Supporting Information for 2D Intrinsic. Ferromagnets from van der Waals. Antiferromagnets

Transcription

1 Supporting Information for 2D Intrinsic Ferromagnets from van der Waals Antiferromagnets Naihua Miao,, Bin Xu, Linggang Zhu,, Jian Zhou, and Zhimei Sun,, School of Materials Science and Engineering, Beihang University, Beijing, 1191, China. Center for Integrated Computational Materials Engineering, International Research Institute for Multidisciplinary Science, Beihang University, Beijing, 1191, China. Physics Department and Institute for Nanoscience and Engineering, University of Arkansas, Fayetteville, Arkansas, 7271, USA. Phone: +86-() Fax: +86-() S1

2 Table of Contents 1. Computational methodologies...[s3] 2. Structural properties of the bulk MOX...[S4] 3. Physical properties of the MOX monolayers and the effect of exchange-correlation functionals and U values...[s6] 4. Calculated Curie temperature, total magnetic moment, magnetocrystalline anisotropy energy of the CrOX monolayers...[s8] 5. Simulated mechanical exfolication process compared with graphite...[s9] 6. Phonon dispersions and AIMD evolutions for the monolayers...[s1] 7. Magnetic configurations for the single-layer supercells...[s12] 8. Electronic structures of the CrOBr monolayer...[s13] 9. Curie temperature of the CrOX monolayers under strains...[s14] 1. Reliability of calculations and predictions...[s16] References...[S17] S2

3 1. Computational methodologies First-principles calculations. All calculations were performed based on spin density functional theory using the projector-augmented wave method S1 as implemented in the Vienna ab initio Simulation Package (VASP). S2 For the exchange-correlation functional, the generalized gradient approximation (GGA) with Perdew-Burke-Ernzerhof parametrization S3 and corrections on strong correlation of 3d electrons (GGA+U) was applied using the Dudarev scheme. S4 The density functional dispersion correction (D3-Grimme) S5 was adopted for the van der Waals (vdw) interactions. A cut-off energy of 45 ev was set for the plane wave basis. The k-meshes for the bulk, 2D crystals, and supercells were , , and 2 2 1, respectively. The convergence criteria were ev for the energy difference in electronic self-consistent calculation and ev/å for the residual forces on atoms. A vacuum of 2 Å was constructed perpendicular to the layer plane for the monolayer supercells. Electronic band structures, density of states, charge densities, and magnetic properties were obtained by GGA+U, unless stated otherwise. Phonon dispersions and ab initio molecular dynamics. The phonon calculations were performed using a supercell with the finite displacement method. S6 S8 Ab initio molecular dynamics (AIMD) simulations were performed at 3 K using canonical ensemble (NVT) with Nose-Hoover thermostat. S9,S1 The total simulation time is 2 ps with a timestep of 2 fs. Monte Carlo simulations. Monte Carlo simulations based on Ising spin Hamiltonians were performed with the Wolff algorithm S11 using supercells. For each 2D crystals under various strains, the structure was firstly optimized and then the nearest exchange coupling constant was individually calculated as input for Monte Carlo simulation to calculate the thermal properties. S3

4 2. Structural properties of the bulk MOX (M=Cr, V, and Ti; X=Cl and Br) Table S1 Calculated lattice parameters, electronic band gap E g, magnetic moment µ of TM ions of the bulk crystals using GGA with the D3-Grimme vdw correction. TM refers to the transition metal atom. The available experimental data are listed in parentheses for comparison. S12,S13 Crystal Approach a (Å) b (Å) c (Å) E g (ev) µ (µ B /TM) CrOCl GGA (3.863) (3.182) (7.694) 1.7 (2.8) 2.82 (3.68) CrOBr GGA (3.863) (3.232) 8.18 (8.36) (3.74) VOCl GGA 3.85 (3.78) 3.29 (3.3) (7.91). (2.) 1.71 VOBr GGA (3.775) 3.42 (3.38) (8.425). 1.9 TiOCl GGA 4.7 (3.786) 3.22 (3.361) (8.45). (1.8). TiOBr GGA 4.22 (3.785) (3.485) 7.99 (8.525). (1.65). Table S2 Calculated lattice parameters, electronic band gap E g, magnetic moment µ of the metal ions of bulk crystals using GGA+U with the D3-Grimme vdw correction. The U values of 7, 4, and 3 ev for the 3d electrons of the Cr, V, and Ti ions were carefully determined by comparing their electronic band gaps with available experiments, repsectively. The available experimental data are listed in parentheses for comparison. S12,S13 Crystal Approach a (Å) b (Å) c (Å) E g (ev) µ (µ B /TM) CrOCl GGA+U (3.863) (3.182) (7.694) 2.71 (2.8) 3.22 (3.68) CrOBr GGA+U (3.863) (3.232) (8.36) (3.74) VOCl GGA+U (3.78) (3.3) (7.91) 2.5 (2.) 2.1 VOBr GGA+U 3.88 (3.775) (3.38) (8.425) TiOCl GGA+U (3.786) (3.361) (8.45) 1.77 (1.8).99 7 TiOBr GGA+U (3.785) (3.485) (8.525) 1.7 (1.65).99 9 As shown in Table S1, the GGA functional can not reasonably reproduced the experimental results, especially on the electronic band gap and magnetic moment. While using the GGA+U approaches (Table S2) with careful determinations of U values by comparing their electronic band gaps and magnetic moments with available experimental data, most of the calculated physical properties of the MOX bulks are in better agreement with the experiments than the pure GGA technique, indicating the strong correlation effects of the 3d electrons should be considered. S14 S16 Our calculations also indicate the GGA+U approach (with U J=7 ev) could better reproduce the ground state of bulk CrOX crystals with S4

