Suitability of binary oxides for molecular-beam epitaxy source materials: A comprehensive thermodynamic analysis

Oxides are of enormous interest for a wide range of applications due to the useful behaviors they exhibit, often with property coefficients or figures of merit at or near the very highest. These include magnetoelectrics (e.g., Cr₂O₃),¹ ferroelectrics (e.g., PbZr_0.2Ti_0.8O₃),² ferromagnets (e.g., La_0.7Sr_0.3MnO₃),³ ferrimagnets (e.g., Sr₂FeMoO₆ and BaFe₁₂O₁₉),^4,5 piezoelectrics (e.g., PbZn_1/3Nb_2/3O₃–PbTiO₃),⁶ multiferroics (e.g., BiFeO₃),⁷ superconductors (e.g., HgBa₂Ca₂Cu₃O_8+x),⁸ metal-to-insulator transition materials (e.g., EuO),⁹ and semiconductors (e.g., BaSnO₃^10,11 and CdO¹² with their high mobilities and Ga₂O₃^13,14 with its high figure of merit for power electronics).

To go from properties to technology, it is often desired to combine oxides with other materials to build and investigate the performance of proof-of-principle devices. One technique for producing high-quality oxides in thin film form is molecular-beam epitaxy (MBE). The resulting thin films can be used to establish the intrinsic properties of a material or assess its performance in prototype devices. MBE is widely employed for making high-quality thin films because of its clean, ultra-high vacuum environment allowing film synthesis to be monitored by a variety of techniques during growth as well as its notable absence of highly energetic species. These characteristics allow for the precise customization of thin films with very few impurities and minimal disorder.¹⁵ When it comes to preparing materials that are highly sensitive to crystalline perfection, MBE achieves electrical transport characteristics surpassing all other thin film growth methods, making it the gold-standard technique for preparing oxide heterostructures.^16–19 

MBE traditionally utilizes elemental molecular beams—one for each element in the compound being formed—that all impinge on the substrate to form the desired compound.¹⁵ Challenges arise when this approach is applied to oxides. Consider trying to grow a layer of an oxide containing harder-to-oxidize elements on top of a layer that oxidizes easily and one does not want to overoxidize, as in the case for SrTiO₃/Si or SrTiO₃/GdTiO₃.^20,21 Having an increased variety of molecular beams to choose from, for example, ones that deliver species already oxidized, could enhance the ability to make well-controlled heterostructures involving oxides by MBE.

For multicomponent materials such as oxides, one might be tempted to just evaporate the desired oxide directly. In general, such an approach does not work because when most oxides are heated, they do not simply evaporate as molecules with the same composition as the starting material, i.e., congruently. Instead, some of the constituents have higher vapor pressures and evaporate first; they evaporate incongruently. This leads to the composition of what is left behind changing, resulting in the partial pressures of the species that evaporate from a multicomponent source to change over time, making it useless for the controlled deposition of thin films. There are, however, some well-known exceptions to this rule. For example, SiO is known to evaporate congruently in vacuum as SiO molecules.^22–24 However, how many other exceptions are there among oxides? Answering this question is important for experimentalists seeking suitable source materials for the growth of oxides by MBE.

Stolyarova and Semenov²² summarized the gas species emanating from Knudsen cells containing many different binary oxides. Complete data, however, showing the partial pressure of each gas species as a function of temperature are not available for many of these oxides,²² limiting the usefulness of the information for MBE experimentalists. Lamoreaux et al.^25,26 conducted thermodynamic analyses on the evaporation behavior of the oxides of elements in groups 1, 2, and 12–14 under reducing conditions (O₂ partial pressure, $PO2$ = 10⁻¹⁰ Pa), oxidizing conditions (⁠$PO2$ = 2 × 10⁴ Pa), and congruent evaporation conditions. By providing the partial pressures of the vapor species over the condensed oxides as a function of temperature, their results^25,26 are widely used to determine if an oxide will make a good MBE source material. Nonetheless, much of the periodic table remains to be filled in as data for many of the oxides of transition metals and rare-earth (RE) metals are missing from their studies.^25,26 This lack of knowledge motivates the present thermodynamic analyses in which we analyze the evaporation behavior of 128 binary oxides; see Table S1 in the supplementary material for a complete list.

In the present work, we perform thermodynamic calculations to comprehensively consider the suitability of binary oxides and two-phase mixtures as potential sources for oxide MBE. Most binary oxides are found to be unsuitable because upon heating, they decompose and evaporate a dominant species that contains either pure oxygen, which pollutes the vacuum environment, or the pure metal, which provides no benefit over using the pure metal directly. A few binary oxides are found that evaporate a metal oxide molecule in the vapor phase and are thus suitable for oxide MBE. Some of these are well appreciated and utilized, while others are new and await experimental verification. Several two-phase mixtures are also found to be suitable for oxide MBE.

Equilibrium calculations were performed using the SGTE substance database (SSUB5)²⁷ within the Thermo-Calc software²⁸ to assess the evaporation behavior of binary oxides, i.e., the gaseous species and their vapor pressures. For amorphous SiO(am), which is well-known to evaporate congruently,^22–24 its close chemical relative GeO(am), and compounds including PtO₂, Pt₃O₄, and PtO, which are missing in the SSUB5 database, thermodynamic descriptions were added using the reported enthalpies (and entropies when available) of formation at 298.15 K for GeO(am),²⁹ SiO(am),³⁰ and the Pt–O compounds.³¹ In calculating the evaporation behavior of oxides, the gas species with the highest partial pressure—which with the increasing temperature is the first to reach a vapor pressure of 10⁻¹ Pa—is described as the “dominant” species. The relevance of 10⁻¹ Pa for MBE is that this partial pressure at the source is typically used for thin film growth.^32–34 If two gas-phase species have the same concentration within a factor of 10, they are both considered dominant; this is the case for the binary oxides of CsO₂, PbO, Rb₂O, Sc₂O₃, TcO₂, U₄O₉, and ZrO₂.

In thermodynamic calculations, the components of the system are defined as the element of interest and O₂ with the number of moles of the non-oxygen element fixed to one, i.e., n(M) = 1 mol. It is worth mentioning that there are many oxygen-containing species in the gas phase such as O, O₂, and O₃. At equilibrium, there is only one independent partial pressure for the independent component O, which can be represented by $PO2$⁠, or P_O, or $PO3$⁠, or the partial pressure of any O-containing species, and the partial pressures of other O-containing species are dependent variables and can be calculated. In other words, the number of independent partial pressures (or activity or chemical potential) equals the number of independent components in the system, and the partial pressures of other components (species) can be calculated, but they are not independent.

The partial pressure of O₂ is fixed at 10⁻⁴ Pa, a typical oxygen background pressure in the growth of oxides by MBE, because at higher background pressures, hot filaments and other MBE components are damaged from oxidation and the fluxes from many elemental sources become unstable due to surface oxidation.^35,36 Calculations are made with the gas phase fixed at zero amount as this is considered to represent equilibrium evaporation from an MBE source; when the material evaporates from an MBE source into the open system of MBE, it does not return. It is worth noting that the total pressure in the system equals the vapor pressure of the condensed phases, which varies with temperature in equilibrium with the fixed gas phase and should not be set as a fixed equilibrium condition in thermodynamic calculations. Furthermore, another benefit in using the partial pressure of O₂ as a fixed equilibrium condition is that the oxygen content in the system is self-regulated by the phase stability of the system. This is an open system with respect to O₂, and the amount of O₂ not only is different in different systems but also varies in the same system as the temperature is changed. An example of the macro file (i.e., the tcm file used by the Thermo-Calc software) to calculate the evaporation of a stable oxide source is provided in Table S3 in the supplementary material.

For metastable oxides in equilibrium with gas, the gas compositions are calculated under the same conditions (temperature, O₂ activity, and moles of the element) as those used for the stable oxide sources, but with all condensed phases removed except for the metastable oxide of interest. Two scenarios are possible when the oxide decomposes; these depend on the kinetics of the compound. If the solid oxide decomposes faster into a metal-oxygen containing gas phase than into another oxide or metal condensed phase, the metastable oxide is potentially useful as an oxide MBE source. If, on the other hand, the solid oxide decomposes faster into another condensed phase (oxide or pure element), O₂ gas will be produced and pollute the vacuum. An example of the macro file to calculate the evaporation of a metastable oxide source is provided in Table S3 in the supplementary material.

