# Electric-field guiding of magnetic skyrmions

###### Abstract

We theoretically study equilibrium and dynamic properties of nanosized magnetic skyrmions in thin magnetic films with broken inversion symmetry, where electric field couples to magnetization via spin-orbit coupling. Based on a symmetry-based phenomenology and micromagnetic simulations we show that this electric-field coupling, via renormalizing the micromagnetic energy, modifies the equilibrium properties of the skyrmion. This change, in turn, results in a significant alteration of the current-induced skyrmion motion. Particularly, speed and direction of the skyrmion can be manipulated by designing a desired energy landscape electrically, which we describe within Thiele’s analytical model and demonstrate in micromagnetic simulations including electric-field-controlled magnetic anisotropy. We additionally use this electric-field control to construct gates for controlling skyrmion motion exhibiting a transistor-like and multiplexer-like function. The proposed electric-field effect can thus provide a low energy electrical knob to extend the reach of information processing with skyrmions.

###### pacs:

75.78.Fg, 75.85.+t, 75.78.Cd, 85.75.-d## I Introduction

One of the central challenges faced by the field of spintronics, as an alternate to conventional charge-based information processing, is to encode information in a magnetic configuration (i.e. a collection of spins) which is both stable against thermal fluctuations and electrically manipulated with minimal energy budget. Recently, magnetic skyrmions have emerged as one such candidate configuration, opening a new sub-field of information processing with skyrmions, dubbed as skyrmionics.Nagaosa and Tokura (2013) Skyrmions are topologically nontrivial magnetic textures characterized by a non-zero integer-valued skyrmion number , which counts the number of times the texture wraps a sphere in the spin-spaceRajaraman (1987). Much recent interest has been fuelled by observation of the skyrmion crystal phaseMühlbauer et al. (2009); *Yu2010; *Heinze2011, its emergent electrodynamic propertiesSchulz et al. (2012) and manipulation by currents at low temperatures.Jonietz et al. (2010); *Romming2013

Meanwhile, motivated by room temperature applications, thin films with perpendicular anisotropy have been proposed as hosts for controlled nucleation and manipulation of individual skyrmions Sampaio et al. (2013). Single skyrmions in such films can be stabilized by either Dzyaloshinskii-Moriya interaction(DMI)Dzyaloshinskii (1957); *Moriya or dipole-dipole interaction(DDI) ^{1}^{1}1In the case of DDI, more generic cylindrical magnetic domains, known as bubbles Malozemoff and Slonczewski (1979), can be stabilized, which may or may not have a non-zero skyrmion number Yu et al. (2012). Here we focus only on textures which are topologically non-trivial and hence refer to them as skyrmions following Ref. Nagaosa and Tokura (2013)competing against Zeeman and exchange energies.
Additionally, these films can be typically sandwiched between a dielectric and a high spin-orbit coupling material resulting in broken inversion symmetry along the growth direction. Such film structures, apart from allowing for a finite DMI, have recently been shown to exhibit current-induced spin-orbit torques(SOT) Miron et al. (2010); *Liu2012, which, in turn, result in efficient current-induced motion of chiral magnetic textures. Emori et al. (2013); *ryunnt13 These attractive properties have made such films an ideal candidate for skyrmion-based race-track-like memory applications.Sampaio et al. (2013); *Iwasaki2013 In fact more recently, experimental evidence for skyrmionic configurations stabilized at room temperature in thin films Jiang et al. (2015); *beach_skyrmion and multilayers Moreau-Luchaire et al. (2015) in proximity with heavy metals have been presented. In Ref. Jiang et al., 2015; *beach_skyrmion, current-induced motion has also been observed.

In this article, we propose and show—via a combination of symmetry based phenomenology and micromagnetic simulations—that an electric-field applied across such film structures can dramatically influence current-induced skyrmion motion. In particular, we find that for a fixed current, the electric-field can control both the speed and the direction of skyrmion motion. More importantly, we show that by applying electric-fields in a certain pattern, defined via gated structures, skyrmions can be guided along desired trajectories. We also identify the mechanism for this electric-field control of skyrmion motion, which can be explained in terms of the change in the static properties of skyrmions, brought about by the electric-field induced modification of the micromagnetic free energy. Finally, utilizing this additional electrical knob we show that the realm of skyrmionics can be extended beyond the race-track-like memory applications, to possibly realize logic and computing functionalities.

