Tunable dispersion relations manipulated by strain in skyrmion-based magnonic crystals

Zhao-Nian Jin(金兆年), Xuan-Lin He(何宣霖), Chao Yu(于超),Henan Fang(方贺男), Lin Chen(陈琳), and Zhi-Kuo Tao(陶志阔),†

1Bell Honors School,Nanjing University of Posts and Telecommunications,Nanjing 210003,China

2College of Electronic and Optical Engineering&College of Flexible Electronics,Nanjing University of Posts and Telecommunications,Nanjing 210003,China

Keywords: skyrmion,magnonic crystal,spin wave,dispersion relation

1.Introduction

Spin waves or corresponding quasiparticle magnons, as advanced information carriers, have attracted much attention in recent years.[1,2]Like semiconductors or photonic crystals, periodically modulated magnetic properties are realized in magnonic crystals,which can hold spin waves or magnons.By employing these periodically arranged structures, the dispersion relations of magnonic crystals can be customized to specific requirements.As a new emerging topologically magnetic texture,skyrmion has also shown the potential in realizing magnonic crystals.[3,4]

In 2009, magnetic skyrmion was experimentally observed in chiral magnet MnSi through neutron scattering by Mühlbaueret al.[5]Owing to the novel properties, skyrmion has shown strong potential applications in next-generation spintronic devices such as magnetic memory units with ultra-high density and logic gates with ultra-low power consumption.[6–9]Therefore, lots of researches of the dynamic properties driven by the electric current[10,11]or alternating microwave magnetic field have been conducted.[12,13]In 2012, spin-wave modes excited by microwave magnetic field in skyrmion crystal were predicted through numerical simulation.[12]Then, the magnetic excitations were observed experimentally in the helimagnetic insulator Cu2OSeO3.[13]The resonant modes are related to the external static magnetic field and material parameters such as Dzyaloshinskii–Moriya interaction (DMI) constant, and perpendicular magnetic anisotropy(PMA)constant.[14–16]

Recently,the skyrmion manipulated by using electrically generated mechanical strain has received much attention.[17,18]By considering the magnetoelastic coupling interaction,strain can be utilized to create skyrmions and annihilate skyrmions,and also modulate the configuration of stripe domains and skyrmions.[19–21]Additionally, from the applications of strain-induced surface acoustic waves(SAWs)another efficient method of generating and manipulating skyrmions has been found.[22–24]

Particularly, the periodical skyrmion arrangement can form magnon bands and bandgaps.Furthermore,the magnon band structure can be modulated by external effects such as magnetic fields and electric fields which will change the magnetic configuration of the skyrmion.[25–27]This dynamically modulated magnon band structure provides an advanced platform for studying tunable spintronics devices.In this work,we theoretically investigate the tunable dispersion relations manipulated by strain in skyrmion-based magnonic crystals.To start with, we analyze the phase diagram by changing strains and external static magnetic fields.Next,we calculate the dispersion relation curves under different strains and analyze the propagation properties of spin waves with specific frequencies.

2.Model and method

The dynamic behaviors of spin waves propagating in skyrmion-based magnonic crystals are simulated by the object oriented micromagnetic framework(OOMMF).[28]For a magnetic system, to obtain the dynamic response of its magnetic moment to time(the dynamic process),the Landau–Lifshitz–Gilbert(LLG)equation needs solving.The LLG equation can be given as follows:

whereαis the Gilbert damping parameter,γis the gyromagnetic ratio,µ0is the vacuum permeability,m(r,t) is the unit magnetization vector, andHeffis the effective magnetic field and can be expressed as

whereMsdenotes the saturation magnetization andEtotis the total energy of a magnetic system.With magneto-elastic energy considered,theEtotcan be given by

whereEexis the exchange coupling energy,EPMAis the perpendicular magnetic anisotropy(PMA)energy,EZeis the Zeeman energy,Edemagis the demagnetizing field energy,EDMIis the DMI energy, andEmeis the magneto-elastic energy.Through the competition among these types of energy,different magnetic structures can be formed, and the ground state can be obtained by calculating the minimum value of the total energy.

In this work, the most important energy components are DMI energy and magneto-elastic energy.There are two types of DMI energy, namely, the interface DMI energy and bulk DMI energy.Here, the interface DMI energy is considered,which is given by

whereDdescribes the strength of interfacial DMI.Magnetoelastic energy can be expressed as

Here,εijis the strain tensor, andB1andB2are the magnetoelastic coupling constants.[29]In this work, biaxial in-plane strains are applied to a skyrmion-based magnonic crystal, so the strain tensor can be reduced to

The typical material parameters for Co are used in the simulation, specifically, they being saturation magnetizationMs= 5.8× 105A/m, exchange stiffness constantAex= 1.5× 10−11J/m, perpendicular anisotropy constantK= 8×105J/m3, gyromagnetic ratioγ= 2.211×105m/(A·s), Gilbert damping coefficientα= 0.005,magneto–elastic coupling constantsB1=−1.617×107N/m2andB2= 2.31×106N/m2.[30,31]The Co nano-layer is 2000 nm×100 nm×1 nm.The 2 nm×2 nm×1 nm mesh cell size is used to discretize the model and all simulations are performed at zero temperature.

