A generic theory of skyrmion crystal formation
AA generic theory of skyrmion crystal formation
X. C. Hu,
1, 2
H. T. Wu,
1, 2 and X. R. Wang
1, 2, ∗ Physics Department, The Hong Kong University of Science and Technology (HKUST), Clear Water Bay, Kowloon, Hong Kong HKUST Shenzhen Research Institute, Shenzhen 518057, China (Dated: January 27, 2021)A generic theory of skyrmion crystal (SkX) formation in chiral magnetic films is presented. Wenumerically demonstrate that a chiral film can have many metastable states with an arbitrarynumber of skyrmions up to a maximal value. A perpendicular magnetic field plays a crucial role inSkX formation. The energy of a film increases monotonically with skyrmion number at zero fieldwhile the film with Q m skyrmions has the lowest energy in a magnetic field. Q m first increaseswith the magnetic field up to an optimal value and then decreases with the field. Outside of a fieldwindow, helical states of low skyrmion number densities are thermal equilibrium phases while anSkX is metastable. Within the field window, SkXs are the thermal equilibrium states below theCurie temperature. However, the time to reach the thermal equilibrium SkX states from a helicalstate would be too long at a low temperature. This causes a widely spread false belief that SkXsare metastable and helical states are thermal equilibrium phase at low temperature and at theoptimal field. Our findings explain well the critical role of a field in SkX formation and fascinatingthermodynamic behaviours of helical states and SkXs. Our theory opens a new avenue for SkXmanipulation and skyrmion-based applications. Skyrmion crystals (SkXs) have attracted enormous at-tention in the past decade not only because they estab-lished the notion of skyrmion [1–6] but also because theyare good platform for studying various physics [7–11]such as the emergent electromagnetic fields and topo-logical Hall effect [11–13]. Most of our knowledge arephenomenological and yet to be confirmed. The existingskyrmion zoo consists of three types of circular skyrmionsin chiral magnets [11, 14, 15], namely, Bloch skyrmions,hedgehog skyrmions, and anti-skyrmions. Whether thereexist new members in the zoo is an open question. SkXshave been observed in many magnetic materials withDzyaloshinskii-Moriya interaction (DMI)[1–6] or with ge-ometric frustration [16, 17], but their underneath physicshas not been adequately revealed.SkXs are formed only under the assistance of an opti-mal magnetic field and the temperature to date. Onceformed, however, SkXs can be metastable states in verylarge temperature-field regions [11]. For example, an SkXcan exist at zero magnetic field and a field much higherthan the optimal value by cooling an SkX at the optimalfield to low temperature followed by removal or rise of themagnetic field. An SkX disappears during the zero fieldwarming and high field warming [11, 18, 19]. An SkX isbelieved to be a thermal equilibrium state only in a verynarrow magnetic-field range and near the Curie tempera-ture. Outside of the range, SkXs can only be metastablestates that are far from equilibrium. At zero field andthe temperature much lower than the Curie temperature,the equilibrium phase is believed to be a collection ofstripe spin textures known as a helical state [20, 21]. Allthese strange thermodynamic-path-dependent phenom-ena have not been satisfactorily explained yet.The thermal equilibrium state of a system comes fromthe competition between the internal energy and entropy. Entropy dominates the higher-temperature phase whileinternal energy determines the lower-temperature one.The entropy of a less-organized structure such as a he-lical state is larger than a well-organized structure suchas an SkX. Current views about SkXs are not consis-tent with this principles [11]. For example, as a thermalequilibrium state near the Curie temperature and at anoptimal field, an SkX of low entropy structure must havea lower energy than a helical phase of relative high en-tropy structure. This putative conclusion contradicts thepopular belief that the SkX becomes a metastable at alower temperature and at the same