# Dynamics of surface electrons in a topological insulator: Cyclotron resonance at room temperature

I. Mohelsky, F. Le Mardelé, J. Dzian, J. Wyzula, X. Sun, C. Cho, B. Piot, M. Shankar, R. Sankar, A. Ferguson, et al.

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I. Mohelsky, F. Le Mardelé, J. Dzian, J. Wyzula, X. Sun, et al. Dynamics of surface electrons in a topological insulator: Cyclotron resonance at room temperature. Physical Review B, 2026, 113 (4), pp.L041201. 10.1103/v51q-61b4. hal-05503256

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# Dynamics of surface electrons in a topological insulator: cyclotron resonance at room temperature

I. Mohelsky,¹,²,\* F. Le Mardelé,¹ J. Dzian,¹,³ J. Wyzula,¹,⁴ X. D. Sun,¹

C. W. Cho,¹ B. A. Piot,¹ M. Shankar,⁵ R. Sankar,⁵ A. Ferguson,² D. Santos-Cottin,²

P. Marsik,² C. Bernhard,² A. Akrap,²,⁶ M. Potemski,¹,⁷,⁸ and M. Orlita¹,³,†

¹Laboratoire National des Champs Magnétiques Intenses, EMFL, CNRS UPR3228,
Univ. Grenoble Alpes, Univ. Toulouse, Univ. Toulouse 3, INSA-T, Grenoble, France

²Department of Physics, Université de Fribourg, Chemin du Musée 3, Fribourg, 1700 Switzerland

³Faculty of Mathematics and Physics, Charles University, Ke Karlovu 5, Prague, 121 16, Czech Republic

⁴Scientific Computing, Theory and Data Division, Paul Scherrer Institute, Switzerland

⁵Institute of Physics, Academia Sinica, Nankang, Taipei, 11529, Taiwan

⁶Department of Physics, Faculty of Science, University of Zagreb, 10000 Zagreb, Croatia

⁷Institute of High Pressure Physics, PAS, Warsaw, PL-01-142 Poland

⁸CENTERA, CEZAMAT, Warsaw University of Technology, Warsaw, PL-02-822 Poland

(Dated: January 9, 2026)

The ability to manipulate the surface states of topological insulators using electric or magnetic fields under ambient conditions is a key step toward their integration into future electronic and optoelectronic devices. Here, we demonstrate – using cyclotron resonance measurements on a tin-doped BiSbTe₂S topological insulator – that moderate magnetic fields can quantize massless surface electrons into Landau levels even at room temperature. This finding suggests that surface-state electrons can behave as long-lived quasiparticles at unexpectedly high temperatures.

Exploring the non-trivial topology of electronic states in solids is nowadays pursued as a path to discover new quantum phases of matter, enabling robust electronic and spintronic applications, advancing quantum computing, or uncovering fundamental connections between condensed matter and high-energy physics [1–3]. Despite initial expectations that topological protection might confer a certain robustness against increasing temperature, this research often remains limited to cryogenic temperatures. While conventional thermally induced dissipation processes may affect topological materials in much the same way as other solids, the stability of topological phases at elevated temperatures is far from trivial, as frequently discussed in theoretical studies [4–8].

Experimentally, the ARPES technique applied to topological insulators (TIs) is likely the only technique that provided clear evidence for the existence of topological surface states under ambient conditions [9]. Optical spectroscopy techniques, central to this work, along with methods such as electronic transport, are directly relevant to the applications of topological materials. Although these techniques have been frequently used to probe surface states [10–12], this has primarily been limited to low temperatures. Demonstrating the existence of surface states under ambient conditions using these methods is therefore crucial for advancing electronic and optoelectronic devices based on topological materials.

In this study, we target BiSbTe₂S (BST2S), a topological insulator in which tin-induced in-gap states ensure its bulk-insulating nature [13, 14] while preserving a

high electronic quality. This is illustrated, e.g., by electron transport through surface states in the regime of the quantum Hall effect (QHE), observed at liquid helium temperatures [15, 16]. Here we identify bulk BST2S crystals as a useful testbed for optical and magneto-optical studies of topologically protected surface states in TIs.

