Supporting Information for. Electric-magneto-optical Kerr effect in a. hybrid organic-inorganic perovskite

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1 Supporting Information for Electric-magneto-optical Kerr effect in a hybrid organic-inorganic perovskite Feng-Ren Fan, Hua Wu,,, Dmitrii Nabok, Shunbo Hu, Wei Ren, Claudia Draxl, and Alessandro Stroppa,, Laboratory for Computational Physical Sciences (MOE), State Key Laboratory of Surface Physics, and Department of Physics, Fudan University, Shanghai 2433, China Collaborative Innovation Center of Advanced Microstructures, Nanjing 2193, China Humboldt-Universität zu Berlin, Physics Department and IRIS Adlershof, Zum Groβen Windkanal 6, D Berlin, Germany Department of Physics, and International Center of Quantum and Molecular Structures, Shanghai University, Shanghai 2444, China CNR-SPIN, Via Vetoio, L Aquila, Italy wuh@fudan.edu.cn; alessandro.stroppa@spin.cnr.it I. Illustrative MOKE picture MOKE is an optical effect describing the change of the polarization of light when reflected from the surface of a magnetic material: a rotation of the polarization plane, the so-called Kerr rotation θ K, and a phase difference between the electric-field components perpendicular and parallel to the plane of the incident light, the so-called Kerr ellipticity, η K. In Fig- S1

2 (a) (b) θ K θ K Figure S1: Schematic diagram of MOKE and its switching upon reversal of the magnetization in a ferromagnetic sample. The red arrows denote the propagation of the light. The black (cyan) arrows in black circles are the polarization directions of the light without (with) rotations. The black arrows on the sample surfaces are the magnetization directions. ure S1 we show MOKE in a FM material, where the reversal of the magnetization, usually accomplished by an external magnetic field, leads to a reversal of the Kerr rotation. II. Crystal structure The crystal structure of Cr-MOF is shown in Figure S2. The Cr 2+ ions are connected by HCOO organic ligands, and the guanidinium cation, C(NH 2 ) + 3 are at the A sites of the perovskite structure. The Cr 2+ ions sit at the center of the octahedrons. Since Cr 2+ is a Jahn-Teller active ion the Cr-O bonds are not equal, and they are grouped in two longer and two shorter ones in the ab plane while the two medium ones are along the c axis. (a) (b) Cr O N C A: C(NH 2 ) 3 + (c) H X: HCOO - Figure S2: (a) Crystal structure of [C(NH 2 ) 3 ]Cr[(HCOO) 3 ], blue octahedrons are CrO 6 units. Details of (b) C(NH 2 ) + 3 and (c) HCOO groups are shown on the right. S2

3 III. Methods Calculations for the relaxed atomic positions were performed using the VASP code 1,2 as done in our previous work. 3 The Kerr angle is calculated as implemented in the all-electron full-potential code EXCITING. 4 The wavefunctions are expanded in terms of the (linearized) augmented planewaves plus local orbitals [(L)APW+lo] basis. 5 The muffin-tin radii (R MT ) for Cr, Cu, K, O, C, N and H atoms are 2.2, 2.2, 2.4, 1.18, 1.18, 1.18, and.75 bohr, respectively. The (L)APW+lo basis size is determined by R MT G Max = 3.. The Brillouin zone (BZ) is sampled by a k-mesh. The exchange-correlation effects are treated within the local-density approximation (LDA). 6 The spin-orbit coupling (SOC) term is included using the second-variation method. The complex Kerr angle is calculated using the equation 4 φ K = θ K +iη K = σ xy, (S1) σ xx 1+ 4πiσ ω xx where σ and ω are the optical conductivity tensor and the frequency of the incident light, respectively. In principle, Equation S1 applies to systems with higher than 3-fold rotational symmetry, 7 e.g., cubic systems, in which σ xy = σ yx and the anti-symmetric part σ A xy = (σ xy σ yx )/2 = σ xy. If not, σ A xy should be used, instead of σ xy, in Equation S1. However, for the calculations of MOKE in Cr-MOF, Equation S1 is a reasonably good approximation (see the following Section V for more details). In addition, σ xx = σ yy is assumed, in view of the leading contribution of σxy A. For calculating the optical conductivity tensor, both interband and intraband term (also called Drude term) are considered in our calculations. For the spectra calculations, broadening of.1 Ha was used. For the 49 calculations of the Kerr angles in (P, M) space (Figure 4 and S6), a smaller energy cutoff was used, by changing the R MT of H to 1.. S3

