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††thanks: These authors contributed equally.††thanks: These authors
contributed equally.††thanks: These authors contributed equally.

# SUPPLEMENTARY INFORMATION: Magnetic order in 2D antiferromagnets revealed by
spontaneous anisotropic magnetostriction

Maurits J. A. Houmes  Kavli Institute of Nanoscience, Delft University of
Technology, Lorentzweg 1,
2628 CJ, Delft, The Netherlands  Gabriele Baglioni  Kavli Institute of
Nanoscience, Delft University of Technology, Lorentzweg 1,
2628 CJ, Delft, The Netherlands  Makars Šiškins  Kavli Institute of
Nanoscience, Delft University of Technology, Lorentzweg 1,
2628 CJ, Delft, The Netherlands  Martin Lee  Kavli Institute of Nanoscience,
Delft University of Technology, Lorentzweg 1,
2628 CJ, Delft, The Netherlands  Dorye L. Esteras  Instituto de Ciencia
Molecular (ICMol), Universitat de València, c/Catedrático José Beltrán 2,
46980 Paterna, Spain  Alberto M. Ruiz  Instituto de Ciencia Molecular (ICMol),
Universitat de València, c/Catedrático José Beltrán 2,
46980 Paterna, Spain  Samuel Mañas-Valero  Kavli Institute of Nanoscience,
Delft University of Technology, Lorentzweg 1,
2628 CJ, Delft, The Netherlands  Instituto de Ciencia Molecular (ICMol),
Universitat de València, c/Catedrático José Beltrán 2,
46980 Paterna, Spain  Carla Boix-Constant  Instituto de Ciencia Molecular
(ICMol), Universitat de València, c/Catedrático José Beltrán 2,
46980 Paterna, Spain  Jose J. Baldoví  Instituto de Ciencia Molecular (ICMol),
Universitat de València, c/Catedrático José Beltrán 2,
46980 Paterna, Spain  Eugenio Coronado  Instituto de Ciencia Molecular
(ICMol), Universitat de València, c/Catedrático José Beltrán 2,
46980 Paterna, Spain  Yaroslav M. Blanter  Kavli Institute of Nanoscience,
Delft University of Technology, Lorentzweg 1,
2628 CJ, Delft, The Netherlands  Peter G. Steeneken  Kavli Institute of
Nanoscience, Delft University of Technology, Lorentzweg 1,
2628 CJ, Delft, The Netherlands  Department of Precision and Microsystems
Engineering, Delft University of Technology, Mekelweg 2,
2628 CD, Delft, The Netherlands  Herre S. J. van der Zant  Kavli Institute of
Nanoscience, Delft University of Technology, Lorentzweg 1,
2628 CJ, Delft, The Netherlands  e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Contents

  1. Supplementary Note 1 Density Functional Theory calculations
  2. Supplementary Note 2 Landau theory of second-order phase transitions and spontaneous magnetostriction
  3. Supplementary Note 3 Derivation of anisotropic resonance frequency
  4. Supplementary Note 4 Anisotropic resonance frequency of FePS3 resonators
  5. Supplementary Note 5 Order parameter related frequency difference $\tilde{f}^{2}_{\text{b}}-\tilde{f}^{2}_{\theta}$
  6. Supplementary Note 6 Critical curve fit

##  Supplementary Note 1 Density Functional Theory calculations

This section summarizes the results of orbital-resolved magnetic exchange
analyses based on maximally localized Wannier functions, for both the
crystallographic and optimized structures. These results are included in
Supplementary Tables $7-42$. Supplementary Tables $43-48$ display the hopping
integrals with the S atoms (Supplementary Fig. 4) that mediate the super-
exchange interactions between magnetic centers for each channel.

