Suboxides and subselenides: intermediate reaction products to form Ga₂O₃, Ga₂Se₃, In₂O₃, In₂Se₃, SnO₂, and SnSe₂ during molecular-beam epitaxy

Abstract

The molecular-beam epitaxial (MBE) growth of III-O and IV-O materials (e.g., Ga₂O₃, In₂O₃, and SnO₂) is known to be reaction-limited by complex 2-step kinetics and the desorption of volatile suboxides (e.g., Ga₂O, In₂O, SnO). We find that the different surface reactivities of suboxides and respective elements (e.g., Ga, In, Sn) with active oxygen define the film-growth-windows (FGWs) and suboxide-formation-windows (SFWs) of III-O and IV-O materials, respectively. To generalize, we provide elementary reaction pathways and respective Gibbs energies to form binary III-O, III-Se, IV-O, and IV-Se ground-states as well as their subcompounds during their MBE growth. We apply the 2-step kinetics model established for oxides to identify the subselenide-limited growth of Ga₂Se₃ as the specific example for III-Se materials. Our kinetic and thermodynamic conclusions suggest subcompound-limited growth may be an inherent property for the growth of III–VI and IV–VI thin films by MBE and related epitaxial growth techniques.

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I. Introduction

In ‘classical’ molecular-beam epitaxy (MBE) elemental cations react directly with reactive anions to form the intended compound on a heated single-crystalline substrate.^1–4 This is because the MBE growth of III–V and II–VI materials is governed by simple 1-step reaction kinetics.^3,5–10 This basic surface physics and reaction kinetics has been one of the prerequisites to controllably synthesize functional thin films at the highest crystalline level,^11,12 for example, enabling the discovery of novel physics at thin film interfaces.¹³

On the other hand, the MBE growth of III-O and IV-O materials is more complex and determined by complex 2-step reaction kinetics and limited by the formation of volatile suboxides (e.g., Ga₂O, In₂O, and SnO).^14–19 These complex surface reactions kinetically prohibit the growth of functional III-O and IV-O thin films in their adsorption-controlled growth regimes.^20,21

Based on the common valences between III–VI or IV–VI materials, it is conceivable that the growth kinetics of III-Se and IV-Se is similar to that of III-O and IV-O compounds. Previous MBE studies on Ga₂Se₃ and In₂Se₃ indicate their growth to be limited by the formation of their subselenides Ga₂Se and In₂Se, respectively^22–24—similar to Ga₂O₃ and In₂O₃ being limited by the formation of their suboxides Ga₂O and In₂O. However, the underlying reaction kinetics that form III-Se and IV-Se thin films remains elusive and the lack of microscopically understanding their reaction pathways hinders the full exploration of growth conditions and their impact on the phase formation and material properties of functional selenide-based thin films.^25–29

In this paper, we start with identifying the surface reactivities (η_M) of elemental metal (with M = Ga, In, and Sn) as well as the surface reactivities (η_S) of molecular suboxides (with S = Ga₂O, In₂O, and SnO) reacting with oxygen, and find that η_M ≥ η_S. As a consequence of η_M ≥ η_S, the film-growth-windows (FGWs) of III-O and IV-O materials fundamentally change upon growth conditions; as we explicitly demonstrate by the example of Ga₂O₃. We next model the growth of Ga₂Se₃ by complex 2-step kinetics and obtain a similar result as established for Ga₂O₃ growth.³⁰ To strengthen our model results, we provide elemental reaction pathways and thermodynamic calculations for the Ga₂O₃, Ga₂Se₃, In₂O₃, In₂Se₃, SnO₂, and SnSe₂ growth systems. In all cases, we obtain that suboxides and subselenides are the cationic-like volatile species in each material system and we propose that subcompounds (e.g., suboxides and subselenides) are the intermediate and rate-limiting reaction products for III–VI and IV–VI MBE growth, in general.

