First-principles study on the conventional superconductivity of N-doped fcc-LuH3

Zihao Huo; Defang Duan; Tiancheng Ma; Zihan Zhang; Qiwen Jiang; Decheng An; Hao Song; Fubo Tian; Tian Cui

doi:10.1063/5.0151844

Abstract

Recently, room-temperature superconductivity has been reported in a nitrogen-doped lutetium hydride at near-ambient pressure [Dasenbrock-Gammon et al., Nature 615, 244 (2023)]. The superconducting properties might arise from Fm

\bar{3}

m-LuH_3-δN_ε. Here, we systematically study the phase diagram of Lu–N–H at 1 GPa using first-principles calculations, and we do not find any thermodynamically stable ternary compounds. In addition, we calculate the dynamic stability and superconducting properties of N-doped Fm

\bar{3}

m-LuH₃ using the virtual crystal approximation (VCA) and the supercell method. The R3m-Lu₂H₅N predicted using the supercell method could be dynamically stable at 50 GPa, with a T_c of 27 K. According to the VCA method, the highest T_c is 22 K, obtained with 1% N-doping at 30 GPa. Moreover, the doping of nitrogen atoms into Fm

\bar{3}

m-LuH₃ slightly enhances T_c, but raises the dynamically stable pressure. Our theoretical results show that the T_c values of N-doped LuH₃ estimated using the Allen–Dynes-modified McMillan equation are much lower than room temperature.

Hydrides have received much attention because of their excellent superconductivities under pressure.1–4 In 2014, H₃S was predicted to be a high-temperature superconductor with a T_c of 191–204 K,5 which was confirmed by the experimentally measured T_c of 203 K at 155 GPa.6,7 Following this success, several superhydrides with high T_c values above 200 K, such as LaH₁₀ (260 K, 170 GPa), YH₆ (224 K, 166 GPa), YH₉ (260 K, 201 GPa), and CaH₆ (215 K, 172 GPa), were predicted and synthesized.8–18 Recently, Song et al.19 comprehensively studied the Lu–H system and predicted that Im $\bar{3}$ m-LuH₆ should have a T_c of 273 K at 100 GPa. Extensive experimental research has been conducted on the superconducting properties of lutetium hydrides under pressure. Fm $\bar{3}$ m-LuH₃ has been successfully synthesized and found to have a T_c of 12 K at 122 GPa.20 Moreover, Pm $\bar{3}$ n-Lu₄H₂₃ has been obtained, with a T_c of 71 K at 218 GPa.21 In particular, a very recent experimental study has reported superconductivity in the Lu–N–H system at near-ambient pressure (∼1 GPa), with the highest T_c to date of 294 K,22 thus achieving room-temperature superconductivity near ambient pressure. The authors of this study attributed the room-temperature superconductivity in the Lu–N–H system to Fm $\bar{3}$ m-LuH_3−δN_ε, which can be regarded as Fm $\bar{3}$ m-LuH₃ doped with nitrogen atoms.

This report could represent a landmark event for the scientific community, but many open questions surrounding this important discovery remain unanswered, for example, the exact stoichiometry and the positions of the hydrogen and nitrogen atoms. A subsequent experimental observation indicates that the pressure-induced color change of LuH₂ is similar to that of N-doped lutetium hydride.23 In addition, a similar compound (LuH_2±xN_y) has been synthesized, but no evidence for superconductivity at pressures ranging from 1 to 6 GPa has been found.24 Density functional theory (DFT) calculations have been used to investigate the optical properties of lutetium hydrides25 and the phase diagram of the Lu–N–H system at 0, 5, and 10 GPa,26 but no thermodynamically stable ternary compounds have been found. Therefore, an investigation of the superconducting properties of N-doped lutetium hydrides is necessary.

In the work described in this paper, we performed a comprehensive first-principles study on the Lu–N–H system at 1 GPa. No thermodynamically stable Lu–N–H ternary compounds were found using the structure search method. We then analyzed the dynamic stability and superconducting properties of N-doped Fm $\bar{3}$ m-LuH₃ using the virtual crystal approximation (VCA) and the supercell method. Using the Allen–Dynes-modified McMillan (A-D-M) equation, we found that the estimated highest T_c of N-doped LuH₃ did not exceed 30 K, which is much lower than room temperature. In addition, we found that the doping of nitrogen atoms may slightly enhance T_c but also increase the dynamically stable pressure of LuH₃.

