Observation of the universal magnetoelectric effect in a 3D topological insulator

Abstract

The electrodynamics of topological insulators (TIs) is described by modified Maxwell's equations, which contain additional terms that couple an electric field to a magnetization and a magnetic field to a polarization of the medium, such that the coupling coefficient is quantized in odd multiples of α/4π per surface. Here we report on the observation of this so-called topological magnetoelectric effect. We use monochromatic terahertz (THz) spectroscopy of TI structures equipped with a semitransparent gate to selectively address surface states. In high external magnetic fields, we observe a universal Faraday rotation angle equal to the fine structure constant α=e² /2hc (in SI units) when a linearly polarized THz radiation of a certain frequency passes through the two surfaces of a strained HgTe 3D TI. These experiments give insight into axion electrodynamics of TIs and may potentially be used for a metrological definition of the three basic physical constants.

ARTICLE

Received 24 Oct 2016 | Accepted 6 Mar 2017 | Published 15 May 2017

V. Dziom1, A. Shuvaev1, A. Pimenov1, G.V. Astakhov2, C. Ames3, K. Bendias3, J. Bttcher4, G. Tkachov4, E.M. Hankiewicz4, C. Brne3, H. Buhmann3 & L.W. Molenkamp3

The electrodynamics of topological insulators (TIs) is described by modied Maxwells equations, which contain additional terms that couple an electric eld to a magnetization and a magnetic eld to a polarization of the medium, such that the coupling coefcient is quantized in odd multiples of a/4p per surface. Here we report on the observation of this so-called topological magnetoelectric effect. We use monochromatic terahertz (THz)

spectroscopy of TI structures equipped with a semitransparent gate to selectively address surface states. In high external magnetic elds, we observe a universal Faraday rotation angle equal to the ne structure constant a e2/2E0hc (in SI units) when a linearly polarized THz

radiation of a certain frequency passes through the two surfaces of a strained HgTe 3D TI. These experiments give insight into axion electrodynamics of TIs and may potentially be used for a metrological denition of the three basic physical constants.

DOI: 10.1038/ncomms15197 OPEN

Observation of the universal magnetoelectric effect in a 3D topological insulator

1 Institute of Solid State Physics, Viennta University of Technology, 1040 Vienna, Austria. 2 Physikalisches Institut (EP6), Universitat Wrzburg, 97074 Wrzburg, Germany. 3 Physikalisches Institut (EP3), Universitat Wrzburg, 97074 Wrzburg, Germany. 4 Institut fr Theoretische Physik und Astronomie, Universitat Wrzburg, 97074 Wrzburg, Germany. Correspondence and requests for materials should be addressed to A.P. (email: mailto:[email protected]

Web End [email protected] ) or to G.V.A. (email: mailto:[email protected]

Web End [email protected] ).

NATURE COMMUNICATIONS | 8:15197 | DOI: 10.1038/ncomms15197 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications 1

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms15197

Maxwells equations are in the foundation of modern optical and electrical technologies. To apply Maxwells equations in conventional matter, it is necessary to

specify constituent relations, describing the polarization Pc(E)

and magnetization Mc(B) as a function of the applied electric and magnetic elds, respectively. Soon after the theoretical prediction13 and experimental discovery of two-dimensional (2D) and three-dimensional (3D) topological insulators (TIs)4,5, it has been recognized that the constituent relations in this new phase of quantum matter contain additional cross-terms Pt(B)

and Mt(E) when time-reversal symmetry is weakly broken6.

PtB N

1

2

1

Here N is an integer and aE1/137 is the ne structure constant. The derivation of equation (1) is based on the topological eld theory of time-reversal invariant insulators6. Its intriguing consequences are the universal Faraday rotation angle yF

j j a,

when a linearly polarized electromagnetic radiation passes through the top and bottom topological surfaces6,7, and magnetic monopole images, induced by electrical charges in proximity to a topological surface8. However, experimental verication of these topological magnetoelectric effects (TMEs) has been lacking. As the modied Maxwells equations describing electrodynamics of TIs are applicable in the low-energy limit, optical experiments should be performed at terahertz (THz) or sub-THz frequencies912. Qualitatively, equation (1) applied to the magnetic and electric elds of the primary THz radiation results in a perpendicular polarized secondary THz radiation. The sum of the primary and secondary radiation can be viewed as the rotation of the polarization plane, that is, as the Faraday effect. We would like to note that the quantum Faraday effect and the TME are basically different manifestations of the same axion physics13.

In real samples, the TME may be screened by nontopological contributions1315. In fact, quantized Faraday rotation has been detected in 2D electron gas16 and graphene17 in the quantum Hall effect (QHE) regime. However, in both experiments the Faraday rotation takes a value yF 4a/(1 nsub), that is, it

depends on the refractive index of the substrate nsub and hence

is not fundamental.

Here we report on the observation of the universal Faraday rotation angle equal to the ne structure constant a.

