Inverse magneto-refraction as a mechanism for laser modification of spin-spin exchange parameters and subsequent terahertz emission from iron oxides

Ultrafast non-thermal manipulation of magnetization by light relies on either indirect coupling of the electric field component of the light with spins via spin-orbit interaction or direct coupling between the magnetic field component and spins. Here we propose a novel scenario for coupling between the electric field of light and spins via optical modification of the exchange interaction, one of the strongest quantum effects, the strength of which can reach 1000 Tesla. We demonstrate that this isotropic opto-magnetic effect, which can be called the inverse magneto-refraction, is allowed in a material of any symmetry. Its existence is corroborated by the experimental observation of THz emission by magnetic-dipole active spin resonances optically excited in a broad class of iron oxides with a canted spin configuration. From its strength we estimate that a sub-picosecond laser pulse with a moderate fluence of ~ 1 mJ/cm^2 acts as a pulsed effective magnetic field of 0.01 Tesla, arising from the optically perturbed balance between the exchange parameters. Our findings are supported by a low-energy theory for the microscopic magnetic interactions between non-equilibrium electrons subjected to an optical field which suggests a possibility to modify the exchange interactions by light over 1 %.

heating [18][19] and photo-doping 17,20 were suggested to cause a modification of the exchange interaction; however, these phenomena rely on absorption of light and are neither universal, i.e. only present in rather specific materials, nor direct, i.e. not instantaneous. The time-resolved evolution of the exchange splitting in magnetic metals Ni and Gd subjected to ultrafast laser excitation was measured using photoelectron spectroscopy 21 and angle-resolved photoemission techniques respectively. Both of these techniques, unfortunately, do not allow distinguishing the intrinsic dynamics of the exchange parameters such as J from the demagnetization dynamics. Nevertheless, a direct truly ultrafast effect of the electric field of light on the exchange interaction must be feasible in any material. In a medium of arbitrary symmetry, such an effect may be expressed by introducing an isotropic term in the Hamiltonian of the two-photon interaction between the light and spin system where I opt is the intensity of light; α and β are some scalar and vector coefficients, respectively, which are defined by microscopic parameters. The presence of the interaction Hamiltonian (1) manifests itself as a magnetic refraction, described by an isotropic contribution to the dielectric permittivity 2 IMR aM   that leads to a dependence of the refractive index on the magnetization M [23][24] . The first term in the Hamiltonian describes the intensity dependent contribution, opt I J    , to the symmetric Heisenberg exchange integral J, whereas the second term describes the intensity dependent contribution opt I β D   to the Dzyaloshinskii-Moriya vector D 6 . Recently the isotropic magneto-refraction effect has been utilized to probe the d-f exchange in EuTe 25 . As all other magneto-optical phenomena, the magneto-refraction must be connected with an inverse effect 26 described by the same Hamiltonian (1), i.e. the optical generation of a torque T i acting on a spin S i due to the light-induced perturbation of the exchange parameters where γ is the absolute value of the gyromagnetic ratio.
In a broad class of transition metal oxides the magnetic order is governed by indirect exchange via ligand ions (super-exchange) 5  oscillations of the magnitude of the weak magnetic moment without a change of its orientation 28 .
According to Eqs. (1) and (2), one expects that the ultrafast optical perturbation of the exchange parameters in these weak ferromagnets is an isotropic mechanism, i.e. it can excite the quasiantiferromagnetic resonance independently from the light polarization and propagation direction.
The excited oscillating magnetic dipole in turn leads to the generation of THz radiation which can be measured using THz emission spectroscopy 29 , as has been demonstrated before in experiments with ferromagnetic metals [30][31][32] and antiferromagnetic insulators NiO [33][34][35][36] and MnO 37 . In the present context, observation of THz emission due to laser excitation of the quasi-antiferromagnetic spin resonance via an isotropic mechanism would indicate an ultrafast manipulation of the exchange interactions.
In our experiments we studied a FeBO 3 single crystal plate cut perpendicularly to the zcrystallographic axis. The sample was illuminated by ~100 fs laser pulses with their carrier frequency centered at 1.55 eV (800 nm). We performed time-resolved detection of the THz radiation emitted from the sample in the direction of the z-axis (see Fig. 2 (a)). The waveforms generated at different temperatures are shown in Fig. 2 Figure 2 (e) demonstrates that, in the latter case, the sample emitted radiation at the frequency of ~0.8 THz, which is the frequency of the quasi-antiferromagnetic mode 28 in TmFeO 3 .
