Abstract
Topological insulators are electronic materials with an insulating bulk and conducting surface. However, due to free carriers in the bulk, the properties of the metallic surface are difficult to detect and characterize in most topological insulator materials. Recently, a new topological insulator Bi_{1.5}Sb_{0.5}Te_{1.7}Se_{1.3} (BSTS) was found, showing high bulk resistivities of 1–10 Ω.cm and greater contrast between the bulk and surface resistivities compared to other Bibased topological insulators. Using Terahertz TimeDomain Spectroscopy (THzTDS), we present complex conductivity of BSTS single crystals, disentangling the surface and bulk contributions. We find that the Drude spectral weight is 1–2 orders of magnitude smaller than in other Bibased topological insulators, and similar to that of Bi_{2}Se_{3} thin films, suggesting a significant contribution of the topological surface states to the conductivity of the BSTS sample. Moreover, an impurity band is present about 30 meV below the Fermi level, and the surface and bulk carrier densities agree with those obtained from transport data. Furthermore, from the surface Drude contribution, we obtain a ~98% transmission through one surface layer — this is consistent with the transmission through singlelayer or bilayer graphene, which shares a common Diraccone feature in the band structure.
Introduction
The special features of a topological insulator — insulating bulk and conducting surface^{1,2}, have been attributed to the presence of timereversal symmetry and spinorbit interaction^{2,3}. Earlierdiscovered threedimensional topological insulators such as Bi_{2}Se_{3}, Bi_{2}Te_{3} and Sb_{2}Te_{3} have a high concentration of bulk carriers, resulting in a weaklyinsulating bulk electronic state and making it difficult to detect the metallic surface states^{4,5,6,7}. In BSTS^{8,9}, nonstoichiometryinduced donors and acceptors compensate each other, yielding high bulk resistivities of 1–10 Ω.cm and high contribution (~70%) of the total conductance by surface transport, thus showing a greater contrast between the bulk and surface resistivities compared to Bi_{2}Se_{3}, Bi_{2}Te_{3} and even Bi_{2}Te_{2}Se^{10}. THzTDS is a noncontact farinfrared optical technique suitable for probing the lowenergy excitations of strongly correlated electronic systems such as cuprate superconductors^{11}, pnictide superconductors^{12}, and colossal magnetoresistance manganites^{11}. In this Report, we present the temperaturedependent optical conductivity of BSTS samples at temperatures 5–150 K in the farinfrared regime (0.4–3.0 THz). By modeling the total conductance as the sum of Drude and Lorentz components, we were able to obtain a Drude weight to be 1–2 orders of magnitude smaller than other Bibased topological insulators^{10}, and close to the value for the surface state of Bi_{2}Se_{3} thin films^{13}. This suggests a significant contribution of the topological surface states to the conductivity of the BSTS sample.
Results
Sample growth, electronic transport and ARPES
Samples were grown by the modified Bridgeman method (Methods section)^{14}. Inplane DC resistivity data were taken using a Quantum Design Physical Property Measurement System (PPMS), as shown in Fig. 1. Both the temperature dependence and the absolute values of resistivity are consistent with those in Ref. 9. Scanning Electron Microscopy with Energy Dispersive Xray spectroscopy (SEMEDX) data of the BSTS samples were reported in Ref. 14, giving the chemical composition of the sample, and consistent with previous publications^{8,9}. Evidences for twodimensional surface properties in our samples were given by ambipolar^{14}, as well as angleresolved photoemission spectroscopy (ARPES) data. The inset of Fig. 1 shows ARPES data of Bi_{1.5}Sb_{0.5}Te_{1.8}Se_{1.2} single crystals obtained from a Scienta R4000 electron analyzer at the Surface, Interface and Nanostructure (SINS) beamline at Singapore Synchrotron Light Source (SSLS) (Methods section). The data show the surface state at the vicinity of Γ point, which is an important signature of a sample being a topological insulator. The observed surface state does have a similar feature to that in a previous report of ARPES on Bi_{1.5}Sb_{0.5}Te_{1.7}Se_{1.3}^{15}. Note that, due to the limited energy resolution, many detailed structures are unresolved at this stage. In this Report we focus on the optical properties in the THz regime.
