Thermodynamics

The purpose of this chapter is to develop the basic thermodynamic insights that can be used to obtain an understanding of epitaxial growth. Thermodynamics un- derUes the epitaxial growth process for all techniques, including OMVPE, since epitaxial growth is simply a highly controlled phase transition. Thus, thermody­ namics completely determines the driving force and, hence, the maximum growth rate for all growth processes. In addition, a thermodynamic understanding of epi­ taxy allows, for many growth conditions, the determination of alloy composition as well as the solid stoichiometry. Even though the nonstoichiometry of III/V semiconductors is small, it controls the concentrations of native defects such as vacancies, interstitials, and antisite defects. This, in turn, affects the incorporation of impurities, both those intentionally added to the system as dopants as well as contaminants. It also partially determines the diffusion coefficients of both major constituents and dopants. The thermodynamics of mixing of semiconductor alloys (III/V, 11/VI, and IV/IV) determines many characteristics of the growth process as well as the prop­ erties of the resultant materials. For example, thermodynamic factors may limit the mutual solubility of the two (or more) components of an alloy. Semiconductor alloys are frequently thought of, especially by device designers, as being com­ pletely miscible (i.e., alloys of any arbitrary composition are assumed to be stable). However, when the sizes of the constituent atoms are sufficiently different, miscibility gaps exist. In addition to solid-phase immiscibility in important alloys

17 18 2 Thermodynamics systems such as GalnAsP and GaInN, this size difference also leads to micro­ scopic structures far different than the random, totally disordered state normally expected for alloys. For example, natural superlattices, with a monolayer period­ icity along a particular crystallographic direction in the lattice, are produced in many III/V alloys during epitaxial growth. Interest in the formation of these or­ dered structures is more than academic, since the band-gap energy, phonon prop­ erties, surface states, and free carrier mobilities may be significantly different for the disordered and ordered phases. Thus, both miscibility gaps and deviations from a random distribution of the atoms constituting the lattice affect the electrical and optical properties of semiconductor alloys in ways that are extremely impor­ tant for many types of devices. In other words, specification of the composition of an alloy does not completely determine the properties. The microscopic arrange­ ment of the atoms must also be considered. In addition to the traditional treatment of the thermodynamics of the bulk semi­ conductors, it has become abundantly clear in recent years that the thermodynam­ ics of the surface must also be considered in any effort to understand the growth processes as well as the characteristics of the materials produced epitaxially. For this reason, this chapter will also consider the thermodynamics of the sur­ face, including the bond rearrangement driven by the reduction in surface energy that leads to the various surface reconstructions. These reconstructed surfaces are observed not only for the static surface but also during vapor-phase epitaxial growth. Evidence for surface reconstruction was first observed using reflection high-energy electron diffraction (RHEED) during MBE growth; reconstruction has now also been observed, using both X-ray scattering and optical techniques, during OMVPE growth.

2.1 Basic Thermodynamics of Phase Equilibrium 2.1.1 Equilibrium Conditions We shall begin this chapter with a brief review of the basic concepts and termi­ nology of thermodynamics. An excellent development of the solution thermo­ dynamic concepts treated here may be found in Swalin [1]. A phase is a region (usually homogeneous) that is physically distinguishable and distinct from other phases. Thus, during OMVPE growth, we have the vapor and various condensed phases, including the solid phase being grown. The basic goal of thermodynamics, as applied to epitaxy, is to define the relationship between the compositions of the various phases in an equilibrium system at constant temperature and pressure. Equilibrium is defined as the state where the Gibbs free energy per mole, G, is a minimum. The Gibbs free energy is defined in terms of the enthalpy, //, and 2.1 Basic Thermodynamics of Phase Equilibrium 19 entropy, S: G = H - TS, (2.1) where H = E ^ PV, (2.2) E is the internal energy, V is the volume, and P is the pressure of the system. G, H, S, E, and V are all extensive quantities—that is, they depend on the size of the system. For convenience, they are expressed on a per-mole basis. For a two- phase, a and yS, system, the total free energy is G' = G'^ ^- G'^^, where the prime denotes the total free energy as opposed to the free energy per mole. Since G' is a minimum at equilibrium, the change in G' by moving an infinitesimally small number of moles of component /, dn-, between the two phases causes no change in G! Expressed mathematically, this is written f) -if)' '0 The partial derivative of G' with respect to n- is such an important quantity for thermodynamic calculations that it is given a name, the chemical potential, rep­ resented as /JL-. Thus, the equilibrium condition may be expressed

fjLf = fjiff (2.4) for each component in the system. For a reversible perturbation of the system, it can be shown from Equations (2.1) and (2.2) plus the relationship dE' = TdS' - PdV ih^i dG' = V dP - S' dT, (2.5) one of Maxwell's equations of thermodynamics. For an ideal gas {PV = nRT) at constant T, Equation (2.5) yields for a change in pressure dG' = nRT d InP. (2.6) Hence, for a single ideal gas ^ = RT InP (2.7) and

M = M° + RT In^, (2.8) where />t° and P° represent the chemical potential and pressure of an arbitrary standard state. For an ideal gas mixture,

ytt, = M? + /?r In^, (2.9) Pi 20 2 Thermodynamics where p- is the partial pressure, equal to the mole fraction x^ multiplied by P, and the standard state is usually pure component L For an ideal liquid or solid solution, the same expression holds withp-//?-^ re­ placed by x-/x^. However, the standard state is pure /, so x^ = 1. The form of Equation (2.9) is so useful that it is retained even for a nonideal solution with x- replaced by the activity a-, which may also be considered a product of x^ and a nonideality factor y-, the activity coefficient: /x.-/xf-H/^rina^ (2.10a) fM^ = fM^^^RT\n(x^y.y (2.10b)

2.1.2 Solution Thermodynamics Thermodynamics can be used to describe the driving force for epitaxy and to specify the maximum growth rate, as will be discussed in Section 2.3. In addition, thermodynamic calculations often give an accurate indication of the composition of multicomponent solids grown by OMVPE, as will be described in Section 2.4. The vapor phase is commonly considered to be ideal (i.e., JC- = Pi/p^-), since the source molecules are typically highly diluted in the carrier gas. However, calcu­ lations involving the liquid and/or solid phases must deal with their nonideality.

2.1.2.1 Regular Solution Model The simplest model that can be used to describe the free energy of mixing of semiconductor liquid and solid solutions is the regular solution model. The term regular solution was first used by Hildebrand to describe a class of solutions that are nonideal but consist of a random arrangement of the constituents [2]. The term has since come to designate a more restricted, semiquantitative model for the cal­ culation of the free energy of mixing of multicomponent systems. Two additional assumptions are (1) interactions between the constituent atoms occur only pair- wise—that is, only between nearest neighbor pairs, and (2) the atoms reside on a lattice with each atom surrounded by Z neighbors. For a solution consisting of only A and C atoms, the nearest-neighbor bond energies are designated 7/^^, Hj^^, and H^^^. The bond energies are commonly thought of as being the sum of ''chemical" energies, frequently related to charge transfer due to differences in electronegativity, and "strain" energies related to distortions in the lattice due to differences in the sizes of the constituent atoms. Using these assumptions, it is possible to express the entropy and enthalpy of mixing of a binary solution A + C in simple terms. The entropy of mixing is simply the ideal configurational entropy of mixing: ^S^= -R[x\nx-h{l -jc)ln(l - x)]. (2.11) 2.1 Basic Thermodynamics of Phase Equilibrium 21

The enthalpy of mixing is obtained by summing nearest-neighbor bond energies ^H^ = x(l -jc)a, (2.12) where the interaction parameter, fl, is

a = ZA^n^AC - K^AA + ^cc)]- (2.13) N^ is Avogadro's number. For phase diagram calculations, the free energy of mix­ ing, obtained from Equations (2.11) and (2.12), plus the relation AGM = Af/M- rA5^, (2.14) is the most significant quantity. The activity coefficients in the solution A -h C are obtained from the expressions

RT\ny^ = -- (2.15) dn- and

1 (1--^/)-^ ,.,., lnr/ = — • (2-16) The regular solution model is not expected to provide significant physical in­ sight into the thermodynamics of mixing of semiconductor solutions. However, treated as a purely empirical model, it is useful for the interpolation and extrapo­ lation of phase diagram information in systems for which some experimental data are available. The interaction parameters can be obtained only by fitting the model to experimental free-energy or activity coefficient data [3] or by application of the model to the calculation of a phase diagram with the interaction parameter(s) ad­ justed to provide the best fit to the experimental data [4, 5]. The regular solution model can be used to calculate phase diagrams in binary, ternary, and quaternary semiconductor systems, as will be discussed in Section 2.2. For quaternary alloys of the type Ai_^B^Ci_ D^, the regular solution model must be modified to in­ clude both first and second nearest-neighbor interactions [6, 7]. It is frequently desirable to have a predictive model allowing the calculation of the thermodynamic properties of semiconductor alloys even in systems for which no experimental data exist. This is a difficult theoretical problem since the co­ hesive energies are much larger than mixing enthalpies, which necessitates that the calculated cohesive energies be extremely accurate. Efforts to give physical significance to the regular solution interaction parameters in solutions of non- electrolytes (i.e., those with no coulombic contributions to the bond energies) began with Hildebrand [8], who defined the interactions in terms of solubility 22 2 Thermodynamics parameters meant to represent the AA and CC bond energies, in an A-C solution, in terms of the energies of vaporization of the pure components. A similar model was applied to III/V liquid solutions with the addition of chemical terms related to the Pauling electronegativity difference between the constituents [9]. In the liquid state, the bonding is nearly metallic; thus, the structure is close packed, and the number of nearest neighbors was taken to be twelve. The calculated binary liquidus curves for the common III/V systems were found to agree well with experimental data, as will be discussed in Section 2.2 on phase diagrams.

2A,22 Bonding in Semiconductor Solid Solutions Before attempting to discuss models specifically developed to calculate the en­ thalpy of mixing of semiconductor alloys, it is useful to consider briefly the struc­ ture and bonding in these solids. The semiconductor solid solutions considered here have the diamond cubic, zincblende, or wurtzite crystal structures. The zinc- blende lattice is shown schematically in Figure 2.1. It may be thought of as two interpenetrating face center cubic (f.c.c.) lattices with cations on one lattice and anions on the other lattice, with its origin displaced to (1/4,1/4,1/4)—that is, 1/4 of the distance along the cube diagonal. The diamond cubic lattice is the same structure with all atoms being identical. The wurtzite structure is similar; that is, each atom still forms four bonds, but with hexagonal symmetry. The lattice con­ stants of the III/V and 11/VI compounds are listed in Tables 2.1 and 2.2. The crystal structure is a clear indication of the bonding in the solid. The num­ ber of nearest neighbors is low since each atom requires only four neighbors to form the covalent bonds necessary to complete the bonding orbitals of each atom. The sp^-hybrid orbitals form tetrahedral angles of about 110° with each other. The strong bending forces characteristic of such covalent bonds are responsible for stabilizing these open structures and preventing their collapse into the denser.

Figure 2.1. Illustration of the zincblende structure. (After Stringfellow [5], by permission of the pubhshers, Butterworths & Co., Ltd. ©.) 2.1 Basic Thermodynamics of Phase Equilibrium 23

Table 2.1 Parameters used in the calculation of i\H^ and a comparison of experimental and calculated results for III-V compounds

(-A//?) (--A//}') Experimental Calculated ^^o(A) /.^ (kcal mol^') Best Value Range Note AIN 4.380^^ 0.449 GaN 4.520^^ 0.500 InN 4.980^ 0.578 AlP 5.451 0.303 22.2 39.8 c AlAs 5.662 0.274 17.3 27.8 c AlSb 6.136 0.435 20.3 25 ± 3 23 ± 2.5 d GaP 5.451 0.330 24.8 24.4 17.2-29 c GaAs 5.653 0.313 16.3 20 ± 2 12.3-20.96 d GaSb 6.096 0.265 9.5 10.5 ± 1.5 9.4-11 d InP 5.869 0.432 20.3 22 ±2 21-21.5 d InAs 6.058 0.347 11.5 13 ± 1 12.4-17 d InSb 6.479 0.315 9.0 8 ± 1 6.8-10 d

^tonicity values from Phillips [18]. ^S. Slriteand H. Morkoc [171]. '^N. N. Sirota[172]. "D. D. Wagman, etal. [173]. twelvefold coordinated structures favored by metals and ionic crystals for which the bonding is nondirectional. The III/V compounds are stoichiometric to within the ability to measure nonstoichiometry [5]. As will be discussed later, this sim­ plifies the calculation of the thermodynamic properties of solid alloys and is an­ other indication of the strong covalent bonding in the solid. The 11/VI semicon­ ductors are more ionic and hence less stoichiometric, although the departure from stoichiometry is still measured in parts per million.

Table 2.2 Lattice parameter and ionicity of common II-VI compounds

^o(A) /i^ ZnS 5.4093 0.623 ZnSe 5.6676 0.630 ZnTe 6.089 0.609 CdS 5.8320 0.685 CdSe 6.050 0.699 CdTe 6.481 0.717 HgS 5.85 0.79 HgSe 6.084 0.68 HgTe 6.461 0.65

^Ionicity values from Phillips [18]. 24 2 Thermodynamics

Traditionally, semiconductor alloys have been described in terms of the virtual crystal approximation (VCA), where the lattice on which the atoms are situated is uniform; that is, the individual bonds are distorted to form a microscopically uni­ form solid solution. This was believed to be dictated by the accuracy with which Vegard's law describes the linear dependence of lattice constant on solid compo­ sition. However, we have recently discovered that the virtual crystal model for semiconductor solid solutions is in fact not a good description of the solid. Using extended X-ray absorption fine structure (EXAFS) results, Mikkelsen and Boyce [10] discovered a totally unanticipated feature of semiconductor alloys: the bond lengths in the alloy more nearly resemble the bond lengths in the pure binary compounds than the average values anticipated from the virtual crystal model. The measured bond lengths for the GalnAs system are shown in Figure 2.2 for com­ parison with values expected from the virtual crystal approximation. These results are typical of the behavior of III/V and 11/VI alloys in general. The valence force field (VFF) model [11] can be used to explain this behavior. The interactions between atoms are considered to be due entirely to strain (i.e., the stretching and bending of the bonds). The simplest form of the VFF calcula­ tion [12] for an alloy AC-BC assumes that the lattice is composed of five types of tetrahedra. Each has an atom C at the center, and the apexes are occupied by a combination of A and B atoms. The atoms on the mixed sublattice (A and B) are fixed at positions determined by the VCA, while the atoms of the common ele­ ment (C) are allowed to relax in each tetrahedron to minimize the total strain energy. Thus, in this simple model the lattice is considered to consist of an array of tetrahedra of five types with from zero to four A atoms (in the A^B i _ ^C alloy)

—1 r- T—T- ,J^7\ In-As 2.60 »-*' --••- GAL

2.55 o EXAFS DATA /VIRTUAL 1 / CRYSTAL

2.50 h -A

^mr^ Ga-As 2.A5 ^i>- h I i 1- till ' 0.2 O.A 06 OB 10 COMPOSITION (x in Ga^.^In^As) Figure 2.2 Ga-As and In-As nearest-neighbor bond length versus mole fraction of InAs in Ga,_Jn^As. Closed circles and broken line denote calculated data. Open circles denote values measured by EXAFS [10]. Middle line shows the average nearest-neighbor distance, which follows Vegard's law. (After Fukui [14], reprinted with permission of the American Institute of Physics.) 2.1 Basic Thermodynamics of Phase Equilibrium 25

TYPE 3 TYPE 4 Figure 2.3. Tetrahedral cells in a ternary III-V alloy semiconductor. (After Ichimura and Sa­ saki [12], reprinted with permission of the American Institute of Physics.) at the corners of the tetrahedra and one C atom within, as shown in Figure 2.3. Using this model, Fukui [13, 14] was able to accurately reproduce the bond lengths as a function of solid composition measured by Mikkelson and Boyce [10] in the GalnAs system. The results are reproduced in Figure 2.2. The behavior of bond lengths in 11/VI alloys is apparently similar to that in the III/V alloys [15-17]: the virtual crystal model does not apply. Motta et al. [15] conclude from EXAFS studies of CdZnTe that the bond lengths in the alloy re­ main nearly equal to their values in the pure compounds. We know that the bonding in a semiconductor is due to long-range effects, particularly the distributed electron energy states in the solid. The same valence electrons that determine the optical and electrical properties of the semiconductor also determine the bonding, as well as the elastic constants. This is contrary to the basic assumptions of the regular solution model, which cannot be expected to provide a physically accurate, predictive description of the enthalpy of mixing in semiconductor alloys. Efforts have been made to apply simple models developed to interpret the di­ electric function, band gap, and other optical properties related to the distributed electronic states in semiconductor solids to treat the bonding in semiconductor alloys. The best-known examples are the Phillips-Van Vechten dielectric theory of electronegativity described in detail in Phillips's Bonds and Bands in Semi­ conductors [18] and Harrison's bond orbital model [19].

2.12.3 Phillips-Van Vechten Model The band gap has been widely regarded as a measure of bonding in semiconduc­ tors (see, e.g., references [20-22]). The most coherent approach to demonstrating 26 2 Thermodynamics this effect has been that of PhiUips and Van Vechten. Van Vechten [23] showed that a simple one-gap model yields the bonding energy due to the creation of the band gap. In place of the complex band structure with band gaps at the r(Eo), X(E^), and LiEj) points in the Brillouin zone, a simple E versus k diagram with a single, average band gap is used [24]. The valence band is lowered by one-half the energy band gap, 1/2 £g, at the zone boundary resulting in the reduction of the energy of some valence band states. The integral of the density of states times the change in energy, plus a "bottom of the band" correction, gives the total energy difference between covalent semiconductor and the metal of the same lat­ tice constant. Phillips [25] found that the average energy gap in covalent solids, ^h. depends only on the lattice constant,

^h^V'- (2.17) The III-V compounds have higher average energy gaps for the same a^ due to the partially ionic nature of their bonding [25]. E^ = (El-^C'^y^\ (2.18) where C is the dielectric electronegativity difference between anion and cation. C has been determined empirically for the III-V compounds, but Phillips demon­ strated that it could be calculated using an expression similar to the one com­ monly found for molecular crystals except for the addition of a Thomas-Fermi screening term. The importance of the Phillips-Van Vechten model is that it focuses on the bonding due to all valence electrons in the compound in contrast to previous work that attempted to obtain bonding information from EQ, the band gap at the center of the Brillouin zone, which ordinarily contains only a small fraction of the bond­ ing electrons. Using the dielectric model, Phillips and Van Vechten were able to calculate the ionicity [26], predict the equiUbrium crystal structure at STP [27], predict the high-pressure semiconductor-metal phase transition [28], and calculate the enthalpy, entropy, and temperature of fusion [29] and the heat of formation [30] for many semiconductors, including the group IV, III-V, and 11/VI materials. The elastic constants can also be correlated with these ''chemical" effects and analyzed semiquantitatively using this model [18].

2.12A Calculation of the Enthalpy of Formation Because the calculation of the enthalpy of formation, AZ/J^, of III-V compounds contains the central elements of the other calculations mentioned, it is a good example of the ability of the model to calculate thermodynamic quantities accu­ rately, and will prove useful in later sections, it will be developed here. Three terms are included in the calculation of A//[? that reflect the enthalpy difference between the III-V semiconductor and the pure group III and V ele- 2.1 Basic Thermodynamics of Phase Equilibrium 27 ments. First, the energy change upon formation of the band gap is propor­ tional to (EyEp)\n(EJEp), as described earlier. Since E^ ^ ^o ~ ^ ^F ^ ^o~^' ^^^ A/ZJ? ^ a^^. Second, in the heavier semiconductors, the binding energies of the d subshells and the s-p core subshells are within a few Rydberg of each other. As a result, there is considerable mixing of valence and conduction band levels with d levels. This dehybridization of the sp^ bonding causes a lowering of A//^. Third, A/Zj? is believed to be proportional to the ionicity, defined as

f, = (i;^] • (2-19) El + C^- The resulting expression for A//|? is

A//0 = Kf,a^3 (2.20) where i^ is a scaUng factor determined to be 1.24 X 10"^ kcal A^ mol~'. The excellent agreement between calculated and experimental values of A//^ for the III-V compounds is illustrated in Table 2.1. The values of/ used in the calculation are also included in Table 2.1. The range for conventional III-V compounds (ex­ cepting the nitrides) is seen to be small, from 0.265 to 0.435. The 11/VI com­ pounds are much more ionic, as seen in Table 2.2.

