Massive and massless Dirac fermions in Pb1−xSnxTe topological crystalline insulator probed by magneto-optical absorption

Dirac fermions in condensed matter physics hold great promise for novel fundamental physics, quantum devices and data storage applications. IV-VI semiconductors, in the inverted regime, have been recently shown to exhibit massless topological surface Dirac fermions protected by crystalline symmetry, as well as massive bulk Dirac fermions. Under a strong magnetic field (B), both surface and bulk states are quantized into Landau levels that disperse as B1/2, and are thus difficult to distinguish. In this work, magneto-optical absorption is used to probe the Landau levels of high mobility Bi-doped Pb0.54Sn0.46Te topological crystalline insulator (111)-oriented films. The high mobility achieved in these thin film structures allows us to probe and distinguish the Landau levels of both surface and bulk Dirac fermions and extract valuable quantitative information about their physical properties. This work paves the way for future magnetooptical and electronic transport experiments aimed at manipulating the band topology of such materials.

transport and optical probes, one needs to be able to understand and rule out contributions from bulk states that may contribute similar, if not identical signatures. Additionally, previous magnetooptical 26, 27 and transport studies 25,28,29 have proven difficult the task of probing, identifying and assigning the Landau levels of topological materials. This is mainly due to the fact that the Fermi level in such systems is pinned to the bulk states, and the mobility is limited, hence requiring fields exceeding 15T to achieve clear Landau quantization 30 .
In this work, we performed detailed magneto-optical absorption measurements to map out the Landau level (LL) spectrum of the bulk and surface bands of high mobility (111) epitaxial Pb 1-x Sn x Te (x = 0.45-0.47) films grown by molecular beam epitaxy (MBE). By lightly doping Pb 1-x Sn x Te with Bi (about 10 19 cm −3 ) 31 , we are able to compensate residual defect doping that results from (Pb, Sn) vacancies, and achieve a low carrier density without compromising the mobility. We can, hence, obtain bulk carrier densities that are close to 1 × 10 18 cm −3 and mobilities of 10000 cm 2 /Vs. This results in a Landau quantization at relatively low magnetic fields (1.5T) and allows us to reliably map the LL spectrum of both bulk and surface states.
Magneto-optical measurements reveal a number of strong interband Landau level transitions that are well described by a massive Dirac dispersion model 25 . These transitions are associated with two types of bulk ellipsoidal Fermi surfaces -a longitudinal carrier valley and three-fold degenerate oblique valleys shown in Fig. 1(b). After reliably mapping out all bulk contributions, we are able to identify a cyclotron resonance feature pertaining to a massless Dirac state having a Fermi velocity v f = 7.3 × 10 5 m/s, attributed to the Γ-point Dirac cone. This is reproduced in two samples having slightly different carrier densities. Our results are in agreement with previous studies on the bulk bands in SnTe 32 , Pb 1-x Sn x Te 33 , and PbTe [34][35][36] and recent calculations of the band structure of the (111) surface states in TCI systems 9,14 . ratio that is determined using a quartz microbalance moved into the substrate position. For the present investigations, the Sn composition of the layers was fixed to x Sn = 0.46 ± 0.01, which is well in the non-trivial TCI regime. The thickness was fixed at 2μ m. Bismuth n-type doping was employed to compensate the intrinsic p-type carrier concentration (typically p > 10 19 cm −3 ) arising in Pb 1-x Sn x Te from native Sn and Pb vacancies. Bi was supplied from a compound Bi 2 Te 3 effusion cell. When substitutionally incorporated on group IV lattice sites, Bi acts as singly charged donor and thus, can compensate the p-type background concentration. From a sample series with varying Bi doping levels, selected samples with carrier concentration around 1 × 10 18 cm −3 and carrier mobilities of 10000 cm 2 /Vs are chosen for further investigation. Apart from the electrical measurements, all samples are characterized by high resolution X-ray diffraction (XRD) shown in Fig. 1(c,d) and atomic force microscopy (AFM, see supplement). From XRD ( Fig. 1(c)), a clear Pb 1-x Sn x Te {111} Bragg series can be observed, evidencing a perfect epitaxial relationship between layer and substrate. The layers are fully relaxed with practically zero residual strain as shown by the reciprocal space map around the asymmetric (513) Bragg reflection ( Fig. 1(d)), where the Pb 1-x Sn x Te peak is found to be located exactly on the zero strain line (ε = 0). The composition of the layers derived from XRD using the Vegard's law agrees within ± 1% with the nominal value.