optimal magnetic fieldwhile the helical phase suddenly takes over the positionof the thermal equilibrium state under the same condi-tions from the SkX. Obviously, a good understanding ofSkXs and their formation is yet uncovered.In this letter, we show that, in contrast to the commonbelief that all skyrmions are circular, we show that stripesare the natural shapes of skyrmions when their formationenergies are negative. So-called helical state is just a col-lection of stripe skyrmions, similar to an SkX that is acollection of disk-like skyrmions originated from the com-pression of highly packed skyrmions. A given chiral mag-netic film with an arbitrary number of skyrmions up toa critical value is metastable. The energy and morphol-ogy of theses metastable states depend on the skyrmionnumber and the magnetic field perpendicular to the film.At zero field, the energy increases with skyrmion numberor skyrmion number density. Thus thermal equilibriumphase below the Curie temperature at zero field shouldbe helical states consisting of a few stripe skyrmions. Atnon-zero field, the film with Q m skyrmions has the low-est energy. Q m first increases with the field up to anoptimal value then decreases with the field. Q m near theoptimal field is large enough such that the average dis- a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n tance between two nearby skyrmions is comparable withskyrmion stripe width, and skyrmions form an SkX. Al-though the thermal equilibrium state should be an SkXnear the optimal field and below the Curie temperature,an SkX appears thermodynamically only near the Curietemperature, and it takes too long for the system to reachthis equilibrium state from a helical state at low temper-ature due to topological protection.We consider a magnetic thin film of thickness d in xy plane. The magnetic energy of the film is E = d (cid:90) (cid:90) { A |∇ (cid:126)m | + D [( (cid:126)m · ∇ ) m z − m z ∇ · (cid:126)m ]+ K (1 − m z ) + µ HM s (1 − m z ) } d S, (1)where A , D , K , H , M s , and µ are ferromagnetic ex-change stiffness constant, DMI coefficient, perpendiculareasy-axis anisotropy, perpendicular magnetic field, thesaturation magnetization, and the vacuum permeability,respectively; and (cid:126)m is magnetization unit vector. Theenergy is set to zero E = 0 for ferromagnetic state of m z = 1. The demagnetization effect is included throughthe effective anisotropy K = K u − µ M s /
2, here K u isthe perpendicular magnetocrystalline anisotropy. This isa good approximation when the film thickness d is muchsmaller than the exchange length [22]. It is known thatisolated circular skyrmions are metastable state of energy8 πAd √ − κ when κ = π D / (16 AK ) < ∂ (cid:126)m∂t = − γ (cid:126)m × (cid:126)H eff + α (cid:126)m × ∂ (cid:126)m∂t , (2)where γ and α are respectively gyromagnetic ratioand Gilbert damping constant. (cid:126)H eff = Aµ M s ∇ (cid:126)m + Kµ M s m z ˆ z + H ˆ z + (cid:126)H d + (cid:126)H DM + (cid:126)h is the effectivefield including the exchange field, the anisotropy field,the external magnetic field along ˆ z , the demagnetiz-ing field (cid:126)H d , the DMI field (cid:126)H DM , and a temperature-induced random magnetic field of magnitude h = (cid:112) αk B T / ( M s µ γ ∆ V ∆ t ), where ∆ V , ∆ t , and T are thecell volume, time step, and the temperature, respectively[23, 24].In the absence of energy sources such as an electriccurrent and the heat bath, the LLG equation describesa dissipative system whose energy can only decrease[25, 26]. Thus, solving the LLG equation is an effi-cient way to find the stable spin textures of Eq. (1). Inthis study, we choose A = 0 . D = − . , K = 0 . , M s = 0 . κ > × × µ H=0.4Tµ H=0.1T Q
0 -2-1012 10 skyrmions 50 skyrmions 150 skyrmions E ( - J ) t(ns)
1 -1012 10 skyrmions 50 skyrmions 150 skyrmions µ H=0T ①② ③⑤ ( d1 ) µ H=0T ( d2 ) µ H=0.1T ( d3 ) µ H=0.4T ρ (nm -2 ) m z -5 0 5 10-1.0-0.50.00.51.0 Profile ① ② -5 0 5 10 15-1.0-0.50.00.51.0 Profile ③ ④ ④ ⑥ -5 0 5 10-1.0-0.50.00.51.0 Profile ⑤ ⑥ FIG. 1. Metastable structures of 10 (a1-c1), 50 (a2-c2) and150 (a3-c3) skyrmions in a 200 nm × × µ H = 0T (a1-a3), 0 .
