The ternary phase diagram of nitrogen doped lutetium hydrides can not explain its claimed high Tc superconductivity

This paper presents the results of an extensive structural search of ternary solids containing lutetium, nitrogen and hydrogen. Based on thousands of thermodynamically stable structures the convex hull of the formation enthalpies is constructed. To obtain the correct energetic ordering, the highly accurate RSCAN DFT functional is used in high quality all-electron calculations, eliminating possible pseudopotential errors. In this way, a novel lutetium hydride structure (HLu2) is found that is on the convex hull. An electron phonon analysis however shows that it is not a candidate structure for near ambient superconductivity. Besides this structure, which appears to have been missed in previous searches, possibly due to different DFT methodologies, our results agree closely with the results of previously published structure search efforts. This shows, that the field of crystal structure prediction has matured to a state where independent methodologies produce consistent and reproducible results, underlining the trustworthiness of modern crystal structure predictions. Hence it is quite unlikely that a structure, that would give rise within standard BCS theory to the superconducting properties, claimed to have been observed by Dasenbrock-Gammon et al (2023 Nature 615 244), exists. This solidifies the evidence that structures with high Tc conventional superconductivity, that could give rise to the experimental claims, do not exist in this material.


Introduction
In their recent publication Dasenbrock-Gammon et al [1] claim to have experimentally observed superconductivity in bulk nitrogen doped lutetium hydride (Lu-N-H) at a T c of 294 K and at a pressure of 1 GPa.Since no detailed analysis of the structure, that is claimed to be superconductive at near ambient conditions, is given, an explanation of the mechanism that could lead to the observed superconductivity is missing.The mystery of the exact composition and structure of the putative superconductor has raised great interest into Lu-N-H structures throughout the entire materials science and solid state physics community.
The reaction of the community to the news of another room temperature superconductor from Dias and coworkers was prompt.Already a few days later, Shan et al [2] published their experimental study about pressure induced color changes in LuH 2 .The observed color changes in the samples are similar to the ones presented in [1] but resistivity measurements showed no signs of superconductivity above 1.5 K.One of the first theoretical studies on the Lu-N-H system was conducted by Liu et al [3].Their work also focused on lutetium hydrides.In order to investigate the convex hull of Lu-H the evolutionary structure prediction algorithm from the USPEX [4] package was used.Liu et al found LuH 2 to be the most stable lutetium hydride and they conclude that the LuH 2 is the parent structure when lutetium hydrides are doped with Nitrogen.Dangić et al [5] also investigated lutetium hydrides where they investigated Raman spectra, phonon band structures and optical properties.They find that LuH 2 is the only structure that can explain the color change in many experiments and conclude that it is synthesized in most experiments.Both Huo et al [6] and Xie et al [7] perform a ternary Lu-N-H structure search where only binary structures are found that are on the convex hull.A subsequent electron phonon analysis in [6] shows that no high T c structures where found by Huo et al An overview of the Lu-N-H convex hull can be found in the recent work of Ferreira et al [8] were the they present the results of a detailed structure search at ambient pressure.In the study of Ferreira et al, the configurational space of the ternary Lu-N-H structure was investigated thoroughly using the USPEX [4] evolutionary search method and the AIRSS [9] random structure search method.In the evolutionary search with USPEX, Ferreira et al calculated energy and forces on the density functional theory (DFT) level and in the random structure search with AIRSS, ephemeral data derived potentials [10] were used.An electron phonon analysis of the best candidate structures for room temperature superconductivity from Ferreira et al disagrees with the observation of near ambient superconductivity made by Dasenbrock-Gammon et al [1].Based on their results, Ferreira et al conclude that the observations made by Dasenbrock-Gammon et al [1] cannot be explained with the electron phonon mechanism that describes conventional superconductivity.
Given that Liu et al [3,8,11] have investigated the configurational and compositional space of Lu-N-H thoroughly the excitement about the Lu-N-H superconductor has been damped considerably as the observations made by Dasenbrock-Gammon et al [1] could not be explained using the current state of the art theoretical materials science methods.
There are basically three options that explain this disagreement between theory and experiment: • Dasenbrock-Gammon et al [1] observed unconventional superconductivity.
• There is an error in the experimental setup of Dasenbrock-Gammon et al [1].
• The correct structure was not found in all theoretical structure searches.
In this paper we present the results of an independent structure search in the ternary Lu-N-H phase diagram, ruling further out the last possibility that an important structure was overlooked.All presented final results were obtained with the regularized SCAN (RSCAN) functional [12], which is widely considered to be the most accurate functional for cohesive energies.Well tested pseudo-potentials for this functional are however scarce.To eliminate any pseudo-potential errors we have therefore performed highly accurate all-electron calculations.Therefore, our results are expected to be more accurate than all other previous results.The same approach has recently been used in a large scale structure search [13] for the putative carbonaceous sulphur hydrides superconductor [14].
Our results solidify the conclusions from the previous studies [3,8,11] that no conventionally superconducting structure can be found in the ternary Lu-N-H phase diagram.