<figure><img src="image_2.png" /><figcaption>FIG. 1. The real part of optical conductivity of Sn-BST2S deduced by ellipsometry at *T* = 5, 50, 100, 200 and 300 K. The oscillations below the band gap correspond to a delaminated layer with the thickness of (7 ± 2) µm. The inset shows the low-energy part of the spectrum with two pronounced infrared-active phonons, but no apparent Drude contribution.</figcaption></figure>

<footnote>\* ivan.mohelsky@centrum.cz</footnote>

<footnote>† milan.orlita@lncmi.cnrs.fr</footnote>
We show, experimentally, that the surface states of a TI can be at room temperature controlled by a moderate magnetic field and even the regime of full Landau quantization is achievable. The electron-phonon interaction, otherwise effective for bulk states in narrow-gap semiconductors such as BiSbTe₂S, appears to affect weakly the surface electrons.

To prepare Sn-doped BST2S crystals, the corresponding mixture of high-purity (5N) raw materials were grounded and then sealed in a quartz tube under the vacuum of about $3 \times 10^{-2}$ Torr. The initial powder was calcinated at 990 K for 18 h, reground and then heated at 1100 K for 12 h in a vertical Bridgman furnace to grow single crystals. The final atomic constitution was determined using field-emission electron-probe microanalyzer (EPMA) as Bi₁.₂Sb₀.₇Te₂S with the relative atomic composition of tin reaching 0.006 and its spatial variation up to 20%. This translates into 8-12 tin atom for every 2000 atoms of bismuth. The magneto-transport characterization of one selected BST2S crystal is presented in Supplemental material [17].

The synthetized BST2S crystals were characterized using the ellipsometry technique at selected temperatures (Fig. 1). The observed response is typical of a narrow-gap semiconductor, with an onset of interband absorption (the band gap) around 300 meV at low temperatures. A pronounced tail of sub-gap excitations is also present. At low energies, two pronounced phonon lines are observed at 8 and 20 meV. They are relatively broad, as expected for phonons in a mixed compound. Based on the symmetry analysis [21, 22], we assign them to the $E_u$ infrared-active modes. Importantly, we find no signs of free-charge-carrier (Drude-type) absorption. This is in line with findings of the preceding study by Jiang et al. [23], except for the maximum in the optical conductivity at the absorption edge. This feature was interpreted in terms of a van-Hove singularity in the joint density of states due to Mexican-hat-type profile of bands at the Γ point, but it is completely absent in our ellipsometry data.

Then, a series of magneto-optical optical experiments was performed on bulk crystals. In all presented measurements, the reflectivity was measured in the Faraday or tilted-field configurations. The radiation of a globar was analyzed by the Bruker Vertex 80v Fourier-transform spectrometer and delivered via light-pipe optics to the sample located in a superconducting and resistive coil, below and above 16 T, respectively. A part of the reflected signal was deviated, using a silicon beamsplitter, toward an external bolometer. A set of relative magneto-reflectivity spectra, $R_B/R_{B=0}$, collected on a bulk BST2S sample in the Faraday configuration, with B oriented along the c axis, is presented in Fig. 2a. For these measurements, we used a plate-like crystal with the approximate thickness of 1 mm and probed the area of $3 \times 3 \text{ mm}^2$. Before a series of experiments, the crystal was stored under ambient conditions and its surface was not

protected or treated apart from the initial exfoliation.

The observed response is dominated by a single ex citation that emerges above $B \approx 6$ T at the energy of 40 meV, i.e., above the phonon modes. It monotonically blueshifts up to the highest applied magnetic field (34 T). The B-derivative of the data, $d[R_B/R_0]/dB$, presented as a false color plot in Fig. 2b, reveals additional two lines at higher energies. These are clearly visible in low magnetic fields, but cannot be resolved in the data collected using the resistive coil because of a lower signal-to-noise ratio. Notably, all excitations are observed at energies far below the bulk band gap of BST2S.

