Resonance modes and microwave driven translational motion of skyrmion crystal under an inclined magnetic field
We theoretically investigate the microwave-active resonance modes of a skyrmion crystal on a thin-plate specimen under application of an external magnetic field that is inclined from the perpendicular direction to the skyrmion plane. In addition to the well-known breathing mode and two rotation modes, we find novel resonance modes that can be regarded as combinations of the breathing and rotation modes. Motivated by the previous theoretical work of Wang [Phys. Rev. B 92, 020403(R) (2015).], which demonstrated skyrmion propagation driven by breathing-mode excitation under an inclined magnetic field, we investigate the propagation of a skyrmion crystal driven by these resonance modes using micromagnetic simulations. We find that the direction and velocity of the propagation vary depending on the excited mode. In addition, it is found that a mode with a dominant counterclockwise-rotation component drives much faster propagation of the skyrmion crystal than the previously studied breathing mode. Our findings enable us to perform efficient manipulation of skyrmions in nanometer-scale devices or in magnetic materials with strong uniaxial magnetic anisotropy such as GaVS and GaVSe, using microwave irradiation.
Noncollinear spin structures in magnets such as spirals, vortices, and chiral solitons with finite helicity and/or chirality show nontrivial collective excitations and thus offer intriguing spintronics and magnonics functions. One of the most important examples of such spin structures is magnetic skyrmions realized in magnets with broken spatial inversion symmetry Bogdanov89 (); Bogdanov94 (); Rossler06 (); Nagaosa13 (); Fert13 (); Mochizuki15a (); Seki15 (), in which keen competition between the Dzyaloshinskii-Moriya (DM) interaction and the ferromagnetic-exchange interaction takes place. The skyrmion structures usually appear in a plane that lies perpendicular to an external magnetic field , where the magnetizations at its periphery (center) are oriented parallel (antiparallel) to . The skyrmions are classified into three types, i.e., the Bloch-type, the Neel-type, and the antivortex-type, according to the way of their magnetization rotation [Fig. 1(a)] Bogdanov94 (); Nagaosa13 ().
In bulk specimens, layered skyrmion structures are stacked to form a tubular structure along the direction. The skyrmions often appear in a hexagonally packed form known as a skyrmion crystal, as shown in Fig. 1(b) Muhlbauer09 (); YuXZ10 (); Seki12 (); Adams12 (); Tonomura12 (). These skyrmions show specific microwave-active collective modes Mochizuki12 (); Petrova11 (); LinSZ14 (); Schwarze15 (); Garst17 (). When an external field is applied normal to the plane of the skyrmion, the skyrmion crystal has one (two) resonance mode(s) activated by a microwave magnetic field perpendicular (parallel) to the thin-plate plane. It shows a breathing mode under the perpendicular field, in which the crystallized skyrmions expand and shrink uniformly in an oscillatory manner. In contrast, the two modes that occur under the in-plane field are rotation modes, for which the rotational sense is counterclockwise (clockwise) for the lower-frequency (higher-frequency) mode.
Recent theoretical studies and experiments have revealed that these resonance modes of skyrmions host interesting microwave and spintronics functions Mochizuki15a (); Finocchio16 (), including gigantic microwave directional dichroism Mochizuki13 (); Okamura13 (); Mochizuki15b (); Okamura15 (), induction of spin voltages Ohe13 (); Shimada15 (), generation of spin currents Hirobe15 (), spin-torque oscillator functions LiuRH15 (); ZhangS15 (), microwave sensing functions Finocchio15 (), and magnonic crystal functions MaF15 (); MoonKW16 (); Mruczkiewicz16 (). These phenomena have all been investigated for the three collective modes mentioned above. Situations in which the skyrmion plane and the skyrmion tube are inclined from the direction rarely occur in bulk specimens because they can easily follow the direction. However, when the skyrmions are confined within a quasi-two-dimensional thin-plate specimen [see Fig. 1(c)] Seki12 (); YuXZ11 (), a situation can be realized in which distributions of the magnetizations and scalar spin chiralities have disproportionate weight and are slanted from the skyrmion center, as shown in Fig. 1(d) and (e) LinSZ15 (). Such a situation can also occur in magnets with strong uniaxial magnetic anisotropies, in which the orientations of the skyrmion plane and the skyrmion tubes are fixed, irrespective of the direction.
