Field-Dependent Size and Shape of Single Magnetic Skyrmions
Niklas Romming, André Kubetzka, Christian Hanneken, Kirsten von Bergmann, Roland Wiesendanger
FField-Dependent Size and Shape of Single Magnetic Skyrmions
Niklas Romming, ∗ André Kubetzka, Christian Hanneken, Kirsten von Bergmann, and Roland Wiesendanger
Department of Physics, University of Hamburg, 20355 Hamburg, Germany (Dated: April 8, 2015)The atomic-scale spin structure of individual isolated skyrmions in an ultrathin film is investigatedin real space by spin-polarized scanning tunneling microscopy. Their axial symmetry as well as theirunique rotational sense is revealed by using both out-of-plane and in-plane sensitive tips. The sizeand shape of skyrmions change as a function of magnetic field. An analytical expression for thedescription of skyrmions is proposed and applied to connect the experimental data to the originaltheoretical model describing chiral skyrmions. Thereby, the relevant material parameters responsiblefor skyrmion formation can be obtained.
Skyrmions are spatially localized solitonic magneticwhirls with axial symmetry and fixed rotational sense[1–4]. They have recently been observed in non-centrosymmetric bulk crystals [5–7] as well as in ultra-thin transition metal films on heavy-element substrates[8, 9], in which a sizable Dzyaloshinskii-Moriya interac-tion (DMI) [10, 11] induces their formation due to thelack of inversion symmetry. Magnetic skyrmions are inthe focus of current research because they offer greatpotential as information carriers in future robust, high-density, and energy-efficient spintronic devices [12]: inaddition to their protected topology and nano-scale size,they can easily be moved by lateral spin currents [12–16]and written as well as deleted by vertical spin-currentinjection [9]. In general, skyrmions couple very effi-ciently to spin currents and respond sensitively to spintransfer torques. For instance, skyrmion-based racetrack-type memory concepts [17] would profit from the factthat skyrmions can be moved with spin-current densi-ties being 5 to 6 orders of magnitude smaller than thoseneeded to move magnetic domain walls [12]. The fabrica-tion of such skyrmionic devices using ultrathin films andmultilayers can stay fully compatible with state-of-the-art technology. While numerous theoretical and simula-tion studies have concentrated on individual skyrmionsand their physical properties, no high-resolution experi-mental characterization of the internal spin structure ofskyrmions has been reported up to now, even thoughthe knowledge about their actual size and shape providesthe foundation for predictions about the interactions ofskyrmions with spin currents or their manipulation byexternal fields as envisaged in potential skyrmion-baseddevice concepts.In this letter we investigate the magnetic field depen-dent spin structure of isolated magnetic skyrmions byspin-polarized scanning tunneling microscopy (SP-STM)[18]. In contrast to techniques like magnetic force mi-croscopy [19] and electron holography [7], the spin struc-ture is measured directly rather than the magnetic fieldemerging from it. The combination of atomic-scale res-olution with spin sensitivity and the ability to mea-sure in external magnetic fields as high as several Teslamakes SP-STM an ideal tool to study the magnetic field- dependent spin structure of single nano-scale skyrmions.The evolution of their size and shape as a function of ex-ternal magnetic field is connected to micromagnetic the-ory of skyrmions [2, 3] via a proposed analytical descrip-tion of skyrmions. In this way, the values of the relevantmaterial parameters can be extracted.Our experimental setup is sketched in Fig. 1(a): Amagnetic probe tip with a well-defined spin orientationat the front atom is used in order to be sensitive to thespin-polarized contribution to the tunnel current, whichdepends on the projection of the local sample magneti-zation onto the quantization axis provided by the mag-netization direction of the tip. Here, we use an anti-ferromagnetic bulk Cr tip to avoid magnetic interactionsof the tip with the sample or an applied field [21, 22].As sample system we have chosen the bilayer of PdFeon an Ir(111) single crystal substrate, which shows thetypical magnetic field induced skyrmion lattice phase [9].At the measurement temperature of T = 4 . K the sam-ple exhibits pronounced hysteresis, enabling an investiga-tion of isolated skyrmions in a wide magnetic field range.Fig. 1(b) shows an overview of a PdFe area, exhibitingseveral circular skyrmions and two ◦ domain wall sec-tions remaining from the spin spiral phase (top right).Due to the use of an out-of-plane sensitive SP-STM tip,the axisymmetric character of the skyrmions becomes di-rectly evident.When an in-plane sensitive SP-STM tip is used, the ap-pearance of the skyrmions changes: now two lobes withmaximal and minimal spin-polarized