Light-induced magnetization driven by interorbital charge motion in a spin-orbit assisted Mott insulator alpha-RuCl3

In a honeycomb-lattice spin-orbit assisted Mott insulator {\alpha}-RuCl3, an ultrafast magnetization is induced by circularly polarized excitation below the Mott gap. Photo-carriers play an important role, which are generated by turning down the synergy of the on-site Coulomb interaction and the spin-orbit interaction realizing the insulator state. An ultrafast 6- fs measurement of photo-carrier dynamics and a quantum mechanical analysis clarify the mechanism, according to which the magnetization emerges from a coherent charge motion between different t2g orbitals (dyz-dxz-dxy) of Ru3+ ions. This ultrafast magnetization is weakened in the antiferromagnetic (AF) phase, which is opposite to the general tendency that the inverse Faraday effect is larger in AF compounds than in paramagnetic ones. This temperature dependence indicates that the interorbital charge motion is affected by pseudo-spin rotational symmetry breaking in the AF phase.

The upper panel of Fig. 2(a) shows time evolutions of  (probe energy Epr=0.62 eV, ||ab plane) induced by the circularly polarized pump pulse (pump energy Epu=0.89 eV, intensity Iex=4 mJ/cm 2 ) at 4 K (dashed curves, T < TN) and 17 K (solid curves, T> TN). The helicity ( + ,  -) of the pump pulse and the direction of the rotation angle are defined as those in the righthanded system (upper inset of Fig. 1(b)). The helicity-dependent responses as large as 4.5-5.5 degrees within the time resolution (ca. 100 fs) in Fig. 2(a) is attributed to the Faraday rotation reflecting an ultrafast magnetization perpendicular to the ab-plane (⊥ab) at both 4 K and 17 K (Supplementary 2 [21]) [38]. The observation of such large  without any long-range order at 17 K contrasts with the ultrafast inverse-Faraday effect in AF compounds [11][12][13][14]. The transient transmittance (T/T) in the lower panel of Fig. 2(a) shows concomitant charge dynamics with a finite decay time indicating real excitation of in-gap states. In fact,  normalized by the thickness (ca. 50 m) and Iex (4 mJ/cm 2 ) is 20 times larger than that of a prototypical AF magnet NiO (111) (1.14 deg. for Epu=0.97 eV, Epr=1.59 eV, Iex= 10 mJ/cm 2 , thickness 100 m [14]). It is 400 times larger than that of a paramagnet Terbium Gallium Garnet (0.15 deg. for Epu=Epr=1.55 eV, Iex=3.1 mJ/cm 2 , thickness 1 mm [39]).
The temperature dependence of  at time delay (td) of 0 ps in the upper panel of Fig. 2(b) shows an abrupt change near TN＝7 K. The reduction of  (of ca. 15 %) below TN indicates that the ultrafast magnetization (⊥ab) is weakened in the AF phase, which is opposite to the general tendency that the inverse Faraday effect is larger in AF compounds than in paramagnetic ones. 6 Above 20 K,  is insensitive to the temperature (Supplementary 3 [21]).
The helicity-independent slower component of  (x10 for td>1 ps in Fig.   2(a)) is attributed to a melting of the AF order (Supplementary 4 [21]) [40]. In fact,  at 10 ps drops near TN (lower panel of Fig. 2(b)). In contrast to the temperature sensitive nature of , T/T do not indicate any temperature dependence near TN (as shown in the lower panel of Fig. 2(a) is equal to the energy where  has a peak ( peak: orange arrow), and it is ca. 0.9 eV for Epu=0.30 eV, and ca. 0.6 eV for Epu=0.62 eV. The  = 0 intersection and the  peak correspond to the resonance energy Eres in the magneto-optical spectrum  =  2 ( ) and  = −  2 ( ) (xy: offdiagonal component of the permittivity, n: refractive index, and l: sample thickness) for the single resonance oscillator [41]. The red dots in the upper panel of Fig. 2(c) show the maximum values in the respective  spectra (i.e., the excitation spectrum of ) for Epu=0.30-0.89 eV. Since a probe light with energy higher than 0.9 eV is not transmitted, a true maximum cannot be detected for a pump light with or lower than 0.3 eV. Thus, the excitation spectrum of  is underestimated at Epu= 0.3 eV. Even in this limited situation, the tendency of the excitation spectrum in the upper panel of Fig. 2 These results show that the probe pulse further excites the in-gap states during their lifetime (< 100 fs) to a state just above the Mott gap. This lifetime is comparable with the inverse of the bandwidth (100 meV) of each peak in the 2 spectrum ( Fig. 1(b)) and is supported by the results of the ultrafast measurement using a 6 fs pulse (Fig. 3(e)). The large  in -RuCl3 is characteristic of the real excitation below the Mott gap and the 2-step excitation at Epu+Epr which is resonant to the Mott gap.
