Abstract
Ultrafast nonthermal manipulation of magnetization by light relies on either indirect coupling of the electric field component of the light with spins via spinorbit interaction or direct coupling between the magnetic field component and spins. Here we propose a scenario for coupling between the electric field of light and spins via optical modification of the exchange interaction, one of the strongest quantum effects with strength of 10^{3} Tesla. We demonstrate that this isotropic optomagnetic effect, which can be called inverse magnetorefraction, is allowed in a material of any symmetry. Its existence is corroborated by the experimental observation of terahertz emission by spin resonances optically excited in a broad class of iron oxides with a canted spin configuration. From its strength we estimate that a subpicosecond modification of the exchange interaction by laser pulses with fluence of about 1 mJ cm^{−2} acts as a pulsed effective magnetic field of 0.01 Tesla.
Introduction
The symmetric part of the exchange interaction between spins is responsible for the very existence of magnetic ordering^{1}. It is described by the Hamiltonian , where J is the exchange integral; and are the spins of the ith and jth adjacent magnetic ions. The antisymmetric part , characterized by a vector parameter D and called Dzyaloshinskii–Moriya interaction, gives rise to canted antiferromagnetism^{2,3} in iron oxides.
The ability to control the exchange interaction by light has intrigued researchers in many areas of physics, ranging from quantum computing^{4,5,6} to strongly correlated materials^{7,8,9}. Laserinduced heating^{10,11} and photodoping^{9,12} have been suggested to cause a modification of the exchange interaction. However, these phenomena rely on the absorption of light and are neither universal, that is, they are only present in specific materials, nor direct, that is, not instantaneous. Recently the timeresolved evolution of the exchange splitting in magnetic metals Ni and Gd subjected to ultrafast laser excitation was measured using photoelectron spectroscopy^{13} and angleresolved photoemission^{14} techniques, respectively. Both of these techniques, unfortunately, do not allow to distinguish 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 phenomenologically by introducing an isotropic term in the Hamiltonian of the twophoton interaction between the light and spins
where I_{opt} is the intensity of light; α and β are some scalar and vector coefficients, respectively. The presence of the interaction Hamiltonian (1) manifests itself as a magnetic refraction, described by an isotropic contribution to the dielectric permittivity ɛ_{IMR}∼M^{2} that leads to a dependence of the refractive index on the magnitude of the magnetization M^{15,16}. The first term in the Hamiltonian describes the intensity dependent contribution, ΔJ=αI_{opt}, to the symmetric Heisenberg exchange integral J, whereas the second term describes the intensity dependent contribution, ΔD=β I_{opt}, to the Dzyaloshinskii–Moriya vector D. Recently the effect of isotropic magnetorefraction has been used to probe d–f exchange in EuTe^{17}. As for other magnetooptical phenomena, isotropic magnetorefraction must be connected with an inverse effect^{18} described by the same Hamiltonian (1), that is, the optical generation of a torque T_{i} acting on a spin S_{i} due to the lightinduced perturbation of the exchange parameters
where γ is the absolute value of the gyromagnetic ratio. The torque (2) is zero in materials with collinear magnetic configurations since . In contrast to the torques exerted by the optical perturbation of the spinorbit interaction^{19,20,21} or transient magnetic field^{22,23}, it is independent of the light polarization.
In a broad class of transition metal oxides the magnetic order is governed by indirect exchange via ligand ions (superexchange)^{1} and is defined by virtual chargetransfer transitions of electrons between ligands and magnetic ions. Hence, one can anticipate the feasibility of a direct effect of the electric field of light on the exchange energy via virtual or real excitation of specific optical transitions that modify the hopping of the electrons between electronic orbitals centred at the transition metal ions and oxygen ligands, respectively.
