Discovery of intrinsic two-dimensional antiferromagnets from transition-metal borides

Shiyao Wang a, Nanxi Miao a, Kehe Su b, Vladislav A. Blatov c and Junjie Wang *a
aState Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, People's Republic of China. E-mail: wang.junjie@nwpu.edu.cn
bSchool of Chemistry and Chemical Engineering, Nowthwestern Polytechnical University, Xi'an, Shaanxi 710072, People's Republic of China
cSamara Center for Theoretical Materials Science (SCTMS), Samara State Technical University, Molodogvardeyskaya St. 244, Samara, 443100, Russia

Received 18th February 2021 , Accepted 19th March 2021

First published on 20th March 2021


Abstract

Intrinsic two-dimensional (2D) magnets are promising materials for developing advanced spintronic devices. A few have already been synthesized from the exfoliation of van der Waals magnetic materials. In this work, by using ab initio calculations and Monte Carlo simulation, a series of 2D MBs (M = Cr, Mn or Fe; B = boron) are predicted possessing robust magnetism, sizeable magnetic anisotropy energy, and excellent structural stability. These 2D MBs can be respectively synthesized from non-van der Waals compounds with low separation energies such as Cr2AlB2, Mn2AlB2, and Fe2AlB2. 2D CrB is a ferromagnetic (FM) metal with a weak in-plane magnetic anisotropy energy of 23.6 μeV per atom. Metallic 2D MnB and FeB are Ising antiferromagnets with an out-of-plane magnetic easy axis and robust magnetic anisotropy energies up to 222.7 and 482.2 μeV per atom, respectively. By using Monte Carlo simulation, the critical temperatures of 2D CrB, MnB, and FeB were calculated to be 440 K, 300 K, and 320 K, respectively. Our study found that the super-exchange interaction plays the dominant role in determining the long-range magnetic ordering of 2D MBs. Moreover, most functionalized 2D MBTs (T = O, OH or F) are predicted to have AFM ground states. Alternating transition metals or functional groups can significantly modulate the magnetic ground state and critical temperature of 2D MBTs. This study suggests that the 2D MBs and MBTs are promising metallic 2D magnets for spintronic applications.


Introduction

The electron spin degree of freedom in two-dimensional (2D) magnetic materials is extremely attractive for the development of next-generation solid-state spintronic devices with low energy consumption and high transmission speed desirable for data storage and magnetic sensing.1–3 However, according to the Mermin–Wagner theorem, an intrinsic 2D magnet could hardly exist because no long-range magnetic order is possible in an isotropic 2D magnet due to thermal fluctuations.4 Very recently, the antiferromagnetism of an FePS3 monolayer (Néel temperature TN = ∼118 K)5 has been experimentally discovered that indicates the beginning of a new era of 2D magnetic materials. It has been shown that thermal fluctuations can be effectively shielded by the magnetic anisotropy energy (MAE) resulting from spin–orbit coupling (SOC).6 Since the above discovery, several materials with intrinsic ferromagnetic (FM) or antiferromagnetic (AFM) order have been realized in the 2D limit, e.g., monolayer CrI3 (Curie temperature TC = ∼45 K),7 bilayer Cr2Ge2Te6 (TC = ∼30 K),8 monolayer Fe3GeTe2 (TC = ∼130 K),9 and bilayer NiPS3 (TN = ∼130 K),10 mostly obtained from peeling their corresponding van der Waals (vdW) compounds. However, most of the above-mentioned 2D intrinsic magnets have a low critical magnetic transition temperature and are unstable in air, which greatly impedes their practical applications.11 Therefore, the search for intrinsically magnetic 2D materials possessing sizeable MAE, high critical temperature, and large magnetic moment, as well as good stability is highly demanded.

The chemical exfoliation of MAX phases (M = transition metals; A = p-block elements; and X = carbon or nitrogen) into 2D transition metal carbides, nitrides, or carbonitrides (named MXenes)12,13 with the composition of Mn+1XnT2 (T = O, OH, and F; n = 1, 2, 3) has opened a door for obtaining novel 2D materials from non-van der Waals compounds with low separation energies. Indeed, it was theoretically found that several bare and surface-terminated MXenes are intrinsically magnetic, e.g., Cr2C,14 Mn2C,15 and Mn2NT2 (T = O, OH, or F).16,17 More recently, the above chemical exfoliation technique has been applied to MAB phases (M = transition metals; A = p-block elements; B = boron) and novel 2D MBs were derived.18–23 The above experiments have triggered many efforts to synthesize novel bulk structures and examine their exfoliation possibilities into 2D structures. For instance, recently, Ti2InB2,18 a layered ternary boride that possesses the combination of the chemical composition of MAB phases and the hexagonal symmetry of MAX phases has been realized and exfoliated into 2D TiB. This study shows that there is still plenty of room in the search for new transition metal-based layered structures and derived 2D structures.22

