ABSTRACT The ability to controllably manipulate magnetic skyrmions, small magnetic whirls with particle-like properties, in nanostructured elements is a prerequisite for incorporating them

into spintronic devices. Here, we use state-of-the-art electron holographic imaging to directly visualize the morphology and nucleation of magnetic skyrmions in a wedge-shaped FeGe

nanostripe that has a width in the range of 45–150 nm. We find that geometrically-confined skyrmions are able to adopt a wide range of sizes and ellipticities in a nanostripe that are absent

in both thin films and bulk materials and can be created from a helical magnetic state with a distorted edge twist in a simple and efficient manner. We perform a theoretical analysis based

GATE-CONTROLLED SKYRMION AND DOMAIN WALL CHIRALITY Article Open access 07 September 2022 STABILIZING MAGNETIC SKYRMIONS IN CONSTRICTED NANOWIRES Article Open access 16 June 2022

INTRODUCTION Magnetic storage in hard disk drive technology is based on the controllable formation of magnetic domains and is approaching its limits1. The ability to manipulate domain walls

instead of domains provides an alternative method for further extending the storage device roadmap2,3. The discovery of topologically stable magnetic skyrmions is of great interest because

of their small size (typically below ∼100 nm) and their high mobility at low-current densities4,5,6,7,8,9,10. In particular, skyrmions are promising candidates for applications in novel data

storage devices based on the race-track memory concept11. The underlying design of such devices relies on the use of skyrmions as data bit carriers, which move along a ferromagnetic

nanostripe that takes on the role of a guiding track. It is therefore important to be able to controllably form and manipulate skyrmions in nanostructured elements. Over the past few years,

many theoretical proposals have been put forward to host and create individual skyrmions in confined geometries12,13,14,15,16,17, including the possibility of locally nucleating an isolated

skyrmion by using a spin-polarized current and driving its motion using a current-induced torque12,13. On the experimental side, skyrmion chains and skyrmion cluster states have been

observed in FeGe nanostripes of fixed width18 and nanodisks19, respectively, using Lorentz transmission electron microscopy (TEM). However, the widths of the samples in these studies were

all larger than the corresponding single skyrmion size. Moreover, recent work has demonstrated powerful skyrmion-based information storage functionalities20, such as the ability to generate

skyrmion bubbles in geometrically confined CoFeB/Ta films using in-plane currents21. Despite these experimental results, the formation and manipulation of skyrmions has never been studied in

detail in confined geometries of sufficiently small dimension. The optimization of information storage requires an optimal width for a nanostripe that hosts a chain of skyrmions, as well as

precise control of their morphology22 and formation in such a confined geometry. Therefore, the next important step is to develop approaches that can be used to control the fine structures

of individual skyrmions and to nucleate them in a simple manner in a nanostripe, whose size is similar to that of the skyrmions themselves. The present study is also directly connected to

the determination of the smallest width of a nanostripe, which is still able to host magnetic skyrmions over a wide range of applied fields and temperatures. Real-space imaging of skyrmions

is essential for addressing these challenges6,20,21,22,23,24. However, the study of individual magnetic skyrmions in nanostructured elements is very difficult because of the required spatial

resolution and sensitivity and the influence of the edge of a nanostructure on the recorded contrast. The latter issue affects the study of skyrmions in nanostructures that have sizes of

below 100 nm using Lorentz TEM because Fresnel fringes, which are formed at the edge of a nanostructure as a result of the need to use an out-of-focus imaging condition, complicate the

interpretation of the recorded magnetic signal18,25. In contrast, the technique of off-axis electron holography (EH) in the TEM allows an electro-optical phase image of a specimen to be

recorded directly, so that unwanted contributions to the signal from the edges of a nanostructure can be subtracted more easily than using other TEM-based techniques. Moreover, the technique

has nm spatial resolution, high phase sensitivity and permits reliable quantification of magnetic states in nanostructured elements26. Digital acquisition and analysis of electron holograms

and image analysis software are then essential, when studying weakly varying phase objects such as skyrmions. The technique has been used to study the three-dimensional structures of