Antiferromagnetic iron oxides possessing weak ferromagnetism, such as iron borate FeBO_{3}, rareearth orthoferrites RFeO_{3} (R stands for a rareearth element) and hematite αFe_{2}O_{3}, are natural candidates for observing such ultrafast optical modification of the superexchange interactions. In these compounds the Fe^{3+} ions (spin quantum number S=5/2 and orbital momentum quantum number L=0) form two magnetic sublattices, the spins of which are antiferromagnetically coupled^{24}. The presence of the Dzyaloshinskii–Moriya antisymmetric exchange interaction leads to a slight canting of the spins from the antiparallel orientation by an angle of ∼0.5–1°. The value of the canting is defined by the ratio D/J between the antisymmetric and symmetric exchange parameters. Thus, one could expect that an ultrafast optical perturbation of the exchange parameters could also change the ratio D/J and thereby trigger, by the torque defined in equation (2), the socalled quasiantiferromagnetic resonance mode. This mode corresponds to oscillations of the magnitude of the weak magnetic moment without a change of its orientation^{25}. According to equations (1) and (2), the ultrafast optical perturbation of the exchange parameters in these weak ferromagnets is an isotropic mechanism, that is, it can excite the quasiantiferromagnetic resonance independently from the light polarization and propagation direction. The excited oscillating magnetic dipole in turn will lead to the generation of terahertz (THz) radiation which can be measured using terahertz emission spectroscopy^{26}, as has been demonstrated before in experiments with ferromagnetic metals^{27,28,29} and antiferromagnetic insulators NiO^{30,31,32,33} and MnO^{34}. In the present context, observation of THz emission due to laser excitation of the quasiantiferromagnetic spin resonance via an isotropic mechanism would indicate an ultrafast manipulation of the exchange interactions. Importantly, to observe the THz radiation the emitting dipole must lie in the plane of the sample and therefore be perpendicular to the propagation direction of light^{26}.
Here we reveal the inverse magnetorefractive effect to be responsible for ultrafast modulation of the superexchange interaction in a very broad class of canted antiferromagnets. Our findings are supported by a lowenergy theory for the magnetic interactions between nonequilibrium electrons subjected to an external timedependent electric field. We present quantitative estimates of the strength and timescale of the optical perturbation of the exchange parameters.
Results
THz emission from weak ferromagnets
Recently we reported measurements of THz emission signals in the rareearth orthoferrites^{26} TmFeO_{3} and ErFeO_{3} which revealed the optical excitation of the highfrequency quasiantiferromagnetic mode in these compounds along with the lowfrequency quasiferromagnetic mode, another form of the antiferromagnetic resonance in canted antiferromagnets which involves the precession of the magnetization with no change in its length^{25}. We also observed unexpected weak modes at ∼0.3 THz and assigned them to paramagnetic impurities^{26}. The measurements suggested that the quasiantiferromagnetic mode must be excited via a polarizationindependent mechanism of coupling between light and spins. However, the data were not sufficient to identify the exact nature of the optomagnetic excitation, in general, and to relate it to an optical perturbation of the exchange parameters D/J via inverse magnetorefractive effect, in particular.
To demonstrate the existence of the inverse magnetorefractive effect described above, and in particular the polarizationindependent ultrafast optical perturbation of the exchange parameters D/J, we have studied the THz emission from a single FeBO_{3} cut perpendicularly to the zcrystallographic axis so that it lacks significant inplane anisotropy of both optical and magnetic properties. The magnetization lying in the plane of the sample was aligned horizontally by a constant bias magnetic field of ∼0.1 T. The sample was illuminated by ∼100fs laser pulses with their photon energy centred at 1.55 eV. We performed timeresolved detection of the THz radiation emitted from the sample in the direction of the z axis (see Fig. 1a). The waveforms generated at different temperatures are shown in Fig. 1b. We observe that the optical excitation of the sample leads to quasimonochromatic emission at a frequency of ∼0.45 THz (Fig. 1c), which corresponds to the frequency of the quasiantiferromagnetic mode in FeBO_{3} (ref. 35). The amplitude of the oscillations gradually decreases as the temperature approaches the Néel temperature T_{N}∼350 K (see Supplementary Fig. 1).
