Evolution of Skyrmion Crystals in Fe 0.5 Co 0.5 Si-Like Quasi-Two-Dimensional Ferromagnets Driven by External Magnetic Field and Temperature
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Evolution of Skyrmion Crystals in Fe . Co . Si-LikeQuasi-Two-Dimensional Ferromagnets Driven byExternal Magnetic Field and Temperature
Zhaosen Liu a,b , Tiantian Huan b , Hou Ian b,c Abstract
Magnetic skyrmions have attracted great research interest in recent yearsdue to their exotic physical properties, scientific merit and potential appli-cations in modern technology. Here, we apply a quantum computationalmethod to investigate the spin configurations of Fe . Co . Si-like quasi-two-dimensional ferromagnetic system with co-existence of Dzyaloshinsky-Moriyaand Heisenberg exchange interactions. We find that within a weak magneticfield perpendicular to the film plane, skyrmion crystal (SkX) of hexagonal-close-packed pattern can be induced, the spin configurations evolve withapplied magnetic field and temperature. This quantum model, if scaled, isable to qualitatively reproduce the experimental results of SkX with longperiodicity. Especially, when the skyrmion size is around a few nano-metersin diameter, or more general, when the characteristic length of the mate-rial approaches the lattice scale , the quantum model is expected to be moreaccurate than the classical ones.
Keywords:
Skyrmion Crystal, Dzyaloshinsky-Moriya Interaction,Quantized Simulation Model Email: [email protected] Email: [email protected]
Preprint submitted to Elsevier February 4, 2020 . Introduction
Skyrmions, a novel type of topological magnetic structures observed insolids, have attracted intensive research interest in recent years [1, 2, 3, 4].The concept of skyrmion was originally introduced in nuclear physics to de-scribe a localized, particle-like configuration in field theory about half cen-tury ago [5]. However, these topological structures were lately predicted toexist in magnetic materials based on micromagnetic model [6, 7], and ex-perimentally observed in bulk materials, such as MnSi with the B20 crystalstructure [8, 9], Fe − x Co x Si and FeGe thin films [10, 11], multi-layer systems[12, 13, 14, 15, 16, 17], as well as insulating magnet Cu OSeO [18]. Magneticskyrmion crystal (SkX) textures are mainly induced by the Dzyaloshinsky-Moriya interaction (DMI) [19, 20]. This chiral interaction exists in the sys-tems with broken structural inversion symmetry, and is present at the sur-faces of thin films or the interfaces of multi-layers. Since transition-metalinterfaces are essential in spintronic devices, the discovery of SkX on theseinterfaces [14, 15, 16] has aroused great research interest in using them asnovel data-storage device concepts [21, 22].In three-dimensional (3D) materials, SkX is usually stable in a rather nar-row T -region immediately below the transition temperature as observed inneutron scattering experiment conducted on MnSi bulk magnet [9]. Howeverin two-dimensional (2D) systems, SkX can exist over a wide temperature re-gion [11, 23]. For instance, using Lorentz transmission electron microscopy,Yu et al. observed that when Fe . Co . Si thin film is placed in external mag-netic fields, the SkX appears and persists down to almost zero temperature[10]. These skyrmions crystalize in the hexagonal-close-packed (HCP) pat-tern with a periodicity of about 90 nm.So far, various descriptions for magnetic skyrmions have been proposed.Most of them include the notion of topology as defined in micromagneticswhere the continuous model is applied. There, a magnetic skyrmion can bedescribed with a non-zero integer value of the topological index, referred toas topological charge, or winding number. The topological charge of this spin2exture can be expressed as Q = 14 π Z Z ~m · ( ∂ x ~m × ∂ y ~m ) dxdy . (1)where ~m is a unit vector specifying the direction of the local magnetic mo-ment, and the integral is taken over the space occupied by the skyrmion.However, we must bear in mind that the concept of topology can onlybe rigorously applied in continuous models to infer the stable spin structures[24]. At a size scale less than a few nanometers, or more general, as thespin-spin correlation length of the material approaches the unit cell scale ,the spin textures of magnetic materials become noncontinuous due to thediscretization of the atomic lattice. For instances, the skyrmions formed atinterfaces are often a few nanometers in diameter [14, 17, 25, 26]. The realspin structures of these skyrmions naturally differ from those derived fromthe continuous model based on classical physics.For the above reasons, we carry out simulations here for a Fe . Co . Si-likethin film by means of a quantum computational method which we developin recent years [27, 28, 29, 30, 31, 32, 33]. Following the similar methodology[10], we model the thin film as a 2D magnetic monolayer, and impose theperiodic boundary conditions to mimic the quasi-infinite size of the system.In brief, the phenomena observed in our simulations can be summarized asbelow: In the absence of external magnetic field, the helical texture is theground state.