Strained HgTe layers grown on CdTe, that are investigated in the present work, are shown to be a 3D TI18 with surface-dominated charge transport19 that was observed in THz experiments as well9,11,20. To eliminate the material details, we perform measurements under antireection conditions, such that the transmission through the CdTe substrate is approaching 100% (ref. 21).

ResultsTHz Faraday effect in 2D systems. The observed Faraday rotation angle yF

j j a (for N 0) comes from two spatially

separated topological surfaces in a 3D TI. This corresponds to the half-quantized Hall conductivity e2/(2h) per surface or, equivalently, to the TME occurring at each surface separately. Therefore, the observed Faraday effect 2(N 1/2)a is intimately

related to the TME, which distinguishes qualitatively our 3D TI from 2D or quasi-2D materials. There is also a quantitative difference. Even without the substrate (nsub 1), the Faraday

rotation in graphene would be quantized as 4(N 1/2)a, including

the spin and valley degeneracies. The minimum Faraday rotation

angle is then yF

j j 2a (for N 0)17. For a conventional

2D electron gas, such as hosted in GaAs/AlGaAs hetero-structures, the quantization of the Faraday angle is expected to be 2Na, where the factor of 2 comes from the equal contributions of the up- and down-spin subsystems, which independently exhibit the integer QHE. This is because in GaAs/AlGaAs heterostructures, the Zeeman splitting for magnetic elds below 10T is negligible compared to the THz photon energy. Therefore, the minimum Faraday rotation angle would be also yF

j j 2a (for N 1)16, which

is twice larger than our result.

Sample details. The strained HgTe lm is a 58 nm thick HgTe layer embedded between two Cd0.7Hg0.3Te layers (Fig. 1a).

The Cd0.7Hg0.3Te layers have a thickness of 51 nm (lower layer) and 11 nm (top/cap layer), respectively. The purpose of these layers is to provide the identical crystalline interface for top and bottom surface of the HgTe lms as well as to protect the HgTe from oxidization and adsorption. This leads to an increase in carrier mobility with a simultaneous decrease in carrier density compared to uncaped samples18. The transport characterization on a standard Hall bar sample shows a carrier density at 0 V gate of 1.7 1011 cm 2 and a carrier mobility of

2.2 105 cm2 V 1 s 1. The optical measurements are carried

out on a sample tted with a 110 nm thick multilayer insulator of SiO2/Si3N4 and a 4 nm thick Ru lm. The Ru lm (oxidized in the air) is used as a semitransparent top-gate electrode22. The gate leads to B15% suppression of the transmission signal, which can be taken into account as eld-independent contribution to the conductivity. The properties of the gate material have been investigated in a separate experiment.

THz spectroscopy. The transmittance experiments at THz frequencies (0.1 THzovo1 THz) are carried out in a MachZehnder interferometer arrangement23,24, allowing measurement of the amplitude and the phase shift of the electromagnetic radiation in a geometry with controlled polarization (Fig. 1a). The monochromatic THz radiation is provided by a backward-wave oscillator. The THz power on the sample is in between 10 and 100 mW with the focal spot of 0.2 cm2. Using wire grid polarizers, the complex transmission coefcient t t

j jeif is obtained both

in parallel tp (Fig. 1b) and cross tc (Fig. 1c) polarization geometries, providing full information about the transmitted light. External magnetic elds B 7 T are applied using

a split-coil superconducting magnet. The experiments are carried out in Faraday geometry, that is, with B applied parallel to the propagation direction of the THz radiation. The ac conductivity tensor ^

so at THz angular frequency o 2pn

is obtained from the experimental data by inverting the Berreman equations25 for the complex transmission coefcient through a thin conducting lm on an insulating substrate. The explicit expressions used in the calculations are given in Methods section.

In general case, the light propagating along the z direction can be characterized by the orthogonal x and y components of the electric and magnetic elds, which can be written in the form of a 4D vector V. The interconnection between vectors V1 and V2, corresponding to different points in space separated by a distance , is given by V1

^

MV2. Here

a 2p B

MtE N

1

2

a2p E:

M is a 4 4 transfer matrix.

For an insulating substrate of thickness and dielectric constant e, this is the identity matrix ^

MCdTeI provided 2 e

^

p n=c is an integer. We nd in a separate experiment on a bare CdTe substrate that this condition is fullled for vE0.35 THz, and all the measurements presented here are performed at this frequency to minimize the contribution to the Faraday signal from the substrate. The corresponding photon energy of 1.4 meV is much

2 NATURE COMMUNICATIONS | 8:15197 | DOI: 10.1038/ncomms15197 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms15197 ARTICLE

a b

0.85

Parallel polarizations, |t p|

Crossed polarizations, |t c|

0.35 THz BWO

Ru(4 nm)

SiO2/Si3N4 (110 nm)

0.80

0.75

Y

0.70

X

0.65

UG

0.60

Cd0.7Hg0.3Te (11 nm)

Cd0.7Hg0.3Te (51 nm)

c

0.15

HgTe58 nm) CdTe

0.10

B

[afii9825]

0.05

0.00

0 1 2 3 4 5 6 7

Magnetic field (T)

