(Fe,Zn)2Mo3O8 with Axion

Supplemental material for Optical Magnetoelectric Resonance in a Polar Magnet (Fe,Zn)2Mo3O8 with Axiontype Coupling T. Kurumaji1*, Y. Takahashi1, 2, 3, J. Fujioka2, R. Masuda2, S. Shishikura2, S. Ishiwata2, 3, and Y. Tokura1,2

1. Magnetic and magnetoelectric properties of (Fe1-xZnx)2Mo3O8. 2. Terahertz time-domain spectroscopy for x = 0.25 and 0.4. 3. Data analysis, derivation of Eq. (2), and odd-parity nature of gyrotropic birefringence.

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1. Magnetic and magnetoelectric properties of (Fe1-xZnx)2Mo3O8. To determine the composition dependence of the magnetic property of (Fe1xZnx)2Mo3O8,

we measured the temperature dependence of the magnetization (M) with

H//c for different values of x (Fig. S1(a)).

Figures S1(b) and S1(c) show the x

dependence of the transition temperature and the saturated magnetic moment (Msat) at 2 K, respectively.

The transition temperature systematically decreases owing to the

dilution of the magnetic interaction between Fe ions through substitution with nonmagnetic Zn doping. The observed linear increase of Msat, as shown in Fig. S1(c), is consistent with the fact that Zn2+ selectively substitutes Fe at tetrahedral sites to reduce the amount of spin moments in the opposite direction to that of the macroscopic magnetization [29].

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FIG. S1. (a) Temperature dependence of magnetization for different x value under H//c.

(b)-

(c) Composition dependence of (b) transition temperature (Neel temperature, TN, or Curie temperature, TC) and (c) saturated magnetization (Msat), estimated from M at 2 K.

To characterize the magnetoelectric (ME) properties of (Fe0.6Zn0.4)2Mo3O8, we measured the external magnetic field (H) dependence of M and the electric polarization (P) with H//c, as shown in Figs. S2(a)-(b).

The H dependence of M at 5 K (Fig. S2(a))

shows a hysteresis loop indicating spontaneous magnetization along the c axis in zero field.

In agreement with the magnetization flop, the butterfly-type hysteresis is

observed in the P-H scan at 5 K (Fig. S2(b)), which is the typical behavior of P for a linear-ME material with spontaneous magnetization [35,36].

At temperatures 10 K-

25 K, the P-H curve shows a cusp near the zero field, suggesting an increase of 𝛼𝑧𝑧 when approaching TC.

By considering the H derivative of P around the zero field, we

deduced the diagonal ME susceptibility, 𝛼𝑧𝑧 .

For the other component 𝛼𝑥𝑥 , we

measured P perpendicularly to the c axis for H parallel to the P direction (P⊥c, H⊥c, and P//H), as shown in Fig. S2(c).

Prior to the measurement, the sample was cooled

from a temperature above TC with 0H = 9 T to polarize the magnetization domain by a slight misalignment of H (~ 1o) towards the c axis. of P at 5 K (Fig. S2(c)).

We observed an H-linear change

The inversion of the slope was also confirmed by an

independent experiment with the opposite cooling field, 0H = -9 T (see Fig. 1(f)).

At

10 K or higher temperatures, the P-H curve shows a butterfly-type hysteresis similar to the case of H//c but with a larger coercive field.

This is because the c component of

H exceeds the coercive field, which rapidly declines with the increase of the temperature,

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as shown in Fig. S2(a). plotted in Fig. S2(d).

The estimated 𝛼𝑧𝑧 and 𝛼𝑥𝑥 at various temperatures are

The figure shows that 𝛼𝑧𝑧 exhibits a divergent behavior near

TC ≈ 30 K and suddenly becomes zero for T > TC.

A similar behavior is observed for

the isostructural compound Mn2Mo3O8 [37], which was attributed to the divergent increase of the magnetic susceptibility near TC.

FIG. S2. H dependence of (a) M for H//c, (b) change of P along the c axis for H//c, and (c) P perpendicularly to the c axis with H parallel to the P direction. for clarity.

The data are vertically shifted

(d) Temperature dependence of the diagonal components of the ME susceptibility,

zz and xx, estimated from (b) and (c), respectively.