When a weak magnetic field is applied normal to the mono-layer, SkX textures of hexagonal-closely-packed (HCP) pattern are induced[10], and the periodic distances for the helical and SkX states agree roughlywith the theoretical values [10, 34]. As the field strength is enhanced andtemperature changes, the SkX textures can be rotated or/and deformed, andtheir periodicitic distances increases. When the external field is sufficientlystrong, a skyrmion plus bimeron phase appears, which still looks symmetricgeometrically. Within further increased external magnetic field, the mag-netic system finally becomes ferromagnetic as expected. Very interestingly,the calculated topological charge density for every SkX also forms period-3cal and symmetric lattice which is almost identical to the correspondingmagnetic SkX, demonstrating the correctness of our simulations. This quan-tum model, when scaled, is able to qualitatively reproduce the experimentalresults for SkX of large spacing distance. On the other hand, when theskyrmion size is around a few nano-meters in diameter, or more general, asthe spin-spin correlation length of the material is in the scale of lattice con-stan t, the quantized discrete model is expected to be more accurate than theclassical ones.
2. The Quantum Computational Method and Related Theory
To simplify the model, a quasi-2D system like Fe . Co . Si thin film canbe modelled as a monolayer ferromagnet with the square crystal structure[10], and its Hamiltonian be expressed as H = − P h J ij ~S i · ~S j − D ij ~r ij · ( ~S i × ~S j ) i − K A P i (cid:16) ~S i · ˆ n (cid:17) − µ B g S ~B · P i ~S i . (2)Here, the first two terms represent the Heisenberg exchange and DM interac-tions with the strengths of J ij and D ij between a pair of spins at the i - and j -th lattice sites, respectively, and < ij > means that these interactions arelimited between the nearest neighboring spins; the third term stands for theuniaxial anisotropy assumed to be perpendicular to the monolayer plane, andthe last one denotes the Zeeman energy of the 2D system within an appliedmagnetic field.The quantum computational method employed in the present work hasbeen described in details in our published papers [27, 28, 29, 30, 31, 32, 33]. Sothe spins appearing in the above Hamiltonian are quantum operators insteadof classical vectors, and all physical quantities are calculated with quantumformulas. In the Heisenberg representation, when S = 1 for example as it isassumed in our simulations, the matrices of the three spin components are: S x = √ √ √ √ , S y = 12 i √ −√ −√ √ , (3) z = − , respectively.In light of molecular field theory, the i -th spin is considered to be underthe interaction of an effective magnetic field B Mi generated by the neighboringmagnetic moments. As a result, if the lattice chosen for simulations consistof N spins, the Hamiltonian of the whole magnetic system shown above canbe decomposed into N coupled Hamiltonians. Each of them is for a spin,and that for the i -th spin is given by H i = − P h J ij ~S i ·h ~S j i − D ij ~r ij · ( ~S i ×h ~S j i ) i − K A P i (cid:16) ~S i · ˆ n (cid:17) − µ B g S ~B · P i ~S i . (4)This Hamiltonian of a single-spin can be easily diagonalized, and the thermalaverage of any physical variable A at temperature T can be calculated with h A i = Tr h ˆ A exp( β H i ) i Tr [exp( β H i )] , (5)where β = − /k B T .All of our recent simulations are started from a random magnetic config-uration above the transition temperature, then carried out stepwise down tovery low temperatures with a reducing temperature step ∆ T <