In order to investigate the isotropic character of the excitation mechanism, we performed systematic measurements on the fluence and polarization dependence and found that the oscillation amplitudes 6 depend linearly on the intensity of the pump (see Supplementary note 1) and are insensitive to the pump polarization (see Supplementary note 2). These observations are in perfect qualitative agreement with the anticipated features of an isotropic mechanism of optical modification of the exchange interaction described by Eq. (1). We have also observed similar polarization-insensitive ultrafast optical excitation of the quasi-antiferromagnetic mode in the y-and x-cut samples of ErFeO 3 , the x-cut YFeO 3 , and in hematite α-Fe 2 O 3 (see Supplementary note 3). Importantly, in all these compounds the effect was clearly seen even at room temperature. The phase of the observed oscillations changed over π with the reversal of the magnetization direction, confirming the magnetic origin of the signals (see Supplementary note 4).
The consistent observation of the photo-excitation of the quasi-antiferromagnetic mode in a range of compounds clearly indicates that this effect originates from the perturbation of the D/J ratio. Indeed, the observation of this effect in FeBO 3 , which lacks significant in-plane anisotropy, rules out any substantial contribution from an optical modification of the magneto-crystalline anisotropy. At the same time, the isotropic and polarization insensitive character of the excitation rules out mechanisms based upon the inverse Faraday effect 1 which is sensitive to the ellipticity of the pump or the inverse Cotton-Mouton effect 39 which is sensitive to the polarization of the pump with respect to the magnetization direction. It is important to note here that the THz emission observed from antiferromagnets NiO [33][34][35][36] and MnO 37 did not contain a contribution isotropic with respect to the pump. Indeed the Dzyaloshinskii-Moriya antisymmetric exchange interaction is not allowed in these cubic insulators NiO and MnO and the torque (2) is equal to zero in accord with our model. In order to specify the possible optical transitions responsible for our observations, we note that the excitation photon energy is not in resonance with any of the weak localized d-d crystal-field transitions in the Fe 3+ ions in FeBO 3 and orthoferrites [40][41] . However, the dispersion of the refraction coefficient for all these compounds is dominated by the off-resonant susceptibilities related to the electric-dipole allowed charge-transfer transitions between the 2p orbitals of oxygen and the 3d orbitals of the Fe 3+ ions above 3 eV [42][43] . During the laser-pulse duration and the time of optical decoherence, the collective electron wave-functions are coherent superpositions of the wavefunctions of the ground and excited states. Such ultrafast modification of the wave-functions affects the exchange interaction between the spins of the neighbouring Fe 3+ ions and thus changes the energy of the super-exchange interaction (see Fig. 1). One can therefore expect that the observed effect of light on the exchange interaction is inherent to all magnetic materials, the magnetic order of which is governed by the super-exchange. However, only when the spins are canted either by the Dzyaloshinskii-Moriya interaction or by an applied magnetic field, such an ultrafast change of the exchange interaction will lead to excitation of the antiferromagnetic resonance and the subsequent emission of THz radiation in accord with Eq. (2). In materials with collinear magnetic configurations the torque (2) . By gradually switching on an oscillating off-resonant electric field we observe an enhancement of the exchange interaction proportional to the intensity of the laser pulse (see Supplementary note 6, Fig. S11). To further understand the dependence of the super-exchange on the laser field, we studied analytically a periodically driven cluster model. The shift of the energy levels under the periodic driving field can be understood within the Floquet theory (see Supplementary note 6), which gives an analytical expression for the change of the exchange interaction: Here, To determine whether laser excitation leads to a decrease or an increase of the ratio D/J we take advantage of the strong temperature dependence of the magnetic anisotropy, which is characteristic for many orthoferrites. For instance, heating of TmFeO 3 from 80 K to 90 K leads to a change of the equilibrium orientation of the weak magnetic moment from the x to the z-axis. If the equilibrium orientation is changed as a result of a sudden heating by a femtosecond laser pulse, such a change is followed by oscillations of the weak magnetic moment in the (xz)-plane at the frequency of the quasi-ferromagnetic mode (~ 100 GHz) 47 . Our measurements clearly reveal that, in the range between 55 K and 68 K, such low-frequency oscillations corresponding to the quasi-ferromagnetic mode are observed in addition to the high-frequency quasi-antiferromagnetic oscillations (see Fig. 3).

