Phono-magnetic analogs to opto-magnetic effects

The magneto-optical and opto-magnetic effects describe the interaction of light with a magnetic medium. The most prominent examples are the Faraday and Cotton-Mouton effects that modify the transmission of light through a medium, and the inverse Faraday and inverse Cotton-Mouton effects that can be used to coherently excite spin waves. Here, we introduce the phenomenology of the analog magneto-phononic and phono-magnetic effects, in which coherently excited vibrational quanta take the place of the light quanta. We show, using a combination of density functional theory and phenomenological modeling, that the effective magnetic fields exerted by these phono-magnetic effects on the spins of antiferromagnetic nickel oxide yield magnitudes comparable or larger than those of the opto-magnetic effects.


I. INTRODUCTION
Since more than two decades, it has been possible to use light to control the magnetic order of materials on timescales smaller than a picosecond, promising application in advanced data storage and data processing in spintronic devices, and in quantum computation [1][2][3][4][5]. One ingredient is the coherent excitation of spin waves, which can be achieved by resonantly driving the spins with the magnetic-field component of an ultrashort laser pulse [6,7]. Another route is by coupling the spins to the electric field component of light through an impulsive stimulated Raman scattering process. Coherent excitation of magnons via Raman scattering has been achieved using the two most prominent mechanisms, the inverse Faraday effect and the inverse Cotton-Mouton effect in recent years [8][9][10][11][12].
In contrast to its recent application to the excitation of magnons, impulsive stimulated Raman scattering has been used for the coherent excitation of phonons for more than 30 years [13,14]. In addition, ionic Raman scattering, in which two coherently excited infrared-active phonons scatter with a Raman-active phonon, has been established as an alternative mechanism both experimentally and theoretically during the last decade [15][16][17]. The mechanism is less dissipative and reduces parasitic electronic effects compared to impulsive stimulated Raman scattering due to the lower energy of the excitation at mid-infrared wavelengths compared with visible light [17,18]. The first demonstration of an excitation of a magnon through coherently-driven phonons [19] has been proposed to result from time-reversal symmetry breaking by the elliptically polarized phonons [19,20]. Coupling of coherent phonons to magnetic order has otherwise been discussed mostly in the context of transient changes of the crystal structure, however [21][22][23][24].
Here, we present a framework to utilize ionic Raman scattering for the coherent excitation of magnons. We show that the magneto-optical and opto-magnetic effects have magneto-phononic and phono-magnetic analogs when we replace the photons by coherently excited infrared-active phonons in the scattering process, as illustrated in Fig. 1. We calculate the corresponding interaction strengths for the paradigmatic antiferromagnet nickel oxide (NiO) from first principles, and we compare the induced effective and real magnetic fields for the different optical and phononic drives. We show that ionic Raman scattering complements impulsive stimulated Raman scattering for the excitation of magnons similarly to the way that it does for Raman-active phonons.

A. Opto-magnetism
We begin by reviewing the classical interaction of light with magnetic matter, which can be written, from the point of view of the opto-magnetic effects, as [3,12] whereε ij = −ε ij /V c , ε ij is the frequency-dependent dielectric tensor, V c is the volume of the unit cell, E i and E * j are the complex electric-field components of light, and the indices i and j denote spatial coordinates. (We use the Einstein notation for summing indices.) In a general antiferromagnet, the dielectric tensor is a complex function of the ferro and antiferromagnetic vectors M = M 1 + M 2 and L = M 1 − M 2 that consist of the sum and the difference of the sublattice magnetizations M 1 and M 2 , respectively. An expansion of ε ij up to second order in M and L yields [3,10,12]: Linear polarization where α ( ) and β ( , ) are the first and second-order magneto-optical coefficients (or magnetic Raman tensors). The different magneto-optical and opto-magnetic effects can be classified according to their role in the expansion of the dielectric tensor, and we show three examples that are relevant for our study in Table I, adapted from Ref. [3].
Magneto-optical effects occur as a result of the modulation of the dielectric function by magnetic order or fluctuations in the material and lead to circular birefringence, as described by the Faraday effect (first order in the magnetic components), and linear birefringence, as described by the Cotton-Mouton effect (second order in the magnetic components).
In the reciprocal opto-magnetic effects, the electricfield component of light acts as an effective magnetic field for the spins in the material, which is given by With an impulsive electric field provided by an ultrashort laser pulse, the effective magnetic field coherently ex-cites magnons via impulsive stimulated Raman scattering [3,8,9,11,12]. In the inverse Faraday effect (first order in the magnetic components), two photons scatter with one magnon, and the light has to be circularly polarized as in Fig. 1(a) in order to transfer angular momentum to or from the spin. In the inverse Cotton-Mouton effect (second order in the magnetic components), two photons scatter with two magnons. In this case the light is linearly polarized as in Fig. 1(c), and the two magnons have angular momenta with equal magnitudes and opposite signs.

