Real-space determination of the isolated magnetic skyrmion deformation under electric current flow

Significance

Magnetic skyrmions have long been heralded as candidates for information carriers in future electronic devices due to their topological protection and low-energy manipulation. It is a common belief that skyrmions are rigid under applied currents, and the majority of skyrmionics publications and proposed applications using skyrmion motion rely on this assumption. We found that skyrmions deform and expand under an applied current with surprising severity even with small currents, highlighting their robust nature. This has a tremendous impact on the skyrmionics and spintronics field, and various ideas need to be reevaluated, whereas other unconventional fields such as reservoir computing rely on such a property and will certainly use these findings to guide a plethora of follow-up studies.

Abstract

The manipulation and control of electron spins, the fundamental building blocks of magnetic domains and spin textures, are at the core of spintronics. Of particular interest is the effect of the electric current on topological magnetic skyrmions, such as the current-induced deformation of isolated skyrmions. The deformation has consequences ranging from perturbed dynamics to modified packing configurations. In this study, we measured the current-driven real-space deformation of isolated, pinned skyrmions within Co₁₀Zn₁₀ at room temperature. We observed that the skyrmions are surprisingly soft, readily deforming during electric current application into an elliptical shape with a well-defined deformation axis (semimajor axis). We found that this axis rotates unidirectionally toward the current direction irrespective of electric current polarity and that the elliptical deformation reverses back upon current termination. We quantified the average distortion δ, which increased by ∼90% during the largest applied current density |j| = 8.46 ×10⁹ A/m² when compared with the skyrmion’s intrinsic shape ($j=0$). Additionally, we demonstrated an approximately 120% average skyrmion core size expansion during current application, highlighting the skyrmions’ inherent topological protection. This evaluation of in situ electric current–induced skyrmion deformation paints a clearer picture of spin-polarized electron–skyrmion interactions and may prove essential in designing spintronic devices.

Electron spins are the fundamental building blocks of magnetic spin textures called skyrmions (1–4), which have been heralded as potential information carriers within future spintronics devices due to their topological protection and low energy manipulation by means of external stimuli such as an electric current or magnetic field (5). Magnetic skyrmions arising from the Dzyaloshinskii–Moriya interaction (DMI) (6, 7) are stabilized in systems in which space inversion symmetry is broken. Since their experimental observation a little over a decade ago (1, 2), tens of skyrmion hosting materials have been identified (8), providing myriad options to design experiments or build devices (9, 10).

Spin transfer torque (STT) enables electric current-driven skyrmion motion and has been modeled via the Landau–Lifshitz–Gilbert (LLG) equation (SI Appendix, Eq. S1). This equation describes the motion of all spins in the system based on material parameters and external stimuli. The pinning potential for a skyrmion may be modeled in general by intrinsic material fluctuations such as local variations in the exchange stiffness or DMI strength (11) or by local extrinsic variations such as atomic defects (12, 13), and leads to a threshold current density, above which the skyrmion is depinned and moves with a velocity oriented at an angle from the current axis direction, called the skyrmion Hall angle. The skyrmion–current interaction has previously been experimentally probed in reciprocal space studies of skyrmion lattice (SkyL) rotation (14) and electronic transport measurements (15) as well as by real-space observations of the current induced motion (16, 17). Therein, the skyrmions were treated as two-dimensional rigid circular bodies, as shown in Fig. 1A, and the Thiele equation, a steady-state approximation of SI Appendix, Eq. S1, could be applied to describe their center-of-mass motion (18, 19).

Fig. 1.

Schematics of (A) a circular shaped and (B) a deformed skyrmion, and (C) various magnetic states observed in a Co₁₀Zn₁₀ thin plate. We measured each data point by increasing μ₀H_ext after cooling with μ₀H_ext = <10 mT to the desired T from above T_C. Inset illustrates the chiral, cubic crystal structure, where Co atoms are shown in blue, and Zn are shown in gray. The Right panels in (C) are example LTEM micrographs of Iso Sky, SkyB, SkyL, and Hel from top to bottom, respectively. Scale bar = 200 nm.

Later, theoretical studies predicted the current-induced dynamics of three-dimensional (3D) skyrmions, known as skyrmion strings (20, 21), as well as skyrmions with distortions from their intrinsic circular profiles (20, 22–27) that arise from SI Appendix, Eq. S1. Noncircular skyrmion profiles may in turn affect the skyrmion’s current-induced dynamics, and indeed, it was the primary candidate for the discrepancy between the expected and measured spin-orbit-torque-driven skyrmion Hall angle recently reported by X-ray microscopy in a multilayered thin film (26). For translational motion to occur, however, a depinning threshold current density must be reached, below which the electric current and pinning potential tug at the skyrmion, deforming its shape. The resulting elliptically shaped skyrmions are similar to those seen in systems with anisotropic DMI (28–33). This elliptical deformation, depicted in Fig. 1B, has since been the topic of several theoretical studies (23, 25, 34–36), but direct experimental verification of the current-induced deformation of the skyrmion’s intrinsic circular shape has yet to be reported.