Structure search with minima hopping
The phase diagram of the ternary Lu-N-H structures were explored using the minima-hopping method [15][16][17][18][19][20].Minima hopping is a method that reliably finds the global minimum of potential energy surfaces using a combination of variable cell shape molecular dynamics [21] along soft modes of the potential energy surface and variable cell shape geometry optimization [22].Since it is not supposed to generate a thermodynamic distribution, it can escape from any funnel by crossing high energy barriers.Because of that, minima hopping will always find the global minimum given a sufficiently long simulation.Other methods such as evolutionary search algorithms introduce moves to generate new structures which can be insufficient to escape from a deep funnel.
In the Minima Hopping runs, energies, forces and the stress tensor were calculated on the DFT level with the standard PBE functional [23] and the SIRIUS library [24,25] which is a GPU accelerated and MPI parallelized plane wave code.Ultrasoft pseudo potentials [26] were used to eliminate the core electrons.A plane wave cutoff of 1400 eV was used and a tight 4 × 4 × 4 Monkhorst-Pack [27] k-point grid was chosen.
In total 108 different stoichiometries were sampled at a pressure of 1 GPa.To ensure convergence of the minima hopping method, the search was only stopped after 25 000 distinct local minima were found.On average 230 different minima were found for every stoichiometry.

All electron calculations
In order to increase the accuracy of the DFT calculations from section 2.1 the 20 lowest minima that are in the energy range of 50 meV per atom compared to the ground state of each stoichiometry were recalculated with a more precise DFT method.With these criteria, 1600 structures were selected for further processing.The error introduced by the pseudopotentials was eliminated by performing an all electron calculation and the error from the PBE functional was reduced by using the accurate RSCAN exchange correlation functional [12,28,29].
FHI-aims [30][31][32] was used for a geometry optimization of the 1200 systems with the previously mentioned settings, a 5 × 5 × 5 Γ centered k-point grid and the tier 2 basis set.The resulting energies are our most accurate ones.They reduce errors in the energetic ordering of the 1200 lowest systems and therefore the chance of finding the wrong ground state.All energies used in the convex hull plots of this paper were obtained using this procedure.

Comparison between the high-performance and high-accuracy DFT calculations
The energies of the most promising structures that were found using the minima hopping method were recalculated with all electron DFT simulations that used the RSCAN functional.The difference in formation enthalpy of the plane wave calculations with PBE functional and the all electron calculation with the RSCAN functional is displayed in figure 2. For most stoichiometries, the error is between 50 meV per atom and 100 meV per atoms.The energy difference between the pseudopotential and the all electron calculation is rather large.Nevertheless, errors of this magnitude are not too uncommon in DFT calculations.The good correlation in formation enthalpy of the nitrogen hydrides indicates that lutetium is responsible for a large part of the energy error.
Even though the energetic error of the pseudopotential calculations is rather large, the energetic ordering of the structures on the convex hull is surprisingly good and the ground state structures were predicted correctly by the pseudopotential calculations.

Lutetium hydrides
In order to get an initial impression for the Lu-N-H system, a structure search for binary lutetium hydrides was first conducted and the formation enthalpies were also verified using highly accurate all electron  calculations.The convex hull of the binary system is displayed in figure 3. H 3 Lu, H 2 Lu and HLu 2 all lie on the convex hull of formation enthalpies which makes them thermodynamically stable.HLu is only 16 meV per atom above the convex hull which is within the typical uncertainty of DFT calculations.Therefore, HLu may also be thermodynamically stable.The high accuracy DFT calculations presented in section 3.1 show that HLu 2 is on the convex hull.The structure and the electronic density of states of H 3 Lu, H 2 Lu and HLu 2 is displayed in figures 4(a)-(c) and (e).Because neither Liu et al [3] or Ferreira et al [8] have found an HLu 2 structure that is on the convex hull we decided to calculate the T c using Quantum Espresso [33] and the Allen-Dynes equation [34].The calculations were done using the same procedure as in [13].To calculate the T c of HLu 2 , a plane wave cutoff 1370 eV, a 16 × 16 × 16 k-grid (a spacing of ∼0.03A −1 ) and a 4 × 4 × 4 q-grid (a spacing of ∼0.13A −1 ) were used.The resulting T c is 0.5 K which is obtained using the value 0.1 for µ * .In order to provide more information about the HLu 2 structure, its electronic density of states and phonon dispersion is shown in figures 5 and 6.