2.1.2.5 Delta Lattice Parameter (DLP) Model for Enthalpy of Mixing An important quantity needed for the calculation of solid-solid, solid-liquid, and solid-vapor phase equilibria is the heat of mixing in the solid, A//^. This coupled with the assumption of a random distribution of constituents on their respective sublattices allows the calculation of the free energy of mixing of the solid alloy. Pseudobinary phase diagrams, which will be discussed in the next section, are ordinarily calculated using the regular solution model for the solid and liquid, with fl^ in Equation (2.12) adjusted until the calculated phase diagram agrees with the pseudobinary solidus data. Preliminary attempts at systematizing data obtained in III-V alloys indicated fl^ to be a function of the lattice parameter difference be­ tween A and B [6]. Several authors [20-22, 31] have suggested that the bonding energy in semi­ conductors is linearly related to the band gap. The work of Phillips and Van Vech- ten suggests that the average band gap should be used in this relationship. Since it varies as a^-^ in semiconductors that are nearly covalent such as the III-V com­ pounds, A//^^ which is used as a measure of bonding energy, might be written ^H^' = Ka^^\ (2.21) 28 2 Thermodynamics

Considering the zero of enthalpy to be infinitely separated atoms, the interaction parameter can be calculated from the enthalpy of mixing at jc = 1/2, yielding [6,32] -2.5 ^A + ^B a^ = 4/^ ^-^+^32-^ 4^^ A (2.22)

(^A cinV ^99K using Vegard's law to obtain the lattice constant at jc = 0.5. The value of K was obtained by making a least-squares fit of Equation (2.22) to available experimen­ tal values of a^ that are listed in Table 2.3. The value 1.15 X 10^ (cal mol-^A^^)

Table 2.3 Comparison of interaction parameters calculated using various models with experimental data*

a^ a^ a^ Ct' (Mod n^ist Phase Alloy (exp)^ (DLP)^ (VFF)b VFF)" Ppls)'^ Separation?^

AllnP 5.01 3.35 GalnP 3.25, 3.50 3.65 5.59 3.83 3.07 AlGaAs 0 0 0 0 0.30 AlInAs 2.50 2.81 4.38 3.06 GalnAs 1.65,2.00, 1.85 4.39 3.03 2.35 2.80, 3.00 AlGaSb 0 0.02 0.03 0.02 AllnSb 0.60 1.46 2.53 1.73 GalnSb 1.48, 1.90 1.85 3.09 2.15 AlAsP AlAsSb 6.53 4.69 AlPSb GaAsP 0.40, 1.00 0.99 1.38 1.01 0.86 GaAsSb 4.00, 4.50 3.36 5.49 3.99 3.96 Yes GaPSb M. Gap 7.91 12.53 9.09 Yes InAsP 0.40 0.58 0.90 0.58 InAsSb 2.25, 2.90 2.29 3.67 2.40 InPSb M. Gap 5.19 8.27 5.35 Yes AlGaN 1.19^ 1.34 0.87 AlInN 17.45" 18.10 11.44 GaInN 9.60 ^^ 9.62 5.98 Yes AlPN 19.68'' 60.79 36.56 AlAsN 57.93'' 85.33 53.42 GaPN 23.00 28.90'^ 42.43 27.38 Yes GaAsN 42.78^^ 59.09 36.84 Yes InPN 19.68'' 29.09 16.33 Yes InAsN 26.71" 39.14 21.87 Yes

*All values are in Kilocalories/mole. *From references 5 and 6. •'From Ho [102]. '^From reference 174. 2.1 Basic Thermodynamics of Phase Equilibrium 29 obtained in this way agrees approximately with the value of 1.26 X 10'^ (cal mol~'A-^) obtained by fitting Equation (2.21) to experimental values of A//^^ The experimental and calculated values are compared in Table 2.3. The standard deviation between experimental and DLP model-calculated results is only 412 cal mol~' for the conventional III/V alloys where reliable experimental values are available. The DLP calculation also appears to be quite accurate for the III/V nitride alloys [33-35]. A similar calculation of interaction parameters in 11/VI solid solutions is also possible, as discussed later [36]. The remarkable accuracy of the DLP model for determination of 0^ must be attributed to a cancellation of errors in the calculation of A//^^ (x = 0.5) and (^H% -f- ^H^)/2. It can be shown that errors linear in jc—that is, ^H^' - A//^^,^ = a + bx—completely cancel and do not affect the calculated value of fl^ Thus, the important feature of the calculation is an accurate description of the compositional dependence of ^H^\ A striking feature of the DLP model is that the interaction parameter, hence the enthalpy of mixing, is always positive. As seen in Table 2.3, the experimental data verify this prediction; not a single III/V alloy has a negative enthalpy of mixing. This is also true for elemental and 11/VI alloys. Remarkably, for systems with end components having the same interatomic spacing, such as GaAs-AlAs, the solid solutions are nearly ideal.

2.1.2.6 Strain Energy Models In the traditional regular solution model, the uniformly positive values of enthalpy of mixing strongly suggests that the enthalpy of mixing is due to strain, rather than chemical factors. Fedders and Muller [37] performed the strain calcula­ tion, which yielded interaction parameters, calculated assuming the virtual crystal model, approximately a factor of four too large. This was interpreted as being due to the failure of the VGA, as described earlier; that is, the bond distortions are considerably smaller than deduced from the virtual crystal model. As observed by Fedders and Muller, this convergence of two apparently divergent approaches is perhaps not surprising considering that both elastic properties and bonding are related to the electron energy states in the solid. Chen and Sher [38] and Sonomura et al. [39] demonstrated that the adjustable parameter in the DLP model could be calculated simply in terms of the strain energy using more realistic models for the bond distortions than the virtual crystal approximation. The mixing enthalpy can also be estimated using the simplified VFF model described in Section 2.1.2.2. The solid is considered to be made up of identical tetrahedra (Figure 2.3) with the position of the central atom, located on the sub- lattice with no mixing, allowed to relax to the position giving the lowest strain energy, considering both stretching and bending distortions. The strain energy due to the stretching and bending of the bonds in each type of tetrahedron is summed 30 2 Thermodynamics over the five types of tetrahedra weighted by the distribution probabiUty (a random arrangement was not assumed in reference [12]). The two terms are coupled and must be solved simultaneously [12]. This approach allows a calculation of the free energy of mixing. The results are similar to those for the DLP model, but with no adjustable parameter. Assuming a random alloy uncouples the energy and distri­ bution terms and so further simplifies the VFF calculation. There are two major drawbacks to the simple forms of the VFF model de­ scribed here. First, when the lattice is assumed to be made up of tetrahedra where the corner atoms take the VGA positions, one of the sublattices is not relaxed. This causes a significant overestimation of the total strain energy. Second, the difference in energy between the several tetrahedra types is much greater than kT for many III/V alloys. This, of course, gives a nonrandom distribution of the five types of tetrahedra. Taking into account the effects of the resulting short- range order (SRO) makes the calculation of the mixing enthalpy difficult, since it couples the two factors [12]. The first difficulty can be surmounted by considering a large ensemble of sev­ eral hundred atoms with the positions of each allowed to relax [40]. However, this involves large-scale calculations and does not address problem 2, although attempts have been made to deal with the nonrandomness by using, for example, the quasi-chemical approximation [41]. This dilemma is easily resolved while maintaining a relatively simple calculation by considering only the dilute limit, where the effect of the SRO is negligible [33, 34, 42]. With this limitation, a full relaxation of the entire lattice can be considered without undue complexity, al­ though the calculation must be done numerically. This approach was developed specifically for dealing with systems with very low solubility limits, in particular for the solubility of the very small N atom in conventional III/V semiconductors such as GaAs, InP, GaP, and so forth [33, 34]. It has also been used to estimate the magnitude of the miscibihty gap in the GaInN alloys [42, 43] of importance for blue-emitting diodes and injection lasers. The resulting values of O^ are included in Table 2.3 for comparison with the experimental results as well as the results of the DLP calculation. Surprisingly, the results of the VFF calculation are closely approximated by the results of the much simpler DLP calculation, which is valid for nondilute solid solutions. This is especially remarkable when it is considered that the adjustable parameter in the DLP calculation did not include data for the N-containing alloys.

2.12.7 First-Principles Models In recent years, advances in fundamental insights relating to the energy of a semi­ conductor lattice, the methodology of solving mathematical problems involving extremely large matrices, and the increasing availability of high-powered com­ puters has allowed first-principles local density self-consistent total energy mini- 2.1 Basic Thermodynamics of Phase Equilibrium 31

mization calculations in semiconductor alloy systems [44]. Using these quantum mechanical calculations, the thermodynamics of semiconductor solid solutions can be calculated without any of the extreme simplifying approximations neces­ sary to obtain simple analytic models. The total energy minimization calculations are based on the entire complex band structures. The results from such calcula­ tions are included in Table 2.3. The mixing enthalpies have also been calculated for InGaN, InAlN, and AlGaN alloys using a pseudopotential perturbation ap­ proach [35].

2.1.2.8 Phase Separation The large positive enthalpy of mixing for systems with a large difference in lattice constant can overwhelm the negative entropy of mixing for temperatures below the critical temperature, resulting in a free energy versus composition curve shown schematically in Figure 2.4, with an upward bowing in the center. This dictates that at equilibrium, a random alloy with composition between points A and B will decompose into a mixture of two phases, denoted A and B in Fig­ ure 2.4. This is frequently observed in III/V system, as will be described in detail in the phase diagram section to follow.

Solid Composition (x) Figure 2.4. Free energy versus solid composition for a hypothetical semiconductor alloy having a large positive enthalpy of mixing. The points A and B are the binodal points, and points C and D represent the spinodal points. 32 2 Thermodynamics

Table 2.4 Interaction parameters for ternary II-VI alloys

n (strain energy model) ^ a(DLPmodel)^ Compound ^ala (cal/mol) (cal/mol)

CdSeTe 0.068 2,430 2,340 ZnSeTe 0.074 2,930 2,530 ZnSeS 0.047 980 1,060 ZnSTe 0.121 9,800 7,490 HgSeTe 0.060 1,750 1,680 MnSeTe 0.086 4,230 CdZnTe 0.060 1,740 1,730 CdHgTe 0.003 I 25 CdMnTe 0.022 140 ZnHgTe 0.057 1,550 1,560 ZnMnTe 0.038 580 HgMnTe 0.019 100 CdZnSe 0.065 2,150 2,024 CdHgSe 0.006 6 16 CdMnSe 0.039 610 ZnHgSe 0.071 2,650 2.290 ZnMnSe 0.025 220 HgMnSe 0.045 860

^Kisker and Zawadzki [36].

Less experimental data for phase equilibria are available for 11/VI systems. However, Kisker and Zawadzki [36] have applied both the DLP model and the Fedders and Muller strain energy approach to the calculation of interaction pa­ rameters for systems involving Cd, Zn, Hg, and Mn on the cation sublattice com­ bined with the anions S, Se, and Te. The results are presented in Table 2.4. The interaction parameters calculated using the two approaches are nearly equal. They range from near zero, for alloys combining Hg and Cd, including the ex­ tremely important infrared device materials system HgCdTe, to very large values for alloys involving S and Te on the anion sublattice. Positive values for the enthalpy of mixing in 11/VI solids are obtained by other calculations and by analysis of the pseudobinary liquid-solid phase diagram. Cal­ culations of Sher et al. [45], which include both strain and chemical effects, indi­ cate that ZnCdTe, HgCdTe, HgZnTe, and CdSeTe all have positive enthalpies of mixing: all are expected to have miscibility gaps at low temperatures. The cal­ culated free energies of mixing, including the nonrandom entropy of mixing, are shown for CdZnTe and CdSeTe in Figure 2.5. Brebrick [46] was able to fit the HgTe-CdTe pseudobinary phase diagram with a temperature-dependent solid interaction parameter of 1,384-0.8452 T. This gives a critical temperature of 287° K. Using a different model for the liquid solu­ tion, which has a small effect on the resultant solid-phase interaction parameter. 2.1 Basic Thermodynamics of Phase Equilibrium 33

T(K) 0 00 T •T—— 1 Iv ^"^ .yCdyTe 300 ^01 SOO^X'

-0 02 - \ \

^ ^.03 / 1000/

^ -0.04 > UL '^ -0.05 H 150o/

-0 06 i

-0.07 \

-0 08 a| 0 0 0.5 1.0 ZnTe CdTe

O -0.04 \-

-0.05

Figure 2.5. Excess-free energy as a function ofy at different temperatures for ZnCdTe and CdSeTe. (After Sher et al. [45], reprinted from the Journal of Crystal Growth with permission from Elsevier Science.)

Marbeuf et al. [47] obtained a somewhat different temperature-dependent inter­ action parameter of 782-1.064 T by fitting the pseudobinary HgTe-CdTe phase diagram, as will be shown in Section 2.2.2. This interaction parameter results in a slightly lower value for the critical temperature. In any case, phase separation is 34 2 Thermodynamics not expected to be a problem during OMVPE growth that occurs at temperatures higher than the critical temperature. Since the self-diffusion coefficients are large for 11/VI solids, the question of whether the miscibility gap will result in cluster­ ing in the solid is relevant. Based on the analysis of III/V solid solutions, to be discussed later, the coherency strain energy is expected to stabilize the homo­ geneous solid even at room temperature.

2.1.2.9 Ordering in III/V Alloys Unexpectedly, long-range ordering has been observed in III/V solid solutions, including those with large positive deviations from ideality. The spontaneous for­ mation of such ordered structures during growth is also well documented for Si-Ge alloys, and some evidence for ordering has been observed for 11/VI alloys [43]. The superficial application of the regular solution model (Sec. 2.1.2.1) sug­ gests that clustering and phase separation will occur in systems with positive in­ teraction parameters, since the AC bonds are less stable than AA and CC bonds [1, 48]. On the other hand, a negative interaction parameter, when AC bonds are more stable, is predicted to lead to both short- and long-range ordering. Short- range ordering simply means that more AC bonds exist than for a random alloy. In long-range ordering, the AC bonds form a pattern with formation of a new superlattice crystal structure having translational symmetry with a period that is usually double that of the normal lattice in a particular crystallographic direction. This doubling of the periodicity is easily detected using transmission electron diffraction (TED). Examples are shown in Figure 2.6. The [001] electron diffrac­ tion pattern for a disordered GaAso^Sbo^ epitaxial layer showing only the spots with unmixed indices typical of the zincblende crystal structure is shown in Figure 2.6a. The electron diffraction pattern for a GaAs^ 3 Sb^ <^ layer grown by OMVPE [49, 50], Figure 2.6b, contains extra diffraction spots indicative of for­ mation of ordered structures. In this particular case, several ordered structures are formed. For two variants of the LIQ (CUAU) ordered structure, the periodicity is doubled along the two (100) directions perpendicular to the growth axis. An ad­ ditional ordered structure, termed El, (chalcopyrite), is also formed with double the periodicity of the normal lattice along the (210) directions. Figure 2.6c is the [110] TED pattern for a disordered GalnP layer. The extra spots seen in the elec­ tron diffraction pattern shown in Figure 2.6d are indicative of ordering along two (111) directions. This is due to formation of the LI, (CuPt) ordered structure. The occurrence of these ordered structures will be discussed more fully in Chapter 9. Such ordered structures have also been observed in essentially all III/V alloys, as summarized in Table 2.5. Ourmazd and Bean [51] have also reported the forma­ tion of {111} ordered structures composed of alternating Ge and Si atomic pairs in Si-Ge alloys grown on (001) Si substrates by MBE. The solution to the apparent dichotomy of ordered structures in systems with 2.1 Basic Thermodynamics of Phase Equilibrium 35

Figure 2.6. [001] pole electron diffraction patterns for disordered (a) and ordered (b) GaAsSb and [110] pole electron diffraction patterns for disordered (c) and ordered (d) GalnP. large positive deviations from ideality is suggested by early observations for or­ dered structures in metal systems. Swalin [1] suggests that the regular solution model conclusions related to the preference of a system for clustering or ordering, for a positive or negative interaction parameter, respectively, apply only when the bond energies, /^AA' ^CC ^^^ ^AC ^^^ ^^^ interaction parameter, are due to chemi­ cal factors. If the positive deviation from ideality is due to strain, the atoms tend to be arranged v^ith a larger-than-random number of unlike nearest-neighbor pairs. Four decades ago Hume-Rothery recognized the obvious, that clustering in Table 2.5 Summary of ordering observed in semiconductor alloys

Growth Substrate Ordered Material Technique Orientation Structure References

AlGaAs OMVPE (110) CuAu Kuan et al. [60] &MBE GaAsSb OMVPE (110) CuAu Jen et al. [49] GalnAs MBE (110) CuAu Uedaetal. [176] GalnAs MBE (110) CuAu Kuan etal. [175] GalnAs OMVPE, (110) CuAu Nakataetal. [177]; Chin MBE etal. [178] GaAsSb OMVPE (001) CuAu + CH Jen etal. [49] GaAsSb MBE (001) CH -H CuPt Murgatroydetal. [179, 180] GalnAsSb OMVPE (001) CuAu + CH Jen etal. [181] AlInAs MBE (001), (2 X 3) {llU-3 period Gomyoetal. [161] reconstruction AllnP MBE (001), (2X2) CuPt-A Gomyoetal. [160] reconstruction AlGalnP OMVPE (001) CuPt Gavrilovic et al. [54]; Kondow etal. [182]; Suzuki etal. [183]; Nozakietal. [184] AlGalnP OMVPE (001) CuPt-B Chen etal. [185] AllnP OMVPE (001) CuPt Kondow etal. [186] AlInAs MBE (001) CuPt Norman etal. [187]; Ueda etal. [188] GaAsP OMVPE (001) CuPt-B Jen etal. [189]; Chen et al. [190]; Piano etal. [55] GaAsSb MBE (001) CuPt-B Murgatroid et al. [179, 180] GaAsSb MBE (001) CuPt-B Ihm etal. [191]; Murgatroyd et al. [192]; Uedaetal. [193] GalnAs OMVPE (001) CuPt Chu etal. [194] GalnAs VLE (001) CuPt Shahidetal. [195] GalnAsP VLE (001) CuPt Shahid etal. [195] GalnAsP OMVPE (001) CuPt Piano et al. [55] GalnP OMVPE (001) CuPt-B Uedaetal. [188]; Suzuki etal. [183]; Moritaetal. [197]; Bellon etal [198] GaInN OMVPE (0001) CuPt-type Ruterana et al. [241] GalnSb OMVPE (001) CuPt Shin etal. [199] InAsSb OMVPE (001) CuPt Jen etal. [166] InAsSb MBE (001) CuPt Seong et al. [200] InPSb OMVPE (001) CuPt Stringfellow[201] GaASo 25Sbo.75 OMVPE (001) Luzonite Jen et al. [57] GalnAs LPE (001) Famatinite (?) Nakayama and Fumita [203] Si-Ge MBE (001) CuPt-like Ourmazd and Bean [51] ZnFeSe CuAu Park et al. [63]; Salabanca- Riba et al. [64] HgCdTe LPE CuPt Chang and Goo [65] 2.1 Basic Thermodynamics of Phase Equilibrium 37 systems with a large difference in atomic size resulted in large strain energies [52]. He suggested that a size difference would drive both short- and long-range order­ ing. More recently, Ichamura and Sasaki [12] reached a similar conclusion using VFF model calculations. They resolved the apparent contradiction by noting that the phase separation resulting from a large positive enthalpy of mixing involves the formation of two completely incoherent phases. Clustering, on the other hand, involves the formation of coherent regions with dissimilar compositions. The co­ herency strain energy prevents clustering, as well as spinodal decomposition [53], as will be discussed later, and, in fact, leads to an increase in the number of bonds between dissimilar atoms (i.e., short-range ordering). Thermodynamic calculations using the models described earlier provided the first detailed rationalization of the ordered structures observed in III/V alloys. The total energy of a Gao5lnQ5P solid was simply found to be lower for the LIQ and Ell ordered structures, with ordering on {100} and {210} planes, respectively, than for a disordered mixture with a random atomic arrangement on the group III sublattice [44]. The LIQ and El, structures, both consisting entirely of tetrahedra with two In atoms and two Ga atoms, type 2 in Figure 2.3, are calculated to have similar energies. Obviously, the type 2 tetrahedra will have the lowest strain en­ ergies of the five types of tetrahedra, thus explaining the stability of these ordered structures in the bulk. However, the energy of the most commonly observed CuPt structure is much higher. This ordered structure is composed entirely of the type 1 and type 3 tetrahedra in Figure 2.3; thus, the strain energy is not relaxed as effi­ ciently as for the LIQ and El, ordered structures. As a consequence, it is not stable in the bulk. As discussed in Section 2.6.2, this dilemma is resolved by considera­ tion of the stabilities of the various ordered structures at the reconstructed sur­ faces. Thus, formation of the CuPt structure is due to surface thermodynamic factors. Annealing studies indicate that CuPt ordering is destroyed by annealing, in agreement with the conclusion that the CuPt structure is not stable in bulk GalnP alloys [54, 55]. Srivastava et al. [56] predicted ordering for alloys with compositions in the ratios 1:3, and 3:1, in addition to the 1:1 ratio discussed earlier. This type of ordering has also been observed in the GaAsSb system [50, 57] with the luzonite (LI3) superlattice structure formed. Theoretical analysis indicates the famatinite structure (DO22) to be more stable [58, 59]. An interesting feature of both the VFF and first-principles calculations is that the ordering is shown to be due to strain factors. This suggests stronger ordering for alloys with large differences between the lattice constants of the binary con­ stituents, in accordance with the simple analysis of Hume-Rothery [52]. However, even for AlGaAs alloys where AlAs and GaAs have the same lattice constant, ordering has been observed in material grown by both OMVPE and MBE [60]. This is not expected from the first-principles calculations [61] and is still not understood. 38 2 Thermodynamics

Wei and Zunger [62] performed first-principles total energy minimization cal­ culations to determine whether ordered structures are energetically stable in HgZnTe, CdZnTe, and HgCdTe solid solutions. Their calculations indicate the (100), CuAu ordered structure, one of the lowest-enthalpy ordered structures for III/V alloys, is unstable in the bulk, relative to the unmixed binary compounds for all three 11/VI alloys. This is attributed to the repulsion between p and d elec­ trons. They also calculated the bond lengths in the alloys and found they were essentially independent of alloy composition; that is, they retained the values for the binary compounds. Nevertheless, the CuAu structure has been experi­ mentally observed in ZnFeSe by Park et al. [63] and Salamanca-Riba et al. [64]. CuPt ordering was also reported for mercury cadmium telluride alloys grown by LPE [65]. Kinetic effects are also observed. For example, as the growth rate increases, the degree of order decreases [66]. This and other kinetic aspects of ordering are dis­ cussed in Chapter 5.