II. Suboxide-formation-window (SFW) versus film-growth-window (FGW)

To understand the origin of different surface reactivities between adsorbed metals (e.g., Ga) and formed suboxides (e.g., Ga₂O) reacting with oxygen, Fig. 1(a)–(g) collect published growth rate (Γ) data of Ga₂O₃,^15,31–33 In₂O₃,³¹ and SnO₂,¹⁴ normalized by their respective nominal oxygen flux, ϕ_O. It depicts the fundamental Γ evolutions depending on the metal-to-oxygen flux ratio, R = ϕ_M/ϕ_O, at given growth temperature, T_G, and metal flux, ϕ_M.

Fig. 1 (a)–(d) Γ normalized by ϕ_O (Γ/ϕ_O) of β-Ga₂O₃ (2̄01) as a function of the Ga-to-O ratio (R = ϕ_Ga/ϕ_O), measured at different T_G. Data is taken from ref. 31. (e) Γ/ϕ_O as a function of R of β-Ga₂O₃(010) [squares], β-Ga₂O₃(2̄01) [discs, same data as shown in panel (c)], β-Ga₂O₃(001) [diamonds], and β-Ga₂O₃(100) [hexagons]. Data is taken from ref. 15 and 31–33. (f) Γ/ϕ_O of bixbiyte In₂O₃(111) as a function of the In-to-O ratio (R = ϕ_In/ϕ_O), measured at different T_G. Data is taken from ref. 17. (g) Γ/ϕ_O of SnO₂(101) as a function of the Sn-to-O flux ratio (R = ϕ_Sn/ϕ_O), obtained at different ϕ_O. Data is taken from ref. 14. (h) Growth-system-dependent maximum M-to-S formation (solid triangles) [eqn (4)–(6)] and maximum S-to-O formation (i.e., the maximum Γ, open triangles) [eqn (1)–(3)] as a function of T_G. Symbols represent experimental data, solid and dotted lines are numeric models^18,31 serving as guides to the eye.

We start with the observed Γ plateaus: the solid lines in Fig. 1(a)–(g) reflect the film-growth-windows (FGWs) of Ga₂O₃, In₂O₃, and SnO₂, obtained by experimental data (shown by the open symbols). At elevated T_G, a Γ plateau emerges and widens with increasing T_G. We define the value of Γ at the plateau as the value of maximum cation incorporation into the thin film at given T_G. Based on the 2-step kinetics of these materials,^20,31 this Γ value thus gives the maximum available oxygen reservoir for suboxide-to-oxide formation (S-to-O). This also defines the 2^nd reaction step to form the oxides Ga₂O₃, In₂O₃, and SnO₂, via the reactions:³¹

with reaction rate constant, K. Adsorbate and solid phases are denoted as a and s, respectively.

To explain the origin of the Γ plateau as well as the onset of the Γ decrease, we now define the suboxide-reaction-window (SRW) for metal-to-suboxide formation (M-to-S). This defines the 1^st reaction step to form Ga₂O₃, In₂O₃, and SnO₂ through forming the suboxides Ga₂O, In₂O, and SnO, respectively, via the reactions:³¹

with reaction rate constant, κ. The gaseous phase is denoted as g and refers to the volatility of suboxides during growth. The SRW is indicated by the dotted lines in Fig. 1(b)–(e) and (g) and obtained by extending the Γ evolutions from extending the M-rich growth regime (the decreasing Γ with ϕ_M) as well as the O-rich regime (the increasing Γ with ϕ_M) until both lines intersect—always forming a triangular shape. The suboxides formed during the 1^st reaction step, eqn (4)–(6), can be further oxidized to the solid compound through a 2^nd reaction step, eqn (1)–(3), or desorb from the growth surface and limit Γ. As a result, the SRW for M-to-S formation is equal to or wider than the FGW for S-to-O formation, i.e., κ ≥ K, depending on T_G. To illustrate their quantitative differences, the maximum (normalized) formation rates of Ga₂O and Ga₂O₃ as well as of In₂O and In₂O₃ are plotted as a function of T_G in Fig. 1(h). A detailed explanation of this effect is given in Fig. 2. The maximum suboxide formation is defined as the peak value of the SRWs, seen by dotted lines in Fig. 1(b)–(d) and (g). In the case of SnO₂, we obtain the suboxide formation is about 1.4Γ for all ϕ_O at T_G = 650 °C. Overall, the reactivity of Sn > In > Ga with O is higher than the one of SnO > In₂O > Ga₂O with O, respectively. This feature can also be referred to the different vapor pressures and surface reactivities of the respective elements and suboxides.^{16,18,34–38}