In this work, we performed real space observations of isolated Bloch-type skyrmions stabilized in the chiral magnet Co₁₀Zn₁₀. We drove the skyrmions by using 1-s-long electric current pulses with current densities $j$ below the critical current density $j_c$ required to depin the skyrmions. Using live video recording, we captured the current-induced skyrmion deformation and determined it to be elliptical. We quantified the deformation by measuring the deformation axis angle θ_def and the elliptical distortion δ versus $j$. We developed a simplistic theoretical model of skyrmion deformation and found that the experimentally measured relations fit well to the model. Additionally, we report the expansion of the skyrmion core size at high current densities.

Results

The skyrmion-hosting, chiral magnet family Co_10-x/2Zn_10-x/2Mn_x (37–40) with β-Mn type structure (see the Inset in Fig. 1C) is particularly useful due to its high transition temperature between paramagnetic (PM) and helimagnetic states. The $x=0$ compound Co₁₀Zn₁₀ used in this study, for example, exhibits magnetic ordering at T < 462 K as a bulk specimen (37). The metastable magnetic state diagram above room temperature for a Co₁₀Zn₁₀ thin plate determined by the present study is shown in Fig. 1C. We measured each data point by increasing μ₀H_ext at a target temperature T after cooling from above T_C to the target temperature with a remanent field (objective lens turned off) of μ₀H_ext < 10 mT. We found various magnetic states including isolated skyrmion (Iso Sky), skyrmion bunch (SkyB) or skyrmion cluster, a helical ground state (Hel) in which the spins twist in a Bloch-type helix along a wavevector, conical state (Con) with wavevector normal to the thin plate, thermodynamically stable SkyL with sixfold symmetry, ferromagnetic phase (FM), and PM region. Lorentz transmission electron microscopy (LTEM) micrographs of each of these states are exemplified in Fig. 1 C, Right. Note that the LTEM contrasts of the Con, FM, and PM states are all monotonic, as seen in the background of the Iso Sky and SkyB panels, because the in-plane magnetization does not vary in the sample plane. While this diagram varies slightly from that measured in the bulk, the differences are probably due to the thin plate geometry, which has a thickness t ≈ 120 nm as measured via scanning ion microscopy (SI Appendix). As seen in the magnetic state diagram, Iso Sky, SkyB, Hel, and Con are all found at room temperature with an externally applied field of μ₀H_ext ≈ 40 mT. The SkyBs separate above a critical external field, marked by the solid red line in Fig. 1C, and therein only isolated skyrmions are stabilized. Such a transition has been previously found and verified in cubic chiral magnets and indicates that the force between skyrmion transitions from attractive (creation of SkyBs embedded in a Con background) to repulsive in this region of the phase diagram (41). The large phase space containing metastable isolated skyrmions makes Co₁₀Zn₁₀ a highly favorable compound for real-space studies of skyrmion dynamics and shape deformation.

To observe the skyrmion deformation in real space, we used LTEM while applying an electric current through the Co₁₀Zn₁₀ thin plate (see SI Appendix and Materials and Methods for further details). Using this powerful technique, we can observe spin textures as they interact with external stimuli in real time. To stabilize magnetic skyrmions in the Co₁₀Zn₁₀ thin plate, we applied an external magnetic field of μ₀H_ext ≈ 40 mT perpendicular to the thin plate. Under these conditions, we observed several Bloch-type Iso Skys in a view area (2.9 μm × 2.2 μm). To measure the skyrmion deformation, we configured the LTEM magnification to observe five isolated skyrmions of interest simultaneously, shown in the in-plane magnetic inductance map in Fig. 2A and numbered 1–5, together with a magnified image of skyrmion 4 shown in the Inset outlined in solid green. Fig. 2B shows a simulated defocused LTEM image of an elliptically deformed skyrmion drawn to illustrate several observables of note. The skyrmion deformation is indicated by the emergence of a long, semimajor axis a and shorter, semiminor axis b, shown in Fig. 2B (see Materials and Methods for details regarding the quantitative measurement of a and b from the defocused LTEM images). From such axis lengths, one may characterize the severity of deformation, or distortion, $\delta =1-b/a$. Let us also define the skyrmion deformation axis angle θ_def, illustrated in Fig. 2B, which is the angle between the semimajor axis and the current direction (the horizontal axis $\widehat{x}$ in this case).

Fig. 2.