To confirm that a similar mechanism is also present in other weak ferromagnets, we performed more detailed measurements of THz emission from orthoferrites similar to those reported in ref. 26, but for a temperature range in which only the quasiantiferromagnetic mode was excited, making the interpretation of the experimental data less complex. Fig. 1e,d demonstrate that below 55 K the TmFeO_{3} single crystal plate cut perpendicularly to the zcrystallographic axis emits radiation with only one spectral component at the frequency of ∼0.8 THz, which is the frequency of the quasiantiferromagnetic mode^{25} in TmFeO_{3} (see also Supplementary Fig. 1). To check that the observed effect is not due to the specific electronic structure of Tm^{3+} ions, we have performed similar experiments on the YFeO_{3} single crystal cut perpendicularly to the xcrystallographic axis (see Fig. 2). Fig. 2 shows that using an ultrafast optical excitation we are able to excite oscillations at a frequency of ∼0.55 THz, which again corresponds to the frequency of the quasiantiferromagnetic mode in YFeO_{3} (ref. 25) (see also Supplementary Fig. 2). We have also observed similar polarizationinsensitive ultrafast optical excitation of the quasiantiferromagnetic mode in y and xcut samples of ErFeO_{3} (see Fig. 3), xcut and ycut DyFeO_{3} and in hematite αFe_{2}O_{3} (see Supplementary Figs 3 and 4; Supplementary Note 1).
Properties of the THz emission
To determine if the excitation mechanism is isotropic, we performed a set of measurements to systematically investigate its dependence on fluence and polarization of the laser pulse and found that the oscillation amplitudes depend linearly on the intensity of the pump (see Supplementary Figs 1 and 2) and are insensitive to the pump polarization (see Supplementary Fig. 5). By comparing the signals generated in the crystals pumped along different crystallographic directions, such as y and x axes in ErFeO_{3} (shown in Fig. 3) one can see that the excitation mechanism is isotropic with respect to the pump propagation direction as well. The phase of the measured oscillations changed by π with the reversal of the magnetization direction, confirming the magnetic origin of the signals (see Fig. 4). Moreover, this shows that the direction of the lightinduced torque exciting the quasiantiferromagnetic oscillations is determined by the orientation of the spins, and not by the polarization of light. All these observations are in perfect qualitative agreement with the anticipated features of an isotropic mechanism of optical modification of the exchange interaction described by equation (1).
Discussion
The consistent observation of the photoexcitation of the quasiantiferromagnetic mode in a range of compounds clearly indicates that this effect originates from the perturbation of the D/J ratio. The isotropic and polarizationinsensitive character of the excitation rules out mechanisms based on the inverse Faraday effect^{19}, which is sensitive to the ellipticity of the pump, or the inverse Cotton–Mouton effect^{36}, which is sensitive to the polarization direction of the pump relative to the magnetization direction. We note that the THz emission observed from the antiferromagnets NiO (refs 30, 31, 32, 33) and MnO (ref. 34) did not contain a contribution that was isotropic relative to the pump polarization. Indeed the Dzyaloshinskii–Moriya antisymmetric exchange interaction is not allowed in these cubic insulators NiO and MnO and in the absence of an external magnetic field the torque (2) is equal to zero. Moreover, the observed effect cannot be attributed to the laserinduced change of the magnetocrystalline anisotropy as demonstrated in garnets^{37} and orthoferrites^{38,39} since this mechanism can trigger only the lowfrequency quasiferromagnetic mode. This conclusion is further corroborated by the observation of this effect in FeBO_{3}, which lacks significant inplane anisotropy.
