Direct control of the skyrmion phase stability by electric field in a magnetoelectric insulator
Alex J. Kruchkov, J. S. White, M. Bartowiak, I. Zivcovic, A. Magrez, H.M. Rønnow
aa r X i v : . [ c ond - m a t . o t h e r] M a r Direct control of the skyrmion phase stability by electric field in a magnetoelectricinsulator
A. J. Kruchkov, ∗ J. S. White, M. Bartkowiak, I. Zivcovic, A. Magrez, and H.M. Rønnow Laboratory for Quantum Magnetism (LQM), ´Ecole Polytechnique F´ed´erale de Lausanne (EPFL),CH-1015 Lausanne, Switzerland Laboratory for Neutron Scattering and Imaging (LNS), Paul Scherrer Institut (PSI),CH-5232 Villigen, Switzerland Laboratory for Scientific Developments and Novel Materials (LDM), Paul Scherrer Institut (PSI),CH-5232 Villigen, Switzerland Crystal Growth Facility, ´Ecole Polytechnique F´ed´erale de Lausanne (EPFL),CH-1015 Lausanne, Switzerland
Magnetic skyrmions are topologically protected spin-whirl quasiparticles currently considered as promising compo-nents for ultra-dense memory devices. In the bulk they form lattices that are stable over just a few Kelvin below theordering temperature. This narrow stability range presents a key challenge for applications, and finding ways to tunethe SkL stability over a wider phase space is a pressing issue. Here we show experimentally that the skyrmion phasein the magnetoelectric insulator Cu OSeO can either expand or shrink substantially depending on the polarity ofa moderate applied electric field. The data are well-described by an expanded mean-field model with fluctuationsthat show how the electric field provides a direct control of the free energy difference between the skyrmion andthe surrounding conical phase. Our finding of the direct electric field control of the skyrmion phase stability offersenormous potential for skyrmionic applications based on a magnetoelectric coupling. To realise skyrmion-based applications, research into cre-ation, control and stabilisation of skyrmions is in an activephase [1, 6–13]. A clear problem to overcome is that in bulkmaterials, the skyrmion lattice (SkL) is always only stableover a very narrow range of temperature ( T ) and appliedmagnetic field ( µ H ) just below the critical temperature T C [6, 14–18]. In Cu OSeO for example, the skyrmion pocketspreads downwards in T by just 3 .
5% of T C , occupying nomore than 1 % of the total ordered phase space [7, 27]. Thisgenerally limited phase space is observed also in other knownbulk skyrmion hosts [15–17, 38], and significantly restrictsthe scope for the development of industrial applications. Theability to enhance or suppress the skyrmion phase space ina sample can provide a flexible platform for the respectivecreation or destruction of skyrmion states. Here we present asimple and reliable mechanism for the stabilisation and desta-bilisation of the skyrmion phase as that due to electric ( E )fields applied to an insulating material.Up to now, several approaches for skyrmion manipulationwere demonstrated using either moderate electric currents,electric fields, or thermal gradients [6, 8–13, 19–24]. Progresstowards tuning the bulk skyrmion phase stability was alsodemonstrated using both applied uniaxial [25, 26] and hydro-static pressure [27]. For possible applications of the insulatingskyrmion host materials, the use of electric field to manipu-late the skyrmions is a very promising option that remainsstill relatively little explored.Here we report a combined theoretical and experimen-tal study of SkL phase stability under moderate E -fields(kV/mm) in the model insulating skyrmion host Cu OSeO .Theoretically the E -field effect is addressed using first orderperturbation theory around the mean-field solution. Thisresults in a small E -field driven shift of the SkL free energythat is nevertheless comparable with the energy differencebetween the skyrmion and conical phases. Furthermore,we develop a new approach for treating the fluctuativefree energy by adding quasiparticle modes near T C which prove to be pivotal in evaluating the free energy differencesbetween the phases. To verify experimentally the theoreticalexpectations for the skyrmion phase stability under an E -field, we use small-angle neutron scattering (SANS) tostudy microscopically how the E -field controls the extent ofthe equilibrium skyrmion phase in Cu OSeO . Consistentwith our theory, we find that both the magnetic field andtemperature extent of the skyrmion phase either expands orshrinks dependent on the E -field polarity. In addition, weidentify the appropriate experimental conditions for eitherthe enhanced or suppressed skyrmion phase stability in thesample, with similar conditions found in theory by exploringthe free energy density map under positive and negativevoltages. ResultsControlling skyrmion phase stability using electricfields.
