Ab Initio Thermodynamic Evaluation of Ruthenium Tetroxide (RuO₄) Vapor Pressure

Abstract

In this study, the sublimation and vapor pressure characteristics of RuO₄ were systematically investigated using ab initio thermodynamic calculations. Structural optimizations and vibrational frequency analyses were performed for gaseous RuO₄ and four candidate solid phases (monoclinic Cm, P2₁/c, C2/c, and cubic P-43n) within the density functional theory (DFT) framework. Gibbs free energies were evaluated by incorporating electronic energies, zero-point corrections, and entropic contributions from translational, rotational, and vibrational modes. The results identify monoclinic C2/c and cubic P-43n as the most stable solid phases across the studied temperature range. Calculated sublimation temperatures of 322 K at 1 atm and 240 K at 1 × 10⁻³ atm were obtained in good agreement with experimental melting and boiling points. Calculated vapor pressures show reasonable agreement with experimental measurements below the triple point, with deviations at higher temperatures attributable to approximating liquid-gas behavior using solid-gas sublimation data. These findings provide the first theoretical description of RuO₄ vapor pressure and offer a computational framework extendable to other transition-metal ALD precursors.

1. Introduction

Ruthenium (Ru) and ruthenium dioxide (RuO₂) have attracted significant attention in semiconductor technology due to their favorable electrical properties. Both exhibit low bulk resistivities ($\mathsf{\rho }$) (7.1 μΩ·cm for Ru and 35 μΩ·cm for RuO₂) and high work functions (4.7 eV and 5.1 eV, respectively) [1,2,3]. These characteristics make Ru a promising candidate to replace copper (Cu) in nanoscale interconnects, as its shorter electron mean free path (6.6 nm compared to 40 nm for Cu) reduces the impact of surface and grain boundary scattering that severely degrades conductivity at nanoscale dimensions [4,5]. RuO₂ has likewise been investigated as an alternative electrode to titanium nitride (TiN) in dynamic random-access memory (DRAM) capacitors owing to its superior interfacial stability and reduced degradation [6,7]. Figure 1 compares the resistivity of several candidate metals as a function of thickness ($\mathrm{d}$). The metals examined include tungsten (W), platinum (Pt), palladium (Pd), Ru, iridium (Ir), cobalt (Co), and molybdenum (Mo) [8,9,10,11]. The results demonstrate that Ru exhibits a slower increase in resistivity with decreasing thickness than most other metals, highlighting its ability to maintain lower resistance at nanometer dimensions and underscoring its suitability for next-generation interconnect technologies.

As device scaling advances to atomic dimensions, atomic layer deposition (ALD) has become essential for producing ultrathin films with precise thickness control and excellent conformality [12,13,14]. To date, several Ru precursors have been examined, including bis(cyclopentadienyl)ruthenium(II) (Ru(Cp)₂) [15], tris(2,2,6,6-tetramethyl-3,5-heptanedionato)ruthenium(III) (Ru(thd)₃) [16], bis(2,4-dimethylpentadienyl)ruthenium(II) (Ru(DMPD)₂) [17], and bis(ethylcyclopentadienyl)ruthenium(II) (Ru(EtCp)₂) [18]. Despite their widespread use, these precursors exhibit inherent limitations, most notably low vapor pressures (~10⁻⁵–10⁻⁴ atm at ~330–360 K) [19,20,21,22] and poor conformality in high aspect-ratio structures [23]. Ru(Cp)₂ offers stable growth characteristics but suffers from impurity incorporation and limited step coverage [15]. Ru(thd)₃ with oxygen (O₂) yields high-purity films but exhibits extremely low growth rates (0.035–0.040 nm/cycle) and long nucleation delays [16]. Ru(DMPD)₂ offers improved film purity in both thermal and plasma-enhanced ALD (PEALD) yet suffers from notably low PEALD growth rates (~0.025 nm/cycle) that reduce manufacturing throughput and incomplete sidewall coverage (~70%) due to anisotropic radical transport effects [24]. Ru(EtCp)₂ with active oxygen (O) radicals enables radical-assisted growth at low temperatures but is susceptible to etching at elevated temperatures, complicating optimization [18]. Together, these results demonstrate the need for alternative Ru precursors with high volatility and improved deposition performance.

