Semiconducting character of LaN: magnitude of the band gap, and origin of the electrical conductivity

Lanthanum nitride (LaN) has attracted research interest in catalysis due to its ability to activate the triple bonds of N$_2$ molecules, enabling efficient and cost-effective synthesis of ammonia from N$_2$ gas. While exciting progress has been made to use LaN in functional applications, the electronic character of LaN (metallic, semi-metallic, or semiconducting) and magnitude of its band gap have so far not been conclusively determined. Here, we investigate the electronic properties of LaN with hybrid density functional theory calculations. In contrast to previous claims that LaN is semi-metallic, our calculations show that LaN is a direct-band-gap semiconductor with a band-gap value of 0.62 eV at the X point of the Brillouin zone. The dispersive character of the bands near the band edges leads to light electron and hole effective masses, making LaN promising for electronic and optoelectronic applications. Our calculations also reveal that nitrogen vacancies and substitutional oxygen atoms are two unintentional shallow donors with low formation energies that can explain the origin of the previously reported electrical conductivity. Our calculations clarify the semiconducting nature of LaN and reveal candidate unintentional point defects that are likely responsible for its measured electrical conductivity.

performance that is comparable to ruthenium-based catalysts but at a much lower cost. 4 Thus, LaN is a promising nitride material for catalysis and other chemical applications.
While LaN exhibits great promise in functional applications, one fundamental question that has not been fully addressed is whether LaN has metallic, semi-metallic, or semiconducting nature.
Understanding this fundamental electronic character of LaN is critical for its future applications.
Many previous theoretical calculations have attempted to elucidate the electronic band structure of LaN. Early density functional theory (DFT) calculations with the augmented plane wave method (APW) showed that the conduction and valence band of LaN overlap by up to 40 mRy, indicating a semi-metallic nature. 5, 6 Vaitheeswaran et al studied the electronic properties of LaN using tightbinding linear muffin-tin orbitals with the local-density approximation (LDA) to the exchangecorrelation functional and found a metallic nature for LaN. They also estimated the superconducting transition temperature to be 0.65 K. 7 Later calculations with the generalized gradient approximation (GGA) functional observed the overlap between valence and conduction band in the band structure and characterized LaN either as metallic or semi-metallic. 8,9 However, these results contradict to the calculations using the hybrid screened-exchange local density approximation (sX-LDA) functional where an indirect band gap of 0.75 eV was found, suggesting that LaN might be a semiconductor. 10 Recently, more calculations seem to support the semiconducting nature of LaN. Gupta et al used LDA functional and found an indirect band gap of 0.5 eV. 11 GGA+U SIC functional was employed by Meenaatci et al and an indirect band gap of 0.65 eV was found. 12 However, by using the LSDA+U functional, Larson et al discovered a small direct band gap of 0.4 eV for LaN. 13 Similarly, a direct band gap of 0.6 eV was recently obtained by MBJLDA functional for both wurtzite and rocksalt LaN. 14,15 In addition, Sreeparvathy et al found a direct band gap of 0.814 eV using full potential linearized augmented plane wave (FP-This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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LAPW) method with TB-mBJ functional, 16 which seems to agree with the experimental band gap of 0.82 eV measured from optical absorption. 17 Despite the progress from previous theoretical studies, the nature (direct or indirect) and the magnitude of the band gap are not conclusively determined for LaN due to different methods and functionals employed in the calculations, which necessitates a re-investigation of the electronic properties of LaN with modern electronic structure calculations.
Since LaN is found by the more advanced functionals to exhibit a band gap, and thus to present a semiconducting rather than a semi-metallic character, its electrical conductivity must originate from intrinsic or unintentional dopants. However, to the best of our knowledge, there is no theoretical investigation into the thermodynamics of the intrinsic defects and common impurities of LaN, which is the key to understand the origin of its electrical conductivity. This knowledge is also necessary in order to rationally tune its conductivity by controlling the defect formation and doping in experiments. Thus, theoretical insights into the intrinsic defect formation and ionization energies are crucial to enable the adoption of LaN in wider electronic and catalysis applications.
In this work, we study the electronic properties (band structure, effective masses, dielectric constants, etc.) and defect thermodynamics of LaN using first-principles calculations based on DFT with the HSE06 hybrid functional, 18,19 which predicts accurate electronic properties for a wide range of materials. We find that LaN is a direct-band-gap semiconductor with a band gap of 0.62 eV at the X point of the Brillouin zone. Our defect calculations attribute the origin of its electrical conductivity to the unintentional formation of N vacancies or substitutional O impurities.
Our calculations clarify the semiconducting nature of LaN and reveal candidate defects that are the likely origin of its measured electrical conductivity.
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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DFT calculations were performed using the Vienna ab-initio Simulation Package (VASP). 20 GW-compatible Perdew-Burke-Ernzerhof (PBE) pseudopotentials 21 for La and N and a planewave energy cutoff of 500 eV were employed in all calculations. The La 5s 2 5p 6 5d 1 6s 2 and N 2s 2 2p 3 were treated as valence electrons. In order to get accurate electronic properties, we used the HSE06 hybrid exchange-correlation functional with a standard mixing parameter of 0.25. 18,19 The electronic band structures of LaN were calculated with a fully relaxed 2-atom rocksalt primitive cell (Space group Fm-3m) and a Γ-center 8×8×8 Brillouin zone sampling grid. 22 The special kpoint path for plotting the band structure followed the convention of Setyawan and Curtarolo. 23 Spin-orbit coupling effects were included in the band structure. The static and high-frequency dielectric constants were calculated by the self-consistent response to finite electric field at the HSE level using modern theory of polarization. [24][25][26][27] Electron and hole effective masses were extracted by fitting the HSE band structure with the hyperbolic equation, where 0 is the energy of the band extremum, * is the effective mass, and is a fitting parameter to characterize the non-parabolicity of the band. The band alignment of LaN was obtained by aligning the bulk average electrostatic potential to the vacuum level. This was done by performing HSE calculations for the LaN (100) and (110) slabs without surface relaxation. The alignment results for the two slabs differ by only 70 meV. Defect calculations 28 were performed with 2×2×2 supercells built from the 8-atom rocksalt unit cell (i.e., 64 atoms) and the Brillouin zone was sampled with a Γ-center 2×2×2 grid. The defect formation energies and charge-transition levels for the ON defect are found to change by less than 0.05 eV for a larger supercell with 128 atoms (Table S1). Thus, the 64-atom supercell was employed for all subsequent defect studies to reduce the computational cost. All defect supercells were relaxed with HSE06 by allowing ion This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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displacements until the forces on the ions were less than 0.02 eV Å -1 . Spin polarization was included in calculations with unpaired electrons. We employed the scheme of Freysoldt et al. 29 and our calculated static dielectric constant for LaN to correct the artificial periodic charged-defect interactions. The competing phases we considered in the thermodynamic analysis of defect formation are La2O3, NO2, NH3, and all stable elemental phases.
We first investigate the band structure of LaN to determine its electronic character (metal or semiconductor) and the magnitude of its band gap, if one exists. Previous DFT calculations using the GGA functional characterized LaN as a semimetal. 8,9 Indeed, our own PBE bandstructure calculations ( Figure S1) reveal that the conduction and valence bands overlap by 115 meV at the X point, which leads to the conclusion that LaN is a semimetal and agrees with many previous theoretical studies. However, calculations with semilocal exchange-correlation functionals such as LDA or PBE severely underestimate the band gaps of materials and may erroneously lead to a closed gap in LaN. Moreover, we find that even after one-shot GW corrections on top of PBE, the band gap is still extremely small (0.05 eV from evidence that LaN is a semiconductor rather than a semimetal. We therefore attribute the miscategorization of LaN as a semimetal in previous studies to the systematic band-gap underestimation problem of LDA and GGA. 0.5 indirect Γ-X GGA+U 12 0.65 indirect Γ-X LSDA+U 13 0.4 direct X-X MBJLDA 14,15 0.6 direct X-X FP-LAPW, TB-mBJ 16 0.814 direct X-X This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.  The valence band of LaN is mainly derived from N 2p orbitals and the conduction band is from the spatially extended unoccupied La 5d orbitals, as can be seen from the orbital projected band structure in Figure S2. Spin-orbit coupling has a profound effect on the structure of the valence band, particularly at the high symmetry points (Γ, X, and W). This arises from the contribution of La p orbitals near those points ( Figure S2). After including spin-orbit effect, the band gap decreases from 0.75 eV to 0.62 eV. The 3-fold degenerate N p orbitals at Γ split into 2-fold degenerate p3/2