0. This trickis very important, since at high temperatures, the effective magnetic field isrelatively weak, the thermal interaction is strong enough to help the spinsovercome the energy barriers, so that the code can avoid being trapped inlocal energy minima, and finally converge down to correct equilibrium statesspontaneously. Obviously, for this purpose, | ∆ T | cannot be too large.At a given temperature, the spins are selected one by one successively toevaluate their thermally averaged values. After all spins in the sample havebeen visited once, an iteration is completed. The iterations are repeated in5 self-consistent manner. When the ratio ( |h ~S ′ i i − h ~S i i| ) / |h ~S i i| between twosuccessive iterations for every spin in the lattice is less than a very smallgiven value τ , convergency is considered to be reached.The periodical length λ of the SkX observed in the Fe . Co . Si film planeis around 90 nm [10], and the lattice constant a is about 0.45 nm [35]. Thus,the nearest pair of skyrmions are about 200 a apart. In order to displaythe calculated SkX texture for the Fe . Co . Si film, the spin wavelength λ was assigned to 10 by Yu et al. in their Monte Carlo simulations [10]. Inthis scaled model, λ is measured in the unit of the side-length of a grid, andeach grid contains n × n spins where n = λ / ( λa ) if both λ /a and λ aremuch larger than 1 [36]. Moreover, the periodic distance of the chiral texturecan be determined by the relative strengths of Heisenberg exchange and DMinteractions [10], tan (cid:18) πλ (cid:19) = D √ J . (6)So, once λ is given, the ratio D/ J can be estimated with this formula.For a discrete spin model, the winding number density of a skyrmion atthe i -th site can be expressed as ρ i = 116 π ~S i · h ( ~S i +ˆ x − ~S i − ˆ x ) × ( ~S i +ˆ y − ~S i − ˆ y ) i . (7)Here an additional prefactor 1 / introduced , since the spacial dis-tances between the spin pair ( ~S i +ˆ x , ~S i − ˆ x ) in the x -direction, and anotherpair ( ~S i +ˆ y , ~S i − ˆ y ) in the y -direction, are both equal to 2 a . Moreover, all spinsmust be normalized before being inserted to this formula. Afterwards, theaveraged winding number per skyrmion can be estimated by dividing thesum over the lattice, P i ρ i , with the total number of skyrmions observed inthe lattice. To check the correctness of above formula, the winding numbershave also been calculated with a formula employed by previous authors [37],which gives the same results.On the other hand, the helicity of a chiral spin texture is defined as [37, 38] γ = 1 N X i (cid:16) ~S i × ~S i +ˆ x · ˆ x + ~S i × ~S i +ˆ y · ˆ y (cid:17) . (8)6his quantity is attributed to the relativistic spin-orbital coupling, its signand the magnitude reflect the swirling direction (right-handed or left-handed),and degree of the chirality.
3. Computational Results
When the spin wavelength λ is assigned to 10, the ratio D/ J is deter-mined to be 1.027 from Eq.(6). This value is used in all our simulations thatare performed on a 30 ×
30 square lattice, and the periodical boundary con-ditions are imposed to mimic the quasi-infinity of the 2D magnetic system.For simplicity, we further assign J /k B = 1 K. That is, all physical quantitiesare scaled with the Heisenberg exchange strength J and k B . We notice from our calculated results that, in the absence of externalmagnetic field, the helical texture is spontaneously formed below the transi-tion temperature, T M = 3.3 J . In Figure 1, the xy component and z -contourof the calculated spin structure are both projected onto the film plane. It is (a) B = 0, T/J = 3.2 Y X Y X -0.2660-0.1995-0.1330-0.0665000.066500.13300.19950.2660 (b) B = 0, T/J =3.2 S z Figure 1.
