Electric-Field Control of Magnon Gaps in a Ferromagnet using a Spatially-Periodic Electric Field

The frequencies and linewidths of spin waves in one-dimensional (1D) and two-dimensional (2D) periodic superlattices of magnetic materials are found, using the Landau–Lifshitz–Gilbert equations. The form of the exchange ̄eld from a surface-torque-free boundary between magnetic materials is derived, and magnetic-material combinations are identi ̄ed which produce gaps in the magnonic spectrum across the entire superlattice Brillouin zone for hexagonal and square-symmetry superlattices. The magnon gaps and spin-wave dispersion properties of a uniform magnetic material under the in°uence of a periodic electric ̄eld are presented. Such results suggest the utility of magnetic insulators for electric̄eld control of spin-wave propagation properties.


Introduction
Advancements in the control of spin-wave propagation and dynamics 1 have led to the demonstration of magnonic bose condensation 2 and coupling of electronic spin currents to spin waves in hybrid systems. 3Such e®ects, along with theoretical proposals 4,5 and experimental demonstrations 6 of electrical control of spin-wave propagation in insulators, may provide the foundation for an informationprocessing technology based on spin waves. 7,8Any such technology would bene¯t from magnetic materials with designed spin-wave dispersion relations, group velocities, and linewidths.A common method of designing such features is the fabrication of a superlattice of di®erent constituent materials, used to design electronic band structures in semiconductor superlattices, 9 and photonic band structures [10][11][12] in dielectric superlattices.
Passive control of spin current propagation by opening gaps in the spin-wave dispersion, such as with a magnonic crystal, [13][14][15][16][17][18][19][20][21][22] has been achieved, especially for quasi-one-dimensional structures.Active manipulation of the spin currents, especially within insulating materials such as ferrites, has proved much more di±cult, and e®orts have focused on modi¯cations from internal currents, 17,23 external magnetic ¯elds, 19,20,24 or spin currents. 25A simple active method of controlling the propagation of spin currents, especially without substantial intrinsic dissipation, would have broad utility in the fundamental studies of spin-wave propagation, such as by allowing time-dependent studies of spin current propagation and dissipation that would illuminate many of the fundamental processes involved in spin current dynamics.Recent predictions 4 and demonstrations 6 of a modi¯ed spin-wave dispersion in yttrium iron garnet (YIG) by an electric ¯eld due to the spin-orbit interaction (via the Dzyaloshinskii-Moriya interaction, or DMI) produced phase shifts in propagating spin waves, but would not open or close a spin-wave gap.
Here we focus on the e®ect of one-dimensional (1D) and two-dimensional (2D) superlattices of magnetic materials on the magnonic frequencies and linewidths, obtained from a reciprocal-space solution to the Landau-Lifshitz-Gilbert (LLG) equation. 26We consider two signi¯cant examples to illustrate the range of potential scenarios that might be possible for spin wave control.First, we consider in¯nite cylinders of one magnetic material that are embedded in a second magnetic material, in a periodic arrangement corresponding to a 2D square lattice or hexagonal lattice.Large gaps within the spin-wave spectrum are obtained when the exchange constants and saturation magnetization of the two materials di®er greatly; thus the gaps are considerably larger for cylinders of iron embedded within YIG than for iron embedded within nickel.For iron embedded in YIG, we demonstrate the existence of a gap throughout the superlattice Brillouin zone in the magnon spectrum for both square and hexagonal symmetry magnonic crystals.In photonic crystals, such a feature forms an essential element of photonic band gap materials, [10][11][12] and permits the control of spontaneous emission of emitters embedded within the photonic crystal; here similarly the spontaneous emission of magnons from a source such as a spin-torque nanooscillator could be suppressed by embedding this spin-wave emitter in a fully-gapped magnonic crystal.For the second example, we consider a uniform magnetic material (YIG) under the in°uence of a periodic electric ¯eld.The presence of this periodic electric ¯eld opens a substantial and tun-able magnon gap within the dispersion and thus permits on/o® voltage-based control of spin-wave propagation.The spatially-varying electric ¯elds can be con¯gured to generate an arti¯cial 1D magnonic crystal structure with su±cient in°uence on the spin waves to open a gap orders of magnitude larger than the linewidth of the spin waves, corresponding to quality factors in excess of 100.Advancements in fundamental studies of spin-wave dynamics are enabled by this theory and these predictions, including electrically-modulated spinwave pump-probe techniques.In addition, this work may improve spin-wave interconnects and other spin-wave devices. 27,28n both of the examples considered here a careful consideration of the boundary conditions between the two e®ective magnetic materials, whether they consist of di®erent materials or of the same material with a di®ering electric ¯eld, is required to obtain the proper spin-wave properties.For a magnonic superlattice the exchange ¯eld that enters into the LLG equations is discontinuous at the boundary, and that discontinuity strongly in°uences the spin wave dynamics.Two distinct forms for this exchange ¯eld in the presence of inhomogeneous material parameters (saturation magnetization and exchange constant) have been described in the literature, 14,[29][30][31][32][33][34] although to our knowledge it has not been pointed out that the di®erences in the solutions of the LLG equation obtained from the two e®ective ¯elds produce large quantitiative di®erences in the spin-wave dispersion and lifewidths, nor has a derivation been presented of the correct form.In Sec. 2 we present an explicit derivation of the correct form of the exchange ¯eld, followed by spin wave frequencies and linewidths for several magnetic material combinations in Sec. 3. In Sec. 4 we show that solutions to the LLG equations for the incorrect forms of the exchange ¯eld di®er greatly from those for the correct form.We note that for both the correct and the incorrect forms of the exchange ¯eld the presence of the dipolar ¯eld generated by the periodic magnetic material breaks the anticipated point-group symmetry of the 2D lattice.The extent of this symmetry breaking, however, di®ers greatly for the two forms of the boundary conditions.In Sec. 5 we present solutions for a magnetic material, YIG, with a periodically-alternating electric ¯eld and describe the spin-wave dispersion relations for the system.