We expect emergence of characteristic resonance modes for skyrmion crystal under application of an inclined field. However, such modes have not been studied systematically to date, although novel modes are expected to host previously unrecognized functions and phenomena. Indeed, a recent theoretical study proposed that translational motion of skyrmions can be driven by application of a microwave magnetic field to a skyrmion-hosting two-dimensional system under an inclined field WangW15 (). In addition, it was found that a skyrmion crystal appears always on the (111)-plane, irrespective of the direction in insulating vanadates GaVS and GaVSe with lacunar spinel structure because of their strong uniaxial magnetic anisotropy Kezsmarki15 (); Ehlers16 (); Ehlers17 (). In insulating skyrmion-hosting materials of this type, specific resonance modes can be sources of interesting microwave magnetoelectric phenomena owing to their multiferroic nature with magnetically induced electric polarizations. Under these circumstances, clarification of the microwave-active modes and the microwave-related phenomena of skyrmions under application of an inclined field becomes an issue of major importance.
In this paper, we theoretically investigate the microwave-active resonance modes of a skyrmion crystal in a two-dimensional system under an inclined field. By numerically solving the Landau-Lifshitz-Gilbert (LLG) equation, we trace dynamics of the magnetizations that constitute the skyrmion crystal to calculate microwave absorption spectra and obtain real-space snapshots for each eigenmode. In addition to the well-known breathing and two types of rotation modes, we find that characteristic modes appear, which can be regarded as combinations of the breathing and rotation modes. Using micromagnetic simulations, we demonstrate that continuous excitation of these resonance modes via microwave application results in propagation of the skyrmion crystal where its direction and velocity sensitively depend on the excited mode (or the microwave frequency) and the microwave intensity. Furthermore, we find that a mode with a dominant counterclockwise-rotation component drives much faster propagation of the skyrmion crystal than the previously examined breathing mode. The knowledge of these resonance modes and the microwave-driven motion of skyrmion crystals under an inclined magnetic field lead to techniques to manipulate skyrmions using microwaves and provide a means to realize unique skyrmion-based devices.
Ii Spin Model and Method
We employ a classical Heisenberg model on a square lattice to describe the magnetism in a thin-plate specimen of a skyrmion-hosting magnet, which contains the ferromagnetic exchange interaction, the DM interaction among the normalized magnetization vectors , and the Zeeman coupling to the external magnetic field Bak80 (); YiSD09 (). The Hamiltonian is given by,
Types of the skyrmion are determined by a structure of the Moriya vectors (), i.e., and produce the Bloch-type skyrmion, and produce the Neel-type skyrmion, and and produce the antivortex-type skyrmion. We adopt =1 for the energy units and set =0.27. The external magnetic field is inclined from the perpendicular direction () towards the direction as = with , where is the inclination angle [see Fig. 1(c)]. Figure 1(f) shows a theoretical phase diagram of this spin model at =0 as a function of when is applied normal to the two-dimensional plane (=0). This phase diagram exhibits the skyrmion-crystal phase in a region of moderate field strength sandwiched by the helical phase and the field-polarized ferromagnetic phase. The unit conversions when =1 meV are summarized in Table 1.
|Exchange int.||1 meV|
|Magnetic field||8.64 T|
The LLG equation is given by,
Here (=0.02) is the Gilbert-damping coefficient. The effective magnetic field acting on the local magnetization on the -th site is calculated from the Hamiltonian =+ in the form
The first term, , is the model Hamiltonian given by Eq. (1). The second term, , represents the coupling between the magnetizations and a time-dependent magnetic field in the form
The calculations are performed using a system of sites where the periodic boundary condition is imposed.
Iii Results for the Resonance Modes
To identify the resonance modes of a skyrmion crystal under application of an inclined field, we first calculate dynamical magnetic susceptibilities,
Here and are Fourier transforms of the time-dependent magnetic field and the simulated time-profile of the total magnetization with . For these calculations, we use a short rectangular pulse for the time-dependent field whose components are given by,
where is the dimensionless time with being the real time. An advantage of using the short pulse is that for a sufficiently short duration with , the Fourier component becomes constant being independent of up to the first order of . The Fourier component is calculated as
As a result, we obtain the relationship .