Figure 1 | THz magnetooptics of a strained HgTe 3D TI. (a) A scheme of the experimental set-up (only one arm of the MachZehnder interferometer is shown). The strained HgTe layer, which is a 3D TI, is sandwiched between (Cd,Hg)Te protecting layers. The top-gate electrode, consisting of a SiO2/Si3N4 multilayer insulator and a thin conducting Ru lm, is semitransparent at THz frequencies. The THz radiation (v 0.35 THz) is linearly polarized, and the

Faraday rotation (yF) and ellipticity (ZF) are measured as a function of the magnetic eld strength B for different gate voltages UG. (b,c) Transmission spectra in the parallel tp

and crossed tc

j j polarizer congurations, respectively. The gate voltage is colour-coded, and the experimental curves are shifted

for clarity. Notations in b,c: e denotes the CR of the topological surface states of electron character, s1 and s2 denote extra resonances with opposite phase to that of the e-CR as discussed in the text.

smaller than the energy gap in strained HgTe (above 10 meV)18, and equations (1) are a good approximation.

For normal incidence, the elds across the conducting interface are connected by the Maxwell equation r H^

sE. Here

the e iot time dependence is assumed for all elds. As the wavelength of 856 mm for v 0.35 THz is much larger than

the HgTe layer thickness, we use the limit of thin lm, and the corresponding transfer matrix ^

MHgTe ^

s

is determined by

the diagonal (sxx) and Hall (sxy) components of the conductivity tensor ^

s. Within the Drude-like model, these components for one type of charge carriers can be written in the form13,26

sxx

syy

1 iot

1 iot

2 Oct

2

s0; 2

sxy

syx

Oct

s0: 3

Here Oc is the cyclotron resonance (CR) frequency, s0 is the dc conductivity, and t is the scattering time. For classical conductors, the CR frequency is written as Oc eB/me, where

me is the effective electron cyclotron mass.The total transfer matrix ^

M

1 iot

2 Oct

2

MCdTe ^

^ MHgTe relates vectors V on both sides of the sample and hence contains full information about the transmission and reection coefcients. Thus, when ^

MCdTe is the identity matrix, the inuence of the substrate is minimized, and the THz response is dominated by the ac transport properties of the HgTe layer, in accord with equations(2) and (3). The calculation of the complex transmission coefcients tp and tc based on the transfer matrix formalism as well as the exact form of the transfer matrices are presented in Methods section.

Magnetic eld dependence of the THz transmission is dominated by a sharp CR of surface electrons (e) Oce at

Be0:4 T (Fig. 1b,c). Below we demonstrate their Dirac-like

character and that they are responsible for the universal Faraday rotation. Remarkably, the observation of the CR both in tp and tc indicates a high purity of our HgTe layer. The scattering time is signicantly longer than the inverse THz frequency ot 1,

and according to equations (2) and (3) the ac conductivity reveals a resonance-like behaviour sxx, sxyp1/(O2ce o2).

Further features are broad resonances at Bs13:7 T and at

Bs22:2 T indicated in Fig. 1 as s1 and s2, respectively. The phase

of the corresponding THz transmission coefcient fc in the vicinity of these resonances has the opposite sign with respect to that of the e-CR. Remarkably, the s1 and s2 resonances disappear with applying positive gate voltage (Figs 1 and 2b). We associate them with either interband Landau-level transitions or thermally activated states as discussed below.

Band structure analysis. To understand the origin of the experimentally observed resonances, we analyse the band structure of tensile strained Cd0.7Hg0.3Te/HgTe layer as shown in Fig. 2a. It is obtained similar to ref. 19 within the tight binding approximation of the 6 6Kane Hamiltonian27,28. Due

to reduced point symmetry at the boundary between the Cd0.7Hg0.3Te and HgTe layers, an additional interface potential is allowed in the Hamiltonian29. This potential is used to shift the Dirac point closer to the valence band edge, so that the tight binding results are in good agreement with recent angle-resolved photoemission spectroscopy (ARPES) experiments18,30 and ab initio calculations31 on HgTe. Final position of the Dirac point is at energy around 40 meV (that is, it is burried in the the

valence band) in agreement with18,30,31. The Dirac-like surface states are located in the band gap between the light-hole (LH, conduction) and heavy-hole (HH, valence) subbands (see the red

NATURE COMMUNICATIONS | 8:15197 | DOI: 10.1038/ncomms15197 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications 3

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms15197

a b

40

1

20

0

Subband energies, E (meV)

2D dcconductivity, [afii9846] 0Z 0

0

1

20

2

c/ B (cm

40

15

3

0 0 1 2

60 0.0 0.2 0.4 0.0 0.5 1.0 1.5 2.0

Wave vector, k (nm1)

Gate voltage, UG (V)