Fig. 2 Γ evolutions for III-O compounds, explicitly drawn for Ga₂O₃. Four distinct growth regimes (i)–(iv) are identified (details provided in the text). (a) The gray and purple areas reflect the modelled film-growth-windows (FGWs) for Ga₂O₃ based on the data obtained at T_G = 500 °C [Fig. 1(a)] and T_G = 600 °C [Fig. 1(c)], respectively. The difference (Δ) between SFW and FGW for Ga-to-Ga₂O formation [eqn (4)] and Ga₂O-to-Ga₂O₃ formation [eqn (1)] is drawn as the pale purple area at T_G = 600 °C. At T_G = 500 °C this difference is zero, i.e., Δ = 0. The length of the Γ plateau is given by λ [regime (ii)] and the width of regime (iii) is defined as ω. All parameters shown here are collected in Table 1. (b) FGWs of Ga₂O₃ as a function of R modelled for 400 °C ≤ T_G ≤ 1200 °C (the pale blue lines), using a numerical approach based on the models given in ref. 18 and 39. At low T_G = 400 °C, the triangular shape defines the maximum accessible η_M = η_S ⇒ SRF = FGW (the dark blue line) divided in regimes (i), (iii), and (iv). With increasing T_G, regime (ii) emerges and the ‘shape’ of the FGW fundamentally changes to a trapezoidal shape due to η_M > η_S ⇒ SRF > FGW (the dark blue line in the center). The fact that SRF ≥ FGW, changes λ, Δ, and ω, depending on T_G. As a consequence, regime (iii) becomes narrower until it vanishes at high T_G. The black dashed lines correspond to the solid black model lines in Fig. 1(a)–(d) and serve as a guide to the eye.

Fig. 1(f) and (g) depict the Γ evolutions of In₂O₃ and SnO₂, respectively, as a function of R, showing qualitatively the same kinetic behaviour as observed for Ga₂O₃. The quantitative differences in Γ between III-O and IV-O materials arise from the different group III-O and IV-O suboxide stoichiometries as well as their different surface reactivities.^16,37

Note, for the sake of simplicity, reactions (1)–(6) are selected as specific examples but may be generalized for other III-O and IV-O materials. For example, the knowledge of the 2-step reaction kinetics was used to form Al₂O₃,⁴⁰ rutile GeO₂,⁴¹ or amorphous GeO₂⁴²via the formation of their suboxides Al₂O and GeO, respectively. We further note that a ‘direct reaction’ to form the solid-state compound, e.g., via 2Ga + 3O → Ga₂O₃, can be kinetically excluded. This assumption is reasonable as the formation of complex compounds can be (usually) described by a set of elementary reactions rather than by non-elementary reactions.⁴³ In other words, forming the oxide thin film via a set of multiple elementary surface reactions via a suboxide formation step is kinetically preferred over a single non-elemental surface reaction step. As the suboxide itself may also undergo a multi-step reaction pathway, we propose a general reaction scheme to form binary III–VI and IV–VI materials and sketch their possible reaction pathways in Fig. 5 (see below).

A. Surface-orientation Γ dependence of Ga₂O₃

Fig. 1(e) shows the comparison of β-Ga₂O₃ FGWs for different surface orientations of β-Ga₂O₃(010), β-Ga₂O₃(2̄01), β-Ga₂O₃(001), and β-Ga₂O₃(100) as a function of R, at otherwise similar growth conditions.^15,17,32,33 At given T_G ≈ 600 °C,^15,31–33 the orientation dependence of Γ on the (hkl) plane, Γ_(hkl), for β-Ga₂O₃ is quantified as

Γ₍₀₁₀₎ ≈ 1.5Γ_(2̄01) ≈ 3.7Γ₍₀₀₁₎ ≈ 7.7Γ₍₁₀₀₎. (7)

