Adsorption and Dissociation of Trimethylindium on an Indium Nitride

J. Phys. Chem. C 2009, 113, 21765–21778

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Adsorption and Dissociation of Trimethylindium on an Indium Nitride Substrate. A Computational Study Beatriz H. Cardelino*,† and Carlos A. Cardelino‡ Chemistry Department, Spelman College, Atlanta, Georgia 30314-4399, and School of Earth & Atmospheric Sciences, Georgia Institute of Technology, Atlanta, Georgia 30332-0340 ReceiVed: August 1, 2009; ReVised Manuscript ReceiVed: NoVember 3, 2009

A computational study of the adsorption and dissociation of trimethylindium (TMI) on an indium nitride (InN) substrate was performed, as an essential component of InN epitaxy in high-pressure chemical vapor deposition. Gibbs free energies of reaction and reaction rate constants were calculated for the adsorption/ desorption of TMI and its derivatives onto four model substrates of InN, for temperatures between 300 and 1400 K, and pressures between 1 and 100 atm. Similar properties were calculated for the methyl dissociation of the adsorbed species. The four model substrates were designed to represent the nitrogen surface of InN with various degrees of coverage. The model-substrates were constructed as four-shell clusters, which were structurally optimized by energy minimization. The electronic energies for the gaseous species were estimated using density functional theory, whereas, for the adsorbate-cluster adducts, the hybrid approach ONIOM was used, based on semiempirical molecular-orbital calculations and density functional theory. Thermodynamic properties and reaction rate constants were estimated using Debye’s approximation for solids, vibrational frequencies obtained from normal vibrational analyses, statistical thermodynamics, and transition-state theory. Validation of the clusters was based on the indium-nitrogen distances and on heat capacity. Systems of coupled differential equations were solved to compute the time progression of the species concentrations at various temperatures (from 500 to 1300 K) and pressures (from 1 to 25 atm). Introduction Trimethylindium [TMI or In(CH3)3] is the principal precursor in the synthesis of the semiconductor material indium nitride (InN). The potential use of InN in field-effect transistor (FET) devices was acknowledged when a maximum room-temperature electron mobility of 2700 cm2 V-1 s-1, with a carrier concentration of 5.3 × 1016 cm-3, was measured in a film produced by radio frequency sputtering of metallic indium in a nitrogen atmosphere.1 The fundamental parameters of InN, particularly its bandgap, have shown wide variations in measured values.2 The direct bandgap has now been established to be near 0.7 eV, a value that was first reported in 2002.3 InN has an effective electron mass between 0.04 and 0.07 m0.4 InN has been proposed for applications in a wide variety of devices, including metal-insulator-semiconductor field-effect transistors (MISFETs),5 gas/liquid sensors,6 ohmic contacts,7 transparent conducting window materials for heterojunction solar cells,8 InN/Si p-n heterojunctions,9 anodes for Li-ion thin film batteries,10 thermoelectric devices,11 terahertz radiation devices,12 and photonic devices.13 Thin polycrystalline films of InN were found to be superconductive at helium temperatures, with superconducting transition temperatures below 3.3 K, dependent on the film structure, and with superconductivity persisting under high magnetic fields.14 Alloyed with gallium nitride (GaN), the ternary system In1-xGaxN was determined to show direct bandgaps that span from the infrared (0.7 eV) to the blue (3.4 eV), depending on the proportion of In.15 The earliest success in the growth of InN with good electrical properties16 was obtained in 1972 by reactive radio frequency * Corresponding author. E-mail: [email protected]. † Spelman College. ‡ Georgia Institute of Technology.

sputtering.17 Epitaxial growth of single crystalline and good quality InN films was widely studied in the 1990s, by metal-organic vapor phase epitaxy (MOVPE) and by molecular beam epitaxy (MBE), which became the most popular growth techniques.16 The source of indium commonly used in MBE is solid indium (in gas-source MBE18) or a metal-organic compound such as TMI [in metal-organic molecular beam epitaxy (MOMBE)19]; the source of nitrogen is usually nitrogen gas (N2) or ammonia (NH3). The source of indium generally used for MOVPE is TMI, and NH3 is the source of nitrogen.20 The epitaxial growth of InN has been particularly challenging, when compared to other group III nitrides [i.e., aluminum nitride (AlN) or GaN]. Because of the low InN dissociation temperature and high equilibrium N2 vapor pressure over the film,21 InN needs to be grown at low temperatures. However, NH3 requires high temperatures to achieve acceptable decomposition rates, and growth of InN at low temperatures has been dominated by the formation of metallic indium droplets due to the shortage of reactive nitrogen.22 On the other hand, enhancement of NH3 decomposition by higher temperatures results in an increase of the hydrogen (H2) partial pressure, which promotes etching of InN.23 A breakthrough in InN epitaxy has been the development of high-pressure chemical vapor deposition (HPCVD)24 reactors. HPCVD counteracts the decomposition of the InN surface at the optimal growth temperatures by increasing reactor pressures, which could go as high as 100 bar. High N2 pressures can stabilize the InN surface at temperatures as high as 1100 K.25 Thus, HPCVD extends the processing parameters beyond those accessible by MBE and MOCVD.26 The technique is used in a high-pressure flow channel reactor, with real-time optical characterization capabilities, at 15 bar and 1073-1173 K,27 where high-structural-quality InN epilayers have been grown,

10.1021/jp907426r  2009 American Chemical Society Published on Web 12/07/2009

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with hexagonal symmetry. The absorption spectra indicated a shift of the absorption edge from 1.8 eV down to approximately 0.7 eV as the NH3-TMI flow ratio decreased. A recent study has shown an exponential dependence between the optical absorption edge and free carrier concentration, and an indication of the existence of a surface layer with higher free carrier concentration than an inner layer.28 In the HPCVD experiments described above, a pulsedinjection scheme is utilized.29 For the precursor injection into the reactor, a two-step process is followed in which a reservoir is filled at slightly above atmospheric pressure, and then the reservoir is compressed with carrier gas to the reactor pressure and injected into the reactor. Moreover, injection of the precursors is separated by injection of carrier gas. TMI in InN Epitaxy. The growth of InN at high pressures requires that the gas phase reactions be minimized so that all the chemistry happens when the precursors come in contact with the surface. Estimates of the thermodynamic and chemical kinetic parameters for the adsorbed precursors are required for the development of reactor models. The process of dissociation of TMI adsorbed to InN is technologically important to understand InN growth. If the molecules adsorbed onto a surface are not tightly bound, they can move laterally over the surface with two degrees of translational freedom, loosely attracted by van der Waals interactions. In such case, the adsorbate cannot be considered to be localized at specific sites. Such systems can be studied using two-dimensional partition functions and adsorption energies.30 In the case of the present study, the adsorbates attach to the substrate with binding energies on the order of a few hundred kilojoules per mole, and they are not free to move laterally. Crystalline solids can be studied by quantum ab initio methods that combine periodic boundary conditions with plane-wave expansions.31 For periodic systems, the information of one unit cell is repeated, whereas for localized aperiodic systems, a supercell containing a portion of the system is repeated in space.32 Typically, plane-wave expansions and periodic boundary conditions are used within the framework of density functional theory (DFT).33 The latter has recently been applied to study the properties of InN and other group III nitrides and phosphides.34 A different approach consists in representing the substrate by an imaginary molecule or cluster, which retains some of the properties of the crystal or film. Clusters are made of a reduced number of atoms, arranged in a similar way as they appear in the crystal, and usually truncated by hydrogen atoms. Once the precursor is attached to the cluster, the change in the electronic energy due to the interaction between the substrate and the adsorbate can be evaluated. The cluster method has been successfully used to predict adsorption parameters for the growth of GaN from trimethylgallium (TMG) and NH3, during chemical vapor deposition,35 where the gallium-nitrogen distances in the clusters were fixed to that of the wurtzite crystal. Formation of the first monolayer of InN from the reaction of hydrazoic acid (HN3) and TMI, coadsorbed on a cluster of TiO2 rutile (110), was modeled using DFT and the pseudopotential method.36 Statistical thermodynamics can be used to predict thermodynamic and chemical kinetic properties for the interaction between the substrate and the adsorbate. In the classical treatment, homogeneous substrates are described by a given adsorption area, and heterogeneous substrates are described by a distribution of areas with different adsorption energies. In the quantum mechanical description, the energy spectrum becomes more complex as the internal structure of the molecules is taken

