Molecular cluster studies of the crystal growth process in MBE and MO-MBE

373

Surface Science 204 (19X8) 2733288 North-Holland. Amsterdam

MOLECULAR CLUSTER STUDIES OF THE CRYSTAL GROWTH PROCESS IN MBE AND MO-MBE 1. Structure and thermodynamics of gas phase reactants [As,, As4, Ga(CH,), Aldo AMORE Paolo NOTA,

BONAPASTA, Maria Rita BRUNI. Guido SCAVIA

Andrea

J

LAPICCIRELLA.

Istituto dt Teorru, Strtttturu Elettronicu e Comportumento Spettrochimrco der Compostr di Coordimxione, Consiglio Na-_ionale delle Ricerche, Areu deliu Ricerc,u di Romu, LW Salurru Km. 29.5, CPIO 00016 Monterotondo Scala Itu(,

and Norberto

TOMASSINI

Istttuto di Metodologie Arwt:ute

Inorguniche. Consrglio Nu:ionale delle Ricerche, Areu dellu R~crrcu di Romu, wu Sularia Km. 29.5. CPIO 00016 Monterotondo Sculo, Itulr

Received

23 November

1987: accepted

for publication

27 May 1988

The structure and thermodynamics of some gas phase molecular reactants [As:. As,. Ga(CH,)?] used in MBE (molecular beam epitaxy) and MO-MBE (metal-organic molecular beam epitaxy) has been studied by Hartree-Fock-Roothaan molecular-orbitals linear-comhination-of-atomic-orbitals self-consistent-field (HFR-MO-LCAO-SCF) ah initio methods. A first attempt to simulate the sticking of a gallium atom at the (100) surface of GaAs has also been made.

1. Introduction In recent years the relevance of ultra-high vacuum (UHV) epitaxial crystal growth techniques such as MBE (molecular beam epitaxy) and MO-MBE (metal-organic MBE) has grown rapidly either because of their importance in the developments of the technology or because of their ability of generating artificial crystals possessing novel physical properties [l-3]. The process of growing epitaxial layers by means of an MBE machine is an art and the conditions of growth are still chosen according to a very long and expensive trial and error procedure. Theoretical investigations. which were able to model and to rationalize the growth process. could have a beneficial 0039-6028/88/$03.50 Q Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

feedhack on the crystal growth work, mainly because of the clean and very controlled conditions (UHV) in which the growth itself occurs [l]. A perfect example of these kind of studies is given by the pioneering work of Madhukar and coworkers [4-181 who have investigated the growth process by Monte Carlo techniques. These authors were able to reach a good understanding of the kinetics of growth by mimicking in a very satisfactory way the experimental oscillating RHEED (reflection high energy electron diffraction) intensities. In particular Madhukar has proposed several atomistic models for the growth [7] of III-V epitaxial artificial crystals, based on earlier suggestions by Arthur [19] and by Foxon and coworkers [20-251, and these models constitute the core of a surface reaction scheme divided into the following steps [6]: random impingement. molecular physisorption (for As, and As, species), surface migration of physisorbed species, chemisorption, surface migration of chemisorbed species, incorporation. Each one of the above-mentioned steps implies the falling of the reacting species into a minimum of the potential energy surface and the progression from one step to the next implies the overcoming of an energy barrier. The knowledge of the energies involved in these barriers and, even, a qualitative assessment of the general topology of the potential energy surface felt by the MBE reacting species near the growing crystal, can give useful hints for the construction of an unified picture of the phenomena happening during the MBE growth. The methods of ab-initio molecular quantum chemistry can be very valuable when dealing with structural and thermodynamical properties of atoms and molecules reacting at the solid surface if small clusters simulating the reacting site [26-431 can be hypothesized. It could, then, be possible to try in the computer the proposed reaction models [7,20-251 and to derive from these calculations the above-mentioned energy parameters thus generating useful information which could be used in the Monte Carlo work. Furthermore all these theoretical predictions could help. in principle. to interpret several in situ experimental measures (e.g. Raman. RHEED etc.) whose accuracy is greatly enhanced by the UHV conditions. This paper represents a first step along the above-quoted lines and it presents an ab initio HFR-MO-LCAO-SCF study of the gas phase reactants used in the GaAs MBE and MO-MBE growth: Asl, As, and Ga(CH,),. This study has a twofold goal: (a) The choice of the basis set which conciliates at the highest possible degree the accuracy of the results with the economy of the calculations (proportional to the inverse of the number of basis functions employed). This study will give useful information on the basis sets to be used in large cluster calculations which will mimic the reaction at one or more surface active sites. (b) The study of the minimum energy gas phase geometry of the metal-organic molecule. Ga(CH,),, which is known to react directly on the

GaAs surface [44-491. As a matter of fact the knowledge of the gas phase conformation of such a molecule could help in the formulation of a reliable model of the docking of Ga(CH,)2 at the solid surface. A first attempt to mimic the sticking of a Ga atom at the unreconstructed GaAs(lOO) surface is also presented.