2.2 Phase Diagrams

Phase diagrams are extremely important for any crystal growth technique. Al­ though not generally recognized, this includes OMVPE [67], CBE, and MBE [68]. Phase diagrams specify the number and compositions of the phases present when the intensive quantities such as temperature, pressure, and overall composi­ tion are specified. In a practical sense, they define the results to be expected when we perform an OMVPE (or CBE or MBE) growth experiment where we specify the typical growth parameters such as temperature and the partial pressures of the gas-phase constituents. As discussed in Chapter 1, for OMVPE the partial pres­ sures change markedly within the gas phase as a result of mass-transport limita­ tions. Thus, we have a choice of defining the number of condensed phases and their compositions in terms of the partial pressures entering the reactor, pf, or the values at the growing interface, p\. We begin with the least ambiguous de­ scriptions, in terms of true equilibrium involving macroscopically homogeneous phases. The application of phase diagrams to typical multiphase situations will be discussed in some detail, since the OMVPE literature is filled with evidence that serious misunderstanding of the simple concepts of phase equilibria is rampant. The Gibbs phase rule is at the heart of any discussion of phase diagrams. It simply states that the number of independent intensive variables or the number of degrees of freedom of the system, F, is a function of the number of components, C and the number of phases, P: F= C- P + 2. (2.23) 2.2 Phase Diagrams 39

The last term represents the temperature and pressure of the system. This is simply a result of applying the equilibrium conditions, of the type given in Equation (2.4), to cover each constituent in all phases of the system. In the simplest case for a two-phase system consisting of a single component, such as steam and water (vapor -f liquid), the system has one degree of freedom. For example, if the pressure is fixed at 1 atm, the temperature is also fixed (at 100°C, the boiling point of water). If the pressure is changed, so is the boiling temperature.

2,2.1 Binary Systems Consider first the so-called liquid-solid equilibrium for a binary III/V system. To be specific, consider the temperature-composition (T-x) phase diagram for the GaAs system seen in Figure 2.7a. For the simplest case of pure Ga, the melting point is fixed on this diagram. This is because we have assumed a pressure of 1 atm. As for the case of water, considered earlier, no vapor is present. However, if the Ga is contained in a typical LPE apparatus—for example, it is covered by an ambient of an "inert" gas, such as hydrogen—this produces a subtle change in the application of the phase rule. Strictly speaking, we have introduced an ad­ ditional component into the system as well as an extra phase, the vapor. Thus, the value of F is unchanged by including this complication. Often, the T-x diagram is used to analyze the LPE growth of GaAs, ignoring the presence of the vapor (i.e., P is considered to be 2—the liquid plus the solid). However, this may cause prob­ lems even for the analysis of LPE growth. For example, the hydrogen may dis­ solve in both the liquid and the solid phases but be ignored. For the case of pure Ga, a finite concentration of Ga is also present in the vapor phase, equal to the vapor pressure of Ga. Thus, it is best to include the vapor in the analysis. At the melting point of Ga, C = 2, F = 3 and, thus, F = L We have assumed that the total pressure is fixed at 1 atm; thus, the system is invariant. This occurs only when the temperature is at the "melting point" of Ga (at this particular pressure). For a typical LPE growth experiment, the temperature is well below the melting point of GaAs, the overall composition is somewhat on the Ga rich side of stoi­ chiometric GaAs, and the pressure is 1 atm. With no cover gas, the system consists of only two phases, the Ga-rich liquid and GaAs solid. There is no vapor phase, because the vapor pressures of Ga and As under these conditions are both below the system pressure of 1 atm. Thus, normally, the vapor is ignored in these phase diagrams, as they are used to analyze LPE growth. However, in reality, LPE growth is performed under a hydrogen (or other nonreactive gas) ambient. Thus, it is proper to consider the additional component (the inert gas) and the additional (vapor) phase. This does not change the value of F in Equation (2.23), but it does allow the specification of the partial pressures of both Ga and As above the 40 2 Thermodynamics

80 X' ^00 As

1 1 1 III 1 ^^,.^4^

10-' 10-2 -\\ \^,

10-3

o - \\ B 10-5 x^^^^ a 10-6 \Ga ^ 10-7 >^'2 io-« - 10-' 10-10 \- h 10-"

10-12 1 1 1 1 X l\ N. ^>v 9,0 10.0

10*/T, •<"'

Figure 2.7, GaAs phase diagrams: (a) temperature-composition, (b) pressure-temperature, and (c) pressure-composition. The latter is drawn at a temperature Tj of approximately 980°C, where the Ga- and As-rich liquidus compositions are X2 and x^. 2.2 Phase Diagrams 41

10^ p(A5j y" 10° As2 ---r^"'

1 ''" p(A$J r ''"As"" t 10-* p(Ga) a. - plGa) 10-8

in-10 1 1 1 1 1 1 0.0 X2 0-2 0-4 0.6 0.8 x'2 TO Ga GaAs As Composition C Figure 2.7.— Continued liquid + solid mixture, as indicated by Figures 2.7b (the pressure-temperature [P-T] diagram) and 2.7c (the pressure-composition [P-x\ diagram). When both the liquid and solid phases are present, as well as the vapor, P = 3. Including the inert gas, C = 3. Thus, when the pressure is 1 atm, fixing the temperature totally defines the system thermodynamically. The composition of the liquid phase is fixed, the composition of the solid phase is fixed (it is not precisely 0.5), and the pressures of Ga and all As species in the vapor are also fixed. The phase diagrams specify all of these parameters. GaAs is not a single perfectly stoichiometric component, as is commonly as­ sumed. The solid composition of GaAs can vary slightly, as seen in the inset of Figure 2.7a where the composition scale is magnified by a factor of approximately 10^. The solid composition can, in fact, vary from 0.5 by an extremely small amount, measured in parts per million. This solid nonstoichiometry has profound consequences for the properties of the solid, as will be discussed later. The GaAs melting point is defined to be the single point where the solidus and liquidus curves touch. This is normally not at precisely x = 0.5. However, as seen in Fig­ ure 2.7b, the As and Ga pressures are defined; thus, the system is invariant. Notice that the As in the vapor phase consists of both the dimer. As 2, and the tetramer, AS4. The relative amounts of the two species are determined by equilibrium ther­ modynamics; thus, they are shown on the P-Tand P-x phase diagrams. Elemental As also exists in the vapor, but at concentrations low enough to neglect. Mon- atomic Ga is the only significant Ga species. Two additional invariant temperatures indicated on the phase diagrams are the two eutectic points, where two solid phases and a single liquid phase are in equi­ librium with the vapor phase. The number of phases is four, and the number of components is three: F must be unity, according to the phase rule. Thus, fixing the O U 72. ^ o .2 .2^ c/5 ^b -^ _^

o .2 o II ^ ^ II ..- Z ^ b .^.

O rn

II =^ £: II ^^; II =^ ".

CX

^^ E^ dC II _^ ^ II ^ "^ a. t*, ^ Ci. ^ ^ c^

."2 !—

CD Z c\i ^Q H H c^i n 00 o 2.2 Phase Diagrams 43 total pressure defines the eutectic temperatures in exactly the same way that it fixes the melting points of the various pure constituents. At the temperature and overall composition designated Tj and X2 (or x^) in Fig­ ure 2.7a, the only phases are the liquid and vapor; thus, there are two degrees of freedom. This single condensed phase field is represented as an area on the phase diagram. Both liquid composition and temperature can be varied independently, and the same two phases will remain. Of course, the vapor composition is a de­ pendent variable in this case. Alternately, the vapor composition and temperature could be varied independendy, and the liquid composition would be determined. The results of this and the other situations to be described here are summarized in Table 2.6. A situation more related to epitaxy is represented by the same overall compo­ sition at a lower temperature designated Tj in Figures 2.7a and 2.7b. This is typical of LPE growth where the temperature of the liquid is lowered until a solid phase is precipitated. This situation is included in our discussion of OMVPE since it represents the simplest case for the determination of nonstoichiometry in the solid. Since we have three phases—the vapor, liquid, and solid—there is only a single degree of freedom, when the total pressure is fixed: the point falls on the liquidus line. When the temperature changes, so must the liquid composition in order to retain the three-phase mixture. The temperature also defines all of the pressures in the vapor as shown by the lines in Figure 2.7b. Notice that two values of each pressure are specified for each temperature. One represents the vapor pressure in equilibrium with the solid/liquid mixture on the Ga-rich side of the T-x diagram, as illustrated by the value of X2 in the T-x diagram. Alternately, we could have chosen a value of x > 0.5 intersecting the liquidus line on the As-rich side of the diagram at x'2. This would produce the higher As and lower Ga partial pressures. The solid in equilibrium with the liquid with composition X2 is that on the very most Ga-rich side of the range of stoichiometry. It would have the lowest number of Ga vacancies, interstitial As atoms, and As antisite defects, As^^, and the maxi­ mum number of As vacancies, interstitial Ga atoms, and Ga antisite defects, Ga^^. On the other hand, growth on the As-rich side of stoichiometry would reverse the situation, producing the maximum number of As antisite defects, Ga vacancies, and so forth. An understanding of the solid stoichiometry will become important when we discuss the incorporation of impurities and defects in epitaxial layers. A summary of three-phase equilibrium is included in Table 2.6. Another instructive situation is when the overall composition of the system is specified to be very nearly x = 0.5 at a temperature lower than the melting point of GaAs—T2, for example. In this case, the overall composition of the system is within the very narrow solid-phase field. The equilibrium phases are then the vapor and the solid with no liquid present. Since the number of phases has de­ creased by one, the phase rule dictates that F be increased by one to a value of two. When the total pressure is 1 atm, this is represented by an area in the phase 44 2 Thermodynamics

0.5003 0.^002 0.5001 0.5000 0.4999 0.4998 • 1 1 ,. ^ y- , 5001_ MAX MELTING POINT

1400 (. \ 1 - \* \ • J • GoP (s) -»- -^ 1 1300 .* \ GoP (s) i / •\ ; - CALC. 1 1 • 1200 - B^X Neutral V Fit ] - • •' Jordan, et^ al. A LEG SEED.UNDOPEO,

1100 - C 1 \ \ POWDERED \ V LEG SEED. UNDOPED - y • LEG SEED. DOPED O LEG TAIL.UNDOPED "^ \ • LEG TAIL. DOPED Gap ($)>•£ \ L • LEG ANNEALED o SO

900 ; . \ X 1 A -J 1 X 1 1 0.4997 0.4998 0.4999 0.5000 0.5001 0.5002

Figure 2.8. Experimental and theoretical existence curves for GaP. The data are from Jordan et al. [70]. (After Van Vechten [71], reprinted with permission from Journal of Electronic Materials, Vol. 4, 1975, a publication of the Metallurgical Society, Warrendale, Pennsylvania.) diagram where temperature and solid composition, the stoichiometry in this case, can be varied independently. Naturally, the range over which the solid compo­ sition can be varied is extremely limited; in other words, the range of non- stoichiometry of the covalently bonded III/V semiconductors is small, at most a few parts per million. This is illustrated using the predicted solid-phase field in GaAs in the inset of Figure 2.7a [69]. In Figure 2.8, experimental data [70] are compared with calculated results for GaP [71], the only III/V system for which such data exist. The congruent point is seen to be on the opposite side of exact stoichiometry than for GaAs in Figure 2.7a. The range of nonstoichiometry for the 11/VI compounds is much larger but is still measured in tens of parts per million. The significant feature of this example is that the stoichiometry of the solid completely specifies the pressures of the Ga and As species, as seen in Figure 2.7c. Also worth noting are the large changes in both As and Ga pressures associated with an extremely small change in solid composition. This example is particularly important since it represents the situation during OMVPE growth: only a vapor and solid are present. In this thermal equilibrium 2.2 Phase Diagrams 45 case, the Ga and As partial pressures in equilibrium with the solid are linked and completely specify the stoichiometry of the solid. During LPE, the GaAs stoichi- ometry is fixed on the Ga-rich side of the solid-phase field. During OMVPE, the As partial pressure is normally an independent variable: it determines both the Ga pressure and the solid stoichiometry. This will be dealt with in more detail in Section 2.3.4. MBE growth of semiconductor materials can also be analyzed using the phase rule [68]. As the pressure of the system is reduced into the regime of MBE growth, the free energy of the vapor phase is reduced, which results in major changes in the phase diagram, as discussed in detail by Tsao [68]. For example, the GaAs temperature-composition phase diagram at a total pressure of 10"^ Torr has a fairly extensive vapor -f GaAs solid region. In this case, C~l and only the vapor and solid phases are present, so P = 2 as well. There is, of course, no carrier gas to worry about in this case. The phase rule determines the system to have two degrees of freedom. To specify the system completely, the temperature and the partial pressure of arsenic at the interface (or the As flux) are typically specified. Thermodynamically, the system resembles OMVPE growth. If the stoichiometry of the system is slightly Ga-rich, a liquid phase, essentially pure Ga, is formed. Thus, similar to OMVPE, MBE growth of GaAs is typically carried out with an excess of As in the vapor phase. Very recently, the As-rich GaAs grown at low temperatures by MBE has be­ come of some interest because of its semi-insulating properties. Because of the extremely low growth temperatures, <200°C, an As-rich second phase can be formed. Thus, growth occurs in a two-condensed-phase region of the phase dia­ gram and P = 3. This reduces the value of F to unity: the solid is always on the As-rich side of the region of solid stoichiometry. Only the temperature is an in­ dependent variable. The entire P-T-x diagram can be calculated using elementary thermodynamics. First consider the calculation of the liquidus line in the T-x diagram. In this case. Equation (2.4) takes the form MAc = /^k + M[:, (2.24) where fx- is given by Equations (2.10a) and (2.10b). The solid AC is the pure standard state, so a^^ ~ 1 and Equation (2.24) can be rewritten KT\xv{a\a'^) + (/x^^ 4- yfi^ - /JL^^^) = 0. (2.25) The second term is (AG^^^ - AG^fc)* the Gibbs free energy of fusion of AC minus the free energy of mixing of the stoichiometric liquid. It represents the free energy change upon melting a mole of AC and then separating the liquid with ^A ~ -^c ^ ^/2 into one gram-atomic weight, each of pure components A and C. In addition, AG^^ ^^V be written in terms of the entropy of fusion, A5'^c' ^Glc = ^slc(Tlc-n (2.26) 46 2 Thermodynamics by assuming A5'^c and A/Z^^ ^^ ^^ temperature-independent. Assuming the liquid to be a regular solution, for which AG^fc rnay be obtained from Equa­ tions (2.11), (2.12), and (2.14), and using Equation (2.16) to obtain the activity coefficients in solution. Equation (2.25) may be rewritten in the form [72]

ln[4.,(l -.,)] + ^; =-^iL^-,^^ (2.27)

This expresses the composition of the liquid in equilibrium with pure solid AC as a function of temperature. Values of A^^ and T^ are available from the literature [4]. The values of O^ may be determined by fitting Equation (2.27) to experimen­ tal data using O' as an adjustable parameter. The resulting interaction parameters generally vary between 0 and -6,000 cal mol "^ for III/V liquid solutions [4, 6], which is indicative of fairly weak interactions characteristic of the metallic bond­ ing in III/V liquids [5]. Models for the thermodynamic properties of metal alloys were developed over sixty years ago [73]. By adding a term due to the screened electronegativity dif­ ference between the group III and the group V elements in the liquid, the theory of metal solutions was extended to describe the thermodynamic properties of liquid III/V solutions [9]. This allows the partial pressures to be calculated yield­ ing the P-T and P-x diagrams, of which Figures 2.7b and 2.7c are examples. A complete compilation of III/V phase diagrams can be found in reference 74.

2.2.1.1 OMVPE Phase Diagrams An important question that must be considered before proceeding to more detailed thermodynamic analysis of OMVPE is. How does thermodynamics relate to epi­ taxial growth for extremely nonequilibrium techniques such as OMVPE? This question also involves the applicability of thermodynamics to CBE and MBE. The connection between thermodynamics and growth results has long been recognized for LPE, where the thermodynamic driving force for growth (i.e., the chemical potential difference between the liquid and solid phases) is relatively small. For LPE, growth proceeds at near-equilibrium conditions throughout the system; that is, the Gibbs free energy driving force for growth is small, as seen in Figure 2.9. Here, it seems natural to treat the growth process using thermodynamics, and, in fact, phase diagrams are well known to give important information about alloy composition, solid stoichiometry, and dopant incorporation, as discussed later. However, even for the vapor-phase growth techniques, such as OMVPE and MBE, where the input vapor is at a very much higher chemical potential than the solid produced, as seen in Figure 2.9, powerful thermodynamic factors still control much of the growth process. This is because, even for a system with a high super- saturation of the input vapor phase, near-equilibrium conditions may prevail near 2.2 Phase Diagrams 47

OMVPE MBE TMGa + Ga + Arsine Arsenic 80

I 60

20

Growth Technique

Figure 2.9. Estimated thermodynamic driving force, Gibbs free energy difference between reac- tants and products, for several epitaxial growth processes. All calculations are for the growth of GaAs at 1,000° K. (After Stringfellow [238], reprinted from the Journal of Crystal Growth with permission from Elsevier Science.) the solid/vapor interface, the topic of Section 2.3.2. However, it may prove nec­ essary to consider the thermodynamic properties of the surface phase(s) in addi­ tion to the vapor and bulk solid phases, as discussed in Section 2.6. The principles used to establish the phase diagram shown in Figure 2.7 can be used to generate a phase diagram having a slightly different form that is directly applicable to the OMVPE growth process. The independent variables for OMVPE growth of GaAs are typically the temperature, input group III partial pressure, and input group V partial pressure. We will assume the TMGa and ASH3 decompose completely to form Ga and AS4, the most stable As species at the temperatures and pressures normally used for OMVPE. These assumptions will be reconsidered in Chapter 5 dealing with kinetic factors. However, only minor complications are involved in including other species in the vapor [75, 76]. To complete the thermo­ dynamic description of the OMVPE process, we will also assume equilibrium at the vapor/solid interface. This most important conceptual step will be discussed in more detail later. One of the first empirical observations for the OMVPE growth of GaAs was that unless the input ASH3 partial pressure is greater than the input TMGa partial pressure, severe surface morphological problems occur, related to the presence of a Ga-rich liquid phase in addition to the GaAs solid. As indicated in Figure 2.9, the input vapor is highly supersaturated relative to the solid. In addition, As and Ga are taken from the vapor in a 1:1 ratio since only highly stoichiometric GaAs is being produced. This indicates that one (or both, in some cases) of the elements 48 2 Thermodynamics is nearly completely depleted at the growing interface to establish equilibrium. The input V/III ratio, the ratio of partial pressures of the group V element to the group III element in the input vapor, determines which element is depleted. When the V/III ratio is greater than unity, the Ga is depleted, and the stoichiometry of the solid is slightly As-rich. When the V/III ratio is unity, both will be depleted. If the As pressure were extremely high, equaling the As pressure at the liquidus from Figure 2.7b at the growth temperature, a second condensed phase would be formed, the As-rich liquid composition on the liquidus curve of Figure 2.7a. The As pressures along the As-rich liquidus in Figure 2.7b are lower than, but approxi­ mately equal to, the As pressures over pure elemental As. This is illustrated in Figure 2.10 where at high V/III ratios (low III/V ratios in the figure) and input As partial pressures below the vapor pressure of elemental As at the temperature for which the diagram is drawn, a single condensed phase, solid GaAs, is produced. The other case, where a second phase is formed at high-input As pressures, would virtually never occur during OMVPE growth since the vapor pressure of As is extremely large, exceeding 1 atm at ordinary growth temperatures. The situation is even simpler if the input III/V ratio exceeds unity. In that case, the As is de­ pleted from the vapor, and the Ga pressure at the interface is only slightly below the input TMGa partial pressure. The Ga partial pressures along the liquidus in Figure 2.7b are extremely small, practically always orders of magnitude below the input Ga partial pressure. Thus, a second condensed phase is again formed, the Ga-rich liquid in this case. At extremely high III/V ratios, only a single condensed phase would form, in this case liquid Ga with As dissolved. Translated into the simple terms of Figure 2.10, a single condensed phase, GaAs solid, is formed if the input III/V ratio is less than unity. Otherwise, two condensed phases are formed. This is independent of the input As pressure. The OMVPE phase diagrams for GaN, InN, and GaInN alloys have also been

GaAs(s)*Ga (I)

1.0 GaAs (s) GaA${s}*Ai{s)

K.»/^ Pf* a(28atm) Figure 2.10. GaAs phase diagram appropriate for OMVPE growth. The regions containing a single solid GaAs phase and the two two-phase regions are shown. (After Stringfellow [204].) 2.2 Phase Diagrams 49 calculated [77]. Here three regions are observed. First, of course, with the proper combination of temperature and input V/III ratio, a single semiconductor solid phase is formed. Second, at low temperatures and low V/III ratios, a group Ill- rich second phase is predicted, in agreement with experimental observations. Higher V/III ratios are required to avoid this for InN, as discussed in Chapter 8. Finally, at high temperatures and low V/III ratios, etching of the solid occurs. The GaAsSb ternary system contains Sb, a less volatile group V element. In this case it is indeed possible to use input Sb, in the form of TMSb, partial pres­ sures exceeding the vapor pressure of elemental Sb. Thus, the Sb-rich liquid phase can be experimentally observed, as seen in Figure 2.11. The presence of the liquid phase causes the vapor-liquid-solid (VLS) growth of whiskers, as seen in the scan­ ning electron microscope micrograph of Figure 2.12. For the 11/VI semiconductors, the group II element is normally more vola­ tile than the group VI element; thus, equilibrium with a single condensed phase normally occurs with a VI/II ratio of less than unity and the sohd produced is normally on the group Il-rich side of perfect stoichiometry.

