Landau level spectroscopy of the PbSnSe topological crystalline insulator

We report on an infrared magnetospectroscopy study of ${\mathrm{Pb}}_{1-x}{\mathrm{Sn}}_x\mathrm{Se}$, a topological crystalline insulator. We have examined a set of samples, all in the inverted regime of electronic bands, with the tin composition varying from $x=0.2$ to 0.33. Our analysis shows that the observed response, composed of a series of interband inter-Landau-level excitations, can be interpreted and modeled using the relativistic-like Hamiltonian for three-dimensional massive Dirac electrons, expanded to include diagonal quadratic terms that impose band inversion. In our data, we have not found any clear signature of massless electron states that are present on the surface of ${\mathrm{Pb}}_{1-x}{\mathrm{Sn}}_x\mathrm{Se}$ crystals in the inverted regime. Reasons for this unexpected result are discussed.

Figure 1

Bulk Brillouin zone for a rocksalt lattice. Red points mark the positions of the four nonequivalent $L$ points at the border of the zone. The blue hexagon shows the surface Brillouin zone in the (111) direction. The relativistic-like conical bands appear at the $\overline{M}$ and $\overline{\mathrm{\Gamma }}$ points, which are surface projections of the bulk $L$ points [21].

Figure 2

(a) The stacked-plot of relative magnetotransmission data, $T_B/T_0$, collected on sample C at $T=2$ K at magnetic fields up to $B=33\mathrm{T}$ with steps of 1 T. For clarity, individual spectra were shifted vertically by $B\left(\text{T}\right)\times 0.24$. (b) False color plot of relative magnetoabsorbance, $A_B=-ln(T_B/T_0)$, measured on sample C. The white lines separate the areas of the data collected using a superconducting and resistive coil.

Figure 3

(a) Fan chart of inter-LL excitations determined experimentally for sample C using a superconducting (squares) and resistive coil (circles). The theoretically calculated energies of inter-LL excitations are plotted as solid lines for parameters deduced for sample C: $2\mathrm{\Delta }=55\mathrm{meV}, M=-20\mathrm{eV}{\AA }^2$, and $v=4.4\times {10}^5$ m/s (Table 2). (b) Landau level spectrum (at $q_z=0$) calculated for the parameters deduced for sample C. The levels plotted in dark cyan and violet correspond to the $\alpha =1$ and $-1$ LL series, respectively. The blue line shows the calculated position of the Fermi energy assuming that the number of electrons is independent of $B$. The vertical arrows show the interband inter-LL excitations active in the electric-dipole approximation. The color coding in both panels has been chosen to facilitate the identification of corresponding transitions.

Figure 4

Band-structure parameters extracted from the magneto-optical data as a function of the nominal tin concentration: (a) energy band gap $E_g=2\mathrm{\Delta }$, (b) velocity parameter $v$, and (c) inversion parameter $M$. The symbols connected by dashed lines indicate the band gap and velocity parameter values extracted in Refs. [22] and [23].

Figure 5

Relative magneto-transmission spectra $T_B/T_0$ at selected values of the applied magnetic field compared to those calculated using the linear-response theory (4). To calculate $T_B$ and $T_0$, the Landau levels up to the index 20 and 200 were included, respectively. The contribution of excitation at nonzero momenta were taken into account up to $q_z=1.25{\mathrm{nm}}^{-1}$. Two cases have been considered to account for the disorder in the system: $\gamma =2\mathrm{meV}$ (red curve) and $\gamma =0.05\mathrm{meV}+0.0125\hslash \omega$ (meV) (blue curve).

Figure 6

(a) Landau level spectrum of surface and bulk states and the expected Fermi energy in sample C as a function of $B$. The dashed and solid lines used for the LLs reflect the spin parameter $\alpha =-1$ and 1, respectively. The vertical arrows show three inter-LL excitations with lowest energies expected in the spectrum: the bulk interband 0 $\to$ 1 transition, and fundamental CR resonances of bulk and surface electrons. (b) False-color plot of relative magnetoabsorbance measured on sample C compared to the theoretically expected positions of bulk (black and green) and surface (dark cyan) inter-LL resonances. The dotted lines correspond to the values of the magnetic field at which the transitions are not active due to occupation of the final-state LL (Pauli blocking). The vertical white line separates the data measured at low magnetic fields using a superconducting coil from the high-field data obtained with a resistive coil.

Figure 7

Differential magnetotransmission spectra, $T_B/T_{B-\delta B}$, plotted for all explored samples A, B, C, D, E, and F in parts (a)–(f), respectively, using the difference $\delta B=1\mathrm{T}$. The spectra are shifted vertically, with the offset scaling linearly with $B$. The energies of three excitations having the lowest energy in the quantum limit of bulk and surface states are marked by the vertical arrows [cf. Fig. 6]: green, the bulk fundamental CR mode; dark cyan, the fundamental CR mode of surface electrons; and black, the lowest interband inter-LL transition ($n=0\to 1$). Transitions are expected to emerge in the spectra (following the occupation effect) when transparent arrows change into solid ones. The dotted cyan line is the theoretically expected response, the differential magnetotransmission, due to the fundamental CR of surface electrons, calculated using the transfer-matrix method; see the text. Neither bulk inter-LL resonances nor the presence of the reststrahlen band is included in this model for simplicity.