Growth and characterization.
Magnetooptics-Bulk States. Infrared magneto-optical spectroscopy experiments are performed on large pieces of two samples (S1 and S2), at 4.5K and up to B = 15T. The samples are exposed to radiation from a mercury lamp in the Faraday geometry, with the magnetic field parallel to the [111] direction. The transmitted signal is then collected using a Si composite bolometer and analyzed by a Fourier transform spectrometer. Figure 2(a,b) show typical infrared transmission spectra taken at different magnetic fields between 55 and 400 meV in both samples. A number of strong absorption minima that disperse with increasing magnetic field can be clearly seen. The strongest series marked by black dots is associated with the bulk longitudinal valley. Several other transitions marked by the red circles, can also be resolved and are attributed to bulk oblique valleys. Note that the mere fact that a strong and clear modulation is seen versus energy at fields as low as 1.5T is unambiguous evidence of the high mobility and low carrier density of the films. Figure 3(a,b) respectively show the detailed analysis performed for S1 and S2, whereby absorption energy minima are identified ( Fig. 2(a,b)) and then plotted as a function of magnetic field in order to construct a Landau fan diagram. A massive Dirac model is then used to fit the data for both types of valleys 25,37,38 . The LL energies at = k 0 z (B and z||[111]) are sufficient to describe the magnetooptical absorption spectra since the joint density of states is optimal for = k 0 z . Taking the zero energy at the midgap, the LL energies given by the massive Dirac model are: Magneto-optical transmission spectra (calculated by dividing the raw spectra by the spectrum at B = 0T) as a function of energy, plotted for magnetic fields between 1.5T and 15T in sample S1 and 2.5T and 15T in S2. The full black circles and empty red circles respectively denote absorption minima corresponding to the longitudinal and oblique valleys. The curves are shifted vertically for clarity purposes. The slope in the baseline of the spectra originates from the response of the detector to the applied magnetic field. Its impact on the position of the minima is negligible, since it is smaller than our reported error bars.
Scientific RepoRts | 6:20323 | DOI: 10.1038/srep20323 The ± sign refers to the conduction ( ) E N c and valence ( ) E N v band levels respectively and ∆ is equal to half of the band gap E g , ∆ = E 2 g . The mass of the Dirac Fermions is then given by where v f is the Fermi velocity. N denotes the Landau index, e is the fundamental electronic charge, and ħ is the reduced Planck constant. The absorption energies for interband transitions are then given by: The cyclotron resonance (CR) as a function of field satisfies the intraband equivalent: Note that a similar expression can also be derived from a two-band k.p model that includes spin degeneracy 36,37,[39][40][41] . For the Sn content that we considered in this work (x ≈ 0.46) 42 , a two-band approach is justified, since the conduction (L 6+ ) and valence (L 6− ) bands are mirror images of each other and the far-band contributions 13,38 can be neglected for an energy gap of 30meV. In this case, the LL spin splitting is known to be almost equal to the cyclotron energy, so that the N th LL of one spin component coincides with the (N+ 1) th LL of the other 13,36,43 . This also follows directly from the Dirac Hamiltonian 37 and is even well known to be valid in general for Dirac electron propagating in vacuum. The spin degeneracy will therefore not give additional transmission minima as discussed in more detail in the supplement.
