Xiao-Wei Sun*ab,
Meng-Ru Chena,
Xi-Long Doub,
Ning Lib,
Tong Wangb and
Ting Songb
aSchool of Mechanical Engineering, Lanzhou Jiaotong University, Lanzhou 730070, PR China. E-mail: sunxw_lzjtu@yeah.net
bSchool of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, PR China
First published on 3rd January 2024
The potential applications of Ir2P are promising due to its desirable hardness, but its fundamental properties are still not fully understood. In this study, we present a systematic investigation of Ir2P's structural, electronic, superconducting, optical, and thermodynamic properties of Ir2P under pressure. Our calculations show that Ir2P has a Fmm structure at ambient pressure, which matches well with experimental data obtained from high-pressure synchrotron X-ray diffraction. As pressure increases, a transition from the Fmm to the I4/mmm phase occurs at 103.4 GPa. The electronic structure and electron-phonon coupling reveal that the Fmm and I4/mmm phases of Ir2P are superconducting materials with superconducting transition temperatures of 2.51 and 0.89 K at 0 and 200 GPa, respectively. The optical properties of Ir2P indicate that it has optical conductivity in the infrared, visible, and ultraviolet regions. Additionally, we observed that the reflectivity R(ω) of Ir2P is higher than 76% in the 25–35 eV energy range at different pressures, which suggests that it could be used as a reflective coating. We also explored the finite-temperature thermodynamic properties of Ir2P, including the Debye temperature, the first and second pressure derivatives of the isothermal bulk modulus, and the thermal expansion coefficient up to 2000 K using the quasi-harmonic Debye model. Our findings offer valuable insights for engineers to design better devices.
TMPs are a group of compounds that possess unique chemical and physical properties. These properties make them highly attractive for potential applications in various fields, such as photonics, electronics, magnetism, hard materials, and catalysis.24–26 Systematic computational investigations have shown that phosphorus atoms in TMPs play a crucial role in hydrogen evolution reactions.27 Phosphorus atoms with higher electronegativity can draw electrons from metal atoms, and negatively charged phosphate groups can act as a base in electrochemical hydrogen evolution reactions. Generally, TMPs with a higher metal content exhibit more metallic characteristics because metal atoms contribute more electrons to the compound, leading to a lower overall electronegativity and a more metallic nature. Additionally, as the atomic ratio of phosphorus to metal increases, the bonding between the phosphorus and metal atoms becomes weaker, which can also promote metallic behavior.28 Silica-supported palladium and ruthenium phosphide catalysts, such as Pd3P, Pd5P2, Ru2P, and RuP, were recently synthesized and studied by Bowker et al.29 to investigate the hydrodesulfurization properties of dibenzothiophene. The properties of these phases were compared with those of the sulfides of the noble metals. Although Ir2P was first reported in 1935 by Blatz et al.,30 there have been comparatively few studies of its synthesis routes and crystal structures. Rundqvist et al.31 examined the crystal structures of three phosphides, namely, Rh2P and Ir2P with the anti-fluorite structure and PtP2 with the pyrite structure. Raub et al.32 discovered that only the Ir2P exhibited a metallic behavior among the family of Ir2X and IrX (where X = P, As, Sb, and Bi) Pt-metal alloys. Sweeney et al.33 explored the feasibility of hydrogen reduction annealing of metal phosphates as a selected pathway to phosphides for rhodium and iridium. Unfortunately, there is limited information available on the high-pressure properties of Ir2P due to the difficulty in synthesizing it, even though high-pressure synthesis is a powerful method for preparing novel materials with unique electrical and mechanical properties. In 2016, Ir2P was successfully synthesized under high-pressure conditions, and it was found to have an anti-fluorite structure with space group Fmm, as determined from synchrotron X-ray diffraction (XRD) pattern analysis. The structure remains stable at room temperature and pressures ranging from 0 to 40.6 GPa. In this structure, each Ir atom is surrounded by four P atoms, forming [IrP4] tetrahedrons, which are located at the edges of the unit cell.34 Research indicates that Ir2P has a significant bulk modulus (B) (320 and 342 GPa) and a relatively modest shear modulus (G) (42 and 64 GPa) derived from generalized gradient approximation (GGA) and local density approximation (LDA) calculations respectively. This indicates a complex bonding profile for Ir2P, encompassing metallic, ionic, and covalent characteristics. Subsequently, the ground-state structure of Ir2P is predicted to be Fmm, while the high-pressure structure is expected to be I4/mmm. It was shown that the predicted ground-state structure is consistent with the experimentally obtained structure and its phase transition pressure is 86.4 GPa.35
We have conducted a comprehensive theoretical analysis of Ir2P, a material about which limited information is available. Our analysis is based on first-principles calculations within the framework of DFT using GGA and LDA. We have optimized its geometric structure and explored its energy-volume equation of state (EOS), as well as delving into its electronic band structures, elastic properties, and superconductivity characteristics. We have also predicted the complex dielectric function, reflectivity, absorption coefficient, optical conductivity, and loss function of Ir2P to obtain the regular behavior of these optical parameters with pressure. Additionally, we have predicted the finite-temperature thermodynamic properties of cubic Ir2P, including the isothermal bulk modulus, and its first and second pressure derivatives, thermal expansion coefficient, and Debye temperature at the atomic level by means of density functional total energy calculations in combination with the quasi-harmonic Debye (QHD) model. These findings can potentially guide further research and help in the development of practical applications for this material.
