the local configuration. The event corresponding to the minimum time is assumed to occur, the configuration is changed and the cycle is repeated 10’ to lo6 times. It should be noted that, after each step, the various probabilities of atoms in the neighborhood of the event have to be recalculated as a result of changes that have occurred in the local configuration and in the local strain. Let us now look into details of interlayer migrations which involve atomic motions on large distances, particularly in the case of compound semiconductors where the jump is over two atomic layers. Fig. 1 schematically represents such a case where an atom is initially in its normal position (I), near a step edge. The final position (4) is two atomic layers below and can be reached by combining several elementary events. First, a simple intralayer migration may bring the atom to position (2) where it cannot be actually adsorbed since the site noted (v) is vacant. The atom is then adsorbed in position (3) two layers below, which is still unstable since it corresponds to an interstitial position of the zincblende lattice. A final intralayer migration is therefore necessary for the atom to reach the stable substitutional position (4) of the lattice. This interlayer migration is therefore completed through an intermediate position in which the atom is still doubly bounded to the substrate but where the strain energy is large because of the bending of the covalent bonds. In all our simulations, each atom is allowed to occupy either a substitutional or interstitial position, and transitions between these two states are counted as independent events, in addition to normal intralayer and interlayer migrations. Simulations are carried out on (100) oriented substrates of (15 x 15) to (50 x 50) atoms. In the majority of cases, the (30 x 30) atoms size is used, with a lattice mismatch of 14.6% corresponding to the case of CdTe-GaAs. The reason for this choice is the large lattice mismatch which enables us to use relatively small substrate sizes. The elastid constants of GaAs and CdTe are also used in the calculations.
Fig. 1. Schematic representation of an interlayer migration, from the initial position (1) to the final position (4), two atomic layers below, through the interstitial position (3).
3. Energy model The purpose of the energy model is to determine the stress and the strain in the deposited layer which cannot be obtained with the assumption of chemical bonds. However, quantum mechanical treatments or sophisticated classical potentials lead to prohibitive computation time since, in our simulation procedure, the total energy has to be calculated and minimized, with respect to atomic positions, 10’ to 10t times. This requirement justifies the choice of a simple energy model. We have evaluated the strain energy within a Valence force field (VFF) approximation where the total deformation energy is expressed as a function of bond elongations and bond angle variations [13]. The two body stretching term is written as Z bondsl~,(A~*)2,and the three body bending term as C bondens,cslio(l/3 + cos.0)’ which is a simplified Stillinger-Weber potential [14]. A first simplification is in the neglect of an exponential term in the Stillinger-Weber potential which limits the range of the potential and presents smooth derivatives at this limit. This requirement is fundamental in Molecular Dynamics simulations with highly disordered systems where the nearest neighbors are not well defined. It becomes less important in Monte Carlo calculations with nearly ordered systems and well defined neighbors. We have made a second simplification by developing the Stillinger-Weber expression in a second order power series of atomic displacements [15]. As a result, the energy minimization, which is carried out using the Newton-Raphson method, can be executed in a single run and an iterative procedure is not needed. The above two simplifications lead to insignificant computation time devoted to the energy calculations and minimizations, but limit the validity of the model to small atomic displacements. This is not always the case with large lattice mismatches used in our simulations. In fact, the calculated strain energies are larger than the actual ones leading to an overacceleration of events related to larger strain energies. However, this is not really crucial in our Monte Carlo approach since the important result is the sequence of events and not the accurate values of the time at which the events should occur. The global time evolution is always monitored by the slowest movements corresponding to small strain energies which are correctly calculated. This overacceleration occurs particularly when comparing the movements of surface atoms respectively in substitutional and interstitial configurations. The latter position is associated with a larger value of the strain energy and the corresponding movements are several orders of magnitude faster than the movements of atoms in normal substitutional positions. One can therefore speak of liquid-like atoms highly mobile on a surface constituted of solid-like atoms.
M. Djafari Rouhani et al. / Materials Science and Engineering B37 (1996) 25-29
27
Fig. 2. Perspective view showing the morphology of a portion of the growing surface.
In our simulations, the various activation energies related to different events are calculated by adding the chemical binding energies of the broken bonds [16] and the strain energy released once the bonds are broken.
4. Results and discussion Simulations have been carried out using the parameters of Section 2, and especially a lattice mismatch of 14.6%. Under these conditions, it has been shown [17] that atoms situated at cluster boundaries are almost completely relaxed. The strain energy increases towards the center of the cluster and reaches a maximum after a few interatomic distances. This means that atoms at the cluster center may rapidly migrate to relax the stress. However, at the cluster center, intralayer migrations are prohibited since all neighboring positions are occupied. The tendency is therefore towards interlayer migrations which lead to the roughening of the deposited layer. Obviously, the kinetic of this roughening is in competition with the kinetic of island formation, both being governed by the binding energies of various species on the surface. One can observe either a roughening or a flattening of the growing surface resulting from a global upward or downward motion of atoms [18]. With the parameters used in our simulations, and in the case of compound semiconductors, a roughening of the surface is observed in the very first stages of the growth, i.e. as soon as islands are formed on the second atomic level allowing upward interlayer migrations. Fig. 2 shows the morphology of a portion of such a growing layer. In this figure, only the atoms in normal substitutional positions are represented and the faces are relatively stable. The highly mobile interstitial atoms moving along these faces are not represented. It can be seen on Fig. 2 that the hills and valleys are oriented in the [l IO] directions and that the facets are (11 l), in agreement with experimental results [9-l 11.
