Quantized Faraday and Kerr rotation and axion electrodynamics of a 3D topological insulator

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Science  02 Dec 2016:
Vol. 354, Issue 6316, pp. 1124-1127
DOI: 10.1126/science.aaf5541

Shining light on a peculiar coupling

One of the long-standing predictions regarding topological insulators is the magnetoelectric effect, a coupling between a material's magnetic and electric properties. Thanks to this coupling, Maxwell's equations inside topological insulators are modified, resulting in so-called axion electrodynamics. Wu et al. used time-domain terahertz (THz) spectroscopy to observe signatures of these unusual electrodynamics in a thin film of Be2Se3. They detected tiny changes to the polarization of THz light after it passed through the thin film, confirming the expected quantization of the magnetoelectric coupling.

Science, this issue p. 1124


Topological insulators have been proposed to be best characterized as bulk magnetoelectric materials that show response functions quantized in terms of fundamental physical constants. Here, we lower the chemical potential of three-dimensional (3D) Bi2Se3 films to ~30 meV above the Dirac point and probe their low-energy electrodynamic response in the presence of magnetic fields with high-precision time-domain terahertz polarimetry. For fields higher than 5 tesla, we observed quantized Faraday and Kerr rotations, whereas the dc transport is still semiclassical. A nontrivial Berry’s phase offset to these values gives evidence for axion electrodynamics and the topological magnetoelectric effect. The time structure used in these measurements allows a direct measure of the fine-structure constant based on a topological invariant of a solid-state system.

Topological phenomena in condensed matter physics provide some of the most precise measurements of fundamental physical constants. The measurement of the quantum conductance Embedded Image from the quantum Hall effect (1) and the flux quantum from the Josephson effect (2, 3) provide the most precise value for Planck’s constant Embedded Image. More recently, topological insulators have been discovered (46), in which topological properties of the bulk wave functions give rise to a topologically protected surface metal with a massless Dirac spectrum. It has been proposed that topological insulators are best characterized not as surface conductors but as bulk magnetoelectrics (7, 8) with a quantized magnetoelectric response coefficient whose size is set by the fine-structure constant Embedded Image. Such a measurement could provide precise values for three fundamental physical constants: the electric charge Embedded Image, Planck’s constant Embedded Image, and the vacuum impedance Embedded Image in a solid-state context.

Magnetoelectrics (ME) are materials in which a polarization can be created by an applied magnetic field or a magnetization can be created by an applied electric field (9); representative examples are Cr2O3 (10) with an ME coupling of the Embedded Image form and multiferroic BiFeO3 (11), where the ME coupling can be expressed (in part) in a Embedded Image form. Topological insulators (TIs) can be characterized as special Embedded Image magnetoelectrics (7, 8), which in the topological field theory can be shown to be a consequence of an additional term Embedded Image added to the usual Maxwell Lagrangian (7). Here, Embedded Image is the fine-structure constant, and Embedded Image and Embedded Image are the permittivity and permeability of free space.