 
We applied a low pass filter to the data (cut-off frequency 250 GHz) to isolate the quasiferromagnetic mode and a high frequency filter (cut-off frequency 650 GHz) to isolate the quasiantiferromagnetic mode. It is seen from Fig. 3 that the high-frequency mode measured at 60 K is in phase with that observed at 40 K. One can also see that the initial phases of the low-frequency quasi- To summarize, the demonstrated feasibility of sub-picosecond modification of the fundamental exchange parameters J and D and the ratio between them opens novel prospects for optical control of magnetically ordered materials. The suggested mechanism is not restricted by any requirement on the crystal symmetry and must thus be applicable to other classes of magnetic materials. Given that in some of them the isotropic magneto-refraction can be significantly larger than in iron oxides, we foresee many opportunities to enhance the effects reported here. We also anticipate that by tuning the wavelength of light, one should be able to affect selectively different exchange parameters in magnetic materials.             To further confirm the magnetic nature of the emitted radiation we checked that the phase of the observed oscillations is shifted over π as the magnetization reverses its polarity by the bias magnetic field ±1 kG (Fig. S8). Such a reversal of the sign of the signals proves that the THz oscillations arise from the spin precession. The magnetocrystalline anisotropy of TmFeO 3 and ErFeO 3 is characterized by a strong temperature dependence in the ~ 80 -100 K temperature interval [4]. In this temperature range the spin configuration of the iron sub-lattices continuously rotates in the (xz) plane, while keeping the weak ferromagnetic moment in the same plane. Thus one might anticipate a strong temperature dependence of the THz emission in the vicinity of the spin-reorientation temperature interval. Indeed, along with the quasi-antiferromagnetic mode, another mode at ~ 100 GHz appears in the spectra of emission generated in TmFeO 3 and ErFeO 3 in vicinity of the spinreorientation temperature range. We attribute this second mode to the quasi-ferromagnetic precession of spins.

Samples
The quasi-ferromagnetic resonance excitation under optical excitation of TmFeO 3 and ErFeO 3 near the spin reorientation temperature region has been reported before and assigned to the thermally induced change of the anisotropy [5][6][7]. This picture concurs with the fact that in YFeO 3 and FeBO 3 the low frequency mode has not been observed due to the absence of a spin reorientation in this material.
The analysis of the waveforms generated in TmFeO 3 at the temperatures of the photoinduced spin-reorientation allowed us to determine the absolute sign of the signal as discussed in the main text of the paper. It is instructive to check whether the sign of the quasiantiferromagnetic oscillation generated in another compound exhibiting spin-reorientation, erbium orthoferrite, is consistent with the obtained result. We applied the same analysis to the signals generated in x-cut ErFeO 3 as illustrated in Fig. S9. Importantly, in such an oriented crystal the direction of the spin-reorientation is opposite with respect to the one in the z-cut crystalline plate of TmFeO 3 . Thus, the initial phase of the quasi-antiferromagnetic mode must be the same as the initial phase of the low frequency quasi-ferromagnetic mode [ Fig. S9 (a)]. This prediction has been fully validated by the experimental data, as shown in Fig. S9

Supplementary note 6 -Microscopic theory of non-equilibrium exchange
To demonstrate theoretically the feasibility of the modification of the super-exchange interaction, we adapt a quantum theory [8] that was recently developed to describe nonequilibrium magnetic interactions in strongly correlated systems. In particular, we specialize the application of this framework to a simple cluster model that mimics the experimental system (in particular, α-Fe 2 O 3 ) and we solve this model numerically. Furthermore, to provide additional theoretical understanding, we derive analytical results from Floquet theory for the same cluster model. Both the numerical and analytical results demonstrate an enhancement of the exchange interaction that scales linearly with the intensity of the electric field.