The black lines in Fig. 3   The additional minima shown as open red circles in Fig. 2(a,b) are associated with carriers in the oblique valleys. As mentioned before, the Fermi surface of (111) oriented Pb 1-x Sn x Te is highly anisotropic [34][35][36] . The oblique valleys result in an anisotropy factor K = 10, defined as the square of the ratio of the maximum and minimum cross-sectional areas of the 3D Fermi surface ( Fig. 1(b)). This agrees with previous studies on the (Pb, Sn) chalcogenides 32,34-36 .
The red lines in Fig. 3 are the calculated magneto-optical absorption energies for the oblique valleys using the massive Dirac model. A good agreement is found between the data and the model, for a Fermi velocity v' f = 5 × 10 5 m/s for both samples. This corresponds to a ratio of 1.46 between the Fermi velocity of the longitudinal and the oblique valleys, in agreement with the expected Fermi surface anisotropy (see supplement). For ∆ = 15meV this yields a mass equal to 0.011 m 0 . The transition 3 v -2 c is measured down to B = 4T, indicating a Fermi energy E F ≈ 50 ± 5meV below the valence band edge of the oblique valley for both S1 and S2. Note that it is not surprising that the Fermi level be slightly different in different valleys, as the Pb 1-x Sn x Te/BaF 2 (111) system is known to exhibit a strong thermal expansion mismatch at low temperature, that may shift the oblique bands up in energy with respect to the longitudinal bands. This has been observed in IV-VI epilayers and quantum wells grown on BaF 2 (111) [44][45][46] .
Note that the CR splitting observed in S1 above 11T in Fig. 2(a), is due to the simultaneous presence of both the longitudinal and the oblique CR lines. The critical magnetic field, above which the (1 v -0 v ) oblique CR line is observed, is in agreement with the energy at which the N = 1 LL crosses the Fermi level. We are, however, unable to resolve both CR lines in S2. The average peak position is thus reported with a large error bar that takes into account the broadening of the peak as a result of the presence of two transitions at those energies.
Magnetooptics -Surface states. We now turn our attention to a CR feature that seems to evade the expected physics of the bulk longitudinal and oblique valleys. Figure 4(a,b) present a zoomed in view of the spectra at high magnetic fields (11-15T) in the range 55-150 meV. Besides the main absorptions associated with the CR and the first interband transition of the longitudinal bulk valley, an absorption indicated by the blue arrows is measured. . (e) Multi-peak fitting in S1 for spectra taken at 15T (red data) and 12T (blue data). Three Gaussian curves resulting from the fitting procedure are shown. The resultant sum of the three peaks (empty black circles) gives an excellent fit to the data at both fields. The blue arrow refers to the peak attributed to the CR-TSS transition. The bulk-CR data points in S1 are also extracted using this method. This absorption (blue points in Fig. 4(c,d)) is interpreted as the CR of the topological surface states (CR-TSS) at the Γ point. The blue lines are the calculated transition energies for a massless Dirac model using the same Fermi velocity as the bulk longitudinal valley, v f = 7.3 × 10 5 m/s, . This gives: Good agreement is again found between the model and the experimental data for the CR-TSS feature as shown by the blue line in Fig. 4(c,d). Moreover, the fact that we do not observe any CR-TSS transitions below 10T in S1 (12T in S2) agrees with the estimated Fermi levels for bulk bands (40 ± 5 meV below the band edge), as accordingly the TSS Fermi level would be close to 60 meV below the Dirac point. This fact actually reinforces our interpretation. Note, however, that the TSS interband transitions cannot be resolved in our experiments because they are located at the same energy as the intense interband transitions of the longitudinal bulk valley. Indeed, the LL energies of the TSS e v NB . This is the case in our measurements for N ≥ 1. Due to the apparent low intensity of the CR-TSS in the raw signal, we have used a multi-peak fitting scheme to identify the position of the transition minima in each case. This procedure is shown in Fig. 4(e). We can clearly identify three peak features; two strong features resulting from the CR of the bulk states as well as a smaller one indicated by the blue arrow that we attribute to the CR-TSS. The feature marked by the blue arrow is essential to explain the dip observed in the raw signal, and is sufficiently large in intensity to be reliably accounted for in the analysis. At 15T we find the CR-TSS at 100 ± 2 meV, whereas at 12T it is located at 92 ± 2 meV. The blue arrows and the CR data points in Fig. 3 and 4 are identified and placed using this method.