Cardelino and Cardelino into account, as the substrate is allowed to have more than one type of bonding site, and as the adsorbates are allowed to interact and modify the substrate. Following the latter description, for example, the adsorption of CO to a Ni(111) crystal was studied by treating the CO molecules as double harmonic oscillators,37 and by considering separately the dependence of the thermodynamic properties on coverage and temperature. A critical aspect of the calculation of the thermodynamic properties is the determination of the vibrational normal frequencies of the adsorbate. For example, DFT calculations were performed to determine the normal frequencies of ethylidyne (CCH3) on a palladium cluster (Pd3);38 semiempirical calculations were used to calculate frequency shifts for the adsorption of CO2 and CO on NaCl cluster surfaces;39 pyridine adsorbed on group-VIIIA transition metals was studied using DFT to understand surface-enhanced Raman spectra.40 In the present investigation, (a) four different clusters were designed to simulate substrates of InN with different degrees of coverage; (b) normal vibrational analyses were applied to energy-minimized free clusters and adsorbate-substrate adducts; (c) the adsorption of gaseous indium species onto InN was modeled using the designed clusters; (d) the subsequent methyl dissociation of adsorbed TMI and its derivatives was studied; (e) Gibbs-free energies and reaction rate constants were estimated for the adsorption and dissociation reactions, at various temperature and pressure conditions and degrees of coverage; and (f) time-dependence analyses were performed to determine the distribution of chemical species concentrations. Methods This section describes the methods used (a) to model the free gaseous species, (b) to model the InN clusters, and the substrate-adsorbate clusters, (c) to calculate the thermodynamic properties of the gaseous species and the designed clusters; (d) to calculate the reaction rate constants for the adsorption of gaseous indium species and for the dissociation of the indium species on the clusters; and (e) to estimate the relative species concentrations during process propagation. Modeling of Free Gaseous Species. TMI [or In(CH3)3], dimethylindium [DMI or In(CH3)2], monomethylindium (MMI or InCH3), atomic indium (In), ethane (C2H6), and methyl radical (CH3) were the free species considered. DFT was used to perform all calculations,33 with B3LYP hybrid density functionals,41 as implemented in the Gaussian 03W quantum mechanical program.42 The following basis sets provided by Gaussian 03W were selected: 6-311 g basis sets with polarization functions for C, N and H in all molecules, and a 3-21 g basis set with polarization functions for In. The total number of basis functions and primitive-Gaussian functions is shown in Table 1, under gas phase. Since TMI, DMI, and MMI can dissociate methyl groups, as well as hydrogen atoms, Table 2 compares the 0 K enthalpy difference calculated for the two processes, for molecules in the gas phase. The values show that breaking a C-H bond requires between 1.7 and 2.5 times more energy than releasing a methyl group. These results justified the selection of indium species used in the present work. However, it should be mentioned that a study of the pyrolysis of TMI in a hot-wall flow-tube reactor found the decomposition of TMI to be a heterogeneous, autocatalytic process, with an induction period that is carrier-gas dependent.43 In that study, the authors proposed the formation of the gas-phase intermediate HIn(CH3)3. Details of the structure and electronic energy of the indium compounds in the gas phase have been previously reported.44

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TABLE 1: Size of the Basis Sets gas phase and model high atom

model low

number of basis functions

number of primitive functions

number of basis functions

number of primitive functions

6 18

8 32

36

93

1 4 4 4

3 12 12 12

H C N In

TABLE 2: Enthalpy of Dissociation of a Methyl Group or a Hydrogen Atom from TMI, DMI, and MMI reaction

0 K reaction enthalpy (kJ/mol)

In(CH3)3 f In(CH3)2 + CH3 In(CH3)3 f In(CH2)(CH3)2 +H In(CH3)2 f InCH3 + CH3 In(CH3)2 f In(CH2)(CH3) + H InCH3 f In + CH3 InCH3 f InCH2+ H

260 438 117 396 227 433

In that report, the basis set for the indium atom had been augmented with respect to the set shown in Table 2 by using the AKR4 basis set, resulting in 72 basis functions and 190 primitive-Gaussian functions, as implemented within Gaussian 03W.42 Such an enlarged basis set was inappropriate for the present study. Modeling of the InN Substrate and the Substrate-Adsorbate Clusters. InN is a binary crystal with wurtzite structure, where each type of atom is tetrahedrally coordinated to four atoms of the other type. Four clusters were designed to represent the substrate with various degrees of coverage (or dangling bonds). Please refer to Figure 1. Considering only the heavy atoms (N and In), all four clusters could be described as consisting of four shells. The first shell contained only the central nitrogen atom, indicated as 1 in the figure and shown by the pointing arrow. This atom had a dangling orbital, and it was where adsorption occurred. The second shell was composed of three indium atoms bonded to the nitrogen atom of the first shell (atoms 2-4). Each of these indium atoms had three additional bonds to nitrogen atoms. These nine nitrogen atoms formed the third and fourth shells. The third shell consisted of six nitrogen atoms, bonded in groups of two to the indium atoms of the second shell (5 and 6 bonded to 2; 7 and 8 bonded to 3; 9 and 10 bonded to 4). Together with the nitrogen of the first shell, these seven nitrogen atoms (1, and 5-10) constituted the nitrogen surface of the InN substrate. The fourth shell consisted of three nitrogen atoms (atoms 11-13) that were part of the InN bulk. Thus, the four clusters contained 10 nitrogen atoms and three indium atoms (In3N10). The InN clusters were truncated with hydrogen atoms. Each of the three nitrogen atoms of the fourth shell (atoms 11-13) was bonded to three hydrogen atoms, that is, a total of nine hydrogen atoms (In3N10H9). The difference among the four clusters was given by the number of hydrogen atoms attached to the six nitrogen atoms of the third shell (atoms 5-10). Each of the six nitrogen atoms could have a dangling bond, in addition to two hydrogen atoms, replacing the hydrogen atoms marked with an x in Figure 1. When all six nitrogen atoms of the third shell had no dangling bond, there were 18 additional hydrogen atoms. This cluster consisted of 40 atoms (In3N10H27), and it had only one dangling atom at the central nitrogen atom of the first shell. It was represented by the symbol NS40, standing for nitrogen-surface of the InN substrate containing 40 atoms. When all hydrogen atoms marked with an x were eliminated, the central nitrogen atom of the first shell (atom 1) was surrounded by six nitrogen atoms with dangling bonds, resulting on a

nitrogen surface with a total of seven dangling bonds. This cluster, represented as NS34, consisted of 34 atoms (In3N10H21). Two other clusters were selected to have two and four dangling bonds, in addition to the central one, resulting in surfaces with three and five dangling bonds, respectively. They were labeled NS36 (In3N10H23) and NS38 (In3N10H25), respectively. All four clusters had an even number of electrons. Thus, the four clusters represented the nitrogen-surface of InN with different coverage, ranging from fully available (NS34) to fully occupied except for the central nitrogen atom (NS40). Calculations on the four free clusters were performed using a semiempirical approach. The PM3 Hamiltonian45 was selected because it includes parameters for the indium atom. The Gaussian 03W42 program uses minimum basis sets for semiempirical calculations, and the number of functions is shown under “model low” in Table 1. All structures were allowed to fully optimize by energy minimization, and normal vibrational analyses were performed. The vibrational analyses provided the frequency values required for the thermodynamic and chemicalkinetic properties, as well as a means for corroborating that the optimized structures were stable structures. Calculations on the substrate-adsorbate systems were performed using ONIOM,46 a hybrid-type approach implemented in the Gaussian 03W42 program. In ONIOM, the molecular system can be partitioned into two or three portions, which are treated at different levels of approximation. ONIOM was the method of choice in the study of interfacial reactions using cluster models as, for example, the atomic layer deposition of Al2O3 on H/Si(111),47 or the alkylation of benzene with ethylene over faujasite zeolite48 For the present work, a two-level type of computation was selected, dividing the molecular adducts into an “active” part and its “environment”. ONIOM performs high-level and low-level calculations on the “active” part of the system, and a low-level calculation on the complete system, or “real” system. The energy difference between the two low-level calculations corresponds to the energy contribution due to the “environment”, which is added to the high-level energy value to obtain an “extrapolated” energy value. Equation 1 shows how the energy terms are organized in ONIOM calculations, where “Model High Energy” represents the calculation on the active part with the high-level approach, “Real Low Energy” represents the calculation of the complete system with the low-level approach, and “Model Low Energy” represents the calculation of the active part with the low-level approach. If needed, in order to perform the partial energy calculations, ONIOM may add a hydrogen atom to satisfy the valence of the “active” part.

Model High Energy + (Real Low Energy Model Low Energy) ) Extrapolated Energy (1) In the present calculation, the adsorbate was considered to be the “active” site, and the level of approximation selected was DFT (B3LYP41) with large basis sets (6-311g** for H, C, and N; 3-21g** for In). It should be mentioned that DFT (B3LYP) calculations with similar basis sets were also used in the calculation of gaseous molecules. The complete substrate-

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Figure 2. Methyl dissociation energy for TMI, DMI, and MMI obtained from ONIOM calculations where (1) the adsorbate constituted the “active” portion and (2) the adsorbate plus the N atom of the cluster where adsorption occurred constituted the “active” portion.

Figure 1. Representation of the four-shell cluster used to model InN. Color code: blue ) nitrogen atom; red ) indium atom; white ) hydrogen atom. The arrow indicates the position of the central nitrogen atom, where adsorption occurs. The x shows hydrogen atoms that were eliminated in different model clusters.

adsorbate system or “real” system was treated using the semiempirical approach (PM345), the same as was used for the model clusters. This investigation included two types of reactions involving adsorption onto the clusters and methyl dissociation of adsorbed species, as shown in eq 2 and 3, where NS stands for the nitrogen surface of the InN substrates.

