## 1. Introduction

Spin-orbit coupling often leads to the formation of linear bands in solids. Electrons in such bands (the Dirac electrons) manifest themselves in special ways in different experiments [

1,

2,

3,

4,

5]. One of these manifestations is in their optical response: the contribution of a

d-dimensional Dirac band to the inter-band optical conductivity, which is calculated to follow a simple power–law frequency dependence [

6,

7]:

Such optical-conductivity behavior—unusual for conventional materials—has indeed been confirmed for (quasi)-2D electrons in graphene, graphite, and the line-node semimetal ZrSiS, where σ(ω) ≈

const(ω) was reported [

8,

9,

10]. In turn, the 3D Dirac electrons in Dirac and Weyl semimetals, such as ZrTe

_{5}, Cd

_{3}As

_{2}, and TaAs, provide the inter-band optical conductivity to be proportional to frequency, σ(ω) ∝ ω [

11,

12,

13]. The linearity in σ(ω) over a broad frequency range in a 3D electron system is often considered as a ’’smoking gun’’ for Dirac physics. For example, Timusk et al. [

14] suggested the presence of 3D Dirac fermions in a number of quasicrystals, based entirely on the observation of a linear σ(ω) in these materials.

Besides, enormous efforts have been made to investigate the symmetry-protected surface states of topological insulators [

2,

3]. However, the dominant physics of the bulk often obscures the surface properties and hence is generally considered as an obstacle for experiments targeting the surface states. Achieving dissipationless surface spin currents may be of primary importance for potential applications of topological insulators, nevertheless, investigations into bulk electronic properties are essential for understanding the complete picture of the topological-state formation [

15].

Our experiments reveal that the bulk optical conductivity of Bi_{2}Te_{2}Se follows a linear frequency dependence in an appreciably broad spectral range. Based on band-structure calculations, we argue that this linearity is not due to transitions within (a) particular 3D linear band(s), but instead a result of contributions from the transitions between the bands with complex dispersion.

## 2. Materials and Methods

Bi

_{2}Te

_{2}Se bulk crystals were synthesized by a modified Bridgman method [

16]. Highly purified (99.9999%) elemental starting materials (Bi, Te, and Se) (Chimmed, Moscow, Russia) were loaded in quartz ampules inside an inert-gas glove box in the stoichiometric ratio 2:2:1. The sealed evacuated ampules were kept at 850 °C for 24 h with periodic stirring to ensure the homogeneity of the melt, followed by a cooldown to 520 °C with a rate of 5 °C/h. The crystals were then annealed at 520 °C for six days. The typical crystal sizes obtained in this way were in the centimeter range. The crystals were cut into appropriate pieces for X-ray, Hall, and optical measurements (and kept in vacuum until the measurements).

Utilizing an X’Pert Pro Extended MRD X-ray diffractometer (PANalytical, Almelo, the Netherlands) we have confirmed the high structural quality of the crystals, see

Figure 1. The free-carrier concentration and mobility were measured in a standard Hall geometry. Indium-soldered contacts were applied to razor-cut Hall bars with typical dimensions of 2 × 0.5 × 0.2 mm

^{3}. For all samples, the conduction was by

n-type carriers. The properties of the sample, used in our infrared studies, are listed in

Table 1.

Optical reflectivity was measured from the (001) plane on freshly cleaved surfaces. The room-temperature experiments were performed in the mid- and near-infrared spectral ranges (600–8000 cm^{−1}, 75 meV–1 eV) with a Bruker Vertex 80v Fourier-transform infrared spectrometer (Bruker Corporation, Billerica, MA, USA). Freshly evaporated gold mirrors served for reference measurements. We used unpolarized light, because Bi_{2}Te_{2}Se possesses C_{3} rotational symmetry along the [001] direction and hence the (001)-plane response, expressed via a second-rank tensor, such as optical conductivity, is isotropic.

## 3. Results and Discussion

In the top panel of

Figure 2, we plot the raw reflectivity data recorded at 300 K. The reflectivity is very flat between 4000 and 8000 cm

^{−1}. In order to obtain the optical conductivity from the reflectivity data, we first tried to fit the measured spectra using a standard Drude–Lorentz procedure [

18]. However, we found that such flat reflectivity is impossible to fit in an acceptable way with a physically meaningful number of Lorentzians. In an alternative approach, we used Kuzmenko’s variational dielectric function method [

19], which produces optical functions with an accuracy equivalent to Kramers–Kronig. For the sake of convenience, the variable part of the dielectric response function was described by a large number of Lorentzians. Justification and details of this approach can be found in [

20]. Similar to the Kramers–Kronig analysis, this method gives less accurate results near the edges of the experimental window. Thus, the results below approximately 2000 cm

^{−1} and above 7000 cm

^{−1} cannot be considered as accurate.

