Xinru
Li
*^{ac},
Zeying
Zhang
^{bc} and
Hongbin
Zhang
^{c}
^{a}College of Physics and Optoelectronic Engineering, Shenzhen University, 518060 Shenzhen, P. R. China. E-mail: xrli@szu.edu.cn
^{b}College of Mathematics and Physics, Beijing University of Chemical Technology, 100029 Beijing, P. R. China
^{c}Institute of Materials Science, Darmstadt University of Technology, 64287 Darmstadt, Germany
First published on 4th December 2019
We present a high throughput study of the magnetic ground states for 90 transition metal dihalide monolayers TMX_{2} using density functional theory based on a collection of Hubbard U values. Stable geometrical phases between 2H and 1T are first determined. Spin-polarized calculations show that 50 out of 55 magnetic TMX_{2} monolayers are energetically prone to the 1T phase. Further, the magnetic ground states are determined by considering four local spin models with respect to different U values. Interestingly, 23 out of 55 TMX_{2} monolayers exhibit robust magnetic ground orderings which will not be changed by the U values. Among them, NiCl_{2} with a magnetic moment of 2 μ_{B} is a ferromagnetic (FM) insulator, while the VX_{2}, MnX_{2} (X = Cl, Br and I), PtCl_{2} and CoI_{2} monolayers have noncollinear antiferromagnetic (120°-AFM) ground states with a tiny in-plane magnetic anisotropic energy, indicating flexible magnetic orientation rotation. The exchange parameters for both robust FM and 120°-AFM systems are analyzed in detail with the Heisenberg model. Our high-throughput calculations give a systematic study of the electronic and magnetic properties of TMX_{2} monolayers, and these two-dimensional materials with versatile magnetic behavior may have great potential for spintronic applications.
Such versatile electronic or magnetic properties have lead to comprehensive research in various fields like catalysis, opto-electronics, spintronics, electrodes and supercapacitors. In particular, transition metal dichalcogenide monolayers (TMDs),^{11} transition metal trichalcogenides MAX_{3},^{12} and 2D transition metal carbides (nitrides) as so-called MXenes^{13} have been systematically studied. For instance, MoS_{2}, MoSe_{2}, WS_{2} and WSe_{2} monolayers have sizable direct band gaps allowing applications in transistors, photodetectors, and electroluminescent devices.^{14,15} VS_{2} and VSe_{2} monolayers are typically metallic ferromagnets with strain-tunable magnetic ground states.^{5} MXenes with semiconducting band gaps around 0.25–2.0 eV yield very large Seebeck parameters around 100 K allowing high thermoelectric performance.^{4} van der Waals layered Cr_{2}Ge_{2}Te_{6} has been experimentally proved to be a magnetic insulator, which are regarded as a key resource for next-generation spintronic devices.^{16}
Beyond transition metal chalcogenides, carbide and nitride transition metal halides are also found to be interesting.^{17} Recently, layer-dependent ferromagnetism in van der Waals CrI_{3} crystals has been systematically studied. It is demonstrated that the monolayer CrI_{3} is an Ising ferromagnet with an out-of-plane spin orientation.^{18} Soon after, (quasi) 2D ferromagnetic insulators have revealed many important applications in phototronics, spintronics and magnetoelectronics. For instance, Seyler et al. unveiled spontaneous circularly polarized excitation in the CrI_{3} monolayers.^{19} Jiang et al. revealed that a direct-to-indirect band gap transition can be caused by rotating the magnetic direction of CrI_{3} from out-of-plane to in-plane.^{20} Klein et al. detected the magnetic ground state and interlayer coupling in CrI_{3} and uncovered a field-induced metamagnetic transition.^{21} In addition, transition metal dihalides (TMX_{2}) with intriguing simple lattice structures, relative large ionicity, and magnetic diversities have invigorated the enthusiasm of systematic researches. Early in 1971, layered NiCl_{2} and CoCl_{2} separated by one or two layers of graphite was proved to be a two-dimensional Heisenberg ferromagnet.^{22} It was shown that intercalation of graphite can weaken significantly the interaction between the TMX_{2} layers, so that these compounds can be regarded as 2D paramagnets. In recent years, TMX_{2} monolayers, especially TMX_{2} magnets, have attracted more attention. For instance, the ferromagnetic or antiferromagnetic properties of TMX_{2} (TM = 3d transition metals; X = Cl, Br, I) in both the 1T and 2H phase were investigated and the Curie temperature of magnetic transition were evaluated using Monte Carlo simulations.