Magnonic analog of a metal-to-insulator transition in a multiferroic heterostructure Open Access

The voltage/electric field control of ferromagnetic (FM) systems is considered central for future low-dissipation, miniaturized devices.^1–3 The coupling of a ferromagnetic film to a ferroelectric substrate with regular ferroelastic stripe domains is a well known technique to control ferromagnetic properties by voltage or electric fields.^4–7 The combination of strain transfer and inverse magnetostriction in this bilayer system causes the imprinting of magnetic stripe domains in the ferromagnetic layer, and the coupling is proportional to the domain polarization, which can be controlled by an external electric field.^8–11 In general, the manipulation of magnons at the nanoscale through the vertical coupling in multilayers is considered one of the most promising and current research topics, falling into the field of 3D magnonics.^12–15 

Another research effort explores the occurrence of a magnonic frequency gap in magnetic meta-materials as it is fundamental to many applications like filters or mirrors.^16–19 A magnonic bandgap prevents magnon propagation along some direction for some frequencies, acting as a filter. If the gap is independent of the direction, we have a full bandgap and an omnidirectional filter. In a compound system, magnons coming along a waveguide can be perfectly reflected at the interface with a material with a frequency bandgap.^20–22 In analogy with electrons in solids, a magnonic meta-material can be seen as a conductor or insulator for magnon propagation depending on the occurrence of a magnonic gap along the magnon propagation direction. However, in principle, even a uniformly magnetized film is a conductor for magnon propagation, only it has no periodicity and no band structure. Hence, to mark the difference, a magnonic conductor, which is also an artificial crystal (i.e., with a periodic modulation of any of its geometric or magnetic properties), can be referred to as magnonic metal, similarly to an electric conductor when consisting of atoms arranged in a crystal structure.

In a ferroelectric/ferromagnetic (FE/FM) bilayer system, the vertical coupling between the layers is the inverse magnetostriction interaction, the intensity of which can be tuned by varying the FE domain polarization through an electric field.^23–25 

Furthermore, we prove a very challenging effect, namely, the possibility to switch to zero the frequency gap preserving the system periodicity, and at the same time change the dispersion curve at the Brillouin zone (BZ) boundary from parabolic to linear (and vice versa). Note that, in these systems with sinusoidal magnetization, we find that the vanishing of the gap is inevitably concurrent with the linearity of the dispersion curves at zone boundary, which form a distinctive “X” shape, as it happens in graphene at K-point. Since zero gap and linear dispersion curve are traditionally associated to free propagation such as electrons in metals, while the presence of a gap and a parabolic dispersion to propagation blockage such as electrons in insulators, we might address this as a reversible magnonic metal-to-insulator transition.

We will discuss and justify the results of the corresponding simulations within Dirac’s magnon approach,²⁸ which we extend to one-dimensional lattices and continuous medium, finding interesting implications for magnon propagation. Finally, we found that in such a system, the slope of the dispersion curve, i.e., the group velocity $v g$⁠, can be tuned through $K u$⁠, and, in particular, switched to zero at high $K u$ values: the same physical system can, hence, be thought of as a versatile device, which can be switched on-demand and in real time (i.e., while magnon is still propagating) from a waveguide (with propagating waves, $v g≠0$⁠) to a dynamic memory (with stationary waves, $v g=0$⁠).

This property makes our system in line to the so-called dynamic or reconfigurable magnonic crystals,²⁹ where the periodicity of the ferromagnetic medium and its spin waves are controlled (possibly, in real time) by an external parameter (magnetic field,^30–32 current,³³ laser-induced heating,³⁴ surface-acoustic-wave induced strain,^35,36 etc.). In addition, a further type called dynamic electromagnonic crystal was devised in the context of multiferroics, which might recall our work:^5,37–39 here, though, the focus was on the hybridization (coupling) between electromagnetic and spin waves (electromagnons) and for this reason even when showing dispersion curves, the focus was mainly on the wavevector range where the coupling occurs (⁠ $∼ 10 3$ rad/m^40,41), and, hence, (depending on the stripe period) around $10 3$ times smaller than the range we refer to (⁠ $∼ 10 7$ rad/m). Actually, in these papers, the gap sometimes refers to the anticrossing effect (repulsion) within the coupling region, not only the one due to Bragg’s diffraction at zone boundary.³⁷ These gaps are always very small (⁠ $∼$MHz), compared to the ones we found in our simulations (⁠ $∼$GHz). To open (and tune) frequency gaps for electromagnons, hence, large stripe periods are used (within $10-800μ m$⁠),^5,38 while we address the nanometric lengthscale (⁠ $≪1μ m$⁠). Clearly, there are a few similarities with our magnonic framework, but electromagnonic crystals are very different tools with different operating ranges and, hence, with different technological purposes.