The insulating state is affected by the excitation below the Mott gap, as shown in the transient reflectivity (R/R) spectra for Epu=0.6 eV ( Fig. 3(b)) and 0.89 eV (Fig. 3 (c)). The steady-state reflectivity (R) is shown in Fig. 3(a) for reference. The bleaching near the Mott-Hubbard transition at 1.  [43][44][45], the oscillation would be related to the intersite hopping processes including the interorbital one. Note that phonon energies are restricted to a range below 40 meV [15][16][17]34]. Such coherent charge motions are often realized by the simultaneous application of a lightfield force to many electrons that are correlated by the Coulomb interaction [46,47]  Here, we theoretically study photo-induced dynamics of the spin and orbital degrees of freedom to analyse the mechanism for the photo-induced magnetization in α-RuCl3. We consider a Hubbard model consisting of dyz, dxz, and dxy orbitals on each site of the honeycomb lattice [43][44][45]     orbitals (such as dxzdyz). b Imaginary part of the permittivity spectrum in mid-and near-infrared regions [31]. The low-energy part below 0.9 eV (magnified 25 times) is attributed to SOC in-gap states [31][32][33]. The experimental configuration of the opto-magneto measurement (polarization rotation induced by a circularly polarized pump pulse) and the excitation from Jeff=3/2 to Jeff=1/2 states [32] are schematically shown.    We have checked that the volume fraction of 3 ̅ is ca. 3/4 and the magnetic transition occurs dominantly at 7 K in our samples by measuring the XRD and magnetic susceptibility. We also check the steady-state polarization rotation angle  to confirm the TN value. Figure S1 As shown in Fig. S1(a), the R shows an abrupt change near TN with increasing temperature where  / indicates a maximum at 7 K ( Fig.   S1(b)). Then, the small remaining component gradually decreases in magnitude toward zero until T=14 K. Thus, the magnetic transition occurs dominantly at TN =7 K in our samples. We have actually chosen the crystal in which the TN=14 K component is the smallest. Considering that the anomaly in R at 7 K is actually confirmed in the steady state, the shift of the temperature where the abrupt change of  occurs in Fig. 2(b) (which is approximately 1 K lower than TN=7 K) is attributed to the heat accumulation by the pump light in 1 kHz operation.
The optical anisotropy in the ab-plane just reflects the sample quality, i.e., the volume fraction of 3 ̅ , and it is small and non-essential for the present study of the photoinduced magnetization and the photocarrier dynamics.
Thus, we do not discuss the dependence of  on the polarization of the probe pulse following the previous reports [31][32][33][34][35]40].  As shown in Fig. S2(b), the similar result is observed also in the AF phase (4 K). It is noteworthy that the coefficient a (4 K) =0.71 is smaller than that of    Fig. 3(c)) is consistent with the results in ref. [42], although the excitation is made by a linearly polarized pump pulse at RT in ref. [42]. The Mott insulating state of this compound is realized by the synergy of the on-site Coulomb interaction U and the SOI [20]. This fact is confirmed within the theoretical framework of this study [Supplementary 10] as described below. Figures S5(a) and S5(b) show calculated optical conductivity spectra for polarizations ||a (Fig. S5(a)) and ||b (Fig. S5(b)) for different values of the SOI (=0.15 (black circles) (which is equal to the value in ref. [45].), 0.10 (blue curves), 0.06 (green curves), and 0.02 eV (red curves)). The other model parameters including U=3.0 eV, JH=0.6 eV, and U'=1.8 eV are the same as those in ref. [45] and those used in Supplementary 10 for Fig. 4