Skyrmion deformation for various current densities. (A) In-plane magnetic induction map of five skyrmions present in a Co₁₀Zn₁₀ thin plate with an applied external magnetic field of μ₀H_ext = 40 mT at 295 K. Inset shows a vector color map indicating the direction of the measured in-plane magnetic field. Scale bar 500 nm. Inset outlined in solid green shows in-plane vector map of skyrmion 4, scale bar = 50 nm. (B) Defocused LTEM multislice simulated image of an elliptical skyrmion. The fit ellipse is drawn in red, and the major axis 2a, minor axis 2b, and deformation axis angle θ_def are labeled on the image in green, red, and white, respectively. (C–J) Defocused LTEM real space micrographs of skyrmion 4 (C, E, G, I) before and (D, F, H, J) during 1-s electric current pulse applications of (C, D) $j = +7.52 \times {10}^9 \mathrm{A}/{\mathrm{m}}^2$, (E, F) $j=-7.52\times {10}^9 \mathrm{A}/{\mathrm{m}}^2$, (G, H) $j = +8.46 \times {10}^9 \mathrm{A}/{\mathrm{m}}^2$, and (I, J) $j = -8.46 \times {10}^9 \mathrm{A}/{\mathrm{m}}^2$. The current direction is along the horizontal axis. Distortion δ and deformation axis angle θ_def (measured from horizontal) are labeled at the top of each panel. The conic functions fit to the bright domain wall contrast are drawn in (D–I) with solid red lines. Scale bar in C is 200 nm.

During the application of electric current, we observed that the skyrmions enter a steady-state deformed shape before returning to their original shape upon current pulse termination. While these skyrmions may be considered “hard” in the sense that Co₁₀Zn₁₀ is a single-crystalline noncentrosymmetric magnet, they exhibit a flexibility that has yet to be seen in comparable compounds such as FeGe (16) and MnSi (1). This is probably due to the random site occupancy within Co₁₀Zn₁₀ (42), as opposed to the former two compounds, which do not have random site occupancy and therefore exhibit a lower density of pinning sites. The abundance and strength of the pinning sites due to random site occupation allowed for the observation of skyrmion deformation via live recording, since many skyrmions remain pinned during observation. We analyzed the bright intensity pixels in the acquired defocused LTEM images (see SI Appendix for more details), a selection of which is shown in Fig. 2 C–J for skyrmion 4 before (Fig. 2 C, E, G, and I) and during (Fig. 2 D, F, H, and J) the application of 1-s electric current pulses. The electric current was held at a constant current density of $j = +7.52 \times {10}^9 \mathrm{A}/{\mathrm{m}}^2$, $j = -7.52 \times {10}^9 \mathrm{A}/{\mathrm{m}}^2$, $j = +8.46 \times {10}^9 \mathrm{A}/{\mathrm{m}}^2$, and $j = -8.46 \times {10}^9 \mathrm{A}/{\mathrm{m}}^2$ in Fig. 2 D, F, H, and J, respectively. The measured skyrmion distortion and deformation axis angles are listed above each panel, and the panels’ cartoon illustrates the measured deformation axis angle. Clearly, the distortion increases, and the deformation axis angle decreases for increasing current density magnitude |j| while the difference due to switching current polarity is relatively small.

Both semimajor and semiminor axis lengths increase with increasing |j|, as shown in Fig. 3A. The plotted data points are the mean values, and the error bars represent the SD from the mean, σ. The deformation axis angle θ_def, however, decreases unidirectionally with increasing |j|, as shown in Fig. 3B, suggesting that the skyrmion’s semimajor axis may align with the current direction for large current densities, a trend that we analyze further later in this study. The initial θ_def (j = 0) values are plotted for all five skyrmions in Fig. 3B, illustrating the large deviation in the measurements attributable to the differences in the initial pinning site locations and intrinsic deformations, as well as experimental measurement noise, which is amplified when the skyrmions are almost circular, as they are at j = 0. The dashed lines in Fig. 3 A and B are qualitative curves drawn as a guide. The distortion increases with increasing |j| similarly to a and b, as plotted in Fig. 3C. Here, we fit the data to the function

$$\begin{array}{c}\delta \left(j\right)=1-\frac{b}{a+c \left(\left|j\right|+j_0\right)},\end{array}$$ [1]

