Magnetooptical determination of a topological index

When a Dirac fermion system acquires an energy-gap, it is said to have either trivial (positive energy-gap) or non-trivial (negative energy-gap) topology, depending on the parity ordering of its conduction and valence bands. The non-trivial regime is identified by the presence of topological surface or edge-state dispersing in the energy gap of the bulk and is attributed a non-zero topological index. In this work, we show that such topological indices can be determined experimentally via an accurate measurement of the effective velocity of bulk massive Dirac fermions. We demonstrate this analytically starting from the Bernevig-Hughes-Zhang Hamiltonian (BHZ) to show how the topological index depends on this velocity. We then experimentally extract the topological index in Pb1-xSnxSe and Pb1-xSnxTe using infrared magnetooptical Landau level spectroscopy. This approach is argued to be universal to all material classes that can be described by a BHZ-like model and that host a topological phase transition.

The concept of band topology in condensed matter systems has revolutionized our understanding of quantum phases of matter. 1,2,3,4 Fundamentally speaking, it has allowed us to understand how unconventional novel states of matter can emerge in systems where fundamental symmetries are preserved. Typically, the topological character of solids is governed by the orbital and parity ordering of the conduction and valence bands. In a number of materials, band topology is proven to be non-trivial, as the parity of the conduction and valence bands is inverted compared to conventional semiconductors and the sign of the energy gap is said to be negative ( Fig. 1(a)). 2,4,5,6,7 Such energy states of matter are typically attributed a non-zero topological index, that is related to the sign of the energy gap. Under certain symmetry considerations (time reversal symmetry, crystalline symmetries, etc.), systems that have a nonzero topological index host gapless metallic Dirac surface states that disperse in the band gap of the semiconductor. 2,5,6,7,8 These topological surface states (TSS) exhibit striking properties such as a relativistic energy-momentum dispersion, a helical spin texture with electron spin locked to the momentum and a high robustness against disorder. 3 The TSS give rise to novel physical effects such as quantized conductance without magnetic fields, 8 the quantum anomalous Hall effect 9,10 and may provide a basis for the realization of Majorana fermions. 11,12 A considerable number of topological material classes have been thus far identified. Among all, in the Z2 topological insulator (TI) class, the band inversion occurs at an odd number of time-reversal symmetric points 13,14 in the first Brillouin zone (BZ) whereas in topological crystalline insulators (TCI) it occurs at an even number of crystalline mirror symmetric points. 15,16,17,18,19  Both the TI 8,20,21,22,23 and TCI 17,18,24,25,26,27,28,29 states have been thoroughly studied experimentally. Moreover, the topological phase transition ( Fig. 1(a)) from trivial to non-trivial 5,30,31,32,33,34,35,36 has also been extensively investigated for both TIs and TCI. However, no direct measurement of a topological index in 3D condensed matter systems has yet been reported. Typically, the topological index is inferred from angle resolved photoemission spectroscopy measurements that observe helical Dirac surface states in 3D TI, or via the observation of the quantum spin Hall effect in 2D TI. In materials where a topological phase transition from trivial to non-trivial can be induced, the topological index can be also inferred from the observation of a closure and reopening of the energy gap. A direct measurement of the topological index would, however, be interesting to consider fundamentally, as it may allow us to shed light on pending issues concerning the thermodynamics of topological phases. For instance, are topological phase transitions first or second order in nature? 30, Does there exist a gapless 3D Dirac state at the critical inversion point? 30,31,37 What mechanism causes the band-inversion and the gapping of surface-states when the system becomes trivial? Can the gapping of surface-states and their acquisition of mass be described by an analogue of the Higgs mechanism? 31,36