NS + In(CH3)x f NS · In(CH3)x

(2)

NS · In(CH3)x f NS · In(CH3)x-1 + CH3

(3)

In the case of adsorption (eq 2), the reaction energies were calculated using the “extrapolated” energy of the product, the semiempirical energy of the cluster, and the high-level calculation of the gaseous reactant. Since the “extrapolated” energy of the product contained a DFT component arising from the adsorbate, plus a semiempirical component arising from the cluster (or “environment”), these energies were in correspondence with the DFT energy of the adsorbate and the semiempirical energy of the free cluster, respectively. In the case of the dissociations (eq 3), the reaction energies were calculated using the model-high (DFT) energies of the substrate-adsorbate product and the reactant, and the DFT energy of the methyl group. The selection of just the adsorbate as the “active” portion for the ONIOM calculations was done so that energies of adsorption could be calculated from the energy difference between the cluster with the adsorbed species and the free substrate plus the gaseous indium species (eq 2). In the case of methyl dissociation energies (eq 3), a different “active” portion could be selected. To test the effect that selecting a different “active” portion would have on the methyl dissociation energy, calculations were performed on the NS36 cluster with adsorbed species, where the “active” portion was defined as the adsorbate plus the N atom of the cluster where adsorption occurred. The results shown in Figure 2 indicate that the ordering of TMI > MMI > DMI for the methyl dissociation energies is maintained, while the numerical differences between the two approaches are relatively small. Consequently, to preserve the consistency of the method used in this research, the same definition of “active” portion for both types of equations was used. Thermodynamic Properties of the Gaseous Species and of the Clusters. The thermodynamic properties of gaseous species were calculated following a statistical thermodynamic

procedure previously reported,44 that was applied to indium species. That study showed that energies could be calculated within a few kilojoules per mole of experimental values. The calculations were based on values of electronic energy (including nuclear-nuclear repulsion) obtained from DFT calculations and harmonic frequencies obtained from normal vibrational analyses. The contributions were partitioned into translational, rotational, vibrational, and internal rotational. The above procedure was modified in order to calculate the thermodynamic properties of the free clusters and the adsorbatecluster systems. The following assumptions were made: (a) there were no translational nor rotational contributions; (b) the vibrational contributions could be classified into substrate and adsorbate contributions; (c) the vibrational contribution of the substrate was subject to a cutoff frequency determined by the Debye’s cutoff temperature for InN, as shown in eq 4; (d) the vibrational contribution of the substrate to internal energy (Uv), heat capacity (Cp) and zero-point energy could be calculated using Debye’s approximation for solid state (eqs 5-7); and (e) the expressions for Uv and Cp could be solved by numerical integration. The Debye’s cutoff vibrational frequency (νD) was defined as

νD )

kB T h D

(4)

where TD is Debye’s temperature of the solid (660 K for InN49), h is Planck’s constant, and kB is Boltzmann’s constant; thus, νD is 459 cm-1 for InN. The Debye vibrational contribution to internal energy (UvD) was given by

UvD)ncell

9NA

h 3

νD

∫0ν

D

ν3 dν ex - 1

x≡

hν kBT

(5)

where ncell is the number of interspersed atomic cells (2 for InN), NA is Avogadro’s number, ν stands for vibrational frequency, and T is temperature. The corresponding Debye’s equations used for heat capacity (CpD) and zero-point energy (ZPED) were

( )

x4ex dx (ex-1)2 9NA ZPED ) ncell hνD 8

CpD)ncell9NAkB

T TD

3

∫0ν

D

x≡

hν kBT

(6) (7)

Reaction Rate Constants for the Adsorption of Gaseous Indium Species onto the Substrates and for the Dissociation of Indium Species Adsorbed onto the Clusters. The reverse reaction rate constant (kr) for the adsorption process (that is, eq 2 in the opposite direction, or the desorption process) and the forward reaction rate constants (kf) for the dissociation of adsorbed indium species (eq 3) were calculated using a

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semiclassical approach previously described.50 This method, applicable to homolytic dissociations in the gas phase, was modified to make it appropriate for reactions that include solids. The kr values for eq 2 and the kf values for eq 3 were obtained from the Gibbs free energy of the undissociated reactant and of the critical configuration, using transition-state theory. The semiclassical approach was based on quantum mechanics and transition-state theory, with the critical configuration determined from (a) linear interpolations for the geometry of the intermediate structures, (b) Morse potentials for the intermediate electronic energies (Er, eq 8), and (c) Hase’s relationship for the vibrational frequencies that become annihilated (νr, eq 9). The Morse potential (Er) as a function of bond distance (r) is given by

(

( ( )

Er)Ed 1 - exp -πν0

2µ Ed

0.5

))

(r - r0)

2

(8)

where Ed is the energy difference between the dissociating products and the unperturbed molecule determined from quantum mechanics, ν0 is the vibrational frequency of the dissociating bond determined from normal vibrational analyses, µ is the reduced mass between the dissociating portions, and r0 is the initial distance between the dissociating atoms. The vibrational frequency that is to become annihilated (νr), as a function of bond distance (r), is given by

( ( )

νr)νi exp -πν0

2µ Ed

0.5

)

(r - r0)

vibrational contributions from UM were classified into the following four groups: (a) substrate vibrations of UM that correlate with substrate vibrations of MA, (b) adsorbate vibrations of UM that correlate with adsorbate vibrations of MA, (c) adsorbate vibrations of UM that correlate with vibrations of MI, and (d) uncorrelated vibrations of UM. The classification of frequencies was based on the magnitude of the vibrational frequency and on the atom displacements of the vibrational mode. One of the uncorrelated vibrations of UM was selected as that of the dissociating bond (ν0 in eq 8 and 9). It should be pointed out that the calculation of the unimolecular rate constants (kuni) was based on the calculation of a rate constant independent of pressure (k∞) and a rate constant first-order on pressure (k1), according to the following equation:

(9)

where νi is the vibrational frequency of the unperturbed molecule that will become annihilated. The forward reaction rate constants (kf) for the adsorption reactions and the reverse reaction rate constants (kr) for the dissociation reactions were calculated using pseudo equilibrium constants for the reactions (K), calculated from the reaction Gibbs free energy. The word pseudo is utilized here because the calculation of the equilibrium constant included the Gibbs free energy of the clusters, even though they represented solids. However, using such equilibrium constants was consistent with the use of the Gibbs-free energy of the clusters in the calculation of the rate constants. Several modifications were implemented to make the procedure appropriate for solids. In what follows, UM stands for the adsorbate-cluster adduct (or undissociated molecule), MA represents the major product (the free cluster in the case of desorptions; the cluster with the dissociated adsorbate in the case of dissociations), and MI represents the leaving group (or minor product). The following modifications were included: (a) there were no translational nor rotational contributions from UM or MA; (b) the vibrational contributions from UM could be separated into three groups for desorption reactions and into four groups for dissociation reactions, depending on how they could be correlated with the products; (c) the substrate vibrational contributions of UM and MA were subject to a cutoff frequency (νD) determined by Debye’s cutoff temperature (TD) for InN, as shown in eq 4; (d) the vibrational contribution of the substrate to internal energy (UvD), heat capacity (CpD), and zero-point energy (ZPED) were calculated using Debye’s approximation for solid state (eqs 5-7); (e) the expressions for UvD and CpD were solved by numerical integration. In the case of the desorption reactions (eq 2), the vibrational contributions from UM were classified into the following three groups: (a) substrate vibrations of UM that correlate with substrate vibrations of MA, (b) adsorbate vibrations of UM that correlate with vibrations of MI, and (c) uncorrelated vibrations of UM. In the case of the dissociation reactions (eq 3), the

kuni )

k∞

(10)

k∞ 1+ k1P

The rate constant independent of pressure (k∞) was calculated using the Rice-Ramsperger-Kassel-Marcus (RRKM) method applied to thermal activation,51 and the rate constant first-order on pressure (k1) was calculated following Troe’s approach.52 Relative Species Concentrations As a Function of Process Propagation. The following simplified set of reactions (eqs 11-13) was chosen to model the dissociation of adsorbed TMI, where NS stands for the nitrogen surface of the InN substrate:

NS · TMI f NS · DMI + CH3

(11)

NS · DMI f NS · MMI + CH3

(12)

NS · MMI f NS · In + CH3

(13)

In order to simulate the entire process of adsorption and methyl dissociation, the dissociation of TMI in the gas phase was also considered (eqs 14-16), as well as the adsorption of the gaseous species onto the clusters (eq 17-20).