2. Methods of calculation All the ab-initio Hartree-Fock-Roothaan molecular-orbital linear-combination-of-atomic-orbitals self-consistent-field (HFR-MO-LCAO-SCF) calculations were carried out by means of GAUSSIAN 80 [50] on the IBM 4361/4 of the lstituto di Teoria e Struttura Elettronica, Consiglio Nazionale delle Ricerche (Italy) and on the IBM 3090/VF of the European Centre for the Scientific and Engineering Computing (ECSEC), IBM Rome, Italy. The minimal basis set of Huzinaga [51] contracted as (4333/433/4) was used for the arsenic and gallium atoms (hereafter referred to as MINB); a second basis set for those two atoms was derived by splitting the valence shell of the above-mentioned basis, after a suggestion of Huzinaga [52], thus obtaining a (43321/4321/4) set (hereafter referred to as SV, split valence). The hydrogen basis set is the Slater-type orbital, STO-3g, of Hehre et al. [53] for the MINB set and the extended 3-lg hydrogen basis by Ditchfield et al. [54] was chosen for the SV set. The fluorine basis set is the one by Huzinaga [51] contracted as (43/4) and (421/31) for the MINB and SV sets respectively. The d-shell was described by five pure d functions, the use of six Cartesian d functions was found to have no influence on the final results and their use was discarded in order to limit the number of basis functions. The total energy minimization procedure adopted is the one of Fletcher and Powell [55] and the total energy derivatives with respect to the free geometrical parameters were obtained by a numerical procedure. The Pople and Nesbet [56] open shell treatment was employed where unpaired electrons were found.

3. Results and discussion 3.1. Studll of the reference molecules: ASH,,

GaH, GaHz and GaF,

Several well known molecules have been considered as bench marks against which the performance of the two different version of the basis set, SV and MINB, was tested. Particular attention was given to the consistency with which the two basis sets are able to reproduce different molecular properties

276

A. Amore Bonuposta et ul. / Molecular cluster studies of cgstal

grou,th in MBE

AS

063 H

Fig. 1. Perspective

drawing

of the ASH,

pyramidal

H C?, molecule

such as structure, ionization potentials, vibrational frequencies, charge distribution and dipole moments. The first calculations were performed on ASH,, fig. 1, and the structural properties of the fully energy minimized molecules are shown in table 1 for the SV and MINB basis sets. The computed MINB As-H bond length is slightly smaller than the experimental value, while the SV set produces an As-H bond longer than the experimental one. The computed H-As-H angles are larger than the experimental ones for both calculations and, in this case, splitting of the basis improves agreement with experiment. The overall accuracy of the basis sets in computing these geometrical parameters is comparable with what is known about HF computations [28]. The ASH, ground electronic state is ‘A, and the valence shell electronic structure is shown in table 2 where the ionization potentials were computed according to Koopmans‘ theorem. The agreement is satisfactory for the highest occupied molecular orbital (HOMO) only, because electron relaxation effects probably begin to play a major role in determining the remaining inner ionization potentials. The difference of performance in reproducing the first ionization potential between the two basis sets used favours the SV set. Table 1 As-H bond distance (in A) and H-As-H angle (in degrees) computed MINB and SV basis sets (experimental values from ref. [57])

As-H H-As-H

MINB

sv

Exptl.

1.501 96.95

1.528 94.65

1.511 92.1

Table 3 Ionization potentials (in eV) for the valence orbitals of ASH, and SV basis sets (experimental values from ref. [58])