B. Phono-magnetism
We now introduce an analogous description for the interaction of spins with coherent lattice vibrations, in which coherently excited infrared-active phonons take the place of photons. The normal coordinate (or amplitude) Q of the phonon then takes the place of the electric-TABLE I. Classification of magneto-optical and opto-magnetic effects, adapted from Ref. [3]. Shown are the interaction Hamiltonians H, the components of the expansion of the dielectric tensor εij, and the effective magnetic fields B eff . Indices i, j, k, and l denote spatial coordinates.

Contribution to
Hamiltonian Magneto-optical effect Opto-magnetic effect field component of the photon, and the interaction can be written, from the point of view of the phono-magnetic effects, as Here, D ij is the projected dynamical matrix that takes the place of the dielectric tensor as the coefficient. It is given by D ij = q T i Dq j , where D is the dynamical matrix and q i/j are the phonon eigenvectors, with the indices i and j denoting the band number of the phonon. The case i = j returns the eigenfrequency of phonon mode i, where amu is the atomic mass unit. The projected dynamical matrix can, analogously to the dielectric tensor, be expanded as a complex function of M and L: where a ( ) and b ( , ) are the first and second-order magneto-phononic coefficients (or magnetic ionic Raman tensors). We classify the different magneto-phononic and phono-magnetic effects according to their role in the expansion of the projected dynamical matrix, and we show three examples that are relevant for our study in Table II.
Magneto-phononic effects occur as a result of the modulation of the interatomic forces, and therefore the dynamical matrix, by the magnetic order and magnetic fluctuations in the material. Different manifestations of the magneto-phononic effects have been described in literature throughout the past century: The first-order effect of the phenomenology is responsible for the well-known Raman process in spin relaxation, where a spin relaxes under the absorption and emission of two phonons [25][26][27][28]. The underlying interaction, called Raman spin-phonon interaction in recent literature [29], is further responsible for the splitting of phonons in a paramagnetic material in an external magnetic field [30][31][32], as well as for the phonon Hall effect [33][34][35]. The second-order effect of the phenomenology describes the ionic contribution to the known magnetodi electric effect [36,37], in which the ionic part of the dielectric function, and therefore the dynamical matrix, is modulated by the magnetic order. Because of the analogy to the magneto-optical effects, these effects have been discussed as the Faraday and Cotton-Mouton effects for phonons in early studies [38][39][40].
We now propose that in the reciprocal phono-magnetic effects the phonons act as an effective magnetic field on the spins in the material, which is given by B eff = ∂H phon /(∂M) or ∂H phon /(∂L). When the phonons are coherently excited by an ultrashort pulse in the terahertz or mid-infrared spectral range, they in turn coherently excite magnons via ionic Raman scattering. The mechanism in first order of the magnetic components is the phonon analog to the inverse Faraday effect, and we therefore simply call it phonon inverse Faraday effect. Two coherently excited phonons scatter with one magnon, and the phonons have to be circularly polarized as in Fig. 1(b) in order to transfer angular momentum to or from the spin. The phonon angular momentum in the time domain is given by (Q iQj − Q jQi ). The mechanism in second order of the magnetic components is the phonon analog to the inverse Cotton-Mouton effect, and we analogously call it phonon inverse Cotton-Mouton effect. Two coherently excited infrared-active phonons scatter with two magnons, where the phonons are linearly polarized as in Fig. 1(d), and the two magnons have angular momenta with equal magnitudes and opposite signs.
The mechanism in the second row of Table II has no opto-magnetic equivalent and we thus discuss it separately in the following. Here, B is a real (not effective) magnetic field that is generated by the orbital magnetic moment of a circularly polarized phonon mode, which can be described within the recently-established dynamical multiferroic effect [20,41,42]. The coefficient g ijk is given by the gyromagnetic ratio of the phonon, not the spin-phonon interaction, and does not require the presence of magnetic order. The phonons can, however, act on (and be acted on by) the spins of a magnetic material through this coupling in an equivalent manner to the Ra-TABLE II. Classification of magneto-phononic and phono-magnetic effects. Shown are the interaction Hamiltonians H, the components of the expansion of the projected dynamical matrix Dij, and the effective and real magnetic fields B eff and B. Indices i and j denote the band index of the phonon modes, indices k, l denote spatial coordinates.