The real part of the optical conductivity obtained from this fit is plotted in the bottom panel of

Figure 2. The eye-catching feature of the figure is the linear increase in σ(ω) at 4000 to 7000 cm

^{−1} (~0.5–0.8 eV).

Let us first argue that the observed optical conductivity originates from the bulk of Bi

_{2}Te

_{2}Se. In Bi

_{2}Te

_{2}Se, the surface Dirac point lies inside the bulk band gap [

21,

22] and metallic surface states have been experimentally confirmed [

21,

22,

23,

24,

25,

26,

27]. Nevertheless, Bi

_{2}Te

_{2}Se samples usually possess a significant concentration of bulk charges due to the basically unavoidable presence of defects, the so-called self-doping [

26,

27,

28,

29,

30]. This is also the case for our sample—its bulk carrier concentration is rather large, as shown in

Table 1. Furthermore, the skin depth, calculated from the complex optical conductivity, is above 30 nm at any measurement frequency, while the thickness of the topologically non-trivial surface layer is believed to be around 1 nm [

3]. Hence, the response detected by our optical measurements is due to the bulk.

Let us also note that, at elevated temperatures, the optical detection of surface carriers in Bi

_{2}Te

_{2}Se, as well as in similar compounds, such as Bi

_{2}Te

_{3} and Bi

_{2}Se

_{3}, remains so far elusive, while bulk carriers clearly manifest themselves in the optical response of Bi

_{2}Te

_{2}Se [

16,

28,

29,

30] and related compounds [

31,

32,

33,

34,

35]. Reijnders et al. have reported on a mixed (surface plus bulk) optical response in Bi

_{2}Te

_{2}Se for low frequencies at temperatures below some 40 K [

30]. However, at room temperature, as well as at frequencies above 2000 cm

^{−1}, their data are perfectly reconciled with entirely bulk response.

Coming back to the linear σ(ω), it is tempting to interpret it in terms of Equation (1), namely, as a signature of a 3D Dirac band (because our σ(ω) reflects the bulk response). Such a band, however, is not expected to appear in the bulk of Bi

_{2}Te

_{2}Se [

36]. We would like to point out that all the available optical conductivity spectra (ours and those previously reported in [

16,

28,

29,

30]) are rather similar to each other, although the linearity of σ(ω) is most apparent in our data. The deviations between the data sets can be assigned, for example, to the abovementioned difference in the exact Fermi-level position in different samples of Bi

_{2}Te

_{2}Se. In order to check the origin of the linear frequency increase in σ(ω), we performed band-structure calculations for Bi

_{2}Te

_{2}Se and then calculated its inter-band optical conductivity.

The band-structure and optical-conductivity calculations were performed using the full potential linear augmented plane-wave method, as implemented in the WIEN2k code [

37]. The exchange-correlation functional is parameterized using the GGA approximation [

38]. The self-consistent charge-densities and optical-conductivity calculations were done with 400 and 2000 k-points in the whole Brillouin zone, respectively. The results of the calculations are shown in

Figure 2 and

Figure 3. The obtained band structure is basically identical to the one reported in [

36]. In order to be reconciled with the bulk electron concentration (the self-doping problem mentioned above), the Fermi level needs to be shifted upwards, as compared to the undoped situation, as shown in

Figure 3. From the figure, it is apparent that there is no truly Dirac band in the bulk of Bi

_{2}Te

_{2}Se.

The calculated optical conductivity is shown as a dashed line in

Figure 2. Taking into account the generally poor reproducibility of the experimental infrared optical conductivity by first-principles calculations (cf., e.g., in [

39,

40]), the agreement between theory and experiment can be considered as fairly good. Further, we should point out that the computed σ(ω) has no intra-band (free-carrier) contribution. Thus, it is not surprising that the low-frequency experimental σ(ω) is larger than the theoretical line. Additionally, the effect of temperature broadening is absent in the calculations. Such broadening would make the smooth step at around 3000 cm

^{−1} even broader [

10].Taking into account the mentioned issues in the computations of σ(ω) is outside of our capacity and beyond the scope of the paper. The important result of our computations is that the linear σ(ω) is nicely reproduced at 4000 to 6000 cm

^{−1} (~0.5–0.75 eV). Thus, we can conclude that this linearity comes as a cumulative effect of transitions between the bands, which do not have a simple linear dispersion. We note that recent measurements of BaCoS

_{2} and GdPtBi provide other examples of linear σ(ω) not due to a simple 3D Dirac band [

41,

42].