Magneto-optical evidence of the topological phase transition in (111)-Pb1-xSnxTe

Recently, the study of topological crystalline insulators has become of great interest in condensed matter physics. In this work, we present infrared magneto-optical investigation of such novel quantum states in narrow gap rocksalt IV-VI semiconductor Pb1-xSnxTe which exhibits a topological phase transition at a critical value xc≈0.40. High-mobility Bi-doped trivial and non-trivial (111)-Pb1-xSnxTe (0≤x≤0.56) films grown on BaF2 substrates by MBE were examined at 4.5K and magnetic fields B=0-15T. Massive and massless Dirac fermion models are used to analyze transmission spectra. We are able to determine the band parameters of the bulk and the topological surface states of such material.


Introduction
The search for topological states of matter beyond the existing Z 2 topological insulators (TIs) protected by time-reversal symmetry has recently become one of the major topics in physics of semiconductors. A novel topological state called topological crystalline insulator (TCI) protected by mirror symmetry of the crystal has been discovered in narrow gap rocksalt IV-VI semiconductors such as Pb 1-x Sn x Te and Pb 1-x Sn x Se alloys [1]. Such materials were shown to exhibit semiconducting bulk states accompanied by gapless Dirac surface states. In Pb 1-x Sn x Te a band inversion occurs at four equivalent L-points in the three-dimensional Brillouin zone (3DBZ) when the Sn content is above the critical value x c ≈0.40, leading to a trivial to non-trivial topological phase transition for x≥x c [2]. Since the L-points are mirror symmetric with respect to six equivalent {110} diagonal crystallographic planes, this band inversion results in an even number of surface Dirac cones at points of (001), (110) or (111) surfaces of the crystal where the L-points are projected [3].
In this work, (111)-oriented Pb 1-x Sn x Te films were studied. Pb 1-x Sn x Te crystallizes in a facecentered-cubic structure as NaCl. The small direct band gap of the compound is located at the L-points of the 3DBZ shown in figure 1a. (111)-Pb 1-x Sn x Te possesses four ellipsoidal bulk carrier pockets at the L-points: one bulk longitudinal valley (black) of which the major axis is parallel to the [111] direction and three bulk oblique valleys (red) of which the major axes are tilted by ±70. 5  illustrated in figure 1b. In the topological trivial phase, the lowest conduction and the highest valence bands are labelled respectively by and . The energy gap (E g >0) initially decreases with increasing Sn content, closes at x=x c , and then re-opens when x>x c (E g <0). In the topological nontrivial phase (inverted regime), and states form respectively the conduction and valence bands. This band inversion results in the emergence of the topological surface states (TSS) at four different points on the corresponding two-dimensional Brillouin zone (2DBZ). As can be seen in figure 1a, the projections of the L-points are located at the and points of the (111) 2DBZ. The bulk longitudinal valley results in the -point Dirac cone (blue), while the three oblique valleys give rise to the -point Dirac cones (green).

Methods
Bi-doped (111)-Pb 1-x Sn x Te (0≤x≤0.56) films were grown by molecular beam epitaxy (MBE) on BaF 2 substrates. PbTe and SnTe effusion cells were employed to control the layer composition [4]. Light Bi-doping is necessary for high Sn content Pb 1-x Sn x Te films for compensating the intrinsic p-type carrier concentration (p>10 19 cm -3 ) originating from native (Pb,Sn) vacancies [5]. Bulk carrier densities ~1×10 18 cm -3 and high mobilities ~10000 cm 2 /Vs are obtained from transport measurements at 77K. Far-and mid-infrared (FIR and MIR) magneto-optical absorption measurements in the Faraday geometry were performed on Pb 1-x Sn x Te samples at 4.5K and magnetic fields B=0-15T. The spectral range is 4-450 meV. The sample pieces were placed in a superconducting coil. B is applied parallel to the [111] direction. Transmission spectra were acquired and analyzed by a Bruker Fourier transform infrared spectrometer. The relative transmission at a fixed B is defined as the normalization of the sample transmission T(B) by the zero-field transmission T(0). Further information about the sample growth, characterization and magneto-optical absorption experiment is given in [6]. Figure 2(a,b) show typical IR transmission spectra obtained for x=0.11 and x=0.56 at 4.5K. Strong transmission minima dispersing as a function of B can be clearly seen. Two series of transitions are associated with different bulk valleys, known to have a massive Dirac dispersion. The strongest series denoted by black arrows is attributed to the bulk longitudinal valley. Other transitions marked by red arrows result from the bulk oblique valleys.

Bulk states
The transmission minima are plotted as a function of B in figure 2(c,d) in order to construct transition fan diagrams. Full black circles and empty red circles denote respectively the transitions in the longitudinal and oblique valleys. The green rectangle represents the BaF 2 substrate reststrahlen band (22-55 meV) where no absorption can be observed. A massive Dirac model is used to fit the data for both types of bulk valleys. As the joint density of states is optimal for =0 ( and ǁ[111]), the Landau levels (LLs) at =0 are sufficient to describe the observed magneto-optical interband transitions. The interband transition ( ) energies, for p-type materials, in the massive Dirac model are given by [6] and [7]: The conduction and valence energies of a LL index are denoted by and , respectively. is defined to be half of the energy gap , .
is the Fermi velocity. The band edge mass of Dirac fermions is given by . and have their usual meaning. The ground cyclotron resonance (CR) can be written as: The black and red solid lines in figure 2(c,d)

Topological surface states
Besides the CR-L and the first interband transition associated with the bulk longitudinal valley, additional weaker transmission minima indicated by blue arrows in figure 2b are observed in Pb 0.44 Sn 0.56 Te. Such minima can be interpreted as the CR of the TSS (CR-TSS) and are seen to unambiguously satisfy a massless Dirac model. The surface LL transition energies can be simply obtained by replacing =0 in equation 2. The energy of the CR-TSS is then given by: The blue solid line in figure 2d is the calculated CR-TSS transition using the same Fermi velocity as the bulk longitudinal valley of Pb 0.44 Sn 0.56 Te, =(7.1±0.1)×10 5 m/s. Since the Fermi velocities of the longitudinal bulk states and the TSS are found to be equal, this indicates that the observed TSS occur at the -point of the (111) 2DBZ. These observations are in good agreement with recent theoretical calculations [3]. Moreover, since no CR-TSS transition below 8T can be observed, one can deduce a Fermi level in the -Dirac cone ~75 meV measured from the Dirac point. Such a CR-TSS is systematically measured in the non-trivial samples and is not observed in the trivial regime. Note that the TSS interband transitions are nearly located at the energy of the strong interband bulk transitions. This can be easily seen by considering the square root terms in equation 1. For ≥1 and B>1T, , so that the TSS interband transitions cannot be experimentally resolved.

Conclusion
FIR and MIR magneto-optical absorption spectroscopy allows us to map out the LL spectrum in highmobility Bi-doped (111)-oriented Pb 1-x Sn x Te (0≤x≤0.56) films grown on BaF 2 substrates using MBE. The bulk band parameters ( , , ) of all samples can be accurately determined using the massive Dirac model. Topologically non-trivial samples (x>0.40) are experimentally demonstrated to exhibit massless Dirac TSS occurring at the -point. This is one of the first observations of TSS-related features by magneto-optics. In topologically trivial samples (x<0.40), only bulk band transitions can be observed. This study paves the way for further work on the optical properties of topological Dirac fermions that are predicted to hold great promise for quantum electronic devices and data storage applications.