Ruthenium tetroxide (RuO₄), characterized by lower molecular weight and inherently higher vapor pressure, represents a highly promising candidate. Experimental studies have reported successful ALD of Ru film using RuO₄/hydrogen (H₂) processes at low temperatures (373 K) with high growth rates (0.1 nm/cycle), minimal nucleation delays, and favorable substrate selectivity [25,26]. Accelerated Ru growth on tantalum pentoxide (Ta₂O₅) has also been observed, attributed to substrate-mediated effects involving partial reduction and cation incorporation [27]. Processes based on RuO₄ have further enabled immediate nucleation of RuO₂ and tunable growth rates, as well as area-selective ALD (ASALD) at low temperatures [28,29]. Nevertheless, despite these advances, vapor pressure data for RuO₄ remain limited to a few experimental measurements [30,31,32], which are not always consistent.

Computational thermodynamics provides an opportunity to address these limitations. Ab initio methods have been successfully applied to other transition-metal precursors, such as molybdenum oxyhalide (MoO₂Cl₂) [33] and molybdenum pentachloride (MoCl₅) [34], to clarify sublimation energetics and vapor pressure behavior. However, to date, RuO₄ has not been systematically studied from this perspective.

This study aims to investigate the sublimation behavior and vapor pressure by performing ab initio thermodynamic calculations. By combining density functional theory (DFT) with vibrational thermodynamics, Gibbs free energies of solid and gaseous RuO₄ phases are evaluated, and equilibrium vapor pressures are derived over relevant temperature ranges. The computational results are compared with available experimental data [30,31,32], providing complementary insights that support more effective precursor selection for ALD applications demanding atomic-scale precision.

2. Calculation Details

Density functional theory calculations were carried out using the Vienna Ab initio Simulation Package (VASP version 6.3.2) [35,36,37]. The projector augmented-wave (PAW) method [37,38] was employed in conjunction with the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) for exchange-correlation interactions [39]. Van der Waals interactions were included through the DFT-D3 dispersion correction method [40] to accurately describe the weak intermolecular forces and lattice parameters in RuO₄. A plane-wave cutoff energy of 500 eV was selected to balance computational efficiency and accuracy.

For gaseous RuO₄ (point group T_d), structural optimizations were performed in a cubic box of 15 × 15 × 15 ų with Monkhorst-Pack k-point sampling to minimize interactions caused by periodic boundary conditions in DFT calculations [41]. For solid RuO₄, four candidate crystal structures—monoclinic Cm, P2₁/c, C2/c, and cubic P-43n—were considered. The Brillouin zone was sampled with Monkhorst–Pack k-point meshes of 3 × 4 × 3 for Cm, 3 × 3 × 3 for P2₁/c and C2/c, and 2 × 2 × 2 for P-43n [41]. To ensure adequate system size for phonon calculations, supercell expansions of 1 × 2 × 2, 2 × 1 × 2, and 1 × 2 × 1 were constructed for Cm, P2₁/c, and C2/c, respectively, while the primitive unit cell was retained for P-43n. The optimized geometries of gaseous RuO₄ and the four candidate solids are presented in Figure 2, illustrating the molecular configuration and coordination environment of Ru in each phase.

Phonon calculations were performed using the finite-displacement (frozen-phonon) method implemented in the phonopy package (version 2.43.2) [42,43], based on force constants obtained from the DFT supercell calculations. The resulting vibrational properties were used to compute temperature-dependent thermodynamic quantities of each solid phase. Phonon dispersion relations for the four RuO₄ polymorphs were also evaluated to examine their dynamic stability. Zero-point energies and temperature-dependent enthalpic corrections were derived from vibrational frequencies. The Gibbs free energy ($\mathrm{G}$) was evaluated within the framework of ab initio thermodynamics according to

$$\mathrm{G}=\mathrm{H}-\mathrm{T}\mathrm{S}=\mathrm{E}+\mathrm{P}\mathrm{V}+\mathrm{Z}\mathrm{P}\mathrm{E}+\int {\mathrm{C}}_{\mathrm{P}}\mathrm{d}\mathrm{T}-\mathrm{T}\left({\mathrm{S}}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}}+{\mathrm{S}}_{\mathrm{r}\mathrm{o}\mathrm{t}}+{\mathrm{S}}_{\mathrm{v}\mathrm{i}\mathrm{b}}\right)$$