In conclusion, under our knowledge, a direct focus on the magnetization undulation and its tunability, the possibility of gradually and reversibly varying the frequency gap from zero to large values (GHz) in a same, unpatterned system, the magnon dispersion curvature (effective mass) were never reported in correlation to each other, particularly in the context of a voltage-control of magnon propagation. For these reasons, we believe that the method of the undulated magnetization offers a solid contribution to the issue of voltage-controlled magnon propagation: our results suggest the use of a magnetization undulation to tune the magnon dispersion curvature and the frequency gap, useful for conceiving versatile spinwave filters and mirrors, waveguides and memories, and potentially any other innovative magnonic logic devices, enriching the long-established thread of the dynamic magnonic crystals.

We perform micromagnetic simulations with the GPU-accelerated micromagnetic simulation program Mumax3.⁴² We consider an infinite ferromagnetic film, 5 nm thick, implemented through a square periodic primitive cell with lattice constant $a=256$ nm and 800 repetitions along the in-plane coordinates (quasi-periodic boundary conditions). The film is discretized with $4×4×5$ nm $3$ micromagnetic elemental cells. We use permalloy magnetic parameters, namely, saturation magnetization $M S=800$ kA/m, exchange stiffness constant $A=1.0× 10 − 11$ J/m, and gyromagnetic ratio $γ=185$ rad GHz/T. We use a fictitious large damping (⁠ $α=0.9$⁠) to quickly relax to the equilibrium magnetization, while for the excitation process, we use zero damping to allow long lasting precession and, hence, large Fourier coefficients. We divide the primitive cell into four equivalent stripes, where we set a uniaxial magnetic anisotropy with the same magnitude $K u$⁠, but different orientation of the easy axis according to the following (periodic) order: $45 °$⁠, $0 °$⁠, $− 45 °$⁠, $0 °$ [Fig. 1(a)]. In order to obtain frequencies around 10 GHz, we also apply a uniform bias magnetic field $B 0=0.1$ T. The film, initially prepared saturated along the $x$-direction, relaxes to the presence of the bias field and the periodic magnetic anisotropy, and at equilibrium displays a continuous sinusoidal magnetization distribution. This result is found for $K u≤ 10 5 J / m 3$⁠: beyond this value, no regular sine function is found for the magnetization, and instead magnetic domains form inside the primitive cell, with a magnetization almost aligned to the local anisotropy axis, and a sharp discontinuity across each domain. Hence, our investigation is limited to $K u≤5× 10 4 J / m 3$⁠, which corresponds to an anisotropy field $B anis≤62.5$ mT (in some works, the anisotropy is preferably seen through the associated magnetic field, usually smaller than this value^43,44).

To compute the dispersion relations, we use a supercell of 200 copies of the primitive cell along the direction of the magnetization undulation (⁠ $x$ axis), obtaining a wavevector resolution $δ k x≃0.012× 10 7$ rad/m. Then, we apply a space–time sinc magnetic field with amplitude 1 mT, cutoff frequency 40 GHz, and wavevector band $k x=0.0490$ rad/nm, and save the magnetization map every $δt=25$ ps. Then, we compute the space–time Fourier transform, resulting in the dispersion relation of magnons, with a Nyquist frequency $f N=20$ GHz (maximum frequency) and a frequency resolution $δf=0.01$ GHz. On the same magnetization maps, we also compute the time Fourier transform and obtain the space-resolved Fourier coefficients for each frequency and wavevector, i.e., the spin wave (SW) mode profiles. Once chosen a specific frequency, with reference to a point in the dispersion curve map, we plot the real part of the $z$-component⁴⁵ of the dynamic magnetization to visualize the corresponding SW profile.