Analysis of time evolutions of T/T and R/R by multi-exponential function
We have analyzed the time profile of T/T (Fig. 2(a) lower panel)) and those of R/R (Fig. 3(d)) by using the following equations.   Thus, the short-lived oscillation is reasonable and essential in this compound.
Note that the optical transitions between Jeff=1/2 and Jeff=3/2 involving the in-gap states are realized by the intersite charge hopping. Therefore, it is quite reasonable to conclude that the intersite charge hopping and the SOC interplay. In fact, the energies of the intersite hopping and the SOI are comparable: 0.1-0.2 eV.

Details of theoretical calculations: Model and parameters
We consider a Hubbard model consisting of dyz, dxz, and dxy orbitals on each site of the honeycomb lattice [43][44][45] using the model parameters employed in ref. [45]. The hopping term is limited to the nearest neighbors because the 6site system is treated by the exact diagonalization method. Numerically we consider a minimum-size system that has three unit cells, each of which consists of two sites A and B, and use periodic boundary conditions shown in Fig. 4(a) (X1, Y1, and Z1 connect nearest-neighbor sites, and X2, Y2, and Z2 connect next-nearest-neighbor sites) to maintain the threefold symmetry.
With one hole per site, the three-orbital Hubbard model is written as, which consists of the kinetic term, the crystal-field term, the spin-orbit coupling, and Coulomb interactions, respectively. With the use of where , , † creates a hole in orbital a with spin  at site i, the kinetic term is written as Here, 2×2 is the 2 × 2 identity matrix and is the hopping matrix defined for each bond connecting nearest-neighbor sites i and j. The latter is one of 1 X , 1 Y , and 1 Z for the X1, Y1, and Z1 bonds, respectively, shown in Fig. 4(a): ) .
The values of the transfer integrals and the other parameters below are taken from ref.
where is the crystal-field tensor given by with Δ 1 = −0.0198 eV, Δ 2 = −0.0175 eV, and Δ 3 = −0.0125 eV. The spin-orbit coupling is written as Here, the intersite distance is set to be unity. The optical conductivity spectra are calculated for the ground state as in [48]. The spectrum on the highenergy side of the Mott gap (~1 eV) is discrete because of the finite-size effect.
The system treated by the exact diagonalization method consists of six sites (i.e., three unit cells); thus, the finite size effect is apparent. Here, Aa and Ab stand for the components of the vector potential A(t), which is shown above in the formula for the Peierls phase, along the aand b-axes shown in Fig. 1(a). The photon helicity is described by FL and FR. Schrödinger equation is numerically solved as before [49]. The decay component is insignificant in Fig. 4(c) because no relaxation process is considered and the system is too small to act as a heat bath for this field amplitude. 38 Supplementary note 11

Details of theoretical analysis: High-frequency expansion in Floquet theory
In general, we need the non-resonant condition to avoid thermalization. (If the non-resonant condition were not satisfied, the system would be thermalized after the application of a continuous wave and its temperature would finally go up to infinity). The frequencies used in the present experiment and theory are below the Mott gap; thus, harmful doubly-occupied sites are hardly produced. Furthermore, the pulse duration is sufficiently short to avoid thermalization. Therefore, quantum Floquet theory is applicable to the present situation.
Thus, continuous waves are considered here: which implies the application of an effective magnetic field to the effective angular momenta in the direction perpendicular to the honeycomb lattice.
Because of the inequality ( 2 − 4 )[ 2 − 4 + 2( 3 − 1 )] < 0 owing to the opposite signs of the dominant transfer integrals 2 and 3 , the effective magnetic field originating from F (2) points to the direction of (1,1,1) for lefthand circular polarization and to the direction of (−1, −1, −1) for right-hand circular polarization. Note that this effective magnetic field emerges without relying on the spin-orbit coupling.