1. Introduction
Group III-nitrides are materials of great importance suitable for fabrication of green, blue and ultraviolet light emitting diodes (LED), laser diodes Reference Nakamura and Fasol[1], and various electronic devices Reference Shur[2]. Progress in these fields are closely related to obtaining epitaxial layers and heterostructures of device quality. The growth techniques widely employed for this purposes are Molecular Beam Epitaxy (MBE) and Metal Organic Vapor Phase Epitaxy (MOVPE) Reference Ambacher[3] (the latter is the basic growth technique for large-scale production of blue LEDs).
Despite considerable empirical knowledge accumulated by many research groups, clear understanding of the relative role of different physical mechanisms governing the growth process is not yet reached. This may be related to complex surface kinetics occurring in the case of group-III nitrides that differentiates these materials from other III-V compounds.
It was found in early studies that thermal decomposition of GaN is of strong non-equilibrium character Reference Munir and Searcy[4], Reference Schoonmaker, Buhl and Lemley[5]. Extremely high thermal stability of group-III nitrides Reference Munir and Searcy[4] and catalytic effect of the liquid phase on GaN vaporization rate Reference Schoonmaker, Buhl and Lemley[5] were evidences that kinetic effects play a significant role in the processes on the surfaces of group-III nitrides.
The most promising results in the field of group-III nitride growth either by MOVPE or by GSMBE were achieved using NH3 as the nitrogen precursors. In spite of the fact that efficiency of NH3 is several orders of magnitude greater than that of N2 , it is still low and, therefore, requires using high V/III ratios. The recent studies Reference Held, Crawford, Johnston, Dabiran and Cohen[6], Reference Mesrine, Grandjean and Massies[7] carried out under MBE conditions show that the ammonia reaction efficiency is about several per cent under typical growth temperatures (in particular, ~ 4% at 870 °C Reference Mesrine, Grandjean and Massies[7] ).
Thermodynamic analysis of various nitrogen precursors used for GaN growth by MOVPE was reported in paper Reference Beaumont, Gibart and Faurie[8]. The calculations show that ammonia should almost completely dissociate into inactive N2 and H2 at temperatures as high as ~400°C. In this case growth of GaN would not be favored due to the lack of reactive nitrogen precursor. This fact contradicts experimental observations. This means that the pure thermodynamic consideration is not applicable to analysis of the real growth situation.
Attempts to take into account the non-equilibrium nature of the surface processes were made in a few works Reference Koukitu and Seki[9], Reference Koukitu, Takahashi, Taki and Seki[10], Reference Lee, Yuri, Uedo and Harris[11]. Koukitu and Seki Reference Koukitu and Seki[9] calculated a dependence of the GaN growth rate on temperature that correlated with the experimental data on MBE growth of GaN. However, their results were obtained by excluding from consideration the dominant pathway of nitrogen desorption from the surface – associative desorption of N2 (actually in ref. Reference Koukitu and Seki[9], it was assumed that only atomic nitrogen can desorb from the surface of GaN). Obviously, this assumption disagrees with the numerous data of mass-spectrometric measurements (see, for example, Reference Munir and Searcy[4]).
A modification of the thermodynamic approach was proposed to study growth of InGaN ternary compounds by MOVPE Reference Koukitu, Takahashi, Taki and Seki[10] and GaN by Hydride Vapor Phase Epitaxy (HVPE) Reference Lee, Yuri, Uedo and Harris[11]. To take into account the non-equilibrium nature of the ammonia decomposition on the growth surface an adjustable parameter, X, was introduced as the stoichiometric coefficient into the heterogeneous cracking reaction of NH3 :
Then a fitting procedure was applied to determine the value of X . Obviously, such an approach can be valid only for a narrow range of process parameters, because it is not related to any realistic physical base. In addition it does not allow one to get insight into mechanisms underlying the growth process of group-III nitrides.