Quantum theory of non-equilibrium magnetic interactions
Our theory exploits the non-equilibrium Green-function formalism developed by Schwinger [9], Keldysh [10], Kadanoff and Baym [11]. The electronic partition function Z is written as a path integral over fermionic (Grassmann) fields of the exponential of a non- (1) The effective action describes the system in equilibrium for t < 0, going out of equilibrium for interactions. The coefficients describing the magnetic interactions are expressed in terms of nonequilibrium electronic Green's functions and related self-energies, which generalize the equilibrium formalism [12] to include the effects of an external time-dependent field.
Assuming that the spin dynamics is slow with respect to electronic hopping processes, we obtain the following formula for the non-equilibrium exchange coupling between sites a and b: where if the ground-state spin correlation function of sites a and b is antiferromagnetic (ferromagnetic), and the quantities to be computed are the following (see Eqs. (107) and (124) in [8]): is obtained from this equation by exchanging > and < in the right-hand side, and for b a  , The above quantities are expressed in terms of the non-equilibrium Green's functions, For the numerical calculations, it is convenient to work out the derivatives analytically, which yields the lesser and greater components of 1 Here we have introduced the quantities is the self-energy, which accounts for the electron correlations.
It must be noticed that the effective exchange interaction in Eq. (2) is not the bare exchange interaction between magnetic moments in sites a and b, since it also includes a term which describes the variation with time of the magnitude of the local magnetic moments (non-Heisenberg effects). It can be shown that, in the absence of symmetry breaking (which is the case we will consider here), the bare exchange parameters are obtained as where     , where ± indicate times on the upper and lower branches of the Keldysh contour, respectively. Note that this is different from the conventional definition of lesser and greater components of Keldysh functions.

Minimal model for super exchange
As minimal model for super-exchange we consider a chain of three atoms denoted 0, 1 and 2 [13,14]. Atoms 0 and 2 correspond to transition metal sites with one partially filled d-orbital and atom 1 contributes one filled (oxygen) p-orbital. The Hamiltonian consists of a local part loc H and a time-dependent hopping term  The time-dependent electric field E(t) is included in ) (t H  (see Eq. 9b) by means of the time-dependent Peierls substitution [15][16][17], which is equivalent to multiplying the equilibrium hopping by a time-dependent phase factor. In the Coulomb gauge (zero scalar potential) and for a spatially uniform vector potential the Peierls phase becomes where   t A || is the component of the vector potential parallel to the chain and a is the lattice spacing. The electric field is then related to the vector potential as     where the amplitude is given in terms of the dimensionless parameter For the numerical solution of the periodically driven cluster model we slowly switch on the Peierls phase using an error function envelope where (for a given ω) the parameters α and t 1 are chosen such that   0 0   and the rise time takes about 10 oscillation periods.

Numerical computation of non-equilibrium exchange parameters
A numerically exact solution of the 3-site super-exchange model out of equilibrium is obtained by solving the time-dependent Schrödinger equation using exact diagonalization. From the time evolution of the states we evaluate the following correlation functions using the Lehmann representation: where C Tˆ is the time-ordering operator along the Keldysh contour and U a is the Coulomb interaction at site a. By computing products like and by combining all terms we obtain the quantity In addition, we evaluate the spin-spin correlation function as For the cluster model we consider a filling with 4 electrons and total spin S z = 0, and prepare the system at t = 0 in the ground state at low temperature.
In equilibrium the exchange interactions are extracted at zero electric field by evolving the system along the real time axis to extract The result is found to be independent on * t in the quasi-equilibrium state formed after slowly ramping up the Peierls phase. conditions. We also note that in the strict limit 0   (which is not relevant for experiments since it requires very long laser pulses) a small negative change ΔJ appears when 0   .
Let us remark that Floquet theory makes predictions that will not be further discussed in the current manuscript. For example, the expression Eq. (24) shows that the effect of the electric field on the exchange interaction is strongly enhanced close to the resonance (where in a solid, however, one has strong absorption). In addition, different effects may occur in the nonperturbative regime, where the coupling to Floquet higher sectors is not negligible. Finally, the current model is clearly just a minimal model for laser-controlled super-exchange. A much richer behavior can be expected if more orbitals are included in the description.