^{23} Moreover, FeX_{2} were proven to be half-metals^{24} with sufficient stability to be exfoliated from bulk layered compounds.^{6} Despite the achievements made on 3d transition metal dihalide monolayers, more pending questions still need to be investigated in a deeper way: (1) since fullerene-like CdCl_{2} closed cage structures have been successfully synthesized,^{25} there is a high possibility to achieve 4d or 5d TMX_{2} monolayers. In this case, the exotic electronic or magnetic properties of 4d or 5d TM dihalide monolayers are open to be explored. (2) Most current magnetic analyses of TMX_{2} monolayers are focusing on collinear magnetic ordering. However, as the TM atomic arrangements in TMX_{2} monolayers of both the 1T and 2H phase are triangular lattices, geometric frustration on the spin configurations is inevitable. Thus, non-collinear antiferromagnetic ordering needs to be further identified.^{26} (3) Previous density-functional-theory (DFT) calculations on TMX_{2} either neglected the local Coulomb interactions (U values) or used specific U value to estimate the electronic or magnetic properties, which may not be in accordance with future experimental results. In this regard, investigating TMX_{2} with a batch of U values to identify property–U relations is instructive to practical experiments.
In this work, we implement a high throughput method to identify the magnetic ground states of 90 TMX_{2} (TM = transition metals from Sc to Hg; X = Cl, Br, I) with four magnetic ordering configurations based on first principles calculations. 55 out of 90 TMX_{2} monolayers are magnetic cases and 50 magnetic TMX_{2} are stable in the 1T phase instead of the 2H phase. Due to the lack of experimental data when describing the correlations on d electrons of TM atoms in TMX_{2} monolayers, the magnetic ground states for magnetic TMX_{2} systems are systematically investigated with respect to Hubbard U values from 0 to 5 eV. We predict that the NiCl_{2} is a robust ferromagnetic insulator with magnetic moments of 2 μ_{B} and Curie temperature of 57 K. Besides, 8 TMX_{2} monolayers are stable in noncollinear antiferromagnetic ground states with different Hubbard U values, and the in-plane magnetic anisotropy energy is estimated to be smaller than 10 μeV indicating an easily overcome thermal fluctuation and a high-flexibility of magnetic orientation.
On the other hand, the dynamic stability of 1T monolayers with a space group symmetry of pm1 has been studied thoroughly by previous high throughput studies. Haastrup et al. introduced the Computational 2D Materials Database (C2DB) containing 1500 2D materials distributed over 30 different crystal structures, including the 1T-CdI_{2} prototype. To determine the local minimum of the potential energy surface, the dynamical stabilities are analyzed by calculating the Γ-point phonons.^{37} It shows that 90 out of 315 1T-CdI_{2} prototype monolayers are of high stability. Besides, phonon dispersions without imaginary frequencies of first row 1T-TMX_{2} (TM = V, Mn, Fe, Co, Ni; X = Cl, Br, I) monolayers have been investigated by Kulish et al., indicating high dynamic stabilities of the 1T-TMX_{2} monolayers.^{23} Furthermore, since the most stable structure generally locates on the global free energy minimum of the potential energy surface, the corresponding potential energy surface for each TMX_{2} will be explored in detail in our further studies by CALYPSO software using the particle swarm optimization (PSO) algorithm.^{38}
Based on the magnetic structure prototypes, DFT calculations are carried out for four such spin configurations of all 90 TMX_{2} monolayers. According to the total energies of different magnetic orderings obtained by the GGA+U (U = 2 eV) method, the most favorable configurations of each TMX_{2} in the corresponding stable geometrical phase are listed in Fig. 3. There are 55 out of 90 systems with a total magnetic moment larger than 0.2 μ_{B} in the FM configuration, which will be considered as magnetic. In accordance with Fig. 1, the stable 2H phases are mainly concentrated in the nonmagnetic (NM) TMX_{2} monolayers with TM atoms with d^{1}–d^{2} configuration, while magnetic FeX_{2} (X = Cl, Br, I) and TcX_{2} (X = Br, I) monolayers with a stable 2H phase are exceptions to this rule. Besides, the Zn-group monolayers with a stable 1T phase are also nonmagnetic.