Among known MAB phases,24 M2AB2 (M = Cr, Mn, and Fe; A = aluminium; and B = boron) crystallize in an orthorhombic Cmmm structure possessing intrinsic magnetism. They are considered as potential rare-earth free materials for magnetocaloric and soft magnetic applications.25–28 The very recent synthesis of 2D MoB29,30 and CrB31 from the corresponding M2AlB2 compounds has further ignited extensive interest for investigating 2D MBs.32,33

In this work, we have systematically investigated the magnetic and electronic properties of 2D MBs by performing high-throughput spin-polarized ab initio calculations. We have found five intrinsic 2D magnets (CrB, MnB, FeB, CoB, and RuB) with good structural stability. Using non-collinear magnetic calculations with spin–orbit coupling (SOC), we predicted that 2D CrB exhibits ferromagnetism with an in-plane magnetic easy axis, while 2D MnB and FeB are Ising-type antiferromagnets with out-of-plane magnetic easy axes. The calculated magnetic anisotropy energies of 2D CrB, MnB, and FeB are 23.6, 222.7, and 482.2 μeV per atom, respectively. Moreover, we predicted the magnetic ground states of 2D CrB, MnB, and FeB using Monte Carlo simulations and found that the corresponding critical temperatures can be as high as 440 K, 300 K, and 320 K, respectively. Our calculations further revealed that most of the functionalized 2D MBT (M = Cr, Mn, and Fe; B = boron; and T = O, OH, and F) structures show AFM ground states. The critical temperature of 2D MBTs can be effectively modulated by both transition metals and functional groups. Therefore, it is highly expected that 2D MBs and MBTs are promising magnetic materials for building practical spintronic devices.

Calculation details

Spin-polarized density functional theory (DFT) calculations as implemented in the Vienna ab initio simulation package (VASP) were performed.34 The projector augmented wave (PAW) pseudopotentials were applied to describe the interactions between valence electrons and ionic cores.35 A plane-wave basis set with a cutoff of 520 eV was employed to expand the wave functions. The Perdew–Burke–Ernzerhof (PBE) form of the generalized gradient approximation (GGA) was used to describe electronic exchange and correlation.36 To describe the strong-correlation effect in transition-metal atoms, on-site Coulomb interactions were introduced using the Dudarev's approach PBE+U (Ueff = U − J).37 The Ueff value of Cr, Mn, and Fe atoms is 2.0 eV because previous studies have indicated that a small Ueff value is adequate to obtain the appropriate magnetic solution of metallic transition metal systems.16,17,38 The magnetic ground states of 2D MBs (M = Cr, Mn, and Fe) were further confirmed by using the hybrid functional HSE06.39 The impact of Ueff on the magnetic ground states of 2D FeB, MnB, and CrB was also studied by changing the Ueff value from 1.0 eV to 5.0 eV and the validity of Ueff = 2.0 eV was confirmed. An 18 Å vacuum region along the c-direction was set to avoid the interaction between neighboring atomic layers. The Γ-centered k-point sampling of 12 × 12 × 1 was performed using the Monkhorst–Pack method.40 Both lattice constants and atomic positions of 2D MBs were optimized by using the conjugate gradient algorithm until the maximum force on a single atom is less than 0.01 eV Å−1. The geometry relaxations of 2D structures were performed with PBE and the magnetic and electronic properties were calculated with PBE+U and HSE06.

Phonon dispersion calculations were performed to confirm the dynamic stabilities of 2D MBs using the Phonopy code41 in the framework of density functional perturbation theory (DFPT).42 A 4 × 4 × 1 supercell of 2D MB was employed to get a converged result of phonon calculations. Born-Oppenheimer ab initio molecular dynamics (AIMD) simulations were performed to evaluate the thermal stability of 2D MBs based on a 4 × 4 × 1 supercell at 600 K for 5 ps with a time step of 1 fs, and the algorithm of Nosé was applied to control the temperature.43

Fig. S1 of the ESI shows that a series of magnetic configurations of 2D MBs including one FM and 10 diverse AFM spin states were chosen from a total of 70 possible configurations based on the symmetry analysis of a 2 × 2 × 1 MB supercell using the Supercell package.44 The spin–spin correlation and critical temperature of the studied 2D structures were evaluated by employing the EspinS package45 in the frame of Monte Carlo (MC) simulations. A 50 × 50 × 1 lattice was adopted in the MC simulations and each spin can rotate randomly in all directions. To ensure calculation accuracy, 1 × 106 MC steps per spin for equilibration and 1 × 106 MC steps for sampling were done. To reduce the correlation between the data, five MC steps of the data collection were skipped. The specific heat CV was calculated at a given temperature when the system reaches its equilibrium state. The critical temperature can be directly extracted from the peak of the thermodynamic specific heat CV plot.46