GaAsSbfs}* Golf) • Droplats on Surface O No Droplets

1.50 •

< ^ 1.25 o 0 o 1.00 1.0 GaAsSbis) "T GaA55b(s)*Sb(l)

1 O 0.75 I % 1 1 B o > N 0.50 2.0 H l#

0.25 h 1 • • 4.0 U^p oi^er liquid Sb (600* C}

1, t 1 1 1 1 1 4 6 8 10

Figure 2.11. Phase diagram for the OMVPE growth of GaAsj.^Sb^ at 600°C. (After String- fellow [205],' reprinted from the Journal of Crystal Growth with permission from Elsevier Science.) 50 2 Thermodynamics

5 |xm Figure 2.12. Scanning electron microscope micrograph showing the vapor-liquid-solid whiskers produced during OMVPE growth at too low a V/III ratio. (After Stringfellow [204].)

These OMVPE phase diagrams are useful for interpreting experimental data but give less information than the complete P-T-x diagrams, which give, in prin­ ciple, the stoichiometry of the solid and the compositions of the other con­ densed phases. This is not realized in practice, since experimental data are usually lacking.

2.2.2 Ternary Systems An important category of semiconductors contains the alloys formed between bi­ nary compounds. Such alloys allow the tailoring of one or more properties for device applications. We have already briefly discussed the free energy of mixing and miscibility gaps in ternary systems. Here we address the ternai*y phase dia­ grams in much more detail, followed by a discussion of quaternary systems. The phase rule specifies that ternary systems, as compared with binary systems, will have one additional degree of freedom for a given number of phases. For example, a system consisting of only a liquid and vapor phase will have three degrees of freedom: the temperature and two compositional parameters are inde­ pendently variable. To represent this situation, a triangular, three-dimensional phase diagram is most convenient, as shown in Figure 2.13. Again, this diagram shows only the condensed phases, but the composition of the vapor phase is com­ pletely determined once the three independent variables are specified. As the tem­ perature of the system in the single liquid-phase region is reduced, the liquidus surface is eventually reached. At this point a solid phase is produced, reducing the degrees of freedom to two. Thus, the liquidus is represented as a surface. When 2.2 Phase Diagrams 51

L + AxB,.xC L+AxB,.xC

Figure 2.13. Schematic ternary III/V phase diagram. (After Stringfellow [206].) the temperature is specified, one liquid compositional parameter is still indepen­ dently variable. The solid composition always lies within a thin planar region running between the pure compounds AC and BC of Figure 2.13. Of course, since the solid alloys are not entirely stoichiometric, this region has a small, but finite, width. Normally, we ignore the nonstoichiometry in these phase diagrams. If the liquid composition lies in the same plane as the solid (i.e., when JC^. = 0.5), the phase diagram can be represented in two dimensions. Since one degree of freedom is arbitrarily removed, by considering the liquid to have only a single compositional variable, the liquidus representing the relationship between liquid composition and temperature with three phases—the vapor, liquid, and solid—in equilibrium, is a line. The line representing the equilibrium solid compositions for this same set of three phases is called the solidus. Each point on the liquidus line is the end of a horizontal tie line terminating at a point on the solidus curve. The region between the liquidus and solidus is a two-phase region where the liquid and solid coexist. The compositions of both the liquid and solid are invariant when the temperature is fixed. A change in the overall composition of the system within this region results only in a change in the relative amounts of the liquid and solid phases. The pseudobinary phase diagram for the AlAs-GaAs system is shown in Figure 2.14a. 52 2 Thermodynamics

1800

AlAs

(a)

GaAs AlAs

Figure 2.14. Al-Ga-As phase diagram: (a) pseudobinary section and (b) isothermal sections and isosolidus concentration curves for the metal-rich portion of the diagram. The data (O) are from Foster et al. [207]. (After Stringfellow [2061.)

An alternate two-dimensional representation of the three-dimensional ternary phase diagram is obtained by taking isothermal sections. Again, the liquidus is represented by a line that is different for each temperature considered. The set of liquidus isotherms for the Al-Ga-As system are shown in Figure 2.14b. These isotherms give information complementary to the pseudobinary sections. How­ ever, they give no information about the solid compositions in equilibrium with the various liquids lying on the liquidus surface. This is remedied by also includ­ ing, on the same diagram, the so-called iso-solid concentration lines [3], also shown in Figure 2.14b. During OMVPE growth, a single soUd phase is typically in equilibrium with the vapor. Thus, the number of degrees of freedom must be equal to three—the number of components, four (including the ambient gas), minus the number of 2.2 Phase Diagrams 53 phases, two, plus one (since the total system pressure is fixed). For a typical OMVPE experiment, the temperature and the vapor pressures of two components are independent variables. The partial pressure of the third component is deter­ mined by the solid/vapor equilibrium condition at the interface. The solid com­ position is then completely determined, including the stoichiometry. If the V/III ratio is greater than unity, the group III partial pressure(s) is orders of magnitude lower than in the input vapor phase. Thus, the V/III ratio at the interface and the solid stoichiometry are determined by the input group V vapor pressure(s), similar to the case described in the last section for OMVPE growth of binary compounds. In addition, if the input partial pressure of a group V element exceeds the partial pressure of the same element at the liquidus, a second condensed phase forms. This was demonstrated for the ternary GaAsSb system in Figure 2.11 in the last section. The liquidus surface and the solid compositions in equilibrium with each point on the liquidus can be calculated using the same thermodynamic concepts used for the calculation of the binary liquidus curves in the last section. The equilib­ rium conditions for the ternary system may be obtained in exactly the same way as described for binary systems, by equating the chemical potentials of the com­ ponents in the two phases: /xj, + /i». = /xXc (2.28) and

MB + Mc = MBC- (2.29) In addition, the activities in the solid must be taken into account, typically using either the DLP or the regular solution model. In either case, the enthalpy of mixing is expressed in terms of the interaction parameter in the solid. The result is a set of two equations:

in(^krkc) = ln(4

The solid activity coefficients are calculated using Equation (2.16), with fl re­ placed by 11^ The activity coefficients in the regular ternary liquid may be writ­ ten [78] RT In y\ = 0.\^x'l + ^\^x'^ + (Si\^ + ^\^ - ^\^)x\x'^^ (232ei) RT In yi^ = (I'^^x^i + n^x^^ + (H^gc + a\^ - a;,c)^k^[:. (2.32b) 54 2 Thermodynamics and RT In y[ ^^AC^A + a BC^B + (0' + ak, ^^AB)-^A-^P (2.32c) The calculation of the ternary liquidus and solidus can be carried out using values of the three liquid interaction parameters discussed earlier, and (1' obtained from the DLP model. Examples showing how closely the calculated phase dia­ grams describe the experimental data are shown in Figures 2.15 and 2.16. Know­ ing the activities in the liquid, the partial pressures of all vapor species can also be easily calculated. 11/VI phase diagrams are calculated using similar techniques. A complication is the association of anion and cation species in the liquid, which requires the slightly more complex regular associated solution model. The HgTe-CdTe phase

iSOOi

T(X

Figure 2.15. AlAs-GaAs, InP-InAs, and AlSb-InSb pseudobinary phase diagrams. Experimental data from references 208 (O), 209 (O), 210 (O, •), 211 (•) and 212 (A). (After Stringfellow [5], by permission of the publishers, Butterworths & Co., Ltd. ©.) 2.2 Phase Diagrams 55

I5CX)

1400

1300

1200

1100

lOOOh- T(^C) 900h-

8001—

700

600 h-

500

400 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 2.16. GaP-InP, AlSb-GaSb, and GaSb-InSb pseudobinary phase diagrams. Experimental data from references 213 (A), 4 H, 214 (O), 215 (D), 216 (A), 217 (O), 211 (#), 218 (•) and 219 (V). (After Stringfellow [5], by permission of the publishers, Butterworths & Co., Ltd. ©.) diagram calculated using this model for the liquid and the quasi-regular solution model for the solid, where the interaction parameter is allowed to vary with tem­ perature, as discussed in Section 2.1.2.5, is seen in Figure 2.17. Until recently, a part of the pseudobinary phase diagram in addition to the liquidus and solidus lines was ignored. Because the enthalpy of mixing for group IV, III/V, and 11/VI semiconductor solid solutions is always positive, as discussed in Section 2.1.2.5, at temperatures below a certain critical temperature {T^ in Fig­ ure 2.18), a single, homogeneous solid phase has a higher free energy than a mix­ ture with two solid phases. Since we again have the coexistence of two condensed phases, the phase rule gives a single degree of freedom. The binodal curve relates the solid compositions and temperature. The free energy versus composition curve at a fixed temperature (Figs. 2.4 and 2.5) is the key to understanding the calcula­ tion of the binodal curve. The region of two solid phases contains horizontal tie lines connecting the two solid phases in equilibrium at a given temperature. The two solid phases (labeled "soUd 1" and "solid 2" in Figure 2.18) represent the 56 2 Thermodynamics

tAC

700 L_

CdTe Figure 2.17. Liquidus and solidus lines in the CdTe-HgTe pseudobinary section. Experimental points are from references 220 (•) and 221 (0). (After Marbeuf et al. [47], reprinted from the Journal of Crystal Growth with permission from Elsevier Science.) two compositions at which a straight Une just touches the free energy versus com­ position curve (points A and B in Fig. 2.4). This construction ensures that equi­ librium prevails. Since the straight line is tangent to the G versus x curve at points A and B, transfer of an infinitesimal amount of material from phase A to phase B results in no change in the total free energy of the system. Two other important points on the G versus energy curve shown in Figure 2.4 are the inflection points lying between A and B. Between these two points the solid so­ lution is unstable against an infinitesimal fluctuation of composition. The spinodal appears on the T-x phase diagram, as indicated in Figure 2.18. In the pseudobinary phase diagram, the boundary of the unstable region is defined by the locus of (d^ Gl dy^)jp = 0 [53], called the spinode. Inside this region, the solid can decompose "spinodally," with no energy barrier. Using the DLP model, the spinodal points are expressed in terms of the difference in lattice parameters of the binary com­ ponents, Aa, if we assume Aa < a, where a is the average lattice parameter [53], d^G RT = /Vv %J5a-^^^al (2.33) _y{i - y) The critical temperature, may be obtained from Equation (2.33), as SJSKAal T = (2.34) 4Ra 4.5 2.2 Phase Diagrams 57

Solid 1 /—-- V — Solid 2

Figure 2.18. Schematic liquid-solid pseudobinary phase diagram.

Using this approximation, the critical temperature always corresponds to a critical solid composition of y^ = 0.5. The calculation can be carried out without that assumption. This yields a value of the critical composition not equal to 0.5 and changes the value of T^ by a minor amount that can usually be ignored. The prior calculation ignores the strain energy caused during spinodal decom­ position of a single crystalline solid solution when the lattice parameter is a func­ tion of solid composition. Spinodal decomposition begins by forming areas with compositions slightly different than the matrix. In the single-crystalline solids considered here, the initial stages of spinodal decomposition will certainly occur coherently (i.e., without the formation of dislocations to relieve the elastic strain). This requires the addition of a term in the free energy due to the resultant strain energy. In a classic paper, Cahn [79] showed that this coherency strain could be included in the stability criterion, yielding [53]

= 0 (2.35) 58 2 Thermodynamics where 77^ = l^aja, E is Young's modulus, and v is Poisson's ratio. The coherency strain energy acts to stabiUze the solid against infinitesimal compositional fluctua­ tions and thus reduces the temperature above which the solution is stable, which we will call T,. T^ does not include strain energy effects, but T^ does. For the DLP model,

J ^ k Lc /2 37) ARa^^ 2(1 - v)RN^' ^^'^'^ Ny is the molar volume of the solid with lattice parameter a^^a. This stabilizing term is very strong in III/V alloys, as seen from the calculated values of T^ and T^ for several III/V systems summarized in Table 2.7. These results suggest that if alloys such as GaAsSb can be grown in the region of solid innmiscibility, they will not decompose. Even at room temperature, there is no driving force for either compositional clustering or spinodal decomposition in most systems [53]. A slightly different aspect of the effect of coherency strain is the stabilization of an alloy in the region of solid immiscibility by the effect of the substrate. Add­ ing the coherency energy due to a lattice parameter mismatch between an epitaxial

Table 2.7 Calculated values of critical temperature with {T^) and without {T^) the inclusion of the coherency strain energy^

^Apor System ^Ac(A) «BC(^) ^..(K) r,(K) TERNARY ALLOYS GaAs^Sb,_^. 5.6536 6.0950 856 * Gajn,_/ 5.4512 5.8696 908 * InAs^,Sb|_^, 6.0590 6.4794 572 * GaJn,_^As 5.6536 6.059 729 * GaJn,_^N 4.520 4.980 1,523 * A1>,-,N 4.380 4.980 2,918 * GaP,N,_, 5.4512 4.520 8591 1820 GaP.Sb,'^^ 5.4512 6.095 1,965 * InP,Sb,_^, 5.8696 6.4794 1,306 * QUATERNARY ALLOYS Ga^In,_^As^,P,_^, 1,081 * GaJn|_^.As^Sb|_^, 1,428 * GaJn,_^P^Sb,_^ 2,470 * InP^As^,Sb,_^_^, 1,319 * GaP^As,Sb,_^l^ 1,996 * AI^Ga,,In,_^J^P 973 * Al^Ga^,In,_^_^As 735 * Al^Ga^In,_^_^Sb 462 *

* Tj. < 0. Alloys stable at all temperatures and concentrations. ^Data from references 42, 80, and 93. 2.2 Phase Diagrams 59 layer and the substrate on which it is grown tends to stabiUze the soUd composi­ tion that produces an exact lattice parameter match [80]. The same effect will stabilize a single solid phase that has the same lattice parameter as the substrate, even though the system has a miscibility gap, indicating a two-phase mixture would have a lower free energy when the coherency strain energy is ignored. This effect was demonstrated experimentally for the AlGaAsSb system by Nahory etal. [81]. At the surface, where the strain energy is reduced, the initial stages of spinodal decomposition apparently do, indeed, occur during both LPE [44, 82] and OMVPE growth [44, 83]. Even when the solid composition is in a region where the free energy is a weak function of composition, fluctuations in solid composi­ tion are expected. The compositional fluctuations and clustering occurring as a result of these phenomena have marked effects on the electrical and optical properties of the semiconductor solids. Fluctuations in solid composition and band gap lead to in­ creased scattering of free carriers and hence lower mobilities. The fluctuations in band gap also lead to broadening of the photoluminescence spectra. In Fig­ ure 2.19, the low temperature photoluminescence half-width, normalized by the compositional dependence of band gap dE^ldx, which yields the effective magni­ tude of the compositional fluctuation, AJC, is plotted versus the ratio of critical

30

25

ro 20 o

15

10

0.2

Figure 2.19. PL half-width normalized by dEJdx versus critical temperature divided by the growth temperature for several alloys grown lattice-matched to the appropriate substrates, including AlGaAs, GalnAs, GalnP, and GaAsSb. The solid line was drawn simply to pass through the data points. (After Cherng et al. [83], reprinted with permission from Journal of Electronic Materials, Vol. 13, 1984, a publication of the Metallurgical Society, Warrendale, Pennsylvania.) 60 2 Thermodynamics temperature to growth temperature. Clearly, even with no miscibility gap, an in­ creasing enthalpy of mixing leads to compositional fluctuations that broaden the PL peak. Benchimol et al. [84] showed that by growing at higher temperatures, the compositional fluctuations due to solid immiscibility could be minimized, re­ sulting in higher electron mobiUties in GalnAsP alloys. Epitaxial layers should also be stable during high temperature cycles necessary for device processing. However, the coherency strain energy is reduced at the surface; thus, some spinodal decomposition may occur at the surface during high- temperature annealing. Phase separation has been verified in a number of III/V alloy systems in situa­ tions where the coherency energy plays no role. An extreme case is the "doping" of GaP with N to produce green LEDs. Early LPE experiments [85, 86] demon­ strated that only a few parts per million of N could be added to GaP before the solubility limit was reached, beyond which no further N could be added. Separate crystals of GaN would precipitate from the solution along with the GaP. Later, it was demonstrated that the solubility limit could be calculated using the DLP model [87]. Subsequent experiments showed that the temperature dependence of the maximum solubility was also correctly anticipated by the DLP calculation [88], as seen in Figure 2.20. Another system with a miscibility gap that prevents

1300 1500 •n

1800

Figure 2.20. Nitrogen concentration in GaP: N as a function of crystallization temperature. Starting temperatures are shown: O, reference 88; A, reference 222; #, reference 223. For the sake of consis­ tency only the starting temperatures are given. ( ) represents the solubility limit calculated by Stringfellow [87]. (After Karpinski et al. [88], reprinted from the Journal of Crystal Growth with permission from Elsevier Science.) 2.2 Phase Diagrams 61

750

700 A rnvi o

T{-C)

650 AA

600

0 0.2 GaSb

Figure 2.21. Binodal curve for the system GaAs,_^Sb^. Data points from Pessetto and String- fellow [89] (A) are compared with those of Takenaka et al. [224] (D) and Gratton et al. [225] (O). Also included are data obtained by OMVPE growth: (•), ref. [226], and (A), ref. [227]. The binodal curve was calculated using the DLP model. (After Cherng et al. [227], reprinted with permission of the American Institute of Physics.)

LPE growth is GaAsSb. Again, LPE growth from certain Ga-rich solutions pro­ duced a two-phase, incoherent, mixture of GaAs-rich and GaSb-rich soUds [89]. The extent of the miscibihty gap was again found to agree quite well with the DLP calculation, as seen in Figure 2.21. The critical temperature is 745°C. The alloys used in blue LEDs and lasers and green LEDs, in particular the GaInN and AlInN alloys, are actually metastable [33, 34, 42]. This may partially account for the difficulties experienced in the OMVPE growth of these materials. Recent experimental data support the theoretical predictions. Singh et al. [90] re­ port evidence of a solid immiscibility for GaInN alloys containing >30% InN grown by MBE. Bedair's group [91] found evidence of spinodal decomposition in GaInN grown by OMVPE with x^^^ values of >28%. This is near the calculated 1200 +

{J 1000 4- ^0 ) 800 I « 0) 600 4- Q. E 400 +

• Single Phase (SAD) 200 • Multiphase (SAD)

0 —4— -H—'—I— 0 0.2 0.4 0.6 0.8 1 GaN InN

Figure 2.22. Binodal (solid) and spinodal (broken) curves calculated by Ho and Stringfellow [42] for the GaInN system. The data points are from Finer et al. [91].