(1)

where $\mathrm{H}$ is the enthalpy, $\mathrm{T}$ is the temperature, $\mathrm{S}$ is the entropy, $\mathrm{E}$ is the DFT total energy, $\mathrm{P}$ is the pressure, $\mathrm{V}$ is the volume, $\mathrm{Z}\mathrm{P}\mathrm{E}$ is the zero-point energy, $\int {\mathrm{C}}_{\mathrm{P}}\mathrm{d}\mathrm{T}$ is the temperature-dependent enthalpic correction, and ${\mathrm{S}}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}}$, ${\mathrm{S}}_{\mathrm{r}\mathrm{o}\mathrm{t}}$, and ${\mathrm{S}}_{\mathrm{v}\mathrm{i}\mathrm{b}}$ are translational, rotational, and vibrational entropies, respectively. Translational entropy was calculated using the Sackur-Tetrode equation, rotational entropy within the rigid-rotor approximation, and vibrational entropy within the harmonic oscillator approximation [44]. For solids, translational and rotational entropy contributions were omitted because of lattice constraints. To prevent overestimation of vibrational entropy, low-frequency modes below 50 cm⁻¹ were adjusted to this cutoff, consistent with established practice [45]. All vibrational modes, excluding imaginary frequencies, were incorporated in the thermodynamic analysis. This methodology ensures reliable Gibbs free energies for both solid and gaseous RuO₄, forming the basis for subsequent evaluation of sublimation behavior and vapor pressure.

3. Results and Discussion

The optimized lattice parameters of the four candidate RuO₄ structures are summarized in

Table 1. Calculated lattice parameters of candidate solid RuO₄ structures (monoclinic Cm, P2₁/c, C2/c, and cubic P-43n) compared with experimental data reported by Pley et al. [46].

		Lattice Parameters
		a (Å)	b (Å)	c (Å)	β (°)
Monoclinic Cm	This work	9.96	3.12	4.78	117.4
Monoclinic P21/c	This work	4.50	8.16	4.58	100.0
Monoclinic C2/c	[46]	9.30	4.40	8.45	116.8
	This work	9.25	4.39	8.38	116.5
Cubic P-43n	[46]	8.51	-	-	-
	This work	8.49	-	-	-

, showing good agreement with previously reported values [46], which confirms the reliability of the present DFT optimization.

Table 2. DFT-calculated total energies of candidate solid RuO₄ structures (monoclinic Cm, P2₁/c, C2/c, and cubic P-43n) compared with literature values from the Materials Project [47] and the Open Quantum Materials Database (OQMD) [48,49]. The results indicate consistent relative stability trends among the polymorphs.

	$\mathbf{E}$ (eV/Formula Unit)
	Materials Project [47]	OQMD [48,49]	This Work
Monoclinic Cm	-	−29.32	−30.46
Monoclinic P21/c	-	−30.80	−33.19
Monoclinic C2/c	−32.23	−31.74	−33.23
Cubic P−43n	−32.22	−31.74	−33.23

lists the total energies of the same structures and demonstrates overall consistency with database references [47,48,49]. The relative Gibbs free energies (Figure 3) indicate that the monoclinic Cm and cubic P-43n phases possess the lowest values across the studied temperature range, indicating nearly indistinguishable thermodynamic stability. Given that RuO₄-based ALD typically operates near 373 K (below its decomposition temperature of 398 K [25,29]), the C2/c phase was selected for subsequent vapor pressure analysis. Phonon calculations for the four polymorphs yielded a small number of imaginary frequencies (3, 3, 3, and 11 for C2/c, P2₁/c, P-43n, and Cm, respectively), suggesting localized dynamical instabilities. The phonon band structures for the four RuO₄ crystal structures are provided in the Supporting Information (Figure S1). These curves show that the C2/c and P-43n phases exhibit only limited imaginary branches near the Γ-point, whereas the P2₁/c phase shows broader regions of negative frequencies with smaller magnitudes, and the Cm phase displays pronounced negative branches across most of the Brillouin zone, confirming its dynamic instability.

To further elucidate phase stability,

Table 3. Thermodynamic contributions to the Gibbs free energy of RuO₄ at 373 K and 1 atm for both solid and gaseous phases. Energy contributions include zero-point energy ($\mathrm{Z}\mathrm{P}\mathrm{E}$), temperature-dependent enthalpy change ($\int {\mathrm{C}}_{\mathrm{P}}\mathrm{d}\mathrm{T}$), and entropy terms including translational (${\mathrm{T}\mathrm{S}}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}}$), rotational (${\mathrm{T}\mathrm{S}}_{\mathrm{r}\mathrm{o}\mathrm{t}}$), and vibrational (${\mathrm{T}\mathrm{S}}_{\mathrm{v}\mathrm{i}\mathrm{b}}$) contributions (all values in eV), illustrating the contrasting entropy partitioning between the two phases.