Note this quantification depends on the adsorption and desorption kinetics of Ga₂O on the respective β-Ga₂O₃ (hkl) growth plane and strongly depends on T_G.^15,30,44 For example, at T_G = 500 °C the relation Γ₍₀₁₀₎ ≈ 2.1Γ_(2̄01) was observed,²⁰ suggesting a different functional dependence of sticking coefficients on the respective Ga₂O₃ growth surface. We thus qualitatively propose, the orientation-dependent Γ evolution of Ga₂O₃ can be explained by an interplay of the corresponding orientation-dependent O sticking coefficients (σ) and suboxide surface reactivities η_S, leading to:

Γ₍₀₁₀₎ > Γ_(2̄01) > Γ₍₀₀₁₎ > Γ₍₁₀₀₎. (8)

A similar orientation-dependent Γ of β-Ga₂O₃ in binary Ga–O and Ga₂O–O systems is reported in ref. 18, 33 and 44.

B. Suboxides limiting the growth domain of III-O compounds

For all compounds, we find that SRW ≥ FGW, thus, the physical origin of changing FGWs and emerging Γ plateaus can now explained.

Fig. 2(a) sketches the Γ evolutions of Ga₂O₃ (2̄01) at T_G = 500 °C (the gray shaded area) and T_G = 600 °C (the purple shaded area), see also Fig. 1(a) and (c). Four regimes are indicated: (i) the O-rich growth regime, i.e., the increasing Γ with increasing Ga flux, ϕ_Ga. In this regime, enough O adsorbates are available to fully oxidize all adsorbed Ga via the consecutive reaction Ga → Ga₂O → Ga₂O₃ [reactions (4) and (1)]. (ii) The ‘pseudo’ O-rich growth regime identified by the width of the Γ plateau, λ. Here, enough O adsorbates are available to oxidize all adsorbed Ga to its suboxide via Ga → Ga₂O [reaction (4)] but not enough O is available to oxidize all formed suboxides to its solid-state compound via Ga₂O → Ga₂O₃ [reaction (1)]—due to thermally-induced suboxide desorption. In other words, the Γ plateau emerges once the formed Ga₂O adsorbate density exceeds the O adsorbate density available for reaction Ga₂O → Ga₂O₃ [reaction (1)].²¹ (iii) The Ga-rich growth regime identified by the decreasing Γ with ϕ_Ga and its width, ω. Here, not enough O is available to oxidize all remaining suboxides to its solid-state compound via Ga₂O → Ga₂O₃ [reaction (1)]. Now, this is due to an O-deficient-induced suboxide desorption mechanism—in addition to the thermally-induced suboxide desorption identified for regime (ii). (iv) The no-growth regime where Ga₂O₃ growth ceases for R ≥ R_m. Here, not enough O is available to oxidize Ga → Ga₂O [reaction (4)] and reaction (1) becomes kinetically forbidden as all available O is consumed in reaction (4).

We next answer the question: Why does Γ start to decrease at R^o [see Fig. 2(a)] and regime (iii) is entered? Each leaving Ga₂O that cannot be oxidized removes 2 × Ga but only 1 × O from the growth front, producing a more O-rich Ga-to-O surface ratio than expected from the nominally supplied ϕ_Ga and ϕ_O. Nevertheless, for R > R^o, ϕ_Ga and resulting Ga adsorbate density exceed a critical value, resulting in a Ga-rich growth surface and thus regime (iii) is entered. In this regime, not enough O is available to oxidize all formed Ga₂O that have remained on the growth surface (i.e., Ga₂O molecules that have not desorbed), and Γ decreases due to the O-surface-deficiency-induced suboxide desorption. As specific example, all parameters and values for Ga₂O₃ are indicated in Fig. 2(a) and collected in Table 1, which, in turn, are extracted from the data plotted in Fig. 1(a)–(d). After considering all desorbing species, it is found that the Ga₂O₃ growth surface becomes stoichiometric once

see Fig. 2 and Table 1.