LLG Formalism for a Quasi Two Dimensional Magnonic Crystal
We consider a magnonic crystal composed of an array of in¯nitely long cylinders of ferromagnetic material A embedded in a second ferromagnetic material B in a square or hexagonal lattice; the structures are shown in Fig. 1 Here, is the gyromagnetic ratio, M s ðrÞ is the spontaneous magnetization, ðrÞ is the Gilbert damping parameter, and r is the three-dimensional (3D) position vector.The e®ective magnetic ¯eld acting on the magnetization Mðr; tÞ consists of three terms: the external ¯eld H 0 , the dynamic dipolar ¯eld hðr; tÞ, and the exchange ¯eld H ex ðr; tÞ.

Derivation of the e®ective magnetic ¯eld
We wish to derive the correct form of H eff ðr; tÞ to use in Eq. (1) for our magnonic crystal.As shown by Gilbert, 26 the exchange ¯eld can be obtained by taking the functional derivative of the exchange energy.For a homogeneous material, the exchange energy is 35 where A is the exchange sti®ness constant.This yields the following exchange ¯eld: For the inhomogeneous crystal considered here the values of the exchange constant and the spontaneous magnetization will di®er for the two ferromagnets, so A and M s become spatially dependent quantities: where ÂðrÞ ¼ 1 in material A and ÂðrÞ ¼ 0 in material B. The exchange energy for this inhomogeneous situation is By approximating the energy with U ex we have neglected nonexchange terms that would give rise to a surface torque (such as terms in the energy associated with surface-induced magnetic anisotropy).The total magnetization will consist of both a time-dependent term and a time-independent term: Mðr; tÞ ¼ M s ðrÞẑ þ mðr; tÞ: Using the linear magnon approximation we assume that the timedependent magnetization is small compared to M s ðrÞ, and therefore we only keep terms up to ¯rst order in mðr; tÞ.With these assumptions the inhomogeneous exchange ¯eld derived from Eq. ( 6) is The exchange ¯eld enters the LLG equation only as a cross-product with the magnetization Mðr; tÞ.The second term is parallel to Mðr; tÞ and thus will not contribute to Eq. ( 1).The third term of Eq. ( 7), which is proportional to mðr; tÞ and parallel to M 0 ¼ M s ðrÞẑ, will only produce terms of second order in mðr; tÞ in Eq. ( 1) and can safely be dropped.Therefore, we can approximate which produces an LLG equation from Eq. ( 1) that is correct to ¯rst order in mðr; tÞ.
We now have the following equation for the e®ective ¯eld: This form is a generalization of the boundary condition obtained at the interface between a ferromagnet and vacuum 36 in the absence of any surface torque, later derived for the boundary condition between dissimilar magnetic materials. 37,38It is also the form used in Refs.29-32.