Although Embedded Image is a generic expression that can be applied even to Cr2O3 [with Embedded Image at low temperature (12)], its form merits additional discussion when applied to TIs. Although it is usually said that one must break both time-reversal symmetry (TRS) and inversion symmetry (IS) to define a magnetoelectric coefficient, this is not formally true. The Lagrangian defines the action Embedded Image, and because all physical observables depend on expEmbedded Image, they are invariant to global changes to Embedded Image of Embedded Image. Therefore, due to the transformation properties of Embedded Image and Embedded Image, if either TRS or IS are present, Embedded Image is constrained to be not just zero (as it is in a nonmagnetoelectric conventional material) but can take on integer multiples of Embedded Image. Three-dimensional (3D) insulators in which either TRS or IS is preserved can be divided into two classes depending on whether Embedded Image is Embedded Image (topological) or Embedded Image (conventional) (7). Here, N is an integer that indicates the highest fully filled Landau level (LL) of the surface if TRS is broken. In either case, Embedded Image can be formulated as a bulk quantity modulo a quantum (here, Embedded Image) in much the same way as the electric polarization P in a ferroelectric can be defined only as a bulk quantity modulo a dipole quantum that depends on the surface charge (13). It is important to note that to support a macroscopic magnetic/electric moment of the sample from an applied electric/magnetic field, macroscopic TRS and IS must both be broken (as they are in conventional magnetoelectrics), but a finite magnetoelectric Embedded Image term is more general than the capacity to support a moment. Because inversion-symmetric Bi2Se3 in magnetic field breaks only TRS, such a sample cannot exhibit a net macroscopic moment from magnetoelectricity unless IS is broken macroscopically through some other means. In the case relevant for our experiment, IS constrains the crystal’s bulk Embedded Image term to be Embedded Image. A net macroscopic moment cannot be generated, but the sample is still magnetoelectric in the sense that Embedded Image still applies. The topological magnetoelectric effect (TME) of this kind has been called “axion electrodynamics” because of an analogy that can be made to the physics of the hypothetical axion particle that was proposed to explain charge conjugation parity symmetry violation (CP violation) in the strong interaction (14).

In the limit where a TRS breaking field is small and the surface chemical potential is tuned near the Dirac point, modified Maxwell’s equations can be derived (section 1 of supplementary text) from the full Lagrangian. The modified Gauss’s and Ampère’s laws readEmbedded Image(1)Embedded Image(2)The consequences of axion electrodynamics are the additional source and current terms in Eqs. 1 and 2 (7, 15). The additional current term gives a half-integer quantum Hall effect (QHE) on the TI surface (7). Although there has been some evidence for half-integer QHE effects in gated TI BiSbTeSe2 exfoliated flakes (16), gated (Bi1­-xSbx)2Te3 thin films (17), and surface charge-transfer doped pure Bi2Se3 films (18) at very high magnetic fields, it is generally not straightforward to observe the QHE in a conventional dc transport-style experiment with leads connected to sample edges, as TIs have a closed surface with no boundaries (19). It is desirable then to use noncontact probes such as Faraday and Kerr rotations (7, 20, 21), which have been predicted to be quantized with a scale set by the fine-structure constant. One can proceed from the modified Ampère’s law Eq. 2 in conjunction with the usual Faraday’s law to derive the reflection and transmission coefficients for a traveling wave incident on a TI surface (section 2 of supplementary text). In an applied magnetic field, one finds that for a TI film on a simple dielectric substrate, the Faraday rotation in the quantum regime isEmbedded Image(3)where n ~ 3.1 is the THz range index of refraction of the substrate (sapphire) and Embedded Image, Embedded Image are the highest fully filled LL of the top and bottom surfaces of the film, which depend on the chemical potential and size of the TRS breaking field.

There have been a number of interrelated challenges in realizing the TME experimentally. First, one must have a negligible level of bulk carriers and a low chemical potential at the surface, but most known topological insulators suffer from inadvertent bulk doping; a metallic gate cannot be used easily in an optical experiment to gate away charge carriers because it would have its own Faraday effect in field. Second, as the topological field theory is derived for the translationally invariant case, one may expect that it will apply only when the TRS breaking perturbation is strong enough to overcome disorder and establish a surface QHE. Third, to reveal the TME, the probe frequencies and temperatures must be well below the Landau level spacing of the surface states, which are given by Embedded Image (where Embedded Image is the Fermi velocity). This puts the relevant frequency in the traditionally challenging sub-THz part of the electromagnetic spectrum. Fourth, THz range experiments with their long wavelengths require large uniform samples of at least a few mm in spatial extent. Fifth, as the size of the effect is set by the fine-structure constant, the rotations are expected to be very small and much smaller than the capacity of conventional THz range polarimetry.