3.Results and discussion

Firstly, our research focuses on examining the stability of a skyrmion-based magnonic crystal.Figure 1(a)illustrates the schematic diagram of our proposed device structure.In this structure, a biased voltage is applied to the piezoelectric layer in order to manipulate the strain in both of the piezoelectric layer and the skyrmion-based magnonic crystal layer.The simulated structure consists of a ferromagnetic (FM) ultrathin nano layer, a heavy metal layer (HM), and a bottom piezoelectric (PE) substrate.The Dzyaloshinskii–Moriya interaction generated by the FM/HM interface mainly results in the generation of skyrmions on the FM layer.Specifically,our proposed structure allows for the achievement of Neeltype skyrmions.In order to investigate the manipulation of skyrmions through strain,a voltage is applied to the PE layer,and the strain is generated owing to the converse piezoelectric effect.The length of the magnonic crystal is 2000 nm,the width is 100 nm, and the thickness of the magnonic crystal layer is 1 nm.

Fig.1.(a) Schematic diagram of the proposed device structure.(b)Magnonic crystals with 20 and 60 skyrmions respectively,and nonperiodic magnetic textures without strain.(c)Phase diagram of strain ε versus magnetic field Hz (D=4 mJ/m2).

By considering the interplay of various types of energy,including the magneto-elastic energy induced by the strain,different magnetic configurations can be achieved.Figure 1(b)shows the typical magnetic distributions.Skrymion-based magnonic crystals with 20 and 60 skyrmions are presented with a periodic arrangement and controlled skyrmion size.Meantime, complex non-periodic magnetic textures can also be obtained by adjusting parameters such as DMI constantD,external static magnetic fieldHz, and strainε.In Fig.1(c),we present the phase diagram of strainεversus magnetic fieldHzwithDfixed at 4 mJ/m2.It can be seen that ferromagnetic states are formed under higher magnetic fieldHzand compressive strain(ε＜0).With the decrease of the compressive strain and the increase of the tensile strain (ε＞0), skyrmion states(or skyrmion-based magnonic crystals) can be formed.Also,non-periodic magnetic texture states are formed under higher tensile strain and lower magnetic fieldHz.It is suggested that the voltage-induced strain can modulate the magnetic total energy via magneto-elastic coupling.On the other hand,the induced strain can also affect the magnetic anisotropy of the FM layer, which will be introduced to re-distribute the magnetic moments.

To excite spin wave in the magnonic crystal,a sine cardinal field,

with amplitudeH'z0= 100 A/m and cutoff frequencyf= 100 GHz, is applied to a central area of 200 nm×100 nm×1 nm as depicted in Fig.2(a).The excited spin waves with frequencies ranging from 0 GHz to 100 GHz travel to both sides of thexdirection,i.e., positive and negativexdirections respectively.Then the fluctuations of magnetization (δmz(x,t) =mz(x,t)−mz(x,0)) along the line from(0 nm,50 nm,1 nm)to(2000 nm,50 nm,1 nm)are collected during 10 ns as shown in Fig.2(b).The color bar denotes the value ofδmz(x,t).To determine the dispersion relation, a two-dimensional double fast Fourier transform is applied to the evolution ofδmz(x,t).

Fig.2.(a) Schematic diagram for obtaining dispersion relations.(b) Fluctuations of magnetization δmz(x,t) without strain (N =60, D=3.5 mJ/m2,H =5×104 A/m).(c) Dispersion relations in skyrmion-based magnonic crystals with strain ε =0.25%, ε =0, and ε =−0.25%, respectively, (N =60,D=3.5 mJ/m2,H=5×104 A/m).

Figure 2(c) displays the dispersion relations observed in the skyrmion-based magnonic crystals under different strains:ε=0.25%,ε=0, andε=−0.25%, which are represented by frequency versus wavenumber.It is known that different branches(modes)are presented in the dispersion relations due to the quantization of the wave vector across the strip width because a standing-wave pattern should form along the width direction of the strip.[25–27]Form Fig.2(c), a periodic nature with folded branches can be seen, suggesting that the branch with the lowest frequencies corresponds to the uniform magnetization dynamics across the width of the strip (mode 1) while the upper one corresponds to the half-wavelength quantization (mode 2).[32]In the meantime, several allowed bands can be observed where spin waves can travel through the magnonic crystals at specific frequencies and wavenumbers.Conversely,frequency ranges falling within the bandgaps(also referred to as forbidden bands, as indicated in Fig.3(c)) prevent the spin waves from propagating through the magnonic crystal.Like the transport of electrons in solid crystals or photons in photonic crystals,the periodic modulation of magnetic properties introduced by the skyrmion lattice contributes to the tailoring of the dispersion relations.

