Association

2.1. The challenge of stabilizing stoichiometric europium oxide 7
increase of Gibbs free energy, we can discriminate whether a reaction will proceed (dG 0). Its differential is given by
dG = −S dT + V dp + μ dN, (2.1)
where S is the entropy, V the volume, and μ the chemical potential. One recognizes that T , p,
and N are the natural state variables of G. The Gibbs free energy is a suitable thermodynamic
measure for chemical reactions at fixed pressure and fixed temperature. Thus, the Gibbs free
energy is perfectly suited to describe the EuO growth by MBE, which is a quasi-equilibrium
system. In thermodynamic equilibrium, there is no total transfer of entropy: ΔS = 0. This
implies in terms of the variables for G for two subsystems A and B (e. g. two competitive EuO
phases):
T A = T B
p A = p B
μ A = μ B
Since the temperature T and pressure p are constant, the only driving force out of the equilibrium
is a possible difference in chemical potential Δμ = μ A − μ B . Provided a constant
atomic flux dN in MBE, the Gibbs free energy (2.1) depends then only on the specific Δμ
of materials. Those material-specific Gibbs free energies are tabulated in databases like the
CRC Handbook or Landolt-Börnstein, 34,35 and a comparison allows for a prediction of quasiequilibrium
reactions as during MBE synthesis.
Ellingham diagrams
Figure 2.2.: Ellingham diagram.
Once the material-specific Gibbs free energies
of formation and the gas lines are
included, one can determine the principle
thermodynamic stability (ΔG r ≷ 0) and regions
of oxidation or separation of metal
and oxygen. These regions are separated
by equilibria at the intersections, which
determine a set of oxygen partial pressure
p(O 2 ) and temperature T .
For oxide growth using MBE, we want to elucidate the question of oxidation reactions, which
is conveniently achieved by Ellingham diagrams. An Ellingham diagram is a compilation of
Gibbs free energies of oxide compounds of interest, with the addition of “gas lines”. One
characteristic is the normalization of G f (T ) to one mole of gaseous oxygen O 2 ; this rule perfectly
describes the adsorption-controlled EuO synthesis, in which the growth progress only
depends on the limited and constant oxygen supply. This weighting of G f (T ) curves to one
Further reading in the field of molecular thermodynamics is found in Dickerson (1969). 36