2. Terahertz time-domain spectroscopy for x = 0.25 and 0.4. Terahertz spectra of the ferrimagnetic phase of (Fe1-xZnx)2Mo3O8 for x = 0 and 0.125 have already been reported [33] for the frequency region of 0.5-2.8 THz.

The

conclusions of Ref. [33] are as follows: (1) the ferrimagnetic phase for x = 0.125 showed a single magnetic-field-active excitation at 2.5 THz at 4.5 K in zero field, which was 4

termed MM2 (here, 3) mode; (2) application of H along the c axis decreased the excitation frequency to 2.3 THz with 0H = 7 T; and (3) the magnetic excitation, corresponding to the 3 mode, was observed for x = 0 in the H-induced ferrimagnetic phase.

In this section, we describe the observation of the magnetic excitations at x =

0.25 and 0.4 in the same frequency region and discuss the effect of Zn-doping on the excitation spectra. Figure S3(a) shows the spectra of the imaginary part of the refractive index  for x 𝜔 𝜔 = 0.25 with an [𝐸in //𝑐, 𝐵in ⊥ 𝑐] geometry and H parallel to the c axis.

In zero field,

three excitation modes are observed, at 1.4 THz (), 2.3 THz (), and 2.5 THz (). Application of H//c shifts the respective excitation frequencies linearly with H, as shown in Fig. S3(c).

The  mode, which corresponds to the MM2 mode observed for

x = 0 and 0.125, shifts downward for H//c.

Unlike these low-doped compounds,

𝜔 𝜔 however, the  mode appears to also respond to [𝐸in ⊥ 𝑐, 𝐵in //𝑐], as shown in Fig. 𝜔 S3(b), suggesting the in-plane electric field activity, 𝐸in ⊥ 𝑐.

To examine the

dependence of the terahertz spectra on Zn doping, we performed the same measurement for x = 0.4; the data are displayed in Figs. S3(d)-(f). Three modes are still observed in this composition although accompanying further modification of their optical activity. 𝜔 In each mode, the activity in 𝐵in ⊥ 𝑐 is maintained, while the  mode is very weakly

detected only with 0H = 7 T (Fig. S3(d)).

𝜔 For the 𝐸in ⊥ 𝑐 geometry (Fig. S3(e)), the

activity of the  mode appears, indicating both electric and magnetic field activity. The mode characteristics and excitation frequencies in zero field for each composition are summarized in Table S1. 5

FIG. S3. (a)-(f) Spectra of  for different light polarizations at 5 K with H//c and the magneticfield dependence of the resonance frequencies for (a)-(c) x = 0.25, and (d)-(f) x = 0.4. The spectral data are vertically shifted for clarity.

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Table S1. Magnetic excitations (1, 2, 3) observed in (Fe1-xZnx)2Mo3O8 for x = 0.125, 0.25, and 0.4.

“EA” (“MA”) indicates the electric (magnetic) field activity, and the excitation

frequency at 5 K in zero field is indicated in parentheses. Mode

x = 0.125

1

-

2

-

3

x = 0.25

x = 0.4

MA

weak EA, MA

(1.4 THz)

(1.4 THz)

MA

MA

(2.3 THz)

(2.2 THz)

MA

EA, MA

EA, weak MA

(2.6 THz)

(2.5 THz)

(2.5 THz)

To associate the excitations for x = 0.4 with the ferrimagnetic state, we measured the temperature dependence of the spectra of the imaginary part of the refractive index

 in zero field for different geometries as shown in Figs. S3(a) and S3(b). For the 𝜔 𝜔 [𝐸in //𝑐, 𝐵in ⊥ 𝑐] geometry, the  and  mode are clearly observed at 5 K, while the  𝜔 𝜔 mode and weak  mode are distinguished for the [𝐸in ⊥ 𝑐, 𝐵in //𝑐] geometry.

peaks of the modes decrease as the temperature increases.

The

Note that a broad peak

profile remains for the 1 and 2 modes even above the transition temperature TC ≈ 30 K (Fig. S3(a)).

This suggests that these newly activated modes are not collective

excitations but crystal-field-like excitations, as observed in Ba2CoGe2O7 [38]. Although the origin of the  and  modes as well as the Zn-induced modification of the optical activity for the corresponding excitations remain unknown, it is safe to say that the  mode is a spin excitation with both electric and magnetic field activity, i.e., magnetoelectric in nature. Hence, this mode causes large gyrotropic birefringence (GB) as described in the main text.