From recent bulk susceptibility measurements ( χ ( E ))of Cu OSeO [22], it was suggested that skyrmions may be”created” or ”annihilated” by applying a dc E -field in suitableparts of the ( T, µ H ) phase diagram. In that study [22] theskyrmion phase is identified as a small drop in χ ( E ), whichserves only as an indirect indication for the existence of theskyrmion phase. Here we use the tool of small-angle neu-tron scattering (SANS) to directly observe the microscopicskyrmionic magnetism in Cu OSeO , and its response to anapplied dc E -field. By SANS the skyrmion lattice phase istypically observed as a sixfold symmetric diffraction pattern,consistent with the so-called multispiral (triple- q ) magneticstructure described by three propagation ( q -)vectors rotatedby 120 ◦ with respect to each other (note that both ± q eachgive a Bragg spot) [6, 15, 28]. To maximise the E -field ef-fect, in our SANS experiments we oriented the sample sothat E || µ H || [111] and, in comparison with the size of theskyrmion phase for E =0, succeeded in observing an expansion(contraction) of the skyrmion pockets for an E -field appliedparallel (antiparallel) to the [111] axis. The changes in thephase diagram are summarised in Figure 1.FIG. 1. Skyrmion phase tuning by electric fields. (a) A sketch of a magnetic skyrmion with the relative direction ofthe applied electric field E , in our experiments E || µ H || [111]. (b) Phase diagram (skyrmion pockets) measured by small-angleneutron scattering (SANS). The skyrmion pocket spreads almost twice in the positive field +5 . − . E < | q | as the Bragg spots in six-fold patterns shown in (e) and (g). In panel(h) no SANS signal is observed above the background level. FIG. 2.
Optimising skyrmion stability in electric fields. (a) Maximum SANS intensity versus temperature along thedirection of the skyrmion pocket growth (dashed line on the inset). The zone favourable for enhancing the skyrmion phasestability (”writing” skyrmions) is 53.9 to 54.9 K, where the skyrmion array population is the highest for
E >
0, while for E = 0the skyrmion phase is absent. For suppressing the skyrmion phase stability (”erasing” skyrmions), it is favourable to place thesample between 55.3 to 56.3 K, where the skyrmion phase is well populated under zero voltage, but becomes strongly suppressedunder E <
0. (b) Total scattered SANS intensity versus temperature for E = 0 and E > T = 54 . µ H at which the SANS intensity is a maximum is plotted in (a). Panel (c) shows similar data as in (b), but for E = 0 and E < T = 55 . T , µ H , E ) conditions, with the sixfold symmetricSANS patterns due to a skyrmion lattice most clearly seen inFigs. 1c,e. In these particular SANS patterns, weaker spotsare also detected to lie between the six strongest spots. Thisindicates the co-existence at various ( T, µ H, E ) conditionsof differently oriented skyrmion lattice domains around the µ H -axis, a phenomenon that has also been reported in otherscattering studies of Cu OSeO [28, 33, 37]. For the patternsshown in Figs. 1d,f,g, each obtained near to an edge of therespective skyrmion phase as determined in the SANS exper-iment, the Bragg spots become ill-defined, and instead the in-tensity appears as azimuthally smeared patches, indicative oforientationally-disordered SkLs (hereafter termed ‘skyrmionarrays’). Since the origin of the SkL disordering is difficultto identify unambiguously, a systematic analysis of all SANSdata is done by evaluating the the total scattered SANS inten-sity observed on the detector within the same annular integra-tion window shown in each of Fig. 1b-g. From this approachwe account for the scattering due to all the skyrmion arraysin the sample when determining the parametric extent of theskyrmion phase.The main result of the SANS analysis is shown in Figure1b. Importantly our results show how it is easier to desta-bilise the skyrmion phase than stabilise it; a positive E -fieldof +5 . E -fieldof only E = − . µm (typical for modern electronics)the skyrmion phase in a sample can be almost entirely desta-bilised (destroyed) or restabilised (restored) with just a coupleof volts - the voltage used in a typical smartphone. Optimum conditions for stabilising and destabilisingthe skyrmion phase
