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1 Sample pepaations Fe 0.5 Co 0.5 Si single cystal was gown by the floating zone technique. The phase puity and cation concentations wee checked by powde X-ay diffaction and Enegy Dispesive X-ay spectoscopy (EDX), espectively. Bulk popeties of the cystal wee confimed to be identical with those epoted in liteatue 1. Electon-tanspaent thin plate with thickness of about 20 nm was pepaed by mechanical polishing and subsequent agon-ion thinning with an acceleation voltage of 4 kv at low tempeatues. Loentz TEM micoscopy and quantitative evaluation of Skymion cystal in Fe 0.5 Co 0.5 Si The schematic of Loentz tansmission electon micoscopy (TEM) is illustated in the supplemental Fig. 1. When the electon beam was iadiated to a feomagnet, Loentz foce induced by the magnetic components nomal to the incident beam, should deflect the electon beam, esulting in the convegence/divegence images of magnetic domain walls at defocused image plane 2. In this study, we changed the objective lens cuent to obtain the vaious magnetic fields along the z-axis, which wee applied to thin plate Fe 0.5 Co 0.5 Si. Moeove, quantitative evaluation of Skymion stuctue, i.e., a lateal magnetization distibution map in Skymion cystal was achieved by using QPt 3, whee thee Loentz micogaphs (unde-, in- and ove-focused images) wee analyzed using the tanspot-of-intensity equation (TIE Eq.) 4-5 2π I( λ = xy [ I( ( ], (1) xyφ that was deived fom Schödinge equation unde the small angle appoximation when the optical wave popagates though a phase object 4. I ( and φ ( stand fo the 1

2 intensity and phase distibutions of popagating optical wave, espectively. TIE Eq. was ecently applied to the simulation fo TEM obsevation of the thin plate 5, which can be also viewed as a weak phase object. On the othe hand, accoding to the Maxwell-Ampée equations, in a magnetic field, φ( and the magnetization M has a elation as e xy φ( = - ( M n) t, h (2) whee t is the thickness of the thin-plate sample, n the unit vecto paallel to the beam diection. In ode to find the magnetization ( M ) distibution in x-y plane, we obtained I(, the change of electon intensity along z-axis, by analyzing the Loentz micoscopic images. Then by substituting I( Finally, we substituted φ( into Eq. (2) and found M. into Eq. (1), we obtained φ (. To get I( fom the expeimental data, we expessed I( appoximately as I I( x, y, z + Δz) - I( x, y, 2Δz 0 z 0 - Δz), (3) whee 2Δz is the distance of the ove-focused and unde-focused planes and Δz << z 0, with z 0 the focus distance of the objective lens. Since the electostatic potential is a function of sample thickness, although it is much smalle than the high electon enegy and the sample is thin enough, it should incease the noises in the phase distibution image. To get a good estimation of the phase distibution, TEM obsevation was caied out below the uppe limit of the defocus distance which depends on the expeimental condition and the initial magnetic popety of the thin plate. 2

3 Supplemental Fig. 2 (a-c) shows thee Loentz micogaphs unde a magnetic field applied nomal to the thin plate sample. The Skymion lattice was obseved in the defocused images (Figs. 2a and 2b). The contast is invesed in those two images, indicating a change in the electon intensity between unde- and ove-focused image planes. The magnetization distibution was obtained by the afoe-mentioned TIE method and is shown in supplemental Fig. 2d. Such a TIE image demonstates that the magnetic components in each spin paticle ae pependicula to the extenal magnetic field expect fo the coe and the peipheal egion of paticle, whee the magnetic component is paallel to the field. In the aay of such hexagonal spin paticles, namely Skymions, magnetic defects (indicated by yellow aows) also exist. The possible oigin of such defects may come fom the layeed disode of Skymion. Compaing the Skymion stuctues of the (001) plane with those of (110) plane (see supplemental Fig. 3), we find that two-dimensional Skymion density, evaluated as Skymion numbe 6, is highe on the (001) plane. We attibute this to the fact that moe Skymion cystal defects have developed in the (110) plane. Supplemental Fig. 4 epesents the fomation of Skymion cystal unde a pependicula magnetic field. A systematic investigation of the magnetic field dependence of the Skymion cystal stuctue eveals that multi-domains wee ceated in the odeed Skymion cystal aound the dislocations of undelying helical stuctue (indicated by white aows in supplemental Fig. 3a). It was found that the Skymions stat to emege aound such dislocations when inceasing the magnetic field up to 20 mt. The coexistence of helical stuctue and Skymion lattice was also obseved in the ange of magnetic field fom 20 mt to 40 mt. By futhe inceasing the magnetic field to 50 mt, Skymion cystals ae geneated and the stipy stuctue is completely eplaced. Howeve, as displayed in supplemental Figs. 3d and 3e, multi-domains and domain 3

4 boundaies ae also obsevable. Inteestingly, the locations of the domain boundaies almost coincide with the afoe-mentioned dislocations. In such Skymion cystal multi-domains, as maked by colo hexagons, the hexagonal lattices of the single domains ae also seen to otate slightly fom each othe. Futhemoe, as the applied magnetic field is high enough above 70 mt, the Skymion density deceases, accompanied by the incease of feomagnetic domain. Accodingly, we conclude that a dislocation in a pope scew stuctue assists the fomation of Skymion cystals but builds multi-domains in them. Monte Calo (MC) simulation Spin inteactions in Fe 0.5 Co 0.5 Si can be modeled by the following thee-dimensional (3D) lattice Hamiltonian H 3D (4) = J S S+ axˆ + S+ ayˆ + S+ azˆ ) K ( S S+ axˆ xˆ + S S+ ayˆ yˆ + S S+ ( azˆ zˆ) H S The -sites span the L L t cubic lattice. Feomagnetic exchange J, Dzyaloshinskii-Moiya anisotopic exchange K, and the Zeeman coupling to the extenal magnetic field H ae included in the model. Assuming slowly vaying spin configuation esults in the continuum model given in the text, 3 ( J / a) H = d ( M ) + ( K / a ) M ( M ) ( H / a ) M 2. (5) The lattice Hamiltonian Eq. (4) is egaded as a lattice adaptation of the continuum Hamiltonian Eq. (5), once appopiate identification between the lattice and continuum 2 3 model paametes is made as ( J / a, K / a, H / a ) ( J, K, H ). The lattice lattice continuum 4