1. 18 y^iooX Cdi _xZnxTe

0.79

/ •^'200"\

0.40

/ / y^—250"^^^

E

0.00

|\ "—asS"'^--^^

-0.39 )\\v^ Vs. 5§r^--.^^^

-0.78 1 r^ 0.00 0.25 0.50 0.75 1.00 mole fraction x Figure 2.23. Gibbs free energy of mixing (kJ/mol) as a function of;: calculated for Cd,_^Zn^Te at different temperatures r(K) below T^. (After Motta et al. [15], reprinted from the Journal of Crystal Growth with permission from Elsevier Science.) 2.2 Phase Diagrams 63

value of 22 percent InN for the spinode at 800° C, the growth temperature [42]. The predicted InN solubihty (the binode) at 800°C is only 6% [42], as seen in Figure 2.22. As mentioned in Section 2.1.2.5,11/VI solids also exhibit a positive deviation from ideality. From the bond distortions measured by EXAFS, Motta et al. [15] were able to calculate the mixing enthalpy of CdZnTe using the VFF model. Their free energy curves are reproduced in Figure 2.23. Naturally, the positive enthalpy of mixing gives rise to free energy curves that predict phase separation at low temperatures. The value of critical temperature is estimated from the data in Fig­ ure 2.23 to be approximately 430° K. The calculated binodal and spinodal curves are given in Figure 2.24. An additional complication for III/V ternary systems is the occurrence of or­ dering during epitaxial growth. The thermodynamic driving force for ordering was discussed in Sections 2.1.2.6 and 2.1.2.7. Surface effects will be discussed in Section 2.6.2. Based on first-principles calculations, Mugabe et al. [92] deter­ mined a phase diagram for the GalnP system, reproduced as Figure 2.25. Ordered structures are seen to be stable for alloy ratios of 3:1, 1:1, and 1:3 in Fig­ ure 2.25a, which represents the calculated phase diagram considering all of the

Lcd,_^n miscibility gap

400 1 \ \ / \ \ / \ \ / \ \ - / ' \ \ ' / ' \ \ ^ 300 —I ' / 1 -/ / J t spinodal ^ \ 1-. 0) 1 1 \ \ j 1 \ \ g 200 \ (D - I \ 1 \ 1 \ } \ - 1 100 i li 1 1 1 1Ml ! . I 1 1 1 1 1 1 1 1 1 III 1 0 0.2 0.4 0.6 0.8 1 mole fraction x Figure 2.24. Calculated binodal and spinodal curves for Cd,_^Zn^Te. (After Motta et al. [15], re­ printed from the Journal of Crystal Growth with permission from Elsevier Science.) 64 2 Thermodynamics

0.0 0.2 0.4 0.6 0.8 1.0 AC Composition X BC

Figure 2.25. Calculated GaP-InP-like phase diagram: (a) first-principles energies; (b) chemical en­ ergies only; (c) elastic-energy dominant. Shaded areas: single ordered phases; dashed: miscibility gap. D = disordered, yS = A,BC4,y = ABC2,5 = AB3C4, respectively. (After Mbaye and Zunger [228].) energy terms. The rather complex phase diagram is seen to obey the phase rule, as discussed for the two-soUd phase regions described earlier.

2.2.3 Quaternary Systems Quaternary systems have become increasingly important because they allow the independent selection of two physical parameters, normally energy band gap and lattice constant. The phase diagrams for quaternary systems (i.e., those containing four elements) are of two types shown schematically in Figure 2.26 [93]. Since the liquid has three compositional degrees of freedom, an isothermal diagram re­ quires all the dimensions. The distinction between the two types of systems is in the mixing in the solid. In the simpler case (Fig. 2.26b), the binary compounds share a common element. These solids are designated A^ByCi_^_yD for mix­ ing solely on the cation sublattice and AB^C^D,_^_^ for mixing on the anion sublattice. Thus, the solid-phase field is represented as a triangle in the phase diagram. For quaternary alloys with mixing on both sublattices, designated 2.2 Phase Diagrams 65

Figure 2.26. Illustrations of the two types of quaternary phase diagrams for III/V and 11/VI alloys: (a) alloys where mixing is on both sublattices, with the solid represented as A,_^B^C,_^D^;(b)alloys with mixing on only one sublattice, with the solid represented as A,_^ B^C D.

A R, .C,D,. the solid-phase field is represented as a square in Figure 2.26a. Both types of quaternary alloy may be thought of as pseudoternary. They have only independent two compositional parameters, x and y, in addition to a third compositional parameter representing the stoichiometry. Application of the phase rule for the system containing only liquid and vapor phases indicates that the system has four degrees of freedom—for example, the three liquid compositional parameters and temperature. Thus, this region is rep­ resented as a volume in the three-dimensional isothermal diagram. 66 2 Thermodynamics

If the liquid is cooled until a solid is formed, as in liquid-phase epitaxial growth, the number of degrees of freedom is reduced to three. Thus, specifying the mole fractions of three elements in the liquid completely defines the system; that is, the temperature and x and y in the solid solution are fixed as well as the solid non- stoichiometry. This situation is summarized in Table 2.6. During OMVPE growth, only the solid and vapor phases are present; thus, the system has four degrees of freedom. In a typical situation, the temperature and the partial pressures of three elements would be specified. This is easily visualized for a III/V system where mixing is entirely on the group V sublattice and the total input molar flow rate for the group V elements far exceeds the input group III molar flow rate. In this case, since nearly equal amounts of group III and group V atoms are removed from the vapor phase to produce the epitaxial layer, the partial pressures of the group V elements at the interface are essentially unchanged from their values in the input gas stream. Specifying these four parameters, the three flow rates and the temperature, determines the group III vapor pressure at the interface and the solid composition, including the nonstoichiometry. The quaternary solid/liquid-phase diagrams, which are particularly useful for LPE growth, can be calculated in a manner similar to the ternary calculations described earlier. In fact, the phase diagrams for III/V type I quaternary solids can be calculated by a direct extension of the ternary calculation described in Section 2.2.2. The two expressions equating chemical potentials in the solid and liquid phases in ternary systems are replaced by three equations involving the chemical potentials of AD, BD, and CD in the A^.B,_^C^D|_^ solid. For the type II quaternary solids, with the chemical formula A^B,_^CyDi_^, the calculation is similar. However, in this case an interesting problem arises. The four equilibrium conditions are written Mk + Ml: = MXC' (2-37a) /^B "^ Mc ^ /^BC (2.37b)

/^A + Ml) = MAD' (2.37C) and /I'B + MD ^ MBD- (2.37d) Since the thermodynamic information to be derived is the same as for the type I systems (i.e., both systems have two solid and three liquid composition parameters in addition to the temperature), one equation is redundant. It can easily be shown that only three of the four equilibrium conditions are independent. The sum of Equations (2.37a) and (2.37d) is identical to the sum of Equations (2.37b) and (2.37c). This provides an interesting constraint, or consistency condition, on either the input values such as the temperatures and entropies of fusion or on the model [6]. Using the regular solution model for treating the free energy of the solid, as developed by direct extension of the ternary calculations by Jordan, Ilegems, and Panish [7, 94] the value of an additional, quaternary interaction parameter is speci- 2.2 Phase Diagrams 67 fied to ensure consistency. A similar result was obtained by Onda and Ito [95] using a more complex model where bond energies in the solid are summed, with the interatomic distances allowed to depart from the simple virtual crystal model, to be consistent with the modern understanding of the structures of actual semi­ conductor alloys discussed earlier. Hence, the bond energies are allowed to vary with alloy composition. The expression for the excess free energy of the type I quaternary solids is identical to the regular solution model expression. However, for the type II quaternary solids, the expression for the excess free energy of the solid is quite different, although the interaction parameters are similar, including the addition of a "quaternary" interaction parameter in addition to the conven­ tional ternary interaction parameters. In the DLP model, since there are no adjustable parameters, only the number of bonds in the solid, which is not specified by the chemical potentials, can be ad­ justed to satisfy the consistency condition, as described by Koukitu and Seki [96]. The physical validity and significance of this assumption are not entirely clear at present. Ichimura and Sasaki [97] have calculated the deviations from an ideal, random arrangement for the atoms in type II solid solutions based on a quasi-chemical equilibrium model that includes both nearest and next nearest neighbor interac­ tions, similar to the model of Onda and Ito described earlier. The deviations from randomness are found to have virtually no effect on the energy band gap or lattice constant. We have already described the positive enthalpy of mixing in ternary alloys, which can lead to phase separation when the size difference between the end com­ ponents exceeds a few percent. In quaternary systems where the solid-phase field is a triangle (i.e., those where mixing occurs on a single sublattice), the critical temperature in the quaternary is the same as that of the pseudobinary with the highest value of T^. The immiscibility problem is accentuated in quaternary solid solutions where mixing occurs on both sublattices. In this case the binodal surface is defined by the free energy versus composition surfaces at each temperature. Two binodal points, connected by a tie line, are defined by a plane surface touch­ ing the free energy versus x and y surface at two points where the derivatives of free energy with respect to both x and y are equal. A calculation along these lines using the DLP model [98] results in an expression for the critical temperature in terms of the lattice parameter differences for the change in group III component, AaA, and group V component, Aa^, for the four binary constituents

(2.38) where K is the DLP constant, defined previously, and the parameters related to the differences in lattice constant are D = a^^^ — a^^ — a^j^ + a^^, A«^ = 68 2 Thermodynamics

V^AD ^BD "^ ^AC a^^)l2, and Aa^^ = {a BC ^BD "^ ^AC a^^)l2. Clearly, the critical temperatures are higher for these quaternary systems since the effects of bond strain energy due to differences in lattice parameter on the two sublattices are additive. The critical temperatures calculated using the DLP model for several III/V quaternary systems are listed in Table 2.7. The significance of the miscibility gap was first recognized by deCremoux et al. for the GalnAsP system [99]. Using the DLP model, they calculated the range of solid immiscibility and found that it included the region needed for the LPE growth of the solid compositions necessary for 1.55 micron lasers lattice-matched to the InP substrate. Figure 2.27 shows the best experimental data for the GalnAsP solid-solid phase diagram, obtained from LPE growth experiments [82] and from long-term annealing experiments in an ambient allowing vapor transport of the materials, and hence more rapid attainment of equilibrium by diffusion through the vapor phase rather than by extremely slow solid-state diffusion [100]. The experimental data are compared with results calculated using the DLP model [93]. Calculated tie lines are included that indicate the compositions of the two solids that would be formed from the decomposition of a solid lying inside the miscibil­ ity gap. The extent of the occurrence of immiscibility in quaternary III/V systems is illustrated by Figure 2.28, the composite diagram calculated using the regular so­ lution model by Onabe [101] showing the spinodal isotherms in all quaternary systems containing Al, Ga, and In combined with P, As, and Sb. The nitrides of Al, Ga, and In have become increasingly important in recent

GaP 0\—

950" K

• "? h

0943"'K Quillec et al,(l98iT~~ ^ ^I053°K Spring Thorpe etol. (1983) 1.0 1.0 0 GoAs InAs

Figure 2.27. Binodal isotherms for the system GalnAsP. The soMd lines were calculated using the DLP model. The data are from SpringThorpe et al. [lOOJ (A) and Quillec et al. [82] (O). (After Stringfellow [93], reprinted from the Journal of Crystal Growth with permission from Elsevier Science.) 2.2 Phase Diagrams 69

Figure 2.28. Spinodal curves for III-V quaternary solid solutions at 400-1,000°C (solid lines). Temperatures are indicated as 4 for 400° C and so on. Dashed lines represent the compositions for lattice matching to GaAs, InP, InAs, and GaSb. Crosses represent the critical points for A,_^B,C,_,D,,quaternaries. (After Onabe [229], © NEC Corporation, 1984.) years, as discussed in Chapters 8-10. The AlGaInN phase diagram is predicted, using a VFF model calculation [42, 102], to have a significant region of solid immiscibility, as seen in Figure 2.29.

AIN

GaN InN Figure 2.29. Calculated solid AlGalnP phase diagram. Heavy solid lines represent the 1,100°C isotherm, and lighter straight lines are the tie lines. (After Ho [102].) ^0 2 Thermodynamics 2.3 Thermodynamic Driving Force for Epitaxial Growth 2.3.1 Equilibrium Conditions For the simple process

A^ B, (2.39) the equilibrium condition is fjil"^ RTIn a% = /ji^^ + RTIn a^, (2.40) where a*^ is the activity at equilibrium. Thus,

a%al = exp (^ '^^^^ ^^ j = ^i = ^^ (2.41) which is the basic law of mass action. AT^ is the equilibrium constant. When the system is not at equilibrium, the thermodynamic driving force to restore equilibrium is A/x = /XQ — /x^, ^/uL = fjil + RT In a%- fjil- RT In a% (lAldi) or

= /?rin(^^^|. (2.42b)

This is the driving force for epitaxy, which is shown for GaAs growth by various epitaxial techniques in Figure 2.9. A nonequilibrium situation is intentionally cre­ ated that drives the system to produce the solid desired. The maximum quantity of this solid that can be produced is simply the amount that would establish equilib­ rium (the supersaturation) and is, thus, fundamentally limited by thermodynamics and the total size of the system (i.e., the total volume of gas passing through the reactor for OMVPE growth) [67]. 2.3.2 Equilibrium at the Solid/Vapor Interface Ordinarily, in the OMVPE system, the growth rate is considerably slower than that calculated from thermodynamics. Kinetics, both surface reaction rates, which will be discussed in detail in Chapter 5, and mass transport through the gas phase, as will be discussed in Chapter 6, are not fast enough to allow equihbrium to be established throughout the system at all times. This situation is illustrated in Fig­ ure 2.30a, where A^t from Equation (2.42) is plotted versus reaction coordinate. This allows the schematic representation of the overall, thermodynamic driving 2.3 Thermodynamic Driving Force for Epitaxial Growth 71

Solid

A^

Reaction Coordinate

(a)

A/x

Reaction Coordinate

(b) Figure 2.30. Schematic diagram of chemical potential versus reaction coordinate, showing the drop in chemical potential required for each step in the growth sequence to keep all rates equal. The difference in individual chemical potentials can alternatively be thought of as ratios of partial pres­ sures of the reactants: (a) the general case and (b) the case of rapid surface kinetics—that is, with A/x^ <^ ^/"-p. (After Stringfellow [67], reprinted from the Journal of Crystal Growth with permission from Elsevier Science.) force for the growth reaction, represented as AyLt*, where * denotes the chemical potential of the input gas phase, calculated with p^ = /7f for all reactants. The growth rate is proportional to the flux of atoms being transported, usually by dif­ fusion, through the gas phase to the interface, which is identical to the flux of atoms crossing the interface into the solid. This diagram shows schematically the driving forces necessary to sustain this flux for the diffusion process (A/Xj^) and the surface reactions (A/x J. Even in cases with a large supersaturation in the input vapor phase (i.e., for A/x* > 0), near-equilibrium conditions may exist at the growing solid surface. This simply requires that the interface kinetics be much more rapid than the diffusion kinetics—that is, that the two processes proceed at the same rate with A/x^ < A/Xi3 [67]. This situation, termed diffusion- (or, more precisely, mass- transport-) limited growth, is shown schematically in Figure 2.29b. Using or­ dinary growth conditions, with temperatures between approximately 550 and 800° C, this is the normal situation for OMVPE growth of GaAs, for example, as deduced from the nearly temperature independent growth rate shown in Figure 1.1 of Chapter 1. For surface kinetically limited processes, the growth rate increases exponentially with increasing temperature. This occurs for the OMVPE growth of 7'2 2 Thermodynamics

GaAs only at temperatures below approximately 550°C. Normally an effort is made to stay within the diffusion-limited growth regime for the growth of high- quality GaAs epitaxial layers. In this situation, the interfacial partial pressures nearly satisfy the equilibrium relationships. Earlier in this chapter we described equilibrium as occurring only between homogeneous phases. Thus, only the vapor in an infinitesimally small volume right at the interface is considered. Another condition for the application of equilibrium principles at the interface is that the processes be reversible. The species considered in the thermodynamic calculations must be able to cross the interface traveling both into the solid as well as into the vapor. This may present a problem in the thermodynamic analysis, since organometallic group III com­ pounds are input into the reactor but probably do not survive to be released from the heated semiconductor surface during growth. The group V sources may, in some cases, also be so unstable that they are also not involved in the actual equi­ librium process occurring at the growing solid/vapor interface. Thus, the species for the equilibrium analysis must be carefully chosen. As an example of the detailed calculation, consider the most common system, TMGa and ASH3 used for the growth of GaAs. The pyrolysis reactions for TMGa and ASH3 separately as well as in combination for GaAs growth will be discussed in detail in Chapter 5. The salient feature for this discussion is that the pyrolysis of the combined reactants is complete by temperatures of approximately 500° C. This somewhat simplifies the choice of reactants for the thermodynamic calcula­ tion. The overall growth reaction can be thought of as consisting of two parts: the group III and group V pyrolysis reactions and the subsequent reactions of the pyrolysis products to form the GaAs solid. The thermodynamic analysis of the TMGa and ASH3 pyrolysis reactions is simple; the reactions go essentially to completion. The TMGa pyrolysis produces Ga, which has a small, but finite, vapor pressure given by the P-T-x diagram in Figure 2.7, discussed in detail ear­ lier. Other species such as GaCH3, GaH2, and GaH are also present at extremely low partial pressures [103]. As we will see, these other Ga species have no effect on the thermodynamic analysis since the partial pressures of all the Ga species at the interface are so low. This is also true for the actual OMVPE growth process. The Ga pressure is, of course, dependent on the As vapor pressure at the interface. A decision about the most appropriate As species is much more complex. At equilibrium the major species is AS4 at the temperatures and pressures in OMVPE reactors during GaAs growth, as seen in Figure 2.7. However, when GaAs is heated, As2 is the major species leaving the solid. In fact, GaAs is sometimes used as an AS2 source for MBE growth. Heated, elemental As gives mainly AS4. Ex­ amining the reverse reaction, early mass spectrometric studies indicated that much more than the equilibrium amount of As 2 (and much less than the equilibrium amount of AS4) is produced during ASH3 pyrolysis [104]. In addition, the extent of hydride pyrolysis was found to be less than that predicted from a thermody- 2.3 Thermodynamic Driving Force for Epitaxial Growth 73

namic analysis, although later studies have shown the pyrolysis to be nearly com­ plete when TMGa is present, as will be discussed in Chapter 5. These problems are examples of kinetic hindrance of the ASH3 pyrolysis process. For the thermo­ dynamic analysis, two choices seem reasonable: use of the thermodynamic equi­ librium products, AS4 (and/or As2 in the thermodynamically determined amount) and Ga, or use of the likely pyrolysis products taking into account kinetic hin­ drance of AS4 formation (i.e., only AS2 and Ga), assuming that one mole of AS2 is formed for every two moles of ASH3 pyrolyzed. Of course, even this is a consid­ erable oversimplification of a complex situation. For example, the tetramers them­ selves apparendy cannot participate in the actual OMVPE growth process. This is not apparent for AS4, but when elemental phosphorus is used, along with TMIn, as a source for the OMVPE growth of InP, no growth occurs until the extremely stable P4 is broken apart [105, 106], by using a plasma, for example. The remainder of the thermodynamic analysis is simple. In the mass-transport- limited case, illustrated schematically in Figure 2.30b, the interfacial partial pres­ sures, pj, nearly satisfy the equilibrium, mass-action equation.

^GaAs 1/4 = ^GaAsGaAs.' (2.43) PCaiPA^ where K^^^s^s is the equilibrium constant for the reaction of elemental Ga plus AS4 to form solid GaAs. Consideration of kinetic hindrance of formation of the tetra- mer would lead to a similar mass-action expression involving AS2, with a higher value for the equilibrium constant, since the dimer is thermodynamically less Stable than the tetramer, which leads to a larger thermodynamic driving force for the growth reaction. For 11/VI systems, the stability of some group VI hydrides, such as H2S, pre­ cludes the assumption that pyrolysis is complete. In this case, Kisker and Zawad- ski [36] derived mass-action equations, similar to those for complete dissociation of the source molecules, based on the monatomic group II species and the group VI hydride being the equilibrium species at the interface. For example, for the growth of ZnS from DEZn and H2S,

;r-3^ = ^z„s- (2.44)

In cases where the group VI pyrolysis is expected to be complete, equations simi­ lar to the mass-action expression for GaAs, Equation (2.43), result. The group VI molecules form higher polyatomic molecules, such as Sg, that are ignored due to kinetic limitations to their formation rates similar to those discussed earlier for the group V tetramer molecules. The processes occurring on the surface can be treated in a manner entirely analogous to the prior discussion. A nonuniform adatom population exists on the ^"4 2 Thermodynamics surface during growth. It is often assumed that a near-equilibrium condition is established at the step edge where adatom incorporation into the lattice occurs. This is approximated as an excess adatom population of zero, which makes the analysis of step-flow growth and two-dimensional nucleation relatively simple [107]. A concentration gradient is formed at the step edge, and diffusion to the step controls the growth rate. Two-dimensional nucleation occurs when the super- saturation on the flat terrace between steps exceeds a critical value. Here the im­ plicit assumption is identical to that for the analysis of the vapor/solid interface, described earlier—namely, that the diffusion process requires a much larger chemical potential difference than is required to drive adatom incorporation at the step. This gives a near-equilibrium adatom condition on the surface near the step edge.