Figure 2 | Charge carriers in strained HgTe. (a) The band structure of the Cd0.7Hg0.3Te/HgTe heterostructure close to the G-point for UG 1.9 V. The

chemical potential (green line) crosses the Dirac-like surface state (red solid line) in the band gap corresponding to the electron CR Oce. The dashed red line shows the linear dispersion of the 2D surface state without hybridization with the heavy-hole band. (b) 2D dc conductivity s0 of different charge carriers (e and Ss s1 s2), obtained by Drude-like ts to equations (2) and (3) of the magnetooptical spectra. The dimensionless values are given relative

to the impedance of free space Z01=cE0 377 Ohm. The inset shows the e-CR in terms of Oce/B as a function of the gate voltage UG.

line in Fig. 2a), and they are not perfectly linear due to the hybridization with the heavy-hole band. The dashed red line in Fig. 2a shows the linear dispersion of the 2D surface state without hybridization with the heavy-hole band. This is a hypothetical curve, since in the realistic material the hybridization with the heavy-hole band is non-zero. The camel back of the heavy-hole band originates from coupling of this band to the electron-like valence band and is therefore a hallmark of the inverted band structure of HgTe. In accordance with previous transport data, the chemical potential crosses the topological surface states for a large range of gate voltages19. The total electron density in Fig. 2a is n 2 1011 cm 2, representing the experimental situation at

UG1:9 V. For simplicity, we assume here the same density at the

top and bottom surfaces. Using the general formula for a quasi-classical CR32 Oc 2peB2 @E k @A, where E(k) is the energy dispersion,

B is the magnetic eld and A is the area enclosed by the wave vector k, we calculate for the topological surface state Oce=B 35 cm 1T 1.

Quantized THz Hall effect. Experimentally, simultaneous t of the real and imaginary parts of tp and tc allows the extraction of all transport characteristics, that is, conductivity, charge carrier density, scattering time and CR frequency21. The inset of Fig. 2b shows experimentally determined electron CR as a function of gate voltage, which perfectly agrees with the theoretical value for the topological Dirac-like surface states. Since only surface states are observed in transport experiments on the similar structures19, a possible explanation of the appearance of additional resonances is interband Landau-level transitions between the HH bulk bands and topological surface states. Such transitions are generally allowed as can be shown

using the Kubo formula. Another possibility would be thermally activated transport between the camel back of the HH bulk band and the surface states. This is generally possible since the THz eld may well induce heating of the carriers, resulting in a higher effective temperature compared to that of the lattice.

From the obtained scattering time and the CR positions in the magnetooptical spectra of Fig. 1b,c, one can calculate the mobility m tOc/B. The surface states demonstrate high mobility

me1:8 105 cm2 V 1 s 1, which agrees with the dc transport

data. Since the e-CR and s1,s2 resonances occur at different magnetic elds, their contributions to the ac transport can be clearly separated, as presented in Fig. 2b. The striking feature of this plot is that the ac conductivity of the surface states dominates at large gate voltages. In what follows, we concentrate therefore on UG41:0 V, while remaining weak contribution from the interband Landau-level transitions/thermally activated transitions are subtracted as explained in Supplementary Fig. 1.

Figure 3a demonstrates the real part of the Hall conductivity sxy. The overall behaviour is provided by the high-eld tail of the classical Drude model, that is, equation (3), resulting in a rapid suppression of sxy with growing magnetic eld. In addition to the classical behaviour, regular oscillations in @sxy/@B can be

recognized, which are linear in inverse magnetic eld (Fig. 3b). The slope of the linear behaviour changes with gate voltage, reecting gate dependence of the electron density per surface. These QHE oscillations are extrapolated to 1/2, indicating Dirac character of the surface electrons33. The oscillations of @sxy/@B in Fig. 3a are superimposed by the 1/B tail of the classical electron CR and therefore are not well pronounced. The visibility can be signicantly improved by inserting the sample in a FabryProt resonator, as we have previously demonstrated for a similar structure11.

4 NATURE COMMUNICATIONS | 8:15197 | DOI: 10.1038/ncomms15197 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms15197 ARTICLE

a

b

6

2 / h

5

THz conductivity of two surfaces, re ([afii9846] xy) in 2e

Plateau number, N

5/2

4

3

3/2

2

1

1/2

Theory

0 1 2 3 4 5 6 7 0 1 2

0

Magnetic field, B (T) Position 1/B (T1)

Figure 3 | THz QHE of the surface states. (a) The real part of the THz Hall conductivity sxy of two surfaces in units of 2e2/h, obtained at UG1:9 V

(symbols). The vertical solid lines indicate the positions of the Hall plateaus, estimated from the maxima in qsxy/qB. Theoretical calculations represented by the thin line are performed as explained in the text. The inset presents the same experimental and theoretical curves in the whole magnetic eld range, including the surface carrier CR at 0.4 T. (b) The plateau number as a function of the inverse magnetic eld for different gate voltages. The lines are linear ts to N hna/eB 1/2, where na corresponds to the carrier density on a single surface.