4 θ K (degree) Cr-MOF Zn-MOF GaAs Energy (ev) Figure S3: Variation of the light rotation as a function of the incident photon energy for Cr-MOF and Zn-MOF with λ=1, and also for GaAs. Zn-MOF is an artificial analog to Cr-MOF. The inversion symmetry is broken in all of them; the time reversal symmetry is preserved in Zn-MOF and GaAs, but not in Cr-MOF. IV. MOKE without time-reversal symmetry breaking Besides the most concerned MOKE in this work, one may consider an influence of the electric Pockels effect. The electric Pockels effect is also related to the inversion symmetry breaking, and it is linearly proportional to the electric field. It produces a polarization rotation for a light to propagate through an electric-optic crystal. It is important to note that although our theoretical approaches and the used EXCIT ING code do not account for the electric Pockels effect, this does not affect our conclusions on the EMOKE in this study, as proven by the following two computational experiments. First, we substitute Zn for Cr in Cr-MOF at λ=1 and then obtain an artificial nonmagnetic material Zn-MOF in a polar structure. Therefore, in Zn-MOF the time-reversal symmetry is preserved, but the inversion symmetry is still broken. As well known, without breaking the time-reversal symmetry, the Kerr rotation should always be zero. In Figure S3, we show our calculated light rotations for both Cr-MOF and Zn-MOF: the Kerr rotation for Zn-MOF is indeed zero at any energy but it is non zero for Cr-MOF. This infers that our approach does not account for the electric Pockels effect (as expected in the nonmagnetic but polar Zn-MOF) but truly describes the Kerr effect present in Cr-MOF. To further confirm this point, secondly we have calculated the light rotation according to our formalism for zinc-blend GaAs, a typical Pockels material. Again, S4

5 the calculated light rotation is zero for GaAs (Figure S3). With these two test calculations for nonmagnetic Zn-MOF and GaAs (both with the broken inversion symmetry), all the present results clearly show that the light rotation obtained for Cr-MOF is not due to the electric Pockels effect but to EMOKE. For a better accuracy, the electric Pockels effect shall be included in a future study. V. MOKE from anti-symmetric part of the optical conductivity tensor θ K (degree) P, M = P =, M = P, M = (σ xy -σ yx )/2 σ xy Energy (ev) Figure S4: Variation of the Kerr rotation as a function of the incident photon energy at (+P, M=), (P=, M=), and ( P, M=). The solid lines are the results calculated using (σ xy σ yx )/2, and dashed lines using σ xy. The existence of MOKE can be explained by the form of the conductivity tensor σ(ω), see Equation S1. For polar case, if there is MOKE, the off-diagonal components σ xy and σ yx are different from zero. More precisely, MOKE comes from the non-zero anti-symmetric part of the conductivity tensor, i.e. σxy A = (σ xy σ yx )/2. So, we have also calculated the Kerr rotation using the following Equation S2, which replaces σ xy in Equation S1 with σxy, A i.e. φ K = θ K +iη K = (σ xy σ yx )/2. (S2) σ xx 1+ 4πiσ ω xx The results for three typical cases, (+P, M=), (P=, M=), and ( P, M=), are shown in Figure S4 (solid lines). Compared with the results calculated using Equation S1 (dashed S5

