The Surface Free Energy

In this section, we focus on different surface phases. We are interested in the surface phase in equilibrium with its environment (e.g., background gas or the silicon carbide (SiC) bulk). This means that the environment acts as a reservoir, because it can give or take any amount of atoms to or from the sample without changing the temperature or pressure. The Gibbs free energy (Sec.4.1) is the appropriate thermodynamic potential to describe such a system. The Gibbs free energyG(T,p,N_i)depends on the temperature and pressure, and on the number of atoms Niper speciesiin the sample. The most stable surface phase and geometry is then the one that minimises the surface free energy. For a surface system that is modeled by a slab with two equivalent surfaces the surface free energy is defined as:

Definition 4.2.1. The surface free energy (γ) the surface under consideration, N_iis the number of atoms of element i andµi the element’s chemical potential and Nc the total number os different chemical species.

The Gibbs free energy is given by Eq.4.1with Eq.4.2as an approximation to the internal energy U.

In this work, we are interested in surface phases in equilibrium with the bulk phase. In a solid likeSiCthe compressibility is extremely low¹. Therefore, the contribution of thep·Vterm in Def.4.2.1to the surface free energy will be neglected. In general, vibrational energy contributions can be included and may lead to small shifts. However, the necessary phonon calculations of a large surfaces including more than 1000 atoms are computationally demanding and not always feasible. We neglect vibrational free energyF^vib and configurational entropy contributionT·S^{con f} to the surface free energy.

Only the total energy termE^BOis left as the predominant term. This allows us to rewrite Def.4.2.1to

γ = ¹

For the input energiesE^BOwe use total energies obtained fromDFT calcu-lations.DFTtotal energies are related to a thermodynamical quantity only in a restricted way. They correspond to the internal energyUEq.4.2at zero temperature and neglecting zero-point vibrations.

Range of the chemical potential

The limits of the chemical potentials are given by the formation enthalpy of

1The volume of cubic silicon carbide (3C-SiC) changes by a factor ofV/V₀ = 0.927 at a pressure of 21.6 GPa and temperature ofT =298 K [300,313]. In this work we are interested in a low-pressure regime (<1MPa). In this pressure regime the change of the3C-SiCbulk volume is negligible.

SiC. The enthalpy of a system is

H(T,p) =G(T,p) +T·S^{con f} =U+p·V, (4.10)

The enthalpy of formation∆Hf is the energy released or needed to form a substance from its elemental constituents in their most stable phases.

Definition 4.2.2. The enthalpy of formation∆Hf

The enthalpy of formation∆H_f is the difference between the standard enthalpies of formation of the reactants and of the products.

∆H_f(T,p) =

where Npis the total number of products involved in the process, H_iis the enthalpy of the product i and its respective stoichiometric coefficient, N_i, and Nr is the total number of reactants, Hjis the enthalpy of the reactant j and Njits stoichiometric coefficient.

=0 . . .reactants and products are in equilibrium

>0 . . .products precipitate into its reactants

As before for the surface free energy (Def.4.2.1), we will neglect the p·V and temperature and vibrational contributions to the internal energyUin Eq.4.10. Then, we can express the enthalpy of formation∆Hf in terms of total energies obtained fromDFTcalculations.

Throughout this work, we addressSiCsurfaces grown by high temperature silicon (Si) sublimation. Here, the SiC bulk is in equilibrium with the surrounding gas phase. The gas phase is atomic carbon (C) andSireservoirs, which supplies or takes away atoms from the surface. The stability of the SiC bulk dictatesµ_Si +µC = E^bulk_SiC where E^bulk_SiC is the total energy of aSiC bulk 2-atom unit cell.

As a next step the limits of the chemical potential have to be defined. The allowable range of the chemical potentials µSiand µCunder equilibrium conditions are fixed by the elemental crystal phases of SiandC. The dia-mond structure for Si is the appropriate bulk phase, but for C, there is a close

competition between diamond and graphite [23,347,348]. Both reference systems will be included in our analysis. In Ch.6and AppendixAa detailed discussion on the competition between the two C phases is presented.

Finding the maximum of the chemical potentials:

The bulk phases define the upper chemical potential limits. Above these limits the SiC crystal decomposes into Si or C

max(_µSi) = ¹

2H_Si^bulk(T,p) max(_µC) = ¹

2H_C^bulk(T,p).

(4.12)

Applying the approximations to the enthalpy of the products and reactants in Def.4.2.2discussed above, we will use total energies Eobtained from DFTcalculations in the following derivation of the limits of the chemical potentials, leading to

max(_µSi) = ¹

2E^bulk_Si , max(_µC) = ¹

2E_C^bulk (4.13) for the maximum chemical potential ofSiandC.

Finding the minimum of the chemical potentials:

Using the equilibrium condition of Def.4.2.2µ_C+µ_Si =E^bulk_SiC , we find for the minimum chemical potential forC

min(µC) = E^bulk_SiC −¹

2ESi. (4.14)

At the end we find for the limits of the chemical potentialµC: E_SiC^bulk−¹

2ESi ≤µC≤ ¹

2E^bulk_C . (4.15) Within the approximations to the enthalpy Eq.4.10, the limits can be rewrit-ten as

1

2E_C^bulk+_∆Hf(SiC) ≤µC≤ ¹

2E^bulk_C (4.16) with∆Hf(SiC) = E_SiC^bulk−¹₂E^bulk_C −¹₂E^bulk_Si .

The chemical potentials,µCandµSi, can be experimentally manipulated for

example through the substrate temperature and background pressure of gases that supply Si or C [259,329,67,302]. A precise control of the reser-voirs provided by the background gases (for instance, disilane (Si2H8) [329]) is desirable, but calibration variations [202] may require exact (T,p) ranges to be adjusted separately for a given growth chamber.

It is important to note, that small changes in the chemical potentials do not necessarily correspond to small changes in the experimental conditions (temperature and pressure). For example, a drastic change in the number of Si (N_Si) and C (N_C) atoms can correspond to a small change of the corres-pondingµ. To cross beyond the carbon rich limit of graphite (or diamond) in equilibriumall Si has to be removed from the SiC crystal.

In this chapter, a bridge betweenDFTcalculations and macroscopic phase stability was built. Theab initioatomistic thermodynamics approach is a tool to compare different systems, bulk or surfaces, that are in thermodynamic equilibrium. However, it cannot give any information about the process of the phase formation or kinetic effects during the growth process. Non-etheless, a first very important step when dealing with systems in different phases, is to identify the thermodynamic equilibrium state.