At 1 GPa, we performed a variable-composition crystal structure search in the Lu–N–H system with ∼10 000 structures using the Ab Initio Random Structure Searching (AIRSS)27 code. We then re-optimized the structures using the ab initio calculation from the Cambridge Serial Total Energy Package (CASTEP).28 An on-the-fly ultrasoft pseudopotential with valence electrons 1s¹ for H, 2s²2p³ for N, and 4f¹⁴5s²5p⁶5d¹6s² for Lu was used, with a kinetic cutoff energy of 800 eV. The Brillouin zone was sampled using a k-point mesh of 2π × 0.03 Å⁻¹ to make the enthalpy calculations converge well to less than 1 meV/atom. Structural relaxations were performed using projector-augmented wave (PAW)29,30 potentials, as implemented in the Vienna Ab Initio Simulation Package (VASP),31 with a cutoff energy of 600 eV. The exchange-correlation functional was described using the Perdew–Burke–Ernzerhof generalized gradient approximation.32

We then prepared a conventional cell of Fm $\bar{3}$ m-LuH₃, including four formula units (f.u.). The calculations were performed for a range of pseudostoichiometric Lu₄H_mN_12−m, including Lu₄H₁₁N, Lu₂H₅N, Lu₄H₉N₃, and LuH₂N, which correspond to nitrogen-to-hydrogen atomic ratios of 0.09, 0.2, 0.33, and 0.5, respectively. Next, we calculated the formation enthalpies and superconducting properties of these structures at 1, 10, and 50 GPa. The formation enthalpy of Lu₄H_mN_12−m was calculated using the following equation: $H^{f} = H (L u_{4} H_{m} N_{12 - m}) - 4 H (L u H_{3}) - m H (N) + m H (H) .$ (1)

We investigated the pressure dependence of the superconductivity of Fm $\bar{3}$ m-LuH₃ at 0.5%–2% doping with N using the VCA at pressures below 100 GPa. The VCA was performed by generating a virtual pseudopotential V_VCA of H_1−xN_x, where V_VCA = (1 − x)V_H + xV_N. Furthermore, we calculated the equation of state for LuH_2.97N_0.03, which is close to the elemental analysis data from previous experiments,22 using DFT and DFT + U (with U = 5.5 eV, which has been used for metal mononitrides22). The result is shown in Fig. S1 (supplementary material). The DFT result without the Hubbard U effect fits better with the experimental data. Therefore, the DFT level calculation for N-doped LuH₃ is acceptable.

Electronic structures, phonon spectra, and electron–phonon coupling (EPC) were calculated using the QUANTUM ESPRESSO (QE)33 package. The PAW pseudopotentials with valence electrons 1s¹ for H, 2s²2p³ for N, and 5s²5p⁶5d¹6s² for Lu were used in the QE package. The self-consistent electron density was evaluated using a k-mesh of 20 × 20 × 20. The phonon spectra and EPC were calculated using a q-mesh of 5 × 5 × 5. The conventional superconducting transition temperature was estimated using the A-D-M equation34 with correction factors and a Coulomb pseudopotential35 with μ* = 0.10 or 0.13.

We performed a random structure search in the Lu–N–H system and constructed the ternary phase diagram (convex hull) at 1 GPa [Fig. 1(a)]. Some of the binary compounds were adopted from previous papers.36–39 Notably, all predicted potential ternary compounds lie above the convex hull at 1 GPa. Thus, no ternary Lu–N–H compounds can remain thermodynamically stable at this pressure, which is consistent with the main results of Xie et al.26 In addition, P $\bar{3}$ m1-Lu₂H₂N was found with an enthalpy of ∼3 meV/atom above the convex hull. Detailed information about the structural parameters is listed in Table S1 (supplementary material). According to the inorganic crystal structure database, 20% of experimentally synthesized materials are metastable.40,41 Therefore, we calculated the x-ray diffraction (XRD) pattern for this metastable compound. Figure 1(b) shows a comparison of experimental and calculated XRD patterns. The calculated XRD pattern of P $\bar{3}$ m1-Lu₂H₂N deviates clearly from the experimental one,22 indicating that this compound does not occur in high-pressure experiments on N-doped lutetium hydride superconductors.

Figure 1.(a) Ternary phase diagram (convex hull) of the Lu–N–H system at 1 GPa. The values of the enthalpy from the convex hull are shown next to the colored circles. The circles with black solid boundaries denote the stable phases. (b) Comparison of the XRD pattern from reported experimental data²² (lower black curve) and the pattern calculated for P

\bar{3}

m1-Lu₂H₂N at ambient pressure (upper red curve). The inset shows the crystal structure of P

\bar{3}

m1-Lu₂H₂N, with green, gray, and pink spheres depicting Lu, N, and H atoms, respectively.