1T (b1-b3) and 0 . Q (the left y − axis and the red curves) and E (the right y − axisand the blue curves) of 10 (the dash lines), 50 (the solid lines),and 150 (the dot-dash lines) skyrmions vary with time (in thelogarithmic scale) for µ H = 0T (d1), 0 .
1T (d2) and 0 . not depend on the Gilbert damping constant. We use alarge α = 0 .
25 to speed up our simulations.Figure 1 plots one metastable structure each for 10, 50,and 150 skyrmions in a 200nm ×
200 nm × µ H = 0 , . . m z = − m z = 1. The domains of 5nm in diam-eter each are arranged in a square lattice initially. Fig.1(a1-a3) are zero field structures of 10 (a1), 50 (a2), and150 (a3) skyrmions characterized by skyrmion number Q defined as Q = (cid:82) ρdxdy , here ρ = π (cid:126)m · ( ∂ x (cid:126)m × ∂ y (cid:126)m )is the skyrmion charge density. The colour encodes theskyrmion charge distribution. Each small domain of zeroskyrmion number initially becomes a skyrmion of Q = 1.Interestingly, both positive and negative charges appearin a stripe skyrmion while only positive charges exist in acircular skyrmion. Figure 1(d1) shows how total Q (theleft y − axis and the red curves) and energy E (the right y − axis and the blue curves) change with time. Withina very short time of order of sub-picoseconds, Q reachesits final stable values while it takes nanoseconds for theenergy to decrease to their minimal values. The negativeskyrmion formation energy explains well the stripe mor-phology of skyrmions that try to fill up the whole film inorder to lower its energy, in contrast to circular skyrmionsfor positive energy [22]. Unexpectedly, the film can hostan arbitrary number of skyrmions up to a large value (seealso Fig. 2 below). At a low skyrmion number of Q = 10,the film is in a helical state with ramified stripe skyrmionsof well-defined width of 8 . Q = 50while it is an SkX of triangular lattice at Q = 150. When Q = 150, or skyrmion number density, is so large suchthat the distance between two neighbouring skyrmionsis comparable to the stripe width, skyrmion-skyrmionrepulsion compress each skyrmions into a disk-like ob-ject. Skyrmions at high skyrmion number density prefera triangular lattice as shown in (a3) instead of the initialsquare lattice.Figure 1(b1-b3 and c1-c3) plot the metastable struc-tures under fields µ H = 0 .
1T (b1-b3) and µ H = 0 . m z > H expandwhile those of m z < H shrink. This is showed in Fig. 1 (b1, b2, c1, and c2)and spin profiles in the insets (see Supplementary Ma-terials). Moreover, the amount of increase and decreaseof white and grey stripes are not symmetric such thatskyrmion-skyrmion repulsion is enhanced by the field,and SkXs tend to occur at lower skyrmion number den-sity. This trend can be clearly seen in (c2) and (c3). Fig.1(d2) and (d3) show similar behaviour for µ H = 0 . .
4T (d3) as their counterparts (d1) at zerofield: Skyrmion number Q grows to final values rapidly insub-picoseconds and E monotonically decreases to theirminimum in nanoseconds. Figure 1 shows unexpectedlythat the nature shape of skyrmions are various types ofstripes when κ > Q far below its maximal value of more than500. Physics around the maximal Q is not our currentconcern, and it will be investigated in future. Q nucle-ation domains of 5 nm in diameter each arranged into aperiodical lattice is used as the initial configuration togenerate a steady structure of Q skyrmions with energy E . The left panel of Fig. 2 is the numerical results for µ H = 0 , E increases monotonically with skyrmion number.Thus, states with few skyrmions or low skyrmion num-ber density are preferred. Since long stripe skyrmionshave more way to deform than disk-like skyrmions, theentropy of a helical phase is larger than an SkX. Thus,helical states should always be the thermal equilibriumphase below the Curie temperature, and an SkX can bea metastable state at most. E is not sensitive to Q , thusthe thermal equilibrium helical states can have variousforms or morphology and contains different number of ①③ ④ ⑥ E ( - J ) Q H=0T H=0.1T H=0.2T H=0.3T H=0.39T H=0.5T H=0.6T H=0.7T (cid:4500)(cid:4501) (cid:4502) (cid:4503)(cid:4504)(cid:4505)(cid:4506) m z FIG. 2. Skyrmion-number dependence of energy at var-ious magnetic fields of µ H = 0T (the squares and theblack curve), 0 .
