Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state

Abstract

Electrodynamic responses from three-dimensional topological insulators are characterized by the universal magnetoelectric term constituent of the Lagrangian formalism. The quantized magnetoelectric coupling, which is generally referred to as topological magnetoelectric effect, has been predicted to induce exotic phenomena including the universal low-energy magneto-optical effects. Here we report the experimental indication of the topological magnetoelectric effect, which is exemplified by magneto-optical Faraday and Kerr rotations in the quantum anomalous Hall states of magnetic topological insulator surfaces by terahertz magneto-optics. The universal relation composed of the observed Faraday and Kerr rotation angles but not of any material parameters (for example, dielectric constant and magnetic susceptibility) well exhibits the trajectory towards the fine structure constant in the quantized limit.

ARTICLE

Received 26 Jan 2016 | Accepted 15 Jun 2016 | Published 20 Jul 2016

Ken N. Okada1,2,*, Youtarou Takahashi1,2,3,*, Masataka Mogi2, Ryutaro Yoshimi1,2, Atsushi Tsukazaki3,4, Kei S. Takahashi1,3, Naoki Ogawa1, Masashi Kawasaki1,2 & Yoshinori Tokura1,2

Electrodynamic responses from three-dimensional topological insulators are characterized by the universal magnetoelectric term constituent of the Lagrangian formalism. The quantized magnetoelectric coupling, which is generally referred to as topological magnetoelectric effect, has been predicted to induce exotic phenomena including the universal low-energy magneto-optical effects. Here we report the experimental indication of the topological magnetoelectric effect, which is exemplied by magneto-optical Faraday and Kerr rotations in the quantum anomalous Hall states of magnetic topological insulator surfaces by terahertz magneto-optics. The universal relation composed of the observed Faraday and Kerr rotation angles but not of any material parameters (for example, dielectric constant and magnetic susceptibility) well exhibits the trajectory towards the ne structure constant in the quantized limit.

DOI: 10.1038/ncomms12245 OPEN

Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state

1 RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan. 2 Department of Applied Physics and Quantum Phase Electronics Center (QPEC), University of Tokyo, Tokyo 113-8656, Japan. 3 PRESTO, Japan Science and Technology Agency (JST), Chiyoda-ku, Tokyo 102-0075, Japan.

4 Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan. * These authors contributed equally to this work. Correspondence and requests for materials should be addressed to K.N.O. (email: mailto:[email protected]

Web End [email protected] ) or to Y.T. (email: mailto:[email protected]

Web End [email protected] ).

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Topological quantum phenomena have been attracting increasing attention in condensed matter physics, because the system determined by the topological structure

exhibits quantized observables, such as magnetic ux in superconductors and Hall conductance in quantum Hall effect. Magnetoelectric coupling, which has been a fundamental concept for contemporary physics including spintronics and multi-ferroics1, is predicted to be quantized in the recently discovered three-dimensional (3D) topological insulators (TIs)210. More specically, quantized magnetoelectric responses are predicted on the quantum anomalous Hall (QAH) state induced by the magnetic mass-gap on the surface Dirac cone under the broken time-reversal symmetry. In the QAH state, as experimentally conrmed recently1118, the surface states exhibit quantized Hall conductance (sxy e2/h and sxx 0) without external

magnetic eld.

In the electromagnetic eld in 3D TIs, the Lagrangian includes the axion term a

4p2

yE B,

which is characterized by the ne structure constant a2pe2=hc (refs 4,19). Under the presence of

time-reversal symmetry the term y is equal to p in TI, while zero in the vacuum or ordinary insulator. When the time-reversal symmetry is broken at the TI surface, Maxwells equations are

modied, resulting in an unusual quantized magnetoelectric effect at the TI surface, referred to as topological magnetoelectric (TME) effect. This novel TME effect is ensured in a low-energy region below the magnetic mass gap at the Dirac point. Since the modied Maxwells equations provide a quantized transverse current, the QAH effect with sxy e2/h can be viewed as a zero-

frequency limit of TME effect. On the other hand, for the optical process the TME effect produces the quantized Faraday and Kerr rotation angles4,6,7, which represent polarization rotations for transmission and reection geometries, respectively. Accordingly, the relation denoted with rotation angles (yF and yK) always leads to the ne structure constant a ( 2pe2/hcB1/137), irrespective

of material parameters such as dielectric constant and magnetic susceptibility, whereas the magneto-optical rotation angles of a thin lm on substrate are substantially modied from those of the free-standing lm in vacuum (yF aB7.3 mrad and

yK

p/2 rad) (refs 4,6,7). Thus the observation of Faraday and

Kerr rotations on QAH state provides a direct measure of a. As a result the rotation angles of electromagnetic wave in the low-energy region are scaled by the d.c. Hall conductance, in accord with the development of the QAH state. It should be emphasized that in spite of the identical origin, those two

a

d

F

BST 1 nm

Substrate (InP)