In this study, a convergence threshold was set at 5.0 × 10−7 eV per atom, to determine if the iterative process of updating the electronic wave functions and potentials has reached a stable solution, for the self-consistent progress. The convergence criterion for energy, force, ionic displacement, and stress were set to 5.0 × 10−6 eV per atom, 0.01 eV Å−1, 5.0 × 10−4 Å, and 0.02 GPa, respectively. When calculating elastic properties, the force on the atom refers to the force applied to the atom during the simulation, while the atomic position displacement refers to the change in the position of the atoms between computational cycles. To prevent excessive strain on the atom and maintain the stability of the simulation, these values are limited to 0.002 eV Å−1 and 1.0 × 10−4 Å, respectively. The maximum strain amplitude is set to be 0.003, which is the maximum amount the atoms can be deformed during the simulation. Six distorted structures were generated within this limit, likely for further analysis or study. These parameters and settings were chosen to ensure that the calculated elastic properties are accurate and reliable, while also maintaining computational efficiency.
The strained energy for cubic and tetragonal Ir2P is expressed by the law from ref. 42. The mechanical stability criteria obtained from elastic stiffness coefficients can be found in ref. 43 or 44. Currently, the stress–strain method is being used. The phonon dispersions for Fmm and I4/mmm structures were calculated on 2 × 2 × 2 supercells which contain 48 and 96 atoms, respectively, using density-functional perturbation theory (DFPT) method implemented in the PHONOPY code.45 The forces result from VASP.
The electron-phonon coupling (EPC) calculations for Ir2P were performed using the QUANTUM-ESPRESSO package46 with ultrasoft pseudopotential for core-valence interaction. The kinetic energy cutoff of 60 Ry was chosen as the plane-wave expansion. The k meshes and q meshes of 12 × 12 × 12 and 3 × 3 × 3, 24 × 24 × 12 and 3 × 3 × 2 were selected as the first Brillouin zone for Fmm and I4/mmm of Ir2P, respectively. Taking into account that the EPC parameters λ of I4/mmm for Ir2P is smaller than 1.5, the superconducting transition temperature (Tc) is estimated through McMillan equation47
(1) |
(2) |
(3) |
The complex dielectric function ε(ω) is often used to characterize the optical properties of a material. Due to the metallic nature of Ir2P, a semi-experiential Drude term and Gaussian smearing of 0.5 eV are used to calculate the frequency-dependent dielectric constants48
ε(ω) = ε1(ω) + iε2(ω). | (4) |
In the formula above, the complex dielectric function, ε(ω), is a complex number representing the ratio of the electric field to the electric displacement in a material. The symbol ω refers to the frequency of the incident photon, while ε1(ω) and ε2(ω) refers to the real and imaginary parts of the complex dielectric function, respectively. The ε2(ω) value is calculated from the momentum matrix element between the occupied and unoccupied electronic states, and the ε1(ω) is derived from the imaginary part using the Kramers–Kronig relation.