1T (the circles and the red curve), 0 .
2T (theup-triangles and the blue curve), 0 .
3T (the down-trianglesand the green curve), 0 .
39T (the diamonds and the violetcurve), 0 .
5T (left-triangles and the yellow curve), 0 .
6T (right-triangles and the cyan curve), and 0 .
7T (the stars and thebrown curve) (the left panel). The film size is the same as inFig. 1. (cid:172) - (cid:178) label the Q m for µ H = 0 , . , . , . , . , . . Q = 11 for (cid:172) at µ H = 0 T , to the mixture ofhelical order and SkX of Q = 31 for (cid:174) at µ H = 0 . T , and tobeautiful SkX of Q = 141 for (cid:175) at µ H = 0 . T . Equally niceSkX of (cid:177) - (cid:178) at a smaller Q m demonstrates that a field helpsthe formation of an SkX. irregular skyrmions. Things are different when a mag-netic field is applied. E of fixed Q increases with H ,and E is minimal at Q m for a fixed field below a criti-cal value. Q m first increases with H up to an optimalfield of around µ H = 0 . T and then decreases with H .Above µ H = 0 .
7T (the stars and the brown), positive E means that ferromagnetic state of m z = 1 is the groundstate with E = 0 and should be thermal equilibrium statebelow the Curie temperature when µ H > . Q m ,at the optimal field of µ H = 0 . /µm at which two nearby skyrmions are in contact.All skyrmions are compressed into a disk-like object andform an SkX. Strictly speaking, they are not circular,as evident from our simulations. All our results agreequantitatively with SkX experiments on MnSi [2, 3].To substantiate our assertion that topological protec-tion prevent a metastable helical state from transform-ing into a thermal equilibrium SkX state at µ H = 0 . K and 29 K , respectively below and near theCurie temperature of T C = 33K. Fig. 3(a) and (b) showthe structures after 30ns evolution. Thermal fluctuationsat 20K are not strong enough to neither create enoughnucleation centres nor cut a short stripe into two thatare the process of breaking the conservation of skyrmionnumber. The system is still in a helical state with Q = 11skyrmions, one more than the initial value (see Movie 1of Supplementary Materials). However, at 29K, a heli-cal state transforms to the final thermal equilibrium SkXstate with Q = 132 skyrmions through cutting stripesinto smaller pieces and creating nucleation centers to gen-erate more skyrmions thermodynamically (see Movie 2of Supplementary Materials). For a comparison, we havealso started from the SkX shown in Fig. 1(c3) with 150skyrmions at µ H = 0 . Q = 136 skyrmions as shown in Fig.3(c) after 10ns evolution (see Movie 3 of SupplementaryMaterials), 14 less than the starting value. This is ex-pected because the average skyrmion number at thermalequilibrium should be smaller than Q m = 141 accordingto the energy curve of µ H = 0 .
3T in Fig. 2.We demonstrate below that an SkX at zero field is not athermal equilibrium state by showing the disappearanceof an SkX in both zero field cooling and warming. Start-ing from the SkX in Fig. 1(c3) and gradually increasing(decreasing) the temperature from 0K (30K) to 30K (0K)at sweep rate of 1K/ns with H = 0 (see Movies 4 and 5 ofSupplementary Materials), final structures shown in Fig.3(d) (zero field cooling) and (e) (zero field warming) arehelical state consisting of stripe skyrmions. In contrast,field cooling at the optimal field of µ H = 0 .
3T from 30Kto 0K at same sweep rate does not change the nature ofthe SkX. This is consistent with our assertion that SkXsare the thermal equilibrium states at µ H = 0 .
