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Electric-field control of spin waves at room temperature in multiferroic BiFeO3

To face the challenges lying beyond current CMOS-based technology, new paradigms for information processing are required. Magnonics proposes to use spin waves to carry and process information, in analogy with photonics that relies on light waves, with several advantageous features such as potential operation in the THz range and excellent coupling to spintronics. Several magnonic analog and digital logic devices have been proposed, and some demonstrated. Just as for spintronics, a key issue for magnonics is the large power required to control/write information (conventionally achieved through magnetic fields applied by strip lines, or by spin transfer from large spin-polarized currents). Here we show that in BiFeO3, a room-temperature magnetoelectric material, the spin wave frequency (>600 GHz) can be tuned electrically by over 30%, in a non-volatile way and with virtually no power dissipation. Theoretical calculations indicate that this effect originates from a linear magnetoelectric effect related to spin-orbit coupling induced by the applied electric field. We argue that these properties make BiFeO3 a promising medium for spin wave generation, conversion and control in future magnonics architectures.

To face the challenges lying beyond current CMOS-based technology, new paradigms for information processing are required. Magnonics 1 proposes to use spin waves to carry and process information, in analogy with photonics that relies on light waves, with several advantageous features such as potential operation in the THz range and excellent coupling to spintronics 2 . Several magnonic analog and digital logic devices 3 have been proposed, and some demonstrated 4 . Just as for spintronics, a key issue for magnonics is the large power required to control/write information (conventionally achieved through magnetic fields applied by strip lines, or by spin transfer from large spin-polarized currents). Here we show that in BiFeO3, a room-temperature magnetoelectric material 5 , the spin wave frequency (>600 GHz) can be tuned electrically by over 30%, in a non-volatile way and with virtually no power dissipation. Theoretical calculations indicate that this effect originates from a linear magnetoelectric effect related to spin-orbit coupling induced by the applied electric field. We argue that these properties make BiFeO3 a promising medium for spin wave generation, conversion and control in future magnonics architectures.
Magnetoelectric multiferroics possess coexisting magnetic and ferroelectric phases, with cross-correlation effects between magnetic and electric degrees of freedom 6 . As such, they can potentially be used to control spinbased properties by electric fields 7 , with very low associated power dissipation. This feature appears promising not only for spintronics, in which information is encoded by the spin-polarization of the electrical current 8 , but also for magnonics that use magnetic excitations (spin waves) for information processing 1 . Indeed, just as coupling between magnetic and ferroelectric order parameters exists in multiferroics, coupled spin and lattice excitations termed electromagnons have also been demonstrated 9 . Such mixed excitations exist at low temperature in multiferroic manganites 9 and are suspected at room temperature in BiFeO 3 10 . Although in the former systems, electromagnons have been shown to depend on magnetic fields 11 , their predicted sensitivity to electric fields, at the heart of their potential applicative interest, remains to be demonstrated.
In this paper, we focus on BiFeO 3 (BFO), a special case among multiferroic materials. Indeed, BFO has a very high ferroelectric polarization reaching 100 µC/cm 2 [12] below T C = 1143 K and becomes an antiferromagnet below T N = 643 K. The spins form a cycloid structure with a wavelength of λ 0 = 62 nm and an associated cy-cloid wavevector equal to Q = 2π/λ 0 13-15 . The magnetic cycloid lies in the (-12-1) plane, formed by the ferroelectric polarization [111] and the cycloid wavevector Q [10 − 1] (Fig. 1a). An electric field can simultaneously induce the flop of the polarization and the cycloidal plane 16 . The electrical switch of antiferromagneticferroelectric domains in BFO has been observed 17,18 and controlled in a ferromagnet-multiferroic heterostructure 7 . Here we report direct electric-field control of spin wave states in BFO at room temperature.
In BFO, inelastic light scattering reveals two species of spin wave excitation modes. These are observed in Raman spectra as two series of sharp peaks labelled cyclon (φ n ) and extra cyclon modes (ψ n ) below 70 cm −1 , the frequency of the lowest phonon mode 10,19,20 . The cyclon φ n and extra cyclon ψ n modes correspond respectively to the spin oscillations in and out of the cycloidal plane (Fig. 1a) and their Raman scattering signature are reported in Fig. 1c. These two species of magnons originate from the translational symmetry breaking 21 of the cycloidal ground state. The spin wave momentum is no more conserved, and can be increased or decreased by a multiple of the cycloid wavevector Q. This leads to magnon zone folding at the Brillouin zone center, where φ n and ψ n correspond to magnon wavevector equal to k = nQ. The experimental set-up used to apply an electric field and to detect the optical excitations of spin waves is presented in Fig. 1b. The device consists of bottom and top (transparent) electrodes that enable the application of an electric field along the [010] and [0 − 10] directions of a bulk BFO sample. The effect of the external electric field on the spin wave modes is tracked in Fig. 1c. An external electric field applied along the [010] direction induces a blue shift of the ψ n modes (ψ 2 and ψ 3 ) and a red shift of the φ n modes (φ 2 , φ 3 , and φ 4 ). The promise of this electric-field control depends on our ability to selectively tune the spin wave states using the polarization hysteresis cycle and the switching of the polarization vector P . field (E [0-10]), the frequency of the φ 2 mode (blue star) increases up to the flip of P along the [1-11] direction (P 2 ) at critical electric field E c ≈ −35 kV/cm 12 . Beyond E c , φ 2 's frequency decreases. The hysteresis cycle can be completed along paths 3 and 4 leading to frequency shifts that are symmetric to the ones along 1 and 2. The ψ 2 mode is also directly connected to the polarization loop, but exhibits an opposite behaviour with respect to the φ 2 mode (Fig. 2b). The extra-cyclon mode ψ 2 presents a much stronger frequency shift under the effect of an electric field of 50 kV/cm, ∆ω ψ2 = (+2.1 ± 0.1) cm −1 .