6 lines in Figure S4), the values of the Kerr rotations somehow change, but mostly at the low energy (E<2.5 ev). However, it is interesting to point out that the results for (+P, M=) are always opposite to those for ( P, M=), independent of the two different formulations of the Kerr rotation. Thus, these test calculations once again confirm our conclusions about the EMOKE made in the main text. VI. The role of the A-site: trilinear coupling and cation substitution The Cr-MOF shows a trilinear coupling among two unstable non-polar modes and a stable polar mode: the two-non polar modes correspond to the tilting of the A-site (X 1 ) and to the pseudo-rotation at B site (X 4 ) due to the orbital ordering of the JT orbitals. 3 This implies that Kerr switching should occur either by switching the JT orbital ordering or by switching the tilting of the A-sites. The former has been discussed in the main text (Figure 2(a)) which basically corresponds to the switching of the Kerr angle by the electric polarization and corresponding JT orbital ordering. Here we discuss the effect of the tilting of A-sites (X 1 mode) and, in particular, we show that the reversal of the tilting also reverses the sign of the Kerr angle. In Figure S5(a) we report the calculated Kerr angle for two energyequivalent structures but with different tilting of the A site, i.e. for +X 1 and X 1, where X 1 is the symmetric mode related to the A-site tilting using the irreducible representation of the high symmetric space group, Imma. 3 The change of the tilting angle is shown in the inset of Figure S5(a). The switching of the Kerr angle is accompanied by a reversal of M z. Therefore, the tilting of the organic cation at the A site has an important role in shaping the Kerr spectra through the mechanism of hybrid improper ferroelectricity. It is conceivable that a judicious choice of a different organic cations may give a different tilting and therefore induces changes in the Kerr spectra. We further explore the role of the A site by considering an extreme case where we replace the guanidinium cation with a simple spherical atom such as K +. Unexpectedly, the Kerr spectra changes (Figure S5(b)). This further highlights the possibility of tuning of the Kerr S6

7 (a) θ K (Deg) (b) θ K (Deg) (c) θ K (Deg) X 1 X 1 X 4 X 1 X 4 X 1 Cr-MOF K@A -.5 Cr-MOF -.1 Cu-MOF Energy(eV) Figure S5: Variation of the Kerr rotation as a function of the incident photon energy for different cases. (a) Two trilinear modes X 4 X 1 (red solid) and X 4 -X 1 (blue dashed); (b) Cr- MOF (red solid) and K-replaced A site Cr-MOF (blue dashed); (c) Cr-MOF (red solid) and Cu-MOF (blue dashed). angle by proper choice of the A-site cations, which, in turn, modifies the molecular tilting at the A site and the ferroelectric polarization of the unit cell, opening new possibilities for the Kerr-angle engineering in complex hybrid materials. While these are certainly new directions to explore, it is beyond the scope of this paper to go in further details. A more appropriate discussion will be considered in a forthcoming publication. VII. Cr-MOF and Cu-MOF Last but not least, although we have started our calculations by considering the Cr-MOF, to the best of our knowledge, this compound has not been synthesized yet. However, a similar compound, a Cu-MOF has been grown and experimentally characterized. 8 According to previous studies, 3,9 Cu-MOF and Cr-MOF have very similar properties. In particular, both S7

8 of them show trilinear coupling as source of ferroelectricity and they are magneto-electric. All these properties have been experimentally verified for the Cu-MOF. Therefore we have also calculated the Kerr spectra in this case. Although the calculated weak-fm component is smaller in the Cu-MOF with respect to the Cr-MOF i.e..19 µ B /Cu and 1.4 µ B /Cr respectively, the ferroelectric polarization is larger in the Cu-MOF than in the Cr-MOF, i.e..46 and.23 µc/cm 2 respectively. 3,9 In Figure S5(c) we show the Kerr spectra for both the Cu-MOF and the Cr-MOF in the two equivalent ferroelectric states. First of all, in both cases, the Kerr reverses its sign upon reversal of both P and M. Interestingly, the Cu-MOF shows the largest Kerr angle in the explored energy range. This is probably related to the larger polarization in the Cu-MOF despite its small net magnetization. This highlights the interesting role of the α, β coefficients which can be strongly depending the specific material. A more extended analysis will be presented in a forthcoming publication. VIII. Fitting In order to test how the Kerr rotation depends on P and M, we perform 49 calculations on a grid of points in (P, M) space. As a result, we have a distribution of Kerr angles in (P, M) space at each energy point, e.g. E=3.82 ev (Figure S6(a)). All these computational experiments show that the Kerr rotation can be switched only by switching both P and M at the same time. Based on the 49 calculations, we can fit Kerr rotations in (P, M) space at each energy point. Because of the centro-symmetry of the Kerr rotation in (P, M) space, we use a target function, that is where n is a positive integer. i θ K (P,M) = αj i P(i j) M j i=2n 1 j= (S3) If only the first order terms are considered, Equation S3 becomes a multilinear equation θ K (P,M) = α 1 P +α1 1 M (S4) S8