## Ii Electric-field coupling: Phenomenology

We begin by constructing a general phenomenological model describing electric-field coupling to the magnetization dynamics based on the symmetries of the structure. We are interested here in two dimensional ferromagnetic films considered to have the following symmetry properties (see Fig. 1): broken inversion symmetry along the growth axis (oriented along ), by interfacing it with different materials on both sides and application of an electric field, which is assumed to be applied along the axis via gates. While for simplicity, the structure is considered isotropic about the axis ^{2}^{2}2As drawn, the isotropicity about the axis is broken globally by the gates, however, here we are interested in electric-field coupling terms which result in local modification of magnetization dynamics.. In addition, we are interested in the temperature regime well below the Curie temperature of the ferromagnet, where, suppressing the magnitude fluctuations of magnetization, the magnetization dynamics can be expressed within the Landau-Lifshitz-Gilbert framework asPitaevskii and Lifshitz (1980) :

(1) |

Here is a unit vector along the magnetization (with magnitude ), i.e. , while and label gyromagnetic ratio and Gilbert damping parameters, respectively. Moreover, we have defined , with representing the effective field contribution derivable from a micromagnetic free energy , and denoting the contribution due to application of current(having a current density, ) and electric-field(). In its minimalistic form, the micromagnetic free energy is comprised of the classical dipole-dipole interaction , the exchange interaction parameterized by an exchange constant , and the Zeeman energy due to an applied magnetic field . In addition, the breaking of inversion symmetry about the axis in the presence of spin-orbit interaction gives rise to interfacial perpendicular anisotropy and DMIDzyaloshinskii (1957); *Moriya, whose strengths are parameterized by and , respectively, resulting in:

(2) | |||

Here, . Similarly, the current and electric field-induced spin-orbit fields, obtained upon imposing the above mentioned structural symmetries on Eq. 1 (noting and are even, while , and are odd under inversion about axis) and, in the spirit of linear response, retaining terms linear in and , are given by:

(3) | |||

The terms proportional to and represent the dissipative components, i.e. which cannot be derived from a free energy, while terms proportional to and represent current and electric-field-induced modification of . In particular, the terms proportional to current density represent the field-like and anti-damping like components of spin-orbit fields, microscopically derived from spin-Hall and Rashba/Edelstein effectsHirsch (1999); *rashba; *Edelstein. Additional current-dependent terms representing corrections due to angle-dependence of spin-orbit field and symmetry-allowed dissipative torques inexpressible in the simple anti-damping like form Garello et al. (2013); *Yu2014 are not included for simplicity. On the other hand, the parameter describing coupling of electric-field to magnetization, denoted by , can be identified as electric-field dependent perpendicular magnetic anisotropy. Microscopically, electric-charge- Duan et al. (2008); *Nakamura2008; *maruyamaNATNANO09, strain- Eerenstein et al. (2006) and ionic movement-induced Bauer et al. (2014) anisotropy modification are few examples which are gaining increased attention in the spintronics community for exploring, both, fundamental physics and potential low-power technological applications Amiri and Wang (2012). Particularly for domain-wall based devices, the electric-field dependent anisotropy has been shown to trap domain wallsBauer et al. (2013) and proposed to induce domain-wall motionUpadhyaya et al. (2013). Finally, the dissipative coupling of electric-field should become relevant for very thin dielectrics Hoffman et al. (2012) which, to the best of our knowledge, has not yet been measured experimentally.