In Fig 2(a), we present calculated microwave absorption spectra, i.e., imaginary parts of the dynamical magnetic susceptibilities Im and Im for the in-plane microwave fields and , respectively, as functions of microwave frequency for several values of inclination angle . Here, we fix =0.036. Note that while these spectra are calculated for the Bloch-type skyrmion crystal, we confirmed that the Neel-type and the antivortex-type show perfectly equivalent spectra. We find that while only two rotation modes with counterclockwise and clockwise rotation senses (referred to as Modes 1 and 3, respectively) are active under the perpendicular field with =0, novel modes (Modes 2 and 4) appear when the field is inclined with . Mode 2 appears between the two original rotation modes (Modes 1 and 3) in frequency, whereas Mode 4 has a higher frequency than Mode 3.
Intensities of these novel modes grow as the inclination angle increases. As will be discussed below, Modes 2 and 4 can be regarded as combinations of the breathing and clockwise rotation modes, where the former (latter) component is dominant for Mode 2 (Mode 4). On the other hand, intensity of Mode 3 is either enhanced or suppressed with increasing , depending on the polarization of the microwave field . When the microwave field is oriented parallel (perpendicular) to the direction toward which the field is inclined, i.e., (), the intensity becomes suppressed (enhanced) as increases.
Figure 2(b) shows imaginary parts of the dynamical magnetic susceptibilities Im calculated for the out-of-plane microwave field for several values of . Again, we find that novel modes (Modes 1, 3 and 4) appear when the field is inclined, whereas the single breathing mode (Mode 2) alone exists under the perpendicular field. Intensities of these novel modes increase as increases, while intensity of the original Mode 2 decreases. As will be discussed below, Mode 1 (Modes 3 and 4) can be regarded as combined oscillations of the dominant counterclockwise (clockwise) rotation component and the subsequent breathing component.
Comparison of the calculated dynamical magnetic susceptibilities in Fig. 2(a) and (b) shows that four types of collective modes activated by , have identical resonance frequencies with four corresponding collective modes activated by , indicating that both the in-plane microwave fields , and the out-of-plane microwave field activate equivalent modes under application of the inclined field.
In Fig. 3, snapshots of the magnetization distributions for each mode focusing on a skyrmion constituting the Bloch-type skyrmion crystal are shown for (left panels) and (right panels) when =0.036 and =30. They are simulated via application of a microwave magnetic field (=) with a corresponding resonance frequency as the time-dependent magnetic field in Eq. (4). In the simulations of these eigenmode dynamics, we have monitored time profiles of net magnetization and their Fourier transforms to confirm that a pure eigenmode with a single-frequency component is excited while other modes are absent. Noticeably, Mode 2 under and Mode 2 under at =0.0666 are identical, which are regarded as a breathing mode. It is also found that Mode 1 (Mode 3) under and Mode 1 (Mode 3) under at =0.0494 (=0.0872) are again identical, and can be regarded as combined oscillations of the dominant counterclockwise (clockwise) rotation and the subsequent breathing component. While the absorption intensity is too weak to enable snapshots of Mode 4 to be obtained under , we expect that they would be equivalent to those of Mode 4 under .
It is known that a single skyrmion in a constricted geometry exhibits quantized higher harmonic radial and azimuthal spin-wave modes when the system is activated by a sinusoidal AC field, especially in the presence of relatively strong damping effects Kim14 (); Beg17 (). In the present calculations, we used an infinite and uniform system with the periodic boundary conditions, and thus such higher harmonics should be absent. However, it might be important to check their absence. For this purpose, we calculate the dynamical magnetic susceptibilities by applying a sinusoidal AC magnetic field. Figure 3(b) [(c)] shows the calculated spectrum for  for , which has peaks at the same frequencies with that in Fig. 2, indicating the absence of higher harmonic modes and the validity of the identified eigenmodes. Note that calculations of the spatial distributions of power and phase may provide useful information to identify the eigenmodes Kumar11 ().