The interpretation of our data is greatly facilitated by the knowledge accumulated in preceding magneto-optical studies of electrons in conical bands. These are found, e.g., in graphene, but also in many other materials [24–31]. The energy spacing of the observed transitions, $1 : 2 : 1 + \sqrt{2}$, as well as their characteristic $\sqrt{B}$-dependence allow us to assign them to particular excitations within the Landau level (LL) spectrum of Dirac electrons, $E_n^\pm = \pm\sqrt{(\Delta/2)^2 + 2v^2e\hbar nB}$, $n = 0, 1, 2, ...$, in which the asymptotic velocity reaches $v = (4.4 \pm 0.2) \times 10^5 \text{ m/s}$ and the gap parameter $\Delta$ vanishes (with the upper limit of several meV). The most pronounced line in our data corresponds to the fundamental CR mode, $0(1) \rightarrow 1(0)$, see the lower inset of Fig. 2c.

The conical bands are typical of many topological materials. Notably, they represent the most salient part of bulk electronic band structure of three-dimensional Dirac and Weyl semimetals [3]. In contrast, in topological insulators – which behave as conventional narrow-gap semiconductors in their bulk – conical bands can only emerge at the surface. This allows us to associate the observed response with the topological protected gapless surface states in BST2S. The magneto-reflectivity experiments performed in the tilted-field configuration, see the upper inset of Fig. 2c, are in line with this assignment, thus illustrating a response sensitive only to the magnetic-field component perpendicular to the sample surface. Importantly, the vanishing parameter $\Delta$ excludes that the observed inter-LL resonances might stem from massive (gapped) Dirac states on the surface, referred to as Volkov-Pankratov states [32, 33]. Such a low value of $\Delta$ might only emerge at a very smooth interface between a topological and normal insulator [33], not at the atomically sharp surface.

The characteristic pattern due to the Landau-quantized massless electrons on the surface of BST2S was observed in reflectivity, but also in transmission discussed later on. In particular, the most pronounced $0(1) \rightarrow 1(0)$ excitation appeared in a series of explored samples, with the same characteristic $\sqrt{B}$ dependence, and within the experimental linewidth, also at the same position in the spectra. However, the integral intensity varied. For certain specimens, the response due to surface electrons was not resolved at all. We interpret this in terms of an occupation effect. The distance of the Fermi energy
<figure><img src="image_3.png" /><figcaption>FIG. 2. Low-temperature infrared magneto-reflectivity of BST2S: (a) Stack-plot of magneto-reflectivity spectra, $R_B/R_0$, and (b) false-color plot of the corresponding derivatives, $d/dB[R_B/R_0]$. (c) The energies of observed inter-LL excitations as a function of the magnetic-field component perpendicular to the sample ($B_{perp}$), extracted as the minima in the $R_B/R_0$ spectra collected in the tilted-field configuration (panel c, upper inset) for the angles $\theta = 0, 50, 60$ and $70$ deg. The error bars are given by the size of symbols. Black lines are the fits of the inter-LL transitions, using the model of massless Landau-quantized electrons, yielding the velocity parameter $v = (4.4 \pm 0.2) \times 10^5$ m/s. The observed inter-LL excitations are indicated in the lower inset of panel c.</figcaption></figure>

from the Dirac point is not a directly controlled param-
eter in our experiments. Therefore, the carrier density
in the surface states may vary among different samples
and even at different locations of one sample. We expect
that it is defined by the properties of underlying bulk
(band bending effects, in-gap states, charged impurities),
as well as by the interaction of the crystal surface with
environment (e.g., adsorbed atom/molecules). Similar
to graphene [34], electron-hole puddles, or in a broader
sense, the spatial variation of the charge density in the
surface states of TIs are regularly observed [35, 36].

Preceding transport studies [13, 15] indicated that the
surface states exhibit an electron-type conductivity. The
appearance of the electron-like $0 \rightarrow 1$ resonance around
$B \approx 6$ T (Fig. 2) indicates that $n = 1$ LL is no longer fully
occupied, i.e., the filling factor dropped below $v = 3/2$
at least in certain locations. There, the electron density
is as low as $v \cdot \zeta \approx 2 \times 10^{11} \text{ cm}^{-2}$ where $\zeta = eB/h$ is
the LL degeneracy. With a further increase of $B$ – when
the filling factor decreases below $1/2$ – also the hole-like
$1 \rightarrow 0$ transition becomes observable (cf. inset in Fig. 2).