We would like to note that our demonstration of an ultrafast change of the ratio between the exchange parameters is based on the observation of the femtosecond excitation of the quasiantiferromagnetic mode of spin resonance. Despite several optical pump–probe spectroscopy experiments on femtosecond laser excitation of spins in the orthoferrites and iron borate, the optical excitation of the quasiantiferromagnetic mode has been very rarely observed. The very first observation of ultrafast laser excitation of both quasiferromagnetic and quasiantiferromagnetic modes was reported for DyFeO_{3} in ref. 19 and later confirmed by Satoh et al.^{40} It was found, however, that for the chosen crystallographic orientation of the crystals the mechanisms of the excitation were dominated by the polarization dependent inverse Faraday and inverse Cotton–Mouton effects. Due to the fact that DyFeO_{3} was a strongly anisotropic material, discerning the helicity independent contribution from the data were not possible. Later studies only revealed the possibility of femtosecond helicity dependent excitation of the quasiferromagnetic mode in TmFeO_{3} (ref. 39), HoFeO_{3} (ref. 41), FeBO_{3} (ref. 36), ErFeO_{3} (ref. 42), and SmPrFeO_{3} (ref. 43). As a result of laserinduced heating and a subsequent spinreorientation phase transition, an ultrafast excitation of again the quasiferromagnetic mode was reported for TmFeO_{3} (ref. 38), ErFeO_{3} (ref. 42) and SmFeO_{3} (ref. 44). No optically induced spin dynamics was reported for hematite.
The main reason why the isotropic, polarizationindependent effect, reported here has not been observed before is that the detection in the aforementioned experiments was based on the magnetooptical Faraday effect which probes the spins indirectly that is, it strongly relies on the magnetooptical susceptibility and does not provide a direct picture of spin dynamics. Using THz emission spectroscopy, which is a more direct probe of the oscillating magnetization^{26}, we have been able to identify the isotropic contribution to the optical excitation of the quasiantiferromagnetic spin resonance, which is the principal result of this paper. We also point out that the excitation of the quasiantiferromagnetic mode via the inverse Faraday effect is possible only in samples with the magnetization pointing outofplane. However in this geometry the THz waves cannot be emitted from the sample, hence we do not observe inverse Faradaylike effects in our THz signals.
To specify the possible optical transitions responsible for our observations, we note that the dispersion of the refraction coefficient for all these compounds is dominated by the offresonant susceptibilities related to the electric dipole allowed chargetransfer transitions between the 2p orbitals of oxygen and the 3d orbitals of the Fe^{3+} ions above 3 eV (refs 45, 46, 47). During the laser pulse duration and the time of optical decoherence, the collective electron wavefunctions are coherent superpositions of the wavefunctions of the ground and excited states. Such ultrafast modification of the wavefunctions affects the exchange interaction between the spins of the neighbouring Fe^{3+} ions and thus changes the energy of the superexchange interaction (see Fig. 5). 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 superexchange. 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 equation (2).
Our data are in excellent agreement with the phenomenology of equation (1) that gives the simplest and most plausible explanation. A possible microscopic scenario underpinning the phenomenology of our results can be understood in the framework of a recently developed formalism^{48} for microscopic magnetic interactions out of equilibrium (see Methods sections and Supplementary Note 2). To demonstrate the effect of a femtosecond laser pulse on the superexchange interaction we numerically evaluated the timedependent exchange for a 3ion Fe^{3+}–O^{2−}–Fe^{3+} cluster, which is characterized by a strong onsite Coulomb interaction U on the Fe^{3+} ions, an energy level shift Δ between the Fe^{3+} and O^{2−} ions, and an equilibrium hopping amplitude t_{0} between Fe and O ions. For a small ratio t_{0}/U, the leadingorder expression for the equilibrium superexchange in this system reads^{49} , where U_{1}=U+Δ. By gradually switching on an oscillating offresonant electric field we observe an enhancement of the exchange interaction proportional to the intensity of the laser pulse (see Supplementary Figs 6 and 7; Supplementary Note 2). To further understand the dependence of the superexchange 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 Floquet theory^{50} (see Supplementary Note 2), which gives an analytical expression for the change of the exchange interaction:
Here, ɛ=eaE_{0}/ħω is the amplitude of the vector potential that describes the electric field in the Coulomb gauge with amplitude E_{0,} e and a are the unit charge and lattice constant, respectively, and ω is the frequency of the optical field. The terms dependent on±ω are the photonassisted charge transfer excitations, while the last two terms describe a laserinduced decrease of the effective hopping amplitude within the Fe^{3+}–O^{2−}–Fe^{3+} cluster by a coherent destruction of tunnelling^{51}. We obtain excellent quantitative agreement of ΔJ/J between equation (3) and the numerical results obtained from the general theory (see Supplementary Note 2). In the experiment we typically have ħω∼U_{1}/2, from which we conclude that the strengthening of the exchange interaction is caused by a photonassisted chargetransfer excitation, as illustrated in Fig. 5. Using typical experimental parameters U=3 eV, Δ=0.25 eV, t_{0}=0.5 eV and ħω=1.5 eV, we find that an optical pulse with a fluence of 1 mJ cm^{−2} and a corresponding electric field amplitude E_{0}=0.12 V Å^{−1} should induce an increase of the exchange integral ΔJ/J of over 1%. Our model analysis neither incorporates multiorbital effects nor a description of the nonequilibrium Dzyaloshinskii–Moriya interaction, which certainly would be beyond the scope of this report. Importantly, we have shown theoretically that the optical manipulation of magnetic interactions is feasible already in the elementary superexchange model defined by the Fe–O–Fe cluster.