current flow are im-aged per skyrmion (Fig. 1(c)) as a result of a positive ornegative projection of the local magnetization directionof the skyrmion onto the spin direction of the tip. Whentip and sample magnetization directions are orthogonalto each other, the spin-polarized contribution to the tun-nel current, and thus the magnetic signal, vanishes: forthe image in Fig. 1(c) this is true close to the center ofthe Skyrmion and above and below the center. The twoskyrmions in the sample area of Fig. 1(c) appear iden-tical, which is always the case for all skyrmions imagedwith a given SP-STM tip. This implies that they ex-hibit indeed a unique rotational sense [23]. According tothe symmetry selection rules of the DMI, these interface- a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r FIG. 1. Spin structure of individual skyrmion inPdFe/Ir(111). (a) Sketch of the experimental setup of a spin-polarized STM tip probing a magnetic skyrmion. (b) Topo-graphic constant-current SP-STM image measured with out-of-plane sensitive magnetic tip; each blue circular entity is askyrmion ( U = +200 mV, I = 1 nA, T = 2 . K, B = − . T).(c) Magnetic signal (methods [20]) of two skyrmions mea-sured with an in-plane magnetization of the tip, (cid:126)m T , re-vealing a two-lobe structure ( U = +250 mV, I = 1 nA, T = 4 . K). (d) Same area as in (c) with inverted mag-netic field; due to the preserved rotational sense, the con-trast is inverted. (e), (f) Line profiles across a skyrmionalong the rectangles in (c), (d), respectively, and fits withEq. (1) ((e) c = (0 . ± . nm, w = (1 . ± . nm; (f) c = (0 . ± . nm, w = (1 . ± . nm) and correspond-ing calculated out-of-plane magnetization m z . The sketchesshow spins with atomic distance, colorized according to theSP-STM contrast. induced skyrmions are expected to be cycloidal (sketchin Fig. 1(a)) [11, 24], in agreement with recent densityfunctional theory (DFT) calculations and Monte Carlosimulations for this system [25]. When the external mag-netic field which induces the skyrmions is applied in theopposite direction, the contrast of the two lobes of theskyrmions is inverted (Fig. 1(d)) since each spin in the FIG. 2. Evolution of the size and shape of skyrmionsin PdFe/Ir(111) as a function of external magnetic field.(a)-(d) Magnetic signal of SP-STM differential conductancemaps (methods [20]) with in-plane magnetized tip ( U =+20 mV, I = 3 nA, U mod = +2 . mV, T = 4 . K) at themagnetic fields as indicated (Supplementary Movie 1 showsfull data set [20]). (e) The size and shape of the skyrmion in-dicated by the box in (a) is evaluated by a fit with Eq. (2) asa function of magnetic field. Inset shows geometrical meaningof c , w and of d , which is numerically calculated; dashed blueline is a fit to d with / ( B − B ) . Solid black lines are ob-tained theoretically for the fitted set of material parameters A , D , and K (see text). Error bars correspond to standarddeviation of fit parameters. sample is inverted while the spin structure of the anti-ferromagnetic tip remains unchanged, providing an ad-ditional proof for the unique rotational sense caused bythe DMI. The two skyrmions in Fig. 1(c) and (d) appearat identical positions due to pinning at atomic defects.To characterize the size and the shape of a skyrmionwe take height profiles across the center (see black rect-angles in Fig. 1(c),(d)). Since there is no exact analyticalexpression to describe skyrmion profiles, we approximatethe cross-section of a skyrmion using a standard ◦ domain wall profile [26, 27], θ ( ρ, c, w ) = (cid:80) + , − (cid:104) arcsin (cid:16) tanh − ρ ± cw/ (cid:17)(cid:105) + π | B z > (cid:80) + , − (cid:104) arcsin (cid:16) tanh − ρ ± cw/ (cid:17)(cid:105) | B z < , (1)where θ defines the polar angle of the magnetization atposition ρ , and c and w resemble the position and widthof two overlapping ◦ domain walls, respectively. Toevaluate the measured data we include the projection oftip and sample magnetization in the fitting procedure(methods [20]).The agreement between experimental data and fit, seeblack data points and red fit lines in Fig. 1(e),(f), jus-tifies the chosen description. Furthermore, a compari-son to numerically calculated skyrmion profiles leads tothe conclusion that Eq. (1) is an excellent approximationfor a wide range of material parameters and field values(Fig. S1 [20]). From Eq. (1) it is straightforward to deter-mine the perpendicular magnetization component m z ( x ) ,see blue dashed lines in Fig. 1(e),(f), and the diameter ofthe skyrmion d , which we define as the diameter of thecircle with m z = 0 . Exploiting the axial symmetry, thespin structure of an isolated skyrmion in two dimensionsis then described by: (cid:126)S ( x, y ) = − sin( θ ( ρ, c, w )) · x/ρ − sin( θ ( ρ, c, w )) · y/ρ cos( θ ( ρ, c, w )) , (2)where ρ = (cid:112) x + y is the radial distance from the centerof the skyrmion located at the origin. Note