Abstract
Electrodynamic responses from threedimensional 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 lowenergy magnetooptical effects. Here we report the experimental indication of the topological magnetoelectric effect, which is exemplified by magnetooptical Faraday and Kerr rotations in the quantum anomalous Hall states of magnetic topological insulator surfaces by terahertz magnetooptics. 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.
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Introduction
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 flux in superconductors and Hall conductance in quantum Hall effect. Magnetoelectric coupling, which has been a fundamental concept for contemporary physics including spintronics and multiferroics^{1}, is predicted to be quantized in the recently discovered threedimensional (3D) topological insulators (TIs)^{2,3,4,5,6,7,8,9,10}. More specifically, quantized magnetoelectric responses are predicted on the quantum anomalous Hall (QAH) state induced by the magnetic massgap on the surface Dirac cone under the broken timereversal symmetry. In the QAH state, as experimentally confirmed recently^{11,12,13,14,15,16,17,18}, the surface states exhibit quantized Hall conductance (σ_{xy}=e^{2}/h and σ_{xx}=0) without external magnetic field.
In the electromagnetic field in 3D TIs, the Lagrangian includes the axion term , which is characterized by the fine structure constant (refs 4, 19). Under the presence of timereversal symmetry the term θ is equal to π in TI, while zero in the vacuum or ordinary insulator. When the timereversal symmetry is broken at the TI surface, Maxwell’s equations are modified, 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 lowenergy region below the magnetic mass gap at the Dirac point. Since the modified Maxwell’s equations provide a quantized transverse current, the QAH effect with σ_{xy}=e^{2}/h can be viewed as a zerofrequency limit of TME effect. On the other hand, for the optical process the TME effect produces the quantized Faraday and Kerr rotation angles^{4,6,7}, which represent polarization rotations for transmission and reflection geometries, respectively. Accordingly, the relation denoted with rotation angles (θ_{F} and θ_{K}) always leads to the fine structure constant α (=2πe^{2}/hc∼1/137), irrespective of material parameters such as dielectric constant and magnetic susceptibility, whereas the magnetooptical rotation angles of a thin film on substrate are substantially modified from those of the freestanding film in vacuum (θ_{F}=α∼7.3 mrad and θ_{K}=π/2 rad) (refs 4, 6, 7). Thus the observation of Faraday and Kerr rotations on QAH state provides a direct measure of α. As a result the rotation angles of electromagnetic wave in the lowenergy 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 phenomena, QAH effect and topological Faraday and Kerr rotations, are observed as quantization of different physical quantities; σ_{xy}=e^{2}/h and θ_{F} =α=2πe^{2}/hc. On the other hand, the difficulties in experimental verification of TME effect have been indicated since the early stage of theoretical predictions^{4,6,7}. This is because the observation of TME effect requires the Fermi energy within the magnetic massgap 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, sufficiently lower than the magnetic massgap to avoid the responses from real electronic transitions.
The QAH state on the surface of TI is stabilized by the magnetic massgap, while accurate size of the gap energy may depend on sample form (film/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: Sb_{2}Te_{3} thin film^{20} to 50 meV for a Cr: (Bi, Sb)_{2}Te_{3} bulk single crystal^{21}. Therefore, the magnetooptics by terahertz (THz) spectroscopy probing the lowerenergy range, for example, 1–8 meV in the present experiment, is suitable for the observation of the possible emergence of topological Faraday/Kerr rotations. Furthermore, the recently developed magnetic modulationdoping in Cr: (Bi, Sb)_{2}Te_{3} thin film^{17} 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 lowenergy magnetooptical responses have been intensively studied^{22,23,24,25,26,27,28,29} 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 films by THz timedomain spectroscopy (TDS). The trajectory towards the fine structure constant α is unveiled by the measurements of THz Faraday and Kerr rotation angles for the surface QAH state.