Table 1 Stoichiometric M-to-O flux ratio, R*. Maximum suboxide formation rate, Γ*, for Ga-to-Ga₂O oxidation. The growth rate value at the plateau, Γ^p. Maximum flux ratio where Ga₂O₃ growth is possible, R^m. The flux ratio at the beginning of the plateau, Rⁱ (with ‘i’ for in). The flux ratio at the end of the plateau, R^o (with ‘o’ for out). The maximum Ga adsorbates, n_Ga, and the maximum O adsorbates, n_O. The difference Δ between the Ga₂O SRW and the Ga₂O₃ FGW. The length λ of the plateau and the width ω of the Ga-rich regime (ii). Finally, the Ga-to-O adsorbate ratio at the end of the plateau, n_Ga/n_O = x/y = constant ≤ ϕ_Ga/ϕ_O [see eqn (9)]. Parameters are indicated in Fig. 2(a) and values are extracted from the data shown in Fig. 1(a)–(d). This example may serve as a blueprint for all discussed III–VI and IV–VI materials

T G (°C)	R* = 2Γ* = 1/3R^m	R ^i = nGa = 2Γ^p	R ^o	λ = R^o − R^i	n O = 1/2(3R* − λ)	Δ = Γ* − Γ^p	ω = 3R* − R^o	n Ga/nO
500	2/3	2/3	2/3	0	1	0	4/3	=2/3
550	0.58	0.48	0.79	0.31	0.72	0.05	0.97	≈2/3
600	0.52	0.36	0.85	0.49	0.54	0.08	0.71	≈2/3
675	0.48	0.24	0.97	0.73	0.36	0.12	0.42	≈2/3

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The above findings have fundamental consequences for the Γ evolution of III–VI and IV–VI compounds. For example, λ increases and ω decreases with increasing T_G because of the enhanced thermally-induced suboxide desorption. As a result, the ‘shape’ of the accessible FGW changes upon growth conditions. Based on the data plotted in Fig. 1(a)–(d), Fig. 2(b) now depicts such an Γ evolution as a function of R and different T_G (here of Ga₂O₃).

At low T_G, the triangular shape of the modeled Γ defines the maximum possible FGW for these materials. With increasing T_G, the Γ plateau emerges and the growth domain becomes trapezoidal and narrows until growth eventually ceases. This finding reveals that the growth of Ga₂O₃ in the Ga-rich regime and elevated T_G is hardly possible, associated with extremely slow Γ. Nevertheless, these growth conditions are desired to improve the crystallographic and transport properties of Ga₂O₃ grown by conventional MBE.²⁸ The same argument holds for In₂O₃, for example.

A solution to overcome these intrinsic and detrimental growth limits for group III and group IV oxides is the use of recent advances in their thin film synthesis, such as suboxide MBE (S-MBE),^20,21,45 metal-exchange catalysis (MEXCAT)^18,37 with metal-oxide-catalyzed epitaxy (MOCATAXY),^17,46 thermal laser epitaxy (TLE),⁴⁰ or hybrid MBE (hMBE).^47,48

These findings can also be transferred to IV-O materials, e.g., SnO₂. In contrast to the ‘asymmetric’ Γ plateau observed for III-O materials (Fig. 2), the Γ plateau observed for IV-O materials is ‘symmetric’ as plotted for SnO₂ in Fig. 1(g). The occurrence of an asymmetric Γ plateau (III–VI) or symmetric Γ plateau (IV–VI) can be explained by the different stoichiometries of III-O and IV-O suboxides. For example, during the growth of SnO₂ the desorption of SnO removes 1 × Sn and 1 × O, resulting in the observed symmetric Γ plateau. In contrast to SnO₂, during the growth of Ga₂O₃ the desorption of Ga₂O removes 2 × Ga and 1 × O, resulting in the observed asymmetric Γ plateau. Another quantitative consequence of the different suboxide stoichiometries is the differently observed slopes in their M-rich regimes for III-O and IV-O materials with

respectively. For example, see Fig. 1(a) for III-O and Fig. 1(g) for IV-O materials.

III. Subselenide-limited growth of Ga₂Se₃

The formation of suboxides has been experimentally reported^14–16 and identified as the growth-limiting step for III-O and IV-O materials and reaction-rate models describing the complex 2-step kinetics for these materials have been developed.^18,31 We anticipate the same kinetics and models can be applied to other III–VI and IV–VI compounds. Therefore, we now apply the 2-step model to the growth of III-Se materials, explicitly, to the growth of Ga₂Se₃.