The xy texture (a) and z -contour (b) of the helical spin structurecalculated at T / J = 3.2 in the absence of external magnetic field. interesting to find there that the periodical lengths in the x and y directionsof the helix state are all equal to 10 in the unit of grid side length as theoret-ically predicted. In every spatial period, there exist two spin strips which areanti-parallel in the [-110] direction on the xy -plane, and across each period,7he z -components of the spins change gradually from the minimum with themost negative value to the maximum, then falls down gradually. This he-lical texture remains unchanged down to almost zero temperature, but themagnitudes of all three components increase with decreasing temperature asexpected. When an external magnetic field is applied perpendicular to the mono-layer plane, the helical structure persists until the field is increased to 0.1Tesla. However, within the magnetic field of this strength, magnetic SkX Y -0.1880-0.1192-0.050500.018250.087000.15580.22450.29330.3620 (a) B = 0.1 Tesla, T/J = 3.2 S z X -0.3580-0.2542-0.1505-0.046750.057000.16080.26450.36830.4720 Y X (b) B = 0.1 Tesla, T/J = 0.31 S z Figure 2.
The SkX textures calculated at (a)
T / J = 3.2, and (b) T / J = 3.1in an external magnetic field of 0.1 T applied normal to the monolayer plane. can be induced at T / J = 3.2, which is immediately below the ferromagneticphase, as displayed in Figure 2(a). There, 18 skyrmions (printed in blue)are observed on the monolayer plane in a pattern of hexagonal-close-packed(HCP) crystal structure, and they are all separated by shallow vortices (dis-played in red). The two sides of such a hexagon are parallel with the x -axis,so it is referred as ’regular’ to distinguish it from those rotated and deformedhexagons. However, the external magnetic field is now too weak to stabi-lize the SkX structure. When temperature falls down to T / J = 0.31, theskyrmions and vortices are all deformed, they are elongated in the [-110]direction, but the chiral spin texture still keeps the regular HCP patternas shown in Figure 2(b). As the temperature drops further, the SkX tex-ture disappears, being replaced by the helical structure afterwards. Hou et l. have reported the similar phenomenon before [39]. In their Monte Carlosimulations done for a 2D chiral magnet, a surprising upturn of the topologi-cal charge was identified at high temperatures. They attributed this upturn tospin fluctuations, that is, the topology was believed to be thermally induced.3.3. Typical SkX State Induced by Moderate External Magnetic Field As the external magnetic field is further increased, a stable SkX phaseappears over a broad temperature range. For instance, when B z = 0.11 T,SkX is observed in the low temperature region T /
J ≤
T / J = 3.2 and 0.1 respectively withinthis field. There, every skyrmion is surrounded by four non-fully developed -0.1460-0.08325-0.020500.042250.10500.16770.23050.29330.3560 (a) B = 0.11 Tesla, T/J = 3.2 K Y X -1.000-0.7500-0.5000-0.250000.25000.50000.75001.000 (b) B = 0.11 Tesla, T/J = 0.1 Y X Figure 3.
The SkX textures calculated at (a)
T / J = 3.2, and (b) T / J =0.1, within an external magnetic field of 0.11 T applied perpendicular to themonolayer plane. vortices. Since D is positive, these two sorts of spin textures are both right-handed. They appear alternatively in the lattice plane. All skyrmions curlclockwise, while the vortices swirl anti-clockwise, so as to minimize the total(free) energy of the whole system. However, in Ref.[10], no surrounding vortices were observed in experi-mental results and classical Monte Carlo (CMC) simulations. The reasonsare given below. To simulate the Fe . Co . Si-Like 2D system, a scaled modelis used here. In this model, one spin represent several hundreds of latticesites. Thus, vortices curling in the opposite direction must appear to reducethe sudden energy increase caused by the closely packed skyrmions. However, n real Fe . Co . Si film, the skyrmions are actually quite large, so the spinscan change their directions continuously, thus much smaller or no surround-ing vortices are required for the spin texture to relax. On the other hand, inthe interstitial areas between the skyrmions, the spins have been considerablyrotated by the external magnetic field to the out-of- plane direction, whereasthe on plane components are greatly reduced. Consequently, even the vorticesappear in the interstitial areas, they are hardly to be observed in experiments.In contrast, before our simulated spin textures are plotted, the on-plane com-ponents of all spins have been normalized, so that the vortices can be easilyseen. In Figure 1(b) of Ref.[10], we can see that interstitial areas between theskyrmions are quite large. However, the vortices are missing in these areas.The SkX shown there was obtained in their CMC simulations. We guess thatthe authors did not do the normalization as we do here.