skyrmions in Fe0.5Co0.5Si films, for which a low temperature of ∼10 K, far below the Curie temperature _T_c∼35 K, was necessary to obtain a high quality signal27. Here, we use off-axis EH to

study a wedge-shaped nanostripe, whose width reaches only 40–150 nm, at several different temperatures, in order to investigate the fine structure and formation of highly geometrically

confined skyrmions. We observe the high flexibilities of individual skyrmions and a unique field-driven helix-to-skyrmion transition directly. RESULTS THEORETICALLY PREDICTED SKYRMION

MORPHOLOGY IN A NANOSTRIPE We first discuss the expected skyrmion morphology in a nanostripe. In general, the stability of skyrmions in chiral magnets, such as MnSi, Fe_x_Co1_−x_Si, FeGe and

other B20 alloys, is governed by the Dzyaloshinskii–Moriya interaction (DMI)8. Competition between the DMI and ferromagnetic exchange coupling results in a homochiral spin helix ground

state with an equilibrium period _L_D, which is determined by the ratio of the energy contributions of these two interactions. In a bulk crystal, the spin helix generally evolves into a

conical phase and then into a field-saturated ferromagnetic state in the presence of an increasing external magnetic field. Skyrmions appear in the form of a lattice and occupy only a tiny

pocket in the magnetic field _H_ and temperature _T_ phase space, as the temperature is slightly below the Curie temperature _T_c. In a thin crystal of a chiral magnet, the stability of a

skyrmion lattice phase will be significantly enhanced6,25 due to uniaxial anisotropy28 or spatial confinement29,30. In both bulk compounds and thin films, skyrmion crystals have a fixed

lattice constant and adopt approximately circular shapes. Both theoretical analysis31 and experimental observations6,25 suggest that elliptical distortions of skyrmions in extended systems

are associated with a loss of stability, which is referred to as an elliptical or strip-out instability. However, because the twists of spins within a skyrmion give rise to a non-trivial

magnetization topology, it should be possible to tune its geometrical morphology without changing its topological class9. Recent Lorentz TEM observations of a FeGe nanostripe with a fixed

width of 130 nm suggest that elliptical skyrmions can be supported in such a confined geometry18. Simulations performed within the framework of a general model for a three-dimensional

isotropic chiral magnet32 in nanostripes further confirm this hypothesis (see the Methods section). In particular, for a long nanostripe of width _W__y_ on the order of the skyrmion size and

length , the theoretical model predicts a loss of radial symmetry of skyrmions without a loss of their stability (Fig. 1a). The skyrmion morphology in a single chain is predicted to depend

on nanostripe width _W__y_ for a certain range of applied magnetic fields. The dimensions of elliptical skyrmion shapes can be described in terms of their semi-axes _a_ and _b_ along and

perpendicular to the nanostripe, respectively. For nanostripe widths _W__y_ close to a critical value , skyrmions exhibit circular shapes (_a=b_), while for and they are predicted to show

longitudinal (_a>b_) and transverse (_a_<_b_) ellipticity, respectively. On further increasing the nanostripe width above a second critical value , the skyrmions are expected to

arrange themselves in the form of a zigzag chain at equilibrium. EXPERIMENTAL RESULTS To test these predictions, we carefully fabricated a wedge-shaped nanostripe from a bulk crystal of FeGe

using a lift-out method (Supplementary Fig. 1). Bulk FeGe possesses a high Curie temperature, _T_c∼278 K (ref. 25). The nanostripe had a thickness _L_ of 110 nm, a length _W__x_ of ∼2.6 μm

and a width _W__y_ that varied linearly from ∼10 to ∼180 nm (Fig. 1b and Supplementary Fig. 2). This range of widths is designed to include the helical period _L_D=70 nm of bulk FeGe (ref.