Results and discussion

Structural model and stabilities

The workflow for the discovery of 2D magnetic MBs is schematically shown in Fig. 1. A set of M atoms including the elements of the third-row (3d) (M = Sc, Ti, V, Cr, Mn, Fe, Co, and Ni), the fourth-row (4d) (M = Y, Zr, Nb, Mo, Tc, Ru, Rh, and Pd), and the fifth-row (5d) (M = Hf, Ta, W, Re, Os, Ir, and Pt) of transition metals were adopted to construct MAB phases. Previous studies have proved the dynamic stability of many M2AlB2 compounds (M = Sc, Ti, Zr, Hf, V, Cr, Mo, W, Mn, Tc, Fe, Rh, and Ni).19 It was suggested that 2D MBs can be obtained by the exfoliation of M2AlB2 phases31 because the calculated exfoliation energies (Eexf) of M2AlB2 are comparable with those of reported MAX phases (0.086–0.205 eV Å−2).47 All 2D MBs in the present work show an orthorhombic symmetry Pmma (no. 51). Fig. S2 of the ESI shows that the electron localization function (ELF) of B zigzag chains in 2D MBs is close to 1.0, corresponding to strong covalent B–B bonds. The ELF = 0.0 of the region around M atoms (M = Cr, Mn, and Fe) in MBs indicates their electron deficiency. Eight obtained 2D MBs (M = V, Cr, Mn, Fe, Co, Tc, Ru, and Os) show a calculated magnetic moment over 1.0μB per formula unit (μB per f.u.) and three of them (M = Cr, Mn, and Fe) have a large magnetic moment over 2.0μB per f.u., which can be regarded as possible candidates for robust 2D magnets (Table S1, ESI). The detailed structural parameters for those eight 2D MBs are listed in Table S1 of the ESI.
image file: d1nr01103k-f1.tif
Fig. 1 Computational workflow for the discovery of 2D magnetic transition-metal borides (2D MBs). All MAB phases were produced by substituting 23 M atoms into known MAB phases. Collinear energy calculations were used to determine the ground magnetic coupling constants of 2D MBs. Non-collinear energy calculations were used to obtain the magnetic anisotropy energy of 2D MBs. The critical transition temperature of 2D MBs was computed by Monte Carlo simulation.

From the preliminary screening procedure shown in Fig. 1, the stabilities of the above-mentioned eight 2D MBs were evaluated by performing a set of calculations including phonon dispersion, molecular dynamics, elastic constant, and cohesive energy using GGA. The phonon dispersions of 2D MBs at 0 K were calculated to determine their dynamic stability, Fig. S3 of the ESI. No imaginary vibrational frequencies in the phonon spectra of five of the eight 2D MBs (M = Cr, Mn, Fe, Co, and Ru) can be found. Furthermore, the thermal stabilities of these five 2D MBs at 600 K were evaluated by performing AIMD simulations. The final snapshots for the above five MBs (Fig. S4, ESI) show that no evident structure disruption of 2D MBs was observed at 600 K. It is suggested that the good thermal stability of 2D MBs is possible owing to the strong B–B covalent bonds as shown in Fig. S2 of the ESI. In addition, the mechanical stabilities of the above 2D MBs were confirmed by calculating the corresponding elastic constants. Generally, a mechanically stable 2D structure must meet the criteria of C11 > 0, C11·C22 > C122, C44 > 0.48 The elastic constants of 2D MBs shown in Table S2 of the ESI satisfy the above criteria suggesting their mechanical stability. The cohesive energies of those 2D structures that are defined as Ecoh = (2EM + 2EBEMB)/4 were also calculated. EMB, EM, and EB are the total energies of the 2D MB, single transition-metal atom, and single boron atom, respectively. The Ecoh values of the five 2D MBs are in the range from 4.72 to 6.03 eV per atom (Table S2, ESI), comparable with known 2D materials, e.g., MoS2 (5.02 eV per atom)49 and silicene (3.98 eV per atom).50 This result also demonstrates the high thermodynamic stability of the proposed 2D MBs (M = Cr, Mn, Fe, Co, and Ru). The topological analysis51 of the MB layers performed using the ToposPro program package52 showed the equivalence of the M–M, M–B, and B–B bond systems at all MBs. Moreover, the modeled MB layers are topologically equivalent to the layers in the MAB phases and in bulk tetragonal and orthorhombic MoB and WB.53

Magnetic properties

Preliminary stability and magnetic screening show that 2D CrB, MnB, and FeB are not only dynamically stable but also exhibit significant magnetic moments (>2.0μB per f.u. as shown in Table S1 of the ESI). Therefore, the magnetic properties of these three 2D MBs (M = Cr, Mn, and Fe) were investigated in detail. One collinear FM and ten collinear AFM configurations of a 2D MB (Fig. S1, ESI) were considered to determine the magnetic ground state of each structure. By comparing the energy differences of different spin configurations derived from the GGA+U (Ueff = 2.0 eV) calculations (Table S3, ESI), we demonstrated that the magnetic ground states of 2D CrB, MnB, and FeB are FM, AFM-5, and AFM-5, respectively. Furthermore, these magnetic ground states were confirmed by HSE06 calculations (Table S3, ESI). We also studied the impact of Ueff on the magnetic ground state of 2D CrB, MnB, and FeB by changing the Ueff value from 1.0 eV to 5.0 eV (Table S4, ESI). It is proved that the magnetic ground state remains unchanged when the Ueff values are in the range from 1.0 to 3.0 eV, 2.0 to 5.0 eV, and 1.0 to 3.0 eV for CrB, MnB, and FeB, respectively. These calculations can confirm the validity of Ueff = 2.0 eV for describing the electronic and magnetic properties of 2D CrB, MnB, and FeB.