Plane-wave solution to LLG equation for quasi two dimensional magnonic crystal
When solving for magnons of a speci¯c frequency !, we write mðr; tÞ ¼ mðrÞ expðÀi!tÞ and the dipolar ¯eld, hðr; tÞ ¼ ÀrÉðrÞ expðÀi!tÞ, with ÉðrÞ the magnetostatic potential.With the form of the effective ¯eld in Eq. ( 9), the LLG equation (Eq.( 1)) can be written Additionally, since there is no z dependence in the above equations, the 3D position vector r has been replaced with the 2D position vector, R ¼ ðx; yÞ.
0][31][32][33] We take advantage of the crystal's periodicity and use Bloch's theorem to write the magnetization and magnetostatic potential as an expansion of plane waves: Here, G i represents a 2D reciprocal lattice vector of the crystal and k is a wave vector in the ¯rst Brillouin zone.The magnetostatic potential can be rewritten in terms of the magnetization by using one of Maxwell's equations: Replacing hðRÞ with ÀrÉðRÞ, substituting in Eqs. ( 12) and ( 13), and solving for the potential yield Next, we need to be able to write the material properties M s ðRÞ, QðRÞ, and ðRÞ in reciprocal space.Since these have the same periodicity as the crystal lattice, this can be done with a Fourier series expansion: The Fourier coe±cients are obtained by an inverse Fourier transform: where S is the area of the 2D unit cell.
Performing the integration for G ¼ 0 gives the average where f is the fractional space occupied by a cylinder in the unit cell.For G 6 ¼ 0, we have Here J 1 is a Bessel function of the ¯rst kind, and R cyl is the radius of the cylinders.The following in¯nite system of equations in reciprocal space is obtained by substituting Eqs. ( 12)-( 16) in Eqs. ( 10) and ( 11): We solve this by limiting the number of reciprocal lattice vectors in the sum and expressing it as a matrix equation: The LLG equation is now reduced to ¯nding the eigenvalues and eigenvectors for the above equation.
We increase the number of Fourier components in the calculation until the result converges.