Contribution to
Hamiltonian Magneto-phononic effect Phono-magnetic effect man spin-phonon interaction/phonon inverse Faraday effect discussed in the previous paragraphs. The magnetophononic effect then describes an intrinsic phonon Zeeman effect that is caused by the action of the spins' magnetic field on the dynamical matrix. Reciprocally, the phono-magnetic effect describes the action of the orbital magnetic moment of the phonons on the spins of the material. It shall be noted that the effects described above are valid in the regime of excitations, in which the magnetic phase of the material is maintained, and no lightinduced phase transitions occur [43][44][45].

III. RESULTS FOR NICKEL OXIDE
We turn to evaluating and estimating the strength of the opto-magnetic and proposed phono-magnetic effects for the example of NiO, in which coherent magnons have recently been generated via the inverse Faraday and inverse Cotton-Mouton effects [11,12].

A. Opto-magnetic effects
We evaluate the effective magnetic fields, B eff , acting on the magnetic order through the opto-magnetic effects according to the expressions shown in Table I. We use the experimental geometry of Ref. [12], where pulses with full width at half maximum durations of 90 fs at fluences of 80 mJ/cm 2 , corresponding to a peak electric field of 25 MV/cm, were sent onto a (111)-cut NiO crystal under nearly normal incidence, as sketched in Fig. 2. The pulses with a photon energy of 0.98 eV (well below the ∼4.3 eV bandgap [48]) excited the in-plane magnon mode with a frequency of Ω m /(2π) = 0.14 THz through impulsive stimulated Raman scattering. For this geometry, an excitation of the in-plane magnon mode via the inverse Faraday effect requires the pulse to be circularly polarized in the xy plane, and the corresponding optomagnetic coefficient is α xyz . For an excitation via the inverse Cotton-Mouton effect, the pulse has to be linearly polarized in the xy plane, and the corresponding coefficient is β xyyx . β xyyx in turn is zero by symmetry [12].
We compute the opto-magnetic coefficients at the frequency of the laser pulse ω 0 as described in the computational methods section, and we show their values  α xyz (ω 0 ) and β xyyx (ω 0 ) in Table III. The ratio of the two Raman tensor components L 0 β xyyx (ω 0 )/α xyz (ω 0 ) = 0.02, where L 0 is the equilibrium antiferromagnetic vector, is within the same order of magnitude as the ex-   [33,49] perimental estimate from Ref. [12]. For the electric field component of the pulse, we assume a cosine function embedded in a Gaussian carrier envelope: E(t) = E 0 exp(−(t−t 0 ) 2 /(2(τ /2 √ 2ln2) 2 )) cos(ω 0 t+ϕ CEP ), where E 0 is the peak electric field, ω 0 is the center frequency, ϕ CEP is the carrier-envelope phase (set to zero without loss of generality), and τ is the full width at half maximum duration of the pulse [50]. We show the time and frequency-dependent effective magnetic fields produced by the opto-magnetic effects in Figs. 3(a) and (b). The driving force of the magnon mode is the Fourier component of the effective magnetic field that is resonant with its eigenfrequency. We find the relevant 0.14 THz components of the effective magnetic fields to yield B IFE (Ω m ) = 14 mT and B ICME (Ω m ) = 0.3 mT, see also Table III. Note that while the effective magnetic field determines the induced magnetization, the anisotropy of the system determines how much energy ends up in the elliptical precession of the spins. NiO has an anisotropy factor of A = 400 [12], meaning induced magnetizations in ∆L induce an A times larger spin precession than induced magnetizations in ∆M . Therefore the inverse Cotton-Mouton effect dominates over the inverse Faraday effect, as A × B ICME > B IFE .