Examining our SANS analysis moreclosely shows that at various points in the magnetic phase di- agram, the moderate E -fields kV/mm either near-fully desta-bilise or stabilise the skyrmion phase, when compared withdata obtained at the same points but at E = 0. As denotedin Fig. 2a, both of these tendencies are observed to occur at asignificant level over large T windows each roughly 1 K wide,this corresponding overall to nearly 4% of T /T C in Cu OSeO .As a representative example, consider the harsh case atwhen skyrmions are expected to be essentially absent fromthe system, such as on the conical/Skyrmion phase border atthe lower- T boundary of the unperturbed skyrmion pocket( E =0). Fig. 2b shows µ H -scan data obtained for this caseat T = 54.8 K. While at E =0 the lack of SANS intensityindicates the skyrmion phase to be essentially absent, the ap-plication of E = +5 . E -field of opposite sign,and also a slightly higher T . Fig. 2c shows that at T = 55.8 K,a negative E = − . µ H for the unperturbed state ( E =0).The optimum T windows for enhancing or suppressingthe skyrmion phase stability are labelled in Fig. 2a, anddetermined from our µ H -scans of the total scatteredSANS intensity obtained at various ( T , E ) conditions - seeFigs. 2b,c, and also [Supplemental Material]). The totalscattered SANS intensity is a quantity indicative of boththe population and quality of the skyrmion arrays in thesample. The data in Fig. 2a show the µ H at whichthe SANS intensity is a maximum at each T and E -field,and from this the most favourable stability conditions forskyrmion arrays are inferred. We find that the approximate T -window of 53.9-54.9 K is appropriate for demonstratingthe strongest enhancement of the skyrmion phase stabilityunder E = +5 . T -window of 55.3-56.3 K issuitable for demonstrating the suppression of the skyrmionphase stability for E = − . T of 55 K allows todemonstrate both a significant enhancement and suppression of the skyrmion phase by E -fields of opposite polarity. Free energy in electric fields.
The underlying mechanismfor either enhancing or suppressing the skyrmion phase sta-bility by E -fields is mediated by the magnetoelectric (ME)coupling in insulating Cu OSeO . Microscopically, the MEcoupling originates from the the d - p hybridisation mechanism(see Refs. [28–31]). The emergent electric dipole moment P = λ ( S y S z , S z S x , S x S y ) is linked to the underlying spinstructure S ( r ) = ( S x , S y , S z ) with the coupling parameter λ of relativistically small size. Crucially, this effect resultsin a P · E shift of energy in E -field because the skyrmionphase now has a nonvanishing electric-dipole moment. Thisperturbation renormalises the elementary helices upon whichthe skyrmion phase is built, and slightly distorts the skyrmionlattice [6] compared for when E =0.In this study, we apply the ME perturbation to the freeenergy described by the effective Ginsburg-Landau functionalwith Dzyaloshinski-Moriya interaction (DMI), and considerthe critical fluctuations which in bulk samples favour theskyrmion phase with respect to the neighbouring conicalphase (see Methods). Due to the relativistically small sizeof λ , the dimensionless E -field is rather small so that,æ = λE/Dk ≪
1, and we can build the perturbation theoryin æ for the modified free energy neglecting all the terms oforder æ and higher. Our finding is that the perturbationsof fluctuative terms come in only at second order, while themean-field energy already shifts in the first order due tothe direct ME and nonlinear contributions (see Methods).The corresponding shift in free energy of the skyrmionphase depends on the direction of E -field (see Fig. 3a),thus either enhancing the skyrmion phase stability ( E >
E <
0) it. While at first sight it can besurprising that perturbatively small E -fields play a crucialrole here, this is facilitated by the very close competitionbetween the skyrmion and conical phases already in themean-field. Calculation of the phase diagram.