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FIG. S4. (a)-(b) Spectra of  for different light polarizations in zero field. The data are vertically shifted for clarity.

3. Data analysis, derivation of Eq. (2), and odd-parity nature of the gyrotropic birefringence In this section, we describe the data analysis procedure, from the raw data (Ez(t) and Ex(t)) to the spectra () and (). The derivation of the theoretical formula of Eq. (2) and experimental evidence for the odd-parity nature of the signal of the GB are also provided here.

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FIG. S5. Time evolution of THz pulse through the sample in zero field after the poling procedure of the magnetic moment.

The incident light is polarized parallelly to the c (z) axis.

(a) Ez and (b) Ex denote transmitted light polarized parallelly and perpendicularly to the incident light, respectively. (c) Time-reversal-odd component of Ex, obtained by antisymmetrization with Ex, for the (+Ps, +Ms) configuration (red curve in (b)) and for the (+Ps, -Ms) configuration (blue curve in (b)).

According to the phenomenological theory presented in Refs. [18-22], for light propagating in a gyrotropic birefringent material with k perpendicular to the bc (xz) 9

plane (Fig. 1(b)), the principal optical axes are rotated with respect to the b and c axes by angles 𝜃𝑥 and 𝜃𝑧 , respectively:

𝜇0

where 𝑍0 = √

𝜖0

tan 𝜃𝑥 = ±𝑍0 𝜇̃𝑥 𝛼GB

√𝜖̃𝑥 𝜇̃𝑧 , 𝜖̃𝑥 𝜇̃𝑧 − 𝜖̃𝑧 𝜇̃𝑥

tan 𝜃𝑧 = ±𝑍0 𝜇̃𝑧 𝛼GB

√𝜖̃𝑧 𝜇̃𝑥 , 𝜖̃𝑧 𝜇̃𝑥 − 𝜖̃𝑥 𝜇̃𝑧

(S1)

(S2)

is the vacuum impedance, 𝜖̃𝑥 (𝜇̃𝑥 ) and 𝜖̃𝑧 (𝜇̃𝑧 ) are the complex

dielectric constants (relative permeability) of the crystal along the 𝑥 and 𝑧 axis, respectively, and 𝛼GB = 𝛼𝑥𝑥 − 𝛼𝑧𝑧 .

The sign ( ± ) corresponds to the light

propagation direction ±𝑘 𝜔 //y. First, we deduce Eq. (2).

We begin with a configuration of monochromatic light

linearly polarized along the c (z) axis that is incident to the sample with k perpendicular to the bc plane (see Figs. 1(b) and 2(c)).

The time evolution of the electric field 𝑬𝑖𝑛

on the vacuum side of the incident sample surface is given as 𝑬in = 𝐸0 exp(−i𝜔𝑡) 𝒛,

(S3)

where E0 is the absolute value of the electric field, 𝜔 is the frequency of light, and z is the unit vector along the z axis.

As mentioned, the radiation field in the crystal is

described by the two eigen-states, i.e., linearly polarized light along the unit vectors 𝒙′ and 𝒛′ as follows 𝒙′ = cos 𝜃𝑥 𝒙 + sin 𝜃𝑥 𝒛 𝒛′ = sin 𝜃𝑧 𝒙 + cos 𝜃𝑧 𝒛.

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(S4)

The incident light propagates in the sample after separating into x’- and z’-polarized lights, which are modified by the transmission through the interface between the vacuum and the crystal, i.e., the amplitude is multiplied by crystal and by

2𝑛̃ 𝑛̃+1

2 𝑛̃+1

when entering the

when exiting the crystal, where 𝑛̃ is the complex refractive index

for the x’- or z’-polarized light.

The electric field of the transmitted light 𝑬out at the

end surface of the sample is expressed as follows

𝑬out

𝐸0 4𝑛̃𝑥′ =− exp(−i𝜔𝑡 ) [ ′ sin 𝜃𝑧 exp(i𝜔𝑛̃𝑥′ 𝑑/𝑐) 𝒙′ (𝑛̃𝑥 + 1)2 cos(𝜃𝑧 + 𝜃𝑥 )

(S5)

4𝑛̃𝑧′ − ′ cos 𝜃𝑥 exp(i𝜔𝑛̃𝑧′ 𝑑/𝑐) 𝒛′]. (𝑛̃𝑧 + 1)2 Here, 𝑛̃𝑥′ and 𝑛̃𝑧′ are complex refractive indices for the x’- and z’-polarized lights, respectively; they coincide with √𝜖̃𝑥 𝜇̃𝑧 and √𝜖̃𝑧 𝜇̃𝑥 to the first order of approximation of the ME susceptibility [19-20].