SUPPLEMENTARY FIG. 1: Top view of a single-layer of MPS3 for different magnetic configurations, namely stripy, Néel, zigzag and ferromagnetic (FM). Blue and red balls represent transition metal atoms with spin up and down components, respectively. SUPPLEMENTARY FIG. 2: Electron configuration of the Co2+ magnetic ions connected by J1 (a) and J2 (b), showing parallel and antiparallel spin orientations, respectively. Green arrows illustrate the most relevant ferromagnetic superexchange channels, namely dyz-dyz and dxy-dxy (cancelled), for J1 and J2 respectively SUPPLEMENTARY FIG. 3: Electron configuration of the Ni2+ magnetic ions connected by J1 (a) and J2 (b), showing parallel and antiparallel spin orientations, respectively. Green arrows illustrate the most relevant ferromagnetic superexchange channels, namely dyz-dyz (cancelled) and dxy-dxy (cancelled), for J1 and J2 respectively. SUPPLEMENTARY TABLE 1: CoPS3, FePS3 and NiPS3 lattice parameters of the crystallographic non-magnetic (NM) and Stripy antiferromagnetic (AF) configurations | CoPS3 | FePS3 | NiPS3
---|---|---|---
Lattice parameter (Å) | a | b | a | b | a | b
Crystallographic NM | 5.895 | 10.190 | 5.947 | 10.301 | 5.812 | 10.070
AF-stripy | 5.772 | 10.003 | 5.978 | 10.172 | 5.826 | 10.084
Change (%) | -2.086 | -1.835 | +0.521 | -1.252 | +0.241 | +0.139
SUPPLEMENTARY TABLE 2: CoPS3, FePS3 and NiPS3 lattice parameters of the crystallographic non-magnetic (NM) and Néel antiferromagnetic (AF) configurations. | CoPS3 | FePS3 | NiPS3
---|---|---|---
Lattice parameter (Å) | a | b | a | b | a | b
Crystallographic NM | 5.895 | 10.190 | 5.947 | 10.301 | 5.812 | 10.070
AF-Néel | 5.885 | 10.187 | 5.947 | 10.303 | 5.834 | 10.106
Change (%) | -0.169 | -0.029 | +0.000 | +0.019 | +0.378 | +0.357
SUPPLEMENTARY TABLE 3: CoPS3, FePS3 and NiPS3 lattice parameters of the crystallographic non-magnetic (NM) and ferromagnetic (FM) configurations. | CoPS3 | FePS3 | NiPS3
---|---|---|---
Lattice parameter (Å) | a | b | a | b | a | b
Crystallographic NM | 5.895 | 10.190 | 5.947 | 10.301 | 5.812 | 10.070
FM | 5.836 | 10.103 | 5.932 | 10.278 | 5.824 | 10.089
Change (%) | -1.001 | -0.854 | -0.252 | -0.223 | +0.207 | +0.189
SUPPLEMENTARY TABLE 4: CoPS3, FePS3 and NiPS3 distances (d and d’) of the crystallographic non-magnetic (NM) and zigzag antiferromagnetic (AF) configurations. | CoPS3 | FePS3 | NiPS3
---|---|---|---
Distance (Å) | d | d’ | d | d’ | d | d’
Crystallographic NM | 3.397 | 3.402 | 3.433 | 3.433 | 3.357 | 3.356
AF-zigzag | 3.723 | 3.192 | 3.626 | 3.316 | 3.353 | 3.359
Change (%) | +9.596 | -6.172 | +5.621 | -3.408 | -0.119 | +0.089
SUPPLEMENTARY TABLE 5: CoPS3, FePS3 and NiPS3 distances (d and d’) of the crystallographic non-magnetic (NM) and Stripy antiferromagnetic (AF) configurations. | CoPS3 | FePS3 | NiPS3
---|---|---|---
Distance (Å) | d | d’ | d | d’ | d | d’
Crystallographic NM | 3.397 | 3.402 | 3.433 | 3.433 | 3.357 | 3.356
AF-stripy | 3.337 | 3.332 | 2.670 | 3.843 | 3.373 | 3.357
Change (%) | -1.766 | -2.058 | -22.225 | +11.943 | +0.477 | +0.029
SUPPLEMENTARY TABLE 6: CoPS3, FePS3 and NiPS3 distances (d and d’) of the crystallographic non-magnetic (NM) and Néel antiferromagnetic (AF) configurations. | CoPS3 | FePS3 | NiPS3
---|---|---|---
Distance (Å) | d | d’ | d | d’ | d | d’
Crystallographic NM | 3.397 | 3.402 | 3.433 | 3.433 | 3.357 | 3.356
AF-Néel | 3.393 | 3.398 | 3.436 | 3.433 | 3.369 | 3.369
Change (%) | -0.118 | -0.118 | +0.087 | +0.000 | +0.357 | +0.387
SUPPLEMENTARY TABLE 7: CoPS3, FePS3 and NiPS3 distances (d and d’) of the crystallographic non-magnetic (NM) and ferromagnetic (FM) configurations. | CoPS3 | FePS3 | NiPS3
---|---|---|---
Distance (Å) | d | d’ | d | d’ | d | d’
Crystallographic NM | 3.397 | 3.402 | 3.433 | 3.433 | 3.357 | 3.356
FM | 3.368 | 3.368 | 3.429 | 3.424 | 3.363 | 3.362
Change (%) | -0.854 | -0.999 | +0.116 | +0.262 | +0.179 | +0.179
SUPPLEMENTARY TABLE 8: Orbital resolved exchange parameters for the experimental structure of FePS3. J1 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -0.467 | -0.154 | -0.157 | -0.047 | -0.027
dxz | -0.154 | 7.091 | 0.063 | -0.307 | 0.288
dyz | -0.157 | 0.063 | 7.215 | -0.103 | 0.109
dx2y2 | -0.047 | -0.307 | -0.103 | -0.194 | -0.017
dxy | -0.027 | 0.288 | 0.109 | -0.017 | 1.612
SUPPLEMENTARY TABLE 9: Orbital resolved exchange parameters for the experimental structure of FePS3. J2 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -0.045 | -0.013 | -0.011 | -0.008 | -1.18
dxz | -0.013 | 0.334 | 0.14 | 0.0 | 0.791
dyz | -0.011 | 0.14 | 0.418 | 0.036 | 0.634
dx2y2 | -0.008 | 0.0 | 0.036 | -0.783 | 0.086
dxy | -1.18 | 0.791 | 0.634 | 0.086 | 11.016
SUPPLEMENTARY TABLE 10: Orbital resolved exchange parameters for the experimental structure of FePS3. J3 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | 0.01 | -0.058 | -0.065 | -0.028 | -0.011
dxz | -0.058 | -0.02 | -0.037 | -0.101 | -0.076
dyz | -0.065 | -0.037 | -0.022 | -0.122 | 0.089
dx2y2 | -0.028 | -0.101 | -0.122 | -0.004 | -0.041
dxy | -0.011 | -0.076 | -0.089 | -0.041 | 0.405
SUPPLEMENTARY TABLE 11: Orbital resolved exchange parameters for the experimental structure of FePS3. J4 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -0.121 | 0.002 | 0.003 | 0.013 | -0.234
dxz | 0.002 | -0.093 | -0.022 | -0.005 | -0.146
dyz | 0.003 | -0.022 | -0.081 | 0.002 | 0.122
dx2y2 | 0.012 | -0.005 | 0.002 | -0.01 | -0.18
dxy | -0.234 | -0.146 | -0.122 | -0.18 | -2.509
SUPPLEMENTARY TABLE 12: Orbital resolved exchange parameters for the experimental structure of FePS3. J5 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -2.323 | 0.019 | 0.015 | -1.175 | 0.148
dxz | 0.019 | 0.035 | 0.024 | -0.01 | -0.239
dyz | 0.015 | 0.024 | 0.022 | -0.032 | 0.154
dx2y2 | -1.175 | -0.01 | -0.032 | -0.154 | -0.138
dxy | 0.148 | -0.239 | -0.154 | -0.138 | -2.491
SUPPLEMENTARY TABLE 13: Orbital resolved exchange parameters for the experimental structure of FePS3. J6 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -0.156 | 0.03 | 0.025 | -0.0 | -0.05
dxz | 0.03 | -0.012 | 0.004 | 0.053 | -0.073
dyz | 0.025 | 0.004 | -0.01 | 0.04 | 0.042
dx2y2 | -0.0 | 0.053 | 0.04 | -5.914 | -0.044
dxy | -0.05 | -0.073 | -0.042 | -0.044 | 0.325
SUPPLEMENTARY TABLE 14: Orbital resolved exchange parameters for the spin polarized optimized structure of FePS3. J1 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -0.899 | -0.192 | -0.188 | -0.127 | -0.017
dxz | -0.192 | 8.323 | 0.035 | -0.29 | 0.306
dyz | -0.188 | 0.035 | 8.383 | -0.088 | 0.12
dx2y2 | -0.127 | -0.29 | -0.088 | -0.28 | -0.044
dxy | -0.017 | 0.306 | 0.12 | -0.044 | 1.823
SUPPLEMENTARY TABLE 15: Orbital resolved exchange parameters for the spin polarized optimized structure of FePS3. J2 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -0.104 | -0.021 | -0.019 | -0.009 | -2.114
dxz | -0.021 | 0.161 | 0.019 | -0.005 | 0.5
dyz | -0.019 | 0.019 | 0.182 | 0.002 | 0.432
dx2y2 | -0.009 | -0.005 | 0.002 | 0.128 | 0.015
dxy | -2.114 | 0.5 | 0.432 | 0.015 | 5.022
SUPPLEMENTARY TABLE 16: Orbital resolved exchange parameters for the spin polarized optimized structure of FePS3. J3 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | 0.012 | -0.083 | -0.093 | -0.059 | -0.013
dxz | -0.083 | -0.025 | -0.044 | -0.088 | -0.059
dyz | -0.093 | -0.044 | -0.027 | -0.105 | -0.064
dx2y2 | -0.059 | -0.088 | -0.105 | 0.007 | -0.038
dxy | -0.013 | -0.059 | -0.064 | -0.038 | 0.278