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 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 quasiferromagnetic mode (∼100 GHz)^{38,39}. As discussed in ref. 26 in the range between 55 and 68 K, such lowfrequency oscillations corresponding to the quasiferromagnetic mode are observed in THz emission spectra together with the highfrequency quasiantiferromagnetic oscillations (see Fig. 6). We applied a low pass filter to the data (cutoff frequency 250 GHz) to isolate the quasiferromagnetic mode and a highfrequency filter (cutoff frequency 650 GHz) to isolate the quasiantiferromagnetic mode. Such a choice of the cutoffs ensures the filtering out of the impurity modes which complicate the dynamics^{26}. It is seen from Fig. 6 that the highfrequency mode measured at 60 K is in phase with that observed at 40 K. One can also see that the initial phases of the lowfrequency quasiferromagnetic and highfrequency quasiantiferrimagnetic modes are ∼180° apart. Note that for the zcut TmFeO_{3} sample, with a net magnetic moment oriented upwards, a laserinduced spinreorientation transition should trigger the quasiferromagnetic mode in such a way that the M_{x} component of the magnetization decreases. The observed difference in the phases between the two oscillations shows that the quasiantiferromagnetic mode is triggered in such a way that the M_{x} component increases, which means that the canting angle becomes larger. Such a behaviour can only be explained by assuming that the quasiantiferromagnetic oscillations are triggered by an increase of the ratio of the exchange parameters D/J. If this conclusion is true, in the xcut sample the initial phases of the two modes must be the same, since the spin reorientation in this sample proceeds in the opposite direction. Measurements in the vicinity of the spinreorientation temperature in ErFeO_{3} cut perpendicular to the x axis confirm this conclusion (see Supplementary Fig. 8; Supplementary Note 3). Interestingly, the increase of the ratio D/J cannot be explained on the basis of the simplistic model defined by the Fe–O–Fe cluster that predicts an increase of J and does not evaluate the change of D. However, the calculation of ΔJ demonstrates the plausibility of the proposed mechanism of optical manipulation of the symmetric exchange interaction in principle.
To deduce the magnitude and timescale of the exchange modification from the experimental data, we have solved the Maxwell equations for a slab of a material with an oscillating magnetization triggered by a perturbation of the ratio D/J and calculated the electromagnetic radiation emitted by the slab into the free space. A quantitative analysis supports the subpicosecond impact on the spin system (see Supplementary Fig. 9; Supplementary Notes 4 and 5). The absence of any significant broadband THz emission, which must accompany a laserinduced ultrafast demagnetization^{27,28} in iron borate and the orthoferrites (Fig. 1), supports the claim that femtosecond changes of the net magnetic moment can be neglected. The fact that the observed spin dynamics do not arise from the laserinduced heating is evidenced by the absence of a correlation between the strength of the observed signals and the specific heat and thermal conductivity of the studied materials. For example, the specific heat of YFeO_{3} below 100 K grows rapidly as the temperature increases while its thermal conductivity exhibits a pronounced peak around 30 K (ref. 52). At the same time the efficiency of the quasiantiferromagnetic mode excitation in this compound does not depend on temperature at all (see Supplementary Fig. 2a). The observation of the very same effect of comparable strength in hematite with high optical absorption ∼2,000 cm^{−1} at 1.55 eV (ref. 53), in the orthoferrites with moderate optical absorption ∼200 cm^{−1} at 1.55 eV (ref. 45) and in virtually transparent iron borate with absorption <100 cm^{−1} at 1.55 eV (ref. 54) shows that the optical modification of the D/J does not rely on laser heating due to optical absorption.