Results
Before analyzing the effect of an electric field out of equilibrium, we study in equilibrium the quality of the general formulas Eq. (8)

Supplementary note 7 -Macroscopic theory of the quasi-antiferromagnetic mode excitation via optical perturbation of the exchange interaction
The equilibrium orientation of the iron spins in canted antiferromagnets is given by the minimum of the thermodynamic potential 2 : where S 1 and S 2 are the vectors that characterize the spins of the iron ions in the two magnetic sublattices, J is the nearest neighbor isotropic exchange interaction constant; D is a constant vector pointing along the y-axis and describing the Dzyaloshinskii-Moriya antisymmetric exchange interaction; K x , K y , K z , K 4 are the constants of the effective anisotropy. The K x , K y , K z , K 4 are purely phenomenological and "effective" parameters in a sense, that they do not necessarily have a clear physical meaning, being a combinations of the single ion anisotropy terms and "hidden" exchange coupling parameters (see Ref. 22 for a detailed discussion).
It is instructive to rewrite the potential (25) Here γ is the absolute value of the gyromagnetic ratio of an electron and Taking into account that in canted antiferromagnets J>>D>>K x , K y , K z >>K 4 [1], the equilibrium magnetic configuration reads Here D/J defines the canting angle which is a small parameter and in the subsequent derivation all terms smaller than the canting angle are neglected.
The spin dynamics is described by the Landau-Lifshitz equations for ferromagnetic and antiferromagnetic vectors 2 The formulae in this file are written in the Gaussian system of units, but the final answers are converted to SI units. 25 , .
The vectors M and L can be represented as a sum of the equilibrium M 0 and L 0 and the timevarying m(t) and l(t) components. In the case of small deviations from equilibrium (m <<M 0 and l << L 0 ) one can obtain a linearized system of equations describing the quasi-antiferromagnetic resonance: Note, that here we restrict ourselves to the terms of first order in smallness with respect to ΔJ, ΔD ,l y , l x and m z . Moreover, since 2 0 2 0

M L 
we can neglect the last term on the right side of Eq. (33). Let us also introduce some phenomenological damping into the equation of motion (33) as where ν is the damping parameter.
Applying the Fourier transformation to Eq. (34) with respect to time t we get where    y l~ is a Fourier transform of l y (t) By rearranging Eq. (35) we obtain The function    m has been already derived above and given by Eq. (38). torque does not overlap with the resonance frequency anymore. Importantly, to keep the amplitude of the excited quasi-antiferromagnetic oscillation of the same amplitude one has to increase the peak amplitude of the torque as its duration becomes longer.
However, even for the case of the shortest torque (100 fs) both calculated and measured signals demonstrate a finite rise-time(~ 1 ps). This effect is due to the fact that the samples act as In FeBO 3 and α-Fe 2 O 3 the period of quasi-antiferromagnetioc oscillation is several times smaller than in the orthoferrites. Therefore, excitation is possible for longer torques. However, it is natural to assume that the mechanism of excitation and its timescale is the same in these materials as in the orthoferrites. states. The short single-cycle THz pulses of magnetic field which are shown to excite spin dynamics can achieve a peak amplitude of 0.1 Tesla [24]. Unfortunately, to generate such a pulse one needs extremely high pump fluences and dedicated laser systems not widely available.
Indeed, the pulses of magnetic field used in Ref. 24 were generated with a help of optical pulses with energy of 5 mJ, that is larger than the total energy of pulses generated in a normal amplified Ti:sapphire laser used in our measurements.
The energy of the interaction between light and magnetic system can be estimated as where V is the optically excited volume of the material (~100 µm×1 mm×1 mm in our measurements). Using Eq. (50) and taking M 0 ≈ 10 emu/cm 3 we estimate the energy ΔW is of the order of 1 µJ per excited area of 1 cm 2 .
Finally we note that the 0.01% change of the ratio of the two exchange constants represents a difference of the relative changes of each of them. This means that if the changes are 33 of the same sign (which is quite likely) then each of them could in fact be much greater than 0.01% in agreement with the prediction of the microscopic theory.