At this point we mention for completeness that terms, allowed by symmetry, where electric-field and current couple to gradients of magnetization, which could themselves be interesting to explore in proposed experimental structures, have been omitted from Eq. II. For example, the electric-field could generate a term of the form , describing microscopic electric-field dependence of DMI. Such electric-field-induced DMI has been proposed to induce a transverse ME effect in spiral spin magnets Siratori and Kita (1980). Motivated by including minimal ingredients to stabilize skyrmions and study non-trivial effects of electric-field on their dynamics based on the available experimental data, we set in the following sections. However, the qualitative features of electric-field control and its mechanism are expected to be extended for the case when some of these parameters are included as well.

## Iii Electric-field effect: statics

The dynamic control of skyrmions, as will be explained later, can be understood in terms of the modification of their static properties. To this end, we discuss first the electric-field effect on the static properties of a skyrmion. In general, two separate regimes can be defined based on the stability of the skyrmion. In the first regime, the skyrmion remains stable with a smooth change in its diameter , and the domain-wall width , surrounding the skyrmion. This change occurs in order to minimize the micromagnetic energy corresponding to the electric-field induced modification of perpendicular anisotropy: the energy cost, due to deviation of magnetization from the easy axis in the wall region, increases with increasing . In an attempt to reduce this energy cost, the domain wall area () is reduced resulting in decrease of both diameter of skyrmion and the wall-width. In the second regime, on the other hand, the skyrmion itself becomes unstable. This instability occurs close to the out-of-plane to in-plane transition of magnetization when the perpendicular anisotropy is lowered to a value, such that, the perpendicular anisotropy energy is overwhelmed by the dipole-dipole interaction, due to the out of plane component of the magnetization Malozemoff and Slonczewski (1979).

To demonstrate the qualitative picture presented above in a specific numerical model, we perform micromagnetic simulations for a DDI-stabilized skyrmion(i.e. with DMI strength set to zero) with saturation magnetization, emu/cm and exchange stiffness erg/cm. In order to take electric-field-induced modification of perpendicular anisotropy into account, the strength of perpendicular anisotropy was varied in the range - erg/cm. This range corresponds to films with erg/cm, Tsunashima et al. (1989) in the absence of electric-field, and a maximum value of electric-field being MV/cm, for a coupling constant erg/(Vcm) Kim et al. (2010). These material parameters are typical for strain-based ferromagnet/dielectric heterostructures such as CoPd(alloys)/PMN-PT, which are chosen as they exhibit one of the largest values of magneto-electric coupling (i.e. ) at room temperature Kim et al. (2010). In addition, a DDI skyrmion state has been found to be stable in finite-sized nanodisks of cobalt- and iron-based ferromagnets with similar material parameters Hehn et al. (1996); *Moutafis2007. The simulated magnetic films are assumed to be two-dimensional with periodic boundary conditions in the film plane and a thickness of nm. Additionally, an external magnetic field of strength with , is applied along the axis, which is needed to stabilize a DDI skyrmion for films of infinite extent Malozemoff and Slonczewski (1979). All micromagnetic simulations are performed using LLG Micromagnetic Simulator LLG () with temperature set to zero.

To study the static properties of a skyrmion, the initial magnetization was set close to a film with a single skyrmion state, i.e. with a central region of sufficiently large diameter pointing along and the rest of the film oriented along . This initial configuration was then allowed to relax to equilibrium for each value of perpendicular anisotropy. The result of such a procedure is summarized in Fig. 2. For anisotropy strength erg/cm, the single skyrmion state is unstable, favoring a multi-domain (for relatively strong , i.e erg/cm) or in-plane magnetization (for still weaker anisotropies, i.e. erg/cm). Such a behavior is typical for films, where the out-of-plane magnetization-induced dipole energy () starts dominating the perpendicular anisotropy energy (). This picture can be further confirmed by calculating the so-called quality factor which has a value of , i.e. close to one, at the critical anisotropy, . On the other hand, for in the simulations, the resulting equilibrium configuration is that of a single skyrmion (shown in the inset of Fig. 2)^{3}^{3}3A second transition corresponding to the collapse of skyrmion is expected to occur on reaching a critical radius for higher anisotropies as observed for bubbles (Malozemoff and Slonczewski, 1979). To see that transition in micromagnetic simulations finer grid and higher anisotropy values need to be explored . As expected, in order to minimize the dipole-dipole interaction energy in the wall, the DDI skyrmion is “Bloch-like” with skyrmion number . Moreover, increasing anisotropy results in decreasing diameter and wall-width surrounding the skyrmion, supporting the qualitative argument presented above.