3T belowthe Curie temperature.The nucleation centres can also be thermally gener-ated near the Curie temperature such that skyrmionscan develop from these thermally generated nucleationcentres, rather than from artificially created nucleationdomains. To substantiate this claim, we carried out aMuMax3 simulation at 29K under the perpendicular fieldof µ H = 0 . Q makes such a transfor-mation difficult because of the conservation of skyrmionnumber under continuous spin structure deformation.This study shows that, similar to liquid drop formation,new skyrmions can be generated only from nucleationcentres or by splitting a stripe skyrmion into two. Theseprocess require external energy sources such as the ther-mal bath and result in topological protection and energybarrier between states of different skyrmion numbers. Al-though the energy of Q = 141 SkX has the lowest energyat µ H = 0 .
3T at zero temperature (Fig. 2), an initialstate with a few stripe skyrmions would not assume thelowest energy state at low temperature (Fig. 3(a)). Thisdemonstrates the multiple metastable states of various Q and topological protection to prevent SkXs and helical (a) (b) (c)(d) (e) m z y m x (f) FIG. 3. (a, b) Structures after 30ns evolution at µ H = 0 . T = 20K (a) and 29K (b), starting from the initial con-figuration of Fig. 1(c1). (c) Structures after 10ns evolutionat µ H = 0 .
3T and T = 29K, starting from the initial con-figuration of Fig. 1(c3). (d,e) Structures after 30ns zero fieldwarming (d) and warming (e), starting from the SkX in Fig.1(c3). (f) An SkX of 133 skyrmions at µ H = 0 .
3T and 29Kcomes from the thermally generated nucleation centres. states from relaxing to the thermal equilibrium phases.This new understanding can perfectly explain those fasci-nating appearance and disappearance of SkX and helicalstates along different thermodynamic paths [18, 19]. Forexample, the persistence of an SkX stability in the fieldcooling at the optimal field is because the SkX is the ther-mal equilibrium state below the Curie temperature, nota metastable state far from the equilibrium as commonwisdom believes. The disappearance of an SkX in zerofield warming and high field warming is because helicalstate is the thermal equilibrium phase. At high enoughtemperature below the Curie temperature, thermal fluc-tuations can spontaneously generate enough nucleationcentres such that the system can change its skyrmionnumber and reach its thermal equilibrium phase of ei-ther helical state of low skyrmion number density or SkXstate of high skyrmion number density.In conclusion, both helical states and SkXs are the col-lections of skyrmions. Stripes, not disks, are the naturalshapes of skyrmions when their formation energy is nega-tive. The distinct morphologies of helical and SkX statescome from the skyrmion number density. Skyrmions be-come disk-like objects and in a triangles lattice due tothe compression and strong repulsion among the highlypacked skyrmions. Unexpectedly, there are enormousnumber of metastable states with an arbitrary numberof skyrmions up to a critical value of above 500 at zerotemperature. The physics around this value needs fur-ther studies. The energy of these states depends on theskyrmion number and the magnetic field. The role of amagnetic field in SkX formation is to create the lowestenergy state being a skyrmion condensate with a highnumber of skyrmions. Our findings have profound impli-cations in skyrmion-based applications.This work is supported by the NSFC Grant (No.11974296 and 11774296) and Hong Kong RGC Grants(No. 16301518 and 16301619). Partial support by theNational Key Research and Development Program ofChina (Grant No. 2018YFB0407600) is also acknowl-edged.X. H. and H. W. contributed equally to this work. ∗ Corresponding author: [email protected][1] S. M¨uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A.Rosch, A. Neubauer, R. Georgii, and P. B¨oni, Science , 915-919 (2009).