The spin dynamics of a FM film with sinusoidal magnetization is rather simple, as far as the undulation amplitude $Y 0$ is small, and consists of Bloch waves with a cell function profile having $m$ (or $n$⁠) nodal lines either perpendicular (backward spin waves, $m$-BA) or parallel (Damon–Eshbach modes, $n$-DE) to the undulation direction (⁠ $x$ axis).^47–50 Mixed modes, labeled $(m,n)≡(m$-BA $×n$-DE), are also possible. In the reduced scheme of the BZ, the number of nodes (or, equivalently, the mode label) represents the band index for any given wavevector $k$⁠.^19,46 Since the magnetization flux $M$ oscillates along $x$ axis, we consider only the dispersion along that axis (i.e., backward waves only).

We computed the dispersions at different values of $K u$⁠, encompassing four orders of magnitude, from $1× 10 1$ to $5× 10 4 J / m 3$⁠. These values are in line with the range reported in the literature about magnetic anisotropy induced by inverse magnetostriction.^9,10,51–54

In sinusoidally magnetized films, the spin dynamics is invariant with respect to a mirror operation of the magnetization map around the $x$ axis (called $σ h$ in Ref. 55), i.e., invariant with respect to the inversion of the magnetization component $m y$⁠. Hence, even if to plot a sinusoidal distribution of the magnetization we had to consider a geometric primitive cell with side $a=256$ nm, the actual, physical primitive cell emerging in the dynamics is only a half of the designed one, namely, $d=128$ nm. Consequently, the physical Brillouin zone boundary occurs at $2× π a=2.45× 10 7$ rad/m (twice as big as the geometric one). The output of the simulations confirmed this point (dashed vertical lines in Fig. 2).

In Fig. 2, we show six indicative cases of dispersion curves to illustrate the metal-to-insulator transition and the variation of the gap as $K u$ is increased.

For any $K u<0.04 kJ / m 3$ (not shown), the magnetization maintains the uniform saturated state, and the dispersion curve does not show any trace of folding to the reduced Brillouin zone, namely, no periodicity is imprinted on the film nor can its dispersion be distinguished from that of the uniform film (⁠ $K u$ = 0). It is a sort of hysteresis under $K u$ variations: in-plane exchange interaction and the strong out-of-plane dipolar fields are stronger than the anisotropy and continue to keep the film magnetization uniform. This is a remarkable result, since it defines the limit of $K u$⁠, below which it is ineffective as a perturbation of the film uniform magnetization: in other words, our simulations suggest that to have evidence for periodicity in the experiments with a real permalloy film (either with metal or insulator behavior), it is necessary to design the ferroelectric overlayer such that it can induce, by vertical coupling, a magnetic anisotropy $K u≥0.04 kJ / m 3$⁠. Below that value, the system behaves like an ordinary film without “bands”: note that only with “bands” it is possible to increase the magnon frequency by concurrently decreasing the wavevector, which is impossible in the single-valued dispersion curve of a uniform film.

Within the range $0.04≤ K u≤0.4 kJ / m 3$ [Fig. 2(a)], the dispersion curve follows that of the uniformly magnetized film, but now showing the characteristic periodic folding to the reduced Brillouin zone, with no appreciable gap (<0.01 GHz): the dispersion curves cross each other at the Brillouin zone boundary in the distinctive “X” shape, where they show a clear linear behavior. In this condition, magnons can continuously change their frequency (and wavevector) for any arbitrarily small variation of the external driving force operated by the (microwave) antenna: we might address them as “free magnons.” This aspect in some way recalls what happens to “free electrons” in metals, where electrons can be put into motion by an arbitrarily small voltage.

Remarkably, variations of $K u$ in the range at which the system behaves like a magnon metal are not affecting the dispersion curve at all, nor changing it with respect to the uniformly magnetized case (except for the periodic folding). We might draw a first conclusion, that in systems with a sinusoidal magnetization, the group velocity, and the consequent magnon mobility, is robust with respect to $K u$ variations.

In a narrow interval across the BZ boundary, the slope of the two crossing curves is the same, because of the space inversion symmetry of the magnetization undulation along $x$ axis.