SUPPLEMENTARY TABLE 17: Orbital resolved exchange parameters for the spin polarized optimized structure of FePS3. J4 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -0.11 | 0.002 | 0.003 | 0.036 | -0.212
dxz | 0.002 | -0.112 | -0.039 | -0.014 | -0.061
dyz | 0.003 | -0.039 | -0.105 | 0.0 | -0.066
dx2y2 | 0.036 | -0.014 | 0.0 | -0.004 | -0.164
dxy | -0.212 | -0.061 | -0.066 | -0.164 | -1.186
SUPPLEMENTARY TABLE 18: Orbital resolved exchange parameters for the spin polarized optimized structure of FePS3. J5 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -2.613 | -0.016 | -0.014 | -1.223 | 0.154
dxz | -0.016 | -0.008 | 0.032 | -0.059 | -0.105
dyz | -0.014 | 0.032 | -0.016 | -0.075 | -0.06
dx2y2 | -1.223 | -0.059 | -0.075 | -0.181 | -0.122
dxy | 0.154 | -0.105 | -0.06 | -0.122 | -1.114
SUPPLEMENTARY TABLE 19: Orbital resolved exchange parameters for the spin polarized optimized structure of FePS3. J6 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -0.196 | 0.064 | 0.056 | -0.001 | -0.236
dxz | 0.064 | -0.026 | 0.005 | 0.083 | -0.092
dyz | 0.056 | 0.005 | -0.022 | 0.083 | -0.071
dx2y2 | -0.001 | 0.083 | 0.083 | -5.386 | -0.025
dxy | -0.236 | -0.092 | -0.071 | -0.025 | 1.036
SUPPLEMENTARY TABLE 20: Orbital resolved exchange parameters for the experimental structure of CoPS3. J1 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | 0.168 | -0.907 | -0.922 | 0.108 | -0.221
dxz | -0.907 | 7.514 | 0.105 | -0.977 | -0.417
dyz | -0.922 | 0.105 | 7.465 | -1.266 | -0.066
dx2y2 | 0.108 | -0.977 | -1.266 | -0.094 | 0.073
dxy | -0.221 | -0.417 | -0.065 | 0.073 | -0.36
SUPPLEMENTARY TABLE 21: Orbital resolved exchange parameters for the experimental structure of CoPS3. J2 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -0.371 | 0.078 | 0.066 | -0.002 | -0.242
dxz | 0.078 | 0.524 | 0.022 | 0.115 | -0.093
dyz | 0.066 | 0.022 | 0.54 | -0.106 | 0.126
dx2y2 | -0.002 | 0.115 | -0.106 | -0.687 | -0.034
dxy | -0.242 | -0.093 | 0.126 | -0.034 | -0.068
SUPPLEMENTARY TABLE 22: Orbital resolved exchange parameters for the experimental structure of CoPS3. J3 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | 0.069 | -0.117 | -0.114 | -0.047 | -0.004
dxz | -0.117 | -0.06 | -0.024 | -0.326 | -0.003
dyz | -0.114 | -0.024 | -0.059 | -0.367 | 0.01
dx2y2 | -0.047 | -0.326 | -0.367 | -0.17 | 0.003
dxy | -0.004 | -0.003 | 0.01 | 0.004 | -0.022
SUPPLEMENTARY TABLE 23: Orbital resolved exchange parameters for the experimental structure of CoPS3. J4 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -0.025 | -0.011 | -0.001 | 0.015 | 0.046
dxz | -0.011 | -0.062 | -0.004 | 0.002 | 0.047
dyz | -0.001 | -0.004 | -0.05 | -0.006 | 0.046
dx2y2 | 0.015 | 0.002 | -0.006 | -0.082 | -0.013
dxy | 0.046 | 0.047 | 0.046 | -0.013 | 0.055
SUPPLEMENTARY TABLE 24: Orbital resolved exchange parameters for the experimental structure of CoPS3. J5 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -3.482 | -0.181 | -0.163 | -0.862 | 0.13
dxz | -0.181 | -0.018 | 0.001 | -0.263 | 0.034
dyz | -0.163 | 0.001 | -0.044 | -0.18 | 0.004
dx2y2 | -0.862 | -0.263 | -0.18 | -0.016 | 0.097
dxy | 0.13 | 0.034 | 0.004 | 0.097 | 0.051
SUPPLEMENTARY TABLE 25: Orbital resolved exchange parameters for the experimental structure of CoPS3. J3 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -0.321 | 0.054 | 0.045 | 0.001 | 0.114
dxz | 0.054 | -0.018 | -0.023 | 0.138 | 0.072
dyz | 0.045 | -0.023 | -0.01 | 0.224 | 0.01
dx2y2 | 0.001 | 0.138 | 0.224 | -3.745 | -0.026
dxy | 0.114 | 0.072 | 0.01 | -0.026 | 0.029
SUPPLEMENTARY TABLE 26: Orbital resolved exchange parameters for the spin polarized optimized structure of CoPS3. J1 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -0.029 | -0.889 | -0.868 | -0.449 | -0.182
dxz | -0.889 | 10.043 | 0.01 | -0.658 | 0.055
dyz | -0.868 | 0.01 | 10.011 | -0.76 | 0.152
dx2y2 | -0.449 | -0.658 | -0.76 | 0.024 | -0.03
dxy | -0.182 | 0.055 | 0.152 | -0.03 | -0.173
SUPPLEMENTARY TABLE 27: Orbital resolved exchange parameters for the spin polarized optimized structure of CoPS3. J2 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -0.213 | 0.02 | 0.018 | 0.0 | -0.113
dxz | 0.02 | 0.106 | -0.008 | -0.014 | 0.122
dyz | 0.018 | -0.008 | 0.109 | -0.038 | 0.128
dx2y2 | 0.0 | -0.014 | -0.038 | -0.035 | 0.0
dxy | -0.113 | 0.122 | 0.128 | 0.0 | -0.199
SUPPLEMENTARY TABLE 28: Orbital resolved exchange parameters for the spin polarized optimized structure of CoPS3. J3 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | 0.029 | -0.135 | -0.136 | -0.162 | -0.027
dxz | -0.135 | -0.049 | -0.018 | -0.2 | 0.051
dyz | -0.136 | -0.018 | -0.049 | -0.215 | 0.051
dx2y2 | -0.162 | -0.2 | -0.215 | -0.089 | -0.007
dxy | -0.027 | 0.051 | 0.051 | -0.007 | 0.011
SUPPLEMENTARY TABLE 29: Orbital resolved exchange parameters for the spin polarized optimized structure of CoPS3. J4 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -0.014 | 0.002 | 0.005 | 0.068 | -0.048
dxz | 0.002 | -0.019 | -0.027 | -0.002 | 0.009
dyz | 0.005 | -0.027 | -0.014 | -0.013 | 0.016
dx2y2 | 0.068 | -0.002 | -0.013 | -0.071 | -0.021
dxy | -0.048 | 0.009 | 0.016 | -0.021 | 0.002
SUPPLEMENTARY TABLE 30: Orbital resolved exchange parameters for the spin polarized optimized structure of CoPS3. J5 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -3.726 | -0.232 | -0.228 | -1.283 | 0.014
dxz | -0.232 | -0.055 | 0.0 | -0.211 | 0.002
dyz | -0.229 | 0.0 | -0.058 | -0.183 | -0.004
dx2y2 | -1.283 | -0.211 | -0.183 | -0.06 | 0.002
dxy | 0.014 | 0.002 | -0.004 | 0.002 | 0.002
SUPPLEMENTARY TABLE 31: Orbital resolved exchange parameters for the spin polarized optimized structure of CoPS3. J6 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -0.483 | 0.099 | 0.087 | 0.001 | 0.316
dxz | 0.099 | -0.068 | 0.0 | 0.08 | 0.064
dyz | 0.087 | 0.0 | -0.066 | 0.108 | 0.041
dx2y2 | 0.001 | 0.08 | 0.108 | -3.258 | -0.01
dxy | 0.316 | 0.064 | 0.041 | -0.01 | 0.196
SUPPLEMENTARY TABLE 32: Orbital resolved exchange parameters for the experimental structure of NiPS3. J1 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -0.055 | -0.008 | -0.012 | -0.038 | 0.002
dxz | -0.008 | 0.014 | -0.002 | 0.011 | 0.001
dyz | -0.012 | -0.002 | 0.012 | 0.007 | 0.0
dx2y2 | -0.038 | 0.011 | 0.007 | -0.312 | -0.004
dxy | 0.003 | 0.001 | 0.0 | -0.004 | -0.002
SUPPLEMENTARY TABLE 33: Orbital resolved exchange parameters for the experimental structure of NiPS3. J2 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -0.161 | -0.012 | -0.01 | 0.0 | -0.011
dxz | -0.012 | -0.006 | 0.001 | 0.001 | 0.005
dyz | -0.01 | 0.001 | -0.006 | 0.001 | 0.005
dx2y2 | 0.0 | 0.001 | 0.001 | -0.1 | -0.008
dxy | -0.011 | 0.005 | 0.005 | -0.008 | 0.017
SUPPLEMENTARY TABLE 34: Orbital resolved exchange parameters for the experimental structure of NiPS3. J3 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | 0.015 | -0.0 | 0.0 | 0.004 | 0.001
dxz | -0.0 | -0.0 | -0.001 | 0.011 | -0.0
dyz | 0.0 | -0.001 | -0.0 | 0.012 | -0.0
dx2y2 | 0.004 | 0.011 | 0.012 | 0.053 | 0.007
dxy | 0.001 | -0.0 | -0.0 | 0.007 | -0.0
SUPPLEMENTARY TABLE 35: Orbital resolved exchange parameters for the experimental structure of NiPS3. J4 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | 0.204 | -0.001 | -0.0 | 0.024 | 0.003
dxz | -0.001 | -0.001 | 0.0 | -0.0 | 0.0
dyz | -0.0 | 0.0 | -0.001 | 0.0 | 0.0
dx2y2 | 0.024 | -0.0 | 0.0 | -0.028 | -0.009
dxy | 0.003 | 0.0 | 0.0 | -0.009 | -0.001
SUPPLEMENTARY TABLE 36: Orbital resolved exchange parameters for the experimental structure of NiPS3. J5 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -5.191 | -0.074 | -0.061 | -2.334 | -0.042