25). In this way, a comprehensive _H_-_W__y_ phase diagram for a wide range of nanostripe widths was constructed from a single sample. Figure 2a,e shows bright-field TEM images of the

fabricated wedge-shaped nanostripe. The regions marked by two white frames were selected for detailed analysis using off-axis EH at temperatures of 220 and 95 K, respectively. Figure

2b–d,f–k shows quantitative magnetic induction maps of the spin texture measured experimentally in the presence of different external magnetic fields using off-axis EH, with the directions

and magnitudes of the projected in-plane magnetization fields shown in the form of composite contour-colour maps. It should be noted that the recorded magnetic signal extends to the very

edges of the nanostripe and is not affected by imaging artifacts, uniquely providing the opportunity to test the predicted existence of twisted edge states14,15,16,17. The imaging conditions

and analysis procedure are described in Supplementary Figs 3 and 4. When the nanostripe was zero-field-cooled from room temperature to 220 K (Fig. 2b–d), it was initially found to contain a

complicated magnetic ground state comprising a mixture of regular, curved and vortex-like magnetic helices with distorted edge spins (Fig. 2b). When a magnetic field was applied normal to

the nanostripe plane, the spin helices in the nanostripe evolved in a complex process to form a single skyrmion chain (SSC) at _μ_0_H_∼148 mT (Fig. 2c). Significantly, the skyrmions can be

seen directly to adopt a sequence of compressed, regular and stretched morphologies with increasing nanostripe width, fully consistent with the theoretical prediction shown in Fig. 1a.

Similar distorted skyrmions were recently reported in two-dimensional FeGe1_−x_Si_x_ samples as a result of local lattice disorder around the edge of a crystal grain24. The distortions

become less pronounced when the applied magnetic field is increased. For example, at _μ_0_H_∼222 mT the skyrmions decrease in size, adjust their positions and adopt more circular shapes

(Fig. 2d). Meanwhile, skyrmions in the narrower part of the nanostripe disappear or migrate, while those in the wider part of the nanostripe form a zigzag skyrmion chain (ZSC) (dotted

rectangle in Fig. 2d). Notably, the skyrmion state is always accompanied by a complete chiral edge twist, which is observed directly in the magnetic induction maps and is characterized by a

single-twist rotation of the magnetization and nearly in-plane spins around the edge of the nanostripe (short white arrows in Fig. 2c)33. According to theoretical analysis, such an edge spin

configuration can be regarded as a type of surface state in a chiral magnet, which preserves the magnetic chirality of the spin texture around the edge14,15,16,17 and has been anticipated

to play a key role in current-induced skyrmion motion in nanostripes12,13. Experimentally, such chiral edge twists have been inferred in our previous Lorentz TEM imaging of magnetic

structure in both nanostripes18 and nanodisks19. However, the contrast around the edge associated with Fresnel fringes from the rapid change in specimen thickness, as discussed in the

introduction, complicated the observed images, especially for sample sizes of below 100 nm. Here, we give the first clear and convincing images of the edge magnetic state of a FeGe

nanostripe. The elliptically distorted skyrmions also persist at lower temperature. However, their formation process is significantly different from that at higher temperature. Previous

investigations have established that low temperatures are not beneficial for the formation of skyrmions, with the helix-to-skyrmion transformation depending on the initial helical

state18,25. After applying the same zero-field cooling procedure to 95 K, the nanostripe contains a mixed helical state, whose wave vector K is parallel and perpendicular to its long axis in

the narrower and wider part of the nanostripe, respectively (Fig. 2f). At _μ_0_H_∼217 mT, the helices transform into a SSC state only where K is parallel to the long axis (Fig. 2g). In this

case, the helix-to-skyrmion transformation has a precise one-to-one correspondence18, with each helix corresponding to a single skyrmion, as indicated by the curved dotted white lines in