The lines at higher energies are interband inter-LL ex-
citations: namely, the twice degenerate $1(2) \rightarrow 2(1)$ tran-
sition accompanied by the $1 \rightarrow 1$ line that is symmetric
across the $n = 0$ LL. Interestingly, this latter line is not

expected among the transitions following the basic set of
selection rules, $n \rightarrow n \pm 1$, for electric-dipole-active inter-
LL excitations. Nevertheless, it has been observed in
the response of high-quality Landau-quantized graphene
and interpreted as due to point defects, strain, random
Coulomb potential, electron-electron interactions or a
combination of these factors [37]. The observed response
shows no indication of any electron-hole asymmetry or a
deviation of the conical band from linearity. The former
would lift the double degeneracy of the $n \rightarrow n + 1$ and
$n + 1 \rightarrow n$ transitions ($n = 0$ and $1$, in our case) and the
latter would imply a departure from the $\sqrt{B}$ dependence,
respectively, see Ref. [38].

Let us now turn our attention to another set of
magneto-optical data presented in Fig. 3. These were
collected at temperatures ranging from 5 to 280 K, us-
ing the experimental arrangement described in Ref. [39].
To maximize the signal in the reflectivity configuration, a
thin slab (thickness ≈ 7 µm) was deposited on a gold mir-
ror and the collected data thus may be viewed as double-
pass magneto-transmission spectra. The data collected
at selected temperatures are presented in a form false-
color plots (Figs. 3a-e). Each color plot is accompanied
by a set of magneto-transmission spectra measured at
several values of the applied magnetic field (Figs. 3f-j).
<figure><img src="image_4.png" /><figcaption>FIG. 3. Temperature evolution of the fundamental CR line: (a-e) Relative (double) magneto-transmission of a 7-µm-thick BST2S layer, $T_B/T_0$, for selected temperatures plotted in a form of false-color plots. (f-j) Magneto-transmission spectra at selected temperatures and magnetic fields applied.</figcaption></figure>

The magneto-optical response traced in Fig. 3 is dominated by a single magneto-transmission resonance. Its characteristic $\sqrt{B}$ dependence (see the further analysis in [17]), as well as its position – within the linewidth the same as in Fig. 2 – allows us to identify it as the 0(1) → 1(0) transition. Hence, the fundamental CR mode of Landau-quantized electrons on the surface of BST2S can be at moderate magnetic fields observed up to room temperature. A small redshift of the CR line is observed with increasing $T$. This indicates a small decrease, approximately by 5%, of the velocity parameter, as compared to its low-temperature value (see also the analysis in [17]).

The width of the transition (FWHM ≈ 15 meV) is smaller than its position in the spectrum. The visibility of the line is thus dominantly governed by the occupation effect. The transition does not broaden visibly with $T$ and its integral intensity reaches the maximum between 40 and 120 K. The observed changes in the transition intensity are likely due to the thermal redistribution of carriers between LLs. Notably, at higher temperatures the

spread of the Fermi-Dirac distribution ($4k_BT$) exceeds even the CR line energy (energy separation between the $n=0$ and 1 LL). For the sake of completeness, we note that the position and width of the fundamental CR line do not vary among explored samples, but the integral intensity does. This might be also due to the electron density of surface electrons varying among the samples investigated.

While CR absorption [40] is a phenomenon regularly observed at low temperatures in semiconducting and semimetallic systems – either bulk or 2D ones – the observation of a well-defined CR absorption at room temperature is not common, unless extreme conditions of tens or even hundreds teslas are approached, see, e.g., [41–44]. In the overwhelming majority of cases, comprising also the conventional semiconductors such as silicon,germanium or GaAs [45–47], the associated CR line broadening at room temperature exceeds in moderated magnetic fields the CR energy. In this respect, InSb is a notable exception, where CR absorption was observed well above the room temperature [48, 49]. In this particular sys-
tem, it is the unusually low effective mass of electrons, and therefore, a large cyclotron energy which overcomes the intrinsic broadening due to electron-phonon interaction, otherwise particularly strong in polar semiconductors [50, 51]. Graphene, with its particularly large energies of optical phonon modes, is another remarkable example of materials with CR absorption observed at room temperature and low magnetic fields [25, 52].