For the cases of gas species evaporated from a two-phase mixture of M_mO_n + M with an overall composition M_xO_y, the conditions for thermodynamic calculations are n(M) = 1 mol, n(O₂) = 0.5y/x mol, and the amount of gas phase is fixed at zero, while the partial pressure of O₂ is determined by the three-phase equilibrium of the two condensed phases and the gas phase. The pressure of the gas phase is much lower than the 10⁻⁴ Pa typical oxygen background pressure used in the growth of oxides by MBE that is mentioned above due to the presence of the pure metal in equilibrium at low temperatures. Nevertheless, when the pure metal becomes metastable at high temperatures, the partial pressure of O₂ can increase significantly as shown in the discussion of Ga₂O₃ and relevant to suboxide-MBE (S-MBE). An example of the macro file to calculate the evaporation of a two-phase mixture case is provided in Table S3 in the supplementary material.

These two types of calculations, i.e., the cases with the fixed partial pressure of O₂ vs the two-phase mixture, represent the two different thermodynamic constraints on the system, viz., an open system vs a closed system. Even though the MBE chamber is an open system, in practice because gas is exchanged between the MBE chamber and the surroundings via the input of gases and its departure through several vacuum pumps, it can behave like either an open or a closed system depending on the fluxes of metallic elements, the partial pressure of O₂, and temperature as demonstrated in the discussion of the thermodynamics of MBE (TOMBE) diagram for ternary systems.¹⁹ In the TOMBE diagram, O₂ partial pressure is plotted against temperature and the diagram is labeled with the stable phases with the gas phase always present. For a binary system, in the closed-system regions of the TOMBE diagram, two condensed phases and the gas phase are in equilibrium with one degree of freedom (independent number of potentials) based on the Gibbs phase rule.³⁷ This is the same situation as typical Ellingham diagrams, such as the two-phase mixture in the present work of two components, with the temperature being the independent potential and the chemical potentials being dependent potentials. Meanwhile in the open-system regions, one condensed phase and the gas phase are in equilibrium with two degrees of freedom. This is the case for a fixed partial pressure of the non-oxygen element with both the temperature and the chemical potential of O₂ being independent potentials in maintaining the two-phase equilibrium. For evaporation, both scenarios are useful as demonstrated in the present work, while for MBE growth of thin films, the open-system scenario with one condensed phase is ideal as demonstrated in the MBE growth of Sr₂RuO₄, SrRuO₃, and CaRuO₃ epitaxial films.^17,19

Four possible scenarios for the evaporation of oxide sources are shown in Fig. 1. These are each described in detail below, together with one additional possibility (scenario 5) consisting of the evaporation of mixtures of two condensed phases. Note that we ignore the complex reaction products in scenarios 2–4 for the sake of simplicity; for example, we ignore the possible products of new oxide + O₂, or new oxide + M, or new oxide + additional oxide, or any combination of them other than the mentioned product M_mO_n.

• Scenario 1: Nearly congruent evaporation of metal-oxygen gas species [M_xO_y(s or $ℓ$) → M_xO_y(g); see S1 of Fig. 1 ]. Here, the letters s, $ℓ$⁠, and g in the parentheses indicate solid, liquid, and gas phases, respectively. In congruent evaporation, the species that evaporates into the gas phase has the same stoichiometric ratio as the solid or liquid source from which it comes; see additional details in Sec. II C. Two attributes make congruent evaporation the best scenario for MBE sources: (i) when the oxide is heated in the crucible and evaporates, metal-oxygen containing gas species of known composition will traverse the MBE chamber and deposit onto the substrate, and (ii) the metal-oxygen gas species M_xO_y(g) has the same stoichiometric ratio as the source material M_xO_y(s or $ℓ$⁠), so the composition of the source material will remain constant over time, making it easier to provide a stable flux of this molecular beam of known composition.

• Scenario 2: Incongruent evaporation of metal-oxygen gas species [M_xO_y(s or $ℓ$) → M_mO_n(g); see S2 of Fig. 1 ]. In incongruent evaporation, the species that evaporate have a different overall stoichiometric ratio than the solid or liquid source from which they emanate. Like congruent evaporation, incongruent evaporation can work for the MBE process, but it is less ideal. Although metal-oxygen containing gas species will be deposited onto the substrate, the gas species coming off of the oxide source [M_mO_n(g)] has a different stoichiometric ratio than the source [M_xO_y(s or $ℓ$⁠)]. This difference in chemistry will cause the source material composition to change over time, making it more difficult to control the flux and composition of the molecular beam.

• Scenario 3: Evaporation of oxygen species [M_xO_y(s or $ℓ$) → O_z(g); see S3 of Fig. 1 ]. For some oxides, oxygen species [O_z(g)] will be the dominant species in the vapor. The dominant evaporation of oxygen species from the metal-oxide source is undesirable for an MBE process. In oxide MBE, it is desired to control the oxidant species (often an activated species such as ozone or those emitted from an oxygen plasma) and its flux directly. Oxygen species [O_z(g)] coming from the metal-oxide sources effectively pollute the ultra-high vacuum and cause a loss of the independent control of growth parameters that is desired in MBE.

• Scenario 4: Evaporation of elemental metal species [M_xO_y(s or $ℓ$) → M_z(g); see S4 of Fig. 1 ]. An elemental metal [M_z(g)] can also be the dominant species given off by the oxide source. Oxides that evaporate elemental metal species are also undesirable for MBE. Evaporation of the elemental metal causes the oxide source composition to become oxygen rich over time. In addition, using such an oxide source to generate a flux of metal species generally offers no advantage over the use of a pure metal source.

• Scenario 5: Two-phase mixture of M_mO_n + M as sources. In addition to the above four scenarios, we also examine one more scenario to produce gas species M_xO_y from a two-phase mixture M_mO_n + M with an overall composition M_xO_y. This scenario builds upon experimental work, showing that this is a viable approach.^38,39

Once the gas compositions emanating from the binary oxides are calculated, each binary oxide is evaluated against three criteria based on MBE chamber conditions necessary to produce high-quality thin films. A binary oxide source is identified as the most ideal for oxide MBE if it meets all three of the following criteria:

• Criterion 1. The solid or liquid metal-oxide source is stable under the given conditions. Stable metal-oxide sources are ideal because they do not decompose and release oxygen, which would pollute the MBE vacuum. Some elements in the SSUB5 database²⁷ such as actinium, astatine, californium, curium, einsteinium, fermium, francium, radium, and protactinium do not have solid or liquid oxides, so these are disqualified. Oxides of some transition metal are disqualified because the stable phase under the given conditions is a pure metal in solid or liquid form. Examples include Ag(s), Au(⁠$ℓ$⁠), Co(s), Cu(s), Ir(s), Os(s), Pd(s), Rh(s), and Ru(s). A select few of the elements are stable as liquid oxides under the given conditions such as Bi₂O₃(⁠$ℓ$⁠), FeO(⁠$ℓ$⁠) (i.e., the liquid Fe-oxide phase or the slag phase), Nb₂O₅(⁠$ℓ$⁠), Rb₂O(⁠$ℓ$⁠), Ta₂O₅(⁠$ℓ$⁠), and Ti₄O₇(⁠$ℓ$⁠). If an element is able to form a metastable oxide, the metastable oxide is investigated. For example, Ir(s) is the most stable phase under the given conditions, but iridium also forms IrO₂. Hence, IrO₂ is investigated as a metastable source. Note that the liquid state of FeO(⁠$ℓ$⁠) is a slag phase with a mixture of all iron oxides (such as FeO, Fe₃O₄, and Fe₂O₃) in arbitrary ratios. Usually, melting processes are accompanied by a change in the metal-oxygen stoichiometry to lower the Gibbs energy by mixing, but not in the solid state.

• Criterion 2. The oxygen-metal containing gas species is dominant. A metal-oxygen gas species [M_xO_y(g)] is ideal as it provides pre-oxidized metals with the known oxygen content to the growing film when it reaches the substrate. Binary oxides for which the dominant evaporant is the oxygen species or metal species are disqualified as MBE sources for not meeting this second criterion.

• Criterion 3. The oxide source evaporates nearly congruently. Congruent evaporation ensures that the oxide source provides a constant flux of the desired species at a fixed temperature. This third criterion distinguishes between the most ideal sources for MBE and those that are less ideal because they evaporate incongruently.