Fig. 2f–h. This special helix-to-skyrmion transformation has been observed in nanostripes with widths above 130 nm (ref. 18). Here, we confirm that the same behaviour is followed in a much

narrower (45–150 nm) nanostripe. At higher magnetic fields, the number of skyrmions remains unchanged and they show a similar behaviour to that at _T_∼220 K, with a reduced size and

ellipticity (Fig. 2h). Moreover, by applying a cyclical magnetization process at _T_∼220 K and then cooling the nanostripe to 95 K in zero field, we were able to control the initial helical

state and to ensure that the direction of the wave vector K was almost completely along the long axis of the nanostripe (Fig. 2i). In response to an applied magnetic field, the

helix-to-skyrmion transition again followed a one-to-one behaviour, leading to similar skyrmion morphologies (Fig. 2c,j). The distinctive nucleation process that we observe for magnetic

skyrmions at low temperature from a helical spin spiral is reproduced in atomistic simulations based on direct energy minimization (see the Methods/Numerical simulations section) for a

nanostripe of fixed width at zero temperature (Supplementary Fig. 5). Notably, both the simulation and the experimental results indicate that the period of the spin spiral remains almost

unchanged over the complete range of their existence. This finding is remarkably different from observations made on spatially extended thin films and bulk crystals34, in which the helical

period increases monotonically with applied field. The reason for this difference is that the adjustment of the spin spiral period to an equilibrium value requires an unwinding of the

twisted edge state and is associated with a high energy barrier. The resulting one-to-one helix-to-skyrmion transition means that the period of the skyrmion chain inherits the periodicity of

the parent spin spiral, resulting in a difference between the period of a single skyrmion chain _P_SkCh and the equilibrium distance _P_SkL in a skyrmion lattice in an extended film

(Supplementary Fig. 5). We further quantified the effects of confinement on skyrmion morphology using both our experimental results and a theoretical model. The results obtained at _T_∼220 K

were used to understand the influence of nanostripe width on skyrmion morphology and to build a width-field phase diagram (Fig. 4). Figure 3a,b show a selection of experimental off-axis EH

images recorded at _T_∼220 K and _μ_0_H_∼148 mT. The deformation of each skyrmion was well estimated by fitting an elliptical shape with semi-axes _a_ and _b_ to its magnetic contrast

profile obtained using off-axis EH. Recent experimental studies using differential phase contrast (DPC) imaging performed in a scanning TEM revealed an intrinsic six-fold symmetry of the

internal structure of skyrmion lattice cells in two-dimensional FeGe crystals35. We have also observed this hexagonal skyrmion structure in two-dimensional FeGe using off-axis EH36. Such

observations confirm that the shapes of isolated skyrmions can be modified significantly in confined geometries. Here, we describe the skyrmion ellipticity by defining a parameter

_f=_1−_a_/_b_, which is zero when a skyrmion is circular and takes negative and positive values when it has longitudinal and transverse ellipticity in the _x_ direction, respectively. In the

experimental plot shown in Fig. 3b, the semi-axis _a_ exhibits a non-monotonic dependence on nanostripe width, while the semi-axis _b_ increases continuously with nanostripe width and is

almost twice as large in the wider part of the wedge as in the narrower part. These results are reproduced in a theoretical calculation by means of a similar analysis, with the critical

nanostripe width at which _a_=_b_ determined to be ∼110 nm. It should be noted that the formation of elliptical skyrmions in a strongly confined geometry is distinctly different from the

formation of anisotropic distorted skyrmions in a two-dimensional FeGe sample in the presence of a strain-induced modification of the intrinsic magnetic interactions37. WIDTH-FIELD MAGNETIC

PHASE DIAGRAM Based on all of our experimental results (Supplementary Figs 6 and 7), we constructed a width-field (-_H_) magnetic phase diagram for the FeGe nanostripe (Fig. 4a). At a low

value of applied magnetic field, a distorted helical spin spiral appears. It then transforms into a pure edge twist, a single skyrmion chain or a zigzag skyrmion chain in an applied magnetic

field of ∼75 mT, depending on the width of the nanostripe . We identified a limiting range of nanostripe widths between 79 and 140 nm, within which a single skyrmion chain is supported. For

 nm, no complete skyrmions survive, while for  nm either a SSC or a ZSC forms, depending on the applied field. The appearance of a ZSC reflects the tendency of interacting skyrmions to

condense into a hexagonal lattice when they are densely packed8. In the narrow part of the nanostripe, at _μ_0_H_∼105 mT the helix-to-skyrmion transition is characterized by the formation of

an incomplete skyrmion, with a half-skyrmion attached to one edge and a twisted state on the other edge (small magenta domain in Fig. 4a; see also Supplementary Fig. 6b). In Supplementary