Results
From Eq. ( 22), we calculate the complex eigenvalues corresponding to the frequencies of magnons in 2D magnetic superlattices of Fe, Co, Ni, and YIG.The real part of n ðkÞ is the magnon frequency of branch n for the wave vector k and the imaginary part is the inverse spin-wave lifetime.To focus on the dependence of these properties on magnetic material combinations, we consider superlattices with a lattice constant a ¼ 10 nm, an external ¯eld 0 H 0 ¼ 0:1 T, and a ¯lling fraction f ¼ 0:5.The material properties, M s , A, and , are listed in Table 1, and are obtained from Refs.39-42.
Figure 2 shows the empty-lattice band structures obtained from the LLG equation for homogeneous crystals of Fe, Co, Ni, and YIG in a square lattice.
Electric-Field Control of Magnon Gaps in a Ferromagnet using a Spatially-Periodic Electric Field Figure 3 shows the same for a hexagonal lattice.As the empty-lattice features are governed by the lattice symmetry and the material's spin wave velocity, these plots depend on material only in setting the frequency scale of the features, which is demonstrated in both Figs. 2 and 3.
Figure 4 shows spin-wave dispersions when Fe is embedded in a square lattice within a host of Co, Ni or YIG, and also for Co, Ni or YIG cylinders embedded in Fe.The change in band structure from the homogeneous case (upper left of Fig. 2) is more substantial when there is a greater di®erence in the     spontaneous magnetization between the two materials.For example, the magnetic properties of Fe and Co are fairly similar, and so for a crystal composed of these materials, the band structure di®ers little from the homogeneous case, with only some small splittings of the spin-wave dispersion curves occurring.However, the magnetization of YIG di®ers from that of Fe by more than a factor of ten, so for crystals of YIG embedded in Fe the magnonic modes are almost completely di®erent from the homogeneous crystal.Furthermore, wider band gaps are opened in the spin-wave dispersion in these structures if the magnetization is larger in the cylinders than it is in the host.For a crystal with Fe cylinders embedded in YIG in a square lattice (bottom left of Fig. 4), there are four gaps occurring within the lowest nine spin-wave modes, whereas if instead YIG cylinders are embedded in Fe (bottom Re(ω) Fig. 6.The lowest nine spin wave frequencies (in THz) for a square lattice magnonic crystal composed of Fe cylinders in Ni with a ¯lling fraction of f ¼ 0:5, and a lattice constant of a ¼ 10 nm.These results were obtained using the exchange ¯eld in Eq. ( 8).The square ¯gures show the entire Brillouin zone of the lattice.Electric-Field Control of Magnon Gaps in a Ferromagnet using a Spatially-Periodic Electric Field right of Fig. 4), there is only one small gap between the ¯rst and second spin-wave modes.Similar in-°uence on cylinder composition was reported in Ref. 29 and ascribed to both the larger exchange constant and larger magnetization of Fe versus YIG; we ¯nd it is due to the larger magnetization, and the e®ect of the larger exchange constant is negligible.Similar results are apparent in Fig. 5 for hexagonal arrangements of cylinders embedded in a magnetic host.The gaps in spin-wave dispersion are greater when the cylinders have larger magnetization; for Fe cylinders embedded in YIG the spinwave dispersion curves are almost °at (bottom left of Fig. 5), whereas there are no spin-wave gaps apparent for YIG embedded in Fe (bottom right of Fig. 5).
We now show the detailed dispersion curves (Fig. 6) and spin-wave relaxation rates (Fig. 7) for a square lattice of Fe cylinders in Ni.Plotted are the lowest nine spin-wave modes (ReðÞ) in the entire ¯rst Brilllouin zone as well as the corresponding inverse spin wave lifetimes (ImðÞ).Quality factors for these modes, corresponding to the ratio of the relaxation rate to the mode frequency, can exceed 100 for such spin waves, especially for the lowestfrequency modes.We note that these plots exhibit the correct square symmetry of the lattice.The linewidths of the lowest-frequency mode are found Fig. 8.The lowest nine spin-wave frequencies (in THz) for a hexagonal lattice magnonic crystal composed of Fe cylinders in Ni with a ¯lling fraction of f ¼ 0:5, and a lattice constant of a ¼ 10 nm.These results were obtained using the exchange ¯eld in Eq. ( 8).The ¯gures include the entire hexagonal Brillouin zone of the lattice.to be smallest at the zone boundary, although this result does not extend to higher-frequency spin waves of the lattice.The detailed dispersion curves (Fig. 8) and spinwave relaxation rates (Fig. 9) for a hexagonal lattice of Fe cylinders in Ni show similar features, although here the linewidths for the lowest spin-wave modes are small at the origin as well as at the zone boundary.The overall linewidths, however, tend to be larger than those in the square lattice.