THzTDS
Details of the THzTDS setup are given in the Methods section. By finding the ratio between the sample and reference spectra, one is able to obtain the complex transmittance, . Figure 2 shows the amplitude of the experimental transmittance of BSTS, , at various temperatures. A periodicity due to multiple reflections within the sample is observed at low temperatures and gets washed out with increasing temperature above ~100 K. We account for these reflections in Eq. (1) below when we extract the optical conductivity. In addition, a drop in to zero has been observed at ~1.7 THz and after which, rises again at ~2.8 THz. This corresponds to an optical phonon mode at 1.9 THz, to be described later. By fitting the experimental to the theoretical expression one is able to obtain the complex refractive index of the sample, . Here d ( = 60 μm) is the sample thickness, c is the speed of light in vacuum, and the summation in Eq. (1) accounts for multiple reflections of the THz pulse inside the sample, with the number of reflections M = 6. Note that, due to the considerable thickness of the sample, we could not assume infinite reflections of the THz pulse inside the sample, as was usually assumed for nanometerthickness thin films.
DrudeLorentz model
The complex refractive index is then used to calculate the complex conductivity , where and , being the permittivity of free space, and the highfrequency dielectric constant. As an unknown quantity, was initially set to 1 when we calculate the experimental conductance, via the expression (t_{bulk} = 60 μm); then it will be used as a fitting parameter^{14} in the conductance fitting for each temperature. We model the conductance to contain a Drude (D) freecarrier contribution averaged over the surface and bulk, and a Lorentz (L) contribution from the insulating bulk, similar to the analysis carried out for Bi_{2}Se_{3}^{14} and given by , where The fitting parameters ω_{pD} and γ_{D} denote the plasma frequency and scattering rate of the Drude component, respectively. ω_{pL}, ω_{oL} and γ_{L} are the plasma frequency, oscillator frequency and the scattering rate of the Lorentz component, respectively. The temperature dependence of these fitting parameters will be discussed later. Figure 3(a) shows the real and imaginary components of the experimental complex conductance of the sample at 30 K, alongside the fit to the DrudeLorentz model, and the respective Drude and Lorentz contributions. Compared to other Bibased topological insulators^{10}, the lowfrequency response of our BSTS sample is Lorentzlike with a very small Drude offset (see Fig. 3(a)). This is consistent with our BSTS sample having a large bulk resistivity.
We estimate how dc conductivity of BSTS varies with temperature by plotting the lowfrequency σ_{1} (ω = 0.4 THz) versus temperature, then fitting it to the thermallyactivated hopping model given by^{16} from 5 K to 150 K, where A is a constant, D is a nonzero offset at 0 K, Δ is the activation energy, and k_{B} is the Boltzmann constant. Figure 3(b) shows the lowfrequency conductivity together with the fit to Eq. (3), with D = (2.80 ± 0.01) (Ω.cm)^{−1}, and activation energy Δ = (26.7 ± 0.2) meV. The fitted value of Δ is consistent with the results by Ren et al. that vary between 22 and 53 meV^{17}. A similar energy scale of 30–40 meV has also been found in Bi_{2}Te_{2}Se, which has been attributed to transitions from the impurity bound states to the electronic continuum^{10}.
We next discuss factors that contribute to the frequency and temperature dependence of the surface and bulk conductance, by looking at the temperature dependence of the fitting parameters in Eq. (2). Figures 4(a) and (b) display the temperaturedependent Drude plasma frequency (ω_{pD}) and Lorentz plasma frequency (ω_{pL}) respectively, while Figs. 4(c) and (d) show the Drude (γ_{D}) and Lorentz (γ_{L}) scattering rates respectively. First note that the absolute values of these fitting parameters are consistent with those obtained for Bi_{2}Se_{3} thin films^{13,18}. We found ω_{pD} to increase significantly with temperature, while γ_{D} (proportional to charge carrier mobility) shows marginal increase with temperature. As for bulk carrier dynamics, both ω_{pL} and γ_{L} exhibit marginal correlation with temperature. Since the square of plasma frequency ω_{p}, is directly proportional to the charge carrier density N via the relation , where e denotes the elementary charge and m* is the effective mass of the charge carrier, this suggests the increase in the surface charge density with temperature while the scattering rate changes only marginally. This is unlike the case of conventional metals, where the increase in resistivity with increasing temperature originates from the increase in scattering rate, with the charge density remaining temperature independent. The increase in surface charge carrier density may be due to the contribution of bulk carriers from the donor impurity band to the topological surface states (Fig. 5). This band, situated ~30 meV ( = activation gap Δ) below the Fermi level, starts to contribute carriers to the Dirac cone above 60 K, leading to a rise in the Drude spectral weight . This relationship between Δ and is supported by our fit of (in Fig. 4(a)) with Eq. (3) — a good fit was obtained with Δ = (31 ± 4) meV (solid line in Fig. 4(a)), consistent with the earlierobtained value of Δ = (26.7 ± 0.2) meV from the σ(ω = 0.4 THz) fit. This is not surprising, since, from Eq. (2), . Figure 5 shows a schematic of the relative positions of the valence band, Dirac point, impurity band, chemical potential, and conduction band in BSTS — they are consistent with that obtained from transport data of BSTS with similar composition^{9}. Note that in Ref. 9, though the Fermi level is pinned to the impurity band in the bulk, the impurity band bends downwards at the surface, such that the impurity band now lies below the Fermi level at the sample surface.