In the SSUB5 database,²⁷ 58 elements have stable solid oxides and 14 elements have metastable oxides under the given conditions. To make the present study more comprehensive, additional metastable oxides under the given conditions are included. The list of all evaluated 128 stable and metastable oxides and 17 elements under the given conditions is available in Table S1 in the supplementary material. In addition, 27 two-phase mixtures of M_mO_n + M are examined, where the elements M considered are rare-earth elements and other elements of practical interest; see Table S2. Note that we do not plot phase stability as a function of temperature for these oxides. Instead, we note in Figs. 2 and 4 and each supplemental figure caption (Figs. S1-S81) the stable phases as a function of temperature for each system.

16 solid and liquid oxides meet the criteria to be classified as ideal MBE sources. These are As₂O₃, B₂O₃, BaO, MoO₃, OsO₄, P₂O₅, PbO, PuO₂, Rb₂O, Re₂O₇, Sb₂O₃, SeO₂, SnO, ThO₂, Tl₂O, and WO₃; see Table I. Some of these oxides have been well studied in Knudsen cell evaporation experiments. Some of the identified oxide sources have been used to make thin films via various deposition methods, but only As₂O₃, B₂O₃, MoO₃, PbO, Sb₂O₃, SeO₂, and WO₃ have been used successfully for MBE growth as described in below. We discuss these oxides in Secs. III A 1–III A 13 in alphabetical order except for As₂O₃ and SnO, which are discussed with Sb₂O₃ in Sec. III A 9, and PuO₂, which is discussed with ThO₂ in Sec. III A 11.

Figure 2 shows the calculated evaporation behavior of solid B₂O₃ (the amorphous/glass phase) as a function of temperature under $PO2$ = 10⁻⁴ Pa, where the B₂O₃ gas species is dominant above 1160 K in equilibrium with solid B₂O₃(s); see also Table I. Figure 2 shows that nearly congruent evaporation of B₂O₃(s), defined by $ΔxO2<0.01$ [see Eq. (1)], occurs at T > 1255 K. As an ancillary test, the ideal congruent evaporation of B₂O₃(s) occurs at T > 1398 K under a fixed total pressure, P_tot = 0.1 Pa. The thermodynamics of B₂O₃(s) volatility was investigated by Lamoreaux et al.,²⁶ and it was shown to evaporate congruently as B₂O₃(g). Stolyarova and Semenov²² also noted that the dominant vapor species of B₂O₃(s) thermally evaporated by a Knudsen cell is B₂O₃(g). B₂O₃ sources in MBE are well known, though they have only been used as doping sources for the MBE growth of silicon.^40,41

Although B₂O₃ is ideal for oxide MBE (see Fig. 2), and thin films have been successfully fabricated using MBE, the high reactivity of B₂O₃ with water to form H₃BO₃ under ambient conditions precludes it from many practical applications.⁴² Putkonen and Niinistö⁴² applied a protective Al₂O₃ capping layer using atomic layer deposition (ALD) to prevent the B₂O₃ thin film from reacting. When patterned, however, the film immediately suffered degradation from the exposed B₂O₃ edge. Proposed applications of B₂O₃ thin films include transistors, ultra-wide bandgap amorphous oxide semiconductors, and optoelectronics when combined with other oxides.^43–45 Glassy thin films such as B₂O₃ also have potential as electrolytes used in batteries, as electrochemical sensors, as supercapacitors, and as electrochromic sensors.⁴⁵ 

The calculated evaporation behavior of BaO is shown in Fig. S8 along with that of BaO₂; see also Table I. It can be seen that the primary species evaporating from BaO₂ is BaO, so a BaO₂ source works the same as BaO. The dominant BaO species in the molecular beam is relevant to making high dielectric constant perovskites such as BaTiO₃ and (Ba, Sr)TiO₃, which have applications in high-K (K is the dielectric constant) memory,⁴⁶ tunable dielectrics,^47,48 optoelectronics,^49–51 and fuel cells.^52,53 Another important barium-containing oxide is BaSnO₃—a semiconductor with high mobility at room temperature—which offers tantalizing properties for transparent electronics.^10,11,54,55 Barium hexaferrites including BaFe₁₂O₁₉ are ferrimagnetic insulators with excellent performance at high frequency.^5,56–58

In addition to complex oxides containing barium, BaO thin films themselves have applications in high current density cathodes, thermionic energy converters, and optical devices.⁵⁹ BaO films have been grown by many groups using MBE with separate beams of barium and O₂.^60–62 Less common, but nonetheless demonstrated, is the growth of BaO films using a BaO₂ source⁶³ or a BaO source.^64,65 A BaO source could prove particularly advantageous when it is desired to not introduce any excess oxygen, e.g., when growing BaO on silicon and it is ideal to have all of the barium oxidized, yet not to oxidize any of the silicon (which can easily occur if excess oxygen is present).^61,62

Several studies have been carried out on the evaporation of MoO₃ from a Knudsen cell. In these studies, (MoO₃)₃, (MoO₃)₄, and (MoO₃)₅ are all observed in the gas phase with (MoO₃)₃ being most abundant at 850 K.^66,67 Our calculations (Fig. S41, see also Table I) indicate congruent evaporation of MoO₃ as well (T > 768 K at $PO2$ = 10⁻⁴ Pa) also with Mo₃O₉ and Mo₄O₁₂ being most dominant followed by Mo₅O₁₅. Du et al.⁶⁸ grew MoO₃ thin films by MBE on SrTiO₃ substrates by evaporating MoO₃ powder from an effusion cell onto a substrate held at 673 K–823 K and immersed in activated oxygen species from an oxygen plasma operating at $PO2$ = 4 × 10⁻⁴ Pa. MoO₃ molecular beams could be used as a route to grow SrMoO₃ thin films, which are being studied for use as transparent conductors because of their very high conductivity.^69,70 MoO₃ thin films are also used as electrochromic sensors and in lithium batteries.⁷¹ 

Solid OsO₄(s) is a stable phase at low temperature, T < 304 K. With the increasing temperature, liquid OsO₄(⁠$ℓ$⁠) becomes stable in a narrow temperature range from 304 K to 314 K. Above 314 K, OsO₂(s) is the stable phase and remains so up to 832 K where solid Os(s) becomes stable. Figure S48 shows that the dominant gas species is OsO₄(g) at low temperatures, i.e., T < 1028 K in the plot of OsO₂(s) as an oxide source as well as in the plot of OsO₄(s) as an oxide source. These results indicate that OsO₄(s, $ℓ$⁠) is an ideal source for MBE oxide when T < 314 K at $PO2$ = 10⁻⁴ Pa; see also Table I.

The calculated evaporation behavior of solid P₂O₅ with an orthorhombic structure is shown in Fig. S47; see also the data in Table I. P₂O₅ has exciting uses as a bioactive thin film material to modify surface properties of biomedical devices, to increase corrosion resistance, or for osteogenic applications.⁷² The P₂O₅ thin films are commonly fabricated using a sol–gel method instead of physical vapor deposition. Stolyarova and Semenov²² recorded the congruent evaporation of P₄O₁₀ to P₄O₁₀ gas species from a P₂O₅(s) source at 500 K. No publication in the literature was found using P₂O₅ in an MBE process.

PbO(s) was shown by Lamoreaux et al.²⁶ to have dominant species Pb(g) under reducing conditions and PbO(g) under oxidizing conditions. Experiments by Lopatin et al.⁷³ found that the dominant gas species in PbO(s) evaporated at 900 K–1150 K are Pb_nO_n(g), Pb(g), and O(g). The Pb_nO_n(g) species comes from the congruent evaporation of $nPbO(s)→PbnOn$ with n = 1–6; the Pb(g) and O(g) species come from a partial dissociation of $PbOs→Pbg+12O2.$ The polymeric Pb_nO_n gas species observed by Lopatin et al.⁷³ were also included in the evaporation analysis by Lamoreaux et al.,²⁶ but these Pb_nO_n gas species are not included in the SSUB5 database and can thus not be seen in Fig. S50. Although the present work does not take into account the Pb_nO_n (n = 2–6) species, the yellow orthorhombic phase PbO_yellow is dominant below 1163 K under the conditions of our thermodynamic calculations (not shown), which agrees with Lopatin’s assessment that PbO is dominant between 900 K and 1150 K.