Fig. 3 Overview of ground geometrical structure and magnetic states for all 90 transition metal dihalides with U = 2. |
Since the majority of magnetic TMX_{2} monolayers are more stable in the 1T phase, in the following discussions the magnetic and electronic structures are mainly discussed in 1T-TMX_{2} monolayers. As for FeX_{2}, the energy differences between the 1T and 2H phase range from 10 to 63 meV per atom under different U, which are quite small, and 1T FeX_{2} monolayers have been proved to be stable in 2D materials database.^{6,40} Thus, our analysis of the electronic and magnetic properties of FeX_{2} are discussed in the 1T phases. Tc is a highly radioactive element which needs to be avoided, thus no further discussions will continue. From the magnetic point of view, it can be clearly noticed by comparing TMX_{2} monolayers in the sequence from 3d to 4d to 5d elements that the tendency of magnetic systems has a remarkable right-shift. It can be well explained by d band broadening which will prevent higher d transition metal based compounds from being spin polarized, i.e. magnetic. The argument is in good accordance with the magnetic behavior of d electrons in transition metal monolayers on noble metal (001) substrates in ref. 41.
At current stage, we have obtained an overall landscape of 90 TMX_{2} monolayers with both geometrical and magnetic ground states clarified under the condition of U = 2. However, one threatening problem is that there is no experimental result for such 2D systems. In this case, the effect of the correlations on d electrons of TM atoms may not be described precisely without supportive experimental data. To solve these problems and to make the theoretical study more instructive, we implement HTP calculations on 55 magnetic 1T-TMX_{2} systems with sequential U values from 0 to 5 in a batch mode to determine the relations between U values and magnetic ground states. Magnetic ground states are evaluated by the energy difference between FM and the most stable AFM states (ΔE). The concrete classification is shown in Fig. 4a. We can clearly see from the pie-chart diagram that 55 out of 90 are magnetic systems. Taking magnetic TMCl_{2} monolayers for example, the energy difference as shown in Fig. 4b can intuitively reflect the changing tendency of magnetic ground state with respect to different U values from 0 to 5. Among all magnetic systems, the spin configurations are classified into two cases with respect to U values: first, 23 U-independent magnetic systems which indicate that the magnetic ground states will not be changed by different U values exhibiting robustness. The 23 TMX_{2} with robust magnetic ground states include TMCl_{2} (TM = V, Cr, Mn, Ni, Ag, and Pt), TMBr_{2} (TM = Ti, V, Cr, Co, Ni, Ag, and Pt), and TMI_{2} (TM = Sc, Ti, V, Mn, Fe, Co, Ni, Pd, and Au). Results indicate that U-independent magnetic TMX_{2} monolayers are mainly dominated by 3d TM dihalides, which can be explained by weaker exchange correlation potentials for lighter TM atoms. Besides, TM dihalides with IB and IIB TM atoms can also be robust magnetic systems against Hubbard U values. Second, 32 U-dependent magnetic systems which show that the magnetic properties will change among different U. Among 32 U-dependent TMX_{2} monolayers, there are 11 systems (U-regular cases) with magnetic ground states which can be tuned regularly by enlarging U values by the linear change of the energy difference. Meanwhile the magnetic ground states of the other 21 systems (U-irregular cases) cannot be linearly tuned by enlarging the U values.