An important finding in this study is that the dispersion relations can be manipulated by strain.As shown in Fig.2(c),the dispersion relations shift toward higher frequencies when strainεchanges from−0.25%to 0.25%.Moreover,the ranges of frequencies within the allowed bands also vary with strain.This indicates that the strain has the ability to manipulate the distribution of magnetic texture through magneto-elastic coupling.In our proposed device, the configuration of periodically arranged skyrmions can be modulated by strain.Thus,the travelling characteristics of spin waves can be manipulated.Consequently, the strain-induced manipulation results in tunable dispersion relation,showcasing the potential for strain as a tool for manipulating and controlling the properties of spin waves.

Furthermore,we summarize the strain-dependent dispersion relations.Figures 3(a) and 3(c) illustrate the straindependent band characteristics inferred from dispersion relations for specific parameter values:K= 8×105J/m3,H=5×104A/m,D=3.5 mJ/m2; andK=2×105J/m3,H=1×106A/m,andD=4 mJ/m2respectively.Correspondingly, figures 3(b) and 3(d) show the variations of frequency variation range(Δf)with strainεof the allowed and forbidden bands.It is evident that the allowed bands shift toward higher frequencies as the strainεchanges from−0.75%to 0.5%.In addition,the skyrmions are annihilated with strainε=0.75%,as also depicted in Fig.1(c).Furthermore, for different values ofD, as shown in Figs.3(a) and 3(c), the allowed bands exhibit the same strain-dependent characteristics.

In the meantime,it is worth noting that the strain not only causes the bands to shift toward higher frequencies,but also allows for manipulation of the frequency variation range within each band.As shown in Fig.3(b), Δfof allowed band II decreases from about 9.8 GHz to 2.5 GHz with strainεchanging from−0.75%to 0.5%.Similarly,Δfof allowed band III decreases from about 4.8 GHz to 2.3 GHz.On the other hand,Δfof forbidden band I increases from about 5 GHz to 16 GHz withεchanging from−0.75% to 0.5%, whereas Δfof forbidden I decreases from about 2.6 GHz to 1.5 GHz withεchanging from−0.75% to−0.25% and keeps stable withεincreasing to 0.5%.Figure 3(d) presents similar characteristics for a different configuration of magnonic crystal.

Fig.3.Strain-dependent band characteristics inferred from dispersion relations for(a)K=8×105 J/m3,H =5×104 A/m,D=3.5 mJ/m2,and (c) K =2×105 J/m3, H =1×106 A/m, D=4 mJ/m2.Panels (b) and (c) display the corresponding frequency ranges of allowed and forbidden bands.

Next,we investigate the traveling behaviors of spin waves manipulated by strain.Figure 4(a) shows a schematic diagram for illustrating the process of exciting spin waves and capturing traveling behaviors.The spin wave is excited by a sinusoidal magnetic field along thezaxis at frequencyf=67 GHz, confined into the region 0 nm＜x ＜100 nm, and then the evolution of average magnetization fluctuationδmzatx=2000 nm is collected for different strains:ε=−0.25%,0%, and 0.25%.In Fig.4, it can be observed that fluctuation of magnetizationδmzatx=2000 nm begins to exhibit significant variations,respectively,at about 1.5 ns withε=−0.25%and at about 1.7 ns withε=0.25%.The corresponding velocities of spin waves are calculated to be 1267 m/s and 1117 m/s,respectively.This indicates that the strain can also manipulate the velocity of the spin wave.Additionally,very tiny magnetization fluctuations are observed in the absence of strain.This is consistent with the characteristics described in Fig.3(a),where the frequency of 67 GHz falls within allowed band I forε=−0.25%and allowed band III forε=−0.25%,while it falls within forbidden band II in the absence of strain.

Fig.4.(a)Schematic diagram for obtaining the traveling behaviors of spin waves.The fluctuations of magnetization δmz at H=5×104 A/m,D=3.5 mJ/m2,x=2000 nm,and ε =−0.25%(b),0%(c),and 0.25%(d).

4.Conclusions

In summary, we investigated the tunable dispersion relations manipulated by strain in skyrmion-based magnonic crystals.Firstly,we calculated the phase diagrams of ground states under strains and external static magnetic fields, which show that the strain can induce different configurations such as FM state,skyrmion state,and complex non-periodic magnetic textures.Then, we studied the propagation properties of spin waves and calculated dispersion relation curves under different strains.It is found that the strain can manipulate the dispersion relations of skyrmion-based magnonic crystals through magneto-elastic coupling.Finally,we confirm that spin waves with specific frequencies can pass through the magnonic crystal or be blocked and that the on-off characteristics can be manipulated by strain.The results may provide a new idea for designing tunable spin wave devices based on skyrmion.