An approach allowing one to account for the surface kinetics related to group III-nitrides was developed in our previous studies and applied to modeling of GaN growth by MBE Reference Averyanova, Karpov, Makarov, Przhevalskii, Ramm and Talalaev[12] Reference Karpov, Makarov and Ramm[13]. The approach is based on introducing a temperature dependent sticking/evaporation coefficient of molecular nitrogen on group-III nitride surfaces into the quasi-equilibrium model of the MBE growth process. The values of the sticking coefficients were extracted from independent experimental data on Langmuir evaporation of binary nitrides in vacuum. Such a way allowed us to get adequate understanding of the main growth trends reported in the literature. But generalization of this approach to the case of MOVPE growth appears to be non-trivial procedure.
In this paper we report on the novel quasi-equilibrium model of MOVPE growth of binary and ternary group III-nitrides where the non-equilibrium effects discussed above are taken into account. It should be noted that the proposed approach has a general character which makes the model applicable to description of the growth processes where kinetic limitations are important at the stage of adsorption/desorption of the gas phase species at the surface of the growing crystal.
2. Model of group III-nitride growth by MOVPE
Modeling of an MOVPE process requires description of various physical phenomena such as gas mixture flow , heat transfer and species mass transport from the reactor inlet to the substrate. The mathematical models of these processes have been developed for many years and are widely described in the literature (see, for instance, Reference Kleijn and Meyyappan[14]). In this paper we focus on the interaction of a gas-phase species with the growing surface of the crystal. As discussed in 1 this topic is a bottleneck in understanding of specific features of the growth process.
Our approach is based on two main assumptions:
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A. The atoms in the adsorption layer are nearly in thermodynamic equilibrium with the crystal. This means that the rate of atomic incorporation from the adsorption layer into the crystal (occurring at the step edges) and the rate of the reverse process – crystal decomposition (with atomic release back into the adsorption layer) are much higher than their net difference equal to the growth rate of the crystal.
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B. Kinetic effects occurring at the stage of adsorption/desorption can be accounted for by introducing the sticking/evaporation coefficients of the individual species. Generally, these coefficients are dependent on temperature and partial pressures of the species.
Below the formulation of the model is given separately for binary and ternary compounds.
2.1 Binary compounds
Let us consider N c gaseous species, named as S i , taking part in the growth process and transportation in the gas phase of N e elemental components. Let N r independent heterogeneous chemical reactions occur on the growth surface (generally, N r =N c −N e ). These reactions can be written in a symbolic form
where ν ji is the matrix of stoichiometric coefficients (positive for the reagents and negative for the products of reactions). The summation in Equation (1) is extended to N c +1 reactants since formation of the solid nitride compound (i = N c +1) has to be accounted for in the chemical reactions as well.
We assume that the adsorption/desorption process can be described by using the Hertz-Knudsen equations, i.e. that the net adsorption rate of the i-th species at the surface is expressed as:
Here α i is the sticking/evaporation coefficient accounting for kinetic limitations in the adsorption/desorption process, β i = (2πm i kT)−1/2 is the Hertz-Knudsen factor for the i-th species (m i is the mass of the molecule, k is the Boltzmann constant), p i is the partial pressure of the i-th species near the growing surface,
Strictly speaking, use of Equation (2) is well justified for the case of single-component crystals and a single-component gas phase. In this case,
Taking into account assumption A, the following mass action law equations could be used if all the species are considered as the ideal gases
Here K j is the equilibrium constant of the j-th heterogeneous reaction which is calculated using standard Gibbs energies
with R being the universal gas constant.
We assume also that growth of nitride compound occurs stoichiometrically. This requires the species fluxes to met certain mass balance equations which have the following general form
Here f ki is the matrix showing how many atoms of the k-th elemental component is contained in the molecule of the i-th species, x k is the vector showing how many atoms of the k-th elemental component is contained in the solid phase, Ω s = M s /ρ s N A is the molecular volume of the crystal, M s is the molar mass, ρ s is the density of the crystal, N A is the Avogadro number, and V g is the growth rate of the crystal.
In accordance with assumption B the Equation (2)-Equation (5) should be added by the relationships between the sticking coefficients α i , growth temperature T and the partial pressures p i . In principle such relationships could be derived from the detailed kinetic model of the surface processes. However, at the moment there is a lack of experimental information needed for development of such a model. Therefore, in this paper the only temperature dependent sticking coefficients are used either extracted from independent experiments on free evaporation of binary nitrides in vacuum are used Reference Averyanova, Przhevalsky, Karpov, Makarov, Ramm and Talalaev[15], or directly measured Reference Mesrine, Grandjean and Massies[7].
