Dirac parameters and topological phase diagram of $\mathrm{P}{\mathrm{b}}_{1-x}\mathrm{S}{\mathrm{n}}_x\mathrm{Se}$ from magnetospectroscopy

$\mathrm{P}{\mathrm{b}}_{1\text{−}x}\mathrm{S}{\mathrm{n}}_x\mathrm{Se}$ hosts three-dimensional (3D) massive Dirac fermions across the entire composition range for which the crystal structure is cubic. In this paper, we present a comprehensive experimental mapping of the 3D band structure parameters of $\mathrm{P}{\mathrm{b}}_{1\text{−}x}\mathrm{S}{\mathrm{n}}_x\mathrm{Se}$ as a function of composition and temperature. We cover a parameter space spanning the band inversion that yields its topological crystalline insulator phase. A nonclosure of the energy gap is evidenced in the vicinity of this phase transition. Using magneto-optical Landau level spectroscopy, we determine the energy gap, Dirac velocity, anisotropy factor, and topological character of $\mathrm{P}{\mathrm{b}}_{1\text{−}x}\mathrm{S}{\mathrm{n}}_x\mathrm{Se}$ epilayers grown by molecular beam epitaxy on $\mathrm{Ba}{\mathrm{F}}_2$ (111). Our results are evidence that $\mathrm{P}{\mathrm{b}}_{1\text{−}x}\mathrm{S}{\mathrm{n}}_x\mathrm{Se}$ is a model system to study topological phases and the nature of the phase transition.

Figure 1

(a) Rocksalt crystalline structure of the lead salt compounds. In blue are the cations and in yellow the anions. (b) Composition-induced topological phase transition of $\mathrm{P}{\mathrm{b}}_{1\text{−}x}\mathrm{S}{\mathrm{n}}_x\mathrm{Se}$ and $\mathrm{P}{\mathrm{b}}_{1\text{−}x}\mathrm{S}{\mathrm{n}}_x\mathrm{Te}$ compounds from the normal insulating (NI) regime (orange shaded) to TCI phase (blue). The band parity at the four $L$ points is inverted in the topological regime ($x\gtrsim 0.16\mathrm{at}4.2\mathrm{K}$ for $\mathrm{P}{\mathrm{b}}_{1\text{−}x}\mathrm{S}{\mathrm{n}}_x\mathrm{Se}$). (c) The first Brillouin zone of the rocksalt structure with its 2D projection onto the studied (111) surface. In the topological regime, four Dirac cones occur at the four $L$ points. In (111) orientation, three $L$ points are oriented oblique to the surface (oblique valleys, red), and the other one is perpendicular to the surface (longitudinal valley, black). The constant-energy ellipsoids are represented in (d) for longitudinal valleys and (e) for oblique valleys. The main ellipsoid axis of the longitudinal valley is parallel to [111] direction, while for oblique valleys it is tilted by $70.5^{\circ }$.

Figure 2

(a)–(d) Transmission spectra of $\mathrm{P}{\mathrm{b}}_{1\text{−}x}\mathrm{S}{\mathrm{n}}_x\mathrm{Se}$ with $x$ ranging from $0.10$ to $0.25$ at different magnetic fields normalized by the spectra at $B=0$. Curves are shifted for clarity. Transmission minima, marked by black and red arrows, indicate optical absorption between Landau levels. They are represented by dots in the corresponding Landau fan charts depicted in (e)–(h), where the experimental data (symbols) are fitted by solid lines corresponding to the massive Dirac model of Eq. (1). Transitions are labeled on the right by ${\mathrm{N}}^{\mathrm{c},\mathrm{v}}\text{−}\mathrm{N}+1^{\mathrm{c},\mathrm{v}}$ (N: LL index; c or v are for conduction or valence band). Red stands for oblique valleys and black for longitudinal valley. The gray dots are experimental absorptions attributed to the interband $1^{\mathrm{v}}\text{−}{0}^{\mathrm{c}}$ transition shifted at $k_z\ne 0$ due to the populated $0^{\mathrm{c}}$ level. The green rectangles denote the restrahlen absorption band of the substrate $\mathrm{Ba}{\mathrm{F}}_2$.

Figure 3

(a) Variation of the energy band gap modulus of $\mathrm{P}{\mathrm{b}}_{1\text{−}x}\mathrm{S}{\mathrm{n}}_x\mathrm{Se}$ with $x$. The dashed line is a guide for the eye, from which the minimum value indicates the transition to the TCI regime ($x=x_c$). In (b) the energy band gap is plotted versus Sn content. Experimental data are fitted with Eq. (3) (dashed line) and Eq. (4) (solid lines) which takes into account the jump of the energy gap across the topological phase transition. (c) Dirac velocity for the oblique (red) and the longitudinal valleys (black) versus Sn content. Solid lines are the fits from Eq. (5).

Figure 4

Transmission spectra in the MIR range normalized by the spectra at $B=0$ of $\mathrm{P}{\mathrm{b}}_{1\text{−}x}\mathrm{S}{\mathrm{n}}_x\mathrm{Se}$ with $x=0.19$ (a) and $x=0.25$ (b) at three different magnetic fields and temperatures ($80,120$, and $200\mathrm{K}$). The corresponding fan charts are plotted in (c)–(e) and (f)–(h), respectively. Experimental data (symbols) are fitted by solid lines corresponding to Eq. (1). Black, green, and red colors stand for the results obtained at $80$, 120, and $200\mathrm{K}$ respectively.

Figure 5

(a) Energy gap variation with temperature of $\mathrm{P}{\mathrm{b}}_{1\text{−}x}\mathrm{S}{\mathrm{n}}_x\mathrm{Se}$ for various Sn content $x$ represented in different colors. Data (symbols) are fitted with Eq. (6) (solid lines). Guide for the eyes are shown as dashed lines to highlight the jump of the energy gap across the temperature-induced topological phase transition, represented by the blue and orange regions. (b) Square Dirac velocities plotted versus temperature for different Sn content $x$ represented in the same colors than in (a). Dashed lines are guides for the eyes.

Figure 6

Color map of $\mathrm{P}{\mathrm{b}}_{1\text{−}x}\mathrm{S}{\mathrm{n}}_x\mathrm{Se}$ energy gap versus temperature and Sn content according to Eq. (6). The color scale for $2\mathrm{\Delta }$ is displayed at the bottom. The topological regime (TCI) is delimited from the trivial regime (NI) by the red line calculated by setting $\mathrm{\Delta }\left(x,T\right)=0$ in Eq. (6).

Figure 7

Vegard's law for $\mathrm{P}{\mathrm{b}}_{1\text{−}x}\mathrm{S}{\mathrm{n}}_x\mathrm{Se}$ epilayers grown on $\mathrm{Ba}{\mathrm{F}}_2$ (111) substrates. The experimental value of the lattice parameter between $x=0$ and $x=0.3$ gives the Sn to Pb ratio.