## Iv Electric-field effect: dynamics

In this section we show how the modification of static properties of the skyrmion results in the electric-field control of current-induced skyrmion motion. Corresponding to the two regimes, defined for modification of static properties, there exist two qualitatively different regimes for the dynamic control.(i) When a uniform electric field with a magnitude below a critical value is applied, such that the skyrmion remains stable, the change in and results in change in the magnitude and the direction of skyrmion motion. (ii) On the other hand, defining unstable regions for skyrmions by application of a non-uniform electric-field above the critical value, results in constraining skyrmion motion along desired paths (which we refer to as the “guiding regime” below). To gain more insight into this connection between electric-field-induced static and dynamic control, we begin with an analytical model of current-induced skyrmion motion based on Thiele’s collective coordinate approach Thiele (1973).

Analytical model— In the collective coordinate approach, the relevant degrees of freedom, describing the equation of motion of a magnetic-texture, are given by the so- called soft modes Tretiakov et al. (2008). The most relevant soft modes for a skyrmion in an infinite film are its rigid translations in the film plane. Consequently, the collective coordinates included for describing the motion of a skyrmion are , which label the position of the skyrmion’s center along the and axis, respectively. Within this approximation, LLG Eq. 1 can be reduced to the following equation of motion Thiele (1973); *Bazaliy2008; *Sampaio2013:

(4) |

where is the velocity vector, while with . Inclusion of other modes, such as wall fluctuations, results in deviations from the rigid motion approximation and endows Eq. 4 with a “mass term” Makhfudz et al. (2012), which is not the focus of the current study. The first term describes the “Lorentz force” ^{4}^{4}4strictly speaking the dimension of each term in Eq. 4 is that of a velocity but for the ease of discussion they are referred here as forcesdue to the skyrmion’s non-trivial topology induced fictitious magnetic field Volovik (1987). Furthermore, the integrand in is the local solid angle, making independent of the exact magnetization texture and simply proportional to the skyrmion number, i.e. . The second and third terms originate from the Gilbert damping and current-induced spin-orbit field term in LLG Eq. 1, respectively, with and , where labels the cartesian coordinates. Finally, the fourth term represents the force due to variation of the skyrmion’s “potential energy” , i.e. . The skyrmion’s potential energy is, in turn, defined as , where the integral is performed for the magnetic configuration of a skyrmion with its center located at . In contrast to ; , and depend on the exact functional form of the magnetization.

In general, the functional form of will depend on the interaction of the skyrmion with its “environment” and will be discussed for the two regimes below, while analytical expressions are obtained for the dissipative and spin-orbit forces using Thiele’s ansatz Thiele (1970) for a skyrmion’s magnetic configuration as and . Within this Thiele’s ansatz, the integrand for and is assumed to be non-zero near the wall region, where the magnetization profile, parameterized by the polar() and azimuthal() angles as , is given by . Additionally, in order to compare to micromagnetics, the Bloch-like DDI skyrmion configuration with is used, i.e. with representing the two-dimensional film plane in cylindrical coordinates.

We begin by looking at the regime when a uniform electric-field, having a magnitude below the critical value required to make the skyrmion unstable, is applied to the infinite film. In this case, the skyrmion’s potential energy is independent of its location and hence . Consequently, solving Eq. 4 for velocity, we obtain:

(5) |

where is the skyrmion’s speed and is the angle between the skyrmion’s motion and the current (see inset of Fig. 3 for a schematic). This constitutes one of the main results of the analytical model, relating the skyrmion’s static and dynamic properties. The electric-field control of skyrmion’s motion in this regime is thus expected to enter via modification of and , as discussed in section III, and is shown in Fig. 3 for the diameter and domain wall width extracted from static micromagnetic simulations. These analytical results are compared later against the velocities and angles extracted directly from dynamic micromagnetic simulations.