Al E A,

for ASH, by means

computed

by means

MINB

sv

Exptl

9.57 13.68 22.96

10.22 13.58 23.01

10.51 12.7 19.0

of the

of the MINB

A. Amore Bonapusta et al. / Moleculur cluster studres

of oystal growth

in MBE

277

The SV partial charges, computed by means of the Mulliken population analysis, are just opposite to those computed with the MINB basis (As partial charge, in electrons, +0.171 and -0.141 for the SV and MINB sets respectively, H partial charge, in electrons, - 0.057 and +0.047 for the SV and MINB sets respectively). Furthermore, in both cases, the computed dipole moments are aligned along the C, axis and oriented from the arsenic toward the hydrogens, thus indicating a negatively charged As atom. The discrepancy between the orientation of the dipole moment and the partial charges computed by the SV basis set is, obviously, an artifact of the Mulliken population analysis by means of which the partial charges themselves are computed [59]. The computed dipole moments are, anyway, both larger than the experimental one (SV 0.876 debye, MINB 1.924 debye, experimental 0.22 debye [59]). The difference in dipole magnitudes indicates that the SV basis exaggerates the charge separation to a lesser extent than the MINB set and, therefore, it is able to produce better agreement with experiment. The present results compare in a satisfactory way with similar calculations done on the same molecule [59,60]. The GaH molecule was adopted as test system for the Ga containing compounds. The computed minimum energy equilibrium distance of the iZf GaH ground state is equal to 1.707 and 1.676 A for the SV and MINB basis sets respectively. Comparison with the experimental value, 1.679 A [61] gives a similar situation to that already obtained in the ASH, case. The fundamental frequency of vibration, computed by numerical derivation of the analytical gradient, is equal to 1545 * 50 cm-’ and 1791~~ 50 cm-’ for the SV and MINB basis sets respectively. The experimental frequency lies in between the two computed values, 1604 cm-’ [61], but it is nearer to the one computed by the SV set. The computed first ionization potentials are very similar, 7.88 eV (SV) and 7.75 eV (MINB), and they are slightly larger than a theoretical estimate obtained by means of a relativistic model, 7.15 eV [62]. The computed charge separation agrees qualitatively for both basis sets (Ga: t-O.30 electrons (Sv), t-O.021 electrons (MINB)). The computed dipole, orientation, on the contrary to what was happening for ASH,, shows a discrepancy with the computed partial charges in the case of the MINB calculations. The MINB dipole (1.4795 debye), indeed, is oriented along the Ga-H bond from the gallium toward the hydrogen thus indicating a negative Ga atom. The GaH charge distribution, deduced from the MINB dipole, is not plausible when it is compared with the one which could be hypothesized on the basis of the electronegativity values attributed to the gallium and hydrogen atoms respectively. The SV dipole is much smaller, 0.0521 debye, and oriented in the opposite way thus correctly indicating a positive Ga atom. Two other molecules (GaH, and GaS), containing the gallium atom, have been considered in order to have a better insight in the performance of the two basis sets. Unfortunately little is known experimentally about these two molecules and few comparisons are made with previous theoretical work [63] whenever it was possible.

278

A. Amore Bonapasta et ai. / Molecular cluster studies

Fig. 2. Perspective

drawing

of the GaH,

of mystulgrowth in MBE

planar

I&,, molecule.

The fully energy minimized structure for GaH, shows a planar molecule, fig. 2, (point symmetry group D,,) with a Ga-H bond distance of 1,586 and 1.601 A for the SV and MINB basis sets respectively (the H-Ga-H angle is set at 120” ). The experimental bond length of the GaH, molecule is not known and a previous theoretical work assumed in the calculations a Ga-H distance of 1.551 A 1631. The GaH, ground electronic state is ‘A,; the valence ionization potentials are shown in table 3 and they fall in the same energy range for both basis sets. The partial charges computed by both basis sets are practically inverted (Ga: +0.534 (SV), -0.125 (MINB); H: - 0.178 (SV), -t-O.041 (MINB)). The MINB Mulliken charges follow the incorrect trends already shown by the GaH MINB dipole. The GaF, molecule was, then, examined in order to study the behaviour of the two basis sets once the Ga atom was coupled with a strongly electronegative element. The fully energy ~nimized GaF, molecule shows a planar D,, structure, characterized by a Ga-F bond distance of 1.721 and 1.726 A for the SV and MINB basis sets respectively. The bond angle is set at 120 *. The experimental bond length of the GaF molecule is 1.775 A [64], a value with respect to which the computed Ga-F bond distance in GaF, shows the expected contraction. The partial charges computed by mean of the two basis sets do not show the inversion found for ASH, and GaH,: Ga + 1.41 (SV), + 1.60 (MINB), F: -0.470 (SV), -0.533 (MINB). Once more the MINB basis set tends to exaggerate the charge separation because of its smaller ~exibility.

Table 3 Vaience ionization basis sets

E’

A’,

potentials

(in eV) computed

for the IA, GaH,

by means of the MINB

MINB

sv

11.37 16.88

11.55 16.36

and SV

A. Amore Bonapasta et al. / Molecular cluster studies of crystal growth in MBE Table 4 Valence ionization

potentials

(in eV) computed

for the ‘A, GaF,

by means

of the MINB

279

and SV

basis sets

E’ E” A; A‘; E’ A; E’ A;