2.3.3 Growth Rate We continue the thermodynamic analysis of OMVPE growth of GaAs to see what practical information can be obtained. Since the input vapor is highly supersaturated, Pl.ip\.y'>Ph.iP\.X"- (2.45) This is equivalent to stating that Ayit* > 0. For the typical case, Pl. < 4/;*,/, (2.46) that is, the V/III ratio is >l. This means that the Ga is nearly depleted at the interface, Ph.

PAS.^PU- (2.48) since the same number of As and Ga atoms are removed from the vapor phase to produce GaAs. This situation makes the analysis of growth rate and solid com­ position particularly simple. The growth rate is proportional to the flux of Ga and As atoms arriving at the interface. For the sake of simplicity, let us assume that the mass-transport process is simple diffusion through a fictitious laminar-flow boundary layer of thickness 8Q. This will be discussed more accurately in Chapter 6, with similar results. The two fluxes are equal, since stoichiometric GaAs is the only product. The flux may be expressed,

/ = ^Ga(/^Ga ~~ PG2) /2 ^(^\ RT8, 2.3 Thermodynamic Driving Force for Epitaxial Growth 75

where D^^ is the diffusion coefficient of Ga, in whatever form it may occur while diffusing through the boundary layer. The completely accurate description of mass transport limited growth must yield an expression similar to Equation (2.49) since the mass transport rate is proportional to the concentration gradient. The other factor, DQJRTSQ, may be considered to be an effective mass-transport coefficient. In either case, the Ga flux, hence the GaAs growth rate, is proportional top%^ only if p^^ <^ p%^. This is an accurate representation of the experimental observations for the OMVPE growth of essentially all III/V systems. Equally clear is that the ratio of the concentrations of A and B for alloys with mixing on the group III sublattice, A^Bi_^C, will be the same as the ratiop\lp%, assuming the diffusion coefficients for the A and B species are nearly equal. This will be discussed in more detail later. A better assumption for the thermodynamic analysis of GaAs growth might be that As2 is the important arsenic species. The change from the tetramer to the dimer in the analysis to this point would cause no perceptible change in the results. However, we have yet to discuss the issue of nonstoichiometry of the GaAs or the growth of alloys with mixing on the group V sublattice. In these cases, the assumption of which group V species to use is more important. The analysis of growth rate in 11/VI systems is more complex, since the vapor pressures of the group II elements are fairly high at typical growth temperatures. An example of the effect of the high group II vapor pressures is the ability to grow 11/VI compounds at cation/anion ratios in the input vapor phase exceeding unity. Lichtmann et al. [108] report the growth of CdTe, using DHTe and DMCd, with 11/VI ratios varying from 3 to 35. This is very unlike the results for III/V semi­ conductors where these high values of cation/anion ratio would result in growth of a two-phase, liquid + solid, mixture as shown in Figures 2.10 and 2.11. As a result of the high group II vapor pressures, the analysis of both growth rate and the cation distribution coefficient must be modified to include the possibility that the group VI element is completely depleted at the interface. This will obviously result in the control of growth rate by the group VI molar flow rate. The CdTe growth rate in the work of Lichtmann et al. [108] was found to be a linear function of DHTe flow rate, as seen in Figure 2.31. Similarly, Parsons et al. [109] grew HgTe and CdTe using MATe, DMCd, and DMHg at values of cation/anion ratio of greater than unity and again report the growth rate to be proportional to the molar flow rate of the Te source. This is true of essentially all Te precursors, as discussed in Chapter 7.

2.3.4 Solid Nonstoichiometry The V/III ratio is commonly varied in an effort to study and/or control the stoi- chiometry of the compound semiconductor. This is important because the stoichi- ometry affects dopant and impurity incorporation as well as the concentrations of 76 2 Thermodynamics

10 100 1000 FLOW RATE OF DILUENT H2 THROUGH DHTe (seem)

Figure 2.31. Temperature-independent growth rate of CdTe as measured on (100) InSb substrates as a function of carrier HT flow rate through a subUmer containing solid chunks of DHTe. (After Lichtmann et al. [108], reprinted from the Journal of Crystal Growth with permission from Elsevier Science.) native defects (i.e., those involving interstitial atoms and vacancies). However, the V/III ratio must be used with some care. In fact, it is very frequently misused and misinterpreted in the literature. An example is when the V/III ratio is varied by changing the group III input partial pressure in cases where the V/III ratio is greater than unity. As will be demonstrated, this changes the growth rate, but not the solid stoichiometry. Solid stoichiometry is controlled by the partial pressures of the group III and group V elements at the growth interface. In our notation, the factor affecting stoichiometry is Pylpiu, not p%lpXii. For the conditions specified in Equations (2.46)-(2.48) (i.e., an input V/III ratio of greater than unity, generalized to include all III/V semiconductors), the group III partial pressure at the interface is com­ pletely determined by the input group V partial pressure [110],

MII/V 1 Pm = (2.50) ^111/V P V Thus, the V/III ratio at the interface may be written Py_ K, P\i' (2.51) Pill 2.4 Solid Composition 11

We see that p\^ has absolutely no effect on solid stoichiometry and depends not on p%, as might be naively expected, but on {p%) ^. This is so often misunderstood that it is worthwhile stressing that even in situations where the basic assumption of near equilibrium at the interface is invalid (i.e., at very low growth tempera­ tures), the input group V partial pressure, corrected for incomplete pyrolysis if necessary, will be the determining factor for the V/III ratio at the interface. Since equal quantities of the group III and the group V elements are depleted from the vapor, the group V partial pressure at the interface will be approximately that in the input gas stream, and the group III partial pressure will be orders of magnitude lower than those in the input gas stream. Simply stated, if the growth rate is pro­ portional to the input group III flow rate, the group III element is essentially used up at the interface. For 11/VI semiconductors, the analysis of solid stoichiometry is similar, with the complication that the vapor may be either group II- or group Vl-rich. In the former case, the group II input partial pressure controls stoichiometry. The latter case is exactly like the analysis for III/V semiconductors: the input anion partial pressure controls the solid stoichiometry.

2.4 Solid Composition

Using thermodynamic considerations, the solid composition can be analyzed in terms of the partial pressures in the vapor phase and the substrate temperature during growth. The equilibrium conditions for OMVPE growth of ternary, or, more precisely, pseudobinary, alloys may be written in exactly the same form as those for the solid/liquid equilibrium, Equations (2.28) and (2.29). For the solid/ vapor equilibrium, /xX + A^c = A^AC (2.52) and MB + M£ = /^k- (2.53) This leads to two mass action expressions, similar to Equations (2.43) and (2.44), for the partial pressures of the reactants at the solid/vapor interface. As for the binary case, the chemical potential differences necessary to drive the surface pro­ cesses are assumed to be small, allowing the approximation of thermodynamic equilibrium at the interface. These mass action expressions can be used along with conservation conditions to calculate the solid composition.

2.4.1 Mixing on the Cation Sublattice For alloys of the type A^B^.^C, where mixing occurs on the cation sublattice, the analysis of solid composition in terms of vapor composition is particularly simple. 78 2 Thermodynamics

Consider the case of the III/V semiconductors. Under normal growth conditions, the input vapor is highly supersaturated and the input V/III ratio is larger than unity. Thus, the partial pressures of both group III components at the interface are nearly zero. Essentially all of the group III atoms reaching the growing solid are incorporated. Thus, the solid composition is determined by the rate of mass trans­ port of each group III element to the interface. If we allow that the mass transport coefficients are proportional to the diffusion coefficients, D^ and DQ , the solid composition can be calculated.

^AP*A (2-54) J^^h D^pl + D^pl If the two diffusion coefficients are approximately equal, the distribution coeffi­ cient, defined as the ratio of the concentration of A to B in the solid divided by that in the vapor. xl{\ - x) k = (2.55) PVP is approximately unity. For the AlGaAs, InGaAs, and AlGaSb systems, the distri­ bution coefficients are, indeed, nearly unity, as seen by the data reproduced in Figure 2.32. However, at higher temperatures, the group III elements can have

O AI^Gai.xAs (Mori et

0.6 h

0.4 h

0.2

0.2 0.4 0.6 0.8 1.0

Figure 2.32. Solid versus vapor concentration for the III/V alloys: (O) Al^Ga|_^As (data from Mori and Watanabe [230]); (A) In,Ga,_,As (data from Ludowise et al. [231]); (D) Al,Ga,_,Sb(data from Cooper et al. [232].) 2.4 Solid Composition 79 significant volatility, which allows for group III distribution coefficients that are not equal to unity. Since In is the most volatile of the common group III elements used in III/V semiconductors. In distribution coefficients of less than unity are observed at high temperatures. An excellent example comes from the growth of GaInN, where high growth temperatures are common. Matsuoka et al. [111] grew high-quality GaInN alloys by low-pressure OMVPE using TMIn, TMGa, and am­ monia on sapphire substrates. The distribution coefficient of In was found to be approximately unity at a growth temperature of 500° C and 0.1 at 800° C. In this case, the distribution coefficient can still be described thermodynamically using the DLP model to describe the enthalpy of mixing [77]. Even the strain energy due to lattice mismatch can affect the group III incorporation into the solid, as discussed in the following section.

2.4.1.1 Effect of Strain As discussed earlier, the group III distribution coefficient is not unity for growth at high temperatures where the group III atoms can rapidly evaporate from the surface. This reveals an interesting phenomenon—lattice latching or lattice pulling. For growth of a thin epitaxial layer of an alloy having an equilibrium lattice constant differing from that of the substrate, the layer is stretched in two dimen­ sions to match the lattice spacing of the substrate. This produces a strain energy in the epitaxial layer that increases the total free energy of the system. As dis­ cussed in more detail in Chapter 9, the system attempts to reduce the total energy in various ways. One possibility for reducing the total free energy is to change the alloy composition to decrease the lattice mismatch. This, of course, reduces the strain energy while increasing the chemical free energy. The minimum free energy can be calculated by simply including the strain energy in the total free energy of the system. The effect of lattice mismatch on solid composition was first studied in 1972 for the LPE growth of Gain? layers on GaAs substrates [80]. It was observed that as the composition of the liquid was gradually changed, from run to run, the com­ position of the unconstrained, noncoherent platelets growing around the edge of the substrate changed as expected from the phase diagram. However, the compo­ sition of the epitaxial layer remained nearly constant. Based on the prior discussion of the enthalpy of mixing for III/V alloys, it should not be surprising that the macroscopic strain energy is large enough to have a significant effect on the solid composition. In systems near the critical tempera­ ture, such as GalnP, the free energy versus composition curve is nearly flat, so a perturbation due to the elastic energy will have a dominant effect. This "lattice latching'' was confirmed for the LPE growth of GalnP (see, e.g., references 112 and 113), GalnAs [114, 115], GalnAsP [116], and AlGaAsSb [117]. Similar effects have been observed for the MBE growth of AlInAs [118, 119], 80 2 Thermodynamics but only at high temperatures where In can rapidly evaporate from the surface. Bugge et al. [120] clearly observed a reduced In incorporation in highly strained GalnAs/GaAs quantum wells grown by OMVPE. As for the MBE results, the In is reevaporated from the surface. Some evidence of lattice latching has also been observed for GaInN layers grown by OMVPE [121]. The same type of phenomena can also stabilize metastable alloys. The occur­ rence of miscibility gaps in semiconductor systems was described earlier. How­ ever, when grown coherently on a substrate, the strain energy associated with the growth of the alloys at the binodal points, not including strain energy, may make them less stable than a single, lattice-matched alloy. This phenomenon is similar to the decrease in the critical temperature for coherent spinodal decomposition, described earlier. A summary and theoretical analysis of lattice latching and the strain stabilization effect can be found in a review by Zunger [122]. 2.4.1.2 II/Vl Alloys The results for 11/VI alloys with mixing on the group II sublattice are similar to those for the III/V alloys with mixing on the group III sublattice. Wright et al. [123] grew epitaxial layers of CdZnS using the reactants DMCd, DMZn, andH2S. They obtained the relationship between solid and vapor composition shown in Figure 2.33. The distribution coefficient is approximately unity at the growth temperature of 400° C. Superficially, this appears to be exactly the same as for the III/V alloys, since the values of VI/II ratio during growth were greater than unity. The group II species should be depleted at the interface because of the high degree of supersaturation in the input vapor phase. However, since the vapor pressures of the group II elements are fairly high at normal growth temperatures, 11/VI alloys can also be grown at VI/II ratios of less than unity. This should lead to an anion distribution coefficient of approximately unity. In that case, the analysis would have to be performed in a manner similar to the calculation in the next section to give a meaningful description of the cation distribution coefficient. For example, the higher thermodynamic stability of CdTe (A//^ = -24.5 kcal/mol) as com­ pared with HgTe (A//^ = -8.1 kcal/mol) apparently results in high Cd distribu­ tion coefficients [124]. 2.4.2 Mixing on the Anion Sublattice The analysis of solid composition for alloys of the type AC^D,_^, with mixing on the anion sublattice, is much more interesting. This is partially due to the use of input anion/cation (V/III or VI/II) ratios in the input vapor phase of greater than unity, which is possible because of the high volatility of the anions, as dis­ cussed in Section 2.2.1.1 in conjunction with the OMVPE phase diagrams. The other factor making the analysis of mixing on the anion sublattice more difficult is the incomplete pyrolysis of the anion source molecules and the problem of 2.4 Solid Composition 81

Figure 2.33. Composition of Cd^Zn,_^S layers as a function of the organometallic compound gas- phase ratio [DMCd]/([DMCd] + [DMZn]). Results for 400°C. (After Wright et al. [123], reprinted from the Journal of Crystal Growth with permission from Elsevier Science.) establishing the gas-phase species to be considered in the thermodynamic analy­ sis. The case of mixing on the cation sublattice discussed in the last subsection is independent of the anion species providing the anion/cation ratio in the input vapor is large. As we will see, this is not true for mixing on the anion sublattice. Consider first the case where the pyrolysis rate for the anion sources is rapid. A useful example is the OMVPE growth of InAsi_^Sb^. The experimental data of Fukui and Horikoshi [125] are plotted in Figure 2.34. The thermodynamic cal­ culation of the Sb distribution coefficient, defined as k^^ = {xl^/x%^)/(xUx\^), where x^i, "^ PTESb^CPirESb "^ P ASH3 )' i^ Q^i^^ simple. We assume that the pyrolysis of the source molecules—TEIn, TESb, and AsHg, in this case—is complete. This allows the thermodynamic calculation in terms of the partial pressures of In, Sb4, and AS4 in the vapor phase at the interface, if we also assume the thermodynami- cally more stable tetramers to be the predominant group V species. Naturally, we also make the approximation of thermodynamic equilibrium at the solid/vapor interface, which yields the two mass-action expressions.

[/4 ^InSb (2.56a) P'lniPsb,) 82 2 Thermodynamics

1.0 , Q O Fukui a HorlkoshI (1980) T-500*C V/m ~ 10 0.1

0.6

0.4

0.2

1.0 ^° TESb/(f»0 TESb ^ f'^^AsHa) Figure 2,34. Solid versus vapor composition for the alloy InAs,__^Sb^. The data are from Fukui and Horikoshi [125]. The solid line was calculated with no adjustable parameters. (After String- fellow [233], reprinted from the Journal of Crystal Growth with permission from Elsevier Science.) and

*InAs „i („i ^1/4 ^InAs- (2.56b) FlnVFAs4>' Two additional conservation constraints are imposed, one on composition

Psb^ PshA (2.57) PSbA Psb4 "^ PAS4 PAS4 and one on stoichiometry

Pt - Pin = ^(Psb4 ~ Psb, + PAS4 PASJ- (2.58) Together, this gives four equations and four unknowns—x, Pca^Psb^^ ^^^PAS^— for a given temperature and specified input gas flow rates or partial pressures. The only further consideration is the treatment of the nonideality of the solid. We simply use the regular solution model expression, Equation (2.16) for the activity coefficient in the InAs,_^Sb^ solid using a value of 2,250 cal/mol for the inter­ action parameter, from Table 2.3. The values of ATi^s^, ^^^ ^inAs ^^^^ determined from the compilation of Brebrick [126]. These values are for the liquid, rather than the vapor, in equilibrium with the solid. Thus, they must be corrected using 2.4 Solid Composition 83

the liquid to vapor transition energies listed in Stull and Sinke [127]. The curve in Figure 2.34 was calculated in this manner, with absolutely no adjustable pa­ rameters. The excellent description of the experimental data indicates clearly that thermodynamic considerations control alloy composition in this system. It is in­ teresting to note that the incorporation of Sb into the solid is suppressed. From simple kinetic (i.e., ^'sticking coefficient") arguments, we might have concluded that the less volatile element would be preferentially incorporated. However, the thermodynamic calculation is simply a quantitative statement that the In-As bond is stronger than the In-Sb bond, which leads to preferential incorporation of As into the solid. This example illustrates the power of the thermodynamic analysis, which assumes equilibrium at the growing interface. A still more complex and interesting system is GaAs,_^Sb^. The two mass- action expressions, one for GaAs and one for GaSb, are solved simultaneously with the two conservation equations for solid composition and solid stoichiometry similar to Equations (2.57) and (2.58). The approximations described earlier are also made for this system, and the activity coefficient in the solid is calculated [128] using the DLP model, which is nearly equivalent to using a solid-phase GaAs-GaSb interaction parameter of 4,000 cal/mol in the regular solution model. The solid composition is plotted versus vapor composition in Figure 2.35, where

1.0 IdOCooper et al,(l982) (OMVPE) ^\ • Present work T«600"»C ///\ 0.8h— ^///\ <* / / \ o y / 1

0.6h1 » » //I (0 / o 0 ^ \ 04

// in/v>i z'' 0.2 hh V • O / ' y/m/V = 0.5-;^^X^

•^"^^^ 1 1 1 \ J 0.2 0.4 0.6 0.8 1.0

Figure 2.35. Solid versus vapor composition for the alloy GaAs, _^Sb^. The data are from the work of Cooper et al. [226] for V/III = 2.0 (O), and V/III = 0.5 (3) and from the work of Stringfellow and Cherng [234] (#). The curves were calculated for various V/III ratios. The broken sections of each curve represent the calculated regions of solid immiscibility. (After Stringfellow and Cherng [234].) 84 2 Thermodynamics experimental data are compared with the calculated results. Several important as­ pects of OMVPE are illustrated in this rather complex figure. First, consider the open data points, obtained for an input V/III ratio of 2.0. Notice that the calculated curve for V/III = 2.0 fits the data well. The Sb distribution coefficient is seen to be less than unity. This accords with our discussion of the Sb distribution coeffi­ cient in the InAsSb system. An additional important point is that the calculation for V/III < 1 yields an antimony distribution coefficient of unity. As discussed in Section 2.4.1 for the case of alloys with mixing on the group III sublattice, when V/III > 1, essentially all of the group III elements reaching the interface are incorporated. The case of GaAsSb with mixing on the group V sublattice with V/III < 1 is completely analogous. The establishment of equilibrium at the inter­ face while the input vapor is highly supersaturated requires that the group V ele­ ments must be virtually exhausted at the interface. A final point relative to Figure 2.35 is the solid-phase miscibility gap where two solid phases are present. The binodal curve for the GaAsSb system was discussed in Section 2.2.2 and is shown in Figure 2.21. However, when the V/III ratio is less than unity, the As and Sb atoms arriving in a random pattern at the surface do not have time to redistribute themselves into GaAs- and GaSb-rich areas before being covered over by the next layer. Thus, owing to kinetic effects, it is possible to grow metastable GaAs,_^Sb^ alloys throughout the entire range of solid compo­ sition, as shown by the solid data points in Figure 2.35. These data were obtained by growth on a sapphire substrate, so stabilization by epilayer/substrate mismatch, discussed in Section 2.4.1.1, is eliminated. They were obtained in a single run in a reactor with incomplete mixing. The V/III ratio varied from one side of the susceptor to the other, causing the change in solid composition fromx|^, < 0.1 to 0.5 fora valueof jc^j^of 0.5. The effect of input V/III ratio on solid composition for GaAs, _,Sb^ is dramatic [128], as seen in Figure 2.36. As already discussed, for values of input V/III ratio that are much greater than unity, the higher stability of GaAs yields a small Sb distribution coefficient. For values of input V/III ratio approximately equal to unity, all As and Sb reaching the interface are incorporated so k^^ = 1. The solid line in Figure 2.36, calculated as described earlier, with no adjustable parameters, gives an excellent description of the experimental data. Significantly, the quality of the OMVPE-grown GaAsg .^ Sb^ ^ layers lattice- matched to the InP substrate was shown to be excellent. The epilayers have excel­ lent surface morphologies and photoluminescence emission that is strong, but somewhat broadened, apparently by compositional fluctuations at the surface dur­ ing growth, as discussed in Section 2.2.2. Other, even more metastable alloys have also been grown by OMVPE employ­ ing the "trick" of using near-unity values of V/III ratio. This resulted in the growth, for the first time, of layers of the quaternary GalnAsSb alloy, having a value of T^ of 1,467°C, throughout the entire range of solid composition. The 2.4 Solid Composition 85