It is worth mentioning that among the 23 magnetism-robust systems, 4 of them are robust ferromagnetic monolayers including 1 ferromagnetic insulator (NiCl_{2}) and 3 half metals including AgCl_{2}, AgBr_{2} and AuI_{2}, while 8 out of 19 U-independent cases are noncollinear AFM monolayers. Our results for the ground states of all magnetic TMX_{2} monolayers are summarized in the Table in the ESI,† corresponding to the classification shown in Fig. 4a. We will make explicit analysis for these 12 1T-TMX_{2} monolayers with robust magnetic orderings with respect to U values in the following discussions.
Band structures with both spin-up and spin-down channels for four robust FM 1T-TMX_{2} monolayers with U = 2 are as shown in Fig. 5. Band curve dispersions in the whole Brillouin zone indicate similar state distributions due to the influence of the same crystalline field. The spin splitting near the Fermi level of NiCl_{2} is about 2.3 eV, that is, larger than the spin splittings of AgCl_{2}, AgBr_{2} and AuI_{2}, which are in the range of 0.2 to 0.5 eV. The magnetization for NiCl_{2}, dominated by the Ni atom, is close to 2 Bohr magneton (μ_{B}) per atom indicating 8 electrons in d orbital filling with high-spin states. Meanwhile for the other Au or Ag based dihalides, the magnetization is distributed and delocalized in both Au (Ag) and halide atoms smaller than 0.5 (μ_{B}) per atom, which can also be explained from particle density of states as shown in Fig. 5. For AgCl_{2}, AgBr_{2} and AuI_{2} monolayers, the d orbitals of Au or Ag are coupled with halide p orbitals which result in strong interaction between adjacent atoms. Besides, AgCl_{2}, AgBr_{2} and AuI_{2} monolayers are half metals with a Fermi level only crossing one spin channel with 100% spin polarization. To ensure the reliability of the band structures of TMX_{2}, the screened hybrid functional Heyd–Scuseria–Ernzerhof (HSE) is further implemented.^{42} Results indicate that both GGA+U and HSE functional do not change the energy dispersions, and orbital occupations remain similar with the two methods. An exception is represented by the band gap which in the HSE functional is larger with respect to GGA+U results. As for NiCl_{2}, the band gap in the HSE functional is 4.48 eV. Our results are consistent with previous studies.^{23} However, this difference does not change other electronic and geometric properties, including exchange interactions and magnetic orderings.
Fig. 5 Spin polarized band structures with U = 2 for 4 U-independent FM cases: (a) NiCl_{2}, (b) AgCl_{2}, (c) AgBr_{2} and (d) AuI_{2}, respectively. |
It is known that ferromagnetic insulators are in high demand in the development of next-generation quantum-spintronic devices, in particular those high symmetric systems without chemical doping or tensile strain that can only work at very low temperatures (below 16 K).^{43} Thus, studies of the magnetic mechanism of ferromagnetic insulator NiCl_{2} monolayers are further required. Since the magnetic moments are mostly located on Ni atoms, the total energies can be mapped onto a Heisenberg model:
(1) |
E_{FM} − E_{211AFM} = −8(J_{1} + J_{2})S^{2} | (2) |
E_{FM} − E_{411AFM} = (−4J_{1} − 8J_{2} − 8J_{3})S^{2} | (3) |
E_{FM} − E_{120°AFM} = −6(J_{1} + J_{3})S^{2} | (4) |
Fig. 6 Schematic diagrams of four representative magnetic ordering arrangements corresponding to Fig. 2. Blue balls indicate TM atoms which form six trigonal lattices with nearest neighbors. The black circles identify the exchange coupling J_{1} between nearest neighbors (NN) of the central TM atom, the green circles identify the exchange coupling J_{2} between second nearest neighbors (SNN) and the orange circles identify the exchange coupling J_{3} between the third nearest neighbors (TNN). |
By combining these three equations, the three exchange parameters of ferromagnetic NiCl_{2} are calculated as shown in Table 1. The exchange parameters: J_{1} = 5.46 meV, J_{2} = −0.23J_{1}, and J_{3} = 0.14J_{1} which notably indicate strong ferromagnetic ground state orderings. The Curie temperature of NiCl_{2} is further evaluated by mean field theory using , where .^{44,45} Results show that NiCl_{2} is a U-independent ferromagnetic insulator with a Curie temperature of 57 K.