Comparison with Alternate E®ective Field
Some calculations of magnonic crystals dispersion curves used a di®erent exchange ¯eld 14,33 di®ers by the positioning of one factor of M s ðrÞ À1 outside the gradient operators.A comparison of the band structures obtained for the two di®erent exchange ¯elds is shown in Fig. 10.An examination of the band structure for a homogeneous material composed of Fe or Ni (Fig. 2) indicates that the results for the correctly derived exchanged ¯eld (Eq.( 8)) produce a band structure that is appreciably di®erent from the homogeneous case, whereas the band structures produced by the alternate exchange ¯eld (Eq.( 26)) are very similar to the homogeneous crystal.In Fig. 11, we show the lowest spin-wave mode and corresponding relaxation rate obtained for Fe cylinders in Ni when using Eq. ( 26) as the exchange ¯eld.When looking at these contours, we might expect them to have the same symmetries as the real space lattice.For a square lattice, that would be symmetry under rotations of 90 and symmetry under re°ections about either axis.However, the nature of the dipolar ¯eld breaks the point group symmetry.The size of the point group symmetry breaking from the dipolar ¯eld is very di®erent for the two boundary conditions; the asymmetry in Fig. 11 is much larger than that in Figs.7 and 9 (which use the correct boundary conditions).

Formation of Magnonic Band Gaps from Spatially-Periodic Electric Field
We consider a slab of YIG with magnetization in plane and a periodic electric ¯eld perpendicular to both the magnetization and the spin-wave propagation direction (Fig. 12).This system is described with the LLG equation, Eq. (1).For the above geometry the e®ective magnetic ¯eld H eff ðr; tÞ of Eq. ( 2) acting on the magnetization consists of the following terms: where ex is the exchange length.H 0 is the magnitude of the external ¯eld (pointing along ẑ), H demag is the demagnetizing ¯eld, 43 hðr; tÞ is the dynamic dipolar ¯eld obtained by satisfying Maxwell's equations, and 2 ex r 2 M is the e®ective exchange ¯eld obtained from the exchange energy. 26he ¯nal term H DM ðrÞ is an e®ective magnetic ¯eld that appears as a result of the DMI in the presence of an electric ¯eld and the spin-orbit interaction.The form of the DMI ¯eld has previously been obtained for a uniform electric ¯eld applied perpendicularly to a YIG slab, 5 where is the DM vector.For a periodic electric ¯eld its form can be obtained as in Ref. 5, from the DMI energy, assuming a spatially dependent electric ¯eld: We now simplify the matrix expressions from Sec. 2 for the 1D case to yield the following expressions: where Here G m ¼ 2m=a is a reciprocal lattice vector of the system in the x-direction, DðGÞ is the Fourier transform of DðrÞ, and n and m are integers.The boundary conditions for spin waves in the transverse directions are obtained from the general exchange boundary conditions 44 : where y 0 and z 0 are the widths of the slab in the y and z directions, and is the pinning parameter of YIG.In order to ensure that these transverse modes do not interfere with the band gaps produced by the electric ¯eld, we set the widths to be small enough so that if there is a transverse component, the lowest spin-wave band is shifted to above the frequency of interest.Using Eqs. ( 33) and ( 34) with 45 ¼ 10 7 m À1 , a width of 50 nm or less will shift the dispersions by more than 2 GHz for either transverse mode, putting the lowest mode above the frequency where the band gap will open.
The frequencies and linewidths are obtained from the complex eigenvalues of Eq. (31).We consider an external ¯eld 0 H 0 ¼ 0:27 T, saturation magnetization M s ¼ 1:4 Â 10 5 A/m, exchange length ex ¼ 16:1 nm, and damping parameter ¼ 0:0006.From the dispersion relations (Fig. 13), we see that the YIG slab has no band gaps when there is no electric ¯eld.However, when a periodic electric ¯eld is applied, the band levels begin to split, opening multiple band gaps with the ¯rst few modes with a quality factor of the ¯rst mode on the order of 100.An additional e®ect is that the wave vector of the dispersion curves is phase shifted due to the electric ¯eld breaking the rotational symmetry of the system.By looking at the density of states of the spin waves (Fig. 16), we see that the appearance of the band gaps relies on the electric ¯eld having a periodic variation.A uniform electric ¯eld only shifts the band levels to a lower frequency.
Both the width and location of the band gaps can be controlled by adjusting various parameters of the system (Fig. 15).Since the electric ¯eld introduces the band gaps into the system, increasing its magnitude has the expected e®ect of increasing the width of the band gap.Additionally, due to the overall e®ect of the electric ¯eld lowering the spinwave frequency (Fig. 16), increasing the strength of the electric ¯eld will also lower the position of the band gap.The lattice constant has a comparatively smaller e®ect on the band gap width, but can have a signi¯cant impact on its location, with the frequency rapidly increasing when the lattice constant is below 100 nm.

