Thickness-dependent electronic structure in ${\mathrm{WTe}}_2$ thin films

We study the electronic structure of ${\mathrm{WTe}}_2$ thin films with different thicknesses. High-quality thin-film samples are obtained with carrier mobility up to 5000 ${\mathrm{cm}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$, which enables us to resolve the four main Fermi pockets from Shubnikov–de Haas (SdH) oscillations. Angle-resolved SdH oscillations show that the ${\mathrm{WTe}}_2$ thin films cross from three-dimensional to two-dimensional electronic systems at a thickness of $\sim 20$ nm. Using the field effect, the nature of the Fermi pockets in thin-film ${\mathrm{WTe}}_2$ is identified, and the evolution of SdH oscillation frequencies is traced over different sample thicknesses. It is found that the frequencies dramatically decrease at a thickness of approximately 12 nm, which indicates the onset of finite-size effects on the band structure. Our work pins down two critical length scales of the thickness-dependent electronic structure in ${\mathrm{WTe}}_2$ thin films.

Figure 1

Characterization of the thin-film quality. (a)–(e) Shubnikov–de Haas (SdH) oscillations of the thin films with thicknesses of 30 nm (S70), 20 nm (S65), 15 nm (S69), and 10 nm (S68). $\mathrm{\Delta }{R}_{xx}$ are obtained by subtracting polynomial backgrounds from magnetic-field-dependent $R_{xx}$. The insets are optical images for each device. (f) The magnetic-field-dependent magnetoresistance (MR) of corresponding thin films in (a)–(e). (g) Residual resistance ratio (RRR) and mobility of the thin films with different thickness.

Figure 2

Crossover to two-dimensional electronic systems. (a)–(d) Angle-dependent fast Fourier transform (FFT) frequencies of SdH oscillations of the 10-, 15-, 20-, and 30-nm ${\mathrm{WTe}}_2$ thin films. The squares, circles, triangles, and diamonds represent the frequencies $F_1$, $F_2$, $F_3$, and $F_4$ from experiments, respectively. The FFT frequencies in (b) also include the data measured at 200 mK and magnetic field up to 15 T (see Fig. 6 in Appendix pp1). The solid lines are angle-dependent FFT frequencies $F$ expected from 2D systems where $F=F_0/cos\left(\theta \right)$ and $F_0$ are the frequencies at $\theta =0^{\circ }$. The insets in (a) and (b) are the schematic diagram of the angle-dependent measurement configuration and a typical FFT spectrum with $\theta ={90}^{\circ }$, respectively. The error bars represent the uncertainty to determine the frequencies in FFT spectra due to FFT resolution. (e) $\mathrm{\Delta }{R}_{xx}$ of the 15-nm thin-film sample (S69) at ${90}^{\circ }$ and $0^{\circ }$ after the polynomial background subtraction.

Figure 3

Field effect on a 12-nm ${\mathrm{WTe}}_2$ film (S67). (a) Back-gate voltage-dependent $R_{xx}$ at 200 mK and 0 T. (b) Plot of SdH oscillations at different $V_{bg}$. The color scale indicates the amplitude of the SdH oscillations. The measurement is performed at 200 mK and magnetic field up to 15 T. (c) Color map of the FFT spectrum of SdH oscillations at different $V_{bg}$. The cyan dashed lines are a guide to the eye showing how each frequency changes with $V_{bg}$. (d) FFT spectra of SdH oscillations at different $V_{bg}$.

Figure 4

Reduced overlap of conduction and valence bands in ${\mathrm{WTe}}_2$ thin films. (a) FFT spectra of SdH oscillations of ${\mathrm{WTe}}_2$ samples with different thicknesses at $\theta ={90}^{\circ }$. The FFT spectrum of the 12-nm sample is measured at 200 mK and magnetic field up to 15 T. The FFT of the 10-nm sample is an average of $Fcos\theta$ over 11 angles from $0^{\circ }$ to ${25}^{\circ }$ to improve the signal-to-noise ratio. (b) Thickness dependence of FFT frequencies. The thicknesses of bulk samples from Refs. [3, 4, 5, 11] are set as infinite compared with the thin-film samples. The error bars represent the uncertainty to determine the frequencies in FFT spectra due to FFT resolution. (c) Schematic diagrams showing the thickness-dependent Fermi pocket size in half of the Brillouin zone (top panels) and the band structure near the Fermi level along the $\mathrm{\Gamma }\text{−}X$ direction (bottom panels). Note that the schematic diagram shows only the relative change in the Fermi pocket size and the band overlap; the change in the band curvature and the Fermi pocket position cannot be measured in the current experiment. (d) The thickness-dependent quantum oscillation obtained from a minimal model of ${\mathrm{WTe}}_2$ (see Appendix pp6).

Figure 5

$\mathrm{\Delta }{R}_{xx}$ vs $B$ at different angles for (a) 15-, (b) 20-, and (c) 30-nm samples. The data are vertically offset for clarity. (d)–(f) FFT spectra corresponding to the SdH oscillations in (a)–(c), respectively.

Figure 6

(a) $\mathrm{\Delta }{R}_{xx}$ vs $B$, (b) the corresponding FFT spectra, and (c) color map of FFT spectra at different angles for the 10-nm sample (S68). The dashed lines in (c) are the angle-dependent FFT frequencies expected from 2D systems, where $F=F_0/cos\left(\theta \right)$ and $F_0$ are the frequencies at $\theta =0^{\circ }$. (d) $\mathrm{\Delta }{R}_{xx}$ vs $B$ at different angles measured at 200 mK and a magnetic field up to 15 T. (e) The corresponding FFT spectra; the dashed lines indicate the frequencies $F_2$ and $F_3$. The data in (a), (d), and (e) are vertically offset for clarity.

Figure 7

Angle-dependent FFT frequencies extracted from (a) bulk reference samples [3] and our (b) 30-nm and (c) 20-nm thin-film samples. The squares, circles, triangles, and diamonds represent the frequencies $F_1$, $F_2$, $F_3$, and $F_4$ from experiments, respectively. The solid lines are angle-dependent FFT frequencies $F$ expected from 2D systems, where $F=F_0/cos\left(\theta \right)$ and $F_0$ are the frequencies at $\theta =0^{\circ }$.

Figure 8

(a) $R_{xx}$ vs $B$ at different $V_{bg}$ for the 12-nm sample (S68) and (b) the corresponding $\mathrm{\Delta }{R}_{xx}$ after polynomial background subtraction.

Figure 9

(a) Color map of the FFT spectra of the 10-nm sample (S68) at different angles after converting $F$ to $Fsin\theta$. (b) FFT spectrum after summing $Fcos\theta$ over 11 different tilt angles from $0^{\circ }$ to ${25}^{\circ }$ in (a).

Figure 10

Color map of FFT spectra of the 20-nm-sample (S65) at different angles after converting $F$ to $Fsin\theta$. The dashed lines indicate the four main frequencies, and the dotted lines indicate the frequencies from higher 2D subbands.