Preview
In this work, we show that the topological index can be directly measured in systems exhibiting topological phase transitions using infrared (IR) Landau level spectroscopy. Starting from the Bernevig-Hughes-Zhang (BHZ) Hamiltonian, we show that to order k 2 an energy dispersion that is identical to that of massive Dirac fermions can be obtained, with a modified Dirac velocity that depends on the topological index. Using magnetooptical IR Landau level spectroscopy, we are able to measure this effective velocity with high precision in TCI Pb1-xSnxSe and Pb1-xSnxTe and thus experimentally extract the topological index. This is a first proof of concept that topological indices can be experimentally measured, and not just inferred from the observation of edge/surface states. We argue that our result is expected to be valid for other systems that exhibit a topological phase transition such as (Hg,Cd)Te, 38 Cd3As2, 39 or BiTlS1-δSeδ. 36,35 Theory The topological index in the BHZ Hamiltonian. Let us start by solving the eigenvalue problem for a system described by the general 40 BHZ effective Hamiltonian. 2 Since we define z to be the direction of the applied magnetic field, cyclotron motion in the plane perpendicular to the field will be considered. We can thus set kz=0, which gives: The case of Pb1-xSnxSe and Pb1-xSnxTe. In order to experimentally study the topological phase transition, and attempt to measure the topological index, we investigate the case of Pb1-xSnxSe and Pb1-xSnxTe IV-VI TCI, where the transition occurs ( Fig. 1(a)) as a function of changing Sn content, at four L-points in the Brillouin zone ( Fig. 1(b)). 30,31,43 ,44,45 The bulk Fermi surface of Pb1-xSnxSe and Pb1-xSnxTe is shown in Fig. 1 It is degenerate and possesses an ellipsoidal bulk carrier valley oriented parallel to the [111] axis referred to as the longitudinal valley as well as the three tilted valleys referred to as the oblique valleys. The fourfold bulk valley degeneracy combined with the mirror symmetric character of the rock-salt crystal structure of Pb1-xSnxSe and Pb1-xSnxTe yields a TCI state that hosts four-fold degenerate TSS. 5,16,25,46,47,48,49,50,51 Despite this subtlety, the band structure and Landau levels of IV-VI semiconductors is ideal to study under the proposed scope, due to their relative simplicity 24,52 compared to other materials such as II-VI and V-VI systems that have highly asymmetric conduction and valence bands. It can actually be shown that a description similar to that of BHZ can be applied to describe the band structure of Pb1-xSnxSe and Pb1-xSnxTe. Starting from a 2-band k.p Hamiltonian where Δ is defined to be the half-band-gap again, and where m is a far-band correction mass term. 43,45,52,53,54,55,56 Interestingly, in Pb1-xSnxSe and Pb1-xSnxTe, 1 M turns out to be relatively small. 57 This theoretical treatment then yields a massive Dirac-like energy dispersion identical to (4) with an effective Dirac velocity given by: This is equivalent to writing the band-edge mass as: Δ changes sign through the topological phase transition, as the conduction and valence bands swap, however, m does not change sign. Generally, m is due to interactions between the valence/conduction band states, and other bands lying far away from the band gap in energy, referred to as far-bands. The ordering of far-bands does not invert when the valence and conduction bands invert, thus the sign of m does not change for fundamental reasons. 45,58 The Landau level spectrum in this case can be determined via the standard procedure of performing a Peierls substitution and using a ladder operator formalism, following what is done is ref. 54. This is detailed in supplementary section 1; we get: Here, N is the Landau level index and ̃= /̃. ± refers to the effective spin. We highlight that this result is analytically equivalent to the result obtained by Mitchell and Wallis in 1966, 54 and subsequent work on the Landau levels of PbTe and other lead-tin-salts. 45,57 Mitchell and Wallis also neglected terms in B 2 which essentially yields the same result that we have after neglecting k 4 contributions.