	$\mathbf{Z}\mathbf{P}\mathbf{E}$	$\int {\mathbf{C}}_{\mathbf{P}}\mathbf{d}\mathbf{T}$	$\mathbf{T}{\mathbf{S}}_{\mathbf{t}\mathbf{r}\mathbf{a}\mathbf{n}\mathbf{s}}$	$\mathbf{T}{\mathbf{S}}_{\mathbf{r}\mathbf{o}\mathbf{t}}$	$\mathbf{T}{\mathbf{S}}_{\mathbf{v}\mathbf{i}\mathbf{b}}$
Solid RuO4	2.92	1.97	-	-	4.38
Gaseous RuO4	0.34	0.22	0.68	0.40	0.10

presents the Gibbs free energy components for solid and gaseous RuO₄ at 373 K and 1 atm. For the gas phase, significant contributions arise from translational (0.68 eV) and rotational (0.40 eV) entropy terms, reflecting the molecular degrees of freedom absent in solids. Conversely, solid RuO₄ exhibits a much larger vibrational entropy contribution (4.38 eV) due to numerous phonon modes, particularly in the low-frequency region. Zero-point energy (2.92 eV vs. 0.34 eV) and enthalpy contributions (1.97 eV vs. 0.22 eV) are also substantially higher for the solid. These findings highlight the contrasting entropy partitioning between solid and gas phases, which governs their relative Gibbs free energy balance at elevated temperatures.

Figure 4 illustrates the temperature-dependent Gibbs free energies of solid and gaseous RuO₄ at partial pressures of (a) 1 atm and (b) 1 × 10⁻³ atm, the latter representing typical ALD operating conditions. The Gibbs free energy of solid RuO₄ at 0 K was used as a reference. The intersections of the solid and gaseous free energy curves define the sublimation point (${\mathrm{T}}_{\mathrm{s}}$), yielding 322 K at 1 atm and 240 K at 1 × 10⁻³ atm. Experimentally, RuO₄ is known to melt at 298 K and boil at 403 K [32], consistent with the calculated sublimation points falling between these transitions. Direct evaluation of the liquid phase is challenging; therefore, a simple interpolated curve between the experimental melting (${\mathrm{T}}_{\mathrm{m}}$) and boiling (${\mathrm{T}}_{\mathrm{b}}$) points is included only for reference. This comparison indicates that the computational framework reasonably captures the thermodynamic behavior of RuO₄ under relevant conditions.

To extend this analysis, Figure 5 compares the calculated vapor pressure of RuO₄ with experimental measurements over a temperature range [30,31,32]. Below the triple point (T < 299 K), the calculated values reproduce the overall temperature dependence and are in reasonable agreement with experimental sublimation data. Nevertheless, the experimental vapor pressures are systematically lower than theoretical predictions, which can be explained by kinetic limitations in the evaporation process. A recent CALPHAD-type thermodynamic assessment of RuO₄ reported by Nuta et al. [50] reported a very low evaporation coefficient of ~0.027 which indicates that only a small fraction of surface molecules contributes to vaporization, thereby causing consistent underestimation of experimental measurements. At higher temperatures (T > 299 K), the deviation becomes more pronounced because the experimental data reflect liquid-gas vaporization, while the present calculations employ solid-gas sublimation data as an approximation, inevitably leading to systematic overestimation. For comparison, conventional Ru precursors such as Ru(Cp)₂ or Ru(thd)₃ typically exhibit vapor pressures on the order of 10⁻⁵–10⁻⁴ atm near 330–360 K [19,20,21,22], whereas RuO₄ already reaches ~10⁻² atm at its triple point (299 K). This large difference highlights the exceptional volatility of RuO₄. These results confirm that RuO₄ possesses a relatively high equilibrium vapor pressure, consistent with prior experimental observations and supporting its suitability as a volatile precursor for atomic layer deposition.

4. Conclusions

In this work, the sublimation and vapor pressure behaviors of RuO₄ were investigated through ab initio thermodynamic calculations. Structural optimizations and vibrational analyses were performed for gaseous RuO₄ and four candidate solid phases, leading to the identification of monoclinic C2/c and cubic P-43n as the most stable crystalline structures. Thermodynamic analysis revealed sublimation temperatures of 322 K at 1 atm and 240 K at 1 × 10⁻³ atm, values that are consistent with the experimental melting and boiling points of RuO₄.

The vapor pressure calculations show reasonable agreement with experimental sublimation data at temperatures below the triple point. Experimental values are consistently lower than theoretical predictions, with modest deviations explained by kinetic limitations in the evaporation process at lower temperatures and more pronounced discrepancies at higher temperatures, where liquid-gas vaporization dominates experimentally while the present calculations employ solid-gas sublimation data as an approximation.

Overall, this work provides the first theoretical description of RuO₄ vapor pressure based on ab initio thermodynamics. The results confirm the high volatility of RuO₄ and its suitability as an ALD precursor. The computational framework demonstrated here is broadly applicable and can be extended to other transition-metal precursors, thereby contributing to the rational design and evaluation of precursors for future semiconductor technologies.