Fig. 5d, we show qualitatively similar incomplete skyrmions obtained during energy minimization using the NCG method. However, it has to be mentioned that, strictly speaking, such

magnetization states obtained using a direct minimization method are not a true physical realization and only final magnetization configurations corresponding to the equilibrium state can be

compared to the experimental data. At higher magnetic fields, the skyrmions gradually lose their stability and collapse into a conical or field-saturated state. We estimate an optimal width

for the nanostripe of ∼110 nm, which corresponds to . For a nanostripe with , the range of existence of skyrmions in an applied magnetic field is significantly lower than for skyrmions in

an extended film. DISCUSSION Our experimental phase diagram agrees closely with a corresponding theoretical phase diagram (Fig. 4b), in which transitions between magnetic phases were

determined by comparing their energies (see the Methods/Numerical simulations section). The width of the nanostripe was further extended significantly in the simulations. Slight

discrepancies between the experimental and theoretical phase diagrams arise from an experimentally observed weakly hysteretic behaviour of the system, which originates from the presence of

finite energy barriers between magnetic states. In the theoretical phase diagram, corresponds to the critical width for ideally circular skyrmions. Below and above , the skyrmions always

exhibit longitudinal and transverse elliptical distortions, respectively. However, above a certain magnetic field and assuming a fixed skyrmion density, as observed experimentally, the

skyrmions become circular over a wide range of nanostripe widths (see the dashed region in Fig. 4b). Outside this range, the skyrmions are always elliptically distorted. However, at high

magnetic fields _μ_0_H_≳300 mT (_H_≳0.9_H_D in reduced units) and finite temperatures, their elliptical distortions become negligibly small and in practice indistinguishable on a background

of thermal fluctuations for any nanostripe width (see, e.g., Fig. 2d,h,k). The good agreement between our experimental and theoretical results allows us to obtain further insight into the

evolution of elliptical skyrmions and the nucleation process by calculating the equilibrium periods of helicoids and skyrmions as a function of applied magnetic field (Fig. 5). For an

infinitely wide film without edge effects, the helicoid remains metastable, with a period that increases with applied magnetic field up to very high field of ∼210 mT (red solid circles in

Fig. 5)38. In contrast, for the nanostripe, in the presence of edges the period of the helicoid increases with increasing magnetic field up to ∼100 mT (red hollow circles in Fig. 5), above

which it loses its stability. Above a critical field (marked by a star), there are no minima on the energy landscape corresponding to a helicoid. The helicoid then transforms into a single

skyrmion chain or a zigzag skyrmion chain, depending on the nanostripe width (Fig. 4b). It should be noted that the transition field of ∼70 mT in the phase diagram (Fig. 4b) is the field at

which the energies of the spin spiral and the skyrmion chain are equivalent, whereas the instability of the helicoid appears at ∼100 mT. At a finite temperature, the experimentally observed

transition is expected to appear between these two critical fields. Our results are based on the application of state-of-the-art electron holography in combination with advanced computer

simulations based on atomistic models and direct energy minimization. In this context, our experimental results could not have been obtained using other methods such as small angle neutron

scattering4, transport measurements39 or Lorentz TEM6. From a methodological perspective, the experimental approach that we use offers the prospect of allowing two and three-dimensional

magnetic textures in other nanometer-scale spin systems to be quantified40 in future studies. Such a capability is currently lacking using any other technique. In a broader context, our

experimental results and theoretical analysis therefore open avenues for exploring quantum confinement in other complex non-linear spin systems. Our results can be envisioned for several

potential applications. First, the high flexibility of spatially-confined magnetic skyrmions that we observe allows them to adapt their shape and size in a nanostructure whose size is

comparable to or even smaller than the size of an equilibrium skyrmion in an extended film. For nanostripe widths , such skyrmions remain stable even at a relatively high temperature of