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band and 1-fold p1/2 band. Similarly, the 2-fold degeneracy between the second and third valence band at the X point and between the first and second valence bands at the W point is broken, with energy splittings of 340 meV and 301 meV, respectively. Table 2 summarizes the structural and electronic properties of LaN. We obtain a lattice constant of 5.282 Å, which is in excellent agreement with the experimental value of 5.284 Å. 30 The conduction band of LaN is dispersive with a bandwidth of several eV, leading to small electron masses especially along the transverse X-W and X-U directions. The hole effective masses are comparable to or even lighter than the hole effective masses of GaN. 31 The energy splitting between the two topmost valence bands is 85 meV, which is larger than the thermal energy at room temperature of 26 meV. This indicates a much weaker phonon-mediated hole scattering from the valence band maximum to the second-highest valence band, as in the case of strained BAs. 32 The reduced scattering rate coupled with light effective masses may lead to a high hole mobility in LaN, although no p-type conductivity has been observed experimentally so far.  We then turn to the origin of the electrical conduction in LaN that is observed in experiment. [35][36][37] Previous studies attributed the conduction to the semi-metallic character of LaN.
However, this explanation is inconsistent with our accurate band-structure results that find LaN to be a semiconductor with a band gap. An alternative explanation is that the electrical conduction in LaN is caused by unintentional doping either by intrinsic defects or by unintentional impurities incorporated during growth. To shed light on this issue, we calculate the formation energy of intrinsic defects (lanthanum vacancies VLa, nitrogen vacancies VN) and common unintentional impurities (substitutional O and H interstitials). Figure 3 shows their formation energy as a function of the Fermi energy (referenced to the VBM) and growth conditions (N-rich or N-poor).
Our key finding is that the donor-like defects (VN and ON) have significantly lower formation energies (even negative) than acceptor-like defects, which identify LaN to be an intrinsically ntype semiconductor. Nitrogen vacancies act as negative-U double donors and are stable in the +2 charge state for Fermi levels throughout the majority of the band gap. O impurities are singly charged shallow donors, with an ionization energy of 50 meV. Therefore, both of these donor-like defects are candidate origins of the measured conductivity.
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. which the conductivity decreases with temperature due to increased carrier scattering by phonons.

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We further extract the donor activation energy by fitting the Arrhenius equation This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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to the experimental electrical-conductivity measurements by Lesunova et al, 36 shown in Figure 4.
The fitted value for the activation energy is 40 meV, which is in excellent agreement with our calculated ionization energy for shallow donors (39 meV) evaluated with the Bohr model This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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In our calculation, ON has a negative formation energy under both N-rich and N-poor conditions. However, this should not be interpreted as a sign of instability for LaN. In fact, the synthesis chemistry of LaN is well understood, and LaN has been utilized in catalysts for NH3 synthesis 4 and for electrodes in supercapacitors. 37