In magnetic elds above 5 T, the Hall conductivity clearly shows a plateau close to sxy e2/h, corresponding to a value

(1/2)e2/h per surface (Fig. 3a). Another plateau close to (3/2)e2/h per surface is also recognizable at a magnetic eld of 3 T. It is superimposed on the CR tail and therefore tilted. The steps in sxy loose their regularity in lower-magnetic elds, as can be qualitatively explained by a nite THz frequency o in magnetooptical experiments. As mentioned above, the overall behaviour of sxy(o) is provided by the classical curve of equation (3), and the real part of sxy can be approximated as sxy(o)Es0Oce/[(O2ce o2)t], which in the limit Oce o

reduces to the expression sxy ne/B, being a multiple of e2/h.

In low magnetic elds, the CR frequency Oce becomes comparable to the THz frequency o, destroying the regularities in sxy(o).

Since in strained HgTe, the Fermi level lies in the bulk band gap (see Fig. 2a and ref. 19), we attribute the observed THz QHE to the formation of the 2D Landau levels at the top and bottom surfaces of the HgTe layer (Fig. 1a). This interpretation is further substantiated by our theoretical analysis of the ac quantum Hall conductivity sxy(o) calculated from the Kubo formula for both top and bottom surface states within the Dirac model13,34.

Our two-surface Dirac model describes well the surface carrier CR (the inset of Fig. 3a). The lengths of the theoretical Hall plateaus in the high magnetic eld region (Fig. 3a) correlate correctly with the positions of the extrema in the derivative @sxy/@B. However, the model predicts much sharper transitions between the QHE plateaus, as observed in the experiment. One of two possible explanations is the heating of the surface carriers by the THz eld, resulting in a higher effective temperature compared to that of the lattice. Such a heating can occur due to inefcient energy relaxation in the electronic system through the emission of LO phonons at low temperatures35. The best t

of our experimental data is obtained with T 25 K (Fig. 3a).

Another explanation is based on spatial uctuations of the surface carrier densities, which are likely to occur in our samples due to their large lateral sizes compared to the typical Hall bars used in the dc measurements. The experimental data of Fig. 3a can alternatively be well tted assuming cold carriers (T 1.8 K) with density uctuations within 10% relative

to their nominal values. As the ts are nearly indistinguishable, we cannot quantitatively determine the contributions of both mechanisms leading to the smearing of the THz QHE plateaus.

The stronger the eld, the closer the Hall conductivity to the quantized values expected for a two-surface Dirac system

sxy

; 4

where Na,b are the integer numbers of the highest occupied Landau levels at the top and bottom surfaces, with naEnb being

the corresponding carrier densities (F0 h/ e

j j is the magnetic

ux quantum). Upon approaching the CR, the Hall conductivity deviates from the quantized values in equation (4) due to the predominance of the intraband transitions between the Landau levels. From the tting procedure, we extract the nominal carrier densities na0:92 1011 cm 2 and nb1:07 1011 cm 2.

The total surface carrier density na nb agrees well with that

obtained from Drude-like ts of magnetooptical spectra. Another tting parameter is the classical (Drude) surface conductivity sa

sbE50e2/h. Its large value indicates high-surface carrier

mobility, insuring that the condition for the quantum Hall

regime,

2

Na Nb 1

e2h ; Na;b Int

na;bF0

B

j j

p OBta;b 4R0sa;b

B

Here2

p OBvF 2 eB

j j

1=2 is the characteristic Landau-level

spacing for a Dirac system, ta,b are the scattering times of the top

j j=na;bF0

q 41,

is met for B41 T.

NATURE COMMUNICATIONS | 8:15197 | DOI: 10.1038/ncomms15197 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications 5

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms15197

a b

0.00

[afii9840] = 0.35 THz

Faraday rotation, [afii9835] F(rad)Faraday ellipticity, [afii9834] F(rad)

0.00

1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0

0.01

0.01

0.02

Faraday rotation, [afii9835] F(rad)

0.03 1 2 3 4 5 6 7

Magnetic field, B (T)

0.02

c

0.00

0.01

0.03

B = 5 T

0.02

Magnetic field, B (T)

Gate voltage, UG (V)

Figure 4 | Quantized THz Faraday rotation of Dirac fermions. Faraday rotation (a) and Faraday ellipticity (b) in a 3D HgTe TI as a function of the external magnetic eld for different gate voltages (colour-coded). The horizontal solid line in a indicates the universal Faraday rotation angle yF

a 7:3 10 3 rad. (c) Gate voltage dependence of the Faraday rotation in a magnetic eld of 5 T. The dashed lines in a,c provide the calculated

value of the Faraday rotation with realistic parameters of the experiment (including the top-gate electode) and assuming the HgTe conductivity exactly equal to sxy e2/h. The short-dashed line in c indicates the yF 2a level. The error bars are estimated taking into account the accuracy of the original

transmission data and the uncertainties due to subtraction of weak residual contribution (Supplementary Fig. 1).

and bottom carriers, R0 h/(2e2) is the resistance quantum, and

vF is the Fermi velocity of the Dirac surface states.