The thorough description of scattering mechanisms is a complex theoretical task, notably in the case of the Landau quantization, and goes beyond the scope of this paper. Nevertheless, we may compare our experimental observations with behavior of other relevant materials. With the strong optical phonon modes (Fig. 1 and [23]), the BST2S TI can be classified as a polar semiconductor [51]. In such materials, one can expect that the scattering of electrons on optical phonons should dominate at high temperatures over other relevant mechanisms, such as the scattering on charge impurities or acoustic phonons [50]. This is, for instance, the case of zinc-blende semiconductors where the scattering of electrons on LO phonons – mediated by the Frölich interaction – is known as the leading relaxation mechanism in the limit of high temperatures. In GaAs, the electron mobility (relaxation time) drops as $\mu \propto T^{-2.3}$ [53] at ambient conditions. The experimentally measured drop of mobility in GaAs is thus even faster as compared to expectations based on a simple argument about the thermally driven population of phonons ($\mu \propto 1/T$).

The absence of any visible thermal broadening of the CR resonance of electrons on the surface of BST2S TI suggests that scattering of electrons on phonons – acoustic or optical ones – is not the leading relaxation mechanism, even when these phonon modes are largely populated at elevated temperatures. This is consistent with other experimental works [10, 54, 55], as well as with available numerical studies in which an unusually small Frölich constant was found [56]. This enables the existence of long-lived quasi-particles even at elevated temperatures. Having electron-phonon coupling excluded as the main source of scattering, one may speculate about the dominant relaxation mechanism responsible for the observed width of the CR mode. These might be, for instance, neutral point defects.

To conclude, unlike bulk states, the surface states of tin-doped BiSbTe$_2$S exhibit remarkable resistance to thermally induced scattering processes. As a result, electrons in these surface states behave as long-lived quasi-particles even at room temperature. The experimental demonstration of the ability to control electronic surface states in a topological insulator by means of a moderate

magnetic field at room temperature brings us closer to the long-sought development of electronic and optoelectronic devices based on topological materials, but it also opens new paths for curiosity-driven fundamental research of solids at ambient conditions.

In this context, one may imagine, for instance, the realization of magneto-statically biased non-reciprocal optical isolators. Such devices have already been experimentally demonstrated [57], with graphene as an active medium. The BST2S TI, with well-defined surface states, could offer a more practical alternative by eliminating the need for epitaxial growth and subsequent transfer of graphene. Furthermore, the observed room-temperature Landau quantization and the exceptionally weak electron-phonon interaction in BST2S suggest the potential to realize the QHE at room temperature under high, yet experimentally attainable, magnetic fields. To date, room-temperature QHE has been convincingly observed only in high-field experiments on graphene [58], enabled by its massless Dirac-type LL spectrum – also present in BST2S surface states – which features a large separation between the $n = 0$ and 1 levels and thus stabilizes the robust $v = 2$ regime. In graphene, the interaction of electrons with (high-energy) optical phonons can be seen as virtually missing, but the scattering on acoustic phonons has been identified as a limiting factor in high-mobility devices [59].

## ACKNOWLEDGMENTS

We acknowledge discussions with D. M. Basko. The authors also acknowledge the support of the LNCIM-CNRS in Grenoble, a member of the European Magnetic Field Laboratory (EMFL). The work has been supported by the ANR project TEASER (ANR-24-CE24-4830). X.S. and B.P. acknowledge support from ANR Grant No. ANR-20-CE30-0015-01. The work has been supported by the exchange programme PHC ORCHID (50852UC). R.S. acknowledges the financial support provided by the Ministry of Science and Technology in Taiwan under Project No. NSTC (113-2124-M-001-045- MY3 and 113-2124-M-001-003), Financial support from the Center of Atomic Initiative for New Materials (AI-Mat), National Taiwan University, (Project No. 113L900801). and Academia Sinica for the budget of ASI-MATE-113-12. M.P. acknowledges support from the European Union (ERC TERAPLASM No. 101053716) and the CENTERA2, FENG.02.01-IP.05- T004/23 project funded within the IRA program of the FNP Poland, co-financed by the EU FENG Programme.