The dependence of on electric-field suggests another qualitatively different route to control the skyrmion’s dynamics via application of non-uniform electric-field and consequently “engineering” desired forces . Interestingly in multiferroics, electric-field gradient-induced forces have been theoretically shown to induce Hall-like skyrmion motionLiu et al. (2013). Here, we propose the scenario where unstable regions [such as the one shown in the inset of Fig. 4(a)] can be defined for skyrmions, where magnetization deviates from the out-of-plane direction, due to the application of an electric field above the critical value. In this case, becomes non-zero as the skyrmion approaches the boundary between the stable and unstable regions due to dipolar interaction, as explained next. The dipolar fields originating from the “core” (i.e. the region inside the skyrmion’s diameter) point in a direction opposite to the core’s magnetization in the region outside the skyrmion’s core. Deviation of magnetization from this dipolar field direction in the unstable region will thus result in a higher dipolar energy when compared with skyrmions interacting with the film in the absence of unstable regions [as illustrated in Fig. 4(b)]. Consequently, as the skyrmion moves closer to the boundary between the stable and unstable regions, the dipolar energy increases and results in preventing the skyrmion to enter the unstable region. Due to this additional a new steady state is expected. This situation is very similar to the case of a skyrmion driven in nanowires, where in steady state the skyrmion is driven along the nanowire, with the dipolar forces from the boundary of the nanowire balancing the Lorentz forceSampaio et al. (2013). Thus generically it is expected that, for a range of drive currents, the skyrmion can be guided along the boundary between the stable and unstable regions, as depicted schematically in Fig. 4. The advantage here, as compared to the nanowire case, is that this boundary is defined by application of the electric-field and hence can be constructed on the fly, providing a controllable method to “rewire skyrmion current”. A general analytical expression for the velocity depends on the details of , however a simple expression can be derived for the case when the magnetization configuration in the unstable region has translational symmetry along the boundary [such as the one shown in the schematic of Fig. 4(a)]. This translational symmetry demands that is oriented perpendicular to the boundary, whose magnitude is denoted by . Solving for a steady state with the skyrmion moving along the boundary, taking the angle between the current and the normal to the boundary to be , and balancing the forces along and transverse to the boundary [as depicted in the inset of Fig. 4(a)], we obtain for the guiding regime:

(6) |

We can see from the above expression that by orienting the electric field gate boundary with respect to the current, i.e. changing , the skyrmion’s speed can be controlled dramatically. The position of the skyrmion with respect to the boundary, on the other hand, is given by the balance of Lorentz and dipolar forces, with the skyrmion pushed closer to the boundary for larger currents. We will now compare these analytical insights to the micromagnetic simulations and, in particular, prove the existence of the electric-field-controlled skyrmion guiding regime.

Micromagnetics—
The micromagnetic simulations were performed for a DDI skyrmion with the same material parameters as used in the case for static simulations. In addition a current-induced spin-orbit torque, with and Liu et al. (2012), was turned on as the driving force for all the simulations presented in this section.^{5}^{5}5In general the effective spin-orbit field also includes a contribution of form , which cannot drive domain-walls in perpendicular magnets Emori et al. (2013); *ryunnt13 and is thus not considered here Here , and are the reduced Planck’s constant, charge of an electron and current density respectively. Experimentally, this can be achieved, for example, by interfacing the ferromagnet in the ferromagnet/dielectric heterostructure with a heavy metal like Ta, Pt, W etc. ^{6}^{6}6In practice additional spin-torque-induced driving forces will act on skyrmion due to current flowing through the ferromagnet itself, which are neglected here for simplicity The dynamics were calculated by integrating Eq. 1 with a damping parameter . In Fig. 3 we first compare the results for a uniform electric-field with a magnitude below the critical field. These results are obtained for a single skyrmion moving in a film, with periodic boundary conditions along the film plane, and a fixed current density of A/cm, while simultaneously varying the anisotropy throughout the film in the stable range (as obtained in the static simulations, i.e. for erg/cm in Fig.2). As can be seen, both the speed and the angle of the skyrmion with respect to the current direction are modulated by application of the electric-field. Moreover for the range of electric-field magnitudes studied here, the trend predicted by the analytical model fits the obtained and well, both qualitatively and quantitatively, substantiating the analytical picture presented above. For the material parameters used, the electric-field modulation shown in this regime is, however, relatively weak. For a much stronger control of skyrmion dynamics, we next turn towards the other regime, i.e application of a non-uniform electric-field.