Quantized THz Faraday effect. Having established that the THz response of the topological surface states in high magnetic elds B45 T is determined by the conductivity quantum G0 e2/h

(Na Nb 0), we turn to the central result of this work, the THz

Faraday effect. Owing to the TME of equation (1), an oscillating electric Exe iot (magnetic Hye iot) eld of the linearly polarized

THz radiation induces in a 3D TI an oscillating magnetic aHxe iot (electric aEye iot) eld. The generated, in such a way, secondary THz radiation is polarized perpendicular to the primary polarization and its amplitude is a times smaller. This can be viewed as a rotation of the initial polarization by an angle yF

j j arctan a 7:3 10 3 rad. Indeed, Fig. 4a clearly

demonstrates that the Faraday angle in high magnetic elds is close to this fundamental value.

We rigorously characterize the THz Faraday effect, and the Faraday ellipticity ZF is shown in Fig. 4b. It is relatively small ZF

j jo yF

j j in high magnetic elds, but does not reach zero. This

observation indicates that while the TME dominates, the interaction of TIs with THz radiation is not a completely dissipationless process in our samples. Remarkably, the universal value of the Faraday angle remains robust against the gate voltage. This is demonstrated in Fig. 4c, where yF

j jEa for

0:7 VoUGo1:9 V.

DiscussionThe observed terahertz Faraday rotation equal to the ne structure constant a e2/2E0hc e2m0c/2h is a direct consequence

of the TME, conrming axion electrodynamics of 3D topological insulators. We use monochromatic terahertz spectroscopy,

providing complete amplitude and phase reconstruction, which can be applied to investigate topological phenomena in various systems, including graphene, 2D electron gas, layered superconductors and recently experimentally discovered Weyl semimetals36. Picoradian angle resolution can be achieved using a balanced detection scheme37, and the universal Faraday rotation in combination with the magnetic ux quantum F0 h/ e

j j

and the conductivity quantum G0 e2/h are suggested14 to use

for a metrological denition of the three basic physical constants, e, h and c (given that the vacuum permeability is equal exactly m04p 10 7 N=A2).

Note added in proof: Okada et al. and Wu et al. reported on the quantized Faraday rotation in TIs independently to this work and around a similar time in arXiv. We note that Okada et al.38 observed only the trajectory towards the ne structure constant. Wu et al.39 used interface doping to reduce the carrier concentration and put the chemical potential in the bulk.

Methods

Experimental technique. Magnetooptical experiments in the THz frequency range (100 GHzono1,000 GHz) have been carried out in a MachZehnder interferometer arrangement23, which allows measurements of the amplitude and the phase shift in a geometry with controlled polarization of radiation.

Theoretical analysis of magnetooptical spectra. To analyse the experimental transmission spectra, we follow the formalism described by Berreman24,25.

The THz light propagating along the z direction can be characterized by the (orthogonal) components of electric (Ex, Ey) and magnetic (Hx, Hy) elds, which may be combined in form of a four-component vector V (Ex, Ey, Hx, Hy). The

propagation of light between two points in space separated by a distance and characterized by vectors V1 and V2 can be described via 4 4 transfer matrix

^

M

as V2

^

MV1. To provide a simple example, for an isotropic dielectric substrate

6 NATURE COMMUNICATIONS | 8:15197 | DOI: 10.1038/ncomms15197 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms15197 ARTICLE

the transfer matrix is simplied to:

^

MCdTe

cos k

0 0 iZ sin k

0 cos k

iZ sin k

0

0 iZ 1sin k

cos k

0

iZ 1sin k

0 0 cos k

Data availability. The data that support the ndings of this study are available from the corresponding authors upon reasonable request.

References

1. Kane, C. L. & Mele, E. J. Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005).

2. Bernevig, B. A., Hughes, T. L. & Zhang, S.-C. Quantum spin Hall effect and topological phase transition in hgte quantum wells. Science 314, 17571761 (2006).

3. Fu, L. & Kane, C. L. Topological insulators with inversion symmetry. Phys. Rev. B 76, 045302 (2007).

4. Knig, M. et al. Quantum spin Hall insulator state in HgTe quantum wells. Science 318, 766770 (2007).

5. Hsieh, D. et al. A topological dirac insulator in a quantum spin Hall phase. Nature 452, 970974 (2008).

6. Qi, X.-L., Hughes, T. L. & Zhang, S.-C. Topological eld theory of time-reversal invariant insulators. Phys. Rev. B 78, 195424 (2008).

7. Tse, W.-K. & MacDonald, A. H. Giant magneto-optical Kerr effect and universal Faraday effect in thin-lm topological insulators. Phys. Rev. Lett. 105, 057401 (2010).

8. Qi, X.-L., Li, R., Zang, J. & Zhang, S.-C. Inducing a magnetic monopole with topological surface states. Science 323, 11841187 (2009).

9. Hancock, J. N. et al. Surface state charge dynamics of a high-mobility three-dimensional topological insulator. Phys. Rev. Lett. 107, 136803

2011:

10. Valds Aguilar, R. et al. Terahertz response and colossal Kerr rotation from the surface states of the topological insulator bi2se3. Phys. Rev. Lett. 108, 087403 (2012).

11. Shuvaev, A. M. et al. Terahertz quantum hall effect of Dirac fermions in a topological insulator. Phys. Rev. B 87, 121104 (2013).