As expected, though the external magnetic field is applied along the z -axis, the spins inside the skyrmion cores align in the opposite direction. Theincrease in total energy incurred by these spins are offset by the surroundingvortices, where the spins have been greatly rotated by the external field towardits orientation.
Figure 2(a) and Figure 3(a) are obtained at the same temperature, theylook very similar at the first glance. However, the vortices and skyrmionshave actually exchanged their positions within external magnetic field of dif-ferent strengths. Moreover, by comparing Figure 3(a) and (b), it is observedthat as temperature drops down, the cores of skyrmions expand, but thoseof the vortices shrink spatially. This rule in general holds in all later cases. B z = 0.13 T When B z = 0.13 T, an extremely shallow vortex lattice is observed at T / J = 3.2 immediately below the transition temperature as displayed inFigure 4(a). The xy projection of this chiral lattice pattern looks similarto that depicted in Figure 2(a): 18 vortices curl clockwise, other 18 vorticesswirl anti-clockwise, and the two sets of vortices appear alternatively on the xy -plane. It can be inferred that such shallow vortical lattices may also be10 Y X S z (a) B = 0.13 Tesla, T/J = 3.2 X -0.9900-0.7425-0.4950-0.24755.551E-170.24750.49500.74250.9900 S z Y (b) B = 0.13 Tesla, T/J = 1.2 Figure 4.
The (a) vortex lattice at
T / J = 3.2, and (b) SkX texture at T / J =1.2, induced by external magnetic field of 0.13 T perpendicular to the monolayerplane. induced immediately below FM phase by external magnetic field of otherstrengths, but they are easily destroyed by thermal disturbance, thus canhardly be observed in experiments. When T / J drops to 3.1, the regularHCP SkX appears, and maintains until T / J = 1.3 over a wide temperaturerange. While T /
J ≤ o by the effective magnetic field. However, the whole spin structure still keepsHCP pattern as depicted in Figure 4(b). The spin texture rotation just described above seems very strange, readersmay wonder what are the reasons behind. Firstly, the self-consistent algo-rithm has been implemented into our computing program, so the code canconverge spontaneously to the equilibrium states. That is, all spin texturespresented in the paper are self-organized, the rotations can not be phantomsimply caused by artificial intervention.Such phenomena will be observed in following figures as well. Theoreti-cally speaking, as temperature or/and the strength of magnetic field vary, notonly the magnitudes, but also the orientations of all spins change correspond-ingly. Since the spins in the system are correlated directly or indirectly, theprocess may inevitably give rise to spin texture rotation to fit new magneticconfiguration. On the other hand, we will see latterly that enhanced magneticfield usually leads to enlarged periodicity. This may also result in SkX rota-tion so that the spins can adjust themselves to meet the requirements of the ew symmetry and fit the periodical boundary conditions.3.5. Evolution of SkX Texture Driven by Enhanced External Magnetic Field At B z = 0.15 T, the same phenomena just described are observed onceagain. In the temperature range 3.2 > T / J > T / J = 1.9,the SkX texture is not only rotated, but also deformed as shown in Figure5(a). Y X -0.9200-0.6906-0.4613-0.2319-0.0025000.22690.45620.68560.9150 S z (a) B = 0.15 Tesla, T/J = 1.9 -0.9600-0.7150-0.4700-0.22500.020000.26500.51000.75501.000 Y X (b) B = 0.17 Tesla, T/J = 0.1 S z -0.5400-0.4013-0.2625-0.12370.015000.15380.29250.43130.5700 S z (c) B = 0.19 Tesla, T/J = 2.9 Y X Y X -0.6850-0.5100-0.3350-0.16000.015000.19000.36500.54000.7150 S z (d) B = 0.23 Tesla, T/J = 2.6 Figure 5.