TMI f DMI + CH3

(14)

DMI f MMI + CH3

(15)

MMI f In + CH3

(16)

NS + TMI f NS NS + DMI f NS NS · +MMI f NS NS · +In f NS

· · · ·

TMI DMI MMI In

(17) (18) (19) (20)

All 10 reactions were characterized by their forward and reverse rate constants. Equations 13 and 20 led directly to the formation of the indium layer during InN growth in pulsed chemical vapor deposition. It should be mentioned that during actual InN epitaxy, the adsorbed indium species could react with species from the nitrogen source, but these reactions are beyond the scope of this study. In order to obtain the relative concentrations of all species, a system of ordinary differential rate equations was used for the chemical species of eqs 11-20, plus eq 21 for the formation of ethane. This system of differential equations was solved using built-in stiff solvers within MATLAB.53 The numerical stability of the calculations was checked using mass balance equations on indium and carbon. Since no hydrogen dissociation was allowed from the methyl groups, a mass balance equation on hydrogen would have been redundant.

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2CH3 f C2H6

Cardelino and Cardelino

(21)

Results and Discussion This section is organized into four parts: (a) structure and thermodynamic properties of the clusters; (b) properties of the adsorbate-substrate species; (c) thermodynamic and kinetic parameters for the adsorption and dissociation reactions; (d) reaction progression in the growth of the indium layer from the indium source on the nitrogen surface during InN growth. The Structure and Thermodynamic Properties of the Clusters: Validation of the Use of the Clusters for Interfacial Studies. As described under Methods, the InN substrate was modeled using four clusters containing three indium atoms, 10 nitrogen atoms, and 21, 23, 25, and 27 hydrogen atoms, for a total of 34, 36, 38, and 40 atoms, respectively. Two main properties were used to compare the clusters with InN bulk: the InN distances and the heat capacities. All four clusters contained three indium atoms (second shell of the clusters). Each indium atom was attached to four nitrogen atoms, for a total of 12 In-N bond distances. One of the bonds from each indium atom was to the nitrogen atom where adsorption occurred (first shell of the clusters). These three In-N distances were denominated R In-N distances. Two of the bonds from each indium atom were to nitrogen atoms that formed the nitrogen-surface of the InN (or third shell), and one was to nitrogen atoms that constituted the InN bulk (or fourth shell). Depending on the number of hydrogen atoms attached to the nitrogen atoms of the third shell, there were In-N distances to tetra-coordinated nitrogen atoms (β In-N distances) and In-N distances to tricoordinated nitrogen atoms (γ In-N distances). The In-N distances between the indium atoms and the nitrogen atoms of the fourth shell were classified as δ. In total, the clusters had three R bonds, three δ bonds and six more bonds, either β or γ: no β and 6 γ for NS34, 2 β and 4 γ for NS36, 4 β and 2 γ for NS38, and 6 β and no γ for NS40. Figure 3 displays the average In-N bond distances obtained for the four clusters. The crystal cell parameters for InN wurtzite54 are a ) 3.533 Å and c ) 5.693 Å, with an average In-N bond distance of 2.135 Å, a value shown in the figure as a black line. In general, the spread of values within the four bond groups (error bars in the figure) was less for NS34 and NS40 than for NS36 and NS38. This is a consequence of symmetry, since the former structures had higher symmetry than the latter (Cs and quasi C3V, respectively, instead of C1). The R In-N distance was 4% smaller than the average experimental value; the β In-N was 5% larger, the γ In-N was almost the same, and the δ In-N was about 7% larger. R In-N bonds

Figure 3. Average In-N bond distances of the four model clusters, classified into four bond types. Bond-type code: R ) N atom of the first shell with In atom of the second shell; β ) In atom of the second shell with tetracoordinated N atom of the third shell; γ ) In atom of the second shell with tricoordinated N atom of the third shell; δ ) In atom of the second shell with N atom of the fourth shell; exp ) experimental average.54

Figure 4. Heat capacity of the four model clusters, as a function of temperature. Exp ) experimental value56

were expected to be smaller than the experimental average because of the dangling bonds. As expected for tetra-coordinated nitrogen atoms, β and δ average distances were all quite similar. Overall, the average of all In-N distances was less than 2% larger than the experimental value. This result is consistent with semiempirical PM3 bond length calculations on molecules containing even smaller atoms than indium.55 Thus, based on bond distances, the clusters probably simulate correctly bulk InN. The heat capacity of the four clusters was estimated using the Debye approximations (eq 4 and 6). The values were compared with experimental data56 for temperatures between 314 and 978 K. Figure 4 displays the heat capacity values obtained for the four clusters, within that temperature range. For the most part, the calculated values had slightly larger rate of change of heat capacity with temperature than the experimental curve. In the range of interest for InN growth, between 700 and 978 K, the rate of change for NS34, NS36, NS38, and NS40 were about 4.2, 3.0, 3.0, and 3.3 mJ K-1 mol-1, respectively, whereas the rate of change for the experimental data was 1.1 mJ K-1 mol-1. Comparing the absolute values for heat capacity of the clusters with the experimental datum at 700 K, the calculated values were 17%, 3.5%, 2.4%, and 9.2% higher for NS34, NS36, NS38, and NS40, respectively. It is not surprising that NS34 and NS40 presented the larger differences with respect to the experimental value, since NS34 was the cluster with more dangling bonds (seven) and NS40 was the cluster with the largest number of hydrogen atoms (27). Thus, based on heat capacity calculations, the clusters seem to adequately model bulk InN. Comparison of other properties, such as heat of formation and bandgaps, would be beyond the scope of this study. The calculation of heats of formation using quantum mechanics require, for example, semiempirical bond additivity corrections, as was previously applied to indium compounds.57 With respect to the bandgaps, Green’s function methods could be used, which take into account electron-electron interactions, but those calculations would be beyond the scope of the present investigation. Properties of the Adsorbate-Substrate Species: Comparison with the Gaseous Species. Calculations on the four adsorbate-substrate species were performed using ONIOM.46 The species were NS · TMI, NS · DMI, NS · MMI, and NS · In. In the ONIOM calculation, TMI, DMI, MMI, and In constituted the “active” portions of the molecules and were calculated using the higher-level approximation (B3LYP41), as well as the lowerlevel approximation (PM345). The whole molecule or “real” system was computed using only the lower-level approximation (PM3). Consequently, the B3LYP energy of the active portion could be compared with the B3LYP energy of the free gaseous species, to have an estimate of the effect that adsorption had on the species. On the other hand, the difference between the

Adsorption and Dissociation of TMI on an InN Substrate

Figure 5. Energy change due to adsorption. Positive values correspond to the difference between the energy of the adsorbate and the energy of the free gaseous species; negative values correspond to the difference between the energy of substrate in the adduct and the energy of the free cluster. Label code: TMI ) trimethylindium, DMI ) dimethylindium, MMI ) monomethylindium, In ) atomic indium.

TABLE 3: Distance between the in Atom of the Adsorbate and the N Atom of the First Shell of the Model Substrate and In-C Distance in the Gas Phase and in the Adsorbed Species

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Figure 6. Vibrational partition function at 1000 K and 10 atm assigned to the adsorbate portion of the adducts, and for the free species.

TABLE 4: Adsorption Energies at 0 K, in kJ mol-1 Units TMI DMI MMI In

NS34

NS36

NS38

NS40

-275 -637 -59 -370

-270 -547 -511 -700

-341 -560 -644 -799

-333 -599 -474 -736

Adsorbate-Substrate Distance (Å) NS34 TMI DMI MMI In

2.104 2.140 2.795 2.757

NS36 2.120 2.062 2.112 2.076

NS38 2.077 2.068 2.019 2.122

NS40 2.042 2.030 2.026 2.081

Average In-C Distance (Å)

TMI DMI MMI

gas

NS34

NS36

NS38

NS40

2.208 2.227 2.277

2.312 2.397 2.308

2.326 2.343 2.274

2.370 2.369 2.332

2.367 2.379 2.348

PM3 energy of the real system and the PM3 energy of the active portion, gave an estimate of the PM3 energy of the “environment” or, in the present case, the NS substrate. This energy difference could be compared with the PM3 energy of the free NS cluster to estimate the effect that adsorption had on the cluster. Figure 5 displays the 0 K energy changes to the free gaseous species and to the clusters upon adsorption. On the basis of the results shown in Figure 5, the 0 K adsorption of the gaseous species was, overall, an exothermic process because of the energy lowering undergone by the clusters. Except for atomic In, the gaseous species increased their energy upon adsorption; the former showed an energy lowering of only 5 J mol-1, in all four clusters. As expected, since it is a nonradical molecule, TMI was the species affected the most by adsorption, with an average endothermicity close to 130 kJ mol-1; DMI showed an average endothermicity of about 30 kJ mol-1, and MMI of about 5 kJ mol-1. Adsorption resulted on considerable substrate energy lowering, with the exception of the NS34 cluster with MMI adsorption. Also as expected, the two gaseous species with unpaired electrons affected more the substrates: DMI, which is a doublet, showed an average exothermicity of about 615 kJ mol-1, and In, which is also a doublet, about 650 kJ mol-1. The effect of TMI and MMI (which is a singlet) adsorption, was a substrate energy lowering of about 430 kJ mol-1. Table 3 displays the distances between the indium atom of the adsorbate and the nitrogen atom of the first shell of the substrate, for all four clusters. On average, NS36 and NS38 had smaller adsorbate-substrate distances with smaller spread (2.09 ( 0.03 Å and 2.07 ( 0.05 Å, respectively), compared to NS34 (2.4 ( 0.3 Å) and NS40 (2.3 ( 0.2 Å). In terms of the

indium species, TMI and DMI had smaller adsorbate-substrate distances with smaller spread (2.09 ( 0.04 Å and 2.07 ( 0.04 Å, respectively), compared to MMI (2.2 ( 0.2 Å) and In (2.3 ( 0.2 Å). Table 3 also displays the average In-C distances in the gas phase and after adsorption. In the gas phase, the three methylated indium species had an average In-C distance of 2.24 ( 0.03 Å. Adsorption increased the In-C distance to 2.34 ( 0.07 Å. The larger In-C distances were consistent with the fact that the adsorbed species had higher energy than in the gas phase. Adsorption energies at 0 K are displayed in Table 4. Roughly, energies became more exothermic (or more negative) down the series TMI > DMI > MMI > In. MMI was an exception for all clusters but NS38; In was an exception for NS36. TMI and DMI adsorptions were exothermic by about 500 and 590 kJ mol-1, respectively, but MMI adsorptions ranged between 60 and 640 kJ mol-1 and those of In ranged between 370 and 800 kJ mol-1. Figure 6 displays, using a logarithmic scale, the values of the vibrational partition functions assigned to TMI, DMI, and MMI as adsorbates, in the four adducts, at 1000 K and 10 atm. Dotted lines show the corresponding values for the gaseous species. In all cases, adsorption increased the value of the property. TMI was the most affected species, in particular with the NS40 substrate. The partition functions increased by factors as high as 4, except for the latter, which was a factor of 13. Since this property is an important component of entropy, an increase in the vibrational partition function lowers Gibbs free energy. Nevertheless, as shown in Table 5, the effects became overshadowed by the changes in vibrational partition function that the substrate incurred compared to the free substrate. Except in the cases of NS34-MMI, NS36-MMI, NS40-MMI, and NS40In, the vibrational partition functions associated with the substrate decreased by several orders of magnitude; these were the same exceptions seen for the trends in adsorption energies. Table 5 also displays the vibrational frequencies assigned to the dissociating bond. Comparison of the vibrational frequencies with the adsorbate-substrate distance showed an inverse correlation of 0.999 for the logarithm of the vibrational frequency versus the bond distance for the NS38 cluster. Thermodynamic and Kinetic Parameters for the Deposition and Dissociation Reactions: Effect of Temperature, Pressure and Degree of Coverage. The thermodynamic and chemical kinetic parameters for reactions 11-20, were computed at 18 atmospheric pressures ranging from 1 to 100 atm, and at

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TABLE 5: Vibrational Partition Function at 1000 K and 10 atm Assigned to the Substrate Portion of the Adducts and the Vibrational Frequency Associated with the Adsorbate-Substrate Bond NS34 free substrate TMI DMI MMI In

NS36

1.29 × 10 1.27 × 1025 1.93 × 1023 3.33 × 1032 2.57 × 1028 30

NS38

6.99 × 10 5.39 × 1022 1.35 × 1023 1.65 × 1026 3.70 × 1022

NS40

2.66 × 10 5.32 × 1019 2.17 × 1024 3.30 × 1025 1.50 × 1024

25

27

NS34

6.82 × 10 1.63 × 1018 1.35 × 1020 1.97 × 1028 2.41 × 1024

NS36

NS38

NS40

-1

22

(cm ) 600 780 651 358

517 581 614 552

619 638 710 564

669 717 665 631

TABLE 6: Methyl Dissociation Energies at 0 K, in kJ mol-1 Units TMI DMI MMI

gas

NS34

NS36

NS38

NS40

275 128 241

221 99 241

176 116 239

135 105 239

170 93 232

111 temperatures ranging from 300 to 1400 K. In the calculation of thermodynamic properties of adsorbed species, there were only two main contributions from electronic energy and vibrational energy, since there were no translational nor rotational contributions and the internal rotation contributions were small. It should be noted that, in the calculation of reaction rate constants, there were gaseous components that included translational and rotational terms. Table 6 displays the 0 K dissociation energy for the dissociation reactions; a similar property for the adsorption reaction is shown in Table 4. In all cases, adsorption decreased the methyl dissociation energy. On average, there was an energy reduction of 100 kJ mol-1 for TMI, 25 kJ mol-1 for DMI, and only 4 kJ mol-1 for MMI. Mostly, dissociation energy decreased with increasing cluster coverage (from NS34 to NS40), with some exceptions. Methyl dissociations of DMI and MMI were slightly affected by cluster coverage (10 and 2%, respectively), whereas for TMI, cluster coverage affected as much as 23%. The calculation of reaction rate constants was based on homolytic dissociations. To estimate the rate constant for adsorption, the rate constant for desorption was calculated first. Then, the value of the adsorption rate constant was estimated using the equilibrium constant for the desorption reaction. As an example, Table 7 summarizes the results obtained at 10 atm and 1000 K for the adsorption reactions. Under these conditions, the only nonspontaneous processes were the deposition of MMI and of TMI on NS34. Figure 7 displays the activation energy for the deposition reactions. It can be seen that, on average,

Figure 7. Activation energy for the deposition reactions.

activation energies decrease in the following order: NS34 > NS38 > NS36 > NS40. With regard to the indium species, TMI and DMI have larger activation energies than MMI and In. The average location of the critical configuration and their spread were 2.57 ( 0.05 Å, 2.84 ( 0.07 Å, 2.84 ( 0.14 Å, and 3.72 ( 0.30 Å, for TMI, DMI, MMI and In, respectively. This ordering of values was consistent with the ordering obtained regarding average adsorbate-substrate bond distances. Table 8 summarizes the results obtained at 10 atm and 1000 K for the methyl-dissociation reactions. Under these conditions, all adsorbed DMI dissociations were spontaneous, as well as the dissociation of TMI adsorbed on NS38. The activation energy for the first methyl decomposition in the gas phase (first reaction of Table 8) was the same as the reported average activation energy of 44 ( 6 kcal mol-1 (184 ( 25 kJ mol-1) obtained from hot-wall flow-tube reactors.43 Figure 8 displays the activation energy for the methyl-dissociation reactions of the gas-phase reactions, as well as for the adsorbed species. The activation energy for the dissociation of DMI was negligible, both in the gas phase or adsorbed. In the case of TMI, adsorption considerably reduced the activation energy to about 50%, in the less favorable case. The activation energy for adsorbed MMI was reduced with respect to dissociation in the gas to about 60%. Cluster coverage had an important influence

TABLE 7: Adsorption Reactions: Rate Constants, Equilibrium Constant, Gibbs Free Energy of Activation, and Adsorbate-Substrate Bond Distance of the Critical Configuration, at 10 atm, 1000 K rate constant adsorption reaction NS34 NS36 NS38 NS40 NS34 NS36 NS38 NS40 NS34 NS36 NS38 NS40 NS34 NS36 NS38 NS40

+ + + + + + + + + + + + + + + +

TMI f NS34 · TMI TMI f NS36 · TMI TMI f NS38 · TMI TMI f NS40 · TMI DMI f NS34 · DMI DMI f NS36 · DMI DMI f NS38 · DMI DMI f NS40 · DMI MMI f NS34 · MMI MMI f NS36 · MMI MMI f NS38 · MMI MMI f NS40 · MMI In f NS34 · In In f NS36 · In In f NS38 · In In f NS40 · In

forward (s

-1

M-1)

1.40 × 10 4.38 × 109 1.06 × 105 6.83 × 109 5.76 × 104 8.37 × 108 1.64 × 108 5.42 × 1011 1.28 × 108 3.61 × 1012 4.63 × 1010 1.15 × 1017 2.02 × 108 1.36 × 1010 2.08 × 109 8.71 × 1011 6

reverse (s-1) 4.55 × 10 7.51 × 107 6.56 × 104 1.17 × 105 1.96 × 10-9 2.13 × 10-5 6.80 × 10-6 6.34 × 10-8 5.47 × 1014 7.21 × 10-4 3.52 × 10-10 2.47 × 10-2 1.31 × 10 1.82 × 10-17 8.17 × 10-23 4.08 × 10-19 7

equilibrium constant -2

3.07 × 10 5.83 × 101 1.61 × 10 5.84 × 104 2.93 × 1013 3.92 × 1013 2.41 × 1013 8.55 × 1018 2.33 × 10-7 5.00 × 1015 1.32 × 1020 4.65 × 1018 1.54 × 108 7.47 × 1026 2.54 × 1031 2.14 × 1030

activation energy (kJ mol-1)

critical config. (Å)

148 80 168 75 170 91 105 37 110 21 56 0 102 64 80 30

2.56 2.58 2.62 2.52 2.78 2.92 2.89 2.80 2.88 2.93 2.84 2.70 4.17 3.54 3.76 3.42

Adsorption and Dissociation of TMI on an InN Substrate

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TABLE 8: Methyl-Dissociation Reactions: Rate Constants, Equilibrium Constant, Gibbs Free Energy of Activation, and Bond Distance at the Critical Configurationa rate constant methyl-dissociation reaction TMI f DMI + CH3 DMI f MMI + CH3 MMI f In + CH3 NS34 · TMI f NS34 · DMI + CH3 NS36 · TMI f NS36 · DMI + CH3 NS38 · TMI f NS38 · DMI + CH3 NS40 · TMI f NS40 · DMI + CH3 NS34 · DMI f NS34 · MMI + CH3 NS36 · DMI f NS36 · MMI + CH3 NS38 · DMI f NS38 · MMI + CH3 NS40 · DMI f NS40 · MMI + CH3 NS34 · MMI f NS34 · In + CH3 NS36 · MMI f NS36 · In + CH3 NS38 · MMI f NS38 · In + CH3 NS40 · MMI f NS40 · In + CH3 a

forward (s-1) reverse (s-1 M-1) equilibrium constant activation energy (kJ mol-1) critical config. (Å) 2.54 × 104 3.25 × 108 1.35 × 102 1.70 × 1010 3.44 × 109 4.21 × 1013 4.31 × 1011 5.21 × 1013 1.44 × 1014 6.34 × 1013 4.82 × 1013 7.90 × 108 5.80 × 108 1.07 × 109 5.04 × 108

7.34 × 10-11 5.99 × 10-10 6.32 × 10-11 4.25 × 10-19 7.61 × 10-15 9.91 × 10-13 4.68 × 10-14 1.62 × 10-3 7.70 × 10-11 3.41 × 10-11 3.51 × 10-6 1.47 × 10-19 2.09 × 10-16 1.14 × 10-14 1.44 × 10-19

1.86 × 10-6 1.94 × 10-1 8.56 × 10-9 7.24 × 10-9 2.62 × 10-5 4.18 × 101 2.01 × 10-2 8.42 × 1010 1.11 × 104 2.16 × 103 1.69 × 108 1.16 × 10-10 1.21 × 10-7 1.22 × 10-5 7.24 × 10-11

184 0 154 79 92 13 51 1 2 1 0 94 95 90 97

10.19 3.18 4.66 3.69 7.79 3.17 5.79 2.48 2.92 2.49 2.48 4.04 3.77 3.88 4.05

Values obtained at 10 atm and 1000 K. The gas-phase values have been included for comparison.

Figure 8. Activation energy for the methyl-dissociation reactions.

on the activation energy reduction for TMI dissociation, but it had no effect for MMI dissociation. Reaction Progression in the Growth of the Indium Layer in InN As a Function of Degree of Coverage, Pressure and Temperature. In order to study reaction progression, the eleven equations eqs 11-21 were written as rate equations. All equations were characterized by forward and reverse rate constants, temperature, and pressure dependence. The sets of equations were solved for the four clusters (NS34, NS36, NS38, and NS40) separately, and for all the clusters simultaneously. The studies performed for the individual clusters were done at 1000 K, for pressures of 1, 5, 10, 15, 20, and 25 atm, and at 10 atm for temperatures of 500, 700, 900, 1000, 1100, and 1300 K. When all the clusters were studied simultaneously, calculations were performed at 1000 and 1200 K, and at 10 and 20 atm. The simulations for the individual clusters consisted of 11 chemical eqs (22 rate constants), and 10 differential equations on each of the species: TMI, DMI, MMI, In, NS · TMI, NS · DMI, NS · MMI, NS · In, CH3, and C2H6, where NS represents the cluster considered. The substrate concentration (NS) was considered to be infinite. When solving for all the clusters simultaneously, the simulation consisted of 32 chemical eqs (64 rate constants), and 22 differential equations: four corresponding to the gaseous indium species, 16 corresponding to the four indium species adsorbed on the four clusters, and two for methyl and ethane. The simulations were geared to determine how coverage, pressure, and temperature affected reaction progression, and to allow for the different clusters to compete. CoWerage Effects. The effect of coverage was studied by looking at the reaction progression for the four clusters, separately, at a pressure of 10 atm and a temperature of 1000

Figure 9. Progression of percent concentrations, for the reactions on NS34, at 10 atm and 1000 K.

K. As an example, Figure 9 displays the log10 of the percent concentrations for the major species, as a function of log10 time (in seconds) for the NS34 cluster. Along the time progression, some concentrations increased, went through maxima and then decreased (e.g., NS34 · TMI), while others steadily increased (e.g., MMI). Special features of the results are described below, where percent concentrations are shown in parentheses. • Set of reactions with NS34. At 10-12 s, the major dissociation species were NS34 · TMI (10-4), followed by DMI (10-5) and MMI (10-5). After 10-7 s, MMI surpassed NS34 · TMI and after 10-4 s, it surpassed TMI. After 1 s, the concentrations were 99.86% MMI, 0.13% TMI, and 0.02% DMI. • Set of reactions with NS36. At 10-14 s, the major dissociation species were NS36 · TMI (10-3); DMI, NS36 · MMI, and NS36 · DMI (the three around 10-7); and NS36 · In (several orders of magnitude smaller, 10-12). At 10-10 s, both NS36 · TMI and NS36 · MMI surpassed TMI. At 10-9 s, NS36 · MMI surpassed NS36 · TMI. NS36 · In increased steadily, and at 10-8 s it surpassed TMI and NS36 · DMI; at 10-7, it also surpassed NS36 · TMI. After 5 × 10-3 s, the concentrations were 99.61% NS36 · MMI and 0.39% NS36 · In. • Reactions with NS38. At 10-14 s, the major dissociation species were DMI, NS38 · TMI, NS38 · DMI, and NS38 · MMI (all around 10-7), with NS38 · MMI being the largest, then NS38 · In (10-11) and MMI (10-12). NS38 · MMI was kept the lead throughout the progression. NS38 · In increased fast, and at 10-11 it surpassed NS · DMI and NS · TMI, and DMI at 10-8. At 10-5 s, NS38 · MMI and NS38 · In surpassed TMI. After 1 × 10-3 s, the concentrations were 92.48% NS38 · MMI and 7.52% NS38 · In.

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• Reactions with NS40. At 10-15 s, the major dissociation species were NS40 · TMI (10-3), NS40 · DMI (10-6), NS40 · MMI (10-7), DMI (10-8), and NS40 · In (10-14). At 10-11 s, NS40 · MMI surpassed NS40 · TMI, and at 10-10 s it surpasses TMI. At 10-9 s, NS40 · In surpassed DMI, NS40 · DMI, and NS40 · TMI. The final equilibrium concentrations were 99.9998% NS40 · MMI and 0.0002% NS40 · In. The computations seemed to indicate that when the nitrogen surface of InN was clean (the case of NS34), the reactions tended to equilibrate mainly at gaseous MMI, some undissociated TMI, and some DMI. As the nitrogen surface became partially occupied (the cases of NS36 and NS38), the products went mainly to adsorbed MMI and some adsorbed In. In the case of NS40, where the substrate became fully occupied except for the nitrogen atom where adsorption occurred, all indium species went to adsorbed MMI. Simultaneous Reactions with the Four Clusters. A crude simulation of site competition during vapor deposition (ignoring possible steric hindrance) was performed at 1000 K and 10 atm using the four clusters, simultaneously. The computation reached equilibrium after about 10 ms. At 10-14 s, the species with concentrations over 10-10 percent were NS40 · TMI and NS36 · TMI(10-3);NS40 · DMI(10-6);NS34 · TMIandNS40 · MMI (10-7); NS38 · TMI and DMI (10-8); NS36 · DMI, NS38 · DMI, and NS36 · MMI (10-9); and NS38 · MMI (10-10). Between 10-10 and 10-8 s, all of those species, except for NS40 · MMI and NS36 · MMI, stopped increasing and started to decrease. After 10-2 s, these two species had reached steady percent concentrations of 66 and 39%, respectively. NS36 · In, which initially was less than 10-15 percent, continuously increased to reach a steady percent concentration of 0.32% at 10-2 s. Competition among the gas phase and the four clusters favored the NS40 and NS36 species. When the reactions were run separately for each cluster, NS36 · MMI and NS40 · MMI were the major final products within their set. That was also the result of the run with all four clusters, simultaneously. When the NS36 cluster was run independently, NS36 · In was the second major product. In the complete set, this species became the third major product. Pressure and Temperature Effects for the Separate Clusters. The effects of pressure and temperature on the progression of the reactions were studied at a fixed temperature of 1000 K, and at a fixed pressure of 10 atm, respectively, using the four clusters separately. Several special features of the progressions were selected and monitored. The progression using the NS34 cluster was characterized by (a) the percent concentrations of TMI, DMI, and MMI (major products of the reactions) at 10 ms, (b) the magnitude and time of occurrence of a maximum on the NS34 · TMI percent concentration, and (c) the time of occurrence of the crossover between the percent concentration of MMI and NS34 · TMI. For the other three clusters, adsorbed MMI (NS · MMI) and adsorbed In (NS · In) were monitored, since these were the major products of reaction. In the case of NS36 and NS40, the amount of adsorbed TMI (NS · TMI) was also monitored by means of its maximum value. However, this species did not show a maximum on NS38, and, since NS38 · DMI did, adsorbed DMI (NS38 · DMI) was monitored instead. Table 9 summarizes the results. NS34 was a special case in which the major products of the reactions were gaseous species. For this cluster, increasing pressure increased the percent concentration of MMI at 10 ms, decreasing TMI concentration. Overall, increasing pressure slightly increased the amount of adsorbed TMI (NS34 · TMI),

Cardelino and Cardelino with its maximum occurring earlier, and its crossover with MMI occurring later. On the other hand, increasing temperature decreased the percent concentration of MMI at 10 ms, increasing the amount of undissociated TMI. The amount of adsorbed TMI (NS34 · TMI) decreased with increasing temperature, and its maximum and crossover with MMI occurred earlier. Pressure had a very small effect on the percent concentration of adsorbed MMI and In for NS36 at 0.01 ms (NS36 · MMI and NS36 · In, respectively). This cluster displayed the highest percent concentration of adsorbed TMI (NS36 · TMI). The maximum percent concentration of adsorbed TMI (NS36 · TMI) increased and occurred earlier with increasing pressure. On the other hand, increasing temperature had opposite effects at 0.01 ms on adsorbed MMI and In, reducing the first (NS36 · MMI) and increasing the latter (NS36 · In). Overall, temperature reduced the amount of adsorbed TMI (NS36 · TMI), and its maximum occurred earlier. Increasing either pressure or temperature made the crossover between adsorbed MMI and adsorbed TMI, as well as that between adsorbed In and adsorbed TMI, occur earlier. Increasing pressure had a variable effect on the percent of adsorbed MMI on NS38 (NS38 · MMI) at 0.05 ms. In particular, the lowest pressure of 1 atm favored products in the gas phase. On the other hand, increasing pressure increased the amount of adsorbed In (NS38 · In) at 0.05 ms (NS38 · In). With respect to the maximum of the NS38 · DMI percent concentration, it increased with increasing pressure and it occurred earlier. A dramatic effect with increasing temperature was detected at 0.05 ms on the percent concentrations of the two species monitored (NS38 · MMI and NS38 · In). The concentration of NS38 · MMI was substantially reduced while the NS38 · In concentration increased by the same amount. Between 1100 and 1200 K, and between 1200 and 1300 K, the changes in concentration were as large as 16%. The maximum of NS38 · DMI did not change consistently with increasing temperature. The crossover of concentrations between adsorbed TMI (NS38 · TMI) and adsorbed MMI (NS38 · MMI) and between adsorbed TMI (NS38 · TMI) and adsorbed In (NS38 · In) also tended to occur earlier with increasing pressure or temperature. At 0.05 ms, pressure had almost no effect on the percent concentrations monitored for NS40 (NS40 · MMI and NS40 · In). Pressure increased the small amount of adsorbed TMI (NS40 · TMI), and had almost no effect on the time the maximum occurred. Increasing temperature had no effect on the concentrations of adsorbed MMI (NS40 · MMI) and In (NS40 · In) at 0.05 ms, decreased the value of the maximum for adsorbed TMI (NS40 · TMI), and it was shifted earlier times. Increasing pressure delayed the crossover of concentrations between adsorbed TMI (NS40 · TMI) with adsorbed MMI (NS40 · MMI) and reduced the time for the crossover with adsorbed In (NS40 · In), whereas temperature had the opposite effect. In summary, when the substrate was clean (NS34), increasing pressure increased dissociation of TMI into MMI as well as adsorption of TMI (NS34 · TMI), whereas increasing temperature had the opposite effect. As the substrate became more occupied (NS36 and NS38), pressure had complex effects on the products, but increasing temperature substantially favored adsorbed In (NS36 · In and NS38 · In). With a saturated substrate (NS40), pressure and temperature had small effects. Pressure and Temperature Effects Using, Simultaneously, the Four Clusters. The main results of the four calculations performed using all the clusters simultaneously (at 10 and

Adsorption and Dissociation of TMI on an InN Substrate

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TABLE 9: Monitored Features during Reaction Progression with Different Pressures and Temperatures NS · TMI/MMI crossover

% concentration cluster

P (atm)

NS34 NS34 NS34 NS34 NS34 NS34 NS34 NS34 NS34 NS34 NS34

1 5 10 15 20 25 10 10 10 10 10

T (K)

time (s)

1000 1000 1000 1000 1000 1000 900 1000 1100 1200 1300

-2

1 × 10 1 × 10-2 1 × 10-2 1 × 10-2 1 × 10-2 1 × 10-2 1 × 10-2 1 × 10-2 1 × 10-2 1 × 10-2 1 × 10-2

TMI 5.861 0.948 0.211 0.136 0.129 0.129 1.383 0.211 1.053 2.992 3.863

% concentration cluster NS36 NS36 NS36 NS36 NS36 NS36 NS36 NS36 NS36 NS36 NS36

P (atm) 1 5 10 15 20 25 10 10 10 10 10

T (K) 1000 1000 1000 1000 1000 1000 900 1000 1100 1200 1300

time (s) -5

1 × 10 1 × 10-5 1 × 10-5 1 × 10-5 1 × 10-5 1 × 10-5 1 × 10-5 1 × 10-5 1 × 10-5 1 × 10-5 1 × 10-5

NS · MMI 99.919 99.917 99.920 99.913 99.924 99.914 99.981 99.920 99.762 99.463 99.338

DMI 0.026 0.030 0.021 0.020 0.023 0.026 0.106 0.021 0.027 0.043 0.079

% concentration

94.112 99.022 99.769 99.843 99.848 99.845 98.505 99.769 98.918 96.956 96.030

time (s)

time (s)

-8

2.83 × 10 6.37 × 10-8 1.13 × 10-7 1.69 × 10-7 2.48 × 10-7 3.23 × 10-7 7.39 × 10-6 1.24 × 10-7 1.02 × 10-8 1.37 × 10-9 3.91 × 10-10

8.35 × 10 6.38 × 10-8 6.17 × 10-8 5.98 × 10-8 5.94 × 10-8 5.72 × 10-8 3.49 × 10-7 6.17 × 10-8 1.02 × 10-8 1.50 × 10-9 3.48 × 10-10

NS · TMI/NS · In crossover

time (s)

time (s)

-9

1.53 × 10 1.88 × 10-9 1.61 × 10-9 1.53 × 10-9 1.45 × 10-9 1.39 × 10-9 1.32 × 10-8 1.61 × 10-9 3.16 × 10-10 6.37 × 10-11 2.07 × 10-11

% conc.

-8

NS · TMI/NS · MMI crossover

NS · In 0.081 0.083 0.080 0.087 0.076 0.086 0.019 0.080 0.238 0.537 0.662

MMI

NS · TMI maximum 0.163 0.965 1.972 2.961 3.930 4.883 19.395 1.972 0.156 0.017 0.003

NS · TMI maximum time (s)

-8

% conc.

-9

9.63 × 10 6.39 × 10-8 6.06 × 10-8 5.93 × 10-8 5.91 × 10-8 5.82 × 10-8 6.17 × 10-7 6.06 × 10-8 1.26 × 10-8 7.23 × 10-9 8.46 × 10-9

3.39 × 10 9.99 × 10-10 5.88 × 10-10 4.46 × 10-10 3.57 × 10-10 2.90 × 10-10 7.92 × 10-10 5.88 × 10-10 4.24 × 10-10 2.42 × 10-10 1.46 × 10-10

NS · TMI/NS · DMI crossover NS · TMI/NS · In crossover

26.45 52.15 62.65 68.20 71.84 74.44 86.45 62.65 29.24 7.74 2.14

NS · DMI maximum

cluster P (atm) T (K)

time (s)

NS · MMI

NS · In

time (s)

time (s)

time (s)

% conc.

NS38 NS38 NS38 NS38 NS38 NS38 NS38 NS38 NS38 NS38 NS38

5 × 10-5 5 × 10-5 5 × 10-5 5 × 10-5 5 × 10-5 5 × 10-5 5 × 10-5 5 × 10-5 5 × 10-5 5 × 10-5 5 × 10-5

47.403 94.569 98.504 98.133 97.836 97.673 99.618 98.504 93.191 76.917 60.878

0.169 0.491 1.227 1.850 2.163 2.327 0.376 1.227 6.804 23.082 39.121

1.21 × 10-9 4.40 × 10-10 2.62 × 10-10 1.71 × 10-10 1.39 × 10-10 1.07 × 10-10 2.87 × 10-10 2.62 × 10-10 1.00 × 10-13 5.16 × 10-14 1.18 × 10-12

2.27 × 10-11 6.77 × 10-12 8.02 × 10-12 1.72 × 10-11 1.46 × 10-11 1.29 × 10-11 9.55 × 10-10 8.02 × 10-12 5.74 × 10-12 6.70 × 10-13 1.18 × 10-12

8.80 × 10-5 2.08 × 10-5 1.05 × 10-5 6.65 × 10-6 5.51 × 10-6 3.80 × 10-6 5.68 × 10-6 1.05 × 10-5 9.92 × 10-6 5.11 × 10-6 2.48 × 10-6

1.19 × 10-3 2.46 × 10-3 3.42 × 10-3 4.17 × 10-3 4.79 × 10-3 5.36 × 10-3 7.36 × 10-3 3.42 × 10-3 2.46 × 10-3 2.45 × 10-3 3.03 × 10-3

1 5 10 15 20 25 10 10 10 10 10

1000 1000 1000 1000 1000 1000 900 1000 1100 1200 1300

% concentration cluster NS40 NS40 NS40 NS40 NS40 NS40 NS40 NS40 NS40 NS40 NS40

P (atm) 1 5 10 15 20 25 10 10 10 10 10

T (K) 1000 1000 1000 1000 1000 1000 900 1000 1100 1200 1300

time (s) -5

5 × 10 5 × 10-5 5 × 10-5 5 × 10-5 5 × 10-5 5 × 10-5 5 × 10-5 5 × 10-5 5 × 10-5 5 × 10-5 5 × 10-5

NS · MMI 99.9994 99.9997 99.9998 99.9998 99.9999 99.9999 99.9999 99.9998 99.9996 99.9995 99.9993

NS · TMI/NS · MMI crossover

NS · TMI/NS · In crossover

time (s)

time (s)

NS · In 0.0006 0.0003 0.0002 0.0002 0.0002 0.0001 0.0001 0.0002 0.0004 0.0005 0.0007

-12

3.97 × 10 6.60 × 10-12 1.34 × 10-11 1.54 × 10-11 1.86 × 10-11 1.89 × 10-11 7.92 × 10-11 1.34 × 10-11 1.42 × 10-12 4.63 × 10-13 1.92 × 10-13

20 atm, and at 1000 and 1200 K) are shown in Table 10. In all cases, the four major initial species (NS40 · TMI, NS36 · TMI, NS40 · DMI, and NS34 · TMI) were the same. Similarly, the four major final species (NS40 · MMI, NS36 · MMI, NS36 · In, and NS38 · MMI) were the same. The results can be summarized as follows: (a) Increasing temperature increased the percent concentration of adsorbed MMI on NS40 but decreased it on NS36; this effect was more pronounced at 10 atm than at 20 atm. (b) Increasing temperature decreased adsorbed In at 10 atm, but slightly increased it at 20 atm. (c) Overall, increasing the temperature

-8

1.77 × 10 4.37 × 10-9 2.23 × 10-9 1.56 × 10-9 1.15 × 10-9 8.59 × 10-10 1.82 × 10-9 2.23 × 10-9 2.24 × 10-9 2.55 × 10-9 3.09 × 10-9

NS · TMI maximum time (s) -10

1.44 × 10 3.99 × 10-11 3.36 × 10-11 3.12 × 10-11 3.10 × 10-11 2.68 × 10-11 7.92 × 10-11 3.36 × 10-11 1.80 × 10-11 1.71 × 10-11 2.93 × 10-11

% conc. 0.29 2.39 5.46 8.52 11.41 14.18 27.16 5.46 0.99 0.20 0.05

from 1000 to 1200 K at 10 atm increased adsorbed MMI by 0.16% and decreased adsorbed In by the same amount. (d) Overall, increasing the temperature from 1000 to 1200 K at 20 atm decreased adsorbed MMI by 0.03% and increased adsorbed In by the same amount. (e) Increasing pressure at 1000 K increased adsorbed MMI on NS40 but decreased it on NS36. (f) Increasing pressure at 1200 K decreased adsorbed MMI on NS40 but increased it on NS36. (g) Increasing pressure, at both temperatures decreased adsorbed In. (h) Overall, increasing pressure from 10 to 20 atm, at 1000 K, increased adsorbed MMI by 0.24% and decreased

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TABLE 10: Initial and Final Percent Concentrations for the Four Major Species Obtained from Calculations Using the Four Model Clusters Simultaneously Initial Values pressure (atm) temperature (K) time (s) species NS40 · TMI NS36 · TMI NS40 · DMI NS34 · TMI

10 1000 2 × 10-15

10 1200 9 × 10-16

20 1000 1 × 10-15

20 1200 6 × 10-16

1.39 × 10-3 8.08 × 10-4 9.64 × 10-7 2.86 × 10-7

3.37 × 10-4 2.18 × 10-4 1.02 × 10-6 2.06 × 10-7

1.97 × 10-3 1.14 × 10-3 9.81 × 10-7 4.03 × 10-7

4.77 × 10-4 3.09 × 10-4 1.03 × 10-6 1.47 × 10-7

Final Values pressure (atm) temperature (K) time (s) species NS40 · MMI NS36 · MMI NS36 · In NS38 · MMI total adsorbed MMI total adsorbed In

10 1000 1 × 10-2

10 1200 1 × 10-4

20 1000 1 × 10-2

20 1200 1 × 10-4

62.4257 37.2702 0.3029 0.0005 99.70 0.30

72.1284 27.7311 0.1395 0.0004 99.86 0.14

65.3849 34.5526 0.0614 0.0009 99.94 0.06

71.8751 28.0336 0.0905 0.0005 99.91 0.09

adsorbed In by the same amount. (i) Overall, increasing pressure from 10 to 20 atm, at 1200 K, increased adsorbed MMI by 0.05% and decreased adsorbed In by the same amount. A detailed time-dependent description of the reaction progression is of particular interest for high-pressure InN epitaxy. In HPCVD, the source materials (TMI and NH3) are delivered in separate pulses into the reactor, and separated by pulses of carrier gas. The simulations performed in this investigation indicate that adsorbed MMI is the main indium species to be encountered by the NH3 pulse. Summary and Conclusions This investigation describes the simulation of adsorption and methyl dissociation of the source material TMI on an InN substrate. This process constitutes an essential part of InN epitaxy by HPCVD, using pulsed TMI and ammonia (NH3), as source materials. The InN substrate was modeled using four clusters containing three indium atoms, nine nitrogen atoms, as well as 21, 23, 25, and 27 hydrogen atoms for truncation purposes. The smallest cluster represented a clean nitrogen surface of an InN substrate, with seven nitrogen atoms each containing a dangling bond. The largest cluster simulated a saturated surface, with only a dangling bond on the central nitrogen where the indium species became adsorbed. The other two intermediate clusters had either three or five dangling bonds on the nitrogen surface of the InN substrate, to represent different degrees of surface coverage. TMI and its dissociated species (DMI, MMI, and In) were adsorbed to the central nitrogen atom of the model clusters. The computations were performed with the hybrid program ONIOM,46 using two-level approximations: DFT (B3LYP41) for the adsorbate, and semiempirical molecular orbital calculations (PM345) for the substrate. The quality of the simulation of the InN substrate was evaluated by comparing In-N distances and heat capacities of the clusters with experimental values. The In-N distances adjacent to the adsorption site were found to be around 4% smaller than the average In-N distance in the crystal. The heat capacities of the clusters in increasing size, at 700 K, were 17%, 3.5%, 2.4%, and 9.2% higher than the experimental value, respectively. Larger differences were expected for the clusters with more dangling bonds or more hydrogen atoms.

The calculations allowed for energy partition in terms of the adsorbate and the substrate. Overall, adsorptions were exothermic processes, with the adsorbates undergoing small increases in energy that were surpassed by a decrease in energy of the substrate. TMI was the adsorbate that suffered the largest energy increase, while In was the adsorbate that, in general, caused the largest energy drop on the surface. Since all structures were optimized by energy minimization, normal vibrational analyses were performed. The results from the molecular-orbital calculations and the vibrational analyses were utilized in the statistical-mechanical estimation of the Gibbs free energy for the reactions, and in the semiclassical determination of the reaction rate constants for the adsorption and methyl-dissociation reactions. The energy partition, together with a classification of the vibrational modes, allowed for a different treatment of the adsorbate and substrate portions, where the substrate portions were computed using Debye’s approximations for solid state. In general, adsorption of the indium species onto the InN clusters reduced both reaction energies as well as activation energies for methyl dissociation. On average, adsorption reduced the methyl-dissociation energies for TMI, DMI, and MMI by 100, 25, and 4 kJ mol-1, respectively. The average activation energies for TMI and MMI were reduced by 125 and 60 kJ mol-1, respectively. The activation energies for DMI were close to 0 both in the gas phase and adsorbed. Systems of coupled differential equations representing the rate equations were solved to provide detailed descriptions of the reaction progression for the methyl dissociation of TMI adsorbed to the InN substrate. The simulations showed that, when the nitrogen surface of InN was clean, the reactions favored production of gaseous MMI, but as the nitrogen surface was partially occupied, the main products became adsorbed MMI and some adsorbed In. For a clean surface, increasing pressure favored dissociation of TMI into MMI and adsorption of TMI, whereas temperature had an opposite effect. For partially occupied surfaces, increasing temperature favored adsorbed In. For the most part, increasing pressure increased adsorbed MMI and decreased adsorbed In; increasing temperature increased adsorbed MMI and decreased adsorbed In at 10 atm, but had an opposite effect at 20 atm. Thus, these computations provided useful information on how reaction progression was affected by coverage, pressure, and temperature.

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