Results
QAH effect observed in a magnetic TI thin film
The Cr_{x}(Bi_{0.26}Sb_{0.74})_{2−x}Te_{3} TI film with magnetic modulationdoping^{17}, where magnetic impurities Cr (x=0.57) are doped in two quintuple layers adjacent to the top and bottom (Bi_{0.26}Sb_{0.74})_{2}Te_{3} layers^{17}, is schematically illustrated in Fig. 1a. The evolution of the magnetization induced by the Crdoping gives rise to the QAH state as shown in Fig. 1b,c. The ferromagnetic transition occurs around T_{C}∼70 K with the onset of the anomalous Hall term in σ_{xy} (Fig. 1b). As temperature decreases, σ_{xy} develops and tends to saturate at the quantized value e^{2}/h at around T=0.5 K, while σ_{xx} steeply decreases towards zero, due to the emergence of the dissipationless chiral edge conduction (Fig. 1b). The Hall angle (σ_{xy}/σ_{xx}) becomes as large as 1 around 4 K, indicating the emergence of QAH regime at temperatures more than an order of magnitude higher^{17} than the uniformly Cr or Vdoped TI films^{11,12,13,14,15,16,18}, due perhaps to the enlargement of the magnetic massgap induced by the rich Crdoping and the reduced disorder in the surface states by Cr dopants^{17}. Hysteretic behaviours of Hall conductance further evidence the development of the QAH regime as shown in Fig. 1c. The fully quantized σ_{xy} at the lowest temperature indicates that the Fermi energy locates well within the magnetic massgap of the surface Dirac cone (Fig. 1a) without additional fieldeffect tuning.
THz magnetooptics on the magnetic TI thin film
THzTDS provides magnetooptical measurements with sufficiently lower photon energy (1–8 meV) than the magnetic massgap (20–50 meV (refs 20, 21)) and with high resolution of lightpolarization rotations (<1 mrad). Recently, this technique has been found to be useful to study polarization rotation in THz region on ferromagnetic semiconductors as well^{30}. The measurement configuration of magnetooptics by THzTDS is schematically illustrated in Fig. 1d (see Methods for detail). Depending on the time delay, the monocycle THz pulse can differentiate the directly transmitted pulse (i) and the delayed pulse generated by backreflection 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 films^{22,24} and graphene on substrates^{31}. As shown in Fig. 1e, the temporal waveform of E_{y}component indicates the pronounced rotation of polarization on the first pulse (i) as well as on the second one (ii) due to the presence of the magnetooptical rotations at zero external magnetic field. The first pulse (i) involves the Faraday rotation (θ_{F}), while the second pulse (ii) is composed of θ_{F} plus the Kerr rotation (θ_{K}) at the back surface of the magnetic TI film (Fig. 1d).
The transmittance spectra obtained by E_{x}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 mode^{32}; see also the optical conductance σ_{xx} spectrum at T=4.3 K also shown in Fig. 2a. The negligibly weak absorption, for example, no Drude response, confirms that the Fermi energy locates within the magnetic massgap of the surface Dirac cone (Fig. 1a).
Fourier transformation of the electric field pulses E_{x}(t) and E_{y}(t) (Fig. 1e) provides the complex Faraday and Kerr rotation spectra (Fig. 2b,c), where the real part, θ_{F}(ω) or θ_{K}(ω), and the imaginary part, η_{F}(ω) or η_{K}(ω), represent the rotation angle and the ellipticity, respectively (see Methods for detail). The rotationangle (real part) spectra for θ_{F} and θ_{K} show finite 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 η_{F} and η_{K} are close to zero in the whole photonenergy region (<8 meV). Note that noiselike fringe structures in the Kerr rotation spectra (Fig. 2c,d), in contrast to the almost ωconstant Faraday rotation spectra, come from the inevitable interference due to the temporal overlap with the earlycoming Faraday rotation signals (Fig. 1e). These characters, that is, little frequency dependence and nearzero ellipticity, strongly indicate that the current THz energy window (1–8 meV) is well below the threshold energy for any magnetooptically active real transitions. This is consistent with the fact that the magnetic massgap on the Dirac point (reported to be 20–50 meV by scanning tunnel spectroscopy^{20,21}) is sufficiently large as compared with the energy range of this measurement. Note also that possible cyclotron resonance under magnetic field, which has been observed in previous magnetooptical studies on TIs^{22,23,24,25,26,27,28}, is absent in the present measurement because of zero external magnetic field. 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 σ_{xx} and σ_{xy} obtained by the d.c. transport measurement (Fig. 1c) through the following relations^{26,31};
Here t_{+(−)} and r_{+(−)} represent the transmittance and reflection coefficients of right and lefthanded circularly polarized light, respectively. The admittance Y_{±} is described as Y_{±}=Z_{0}(σ_{xx}±iσ_{xy}) (Z_{0}=377 Ω: the vacuum impedance) and n_{s} is the refractive index of the InP substrate. We determined n_{s} as 3.47 by measuring THz response of the substrate, which well agrees with literature^{33}. For instance, the estimated rotation angles for θ_{F} and θ_{K} at ω=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 exemplifies 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 T_{C} (see also Fig. 3a), in accord with the disappearance of the ferromagnetic state. We also confirmed that the Faraday and Kerr rotations arising from the QAH state are reproducibly observed for the different sample with different heterostructure and T_{C} (∼40 K) (see Supplementary Figs 1 and 2 and Supplementary Note 1).
