Unconventional stoichiometric two-dimensional potassium nitrides with anion-driven metallicity and superconductivity

Abstract

The design of compounds with interesting coordination geometries, exotic oxidation states, and intriguing properties is of fundamental interest in physics, chemistry, and materials science. Herein, first-principle swarm-intelligence structure search calculations identify two unconventional stoichiometric KN₂ and KN₄ monolayers, in which the coordination number of K atoms increases from 6 to 12, becoming two-dimensional (2D) metal nitrides with the highest coordination number. Nitrogen motifs in stable K–N monolayers depend on the nitrogen content (e.g., N₂⁻ dimer in KN₂ and N₂^0.5− dimer in KN₄), and are accompanied by an electronic transition from superconducting to metallic. Intriguingly, the KN₂ monolayer shows a unique superconducting energy gap with a calculated critical temperature (T_c) of 4.3 K under ambient conditions. The superconductivity is mainly derived from strong electron–phonon coupling (EPC) of the N 2p electrons and low- and mid-frequency K and N atomic vibrational modes. Our work provides key insight into 2D metal-bearing nitrides and the correlation between non-stoichiometry and conductivity mechanisms.

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Introduction

Since the discovery of superconductivity in metallic mercury at 4.2 K in 1911, the exploration of superconductors and understanding of microscopic mechanisms have become important and challenging issues in condensed matter physics.^1,2 Inspired by theoretical calculations, great breakthroughs have been achieved in H₃S^3,4 (203 K at 155 GPa) and LaH₁₀^5,6 (260 K at 188 GPa) to fulfill the dream of high-temperature superconductivity. Although these materials display high superconducting transition temperature (T_c), high pressure and complicated experimental equipment greatly limit their practical applications.⁷

In addition to pressure-induced superconductivity, the role of dimensionality reduction on superconductivity has also attracted massive attention. Compared with bulk superconducting materials, 2D superconductors play a critical and irreplaceable role in small portable magnetic resonance imagers, high-precision micro-magnetic field and low-dissipation integrated circuits.^8–10 The 2D superconductivity state has recently been observed in graphene,^11,12 transition-metal dichalcogenides (NbS₂,¹³ MoS₂,¹⁴ NbSe₂,¹⁵ TiSe₂¹⁶), ZrNCl surface,^17,18 single Pb layer,¹⁹etc., with the help of electron/hole doping, intercalation, gating, and substrate proximity effects. Their superconducting transition temperatures were estimated to be 1–30 K. However, the number of 2D intrinsic superconductors is rather limited and there is a pressing task to expand the family of 2D superconductors.

Nitrogen, as one of the major components of the atmosphere,²⁰ is a fascinating element. But the understanding of nitrogen chemistry is very limited due to the extremely high stability of the triple-bond in the N₂ molecule. For a long time, only nitride N³⁻ and azide N₃⁻ nitrogen motifs were known. It was not until 2001 that Kniep et al. demonstrated the diazenides N₂²⁻ in the deprotonated diazene N₂H₂ with a N=N bond.²¹ In the same year, binary diazenides SrN₂ and BaN₂ were synthesized for the first time.²² Subsequently, stable homoatomic ions N₅⁻ and N₅⁺ were successfully discovered,^23,24 greatly expanding the chemical richness of nitrogen. Lately, the study of N-rich nitrides revealed the ubiquity of charged [N₂]^x− (x = 1, 2, 3 or 4) dimers with N–N bond lengths from 1.15 to 1.35 Å in binary metal nitrides (e.g., LiN₂,²⁵ Li₂N₂,²⁶ CrN₂,²⁷ IrN₂²⁸).

Introducing metal elements into nitrogen is an effective way to discover interesting nitrogen configurations and enrich the understanding of nitrogen evolution behavior. Specifically, 2D metal-bearing nitrogen-based materials exhibit diverse geometric structures and unique electronic properties, which are closely correlated with metal species and metal/nitrogen ratio. For 2D alkaline-earth metal nitrides (IIANs), a honeycomb BeN₂ monolayer with direct bandgap and high mobility is expected to be useful for field-effect transistors (FETs).²⁹ Interestingly, Ca₂N, Sr₂N and Ba₂N sheets are electrides, in which the excess electrons are localized at the interstitial region and act as anions.^30–32 On the other hand, 2D IIIANs (AlN, GaN, InN and TlN) with graphene-like honeycomb structures are semiconductors,^33–35 while penta-AlN₂ with Cairo pentagonal tiling geometry is a half-metal and shows ferromagnetism.³⁶ Later, the prediction of 2D alkali metal nitrides (IANs) enriched the family of metal nitrides. In particular, a series of stable cation-deficient 2D K₂N monolayers are p-type ferromagnetic metals or half metals with high Curie temperature of 480–1180 K.³⁷

