Nontrivial Nature and Penetration Depth of Topological Surface States in ${\mathrm{SmB}}_6$ Thin Films

The nontrivial feature and penetration depth of the topological surface states (TSS) in ${\mathrm{SmB}}_6$ were studied via spin pumping. The experiments used ${\mathrm{SmB}}_6$ thin films grown on the bulk magnetic insulator ${\mathrm{Y}}_3{\mathrm{Fe}}_5{\mathrm{O}}_{12}$ (YIG). Upon the excitation of magnetization precession in the YIG, a spin current is generated in the ${\mathrm{SmB}}_6$ that produces, via spin-orbit coupling, a lateral electrical voltage in the film. This spin-pumping voltage signal becomes considerably stronger as the temperature decreases from 150 to 10 K, and such an enhancement most likely originates from the spin-momentum locking of the TSS and may thereby serve as evidence for the nontrivial nature of the TSS. The voltage data also show a unique film thickness dependence that suggests a TSS depth of $\sim 32\mathrm{nm}$. The spin-pumping results are supported by transport measurements and analyses using a tight binding model.

Figure 1

Spin-pumping experiments. (a)–(b) Experiment approach. (c)–(d) Spin-pumping signals measured using opposite fields ($H$) and an input microwave power $P_{\mathrm{in}}=100\mathrm{mW}$. The curves show calculated spin-wave frequencies. (e) Signals measured at different $P_{\mathrm{in}}$, where $V_{H>0}$ ($V_{H<0}$) is the signal measured at $H=400\mathrm{Oe}$ ($H=-400\mathrm{Oe}$). The vertical dashed lines indicate the calculated frequencies of three modes. (f) Amplitude of the (1,0) peak in (e) vs $P_{\mathrm{in}}$.

Figure 2

Evidence for nontrivial TSS. The first and second rows present the data for ${\mathrm{SmB}}_6(20\mathrm{nm})/\mathrm{YIG}$ and ${\mathrm{SmB}}_6(120\mathrm{nm})/\mathrm{YIG}$, respectively. In each row, the color maps present the signals measured for $H>0$ at different $T$, while the rightmost diagram shows the $(V_{H>0}-V_{H<0})/2$ signals measured for $|H|=0.4\mathrm{kOe}$ at different $T$. All the color maps use the same scale shown by the color bar in (a).

Figure 3

Temperature and length characteristics of TSS. (a) Amplitude of the (1,0) mode, $V_{1,0}$, as a function of $T$ for nine samples with different $d$. (b) The same $V_{1,0}$ data in (a) but normalized by the $V_{1,0}$ values at $T=297\mathrm{K}$. (c) and (d) give the $\sigma (V_{1,0}/a)d$ data at 297 and 10 K, respectively, as a function of $d$, with the error bars estimated from the noise level of the signals. The red curves in (c) and (d) are fits to Eq. (2).

Figure 4

Transport measurements confirming ${\mathrm{SmB}}_6$ film quality and TSS depth. (a) Resistance $R$ vs $T$. The curves are guides for the eye. (b) Conductance $G$ vs $T$. The curves are fits to Eq. (3). (c), (d), and (e) The fitting-yielded gap energy $E_g$, bulk conductance $G_b$, and residual conductance $G_r$, respectively. The lines in (c) indicate the values reported previously. The line in (d) is a liner fit to the five points on the right side of the kink. The line in (e) is a linear fit.

Figure 5

Theoretical analysis of TSS in ${\mathrm{SmB}}_6$. (a) and (b) The band structure and wave function of the surface states of a 50-layer film (21 nm), respectively. (c) and (d) The results of a 200-layer film (83 nm).