dxz | -0.074 | -0.006 | -0.0 | -0.031 | 0.0
dyz | -0.061 | -0.0 | -0.006 | -0.024 | -0.0
dx2y2 | -2.334 | -0.031 | -0.024 | -0.38 | -0.003
dxy | -0.042 | 0.0 | -0.0 | -0.003 | -0.001
SUPPLEMENTARY TABLE 37: Orbital resolved exchange parameters for the experimental structure of NiPS3. J6 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -0.058 | -0.002 | -0.002 | 0.0 | 0.028
dxz | -0.002 | 0.0 | 0.0 | -0.003 | 0.0
dyz | -0.002 | 0.0 | 0.0 | -0.004 | 0.0
dx2y2 | 0.0 | -0.003 | -0.004 | -10.997 | -0.092
dxy | 0.028 | 0.0 | 0.0 | -0.092 | -0.008
SUPPLEMENTARY TABLE 38: Orbital resolved exchange parameters for the spin polarized optimized structure of NiPS3. J1 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | 0.099 | 0.009 | 0.007 | -0.05 | 0.007
dxz | 0.009 | 0.004 | -0.002 | 0.044 | -0.0
dyz | 0.007 | -0.002 | 0.002 | 0.038 | -0.0
dx2y2 | -0.05 | 0.044 | 0.038 | -0.219 | -0.003
dxy | 0.007 | -0.0 | -0.0 | -0.003 | -0.004
SUPPLEMENTARY TABLE 39: Orbital resolved exchange parameters for the spin polarized optimized structure of NiPS3. J2 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -0.17 | -0.009 | -0.007 | 0.001 | 0.08
dxz | -0.009 | -0.008 | -0.002 | 0.007 | 0.005
dyz | -0.007 | -0.002 | -0.009 | 0.007 | 0.004
dx2y2 | 0.001 | 0.007 | 0.007 | 0.054 | -0.004
dxy | 0.08 | 0.005 | 0.004 | -0.004 | 0.002
SUPPLEMENTARY TABLE 40: Orbital resolved exchange parameters for the spin polarized optimized structure of NiPS3. J3 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | 0.012 | -0.002 | -0.001 | -0.013 | 0.001
dxz | -0.002 | -0.001 | -0.001 | 0.008 | -0.0
dyz | -0.001 | -0.001 | -0.0 | 0.008 | -0.001
dx2y2 | -0.013 | 0.008 | 0.008 | 0.028 | 0.01
dxy | 0.001 | -0.0 | -0.001 | 0.01 | -0.001
SUPPLEMENTARY TABLE 41: Orbital resolved exchange parameters for the spin polarized optimized structure of NiPS3. J4 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | 0.116 | 0.002 | 0.002 | 0.028 | -0.006
dxz | 0.002 | -0.002 | 0.0 | 0.0 | 0.0
dyz | 0.002 | 0.0 | -0.002 | 0.0 | 0.0
dx2y2 | 0.028 | 0.0 | 0.0 | -0.028 | -0.01
dxy | -0.006 | 0.0 | 0.0 | -0.01 | 0.0
SUPPLEMENTARY TABLE 42: Orbital resolved exchange parameters for the spin polarized optimized structure of NiPS3. J5 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -4.91 | -0.056 | -0.046 | -2.12 | -0.032
dxz | -0.056 | -0.009 | 0.0 | -0.013 | 0.0
dyz | -0.046 | 0.0 | -0.008 | -0.007 | -0.0
dx2y2 | -2.12 | -0.013 | -0.007 | -0.353 | -0.003
dxy | -0.032 | 0.0 | -0.0 | -0.003 | -0.001
SUPPLEMENTARY TABLE 43: Orbital resolved exchange parameters for the spin polarized optimized structure of NiPS3. J6 (meV) | dz2 | dxz | dyz | dx2y2 | dxy
---|---|---|---|---|---
dz2 | -0.04 | -0.002 | -0.002 | 0.0 | 0.048
dxz | -0.002 | 0.0 | 0.0 | -0.003 | 0.0
dyz | -0.002 | 0.0 | 0.0 | -0.004 | 0.0
dx2y2 | 0.0 | -0.003 | -0.004 | -10.334 | -0.07
dxy | 0.048 | 0.0 | 0.0 | -0.07 | -0.012
SUPPLEMENTARY FIG. 4: Top view of a cluster of single-layer FePS3 centered in Fe1 labelling the first Fe atoms neighbours and the S atoms that mediate super-exchange magnetic interactions. SUPPLEMENTARY TABLE 44: Hopping parameters for the crystallographic structure of FePS3. | S1(pz) | S1(px) | S1(py) | S2(pz) | S2(px) | S2(py)
---|---|---|---|---|---|---
Fe1 | dz2 | 1.089 | -0.079 | -0.243 | 0.054 | -0.012 | -0.608
dxz | -0.124 | -0.331 | -0.029 | -0.041 | -0.021 | 0.028
dyz | -0.254 | -0.100 | -0.357 | -0.427 | 0.127 | -0.002
dx2y2 | 0.002 | -0.005 | -0.037 | 0.127 | -0.086 | -0.987
dxy | 0.003 | 0.021 | 0.038 | 0.037 | -0.341 | 0.181
Fe2 | dz2 | -0.054 | 0.012 | 0.608 | -1.089 | 0.079 | 0.243
dxz | 0.041 | 0.021 | -0.028 | 0.124 | 0.331 | 0.029
dyz | 0.427 | -0.127 | 0.002 | 0.254 | 0.100 | 0.357
dx2y2 | -0.127 | 0.086 | 0.987 | -0.002 | 0.005 | 0.037
dxy | -0.037 | 0.341 | -0.181 | -0.003 | -0.021 | -0.038
SUPPLEMENTARY TABLE 45: Hopping parameters for the crystallographic structure of FePS3. | S3(pz) | S3(px) | S3(py) | S4(pz) | S4(px) | S4(py)
---|---|---|---|---|---|---
Fe1 | dz2 | 0.015 | -0.525 | -0.037 | -0.015 | 0.040 | 0.525
dxz | -0.274 | -0.023 | -0.031 | -0.020 | -0.008 | -0.029
dyz | 0.037 | 0.030 | 0.010 | 0.276 | 0.030 | 0.026
dx2y2 | -0.036 | 0.942 | 0.032 | -0.028 | 0.078 | 0.939
dxy | -0.070 | -0.043 | -0.325 | 0.075 | 0.319 | -0.078
Fe3 | dz2 | 0.023 | -0.056 | -0.562 | -0.022 | 0.562 | 0.053
dxz | 0.028 | 0.018 | 0.062 | 0.334 | -0.016 | 0.012
dyz | -0.336 | -0.011 | 0.011 | -0.049 | -0.062 | -0.018
dx2y2 | 0.041 | -0.117 | -1.048 | 0.052 | -1.050 | -0.058
dxy | -0.096 | -0.414 | 0.080 | 0.090 | 0.056 | 0.423
SUPPLEMENTARY TABLE 46: Hopping parameters for the crystallographic structure of CoPS3. | S1(pz) | S1(px) | S1(py) | S2(pz) | S2(px) | S2(py)
---|---|---|---|---|---|---
Co1 | dz2 | 1.019 | -0.281 | -0.178 | 0.032 | -0.174 | -0.581
dxz | -0.229 | -0.339 | 0.098 | -0.069 | -0.032 | 0.039
dyz | -0.284 | 0.040 | -0.341 | -0.431 | 0.024 | -0.056
dx2y2 | -0.003 | 0.003 | -0.043 | 0.059 | -0.405 | -0.902
dxy | -0.001 | 0.017 | 0.036 | -0.089 | -0.35 | 0.381
Co2 | dz2 | -0.032 | 0.174 | 0.581 | -1.019 | 0.279 | 0.178
dxz | 0.069 | 0.032 | -0.039 | 0.229 | 0.339 | -0.097
dyz | 0.431 | -0.025 | 0.056 | 0.284 | -0.041 | 0.341
dx2y2 | -0.059 | 0.406 | 0.901 | 0.003 | -0.003 | 0.043
dxy | 0.089 | 0.350 | -0.382 | 0.001 | -0.017 | -0.036
SUPPLEMENTARY TABLE 47: Hopping parameters for the crystallographic structure of CoPS3. | S3(pz) | S3(px) | S3(py) | S4(pz) | S4(px) | S4(py)
---|---|---|---|---|---|---
Co1 | dz2 | 0.017 | -0.489 | -0.033 | -0.018 | 0.035 | 0.488
dxz | -0.241 | -0.043 | -0.014 | -0.029 | -0.013 | -0.034
dyz | 0.044 | 0.037 | 0.014 | 0.243 | 0.014 | 0.046
dx2y2 | -0.042 | 0.904 | 0.011 | -0.034 | 0.051 | 0.908
dxy | -0.072 | -0.094 | -0.299 | 0.077 | 0.296 | -0.019
Co3 | dz2 | 0.015 | -0.031 | -0.534 | -0.015 | 0.534 | 0.029
dxz | 0.037 | 0.020 | 0.056 | 0.282 | 0.011 | 0.015
dyz | -0.285 | -0.015 | -0.015 | -0.054 | -0.057 | -0.020
dx2y2 | 0.037 | -0.036 | -1.011 | 0.046 | -1.005 | 0.009
dxy | -0.089 | -0.341 | 0.012 | 0.083 | 0.113 | 0.341
SUPPLEMENTARY TABLE 48: Hopping parameters for the crystallographic structure of NiPS3. | S1(pz) | S1(px) | S1(py) | S2(pz) | S2(px) | S2(py)
---|---|---|---|---|---|---
Ni1 | dz2 | 1.023 | -0.279 | -0.112 | 0.009 | -0.116 | -0.553
dxz | -0.288 | -0.316 | 0.066 | -0.055 | -0.038 | 0.047
dyz | -0.166 | 0.053 | -0.352 | -0.367 | -0.011 | 0.015
dx2y2 | 0.032 | 0.025 | -0.022 | 0.044 | -0.237 | -0.890
dxy | -0.017 | -0.008 | 0.053 | -0.057 | -0.320 | 0.262
Ni2 | dz2 | -0.008 | 0.118 | 0.553 | -1.026 | 0.275 | 0.113
dxz | 0.055 | 0.038 | -0.047 | 0.288 | 0.316 | -0.065
dyz | 0.366 | 0.009 | -0.015 | 0.166 | -0.055 | 0.353
dx2y2 | -0.043 | 0.241 | 0.889 | -0.032 | -0.025 | 0.022
dxy | 0.058 | 0.320 | -0.263 | 0.017 | 0.008 | -0.053
SUPPLEMENTARY TABLE 49: Hopping parameters for the crystallographic structure of NiPS3. | S3(pz) | S3(px) | S3(py) | S4(pz) | S4(px) | S4(py)
---|---|---|---|---|---|---
Ni1 | dz2 | 0.034 | -0.526 | -0.061 | -0.035 | 0.064 | 0.525
dxz | -0.331 | -0.053 | -0.030 | -0.011 | -0.016 | -0.031
dyz | 0.039 | 0.036 | 0.019 | 0.333 | 0.030 | 0.057
dx2y2 | -0.069 | 0.971 | 0.090 | -0.057 | 0.160 | 0.955
dxy | -0.068 | 0.003 | -0.389 | 0.080 | 0.369 | -0.170
Ni3 | dz2 | 0.032 | -0.05 | -0.556 | -0.031 | 0.557 | 0.047
dxz | 0.015 | 0.017 | 0.042 | 0.352 | 0.032 | 0.043
dyz | -0.355 | -0.042 | -0.036 | -0.045 | -0.045 | -0.020
dx2y2 | 0.051 | -0.140 | -1.025 | 0.062 | -1.036 | -0.067
dxy | -0.082 | -0.393 | 0.159 | 0.071 | 0.019 | 0.408