Remarkably, the ψ 2 frequency is seen to increase by over 5 cm −1 at maximum E field, corresponding to a 30% shift of its natural frequency. This experiment clearly demonstrates the coupling between the spin waves and the ferroelectric character of the material and that a spin wave frequency hysteresis loop can be created and controlled at room temperature using the ferroelectric hysteresis loop.
There exists other methods for electric-field control of spin waves. Rado  The present experiment shows that the coupling between magnetic and ferroelectric degrees of freedom provides a much more efficient method for electric-field control of spin waves, with frequency shifts over 5 cm −1 (30%), several orders of magnitude larger than with previous methods.
The strong spin wave frequency shifts that we observe may be interpreted as resulting from a combination of magnetoelectric interactions. The component of E along P (denoted E z ) cannot play any role because of the competition with the internal electric field parallel toẑ produced by P . Hence we only consider interactions involving E ⊥ = (E x , E y ). The Dzyaloshinskii-Moriya (DM) interaction that couples E to M × L is allowed by the R3c crystal symmetry of BFO 24 , and also induces a transition from the cycloidal to a homogeneous spin state at large E ⊥ 25 . The DM interaction produces spin wave shifts that are independent of E at low electric field, at odds with the data, and that scale as E 2 ⊥ at larger field 26 . Another class of magnetoelectric interactions, the so called flexoelectric interactions 25,27 couple E to gradients of L. For example, the flexoelectric interaction in a homogeneous magnet leads to an instability towards a spiral state and to spin wave shifts linear in E ⊥ 27 . However, in our case of bulk BFO the magnetic order is a cycloid that is itself induced by terms that couple P to gradients of L and have the same form as the flexoelectric interaction 25,28 . Therefore, there is a competition between these different couplings that leads to a renormalization of the cycloid wavevector. Since both ω φ and ω ψ are directly proportional to the cycloid wavevector, a large E ⊥ will increase all spin wave frequencies by the same amount. Thus, the DM and the flexoelectric interactions are not able to reproduce the experimental data at lower field but might play a role at larger field.
Here, we have found that the spin wave frequency shifts can be interpreted using the Landau free energy model based on an additional kind of magnetoelectric interactions induced by the external electric field. Indeed, the R3c symmetry of BFO allows the presence of two linear magnetoelectric interactions that couple the Néel order parameter L directly with E ⊥ , Here, we have found that the spin wave frequency shifts can be interpreted using the Landau free energy model based on an additional kind of magnetoelectric interactions induced by the external electric field. Indeed, the R3c symmetry of BFO allows the presence of two linear magnetoelectric interactions that couple the Néel order parameter L directly with E ⊥ , with κ 1 and κ 2 phenomenological coupling constants. Physically, F 1 and F 2 model additional magnetic anisotropy energies induced by E ⊥ . Hence their micro- When the applied electric field is zero, the energy of the φn and ψn modes are E φn = nv0Q and E ψn = √ 1 + n 2 v0Q respectively, with index n labelling the modes, v0 = 1.4 × 10 6 cm/s the fundamental cyclon spin wave velocity, and Q the cycloid wavevector 10 . The figure shows the spin wave mode energies normalized to v0Q as a function of the ratio of the applied electric field E divided by the electric field E hs required to destroy the cycloid. When E > E hs the spin ground state becomes homogeneous in space. The closed symbols are the experimental data obtained following path 1 (insert) in the polarization cycle using scopic origin is related to spin-orbit coupling. The first interaction (F 1 ) produces a red shift for both ψ n and φ n modes when E ⊥ is increased. Thus, F 1 cannot explain the experimental observations at low E ⊥ (Fig. 2a,b). On the other hand, the interaction F 2 induces a blue shift for all ψ n modes, and a red shift for φ n modes with n ≥ 2 as found experimentally, see Fig. 3. This interaction drives the system into a homogeneous state when E ⊥ becomes greater than a homogeneous critical field E hs ∝ 1/κ 2 . The calculated curves show that cyclon and extra cyclon mode frequencies converge to two fixed points at the crit-ical field E hs . A comparison between theory and experiment (Fig. 3) gives an estimate of E hs ≈ 160 kV/cm (above the maximum electric field of 125 kV/cm that could be applied to our sample without damage). Calculations are in good agreement with the experimental data in the low field region (see Fig. 3), but a clear deviation occurs at high fields. Interestingly, cyclon and extra cyclon modes seem to merge for E > 0.5E hs . This effect and the difference between theory and experiment at high fields might be due to other magnetoelectric interactions such as the DM and the flexoelectric interactions discussed above.