MINB

sv

14.40 14.12 14.86 15.86 15.86 17.77 36.36 36.64

14.92 15.41 15.50 16.48 16.52 18.25 34.65 34.89

The GaF, ground electronic state is ‘A, and the valence ionization potentials are shown in table 4. The SV and MINB estimates of the first ionization potential fall in the same energy region whilst the estimates of the inner ionization potentials show larger discrepancies between themselves. One point to bear in mind when dealing with molecules containing third-row atoms is to take somehow into account the bias due to the influences of the electron correlation, of the basis set limitations and of the relativistic contributions on the computed molecular observables. The simple HFR calculations are known to reproduce the main structural parameters of a molecule with an error in the range 0.02-0.03 A for the bond lengths and 3”-4.4’ for the bond angles [28,65], and these quantities are similar to the errors found for the SV results. Furthermore most of the above-quoted error ranges must be probably slightly enlarged in the present case because they stem from studies of molecules containing first- and second-row atoms for which the minimal basis sets employed are probably able to give more accurate final wave functions than those produced for molecules containing third-row atoms [59]. The relativistic contributions are known to slightly contract the computed bond lengths (up to - 10%) and to decrease the computed total absolute energies but they do not change significantly the overall shape of the molecular potential surface near the minima (i.e. the frequencies of a vibrational stretching mode computed with and without relativistic corrections are almost equal) [62]. It is, then, safe to say that HF computations performed on molecules containing third-row atoms by means of the SV basis set will, in general, overestimate the geometrical molecular parameters and the errors will be of the order of magnitude of those quoted in the literature [28]. In addition, particular attention must be given, in performing cluster calculations which simulate a III-V solid state situation, to the way in which the two proposed basis sets describe the variation of the charge on the hydrogens on going from the hydrides of the arsenic to the hydrides of the gallium. As a matter of fact the hydrogen is the most commonly used

280

A. Amore Bonapasta et al. / Molecular cluster studies of crystal growth UI MBE

“saturator” [37] in cluster calculations because it terminates the proposed models thus ensuring either the proper hybridization of the heavy atoms employed or the absence of spurious radicals [28,37]. In the realm of the III-V cluster calculations, the hydrogen must play an amphoteric role in order to be classified as a good “saturator”: it must be a donor when it is attached to the arsenic, thus correctly mimicking the missing gallium, and the exact reverse must be true when it is attached to the gallium. This kind of behaviour, either dictated by chemistry or needed by the nature of the III-V cluster calculations, is consistently produced only by the SV basis set, once more reliable information about the charge distribution in hydrides coming from the analysis of the dipole moments [59] is taken into account. The analysis of the above results shows that the SV basis set is able to produce better agreement with the overall structural and electronic experimental properties of the test molecules, even though the MINB set is not performing badly. As a consequence the SV has been adopted in the following. 3.2. Study

of the MBE

arid MO-MBE

gas phase

reactants:

As,,

As,

and

Ga(CW, As, and As, are known to be the major constituents of the group V impinging molecular beams, produced in the MBE growth chamber either by K-cell, with and without a cracker zone [l], or by the gas sources [2]. It is then useful to compute the structural and thermodynamical properties for these two molecules and this was accomplished by using the more extended SV basis set. The total energy of the linear As, molecule was minimized with respect to the bond distance. The minimum was found for the minimum was found for the ‘2; electronic ground state with an As-As bond distance of 2.139 A. This value compares satisfactorily with experiment, 2.103 A [66]. The valence ionization potentials are reported in table 5. It was tried to further increase the basis set expansion by splitting the d orbitals in two shells, thus obtaining a (43321/4321/31) set. This gave an As-As distance of 2.125 A but this 0.6% increase in accuracy was heavily paid in terms of number of basis functions ( +5/As atom) and it was not felt useful to proceed further along this line. Anyway all the computed quantities exceeded the experimental value. The fundamental frequency of vibration. computed by numerical derivation of the analytical gradient, is equal to 421 + 50 cm-‘, a value which is in very good agreement with experiment. 430 cm-’ [66]. The As, molecule was minimized following several pathways of the potential surface and the absolute minimum was found for the tetrahedral arrangement. fig. 3, characterized by an As-As bond distance equal to 2.559 A. The experimental distance is equal to 2.435 t_ 0.004 A [67]. The computed As-As bond distance in As, confirms the above-mentioned general trends but it

A. Amore Bonapasta er al. / Molecular cluster studies of crys~ul growth in MBE Table 5 The valence ionization potentials means of the SV basis set

(in eV) computed

For As,(‘B:

) and As,(‘A,

) molecules

281

by

As4 EL

Orbital

9.35 10.18 16.75 24.01

E

FL 9.01

T2

10.00

Al TZ AI

11.10 19.40 26.81

produces a slightly larger error with respect to experiment than the test cases. The electronic configuration is ‘A, and the orbital energies are given in table 5. During the search for the minimum energy geometry a square planar configuration was carefully explored for the As, molecule. This configuration was found to be a minimum for an As-As distance of 2.444 A; AE = ( Esquare -E tetrahed) is equal to 101 kcal/mol; the order of magnitude of this energy difference clearly suggests that this molecular arrangement is unreachable at the temperatures in which the MBE growth chamber operates (600-700” C)

ia. The experimental dissociation energy of As, into As2 shows large discrepancies in the literature [69] (experimental estimates vary from 52 to 80 kcal/mol). An accurate analysis of the available literature data performed by Tmar et al. [69] fixes the experimental AH & at 54.4 kcal/mol. The theoretical estimates of this quantity come from the difference of the computed total energies of the As, and As, molecules according to the following equation Strictly speaking, two further corrections are AH,,,,, = Eto,(Asq) - 2E,,,(As,). needed in order to correctly compare the theoretical value with experiment: the extrapolation of the experimental AH from 298 K to 0 K and the

nAs As

Fig. 3. Perspective

drawing

of the As, tetrahedral

Td molecule.