Fig. 3 shows the Γ evolution of Ga₂Se₃ as a function of the Se-to-Ga ratio, r. The 2-step model described above for oxides is applied to the data and describes the growth kinetics for Ga₂Se₃ very accurately—indicating its MBE growth is limited by Ga₂Se desorption. A similar Γ-behavior is also reported for the growth of In₂Se₃ by MBE.²³ In ref. 22 and 23, it was speculated that the growth of Ga₂Se₃ and In₂Se₃ ceases in the excess of Ga and In fluxes, respectively, due to the re-evaporation of the Ga and Se compounds. However, the physical origin for the observed Γ evolutions for Ga₂Se₃ and In₂Se₃ remained elusive.^22,23 The black lines in Fig. 3 show numeric model calculations by a subselenide-mediated 2-step model. Three distinct regimes are identified: (I) no growth regime for 0 < r = ϕ_Se/ϕ_Ga ≤ 1/2, because all reactive Se is consumed for Ga₂Se formation, i.e., r = 1/2 defines the stoichiometric flux ratio for 2Ga + Se → Ga₂Se formation, e.g., viareaction (14). (II) The Ga-rich regime is entered for 1/2 < r ≤ 3/2, because not enough reactive Se is available to convert Ga₂Se → Ga₂Se₃, e.g., viareaction (15). The stoichiometric flux ratio for Ga₂Se₃ formation is thus r = 3/2. (III) For r > 3/2 the Se-rich flux regime is entered and enough reactive Se is available to selenize Ga → Ga₂Se → Ga₂Se₃. Γ is now limited by the supplied ϕ_Ga.

Fig. 3 Normalized Γ/ϕ_Ga of α-Ga₂Se₃ as a function of the Se-to-Ga flux ratio, ϕ_Se/ϕ_Ga = r. Data is taken from ref. 22, Ga and Se densities in α-Ga₂Se₃⁴⁹ are used to convert ϕ_Ga, ϕ_Se, and Γ into nm⁻² s⁻¹. The solid black line is the application of the 2-step model to the experimental data. The data and model shown in the main graph and the inset are complementary but displayed for the sake of readability. In the Se-rich rich regime, Γ is maximized and limited by the supplied ϕ_Ga. Note, the horizontal axis in this graph is swapped when compared to the horizontal axes in Fig. 1.

Based on the data shown in Fig. 1–3, we can now generalize eqn (10) for III–VI and VI-VI materials for the anion-rich regime (A), plateau regime (P), and cation-rich regime (C) as

IV. Thermodynamic analysis and surface reactions

To support and strengthen our prediction that suboxides and subselenides limit the growth of oxides and selenides, respectively, we now perform thermodynamical calculations. Equilibrium calculations are performed using the SGTE substance database (SSUB5)⁵⁰ within the Thermo-Calc software⁵¹ to assess the evaporation behavior of cation-like and anion-like species as a function of temperature of the binary oxides Ga₂O₃, In₂O₃, and SnO₂ and the complementary, binary selenides Ga₂Se₃, In₂Se₃, and SnSe₂. The results are plotted in Fig. 4.

Fig. 4 (a) Calculated partial pressures of the gas species C = Ga, In, Sn, A = O, Se, C_y−xA_y−x = GaO, InO, InSe, SnO, SnSe [reaction (12)], and C_xA_y−x = Ga₂O, In₂O, In₂Se [reactions (13) and (14)].