##  Supplementary Note 2 Landau theory of second-order phase transitions and
spontaneous magnetostriction

Magnetostriction is a coupling between magnetic and mechanical parts of our
system. This coupling can be described by an energy term in the total free
energy of our system Landau1984 . We can then write the total free energy as

$F-F_{0}=U_{\rm el}(z)+a(T-T_{\rm
N})^{2\beta}L_{i}L_{i}+BL_{i}L_{i}L_{i}L_{i}-\sigma_{ij}(z)\lambda_{ijkl}L_{k}L_{l}.$
(1)

Here $F$ is the total free energy of in the AF phase at zero magnetic field,
$F_{0}$ is the free energy of paramagnetic phase, $U_{\text{el}}(z)$ is the
elastic energy of a membrane with deflection $z$ at its centre, $T$ is the
temperature of our system and $T_{\text{N}}$ is the Néel temperature, $L_{i}$
are the components of the Néel vector, $\beta$ is a critical exponent, $a$ and
$B$ are positive constants, $\sigma_{ij}(z)$ is the stress tensor and
$\lambda_{ijkl}$ is the magnetostriction tensor. The last term couples the
stress to the Néel vector thereby describing the magnetostriction. If we
assume the Néel vector to be fully aligned with the easy axis, Eq. (1)
simplifies to:

$F-F_{0}=U_{\rm el}(z)+a(T-T_{\rm
N})^{2\beta}L^{2}+BL^{4}-\sigma_{ij}(z)\lambda_{ij}L^{2},$ (2)

where $L$ is the magnetic order parameter (i.e. the magnitude of the Néel
vector). For notational convenience we write $\lambda_{ij}$ in dropping the
third and fourth index of $\lambda_{ijkl}$ as only the component where $kl$
corresponds to the easy axis contributes. The elastic energy in a homogeneous
membrane is given by Landau_vol7

$U_{\rm
el}=\int\int\frac{S_{ijkl}}{2}\sigma_{ij}(x,y,z)\sigma_{kl}(x,y,z)dxdy,$ (3)

where the integration runs over the in plane dimensions of the membrane, $z$
is the membrane deflection at its centre and should not be confused with the
out of plane coordinate. For ease of notation we will not explicitly write the
integration and coordinate dependence from here on. Assuming the membrane
thickness does not vary significantly we can take the out of plane component
stress component to vanish, $\sigma_{zx}=\sigma_{zy}=\sigma_{zz}=0$. Eq. (3)
then simplifies to

$U_{\rm
el}=\frac{S_{xxxx}}{2}\sigma_{xx}\sigma_{xx}+\frac{S_{yyyy}}{2}\sigma_{yy}\sigma_{yy}+S_{xxyy}\sigma_{xx}\sigma_{yy}+S_{xxxy}\sigma_{xx}\sigma_{xy}+S_{yyxy}\sigma_{yy}\sigma_{xy}+\frac{S_{xyxy}}{2}\sigma_{xy}\sigma_{xy}.$
(4)

Taking our coordinates such that $x,y$ correspond with the principle stress
directions all terms containing $\sigma_{xy}$ vanish. This simplifies the
elastic energy further to

$U_{\rm
el}=\frac{S_{xxxx}}{2}\sigma_{xx}\sigma_{xx}+\frac{S_{yyyy}}{2}\sigma_{yy}\sigma_{yy}+S_{xxyy}\sigma_{xx}\sigma_{yy}.$
(5)

Assuming the material has isotropic elastic properties the relevant compliance
tensor components are

$S_{xxxx}=S_{yyyy}=\frac{1}{E}\text{ and }S_{xxyy}=\frac{-\nu}{E}.$ (6)

Substituting this in to Eq. (5) we find

$U_{\rm
el}=\frac{1}{2E}\sigma_{xx}\sigma_{xx}+\frac{1}{2E}\sigma_{yy}\sigma_{yy}-\frac{\nu}{E}\sigma_{xx}\sigma_{yy}.$
(7)

By taking the derivative of the free energy with respect to either $z$ or $L$
we find the forces acting on these degrees of freedom, $\phi_{L}$ and
$\phi_{z}$ respectively to be given by

$\displaystyle-\phi_{L}=\frac{d(F-F_{0})}{dL}=2a(T-T_{\rm
N})^{2\beta}L+4BL^{3}-2\sigma_{ij}(z)\lambda_{ij}L,$ (8)
$\displaystyle-\phi_{z}=\frac{d(F-F_{0})}{dz}=\left(S_{ijkl}\sigma_{ij}(z)+\lambda_{kl}L^{2}\right)\frac{d\sigma_{kl}(z)}{dz}.$
(9)

### Order parameter and critical exponent

In order to find an equation that describes the order parameter as a function
of temperature, we find a solution for equation (8) for the case where
$\phi_{L}=0$. Aside from the trivial solution $L=0$, we find for below the
transition the additional solution:

$L^{2}=-\frac{a}{2B}(T-T_{\rm
N})^{2\beta}+\frac{\sigma_{ij}\lambda_{ij}}{2B},$ (10)

which could be rewritten for $T^{*}_{\rm N}=T_{\rm
N}-(\frac{\sigma_{ij}\lambda_{ij}}{a})^{\frac{1}{2\beta}}$ Siskins2020  as:

$L^{2}=\frac{a}{2B}(T^{*}_{\rm N}-T)^{2\beta}.$ (11)

This equation now describes the temperature dependence of the order parameter
in a critical region near $T_{\rm N}$ with a corresponding critical exponent
$\beta$.