1 1 1 1 1

GaAs/.^Sb^ o '' k ^ 5b = 0.5 T = 600"C 0 x-ray 0 0 \ • EOS(SEM) 0.8 r- » PL A 0 "^

O / 0.6 \-h A

0.4

0.2 -

1 1 1 1 1 1 0.2 0.4 0.6 0.8 1.0 1.2 n/I Ratio Figure 2,36. Sb distribution coefficient (mole fraction GaSb in the solid/ratio of TMSb to total group V in the input vapor phase) versus III/V ratio in the input vapor phase. The data were obtained using various methods for the determination of solid composition. The curve was calculated assuming thermodynamic equilibrium to be established at the growing interface, as described in the text. (After Cherngetal. [227].) largest difference in atomic radius for the common III/V elements is between P and Sb. For this reason, very Uttle effort has been expended on attempting to grow GaPSb and InPSb alloys. Calculated values of T^ are 1,692° and 1,033°C, respec­ tively. However, using the techniques described earlier, each has been grown throughout the region of solid immiscibility. The growth and properties of all of these metastable alloys will be discussed in detail in Chapter 8, where each alloy system is considered separately. To complete this discussion of OMVPE growth of alloys with mixing on the group V sublattice, we discuss the growth of GaAsP and InAsP where the deter­ mination of the actual group V species present at the interface plays an important role. So far, we have gotten away with the hypothesis that pyrolysis is complete and that mainly the thermodynamically most stable tetramers are formed. For the OMVPE growth of these materials, both assumptions are incorrect. Using ASH3 and PH3 as the group V sources, the solid composition is extremely temperature- dependent, as seen in Figure 2.37. At 600° C, the P distribution coefficient is small, approximately 0.05 for InAsP. As the temperature increases, k^ approaches unity at 850°C. Knowing that PH3 is much more difficult to pyrolyze than ASH3, as discussed in Chapter 5, suggests that the phosphorus distribution coefficient is at least partially determined by the kinetics of pyrolysis of the source molecules. 86 2 Thermodynamics

Figure 2.37. Solid versus vapor composition for the [11/V alloys: (O) InAs,P,_, at 600°C (Fukui and Horikoshi [235]); (A) GaAs,_^P^. at 750°C (Ludowise and Dietze [236]); and GaAs,_^.P^. at 650°C (•), 700°C (•), 750°C (T), 800°C (A), and 850°C (•) (Samuelson et al. [237]). (After Stringfellow [233], reprinted from the Journal of Crystal Growth with permission from Elsevier Science.)

Thermodynamically, the phosphides are more stable than the arsenides, so kp should be greater than unity. Smeets [129] obtained excellent agreement with the experimental results for the GaAsP distribution coefficient using the thermodynamic model, considering an equilibrium distribution of As and P among monomer, dimer, and tetramer species in the vapor. The incomplete PH3 pyrolysis was accounted for by assum­ ing an exponential temperature dependence of PH3 pyrolysis, with an activation energy of 30 kcal/mol. Leys et al. [130] determined the lower incorporation effi­ ciency for P to be dependent on both growth rate, or dwell time of ASH3 and PH3 molecules on the surface, and strain in the layer. They attributed these effects to changes in the adsorption/desorption rate constants for ASH3 and PH3 on the surface. Seki and Koukitu [131] calculated the solid-vapor composition diagrams for many important III/V ternary systems. Their analysis follows that described ear­ lier. They included only the atomic group III and both dimer and tetramer group V species in their thermodynamic equilibrium proportions. An interesting test of the hypothesis that PH3 pyrolysis controls P incorporation into the solid is provided by replacing the PH3 with TBP, which pyrolyzes at significantly lower temperatures, as discussed in Chapter 4. Chen et al. [132] grew GaAsP using TMIn, TBP, and ASH3 in an atmospheric pressure OMVPE reactor. 2.4 Solid Composition 87

^ Px Grow t h

Figure 2.38. Solid versus vapor composition for OMVPE growth of GaAs,_^_P^ and InAS|_^P^..- (A) GaAsP grown using TBP at 610°C [132]; (•) GaAsP grown using PH, at 650°C [237]. The solid curves are meant to represent the experimental data. The broken curves are the results of thermo­ dynamic calculations with the major group V species being the tetramers ( ) and the dimers ( ). (After Chen et al. [132], reprinted with permission from Journal of Electronic Materials, Vol. 17, 1988, a publication of the Metallurgical Society, Warrendale, Pennsylvania.)

The experimental dependence of jcf, on the ratio of TBP to TBP + ASH3 in the vapor is shown in Figure 2.38. The phosphorus distribution coefficient was signifi­ cantly increased by replacement of PH3 by TBP Equally interesting, the value of kp is still significantly below the values predicted by the thermodynamic equilib­ rium calculation, indicated as the broken Une. This is because the assumption that the thermal equilibrium species is formed at the interface is incorrect. In fact, if P4 had been formed, very little P would have been incorporated into the solid. As discussed earlier, P4 is so stable that it must be decomposed using a plasma for OMVPE growth using an elemental P source. Here we see evidence of kinetic hindrance of the thermodynamically driven processes. The formation of P4 is much slower than the formation of the dimer. Assuming that all TBP (and ASH3) pyrolysis results in formation of the dimer, the thermodynamic calculation results in the dashed-dotted curve, which agrees with the experimental data very closely. The lattice-latching or -pulling effect, described earlier, also occurs for mixing on the group V sublattice for III/V alloys. Leys et al. [130] observed that As incorporation was markedly suppressed for the OMVPE growth of GaAsP on GaP substrates. The magnitude of the effect was found to exceed the thermodynamic predictions. They suggest that the strain and/or steric effects produce a change in 88 2 Thermodynamics the adsorption and surface kinetic processes. For the OMVPE growth of GalnP at 675°C, Schaus et al. [133] report two phenomena indicative of lattice latching. The solid composition was found to be nearly independent of the TMGa flow rate and to be dependent on the lattice constant of the substrate. The authors explain their results qualitatively in terms of the thermodynamic effects described in Sec­ tion 2.4.1.1. Complicating the situation even further is the effect of the surface reconstruc­ tion (discussed in Chapter 3) on the solid composition. Even when the impinging group V species are monomers, from a cracker cell during CBE growth, a qua­ dratic relation between solid and vapor composition is observed, as discussed in more detail in Section 2.6.4.

2.4.2.1 Ternary II/Vl Systems The calculation of the solid composition as a function of input partial pressures and temperature for 11/VI systems is similar to that described for III/V systems. Again the anion/cation ratio in the input vapor phase plays a key role. A differ­ ence is that the group VI hydrides do not dissociate completely at the relative low temperatures used for OMVPE growth of the 11/VI alloys. Using the system ZnS-ZnSe as an example, the results of thermodynamic calculations by Kisker and Zawadski [36] of the equilibrium species versus temperature for H2Se and H2S pyrolysis are reproduced in Figures 2.39 and 2.40. At typical growth tem-

1 1 1 1 1 1 - H2

- z o HgSe i^^_ * A 4 .—Cr- ——A 2 -^

- o o -8h

-10 11 / 1 1 1 1 1 500 700 900 1100 1300 1500 TEMPERATURE {°K) Figure 2.39. Pyrolysis of H2S in the presence of H2, at 1 atm (solid symbols) and 0.001 atm (open symbols). (A, •) H2:H2S = 1,000. (After Kisker and Zawadzki [36], reprinted from the Journal of Crystal Growth with permission from Elsevier Science.) 2.4 Solid Composition 89

-10 1000 1200 1400 1600 1800 2000 TEMPERATURE (^K) Figure 2.40. Pyrolysis ofH-^Se in the presence of H2, at 1 atm (solid symbols) and 0.001 atm(open symbols). (D, •) H2:H2S = 1,000. (After Kisker and Zawadzki [36], reprinted from the Journal of Crystal Growth with permission from Elsevier Science.) peratures of 300°-500°C, the major group VI species are the hydrides. Thus the thermodynamic conditions are sUghtly different than those used for III/V com­ pounds, where complete group V hydride pyrolysis is assumed. The gas phase at the interface is assumed to consist of Zn, H^S, H2Se, and H2. As discussed in Section 2.3.2, this results in a sUghtly different equilibrium condition, as ex­ pressed in Equation (2.44). The conservation constraints are similar to Equations (2.57) and (2.58). The solid is treated as a regular solution with the interaction parameter calculated using the DLP or strain models, as shown in Table 2.4. The calculated results for the ZnSe^S,_^ are compared with the experimental data in Figure 2.41. The solid line was calculated using parameters appropriate to the filled data points of Wright and Cockayne [133], and the broken line was calcu­ lated with the growth parameters used to obtain the open data point of Fujita et al. [134]. Clearly, the thermodynamic calculation describes the experimental data well assuming incomplete pyrolysis of the group VI source molecules. This clearly implies that the ambient must play a significant role in the growth process, because the substitution of an inert gas for H2 will enhance the equilibrium extent of pyrolysis, since it appears in the mass-action expression. Calculated results for the CdSe^Tei_^ system are shown in Figure 2.42. The VI/II ratio, denoted R in this diagram, is a variable. The solid symbols represent the calculated results using an inert carrier gas, and the open symbols, a hydrogen carrier gas. The line with slope = 1, representing a unity group VI distribution coefficient, was calculated forR= 1. i 1 1 1 1 1 1 0 / V • / 0.8

/ /• / / 9> 0.6 / / - c • / / X / / / >/ / 0.4 ~ / / / - /

0.2 -

0.0 r 1 1 1 1 1 0 0 0.2 0.4 0.6 0.8 1.0

Figure 2.41. Solid-vapor distribution function curves for ZnSe^S,_^ compared with experimen­ tal data of references 133 (•) and 134 (O). For the calculation, H = 980 cal/mol, and VI/II = I, T = 340°C ( ); VI/II = 1.5, T - 400°C ( ); and VI/II = 5, T = 350°C ( ). (After Kisker and Zawadzki [36], reprinted from the Journal of Crystal Growth with permission from Else­ vier Science.)

1 1 1 1 1 R = 10 1.0

/ r^^ /R = 2 0.8

0.6 - o X 0.4 -

R = 2/ /R = IO

0.2

0 0 Y^— T^ 1 1 1 1 0.0 0.2 0.4 0.6 0.8 1.0

H,Se Figure 2.42. Calculated soHd-vapor distribution function curves for the CdSe^Te, _ ^ ternary system assuming T = 350°C and (1 = 2,430 cal/mol. Solid Une: VI/II < 1 for all conditions; solid symbols: inert carrier gas; open symbols: hydrogen carrier gas. (After Kisker and Zawadzki [36], reprinted from the Journal of Crystal Growth with permission from Elsevier Science.) 2.4 Solid Composition 91

2.4.3 Dopant Incorporation Following the analysis of the thermodynamic factors controlling the incorporation of majority constituents, it is natural to consider the incorporation of the minority constituents so critical for the fabrication of structures suitable for electronic and photonic devices. Dopant incorporation is treated using precisely the same approach described earlier [110]. For simplicity, only the III/V semiconductors will be treated here: analogous equations can be written for dopants in 11/VI semiconductor systems. Applying the model to a system involving a single group III element, a single group V element, and a single dopant element yields expressions for the respective fluxes to the interface similar to Equation (2.49). Since the group III element is depleted at the interface, assuming an input V/III ratio of greater than unity, and assuming the diffusion coefficients are approximately equal, yields a simple ex­ pression for the dopant concentration in the solid,

^h = ^ = PLIZD (2.59)

In addition, we have the thermodynamic relationship between the dopant partial pressure right at the interface, pj^, and the solid composition.

rS == kp\,, (2.60) where k is the true, thermodynamic distribution coefficient. This allows us to write the final expression for solid composition,

P III The physical significance of this expression becomes clear by considering the two Umiting cases. In case 1, where the vapor pressure of the dopant is low, essentially all dopant reaching the interface is incorporated (i.e., p\y < p^); thus,

xh = ^' (2-62) Pm In case 2, where the vapor pressure of the dopant is high and most of the dopant is reevaporated, giving pj^ ~ p^,

^h = kp*D' (2.63) Case 1 represents the mass-transport-limited case. In case 2, thermodynamic con­ siderations applied to the input gas phase determine the number of dopant atoms incorporated into the solid. Of course, in both cases the gas phase adjacent to the interface is assumed to be in equilibrium with the solid. 92 2 Thermodynamics

Explicit expressions for the distribution coefficient can be written in each spe­ cific case. When the volatile dopant, such as Zn, resides on the group III sublattice,

P%]PD = ^IUPD^ (2.64) TD- where /^D-V is the equilibrium constant and A:iii is proportional topi;, • For volatile dopants such as S, Se, and Te, which reside on the group V sublattice.

^Ill-n ^nr-v I r'^-s = . f^ ^P'° = ^vPb. (2.65) 7in-D ^iii-v Pv where, again, K represents the equilibrium constant. In this case, k^ is propor­ tional to l/p%. Again, for illustrative purposes, the group V element has been assumed to be the monomer. The characteristics distinguishing the behavior of case 1 and 2 impurities, listed in Table 2.8, can be summarized as follows: The case 1 distribution coefficient is proportional to l/r^ or l/p^fii and independent of both temperature and p%. The case 2 distribution coefficient is independent of both growth rate and p^n. The dependence onp% depends on which sublattice the dopant occupies. For dopants on the III sublattice, it is proportional to p% and to \/p% for dopants on the V sublattice. For case 2 dopants, k is frequently highly temperature-dependent. A simple thermodynamic analysis shows that the temperature dependence of K^^^_Q is reflected in the reciprocal of the vapor pressure of the dopant. This behavior is clearly seen in the temperature dependence of the distribution coefficients of Zn and S in GaAs, as discussed later. In addition, it explains the relative values of the distribution coefficients of the group VI dopants. The least volatile, Te, has the

Table 2.8 Summary of distribution coefficients of donors and acceptors in GaAs'*

Dopant ^g(Pm) T p^ Conclusion Case I i —> -> Case 2 (m) —> i T Case 2 (V) —> i i Zn T(?) i T Case 2 on HI site S T i i Case 2 on V site Se i i Case 2 on V site; '^Se ^ '^s' FSe ^ Fs Si i T —> Case 1; SiH4 (pyroiysis) pyroiysis limited Residual donor T T SiR4, GeR4, CR4 pyroiysis limited Residual acceptor T n ^From Reference 110. Arrows indicate the change in distribution coefficient caused by an increase in the growth parameter: T and i indicate increase and decrease, and —» indicates no effect. 2.4 Solid Composition 93 highest distribution coefficient with a lower value for the more volatile Se and the lowest distribution coefficient for the most volatile, S. The basic concepts, which determine the value of the distribution coefficient, can be illustrated using the behavior of common dopants used for GaAs grown by OMVPE. We will consider first Zn, for which a wealth of data are available on the effects of the major growth parameters on the distribution coefficient. The tem­ perature dependence of the distribution coefficient for Zn in GaAs, using DEZn as the dopant, is illustrated in Figure 2.43 using the data of Glew [135]. The dis­ tribution coefficient decreases by more than 10^ as the growth temperature is in­ creased from 600° to SOO^'C. A relative plot of the reciprocal of the vapor pressure of pure, liquid Zn is included in Figure 2.43 for comparison. The two lines have nearly identical slopes; thus, the Zn distribution coefficient is clearly proportional to the reciprocal of the vapor pressure of elemental Zn. Bass and Oliver [136] found the Zn distribution coefficient to have a similar temperature dependence, to increase slightly with increasing TMGa input partial pressure, and to be propor­ tional to the input ASH3 partial pressure, as illustrated in Figure 2.44. The results are summarized in Table 2.8. With the exception of the dependence on TMGa input partial pressure, the behavior of Zn is exactly as expected for a high vapor pressure (case 2) acceptor residing on the group III sublattice. It should be noted

10^

10"

•G 10--

u 10""

^ 10" aA$(Glew) GaAs (Bass & Oliver) aAs (Bass) 10- '>P(Hsu et a\.)

10-* 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1000/T (1/K) Figure 2.43. Distribution coefficient, normalized by multiplying by the input group III partial pres­ sure, versus reciprocal temperature for the following systems: Zn in GaAs, data points (•) from Glew [135]); S in GaAs, data points (•) from Bass and Oliver [136]; Si in GaAs, data points (X) from Bass [138]; Si in InP, data points (D) from Hsu et al. [139]. Also included in the plot are broken lines indicating the reciprocal vapor pressure of Zn and S using relative scales and the relative growth rate for Si grown using SiH4. (After Stringfellow [110], reprinted from the Journal of Crystal Growth with permission from Elsevier Science.) 94 2 Thermodynamics

10^ • Zn:GaAs-690*C (Bass & Oliver) O 5:GaAs-730*C (Bass & Oliver) • S:GaAs-720'C(Glew) aSe:GaA$-720*C(Glew)

^- 10^

u 10^ h

10' •

10- 1 1 1 10"' lO"-" 10'' 10-' Arsine Partial Pressure Figure 2.44. Distribution coefficients of the dopants Zn, S, and Se in GaAs versus the input partial pressure of ASH3 during OMVPE growth. The data are from Bass and Oliver [136] for Zn (•) and S (0), and Glew [137] for S (•) and Se (D). (After Stringfellow [110], reprinted from the Journal of Crystal Growth with permission from Elsevier Science.) that neither model predicts an increase in distribution coefficient with increasing growth rate, orp^u- The case of S in GaAs can be analyzed in a similar manner. In Figure 2.43 we observe the decrease in sulfur distribution coefficient with increasing growth tem­ perature. The slope is similar to that of the reciprocal of the sulfur vapor pressure plotted, using a relative scale, for comparison. In Figure 2.44 the data of Glew [137] and Bass [138] show that the sulfur distribution coefficient is inversely pro­ portional to the input partial pressure of ASH3. Bass [138] also report an unex­ plained, sublinear increase in k^ with increasing input TMGa partial pressure, similar to the case for Zn. As summarized in Table 2.8, S behaves as a high-vapor- pressure (case 2) donor residing on the group V sublattice. The behavior of Se in GaAs [138] is nearly identical with that of S, except that the distribution coeffi­ cient is more than an order of magnitude larger, presumably due to the lower vapor pressure of Se. It should also be mentioned that the distribution coefficient for Se in GaAs is not clearly defined since the carrier concentration is a superlinear func­ tion of Se partial pressure in the input gas stream. However, the qualitative behav­ ior of Se doping is consistent with that of a high-vapor-pressure dopant residing on the group V sublattice. A clearly dissimilar dopant is Si, using SiH4 as the doping source. Si is a low- vapor-pressure element at the temperatures considered here. As seen from the re­ sults of Bass [138] for GaAs and the data of Hsu et al. [139] for InP, shown in Figure 2.43, the Si doping level increases with increasing growth temperature. This is clearly a situation not covered by the simple model. However, a compari- 2.5 Quaternary Systems 95

son with the temperature dependence of the growth rate of Si from SiH4, shown as a broken Une in Figure 2.43, indicates the explanation. The pyrolysis of SiH4 is incomplete over the range of growth temperatures studied. Thus, the increasing growth rate of Si and the increase in SiH4 doping efficiency in GaAs and InP reflect the more efficient pyrolysis of SiH4 at higher temperatures. This inter­ pretation is supported by the data of Kuech et al. [140], which show a similar temperature dependence for SiH4 doping but give a temperature-independent dis­ tribution coefficient for Si2H5, which pyrolyzes completely over the entire tem­ perature range studied. Bass [138] reported the Si doping level to decrease with increasing TMGa flow rate, with n o^ TMGa"^^. This is qualitatively the behavior expected of a low-vapor-pressure (case 1) dopant. Bass also reported n to be pro­ portional to the ASH3 partial pressure to the power of -0.6. However, more recent work by Kuech et al. [140] failed to find a dependence on ASH3 flow rate, in line with the expectations from the model. Overall, the simple theory, based on the assumption of thermodynamic equi­ librium at the growth interface, gives an accurate qualitative description of many aspects of the doping behavior of the common dopants in GaAs, as seen in Table 2.8. The few features not described by the model have been attributed to homogeneous gas phase reactions and other kinetic effects not included in the simple model [135, 136, 139]. We have also assumed the dopant partial pressure to be independent of the group III and group V flow rates. This may not be accu­ rate if the concentrations of residual impurities are significant. This discussion has neglected the ionization of the impurities when incorpo­ rated into the lattice. This, of course, produces additional species, including elec­ trons and holes, in the mass-action expressions. Thus, dopant incorporation can depend on the position of the Fermi level [141]. Compensation and passivation of dopants have also been ignored. They are treated for individual systems in Chapter 8. The surface structure may also affect dopant incorporation, which is discussed in Section 3.5.1.