where j₀ is an optimal current offset parameter. The fitting parameter c is material dependent and defined in SI Appendix, where Eq. 1 is also derived from the simplistic theoretical model outlined later in the text. The fit is plotted by the solid red line and is valid for $\left|j\right|>\left|j_{c_1}\right| {\text{A/m}}^2$, the reason for which we return to later in the text. We estimated a₀ and b₀ by using the average zero-current values a(j = 0) = 3.43 × 10⁻⁸ m and b(j = 0) = 3.03 × 10⁻⁸ m, measured from the reconstructed in-plane magnetic induction image in Fig. 2A. Assuming a theoretical intrinsic shape that is circular, we set a₀ = b₀ = 3.03 × 10⁻⁸ m. Using the least-squares fitting method, we determined the optimal fit parameters of (c, j₀) = 5.9 × 10⁻¹⁸ A/m³, −4.4 × 10⁹ A/m²). As mentioned before, the distortion, a measure of ellipticity of the skyrmion’s deformed shape, increases with increasing |j| for both current polarities above the threshold of $\left|j_{c1}\right|\approx 6.6\times {10}^9 {\text{A/m}}^2$. These results demonstrate the STT-induced deforming effect on a skyrmion’s shape: an elliptical deformation that increases in severity even before the critical current density required to depin the skyrmions into a “flow” motion state.

Fig. 3.

Quantification of experimentally measured skyrmion deformation. (A, B) Measured mean semimajor and semiminor axis lengths (a and b) and the mean deformation axis angle θ_def versus applied current density j. Qualitative dashed lines are drawn in (A, B) as guidelines for the eye. The mean deformation axis angle θ_def for all five skyrmions is shown at j = 0. (C) Average skyrmion shape distortion δ versus applied current density j with a least-squares fit to the function $\delta \left(j\right)=1-\frac{b}{a+c \left(\left|j\right|+j_0\right)}$, which excludes the $\left|j\right|<\left|j_{c_1}\right|$ data points. (D) Average measured skyrmion core area defined as $\mathcal{A}$ = πab as a function of applied current density. The dashed line in D is a guide for the eye. The two critical current densities, $j_{c_1}$ and $j_{c_2}$, are drawn for all four plots by vertical dash-dot-dash and dotted lines, respectively. Insets show the defocused LTEM real space micrographs of skyrmions 4 and 5 (Sky #4, Sky #5) before (Left) and during (Right) electric current application with density $j = -8.46 \times {10}^9 \mathrm{A}/{\mathrm{m}}^2$. The conic fits are drawn in dashed black/red and solid red, respectively. The fits from j = 0 (Left) are overlaid on the j ≠ 0 panels (Right), and the scale bar = 150 nm. Error bars in (A–D) indicate the SD from the mean.

Although the results thus far characterize the skyrmion deformation, one may also consider the effect of such a deformation on the skyrmion’s core size, defined as the cross-sectional area enclosed by the domain walls of the skyrmion, $\mathcal{A}$ = πab, which is plotted against j in Fig. 3D. We note that the change in all the observables plotted in Fig. 3 is pinning site dependent, and we have plotted the average values taken over all observed sites, where the error bars represent the SD. A dashed line is drawn as a qualitative guide. The average skyrmion core area increases by ∼120%, from $\mathcal{A}$≈ 3.23 ± 0.66 × 10⁻³ μm² to $\mathcal{A}$ ≈ 7.07 ± 1.22 × 10⁻³ μm², and returns to its original area upon electric current termination. The Insets in Fig. 3D show the defocused LTEM images of skyrmions 4 and 5 (labeled “Sky #4” and “Sky #5”) before (Left) and during (Right) electric current application with density j = −8.46 × 10⁹ A/m². The conic functions fit to the bright pixels are drawn in dashed black/red in the j = 0 (Left) and in solid red in the j ≠ 0 (Right) panels. The conic fits for j = 0 are overlaid onto the j ≠ 0 panels for comparison. The skyrmion areas drawn in the j = 0 panels are contained entirely within the corresponding j ≠ 0 skyrmion areas, suggesting that both a and b increased during electric current application as confirmed in the plot of $\mathcal{A}$ against j.

After measuring the skyrmion deformation for 21 current density values ranging from j = −8.46 A/m² to j = +8.46 A/m², we found several notable features. First, all observables measured including the semimajor and semiminor axis lengths, deformation axis angle, distortion, and skyrmion core size vary unidirectionally with increasing current density magnitude. Second, two critical current densities arise, above which the observables begin to rapidly change. One is $\left|j_{c_1}\right|\approx 6.6\times {10}^9 {\text{A/m}}^2$, which indicates a unidirectional decrease in the deformation axis angle θ_def. The next is $\left|j_{c_2}\right|\approx 7.75\times {10}^9 {\text{A/m}}^2$, above which a unidirectional increase in all other observables commences. At j = 0, the skyrmions are pinned, so their deformation away from a circular shape depends on their relaxed structure throughout the thin plate. The pinning site is probably localized as clusters of Co atoms because of random site occupancy, a crystal defect, thickness variation, or surface adatom. Therefore, it is reasonable that the skyrmion string would deform near the pinning potential relative to the rest of its length through the plate. For low current values $\left|j\right|<\left|j_{c_1}\right|$, the skyrmion perturbs away from its intrinsically deformed, pinned shape (deformation axis angle θ_def,0). This may be observed as a rotation of the longer, major axis toward the driving force direction, corresponding to a change in θ_def toward some final value (θ_def,f). However, in this current density regime, the measured rotation is small and contained entirely within the error bars in Fig. 3B; moreover, the change in distortion is negligible and may even be negative as the skyrmion’s deformation axis rotates away from its original deformed shape.