_T_∼0.8 _T_c. Since the efficient coupling of electron currents to skyrmions is related to topological charge and is independent of skyrmion morphology9, we expect that elliptical skyrmions

can be moved as efficiently as circular skyrmions. Moreover, in contrast to the so-called precession and breathing modes that are exhibited by approximate circular skyrmions in extended

films41, we predict unique oscillating and stretch-out excitation modes in alternating magnetic fields for elliptically-distorted skyrmions in nanostripes whose widths are smaller than twice

the period of an equilibrium spin spiral, that is, . A further important aspect revealed by our observations is the direct visualization of a mechanism of skyrmion nucleation that is almost

independent of their initial state and characterized by the conservation of skyrmion number with respect to the number of helical spirals in the nanostripe. In contrast to earlier

proposals12,21,42, the controlled formation of a well-defined number of skyrmions in a nanostripe whose width lies in the range, can be achieved simply by varying the applied magnetic field.

Such an approach provides a simple and efficient method of skyrmion nucleation that is highly suitable for applications in spintronic devices. For instance, by applying a magnetic field

locally to a nanostripe with a fixed length equal to an integer number of spin spiral periods , one can reliably initiate the nucleation of _N_ skyrmions, which then can be pushed onto a

skyrmion track by applying an electric current. Moreover, the range of nanostripe widths can be considered as a range for the width of a skyrmion racetrack device. Indeed, according to our

observations and theoretical modelling, for _W__y_ outside this range we observe an instability of a single skyrmion chain in the form of a collapse of individual skyrmions when

_W__y_<<_L_D or an instability in the form of the formation of a zigzag chain of skyrmions when _W__y_≫2_L_D. METHODS SPECIMEN PREPARATION Polycrystalline B20-type FeGe samples were

synthesized using a cubic anvil-type high-pressure apparatus18. Structural characterization by X-ray diffraction and susceptibility measurements were used to confirm the crystalline quality

of the bulk FeGe. A wedge-shaped nanostripe was prepared for TEM observation using a lift-out method in a focused ion beam (FIB) scanning electron microscope (SEM) dual beam workstation (FEI

Helios Nanolab 600i) equipped with a gas injection system (GIS) and a micromanipulator (Oxford Omniprobe 200+). Details of the sample fabrication procedure are given in Supplementary Fig.

1. OFF-AXIS ELECTRON HOLOGRAPHY Transmission electron microscopy was carried out in an FEI Titan 80-300 XFEG TEM operated at 300 kV. For magnetic imaging, the specimen was placed in magnetic

field-free conditions (Lorentz mode) with the conventional objective lens turned off. The excitation of the objective lens was varied to apply magnetic fields normal to the specimen plane

over a field range of between 0 and ∼1.5 T. A double tilt liquid nitrogen specimen holder (model 636, Gatan Co.) and a temperature controller were used to vary the specimen temperature

between 380 and 95 K. In off-axis electron holography experiments, the biprism voltage was typically set to 90 V to produce an overlap interference width of 1,200 nm and a holographic

interference fringe spacing of 3.6 nm at a nominal magnification of 38,000. For hologram recording, a cumulative acquisition approach was used to record 20 holograms (with an exposure time

of 6 s for each hologram). Off-axis electron holograms were reconstructed numerically using a standard Fourier transform based method with sideband filtering using custom-designed Matlab

codes. Contour lines and colour maps were generated from recorded magnetic phase images to yield magnetic induction maps (see Supplementary Figs 3 and 4 for details). NUMERICAL SIMULATIONS

Theoretical models of skyrmions were constructed using a classical spin Hamiltonian for a simple cubic lattice consisting of only three energy terms: the Heisenberg exchange interaction, the