The maximum value of the oscillating magnetization in the samples is estimated to be ∼1 A m^{−1}. Oscillations with such an amplitude can only be triggered if the laser excitation results in an ultrafast increase of the ratio D/J by >0.01% (see Supplementary Notes 4 and 5). Taking into account the parameters of our experiment, one can find that the subpicosecond laser excitation with a fluence of ∼1 mJ cm^{−2} changes the potential energy of the magnetic system by ∼1 μJ cm^{−2} and acts as an effective magnetic field pulse of ∼0.01 T (see Supplementary Note 6). These values (normalized to the optical fluence) correspond to some of the largest effects of light on magnetic systems observed to date^{19,22}.
To summarize, the demonstrated feasibility of a subpicosecond modification of the fundamental exchange parameters J and D and the ratio between them opens wide 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 materials isotropic magnetorefraction can be significantly larger than that in iron oxides, we foresee many opportunities to enhance the effects reported here. Finally, we anticipate that by tuning the wavelength of light, one should be able to affect selectively different exchange parameters in magnetic materials.
Methods
Samples
The crystals used in the present study were grown by floating zone melting (orthoferrites) and from the gas phase (iron borate and hematite) The orthoferrite samples were 60–100μm thick and cut perpendicularly to the z axis (TmFeO_{3}), the x axis (YFeO_{3}, ErFeO_{3}) and the y axis, (ErFeO_{3}, DyFeO_{3}). The iron borate FeBO_{3} sample (370μm thick) and haematite αFe_{2}O_{3} sample (500μm thick) were cut perpendicularly to the z axis. The lateral size of all plates was ∼5 mm.
Terahertz spectrometer
A conventional timedomain THz spectrometer was used in the measurements. The THz spectrometer was powered by a Ti:sapphire amplified laser, emitting a sequence of optical pulses (800 nm wavelength, 100 fs duration) with the repetition frequency of 1 kHz. Each laser pulse was divided into a stronger pump pulse and a weaker probe pulse. The pump spot size was larger than the aperture in the sample holder (∼2 mm in diameter) to provide a quasiuniform excitation with a fluence of ∼1 mJ cm^{−2}. The electric field of the emitted THz wave was measured by the electrooptical sampling technique. The sample was held inside a closed cycle, helium cryostat (15–300 K, 10^{−4} mbar).
Theory of nonequilibrium exchange interactions
We use a general formalism in which magnetic interactions are obtained from a purely electronic model by introducing small timedependent rotations of the spin quantization axes as was recently described in (ref. 48). For the general nonequilibrium case the evolution of the electronic model is described using the Schwinger–Keldysh/Kadanoff–Baym nonequilibrium action and partition function, with the effective action written in terms of Grassmann fields. By integrating over the electronic degrees of freedom in the rotated reference frame, an effective quadratic spin model is obtained in which the timedependent exchange interaction parameters are identified from the mapping of the effective action to a timedependent classical Heisenberg model. The resulting expressions turn out to be combinations of nonequilibrium electronic Green’s functions and selfenergies, which have to be evaluated numerically to assess the modification of exchange interaction by timedependent perturbations of the electron model.