Drude spectral weight
In an optical conductivity study of a group of Bibased topological insulators that does not include BSTS^{10}, the farinfrared conductivity of its most compensated sample (Bi_{2}Te_{2}Se) is still affected by the extrinsic charge carriers due to nonstoichiometry and doping. In Ref. 10, in order to compare the charge densities due to the (a) topological surface states and (b) free (extrinsic) carriers in the bulk, the optical spectral weight was calculated using where the cutoff frequency, set at Ω_{max} = 500 cm^{−1}, separates the lowfrequency excitations from the interband transitions. We perform the same calculation for our BSTS sample using the Drude fitting parameters. Figure 6(a) displays the spectral weight of BSTS as a function of frequency, calculated at 5 K, and superposed with data from other Bibased topological insulators from Ref. 10. The dashed line indicates the Drude spectral weight of Bi_{2}Se_{3} thin films, which was attributed to the topological surface states^{13}. We see that, compared to the spectral weights of other Bibased topological insulators, the Drude spectral weight of our BSTS sample is one order of magnitude smaller than the most compensated sample — Bi_{2}Te_{2}Se, and is comparable to that of Bi_{2}Se_{3} thin film. This is a clear indication that in BSTS, in comparison to other Bibased topological insulator samples, the effect of the 3D bulk states is further suppressed, and the topological surface states could be more clearly resolved.
Surface and bulk carrier density
Now that we have established the Drude component to have a significant twodimensional surface contribution, we then reexpress the experimental Drude conductance in Eq. (2) as where the first term on the righthandside is the twodimensional Drude conductance arising from a linear (Diraclike) dispersion of the surface carriers^{19}, with the factor 2 accounting for the top and bottom surfaces of the sample. The second term on the righthandside is the threedimensional Drude conductance from the bulk carriers similar to Eq. (2), τ = 1/γ_{D}, N_{surf} the surface electron density (of one surface), N_{bulk} the bulk electron density, ν_{f} = 6.0 × 10^{5} m/s the Fermi velocity^{9}, and m* = 0.32m_{e} the electron effective mass^{9}. By fitting the 10 K data to Eq. (7) from the 60μm sample shown in Fig. 3, and another 90μm sample shown in Fig. 7 later, we obtain two simultaneous equations in N_{surf} and N_{bulk}, which are solved to give N_{surf} = 3.8 × 10^{13} cm^{−2} and N_{bulk} = 1.7 × 10^{16} cm^{−3}. We note that, though the obtained bulk carrier density is comparable to that obtained from transport measurements with N_{bulk} = 2.3 × 10^{16} cm^{−3} ^{9}, the surface carrier density is one order of magnitude larger than that from the transport measurements with N_{surf} = 1.2 × 10^{12} cm^{−2} ^{9}. The origin of this discrepancy may come from the nature of optical techniques versus transport measurements. For a system like BSTS with impurityinduced inhomogeneity, the optical technique not only measures the carriers which directly scatter off impurities, but also free carriers which move back and forth within an area with a linear dimension determined by the mean free path. Both types of carriers contribute to an average Drude conductivity. On the other hand, in transport measurements, only the delocalized carriers throughout the whole region between two electrical contacts are measured. A similar reasoning has also been proposed for the optical study of phaseseparated manganite thin films^{20}. Scanning tunnelling microscopy of Bi_{2}Se_{3} and Bi_{2}Te_{3} revealed inhomogeneity of the local density of states on the surface near the Dirac point, due to charge disorder caused by bulk defects (“defectinduced charged puddles”)^{21}. To lend further credence to the quality of our BSTS THzTDS data, we note that the larger value of N_{surf} from optical spectroscopy compared to transport measurements was also seen in Bi_{2}Se_{3}. For example, THzTDS data on Bi_{2}Se_{3} by Aguilar et al.^{13,18} obtained N_{surf} ~ 2.1–3.3 × 10^{13} cm^{−2}, which agrees with our data for BSTS (3.8 × 10^{13} cm^{−2}). Transport data on Bi_{2}Se_{3} yielded smaller values of N_{surf}: 4.7 × 10^{12} cm^{−2} from Shubnikov de Haas (SdH) oscillations of single crystals^{22}, and 7 × 10^{11} cm^{−2} from SdH oscillations of nanostructures^{23}.