To study the behavior of Ir2P under finite temperature conditions, a QHD model was used to determine the Debye temperature, isothermal bulk modulus, and its first and second pressure derivatives, as well as the thermal expansion coefficient. These properties are important thermodynamic characteristics of materials. The GIBBS software program49 was used to implement the QHD model.
Since Ir is a metal containing d electrons, we explore the magnetim of the Fmm and I4/mmm phases. We perform energy mapping of all conceivable magnetic configurations and calculate the spin-polarized density of states for both phases, as shown in Fig. S1 and S2.† The absence of a non-zero magnetic moment per Ir atom in the considered magnetic configurations, and the symmetric density of states, confirmed that both phases are nonmagnetic. Therefore, this study does not incorporate spin polarization.
Fig. 2 Crystal structures for the (a) Fmm and (b) I4/mmm phases of Ir2P. The gold and purple spheres represent Ir and P atoms, respectively. |
Several geometry optimizations were conducted using the LDA and GGA approximations. Fig. 3 shows the dependence of the calculated relative volume on the external pressure up to 80 GPa at zero temperature for Ir2P. Previous studies have indicated that the pressure dependence of the volume ratio is a more direct measure of compressibility compared to the description of physical properties alone.34 For comparison, the experimental and theoretical data up to 40.6 GPa provided by Wang et al.34 are also plotted Fig. 3, along with the theoretical P–V/V0 data of conventional superhard materials such as diamond and c-BN50 at T = 0 K up to 80 GPa. Our results obtained with DFT calculations match well with that of Wang et al.34 Due to the discrepancy between GGA and LDA for exchange-correlation potentials within the DFT framework, the average of GGA and LDA values is employed as the rescaled estimate. It is evident from Fig. 3 that the anti-fluorite structured Ir2P is more compressible than conventional superhard materials, such as c-BN and diamond. This implies that the anti-fluorite structure may exhibit a higher degree of flexibility or less resistance to pressure, compared to the conventional superhard materials.
Fig. 3 Comparison of the calculated pressure versus volume ratio of 0 K isotherms for the anti-fluorite structure of Ir2P with experiments and other theoretical data. |
Table 1 lists the parameters of the fitted equation of state (EOS) for Ir2P with the cubic anti-fluorite structure. These parameters include a0, V0, K0 and , which were obtained from the 3rd-order Birch–Murnaghan EOS51 using the Eosfit52 software.52 To analyze the results from EOS, we have also included the results from DFT and experiments by Wang et al.34 in Table 1. Unfortunately, we cannot compare the results of equilibrium lattice parameter a0 and equilibrium volume V0 among studies due to the lack of data. However, the values of bulk modulus K0 and its first pressure derivative obtained from DFT calculations are in very good agreement with the theoretical results and are also comparable to the ones obtained from the X-ray diffraction experiment data of Wang et al.34 This indicates that our calculations are accurate, and the methods can be used to predict other properties of Ir2P with the anti-fluorite structure.
In order to better understand the structural stability of cubic Ir2P under strain, it is important to calculate the elastic constants of the material. Table 2 presents the calculated elastic constants C11, C12, and C44, G, Young's modulus (E), and Poisson's ratio (υ) for Ir2P with the anti-fluorite structure at zero pressure and zero temperature. The values for E, G, and υ are derived according to the Voight–Reuss–Hill averaging scheme.53 For any crystal to be mechanically stable, it must have a positive strain energy.54 The elastic constants obtained suggest that the Ir2P with anti-fluorite structure is mechanically stable. However, despite its elastic stability, Ir2P with an anti-fluorite structure has smaller B (316.520 and 343.068 GPa), lower G (117.656 and 129.203 GPa), and larger υ (0.335 and 0.333) when calculated using GGA and LDA in the DFT framework. This suggests that its mechanical properties are inferior in comparison to conventional superhard materials like diamond and c-BN.50
Method | C11 (GPa) | C12 (GPa) | C44 (GPa) | G (GPa) | E (GPa) | υ | |
---|---|---|---|---|---|---|---|
DFT-GGA | 679.459 | 135.050 | 61.741 | 117.656 | 314.055 | 0.335 | This work |
DFT-LDA | 739.766 | 144.719 | 68.134 | 129.203 | 344.377 | 0.333 | This work |
It is well-documented that GGA in first-principles calculations often overestimates volume and underestimates bulk modulus.55 This shortcoming is also extended to the elastic constant calculations at high pressure, like C11, for the Fmm and I4/mmm phases of Ir2P, as shown in Fig. 4. It is found that the effect of the pressure on C11 (679.459 and 1307.119 GPa for GGA and 739.766 and 1373.326 GPa for LDA calculations at 0 and 80 GPa, respectively), which represents elasticity in length, is much larger than that on C12 (135.050 and 360.009 GPa for GGA and 144.719 and 368.275 GPa for LDA calculations at 0 and 80 GPa, respectively) and C44 (61.741 and 117.102 GPa for GGA and 68.134 and 124.536 GPa for LDA calculations at 0 and 80 GPa, respectively), which characterize the elasticity in shape in the pressure range of 0–80 GPa. However, this little discrepancy in the presentation doesn't influence the judgment of the high-pressure structural stability for the cubic Ir2P. The obtained elastic constants exhibited in Fig. 4(b) show that the I4/mmm phase satisfies the Born criteria for tetragonal crystal systems in the pressure range studied.