For Pb1-xSnxSe and Pb1-xSnxTe, we can then evaluate the topological index, as defined by Fu 6 and later used by Juricic et al. 34 for TCIs, as:

Experimental Results
Growth and characterization. In order to study the topological transition in Pb1-xSnxSe and Pb1-xSnxTe, epilayers are grown in the [111]-direction by molecular beam epitaxy (MBE) on freshly cleaved BaF2 (111) substrates. 60,61,62 The Sn concentration x is systematically varied over a wide range, 0≤x≤0.3 for Pb1-xSnxSe and 0≤x≤0.56 for Pb1-xSnxTe. In-situ reflection high energy electron diffraction and ex-situ atomic force microscopy are initially used to characterize the films (see supplementary material S2). X-ray diffraction (XRD) (Fig. 2) is used in order to determine the lattice constant and composition with a precision better than 2%. Fig. 2(a,c) shows the (222) XRD Bragg reflection for a series of Pb1-xSnxSe (Fig. (2a)) and Pb1-xSnxTe (Fig. (2c)) films with different Sn content, illustrating the monotonic shift of the (222) diffraction peaks to higher diffraction angles with increasing Sn content. Epilayers having a thickness > 0.5 µm are proven to be fully relaxed by studying reciprocal space maps showing the (513) Bragg reflection. 24 From the peak positions, the lattice constant a0 and Sn concentration of the ternary material can be directly obtained from Vegard's law ( Fig. 2(b)).
Transport measurements are performed to extract the carrier density and mobility. Carrier densities as low as 10 17 cm -3 are achieved in Pb1-xSnxSe. For Pb1-xSnxTe x>0.25, moderate Bi doping (<10 19 cm -3 ) 63 is used to limit the carrier density to no more than p=2x10 18 cm -3 . The Hall mobility μ of the samples at 77K is measured to be around 30000 to 60000 cm 2 /Vs for Pb1-xSnxSe, and between 5000 and 20000 cm 2 /Vs for Pb1-xSnxTe. These excellent transport properties allow us to observe Landau quantization at low magnetic fields. Magnetooptical Landau-level spectroscopy of the bulk band structure. Magnetooptical IR Landau level spectroscopy is then performed in transmission mode in the Faraday geometry up to 17T, at T=4.5K in the mid-IR range. The relative transmission at fixed magnetic field T(B)/T(B=0) is extracted and analyzed. This technique is highly sensitive to the bulk, yet not blind to the TSS, and is thus an ideal tool to study the inversion of bulk energy bands in topological materials to measure the topological index. Additionally, this technique allows a quantitative assessment of the TSS band structure, via cyclotron resonance measurements. 24 Landau level transitions Fig. 3(a,b) and magnetooptical spectra Fig. 3(c,d) are shown for two Pb1-xSnxSe samples that are respectively trivial (x=0.14) and non-trivial (x=0.19) at T=4.5K. 44,30 Minima observed in Fig. 3(c,d) correspond to absorptions due to the presence of Landau-level (LL) transitions at high magnetic fields (μB>>1). Transitions from a valence band LL to a conduction band LL are referred to as interband transitions. Transitions between two LL of the same band are referred to as intraband transitions, or more simply as cyclotron resonances (CR). A large number of interband transitions is observed (Fig. 3(a,b)) down to low energies evidencing a Fermi level position close to the valence band edge in all samples. Sharp absorption lines can be seen in all spectra with a field onset close to 1T, evidencing a very high mobility. An energy cutoff at 55meV and down to 22meV in the far-IR is due to the Reststrahlen band of the BaF2 substrate.
Due to the many-valley band structure of [111]-oriented IV-VI semiconductors, two Landau level series, (Fig. 3(a,b)) pertaining to the longitudinal (black) and oblique (red) valleys (see Fig. 1(b)) are identified and analyzed for B|| [111]. The oblique valleys are tilted by 70.5° with respect to the longitudinal one. 64 The oblique velocity is lower than the longitudinal velocity. This anisotropy in D v is small for Pb1-xSnxSe but rather large for Pb1-xSnxTe (Supplement (S3)). 24 The interband transitions in Fig. 3(a,b) can be well described using a massive Dirac-like Landau-level spectrum given in Eq. (7). For kz=0 (kz // B), the selection rules for the Faraday geometry result in the transition energies given by: 54 The ground cyclotron resonance energy at kz=0 is given by: 53 The curve fits in Fig. 3(a,b) allow us to precisely determine the energy gap We find 20meV and 25meV from the band edge in x=0.14 and x=0.19 respectively, in agreement with the bulk carrier density (p≈2x10 17 cm -3 and n≈3x10 17 cm -3 , respectively).