TI film

AIOx TI film

x

TI film AIOx

Cr: BST 1 nm

BST 4 nm

AIOx

EF

Substrate (InP)

Cr: BST 1 nm

BST 1 nm

E

b

z

y

2.0

1.5

1.0

0.5

0.00.1 1 10 100 T (K)

B (T)

B = 0.1 T

xx

2 / h )

xx, xy (e

xy

e

c

T = 0.5 K1.5 K4.3 K

1.0

(i)

T = 1.5 K

E y (a.u.)E x (a.u.)

2 / h )

20 K35 K70 K

0.5

0

0.5

0

(ii)

0.5

0.0

xy (e

0.5

(i) (ii)

Substrate (InP)

F F + K

(i)

(ii)

1.0

1.0

300

0.5

0.0

0.5

1.0

0 5 10 15 20Time (ps)

Figure 1 | THz Faraday and Kerr rotation of QAH state on magnetic TI lm. (a) Schematics of the TI lm with magnetic modulation doping and of the band structure of surface states under the presence of time-reversal-symmetry breaking magnetization. (b) Temperature dependence of the longitudinal (sxx) and Hall (sxy) conductances at B 0.1 T. (c) Magnetic eld dependence of sxy at various temperatures. (d) Schematics of the THz magneto-optics for

the magnetic TI lm on an InP substrate. Crossed-Nicol geometry was employed for the detection of the magneto-optical rotation of light polarization. Faraday and Kerr rotations are measured by the rst THz pulse (i) and the second THz pulse (ii), respectively. (e) Time evolutions of THz pulses through the magnetic TI lm at 0 T after the poling of magnetic moment. (See Methods for detail.) Ex and Ey are transmitted light polarized parallel and perpendicular to the incident light, respectively. The rst pulse (i) represents the directly transmitted light through the TI lm and substrate, while the THz pulse once reected at back surface of substrate appears as the second pulse (ii) with a time delay. Ey component of the second pulse includes the Kerr rotation yK at back-surface of TI lm as well as the Faraday rotation yF.

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phenomena, QAH effect and topological Faraday and Kerr rotations, are observed as quantization of different physical quantities; sxy e2/h and yF a 2pe2/hc. On the other hand,

the difculties in experimental verication of TME effect have been indicated since the early stage of theoretical predictions4,6,7. This is because the observation of TME effect requires the Fermi energy within the magnetic mass-gap on the surface Dirac cone, and hence precise Fermi energy tuning is indispensable. In addition, the observation of genuine TME signal is limited to the low energy, that is, sufciently lower than the magnetic mass-gap to avoid the responses from real electronic transitions.

The QAH state on the surface of TI is stabilized by the magnetic mass-gap, while accurate size of the gap energy may depend on sample form (lm/bulk) as well as concentration of magnetic dopants or defects. In fact, the gap energy has been reported to range from 22 meV for a Cr: Sb2Te3 thin lm20 to 50 meV for a Cr: (Bi, Sb)2Te3 bulk single crystal21. Therefore, the magneto-optics by terahertz (THz) spectroscopy probing the lower-energy range, for example, 18 meV in the present experiment, is suitable for the observation of the possible emergence of topological Faraday/Kerr rotations. Furthermore, the recently developed magnetic modulation-doping in Cr: (Bi, Sb)2Te3 thin lm17 can markedly widen the observable temperature region of QAH effect up to several Kelvin, making feasible the optical measurement of QAH state. So far the low-energy magneto-optical responses have been intensively studied2229 mostly for nonmagnetic TIs, in which the cyclotron resonances of the surface states as well as the bulk carriers are reported. However, an experimental demonstration of TME effect on QAH state remains elusive.

In this paper, we show the experimental signature of topological Faraday/Kerr rotations in QAH states on magnetic TI thin lms by THz time-domain spectroscopy (TDS). The trajectory towards the ne structure constant a is unveiled by the measurements of THz Faraday and Kerr rotation angles for the surface QAH state.