For the investigated compounds Ga₂O₃, In₂O₃, SnO₂, In₂Se₃, and SnSe₂, we find that the most volatile, cationic-like species at relevant T_G are the suboxides and subselenides Ga₂O, In₂O, SnO, In₂Se, and SnSe, respectively, and are in accordance with our kinetics findings that their growth is reaction-limited by subcompound formation and their subsequent desorption. Note, for the Ga₂Se₃ system, the subselenides GaSe and Ga₂Se are missing in the SSUB5 and other thermodynamic databases, thus, we use the kinetic data shown in Fig. 3 to identify that the growth of Ga₂Se₃ is reaction-limited by the subselenide Ga₂Se. This is in agreement with all other investigated growth systems. For example, if Ga was the volatile, cationic species limiting the growth of Ga₂Se₃, Γ would reach a plateau in the Ga-rich regime instead, being similar to the growth kinetics observed for binary III-N compounds.⁸ The fact that Γ of Ga₂Se₃ and In₂Se₃ decrease in the Ga-rich and In-rich regimes, respectively, can thus be explained by the desorption of Ga₂Se and In₂Se. We note that the desorption of GaSe and InSe would also explain a decreasing Γ in the cation-rich regimes but with different slopes as given in eqn (11).

In addition, calculated Gibbs energies (ΔG) to form Ga₂Se₃ further strengthen our hypothesis of a 2-step reaction kinetics underlying the formation of III-Se compounds. To unambiguously identify the growth-rate-limiting steps of Ga₂Se₃ and In₂Se₃, in situ line-of-sight mass spectroscopy will reveal which subcompound is formed on the respective growth surface.

To microscopically understand the observed and modeled Γ evolutions (Fig. 1–3) and evaporation of suboxides and subselenides, a general reaction scheme for III–VI and IV–VI compounds is proposed in Fig. 5. It depicts a C_xA_y layer (e.g., Ga₂Se₃), impinging cation flux ϕ_c and anion flux ϕ_a, producing the cation (C), anion (A), and subcompound surface populations C_y−xA_y−x (e.g., GaSe), C_xA_y−x (e.g., Ga₂Se), C_xA_2(y−x) (e.g., Ga₂Se₂). Stoichiometric coefficients for III–VI and IV–VI materials are x = 2 and y = 3 as well as x = 1 and y = 2, respectively. The reaction scheme depicted in Fig. 5 is an extension and refinement of the reaction scheme introduced in ref. 31.

Fig. 5 MBE reaction scheme for binary III–VI and IV–VI materials, showing impinging ϕ_c and ϕ_a, resulting cation n_c, anion n_a, respective subcompound reservoirs, and the final compound C_xA_y. Chemical reactions (12)–(17) are indicated by reaction rate constants k_i.

Consecutive reaction pathways to form III-O, III-Se, IV-O, and IV-Se are:

with reaction rate constant k_i with i = α, β, γ, δ, ε, ζ. In eqn (13), the relation y − 1 = x is used. Note, for the growth of IV–VI compounds, reactions (13) and (16) are forbidden and reactions (15) and (17) are identical due to their stoichiometric coefficients x = 1 and y = 2. Consequently, the surface reaction pathways for IV–VI are not as complex as for III–VI compounds.

It has been shown for III-O materials (e.g., Ga₂O₃) and IV-O materials (e.g., SnO₂) that these compounds can be chemically decomposed (etched) by their respective elemental metal (e.g., Ga or Sn) to form its respective suboxide (e.g., GaO, Ga₂O, or SnO)¹⁶via the reactions

with etching rate constants e_α and e_β to form the subcompounds C_y−xA_y−x (e.g., GaO or GaSe) and C_xA_y−x (e.g., Ga₂O or Ga₂Se), respectively. Note, for III-O materials only reaction (19) has been experimentally observed under MBE conditions.¹⁶

Finally, Fig. 6 plots our calculated ΔG using eqn (20)–(24) as a function of T_G of Ga₂O₃, Ga₂Se₃, In₂O₃, In₂Se₃, SnO₂, and SnSe₂ (calculations are given in the Appendix I). We calculated ΔG for growth and etch reactions (12)–(19) once data were available.⁵² For all investigated materials and relevant T_G, the formation of the suboxide and subselenide is thermodynamically feasible. This is in agreement with the observed III-O and IV-O kinetics and we further calculate that III-Se and IV-Se compounds can be chemically decomposed by their elemental metal to form their respective subselenide through reactions (18) and (19). We thus conjecture that the formation of suboxides and subselenides is kinetically and thermodynamically favorable and the rate-limiting step for a wide-range of III-O, III-Se, IV-O, and IV-Se compound materials.