The Heisenberg Hamiltonian54 of a 2D MB considering five neighbor interactions can be given by the following equation:

 
image file: d1nr01103k-t1.tif(1)
where J1, J2, J3, J4, and J5 are the nearest- (1st), second-nearest- (2nd), third-nearest- (3rd), fourth-nearest- (4th), and fifth-nearest-neighbor (5th) magnetic coupling parameters, respectively (marked by arrows in Fig. S1). The positive and negative values of J refer to the ferromagnetic and antiferromagnetic coupling effect, respectively. S is the spin vector of each M atom; A is the magnetic anisotropy energy parameter; and Sei is the component of the spin vector along the magnetic easy axis. The parameters J1, J2, J3, J4, and J5 can be obtained through mapping the total energies of the FM and collinear AFM-i (i = 0, 1, …, 9) of 2D MB into the Hamiltonian and employing the least-squares method.55 The calculation details and the values of J for 2D CrB, MnB, and FeB obtained from the GGA+U (Ueff = 2.0 eV) are shown in Table S5 of the ESI.

Our calculations show that GGA+U provides reasonable results to determine the magnetic ground state of 2D MBs and is much less time-consuming than HSE06. Therefore, GGA+U was employed in the non-collinear magnetic calculations with spin–orbit coupling (SOC) to determine the magnetic anisotropy energy (MAE) of 2D MBs. The MAE can reflect the difficulty of spin flipping and play a crucial role in the stability of the ordered spin arrangement of 2D materials,6 which is defined as the energy difference between the system with the spin direction along the magnetic hard axis and the system with spin parallel to the magnetic easy axis. Note that the direction of the easy axis (or the hard axis) of magnetization corresponds to the lowest (or the highest) energy of the system. Herein, we plotted the energy E(θ) of 2D CrB, MnB, and FeB as the spin vector of M atom S(θ) rotates with angle θ from 0° to 180° with intervals of 15° through ac and bc planes as shown in Fig. 2a–c. As seen in Fig. 2a, the magnetic hard axis and easy axis of 2D CrB appear at θ = 90° in the ac plane and θ = 90° in the bc plane, respectively, indicating that the MAE of 2D CrB is 23.6 μeV per atom and the magnetic easy axis of 2D CrB is the b (010) axis. Fig. 2b and c show that the energy of 2D MnB and FeB exhibit a strong dependence on the polar θ in the ac and bc planes. The energy of MnB (or FeB) reaches the maximum at θ = 90° in the ac plane, corresponding to the a (100) axis as the magnetic hard axis, while the energy of MnB (or FeB) is the lowest at θ = 0° in the ac plane or the bc plane, corresponding to the c (001) axis as the magnetic easy axis. Therefore, the MAE of 2D MnB (or FeB) is 222.7 (or 482.2) μeV per atom as shown in Fig. 2b and c. The magnetic easy axis of 2D MnB (or FeB) is out-of-plane which indicates that both MnB and FeB are 2D intrinsic Ising magnets. Moreover, the MAE values of 2D MnB and FeB are significantly larger than that of the Fe monolayer deposited on the Rh (111) (∼80 μeV per f.u.) substrate, the Co ultrathin films deposited on the Pt (111) (∼100 μeV per f.u.) substrate,56 and 2D Ising magnet MXene, e.g., Mn2NO2 (63 μeV per atom) and Cr2NO2 (22 μeV per atom).16 As a result, it is promising that the magnetic anisotropy energy of 2D MnB and FeB is robust for enabling high-density storage and quantum spin processing.57


image file: d1nr01103k-f2.tif
Fig. 2 Angular dependence of the calculated relative energy E (μeV per atom) of 2D (a) CrB, (b) MnB, and (c) FeB. (d) The spin vector S is rotated with the polar angle θ through the ac and bc planes, respectively.

Since 2D MBs (M = Cr, Mn, and Fe) possess a uniaxial tetragonal symmetry, the angular dependence of the E(θ) can be described by the following equation:58,59

 
E(θ) = K1[thin space (1/6-em)]sin2[thin space (1/6-em)]θ + K2[thin space (1/6-em)]sin4[thin space (1/6-em)]θ(2)
where K1 and K2 are the anisotropy coefficients and θ is the polar angle relative to the c axis in the ac or bc plane (Fig. 2d). Generally, K1 > 0 suggests that the magnetic easy axis will be along an out-of-plane direction (c axis), whereas K1 < 0 means that the preferred magnetic axis will be perpendicular to the c axis. By fitting the angular dependence of energy using eqn (2), the K1 and K2 values of 2D MnB (or FeB) in the ac plane are 222.7 μeV (or 482.2 μeV) and −0.1 μeV (or 3.2 μeV), respectively, while the K1 and K2 values of 2D MnB (or FeB) in the bc plane are 187.9 μeV (or 230.1 μeV) and −0.3 μeV (or −0.1 μeV), respectively. Clearly, the K1 values of 2D MnB and FeB are positive and predominant, agreeing with the fact that they both possess an out-of-plane magnetic easy axis. Furthermore, the negative K1 value (−6.5 μeV) of 2D CrB in the bc plane indicates that the preferred magnetic axis is perpendicular to the c axis (Table S6, ESI).