Since the magneticlayer thickness in the present heterostructure film is 2 nm in total, the figure of merit of spontaneous Faraday rotation of our sample can be effectively regarded as ∼7 × 10^{5} degree per cm in the offresonant condition with nearzero ellipticity. For comparison, Faraday rotation of most wellknown Faraday rotator Bidoped yttrium iron garnet is 9 × 10^{2} 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.
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 ∼4 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 ω=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 ω=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 fine structure constant α, irrespective of any material parameters such as the dielectric constant and the thickness of the film, the capping layer and the substrate^{6}. In our measurement geometry, the universal relationship between θ_{F} and θ_{K} in the QAH state is obtained from equations (1) and (2);
Here we define the left side of equation (3) as the scaling function f (θ_{F}, θ_{K}). In Fig. 3c, the function f (θ_{F}, θ_{K}) versus d.c. Hall conductance σ_{xy}^{d.c.} is plotted, in which the f (θ_{F}, θ_{K}) is expected to reach α (=2πe^{2}/hc∼1/137) in the quantized limit. With increasing σ_{xy} to the quantized conductance by lowering temperature, the dimensionless f (θ_{F}, θ_{K}) approaches the universal value α, in good agreement with the estimation at ω=0 (line in Fig. 3c), manifesting the trajectory towards the quantized value α determined uniquely and solely by the magnetooptical rotation angles.
The small deviation in θ_{F}, θ_{K} and f (θ_{F}, θ_{K}) 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 field during the terahertz measurements (Fig. 1e). Indeed, the measurements under magnetic field 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 (θ_{F}, θ_{K}) (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 ingap states^{35}. 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 (σ_{xx}) 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 σ_{xx}∼0 is attained.
After submitting the manuscript, two other groups uploaded preprints on an ePrint server reporting quantized magnetooptical rotations, which were observed on nonmagnetic TIs under application of magnetic field^{36,37}. In the present study on the quantum anomalous Hall effect in zero magnetic field, we have observed the almost quantized Faraday and Kerr rotations via exchange interaction with localized magnetic moments instead of external magnetic field, 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 magnetooptical rotation angles shows the trajectory converging to the fine structure constant α with the approach to the QAH state.
Methods
Sample fabrication
The 8nmthick TI films with magnetic modulation doping were grown on bothsidepolished insulating InP substrates by molecular beam epitaxial growth as described in ref. 17. To protect the film from degradation, a 3nmthick AlO_{x} 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 film. Possible modification of rotation angles by the AlO_{x} capping layer is estimated to be as small as 0.01% at most, and hence neglected in the analysis described in the main text.
Magnetooptical terahertz spectroscopy
In the THzTDS, laser pulses with duration of 100 fs from a modelocked Ti: sapphire laser were split into two paths to generate and detect THz pulses. THz pulses were generated by a bowtieshaped antenna and detected by a dipole antenna. The E_{y}(t) component of the transmitted THz pulses (Fig. 1e) was measured by the CrossedNicole configuration by using wiregrid polarizers. The polarization rotation E_{y}(t) at 0 T is defined 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; . The Faraday rotation of the substrate (InP) is smaller than the sensitivity of our equipment (<10 μrad T^{−1}). The Fourier transformation of the first THz pulses E_{x}(t) and E_{y}(t) (Fig. 1e) gives the complex Faraday rotation spectra E_{y}(ω)/E_{x}(ω)=(sinθ_{F}(ω)+iη_{F}(ω)cosθ_{F}(ω))/(cosθ_{F}(ω)−iη_{F}(ω)sinθ_{F}(ω))∼θ_{F}(ω)+iη_{F}(ω) (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 earlycoming 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 σ(ω)=σ_{1}(ω)+iσ_{2}(ω) of TI film;
where t (ω) is the complex transmittance, Z_{0} is the impedance of free space (377 Ω) and n_{s} the refractive index of the InP substrate.
Data availability
The authors declare that the data supporting the findings of this study are available within the article and its Supplementary Information.
Additional information
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. 7:12245 doi: 10.1038/ncomms12245 (2016).
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Acknowledgements
This research was supported by the Japan Society for the Promotion of Science through the Funding Program for WorldLeading 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 GrantinAid for Scientific Research No. 24224009, 24226002 and 26706011. K.N.O. is supported by RIKEN Junior Research Associate Program.
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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 modulationdoped topological insulator thin films 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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Okada, K., Takahashi, Y., Mogi, M. et al. Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state. Nat Commun 7, 12245 (2016). https://doi.org/10.1038/ncomms12245
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DOI: https://doi.org/10.1038/ncomms12245
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