Here, we overcome these challenges by using recently developed low-density and high-mobility Bi2Se3 molecular beam epitaxy (MBE) films (18), in conjunction with a high-precision polarimetry technique (22). Time-domain terahertz spectroscopy (TDTS) is a powerful tool to study the low-energy electrodynamics of topological insulators. Samples are thin films of Bi2Se3 grown by MBE with a recently developed recipe (18) that results in true bulk-insulating TIs with low surface chemical potential. These films were further treated in situ by a thin charge-transfer layer of deposited MoO3 that further decreases the carrier density and puts the chemical potential close to the Dirac point. MoO3 is a semiconductor with a gap of ~3 eV (23) and does not contribute to Faraday rotation. Details of the film growth can be found in (18).

First, as topological surface states have been shown to be the only conducting channel in these films (18), we further verify the low Fermi energy of these films through measurements of their optical conductivity and cyclotron resonance at low field in the semiclassical transport regime (figs. S1 and S2 and section 3 of supplementary text). Next, we explore their low-frequency Faraday rotation (Fig. 1A). The complex THz-range Faraday rotation was measured with the polarization modulation technique (22, 24). The Faraday rotation is a complex quantity in which the real part is the rotation of the major axis of the ellipse and the imaginary part is related to the ellipticity (Fig. 1B). The full-field data of a 10-QL sample is shown in Fig. 1, C and D. At low fields (<4 T), the Faraday rotation shows semiclassical cyclotron resonance, as demonstrated by the shifting of the inflection point (close to the zero value) in the real part and the shifting of the minimum in the imaginary part with fields (24). For the 10-QL sample, above 5 T, the inflection point in the real part of the Faraday rotation moves above our frequency range, and the low-frequency tail becomes flat and overlaps with higher field data. In our TDTS measurements, top and bottom states are measured simultaneously (18, 24, 25), so the quantized Faraday rotation is given by Eq. 3.

Fig. 1 Quantized Faraday rotation of the topological surface states.

(A) Diagram of the Faraday rotation experiment. P1, P2, and P3 are polarizers. The polarization plane (Embedded Image) of the linearly polarized incoming THz beam is rotated by the Faraday angle Embedded Image (into the Embedded Image plane) after passing through Bi2Se3 on a sapphire substrate in a perpendicular magnetic field (Embedded Image direction). The polarization acquires an ellipticity simultaneously, as shown in (B). (C) Real part of Faraday rotation of 10-QL Bi2Se3 films with MoO3 at 4.5 K for various values of the magnetic field [color coded as in (D)]. The dash-dot line is the expectation from Eq. 3. (D) Imaginary part of Faraday rotation. A representative cyclotron frequency is marked by a red arrow for data at 2.5 T. (E) Quantized Faraday rotation for different samples. Dashed black lines are theoretical expectation values assuming certain values for the filling factor of the surface states. (F) dc transport Hall resistance of a representative 8-QL sample.

These data are well described by the prediction for the Embedded Image plateau. Because the resolution of our THz polarimetry is within 1 mrad, we conclude that the 10-QL sample enters the quantized regime when the field is above 5.75 T, with the low-frequency tails falling on the expected value. Similarly, for the 6-, 8-, 12- and 16-QL samples, the low-frequency Faraday rotations fall on the Embedded Image, Embedded Image, Embedded Image, and Embedded Image plateaus, respectively (Fig. 1E). Aside from the filling factor differences, the only qualitative differences between samples is that thicker samples have a narrower magnetic field range where the Faraday rotation is quantized because they have a slightly higher carrier density and filling factor at the same magnetic fields (figs. S3 and S4 and section 3 of supplementary text). Because we measure the top and bottom surfaces of the thin film simultaneously, essential for our interpretation in terms of axion electrodynamics is that we can treat the top and bottom surfaces independently. Previous angle-resolved photoemission spectroscopy work (26) and theory (27) showed that the hybridization gap from top and bottom surfaces was negligible for film thicknesses more than 6 QL (see further discussion in section 4 of supplementary text).