To explain the skyrmion’s current-driven unidirectional deformation and major axis rotation we propose a simplistic theoretical model. We describe the skyrmion as a bound state of two half-skyrmions (meron-like objects), each with the same vorticity. One half-skyrmion is pinned at a defect, while the other is mobile. The attractive interaction between the two halves is approximated by a harmonic pinning potential, V(d) = V₀d². Here, d is the distance between the objects, which are connected by a spring that locally resembles a domain wall pair. In the skyrmion picture, d is the skyrmion’s elongation 2Δa = 2(a_f – a₀), where a₀ and a_f are the initial and final semimajor axis lengths, respectively.

Now, we describe the motion of the mobile half-skyrmion at the relative position R, |R| = d, plus the attached spring, by a modified Thiele equation,

$$\begin{array}{c}-\frac{\gamma }{M_s}\frac{dV\left(R\right)}{d\mathit{R}}=\mathit{G}\times (\dot{\mathit{R}}-{\mathit{v}}_s)+D\left(R\right)\cdot (\alpha \dot{\mathit{R}}-\beta {\mathit{v}}_s),\end{array}$$ [2]

in the steady-state limit when dynamics are absent, $\dot{\mathit{R}}=0$. The other constituents of the Thiele equation, Eq. 2, are the gyromagnetic ratio γ, the saturation magnetization M_S, the gyrovector $\mathit{G}=4\pi Q\widehat{\mathit{z}}$ with the skyrmion winding number $Q=\frac{1}{4\pi }{\displaystyle {\int }_V\widehat{\mathit{m}}}\cdot \left({\partial }_x\widehat{\mathit{m}}\times {\partial }_y\widehat{\mathit{m}}\right)d^{2}r$, the spin velocity ${\mathit{v}}_s\propto \mathit{j}$, the dissipation matrix D with elements $D_{ij}={\displaystyle {\int }_V{\partial }_{r_i}\widehat{\mathit{m}}\cdot {\partial }_{r_j}\widehat{\mathit{m}}} d^{2}r$, the Gilbert damping α, and the nonadiabatic STT parameter β. Because of the elongation d = R of the two-meron object in the direction ${\widehat{\mathit{e}}}_{\theta }={\left(\cos {\theta }_{\mathit{def}},\sin {\theta }_{\mathit{def}}\right)}^T$, the dissipation elements D_ij in the perpendicular direction $\left(\widehat{\mathit{z}}\times {\widehat{\mathit{e}}}_{\theta }\right)$ increase, rendering D asymmetric. We assume that it takes the form D = D_S + D₁d, that is, that it constitutes of the symmetric part D_S ≤ 2π (D_S ≥ 4π for skyrmions) and a nonsymmetric correction of D, which is linear in the elongation d and scales as D₁. Keeping only terms linear in $v_s=\left|{\mathit{v}}_s\right|$, we find

$$\begin{array}{c}d=\sqrt{{\left(\beta D_s\right)}^2+{\left(4\pi Q\right)}^2}\frac{v_s M_s}{2\gamma V_0},\end{array}$$ [3]

where Q ≈ +1/2 is the meron’s winding number (Q = +1 is the complete skyrmion’s winding number). Moreover, we can self-consistently solve for the deformation angle θ_def, which yields

$$\begin{array}{c}{\theta }_{\mathit{def}}=co{t}^{-1}\left(\frac{\beta }{4\pi Q}\left(D_s+d D_1\right)\right),\end{array}$$ [4]

where dD1 is a corrective factor due to the elongation of the skyrmion. Note that in the limit of small elongation, dD₁ << D_S, the angle θ_def is related to the skyrmion Hall angle in the limit α << β by a rotation of π/2 due to the Magnus force induced by the topological charge Q.