[2] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H.Han, Y. Matsui, N. Nagaosa, and Y. Tokura, Nature ,901–904 (2010).[3] X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z.Zhang, S. Ishiwata, Y. Matsui, and Y. Tokura, Nat.Mater. , 106–109 (2011).[4] S. Heinze, K. V. Bergmann, M. Menzel, J. Brede, A. Ku-betzka, R. Wiesendanger, G. Bihlmayer, and S. Bl¨ugel,Nat. Phys. , 713-718 (2011).[5] N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B.Wolter, K. V. Bergmann, A. Kubetzka, and R. Wiesen-danger, Science , 636-639 (2013).[6] Y. Onose, Y. Okamura, S. Seki, S. Ishiwata, and Y.Tokura, Phys. Rev. Lett. , 037603 (2012).[7] W. Munzer, A. Neubauer, T. Adams, S. M¨uhlbauer, C.Franz, F. Jonietz, R. Georgii, P. B¨oni, B. Pedersen, M.Schmidt, A. Rosch, and C. Pfleiderer, Phys. Rev. B ,041203 (2010).[8] R. Ritz, M. Halder, C. Franz, A. Bauer, M. Wagner, R.Bamler, A. Rosch, and C. Pfleiderer, Phys. Rev. B ,134424 (2013).[9] H. Y. Yuan, X.S. Wang, Man-Hong Yung, and X. R.Wang, Phys. Rev. B , 014428 (2019).[10] X. Gong, H. Y. Yuan, and X. R. Wang, Phys. Rev. B , 064421 (2020).[11] C. Back, V. Cros, H. Ebert, K. Everschor-Sitte, A. Fert,M. Garst, Tianping Ma, S. Mankovsky, T. L. Monchesky,M. Mostovoy et al. J. Phys. D , 363001 (2020).[12] A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R.Ritz, P. G. Niklowitz, and P. B¨oni, Phys. Rev. Lett. , 186602(2009).[13] J. Zang, M. Mostovoy, J. H. Han, and N. Nagaosa, Phys.Rev. Lett. , 136804 (2011).[14] A. N. Bogdanov, and U. K. R¨oßler, Phys. Rev. Lett. ,037203 (2001).[15] U. K. R¨oßler, A. N. Bogdanov, and C. Pfleiderer, Nature , 797-801 (2006).[16] T. Okubo, S. Chung, and H. Kawamura, Phys. Rev. Lett. , 017206 (2012).[17] A. O. Leonov, and M. Mostovoy, Nat. Commun. , 8275(2015).[18] K. Karube, J. S. White, D. Morikawa, M. Bartkowiak,A. Kikkawa, Y. Tokunaga, T. Arima, H. M. Rønnow, Y.Tokura, and Y. Taguchi, Phys. Rev. Mater. , 074405(2017).[19] K. Karube1, J. S. White, N. Reynolds, J. L. Gavilano,H. Oike, A. Kikkawa, F. Kagawa, Y. Tokunaga, H. M.Rønnow, Y. Tokura et al. Nat. Mat. , 1237 (2016).[20] S. Rohart, and A. Thiaville, Phys. Rev. B , 184422(2013).[21] M. Uchida, Y. Onose, Y. Matsui, and Y. Tokura, Science , 359 (2006).[22] X. S. Wang, H. Y. Yuan, and X. R. Wang, Commun.Phys. , 31 (2018).[23] W. F. Brown, Phys. Rev. 130, 1677 (1963).[24] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F.Garcia-Sanchez, and B. V. Waeyenberge, AIP. Adv. ,107133 (2014).[25] X. R. Wang, P. Yan, J. Lu, and C. He, Ann. Phys. (NY) , 1815 (2009).[26] X. R. Wang, P. Yan, and J. Lu, Europhys. Lett. , 67001(2009).[27] J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert,Nat. Nanotechnol. , 839 (2013).[28] E. A. Karhu, U. K. R¨oßler, A. N. Bogdanov, S. Kah-waji, B. J. Kirby, H. Fritzsche, M. D. Robertson, C. F.Majkrzak, and T. L. Monchesky, Phys. Rev. B ,094429 (2012).[29] X. R. Wang, X. C. Hu, and H. T. Wu, Stripe skyrmionsand skyrmion crystals, preprint.[30] Spin profile of stripes are well characterized by the width L and wall thickness w . Stripe width depends on ma-terials parameters as L = a ( κ ) A/D , where a is almosta constant when κ >>κ >>