Discussion
In sum, we are able to map out the band structure of Bi-doped Pb 0.54 Sn 0.46 Te in the vicinity of the band gap. Our interpretation is summarized in Fig. 5. The longitudinal bulk valley is found to satisfy a massive Dirac Hamiltonian with a Fermi velocity v f = (7.3 ± 0.1) × 10 5 m/s and an effective mass equal to m* = 0.005 m 0 . This Fermi velocity is in excellent agreement with the k.p matrix element determined in ref. 13. (see supplement). The bulk oblique valleys satisfy that same massive Dirac Hamiltonian with v' f = (5.0 ± 0.1) × 10 5 m/s and an effective mass m'* = 0.011m 0 . Finally, a CR that satisfies a massless Dirac model with v f = (7.3 ± 0.1) × 10 5 m/s is observed and attributed to the TSS Γ-point Dirac cone. We did not see any signal from LL pertaining to the M-point Dirac cones. The reason behind this might be the fact that the Fermi velocity for M-point Dirac cones coincides with that of the oblique bulk valleys. In that case, the interband transitions of the M-point Dirac cone will overlap with those of the oblique bulk valleys. The CR signal for M-point Dirac cones is likely to occur at energies similar to where the bulk-CR lines were observed. It is also expected to be weaker in intensity; we are thus not able to resolve it.
Finally, one has to keep in mind that the Fermi velocity of the oblique valleys is expected to be anisotropic 9 . This is due to the fact that the oblique valleys are tilted by 70.5⁰ with respect to the [111] direction and the applied magnetic field. Our experimental value of v' f = (5.0 ± 0.1) × 10 5 m/s is thus an effective result given by . Here − v M K denotes the Fermi velocity along the − M K direction, expected to be almost equal to that measured at the Γ-point -(7.3 ± 0.1) × 10 5 m/s 9 . One can thus extract . The details of this geometric argument are discussed in the supplement.
In conclusion, we have mapped out the Landau level spectrum of high mobility Bi-doped Pb 0.54 Sn 0.46 Te (111) epilayers grown on BaF 2 . The high mobility and low carrier density achieved in these films lead to a Landau quantization at about 1.5T. This allows us to reliably map out the LL at low energies. The bulk longitudinal and oblique valleys of Pb 0.54 Sn 0.46 Te can be reliably interpreted by a massive Dirac fermion model. The cyclotron resonance of the topological surface Γ-point Dirac cone is also revealed above 10T. Its dispersion is characteristic of massless Dirac fermions having a Fermi velocity v f = 7.3 × 10 5 m/s. The interband transitions corresponding to the TSS cannot be resolved as they overlap those of the bulk longitudinal valley. Our results provide vital information about the effective masses and Fermi velocities of different bands in Pb 0.54 Sn 0. 46 Te, most notably the Γ-Dirac point, and will be of great use to future transport experiments studying quantum oscillations and coherent transport.

Methods
Bi-doped Pb 1-x Sn x Te films were grown by molecular beam epitaxy in a Varian Gen II MBE system on cleaved BaF 2 (111) substrates. High purity PbTe, SnTe and Bi 2 Te 3 source materials were used. High resolution X-ray diffraction measurements were then 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. Preliminary transport measurements were performed at 77K in order to determine the carrier density and mobility. Further transport characterization was performed at 2K and up to 8T in an Oxford Instruments 1.5K/9T cryostat. Magneto-optical absorption experiments were performed in an Oxford Instruments 1.5K/15T cryostat at 4.2K. Spectra were acquired using a Bruker Fourier transform spectrometer.