It is important to point out that this effect is not just the conventional dc quantum Hall effect. We can contrast the quantized optical response with dc transport that has shown quantum Hall resistivity plateaus in these films only above ~24 T, as shown for a typical sample 8-QL film in Fig. 1F. When an external magnetic field is applied perpendicular to the films, top and bottom surface states are gapped because of LL formation, whereas the side surfaces parallel to the magnetic field remain gapless because a small in-plane field will cause only a shift of the Dirac points (28). The dc QHE in conventional 2D electron gas (2DEG) is usually regarded as occurring through ballistic 1D chiral states formed at the edge of the sample. In the present case, the dc QHE is corrupted at low fields by the nonchiral side states (Fig. 2B) . These side surface states can be gapped by an amount Embedded Image through finite size effects (where d is the film thickness), but in order that they do not contribute to dc transport, this gap must be larger than the LL spacing Embedded Image (19). This condition is hard to fulfill with films thick enough to be effectively 3D and with fields large enough to establish a surface QHE. We believe that quantized dc transport is achieved in high fields because the nonchiral side states localize in high magnetic field in the highly disordered edges. In the present experiment, THz radiation is focused onto a local spot far from the edges of the film, so irrespective of their properties they cannot contribute to the spectral response. The Hall response measured here originates in the “bulk” of the sample (topological surface states), and the edge state picture does not apply. Section 5 of the supplementary text provides further discussion on the ac QHE and on how the incompressible bulk responds to an oscillating charge density in an ac experiment.

Fig. 2 Quantized Kerr rotation of the surface states.

(A) Diagram of the Kerr rotation experiment. The black and blue arrows show the optical path for the first and second pulses in the time trace. (B) Bi2Se3 thin film in a magnetic field (substrate not shown). Nonchiral edge states from the side surface states are shown by bidirectional arrows. The circle on the top surface indicates the THz spot where transverse oscillating charge density is built up by the THz electric field. (C) Real and (D) imaginary part of Kerr rotation of 10-QL new Bi2Se3 with MoO3 at 4.5 K. (E) Quantized Kerr rotation for different samples.

Data in Fig. 1 gives evidence for a Faraday rotation set by the fine-structure constant. However, such measurements by themselves are limited, as Eq. 3 shows that the Faraday rotation still depends nonuniversally on the index of refraction of the substrate n, and our ability to measure the fine-structure constant to high precision is limited by our knowledge of n. However, by using the explicit time structure of TDTS we can define and measure a quantity that depends only on the fine-structure constant (and surface filling factors). When THz light is transmitted through a film and substrate, the substrate itself can be used as an optical resonator (29, 30), resulting in a series of pulses that each have different histories of interaction with the film (Fig. 2A). The first peak that is transmitted through the film undergoes a Faraday rotation, whereas the second peak undergoes an additional reflection and Kerr rotation Embedded Image. By subtracting the Faraday rotation, we can measure the Kerr rotation separately. In the quantized regime, one can show (section 2 of supplementary text) that the Kerr rotation (up to factors of the third order in α) isEmbedded Image(4)Representative data for the 10-QL sample for the Kerr rotation is shown in Fig. 2, C and D. Similar to the Faraday rotation, the signatures of cyclotron resonance are inflection points in the real part and dips in the imaginary part. Above 5.75 T, the Kerr rotation of a 10-QL sample is quantized as Embedded Image to within our experimental resolution at frequencies below 0.8 THz. The prefactor of 5 is the same as arrived at in the Faraday rotation. We measured Kerr rotation on samples with different thickness—6, 8, 12, and 16 QL—and in all cases the rotation is given by Embedded Image times the filling factor found in the Faraday rotation experiments (Fig. 2E). Combining Eqs. 3 and 4, one can eliminate the dependence on the index of the substrate and measure the fine-structure constant directly.