The evaporation behavior of PbO_yellow in Fig. S50 disagrees with the analysis by Lamoreaux et al.,²⁶ which shows Pb(g) to be dominant from 700 K to 1800 K at $PO2$ = 10⁻¹⁰ Pa. On the other hand, an experimental attempt at using PbO(s) as an MBE source in an MBE system with no added oxygen (and a chamber background pressure in the 10⁻¹⁰ Pa range) found that PbO(s) decomposed into mainly Pb(g) with very little PbO(g),⁷⁴ which is consistent with the expectations of Lamoreaux et al.²⁶ To see if the disagreement between our calculations and those of Lamoreaux et al.²⁶ is due to the difference in O₂ partial pressure conditions, we redid our calculations as a function of oxygen partial pressure for $PO2$ from 10⁻¹ Pa to 10⁻¹⁰ Pa. Figure 3 shows the temperatures at which Pb(g) becomes the dominant species in the gas phase as a function of $PO2$⁠; the temperature at which Pb(g) becomes dominant over PbO(g) decreases with decreasing $PO2$⁠. At the pressure used by Lamoreaux et al.,²⁶  $PO2$ = 10⁻¹⁰ Pa, the calculations show that the partial pressure of Pb(g) exceeds that of PbO(g) at 725 K, in good agreement with findings of Lamoreaux et al.²⁶ who found that this crossover occurs around 700 K at $PO2$ = 10⁻¹⁰ Pa.

A PbO source was used by Rispens and Noheda to grow high-quality PbTiO₃ films by MBE.⁷⁵ These authors found the use of PbO(s) advantageous over Pb(⁠$ℓ$⁠), which is consistent with the results from Fig. S50 where the dominant species in the gas phase at MBE-relevant growth conditions is PbO. We note that based on the diagram by Lamoreaux et al.,²⁶ PbO is not a good source to use for the growth of PbTiO₃ or other Pb-containing oxides by MBE. This points to the importance of the current comprehensive analysis at a fixed oxygen pressure (10⁻⁴ Pa) that is relevant for oxide MBE and in the case of PbO dramatically changes the dominant species in the gas phase as shown in Fig. 3.

Rb₂O is a liquid at temperatures at which the dominant species evaporated from it have a vapor pressure of 10⁻¹ Pa. At this temperature, there are actually two species with high and comparable vapor pressure in the gas phase: Rb(g) and Rb₂O(g). From the calculated vapor pressures over liquid Rb₂O, Rb₂O is seen to evaporate nearly congruently at higher partial pressures of O₂ (such as $PO2$ = 10⁻⁴ Pa and 10⁻¹ Pa); see Fig. S56 as well as Table I.

At lower $PO2$ (e.g., <10⁻⁴ Pa), the vapor is Rb-rich compared to the Rb₂O source. Although at these lower oxygen partial pressures it is not an ideal MBE source, it should still produce a molecular beam containing a significant fraction of pre-oxidized rubidium in the form of Rb₂O. Rb₂O has been used as an MBE source for the growth of superconducting (Bi, Rb)BaO₃ films by Hellman et al.⁷⁶ These authors noted that the Rb₂O source showed signs of decomposition into a Rb-rich molecular beam, consistent with the present calculations at different $PO2$ values (Fig. S56). They also mentioned that they did not observe any Rb oxide molecules by mass spectrometry. This is inconsistent with the present calculations with higher partial pressures of O₂, in which significant Rb₂O(g) in the gas phase is expected (see $PO2$ = 10⁻⁴ Pa and 10⁻¹ Pa in Fig. S56). A possible reason for this is the strong dependence of the Rb₂O(g) partial pressure on the oxygen partial pressure. Our calculations are done at $PO2$ = 10⁻⁴ Pa (as well as 10⁻¹ Pa and 10⁻⁷ Pa in the present case); at lower oxygen partial pressures, which are likely the conditions under which Hellman et al.⁷⁶ operated their mass spectrometer, Rb(g) becomes the dominant gas species.

Through Knudsen cell studies, Skinner⁷⁷ found that Re₂O₇(s) evaporates congruently as Re₂O₇(g) at $PO2$ = 2 × 10⁻⁴ Pa and T = 404 K. Due to a small amount of O₂ that simultaneously vaporized, some residual ReO₃(s) was left in the cell. The ions exhibiting the highest signal in Skinner’s mass spectrometry study are in the following order: Re₂O₇, ReO₃, ReO₂, Re₂O₆, and Re₂O₅. In the present calculations (Fig. S57 as well as Table I), Re₂O₇ is the dominant gas species followed by Re₂O₆, which shares the stoichiometric ratio of ReO₃. To the best of our knowledge, Re₂O₇(s, $ℓ$⁠) has not yet been used as an oxide source for MBE growth. Like B₂O₃, Re₂O₇ is very hygroscopic, which may limit its applications.⁷⁸ 

Figure S5 indicates liquid As₂O₃(⁠$ℓ$⁠) is a stable phase at higher temperatures (T > 677 K) transformed from solid As₂O₅(s). In the temperature range 677 K–1274 K, As₂O₃(⁠$ℓ$⁠) is a possible ideal MBE source with the dominant gas species of As₄O₆(g) at $PO2$ = 10⁻⁴ Pa; see also Table I.

The behavior calculated for Sb₂O₃ is analogous to that of As₂O₃. At higher temperatures (T > 997 K), liquid Sb₂O₃(⁠$ℓ$⁠) is a stable phase transformed from solid SbO₂(s). Figure S60 as well as Table I shows that Sb₂O₃(⁠$ℓ$⁠) is also a possible ideal MBE source with the dominant gas species being Sb₄O₆(g) in the temperature range of 997 K–1429 K. Nearly congruent evaporation of Sb₂O₃(⁠$ℓ$⁠) is calculated to occur over the temperature range 997 K–1398 K at $PO2$ = 10⁻⁴ Pa.

Similar to As₂O₃(⁠$ℓ$⁠) and Sb₂O₃(⁠$ℓ$⁠), at higher temperatures (T > 1498 K), liquid SnO(⁠$ℓ$⁠) becomes a stable phase. Figure S65 as well as Table I shows that SnO(⁠$ℓ$⁠) is also a possible ideal MBE source with the dominant gas species being Sn₁O₁(g) when T > 1498 K. SnO and SnO₂ are discussed further in Sec. III B 1.

Note that the solid phases of As₂O₃(s), Sb₂O₃(s), and SnO(s) are metastable, but they evaporate nearly congruently based on our calculations; see Figs. S5, S60, and S65 and Table III. Both As₂O₃(s) and Sb₂O₃(s) have been used as source materials in oxide MBE and were observed to evaporate congruently by Stall.⁷⁴ In his study, Stall noted the much lower vapor pressure of Sb₂O₃ (∼10⁻² Torr) compared with As₂O₃ (∼10² Torr) at 673 K.⁷⁴ To provide the desired flux of Sb₄O₆(g), the effusion cell containing Sb₂O₃(s) was operated at a temperature around 750 K, at which our calculations indicate that SbO₂(s) is the stable phase. Nonetheless, Sb₂O₃(s) was observed to behave as an ideal MBE source by providing a molecular beam with the same stoichiometry as the source material [consistent with our calculation that the dominant species in the gas phase is Sb₄O₆(g)].

The calculated evaporation behavior of SeO₂(s) is shown in Fig. S62 as well as Table I. The results indicate that the gaseous SeO₂ species is dominant when SeO₂(s) is heated and agree well with Knudsen evaporation measurements between 360 K and 660 K.^22,79,80

Metal oxides including SeO₂ as well as SnO₂, ZnO, TiO₂, VO, and WO₃ are particularly good gas sensors because the electrical conductivity of the thin film changes with gas adsorption.^81,82 Manno et al.⁸² studied both SeO₂–SnO₂ and SeO₂–In₂O₃ thin films for applications in NO_x sensors. The SeO₂–In₂O₃ thin films were made by evaporating an InSe source and post-annealing the film in O₂. The SeO₂–SnO₂ thin films on the other hand were deposited onto an unheated quartz substrate by vaporizing 99.999% pure SeO₂ and SnO₂ with $PO2$ = 5 × 10⁻⁵ Pa. After deposition, the films were annealed in flowing oxygen at 400 °C. The specific evaporation behavior of SeO₂ and SnO₂ is not discussed by the authors,⁸² but it is likely that SnO_x species were in the vapor since SnO_x species from an SnO₂ source were noted under similar evaporation conditions.¹¹ 

In agreement with the present calculations (Fig. S71 as well as Table I), the dominant gas species given off by a ThO₂ source in a Knudsen cell was shown to be ThO₂(g).²² The evaporation behavior of ThO₂ is, however, not well studied in the literature, and the use of ThO₂ source materials in MBE is not reported. Similar to ThO₂(s), PuO₂(s) has not been reported as an MBE source either. In addition, no Knudsen cell evaporation studies could be found for PuO₂ in the literature.