To confirm the magnetic ground state of 2D CrB, MnB, and FeB, the spin–spin correlation was obtained by using Monte Carlo (MC) simulation45 at a low temperature (T = 5 K). Fig. 3a–c show the average value of the products of the neighboring spins sum(Si·Sj)/N and their absolute values sum(|Si·Sj|)/N for the spin Hamiltonian given by the coupling constants using GGA+U (Ueff = 2.0 eV). N is the size of lattice in the MC simulation. The calculated values 1 and −1 of sum(Si·Sj)/N indicate that the 2D MB is FM and AFM coupling, respectively. Fig. 3a shows that the calculated sum(Si·Sj)/N for the 1st, 2nd, 3rd, 4th, and 5th nearest neighboring spins is 1 for 2D CrB, referring to the FM coupling. Fig. 3b and c show that the calculated sum(Si·Sj)/N for 2D MnB and FeB is 1 for the 1st and 2nd nearest neighboring spins and becomes −1 for the 3rd, 4th, and 5th nearest neighboring spins, which is consistent with the fact that AFM-5 is the ground state configuration of these two structures. For all structures, the calculated value of sum(|Si·Sj|)/N for the 1st, 2nd, 3rd, 4th, and 5th spins is 1, which means that the direction of one spin is parallel or antiparallel to the direction of its neighboring spins. MC simulations confirm that the FM of 2D CrB and the collinear AFM-5 of 2D MnB and FeB were captured at low temperature, further suggesting that the FM, AFM-5, and AFM-5 are the magnetic ground state for 2D CrB, MnB, and FeB, respectively. Note that the 2D MnB was suggested to be ferromagnetic in a previous study because they missed the AFM-5 configuration with the lowest energy.21


image file: d1nr01103k-f3.tif
Fig. 3 Calculated spin–spin correlations of 1st, 2nd, 3rd, 4th, and 5th neighboring magnetic atoms by Monte Carlo simulation at T = 5 K for 2D (a) CrB, (b) MnB, and (c) FeB. Calculated specific heat CV as functions of temperature T for 2D (d) CrB, (e) MnB, and (f) FeB.

Moreover, we studied the critical temperature of 2D CrB, MnB, and FeB by performing MC simulation. The Curie (TC) or Néel (TN) temperature corresponds to the ferromagnetic or antiferromagnetic to paramagnetic phase transition. As shown in Fig. 3d–f, the calculated critical temperatures of 2D CrB, MnB, and FeB are 440 K, 300 K, and 320 K, respectively. Note that although the magnetic anisotropy of 2D CrB is not very significant (23.6 μeV per atom), the large difference in the energy between the FM and AFM states can increase the TC beyond the room temperature.60 By the use of the same approach, our calculated TC for the monolayer CrI3 is around 50 K (Fig. S5, ESI), which is close to the theoretical value of 51 K (ref. 61) and the experimental measurement of 45 K.7 Therefore, our calculated high critical temperatures of 2D CrB, MnB, and FeB are reliable, which make these 2D structures promising for enabling spintronic devices at room temperature (Fig. 1).

The origin of magnetism

Since the magnetic properties of transition-metal compounds are highly correlated with the valence states of transition metal, we performed Bader charge analysis62 for 2D MBs (M = Cr, Mn, and Fe; B = boron) with the most stable magnetic configuration to determine the valence states of Cr, Mn, and Fe by accounting for charge transfer between transition-metal and boron atoms. It is demonstrated that the Cr, Mn, and Fe atoms in 2D MBs transfer 0.8|e|, 0.7|e|, and 0.4|e|, respectively, to the B atom. We performed Bader charge analysis for CrS2 and found that the charge transfer from Cr4+ to S2− is −1.20|e|. We also noted that the valence states of Mn and Fe are +3 and +1 in the recently predicted 2D MnAs59 and Fe3P63 when the charge transfer from Mn3+ to As3− and Fe1+ to P3− is −0.88|e| and −0.28|e|, respectively. The valence states of Fe and B are regarded as +3 and −1, respectively, when the charge transfer from Fe3+ to B1− is −0.87|e| in 2D FeB3.64 Considering the Bader charge analysis normally underestimates the number of charge transfer65 and the electronegativity of S (2.5), P (2.1), and As (2.0) atoms being close to that of the B (2.0) atom, we suggested that the valence states of Cr, Mn, and Fe in 2D MBs can be possibly +2, +2, and +1, respectively, by comparing the results of charge transfer among 2D MBs (M = Cr, Mn, and Fe) with the above-mentioned work.59,63,64 Moreover, the valence electronic configurations of freestanding Cr, Mn, and Fe atoms are [Ar] [3d5, 4s1], [Ar] [3d6, 4s1], and [Ar] [3d7, 4s1], respectively, the remaining four, five, and seven d-electrons of M atoms would fill into d-orbitals when the Cr, Mn, and Fe atoms of 2D MB transfer two, two, and one electron, respectively, to neighboring boron atoms. For 2D Pmma-MB, five d orbitals of the M atom have nondegenerate energies, originating from the B atom induced asymmetric octahedral crystal field around M atoms.21 Following Hund's rules and the Pauli exclusion principle, the theoretical magnetic moment per transition-metal atom of 2D CrB, MnB, and FeB can simply be predicted to be 4.0, 5.0, and 3.0μB, respectively, which are close to the HSE06 calculations (3.20μB, 3.50μB, and 2.75μB for Cr, Mn, and Fe, respectively) as listed in Table S1 of the ESI.