2.5 Quaternary Systems

The solid/vapor phase diagrams useful for OMVPE growth of III/V type I quater­ nary solids can be calculated by a direct extension of the ternary calculation described in Sections 2.4.1 and 2.4.2, using one of the three models for the qua­ ternary solid described in Section 2.1.2—that is, the regular solution model, as developed by Jordan, Ilegems, and Panish [7, 94], the bond model of Onda and Ito [95], or the DLP model [6]. For either the type I or type II quaternary solids, three independent equilibrium conditions are used in addition to the conservation condition for stoichiometry and the two conservation conditions for solid com­ position, one for each sublattices. This totals six equations in six unknowns, the 96 2 Thermodynamics

InGaAsP T = 550'C Y/ffl = ]0 Pffl = 5'10^atm

0.95

inP 0.2 0.4 0.6 0.8 6aP Input Ga/(Ga>ln) Ratio

Figure 2.45. Diagram of solid composition versus input mole ratio for In,_^Ga,As^,P,_^ alloys. The dashed lines indicate the alloy compositions lattice-matched to InP and GaAs substrates. (After Koukitu and Seki [142], reprinted from the Journal of Crystal Growth with permission from Elsevier Science.) four partial pressures at the interface and the two solid composition parameters x and y. Koukitu and Seki [142] have calculated several of the quaternary phase diagrams. For example, two of the phase diagrams for the important GalnAsP and GalnAsSb systems are reproduced in Figures 2.45 and 2.46.

2.6 Thermodynamics of the Surface

The importance of bulk thermodynamics in determining the solid composition and stoichiometry for semiconductors grown epitaxially, as described earlier, has been known for decades. In recent years, it has become apparent that the thermo­ dynamics of the surface often plays the dominant role in many aspects of the OMVPE growth process. After all, the growth occurs entirely at the surface, and diffusion in the solid is slow. Thus, the vapor and surface are nearly in equilibrium 2.6 Thermodynamics of the Surface 97

InGaAsSb .n5 T= 550°C 7/m = 10 Pm = 5«10' atm

Xga inSb 0.2 0.4 0.6 0.8 Ga T r » 1 '^ » 1 C.995 0.99 r J 1 • 1 •

1 <' Ho.8 • • ^

• 0.97 • •

• iO.6 • + / ^ 0.95 y ^^J Ysb 0.4 / 1 0.93 1^ ^j X rO.?! j^5 0.2 ;jjl rO.3 0 91 0.9 1 ^~r—f ^'^i 1 I 1 1 i i Sy InAs 0.2 0.4 0.6 0.8 GaAs Input Ga/(Ga + In) Ratio Figure 2.46. Diagram of solid composition versus input mole ratio for In,_^Ga^As,_^Sb^alloys. The dashed lines indicate the alloy compositions lattice-matched to InAs and GaSb. (After Koukitu and Seki [142], reprinted from the Journal of Crystal Growth with permission from Elsevier Science.) for OMVPE (and, indeed, MBE) growth of semiconductor materials under most conditions. If the entire system were completely at equilibrium, it would not be important to consider the surface, since the bulk solid would also be in equilibrium with the vapor at the interface. However, the diffusion coefficients in these highly stoichio­ metric solids, where the vacancy concentrations are extremely low, are so small that the bulk solid is often not able to attain equilibrium with the surface. In this case, the stoichiometry of the solid is determined by the thermodynamics of the surface phase. This was first realized in regard to dopant incorporation. Since the dopants are incorporated as charged species, the position of the Fermi level is an important factor in dopant incorporation. The Fermi level is frequently pinned at the surface in III/V semiconductors, which gives a different incorporation coeffi­ cient for the surface than for the bulk [141]. However, until recently, little has been known about the actual state of the surface during OMVPE growth. Tech­ niques described in Chapter 3, such as scanning tunneling microscopy (STM) and reflection high-energy electron diffraction (RHEED), have been available for the characterization of the surface under the ultrahigh-vacuum (UHV) conditions of 98 2 Thermodynamics

MBE and CBE, but only the recently perfected optical probes can be used at the higher pressures of OMVPE. Fortunately, they allow the characterization of the bonding at the surface. 2.6.1 Surface Reconstruction The surfaces created by termination of a model zincblende solid are shown in Figure 2.47. For the (100) surface, commonly used for epitaxial growth of III/V semiconductors, two of the four sp"^ bonds in the bulk are unsatisfied for the atoms on the surface. This would be expected to result in very strong binding of adatoms to the surface, resulting in very low surface mobilities and, consequently, very rough surfaces. This will be discussed further in Chapter 3. This appears to be contrary to experimental observations of large diffusion lengths [143] and the ability to grow quantum well structures with abrupt and smooth interfaces. These were the first indications that each surface atom does not have two dangling bonds. Elementary thermodynamic considerations suggest that the surface atoms will re­ arrange to lower the surface energy by reducing the number of dangling bonds as well as changing the bonding geometries. In recent years, first-principles calcula­ tions have given detailed information about the way in which this might occur [144]. RHEED experiments on semiconductor surfaces have given corresponding experimental information about the surface configuration versus temperature and the group V flux arriving at the surface [145]. The results are found to correspond closely to those obtained using optical techniques such as reflection difference spectroscopy (RDS) [146]. The surface reconstructions seen for GaAs under con­ ditions used for growth by atmospheric pressure OMVPE are shown to be surpris­ ingly similar to those for growth by MBE. The most commonly observed structure for GaAs during MBE growth is the (2 X 4) reconstruction, a terminology that

[001

(111) A (110) '^ '^ DANGLING HYBRID

(OOf) Figure 2.47. Atomic configurations on (100), (110), and (lll)A zincblende surfaces, ignoring reconstruction. 2.6 Thermodynamics of the Surface 99

(a) a (2X4) A(t«f Fan»( and Palmstom, (b) ^ (2x4) A«t,f Farrrt and Palmstom, (C) Y (2x4) Aft»f Far«( and Palmstonn t,.., WhMi 1

^[110] ^yX^^A^/^^ Tl

> • As O ° Ga (d) P (2x4) AfterHashimiBj (fat (e) Y (2x4) AftBT Hashimou »t ad

Figure 2.48. Models for (2 X 4) reconstruction on (OOl)-oriented GaAs. The As surface cover­ ages are listed along with the references first suggesting these structures: (a) 0.5 [147], (b) 0.75 [147], (c) 1.0 [147], (d) 0.75 [239], and (e) 1.0 [239] ML. refers to the periodicity of the (001) surface in the two (110) directions orthogonal to the (001) growth axis. Farrell and Palmstrom [147] divided the class of sur­ faces having the (2 X 4) symmetry into three distinct structures, the a, l3, and y (2 X 4) reconstructions shown in Figure 2.48. The a (2 X 4) structure has 50% of the surface sites covered by [T10]-oriented As-As dimers arranged into [110] rows. The 13 (2 X 4) reconstruction has three of four sites in the unit cell occu­ pied by the [110] As-As dimers. The As surface coverage is even higher for the y (2 X 4) structure where four As-As dimers cover the surface of each unit cell, but one is rotated to the [110] orientation. There is some disagreement about the actual atomic arrangements; however, the (2 X 4) reconstruction is certainly a family of structures having a common symmetry in RHEED and X-ray diffraction studies. The other anion-rich surface frequently seen under the conditions used for epitaxial growth is the (4 X 4) structure. The STM images of Biegelsen et al. [148] clearly define this structure for GaAs to consist of a double layer of As. The topmost As-As dimers have the [110] orientation, as seen in the schematic diagram of Figure 2.49. The group Ill-rich (4 X 2) reconstruction consists mainly of [110]-oriented Ga dimers. Since OMVPE is normally carried out with a group V excess in the vapor, this reconstruction is probably of less significance.

2.6,1.1 Surface-Phase Diagram The reconstructed semiconductor surface formed is determined by the extensive thermodynamic variables of the system. Thus, the reconstructions can be repre­ sented by a surface-phase diagram. The surface-phase diagrams, determined using 100 2 Thermodynamics W

[110]

[110]

• As O Ga • As o Ga

Figure 2.49. A model for the c (4 x 4) reconstructed (OOl)-oriented (4 X 4) surface [148]. (Re­ printed with permission of the American Institute of Physics.)

SPA spectroscopy, for GaAs and InP are superimposed in Figure 2.50 [149]. The phase rule, represented by Equation (2.23), appHes to the surface phases in exactly the same way as for the bulk. Thus, when the system consists of a single surface reconstruction (surface phase) plus the vapor, the system has two degrees of free­ dom, whether under the UHV conditions of MBE or at a fixed total pressure in OMVPE. Thus, fixing two independent variables, typically temperature and the group V partial pressure, completely defines the thermodynamic state of the sys­ tem. When two reconstructions coexist, the degree of freedom is unity. Thus, this state of the system is represented by a line in Figure 2.50. The (2 X 4) reconstruction is observed for typical MBE growth conditions, using V/III ratios in excess of unity for the growth of GaAs. X-ray studies indicate that the (4 X 4) reconstruction is formed during GaAs grown by OMVPE, due to the higher As pressures used [150, 151]. One of the family of (2 X 4) reconstruc­ tions is formed during the OMVPE growth of InP [149] and GalnP [152]. The (4 X 4) surface is not typically formed due to the higher volatility of P, especially when the surface contains In. The In-P bond is so weak that the (4 X 4) recon­ struction is not formed on these surfaces. This is discussed in more detail in Sec­ tion 3.2. A second layer of P atoms requires a much higher P pressure than is typically used for OMVPE growth, as indicated in Figure 2.50. The close similarity of surface reconstructions observed during MBE and OMVPE growth with those obtained under static conditions is an indication that 2.6 Thermodynamics of the Surface 101

Ts (°C) 700 600 500 10=

GaAs InP 102

O O CLTD c(4x4) Q. (2x4)Y o like •5 10^ like (2x4) a like o\ GaAs c(4x4) like oob GaAs (2x4) 7 like InP excess P InP marginal •a InP (2x4) like I 10-1 L _L_ 1.00 1.10 1.20 1.30 1.40 103/Ts (K-1) Figure 2.50. Surface phase diagrams for (001) GaAs and InP determined using SPA measurements. (After Kobayashi and Kobayashi [149], reprinted with permission of the Journal of Electronic Mate­ rials, a publication of the Metallurgical Society, Warrendale, Pennsylvania.) the growth processes occur under conditions that give a surface that closely ap­ proximates the surface at thermodynamic equilibrium. This is entirely consistent with the observation that the adatom population on the surface during the MBE growth of GaAs is nearly the same as for the static surface at equilibrium [153], as will be discussed more fully in Chapter 3. Obviously, thermodynamic factors will determine much of what occurs at the surface during the growth process and most of the properties of the resulting materials. All of this strongly supports the validity of the equilibrium model of OMVPE growth described in Section 2.3.2.

2.6.2 Effect of the Surface on Ordering A very clear indication of the importance of surface thermodynamics for III/V alloys (as well as Si-Ge and 11/VI alloys) is the effect of the surface on the micro­ scopic arrangement of the atoms in the solid (e.g., the production of ordered struc­ tures spontaneously during OMVPE growth). VFF calculations indicate that the (2 X 4) reconstruction will favor formation of the B-variants, (Til) and (ill), of the CuPt structure [154], which are not stable in the bulk, as discussed in Sec­ tion 2.1.2.9. The [110] rows of [110]-oriented group V dimers lead to alternating [110] rows of compressive and tensile strain in the third buried layer, because the surface group V atoms move closer together where a dimer bond is formed. For alloys with mixing on the group III sublattice, such as GalnP, this produces the [110] rows of alternating large and small atoms that comprise the CuPt B variants. These calculations also predict that for alloys with mixing on the group V sub- lattice, such as GaAsP, the [110] group V dimer rows also produce the CuPt B variants. This is in agreement with experimental results. Empirically, nearly all 102 2 Thermodynamics

III/V alloys are observed to form the CuPt structure during OMVPE growth using typical growth conditions, as indicated in Table 2.5. Exactly the same variants are seen for alloys with mixing on the group III sublattice as for those with mixing on the group V sublattice [155]. The correspondence between the presence of [TlO] P dimers and CuPt ordering for GalnP layers grown by OMVPE has recently been verified by using the SPA technique for measurements of the nature of the chemical bonding at the surface [156-158]. As discussed earlier, optical techniques such as SPA are the only methods capable of yielding in situ information about the surface reconstruction during OMVPE growth [149]. Optical techniques give information about only the energy and symmetry of electronic transitions involving surface atoms. It is impossible to obtain direct information about the long-range order from such measurements. However, SPA results are seen to correlate closely with RHEED results in UHV systems [149, 151, 159], giving confidence in their value for de­ termining, indirectly, the surface reconstruction during OMVPE growth, as will be discussed in Chapter 3. For example, the only known structures involving [TlO] group V dimers on the (001) surfaces of III/V semiconductors are the family of (2 X 4)-type reconstructions. Murata et al. [156-158] have clearly demonstrated that as the concentration of [TlO] dimers was decreased, by increasing the temperature or decreasing the par­ tial pressure of the P precursor, the degree of CuPt order was also decreased. A monotonic relationship between the [TlO] dimer concentration and the degree of order was demonstrated over a wide range of growth parameters for OMVPE growth on singular (001) substrates, as seen in Figure 2.51. Gomyo et al. [160] grew AllnP layers on (001) substrates by MBE. RHEED measurements indicate the (2 X 2) reconstruction, terminated by a double layer of anions, is formed. In this surface configuration the dimers are rotated by 90° as compared with the (2 X 4) structure and the rows run in the [110] direction. This was found to produce the A variants of the CuPt structure. The results agree with the calculations of Zhang et al. [154] that indicate that the A variants are stabilized by the (2 X 2) reconstruction. Further confirmation of the close relationship between surface reconstruction and ordering is the production of a triple-period structure with ordering along the {111} A planes for the MBE growth of AlInAs layers on (001) InP substrates using conditions giving the (2 X 3) surface reconstruction [161, 162]. Again, Zhang et al. [154] showed that for the (2 X 3) reconstruction, the strain energy of the subsurface layers is minimized by formation of the triple-period ordered struc­ tures observed. Seong et al. [163] observed a similar tripHng of the periodicity, but in the (110} direction, for InAsSb layers grown by MBE on (001) substrates. Naturally, in an accurate phase diagram, either for the bulk or the surface phase(s), the ordered phases will be stable only over a limited range of solid com­ position and temperature [44]. At high temperatures the entropy term in the free 2.6 Thermodynamics of the Surface 103

0.8

0)

0) 0)

•o

0 0.5 1 1.5 2 2.5 3 SPA Signal Difference (%) Figure 2.51. Degree of CuPt order versus the SPA signal due to [TlO] P dimers on the (001) sur­ face. (After Hsu et al. [240], reprinted from the Journal of Crystal Growth with permission from Elsevier Science.) energy leads to stabilization of the disordered phase, as seen in Figure 2.25. This is apart from changes in the surface induced by the temperature change, such as the changes in surface reconstruction with temperature discussed earlier. The range of solid composition over which the ordered structure is stable should also be limited. One expects that the CuPt structure, for example, will be most stable for a 1:1 ratio of Ga to In in GalnP. Experimentally, ordering is found to exist over a wide range of solid compositions in Gain? [164], GaAsP [165], and InAsSb [166] alloys. Further support for the thermodynamic description of ordering is obtained from the results of annealing experiments. As mentioned, annealing metal alloys is a technique used to enhance the degree of order. This occurs because the ordered structure is stable in the bulk. Annealing of III/V alloys gives the opposite result. Piano et al. [55] were able to destroy the order produced in GalnP samples during growth by annealing for four hours at 825°C. Similarly, Gavrilovic et al. [54] were able to eliminate the ordered structure by annealing at 700° C for times as long as approximately 100 hours. This is evidence that, indeed, the CuPt ordered structure is not stable in the bulk. To summarize this section, it is clear that a strong link has been established between the occurrence of CuPt ordering and the surface structure (reconstruc­ tion) during growth. Both theoretical calculations and the results of experimental investigations strongly indicate that the CuPt structure is not stable in bulk GalnP. 104 2 Thermodynamics

It forms at the surface during vapor phase epitaxial growth on (OOl)-oriented sub­ strates that are reconstructed to produce [110] rows of [110] group V dimers on the surface. However, the experimental results discussed in Chapter 5 strongly suggest that other, kinetic factors may also be significant. For example, ordering disappears at high growth rates, suggesting that for rapid growth the time before a layer is cov­ ered by the next layer is insufficient for rearrangement of the surface atoms to form the ordered structure. On the other hand, misorientation of the (001) sub­ strates by a few degrees to produce [110] surface steps is found to enhance the formation of the CuPt ordered structure, while [110] steps are found to retard the ordering process. This suggests that surface steps may play an important role in the kinetic processes leading to the formation of the CuPt ordered structure.

2.6.3 Stoichiometry and Doping The thermodynamics of the surface is also expected to have a direct influence on materials properties other than ordering, such as dopant incorporation, stoichi­ ometry, and the concentrations of native defects. The surface reconstruction will also almost certainly affect the surface mobilities of adatoms and the density and nature of steps and kinks on the surface, all of which will affect surface morphol­ ogies and defect densities. However, understanding of this is limited at this time. Early results clearly indicate that the nature of the surface affects dopant incorpo­ ration [167]; however, this is attributed mainly to kinetic effects, as discussed in Section 3.5.1. The solid solubility of N in conventional III/V semiconductors such as GaAs, GaP, and InP is limited because of the very small size of N, as discussed in Sec­ tion 2.2.2. Calculations indicate that anion dimerization will increase the solubil­ ity near the surface by several orders of magnitude [168]. Since the limited solid solubility is due to the microscopic strain energy caused by the difference in size of the atoms, location near the surface will increase the solubility due to the re­ duction in strain energy. This phenomenon may help to explain the ease with which the solubility of N in conventional III/V semiconductors can be exceeded, as discussed in Section 8.8.1. It is reminiscent of the formation of the CuPt or­ dered structure in the bulk, while it is thermodynamically stable only near the (001) surface, as described earlier. In both cases, metastable arrangements of the atoms persist in the bulk due to the very small-bulk self-diffusion coefficients.

2.6.4 Solid Composition Surface thermodynamics is found to directly affect the solid composition for alloys with mixing on the group V sublattice. For the OMVPE growth of GaAsP using the group V hydrides, bulk thermodynamic treatments of the As/P ratio in References 105

the solid versus vapor composition and temperature agree with the experimental results only when kinetic factors such as the pyrolysis of the group V hydrides is taken into account, as described in Section 2.4.2. The bulk thermodynamic treat­ ment appears to be adequate for the description of the solid composition when the P precursor is the rapidly pyrolyzing TBP, where the species reaching the surface are most likely the As and P dimers. Results obtained by OMMBE are more dif­ ficult to interpret, perhaps because the elemental group V sources produce tetra- mers, which must be broken down on the surface to allow incorporation into the solid. This also introduces kinetic factors in the determination of the solid com­ position [169]. A more recent study has used the cracked hydrides for the CBE growth of GaAsP [170]. In this case. As and P monomers are the species arriving at the interface. The observed quadratic dependence of As in the solid on the vapor phase As concentration has been attributed to the incorporation of As and P into the solid only from As-As and P-P dimer pairs on the surface. The surface dimers are assumed to be in equilibrium with the vapor. Using a semiempirical factor accounting for the greater strength of the P-P bonds, the experimental solid com­ position can be quantitatively described as a function of the As/P ratio in the vapor. An interesting feature of this work is that the As/P ratio in the solid is found to be a strong function of the strain in the epitaxial layer. Cunningham et al. [170] also interpret this factor in terms of a surface thermodynamic phenomenon. An improved match in covalent bond length between As dimers and the tetragonally strained GaAsP lattice results in the formation of more As-As dimers on the sur­ face and a consequent increase in As incorporation into the solid.

References

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