Guided by this result, we plot θ_def as a function of skyrmion elongation $d=2\mathrm{\Delta }a$ in Fig. 4. In Fig. 4A, the raw data are plotted as blue star markers for positive electric current density pulse measurements, j > 0, and the brown circle markers are measurements for j < 0. The gray line is a least-squares fit of Eq. 4 to the data binned into 61 equally spaced bins of $2\mathrm{\Delta }a$ in the same range of measured $\mathrm{\Delta }a$ (solid line, [0, 200]) and in an extended range (dashed line). The green dotted lines are qualitative fits illustrating potential θ_def(2Δa) paths for different starting θ_def(0), representing different initial pinning site positions or deformations. The data points from the current density range $\left|j\right|<\left|j_{c_1}\right|$ and those with elongation $2\mathrm{\Delta }a<0$ are not plotted. The simulated data plotted in Fig. 4B show three distinct regimes: steady state (black triangle markers), transient state (green “+” markers), and severely elongated state (red square markers). Representative examples of each state are shown in the Insets outlined in black (steady state), green (transient), and red (severely elongated), respectively.

Fig. 4.

Deformation axis angle θ_def versus skyrmion elongation 2Δa obtained from (A) experiment and (B) micromagnetic simulations. (A) Raw data are plotted with blue stars for positive electric current density pulse measurements, j > 0, whereas the brown circles indicate measurements for the opposite polarity, j < 0. The black line is a least-squares fit of Eq. 4 to the data binned into 61 equally spaced bins of 2Δa in the same range of measured Δa (solid line) and in an extended range (dashed line). The green dotted lines are qualitative fits illustrating potential θ_def(2Δa) paths for different starting θ_def(0), representing different initial pinning site positions or deformations. The data points from the current density range $\left|j\right|<\left|j_{c_1}\right|$ and those with elongation 2Δa < 0 are not plotted. (B) Simulated data show three distinct regimes: steady state (black triangle markers), transient state (green “+” markers), and severely elongated state (red square markers). Representative examples of each state are shown in the insets outlined in black (steady state), green (transient), and red (severely elongated), respectively.

We find the optimal fitting parameters to be βD_S ≈ 2.88π and βD₁ ≈ 0.04π/nm. Thus, if we assume that DS → [2π, 4π], where D_S = 2π corresponds to the Belavin–Polyakov skyrmion profile (43), then β → [0.7, 1.4], which seems reasonable for a conductor, while D₁ → [0.03 π, 0.06π]/nm As seen in Fig. 4, θ_def approaches 0° asymptotically as j increases. During this rotation, the other deformation observables, including e, a, b, and A, do not significantly change until the second threshold current density, $\left|j_{c_2}\right|\approx 7.75\times {10}^9 \text{A}/{\text{m}}^2$. Above $\left|j_{c_2}\right|$, the skyrmion deformation measured in Fig. 3 A, C, and D grows more severe until it reaches the critical current density for depinning into a skyrmion flow state, indicating that the skyrmions will depin well before θ_def reaches 0°. In fact, we found that |j_c| ≈ 8.55 × 10⁹ A/m² for the majority of skyrmions. Because the skyrmions’ deformation can no longer be captured via conventional LTEM once they enter a flow state, we cannot include data for |j| > 8.46 ×10⁹ A/m². Once depinned, the skyrmion deformation should grow with j², and so it will initially shrink into a less deformed state. Previous theoretical studies have predicted that the depinned skyrmion deformation should continue, accompanied by a skyrmion expansion until skyrmions eventually either annihilate or duplicate at higher current density regimes (23). The specific deformation and annihilation phenomena observed as a function of drive current depend heavily on the material parameters such as the Gilbert damping and magnetocrystalline anisotropy, as well as on the strength and size of the pinning sites.

Discussion

Let us come back to the curves fit to the experimental observations of δ(j), shown in Fig. 3C, where the fit excludes the $\delta (\left|j\right|<|j_{c_1}|)$ data points, which perhaps merits some discussion. At j = 0, pinning-induced deformation would result in a nonzero distortion. The direction and strength of this deformation across multiple skyrmions can be seen as random when compared with the STT-induced deformation, in which the entire length of the skyrmion string interacts with a uniform current. Indeed, as mentioned previously, the skyrmion deforms away from its pinned shape for low current values, $\left|j\right|<\left|j_{c_1}\right|$, which is illustrated by the large spread in θ_def(j = 0) shown in Figs. 3B and 4A. Therefore, it is reasonable to exclude these data points when fitting the δ(j) and θ_def(2Δa) curves.

In addition to the skyrmion deformation theory presented here, which predicts a θ_def(2Δa) curve that asymptotically approaches 0° as the skyrmion elongation increases, we performed micromagnetic simulations of electric current–induced deformation of a pinned skyrmion. These simulated data provide an LLG approach to the skyrmion deformation, and the resulting trend shown in Fig. 4B qualitatively matches the experimental data and fit line quite well. The quantitative differences between experiment and simulation may be attributable to the 3D characteristics of magnetic skyrmions in a real thin plate and their effects on the skyrmion deformation in the magnet. In particular, any skyrmion profile differences along the thickness due to local pinning sites or even demagnetization effects at the surface would probably change the resulting skyrmion’s response to electric current. In contrast, the micromagnetic simulations were performed in two dimensions.

