Supplemental Material: Quantitative subcompound-mediated reaction model for the molecular beam epitaxy of III-VI and IV-VI thin films — applied to Ga2 O3 , In2 O3 , and SnO2 Patrick Vogt1, ∗ and Oliver Bierwagen1 1 Paul-Drude-Institut
für Festkörperelektronik, Leibniz-Institut im Forschungsverbund Berlin e.V., Hausvogteiplatz 5–7, 10117 Berlin, Germany
THERMODYNAMIC CALCULATIONS — TWO-STEP REACTION MECHANISM
In order to asses the feasibility of our model assumptions and results given in the main text, thermodynamically, we perform thermochemical calculations. We determine the Gibbs free energy of formation, ∆G, of all chemical reactions included in our model. Furthermore, we calculate ∆G for other III-VI and IV-VI compounds that we propose to be describable by our model. Plasma-assisted molecular beam epitaxy (PAMBE), the synthesis method we have used for our studies, takes place under isobaric-isothermal growth conditons. Hence, the corresponding Gibbs free energy G(T) at given temperature T can be estimated as G(T) = H(T) − T S(T),
(S1)
with enthalpy H(T) and entropy S(T). Their values, in turn, can be determined as ∫ TG H(T) = HT0 + dT C(T) (S2) T0
and S(T) = ST0 +
∫
TG
T0
C(T) , dT T
(S3)
respectively, with heat capacity C(T) being a function of T. For a given compound, HT0 and ST0 are the standard enthalpy of formation and standard entropy of formation, respectively, at room temperature T0 = 295 K. For our calculations, the integrals of H(T) and S(T) are taken from T0 to growth temperature TG . At given T, C(T) reads as C(T) = a + b 10−3 T + c 106 T −2 + d 10−6 T 2 .
(S4)
For all species discussed, the prefactors a, b, c, and d, as well as HT0 and ST0 , are taken from Ref. [1]. A chemical reaction may proceed spontaneously if ∆G is negative, i.e. ∆G < 0. For a given reaction, with reactants Ri and products P j , it can beÍdetermined by the sum of the Gibbs free energies of P jÍ , j G P j , minus the sum of the Gibbs free energies of Ri , i G Ri , i.e. as Õ Õ ∆G = p j G Pj − ri G Ri . (S5) j
i
The stoichiometric coefficients of Ri and P j are denoted as ri and p j , respectively. The reactions we have taken into account for our thermochemical calculations are: (i) the direct subcompound formation (e. g. suboxide, subselenide, subsulfuride, or subtelluride)—Eq. (1) in the main text given for III-O and IV-O materials, (ii) the subcompound formation due to elemental metal (Me) etching of a growing layer—Eq. (2) in the main text given for III-O and IV-O materials, (iii) and its further reaction to the solid Me compound—Eq. (3) in the main text given for III-O and IV-O materials. Now, the corresponding chemical reactions for all III-VI and IV-VI compounds read as: xMe (g) + (y − x)A (g) −−−−→ Mex Ay−x (g) , (S6) x(y − 1)Me (g) + Mex Ay (s) −−−−→ yMex Ay−x (g), (S7) Mex Ay−x (g) + xA (g) −−−−→ Mex Ay (s),
(S8)
respectively. Cations are named as ‘Me’ (such as Ga, In, or Sn) and anions are denoted as ‘A’ (such as O, Se, S, or Te). The stoichimetric coefficients of Me and A in Mex Ay are denoted as x and y, respectively. We want to stress that we equated the gaseous phase g to the adsorbate phase a (as given for the reactions in the main text) for our calculations here. As a first approach, this assumption is suitable since during PAMBE the adsorbates either accumulate/desorb from/into the gaseous phase. The calculated ∆G’s by Eqs. (S1)–(S5), for reactions (S6)– (S8), are plotted as a function of TG in Fig.-S 1. In the TG regime we are concerned with, and for all materials we have taken into account, ∆G obtained for subcompound formation through reaction (S6) (depicted as filled squares) is lower than the one obtained for subcompound formation via reaction (S7) (shown as open squares). This result suggests that reaction (S6) is energetically preferred over reaction (S7), i. e. thermodynamically favored. In addition, it thermodynamically leads to the same conclusion of a first reaction step, as given kinetically given in the main article—for a wide range of III-IV and IV-VI compounds. Calculating ∆G for the further reaction of the subcompound to the solid Me compound through reaction (S8) (plotted as discs) proposes the thermodynamic feasibility of our concluded second reaction step for all materials considered, since their obtained ∆G < 0 (for all TG ’s employed).
2 0
(a )
-1 -2 -3
S n O 2
-5 0
(c )
-3 -6 -9
G a 2 O 3
-1 2 -1 5
G ib b s f r e e e n e r g y
Δ
G (e V / f.u .)
-4
0
(e )
-3 -6 -9 0
(g )
-2 -4 -6
-8 -1 0
4 0 0
6 0 0
8 0 0
In 2 S 3
-1 -2 -3
S n S e 2
-4 -5 0
(d )
-2 -4 -8
-1 0 0
(f)
-2 -4 -6
In 2 S e 3
-8
(h )
1 0 0 0 4 0 0
In 2 T e 3
T e m p e ra tu re T
6 0 0 G
pound decomposition [10–19], (iii) the predictive power of our model for III-O and IV-O semiconductors (with different Me-to-O stoichiometries), (iv) as well as the supportive thermochemical calculations: strongly confirm the physical accuracy and generality of our growth model, with its stated two-step reaction mechanism, for the thin film synthesis of a wide range of III-VI and IV-VI compounds.
-6
G a 2 S e 3
In 2 O 3
-1 2 -1 5
0
(b )
(°C )
8 0 0
1 0 0 0
-1 0 3 0 -3 -6 -9 -1 2
Fig.-S 1. ∆G as calculated by Eqs. (S1)–(S5) as a function of TG . (i) Filled squares show ∆G obtained for first reaction step, Eq. (S6). (ii) Open squares depict ∆G for layer etching, Eq. (S7). (iii) Filled discs plot ∆G for the second reaction step, Eq. (S8). All values of ∆G are given in eV per formula unit (f.u.).
To conclude, subcompound formation through reaction (S6) is thermodynamically and kinetically favored over reaction (S7). Thus, we assume the same reaction behavior for a wide range of III-VI and IV-VI materials, such as for the ones presented. For our model derivation and its predictions, all chemical reactions we have taken into account, the obtained ∆G’s thermodynamically support all our growth model results. These results strengthen its predictive power and extension to other III-VI and VI-IV compounds, as well. However, we note that these thermochemical calculations are obtained for reactions proceeding in thermodynamic equilibrium. Thus, they can only serve as a guidance since the growth process during PAMBE is (usually) a non-equilibrium one. Finally, we want to summarize our findings, combine them with literature data, and examine their consistency with our derived growth model (as given in the main text): (i) the existence and formation of subcompounds during growth for various III-VI and IV-VI materials [2–11]—indicating to be an inherent property for the synthesis of these materials, (ii) the higher volatility of a subcompound—e. g. as in the case for the predicted In2 Se3 [12]—as compared to other compound constituents, like elemental Me desorption or solid Me com-
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