Research ArticleCONDENSED MATTER PHYSICS

Magnetic Hamiltonian parameter estimation using deep learning techniques

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Science Advances  25 Sep 2020:
Vol. 6, no. 39, eabb0872
DOI: 10.1126/sciadv.abb0872
  • Fig. 1 Schematic diagram of our study.

    (A) Data generation process showing the sampled spin configurations generated through the simulated annealing process. The color wheel indicates the in-plane magnetization directions, and the grayscale indicates the out-of-plane magnetization directions. (B and C) The training and testing processes used in this study. (D) The additional validation process with experimentally observed magnetic domain images.

  • Fig. 2 Noise resistances and network capacities.

    Estimation errors for each of β (A and B), K (C and D), and D (E and F). The networks were trained using noisy (left column) and noiseless (right column) datasets. MAEY is 〈∣YTrueYEst.∣〉εTs where Y stands for β, K, and D. The subscripts True and Est. indicate true and estimated parameters, respectively. Data points are calculated using each noisy test dataset with different εTs values.

  • Fig. 3 Input spin configurations and heatmaps of estimation errors.

    (A) Examples of simulated spin configurations and heatmaps for (B) 〈∣∆β∣〉, (C) 〈∣∆K∣〉, and (D) 〈∣∆D∣〉 represented in KN and DN parameter space when β = 0. (E) Examples of simulated spin configurations and (F to H) heatmaps representations for each 〈∣∆β∣〉, 〈∣∆K∣〉, and 〈∣∆D∣〉 when β = 3D. The noiseless images are used in (A) and (E). The values located at the same (KN, DN) point of each heatmap were averaged. (I) Three-dimensional representation of the estimation errors. Nearly 8000 data points are sampled from the test dataset to be displayed in (I), where blue, red, and green dots are projections onto each plane. Black arrows show the principal components, and the numbers near the tips of arrows show the variance ratios from PCA [the most dominant component is along (0.005, 0.689, and 0.725) with an 89.4% variance ratio].

  • Fig. 4 Estimation results based on SPLEEM images.

    (A) Schematic diagram of the Ni/[Co/Ni]2/Ir/Pt(111) system, where dIr is the thickness of iridium in the unit of monolayer (ML). (B and C) Raw (B) and postprocessed (C) images of observed magnetic domain and domain wall structures by SPLEEM. The field of view for (B) and (C) is 10 and 3 μm, respectively. White and black in the three columns in (B) correspond to magnetizations along +Sx/−Sx, +Sy/−Sy, and −Sz/+Sz, respectively. The color wheel represents the direction of in-plane magnetization in (C). (D) The K, D, and β parameter values estimated by the trained network using SPLEEM images with various Ir thicknesses. The dotted lines in (D) show the mean values for the estimated K and 2πD. The letter B on the Ir thickness axis indicates the case of a bulk Ir crystal. (E) Domain wall (DW) profiles from a SPLEEM image and simulated spin configurations using the estimated parameters when dIr = 2.5 ML. The DW profiles are shifted so that they do not overlap. The dashed lines in (E) are fitting curves using the function Tanh((x − c)/w), where c and w are fitting parameters and x is for a lateral position.

Supplementary Materials

  • Supplementary Materials

    Magnetic Hamiltonian parameter estimation using deep learning techniques

    H. Y. Kwon, H. G. Yoon, C. Lee, G. Chen, K. Liu, A. K. Schmid, Y. Z. Wu, J. W. Choi, C. Won

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