Conclusion
Spin-wave dispersion curves and relaxation rates have been calculated for hexagonal and square 2D lattices of magnetic cylinders embedded in another magnetic material.The correct form of the exchange ¯eld at the boundary between these two magnetic materials has been found, and the di®erence from another form used in the literature has been shown to be signi¯cant.Full-zone magnonic gaps are obtained for lattice materials that di®er substantially in their saturation magnetization, such as Fe and YIG.Quality factors for spin waves can exceed 100, especially for the lowest-frequency spin mode.These results should assist in the design of magnonic crystals that can focus or redirect spin waves due to their e®ective band structure.
We have also calculated the band structures for a slab of YIG in the presence of a periodically varying electric ¯eld.This ¯eld opens band gaps in the dispersion relations whose frequency and width can be adjusted by modifying the strength of the electric ¯eld and the length scale of the periodicity.This principle could be used in the development of a magnon transistor and is more e±cient than other proposed methods of controlling the spin-wave band structure 20,25,46 since power is required only when switching the electric ¯eld on or o®.The band gaps obtained by this method are also much larger than those obtained via other methods.For spin waves with a similar frequency as described here, band gaps of tens of MHz were reported, 20,46 whereas we obtained gap widths on the order of several hundred MHz.

Fig. 2 .
Fig. 2. Empty square lattice band structure obtained from the LLG equation for a homogeneous crystal of Fe (upper left), Co (upper right), Ni (lower left), and YIG (lower right) with lattice constant a ¼ 10 nm.Frequencies are in units of THz.

Fig. 3 .
Fig. 3. Empty hexagonal lattice band structure obtained from the LLG equation for a homogeneous crystal of Fe (upper left), Co (upper right), Ni (lower left), and YIG (lower right) with lattice constant a ¼ 10 nm.Frequencies are in units of THz.

Fig. 4 .Fig. 5 .
Fig. 4. Magnonic band structures for a square lattice magnonic crystal with lattice spacing a ¼ 10 nm and ¯lling fraction f ¼ 0:5.On the left is Fe cylinders embedded in Co (top), Ni (middle), and YIG (bottom).The right is for an Fe host with Co (top), Ni (middle), and YIG (bottom) cylinders.Frequencies are in units of THz.

Fig. 7 .
Fig. 7.The spin wave relaxation rate (in units of GHz) corresponding to the lowest nine spin wave modes from Fig. 6.The square ¯gures show the entire Brillouin zone of the lattice.

Fig. 9 .
Fig. 9.The spin wave relaxation rates (in units of GHz) corresponding to the lowest nine spin wave modes from Fig. 8.The ¯gures include the entire Brillouin zone of the hexagonal lattice.

Table 1 .
Properties of the di®erent materials considered for the magnonic crystals.
than the one derived in Sec. 2. The alternate form,