To calculate the phasediagram in E -field, we use a new approach developed on thebasis of effective models from Refs.[6, 15, 32] (see Methods).In contrast to these earlier studies the new approach is moreself-consistent in the way that it captures phase diagram,provides a deeper understanding of the role of quasiparticlemodes near T C , and covers the path-integral approach forcalculating the fluctuative free energy [15] as a limiting case.We thus treat the first-order perturbation in E -fields on topof the mean-field solution, and add the fluctuative contribu-tions that stabilise the skyrmion lattice in the bulk. The maincontribution to E -field effect here is given by the shift of themean-field free energy difference between the two phases (con-ical and SkL), while the fluctuative shift under voltages canbe considered quadratically small but lead to asymmetricalbehaviour as discussed in the next section.The new approach to the phase diagram calculation allowsus to understand deeper the stability of the skyrmion lat-tice on the intuitive, pictorial level: the critical fluctuations(waves) are superposed on top of the variationally minimisedfree energies. There are three critical modes ω (0 , , k aroundthe mean-field (see Supplemental Material), with ω (0) k soft onthe sphere | k | = k , which means that it cost very little energyto add many such fluctuations if they are coherent with thehelix k . Thus ω (0) k is the so-called ”dangerous” mode since it results in a Van-Hove-like singularity at T C and eventuallybreaks down the ordered phases into the disordered (param-agnetic) phase [32]. Below T c the breaking of symmetry canbe observed by SANS with either a six-fold pattern (skyrmionphase) or two-fold pattern (helical or conical phase), both cir-cumscribed by a sphere | k | = k in reciprocal space. Our cal-culation shows that the skyrmion phase is favoured becauseadding fluctuations costs more entropy in the skyrmion phase.This analysis leads also to a qualitative criterion of the ver-tical breakdown of the ordered phases at T C (see Methods).Asymptotically, the main contribution of the fluctuative freeenergy is given in the short-scale physics, where Cu OSeO is ”almost” a ferromagnet, thus reproducing the surprisingresult of the path-integral approach [15] as a limiting case.The model described here captures the qualitative physics ofthe system, as exemplified by the theoretical phase diagramshown in Fig. 3b.Experimentally, we have observed that the parametric ex-tent (stability) of the skyrmion phase become enhanced un-der E >
0. This observation can be addressed theoreticallyby exploring the free energy density map across the phasespace for different values of E -fields. We find that the depthof free energy minimum deepens with an increasingly positive E -field, as intuitively might be expected. For example, if wesit at T = 54 . E =0, the free energy of the skyrmionphase has a gap with respect to the conical phase (see Figure3a), which means that the skyrmion phase is not favouredat this condition; if we now include the E -field, there is a fi-nite range of µ H where the free energy difference is negativewith respect to the conical phase and the skyrmion lattice cannow exist. In Fig. 3c we label the regions favourable for ei-ther enhancing (stabilising) or suppressing (destabilising) theskyrmion phase stability as spanning approximately half ofthe calculated E -field modified skyrmion pockets. Discussion
In some respects, the observed E -field effect on the skyrmionphase stability resembles that achieved due to either applieduniaxial [25, 26] or hydrostatic pressure [27]. Clearly however,integrating the pressure effect on skyrmion stability into atechnological setting is very challenging. In contrast, the E -field effect proves to be both a versatile and reliable externalparameter; providing an efficient control of both the skyrmionposition [6, 8, 40] and the stability of the phase as a whole asdemonstrated here.We also clarify that the underlying mechanisms govern-ing the two phenomena of the pressure and E -field effectsare different. Namely, in the present E -field study, it is avoltage-induced distortion of the SkL which either enhancesor suppresses the stability of the skyrmion phase with respectto the conical phase. Similar experimental observations asreported here were recently communicated from indirect bulksusceptibility data without any theoretical support[22]. Ourpresent study lays both theoretical and experimental foun-dations for fully exploring alternative µ H - and E -field con-figurations, not only in reciprocal-space measurements likeSANS, but crucially real-space imaging techniques such ascryo-Lorentz transmission electron microscopy (LTEM). Inaddition, there is an urgency for studying the E -field effect onboth equilibrium and metastable skyrmion phases, since eachof these can serve as platforms for exploring single-skyrmioncreation/annihilation processes, and real-time E -field guidingof skyrmions in confined geometries. FIG. 3.