It is widely accepted that a stable crystalline structure requires all phonon frequencies to be positive at zero temperature. To ensure this stability, the phonon dispersion calculation has been performed within the finite displacement theory using the PHONOPY code for the Fmm and I4/mmm phases at 0, 100 and 120, 200 GPa, respectively. As demonstrated in Fig. 5, there are no imaginary frequencies present in the entire Brillouin zone at the selected pressures. This finding indicates that both phases are dynamically stable, meaning that the material remains stable under the applied pressure conditions, as there are no negative phonon frequencies that would suggest an instability in the crystalline structure. The absence of imaginary phonon frequencies also confirms that the material's elastic properties are well-defined and robust, rendering it suitable for a wide range of applications.54 The phonon spectra of the Fmm and the I4/mmm phases are divided into two modes, low and high frequency modes. Both phases demonstrate notable phononic gaps within the frequency ranges of 6.3–9.9 THz and 8.2–11.9 THz, respectively, along the optical branch at 0 GPa and 120 GPa. These broader phononic gaps separate the optical phonon modes into high and low-frequency modes. Moreover, both phases exhibit an increasing phononic gap with increasing pressure, with the value increasing from 3.6 (0 GPa) to 5.9 (100 GPa) for the Fmm phase and 3.7 (120 GPa) to 4 (200 GPa) for the I4/mmm phase. This finding suggests that Ir2P hold potential as materials for phononic devices, including applications in phonon waveguides, cavities, and filters.56–58 Subsequently, the GGA with a correction of the Perdew–Burke–Ernzerhof version, the PBEsol, which is known to yield better results for solids, is used to investigate the electronic and finite-temperature thermodynamic properties of Ir2P with the anti-fluorite structure.
Fig. 5 The calculated phonon dispersion curves of Ir2P for Fmm at (a) 0 and (b) 100 GPa and I4/mmm phases at (c) 120 and (d) 200 GPa, respectively. |
Phase | Pressure | NEf | λ | ωlog | Tc (μ* = 0.1) | Tc (μ* = 0.13) |
---|---|---|---|---|---|---|
Fmm | 0 | 0.91 | 0.63 | 94.00 | 2.51 | 1.83 |
I4/mmm | 200 | 0.95 | 0.38 | 283.71 | 0.89 | 0.35 |
Fig. 8 The projected phonon densities of states PHDOS, Eliashberg phonon spectral function α2F(ω) and integrated electron-phonon coupling λ(ω) of the (a) and (c) Fmm and (b) and (d) I4/mmm. |
Fig. 10 and S5† display the optical properties of Ir2P under pressure in the low-energy and high-energy regions, including the absorption coefficient α(ω), reflectivity R(ω), optical conductivity σ(ω), and loss function L(ω). The starting point of optical absorption coefficient at zero photon energy indicates that Ir2P has metallicity, which is consistent with its electronic structures (see Fig. 6 and 7). In Fig. 10(a) and S5,† the optical absorption spectra of both structures show a broad, continuous absorption characteristic, distinct from the well-defined absorption peaks in semiconductors or insulators. This is attributed to the continuous band structure of metallic materials, lacking a band gap, which allows electrons to move freely, leading to a more uniform absorption of light. The presence of these inconspicuous absorption peaks is associated with electron transition, as incident light imparts energy to electrons, prompting their transition from the ground state to the excited state. Additionally, their values progressively rise with energy, peaking at 14.9 (0 GPa) and 15.4 eV (120 GPa), respectively, implying the materials' proficiency as ultraviolet absorbers. Their overall absorption region and peaks exhibit an increase with rising pressure. These findings suggest that applying pressure can broaden the photoresponse range of Ir2P and improve its light absorption in the ultraviolet region [see Fig. S5(a)†]. Both structures exhibit reflectivity values exceeding 76% in the 25–35 eV, suggesting that Ir2P can serve effectively as a reflective coating within this energy range [see