The skyrmion’s native shape is not only deforming elliptically, but its domain wall spins are also rotating into the surrounding spin background, expanding the skyrmion’s core for the duration of the current pulse. We found that in addition to an increased distortion, the minor axis length increases with increased current density. Such an experimental feature cannot be explained by a pure tilting or bending of the skyrmion string (44, 45), suggesting that the skyrmion’s core truly expands in the plane of the thin plate. The expansion of the skyrmion’s core size may be explained by the interplay between the STT force and the pinning force of the potential well, while the expansion of the minor axis length suggests an expulsive force, perhaps due to an optimal curvature for elongated skyrmions that varies from that of the pure major axis elongation assumed in our model. The skyrmion’s ability to deform and expand during current application and return to its original shape upon current termination demonstrates an intrinsic elasticity of the spin texture. Furthermore, its resistance to annihilation while under large deformation illustrates the topological protection inherent to skyrmions (23).

These results increase our understanding of the STT effect on magnetic skyrmions, including their elastic nature. These real-space observations of skyrmion distortion highlight the topological protection of magnetic skyrmions. Their walls may be stretched or even blown up like balloons, but these external forces do not burst the spin texture. The large deformations reported here have notable consequences for skyrmion-based devices and dynamical studies, since the deformed shape will result in altered dynamics and skyrmion packing, ideal for applications that rely on emergent inertia such as reservoir computing (46). The real-space spin texture deformation must therefore be well characterized and considered while designing future spintronics devices.

Materials and Methods

LTEM Imaging and Electric Current Application.

We performed the defocused LTEM experiments in a transmission electron microscope (JEM-2100F, JEOL) equipped with a double-tilt electrical holder (Protochips, Fusion Select). The sample holder was powered with a Keithley 2450 Source Measure Unit. We applied external magnetic fields to the (110) Co₁₀Zn₁₀ plates by exciting the objective lens current of the JEM-2100F, which generates a field parallel to the incident electron beam. In LTEM, a beam of electrons transmits through a magnetic specimen placed in a field-free region of the microscope. As a free electron passes through flat magnetic specimens, it accumulates a phase cumulatively through the sample thickness due to the magnetic induction field components oriented in the plane of the plate, which manifests itself as an angular Lorentz deflection from the optical axis upon exiting the sample. While this deflection is invisible for in-focus images at the detector, defocusing the image reveals the intersections of domains, or domain walls of the magnet. This technique allows for live video recording of the skyrmion deformation because the defocus can be held at a constant value, rendering Bloch-type magnetic domain walls visible. While invaluable for this study, one must be careful so as not to misinterpret the contrast. In thin plate magnets, the magnetic skyrmions are 3D strings that extend throughout the thickness. Therefore, LTEM images are the summation of the electron–specimen interaction through the thickness projected onto a two-dimensional detector image plane. While holding the defocus constant at Δf = +576 μm and the externally applied magnetic field at μ₀H_ext = 40 mT, we varied the current density from j = 4.69 × 10⁹ A/m² to j = 8.46 × 10⁹ A/m² and applied 10 1-s current pulses at each current density (4-s interval between each pulse), followed immediately by 10 current pulses with opposite polarity. Current density values |j| < 4.69 × 10⁹ A/m² showed no change in LTEM contrast. When two LTEM images of a magnetic spin texture are acquired at over- and under-defocus values, the in-plane magnetic inductance of the sample may be reconstructed via the transport-of-intensity equation, assuming a flat electrostatic distribution (thickness).

Calculation of the True a and b via Quantitative Phase Imaging.

Because the experimentally measured $a_{\mathrm{\Delta }f}$ and $b_{\mathrm{\Delta }f}$ values were measured with LTEM images holding the defocus ($\mathrm{\Delta }f$) value constant, the true $a$ and $b$ values will be $a=a_{\mathrm{\text{Δ}}f}-\ell$ and $b=b_{\mathrm{\text{Δ}}f}-\ell$, where $\ell$ is the real space distance between the true skyrmion domain wall characterized by a maximum in-plane magnetization and the maximum intensity (electron counts) measured by the charge-coupled device camera in the defocused LTEM images. We estimated $\ell (j=0)=\frac{\left(\overline{a}-\overline{a_{\text{Δ}f}}\right)+\left(\overline{b}-\overline{b_{\text{Δ}f}}\right)}{2}(j=0)$, using measurements of $a(j=0)$ and $b(j=0)$ in the in-plane magnetic inductance map calculated via the transport of intensity equation on a series of defocused images, shown in Fig. 2A. Measuring the peak-to-peak distance between in-plane magnetic inductance intensities within the skyrmions’ domain wall, we found that $\overline{a}\left(j=0\right)=34.3\pm 0.3 \text{nm}$, $b(j=0)=30.3\pm 0.2 \mathrm{nm}$, and $\ell =59.3\pm 0.9 \text{nm}$, where the values reported are the mean and SE, respectively. We used this measurement of $\ell$ in the calculation of $a\left(j\right)$, $b\left(j\right)$, $\delta \left(j\right)$, and $\mathcal{A}\left(j\right)$, propagating the measured error in $\ell$ through each calculation in the standard way (47).