B. Phono-magnetic effects
We now evaluate the effective and real magnetic fields, B eff and B, acting on the magnetic order through the phono-magnetic effects according to the expressions shown in Table II, with the same geometry as before. For an excitation of the in-plane magnon mode via the phonon inverse Faraday and dynamical multiferroic effects, the infrared-active B u modes perpendicular to the [111] direction have to be coherently excited with a circularly polarized mid-infrared pulse. The corresponding phono-magnetic coefficients are a ijz and g ijz , where i and j correspond to the nearly degenerate B u modes at 12.06 and 12.08 THz. For an excitation via the phonon inverse Cotton-Mouton effect, one of the infrared-active B u modes has to be excited with a linearly polarized mid-infrared pulse. The corresponding phono-magnetic coefficients are b ijzz and b ijxx , where i and j correspond to the same phonon modes as before.
We compute the coefficients of the phonon inverse Cotton-Mouton effect analogously to the opto-magnetic effects as described in the computational methods section. For the gyromagnetic ratio in the dynamical multiferroic effect, we use a formalism established in previous publications [20,41]. For the phonon inverse Faraday effect, no straightforward first-principles implementation to calculate the imaginary part of the dynamical matrix exists to date, in contrast to the imaginary part of the dielectric tensor that is necessary for the inverse Faraday effect. This process can, in principle, be estimated using crystal field theory with input parameters calculated with quantum chemistry methods, which is beyond the scope of this study. We therefore use an experimentally estimated value of the underlying Raman spin-phonon interaction in dielectrics from Refs. [33,49]. We show the phono-magnetic coefficients a ijz , g ijz , b ijzz , and b ijxx in Table III.
In order to obtain the phonon amplitude Q i , we nu-  merically solve the equation of motion that models the coherent excitation of the infrared-active B u modes by a mid-infrared pulse: Here, Ω i are the eigenfrequencies, κ i the linewidths, and Z i the mode effective charges of the B u modes [51], and E(t) the electric field component of the pulse [50]. In order to generate phonon angular momentum, (Q iQj − Q jQi ), the electric field pulse for Q j is delayed in time by a quarter of the phonon period, t 0 = (4Ω j ) −1 , compared to Q i . We simulate a mid-infrared pulse that is resonant with the phonon modes, ω 0 = Ω i ≈ Ω j , with τ = 2.25 ps and E 0 = 5 MV/cm. This pulse has the same energy as that used to model the opto-magnetic effects. According to the Lindemann stability criterion, a material will melt, when the root-mean square displacements of the atoms exceed 10-20% of the interatomic distance. For the pulse used here, we expect a rootmean square displacement of the oxygen ions of around 0.4Å, which corresponds to 20% of the interatomic distance. Therefore, the values presented here should be seen as an upper limit for the induced magnetic fields. We show the time and frequency-dependent effective and real magnetic fields produced by the phono-magnetic effects in Figs. 3(c) and (d). We find the effective magnetic field components at the frequency of the magnon mode to yield B PIFE (Ω m ) = 9.6 T and B DM (Ω m ) = 2.5 mT, see also Table III. The dynamical multiferroic effect lies in the same order of magnitude as the opto-magnetic effects for comparable pulse energies. Using the estimated common value for the Raman spin-phonon coupling, we calculate the phonon inverse Faraday effect to be three orders of magnitude larger than the dynamical multiferroic and the opto-magnetic effects. The coefficient of the phonon inverse Cotton-Mouton effect for the expansion in L 2 is zero. In contrast, the coefficient for the expansion in M 2 is nonzero, but as the effective magnetic field scales with the equilibrium ferromagnetic vector, M 0 = 0, this effect also vanishes. We therefore expect no second-order phono-magnetic effect for the in-plane magnon mode in NiO.