Transmission through surface layer
We next use the fitting results from Eq. (5) to estimate the transmission change through the surface state of our BSTS sample, with an estimated thickness of ~2 nm^{7,24}. From the twodimensional Drude conductance and thus conductivity , we can obtain the Drude real and imaginary dielectric constant given by (in SI units) and . The extinction coefficient κ(ω) is then given in terms of ε_{1}(ω) and ε_{2}(ω) by At 10 K and 1 THz, κ ≈ 210, hence the absorption coefficient at 1 THz is α(1 THz) = 8.8 × 10^{6} m^{−1}. Finally, the transmission (of 1THz component) through one pass of one (top or bottom) surface layer is exp(−αt_{surf}) ~ 98.3%. This THz transmission of ~98% is consistent with that of singlelayer graphene^{25} or bilayer graphene^{26}. Whether this similarity is due to the common Diraccone feature in their band structures, is an intriguing question.
Effect of aging
Our data might be complicated by two effects. First, in ARPES data of BSTS, although the chemical potential was initially in the bulk gap, the chemical potential on the surface, after 30 hours, has shifted upwards into the conduction band^{15}. However, transport data showed that the surface chemical potential remains in the bulk gap even after one month of exposure to air^{9}. Second, the valence, conduction and impurity bands bend downwards at the surface due to air exposure^{9}. All the data shown in this paper were taken within 24 hours after cleaving. In order to ascertain that the temperature dependences in Fig. 4 are not due to aging or bandbending effects, we took data on another freshly cleaved 90μm BSTS sample, but now in the temperature sequence 10 K → 50 K → 90 K → 150 K → 10 K → 50 K → 90 K → 150 K, all under 2 hours 15 minutes. The conductivity [σ_{1}(ω)] spectra at the respective temperatures is shown in Fig. 7. Our results show that, within the short time span of datataking, the conductivity values at the same temperature were reproducible over different runs. Hence, provided we take our data in a reasonably short time (~hours), the optical properties of BSTS shown in our paper are not sensitive to aging or band bending. This conclusion is consistent with SdH data of BSTS — the SdH oscillations were observed even after long exposure to air (one month), whose magnetic field dependence signified the 2D nature of these oscillations, strongly suggesting the robustness of the surface state against aging^{9}. It is also consistent with the ARPES data of Bi_{2}(TeSe)_{3} topological insulator, where no significant aging effects were found for more than two weeks after cleaving, under a good vacuum condition^{27}.
Phonons
From the conductance fittings, we also obtained the oscillator frequency ω_{0L} = (1.92 ± 0.04) THz, and temperature independent. This oscillator can be attributed to an optical phonon mode, which also agrees with the THzTDS data of Bi_{2}Se_{3} thin films^{18}, where a similar optical phonon has been found at ~2.0 THz. This result is also consistent with the longitudinal optical phonon obtained by pumpprobe^{28,29} and Raman^{30,31} studies on Bi_{2}Se_{3}. A similar phonon mode has also been observed in Bi_{2}Te_{3} thin films at 1.84 THz^{32}.