Encouraged by the aforementioned advance in 2D metal nitrides, we perform a systematic search for 2D N-rich KN_x (x = 1–4) compounds via the particle swarm optimization algorithm.^38,39 We have identified two stable KN₂ and KN₄ monolayers with unconventional stoichiometries, in which all N atoms form quasi-molecular N₂ dimers. In the KN₄ monolayer, the K atom reaches 12-fold hypercoordination, which is the highest one reported in 2D metal nitrides. The electronic properties of 2D KN_x (x = 2, 4) monolayers are closely related to the nitrogen motif, that is, KN₄ contains N₂^0.5−, which is metallic, while KN₂ containing N₂⁻ is superconducting with T_c of 4.3 K.

Computational details

Structural evolution algorithms play a key role in accelerating the discovery of novel 2D materials and broadening their application.^40–45 The stable 2D K–N structural prediction was based on a global minimization of free energy surfaces merging ab initio total-energy calculations as implemented in the CALYPSO code.^38,39,46 A vacuum space of more than 15 Å was added to avoid interaction between a layer and its periodic images along the c-axis. The structural relaxation and electronic property calculations were carried out by employing the Vienna ab initio Simulation Package (VASP),^47,48 with electron–ion interactions described by the augmented wave potentials⁴⁹ (3s²3p⁶4s¹ and 2s²2p³ valence states for K and N atoms, respectively) and plane-wave basis with a kinetic energy cutoff of 600 eV. The Monkhorst–Pack scheme⁵⁰ with a dense k-point grid spacing of 2π × 0.03 Å⁻¹ was chosen to ensure good convergence of the total energy. Dynamic stability was assessed by the supercell method⁵¹ implemented in the Phonopy code.⁵² Electron–phonon coupling (EPC) calculations were performed with density functional perturbation theory (DFPT) using the Quantum ESPRESSO package.^53,54 Anisotropic superconducting properties were investigated by solving the fully anisotropic Migdal–Eliashberg equations as implemented in the electron–phonon Wannier (EPW) code.^55–57 More details can be found in the Supplemental Material.

Results and discussion

In this work, we mainly focus on the lowest-energy structural identification of changes in nitrogen proportion (KN_x, where x = 1–4). We have successfully identified two hitherto unknown 2D thermodynamically and dynamically stable structures, i.e., superconducting KN₂ (space group Cmmm, Fig. 1a) and metallic KN₄ (space group P1̄, Fig. 1b) monolayers with unconventional stoichiometry. Up to now, several 2D non-stoichiometric compounds (e.g., Na₂Cl, Na₃Cl, and CaCl) have been synthesized.⁵⁸ Both KN₂ and KN₄ monolayers have negative formation enthalpies with respect to the elemental solid of K⁵⁹ and the molecular crystal of N₂⁶⁰ (Fig. 2a), indicating that the synthetic reactions by converting K and N₂ into KN₂ and KN₄ are thermodynamically favorable and experimentally feasible. The cohesive energies of KN₂ and KN₄ monolayers are 5.69 and 6.70 eV per atom, respectively, which surpasses the experimentally synthesized silicene (3.48 eV per atom)⁶¹ and phosphorene (3.30 eV per atom)⁶² and is comparable to MoS₂ (5.05 eV per atom).⁶³ Phonon dispersions of KN₂ and KN₄ with no imaginary frequencies in the whole Brillouin zone demonstrate that they are dynamically stable (Fig. S1, ESI†). According to the derived elastic constants (Table S1, ESI†), KN₂ and KN₄ satisfy the criteria for mechanical stability. We also calculate the Young's modulus E(θ) and Poisson's ratio v(θ) (Fig. S2, ESI†), which show anisotropy in both cases. Other stoichiometric formulas of KN_x (x = 1, 3) (see Fig. 1c and 1d for structures) are incompatible with thermodynamic or dynamical stability (Fig. 2a and Fig. S3, ESI†); thus they are not conducive to experimental synthesis and will not be further considered thereafter.

Fig. 1 (a) Top and side views of the KN₂ monolayer, where the primitive- and unit-cell are enclosed with dashed black and blue frames, respectively. Top and side views of (b) KN₄, (c) KN and (d) KN₃ monolayers, where the unit-cell is enclosed with a dashed black frame.