The skyrmion lattices simulated when (a) B = 0.15 T, T / J = 1.9,(b) B = 0.17 T, T / J = 0.1, (c) B = 0.19 T, T / J = 2.9, and (d) B = 0.23 T, T / J = 2.6, respectively. As B z is increased to 0.17 T, the SkX is rotated by the effective magneticfield once it is formed immediately below the polarized FM phase, and thischiral texture persists from T / J = 3.0 down to very low temperature asshown in Figure 5(b). Now, the HCP SkX pattern is not deformed, however,every skyrmion has been considerably enlarged, and the total number ofskyrmions N s reduced to 15. 12s depicted in Figure 5(c), when B z reaches 0.19 T, the SkX texture hasbeen rotated clockwise further, and the total skyrmion number N s reducedto 12 in a wide temperature below T / J =2.9.Very interestingly, when B z is increased to, for examples, 0.23, 0.25 and0.27 T, the spin textures are found to re-assume the regular HCP SkX patternin the low temperature range as displayed in Figure 5(d), and the totalskyrmion numbers are equal to 12 in all these cases. The SkX texture can be maintained until the external magnetic field isincreased up to the critical value B cz = 0.28 T. At T / J = 2.4, which is justbelow the FM phase, the total skyrmion number N s is reduced to 8, but theskyrmions are still distributed in the regular HCP pattern as seen in Figure6(a). -0.7800-0.5869-0.3938-0.2006-0.0075000.18560.37870.57190.7650 (a) B = 2.8 Tesla, T/J = 2.4 S z Y X -0.8150-0.6106-0.4062-0.20190.0025000.20690.41130.61560.8200 (b) B = 2.8 Tesla, T/J = 2.3 K S z Y X Figure 6.
When B z = 0.28 T, (a) SkX, and (b) bimeron textures, are formedat T / J = 2.4 and T /
J ≤
Within the external magnetic field of this strength, as temperature dropsto the range
T /
J ≤
When B z = 0.29 T and T /
J ≤ × D >
0. In comparison with those shown in Figure 6(b),these strips are much longer. This effect becomes more evident when B z isincreased to 0.30 T: the helical strips look much longer along the diagonal ofthe square lattice. Moreover, in the low temperature region T /
J ≤
2, theperiodical distances between the strips become 10 in the x -direction, but 14in the y -direction.This finite-helical-strip texture maintains until B z is increased to 0.35 T.When B z = 0.36 T, four skyrmions are observed at T / J = 1.9, 1.8, and 1.7inside the square lattice. However, the skyrmions are not evenly distributed,since now the 30 ×
30 square lattice does not fit the periodical textures even ifthey exist in reality. Nevertheless, as temperature
T / J falls down below 1.6,the magnetic system re-assumes its finite-helical-strip texture again. Thesestrips can be classified into two groups which are orthogonal to each otheras observed in experiment [10]. Now λ is about 14 in either x or y direction,which mismatches with the lattice size, so that the calculated spin configura-tion does not look very symmetric. Therefore, to obtain accurate calculatedresults, we must adjust the lattice size to find the best one which has thelowest total (free ) energy.While B z is further increased, the spin textures consisting of sparse de-formed skyrmions and vortices can be observed, but they are quite irregular,14robably for the sake just described. These spin textures resemble thosedepicted in Figure 3(g) of Ref.[10].