Theory: skyrmion lattices in electric fields. a) Free energy difference between skyrmion and conical phases at T =55.8 K for three values of E -field ( E = 0, E = ± ). For positive E -field(red), the free energy minimum due to the skyrmion phase is deeper than in the absence of voltage (green) meaning that theskyrmion phase is more stable against perturbations; meanwhile the negative voltage destabilises the skyrmion phase (bluedashed line). (b) Phase diagram (skyrmion pockets) for E -fields ± E || H || [111]. (c) Free energy densitymap for E > E = 0, E < OSeO . Since our observations can be generally expectedto occur in a suitable insulating host material at room tem-perature, our study provides motivation for the theoreticalexploration of skyrmions both over the richer phase space af-forded by the E -field, and, while awaiting suitable real materi-als, the development of insulator skyrmion-based data storageand racetrack memory devices where one needs to copy anderase huge arrays of data. In addition, the theoretical ap-proach herein can be extended towards describing the E -fieldeffect on stable and metastable skyrmion states in thin films,which are also of paramount technological importance. Methods
Small-angle neutron scattering (SANS) . For the SANS ex-periment, we used a single crystal crystal grown using chemicalvapour transport [39]. The crystal was of mass 6 mg and volume3 . × . × .
50 mm with the thinnest axis k [111], and [¯1¯12] vertical.The sample was mounted onto a bespoke sample stick designed forapplying dc E -fields [41]. In our experiments we achieved E -fieldsranging from +5 . − . E -fields outside this range.The sample was loaded into a horizontal field cryomagnet atthe SANS-II beamline, SINQ, PSI. The magnetic field ( µ H ) wasapplied parallel to both the [111] direction of the sample andthe incident neutron beam to give the experimental geometry E k µ H k [111]. In this geometry, the SANS signal is only detectedfrom the skyrmion phase, which typically presents as a hexagonalscattering pattern with propagation vectors q ⊥ µ H . In this ge-ometry, we avoid detecting any SANS signal due to either of theneighbouring helical ( q kh i ) or conical phases ( q k µ H ), sincethe propagation vectors of these phases lie well out of the SANSdetector plane.We used incident neutrons with a wavelength of 10.8 ˚A(∆ λ/λ =10%). The scattered neutrons were detected using a position-sensitive multidetector. The SANS measurements were done byrotating (‘rocking’) the sample and cryomagnet ensemble over an-gles that brought the various SkL diffraction spots onto the Braggcondition at the detector. Data taken at 70 K in the paramagneticstate were used for background subtraction. Before starting each µ H -scan, the sample was initially zero field-cooled from 70 K to atarget temperature, with the E -field applied when thermal equilib-rium was achieved. The E -field was maintained during the µ H -scan. At each T we define the µ H extent of the SkL phase as thatover which SANS intensity is detected. We use this criterion toextract the parametric extent of the SkL phase for ( µ H , T , E ) asshown in Figures 1,2. See Supplemental Material for more details. Mean-field free energy.