Ternary lutetium nitrogen hydrides
The convex hull of formation enthalpies of the Lu-N-H system is displayed in figure 1.There is only one ternary stoichiometry, that is on the convex hull:  Additional electron phonon calculations were done for H 2 Lu 5 N, H 3 Lu 4 N, H 3 Lu 4 N 2 H 4 Lu 4 N 2 , H 3 Lu 9 N and H 5 Lu 9 N because their convex hull distance is smaller than 50 meV per atom and they exhibit a moderately pronounced singularity at the Fermi level.The highest T c that was calculated for these structures is H 3 Lu 4 N with 5 K.
Our results indicate that nitrogen doped LuH x crystals are thermodynamically unstable since they are all above the convex hull of formation ethalpies.Other than that, no ternary Lu-N-H structures with a low convex hull distance and interesting features in the electronic density of states was found in our structure search.Our results therefore suggest that Lu-N-H compositions are not responsible for the high T c measured by Dasenbrock-Gammon et al [1] or that unconventional superconductivity, which cannot be described by our theory, was observed by Dasenbrock-Gammon et al [1]

Conclusion
Theoretical structure prediction is by no means an easy or routine task, especially for complex ternary such as Lu-N-H involving rare elements such as Lu.It is therefore reassuring, that two independent studies based on completely different methodologies and codes come to comparable conclusion.The only notable difference between our study and the previous studies is that we have identified HLu 2 to be on the convex hull.Our calculation of the T c of HLu 2 shows that it is not responsible for the high T c measured by Dasenbrock-Gammon et al [1] Otherwise, very similar structures were found which lead to a comparable convex hull.Only a few (H 2 Lu, H 3 Lu, HLu 2 , LuN and H 5 Lu 4 N 2 ) Lu-N-H stoichiometries lie on this convex hull at 1 GPa of pressure.This is consistent with prior structure searches [3,8,11].Firstly, this highlights the fact, that modern crystal structure prediction methods have reached a level of maturity, where consistent and reproducible results can be expected.Secondly, the fact that two studies, based on independent methodologies, come to the same central conclusion reduces the risk that a bias was introduced during the structure search and that a possible structure with superconducting properties was overlooked.The theoretical results from our study and previous structure searches [3,8,11] disagree with the experimental observation made by Dasenbrock-Gammon et al [1] since no structure was found that can explain the claimed room temperature superconductivity.This allows us to conclude, with a very high certainty, that no conventionally superconducting Lu-N-H and Lu-H structure exists.

Structural data
Low enthalpy Lu-N-H structures, their electronic density of states and convex hull distances can be found in this GitHub repository: https://github.com/moritzgubler/H-Lu-N.The enthalpy of these structures was calculated as described in section 3.1.

Figure 1 .
Figure 1.Formation enthalpy difference to the convex hull in meV per atom at 1 GPa of pressure.The black lines indicate the convex hull.

Figure 2 .
Figure 2. Difference in formation enthalpy for the ground state of each stoichiometry between a PBE pseudopotential calculation and an all electron RSCAN calculation at 1 GPa of pressure.

Figure 3 .
Figure 3.A comparison of the binary convex hull of the Lu-H system calculated with the high throughput and high accuracy DFT methods.The convex hull is displayed by the black lines.
H 5 Lu 4 N 2 .It has an orthorombic lattice with cell parameters a = 3.42 A, b = 5.57A and c = 5.90 A. It is pictured in figure 4(d) and a plot of the electronic density of states can be found in figure 4(e).The density of states has no Van Hove singularity at the Fermi level or any other special features which indicates that it is an ordinary metal.

Figure 4 .
Figure 4.A selection of structures that lie on the convex hull at 1 GPa of pressure and electronic density of calculated with an all electron RSCAN simulation.Hydrogen is pictured in white, nitrogen in blue and lutetium in red and the Fermi level is shifted to zero.

Figure 5 .
Figure 5. Electronic band structure and the density of states of HLu2.The dashed line shows the Fermi energy of 14.54 eV.

Figure 6 .
Figure 6.Phonon dispersion and density of states of HLu2.