We have investigated the electronic structures of 2D CrB, MnB, and FeB with the most stable magnetic configuration using GGA+U (Ueff = 2.0 eV) and HSE06 (Fig. S6, ESI). Both show that 2D CrB, MnB, and FeB are metallic with several partially occupied d bands across the Fermi level. As shown in Fig. S6 of the ESI, the projected band structures and density of states (DOS) on atomic orbitals indicate that the states around the Fermi level are dominated by the M-d orbitals with only minor hybridization with B-p states located at around −2.0 eV due to the M–B ionic bond. From the projected DOS of 2D MBs, it is seen that the asymmetrical density of states of five M-d orbitals mostly contribute to the robust magnetism of 2D CrB, MnB, and FeB as shown in Fig. S7 of the ESI. Additionally, the dispersions of M-d orbitals of MB are wide, indicating the itinerant magnetism of CrB, MnB, and FeB.

To gain some insights into the origin of magnetism for 2D MB, we have calculated the band center (εd) and occupation number (md) of five partial d orbitals, respectively, for a M atom in 2D MB with a ground magnetic structure. Herein, by calculating the density of states (DOS) projected on an atomic orbital using GGA+U (Fig. S7, ESI), the εd and md of partial d orbitals can be respectively obtained using the below equations:66

 
image file: d1nr01103k-t2.tif(3)
 
image file: d1nr01103k-t3.tif(4)
where nd(ε) is the density of states of the partial d orbitals of an M atom at a given energy ε and Ef is the Fermi level which is set to zero. Fig. 4a–c plot the calculated εd and md of the partial d orbitals of an M atom for 2D CrB, MnB, and FeB. It is further found that the d-band center of the M atom in the spin-up state (εup) is below the Fermi level and the d-band center of the M atom in the spin-down state (εdown) gradually degrades. The sum of md for Cr-3d, Mn-3d, and Fe-3d in the spin-up state is 3.77, 4.24, and 4.35 with the contribution of all nondegenerate d orbitals. Moreover, the sum of md for Cr-3d, Mn-3d, and Fe-3d in the spin-down state is gradually increased (0.63, 0.81, and 2.06). Hence, the total occupation number of Cr-d, Mn-d, and Fe-d orbitals is 3.14, 3.43, and 2.29, which is close to the calculated magnetic moment of single Cr, Mn, and Fe atoms in 2D MBs (2.89μB, 3.28μB, 2.41μB, respectively, derived from the GGA+U calculations).


image file: d1nr01103k-f4.tif
Fig. 4 Calculated energy diagram of the d-band center for 2D (a) CrB, (b) MnB, and (c) FeB; calculated spin densities (d and f) and schematic illustration of the super-exchange interaction between M-d and B-p along the b direction (e and g) of 2D CrB (d and e) and FeB (MnB) (f and g) in most stable magnetic states. The d-band centers of up εup and down εdown in (a–c) are obtained by [ε(dxy) + ε(dyz) + ε(dxz) + ε(dz2) + ε(dx2y2)]/5. The average d-band center εa is obtained by (εup + εdown)/2. The occupation numbers md for each partial d-band are labeled in a–c. All Fermi levels in a–c are set to zero. Yellow and blue areas in d and f represent spin-up and spin-down electrons, respectively. The isosurfaces for spin-up and spin-down densities were set as 0.07e per a.u.3 Dashed spin arrows in e and g refer to the excited B-d electrons from B-p electrons. The d and p electrons of B atoms are paired and B atoms are nonmagnetic. Thin arrows of M1 and M5 atoms stand for the corresponding spin orbitals that are not fully occupied. Thick arrows of M1 and M5 atoms stand for the corresponding spin orbitals that are fully occupied.