282

A. Amore Bonapasta et al. / Molecular cluster studies of crystal growth in MBE

subtraction of the zero point energy term [70]. The first of these two corrections was estimated by integrating the equation for the C,‘s [67] giving a final AH: of 54.1 kcal/mol. It was not possible to estimate the zero point energy correction term because it was not available; there is neither a theoretical full vibrational analysis of As, nor the knowledge of experimental frequencies for this molecule. The quantities involved in this term must, anyway, amount to no more than a few fractions of a kcal/mol if one considers the energy equivalent of the stretching vibration of As, (1.2 kcal/mol). a value which heavily underestimates the AH,,,,, is equal to 21.5 kcal/mol, best experimental value of AH: and its difference with experiment is well off the eventual magnitude of the zero point energy correction term. This theoretical estimate, anyway, represents an improvement with respect to previous ab initio calculations performed on the same systems [71]. The present results, even though they are qualitatively valid, are heavily affected by the limited basis set used. Probably the basis set adopted here does not cope with both molecular systems, As, and AS?, at the same degree of accuracy. Then the systematic errors due to the small basis set expansion are not equal in both molecules and they do not cancel out when the total energy difference is taken. A clear symptom of this fact is derived by the analysis of the magnitude of the difference between theoretical and experimental bond lengths obtained for As, and As, respectively: for As, this difference does fall within the known error range whilst for As, it does not. The Ga( CH, ), molecule is one of the gallium based metal-organic reactants used in MO-MBE. This molecule dissociates irreversibly once it hits the reacting surface, thus producing three CH; radicals plus a gallium atom [44-491. The total energy of Ga(CH,), was minimized varying the Ga-C and C-H bond lengths, Ga-C-H valence angles, and the C-Ga-C-H torsion angles. The basis set used was the SV one. The C, axis was maintained throughout the computations but several conformations with different symmetries were tried. The minimum energy geometry is shown in fig. 4 and the relevant computed structural parameters are found in table 6 together with the experimen-

“6

b

H

Fig. 4. Perspective

drawing

of the Ga(CH,),

C,,. molecule

in the most stable conformation

A. Amore Bonapasta et al. / Molerular cluster studies of cqwtal growth in MBE Table 6 Bond lengths (in A) and valence and torsional angles (in degrees) molecule in the C,, symmetry (experimental data from ref. [72])

Ga-C C-H Ga-C-H C-Ga-C-H

for the planar

SV

Exptl.

2.017 1.085 110.83 90.00

1.967 + 0.002 1.083 * 0.003 112.1 +0.8 _

The remaining two torsional angles of the methyl here by successively adding 120 O.

hydrogens

are derived

283

Ga(CH,),

from the one presented

tal data derived by gas phase electron diffraction (GPED) measurements [72]. The planar GaC, moiety has the three Ga-C bonds at 120’ and the methyl groups are rotated so that, for each methyl group, one hydrogen lies on a plane perpendicular to the GaC, one (C-Ga-C-H torsion angle equal to 90 ’ ); the molecular point symmetry group is C,,. The agreement with the GPED data for the overall shape of the molecule (i.e. planar GaC, moiety), the bond lengths and the bond angles is quite good. The experiment does not give a definite value to the C-Ga-C-H torsion angles because, at room temperature, it sees freely rotating methyl groups. The computed barrier of rotation is less than 1.0 kcal/mol and this value confirms the possibility of free rotation for the methyls. Several other conformations have been explored (i.e. pyramidal GaC, moiety, synplanar C-Ga-C-H torsion angles etc.) but all of them had total energies slightly higher with respect to the above-presented geometry. In the minimum energy conformation the molecule has a small dipole moment, 0.053 debye, vertically oriented with respect to the GaC, plane, the gallium atom carries a positive net charge, + 1.097 electrons, which is counterbalanced by three negative methyl groups. The methyl carbon atom is the sink of the negative charge, -0.968 electrons, whilst the hydrogen are positive, average charge +0.2008 electrons. The electronic ground state is the ‘A, singlet. The HOMO has E symmetry (ionization potential 10.06 eV) and it has strong contributions from the Ga-C bonds, whilst the lowest unoccupied molecular orbital (LUMO), A, symmetry, has a strong Gap2 character and, thus, this empty orbital is ready to form a dative bond with an eventual donor molecule. In conclusion, this molecule could be described as a positive center, carrying an empty acceptor orbital, surrounded by three bulky, polarizable and negative substituents. In this form the docking of the molecule at the surface during the physisorption step will surely be very much different than in the case of monatomic Ga. The docking stage could, even, be more favoured because the GaAs surface could more easily “recognize ” the peculiar shape and polarity of the incoming Ga(CH,), molecule.