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Supplemental material for "Dynamics of surface electrons in a topological insulator: cyclotron resonance at room temperature"

I. Mohelsky,¹,²,* F. Le Mardelé,¹ J. Dzian,¹,³ J. Wyzula,¹,⁴ X. D. Sun,¹

C. W. Cho,¹ B. A. Piot,¹ M. Shankar,⁵ R. Sankar,⁵ A. Ferguson,² D. Santos-Cottin,²

P. Marsik,² C. Bernhard,² A. Akrap,²,⁶ M. Potemski,¹,⁷,⁸ and M. Orlita¹,³,†

¹Laboratoire National des Champs Magnétiques Intenses, EMFL, CNRS UPR3228,
Univ. Grenoble Alpes, Univ. Toulouse, Univ. Toulouse 3, INSA-T, Grenoble, France

²Department of Physics, Université de Fribourg, Chemin du Musée 3, Fribourg, 1700 Switzerland

³Faculty of Mathematics and Physics, Charles University, Ke Karlovu 5, Prague, 121 16, Czech Republic

⁴Scientific Computing, Theory and Data Division, Paul Scherrer Institute, Switzerland

⁵Institute of Physics, Academia Sinica, Nankang, Taipei, 11529, Taiwan

⁶Department of Physics, Faculty of Science, University of Zagreb, 10000 Zagreb, Croatia

⁷Institute of High Pressure Physics, PAS, Warsaw, PL-01-142 Poland

⁸CENTERA, CEZAMAT, Warsaw University of Technology, Warsaw, PL-02-822 Poland

(Dated: January 5, 2026)

# I. MAGNETO-TRANSPORT CHARACTERIZATION

## Technique

Magneto-transport technique was used a complementary method to characterize the explored BST2S system. Here we present, see Fig. S1, a set of electron transport data collected as a function of temperature as well as the magnetic field strength and direction (the angle $θ$ between $B$ direction and the $c$ axis) taken on one BST2S sample. These transport measurements have been carried out in a dedicated setup for this purpose, using the standard low-frequency lock-in technique. To this end, we selected a macroscopic BST2S crystal from the same batch as used for our optical experiments, created silver-paste electrical contacts arranged in the Hall-bar configuration, and placed the sample into the variable-temperature insert inside a superconducting coil (up to 16 T).

## Shubnikov-de Haas oscillations

The measured trace of longitudinal magneto-resistance exhibits a clear signature of $B$-induced oscillations. These are well visible in the background-removed trace, $\Delta R_{xx}$, see Fig. S1a. The performed fast Fourier transform analysis implies that the observed oscillations are periodic in $1/B$ and the corresponding spectrum is dominated by a single frequency at $F \approx 36$ T, see Fig. S1b. We interpret this behaviour in term of conventional Shubnikov-de Haas (SdH) oscillations. The gaps between Landau levels at the Fermi energy, as well as the associated effective masses were deduced at four different values of the applied magnetic field from the temperature

damping of ShD oscillations, see Fig. S1c. The field-dependence of these parameters implies that we deal with electrons in a strongly non-parabolic dispersion. A pronounced angle dependence of SdH oscillations ($θ = 0$ corresponds to $B \parallel c$), see Figs. S1d,e, also suggests a 2D character of these electrons, but a highly anisotropic 3D electron gas cannot be either excluded.

Nevertheless, the comparison with the optical data taken at $B = 0$, see Fig. 1 in the main text, allows us to associate the observed SdH oscillations with the surface states of BST2S. This is based on the SdH frequency itself that reaches $F \approx 36$ T, observed at $θ = 0$. If corresponding to bulk electrons, their density would have to exceed $10^{18} \text{ cm}^{-3}$. Such a high density would then be necessarily manifested in the optical response (see Fig. 1 in the main text and the related discussion) through a characteristic Drude-type absorption at low photon frequencies, the strength of which scales linearly with the number of charge carriers. Such Drude response is not seen in our ellipsometry data. In fact, the density of bulk electrons above $10^{18} \text{ cm}^{-3}$ typically makes crystals totally opaque in the far infrared range. In contrast, all explored samples were transparent down to low photon energies, except for regions dominated by phonon-related absorption.