Tl₂O was calculated by Lamoreaux et al.²⁶ to exhibit congruent evaporation behavior under reducing conditions. Tl₂O is more stable than Tl₄O₃ and Tl₂O₃ at low oxygen partial pressures, which makes it ideal for the MBE vacuum environment.^26,83 Tl₂O powder is widely available from commercial distributors but is quite toxic. One concern with Tl₂O is its high reactivity with O₂ and many other elements at elevated temperatures.⁸³ Holstein⁸³ used an argon atmosphere to mitigate this issue and found the vapor pressure of Tl₂O(g) over the Tl₂O(s) source to be 29 Pa at 820 K, which agrees well with the evaporation behavior calculated in the present work (Fig. S73 and Table I). Tl₂O has been used in the growth of thin films of the high-temperature superconductors Tl–Ba–Ca–Cu–O and Tl–Pb–Sr–Ca–Cu–O, but no literature was found showing Tl₂O(s, $ℓ$⁠) used as a source material in MBE.⁸³ 

WO₃(s) is described to vaporize polymerically like MoO₃(s);²² see Fig. S77 vs Fig. S41. A Knudsen cell evaporation study by Blackburn et al.⁸⁴ showed that (WO₃)₃ is the dominant vapor species. WO₂ is also a vapor species but is reported to decompose into (WO₃)₃ and W species.⁸⁴ The latter observation is not reflected in the present calculations (Fig. S77 and Table I), since the partial pressures of WO₂ and W in the present calculations are many orders of magnitude lower than that of W₃O₉ or any of the W_xO_3x species. The results shown in Fig. S77 are consistent with a heated charge of WO₃ that provides a beam of W₃O₉ as described in the Knudsen cell study of Blackburn et al.⁸⁴ Li et al.⁸⁵ used WO₃ powder as an MBE source to successfully grow WO₃ thin films for applications as a photocatalyst. In their experiments, WO₃ powder was evaporated from a high-temperature effusion cell and deposited as an epitaxial WO₃ film at a substrate temperature of 773 K in the presence of activated oxygen species from an oxygen plasma operating at $PO2$ = 4 × 10⁻⁴ Pa.⁸⁵ 

Having established which binary oxides evaporate nearly congruently, we next consider the possibility that a metastable binary oxide (if there were some way to produce it and constrain it from not decomposing in the solid state) would evaporate nearly congruently when heated. Table II summarizes the binary oxides that do not meet criterion 3 for nearly congruent evaporation (see Sec. II C) but evaporate incongruently instead. Such oxides are not ideal as the composition of the source will generally change over time as it becomes depleted in some component, but they might still be useful for MBE if they provide a desired gaseous species. In addition to considering the binary oxides in Table II, the species emanating from them with the highest partial pressure, i.e., the dominant gas species, were separately investigated as metastable sources to see if nearly congruent evaporation exists for those species. TcO₂ was also investigated but is omitted from the present work because it exhibits a decrease in vapor pressure with the increasing temperature in our calculations.

Figure S65 shows the calculated evaporation behavior of SnO₂ as a function of temperature, where the gaseous species SnO is dominant above 1128 K in equilibrium with solid SnO₂ (see also Table II). To provide MBE-relevant fluxes, the temperature of the SnO₂ source is about 1350 K–1400 K, so when an SnO₂ source is used in oxide MBE, SnO is indeed the dominant species in the gas phase. SnO₂ was first used as a source in oxide MBE to achieve n-type doping of Ga₂O₃ with Sn⁴⁺.⁸⁶ More recently, it has been used to grow BaSnO₃^10,11 and SnO.⁸⁷ The La-doped BaSnO₃ films grown by MBE with an SnO₂ source exhibit the highest mobility^10,11 and the best transistor performance⁵⁵ achieved to date, demonstrating SnO₂ as a viable MBE source.

Because SnO(g) is the dominant evaporating species, the use of SnO₂(s) as an MBE source also provides a possible route to grow Sn²⁺-containing compounds such as p-type semiconductors that are predicted to have high mobility including SnO,^87,88 K₂Sn₂O₃,⁸⁷ Rb₂Sn₂O₃,⁸⁷ and Ta₂SnO₆.^88,89 Note that Sn²⁺ is a difficult oxidation state to stabilize. Many thermodynamic phase diagrams, e.g., those for Sn–O⁹⁰ as well as Sn–SnO₂,⁹¹ omit SnO considering it to be metastable and to disproportionate into Sn and SnO₂; this agrees with the present thermodynamic analysis shown in Fig. S65. Being able to deliver a molecular beam of SnO to the substrate surface greatly simplifies the synthetic challenge to preparing Sn²⁺-containing materials. Furthermore, for the synthesis of anti-perovskites such as Sr₃SnO, it is preferable to have SnO arrive to the growth surface pre-oxidized and without any excess oxygen in order to avoid the undesired reaction of Sr₃SnO with excess oxygen to form unwanted SrO.⁹² Finally, the present work (Fig. S65) also indicates that SnO(s) can be a metastable source that nearly congruently evaporates as SnO.

Following the recent experimental work of Hoffmann et al.,³⁹ we also examined the gas species for evaporation from a two-phase mixture of SnO₂(s) + Sn(⁠$ℓ$⁠) that has an overall composition SnO. Figure 4 shows that SnO(g) is the dominant gas species, indicating that a mixture of SnO₂(s) + Sn(⁠$ℓ$⁠) is also an ideal MBE source providing the nearly congruent evaporation of SnO; see a complete list of the two-phase mixtures examined in Table S2. As discussed in Sec. II A, the partial pressure of O₂ in the two-phase mixture is much lower than the typical MBE background value of 10⁻⁴ Pa, e.g., its value for a two-phase mixture of SnO₂(s) + Sn(⁠$ℓ$⁠) at 1200 K is about 10⁻⁹ Pa. The partial pressures of O₂ and all other species in the gas phase are determined by the two-phase equilibrium and are independent of the relative amounts of SnO₂(s) and Sn(⁠$ℓ$⁠) in the system. The use of equal amounts of SnO₂(s) and Sn(⁠$ℓ$⁠) to give the nominal SnO composition of the source may simplify experimental control if SnO₂(s) and Sn(⁠$ℓ$⁠) have similar evaporation rates. If the two condensed phases have very different evaporation kinetics, one may adjust the relative amounts of each phase in the two-phase mixture to provide the desired partial pressures in the MBE beam. Additional impacts of the metal-to-oxide ratio will be discussed in a forthcoming paper where the S-MBE method is more systematically examined.⁹³ 

The present calculations (Fig. S27) show that Ga₂O₃(s) provides a beam of Ga₂O(g). This is in agreement with the experiments performed by Butt et al.⁹⁴ demonstrating that Ga₂O is the dominant gas species over Ga₂O₃(s). We did not investigate the nearly congruent evaporation of Ga₂O because Ga₂O(s) is not available in the SSUB5 database²⁷ [note that Ga₂O(s) is not a stable phase at 0 K according to first-principles calculations in the OQMD database].⁹⁵ 

Ga₂O₃ thin films have several applications, including gas sensing. Its large bandgap, ability to be doped with n-type carriers, relatively high mobility, and high Baliga figure of merit that is second only to diamond also attract tremendous interest for this semiconductor to be used for high-power electronics and ultraviolet (UV) detectors.^13,14 The growth of Ga₂O₃ thin films by MBE is a burgeoning area of research. Although most groups use Ga(⁠$ℓ$⁠) as the MBE source,^86,96–100 both amorphous¹⁰¹ and epitaxial¹⁰² films of Ga₂O₃(s) have been grown by MBE from a Ga₂O₃(s) source.