284

A. Amore Bonapasta et al. / Molecular cluster studies of q~stal growth rn MBE

3.3. Sticking site

process of a gallium atom at the (100) GaAs surface cation lattice

The incorporation of the gallium atom during growth could be described, in simple terms, in the following way: The Ga atom impinges the (100) growing surface and it attaches randomly to any site. It then either evaporates back or migrates to an active kink site and gets incorporated after having gone through several kinetically controlled steps [73]. A first and simple approach to the reproduction of the sticking process is given by the investigation of the simple ASH,-Ga-ASH, cluster, The two ASH, groups are placed at the ideal GaAs(lOO) surface anion sites with the As-H bonds pointing in the direction of the bulk cation sites and the Ga atom is progressively moved toward its ideal location at the crystal surface from the vacuum along the [loo] direction vertical to the As-As axis, fig. 5. The As-H bond lengths are those of the corresponding hydrides thus ensuring the proper hybridization of the As atoms [26,27]. The impinging gallium atom was placed in five different locations progressively nearer to the surface cation site; all the remaining atoms were kept fixed at their original positions during the calculations and no attempt has been made, at this stage, to mimic the dynamics of the surface reconstruction process. The exact location of the minimum was found by interpolating a parabola around the three points defining the well of the potential energy curve. The As-Ga distance at the minimum is equal to 2.88 A which is - 0.4 A longer than the crystalline Ga-As distance, 2.45 A. Such long equilibrium distances indicate that the gallium atom is not strongly bound in this situation and, thus, it could hop easily toward a kink site [73]. In an attempt to reproduce the

H H Fig. 5. Perspective drawing of the ASH 2 -Ga-AsH I reaction pathway. The gallium atom is placed at the ideal cation lattice site. The crosses indicate the four more different gallium positions considered for the energy scan along the [100] direction.

A. Amore Bonapasta ei al. / Molecular cluster studies of qvsfal

H

growth in MBE

285

H

Fig. 6. Perspective drawing of the ASH, -Ga-AsH, reaction pathway. The gallium atom is placed at the ideal cation lattice site. The crosses indicate the four more different gallium positions considered for the energy scan along the [loo] direction.

effect of the strain due to some form of surface reconstruction, the above calculations were repeated after having deformed the H-As-H angles; the minimum was found by the above-quoted method at an even longer Ga-As distance, 3.17 A. The strain and the deformations around the anion site seem to weaken the Ga-As bond further thus favouring an eventual hopping. The ASH,-Ga-ASH, model has an unpaired electron in a doublet electronic configuration. The odd electron is placed mainly on the arsenic atoms all throughout the gallium motion. A second batch of calculations was made on the ASH,-Ga-ASH, cluster which mimics a gallium reaction site surrounded by tri-coordinated arsenic atoms, fig. 6. The minimum of the total energy was found at a longer As-Ga distance than before, 3.17 A. In addition, by interpolating a parabola at the three points defining the minimum of the potential well a rough estimate of the second derivative of the total energy at the minimum was derived, which is indicative of the force constant opposing to the gallium motion. The ASH,-Ga-ASH, quantity is six times smaller than that derived for the ASH,-Ga-ASH, model. The ASH,-Ga-ASH, cluster reacts to the deformation of the ASH, pyramid by displacing the minimum energy position of the gallium atom even further away from the arsenic than in the deformed ASH?-Ga-ASH, cluster (Ga-As distance > 3.3 A). There is another marked difference with the previous case: the unpaired electron is located almost exclusively on the gallium atom all through the reaction coordinate. It is important, at this point, to stress several important factors which make these calculations only a first and crude attempt of investigating the GaAs surface reactions: (a) the model is very small and rigid, the reaction coordinate space is not exhaustively explored and the As-As relaxation effects are not included;

286

A. Amore Bonapasia et al. / Molecular cluster studies of crystal growth in MBE

(b) the basis set is limited; (c) the electron correlation effects are not included; (d) the hydrogens used as “saturators” for the arsenic could influence the final results [37]. Previous studies performed on defects in solids (i.e. phenomena which generate a localized situation) by means of the cluster approach [42,43], together with the above-presented results on the gas phase molecules, have shown that all the above-mentioned factors do not severely alter the fundamental indications derived at the HFR level. Work is in progress in order to accurately evaluate the importance of each of these factors in the realm of III-V cluster calculations. Nevertheless, the above results are certainly arousing and indicative of what can be done by means of this kind of computations. Furthermore, they even tend to confirm some assumptions made about surface migration energy barriers by Singh et al. in their work [73] concerning cation incorporation during the MBE process. In addition, calculations performed on a similar problem (i.e. Ga chemisorption at the GaAs(ll0) reconstructed surface) by means of more accurate theoretical methods give an equilibrium distance for the Ga-GaAs (surf.) bond of the same order of magnitude as the one found in the present investigation [31].

Acknowledgements The gift of large portions of computer time by the ECSEC, IBM Rome ITALY, is gratefully aknowledged. The authors deeply thank Dr. Piero Sguazzero of the ECSEC, IBM Rome, Italy, for his continuous encouragement and for his skillful suggestions about the vectorization of the GAUSSIAN 80 program. Thanks are due to Mr. B. Trabassi of the Research Area of Montelibretti (CNR, Italy) for the glossy prints.