Robust AFM states with respect to U parameters in TMX_{2} are also worth mentioning. Previous studies have shown that antiferromagnets have typically much faster dynamics than ferromagnets, as the reorientation of antiferromagnetic moments will cost exchange energy. In particular, the insulating antiferromagnets have been proved to have significant application in optical devices. Early in 2004, Kimel et al. uncovered that the spins of the antiferromagnet TmFeO_{3} can be tuned on a timescale of a few picoseconds, in contrast to hundreds of picoseconds in a ferromagnet.^{46} Besides, Takei et al. proved theoretically that spin–superfluid transport can be realized in AFM insulator.^{47} In TMX_{2} systems, we filter 8 noncollinear AFM insulators whose magnetic ground state will not change with respect to various U, and the corresponding band structures are listed in Fig. 7. The results show that the band gaps of these noncollinear AFM insulators are in the range between 0.5 and 2.5 eV with U = 2. The NN, SNN and TNN exchange parameters listed in Table 1 are analyzed by the Heisenberg model. The positive J_{1} is 3.8 to 5.6 times larger than the absolute value of J_{2}, which is consistent with the favored noncollinear AFM spin configurations. Besides, the in-plane magnetic anisotropy energy (MAE) for 8 noncollinear AFM are estimated by a rotating magnetic orientation with an interval of 10°. These results indicate that the in-plane MAE is smaller than 10 μeV per unit-cell for each noncollinear AFM TMX_{2}, which shows that the in-plane magnetic orientation can overcome small thermal fluctuation and is of high flexibility. There is one thing worth mentioning for low-dimensional magnetic systems, especially for noncollinear cases, which is that a spin–spiral structure may exist. In previous studies, the exchange interaction parameters of the 3d transition metal chains have been evaluated by using energy dispersion relations of the spin–spiral waves. Results show that the magnetic coupling in V, Mn, Cr chains are frustrated.^{48} The results are in consistent with the frustrated magnetic behaviors in Mn or V based TMX_{2} systems. For a better understanding of the spin–spiral waves and the exchange interaction of transition metal dihalides, further studies need to be explored.
TMX_{2} | J _{1} | J _{2} | J _{3} | T _{C} | μ _{B} | GS |
---|---|---|---|---|---|---|
NiCl_{2} | 5.46 | −1.28 | 0.75 | 57 | 2 | FM |
VCl_{2} | −28.54 | 6.26 | −3.97 | — | 3 | 120°-AFM |
MnCl_{2} | −3.02 | 0.68 | −0.65 | — | 4 | 120°-AFM |
PtCl_{2} | −18.06 | 4.03 | −7.63 | — | 1 | 120°-AFM |
VBr_{2} | −17.07 | 3.91 | −2.32 | — | 3 | 120°-AFM |
VI_{2} | −7.68 | 1.84 | −1.21 | — | 3 | 120°-AFM |
MnI_{2} | −2.93 | 0.52 | −0.87 | — | 4 | 120°-AFM |
CoI_{2} | −2.80 | 0.69 | −2.3 | — | 3 | 120°-AFM |
Footnote |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9na00588a |
This journal is © The Royal Society of Chemistry 2020 |