This method is implemented for the simplest model system that exhibits the physics of superexchange, which consists of a chain of three atoms, labelled as 0, 1 and 2. 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 H_{loc} and a timedependent hopping term H′(t):
Here creates an electron with spin σ={↑,↓} at site j, and is the number operator. The parameters ɛ_{d} and ɛ_{p} are the orbital energies of d and p orbitals, respectively, and U is the local (Hubbard) interaction energy associated with d orbitals. H′(t) accounts for hopping between p and d orbitals; t_{0} is the equilibrium hopping parameter, while ϕ(t) is the timedependent Peierls phase, which absorbs the effect of the timedependent electric field. In the Coulomb gauge and for a spatially uniform vector potential the Peierls phase is given by
where A_{}(t) is the component of the vector potential parallel to the chain and a is the lattice spacing. In the chosen gauge, the electric field is related to the vector potential as . In the experimentally relevant regime the model is characterized by U, U+ɛ_{d}−ɛ_{p}>>t_{0} and a total filling of four electrons. To compute the nonequilibrium functions, this cluster model is solved numerically using exact diagonalization of the timedependent Schrödinger equation. The electric field is taken as the product of an oscillatory component and a gradually changing envelope function with a rising time in the order of ∼10 periods of oscillation. The frequency ω of the oscillating part is below the chargetransfer gap, which prevents direct chargetransfer transitions, consistently with the experimental conditions.
Further details related to the theoretical and numerical methods are discussed in Supplementary Note 2.
Additional information
How to cite this article: Mikhaylovskiy, R. V. et al. Ultrafast optical modification of exchange interactions in iron oxides. Nat. Commun. 6:8190 doi: 10.1038/ncomms9190 (2015).
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Acknowledgements
The research leading to these results was partially funded from the European Commission’s 7th Framework Program (FP7/2007–2013) under Grant Agreement 228673 (MAGNONICS) and from EPSRC of the UK under project EP/E055087/1. The work was partially supported by the Netherlands Organization for Scientific Research (NWO), the Foundation for Fundamental Research on Matter (FOM), the European Union’s Seventh Framework Program (FP7/2007–2013) grants no. NMP3LA2010246102 (IFOX), no. 280555 (GoFast), no. 214810 (FANTOMAS), the European Research Council under the European Union’s Seventh Framework Program (FP7/2007–2013)/ERC Grant Agreement No. 257280 (Femtomagnetism) and ERC Grant Agreement 339813(EXCHANGE). The funding support by the Russian Ministry of Education and Science was realized via the Program ‘Leading Scientist’ Program (Projects 14.B25.31.0025 and 14.750.31.0034). R.V.P. and A.W. acknowledge the support of the joint RFBRNSFC project (No. 120291172a, 51211120184, 51511130037) and Bureau of International Cooperation, CAS. A. W. thanks for the support of NSFC project (No. 51372257, 51572275) and NSFCNWO joint project (No. 51211130596). A.S. and M.I.K. acknowledge support from EU Seventh Framework Program grant agreement No. 281043 (FEMTOSPIN). J.H.M. acknowledges funding from NWO by a Rubicon grant. E.H. is supported by the European Commission (FP7ICT2013613024–GRASP) and EPSRC fellowship (EP/K041215/1).
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R.V.M., V.V.K. and A.V.K. conceived the project. E.H. and R.V.M. designed and built the experimental set up used to study TmFeO_{3} and YFeO_{3} samples at the University of Exeter. R.V.M. designed and built the experimental set up at the Radboud University Nijmegen to study the ErFeO_{3}, DyFeO_{3}, FeBO_{3} and αFe_{2}O_{3} samples. R.V.M. performed all measurements, analysed the data and developed the macroscopic theoretical formalism. A.S. and M.I.K. derived the quantum theory of nonequilibrium exchange. J.M. and M.E. derived the Floquet theory and performed numerical simulations of nonequilibrium exchange. R.V.P. and A.W. prepared the samples. R.V.M. and A.V.K cowrote the paper with substantial contributions from E.H., V.V.K., R.V.P., A.S., J.M., M.E., M.I.K. and Th.R. The project was coordinated by A.V.K.
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Mikhaylovskiy, R., Hendry, E., Secchi, A. et al. Ultrafast optical modification of exchange interactions in iron oxides. Nat Commun 6, 8190 (2015). https://doi.org/10.1038/ncomms9190
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DOI: https://doi.org/10.1038/ncomms9190
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