References [l] E.H.C. Parker, Ed., The Technology and Physics of Molecular Beam Epitaxy (Plenum, New York. 1985). [2] M.B. Parrish. J. Crystal Growth 81 (1987) 249. [3] K. Ploog, J. Crystal Growth 79 (1986) 887. [4] J. Singh and A. Madhukar, J. Vacuum Sci. Technol. 20 (1982) 716. [5] J. Singh and A. Madhukar, Phys. Rev. Letters 51 (1983) 794. [6] A. Madhukar, Surface Sci. 132 (1983) 344. [7] S.V. Ghaisas and A. Madhukar, J. Vacuum Sci. Technol. B 3 (1985) 540. [8] A. Madhukar, T.C. Lee. M.Y. Yen, P. Chen, G.Y. Rim. S.V. Ghaisas and P.G. Newman, Appl. Phys. Letters 46 (1985) 1148. [9] A. Madhukar and S.V. Ghaisas, Appl. Phys. Letters 47 (1985) 247.

A. Amore Bonapasta et al. / Molecular cluster srudres of crystal growth in MBE [lo] [ll] (121 [13] 1141 [15] 1161 [17] 1181 1191 [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] 1311 [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] 1431 [44] [45] [46] (471

287

F.J. Grunthaner, M.Y. Yen, R. Fernandez. T.C. Lee, A. Madhukar and B.F. Lewis, Appl. Phys. Letters 46 (1985) 983. B.F. Lewis, F.J. Grunthaner, A. Madhukar, T.C. Lee and R. Fernandez. J. Vacuum Sci. Technol. B 3 (1985) 1317. P. Chen, J.Y. Kim, A. Madhukar and N.M. Cho, J. Vacuum Sci. Technol. B 4 (1986) 890. T.C. Lee, M.Y. Yen, P. Chen and A. Madhukar, J. Vacuum Sci. Technol. A 4 (1986) 884. T.C. Lee, M.Y. Yen, P. Chen and A. Madhukar, Surface Sci. 174 (1986) 55. M. Thomsen and A. Madhukar, J. Crystal Growth 80 (1987) 275. M. Thomsen, S.V. Ghaisas and A. Madhukar, J. Crystal Growth 84 (1987) 79. M. Thomsen and A. Madhukar, J. Crystal Growth 84 (1987) 98. A. Madhukar, P. Chen, F. Voillot, M. Thomsen, J.Y. Kim, W.C. Tang and S.V. Ghaisas, J. Crystal Growth 81 (1987) 26. J. Arthur, Surface Sci. 43 (1974) 449. C.T. Foxon, M.R. Boudry and B.A. Joyce, Surface Sci. 44 (1974) 69. CT. Foxon and B.A. Joyce, Surface Sci. 50 (1975) 434. C.T. Foxon and B.A. Joyce, Surface Sci. 64 (1977) 293. C.T. Foxon and B.A. Joyce, J. Crystal Growth 44 (1978) 75. C.T. Foxon, B.A. Joyce and M.T. Norris, J. Crystal Growth 49 (1980) 132. C.T. Foxon, J. Vacuum Sci. Technol. B 4 (1986) 867. W.A. Goddard III, J.J. Barton, A. Redondo and T.C. McGill. J. Vacuum Sci. Technol. 15 (1978) 1274. J.J. Barton, W.A. Goddard III and T.C. McGill, J. Vacuum Sci. Technol. 16 (1979) 1178. W.A. Goddard III and T.C. McGill, J. Vacuum Sci. Technol. 16 (1979) 1308. CA. Swarts, W.A. Goddard III and T.C. McGill, J. Vacuum Sci. Technol. 17 (1980) 982. J.J. Barton, C.A. Swarts, W.A. Goddard III and T.C. McGill, J. Vacuum Sci. Technol. 17 (1980) 164. C.A. Swarts, J.J. Barton. W.A. Goddard III and T.C. McGill, J. Vacuum Sci. Technol. 17 (1980) 869. T.A. Swarts, W.A. Goddard III and T.C. McGill, J. Vacuum Sci. Technol. 19 (1981) 360. A. Redondo, W.A. Goddard III, T.C. McGill and G.T. Surratt, Solid State Commun. 20 (1976) 733. A. Redondo, W.A. Goddard III, T.C. McGill and G.T. Surratt, Solid State Commun. 21 (1977) 991. G.T. Surratt and W.A. Goddard III, Phys. Rev. B 18 (1978) 2831. K. Hermann and P.S. Bagus, Phys. Rev. B 20 (1979) 1603. AC. Kenton and M.W. Ribarsky, Phys. Rev. B 23 (1981) 2897. L.C. Snyder and 2. Wasserman, Surface Sci. 71 (1978) 407. L.C. Snyder and Z. Wasserman, Surface Sci. 77 (1978) 52. L.C. Snyder, Z. Wasserman and J.W. Moskowitz, J. Vacuum Sci. Technol. 16 (1979) 1266. A. Amore Bonapasta, C. Battistoni, A. Lapiccirella. E. Semprini, F. Stefani and N. Tomassini. Nuovo Cimento D 6 (1985) 51. A. Amore Bonapasta, C. Battistoni, A. Lapiccirella, N. Tomassini. S.L. Altman and K.W. Lodge, Phys. Rev. B 37 (1988) 3058. A. Amore Bonapasta, A. Lapiccirella, N. Tomassini and M. Capizzi, Phys. Rev. B 36 (1987) 6228. N. Piitz, E. Veuhoff, H. Heinecke, M. Heyen, H. Ltith and P. Balk, J. Vacuum Sci. Technol. B 3 (1985) 671. N. Piitz, H. Heinecke, M. Heyen, P. Balk, M. Weyers and H. Liith, J. Crystal Growth 74 (1986) 292. W.T. Tsang, J. Vacuum Sci. Technol. B 3 (1985) 666. W.T. Tsang. J. Crystal Growth 81 (1987) 261.