Finally, as B z ≥ z -direction, the whole system is polarized to be completely ferro-magnetic. ≤ B ≤ B ≤ B = 0.10, SkX texture appears at T / J =0.32, then it is greatlystretched along the [-110] direction at T / J = 0.31. And in a field range0.29 T ≤ B ≤ With Eq.(7,8) given in Sec.II, the sums of the winding number density, P i ρ i , over the whole lattice and the averaged helicities per site γ for thespin textures are calculated, the corresponding curves are displayed in Figure7(b,c).As shown in above figures, in one SkX, a skyrmion is usually surroundedby four shallow unfully developed and frequently considerably deformed vor-15 .0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.000.050.100.150.200.250.300.35 T/J
HSkX FM H + SkX (a) B ( T ) (b) T/J -0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.10.00.1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 (c)
T/J
Figure 7. (a) Phase diagram of the magnetic system, (b) total winding number,and (c) helicity curves of the spin textures in external magnetic fields of differentstrengths. B z (T) 0.11 0.13 0.15 0.17 0.19 0.23 0.27 N s
18 18 18 15 12 12 12 Q av -0.753 -0.770 -0.771 -0.809 -0.845 -0.845 -0.843Table 1: Averaged winding numbers per skyrmion complex calculated at
T / J =0.1 within external magnetic field of different strengths. tices of appreciable sizes. These skyrmions and their neighboring vorticesare entangled together, so that it is very difficult to identify the boundariesbetween these two sorts of chiral spin textures. For such a SkX, the aver-aged winding number Q av per spin complex, consisting of one skyrmion andthe parts of its neighboring vortices, can be estimated by dividing the sum, P i ρ i , with the skyrmion number N s observed inside the 30 ×
30 square lat-tice. Table 1 lists these averaged numbers obtained at
T / J = 0.1 in externalmagnetic fields of different strengths. We know already that as B z changesfrom 0.11 to 0.27 T, N s decreases from 18 to 12, so Q av is expected to changeaccordingly.In the absence of external magnetic field, the helical structure is theground state of the system, no skyrmion is formed, so P i ρ i = 0, and thehelicity per spin γ = -0.831 while T /J ≤ B z = 0.10 T, the system behaves extremely unusually. In the16emperature region T /
J ≤ . P i ρ i is always equal to zero, that is, noskyrmion can be formed and the helical texture dominates. However, at T / J = 3.2, P i ρ i suddenly falls down to -13.201 ( Q av = -0.7334) where SkXemerges, then immediately increases to -7.568 ( Q av = -0.420) at T / J = 3.1where the SkX is considerably deformed as shown in Figure 2.While B z = 0.28 T and T / J = 2.4, eight skyrmions are observed. Butbelow this temperature, four skyrmions and four bimerons appear. If onebimeron is considered to be composed of two semi-skyrmions, so the totalskyrmion number shown in Figure 6(b) is approximately eight, which gives Q av = -0.830 for one skyrmion complex, that is close to those values tabulatedin Table 1. More interestingly, the topological charge density calculated with Eq.(7)for every SkX also forms periodical and symmetric lattice which looks almostidentical to the corresponding magnetic SkX, as displayed in Figure S1 of thesupplementary part, demonstrating the correctness of our simulations.
Obviously, when the system is polarized by a strong external magneticfield to become completely ferromagnetic, the averaged helicity per site γ is equal to zero as shown in Figure 7(c). In the helical state, γ is around-0.8313, and in the case of SkX texture, the helicity is found to be in theregion -0.7822 < γ < -0.4072. The points with γ = 0 form the boundarybetween the ferromagnetic and chiral phases.