Moreover, the d-band center, that is commonly used as an actively descriptor67 in catalysis, of 2D CrB, MnB, and FeB, can give us an insight into the catalytic activity of MB on accounting for adsorbate–metal interactions. As shown in Fig. 4a–c, the d-band centers of CrB, MnB, and FeB for the spin-up state εup (spin-down state εdown) are −1.41 eV (2.29 eV), −3.05 eV (0.27 eV), and −3.20 eV (−0.55 eV), respectively, and then the average d-band centers (εa) of 2D CrB, MnB, and FeB are 0.44 eV, −1.39 eV, and −1.88 eV, respectively. Clearly, it illustrates that the average d-band center (εa) is gradually far away from the Fermi level when M varies from Cr to Fe (Fig. 4a–c). Since the antibonding states are always above the d states in terms of energy, the downshift of the d-band center of 2D MnB and FeB indicates that the antibonding energy states are lowered and the interaction between the adsorbate and 2D MBs is weakened.68 Therefore, the 2D MnB and FeB can effectively decrease the adsorption energy of hydrogen (H) and facilitate the desorption of H from the catalyst surface. For instance, the adsorption energy of H for 2D CrB is very negative, indicating the low kinetic energy of the release of the hydrogen molecule on its surface.69

Lastly, to understand the long-range magnetic ordering of 2D MBs (M = Cr, Mn, and Fe) in their ground magnetic state, we have plotted the corresponding spin density distributions as shown in Fig. 4d and f. It shows that in 2D CrB, the magnetic moments of all Cr atoms (Cr-1, Cr-2, …, Cr-8) are positive as shown in Fig. 4d. In contrast, the spin direction of Fe-1, Fe-2, Fe-3, and Fe-4 (Mn-1, Mn-2, Mn-3, and Mn-4) atoms in 2D FeB (or MnB) is antiparallel to that of Fe-5, Fe-6, Fe-7, and Fe-8 (Mn-5, Mn-6, Mn-7, and Mn-8) atoms (Fig. 4f). As mentioned above, the M atoms in 2D CrB, MnB and FeB have four, five, and seven d-electrons, respectively. Therefore, the difference of magnetic ordering is possibly due to the occupation of partial M-d orbitals in 2D MBs.

For CrB, the band centers of all d orbitals in the spin-up state and spin-down state are below and above the Fermi level, respectively (Fig. 4a), and the center of spin-up Cr-d orbitals is shallower than those of Mn-d and Fe-d orbitals (Fig. 4a–c). This means that these spin up orbitals are not fully occupied, which is consistent with the Bader charge calculation. According to the super-exchange mechanism70 and Hund's rules, the excited electrons of B-d orbitals from B-p orbitals would parallel with the d electrons of the neighbour M1 atom (Fig. 4e). Then the remaining B-p electrons would have negative interactions with the d electrons of another neighbour M5 atom, which means that the spin direction of B-p electrons and that of M5-d electrons are antiparallel (Fig. 4e). Hence, the interactions of M1 (Cr1) and M5 (Cr5) are FM coupling (Fig. 4d).

For FeB or MnB, the band centers of the partial spin-down M-d orbitals were shifted to below the Fermi level (Fig. 4b and c). In other words, the spin-up M-d orbitals are fully occupied, which is consistent with the fact that there are five and seven electrons in the d-orbitals of M atoms in MnB and FeB, respectively. Consequently, the excited electrons of B-d orbitals from B-p orbitals would be antiparallel to Fe1-d (or Mn1-d) electrons.70 The remanent B-p electrons also have negative interactions with Fe5-d (or Mn5-d) and the spin direction of B-p electrons is antiparallel to that of Fe5-d (or Mn5-d) electrons, as shown in Fig. 4g. Hence, the interactions of Fe1-d (or Mn1-d) and Fe5-d (Mn5-d) are AFM coupling (Fig. 4f). In brief, the magnetic coupling of 2D MBs along the b direction is determined by the super-exchange interactions and the FM or AFM coupling is sensitive to the occupation of the partial M-d orbitals of 2D MBs. In addition, the magnetic coupling of 2D MBs along the a direction can be attributed to the direct-exchange interaction derived from M-d orbitals, in which both are FM (Fig. 4d and f).71

Surface functionalization

It was reported that the functional group T (T = F, OH, and O) can easily adsorb on the surface of MXenes during the preparation process from MAX phases, and can change the physical properties of bare MXenes.72–74 Hence, it is essential to study the influence of surface-terminated 2D MBs (M = Cr, Mn, and Fe; B = boron) on their magnetic and electronic properties. Four different adsorption sites of functional groups on the bare MBs, namely, top, bridge-1, bridge-2, and hollow sites (Fig. 5a), were investigated to find out their most stable adsorption configuration. The adsorption energies of different configurations for 2D MBs were calculated using the below equation:
 
Ead = [E(M2B2T2) − E(M2B2) − 2E(T)]/2(5)
where E(MBT), E(MB), and E(T) are the spin-polarized energies of MBT, MB, and functional group T, respectively. As an example, we have compared the adsorption energies of different configurations for 2D FeB, as shown in Fig. 5b. It is observed that the O atom favors the hollow adsorption site while OH (or F) is mostly adsorbed at the bridge-2 site (or the bridge-1 site). Furthermore, the calculated adsorption energies show that the favorable adsorption site for the O atom is still the hollow site of CrB and MnB, and the F atom and OH favor either the bridge-2 site of CrB or the bridge-1 site of MnB as shown in Fig. S8 of the ESI.