In order to give insight about the control over $ν t$⁠, using the analytical dispersion relation for dipole-exchange backward spin waves, presented in Ref. 59, we plotted the curves for three systems (Fig. 3) differing in geometrical and magnetic parameters, producing the same frequency (10.32 GHz) at zone boundary but with very different $v g$⁠. The black curve reproduces analytically the exact dispersion curve presented in Fig. 2(a) and found by simulations, only without the folding to the reduced BZ, for simplicity. By linear interpolation, we found a group velocity $v g≃314$ m/s, as well as $ν 0=9.095$ GHz and $ν ( π d )≡ν(κ=0)=10.32$ GHz; hence, the hopping rate is in this case $ν t=1.225$ GHz. The red curve is designed to have $ν t=ν ( π d )$ and refers to a 100 nm thick film of a fictitious material with $M s=900$ kA/m, exchange parameter $A=14.0× 10 − 11$ J/m and subject to an applied field $B 0=10$ mT. Its linear approximation (tangent) at zone boundary is also plotted (dashed red line): $ν t≃10.32$ GHz, corresponding to $v g≃2.64$ km/s. The blue curve refers to another fictitious material, designed to have $ν t≫ν ( π d )$⁠. To compare with the previous curves and keep the same frequency (10.32 GHz) at zone boundary, we used the following parameters: film thickness 1.5  $μ$m, $M s=1500$ kA/m, $A=50× 10 − 11$ J/m, gyromagnetic ratio $γ=150$ rad GHz/T, under a bias field of 10 mT. Its linear approximation (tangent) at zone boundary is shown (dashed blue line): $ν t≡ν(π/d)− ν 0=19.6$ GHz (⁠ $ν 0≃−9.3$ GHz), corresponding to $v g≃5.02$ km/s.

Dirac’s magnon picture, applied to our results, discloses the vast consequences of the approach to a voltage-driven magnon dynamics under a sinusoidal magnetization: to have a magnonic metal behavior (i.e., periodicity and bands, with a linear dispersion and no gap at zone boundary), anisotropy is necessary indeed, and in our case, we found that the anisotropy coefficient must be $K u≥0.04 kJ / m 3$⁠. Under these conditions, magnons behave like non-dispersive mass-less wave packets, with a group velocity, and, hence, mobility, which can be designed by an adequate choice of magnetic and geometric parameters (Fig. 3).

The very fact that magnons, in analogy to electrons in solids (and graphene), can be interpreted in wave mechanics in terms of wave packets,⁶⁰ stands at the origin of the extension of Dirac’s magnon picture to our case, characterized by a linear lattice and a continuous medium. Being wave packets, they possess a phase velocity (associated to the pulsation) and an independent group velocity (associated to the propagation), both determined by the involved interactions. For this reason, we suggest to address a ferromagnetic film, subject to an appropriate periodic anisotropy, as a the magnonic analog of a metal material.

In a sense, to hop from one half of the unit cell to the other half [in Fig. 4, moving from (a) to (b)], under an external force, magnons are required an extra amount of energy, corresponding to the gap, so that at the BZ boundary a continuous propagation is inhibited and only a stationary wave behavior is possible. This extra energy is increasing with increasing $K u$ [Figs. 2(c)–2(f)] and is due to the symmetry breaking introduced by the periodic anisotropy: hence, $K u$ can be used to tune the magnon mobility in real time. For instance, when $K u=5× 10 3 J / m 3$⁠, a simple fit with the corresponding dispersion curve data at the BZ boundary of Fig. 2(c), under a parabolic approximation, gives an effective mass $m ∗=5.59× 10 − 27$ kg, i.e. (just to provide a quantitative reference), 6138 times the electron mass. However, if we increase the anisotropy coefficient to $K u=2× 10 4 J / m 3$ [Fig. 2(e)], i.e., 4 times larger, the fit provides $m ∗=11.98× 10 − 27$ kg, i.e., slightly more than the double (13153 electron masses). Hence, as the curvature of the upper dispersion curve decreases for increasing $K u$ (i.e., the parabola becomes wider), the effective mass increases and so does the magnon inertia to speed changes. Note that for $K u=5× 10 3 J / m 3$⁠, the frequency gap is evaluated about 0.380 GHz, while for $K u=2× 10 4 J / m 3$⁠, it is about 0.800 GHz.

The occurrence of the bandgap is the consequence of the interaction of the spin modes with the periodic anisotropy field (which is the simulation of the inverse magnetoelastic effect in a multiferroic bilayer). The spin modes relevant for the dispersions are, in our case, backward modes of order $m$ (⁠ $m$-BA), with $m$ = 0, 1, 2, $…$⁠. In fact, the only propagation direction we consider is $x$⁠, along which on the average the magnetization is oriented (⁠ $k∥ M$⁠). At the Brillouin zone boundary, when $k= π d$⁠, modes $m$-BA and $(m+1)$-BA are out of phase of $π/2$⁠, i.e., shifted of a quarter of their wavelength, and, hence, they experience differently the anisotropy field (e.g., 0-BA mode, i.e., the fundamental mode F, has nodes where the easy axis is at $0 °$⁠, while 1-BA where it is at $± 45 °$⁠, see Fig. 4). This makes them no longer degenerate, and a frequency gap forms. As already remarked, the gap is forming only for $K u≥0.4× 10 3 J / m 3$⁠.

In our simulations, we found that the frequency gap increases with increasing $K u$⁠, and, after a first linear trend, tends to a saturation value of 0.9 GHz (Fig. 5). Being outside the purposes of the present investigation, we speculate that with an appropriate choice of materials, either the slope or the saturation values might be changed for specific purposes.

Finally, we underline an interesting effect: when the magnetic anisotropy becomes very large (e.g., $K u≥2× 10 4 J / m 3$⁠), the lowest frequency dispersion curve becomes progressively flatter, while the higher dispersion curve maintains its parabolic shape. Hence, magnons belonging to the lower band become stationary (non-propagating resonances), while those belonging to the upper band are still propagating waves [Figs. 2(e) and 2(f)]. Once more this recalls analogies with electrons in semiconductors, which are bound and localized in the lower (valence) band and free and delocalized in the upper (conduction) band. The occurrence of either propagating or stationary modes in the same system at a tunable frequency gap is particularly interesting for signal filtering and manipulation.^61,62

In conclusion, we devise a multiferroic magnonic crystal, which can be continuously (and reversibly) turned from a metal (zero gap, linear dispersion) to an insulator (significant gap, parabolic dispersion) for magnon propagation in real time with a single control parameter. The ultimate control parameter is the electric voltage applied to the FE overlayer. The FE domain distribution transfers vertically to the FM film through the inverse magnetoelastic effect, which we simulated through a magnetic anisotropy with an easy axis periodically varying in one direction, producing a sinusoidal distribution of the magnetization acting as a tunable magnonic crystal. The magnon gap width, group velocity, and effective mass can be effectively tuned by the (uniform) anisotropy coefficient in a wide range of values (⁠ $0-5× 10 4 J / m 3$⁠). We extended the application of Dirac’s magnon picture to a linear lattice in a continuous medium under the magnonic analog of metal (gapless) conditions (⁠ $0.04< K u<0.4 kJ / m 3$⁠), and we discussed how to control the hopping rate, which determines the magnon conductivity and is related to the strength of the overall interaction (exchange, dipolar, anisotropy). Under the magnon insulator conditions (⁠ $K u≥0.4 kJ / m 3$⁠), the emerging frequency gap was found to increase with increasing $K u$⁠, linearly at first, then gradually reaching saturation. Furthermore, we correlated the curvature of the parabolic dispersion branch to the magnon effective mass and found that the lower band tends to progressively reduce with increasing $K u$ up to become flat. In this way, films with sinusoidal magnetization at high $K u$ show the simultaneous presence of stationary and propagating magnons at a frequency distance close to 1 GHz: with this property, a suitable magnon can be excited to either carry or (dynamically) store information in the same physical magnonic device. Our results are particularly interesting for conceiving low-dissipation miniaturized devices involving ferromagnetic media and magnons as information carriers. Specifically, new magnonic devices based on voltage-controlled magnetic anisotropy and angular momentum transfer (magnons) would need no or extremely reduced electric currents and, hence, could be further miniaturized, at even large clock speeds, without the limitations due to Joule heating.

P.M. and F.M. acknowledge support by the Department of Physics and Earth Sciences-University of Ferrara Grant Bando FIRD 2023, as well as the CINECA award under the ISCRA initiative, for the availability of high performance computing resources and support (Project SWIM-3D on Leonardo). A.R. and E.I. acknowledge support by the National Science Foundation (NSF) under Grant No. 2205796.

The authors have no conflicts to disclose.

P. Micaletti: Writing – review & editing (equal). A. Roxburgh: Writing – review & editing (equal). E. Iacocca: Writing – review & editing (equal). M. Marzolla: Software (equal); Writing – review & editing (equal). F. Montoncello: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Software (equal); Supervision (lead); Writing – original draft (lead); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.