## Weak antilocalization

A pronounced weak antilocalization (WAL) effect is another indication that the transport is dominated by topological surface states. The WAL is manifested by a sharp conductivity maximum, see Fig. S1f, which weakens and broadens with increasing temperature, and which merges with the strong positive magneto-resistance background above 20 K. Theoretically, the effect can be described using the standard Hikami-Larkin-Nagaoka (HLN) model [1]. For strong spin-orbit coupling, the case relevant for us, the $B$-induced change in conductivity, $\Delta\sigma(B) = \sigma(B) - \sigma(0)$, is expected to follow the

<footnote>* ivan.mohelsky@centrum.cz</footnote>

<footnote>† milan.orlita@lncmi.cnrs.fr</footnote>
dependence parametrized by the coherence length L and the scaling factor α :

$$ \Delta\sigma(B) = \frac{\alpha e^2}{2\pi^2\hbar} \left[ \psi\left(\frac{B_{\perp}}{B} + \frac{1}{2}\right) - \ln\left(\frac{B_{\perp}}{B}\right) \right], $$

where $B_{\perp} = \hbar/(4eL^2)$.

The above expression reproduces well the experimentally observed WAL profile, see solid lines in Fig. S1f. The coherence length L drops from 450 nm at 1.3 K down to 150 nm at 20 K, roughly following $T^{-0.4}$ dependence. The extracted values of L compare well with 350 nm at 3 K deduced for surface states in Bi$_2$Se$_3$ by Stephen et al. [2]. To extract the scaling factor α, the zero-field conductivity, $\sigma_0$, has been roughly estimated using van der Pauw method. The deduced scaling factor $\alpha \approx -0.7$ lies in between the values -1 and -1/2 – the value for an ideal topological insulator and the typical experimentally reported value in the presence of a coupling between the two surfaces via bulk states, respectively [3].

Discussion

Knowing that the SdH signal comes from the topologically protected surface states, the frequency of oscillations allows us to estimate the corresponding carrier density (per surface): $n_{2D} = eF/\hbar \approx 1 \times 10^{12} \text{ cm}^{-2}$. This value is almost an order of magnitude higher as compared to the density inferred from the magneto-optical data presented in the paper. On the other hand, this is perfectly in line with our observation that the carrier density in the surface states varies a lot among the explored samples. Some of them, presumably those with high doping, did not show any magneto-optical signal from the surface states (due to the occupation effect). Hence, the result of our magneto-transport characterization is line, but not directly comparable with the presented Landau level spectroscopy data. The carrier density in the surface states, exposed to the environment, seems to be a unique quantity for each sample and even for each particular optical or transport experiment performed under specific conditions. Let us also note that the density $n_{2D} \approx 10^{12} \text{ cm}^{-2}$ implies, for the velocity parameter deduced by magneto-optics $v \approx 0.4 \times 10^6 \text{ m/s}$, the Fermi

level slightly below 100 meV. The Fermi energy of surface electrons thus remains well within the (bulk) band gap of 300 meV, provided the charge neutrality point of the conical band is not far from the mid-position with respect to the bulk band gap.

II. TEMPERATURE DEPENDENCE OF MAGNETO-TRANSMISSION – COMPLEMENTARY ANALYSIS

To judge the field-dependence and the gap parameter of the transition visible in the magneto transmission data (Fig. 3a-e in the main text), we re-plot them using $E^2$ versus B dependence in Fig. S2. The experimental data are compared (dashed lines) with the theoretical expected energy of the 0→1 transition, $\hbar\omega = v\sqrt{2e\hbar B}$, assuming the velocity parameter $v = 0.40, 0.40, 0.39, 0.38$ and $0.37$ in units of $10^6 \text{ m/s}$ in panels a-e, respectively. The agreement is nearly perfect at low temperatures (below 200 K), fully consistent with theoretical expectations for the fundamental cyclotron mode in a gapless conical band. At higher temperatures, above 200 K, the experimentally observed dependence of the transition weakly deviates from the theoretical expectation. Interestingly, the excitation remains linear in the $E^2$ versus B plot, but a weak negative offset gradually develops. This negative offset cannot be associated with any band gap opening. Instead, it suggests that the dispersion of surface electrons gradually departs, with the increasing temperature, from its nearly ideal conical shape observed low temperatures. Such a deviation from linearity may, in fact, be expected due to surface-bulk correspondence, as the (bulk) band gap visibly decreases with T (see Fig. 1 in the main text).