5 spacing a is the typical size of the block inside which micoscopic spins can be teated as constant. Simulations wee fist pefomed by inceasing the laye thickness t fom 1 to d - the wavelength of the spial spin modulation. Up to this thickness the esulting low-tempeatue spin pattens within each laye ae the same as those of a single two-dimensional plane. Thus we can educe the model to a two-dimensional (2D) one by tuning off the ielevant inteaction fo the z-diection bonds and teating all the H J S S + S ) K ( S S xˆ + S S yˆ) H S. (6) ( 2 D = + axˆ + ayˆ + axˆ + ayˆ spins shaing the same ( x, y) coodinate as a single spin. The paametes of the 2D model ae obtained fom those in the 3D model though multiplying by the laye thicknesst. Each classical spin S in the 2D model is taken to have the unit length, ( S ) 2 = 1. Peiodic bounday conditions wee imposed on a L L lattice and single-flip Metopolis algoithm was used thoughout the calculation. The atio K / J was chosen to yield the spial popagation wave vecto, k = ( k, k) with k detemined by tan( k) = K / 2J. Wavelengths of d = 6 ( k = 2π / 6 ) lattice constants on 36x36 lattice and d = 10 ( k = 2 π /10 ) on 30x30 lattice wee used fo most of the simulation esults. These values of d ae lage enough fo the continuum appoximation Eq. (5) to be justified. Magnetic field oientation was fixed along the z-diection, as in the expeimental setup. Two independent schemes wee employed to check the consistency of the simulation: (1) Tempeatue sweep: At each fixed K/J and H, tempeatues wee gadually loweed down to T/J = 0.01 duing the MC update to each the coect gound state. (2) Field 5

6 sweep: At a fixed tempeatue T, magnetic field stength H was gadually inceased fom zeo to a lage enough value to yield a completely feomagnetic state. Equilibium situation fo a given paamete set was typically eached well within the fist 10 5 MC steps, and the next 10 5 MC configuations wee used to compute physical quantities such as the Skymion numbe. Phase diagam shown in Fig. 3 was obtained fom the combination of tempeatue and field sweep MC pocedues outlined above. The Skymion cystal (SkX) phase bounday was identified fom a combination of the inspection of the eal-space image of a tiangula lattice stuctue of Skymions, the hexagonal Bagg pattens in the Fouie analysis, and a shap incease/decease in the calculated Skymion numbe. The phase bounday sepaating the helical spin (H) phase fom the H+Sk egion in Fig. 3h is chaacteized by the appeaance of isolated Skymions foming defects in the spin stipe stuctue. The defects incease in density upon inceasing the field stength and eventually fill the whole space as a hexagonal aay when SkX phase is established. The egion intevening SkX and feomagnetic phase (FM+Sk in Fig. 3h) is chaacteized by the thinning out of the Skymions as well as disodeing in thei positions. At high tempeatue whee the SkX phase is no longe identifiable, the cossove fom H+Sk to FM+Sk occus in a continuous manne. Refeences [1] Onose, Y., et al., Phys. Rev. B 72, (2006) [2] P. J. Gundy and R. S. Tebble, Adv. Phys. 17 (1968) 153. [3] M.R. Teague, J. Opt. Soc. Am. 73 (1983) [4] K. Ishizuka and B. Allman, J. Electon Micosc. 54 (2005) 191. [5] S. Bajt, et al., Uitamicoscopy 83 (2000) 67. [6] M. Onoda, et al., J. Phys. Soc. Jpn. 73 (2004)

7 Electon beam Thin plate Objective lens B y x I Image plane z Supplemental Figue 1: Sketch of Loentz tansmission electon micoscopy. Unde-focused a Ove-focused b In-focused c TIE d [010] [100] 200 nm Supplemental Figue 2: Skymion cystal stuctue in Fe 0.5 Co 0.5 Si unde an extenal magnetic field of 50 mt nomal to the image plane. a-c: The unde-, ove- and in-focused Loentz TEM images. d: Tanspot-of-intensity equation (TIE) image obtained by using the commecial softwae, QPt, based on such thee Loentz TEM images. The defects of Skymion cystal ae signed by yellow aows. 7

8 a c b [110] [110] [001] d Supplemental Figue 3: Skymion cystal stuctue (a-b) and schematic of hexagonal votex stuctue (c-d) in Fe 0.5 Co 0.5 Si at 25 K unde an applied field of 50 mt. (c) and (d) epesent the schematics of odeed and disodeed votex stuctues, espectively. 10 mt a 20 mt b 30 mt c [010] [100] 50 mt d 60 mt e 70 mt f [010] [100] Supplemental Figue 4: Fomation of Skymion cystal in Fe 0.5 Co 0.5 Si unde extenal magnetic fields which wee applied along the c-axis. Dislocations of the magnetic stuctue and the domain boundaies of Skymion cystal ae shown by white aows and colo dashed lines, espectively. 8