Association

5.2. Thermodynamic analysis of the EuO/Si interface 97
Figure 5.8.: Three growth regimes of EuO, illustrated in a Eu–O–Si Gibbs triangle (inset).
(I) describes the oxygen-rich regime, (III) the Eu-rich EuO regime, and (II) can be referred to,
when EuO is stabilized on (H-passivated) Si, as the stoichiometric EuO regime.
significant shift to lower Gibbs free energy from the Eu oxides to SiO 2 by 260 kJ/mol, which
clearly exceeds energies of the Si surface or surface diffusion kinetics. This energy shift is
nearly temperature independent. Thus we can conclude, that EuO is not only statically stable
on Si, but is also under gaseous oxygen supply and under elevated temperature significantly
more stable than the formation of SiO 2 . We remark, that this prediction is of thermodynamic
nature, thus the result (stable Eu oxides) may be reached after a long time or even mixed up
by interface kinetics.
The Gibbs phase rule 36 connects the number of constituents C and resulting phases P with
the degrees of freedom F of a system under constant pressure as
F = C − P +1. (5.1)
In our system, the constituents are O 2 –Eu–Si at constant and limited O 2 pressure, and the
resulting phases are assumed to be SiO 2 + EuO + EuSi 2 during Eu distillation growth † .For
this three-constituent system, a Gibbs triangle serves as a perfect visualization as depicted
in Fig. 5.8a. The Eu–O–Si phase diagrams are constrained to contain only one, two, and
at most three-phase regions. A one-phase region exists at the vertex of the triangle, e. g.
Si. A two-phase region exists on any tie line where phases thermodynamically meet, e. g.
Si and Eu. The interior of any triangle is a three-phase region, including probabilities for
e. g. EuO + Eu + EuSi 2 in region III. Crossing tie lines are not allowed because at the point
of their intersection the simultaneous coexistence of four phases would contradict the Gibbs
phase rule, this for example negates the simultaneous existence of EuO + Eu + SiO 2 + EuSi 2
in thermodynamic equilibrium, as read from a fictitious crossing of lines SiO 2 → Eu with
EuO → EuSi 2 from Fig. 5.8a. Hence, from the Gibbs triangle, we obtain a limit to three
chemical regimes (I, II, and III) of resulting phases, which limits the number of co-existing
phases to those being in the same subtriangle.
The Gibbs phase rule for constant pressure is sometimes referred to as the “solid state phase rule”. It is
particularly suited for solid systems under low pressures in which one constituent is gaseous.
† as introduced in Ch. 4