By substituting Eq. (S5) with Eq. (S4), the ratio

𝐸𝑥 (𝜔)/𝐸𝑧 (𝜔) to the first order of 𝛼GB gives the complex rotation angle 𝜃(𝜔) + i𝜂 (𝜔), which reproduces Eq. (2). 4𝑛̃𝑧′ 2 (𝑛̃𝑧′ +1)

On the other hand, 𝐸𝑧 (𝜔)/𝐸0 is approximated as

′

𝜔 𝜔 ei𝜔𝑛̃𝑧𝑑/𝑐 , providing the refractive indices n and  for the [ 𝐸in //𝑐, 𝐵in ⊥ 𝑐]

geometry in Fig. 2(a).

𝜔 𝜔 This is also valid for n and  for the [𝐸in ⊥ 𝑐, 𝐵in //𝑐] geometry

in Fig. 2(b). We introduce the quantity 𝐺 (𝜔), which is associated with Eq. (2), as follows

𝐺 (𝜔 ) =

̃𝑥 √𝜖̃𝑧 𝜇

2

[

̃ 𝑥 +1) √𝜖̃𝑥 𝜇 ̃𝑧 (√𝜖̃𝑧 𝜇

2 ̃ 𝑥 −𝜖̃𝑥𝜇 ̃ 𝑧 (√𝜖̃𝑥 𝜇 𝜖̃𝑧 𝜇 ̃ 𝑧 +1) √𝜖̃𝑧 𝜇 ̃𝑥

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exp

̃ 𝑧 −√𝜖̃𝑧 𝜇 ̃ 𝑥 )𝑑 i𝜔(√𝜖̃𝑥 𝜇 𝑐

− 1],

(S6)

where 𝑍0 𝜇̃𝑧 𝛼GB (𝜔)𝐺 (𝜔) gives 𝜃(𝜔) + i𝜂 (𝜔).

The spectra of √𝜖̃𝑧 𝜇̃𝑥 and √𝜖̃𝑥 𝜇̃𝑧

are experimentally obtained from those of n and  (√𝜖̃𝜇̃ = 𝑛 + i𝜅), which are measured 𝜔 𝜔 𝜔 𝜔 with the [𝐸in //𝑐, 𝐵in ⊥ 𝑐] and [𝐸in ⊥ 𝑐, 𝐵in //𝑐] geometries, as shown in Figs. 2(a) and

2(b), respectively.

For example, the calculated spectra of the real and imaginary parts

of 𝐺 (𝜔) at 5 K in zero field for x = 0.4 is shown in Fig. S6.

By combining 𝐺 (𝜔)

and the dispersion of 𝛼GB in Fig. 2(f), we obtain the black curves in Fig. 2(d) and 2(e).

FIG. S6. Spectra of real (orange) and imaginary (blue) part of G() calculated from the spectra of n and  (Fig. 2(a) and 2(b)) at 5 K in zero field for x = 0.4.

Equation (S2) predicts that the signal of the GB (∝ 𝛼GB ) is odd in terms of the ferroelectric order parameter, i.e., the direction of spontaneous polarization (Ps).

We

measured the optical rotation using the same terahertz spectroscopy experimental setup and with Ps in the opposite direction by rotating the sample by 180o around the x axis. 12

Figures S7(a)-(d) show the summary of the spectra of  and  with the corresponding configurations of +Ps and -Ps at different temperatures and magnitudes of H. All data sets for the -Ps configuration (black lines) are clearly the inverted corresponding data in the +Ps configuration (colored lines), confirming the odd-parity nature of the observed optical rotation.

FIG. S7. Spectra of the (a), (c) rotation angle () and (b), (d) ellipticity () at different temperatures and magnetic fields with different directions of the spontaneous polarization ±Ps. The colored (black) lines are obtained by the antisymmetrization of (+Ps, +Ms) and (+Ps, -Ms) ((-Ps, +Ms) and (-Ps, -Ms)).

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