that withinthis model the spin structure of the skyrmion is fullydetermined by only two parameters, c and w .The impact of external magnetic fields onto the sizeand shape of a skyrmion becomes evident in field-dependent SP-STM experiments: Fig. 2(a)-(d) show theidentical sample area imaged at different external mag-netic field strengths. The color scale resembles the mag-netic contribution to maps of differential tunnel con-ductance (d I /d U ), and the use of an in-plane sensi-tive tip leads again to the two-lobe appearance of theskyrmions. The decrease of the skyrmion size with in-creasing field can be directly seen in the displayed im-age sequence (Supplementary Movie 1 shows full dataset [20]). For a quantitative analysis we fit a single iso-lated skyrmion (black box in Fig. 2(a)) with our two-dimensional skyrmion model and obtain the characteris-tic parameters c , w , and d .The evolution of these parameters with external fieldis shown in Fig. 2(e). The diameter of the skyrmionroughly scales with / ( B − B ) , see dashed blue line,in agreement with numerical calculations [3, 28]. While,in the investigated field range, the diameter changes bymore than a factor of two, the effect on the width of FIG. 3. Validation of material parameters via micromagneticsimulations. (a) Internal spin structure of the skyrmion as de-scribed by Eq. (1) for the field values indicated. Inset showsvisualization of spins with atomic distance as parametrizedby Eq. (2); c , w given by fits in Fig. 2(e). (b) Comparisonof experimental and simulated height profiles across an in-dividual skyrmion (box in insets) for several magnetic fieldvalues, and a fit with Eq. (1). Left and right insets showSP-STM experimental data from Fig. 2(a)-(d) and micromag-netic simulations based on the derived material parameters,respectively. the transition region is only about %. Consequently,as can be seen in Fig. 3(a), the skyrmion shape changesqualitatively with magnetic field, leading to a significantdecrease in the number of spins with a component op-posite to the magnetic field. This results from a subtlebalance of all involved energies, where the Zeeman en-ergy leads to a compressing force and the DMI stabilizesskyrmions against collapse to the ferromagnetic state.To assess these underlying interactions for the biatomicPdFe layer in the framework of micromagnetic continuumtheory, we establish a connection to the standard energyfunctional in cylindrical coordinates [1–3, 28–30]: E = 2 πt (cid:90) ∞ (cid:34) A (cid:32)(cid:18) d θ d ρ (cid:19) + sin θρ (cid:33) + D (cid:18) d θ d ρ + sin θ cos θρ (cid:19) (3) − K cos θ − B z M S cos θ (cid:35) ρ d ρ where exchange stiffness A , DMI constant D , uniaxialeffective anisotropy constant K , and saturation magne-tization M S are the material dependent parameters, B z is the external out-of-plane magnetic field and t is thefilm thickness. From DFT calculations [25], we estimatean M S ≈ . MA m − (methods [20]). The magnetiza-tion profile θ ( ρ ) is given by our experimentally verifiedskyrmion model, Eq. (1). Now for each set of A , D , K , B the energy functional can be minimized with respectto c and w . These theoretical curves c ( B ) and w ( B ) arefitted to the experimentally obtained values for c ( B ) and w ( B ) via an error weighted least square fit with A , D , K as fitting parameters (Fig. 2(e)). The solid lines are thecalculated values of d , c , w for A = (2 . ± . pJ m − , D = (3 . ± . mJ m − and K = (2 . ± . MJ m − [31] as a function of magnetic field. These parametersare in the range expected for thin-film systems [25, 32],and the agreement of c and w obtained from theory withthose from a fit to the experimental data is evident.To demonstrate that these derived material param-eters can be used to accurately reproduce the exper-imental data, we perform micromagnetic simulations[20, 30, 33]. Fig. 3(b) shows height profiles across anisolated skyrmion at four different magnetic field val-ues. The SP-STM data (black circles), the skyrmion fitwith Eq. (1) (red line) and the height profile across theskyrmion in a micromagnetic simulation (blue dashed)nicely coincide, and the real-space agreement betweenexperimental data and simulation is demonstrated in theinsets to Fig. 3(b). Additionally, the accurate descriptionof the field-dependent magnetism of the PdFe bilayer bythe derived material parameters extends to lower mag-netic fields (Fig. S2 [20]).The presented SP-STM study provides access to theactual spin structure of an isolated skyrmion, enablinga direct comparison to micromagnetic theory. Sincethe field-dependent evolution of size and shape of singleskyrmions is governed by a balance of the magnetic in-teractions, a detailed experimental characterization canyield the relevant material parameters such as the DMI,which is responsible for the stability of these particle-like states. 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