Another route to produce a molecular beam of Ga₂O(g) is to use a mixture of Ga(⁠$ℓ$⁠) and Ga₂O₃(s) instead of the incongruent evaporation of Ga₂O₃(s).^38,39 Such an approach has the advantage that a much lower temperature is needed to provide the same flux of Ga₂O(g) in the molecular beam from a Ga₂O₃(s) + Ga(⁠$ℓ$⁠) mixture compared to that from just Ga₂O₃(s), see Fig. S27 and Table S2 with the partial pressure of O₂ being 10⁻²² Pa at 1000 K. A prior MBE work used an iridium crucible to contain Ga₂O₃(s) because of the reactivity of Ga₂O₃ at high temperatures (in excess of 1900 K) needed to grow Ga₂O₃ films by MBE from a pure Ga₂O₃(s) source.^101,102 This temperature is considerably higher than that expected from Fig. S27 to yield a Ga₂O(g) partial pressure of 10⁻¹ Pa (∼1500 K is expected). Using a 5:1 mixture of Ga:Ga₂O₃ decreases the temperature of the crucible, i.e., the temperature needed to reach the same vapor pressure of Ga₂O(g), by about 500 °C.³⁸ A more recent MBE study shows similar temperature lowering for a 3.8:1 mixture of Ga:Ga₂O₃.³⁹ Exploiting molecular beams of suboxides such as Ga₂O(g) to grow films of compounds such as Ga₂O₃(s) is a powerful alternative to conventional MBE with many advantages; this approach is termed suboxide-MBE (S-MBE).⁹³ 

TeO₂ thin films have applications as gamma radiation detectors due to their sensitivity to gamma radiation. Sudha et al.¹⁰³ thermally evaporated TeO₂ powder onto a glass substrate at 10⁻³ Pa to form TeO₂ amorphous thin films for gas sensing; these amorphous films could be subsequently annealed to make them crystalline. The congruent evaporation behavior reported in Sudha et al.’s work¹⁰³ disagrees with the incongruent evaporation calculated in the present study of TeO₂(s) sublimation to Te₂O₂(s) at 760 K–892 K, although TeO₂(s) is the dominant gas species below 760 K (Fig. S70). Unfortunately, Sudha et al.¹⁰³ did not provide a temperature for the thermal evaporation of the TeO₂ powder to allow for comparison between their work and the present study. Additionally, Fig. S70 shows that the vapor pressures of Te₂O₂, TeO₂, and Te₂O₄ gas species decrease at temperatures above 913 K, while the vapor pressures of the other gas species continue to increase based on the present thermodynamic calculations.²⁷ 

The present calculations (see the figures in the supplementary material) indicate that the majority of the rare-earth oxides share a similar evaporation behavior that is described below. The majority are most stable in sesquioxide form and provide molecular beams of stoichiometric REO (RE = rare earth), making them incongruent evaporation sources for MBE. There are, however, a few exceptions where the gas-phase species with the highest partial pressure is RE or REO₂:

• Supply REO: Sc₂O₃, La₂O₃, Ce₂O₃ (and CeO₂), Pr₂O₃, Nd₂O₃, Pm₂O₃, Sm₂O₃, Gd₂O₃, Tb₂O₃, Dy₂O₃, Er₂O₃, and Lu₂O₃.

• Supply RE: Eu₂O₃ (and EuO), Tm₂O₃, and Yb₂O₃.

• Supply REO₂: Y₂O₃.

EuO(s), SmO(s), and YbO(s) are known to be stable as RE²⁺ ions in addition to their sesquioxide forms. EuO(s) is calculated to produce a beam of Eu metal when heated as is its sesquioxide counterpart, Eu₂O₃(s); see Fig. S24. Unfortunately, thermodynamic calculations for SmO(s) and YbO(s) as oxide MBE sources could not be performed because they are not included in the SSUB5 database.²⁷ Experimentally, YbO thin films have been grown by MBE, although using separate beams of ytterbium and O₂ to deposit YbO.¹⁰⁴ The same is true for the growth of EuO by MBE: from separate beams of europium and O₂.^105–107 

Many of these rare-earth sesquioxides have been investigated as high-K gate dielectrics for use in metal–oxide–semiconductor field-effect transistors (MOSFETs) because of their high dielectric constant, large bandgap, and thermodynamic stability in contact with silicon.^108–112 To investigate these rare-earth sesquioxides as a replacement for SiO₂ in MOSFETs, researchers have used MBE to deposit Sc₂O₃, Y₂O₃, La₂O₃, Pr₂O₃, Nd₂O₃, Gd₂O₃, and Lu₂O₃ films on silicon from solid oxide sources of these same materials, i.e., from Sc₂O₃(s),¹¹³ Y₂O₃(s),¹¹⁴ La₂O₃(s),¹⁰⁸ Pr₆O₁₁(s),^115,116 Nd₂O₃(s),¹¹⁷ Gd₂O₃(s),^108,114,118 to Lu₂O₃(s).¹⁰⁸ 

For Sc₂O₃ (Fig. S61), Y₂O₃ (Fig. S78), La₂O₃ (Fig. S36), Ce₂O₃ (Fig. S14), Pr₂O₃ (Fig. S53), Nd₂O₃ (Fig. S44), Pm₂O₃ (Fig. S52), Sm₂O₃ (Fig. S64), Gd₂O₃ (Fig. S28), Tb₂O₃ (Fig. S68), Dy₂O₃ (Fig. S21), Er₂O₃ (Fig. S22), and Lu₂O₃ (Fig. S38), our thermodynamic calculations indicate that all of these rare-earth sesquioxides should provide molecular beams of REO(g) when heated to a temperature where the dominant species in the gas phase has a vapor pressure of 10⁻¹ Pa, i.e., the vapor pressure needed for typical oxide MBE growth. Unfortunately, the temperatures needed according to our vapor pressure calculations are near the limit of effusion cells for many of these RE₂O₃(s) sources, making it appropriate to use an e-beam evaporator or laser thermal evaporator to produce REO(g) molecular beams with source longevity.¹²⁰ 

Stolyarova et al.^22,121 listed YO(g) as the dominant vapor constituent of thermally evaporated Y₂O₃(s). In the present calculations (Fig. S78), however, YO₂(g) is shown to be the dominant gas species when a Y₂O₃ source is heated in the temperature range of 1980 K < T < 2680 K, and YO(g) becomes the dominant vapor constituent when T > 2680 K. By omitting YO₂(g), YO(g) will be the dominant vapor constituent when T > 2370 K (Fig. S78). From the present study of the Y–O system, we note that calculations that disagree with experiments could be because certain gas species are added or omitted from the oxide systems in the SSUB5 database.²⁷ This is also seen in other systems such as the Pb–O system where the Pb_nO_n species are omitted from the SSUB5 database but included in the analysis by Lamoreaux et al.²⁶; see Sec. III A 5.

For Eu₂O₃ (and EuO), the present calculations (Fig. S24) reveal that Eu₂O₃(s) and EuO(s) should evaporate as Eu(g), which offers no advantage in using oxide sources over elemental europium. The advantage of europium metal is that it can be melted into the crucible to provide a dense fill with good thermal contact to the crucible (and surrounding thermocouple). Elemental europium has been used in oxide MBE for the growth of EuTiO₃¹²² and EuO.^105–107 

For Tm₂O₃, no literature on the evaporation behavior of Tm₂O₃ could be found to corroborate that Tm₂O₃ evaporates as a gas of the elemental metal thulium. If this is the case, a Tm₂O₃ source would offer no advantage over using elemental thulium metal as an MBE source (see Fig. S74).

Following a recent publication by Hoffmann et al.,³⁹ we also examined the gas species as a function of temperature for a two-phase mixture of RE₂O₃ + RE that has an overall composition of REO. It is shown that the dominant gas species is in general RE(g) and the second dominant gas species is REO(g); see Table S2 and associated figures, including those for the two-phase mixture case with cerium (Fig. S14), dysprosium (Fig. S21), erbium (Fig. S22), europium (Fig. S24), gadolinium (Fig. S28), holmium (Fig. S32), lanthanum (Fig. S36), lutetium (Fig. S38), neodymium (Fig. S44), promethium (Fig. S52), praseodymium (Fig. S53), samarium (Fig. S64), terbium (Fig. S68), thulium (Fig. S74), and ytterbium (Fig. S79). At high temperatures, REO(g) is calculated to be the most dominant gas species, for example, T > 1136 K for CeO(g), T > 1308 K for LaO(g), and T > 2239 K for PrO(g). For other RE elements, however, the temperatures are extremely high (≫3000 K) in order to make the dominant gas species REO(g).