In addition to the resonance modes of the Bloch-type skyrmion crystal, we also examined those of the Neel-type and the antivortex-type skyrmion crystals. We find that the microwave absorption spectra for these three different types of skyrmion crystals overlap perfectly, indicating that the three skyrmion crystals have resonance modes with identical frequencies and identical intensities. However, differences appear in terms of the rotation senses of Modes 1, 3, and 4. The rotation sense of the antivortex skyrmion crystal is always opposite to that of the corresponding modes of the Bloch-type and Neel-type skyrmion crystals. Figure 4 shows snapshots of the calculated magnetization distributions of Mode 1 for the three types of skyrmion crystals. The rotation sense is counterclockwise for the Bloch-type and the Neel-type, whereas it is clockwise for the antivortex-type Nayak17 ().
To close this section, it is worth mentioning that the present study is based on a pure two-dimensional model although real specimens have finite thickness. In a real specimen, magnetic structures at the surfaces might be different from those inside the specimen because magnetizations at the surfaces have neighbors only on one side Rybakov16 (); ZhangSL18 (). Such surface magnetic structures might affect the properties of magnetic resonances quantitatively. However, we expect that our results will not be changed qualitatively or semi-quantitatively even if we adopt a three-dimensional model. In addition, the demagnetization effects due to the magnetic dipole interactions are not incorporated in the present study. A recent study on the microwave-active skyrmion resonances revealed that frequencies and amplitudes of the skyrmion modes vary depending on the shape and thickness of the specimens due to the demagnetization effects Schwarze15 (). However, the results of Ref. Schwarze15 () also indicated that the demagnetization effects do not change the properties of resonant modes qualitatively.
Iv Results of Translational Motion
A recent theoretical study by Wang discovered that translational motion of skyrmions can be driven by an out-of-plane microwave field under an inclined magnetic field through activation of their breathing oscillations WangW15 (). Motivated by this study, we investigate the motions of a two-dimensional skyrmion crystal driven by several different resonance modes under the field inclined towards the direction. In the numerical simulations, we find that the translational motion can be driven not only by the previously examined out-of-plane microwave field but also by in-plane microwave fields , via activation of the rotational oscillations of the skyrmions.
Figure 5 shows snapshots of the skyrmion crystal driven by (right upper panel) and the same skyrmion crystal driven by (right lower panel) at =400 ns after the microwave irradiation commences. The figure also shows the initial configuration of the skyrmion crystal at =0 (left panel) under application of an inclined magnetic field = with =0.036 and =30. Here, the microwave field is given by () with . An area composed of 9696 is magnified in the figure, although the simulations are performed using a system of 96111 sites. The displacement vectors connecting the original position and the position at =400 ns are indicated by the thick arrows shown in the right-side panels. When the microwave field with =0.0494 activates Mode 1 with a dominant counterclockwise-rotation component, the skyrmion crystal propagates in a direction close to the positive direction, whereas the same skyrmion crystal propagates in a direction close to the negative direction when with activates Mode 2 with breathing oscillations. We also find that the travel distance in the former case is much longer than that in the latter case, which indicates that the in-plane microwave field drives much faster motions of the skyrmion crystal than the out-of-plane microwave field .
Next we investigate the microwave frequency dependence of the velocity = of the driven skyrmion crystal under an inclined magnetic field = with =0.036 and =30. In Figs. 6(a) and (b), we plot the simulated -dependence of and , respectively, for different microwave polarizations where the microwave amplitude is set to be =0.0006. We find that the velocities are enhanced to have peaks at frequencies that correspond to the resonant modes, whereas their signs vary depending on the mode. We also find that the velocity is highest when the in-plane microwave field () activates Mode 1 with the dominant counterclockwise-rotation component. In this case, the value of becomes 0.04 m/s. In contrast, the velocity when Mode 2 is activated at is highest in the case of the out-of-plane microwave field , where it reaches m/s. Note that the in-plane microwave field drives the skyrmion crystal approximately four times faster than the out-of-plane microwave field .