Fig. 2 (a) Formation enthalpy of probable K–N structures searched by CALYPSO relative to elemental K and N₂ solids. K–N compounds with filled symbols (red color) are thermodynamically and dynamically stable, whereas other symbols indicate that the structure does not satisfy thermodynamic or dynamic stability. (b) Chemical bonding analysis of the KN₂ monolayer. (c) Spin-dependent PDOS of the KN₂ monolayer.

The primitive cell of the KN₂ monolayer consists of three atoms—one K atom and two equivalent N atoms. The optimized cell lattice parameters are a = b = 4.42 Å, in which the K atom is coordinated by six N atoms, forming planar K-centered nitrogen wheels (K@N₆, Fig. 1a). The K–N bond lengths are 2.76 Å and 2.95 Å, respectively, which are comparable to those in 2D K–N compounds³⁷ (e.g., 2.790 Å in H-K₂N, 2.785 Å in T-K₂N, and 2.922 Å in I-K₂N). The most striking structural feature is that all N atoms exist in quasi-molecular N₂ dimers with a nitrogen–nitrogen distance of 1.18 Å, which is much shorter than the N–N single bond length of 1.40 Å in the crystalline phase of the elemental solid.⁶⁴ The nitrogen–nitrogen bond length of our KN₂ monolayer is between the N≡N triple bond length (e.g., 1.10 Å in hexagonal N₂⁶⁵) and the N=N double bond lengths, such as 1.23 Å in penta-PtN₂ monolayer,⁶⁶ 1.22 Å in IrN₂ monolayer⁶⁷ and 1.24 Å for bulk BaN₂.⁶⁸ N₂ dimers in KN₂ may show an overlap between the double and triple bonds consisting of σ and π bonds, where p_z orbitals form localized π bonds (Fig. 2b). To decipher the nature of bonding and determine the charge state of quasi-molecular N₂, Bader charge analysis was performed. Each K atom loses 0.82 e, having a formal oxidation of +1, while the Bader partial charge is −0.40 e for N. These results confirm that this species is [N₂]⁻ wherein all of the N atoms assume a −0.5 formal oxidation state. Obviously, [N₂]⁻ in KN₂ contains an unpaired p electron that produces a paramagnetic state, which is further confirmed by the analysis of the spin-polarized projected density of states (PDOS) in Fig. 2c.

The N-richer KN₄ (Fig. 1b) consists of K atoms sandwiched between two layers of N atoms. Interestingly, as the N content increases, the coordination environment of the N atom becomes 12-fold hypercoordinated, that is, each K atom is located at the center of twelve surrounding N atoms (K@N₁₂, Fig. 1b). For comparison, in the reported 2D metal nitrides, the Be atom coordinates with three N atoms to form sp² hybridization in the BeN₂ monolayer,²⁹ and the transition metal atoms in 1H-TMN₂ (TM = Ti, Zr, Hf and W) and 1T-TMN₂ (TM = Zr, Hf, Nb, Ta and Mn) coordinate with six N atoms.⁶⁹ To the best of our knowledge, the KN₄ monolayer has the highest coordination number among the known 2D metal nitrides. The K–N bond lengths (2.87–3.25 Å) in KN₄ are moderately longer than those in KN₂ (2.75–2.95 Å), indicating that the K–N interaction is weakened. The nitrogen–nitrogen bond length (1.14 Å) is slightly shorter than that in KN₂ (1.18 Å). The average Mulliken overlap population of the nitrogen–nitrogen bond in KN₄ (1.35) is larger than that in KN₂ (1.20), indicating that the nitrogen–nitrogen bond in KN₄ is stronger. Accordingly, the highest phonon frequency (65.2 THz) is higher than that of KN₂ (57.5 THz) (Fig. S1, ESI†), reflecting stronger nitrogen–nitrogen interaction in KN₄. The calculated Bader partial charge for K (0.85 e) is comparable to 0.80 e in the KN₂ monolayer. Based on the nitrogen–nitrogen distance and Bader charge (Table S2, ESI†), the KN₄ monolayer contains N₂^0.5− motifs. As mentioned above, the introduction of K not only greatly enriches the coordination geometry of N, but also contributes to structural stability. In the successive contents, we mainly discuss the KN₂ monolayer with superconductivity, and a more detailed analysis of the structural information and electronic properties of the KN₄ monolayer is given in the ESI.†