Fig. 6 The Gibbs energy (ΔG) as a function of temperature T. Values of ΔG are in eV per formula unit (f.u.). (a)–(f) ΔG for subcompound formations viaeqn (12) [black solid lines] and eqn (13) [dark-blue dotted lines]. (g)–(l) ΔG for solid thin film formations viaeqn (17) [blue solid lines]. (m)–(r) ΔG for thin film etching to form subcompounds viaeqn (18) [gray dotted lines] and eqn (19) [gray solid lines].

V. Conclusions

We identify by published growth rate (Γ) data of the Ga₂O₃, In₂O₃, and SnO₂ growth systems that the elements Ga, In, and Sn possess a higher reaction efficiency (η_M) with adsorbed O than their corresponding volatile suboxides (η_S) Ga₂O, In₂O, and SnO. We find that η_M ≥ η_S and quantified the fundamental growth domain of III–VI materials whose regimes strongly depend on MBE growth conditions and find that SFW ≥ FGW. In particular, we observe a vanishing M-rich growth regime with increasing T_G, leading to off-stoichiometric growth surfaces concerning their initially adsorbed densities of group III and group VI elements.

By combining a 2-step kinetic model to experimental Γ data of Ga₂Se₃ with our thermodynamic analysis and calculations we find the volatile species for the Ga₂Se₃, In₂Se₃, and SnSe₂ systems are the subselenides Ga₂Se, In₂Se, and SnSe, respectively. We provide a detailed reaction diagram for the growth of III-O, III-Se, IV-O, and IV-Se materials systems, supported by thermodynamic calculations.

The identified thermodynamic and kinetic feasibility of the proposed 2-step reaction mechanisms for III-Se and IV-Se materials let us conclude that III-Se and IV-Se compounds grow via (qualitatively) the same 2-step reaction mechanism—similar to what is established for III-O and IV-O materials.^21,31,38 To unambiguously identify the growth-rate-limiting steps and volatile species of the proposed binary III-O, III-Se, IV-O, and IV-Se growth systems [e.g., eqn (12)–(19)], in situ line-of-sight mass spectroscopy will reveal which subcompound is formed on the respective growth surface.

Author contributions

Patrick Vogt: conceptualization (lead), data curation (lead), formal analysis (lead), investigation (lead), methodology (lead), writing original draft (lead). Shun-Li Shang: methodology (equal), data curation (equal), formal analysis (supporting), investigation (supporting), writing original draft (supporting). Zi-Kui Liu: data curation (supporting), investigation (supporting), methodology (supporting), writing original draft (supporting).

Data availability

Data is available upon reasonable request from the corresponding authors.

Conflicts of interest

The authors have no conflicts of interest to declare.

Appendices

Appendix I

Thin film growth via MBE takes place under isobaric-isothermal conditions. The change in the Gibbs energy ΔG(T) at given temperature T is

ΔG(T) = ΔH(T) − TΔS(T), (20)

with the change in enthalpy ΔH(T) and the change in entropy ΔS(T) determined asrespectively. ΔH₀ and ΔS₀ denote the change in ΔH and ΔS at room temperature, T₀ = 295 K. The heat capacity C(T) is calculated as

C(T) = a + b 10⁻³T + c 10⁶T⁻² + d 10⁻⁶T². (23)

For all discussed species, ΔH₀, ΔS₀, a, b, c, and d are taken from ref. 52. A chemical reaction may occur spontaneously once ΔG < 0. For a given reaction, with reactants R_i and products P_j, it can be determined by the sum of the Gibbs energies of P_j, minus the sum of the Gibbs energies of R_i, , i.e.The stoichiometric coefficients of R_i and P_j are denoted as r_i and p_j, respectively.

Acknowledgements

PV acknowledges the Max Planck Institute for Solid State Research for financial support and Martin Eickhoff and Alexander Karg for fruitful discussions regarding the reaction chemistries of III-O compounds. SLS and ZKL acknowledge the support from Endowed Dorothy Pate Enright Professorship at the Pennsylvania State University.

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