Embedded Image(5)

Measuring these quantities in a single scan and taking ratios also serves to minimize the systematic noise in the output for Embedded Image. Using Embedded Image and Embedded Image for the 10- and 12-QL samples, respectively, we plot the results of Eq. 5 for two samples in Fig. 3, A and B, and find for both that the measured value is close to 1/137 (~7.3) mrad. Averaging over the frequency range that quantized rotation is observed (0.2 to 0.8 THz) for all samples measured, we find a best measured value for Embedded Image of 1/137.9, which is close to the accepted value 1/137.04. This represents a direct measurement of the fine-structure constant based on a topological invariant in a solid-state system. Although the level of precision that we have achieved for Embedded Image is far less than, for instance, its determination via the anomalous magnetic moment of the electron (31), the quantization should be considered quite good. Its deviation from the accepted value is ~0.5Embedded Image, which can be compared favorably to the quantization seen in the quantum spin Hall effect, which was quantized only to the 10Embedded Image level (32). Moreover, the observed quantization is far better than that observed previously in the ac QHE of 2DEG-like GaAs heterostructures and graphene (33, 34). If this measurement could be further refined, it could, along with measures of the Josephson effect and quantized Hall resistances in 2DEG, provide a purely solid-state measure in a redefined conventional electrical unit scheme for the impedance of free space Embedded Image, which would in turn allow c to become a measured quantity in a condensed matter experiment.

Fig. 3 A measure of the fine-structure constant and evidence for axion electrodynamics and the TME.

The measured fine-structure constant of (A) 12-QL and (B) 10-QL new Bi2Se3 with MoO3 at different fields at 4.5 K. (C) Measured quantization index versus filling factor. The solid line is the expectation for quadratic bands, and the dashed line is for two topological surface states.

It is important to distinguish our results from a conventional QHE, as may be observed in a 2DEG. Do we truly probe axion electrodynamics and the TME effect? As discussed above, the TME is characterized by a Embedded Image angle that is Embedded Image or equivalent a half-integer QHE effect. In Fig. 3C, we plot the observed quantization index versus the total filling factor (which can be measured independently, as discussed above and in sections 3 and 6 of the supplementary text). There is a systematic offset of 1 in the position of the plateaus that originates from the Berry’s phase. With our previous results establishing surface state transport from two independent surfaces (18, 24), one must associate a contribution to this offset of 1/2 for each surface by itself. This establishes a Embedded Image value of the axion angle of the topological insulators and the TME.

The quantized response that we find here should not be viewed as a simple manifestation of the quantized quantum Hall transport seen in usual 2DEGs, because TI surface states live on a closed 2D manifold embedded in 3D space. In a formalism in which the TIs are described as bulk magnetoelectrics, this response can be described in the context of a topological magnetoelectric effect and axion electrodynamics. Going forward, the technique may prove to be an essential tool in the discovery of theoretically anticipated states of matter such as fractional topological insulators in the form of a fractional magnetoelectric effect (35, 36).

Supplementary Materials

Materials and Methods

Supplementary Text

Figs. S1 to S4

Movies S1 and S2

References (3745)

References and Notes

Acknowledgments: We thank M. Franz, T. Hughes, A. MacDonald, J. Maciejko, J. Moore, M. Orlita, V. Oganesyan, W.-K. Tse, A. Turner, R. Valdés Aguilar, X. L. Qi, and S.C. Zhang for helpful discussions. Experiments were supported by the Army Research Office Grant W911NF-15-1-0560, with additional support by the Gordon and Betty Moore Foundation through grant GBMF2628 to N.P.A. at Johns Hopkins University. Film growth for this work was supported by the NSF DMR-1308142, EFMA-1542798, and the Gordon and Betty Moore Foundation EPiQS Initiative Grant GBMF4418 to S.O. at Rutgers.
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