The electron localization function (ELF) in KN₂ reveals electronic depletion around the K atoms and accumulation nearby N atoms (Fig. 3a), signifying a certain degree of ionicity between N₂ dimers and K atoms, which is further supported by Bader charge analysis (Table S2, ESI†). On the contrary, there appears prominent electron localization between the N–N bonds as a signature of strong covalent bonding. Moreover, the non-bonding electrons located around the N atoms form lone electron pairs. Inspired by the unusual N-rich stoichiometry, interesting coordination geometries and exotic oxidation states, the projected electronic bands, PDOS, and Fermi surface of KN₂ are also calculated. KN₂ is one electron formula unit per cell short of the insulator electron count. As expected, KN₂ is a metal with one band (denoted as n = 1, Fig. 3b) crossing the Fermi level, which is in sharp contrast with the non-metallic 2D metal nitrides (e.g., BeN₂;²⁹ PtN₂⁷⁰ and Mg₃N₂⁷¹), but similar to the metallic O-YN₂⁷² and H-ReN₂.⁷³ Below the Fermi level, there is a sizeable energy gap from −0.8 eV to −7.8 eV. Hence, the system can be tuned into an insulator by doping one hole per cell (Fig. S4, ESI†). Additionally, the metallicity of KN₂ is further confirmed by more accurate HSE06 calculations (Fig. S5, ESI†). The coexistence of the “flat band-steep band” feature (e.g., Γ → S; Y → Γ and around Y) emerges around the Fermi level, which induces the high density of states near the Fermi level, forming a prominent van Hove singularity (vHs) (Fig. 3b) that is shown to be favorable for the formation of stable Cooper pairs.⁷⁴ PDOS analysis reveals that its metallicity stems mainly from the contribution of N 2p, which is the typical anion-driven (N₂⁻) metallicity. Further decomposition of PDOS indicates that the contribution of N 2p_z to metallicity is much higher than that of N 2p_x and 2p_y (Fig. 3c), which promotes the in-plane conductivity. The Fermi surface plotted in Fig. 3d shows two electron pockets at the Γ and Y points, ascribable mainly to the N 2p orbitals, and the Fermi velocity has small values on the surface surrounding the Y point. In addition, details of Fermi surface (FS) nesting can be calculated by the nesting function ξ(q):

where ε_kn and ε_k+qm are denoted as the Kohn–Sham energy. The large value of the nesting function ξ(q) indicates that the FS is more likely to nest along this vector. It is obvious that the contribution of the FS nesting is largest around the Γ point, but it represents the entire FS into itself, which has no specific physical meaning. ξ(q) also shows that a region of the FS is nested by Γ → S; S → Y and Y → Γ vectors, demonstrating that the FS nesting may occur along these directions, which may be favorable for the electron–phonon coupling (EPC).

Fig. 3 (a) Electron localization function (ELF) of the KN₂ monolayer. (b) and (c) The projected electronic band structures and projected density of states (PDOS) of the KN₂ monolayer calculated at the PBE level. (d) The Fermi surface corresponds to the one band crossing the Fermi level, color coded by the Fermi velocity. (e) Fermi surface nesting function ξ(q) for the KN₂ monolayer.

Motivated by the strong PDOS peak near the Fermi level, we examined the electron–phonon coupling calculations. KN₂ with three atoms per primitive cell has nine phonon modes: three acoustic and six optical branches. Obviously, as shown in Fig. 4b, there is a large phonon gap (∼1400 cm⁻¹) between mid- and high-frequency optical branches, where the vibration modes of K and N atoms together dominate the mid-frequency optical branches, whereas high-frequency phonon modes are merely characterized by the vibration of N atoms (Fig. 4c). In order to better reveal the relative motions of K and N atoms, the vibration modes of six optical branches at the Γ point are investigated (Fig. 4a). The fourth vibration mode (v = 4) at 75.6 cm⁻¹ is mainly derived from the out-of-plane vibrations of K and N atoms. At 155.6 cm⁻¹, the out-of-plane vibration of N atoms governs the fifth mode (v = 5). On the contrary, the optical branch vibrations (v = 6, 7) are mainly derived from the in-plane vibrations of K and N atoms along the a and b directions, respectively. Modes 8 and 9 (v = 8, 9) along the a and b directions are bending and stretching vibrations between N–N, respectively, which is a direct consequence of the large frequency difference between modes 8 and 9.

Fig. 4 (a) The six optical vibrational modes of KN₂ monolayer at the Γ point in the Brillouin zone (the arrows represent the direction of atomic vibration). (b) The calculated phonon dispersion curves and the size of the circles in the phonon spectra are proportional to partial EPC parameter λ_q,v of the KN₂ monolayer. (c) Phonon density of states (PhDOS) for the KN₂ monolayer. (d) Eliashberg spectral function (green area) and frequency-dependent EPC parameters λ (red line) of the KN₂ monolayer. (e) Normalized quasiparticle superconducting density of states (SDOS) at T = 3 K and (f) energy distribution of superconducting gaps Δ_nkversus T for the KN₂ monolayer.