4. Conclusions and Discussion
We have employed a quantum simulation approach to investigate a quasi-2D Fe . Co . Si-like ferromagnetic system. In the absence of external mag-netic field, the helical texture is the ground state. When a moderate externalmagnetic field is exerted perpendicular to the monolayer, SkX textures of theHCP structure are induced. As the magnetic field strength is increased, theSkX textures can be deformed, rotated, and the periodical lengths increased.Afterwards, complicated spin structures, such as skyrmions plus bimerons,17nite-length helical strips, sparse skyrmions surrounded by vortices, appearsuccessively. And finally, the system becomes ferromagnetic as B z ≥ xy -plane, whilethose of the neighboring shallow vortices order anti-clockwise in the sameplane, so that the total (free) energy of the whole system can be minimizedand the SkX texture stabilized. All chiral spin textures are right-handed since D is positives. Moreover, the calculated topological charge density for everySkX also forms periodical and symmetric lattice which is almost identical tothe corresponding SkX, demonstrating the correctness of our simulations.In a zero or weak magnetic field, the periodic wavelengths of the helicaland SkX textures agree roughly with the theoretical values estimated withEq.(6). Late on, we found that [32], when a weak a magnetic field is applied,another formula tan (cid:18) πλ (cid:19) = D J (9)agrees better with the SkX periodicity in the diagonal [110] or [-110] direc-tion, as shown in Figure 2. However, this formula was actually derived forthe helical state based on a continuous model [34]. As indicated by the au-thors, for the skyrmion states, no analytical expression for the periodicitycan be obtained [34]. Nevertheless, if the wavelength estimated from Eq.(6)is projected onto the diagonal direction, it gives a value that is roughly equalto the periodicity evaluated from Eq.(9).As B z is enhanced, the periodicity is increased. Consequently, the latticesize chosen for simulation cannot fit the spin texture any longer, this mis-match can destroy the periodical structures in simulations even they exist inreality. Therefore, in order to generate symmetric and periodic spin texturesand to study the physical properties accurately in strong external magneticfield, the lattice size has to be adjusted. In principle, a lattice which givesthe lowest total (free) energy is expected to produce the best results.An isolated skyrmion can be induced from the helical state when theexternal magnetic field is stronger than the critical value B c = 0 . D / J [35, 36, 42]. Inserting the values of J and D we use here into this formulae18rovides B c = 0.148 T, which is comparable to B c = 0.11 T that we find inour simulations. On the other hand, if B exceeds another critical value B c =0 . D / J , SkX texture disappears, the skyrmion plus bimeron states emergeafterwards [35, 36, 42]. The above two formulas gives the ratio B c /B c =0.25. However, for FeGe and Fe . Co . Si, the ratios of the two transitionfields are observed in experiments to be within a range of (0.3, 0.39). In thepresent case, we find in simulations that B c = 0.28 T, the ratio B c /B c isapproximately 0.39, which agrees well with the experimental observation.In chiral magnetic systems, the SkX wavelengths are usually quite large.To calculate and display their periodical spin textures, the grid model hasto be used, where J and D parameters are scaled, so are other physicalquantities [35, 36]. To do this, we further quantize the grid with a quantumspin. From the calculated results presented above, we can conclude thatthis treatment is quite effective in describing the detailed spin textures ofthe quasi-2D system, as they evolve, driven by changing temperature andexternal magnetic field.Especially, the skymions recently observed at interfaces are only a fewnanometers in diameter [17, 25, 26, 35], and it is these magnetic materialsthat are of great importance in future technology. Obviously, within thesemagnetic systems, the spin vectors cannot change continuously, thus ourdiscrete quantum model is expected to be more accurate and effective thanthose popularly used classical ones. Acknowledgement
Z.-S. Liu acknowledges the financial support provided by National Natural Sci-ence Foundation of China under grant No. 11274177 and by University of Macau.H. Ian is supported by FDCT of Macau under grant 065/2016/A2 and by NationalScience Foundation of China under grant 11404415. eferences [1] A. N. Bogdanov and U. K. R¨o β ler, Phys. Rev. Lett. 87 (2001) 037203.[2] M. Lee, W. Kang, Y. Onose, Y. Tokura, and N.P. Ong, Phys. Rev. Lett.
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