Table III summarizes the metastable oxides that evaporate nearly congruently based on the present calculations. Of these, the oxides IrO₂, PtO₂, and RuO₂ would give the most advantage because their stable counterparts are elemental metals (iridium, platinum, and ruthenium) meaning no other solid (or liquid) binary oxide sources are available to create molecular beams of these oxides. Similar to Table I, Table III also lists the temperature range for nearly congruent evaporation defined by $ΔxO2<0.01$⁠; see Eq. (1).

The concern with using metastable oxides as source materials for MBE is that these compounds could decompose into more stable compounds and in the process pollute the MBE vacuum with O₂ species. For metastable oxides, there are two scenarios that can occur when the oxide is heated, which depend on the kinetics of the compound as discussed in Sec. II A. The kinetics of oxide decomposition is beyond the scope of the present work, but a literature search has been conducted for experimental evidence of the successful use of metastable oxides in MBE; the details are described below. Furthermore, the decomposition of RuO₂ has been studied in the present work by differential thermal analysis with simultaneous thermogravimetry (DTA/TG) to follow the decomposition process under conditions approaching those in the high-vacuum MBE environment.

Figure S19 shows that Cs₂O(s) is a metastable phase, but it evaporates nearly congruently in the temperature range of 705 K–1059 K due to the value of $ΔxO2<0.01$ [see Eq. (1)]; see Table III.

From Table II, it can be seen that the evaporation of SiO₂ and GeO₂ is incongruent and provides molecular beams of SiO and GeO, respectively. Unfortunately, these latter monoxides are not in the SSUB5 database.²⁷ Amorphous SiO(am) is well-known to evaporate nearly congruently;^22–24 its thermodynamic properties³⁰ were added to the database in the present work. In addition, the enthalpy of formation of the amorphous GeO(am)²⁹ was also added. The fact that SiO(am) evaporates nearly congruently makes it of interest for synthesizing Si²⁺-based compounds in MBE, and GeO(am) behaves similarly to SiO(am) and could be a route to Ge²⁺-based compounds.

Figure S63 illustrates the evaporation behavior of SiO(am) calculated at $PO2$ = 10⁻⁴ Pa, showing that SiO(am) gives off SiO₂(g) species at low temperatures (below 1320 K) and SiO(g) species at higher temperatures (above 1320 K). Stoichiometric SiO has been found to evaporate from an SiO(am) source at temperatures in the vicinity of 1520 K in high vacuum (∼10⁻⁶ Pa),^23,24 which is consistent with the results shown in Fig. S63. When the evaporation behavior of SiO(am) is calculated under a stronger vacuum such as $PO2$ = 10⁻¹⁵ Pa, the SiO gas species becomes dominant at lower temperatures (above 990 K) than it does in lower vacuum (e.g., 1320 K as mentioned above). Stolyarova and Semenov²² indicated that SiO(am) evaporates congruently in a Knudsen cell experiment between 1175 K and 1410 K. This is confirmed by the present calculations. Should an interesting Si-containing multicomponent oxide warrant investigation, SiO(am) could be used to provide a molecular beam of SiO(g). One system that might be relevant is the growth of Si-doped Ga₂O₃, where the high oxidant pressures involved in the growth of Ga₂O₃ result in the oxidation of the surface of the traditional Si(s) doping sources used in MBE.³⁴ This oxidation results in the flux emanating from a Si(s) source to change rapidly over time, impeding the use of silicon as a dopant in the growth of oxides by MBE.

The calculated evaporation from a two-phase mixture of SiO₂ + Si with an overall composition SiO (Fig. S63) indicates that SiO(g) is the dominant gas species in the temperature range of study (see also Table S2) in agreement with a recent report.³⁹ Similar to the Si–O case, GeO₂(s) is the stable phase up to 872 K, beyond which GeO(am) becomes stable based on the SSUB5 database and the enthalpy of formation of GeO(am) from the literature;²⁹ see also the note in the figure caption of Fig. S29. GeO(g) is the dominant gas species when the solid/liquid source is GeO₂(s), GeO(am), or a two-phase mixture of GeO₂ + Ge (see also Table S2). Our conclusions for the GeO₂ + Ge mixture are in good agreement with the thermodynamic calculations by Hoffmann et al.³⁹ 

IrO₂(s) and RuO₂(s) are of particular interest for oxide MBE because in elemental form, Ir(s) and Ru(s) sources have such low vapor pressures that today’s MBE effusion cells are unable to evaporate them. Solid iridium and ruthenium can be evaporated by electron-beam evaporation sources, but the stability of the flux from an electron-beam evaporator is inferior to the flux stability provided by effusion cells. IrO₂ and RuO₂ are components of materials with a multitude of interesting properties and electronic structures due to the high spin–orbit coupling and electron correlations present in materials containing these constituents. Examples include the unconventional superconductor Sr₂RuO₄,¹²³ the itinerant metamagnet Sr₃Ru₂O₇,¹²⁴ features in the electronic structure of doped Sr₂IrO₄ that are akin to the cuprate high-temperature superconductors,¹²⁵ the magnetically ordered Mott insulator Na₂IrO₃,¹²⁶ Weyl semimetals in the RE₂Ir₂O₇ pyrochlores,¹²⁷ and heterostructures containing these materials that could, for example, host a two-dimensional gas of magnetic monopoles.¹²⁸ MBE sources that provided stable fluxes of IrO₂ and RuO₂ could thus greatly enhance the ability of oxide MBE to create iridates and ruthenates that are customized with atomic-layer precision.

IrO₂ and RuO₂ are commercially available powders, so it is important to investigate these oxides even though they are metastable under MBE deposition conditions. In the present calculations, metastable IrO₂ and RuO₂ are found to evaporate nearly congruently as shown in Figs. S34 and S59, respectively. IrO₂ and RuO₂ thin films have been fabricated using MBE; however, they were made using elemental iridium or ruthenium sources heated by an electron-beam evaporator with oxidation provided by a separate molecular beam of ozone.^129,130 No literature was found on the evaporation of RuO₂ or the use of RuO₂ as a source in oxide MBE.

As IrO₂, PtO₂, and RuO₂ share the chemical similarity of all being 4+ oxides of platinoids, we performed some preliminary experiments on one of these compounds, RuO₂(s), and found that it decomposes before it evaporates. Specifically, our experiments indicate that when heated, RuO₂(s) undergoes the reaction 2RuO₂(s) → Ru₂O₃(s) + 1$$ 2 O₂(g). The predominant species that RuO₂(s) gives off when heated is thus O₂(g), making it not useful as an MBE source. Figure 5 shows two thermogravimetric (TG) measurements where RuO₂(s) was heated under ambient pressure in an Ar/O₂ mixture or in nominally pure Ar with 99.9999%–99.999% purity, which results in the stated oxygen partial pressures.¹³¹ Between room temperature and 773 K, the changes in the first TG step occur, corresponding to Δm/m = −5.7% (oxidizing atmosphere) or −6.0% (Ar). These observed mass changes would be consistent with the reaction 2RuO₂(s) → Ru₂O₃(s) + 1$$ 2 O₂(g), where Δm/m of the condensed phases would be −6.0%. The decomposition of the remainder becomes significant above 1273 K–1370 K. Since Ru₂O₃(s) is not in the SSUB5 thermodynamic database, this reaction is not predicted by our thermodynamic calculations. Nonetheless, Ru³⁺ is reported in halides, such as RuF₃ and RuCl₃, as well as in the oxide LaRuO₃.¹³² 

NbO₂ has been shown to have NbO₂ species in the gas phase when evaporated from a Knudsen cell at 2050 K.²² Having NbO₂ as the dominant species in the gas phase is expected regardless of whether evaporation occurs from metastable NbO₂(s) or from the stable phase, Nb₂O₅(⁠$ℓ$⁠), at 2050 K. The present work (Fig. S43 and Table III) predicts that (metastable) NbO₂ evaporates nearly congruently between 1768 K and 2283 K at $PO2$ = 10⁻⁴ Pa. NbO₂ has useful metal-to-insulator transition electronic properties motivating its growth by MBE.¹³³ NbO₂(s) could be a viable source for the growth of NbO₂ thin films and is worth trying as an MBE source.