Discussion
In summary, we measured the terahertz conductivity of Bi_{1.5}Sb_{0.5}Te_{1.8}Se_{1.2} as a function of temperature using THzTDS. We have modeled the experimental conductance as a sum of a Drude term and a Lorentz term. By careful analysis of the temperature dependence of the Drude term and the optical spectral weight, we found the topological surface states to be more clearly resolved compared to other Bibased topological insulator samples. The present work has demonstrated for the first time by use of TDSTHz spectroscopy that the bulk insulating character can be enhanced through the compensation of nonstoichiometryinduced donors and acceptors in Bi_{1.5}Sb_{0.5}Te_{1.8}Se_{1.2}. This success is due to the unique capability of TDSTHz in differentiating the surface and bulk contribution to the conductivity of a material like topological insulators. Future thicknessdependent THz conductivity studies of topological insulator thin films, and the identification of relative surface and bulk contributions, will further advance our understanding of the electronic states in these technologically important materials.
Methods
Sample growth
BSTS single crystals were synthesized by melting high purity (99.9999%) Bi, Sb, Te and Se with molar ratio 1.5:0.5:1.8:1.2 at 850°C in an evacuated quartz tube. The temperature was then gradually decreased to room temperature over a span of three weeks^{14}. The BSTS single crystal was then cleaved repeatedly ex situ to a thickness of 60 μm while maintaining a large surface area (~4 × 4 mm^{2}), then placed into a cryostat under 3 minutes.
ARPES
ARPES data were taken with the Scienta R4000 electron analyzer at the Surface, Interface and Nanostructure (SINS) beamline at Singapore Synchrotron Light Source (SSLS). The incoming photon energy was set to be 60 eV with linear polarization, and energy resolution of about 140 meV. The photon energy and Fermi level were calibrated using a clean gold foil which is thoroughly sputtered before measurement. The samples were insitu cleaved inside the chamber, at a pressure of 1–2 × 10^{−10} mbar. Data were taken at 300 K.
THzTDS
THz transmission of the BSTS single crystal was measured using a conventional THzTDS system (TeraView Spectra 3000) incorporated with a Janis ST100FTIR cryostat. The THz signal was generated and detected by photoconductive antennae fabricated on low temperaturegrown GaAs films. The aperture diameter is 3.5 mm, allowing for an accurate measurement of the THz signal down to ~0.4 THz. The timedomain electric field of the THz pulse signal is transmitted through the BSTS sample , while the reference signal is transmitted through vacuum. 1800 THz traces were taken in 60 seconds for each reference or sample run. The sample holder was moved back and forth between the sample and reference positions by means of a vertical motorized stage with a resolution of 2.5 μm. Fast Fourier Transform (FFT) was then performed on the timedomain THz signal to obtain the amplitude and phase of the THz spectra. Since the THzTDS detects both the amplitude and phase of the THz signal, there is no need to use the KramersKronig transformation to extract the real and imaginary components of the material optical parameters.
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Acknowledgements
We thank A. Castro Neto, D. Talbayev, V. Venkatesan, P. Di Pietro and S. Lupi for discussions. L.W. acknowledges funding from Singapore National Research Foundation RCA08/018 and Singapore Ministry of Education AcRF Tier 2 (MOE2010T22059). E.E.M.C. acknowledges support from Singapore Ministry of Education AcRF Tier 1 (RG 13/12), Tier 2 (ARC 23/08), as well as the National Research Foundation Competitive Research Programme (NRFCRP4200804). J.X.Z. is supported by the National Nuclear Security Administration of the U.S. DOE at LANL under Contract No. DEAC5206NA25396, the US. DOE Office of Basic Energy Sciences, and in part by the Center for Integrated Nanotechnologies, a U.S. DOE Office of Basic Energy Sciences user facility.
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Affiliations
Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
 Chi Sin Tang
 , Bin Xia
 , Xingquan Zou
 , Shi Chen
 , Lan Wang
 & Elbert E. M. Chia
NUSNNINanoCore, Department of Physics, National University of Singapore, 117542, Singapore
 HongWei Ou
 & A. Rusydi
Theoretical Division and Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos NM 87545, USA
 JianXin Zhu
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Contributions
B.X. and L.W. prepared the samples and conducted electrical measurements. ARPES measurements were taken by S.C., H.W.O. and A.R. THzTDS data were taken by C.S.T. and X.Q.Z. and analyzed by C.S.T., J.X.Z. and E.E.M.C. Insight into physical mechanism was provided by J.X.Z. The manuscript was prepared by C.S.T. and E.E.M.C. with assistance from J.X.Z. and A.R. The project was initiated by L.W. and E.E.M.C. and led by E.E.M.C.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Lan Wang or A. Rusydi or Elbert E. M. Chia.
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