To demonstrate the skyrmion guiding regime by defining unstable regions for skyrmions, micromagnetic simulations were performed with the value of erg/cm in the stable(unstable) regions marked in Fig. 5. The dynamics was then simulated for a skyrmion driven by current, with the initial state of a single skyrmion in the stable region. In Fig. 5 we show various snapshots of the results of these simulations. First note that, as was discussed in the section III, the magnetization in the unstable region goes to an in-plane state. In the initial stage of the simulation, i.e. when the skyrmion is relatively far from the the unstable region [Fig 5(b)], the skyrmion behaves very similar to case moving at an angle with respect to the current. However, as the skyrmion approaches the boundary between the stable and unstable region [Fig. 5(c)] a non-zero deflects the skyrmion, preventing it from entering the unstable region. After ns a new steady state is reached with the skyrmion moving along the boundary between the stable and unstable region [Fig. 5(d)] and hence proving the existence of the guiding regime. The robustness of this steady state was checked by varying the magnitude of drive current density. For this example, the skyrmion is still driven along the boundary for current density below A/cm. As the drive force increases, with increasing current density, the skyrmion is pushed closer to the boundary according to Eq. 6. However, above A/cm the drive force overcomes , resulting in the skyrmion entering the unstable region where it is annihilated. Motivated by applications, we next discuss how this regime of skyrmion guiding could provide new opportunities for constructing possible spintronic devices.

## V Skyrmionics beyond race-tracks

Nanosized skyrmions can be used as non-volatile information carriers in a spintronic device owing to their stability against thermal fluctuations. The information stored in the skyrmion can, in turn, be read both optically or electrically (by utilizing various magneto-resistance effects such as topological Hall effect Lee et al. (2009), tunneling- Julliere (1975) or giant- Baibich et al. (1988); *PhysRevB.39.4828 magnetoresistance). One such proposed example of a spintronic device is a race-track memory, where information stored in a magnetic texture is moved by current to a desired location where it can be read or writtenParkin et al. (2008). With the discovery of current-induced skyrmion motion, skyrmions in a nanowire are considered as a promising alternate to “conventional” domains in a race-track memory as they show very weak pinning by external impurities requiring orders of magnitude lower critical depinning currents Jonietz et al. (2010). The new possibility of controlling “skyrmion current” by the electrical-field presented here is thus expected to open up novel avenues of information processing with skyrmions. We demonstrate this by specifically showing via micromagnetic simulations that electric-field gates can be used to turn “skyrmion currents” off, i.e. providing a transistor-like operation, and deflect information carried by skyrmions in specific parts of the spintronic circuit, i.e. a multiplexer-like operation.

To demonstrate the transistor-like operation, a nanowire geometry is adopted in the simulations with width of the nanowires set at 200 nm. The anisotropy strength was erg/cm throughout the simulations everywhere in the nanowire except in a region of width nm defined as the gate. To simulate the effect of electric-field applied on the gated region, the anisotropy value was dropped to erg/cm when the gate is in the “ON” state and kept at erg/cm in the “OFF” state. The rest of the parameters are same as for the dynamical simulations presented above. The result of the application of a drive current pulse along the axis of duration ns is shown in Fig. 6. When the gate is in the OFF state, the skyrmion is driven past the gate region along the nanowire (Fig. 6 bottom left). On the other hand, when the same current pulse is applied in the presence of a gate voltage, the skyrmion is blocked by the gate (Fig. 6 bottom right), due to the repulsive dipolar forces from the unstable gated region, thus exhibiting a transistor-like function. This situation is the special case of the guiding regime, i.e. Eq. 6, with .