9 (a) 1 Original data (b) 1 1st-order fitting (c) 1 3rd-order fitting.6 M / M max P / P max P / P max (d) (e) θ K (degree) original 1st order fitting 3rd order fitting Energy (ev) P / P max P / P max (f) M / M max P / P max M / M max -.6 Figure S6: Kerr-rotation distribution at 3.82 ev in the whole (P, M) space. (a) The original 49 grid points, (b) first-order multilinear fitting, and (c) third-order fitting. The black-square line is the magneto-electric path, and the orange line stands for points which give a zero Kerr rotation. (d) Comparison of the original (black), first-order fitting (blue) and third-order fitting (red) of the Kerr rotations at (+P max, +M max ). (e) Comparison of the original (black dots) and first-order fitting of the Kerr rotations at 3.82 ev. (f) Comparison of the original (black dots) and third-order fits of the Kerr rotations at 3.82 ev. We can see that the third order fitting results are much closer to the original data than the first-order ones. where α 1 and α1 1 are the parameters α and β mentioned in main text. The fitting is a multilinear fitting. As long as we have α 1 and α1 1, we can reproduce all the Kerr rotations in the whole (P, M) space at each energy point. We show the Kerr-rotation distribution at E = 3.82 ev in Figure S6(b). Moreover, we have also expanded Equation S3 up to the third order, θ K (P,M) = α 1 P +α1 1 M +α3 P3 +α 3 1 P2 M +α 3 2 PM2 +α 3 3 M3 (S5) We can also reproduce the Kerr-rotation distribution in the whole (P, M) space (Figure S6(c)). We can now compare the two fitting procedures, first vs third order. In S9

10 Figure S6(d), we show the spectra θ K (+P max,+m max ;E). The black, blue, and red line correspond to the original, first-order and third-order fitting results respectively. We can see that both the first and third order fitting describe the Kerr rotation rather well. In Figure S6(e) and (f), we give the comparison at E = 3.82 ev for the first-order and third-order fitting with the original results respectively. As we have seen in Figure S6(d), the third-order result is closer to the original one, as expected. However, in both cases, the fitted values represent very well the real values and, in particular, they reproduce the underlying symmetry in (P, M) space. The fitting procedure confirms the switching properties that we discussed previously, i.e. only when P and M are reversed at the same time, the Kerr angle can be switched. The most important result is that a measure of θ K is proportional to P and M not just only to M as previously believed. Thus EMOKE represents a new property in material science which deserves further investigation both from theory and experiments. References (1) Kresse, G.; Furthmuller, J. Phys. Rev. B 1996, 54, (2) Blöchl, P. E. Phys. Rev. B 1994, 5, (3) Stroppa, A.; Barone, P.; Jain, P.; Perez-Mato, J. M.; Picozzi, S. Adv. Mater. 213, 25, (4) Gulans, A.; Kontur, S.; Meisenbichler, C.; Nabok, D.; Pavone, P.; Rigamonti, S.; Sagmeister, S.; Werner, U.; Draxl, C. J. Phys. Condens. Matter. 214, 26, (5) Singh, D. J.; Nordstrom, L. Planewaves, Pseudopotentials, and the LAPW method; Springer Science & Business Media, 26. (6) Perdew, J. P.; Wang, Y. Phys. Rev. B 1992, 45, (7) Kahn, F. J.; Pershan, P. S.; Remeika, J. P. Phys. Rev. 1969, 186, S1

11 (8) Hu, K. L.; Kurmoo, M.; Wang, Z.; Gao, S. Chem. Eur. J. 29, 15, 125. (9) Stroppa, A.; Jain, P.; Barone, P.; Marsman, M.; Perez-Mato, J. M.; Cheetham, A. K.; Kroto, H. W.; Picozzi, S. Angew. Chem. 211, 123, S11