image file: d1nr01103k-f5.tif
Fig. 5 (a) Top view of the MB and four different adsorption sites. (b) Calculated adsorption energies of functional groups (O, OH, and F) on different adsorption sites of FeB. Calculated spin–spin correlations of the 1st, 2nd, 3rd, 4th, and 5th neighboring magnetic atoms by MC simulation at T = 10 K for 2D (c) FeBO, FeBOH, and FeBF. Calculated specific heat CV as a function of temperature T for 2D (d) FeBO, FeBOH, and FeBF.

It was expected that the electronic and magnetic properties of surface-terminated 2D MBs can be different from the pristine ones. Therefore, we performed a set of DFT calculations and compared the total energies of 11 collinear magnetic configurations to clarify the magnetic ground state of MBs with surface termination. The calculation results (listed in Table S7) show that most functionalized 2D MBs have an AFM ground state, while 2D CrBOH, CrBF and MnBF show ferromagnetic properties. Furthermore, both the magnetic coupling parameters and magnetic anisotropy energies of 2D MBO, MBOH, and MBF were calculated using the same method as for bare 2D MBs (Fig. S1 and Table S8, ESI). Taking the family of 2D FeBT as an example, we found that the ground states of 2D FeBO, FeBOH, and FeBF are AFM-5, AFM-9, and AFM-9, respectively, by comparing the energy differences of different spin configurations (Table S7, ESI). The calculated Néel temperatures of 2D FeBO, FeBOH, and FeBF by employing MC simulations are 910, 930, and 420 K, respectively (Fig. 5c), which are higher than that of bare 2D FeB (320 K). The calculated critical temperatures of other 2D MBTs, which are in the temperature range of 250–560 K, are shown in Figs. S9 and S10. One can see that almost all Néel temperatures of antiferromagnetic 2D MBTs are above room temperature, which is attractive for enabling the application of 2D antiferromagnetic spintronics.

Also, the calculated spin–spin correlations of 2D MBO, MBOH, and MBF at a very low temperature (T = 10 K) by employing MC simulations gave consistent results with the energy calculations (Fig. 5d, S9 and S10). Therefore, our study suggests that the magnetic ground state of 2D MBs (M = Fe, Mn, and Cr) can be effectively modulated by both changing transition metals and functional groups. Finally, most of the functionalized 2D MBTs are metallic as the bare 2D MBs, while FeBOH and CrBO are semiconductors with narrow band gaps of 0.26 and 0.19 eV, respectively (Fig. S11, S12, and S13, ESI).

Conclusions

In this work, we present an extensive study of the stabilities and magnetic properties of 2D borides using a set of first-principles calculations. Our study predicts that 2D MBs (M = Cr, Mn, and Fe) are stable with intriguing magnetic properties. The magnetic ground state of 2D CrB is the FM metal, whereas MnB and FeB are AFM metals. Non-collinear magnetic calculations with spin–orbit coupling show that the 2D CrB displays an in-plane magnetic easy axis and exhibits a magnetic anisotropy energy of 23.6 μeV per f.u., while 2D MnB and FeB are typical Ising antiferromagnets with out-of-plane easy magnetic axes and possess robust magnetic anisotropy energies of 222.7 and 482.2 μeV per f.u., respectively. Moreover, by performing Monte Carlo simulations, the magnetic ground states of all 2D MBs are confirmed and the calculated critical temperatures of 2D CrB, MnB and FeB are 440 K, 300 K and 320 K, respectively. Our study confirms that the super-exchange interaction plays a vital role in determining the long-range magnetic ordering of 2D MBs. More importantly, the super-exchange is sensitive to the occupation of the partial M-d orbitals of 2D MBs. Furthermore, the magnetic properties of functionalized 2D MBTs (M = Cr, Mn, and Fe; T = O, OH, and F) were studied. Intriguingly, most of the functionalized 2D MBTs were predicted to have AFM ground states with a high Néel temperature, which is promising for enabling antiferromagnetic spintronics at the nanoscale. Our research reveals for the first time that there are rich AFM spin structures with high Néel temperatures in 2D transition metal borides and demonstrates that 2D MBs and MBTs are promising candidates for building novel spintronic devices.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grants No. 51872242 and 51761135032) and the Fundamental Research Funds for the Central Universities (No. D5000200142). V.A.B. thanks the Ministry of Education and Science of the Russian Federation for financial support within grant No. 0778-2020-0005.

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Footnote

Electronic supplementary information (ESI) available: Phonon dispersions, elastic constants, and AIMD simulations of 2D MBs. The spin configurations and magnetic and electronic properties of 2D MBs and MBTs. See DOI: 10.1039/d1nr01103k

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