Next, we investigated the doping effect on N-doped LuH₃ using the supercell method. The cubic cell of Lu₄H_mN_12−m was constructed by replacing hydrogen atoms with nitrogen atoms, for m = 8–11. For each concentration, we performed a geometry optimization on the configurations of various octahedral (O) and tetrahedral (T) sites [see Fig. 2(a)] and then calculated the total energy. Figures 2(b)–2(d) show the results for the formation enthalpy of Lu₄H_mN_12−m at different pressures. At 1 GPa, only the formation enthalpy of Lu₄H₁₁N is negative (∼−24 meV/atom), indicating that this compound may be more stable than the others. A similar situation also occurs at 10 GPa (where the formation enthalpy is ∼−32 meV/atom for Lu₄H₁₁N). Thus, only Lu₄H₁₁N is thermodynamically encouraged to be formed below 10 GPa. At 50 GPa, Lu₄H₁₁N, Lu₂H₅N, Lu₄H₉N₃, and LuH₂N are thermodynamically encouraged to be formed. Thus, Lu₄H_mN_12−m with low nitrogen doping is enthalpically favored at low pressure. The doping concentration of nitrogen increases with increasing pressure. In addition, when one nitrogen atom occupies a T site, the formation enthalpy of Lu₄H_mN_12−m at 50 GPa is the lowest. Therefore, one nitrogen atom occupying a T site in Lu₄H_mN_12−m is enthalpically preferred.

Figure 2.(a) Crystal structure of Fm

\bar{3}

m-LuH₃ with different types of H occupancy sites. Green spheres depict Lu atoms; pink and red spheres depict H occupancy of O and T sites, respectively. (b)–(d) Calculated enthalpy of formation of Lu₄H_mN_12−m as a function of T-site occupancy at 1, 10, and 50 GPa, respectively. n(T) is the number of nitrogen atoms in each stoichiometry occupying T sites.

We then calculated the phonon spectra and superconducting properties of Lu₄H_mN_12−m. T_c values were estimated using the A-D-M equation with correction factors (Table S2, supplementary material). At 1 and 10 GPa, Lu₄H_mN_12−m cannot dynamically stabilize, which may be due to the thermodynamical42 and dynamical instability of the parent Fm $\bar{3}$ m-LuH₃ [see Fig. S2(b), supplementary material]. At 50 GPa, only R3m-Lu₂H₅N can dynamically stabilize. The crystal structure, phonon spectrum, and EPC of R3m-Lu₂H₅N are illustrated in Fig. 3. R3m-Lu₂H₅N (Lu₄H₁₀N₂) can be regarded as Fm $\bar{3}$ m-LuH₃ with two nitrogen atoms substituting its T and O sites, respectively. The calculated phonon spectrum and EPC show that the contributions of medium- and low-frequency phonons (13–25 THz) to the EPC are the highest (about 46% to total λ), whereas high-frequency phonons (45–55 THz) make hardly any contribution. Thus, the EPC of R3m-Lu₂H₅N is primarily contributed by medium- and low-frequency phonons. The calculated EPC parameter λ for R3m-Lu₂H₅N at 50 GPa in the harmonic approximation is 0.82. Using the calculated logarithmic average frequency ω_log, along with a Coulomb pseudopotential μ* value of 0.1, the resultant T_c value is 27 K.

Figure 3.(a) Phonon spectra, Eliashberg phonon spectral function α²F(ω), and electron–phonon integral λ for R3m-Lu₂H₅N at 50 GPa. The magnitudes of the phonon linewidths are represented by the sizes of the blue circles. (b) and (c) Crystal structures of R3m-Lu₂H₅N in the framework of Fm

\bar{3}

m-LuH₃ and the conventional cell, respectively. Green, gray, and pink spheres depict Lu, N, and H atoms, respectively.

In the case of low nitrogen doping concentrations, we used the VCA method to investigate the pressure dependence of T_c in N-doped Fm $\bar{3}$ m-LuH₃. Figure 4(a) shows the dependence of the minimum dynamically stable pressure on the N-doping concentration. We find that the minimum dynamically stable pressure of N-doped Fm $\bar{3}$ m-LuH₃ increases from 25 to 70 GPa when the doping concentration increases from 0% to 2%. Thus, doping with N atoms will raise the dynamically stable pressure of LuH₃. Through a softening mechanism,43 pressure can affect T_c by altering the electron–phonon constant λ. Therefore, we calculated the T_c of N-doped LuH₃ with doping concentrations ranging from 0% to 1.5% at 50 GPa [see Fig. 4(b)] to investigate the effects of doping with nitrogen atoms on superconductivity. Our simulations show that the lowest T_c is 4 K for LuH₃ without doping. In addition, T_c increases with increasing N-doping concentration; thus, doping N atoms into LuH₃ will increase T_c. However, the highest T_c in this VCA calculation is 22 K, obtained with 1% N-doping at 30 GPa (see Table S2, supplementary material), which is much lower than room temperature. Notably, the anharmonicity correction to the atomic motions imposed by the large ionic quantum fluctuation will renormalize the phonon frequency in hydrogen-based superconductors. Therefore, the anharmonic effect may potentially decrease the dynamically stable pressure of N-doped LuH₃ and affect T_c, which requires further theoretical investigation in this system.