Dzyaloshinskii–Moriya interaction (DMI) and the energy of the applied magnetic field32,43 in the form: where _J_ is the exchange coupling constant, D_ij_ is the Dzyaloshinskii–Moriya vector

defined as D_ij_=_D_·R_ij_, _D_ is the DMI scale constant, R_ij_ is a unit vector pointing from lattice site _i_ to lattice site _j_, _μ_0 is the vacuum permeability, H is the external

magnetic field and M_i_=M_i_/_M__i_ is a unit vector of the magnetic moment at lattice site _i_. The symbol denotes summation over all nearest-neighbor pairs. Heisenberg exchange and DMI are

assumed to be isotropic in all three spatial directions. Following a standard approach, we ignore the contribution of magnetocrystalline anisotropy32, which in the case of FeGe is

negligibly small with respect to other energy terms, especially at relatively large temperatures as in our experimental setup44. Because of mainly Bloch-type modulations of the magnetization

in FeGe, the stray field effect can be described exclusively in terms of so-called surface magnetostatic charges7, which in the case of a nanostripe produce a closed flux of the stray field

mainly outside the sample. Therefore, we expect a relatively small contribution from magnetostatic interactions and in the first approximation we neglect it. Nevertheless, in the most

general case and for a precise quantitative description of the system the contribution of stray field interactions should be taken into account. Following earlier studies8,32, we introduce

the notation where _L_D is the period of a homogeneous helical spin spiral in zero magnetic field, _H_D is the saturation magnetic field corresponding to the transition between the conical

phase and a field saturated ferromagnetic state for the bulk crystal, and are micromagnetic constants for the exchange interaction and the DMI, respectively, _M_s is the magnetization of the

material and _a_ is the lattice constant. In the present simulations, we used _L_D=70 _a_ and a size for the simulated domain of between 30 and 512 spins in all three spatial directions. A

value for _H_D of 0.325 T was chosen based on both the present results and previous experimental measurements45. The realistically large _L_D/_a_ ratio and the choice of value for _H_D

resulted in good qualitative and quantitative agreement between the simulations and the experimental measurements. Taking into account the expression for _L_D in equation (2) in the

calculations, we set the coupling constants in arbitrary units of energy to be, and the applied field _H_ in units of saturation field _H_D according to equation (3). For direct minimization

of the model Hamiltonian (equation (1)), we used a non-linear conjugate gradient (NCG) method employing adaptive stereographic projections for the magnetization vectors, all implemented on

NVIDIA CUDA architecture32. As an initial state for the helicoid and conical phases, we used an ordinary Bloch-type spiral with a propagation vector along the long side of the nanostripe

(_x_-axis) or perpendicular to it and along the applied field (_z_-axis), respectively. For the skyrmion phase, we used as the initial configuration a simple ferromagnetic state with its

magnetization along the direction of the applied field (positive direction of the _z_-axis) and cylindrical domains with opposite magnetization. At each point on the phase diagram of

magnetic states shown in Fig. 4, we identified the equilibrium period of the spirals and the equilibrium density of skyrmions in the system, which correspond to the lowest energy density of

the particular phase and then compared the energies of all competing phases. Moreover, an NCG method was used to simulate the field-driven magnetization process of skyrmion formation from

the helical state. In this case, the initial magnetic state was set to be spin helices and the external magnetic field was then increased gradually so that a metastable state with a local

minimum energy was obtained (Supplementary Fig. 5). This procedure is different from a conventional Landau–Lifshitz–Gilbert (LLG) dynamics simulation46, in which the magnetization process is

obtained by solving the LLG equation. However, in a certain sense, the NCG method can be referred to as pseudo-dynamic, meaning that at each iteration step it follows the direction of the

lowest energy gradient, but without a pre-defined time step. DATA AVAILABILITY All relevant data are available from the authors on reasonable request. See author contributions for enquiries

about specific data sets. ADDITIONAL INFORMATION HOW TO CITE THIS ARTICLE: Jin, C. _et al_. Control of morphology and formation of highly geometrically confined magnetic skyrmions. _Nat.