ResultsQAH effect observed in a magnetic TI thin lm. The Crx(Bi0.26Sb0.74)2 xTe3 TI lm with magnetic modulation-doping17, where magnetic impurities Cr (x 0.57) are doped in

two quintuple layers adjacent to the top and bottom (Bi0.26Sb0.74)2Te3 layers17, is schematically illustrated in Fig. 1a.

The evolution of the magnetization induced by the Cr-doping gives rise to the QAH state as shown in Fig. 1b,c. The ferromagnetic transition occurs around TCB70 K with the onset of the anomalous Hall term in sxy (Fig. 1b). As temperature decreases, sxy develops and tends to saturate at the quantized value e2/h at around T 0.5 K, while sxx steeply decreases

towards zero, due to the emergence of the dissipationless chiral edge conduction (Fig. 1b). The Hall angle (sxy/sxx) becomes as

large as 1 around 4 K, indicating the emergence of QAH regime at temperatures more than an order of magnitude higher17 than the uniformly Cr- or V-doped TI lms1116,18, due perhaps to the enlargement of the magnetic mass-gap induced by the rich Cr-doping and the reduced disorder in the surface states by Cr dopants17. Hysteretic behaviours of Hall conductance further evidence the development of the QAH regime as shown in Fig. 1c. The fully quantized sxy at the lowest temperature indicates that the Fermi energy locates well within the magnetic mass-gap of the surface Dirac cone (Fig. 1a) without additional eld-effect tuning.

THz magneto-optics on the magnetic TI thin lm. THz-TDS provides magneto-optical measurements with sufciently lower photon energy (18 meV) than the magnetic mass-gap (2050 meV (refs 20,21)) and with high resolution of light-polarization rotations (o1 mrad). Recently, this technique has been found to be useful to study polarization rotation in THz region on ferromagnetic semiconductors as well30. The measurement conguration of magneto-optics by THz-TDS is schematically illustrated in Fig. 1d (see Methods for detail). Depending on the time delay, the monocycle THz pulse can

a b c

T = 1.5 K

1.0

0.8

0.6 0 2 4 6 8

1.0

0.5

0.0

Rotation angle (mrad)

54321010 2 4 6 8

Rotation angle (mrad)Transmittance

T = 4.3 K

F

F

15 12

9 6 3 0 3

T = 1.5 K

10 K15 K54 K62 K

xx (10

Rotation angle (mrad)

3 1 )

K

K

0 2 4 6 8

h (meV)

h (meV)

h (meV)

d

10

8

6

4

2

0

2 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 h (meV)

h (meV)

h (meV)

h (meV)

h (meV)

h (meV)

h (meV)

T = 1.5 K

4.3 K

15 K

20 K

35 K

54 K

70 K

K

F

Figure 2 | THz Faraday and Kerr rotation spectra of the magnetic TI lm. (a) Transmittance (left ordinate) spectra of the magnetic TI lm at various temperatures and optical conductance (right ordinate) sxx spectrum at T 4.3 K. The transmittance close to unity was observed, indicating the

negligible carrier absorptions due to ne-tuning of the Fermi level within the magnetic mass-gap of the surface state. The low-lying optical phonon mode is discerned as a tiny dip around 7 meV and also as the peak of optical conductance spectrum. (b,c) Complex Faraday (b) and Kerr (c) rotation spectra at 1.5 K (see the main text and Methods for denition). The real parts (yF and yK) represent the rotation angle of light polarization. The imaginary parts (ZF and ZK)

represent the ellipticity, which is negligibly small as expected. Rotation angles at o 0 evaluated from the d.c. transport data (Fig. 1c) are also plotted on

the left ordinates. (d) Temperature dependence of the Faraday (red) and Kerr (blue) rotation spectra with the evaluated o 0 values on the ordinates.

The error bars in bd are evaluated by s.e.m. of several runs of measurement.

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differentiate the directly transmitted pulse (i) and the delayed pulse generated by back-reection at the back surface of substrate(ii), as shown in Fig. 1d,e; this enables us to separably measure Faraday and Kerr rotations, as reported for TI thin lms22,24 and graphene on substrates31. As shown in Fig. 1e, the temporal waveform of Ey-component indicates the pronounced rotation of polarization on the rst pulse (i) as well as on the second one(ii) due to the presence of the magneto-optical rotations at zero external magnetic eld. The rst pulse (i) involves the Faraday rotation (yF), while the second pulse (ii) is composed of yF plus the Kerr rotation (yK) at the back surface of the magnetic TI lm (Fig. 1d).

