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Thermodynamics 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].
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