Figure 4.(a) Dependence of minimum dynamically stable pressure on N-doping concentration. (b) Dependence of T_c on N-doping concentration at 50 GPa. The Coulomb pseudopotential uses μ* = 0.10.

We calculated the band structure and density of states (DOS) of LuH₃ and LuH_2.97N_0.03 (1% N-doped) at 50 GPa [Fig. 5(a)]. The finite DOS at the Fermi level indicates the metallic nature of these structures. LuH₃ has a DOS N(ϵ_F) that reaches 0.269 states eV⁻¹ f.u.⁻¹ at the Fermi level. Doping N atoms into LuH₃ will raise the Fermi level and cause the bands near the Γ point to fall on top of the Fermi level. Consequently, the Fermi level can be moved closer to the DOS peak, and N(ϵ_F) will then increase to 0.6 states eV⁻¹ f.u.⁻¹ Therefore, doping by N atoms can significantly enhance the metallic characteristic of LuH₃ by moving the Fermi level closer to the DOS peak.

Figure 5.(a) Electronic band structure and DOS (states eV⁻¹ f.u.⁻¹) of LuH₃ (solid black curves) and LuH_2.97N_0.03 (solid red curves) at 50 GPa. The grey solid horizontal line indicates the Fermi energy. (b) and (c) Phonon spectra, Eliashberg phonon spectral function α²F(ω), and electron–phonon integral λ for LuH₃ and LuH_2.97N_0.03, respectively, at 50 GPa. The magnitudes of the phonon linewidths are represented by the sizes of the blue circles.

We then examined the phonon spectra and EPC of LuH₃ and LuH_2.97N_0.03 at 50 GPa [Figs. 5(b) and 5(c)]. In LuH₃, the electron–phonon constant λ is primarily contributed by the optical branch (λ_opt), which accounts for ∼69% of the total λ. Conversely, the acoustic branch (λ_ac) makes a much smaller contribution to λ, accounting for ∼31% of the total λ. Doping by nitrogen atoms at 1% concentration increases λ_ac from 0.137 to 0.268, while λ_opt remains almost unchanged. Therefore, the increased T_c of N-doped LuH₃ can be attributed to the enhancement of λ_ac. Interestingly, the optical branch with a frequency of ∼21 THz along the W–L–Γ direction in LuH₃ varies slightly with the wave vector q in the Brillouin zone. However, a significant softening of this phonon mode is observed after doping by nitrogen atoms [Fig. 5(c)]. When the pressure decreases to 25 GPa (Fig. S4, supplementary material), the imaginary phonon frequency occurs near the L point, which indicates that the optical branch softening induced by N doping is responsible for the reduced stability of LuH₃.

We performed a first-principles study on the Lu–N–H system and found no stable ternary compounds at 1 GPa. We then analyzed N-doped Fm $\bar{3}$ m-LuH₃ using supercell and VCA methods. The result of the supercell method indicates that R3m-Lu₂H₅N can be dynamically stable at 50 GPa, with a T_c of 27 K. The VCA results indicate that the highest T_c is 22 K, obtained with 1% N-doping at 30 GPa. In addition, doping with nitrogen atoms slightly increases the T_c of Fm $\bar{3}$ m-LuH₃ by enhancing the EPC of acoustic phonons. However, this doping effect also leads to significant phonon softening and increases the dynamically stable pressure of LuH₃. Finally, within the pressure range investigated in our study, the highest T_c of N-doped Fm $\bar{3}$ m-LuH₃ does not exceed 30 K, which is much lower than room temperature. Our theoretical calculations were performed using standard DFT parameters and assuming conventional superconductivity, and calculations considering more corrections, including strong electron correlations, spin–orbit coupling, the anharmonicity effect, and unconventional mechanisms of superconductivity are required in the future.

Acknowledgment. This work was supported by the National Key R&D Program of China (Grant Nos. 2018YFA0305900 and 2022YFA1402304), the National Natural Science Foundation of China (Grant Nos. 12122405, 52072188, and 12274169), the Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT_15R23), and a Jilin Provincial Science and Technology Development Project (Grant No. 20210509038RQ). Some of the calculations were performed at the High Performance Computing Center of Jilin University and on TianHe-1(A) at the National Supercomputer Center in Tianjin.