### Magnetostrictive strain and resonance frequency

To assess the magnetostriction contribution to strain and thus the frequency
of a rectangular membrane resonator, we need to find stiffness of the membrane
from its force-deflection equation. In doing that we analyse equation (9).
First, we describe strain equation for the rectangular membrane at its centre
as:

$\displaystyle\epsilon_{xx}(z)=\epsilon_{0,x}+\frac{c_{1}}{2}\frac{z^{2}}{l^{2}}$
(12a)
$\displaystyle\epsilon_{yy}(z)=\epsilon_{0,y}+\frac{c_{1}}{2}\frac{z^{2}}{w^{2}}$
(12b)

where $c_{1}$ is a geometrical pre-factor that describes the deflection shape
of the fundamental mode of vibration bunch2008mechanical; SiskinsAs2S32019 .
For $w\ll l$ we can neglect the $z$ dependence of $\epsilon_{xx}(z)$. Now, we
substitute (12) to (9), using the relation
$\sigma_{ij}=C_{ijkl}\epsilon_{kl}$, we find

$-\phi_{z}=\left(\frac{E}{1-\nu^{2}}\epsilon_{0,y}+\frac{\nu
E}{1-\nu^{2}}\epsilon_{0,x}-\lambda_{yy}L^{2}\right)\frac{c_{1}}{w^{2}}z+\frac{E}{1-\nu^{2}}\frac{c_{1}^{2}}{2}\frac{z^{3}}{w^{4}},$
(13)

where we used that $C_{xxxx}=C_{yyyy}=\frac{E}{1-\nu^{2}}$ and
$C_{yyxx}=C_{xxyy}=\frac{\nu E}{1-\nu^{2}}$. Eq. (13) becomes

$-\phi_{z}=k_{1}z-\frac{\lambda_{ij}c_{1}}{w^{2}}L^{2}z+k_{3}z^{3}$ (14)

where $k_{1}$ is the elastic linear stiffness and $k_{3}$ is the cubic elastic
stiffness, given by

$\displaystyle k_{1}$
$\displaystyle=\frac{E}{1-\nu^{2}}(\epsilon_{0,y}+\nu\epsilon_{0,x})\frac{c_{1}}{w^{2}}$
(15) $\displaystyle k_{3}$
$\displaystyle=\frac{E}{1-\nu^{2}}\frac{c_{1}^{2}}{2w^{4}}.$ (16)

Assuming small deflections we can neglect the $z^{3}$ contribution in eq. (13)
and find that the linear stiffness is changed with respect to the purely
elastic case. If we consider a rectangular cavity with it’s long axis is
parallel to the crystalline axis $b$ or $a$ respectively we find:

$-\phi_{z\
\text{b,a}}=\left(k_{1}-\frac{c_{1}}{w^{2}}\lambda_{\text{a,b}}L^{2}\right)z\,,$
(17)

where $\lambda_{\text{a,b}}$ are the phenomenological magnetostriction
coefficients, chosen such that to couple $a$ and $b$ crystalline directions
and $L$. That leads to a change in the effective linear stiffness $k_{b,a}$:

$k_{\text{b,a}}=k_{1}-\frac{c_{1}}{w^{2}}\lambda_{\text{a,b}}L^{2},$ (18)

which can be used to write down the frequency equations using
$f_{\text{a,b}}=\frac{1}{2\pi}\sqrt{\frac{k_{\text{a,b}}}{m}}$,

$\displaystyle f_{\text{b}}$
$\displaystyle\approx\frac{1}{2\pi}\sqrt{\frac{1}{m}\frac{c_{1}}{w^{2}}\left[\frac{E}{1-\nu^{2}}(\epsilon_{0,\text{a}}+\nu\epsilon_{0,\text{b}})-\lambda_{\text{a}}L^{2}\right]},$
(19a) $\displaystyle f_{\text{a}}$
$\displaystyle\approx\frac{1}{2\pi}\sqrt{\frac{1}{m}\frac{c_{1}}{w^{2}}\left[\frac{E}{1-\nu^{2}}(\epsilon_{0,\text{b}}+\nu\epsilon_{0,\text{a}})-\lambda_{\text{b}}L^{2}\right]},$
(19b)

where $m$ is the mass of the membrane. And define magnetostrictive strain by:

$\displaystyle\epsilon_{\rm{ms},\text{a}}$
$\displaystyle=\frac{c_{1}}{mw^{2}}\lambda_{\text{a}}L^{2},$ (20)
$\displaystyle\epsilon_{\rm{ms},\text{b}}$
$\displaystyle=\frac{c_{1}}{mw^{2}}\lambda_{\text{b}}L^{2}.$ (21)

Taking a difference of the squares of equations (19) and assuming
$\epsilon_{0,\text{a}}=\epsilon_{0,\text{b}}$, we arrive at the final
equation:

$\displaystyle
f^{2}_{\text{b}}-f^{2}_{\text{a}}=-\frac{1}{4\pi^{2}}\frac{c_{1}}{mw^{2}}\left[\lambda_{\text{b}}-\lambda_{\text{a}}\right]L^{2},$
(22)

which relates antiferromagnetic order parameter $L$ and measured resonance
frequencies of orthogonal resonators aligned to crystalline axes
$f_{\text{a,b}}$ in ordered phase. Finally, one can show that by plugging
equation (11) to (22):

$f^{2}_{\text{b}}-f^{2}_{\text{a}}\propto(T^{*}_{\rm N}-T)^{2\beta},$ (23)

which could be used to fit experimental data to extract critical exponent
$\beta$ near the phase transition temperature $T^{*}_{\rm N}$.

##  Supplementary Note 3 Derivation of anisotropic resonance frequency

Here, we derive the general equation of the resonance frequency of a
rectangular cavity oriented at an angle $\theta$ with respect to the
crystalline axes, as schematically shown in Fig. 5. The global coordinate
system is defined by the crystallographic axes a and b, along which the
material deforms resulting in stresses $\sigma_{aa}$ and $\sigma_{bb}$. The
longest side of the cavity, with length $l$, can be oriented at an arbitrary
angle $\theta$ with respect to b.

SUPPLEMENTARY FIG. 5: Schematic illustration of rectangular membrane
Rectangular membrane of width $w$ and length $l$ oriented with its long side
at an angle $\theta$ with respect to the crystalline direction $b$. The
$x$-$y$ direction refer to the main directions of the rectangular membrane.

Let us first consider a cavity oriented parallel to a crystallographic axis.
Since the membranes are very thin, we can assume that the stress in the
direction perpendicular to the plane is zero, $\sigma_{cc}=0$, the membrane’s
stress tensor can be expressed as

$\bm{\sigma}=\begin{pmatrix}\sigma_{aa}&\sigma_{ab}\\\
\sigma_{ba}&\sigma_{bb}\end{pmatrix}_{ab}\,,$ (24)

where the subscript $\big{(}\big{)}_{ab}$ indicates we expressed the stress
tensor in the basis of the crystallographic coordinate system. If we assume
that there are no shear forces acting on the crystal lattice,
$\sigma_{ab}=\sigma_{ba}=0$, there will be no shear on cavities oriented along
the main crystallographic axes. Now, if we consider a rectangular cavity
rotated by $\theta$ with respect to the crystallographic axes, we can define a
rotated xy-coordinate system oriented along the main axis of the rectangle. To
express $\bm{\sigma}$ in this coordinate system we use the tensor
transformation rule, $\sigma^{\prime}_{ij}=q_{ki}q_{lj}\sigma_{kl}$ where
$q_{ij}$ are components of the rotation tensor transforming the ab-coordinate
system, $\bm{e}$, into the xy-coordinate system, $\bm{e^{\prime}}$ as
$\bm{e}^{\prime}_{i}=q_{ij}\bm{e}_{j}$. We then get:

$\bm{\sigma}=\begin{pmatrix}\sigma_{xx}&\sigma_{xy}\\\
\sigma_{yx}&\sigma_{yy}\end{pmatrix}_{xy}=\begin{pmatrix}\cos^{2}(\theta)\sigma_{aa}+\sin^{2}(\theta)\sigma_{bb}&-\cos(\theta)\sin(\theta)\sigma_{aa}+\sin(\theta)\cos(\theta)\sigma_{bb}\\\
-\sin(\theta)\cos(\theta)\sigma_{aa}+\sin(\theta)\cos(\theta)\sigma_{bb}&\sin^{2}(\theta)\sigma_{aa}+\cos^{2}(\theta)\sigma_{bb}\end{pmatrix}_{xy}\,,$
(25)

Then, the resonance frequency of a rectangular membrane oriented at an angle
$\theta$ with respect to the crystallographic axis can be expressed as

$f_{\theta}\approx\frac{1}{2}\sqrt{\frac{1}{\rho}\left(\frac{\sigma_{xx}}{l^{2}}+\frac{\sigma_{yy}}{w^{2}}\right)}\,.$
(26)

In the case of high-aspect ratio membranes ($w\ll l$), Eq. 26 can be
approximated to

$f_{\theta}\approx\frac{1}{2}\sqrt{\frac{1}{\rho}\frac{\sigma_{yy}}{w^{2}}}=\frac{1}{2}\sqrt{\frac{1}{\rho
w^{2}}\left(\sin^{2}(\theta)\sigma_{aa}+\cos^{2}(\theta)\sigma_{bb}\right)}\,,$
(27)

which is Eq. 3 of the main text.