Next we simulate a “wing-shaped” nanowire junction, with two nanowires, each of width 200 nm, meeting at an angle representing two different arms of a spintronic circuit. A gate in this case is defined on one of the arms as shown in Fig. 7, which also shows snapshots of the simulations. When the gate is OFF, a current pulse along the axis in this case drives the skyrmion to the lower arm (Fig. 7 bottom left). This motion is similar to a skyrmion moving at an angle with the current due to the Lorentz force, as seen for a skyrmion moving in an infinite film. However, when the gate voltage is turned on, the skyrmion driven by the same current pulse is deflected off the gate region into the upper arm (Fig. 7 bottom right). Such structures can thus be used to rewire the skyrmion current into specific parts of the circuit.

## Vi Summary

It is shown that an electric-field provides an additional degree of freedom for effective control of current-driven skyrmion motion. By using a symmetry-based phenomenology we find the general form of spin-orbit-induced electric-field coupling for thin films with broken inversion symmetry about the film plane. In general, this coupling can be classified as dissipative, i.e. not derivable from a free energy, and conservative, i.e. terms which can be understood as electric-field- induced modification of the free energy. The conservative terms can further be identified as electric-field dependent perpendicular anisotropy (when electric field does not couple to gradients of magnetization) and electric-field dependent exchange and Dzyaloshinskii-Moriya interaction (when electric-field couples to gradient in magnetization). Motivated by applications and available experimental data, we specifically studied effects of the electric-field dependent perpendicular ansisotropy in thin films, which also harbor a nanosized skyrmion state stabilized by dipolar interaction. We find that there exist two regimes for controlling the static and hence the dynamical properties of skyrmions: (a) below a critical value, such that the corresponding perpendicular ansitoropy is still strong enough to overcome the out-of-plane dipolar interaction, skyrmions are a stable configuration showing a decrease in diameter and domain wall width surrounding the skyrmion with increasing anisotropy, and (b) above the critical value, the skyrmion itself becomes unstable due to formation of multi-domain/ in-plane magnetized state. The decrease in diameter and domain wall width with increasing anisotropy in regime (a) is, in turn, driven by an attempt to reduce the wall energy. This change of static properties modulates both the speed and the angle of skyrmion motion driven by a current induced spin-orbit force. This modulation can be understood within Thiele’s analytical model of rigid skyrmion motion, where, the static properties enter the velocity via and dependence of spin-orbit and damping forces acting on the skyrmion. On the other hand, corresponding to regime (b), desired unstable regions can be defined for skyrmions using gate voltages, which are observed in micromagnetic simulations to guide skyrmions along the boundary between stable and unstable regions.This regime of skyrmion guiding is established by appearance of additional forces due to variation of skyrmion’s potential energy, caused by the dipolar interactions, as the skyrmion approaches the unstable region. Using this ability to electrically design potential energy landscapes, we demonstrate transistor-like and multiplexer-like spintronic devices controlling skyrmion current by defining electric-field gates. The proposed electric-field control is thus expected to extend the reach of skyrmionics into logic and computing devices.

###### Acknowledgements.

This work was partially supported by the National Science Foundation Nanosystems Engineering Research Center for Translational Applications of Nanoscale Multiferroic Systems (TANMS), and in part by the FAME Center, one of six centers of the Semiconductor Technology Advanced Research network (STARnet), a Semiconductor Research Corporation (SRC) program sponsored by the Microelectronics advanced Research Corporation (MARCO) and the Defense Advanced Research Projects Agency (DARPA). P.U. also acknowledges partial support from the Spins and Heat in Nanoscale Electronic Systems (SHINES), an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under Award # DE-SC0012670. We would like to thank Yaroslav Tserkovnyak, Greg Carman, Axel Hoffman, Wanjun Jiang, and Kin Wong for useful discussions.## References

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