12. Wu, L. et al. A sudden collapse in the transport lifetime across the topological phase transition in (Bi1 xInx)2Se3. Nat. Phys. 9, 410414 (2013).

13. Tse, W.-K. & MacDonald, A. H. Magneto-optical and magnetoelectric effects of topological insulators in quantizing magnetic elds. Phys. Rev. B 82, 161104 (2010).

14. Maciejko, J., Qi, X.-L., Drew, H. D. & Zhang, S.-C. Topological quantization in units of the ne structure constant. Phys. Rev. Lett. 105, 166803

2010:

15. Tkachov, G. & Hankiewicz, E. M. Anomalous galvanomagnetism, cyclotron resonance, and microwave spectroscopy of topological insulators. Phys. Rev. B 84, 035405 (2011).

16. Ikebe, Y. et al. Optical Hall effect in the integer quantum hall regime. Phys. Rev. Lett. 104, 256802 (2010).

17. Shimano, R. et al. Quantum Faraday and Kerr rotations in graphene. Nat. Commun. 4, 1841 (2013).

18. Brne, C. et al. Quantum Hall effect from the topological surface states of strained bulk HgTe. Phys. Rev. Lett. 106, 126803 (2011).

19. Brne, C. et al. Dirac-screening stabilized surface-state transport in a topological insulator. Phys. Rev. X 4, 041045 (2014).

20. Dantscher, K.-M. et al. Cyclotron-resonance-assisted photocurrents in surface states of a three-dimensional topological insulator based on a strained high-mobility HgTe lm. Phys. Rev. B 92, 165314 (2015).

21. Shuvaev, A. M. et al. Giant magneto-optical Faraday effect in HgTe thin lms in the terahertz spectral range. Phys. Rev. Lett. 106, 107404

2011:

22. Shuvaev, A. et al. Room temperature electrically tunable terahertz Faraday effect. Appl. Phys. Lett. 102, 241902 (2013).

23. Volkov, A. A., Goncharov, Y. G., Kozlov, G. V., Lebedev, S. P. & Prokhorov, A.M. Dielectric measurements in the submillimeter wavelength region. Infrared Phys. 25, 369 (1985).24. Shuvaev, A. M. et al. Terahertz magneto-optical spectroscopy in HgTe thin lms. Semicond. Sci. Technol. 27, 124004 (2012).

25. Berreman, D. W. Optics in stratied and anisotropic media: 4x4-matrix formulation. J. Opt. Soc. Am. 62, 502 (1972).

26. Palik, E. D. & Furdyna, J. K. Infrared and microwave magnetoplasma effects in semiconductors. Rep. Prog. Phys. 33, 1193 (1970).

27. Novik, E. G. et al. Band structure of semimagnetic Hg1 yMnyTe quantum

wells. Phys. Rev. B 72, 035321 (2005).28. Baum, Y. et al. Self-consistent k p calculations for gated thin layers

of three-dimensional topological insulators. Phys. Rev. B 89, 245136

2014:

29. Ivchenko, E. L., Kaminski, A. Y. & Rssler, U. Heavy-light hole mixing at zinc-blende (001) interfaces under normal incidence. Phys. Rev. B 54, 58525859 (1996).

30. Liu, C. et al. Tunable spin helical Dirac quasiparticles on the surface of three-dimensional hgte. Phys. Rev. B 92, 115436 (2015).

31. Wu, S.-C., Yan, B. & Felser, C. Ab initio study of topological surface states of strained HgTe. EPL 107, 57006 (2014).

0

B

B

1

C

C

A

: 5

p o=c is the wave vector. In the following, we assume m 1. The Berreman procedure is in general not limited to the case of normal

incidence. However, in such geometry the choice of tangential eld components simplies the treatment. Electric and magnetic elds across the interfaces are connected by the Maxwell equation r H^

Here Z

m=e p

@ and k

me

sE. Here ^

s is the complex conductivity

tensor of the material and the time dependence in form e iot is assumed.

The full transfer matrix ^

M

MCdTe ^

^ MHgTe describes the transmission and reection coefcients, which can be calculated using another basis. In this basis, the vector V consists of (i) the amplitude of the linearly polarized wave (Ex) propagating in the positive direction, (ii) the amplitude of the wave with the same polarization propagating in the negative direction, and (iii) and (iv) of two waves with perpendicular polarization (Ey). The propagation matrix in the new basis is ^

M0

^ M ^

V, with

^

V

1 1 0 0 0 0 1 1 0 0 1 1

1 1 0 0

0

B

B

@

1

C

C

A

: 6

The present experiment is described by a linearly polarized incident wave and by two components of the transmitted (t) and reected (r) waves, respectively. The equation connecting all waves is given by:

tc

0 tp

0

0

B

B

@

1

C

C

A

^

M0

0

B

B

@

0 rc

1 rp

1

C

C

A

: 7

Here the tp and tc are the complex transmittance amplitudes within paralleland crossed polarizers geometry, rp and rc are respective reectivity coefcients. The Faraday rotation y and Faraday ellipticity Z are obtained from the transmission amplitudes tp