The demonstration of electrical control of spin wave states represents a significant step towards making new spin wave-based technologies. The cyclon spin wave dispersion can be written as ω φ = v φ k, and the extracyclon dispersion as ω ψ = v ψ k 2 + Q 2 . For E = 0, the spin wave velocities are equal, v φ = v ψ = v 0 = 1.4 × 10 6 cm/s 10 . However, under an electric field of 100 kV/cm the φ and ψ spin wave velocities are tuned by ∆v φ = −6 × 10 4 cm/s, and ∆v ψ = +4 × 10 5 cm/s respectively. While the speed of electron spin precession in a magnetic field is generally fixed by intrinsic properties of the materials considered, our work shows that in a multiferroic the spin wave speed can be continuously adjusted and in a different way depending on the spin propagating mode by the applied electric fields.
We have demonstrated that the frequency of spin waves can been tuned electrically by over 30% at room temperature in multiferroic BiFeO 3 . This electric-field dependence mimics that of the ferroelectric polarization, providing a handle for the non-volatile, low-power control of spin waves. Thus, aside from its potential for spintronics 29 and photonics 30 , BiFeO 3 emerges as an exciting platform for testing novel magnonic device concepts, which further confirms it as a key multifunctional material for beyond-CMOS technology.  16 . A conducting Indium tin oxide (ITO) layer was grown by pulsed laser deposition on the top of the (010) sample surface to apply a uniform electric field. ITO has no Raman signal in the frequency range of interest. All the measurements were performed in vacuum at room temperature on several samples.
Raman scattering was performed on the (010) sample surface in backscattering geometry using the 647.1 nm laser line. Raman scattering is collected by a triple spectrometer Jobin Yvon T64000 equipped with a CCD. The spot size is about 100 µm 2 and the penetration depth is less than 10 −5 cm. Reproducible Raman measurements have been performed on several points on the sample surfaces.
The φ n and ψ n modes (n index labels the modes from their lowest to highest energy) were selectively observed using parallel and crossed polarizations in the (010) plane, respectively (see Supplementary Information). The polarizations are defined with respect to the projection of the cycloid wavevector Q  in the (010) plane (the direction of the wavevector Q is not directly accessible in the (010) plane). In order to show simultaneously several φ n and ψ n modes, Fig. 1b presents measurements without specific polarization.

Theoretical method
We consider a total free energy given by F = F 0 + F 1 + F 2 , with F 0 the usual model free energy of multiferroic BFO 21,25 , and F 1 , F 2 given by Eqs. (1) and (2) respectively [we omit the purely ferroelectric contributions to Eq. (3) because they play no role in the discussion below]. Here A, r, G, c, and α P are phenomenological constants. The ground state is determined by the condition δF δL = 0; when E ⊥ = 0 the ground state is a harmonic cycloid L 0 = L 0 [sin (Qx)x + cos (Qx)ẑ] with wavevector Q = αP c and amplitude L 2 0 = (−A + cQ 2 )/G. The effect of E ⊥ is to reduce the ground state wavevector Q and to add additional anharmonic contributions to L 0 at odd multiples of Q. When E ⊥ is equal to a critical field E hs = (παP P ) 2 4c 1 κ2 , the cycloid wavevector Q becomes equal to zero, signaling a transition to a homogeneous state.
The spin wave excitation frequencies are determined from the Landau-Lifshitz equations of motion. After combining the equations of motion for the order parameters L and M we get where γ is the gyromagnetic ratio. We find wave-like solutions by plugging in L = L 0 + (δL)e iωt in Eq. (4), with (δL) = φ(x)D + ψ(x)ŷ. The unitary vectorD is tangential to L 0 , i.e. it is orthogonal to L 0 but lies within the cycloid plane. Plugging φ(x) = n φ n e inQx , and ψ(x) = n ψ n e inQx into Eq. (4) leads to a matrix equation whose eigenvalues give the spin wave excitations at k = 0.
We want to underline that the free energy model for BFO Eq. (3) does not contain any magnetic anisotropy. Nevertheless, the present experiment shows that an external E field induces magnetic anisotropy linear in E, see Eq. (2). In other words, the electric field activates a latent anisotropy in the material, leading to a large effect in the case of Eq. (2).