The effective mean-field theory is basedon the coarse-grained magnetisation approach M ( r ) = M s S ( r ) andis sufficiently described in [15]. One starts with the mean-fieldapproach with free energy F [ M ] = h Θ T M + J ( ∇ M ) + D M · ( ∇ × M ) + U M − H · M i (1)where the averaging is h ... i = R dVV ... , and Θ T ∝ α ( T − T C ) near T C , J is the Heisenberg stiffness and D is DMI, H is the mag-netic field, and the higher-order term U grants the formation ofthe crystalline phase [15]. In the mean-field, the interplay be-tween Heisenberg and DMI energies determines the helical vector as k = D/ J . The long-range-ordered hexagonal skyrmion lattice isapproximated as S ( r ) ≃ m + µ P q n S q n e i q n r + iϕ n +c.c., where thesummation runs over the crystalline order vectors q + q + q = 0.In the mean-field, the skyrmion phase is slightly gapped with re-spect to the conical phase, however the two are closely competing.Further details of the mean-field theory described in Ref. [15]. Perturbation theory in electric fields.
The magneto-electriccoupling in Cu OSeO is relativistically small, so the perturbationparameter is æ = λE/ Dk ≪
1. It is sufficient to use the firstorder perturbation theory on top of the non-perturbed free energy.We go to the rotated frame defined by the magnetic field direc-tion along [1 1 1], and re-write the free energy. The first orderperturbation theory gives eigenvectors: | S (æ) k i = | S (0) k i + X n =0 | S ( n ) k i h S ( n ) k | ˆ H E | S (0) k i ε (0) k − ε ( n ) k + O (æ ) , (2)which are now the basis for constructing the distorted skyrmionlattice. For other field ( H , E ) configurations, we re-do the calcu-lations in the new rotated frames. See Supplemental Materials forfurther details. Fluctuation-induced phase stabilisation.
We use a new ap-proach, which captures as a limiting case the fluctuation free energyfrom [15]. The essential physics is captured already in Gaussian(noninteracting) fluctuations with free energy density F fluct = X i | k | < Λ X k ω ( i ) k f ( i ) k − T S fluct , (3)where Λ = 2 π/a is the natural cut-off, f ( i ) k is the critical modesdistribution, and the entropy of Gaussian fluctuations isS fluct = X i | k | < Λ X k { (1 + f ( i ) k ) ln(1 + f ( i ) k ) − f ( i ) k ln f ( i ) k } (4)in the case of bosons. Fluctuations around mean-field are describedby the generalised susceptibility χ − ij ( r , r ′ ) = T δ FδM i ( r ) δM j ( r ′ ) , giv-ing rise to several collective modes (See Supplemental Material).On the local scale (or k ≫ J/D ), the chiral magnet is reminiscentof a ferromagnet, so the modes behave asymptotically ω k ∝ k forlarge k , thus asymptotically F fluct ≃ log βω k ∝ log k , which cov-ers the model of Ref.[15]. The main contribution to (5) is given bythe short length-scale (”ferromagnetic”) physics,∆ F fluct ≃ UπaJ h S − S i T, (5)The electric field also slightly affects the fluctuative energy, becauseit modifies the correlation length near T C and so renormalises J eff ,which is neglected here as a higher-order (æ ) effect. See [Supple-mental Material] for further details. Parameters of the effective model.
For our numerical cal-culations we use T C = 58 K, which approximately sets the Heisen-berg stiffness as J = 4 . × − Jm / A . From the SANS mea-surement we establish directly the modulation period of 60 nm,which estimatively differs by a few percents from the mean-fieldvalue 2 π/k , because the mean-field ordering vector k = D/ J isslightly renormalised by the fluctuations near T C . This sets the”bare” DM interaction entering (1) as D = − . × − J / A .The lattice parameter is a = 8 . × − m, which gives thenatural cutoff Λ = 2 π/a ≈ k . The saturation magnetizationin Cu OSeO is M s = 1 . × A/m and scales with temper-ature as M s ( T ) = M s (1 − ( T/T C ) α ) α , with α = 1 .
95 and α = 0 .
393 [42]. We choose the nonlinear coupling responsi-ble for SkL formation K = 6 . × − Jm − A − and Landauparameter α T = θ T /Jk ( T − T C ) = 3 . − . For the quali-tative phase diagram shown in Fig. 3b, we use a symmetric-response model (æ ), for which the best fit to SANS data is foræ = 0 .