The transmittance spectra obtained by Ex-component at different temperatures are shown in Fig. 2a. The transmittance is close to unity, that is, no discernible absorption, except for the dips around 7 meV indicated by the arrow, which is assigned to the optical phonon mode32; see also the optical conductance sxx spectrum at T 4.3 K also shown in Fig. 2a. The negligibly weak

absorption, for example, no Drude response, conrms that the Fermi energy locates within the magnetic mass-gap of the surface Dirac cone (Fig. 1a).

Fourier transformation of the electric eld pulses Ex(t) and

Ey(t) (Fig. 1e) provides the complex Faraday and Kerr rotation spectra (Fig. 2b,c), where the real part, yF(o) or yK(o), and the imaginary part, ZF(o) or ZK(o), represent the rotation angle and the ellipticity, respectively (see Methods for detail). The rotation-angle (real part) spectra for yF and yK show nite values around 2.6 and 6.9 mrad, respectively, with modest frequency dependence, as shown in Fig. 2b,c. The ellipticity (imaginary part) spectra for ZF and ZK are close to zero in the whole photon-energy region (o8 meV). Note that noise-like fringe structures in the

Kerr rotation spectra (Fig. 2c,d), in contrast to the almost o-constant Faraday rotation spectra, come from the inevitable interference due to the temporal overlap with the early-coming Faraday rotation signals (Fig. 1e). These characters, that is, little frequency dependence and near-zero ellipticity, strongly indicate that the current THz energy window (18 meV) is well below the threshold energy for any magneto-optically active real transitions. This is consistent with the fact that the magnetic mass-gap on the Dirac point (reported to be 2050 meV by scanning tunnel spectroscopy2021) is sufciently large as compared with the energy range of this measurement. Note also that possible cyclotron resonance under magnetic eld, which has been observed in previous magneto-optical studies on TIs2228, is absent in the present measurement because of zero external magnetic eld. Furthermore, the observed Faraday and Kerr rotation angles are quantitatively consistent with the estimated rotation angles at d.c. limit (Fig. 2b,c), which are calculated from sxx and sxy obtained by the d.c. transport measurement (Fig. 1c)

through the following relations26,31;

yF

a

Rotation angle (mrad) Rotation angle (mrad)

10

8

6

4

2

0 0 0.2

xyd.c. (e2/h)

K

c

8

F

f =

6

3 )

0 0 20 40

T (K)

K

60 80

f( F, K) (10

4

b

2

0 0.2 0.4 0.6 0.8 1.0

F

xyd.c. (e2/h)

0.4 0.6 0.8 1.0

0

1

2 arg

! t

21 ns Y

t

t

1

yK

: 2

Here t ( ) and r ( ) represent the transmittance and

reection coefcients of right- and left-handed circularly polarized light, respectively. The admittance Y is described as Y

Z0(sxxisxy) (Z0 377 O: the vacuum impedance) and ns

is the refractive index of the InP substrate. We determined ns as

3.47 by measuring THz response of the substrate, which well agrees with literature33. For instance, the estimated rotation angles for yF and yK at o 0 are around 3.1 and 8.7 mrad at 1.5 K

(indicated with closed squares on the respective ordinates in

Fig. 2b,c). This quantitative agreement with the d.c. QAH state exemplies that the observed THz rotation stems from the TME effect on the TI surfaces. Figure 2d shows the temperature evolution of the Faraday and Kerr rotation spectra. The rotation angles decrease with increasing temperature and vanish at TC (see also Fig. 3a), in accord with the disappearance of the ferromagnetic state. We also conrmed that the Faraday and Kerr rotations arising from the QAH state are reproducibly observed for the different sample with different heterostructure and TC (B40 K) (see Supplementary Figs 1 and 2 and Supplementary Note 1).