The magnitude of the EPC λ_q,v can be calculated by:

where γ_q,v is the phonon linewidth, N(E_F) is the electronic density of states at the Fermi level, and ω_qv is the phonon frequency of mode v and wave vector q. The phonon linewidths γ_qv can be estimated by:where g^v_kn,k+qm is the electron–phonon matrix element.

The Eliashberg spectral function for the electron–phonon interaction and the frequency-dependent EPC can be calculated by:

and

In the KN₂ monolayer, the calculated EPC parameter λ is 0.6, which is comparable to 0.61 in MgB₂.⁷⁵ The low- and mid-frequency vibrations below 300 cm⁻¹ give 97% contribution to λ (Fig. 4d), which is different from the high-frequency H-derived vibrations of high-T_c LaH₁₀^5,6 and YH₁₀.⁷⁶ In particular, based on the phonon dispersion curves with λ weights (Fig. 4b) nearby the Y point contribute significantly to the EPC. The strong EPC originates from its low velocity near the Y point. Low Fermi velocity corresponds to the “flat band” in the band structure, which significantly increases the density of electron states and further facilitates EPC. On the other hand, the strong EPC is also ascribed to phonon softening along S → Y and Y → Γ. Subsequently, we have estimated the superconducting critical temperatures T_c of KN₂ using the Allen–Dynes modified McMillan equation on the basis of Bardeen–Cooper—Schrieffer (BCS) theory with a typical choice of μ* = 0.1.^55,77–79 The T_c of KN₂ is predicted to be 2 K at ambient pressure, suggesting that it is a phonon-mediated superconductor. The present T_c value is comparable to 0.86 K for 1T-TaN₂,⁸⁰ 1.3 K for 1H-TaN₂,⁸⁰ and 3.4 K for Ba₂N monolayer.⁸¹

Compared to the traditional McMillan approximation, solving the Migdal–Eliashberg equation is more reliable for layered or low-dimensional systems.⁸² We also employ anisotropic Migdal–Eliashberg equations with subsequent analytic continuation to the real axis using Padé functions to explore the superconduction of the KN₂ monolayer. The calculated the normalized quasiparticle superconducting density of states at 3 K, which is given in Fig. 4e, showing only a pair of peaks corresponding to a single superconducting gap. This is in sharp contrast with multigap superconductors like MgB₂ solid^83,84 and LiBC⁸⁵ monolayer. The anisotropic superconducting gap vanishes at about 4.3 K (Fig. 4f), which is slightly higher than the value from the McMillan–Allen–Dynes formula (2 K). Similar situations occurred in other systems,^86,87 indicating that our results on the superconductivity of KN₂ monolayer are reasonable. KN₄ monolayer is metallic rather than superconducting, probably because the N₂ dimer in KN₄ (N₂^0.5−) is less charged than that in KN₂ (N₂⁻). Furthermore, the high electronic density of states leads to the emergence of unique vHs near the Fermi level in the KN₂ monolayer, which is conducive to the formation of stable Cooper pairs, playing a crucial role in enhancing the EPC.

Conclusions

To summarize, we uncover two stable non-stoichiometric 2D materials, KN₂ and KN₄ monolayers, using a combination of swarm-intelligence structural search and ab initio calculation. Their negative formation energies, high cohesive energies and dynamical stabilities suggest the synthetic feasibility. As the nitrogen content increases, the ligand N atom increases from 6 to 12. Meanwhile, the nitrogen motifs evolve from N₂⁻ to N₂^0.5−. The unique coordination geometries and exotic oxidation states induce distinct electronic properties. KN₄ is metallic, while KN₂ exhibits superconductivity with a T_c value of 4.3 K. The N 2p electrons and low- and mid-frequency atomic vibrations of K and N atoms are mainly responsible for the strong electron–phonon coupling. These findings not only deepen the understanding of the hypercoordination and oxidation state of nitrogen, but also promote future efforts in exploring the superconductivity of 2D metal nitrides.

Conflicts of interest

The authors declare no competing financial interest.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (12274050 and 91961204) and the Fundamental Research Funds for the Central Universities (DUT22LAB104 and DUT22ZD103). The authors also acknowledge the Supercomputing Center of the Dalian University of Technology and Tianjin for providing the computing resources.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d3tc03718e

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