288 [48]

(491 [50]

1511 1521 [53] [54] 1551 [56] [57]

[58] [59] [60] (611 [62] [63] [64] [65] [66] [67] [68] [69] (701 [71] [72] [73]

A. Amore

Bonapasta

er al. / Moleculur

cluster studies

of c~vstal growth

in MBE

L.M. Fraas. P.S. McLeod. L.D. Partain and J.A. Cape. J. Vacuum Sci. Technol. B 4 (1986) 22. H. Seki and A. Koukitu, J. Crystal Growth 74 (1986) 172. J.S. Binkley, R.A. Whiteside, R. Krisnan, R. Seeger. D.J. De Frees. H.B. Schlegel. S. Topiol. L.R. Kahan and J.A. Pople. GAUSSIAN 80. Quantum Chemistry Program Exchange, Program No. 406, 437 Indiana University (1980). S. Huzinaga. Ed.. Gaussian Basis Sets for Molecular Calculations (Elsevier, Amsterdam. 1984). S. Huzinaga, Ed. in: Gaussian Basis Sets for Molecular Calculations (Elsevier. Amsterdam. 1984) pp. 13-16. W.J. Hehre, R.F. Stewart and J.A. Pople. J. Chem. Phys. 51 (1969) 2657. R. Ditchfield. W.J. Hehre and J.A. Pople. J. Chem. Phys. 54 (1971) 724. R. Fletcher and M.J.D. Powell, Computer J. 6 (1963) 163. J.A. Pople and R.K. Nesbet. J. Chem. Phys. 22 (1954) 571. J.H. Collomon. E. Hirota. K. Kuchitsu, W.J. Lafferty, A.G. Maki and C.S. Pote. in: Structure Data on Free Polyatomic Molecules. Landolt-Bornstein. New Series. Group 2. Vol. 7. Eds. K.H. Hellwege and A.M. Hellwege (Springer, Berlin. 1976). W.C. Prince, H.J. Lempka and T.R. Passmore, Proc. Roy. Sot. A 304 (1968) 53. R.H. Findlay. J. Chem. Sot. Faraday Trans. 72 (1976) 388. W.J. Pietro. B.A. Levi. W.J. Hehre and R.F. Steward. Inorg. Chem. 19 (1980) 2225. M.L. Ginter and K.K. Innes. J. Mol. Spectrosc. 7 (1961) 64. P. PyykkG and J.P. Desclaux, Chem. Phys. Letters 42 (1976) 545. P.E. Stevenson and W.N. Lipscomb. J. Chem. Phys. 52 (1970) 5343. L.E. Sutton, Ed., Chem. Sot. (London) Spec. Publ. 11 (1958). J.A. Pople, in: Application of Electronic Structure Theory. Ed. H.F. Schaefer III (Plenum Press, New York. 1977) p. 11. K.P. Huber and G. Herzberg, Constants of Diatomic Molecules (Van Nostrand-Reinhold. New York, 1979) p. 38. Y. Morino. T. Ukaji and T. Ito. Bull. Chem. Sot. Japan 39 (1966) 64. J.D. Grange, in: The Technology and Physics of Molecular Beam Epitaxy. Ed. E.H.C. Parker (Plenum, New York, 1985) p. 47. M. Tmar, A. Grabriel, C. Chatillon and I. Ansara. J. Crystal Growth 69 (1984) 421. LG. Csizmadia. Ed.. in: Theory and Practice of MO Calculations on Organic Molecules (Elsevier, Amsterdam. 1976) p. 70. Y. Watanabe, Y. Sakai and H. Kashiwagi. Chem. Phys. Letters 120 (1985) 363. B. Beagley. D.G. Schmidling and LA. Steer. J. Mol. Struct. 21 (1974) 437. J. Sing and K.K. Bajaj, J. Vacuum Sci. Technol. B 2 (1984) 576.