Now, let us consider the constitutive equations of the material:

$\displaystyle c_{1}$
$\displaystyle=E(\epsilon_{\text{fab},aa}-\epsilon_{\text{th},aa}-\epsilon_{\text{ms},aa})=E\left(\epsilon_{\text{fab},aa}-\int_{T_{0}}^{T_{1}}\alpha_{\text{a}}(T)dT-\lambda_{\text{a}}L^{2}(T_{1})\right)=\sigma_{aa}(T_{1})-\nu\sigma_{bb}(T_{1})$
(28) $\displaystyle c_{2}$
$\displaystyle=E(\epsilon_{\text{fab},bb}-\epsilon_{\text{th},bb}-\epsilon_{\text{ms},bb})=E\left(\epsilon_{\text{fab},bb}-\int_{T_{0}}^{T_{1}}\alpha_{\text{b}}(T)dT-\lambda_{\text{b}}L^{2}(T_{1})\right)=\sigma_{bb}(T_{1})-\nu\sigma_{aa}(T_{1})\,,$
(29)

where $\epsilon_{\text{fab}}$ is residual fabrication strain at $T=T_{0}$,
$\epsilon_{\text{th}}$ and $\epsilon_{\text{ms}}$ are respectively the thermal
expansion and magnetostriction contributions to strain, $\alpha$ is the
thermal expansion coefficient, $\lambda$ the magnetostriction coefficient and
$E$ is the Young’s modulus, which is assumed to be isotropic. We can thus
write

$\displaystyle\sigma_{aa}$ $\displaystyle=c_{1}+\nu\sigma_{bb}$ (30)
$\displaystyle\sigma_{bb}$ $\displaystyle=c_{2}+\nu\sigma_{aa}\,,$ (31)

which can be combined in the following expressions for $\sigma_{aa}$ and
$\sigma_{bb}$:

$\displaystyle\sigma_{aa}$ $\displaystyle=\frac{c_{1}+\nu c_{2}}{1-\nu^{2}}$
(32) $\displaystyle\sigma_{bb}$ $\displaystyle=\frac{c_{2}+\nu
c_{1}}{1-\nu^{2}}\,.$ (33)

We can now rewrite Eq. 27 in terms of the different contributions to strain,
i.e. residual strain from fabrication ($\epsilon_{\text{fab}}$), thermal
expansion ($\propto\alpha$) and magnetostriction ($\propto\lambda$):

$\displaystyle f_{\theta}(T)$ $\displaystyle=\frac{1}{2}\sqrt{\frac{E}{\rho
w^{2}(1-\nu^{2})}\left[\sin^{2}\theta(c_{1}+\nu
c_{2})+\cos^{2}\theta(c_{2}+\nu c_{1})\right]}$ (34)
$\displaystyle=\frac{1}{2}\sqrt{\frac{E}{\rho
w^{2}(1-\nu^{2})}[(\sin^{2}\theta+\nu\cos^{2}\theta)(\epsilon_{\text{fab},aa}-\epsilon_{\text{th},aa}-\epsilon_{\text{ms},aa})+(\cos^{2}\theta+\nu\sin^{2}\theta)(\epsilon_{\text{fab},bb}-\epsilon_{\text{th},bb}-\epsilon_{\text{ms},bb})]}\,.$
(35)

Which is consistent with Eq. (19). We can eliminate the pretension
$\epsilon_{\text{fab}}$ terms by considering
$\tilde{f_{\theta}}(T)=f_{\theta}(T)-f_{\theta}(T_{0})$. In the following, we
assume that the only anisotropic temperature-dependent contribution to the
total strain comes from magnetostriction, thus we take
$\epsilon_{\text{th},aa}=\epsilon_{\text{th},bb}=\epsilon_{\text{th}}$. By
definition of $\theta$ we have $b\rightarrow\theta=0^{\circ}$ and
$a\rightarrow\theta=90^{\circ}$. From Eq. (35), we then find that
$\tilde{f}^{2}_{\text{a}}-\tilde{f}^{2}_{\text{b}}$ becomes

$\displaystyle\tilde{f}^{2}_{\text{a}}-\tilde{f}^{2}_{\text{b}}$
$\displaystyle=\frac{E}{4\rho
w^{2}(1+\nu)}\big{(}-\epsilon_{\text{ms},aa}+\epsilon_{\text{ms},bb}\big{)}$
(36) $\displaystyle=-\frac{E}{4\rho
w^{2}(1+\nu)}(\lambda_{\text{a}}-\lambda_{\text{b}})L^{2}$ (37)

from which we can directly extract the order parameter. The thermal expansion
contribution to strain $\epsilon_{\alpha}$ is proportional to the integral
over the temperature of the thermal expansion coefficient $\alpha$, which is
proportional to the Debye specific heat, $C_{\text{Debye}}$, via the Grünesen
parameter. Thus, the derivative with respect to temperature of
$f_{\theta}^{2}$

$\displaystyle\frac{df_{\theta}^{2}}{dT}$ $\displaystyle=\frac{E}{4\rho
w^{2}(1-\nu^{2})}\left[(\sin^{2}\theta+\nu\cos^{2}\theta)\left(-\alpha-\lambda_{\text{a}}\frac{dL^{2}}{dT}\right)+(\cos^{2}\theta+\nu\sin^{2}\theta)\left(-\alpha-\lambda_{\text{b}}\frac{dL^{2}}{dT}\right)\right]$
(38) $\displaystyle=\frac{-E}{4\rho
w^{2}(1-\nu^{2})}\left[\alpha(1+\nu)+\left(\sin^{2}\theta(\lambda_{\text{a}}+\nu\lambda_{\text{b}})\frac{dL^{2}}{dT}+\cos^{2}\theta(\lambda_{\text{b}}+\nu\lambda_{\text{a}})\frac{dL^{2}}{dT}\right)\right]\,.$
(39)

can be fitted to $b_{1}C_{\text{Debye}}+b_{2}\dfrac{dL^{2}}{dT}$ where $b_{1}$
and $b_{2}$ are fit parameters, and $\dfrac{dL^{2}}{dT}$ is estimated from Eq.
37. The results of these fits along to measured data of
$\dfrac{df_{\theta}^{2}}{dT}$ are shown in Fig. 6.Polar plot of the resulting
$b_{1}(\theta)$ and $b_{2}(\theta)$ are shown in Fig. 7, which confirm that
the thermal contribution to strain does not exhibit significant anisotropic
behavior.

SUPPLEMENTARY FIG. 6: Angle-resolved $\dfrac{df^{2}}{dT}$: Plot of measured
(light blue dots)$\dfrac{df^{2}}{dT}$ and fit to
$b_{1}C_{\text{Debye}}+b_{2}\dfrac{dL^{2}}{dT}$ (blue full line) for all
angles of a star-shaped array of CoPS3.

From Eq. 42 and 40, the expected angle dependance of the parameter $b_{2}$ is

$b_{2}(\theta)=\left(\frac{E}{4\rho
w^{2}(1-\nu^{2})}\right)^{2}(1-\nu)(\lambda_{\text{a}}-\lambda_{\text{b}})[(\sin^{2}\theta(\lambda_{\text{a}}+\nu\lambda_{\text{b}})+\cos^{2}\theta(\lambda_{\text{b}}+\nu\lambda_{\text{a}})]\,,$
(40)

which we use to fit $b_{2}(\theta)$ in Fig.7 to
$A\sin^{2}\theta+B\cos^{2}\theta$ where

$\frac{A}{B}=\frac{\lambda_{\text{a}}+\nu\lambda_{\text{b}}}{\lambda_{\text{b}}+\nu\lambda_{\text{a}}}\,.$
(41)

The fit yields $A/B=-2.062$ and -1.798 for CoPS3 and $A/B=-5.025$ and -8.695
for FePS3.

SUPPLEMENTARY FIG. 7: Polar plot of fit parameters $b_{1}$ and $b_{2}$: Polar
plot of fit parameters $b_{1}$ and $b_{2}$ from the fits to $df^{2}/dT$ shown
in Fig. 6.

##  Supplementary Note 4 Anisotropic resonance frequency of FePS3 resonators

Figure 8 shows resonance frequency data measured on FePS3 star-cavity
resonators, as presented in Fig. 2 of the main text for CoPS3 and NiPS3
samples. Figure 8a shows the temperature dependence of the resonance frequency
of membranes oriented along the $a$-axis (in red) and $b$-axis (in blue).
Similarly to CoPS3, the opposite strain along $a$ and $b$ arising from
spontaneous magnetostriction results in opposite behaviour of the resonance
frequency near the transition temperature $T_{\text{N}}$. The resulting
anisotropic response is further illustrated in the map plot of resonance
frequency as a function of temperature and angles in Fig. 8b and in the polar
plot of $f_{\text{res}}(T)-f_{\text{res}}(140\,\mathrm{K})$ in Fig. 8c.