Sample Preparation.

Single-crystalline Co₁₀Zn₁₀ was grown by slow cooling in an evacuated quartz tube (40). The single crystals were cut along the (110), (110), and (001) planes with rectangular shapes for focused ion beam (FIB) fabrication of thin plate devices viewable in a transmission electron microscope (TEM). We cut and thinned the crystal into a ~500-nm thin plate by using FIB microsampling technology (48) such that the (110) crystal axis was aligned perpendicular to the plate, and then contacted the thin plate to either side of an 18-μm-wide TEM viewing slit on a commercial Fusion Select Electrical FIB-Optimized E-chip. The E-chip consists of a Si base that is 4 mm × 3.055 mm large with Pt electrodes connecting the chip to the sample holder. The Co₁₀Zn₁₀ thin plate was contacted to the Pt electrodes by W gas deposition within the FIB. We deposited amorphous C onto the thin plate’s top edge to aid in the final thinning cuts made using Ga-ion beam sputtering in the FIB to a final thickness of $t\approx 120 \mathrm{nm}$.

In-Plane Magnetic Induction Field Maps.

The in-plane magnetic induction field distribution was obtained by the transport-of-intensity equation analysis (49) of the overfocus and underfocus LTEM images and displayed by color mapping.

Calculation of Conic Section Fit.

We captured the LTEM images by using a charge-coupled device camera housed in the JEM-2100F, configured to 8-fps live imaging mode. We then used the commercial software Camtasia to record the images to file. We used home-built Python software to crop the images to isolate each magnetic skyrmion, and then Gaussian filtered and threshold binned the data before fitting a conic section with the functional form Ax² + Bxy + Cy² + Dx + Ey + F = 0 via the least squares method, from which we calculated the observables in the manuscript. We manually went through each fit and discarded the fits that clearly did not match the skyrmion contrast due to low image contrast or when two skyrmions combined to form a SkyB, which occurred for two skyrmions at j > 7.8 A/m². Of the 2,000 data points measured, we discarded 82 due to bad fits and 249 due to skyrmion bunching.

Characterization of Joule Heating.

We measured the resistance R over a range of temperatures from T = 250 K to T = 350 K by using a physical property measurement system. We extracted the slope of T(R) and used it to find the change in temperature of the thin plate due to Joule heating across the 1-s electric current pulses and found that the average temperature increased by 3.05 K for the largest current density. The results are shown in SI Appendix.

Micromagnetic Simulations of Electric Current–Induced Skyrmion Elongation.

We performed micromagnetic simulations based on the LLG equation in MuMax3 software (50). The simulated magnet has a size of 480 × 480 × 3.75 nm³ below the stripe-out instability of the skyrmion and 1,920 × 1,920 × 3.75 nm³ above the instability, with voxel lengths of 3.75 × 3.75 × 3.75 nm³ and periodic boundary conditions along the x and y directions. The material parameters were chosen as follows: saturated magnetization M_S = 4.2 × 10⁵ A • m⁻¹, exchange stiffness constant A = 2.8 × 10⁻¹¹ J • m⁻¹, bulk DMI constant D_bulk = 2.4 × 10⁻³ J • m⁻², and external magnetic field B_Z = −156 mT. Dipolar interactions are neglected. The pinning site was introduced as a cylinder with a diameter of 15 nm and a uniaxial anisotropy of K_u = −10⁶ J • m⁻³. The Gilbert damping constant was set to α = 0.16. The nonadiabatic spin transfer–torque parameter is β = 2α, and the spin polarization is for simplicity set to P = 1. We increased the current density smoothly in increments of 1 × 10¹⁰ A m⁻² or 2 × 10¹⁰ A m⁻² with 100 ns simulation time at every step to relax the driven skyrmion to its steady-state distortion. With our choice of parameters, we found that the skyrmion is stable for currents j ≤ 3.0 × 10¹¹ A m⁻². The data above the instability were acquired for j = 3.1 × 10¹¹ A m⁻².

Data, Materials, and Software Availability

The data and code used for analysis that support the plots within this article and other findings of this study are available at figshare at DOI: 10.6084/m9.figshare.20128244 (51). All study data are included in the article and/or supporting information.