Measurement of the topological index.
We systematically study a total of 8 Pb1-xSnxSe samples (0≤x≤0.3) and 20 Pb1-xSnxTe samples (0≤x≤0. 56). Note that in all samples, the magnetooptical transitions can be well interpreted with a massive Dirac model (Eq. 7) in order to extract  2 and D v . The analysis is shown in Fig. 4 for Pb1-xSnxSe and in the supplement (S3) for Pb1-xSnxTe. The longitudinal and oblique velocities extracted from the massive Dirac model are respectively plotted versus x in Fig. 4(a,b). They show a consistent decrease as x is increased for both materials. The band gap  2 is shown in Fig. 4(c). A minimum is observed for x=0.165 followed by an increase when x is increased beyond this concentration.
The smallest measured energy gap is 15±5meV. 30 We can still estimate c v the critical velocity to be equal to 5.0x10 5 m/s for the longitudinal valley and 4.7x10 5 m/s for the oblique valleys ( Fig. 4(a,b)), since the critical composition of Pb1-xSnxSe is known. In Fig. 4(d)  Cyclotron resonance of topological surface states. Finally, we corroborate our findings by further evidencing the observation of topological surface states that appear in the non-trivial regime. Measurements in the far-IR up to 17T are performed and are shown for x=0.14 and x=0.19 in Fig. 5(a,b). The Landau level transitions in the far-IR are shown in Fig. 5(c,d). The CR of the bulk valleys (CR-LO) and the first interband transition can be seen in Pb0.86Sn0.14Se (Fig.5(a,c)). In Pb0.81Sn0.19Se, two transitions can be resolved, the first interband transition, as well as an additional transition marked by a blue arrow in Fig. 5(b,d). It occurs at energies higher than 60 meV where the CR-LO of the bulk bands is expected, as seen in Fig. 5(d). It is observed in Pb0.81Sn0.19Se (Fig. 5(b)), but not in Pb0.86Sn0.14Se (Fig. 5(a)). Its dispersion (blue in Fig. 5(d)) agrees well with that of Landau-levels of massless Dirac fermions having a velocity D v =(4.7±0.1)x10 5 m/s -almost equal to that of the bulk valleys: This transition could thus attributed to the ground-state CR of massless Dirac TSS.  10(a,b)) and massless Dirac model for CR-TSS (Eq. 11). The reststrahlen of BaF2 is shown in green.
This transition is only visible at 15T and above in Pb0.81Sn0.19Se. Accordingly, the Fermi level is estimated to be around 60meV from the Dirac point. We cannot distinguish the  -Dirac cone from the  -Dirac cones (see Fig. 1 (b)) in Pb0.81Sn0.19Se since the two have similar Fermi velocities and hence have overlapping CR. This might also explain the large intensity of the transition attributed to the TSS. Further evidence of the observation of massless TSS in Pb1-xSnxTe (x>0.40) is presented in the supplement (S2) as well as in a previous work on Pb0.54Sn0.46Te. 24 All in all, Fig. 5 shows that a CR resulting from TSS is observed when

Discussion
In conclusion, our result is an analytical and experimental proof that the topological index of 3D topological insulators can be measured by magnetooptical Landau level spectroscopy. Our approach is most suitable for ultra-narrow gap systems, having very light effective masses. It is not as suitable for materials having heavy effective masses such as Bi2Se3.