SUPPLEMENTARY FIG. 8: Angle-resolved resonance frequency data of FePS3
membranes. a Temperature dependence of $f_{\text{res}}$ of a FePS3 rectangular
membrane orientated at 0∘ (blue) and 90∘ (red) with respect to the b
crystallographic axis. Note that $f_{0,0^{\circ}}$ and $f_{0,90^{\circ}}$ are
proportional to the strain $\epsilon_{\text{a}}$ and $\epsilon_{\text{b}}$
respectively (see Eq. 2). The dashed-dotted grey line indicates the transition
temperature $T_{\text{N}}$. b Resonance frequency difference,
$f_{\text{res}}(T)-f_{\text{res}}(140\,\mathrm{K})$, as a function of angle
$\theta$ with respect to $b$-axis and temperature. c Polar plot of
$f_{\text{res}}(T)-f_{\text{res}}(140\,\mathrm{K})$ taken along the red dashed
line in (b).

##  Supplementary Note 5 Order parameter related frequency difference
$\tilde{f}^{2}_{\text{b}}-\tilde{f}^{2}_{\theta}$

In the main text, we have shown how to relate the difference
$f^{2}_{\text{b}}-f^{2}_{\text{a}}$ to the antiferromagnetic order parameter
through the magnetostriction induced strain at the phase transition
$\epsilon_{\text{ms},aa}=\lambda_{\text{a}}L^{2}$ and
$\epsilon_{\text{ms},bb}=\lambda_{\text{b}}L^{2}$. In general (see derivation
in Supplementary Note 2), $\tilde{f}^{2}_{\text{b}}-\tilde{f}^{2}_{\theta}$ is
also proportional to $L^{2}$, where $\tilde{f}^{2}_{\theta}$ is the pretension
corrected resonance frequency of a rectangular cavity oriented at an angle
$\theta$ with respect to the $b$-axis. We show this quantity for the CoPS3 and
FePS3 star-cavity resonators in Fig. 9a,d by plotting
$\tilde{f}^{2}_{\text{b}}-\tilde{f}^{2}_{\theta}$ as a function of angle and
temperature in Fig. 9b,e. Figure 9c,f shows the polar plot of
$\tilde{f}^{2}_{\text{b}}-\tilde{f}^{2}_{\theta}$ taken along the red dashed
line in 9b,e.

SUPPLEMENTARY FIG. 9: Angle-resolved data of
$f^{2}_{\text{b}}-f^{2}_{\theta}\propto L^{2}$ of CoPS3 and FePS3 membranes.
a,c Optical image of the CoPS3 (a) and FePS3 (c) resonators. Scale bar 12
$\mu$m. b Resonance frequency difference,
$f^{2}_{\text{b}}-f^{2}_{\theta}\propto L^{2}$, as a function of angle
$\theta$ and temperature of the CoPS3 sample in (a). c Polar plot of
$f^{2}_{\text{b}}-f^{2}_{\theta}$ taken along the red dashed line in (b). e,f
follows the same structure as (b,c) for the FePS3 sample in (c).

This behaviour is observed for the thicker samples ($t>10$ nm) and it is
exploited to have a better estimate of the critical parameters $\beta$ and
$T_{\text{N}}$ as discussed in Supplementary Note 6. For thinner resonators
every irregularity, like wrinkles or tears, can strongly affect their mode
shapes. In some cases, these imperfections can drastically change the
resonance frequency of the fundamental mode, as well as its temperature
dependence. Therefore, when analysing the critical behaviour of thin flakes,
we choose only the most pristine and unaffected membranes out of all
fabricated out of a single flake to be sure we are not affected by these
irregularities.

##  Supplementary Note 6 Critical curve fit

To extract critical parameters $\beta$ and $T_{\text{N}}$ shown in Fig. 3 and
4 of the main text, we fit the order parameter related difference
$f^{2}_{\text{b}}-f^{2}_{\text{a}}\propto L^{2}$ to the power law
$A_{\theta}(1-T/T_{\text{N}})^{2\beta}$. The experimental determination of
critical parameters is often debated due to the difficulty of extracting from
one set of data, three strongly correlated parameters, $\beta$, $T_{\rm N}$
and $A_{\theta}$. In addition, finite size effects are known to smear the
transition which usually results in a non-zero tail of the order parameter in
the disordered state and makes it harder to unambiguously determine the
critical temperature. Also, the choice of the temperature interval for the fit
is not universal and it is often arbitrary.

In order to have a better estimate of the critical exponents from our
experiments, we compute $f^{2}_{\text{b}}-f^{2}_{\theta}$ for all $\theta$ in
a star and fit the data to $A_{\theta}(1-T/T_{\text{N}})^{2\beta}$. For each
star, we then calculate the average value and standard deviation of the
critical parameters $T_{\rm N}$ and $\beta$ weighted by the error from the fit
$T_{\text{N,err}}$ and $\beta_{\text{err}}$.

We start with an initial guess, $T^{*}_{\text{N}}$, for $T_{\text{N}}$ by
extracting the maximum of the derivative of $f^{2}_{\text{b}}-f^{2}_{\theta}$
with respect to temperature as shown in Fig. 10. We then fit the
$A_{\theta}(1-T/T_{\text{N}})^{2\beta}$ to $f^{2}_{\text{b}}-f^{2}_{\theta}$
over the range $[\alpha T^{*}_{\text{N}},T^{*}_{\text{N}}]$, for $\alpha$
varying between $[0.85,0.95]$ allowing $A,T_{\text{N}},\beta$ to vary. We then
define the total error for each $\alpha$ to be
$T_{\text{N,err}}+\beta_{\text{err}}$, where $T_{\text{N,err}}$,
$\beta_{\text{err}}$ are the standard deviation errors of the fit. We then
take the extracted $T_{\text{N}},\beta$ with corresponding $T_{\text{N,err}}$,
$\beta_{\text{err}}$ to be ones given the fit corresponding to the $\alpha$
minimizing the total error. We repeat this process for each $\theta$ yielding
a distribution of $T_{\text{N}},\beta$. We then extract a weighted mean of
this distribution as follows:

$\overline{\beta}=\frac{1}{N}\sum_{\theta}\beta\frac{\beta_{\text{err,min}}}{\beta_{\text{err}}}$
(42)

Where $N$ is the number of cavity pairs and $\beta_{\text{err,min}}$ the
$\beta_{\text{err}}$ of the pairing with smallest $\beta_{\text{err}}$. We
then fit a normal distribution with $\overline{\beta}$ as mean to the
distribution of $\beta$ where weigh each $\beta$ by
$\frac{\beta_{\text{err,min}}}{\beta_{\text{err}}}$, from which we extract the
standard deviation. Repeating this process for $T_{N}$. The resulting
parameters for each sample are listed in Table 50.

SUPPLEMENTARY FIG. 10: First estimate of $T_{\rm N}$ from the derivative of the order parameter. Material | Sample |  $t$ (nm) | $\beta$ |  $T_{\text{N}}$ (K)
---|---|---|---|---
CoPS3 | 1 | 33 | 0.28 $\pm$ 0.017 | 115.7 $\pm$ 0.38
CoPS3 | 2 | 52 | 0.311 $\pm$ 0.025 | 117 $\pm$ 0.35
CoPS3 | 3 | 52 | 0.298 $\pm$ 0.034 | 116.1 $\pm$ 0.18
CoPS3 | 4 | 8 | 0.195 $\pm$ 0.0446 | 107.8 $\pm$ 2.65
CoPS3 | 5 | 8.6 | 0.218 $\pm$ 0.002 | 102.5 $\pm$ 1.3
FePS3 | 1 | 60 | 0.208 $\pm$ 0.0328 | 112.7 $\pm$ 0.87
FePS3 | 2 | 40 | 0.203 $\pm$ 0.03 | 109.1 $\pm$ 0.37
FePS3 | 3 | 10 | 0.194 $\pm$ 0.023 | 110.2 $\pm$ 0.48
FePS3 | 4 | 7 | 0.206 $\pm$ 0.047 | 107.9 $\pm$ 0.76
NiPS3 | 1 | 48 | 0.218 $\pm$ 0.016 | 150.7 $\pm$ 0.7
SUPPLEMENTARY TABLE 50: Critical exponents of MPS3 samples. Critical
exponents, $\beta$ and $T_{\text{N}}$, for CoPS3, FePS3 and NiPS3 samples of
different thicknesses, extracted following the procedure described in
Supplementary Information Supplementary Note 6

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