Since the magnetic-layer thickness in the present hetero-structure lm is 2 nm in total, the gure of merit of spontaneous Faraday rotation of our sample can be effectively regarded as B7 105 degree per cm in the off-resonant condition with

near-zero ellipticity. For comparison, Faraday rotation of most well-known Faraday rotator Bi-doped yttrium iron garnet is 9 102 degree per cm around 1 eV (ref. 34). Thus the rotation

angle observed here is remarkably large as compared with the conventional Faraday rotation in ferromagnets. Furthermore, we observed the nearly same magnitude of rotation angles on the different heterostructure TI sample with twice the thickness of magnetic layer (4 nm in total) (see Supplementary Figs 1 and 2 and Supplementary Note 1). These results strongly indicate that the polarization rotations observed here intrinsically originate from the response of the TI surface states.

10

8

6

4

2

r

r

Figure 3 | Trajectory towards the quantized TME response. Closed circles correspond to the rotation spectra in Fig. 2bd. Closed triangles represent the data of another different sample. Open symbols represent the data at B 1 T. (a,b) Faraday (red) and Kerr (blue) rotation angles versus

temperature (a) and d.c. Hall conductance sxyd.c. (Fig. 1c) (b). The rotation angles and their error bars in a,b are determined by the mean and s.d.

of the rotation angle spectra belowB4 meV, respectively. The solid lines in b represent the estimation from equations (1) and (2) in the main text.

(c) Evolution of the scaling function f yF; yK

1

2 arg

! r

1 ns Y

1 ns Y

cot y cot y

cot y 2cot y cot y 1 (see the

main text and equation (3)) without any material parameters as a function of d.c. Hall conductance, which is expected to reach the ne structure constant a ( 2pe2/hcB1/137) in the quantized limit, as indicated by a

straight line. The error bars of f (yF, yK) are determined by its total derivative calculated from s.d. of yF and yK presented in a,b.

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Trajectory towards topological Faraday and Kerr rotations. In Fig. 3b the rotation angles at different temperatures, which are measured by averaging the rotation angle below B4 meV (Fig. 2d), are displayed (closed circles) as a function of the d.c. Hall conductance together with the calculated values from equations (1) and (2) at o 0. Note that open circles in Fig. 3

correspond to the rotation angles measured at B 1 T

(see Supplementary Fig. 3 and Supplementary Note 2). We also plot the data for the different sample at T 1.5 K and B 0 T

(1 T) denoted by closed (open) triangles (see Supplementary Figs 1 and 2 and Supplementary Note 1). The observed THz Faraday and Kerr rotation angles show a good agreement with the estimated o 0 value, although small deviations are still

discerned.

The relationship between the Faraday and Kerr rotation angles at the quantized limit is expected to lead to the ne structure constant a, irrespective of any material parameters such as the dielectric constant and the thickness of the lm, the capping layer and the substrate6. In our measurement geometry, the universal relationship between yF and yK in the QAH state is obtained from equations (1) and (2);

cot yF cot yK

cot2 yF 2cot yF cot yK 1

a: 3

Here we dene the left side of equation (3) as the scaling function f (yF, yK). In Fig. 3c, the function f (yF, yK) versus d.c. Hall conductance sxyd.c. is plotted, in which the f (yF, yK) is expected to reach a ( 2pe2/hcB1/137) in the quantized limit. With

increasing sxy to the quantized conductance by lowering temperature, the dimensionless f (yF, yK) approaches the universal value a, in good agreement with the estimation at o 0 (line in Fig. 3c), manifesting the trajectory towards the

quantized value a determined uniquely and solely by the magneto-optical rotation angles.

The small deviation in yF, yK and f (yF, yK) from the estimation based on d.c. Hall conductance are discerned (Fig. 3b,c). One reason for the small reduction of the rotation angles from the d.c. limit may be partial magnetization reversal under zero magnetic eld during the terahertz measurements (Fig. 1e). Indeed, the measurements under magnetic eld of B 1 T, where the

magnetization reversal is totally avoided during the measurement, result in slightly higher values of Faraday and Kerr rotations and hence of f (yF, yK) (open circles and triangles in Fig. 3b,c), although slight deviations from the expectations are still discerned. Another possible cause is a difference of the characters between the d.c. Hall measurement and optical one in the quantum Hall regime. The d.c. Hall measurement detects the conduction of the chiral edges states developing at the sample edge, irrespective of inside small domains or islands where the quantization may remain incomplete due to defects with potential hills/valleys or to residual in-gap states35. On the other hand, since optical measurement detects the conductance averaged over the spot area, the obtained rotation angles might be reduced from the d.c. limit due to those islands. A certain amount of residual d.c. longitudinal conductance (sxx) at the lowest temperature

(1.5 K) of the present optical experiment implies the persistence of such an effect (Fig. 1b), which would be thoroughly cleared up at further lower temperatures where sxxB0 is attained.