5 opposite magnetic moments on the Cr ions in adjacent layers (the energy difference between the ferromagnetic and antiferromagnetic states are -1.1 and -2.6 mev for bulk CrOCl and CrOBr, respectively). In addition, previous investigations also suggest that the main charactors of experimental photoemission spectra could only be well captured by the density of states calculated from the GGA+U calculations. S15 Therefore, in the following study of the MOX bulks and monolayers, the GGA+U approach will be adopted for all calculations, unless stated otherwise. S5

6 3. Physical properties of the MOX monolayers and the effect of exchange-correlation functionals and U values Table S3 Optimized lattice constant a and b (Å), electronic band-gap E g (ev) of majority and minority spins separated by a comma, magnetic moment µ (µ B ), total energy difference E F M AF M (mev/tm) between the ferromagnetic (FM) and antiferromagnetic (AFM) states, nearest exchange coupling constant J (k B *K), ground state configuration G.S., and dynamical stability. TM refers to the transition-metal atom. 2D Crystal a b E g µ E F M AF M J/link G.S. Stability CrOCl , FM Stable CrOBr , FM Stable VOCl AFM Unstable VOBr AFM Unstable TiOCl AFM Unstable TiOBr AFM Unstable Table S4 Total energy difference E F M AF M (mev/tm) between the ferromagnetic and anti-ferromagnetic states calculated from GGA+U with various U values and the LDA+U approaches. E F M AF M GGA+3eV GGA+5eV GGA+7eV LDA+7eV CrOCl CrOBr Note that, to study the effect of U values and the choice of exchange-correlation functionals on the magnetic ground-state of the CrOCl and CrOBr monolayers, we also performed various DFT calculations using the GGA+U with different U values and the local density approximation S17 plus U approaches (LDA+U). As shown in Table S4, using the GGA+U with U of 3, 5, and 7 ev, the ground state of single-layer CrOCl and CrOBr with the lowest energy is always ferromagnetic and the deviation of the E F M AF M from different U is within 1 mev/tm, suggesting that the total energy difference E F M AF M is only slightly influenced by the U values. Within the LDA+U approach, the ground state of both 2D crystals is also ferromagnetic, which is in consistent with the result from GGA+U calculations. Therefore, all these calculations further confirm the 2D robust intrinsic ferromagnetism in the CrOX S6

7 monolayers and good reliability of our work as well. S7

8 4. Calculated Curie temperature, total magnetic moment, magnetocrystalline anisotropy energy of the CrOX monolayers Table S5 Calculated T C (K), total magnetic moment µ T otal (µ B /f.u.), magnetocrystalline anisotropy energy E [1] -E [1] and E [1] -E [1] (mev/cr), easy magnetization axis EMA, and total energy E (ev/cell) with magnetization along different axes. The [1] is the out-of-plane direction. 2D Crystal T C µ T otal E [1] -E [1] E [1] -E [1] EMA E [1] E [1] E [1] CrOCl [1] CrOBr [1] Table S6 Calculated magnetocrystalline anisotropy energy E [1] -E [1] and E [1] -E [1] (mev/cr) of the CrOBr monolayer using different k-mesh and cut-off energy for the basis set. EMA refers to the easy magnetization axis. k-mesh Cut-off E [1[ -E [1] E [1] -E [1] EMA ev.29.3 [1] ev [1] ev [1] To confirm the convergence and reliability of our results, we have carefully double-checked the MAE of the CrOBr monolayer at different k-mesh and cut-off energy. As shown in Table S6, the MAE values calculated using larger cut-off energy and denser k-mesh are very close and all of them suggest an easy magnetization axis along the [1] direction, consistently supporting our conclusions. S8

9 5. Simulated mechanical exfolication process for the MOX crystals compared with graphite.35.3 Exfoliation Energy (J/m 2 ) CrOCl CrOBr VOCl VOBr Graphite d-d (Å) Figure S1 Calculated exfoliation energy vs separation distance d d in comparison with graphite, where d is the vdw gap between adjacent layers of the bulk crystals. S9

10 6. Calculated phonon dispersions and AIMD evolutions for the monolayers E 3.5 µ Y X S (c) SL CrOBr Phonon Total Energy E (ev/atom) Y X S.5 (e) SL VOCl Phonon Frequency (cm-1) X S (g) SL TiOCl Phonon 6 6 Frequency (cm-1 ) X S X S Y Y (h) SL TiOBr Phonon Time (ps) 5 Y 5 (f) SL VOBr Phonon K Frequency (cm-1) 1 Time (ps) E µ (d) SL CrOBr AIMD Frequency (cm-1) K Magnetic Moment (µ B/f.u.) Total Energy E (ev/atom) Frequency (cm-1) 6 Frequency (cm-1) (b) SL CrOCl AIMD -6.9 Magnetic Moment (µ B/f.u.) (a) SL CrOCl Phonon 7 X S Y Figure S2 Calculated phonon dispersions for all MOX monolayers and AIMD evolutions of total energy and magnetic moments for the CrOX monolayers. The insets show the snapshots of single-layer CrOX structures at 3 K after 2 ps AIMD simulations. As shown in Figure S2, apparently the CrOCl and CrOBr monolayers are dynamically stable as no imaginary modes are observed. For the other MOX monolayers, there are imags1

11 inary frequencies in the phonon dispersions, indicating that they are dynamically unstable. In addition, according to the AIMD trajectories, both CrOCl and CrOBr monolayers are also thermally stable. Therefore we focused our work on single-layer CrOCl and CrOBr as they possess both great thermal and dynamical stabilities. S11

12 7. Considered magnetic configurations for the supercells of the MOX monolayers (a) FM (b) AFM-1 (c) AFM-2 (d) AFM-3 (e) AFM-4 (f) AFM-5 y x y x Figure S3 Various spin ordering including one ferromagnetic state (a) and five antiferromagnetic states (b-f). The up-spins and down-spins on the Cr ions are denoted as purple and black arrows, respectively. We have considered six different magnetic configurations for the the supercells of the MOX monolayers, including one ferromagnetic and five antiferromagnetic states. For the CrOCl and CrOBr monolayers, the ferromagnetic state is always the ground state with the lowest total energy. S12

13 8. Calculated electronic structures of the CrOBr monolayer 6 (c) (b) (a) 4 (d) 4 Total Cr 3d O 2p Br 2p 2 Energy (ev) S (e) (f) z Y X S Density of States Y X Figure S4 Calculated electronic structures of the CrOBr monolayer with (a) majority and (c) minority spins, and (b) the spin-resolved projected density of states. The (d) and (e) are the spin-dependent charge densities, while (f) depicts the spin wave functions. The Fermi level is denoted by a dashed line at ev. S13 x

14 9. Calculated Curie temperature of the CrOX monolayers under various external strains (a) Single-layer CrOCl 2 Curie Temperature T C (K) 15 1 CrOCl, Biaxial strain CrOCl, Uniaxial strain along x CrOCl, Uniaxial strain along y Strain (%) (b) Single-layer CrOBr 2 Curie Temperature T C (K) 15 1 CrOBr, Biaxial strain CrOBr, Uniaxial strain along x CrOBr, Uniaxial strain along y Strain (%) Figure S5 Calculated T C of single-layer CrOX under biaxial strains and uniaxial strains (±5%) along the x and y axis. As illustrated in Figure S5, the applied compressive (-5%) and tensile (+5%) strains have a significant effect on the Curie temperature of the CrOX monolayers and show similar trend in both 2D crystals. The T C of both 2D crystals shows monoclinic increase under biaxial S14

15 strains and uniaxial strains along y axis from compression to tension, while it decreases slightly under uniaxial strains along the x direction. This behavior can be understood by the very anisotropic magnetic interactions of Cr ions along the x and y directions as also indicated by the different MAE values (Table S5) of the CrOX monolayers. The stronger magnetic Cr-Cr interactions along the y direction result in more subtle response to strain. While the magnetic interactions along x is much weaker and thus it is less sensitive to deformations as consistently suggested by the strain response of Curie temperatures in Figure S5. For the single-layer CrOX, their T C can be flexibly tuned between 5 and 24 K under various strains as which greatly change the magnetic energetics of the systems. Under 5% biaxial tensile strain, the T C of the CrOCl monolayer increases to 24 K, over the previous record (19 K) of the DMS, S18 indicating its great promise for future spintronics. S15

16 1. Reliability of calculations and predictions CrI 3 J=2.2 mev CrI 3 J=2.7 mev Specific Heat C V T (K) Figure S6 Calculated T C of single-layer CrI 3 with a J of 2.2 and 2.7 mev. To assess the reliability of the method for the prediction of Curie temperature, we also calculated the T C for the CrI 3 monolayer. Using the J values of 2.2 mev from Ref. S19 and 2.7 mev from Ref. S2 the calculated T C are around 51 K and 72 K, respectively, where the former is in excellent agreement the experimental measurement of 45 K S21 and the latter is higher but in the same order of magnitude, suggesting that the J is crucial for the estimation of T C. As shown in Table S4, our calculated E F M AF M is only slightly influenced by different U values within a discrepancy of 6%, indicating that our calculations are reliable. And different functional like LDA+U also gives quantitatively consistent result. S16

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