,

tc

j j and phase shifts fp, fc as tan2y

2 tp

j jcos fp fc

tp 2

tc

tc

j j2

;

8

tc

sin2Z

2 tp

j jsin fp fc

tp 2

tc

j j2

:

To interpret the experimental data, we use the ac conductivity tensor ^

so obtained in

the classical (Drude) limit from the Kubo conductivity of topological surface states7,15. We note that for a 2D conducting lm on an isotropic dielectric substrate, the complex transmission can be obtained analytically. For a substrate with permittivity e, the nal equations for the spectra in parallel (tp) and crossed (tc) polarizers are given by:

tp

2axx

a2xx a

2xy ; 9

tc

2axy

a2xx a

2xy ; 10

where

axx 1 sxxZ0

cosk iZ sink

cosk iZ 1sink

;

11

axysxyZ0 cosk iZ sink

: 12 Here is equal to the substrate thickness, Z0E377 Ohm is the impedance of free space,

Z 1= e

p is the relative impedance of the substrate and k e

p o=c is the wave vector of the radiation in the substrate. The components of the conductivity tensor, sxx(o),

and sxy(o), are given by equations (2) and (3). As can be clearly seen, equations (9) and(10) can be inverted analytically to get the solution for the complex conductivity thus avoiding the numerical procedure.

As has been discussed previously6, in case of a lm on a substrate the universal value of the Faraday rotation angle should be modied by the refractive index of the substrate. We note that an innite substrate has been assumed in these calculations. In the present case of nite substrate, exact transmission matrix formalism has been utilized thus automatically taking into account the inuence of the substrate and of the Ru-gate. Moreover, monochromatic radiation has been used in the experiments and the frequency of this radiation has been selected close to a maximum of the FabryProt resonances in the substrate (sink ! 0). In

this case, equations (11) and (12) can be simplied to

axx2 sxxZ0 13

axysxyZ0 14

and the inuence of the substrate is minimized.

NATURE COMMUNICATIONS | 8:15197 | DOI: 10.1038/ncomms15197 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications 7

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms15197

32. Ashcroft, N. W. & Mermin, N. D. Solid State Physics (Harcourt College Publishers, 1976).

33. Buttner, B. et al. Single valley Dirac fermions in zero-gap HgTe quantum wells. Nat. Phys. 7, 418422 (2011).

34. Tkachov, G. & Hankiewicz, E. M. Spin-helical transport in normal and superconducting topological insulators. Phys. Status Solidi (b) 250, 215232 (2013).

35. Kiessling, T. et al. Spatially resolved photocarrier energy relaxation in low-doped bulk GaAs. Phys. Rev. B 86, 161201 (2012).

36. Xu, S.-Y. et al. Discovery of a Weyl fermion semimetal and topological Fermi arcs. Science 349, 613617 (2015).

37. Crooker, S. A. et al. Spin noise of electrons and holes in self-assembled quantum dots. Phys. Rev. Lett. 104, 036601 (2010).

38. Okada, K. N. et al. Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous hall state. Nat. Commun. 7, 12245 (2016).

39. Wu, L. et al. Quantized Faraday and Kerr rotation and axion electrodynamics of a 3D topological insulator. Science 354, 11241127 (2016).

Acknowledgements

This work was supported by Austrian Science Funds (I1648-N27, W-1243, P27098-N27), as well as by the German Research Foundation (DFG) through SFB 1170 ToCoTronics , the SPP 1666 and TK60/4-1, the Bavarian ENB Graduate School on Topological Insulators and the ERC (AG project 3-TOP).

Author contributions

V.D., A.S. and A.P. contributed to the THz experiments. C.A., K.B., C.B. and H.B. grew the samples and fabricated the gate electrodes. J.B., G.T. and E.M.H. performed the theoretical analysis. A.S., A.P., G.V.A., C.B., H.B. and L.W.M. conceived the experiment. G.V.A., A.P. and L.W.M. wrote the experimental part of the paper. G.T. and E.M.H.

wrote the theoretical part of the paper. G.V.A. coordinated the research project. All authors participated in the interpretation of the experiments.

Additional information

Supplementary Information accompanies this paper at http://www.nature.com/naturecommunications

Web End =http://www.nature.com/ http://www.nature.com/naturecommunications

Web End =naturecommunications

Competing interests: The authors declare no competing nancial interests.

Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions/

Web End =http://npg.nature.com/ http://npg.nature.com/reprintsandpermissions/

Web End =reprintsandpermissions/

How to cite this article: Dziom, V. et al. Observation of the universal magnetoelectric effect in a 3D topological insulator. Nat. Commun. 8, 15197 doi: 10.1038/ncomms15197 (2017).

Publishers note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afliations.

This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the articles Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Web End =http://creativecommons.org/licenses/by/4.0/

r The Author(s) 2017

8 NATURE COMMUNICATIONS | 8:15197 | DOI: 10.1038/ncomms15197 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications