Shiyao
Wang
^{a},
Nanxi
Miao
^{a},
Kehe
Su
^{b},
Vladislav A.
Blatov
^{c} and
Junjie
Wang
*^{a}
^{a}State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, People's Republic of China. E-mail: wang.junjie@nwpu.edu.cn
^{b}School of Chemistry and Chemical Engineering, Nowthwestern Polytechnical University, Xi'an, Shaanxi 710072, People's Republic of China
^{c}Samara 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

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 Cr_{2}AlB_{2}, Mn_{2}AlB_{2}, and Fe_{2}AlB_{2}. 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.

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 M_{n+1}X_{n}T_{2} (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., Cr_{2}C,^{14} Mn_{2}C,^{15} and Mn_{2}NT_{2} (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, Ti_{2}InB_{2},^{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} M_{2}AB_{2} (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 MoB^{29,30} and CrB^{31} from the corresponding M_{2}AlB_{2} 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.

Phonon dispersion calculations were performed to confirm the dynamic stabilities of 2D MBs using the Phonopy code^{41} 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 package^{45} 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 × 10^{6} MC steps per spin for equilibration and 1 × 10^{6} 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 C_{V} 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 C_{V} plot.^{46}

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 C_{11} > 0, C_{11}·C_{22} > C_{12}^{2}, C_{44} > 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 E_{coh} = (2E_{M} + 2E_{B} − E_{MB})/4 were also calculated. E_{MB}, E_{M}, and E_{B} are the total energies of the 2D MB, single transition-metal atom, and single boron atom, respectively. The E_{coh} 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., MoS_{2} (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 analysis^{51} of the MB layers performed using the ToposPro program package^{52} 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}

The Heisenberg Hamiltonian^{54} of a 2D MB considering five neighbor interactions can be given by the following equation:

(1) |

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 a–c and b–c 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 a–c plane and θ = 90° in the b–c 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 a–c and b–c planes. The energy of MnB (or FeB) reaches the maximum at θ = 90° in the a–c 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 a–c plane or the b–c 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., Mn_{2}NO_{2} (63 μeV per atom) and Cr_{2}NO_{2} (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}

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) = K_{1}sin^{2}θ + K_{2}sin^{4}θ | (2) |

To confirm the magnetic ground state of 2D CrB, MnB, and FeB, the spin–spin correlation was obtained by using Monte Carlo (MC) simulation^{45} at a low temperature (T = 5 K). Fig. 3a–c show the average value of the products of the neighboring spins sum(S_{i}·S_{j})/N and their absolute values sum(|S_{i}·S_{j}|)/N for the spin Hamiltonian given by the coupling constants using GGA+U (U_{eff} = 2.0 eV). N is the size of lattice in the MC simulation. The calculated values 1 and −1 of sum(S_{i}·S_{j})/N indicate that the 2D MB is FM and AFM coupling, respectively. Fig. 3a shows that the calculated sum(S_{i}·S_{j})/N for the 1^{st}, 2^{nd}, 3^{rd}, 4^{th}, and 5^{th} nearest neighboring spins is 1 for 2D CrB, referring to the FM coupling. Fig. 3b and c show that the calculated sum(S_{i}·S_{j})/N for 2D MnB and FeB is 1 for the 1^{st} and 2^{nd} nearest neighboring spins and becomes −1 for the 3^{rd}, 4^{th}, and 5^{th} 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(|S_{i}·S_{j}|)/N for the 1^{st}, 2^{nd}, 3^{rd}, 4^{th}, and 5^{th} 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}

Moreover, we studied the critical temperature of 2D CrB, MnB, and FeB by performing MC simulation. The Curie (T_{C}) or Néel (T_{N}) 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 T_{C} beyond the room temperature.^{60} By the use of the same approach, our calculated T_{C} for the monolayer CrI_{3} 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).

We have investigated the electronic structures of 2D CrB, MnB, and FeB with the most stable magnetic configuration using GGA+U (U_{eff} = 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 (ε_{d}) and occupation number (m_{d}) 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 ε_{d} and m_{d} of partial d orbitals can be respectively obtained using the below equations:^{66}

(3) |

(4) |

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 ε_{up} and down ε_{down} in (a–c) are obtained by [ε(d_{xy}) + ε(d_{yz}) + ε(d_{xz}) + ε(d_{z2}) + ε(d_{x2−y2})]/5. The average d-band center ε_{a} is obtained by (ε_{up} + ε_{down})/2. The occupation numbers m_{d} 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 descriptor^{67} 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 ε_{up} (spin-down state ε_{down}) 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 (ε_{a}) 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 (ε_{a}) 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 mechanism^{70} 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}

E_{ad} = [E(M_{2}B_{2}T_{2}) − E(M_{2}B_{2}) − 2E(T)]/2 | (5) |

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†).

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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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