Given the morphology of the surface, the growth of islands on the top layer is inhibited after a certain time. The incoming atoms therefore accumulate as interstitials on the (111) facets. This effect can be seen on Fig. 3 where the coverage of the substrate surface and the number of atoms in interstitial positions are reported vs. time. The coverage of the surface is represented by the number of atoms in the first layer of the deposited film. It can be observed that while the coverage has reached a stable value, the number of interstitials is still increasing. At this stage, the morphology of the surface is almost stabilized and the growth continues by incorporation of atoms into interstitial positions. These atoms cover the (111) facets forming the V-grooves and tend to fill these grooves with a liquid-like material. Fig. 4, where the accumulated number of intralayer and interlayer migrations over the number of deposited atoms are reported vs. time, shows this liquid-like behavior. The rate of migrations at a given time is represented by the derivative of the curves in Fig. 4. The curve (a) of this figure reaches a maximum with a zero derivative which is the result of a stabilized morphology in terms of atoms in normal substitutional
TIME (in Monolayer Deposition) Fig. 3. The coverage of the substrate surface (curve (a)) and the number of interstitial atoms in the deposited film (curve (b)), vs. time.
M. Djafari Rouhani et al. / Materiab Science and Engineering B37 (1996) 25-29
28
035
130
13
WJ
TIME (in Monolayer Deposition)
170 195 0,5 TIME (in Monolayer Deposition)
Fig. 4. The accumulated number of intralayer (curve(a)) and interlayer migrations (curve(b)) over the total number of depositedatoms, vs. time.
Fig. 6. The coveragesof the secondlayer of the depositedfilm (curve (a)) and of the bottom of V-groovesby interstitial atoms (curve(b)), vs. time.
positions. In contrast, interlayer migrations (curve (b)) are almost absent during a relatively long period which nearly coincides with the period of formation of the V-grooves. Once interlayer migrations are initiated (see Fig. 5) because of the formation of (111) facets, their numbers grow monotonically with a large derivative. This shows their much higher mobility compared with the mobility of atoms in substitutional positions, and their liquid-like behavior. Similar conclusions can be drawn from Fig. 5 where the number of various types of event are reported vs. time. The intralayer migrations start from the beginning but reach a maximum, while the interlayer migrations and substitutional-interstitial transitions are only
surface due to its high surface energy. In contrast, the normal positions at the bottom of the grooves are also associatedwith a high strain energy and are unstable positions. Atoms at the bottom of the grooves are therefore in interstitial positions between the substrate atoms and the atoms in the secondatomic layer of the deposited film. This effect is illustrated in Fig. 6 which compares the number of these atoms in interstitial positions and the number of atoms in the second layer of the film, vs. time. As can be seen, the two curves follow each other very closely, showing that the bottom of the grooves are completely filled with atoms that cannot be incorporated in stable positions. The global result is that the grooves will normally be filled from the edges and not from the bottom. However, due to the (111) orientation of the facets, single atom migration and incorporation cannot fulfill this requirement. The only possibility for the incorporation of solid-like stable atoms in the grooves is through collective motions involving two or more atoms at a time. This type of collective motion is presently under investigation and is not included in the above simulations.
initiated after a certain period of time but are continuously growing. In addition, the large number of substitutional-interstitial transitions shows the difficulty for each single atom in an interstitial position to be stabilized via its incorporation
in a new substitutional
posi-
tion, despite very frequent trials. The strain energy in thesenew positions is such that the reverse event occurs almost immediately. This results partly from the morphology of the surface which presents V-grooves with
(111) facets.The incorporation of atoms on thesefacets is indeed difficult because of the stability of the (111)
5. Conclusion
TJME (in Monolayer Deposition)
j
Fig. 5. Number of varioustypesof event:intralayer migrations (curve (a)), substitutional-interstitial transitions (curve (b)) and interlayer migrations (curve(c)), vs. time.
Taking into account the enhancement of atomic movements by the local stress, we have shown the roughening of the deposited lay&r during the heteroepitaxial growth of lattice mismatched semiconductors. Using a set of parameters corresponding to the case of CdTe-GaAs, we have observed the formation of Vgrooves presenting (111) facets, This is rapidly followed by a stabilization of the surface morphology and the appearance of low and high mobility atoms showing respectively solid-like and liquid-like behaviors. The final incorporation of atoms into stable positions cannot be properly described by single atom motions and may only be achieved through collective motions in-
M.
Djafari
Roullmzi
et al. J Materials
volving several atoms. This type of event, presently under investigation, may best be represented by reactions between atoms. It is expected that the filling of the grooves by incorporation of atoms in stable positions, following the reactions, will still leave empty spaces associated with large stresses that would act as nuclei for misfit dislocations.
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