In [111]-oriented Pb1-xSnxSe and Pb1-xSnxTe, we have experimentally shown that a measurement of D v relative to the critical c v allows one to experimentally determine the topological character of a material.
This work is thus a proof of concept that the topological index can be measured using a technique that quantifies the bulk band parameters, and not just inferred from the observation of surface states. Our results are further supported by the observation of a cyclotron resonance from massless Dirac topological surface-states in the non-trivial regime. We believe this approach can be applicable to any system supporting massive or massless Dirac fermions that go through a topological phase transition and that can be described by a BHZ Hamiltonian. 36,35,38,40,65,20,66,67,68,69,70 Accordingly, we propose to further study under the same scope compound series such as Hg1-xCdxTe (see supplement 5), 38 BiTlS1-δSeδ, 36,35 Bi2-xInxSe3, 68 topological Heusler materials 69 as well as Dirac semimetals such as Na3Bi and Cd3As2, 39,70 all expected to have a trivial to non-trivial topological phase transition.

MBE growth.
Epitaxial growth of (111) Pb1-xSnxSe and Pb1-xSnxTe 60,61 films on BaF2 (111) substrates is performed using a Riber 1000 and a Varian GEN-II molecular beam epitaxy setup respectively. Samples are grown under UHV conditions better than 5 × 10 -10 mbar. Effusion cells filled with stoichiometric PbSe, PbTe, SnSe and SnTe are used as source material. The chemical composition of the ternary layers is varied over a wide range by control of the SnSe/PbSe (SnTe/PbTe) beam flux ratio that is measured precisely using a quartz microbalance moved into the substrate position. The growth rates are typically 1 µm/hour (~1 monolayer/sec) and the growth temperature is set to 380°C as checked by an IRCON infrared pyrometer. The thickness of the films is in the range of 1-3 µm. In order to obtain a low free carrier concentration (<10 18 cm -3 ), and compensate the native background hole concentration that increases strongly for higher Sn concentration in Pb1-xSnxSe and Pb1-xSnxTe, extrinsic n-type Bi-doping was provided by Bi2Se3 or Bi2Te3 doping cells. 63 The growth is monitored in-situ using reflection high energy electron diffraction (RHEED). X-ray diffraction. X-ray diffraction measurements are performed using Cu-Kα 1 radiation in a Seifert XRD3003 diffractometer, equipped with a parabolic mirror, a Ge(220) primary beam Bartels monochromator and a Meteor 1D linear pixel detector. Transport characterization. Transport measurements are performed at 77K using a van der Pauw geometry in order to determine the Hall carrier density and mobility. Magnetooptical absorption spectroscopy. Magnetooptical absorption experiments are performed in an Oxford Instruments 1.5K/17T cryostat at 4.5K. Spectra are acquired using a Bruker Fourier transform spectrometer. All measurements are made in Faraday geometry. Two different broadband sources are used for different IR ranges: a far-IR source (3-900cm -1 ) and a mid-IR source (400-3600cm -1 ). Measurements are performed at fixed magnetic fields between 0 and 15T and occasionally up to 17T. A He cooled bolometer, is used to detect the transmitted signal. The relative transmission at fixed magnetic field T(B)/T(B=0) is extracted and analyzed. Baseline signal contributions from the bolometer's response variations in the magnetic field are negligible in this experiment, due to the large amplitude of the observed transitions.

S1. Landau quantization in the Mitchell-Wallis description
We now will treat the problem of Landau quantization for the longitudinal valley of (111)-oriented IV-VI semiconductors using the Mitchell ). ̃ and ̃ are the far-band contributions. The effective g factors of the conduction (gc) and valence (gv) bands are given by = +̃ and = − +̃, where the tilde terms represent the far-band contributions. Also note that terms that vary as B 2 in the square root are explicitly neglected in the Mitchell and Wallis paper. This is equivalent to neglecting k 4 terms in the BHZ eigenvalue.