Combining Equation (2)- Equation (5) we get the equation array which can be considered as the boundary conditions relating the species fluxes to their concentrations at the growth surface.
2.2 Ternary compounds
A ternary compound
Here W is the interaction energy of binary constituents in the solid phase.
In the case of the ternary compound one should consider N r =N c −N e +2 heterogeneous chemical reactions
with two solid products (i = N c +1, N c +2) being the binary constituents A3N and B3N respectively. Then Equation (8) should be substituted by the following equation
Here
and the activity coefficients γ 1 and γ 2 are determined from the equations
As previously, Equation.(9)-Equation (11) should be added by the mass balance equations
where x k l are vectors (l = 1,2) showing how many atoms of the k-th elemental component is contained in the binary compounds A3N (l = 1) and B3N (l = 2). In addition, one has to use the equation
accounting for normalization of the sum of the concentrations of binary constituents in the ternary compound. And, in accordance to the Vegard law the molecular volume of the solid phase can be calculated as
where Ω1 and Ω2 are the molecular volumes of the binary constituents – A3N and B3N respectively.
3. Analysis of MOVPE growth of GaN and InGaN
To demonstrate the workability of the model we apply it to analysis of growth of GaN and InGaN in a horizontal tube MOVPE reactor. Since the main goal of the work is evaluation of the role of non-equilibrium kinetics in the growth process, a simplified treatment of mass transport in the reactor is used below. In order to relate the partial pressures of species at the inlet of reactor p in and at the growth surface p i to the flux of the i-th species at the surface J i , theso-called “diffusive conductivity” B i is introduced by the following way:
Obviously, B i as well as p i is dependent on substrate position, reactor geometry, diffusion coefficients of the species, gas flow velocity etc. According to the analytical model of GaAs growth in a horizontal tube reactor, proposed in Reference van Sark, Janssen, de Croon and Giling[17]
where D i is the diffusion coefficient of the i-th species, h is the height of the reactor, z is the local coordinate along the susceptor measured from the beginning of the deposition zone, V F is the average gas flow velocity.
Combining Equations (2) and (16) the values of the partial pressures near the growth surface p i as well as the flux of the i-th species at the surface J i can be derived as a functions of the inlet pressures
Combining Equation (17) with Equation (2)-Equation (5) in case of binary nitride and with Equation (9)-Equation (14) in case of ternary nitride, a closed set of nonlinear equations is obtained in respect to equilibrium partial pressures
In our analysis we use the well-known Wilke approximation (see, for instance, Reference Kleijn and Meyyappan[14] ) to calculate the diffusion coefficients of gaseous species D i . Standard Gibbs energies
First, the proposed model is used to analyze the GaN growth by MOVPE in horizontal tube reactor. The process parameters are taken from Ref. Reference Chen, Liu, Steigerwald, Imler, Kuo, Craford, Ludowise, Lester and Amano[19]. The following gas phase species are considered: trimethylgallium (TMGa) and ammonia (NH3) as the gallium and nitrogen precursors respectively, H2 as the carrier gas, monomethylgallium (MMGa) and CH4 as the products of TMGa gas-phase decomposition, N2, Ga and GaH as the products of surface reactions.
In order to relate the equilibrium pressures
The possibility of adduct formation due to pre-reaction between group-III metalorganic precursors and ammonia is discussed in several papers Reference Chen, Liu, Steigerwald, Imler, Kuo, Craford, Ludowise, Lester and Amano[19] Reference Thon and Kuech[20]. This reaction leading to depletion of group-III precursors is generally assumed to be important during MOVPE of AlN and AlGaN but its relative contribution into the GaN and InGaN growth process is not clearly understood now. Moreover there is lack of reliable data on adduct formation kinetics as well as on the nature of adduct-derived species. That is why we does not include adduct formation into our model. In general this reaction should not be neglected but at present stage of our study we do not take it into account due to the above reasons.