In our further analysis, we extracted, from the data in Fig. S2, the position of the fundamental CR line as a function of the applied magnetic field and temperature, see Fig. S3. The inset in Fig. S3 shows the evolution of the linewidth (FWHM) and integral intensity with temperature at $B = 16$ T. To this end, the magneto-transmission spectra were fitted using a single Lorentzian, see Fig. S4. While the integral intensity seems to reach its maximum between 40 and 120 K, no clear tendency is observed, within the error bars, for the linewidth.

[1] S. Hikami, A. I. Larkin, and Y. Nagaoka, Spin-orbit interaction and magnetoresistance in the two dimensional random system, Progress of Theoretical Physics 63, 707 (1980).

[2] G. M. Stephen, O. A. Vail, J. Lu, W. A. Beck, P. J. Taylor, and A. L. Friedman, Weak antilocalization and anisotropic magnetoresistance as a probe of surface states in topological Bi$_2$Te$_x$Se$_{3-x}$ thin films, Scientific Reports 10, 4845 (2020).

[3] R. Gracia-Abad, S. Sangiao, C. Bigi, S. Kumar Chaluvadi, P. Orgiani, and J. M. De Teresa, Omnipresence of weak antilocalization (WAL) in Bi$_2$Se$_3$ thin films: A review on its origin, Nanomaterials 11, 1077 (2021).
<figure><img src="image_5.png" /><figcaption>FIG. S1. Magneto-transport characterization of BST2S: (a) temperature dependence of background-removed longitudinal magneto-resistance, $\Delta R_{xx}(B, T)$, with pronounced Shubnikov-de Haas oscillations. (b) the fast-Fourier transform of $\Delta R_{xx}$ at $T = 1.4$ K, showing the dominant frequency of $F = 36$ T. (c) the extracted Landau gaps and the corresponding effective mass deduced using Lifshitz-Kosevitch formula from the temperature-driven damping of oscillations. The angle dependence of $\Delta R_{xx}(B, T)$ for $T = 1.4$ K is shown in (d) and (e). Temperature dependence of the WAL signal presented in the form of the relative change of conductivity at low magnetic fields is shown in (f). The relative change in conductivity was obtained using an approximate formula: $\Delta\sigma(B)/\sigma_0 = R_{xx}(B)/R_0 - 1$.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>FIG. S2. Temperature dependence of magneto-transmission data in Fig. 3 of the main text plotted in a form $E^2$ versus $B$ dependence. The dashed lines are energies expected for the fundamental cyclotron resonance mode ($\hbar\omega = v\sqrt{2e\hbar B}$) for the velocity parameter $v = 0.4, 0.4, 0.39, 0.38$ and $0.37$ in units of $10^6$ m/s, in panels a-e, respectively.</figcaption></figure>
<figure><img src="image_7.png" /></figure>

FIG. S3. Positions of the fundamental CR mode extracted from the data in Fig. S2. Individual temperatures, $T = 5, 40, 120, 200$ and $280$ K are distinguished using the color-coding. The dashed lines, showing the theoretically expected positions of the fundamental CR line at selected velocity parameters, are taken from Fig. S2, panels a-e. The inset shows the extracted linewidth (FWHM) and integral intensity at $B = 16$ T as a function of $T$, together with the corresponding estimated error bars.

<figure><img src="image_8.png" /></figure>

FIG. S4. Relative magneto-transmission of BST2S at $B = 16$ T and selected temperatures $T = 5, 40, 120, 200$ and $280$ K. The spectra are taken from Fig. 3 of the main text. The red parts of them were fitted using a single Lorentzian curve to extract parameters in Fig. S3.