C. Discussion
We have shown that for comparable pulse energies, the phono-magnetic effects that are based on ionic Raman scattering generate effective and real magnetic fields that are comparable or larger than those generated by the opto-magnetic effects that are based on impulsive stimulated Raman scattering. The pulse frequencies that are required to coherently excite phonons for the phonomagnetic effects lie in the terahertz and mid-infrared region (∼tens of meV), where parasitic electronic excitations are reduced compared to the pulses commonly used to trigger opto-magnetic effects, which use much higher photon energies (∼eV). This makes the phono-magnetic -6. -3.
-1. effects more selective than the opto-magnetic effects and opens the possibility to generate coherent magnons in the electronic ground state in small band gap semiconductors.
The phonon inverse Faraday effect, for which we have used a literature value for the coupling strength, yields the largest effective magnetic field and is the most promising candidate for exploring coherent magnon excitation in the future. In NiO, the estimate dominates over the dynamical multiferroic effect by several orders of magnitude. This suggests that the phonon inverse Faraday effect is the dominant mechanism responsible for the magnon excitation in the first demonstration of coherent phonon-magnon coupling in erbium ferrite (ErFeO 3 ) in Ref. [19], contrary to the originally proposed dynamical multiferroic effect [19,20]. Quantitative calculations of the effect will be necessary in the future in order to confirm this statement and to predict materials in which the effect can be maximized. We see no second-order phono-magnetic effect for antiferromagnetic NiO, in contrast to the well-known second-order opto-magnetic inverse Cotton-Mouton effect, and despite the observation that the frequency of some phonon modes depends on the magnetic order [52]. We expect that ferromagnets or weak ferromagnets, and other antiferromagnets may have the appropriate symmetry for the required coupling to occur.
Finally, we discuss the effect of the pulse duration on the effective magnetic field. We show B(Ω m ) for different values of the full width at half maximum pulse duration τ in Fig. 4, where we adjust the peak electric field E 0 such that the pulse energy remains constant. The data sets for the 0.98 eV and mid-infrared pulses are normalized to the value of the shortest pulse. In the opto-magnetic effects, the 0.14 THz component of the effective magnetic field is nearly independent of τ for the 10 fs to 200 fs range shown here, because Ω m ω 0 can be regarded as an effectively static component of the difference-frequency mixing of the electric field components in the pulse. In the phonomagnetic effects in contrast, the effective magnetic field depends strongly on the pulse duration, because the amplitude of the infrared-active mode depends strongly on it. The coherent excitation of the infrared-active mode is a resonant process and its efficiency increases with increasing driving time, until the phonon decay starts dominating the dynamics. This dependence on the pulse duration makes it possible fine-tune the efficiency of the phono-magnetic effects.

IV. COMPUTATIONAL METHODS
We calculate the opto and phono-magnetic coefficients, phonon eigenvectors, and phonon eigenfrequencies from first-principles using the density functional theory formalism as implemented in the Vienna Ab initio Simulation Package (VASP) [53,54] and the frozen-phonon method as implemented in the Phonopy package [55]. We use the VASP projector-augmented wave (PAW) pseudopotentials with valencies Ni 3p 6 3d 9 4s 1 and O 2s 2 2p 4 and converge the Hellmann-Feynman forces to 25 µeV/Å using a plane-wave energy cut-off of 800 eV and a 8×8×8 gamma-centered k-point mesh to sample the Brillouin zone. We take spin-orbit coupling into account for every calculation. For the exchange-correlation functional, we choose the Perdew-Burke-Ernzerhof revised for solids (PBEsol) form of the generalized gradient approximation (GGA) [56]. We apply an on-site Coulomb interaction of 4 eV on the Ni 3d states that well reproduces both the G-type antiferromagnetic ordering and lattice dynamical properties [57]. The volume of the fully relaxed primitive unit cell is V c = 35.59Å 3 and the magnitude of the equilibrium antiferromagnetic vector projected into Wigner-Seitz spheres L 0 = 3.24 µ B /V c . We find the mode effective charges of the infrared-active B u modes to be 0.84e, where e is the elementary charge, and we use a phenomenological values for the phonon linewidths in oxides of κ i ≈ 0.05 × Ω i /(2π) [58]. We vary the angle of the noncollinear spins of NiO along the directions of the in-plane magnon mode coordinates as displayed in Fig. 2 in increments of 5 • between ±90 • and calculate the frequency-dependent dielectric function ε ij and the projected dynamical matrix D ij for each step. We then fit the functions ε ij (M, L) and D ij (M, L) to quadratic polynomials in M and L as shown in Fig. 5 to obtain the magneto-optical and magneto-phononic coefficients. A similar method has been used to obtain nonlinear phonon-phonon couplings in a recent study [24]. Note that the coefficients for the magneto-dielectric/phonon inverse Cotton-Mouton effects can, in principle, also be obtained by calculating the phonon-amplitude dependent exchange interactions instead, see, e.g., Ref. [22].

V. CONCLUSION
As terahertz and mid-infrared sources are becoming widely available [59], we anticipate that future exper-iments will utilize ionic Raman scattering to generate coherent spin-waves. At the same time, recent improvements in ab-initio spin-lattice dynamics [60][61][62][63][64][65] will accelerate the prediction of phono-magnetic effects in a variety of materials. We believe that our analysis provided in this study makes the first step towards utilizing the physics of coherent optical phononics [18,66] in the field of spintronics [4,5].