Figure 1 shows the GaN growth rate as a function of growth temperature compared to the experimental data taken from Reference Chen, Liu, Steigerwald, Imler, Kuo, Craford, Ludowise, Lester and Amano[19]. One can see that in the wide temperature range (up to ~1100°C) growth rate is nearly independent of temperature and growth of GaN is limited by mass transport of gallium-containing species to the growing surface. At higher temperatures growth rate sharply decreases due to increase of gallium desorption from the surface. One can see that starting from ~700°C the calculated and measured values are in a good agreement. The low-temperature decrease of the GaN growth rate observed in experiment is not reproduced by our model. Under these conditions other mechanisms play important role, and more detailed theoretical consideration of the surface processes is required to describe the growth rate suppression at T < 700°C.
GaN growth rate as a function of substrate temperature. Solid line - modeling results, circles - experimental data from Reference Chen, Liu, Steigerwald, Imler, Kuo, Craford, Ludowise, Lester and Amano[19].
The dependence of the species partial pressures p i at the surface on the temperature is shown on Figure 2. One can see that at elevated temperatures a remarkable amount of N2 is generated both due to GaN evaporation and ammonia decomposition at the surface. It should be noted that the equilibrium pressure of nitrogen
Partial pressures of species near the/ growing surface as a function of substrate temperature. Dashed line - thermodynamic pressure of ammonia.
Modeling of InGaN growth by MOVPE is based on the experimental data taken from Ref. Reference Talalaev, Yakovlev, Karpov, Makarov, Schoen, Heuken, Strauch and Juergensen[22]. The following set of the gas phase species is taken into account: trimethylindium (TMIn), triethylgallium (TEGa) and ammonia (NH3) as the indium, gallium and nitrogen precursors respectively, N2 as the carrier gas, In ,Ga, CH4 and C2H5 as the products of group-III precursor pyrolysis, H2, InH, GaH as the products of surface reactions. In Reference Talalaev, Yakovlev, Karpov, Makarov, Schoen, Heuken, Strauch and Juergensen[22] the horizontal tube AIX-200 reactor with the so called “hot-zone mixing” is used where the gallium and nitrogen precursors are mixed near the substrate. The gas temperature is there sufficiently high, and adduct formation seems not to occur.
We consider the following independent heterogeneous reactions occurring at the surface while InGaN growing:
The interaction energy of binary constituents is taken after Reference Karpov, Makarov and Ramm[23] equal to 25700 J/mole.
The InN composition in the InGaN ternary compound as a function of substrate temperature under different TMIn/(TMIn+TEGa) input ratios is shown in Figure 3. At low temperatures efficiencies of indium and gallium incorporation into the epitaxial layer are equal, and the InN composition is nearly the same as in the gas phase. But at higher temperatures the indium desorption rate becomes larger than that of gallium and the InN composition decreases. Such decrease was observed in experiment Reference Talalaev, Yakovlev, Karpov, Makarov, Schoen, Heuken, Strauch and Juergensen[22]. Taking into account that no fitting procedure was used in our model, the agreement between experimental data and theoretical predictions looks quite reasonably.
Indium content in InGaN layer as a function of substrate temperature. Violet(dashed) line corresponds to TMIn/(TMIn+TEGa) ratio = 0.3, red line corresponds to TMIn/(TMIn+TEGa) ratio = 0.2, green (dashed-dotted ) line corresponds to TMIn/(TMIn+TEGa) ratio = 0.1. Points are the experimental data from Reference Talalaev, Yakovlev, Karpov, Makarov, Schoen, Heuken, Strauch and Juergensen[22] (TMIn/(TMIn+TEGa) ratio = 0.2).
4. Conclusion
A novel theoretical approach to modeling of growth of III-group nitrides by MOVPE is proposed accounting for specific kinetics of nitrogen and ammonia interaction with the growth surface. The approach was applied to analysis of growth of GaN and InGaN compounds in a horizontal tube reactor. Good agreement between theoretically predicted and experimentally measured GaN growth rate and indium composition in InGaN layers is obtained without any fitting procedure.
