Possible origin of linear magnetoresistance: Observation of Dirac surface states in layered ${\mathrm{PtBi}}_2$

The nonmagnetic compounds showing extremely large magnetoresistance are attracting a great deal of research interest due to their potential applications in the field of spintronics. ${\mathrm{PtBi}}_2$ is one of such interesting compounds showing large linear magnetoresistance (MR) in both the hexagonal and pyrite crystal structure. We use angle-resolved photoelectron spectroscopy and density functional theory calculations to understand the mechanism of liner MR observed in the layered ${\mathrm{PtBi}}_2$. Our results uncover linear dispersive surface Dirac states at the $\overline{\mathrm{\Gamma }}$ point, crossing the Fermi level with a node at a binding energy of $\approx 900$ meV, in addition to the previously reported Dirac states at the $\overline{M}$ point in the same compound. We further notice from our dichroic measurements that these surface states show an asymmetric spectral intensity when measured with left and right circularly polarized light, hinting at a substantial spin polarization of the bands. Following these observations, we suggest that the linear dispersive Dirac states at the $\overline{\mathrm{\Gamma }}$ and $\overline{M}$ points are likely to play a crucial role for the linear field dependent magnetoresistance recorded in this compound.

Figure 1

Electronic structure of ${\mathrm{PtBi}}_2$. (a) Hexagonal crystal structure projected onto the $ab$ plane. (b) Three-dimensional (3D) view of the Brillouin zone with the high-symmetry points. (c) Schematic representation of the measuring geometry in which the $s$- and $p$-plane polarized lights are defined with respect to the analyzer entrance slit (ES) and the scattering plane (SP). (d) Fermi-surface map extracted from the experiment using $p$-polarized light with a photon energy of 100 eV. For a direct comparison, Fermi contours from the DFT calculations are overlapped on (d). (e) Fermi-surface map contributed from the bulk derived from the DFT calculations. (f) Energy distribution map taken along the $A\text{−}L$ high-symmetry line. (g) Energy-momentum ($E-k$) plot from the density functional theory (DFT) calculations along the $A\text{−}L$ high-symmetry line. In the figure, the blue dashed lines represent Dirac surface states.

Figure 2

Photon energy dependent ARPES data. The data are measured using $p$-polarized light. (a) $k_z$ Fermi surface map taken in the $\mathrm{\Gamma }MLA$ plane. (b) Energy distribution map taken along the $\mathrm{\Gamma }-A$ symmetry line. (c) Energy distribution curves as a function of photon energy. (d) $E-k$ plot derived from the DFT calculations in the $\mathrm{\Gamma }\text{−}A$ high symmetry line. (e)-(h) Photon energy dependent EDMs taken along the $\mathrm{\Gamma }\text{−}M$ direction. (j)-(m) second derivatives of (e)-(h). (i) EDM measured using $p$-polarized light along the $\mathrm{\Gamma }\text{−}M$ symmetry line with a photon energy of 80 eV from a different cleavage plane than the rest of the data. (n) second derivative of (i). In the figure, the blue and green dashed lines represent the surface states and the red dashed lines represent the bulk states. Dirac point (DP) of the surface states at the $\overline{\mathrm{\Gamma }}$-point is at around 900 meV and at the $\overline{M}$-point is at around 150 meV below $E_F$.

Figure 3

Circular dichroism. (a) EDM measured using left circular ($I^{-}$), (b) right circular ($I^{+}$), and (c) the resultant EDM after the spectral intensity subtraction ($I^{-}-I^{+}$) measured with a photon energy of 17 eV. MDC curve taken at $E_F$ is placed on top of (c). (d) EDCs extracted from (c). (e)–(l) Similar data to (a)–(c) but measured with 40- and 100-eV photon energies, respectively.