ReO₃(s) is a metastable phase in the Re–O system, and it is expected that it could evaporate nearly congruently. In our thermodynamic calculations using $PO2$ = 10⁻⁴ Pa, however, the gas phase does not satisfy the requirement of having zero amount. So we performed ancillary thermodynamic calculations of ReO₃(s) at a very low oxygen partial pressure of $PO2$ = 10⁻¹⁵ Pa. The results of these calculations, shown in Fig. S57, indicate that ReO₃(s) is a metastable oxide that evaporates nearly congruently at $PO2$ = 10⁻¹⁵ Pa, i.e., the dominant gas species is Re₂O₆ when T > 848 K; see also Table III. It is expected that ReO₃(s) could be a viable source for the growth of ReO₃ thin films albeit our calculations were performed at $PO2$ = 10⁻¹⁵ Pa. ReO₃ is also discussed in Sec. III A 8.

TiO₂ has been shown to evolve TiO species when evaporated from a Knudsen cell at 1920 K.²² This result is unexpected from our thermodynamic calculations. If the initial TiO₂ loaded into the Knudsen cell decomposes into the stable phase, on heating to 1920 K, it would be Ti₄O₇(s) and the dominant species evaporated at 1920 K would be TiO₂(g). On the other hand, if TiO₂ loaded into the Knudsen cell remained in the metastable rutile polymorph at 1920 K, then the dominant species evaporated at 1920 K would also be TiO₂(g). Our calculations only show TiO to be the dominant vapor species at temperatures above 2220 K for both anatase and rutile TiO₂, which are both metastable at this temperature, at the 10⁻⁴ Pa O₂ partial pressure of the present work; see Fig. S72 as well as Table III.

Solid UO₃(s) is a stable phase when T < 525 K; at higher temperatures, the solid U₃O₈(s) becomes stable (>525 K). Figure S75 shows that U₂O₆(g) is the dominant gas species at temperatures T > 517 K and $PO2$ = 10⁻⁴ Pa [O₂(g) is dominant at T < 517 K]. Although U₂O₆(g) is the dominant species in the gas phase, the value of $ΔxO2>0.01$ [see Eq. (1)] in the temperature range of 517 K–525 K due to the nonnegligible concentration of other species in the gas phase. This disqualifies UO₃(s) as an ideal MBE source from the definition of nearly congruent evaporation that we have used. It is worth mentioning that at an extremely low partial pressure of O₂(g), such as $PO2$ = 10⁻¹⁵ Pa, it is possible to make $ΔxO2<0.01$ for UO₃(s).

As a metastable phase, the present calculations show that UO₃(s) evaporates nearly congruently in the temperature range of 544 K–1900 K at $PO2$ = 10⁻⁴ Pa; see Table III.

Table IV shows the metastable oxide sources that exhibit incongruent evaporation. Based on the dominant species in the molecular beams from the stable sources that exhibit incongruent evaporation, these metastable oxides are calculated in the present work to see if they would exhibit nearly congruent evaporation. Unfortunately, they all exhibit incongruent evaporation as can be seen in the calculated figures in the supplementary material. Specifically, see Fig. S14 for the binary oxides containing cerium, Fig. S19 for those containing cesium, Fig. S50 for those containing lead, Fig. S53 for those containing praseodymium, Fig. S67 for those containing tantalum, and Fig. S68 for those containing terbium. These metastable incongruent evaporants offer no advantage over their stable incongruent evaporant counterparts and hence are not discussed further.

A comprehensive thermodynamic investigation regarding the evaporation behavior of 128 binary oxides has been performed to evaluate their suitability as MBE source materials. Based mainly on the SSUB5 database used in the present work,²⁷ we conclude that 16 solid or liquid oxides are most ideal for MBE, i.e., As₂O₃, B₂O₃, BaO, MoO₃, OsO₄, P₂O₅, PbO, PuO₂, Rb₂O, Re₂O₇, Sb₂O₃, SeO₂, SnO, ThO₂, Tl₂O, and WO₃. Of these, As₂O₃, B₂O₃, BaO, MoO₃, PbO, Sb₂O₃, and WO₃ have been utilized to date as MBE source materials. The remaining nine oxides await verification as good oxide source materials in MBE. The use of PbO(s) as a source material may be limited by the allowable O₂ partial pressure in the MBE chamber. Of the solid oxide sources that show incongruent evaporation, SnO₂ has been shown to work experimentally and produces a molecular beam of mainly SnO(g). Although they do not meet the third criterion to be ideal MBE sources (i.e., nearly congruent evaporation; see Sec. II C), some oxides that meet the first criterion (i.e., solid or liquid metal-oxide source that is stable under the given conditions; see Sec. II C) may also work as oxide MBE source materials, like SnO₂ does. The amorphous and metastable phase SiO(s) is known to evaporate before it decomposes, making it suitable as an MBE source. The same may be true for other metastable phases including amorphous GeO(s) and SnO(s). Finally, a likely route to achieve the desired nearly congruent evaporation characteristic of an ideal MBE source—a characteristic that keeps the fluxes of the species in the molecular beams constant, because the composition of the source itself is not changing over the life of the source—is to use a two-phase mixture. Potential MBE sources of this mixture type are, for example, (Al₂O₃ + Al) to produce a molecular beam of Al₂O(g), (Ce₂O₃ + Ce) to produce CeO(g), (Ga₂O₃ + Ga) to produce Ga₂O(g), (GeO₂ + Ge) to produce GeO(g), (HfO₂ + Hf) to produce HfO(g), (In₂O₃ + In) to produce In₂O(g), (La₂O₃ + La) to produce LaO(g), (Pr₂O₃ + Pr) to produce PrO(g), (SiO₂ + Si) to produce SiO(g), (SnO₂ + Sn) to produce SnO(g), (Ta₂O₅ + Ta) to produce TaO₂(g), and (ZrO₂ + Zr) to produce ZrO(g); see the 27 cases examined in Table S2. Our conclusions for two-phase mixtures are in agreement with the recent thermodynamic calculations by Hoffmann et al.³⁹ for Al₂O(g), Ga₂O(g), GeO(g), In₂O(g), LaO(g), PrO(g), SiO(g), and SnO(g). Our predictions also identify four additional two-phase sources of interest for producing molecular beams of CeO(g), HfO(g), TaO₂(g), and ZrO(g).

Several trends are evident in the evaporation behavior of binary metal oxides. These are shown in Fig. 6. Alkali-metal oxides, alkaline-earth metal oxides, and some of the transition-metal oxides evaporate mainly elemental metal gas species and thus are not ideal for MBE. The rare-earth oxides and some of the transition metals surrounding them as well as the metalloid oxides exhibit incongruent evaporation. Oxides ideal for MBE are the column IVA oxides SiO, GeO, SnO, and PbO, although the first three of these are metastable (amorphous). In addition, there are disruptions to certain trends such as the evaporation of elemental metals from EuO, Tm₂O₃, and Yb₂O₃, which do not follow the incongruent evaporation trend of the other rare earths.

See the supplementary material for an alphabetical list of 128 solid and liquid oxides together with a list of 17 elements of which the most stable phase is not an oxide (Table S1), a list of 27 two-phase mixtures of a solid (or liquid) of pure element M in combination with one of its binary oxides yielding an overall composition of MxOy (Table S2), three illustrative examples of macro files to perform some of the Thermo-Calc calculations presented (Table S3), and an alphabetical list of supplemental figures (Figs. S1–S81) showing the calculated partial pressures of gas species over the phase(s) of interest.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

We acknowledge financial support from the National Science Foundation (NSF) through Grant No. CMMI-1825538 and the Platform for the Accelerated Realization, Analysis and Discovery of Interface Materials (PARADIM) under Cooperative Agreement No. DMR-1539918. B.J.B. was supported by a NASA Space Technology Research Fellowship (Grant No. 80NSSC18K1168). This material is based on the work supported by the Air Force Office of Scientific Research under Award No. FA9550-18-1-0529. D.G.S. acknowledges stimulating interactions with O. Bierwagen, W. Braun, M. Budde, G. Christiani, R. Droopad, G. Hoffmann, J. P. Maria, and P. Vogt regarding oxide MBE utilizing oxide sources.