After submitting the manuscript, two other groups uploaded preprints on an e-Print server reporting quantized magneto-optical rotations, which were observed on nonmagnetic TIs under application of magnetic eld36,37. In the present study on the quantum anomalous Hall effect in zero magnetic eld, we have observed the almost quantized Faraday and Kerr rotations via exchange interaction with localized magnetic moments instead of

external magnetic eld, that is, without any contribution from the cyclotron motion of conduction electron.

In conclusion, we have experimentally investigated the TME effect on the QAH state of surface state of TI by measurements of Faraday and Kerr rotations in THz region. The observed Faraday and Kerr rotation angles show quantitative agreement with the estimation from the d.c. transport results. The universal relationship with the magneto-optical rotation angles shows the trajectory converging to the ne structure constant a with the approach to the QAH state.

Methods

Sample fabrication. The 8-nm-thick TI lms with magnetic modulation doping were grown on both-side-polished insulating InP substrates by molecular beam epitaxial growth as described in ref. 17. To protect the lm from degradation,a 3-nm-thick AlOx layer was immediately deposited with ex situ atomic layer deposition. Transport measurement and optical THz spectroscopy were performed on different samples from the same batch for each lm. Possible modication of rotation angles by the AlOx capping layer is estimated to be as small as 0.01% at most, and hence neglected in the analysis described in the main text.

Magneto-optical terahertz spectroscopy. In the THz-TDS, laser pulses with duration of 100 fs from a mode-locked Ti: sapphire laser were split into two paths to generate and detect THz pulses. THz pulses were generated by a bow-tie-shaped antenna and detected by a dipole antenna. The Ey(t) component of the transmitted

THz pulses (Fig. 1e) was measured by the Crossed-Nicole conguration by using wire-grid polarizers. The polarization rotation Ey(t) at 0 T is dened by the antisymmetrized waveform to eliminate the background signal, which is the difference between signals with magnetization for z directions after the poling of the magnetization at 1 T; EytE Myt E My t

=2. The Faraday rotation of the

substrate (InP) is smaller than the sensitivity of our equipment (o10 mrad T 1). The Fourier transformation of the rst THz pulses Ex(t) and Ey(t) (Fig. 1e) gives the complex Faraday rotation spectra Ey(o)/Ex(o) (sinyF(o) iZF(o)cosyF(o))/

(cosyF(o) iZF(o)sinyF(o))ByF(o) iZF(o) (Fig. 1b) for the small rotation

angles. The rotation spectra obtained by the second pulses give the sum of the Kerr and Faraday rotation spectra. In Kerr rotation spectra, inevitable interference with early-coming Faraday rotation signal in time domain (Fig. 1e) results in the fringe structures. We carefully examined the interference in the raw data and extracted Kerr signal to minimize them. Transmittance spectra were obtained by comparison between the transmission of sample and bare substrate. We applied the following standard formula to obtain the complex conductance s(o) s1(o) is2(o)

of TI lm;

t o

1 ns

1 ns Z0s o

;

where t (o) is the complex transmittance, Z0 is the impedance of free space (377 O) and ns the refractive index of the InP substrate.

Data availability. The authors declare that the data supporting the ndings of this study are available within the article and its Supplementary Information.

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Acknowledgements

This research was supported by the Japan Society for the Promotion of Science through the Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program) on Quantum Science on Strong Correlation initiated by the Council for Science and Technology Policy and by JSPS Grant-in-Aid for Scientic Research No. 24224009, 24226002 and 26706011. K.N.O. is supported by RIKEN Junior Research Associate Program.

Author contributions

Y. Tokura conceived the project. K.N.O. and Y. Takahashi carried out optical terahertz spectroscopy and analysed data. M.M., R.Y., K.N.O., A.T. and K.S.T. prepared the modulation-doped topological insulator thin lms and performed the structural and transport characterizations. The results were discussed and interpreted by K.N.O.,Y. Takahashi, N.O., M.K., A.T. and Y. Tokura.

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How to cite this article: Okada, K. N. et al. Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state. Nat. Commun. 7:12245doi: 10.1038/ncomms12245 (2016).

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