Anomalous three-dimensional bulk ac conduction within the Kondo gap of ${\mathrm{SmB}}_6$ single crystals

The Kondo insulator ${\mathrm{SmB}}_6$ has long been known to display anomalous transport behavior at low temperatures, $T <5$ K. In this temperatures range, a plateau is observed in the dc resistivity, contrary to the exponential divergence expected for a gapped system. Recent theoretical calculations suggest that ${\mathrm{SmB}}_6$ may be the first topological Kondo insulator (TKI) and propose that the residual conductivity is due to topological surface states which reside within the Kondo gap. Since the TKI prediction many experiments have claimed to observe high mobility surface states within a perfectly insulating hybridization gap. Here, we investigate the low energy optical conductivity within the hybridization gap of single crystals of ${\mathrm{SmB}}_6$ via time domain terahertz spectroscopy. Samples grown by both optical floating zone and aluminum flux methods are investigated to probe for differences originating from sample growth techniques. We find that both samples display significant three-dimensional bulk conduction originating within the Kondo gap. Although ${\mathrm{SmB}}_6$ may be a bulk dc insulator, it shows significant bulk ac conduction that is many orders of magnitude larger than any known impurity band conduction. The nature of these in-gap states and their coupling with the low energy spin excitons of ${\mathrm{SmB}}_6$ is discussed. Additionally, the well-defined conduction path geometry of our optical experiments allows us to show that any surface states, which lie below our detection threshold if present, must have a sheet resistance of $R/\text{square}\ge$ 1000 $\mathrm{\Omega }$.

Figure 1

(a) Transmitted electric field of an optical floating zone grown ${\mathrm{SmB}}_6$ single crystal mounted to an ${\mathrm{Al}}_2{\mathrm{O}}_3$ substrate as a function of time at 3 and 30 K. The 30-K signal is identified as background “light leak” signal which is caused by diffraction of light around the sample. (b) Transmitted electric field as a function of temperature once the background signal shown in (a) is removed by subtraction. (c) Transmitted electric field of the aluminum flux grown ${\mathrm{SmB}}_6$ mosaic at 3 and 30 K. The light leak in this case is much larger due to diffraction between neighboring samples within the mosaic. The two largest signals shown stem from light only transmitted through the ${\mathrm{Al}}_2{\mathrm{O}}_3$ substrate and must be subtracted from the reference substrate's transmitted electric field. The inset shows an expanded view of the time window in which the signal from light transmitted through the ${\mathrm{SmB}}_6$ mosaic is observed. (d) Transmitted electric field as a function of temperature once the background signal shown in (c) has been removed by subtraction. See text for more details.

Figure 2

(a) and (b) Magnitudes of the complex transmissions, as defined in Eq. (2), as a function of frequency and temperature for representative samples grown by both (a) optical floating zone and (b) aluminum flux methods. The two samples had thicknesses of 22 $\mu \mathrm{m}$ and 62 $\mu \mathrm{m}$, respectively. (c) and (d) Real part of the optical conductivity. ${\sigma }_1(\omega ,T)$, calculated from the transmissions shown in (a) and (b).

Figure 3

(a) Thickness dependence of the optical conductivity at $T$ = 3 K $\pm$ 0.1 K in the frequency range of the highest signal to noise of our measurement. Data from two different optical floating zone samples are presented as well as data from the aluminum flux grown mosaic. The colored regions represent the estimated experimental uncertainty of our measurement for each sample. One can see that no systematic dependence on thickness is observed, indicating 3D bulk conduction. (b) Change in optical conductivity expected if surface states were present with the given sheet resistance as derived from our reffit trilayer model. The two lower curves are the conductivities from the optical floating zone sample No. 2 presented in (a) with the average of the two conductivities subtracted. The gray box represents our estimated measurement uncertainty. From this we conclude that if surface states are present then they must have a sheet resistance $R/\text{square}\ge$ 1000 $\mathrm{\Omega }$. See text for details.