Fig. S5(b)†]. With increasing energy, their optical conductivity initiates an ascent from a photon energy of zero, attributed to the absence of a bandgap in Ir2P, and culminates in a maximum in the ultraviolet region [see Fig. 10(c) and S5(c)†]. The research reveals that Ir2P exhibits optical conductivity across the infrared, visible, and ultraviolet wavelength ranges. Based on Fig. 10(d) and S5(d),† it is evident that the Fmm and I4/mmm phases exhibit smaller loss function in the visible region, and their maximum energy loss peaks occur at 26.2 eV (0 GPa) and 31.3 eV (120 GPa), corresponding to their plasmonic frequencies, respectively. When the incident photon energy surpasses its plasma frequency, the absorption coefficient and reflectivity decrease drastically, and the Ir2P becomes transparent, implying that the Ir2P changes from a metal response to a dielectric response. Moreover, both structures exhibit a blueshift in reflectivity, optical conductivity and loss function with increasing pressure towards the high-energy region.
Fig. 11 The calculated volume and bulk modulus for the Fmm phase of Ir2P (a) as a function of pressure at different temperatures and (b) as a function of temperatures at various pressures. |
The B value of Ir2P changes regularly with temperature and pressure. However, the first and second pressure derivatives of B, and show some variation with changes in pressure and temperature, as shown in Fig. 12. Li et al.59 noted that the and derived from the three-parameter Vinet or Rose EOS often deviate significantly from experimental observations, indicating the need for further development of the EOS with improved performance. As shown in Fig. 12(a) and (b), increases with the temperature throughout the pressure range of 0–80 GPa, while decreases rapidly at low pressures, and moderately at higher pressures with an increase in temperature. For Ir2P, reducing pressure has a similar effect on as increasing temperature. When the pressure is less than 30 GPa, changes sharply, and gradually becomes gentler with an increase in pressure, remaining relatively stable after the pressure exceeds 40 GPa. Therefore, under high-pressure conditions, the response of in Ir2P to temperature change is relatively slow.
Fig. 12 (a) Pressure dependence of the and at different temperatures and (b) temperature dependence of the and at various pressures for Fmm phase of Ir2P. |
In the quasi-harmonic approximation, anharmonicity is limited to thermal expansion. Fig. 13 shows how the volume thermal expansion coefficient α changes with temperature under different pressures. As seen in Fig. 13(a) and (b), α decreases significantly with increasing pressure at various temperatures, and increases with rising temperature at different pressures. This indicates that anharmonic effects have a significant impact on the antifluorite structure of Ir2P under low-pressure and high-temperature conditions.
Fig. 13 (a) The pressure-dependent thermal expansion coefficient α at various temperatures and (b) the temperature-dependent α at various pressures for Fmm phase of Ir2P. |
Fig. 14 illustrates the relationship between the Debye temperature, pressure and temperature for Ir2P with the anti-fluorite structure. As displayed in Fig. 14(a) and (b), it can be observed that θD values increase with the rise in pressure. At low pressures, the θD value undergoes a significant reduction when the temperature varies from 0 to 2000 K. Nonetheless, as the pressure increases, the extent of this reduction becomes less significant. To the best of our knowledge, this is the first quantitative theoretical prediction for Ir2P concerning the pressure and temperature dependence of the Debye temperature. Further research is needed to experimentally confirm these findings.
Fig. 14 (a) Pressure dependence of the θD at various temperatures and (b) temperature dependence of θD at various pressures for Fmm phase of Ir2P. |
Footnote |
† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d3ra07464a |
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