Three-dimensionality of the bulk electronic structure in ${\mathrm{WTe}}_2$

We use temperature- and field-dependent resistivity measurements (Shubnikov–de Haas quantum oscillations) and ultrahigh-resolution, tunable, vacuum ultraviolet laser-based angle-resolved photoemission spectroscopy (ARPES) to study the three-dimensionality (3D) of the bulk electronic structure in $\mathrm{WTe}{}_2$, a type II Weyl semimetal. The bulk Fermi surface (FS) consists of two pairs of electron pockets and two pairs of hole pockets along the $X\text{−}\mathrm{\Gamma }\text{−}X$ direction as detected by using an incident photon energy of 6.7 eV, which is consistent with the previously reported data. However, if using an incident photon energy of 6.36 eV, another pair of tiny electron pockets is detected on both sides of the $\mathrm{\Gamma }$ point, which is in agreement with the small quantum oscillation frequency peak observed in the magnetoresistance. Therefore, the bulk, 3D FS consists of three pairs of electron pockets and two pairs of hole pockets in total. With the ability of fine tuning the incident photon energy, we demonstrate the strong three-dimensionality of the bulk electronic structure in $\mathrm{WTe}{}_2$. The combination of resistivity and ARPES measurements reveals the complete, and consistent, picture of the bulk electronic structure of this material.

Figure 1

Quantum oscillation analysis on ${\mathrm{WTe}}_2$. (a) Magnetoresistance measured at $T=1.8$ K, 2.5 K, 4 K, 6 K, 8 K, and 10 K. (b) Shubnikov–de Haas oscillation after subtracting the background. (c) Fast Fourier transform (FFT) analysis of quantum oscillation. (d) Temperature dependence of the oscillation amplitude as a function of temperatures for the peak, $F^{*}$. The closed circles are the data, and solid line is the fitted line of Lifshitz-Kosevich formula.

Figure 2

Fermi surface plots and band dispersion measured at different photon energies. (a)–(d) Fermi surface plots measured at photon energies of 6.70, 6.36, 6.05, and 5.77 eV, respectively. The red solid circles (from left to right) correspond to the quantum oscillation frequencies of $F^3$, $F^2$, $F^4$, and $F^1$, respectively. The red dashed circles correspond to the the quantum oscillation frequency of $F^{*}$. (e)–(h) Band dispersion follows the black dashed lines in (a)–(d), respectively. The red dashed lines are guides to the eye.

Figure 3

Band dispersion, momentum dispersion curves, and energy dispersion curves measured at different photon energies. (a)–(d) Band dispersion along cut #1 in Fig. 2 measured at photon energies of 6.7, 6.36, 6.05, and 5.77 eV, respectively. (e) Momentum dispersion curves at the $E{}_F$ of (a)–(d). (f)–(i) Band dispersion along cut #2 in Fig. 2 measured at photon energies of 6.7, 6.36, 6.05, and 5.77 eV, respectively. (j) Momentum dispersion curves at the $E{}_F$ of (f)–(i). (k)–(n) Band dispersion along cut #3 in Fig. 2 measured at photon energies of 6.7, 6.36, 6.05, and 5.77 eV, respectively. (o) Energy dispersion curves along the red dashed lines in (k)–(n). Black arrows point to the locations of the lower hole bands in (k)–(n). (p) Diameters of the electron and hole pockets measured at different photon energies extracted from the momentum dispersion curves in (e) and (j). $k_z$ values are calculated using a published value of the inner potential $V_0=-6.5\text{eV}$ [43].