We then investigate -dependence of the velocity of the skyrmion crystal when driven by Mode 1 and Mode 2, where is the inclination angle of the external magnetic field = with =0.036. In Fig. 7(a), the calculated absolute values of velocity = are plotted for Mode 1. The velocity increases noticeably as increases and seems to become saturated. Figure 7(b) shows the direction of propagation for different microwave polarizations. We find that the skyrmion crystal under the in-plane microwave fields , moves approximately in the positive direction, irrespective of the value of . In contrast, the skyrmion crystal under the out-of-plane microwave field moves approximately in the positive direction when is small, whereas the propagation direction becomes slanted towards the positive direction as increases.
Figure 7(c) shows calculated speeds of = for Mode 2 activated under different microwave polarizations. For , the velocity initially increases as increases to reach a maximum at 15 and, subsequently, decreases gradually with increasing . In contrast, this type of peak-maximum behavior is not clear in the , case, but the velocity shows saturation behavior or slight decrease after the initial increase in the small region. Figure 7(d) indicates that the direction of propagation under the in-plane microwave fields , is approximately in the negative direction. In contrast, the direction of propagation under the out-of-plane microwave field is approximately in the negative direction and is slanted slightly towards the positive direction. In both cases, the changes in the propagation direction upon variation of are small.
Note that each of the velocity data are measured at the resonant frequency of the mode, which varies depending on the inclination angle . The -dependence of the resonant frequency for each mode is summarized in Fig. 2(e). It should also be noted that the data shown in Fig. 7 are calculated for the Bloch-type skyrmion crystal, but we have examined the other two types of skyrmion crystals as well and have found that the absolute speed data plotted in Figs. 7(a) and (c) do not alter. In contrast, the direction of propagation changes depending on the type of skyrmion, but they are related to each other. The plots in Fig. 7(b) and (d) also hold for the Neel-type and antivortex-type skyrmion crystals if we replace the definitions of in the insets. These definitions should be replaced with those in Fig. 8(a) for the Neel-type skyrmion crystal and those in Fig. 8(b) for the antivortex-type skyrmion crystal.
Finally, we examine the -dependence of the velocity. Figure 9(a) shows the calculated absolute velocity = of skyrmion crystal when driven by a microwave field through activation of Mode 1. Here is a perpendicular component of the inclined magnetic field = with =30. The microwave amplitude is again fixed at =0.0006. The plots for , show a maximum at 0.35, which deep inside the skyrmion crystal phase, whereas the plot for monotonically increases with increasing . In contrast, the plots for Mode 2 in Fig. 9(b) show more complex behavior that is dependent on the microwave polarization. Clarification of a microscopic mechanism of these characteristic behaviors is an issue of importance, which should be clarified in a future.
Note that the present study is based on a pure two-dimensional model. The absolute velocities may slightly vary depending on thickness of real thin-plate specimens because the relative weight of influences from the surface magnetic structures and pinning effects due to impurities, grains and defects must differ depending on the sample thickness. A quantitative investigation on the skyrmion velocities in real three-dimensional specimens is left for future studies.
In summary, we have theoretically studied the microwave-active resonance modes of skyrmion crystal on a thin-plate specimen under a perpendicular or inclined external magnetic field . We have found that while only two rotation modes or a single breathing mode are active under the perpendicular field, novel microwave-active modes emerge when the field is inclined. The modes that exists originally under the perpendicular field are enhanced or suppressed depending on the polarization of the microwave field . Recent studies revealed that the collective modes of skyrmions host rich physical phenomena and may provide potentially useful device functions, which are attracting a great deal of research interest. As an example of these phenomena, we have investigated the microwave-driven translational motion of a skyrmion crystal under an inclined field WangW15 (). Our numerical simulations have demonstrated that the propagation velocity of skyrmion crystal is enhanced at resonant frequencies of the modes, while the velocity and direction of the skyrmion propagation are sensitively dependent on the activated modes. Importantly, the in-plane microwave field that activates the dominant counterclockwise rotation drives the skyrmion crystal much rapidly than the out-of-plane microwave field that activates the breathing mode studied in Ref. WangW15 (). The knowledge obtained in this study will help to open novel research into the spintronics and magnonics functions of skyrmions based on microwave irradiation.
This work was supported by JSPS KAKENHI (Grant No. 17H02924), Waseda University Grant for Special Research Projects (Project Nos. 2017S-101, 2018K-257), and JST PRESTO (Grant No. JPMJPR132A).
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