Commun._ 8, 15569 doi: 10.1038/ncomms15569 (2017). PUBLISHER’S NOTE: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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ACKNOWLEDGEMENTS This work was supported by the Natural Science Foundation of China, Grant Nos. 51622105, 11474290, 11374302 and U1432251; the Key Research Program of Frontier Sciences,

CASOQYZDB-SSW-SLH009; the Youth Innovation Promotion Association CAS No. 2015267; the CAS/SAFEA international partnership program for creative research teams of China and the National Key

Basic Research of China. Work by F.N.R. was carried out within the state assignment of FASO of Russia (theme Quantum No. 01201463332). The research leading to these results has received

funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement number 320832. AUTHOR INFORMATION AUTHORS AND

AFFILIATIONS * The Anhui Key Laboratory of Condensed Matter Physics at Extreme Conditions, High Magnetic Field Laboratory, Chinese Academy of Science (CAS), Hefei, 230031, Anhui Province,

China Chiming Jin, Haifeng Du, Mingliang Tian & Yuheng Zhang * Department of Physics, University of Science and Technology of China, Hefei, 230031, Anhui Province, China Chiming Jin *

Ernst Ruska-Centre for Microscopy and Spectroscopy with Electrons and Peter Grünberg Institute, Forschungszentrum Jülich, Jülich, 52425, Germany Zi-An Li, András Kovács, Jan Caron, Fengshan

Zheng & Rafal E Dunin-Borkowski * Faculty of Physics and Center for Nanointegration (CENIDE), University of Duisburg-Essen, Duisburg, 48047, Germany Zi-An Li & Michael Farle *

Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, China Zi-An Li * M.N. Miheev Institute of Metal Physics of Ural Branch of Russian Academy of Sciences, Ekaterinburg,

620990, Russia Filipp N. Rybakov * Ural Federal University, Ekaterinburg, 620002, Russia Filipp N. Rybakov * KTH Royal Institute of Technology, Stockholm, SE-10691, Sweden Filipp N. Rybakov

* Peter Grünberg Institute and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, Jülich, 52425, Germany Nikolai S. Kiselev & Stefan Blügel * Collaborative Innovation

Center of Advanced Microstructures, Nanjing University, 210093, Jiangsu Province, China Haifeng Du, Mingliang Tian & Yuheng Zhang * Center of Functionalized Magnetic Materials, Immanuel

Kant Baltic Federal University, 236041, Kaliningrad, Russian Federation Michael Farle Authors * Chiming Jin View author publications You can also search for this author inPubMed Google

Scholar * Zi-An Li View author publications You can also search for this author inPubMed Google Scholar * András Kovács View author publications You can also search for this author inPubMed 

Google Scholar * Jan Caron View author publications You can also search for this author inPubMed Google Scholar * Fengshan Zheng View author publications You can also search for this author

inPubMed Google Scholar * Filipp N. Rybakov View author publications You can also search for this author inPubMed Google Scholar * Nikolai S. Kiselev View author publications You can also

search for this author inPubMed Google Scholar * Haifeng Du View author publications You can also search for this author inPubMed Google Scholar * Stefan Blügel View author publications You

can also search for this author inPubMed Google Scholar * Mingliang Tian View author publications You can also search for this author inPubMed Google Scholar * Yuheng Zhang View author

publications You can also search for this author inPubMed Google Scholar * Michael Farle View author publications You can also search for this author inPubMed Google Scholar * Rafal E

Dunin-Borkowski View author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS H.D., C.J. and M.T. conceived the project and prepared the samples. Z.-A.L.

performed the electron holography experiments and analysed the holographic data with assistance from A.K., J.C. and F.Z. S.K. and F.N.R. developed the model calculations. The manuscript was

prepared by H.D., Z.-A.L. and N.S.K., with contributions from M.F., R.E.D.-B., S.B. and Y.Z. All authors discussed the results and contributed to the manuscript. CORRESPONDING AUTHORS

Correspondence to Zi-An Li, Nikolai S. Kiselev or Haifeng Du. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no competing financial interests. SUPPLEMENTARY INFORMATION

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