Bulk Cyclotron Resonance in the Topological Insulator Bi₂Te₃

Abstract

We investigated magneto-optical response of undoped Bi₂Te₃ films in the terahertz frequency range (0.3–5.1 THz, 10–170 cm⁻¹) in magnetic fields up to 10 T. The optical transmission, measured in the Faraday geometry, is dominated by a broad Lorentzian-shaped mode, whose central frequency linearly increases with applied field. In zero field, the Lorentzian is centered at zero frequency, representing hence the free-carrier Drude response. We interpret the mode as a cyclotron resonance (CR) of free carriers in Bi₂Te₃. Because the mode’s frequency position follows a linear magnetic-field dependence and because undoped Bi₂Te₃ is known to possess appreciable number of bulk carriers, we associate the mode with a bulk CR. In addition, the cyclotron mass obtained from our measurements fits well the literature data on the bulk effective mass in Bi₂Te₃. Interestingly, the width of the CR mode demonstrates a behavior non-monotonous in field. We propose that the CR width is defined by two competing factors: impurity scattering, which rate decreases in increasing field, and electron-phonon scattering, which exhibits the opposite behavior.

1. Introduction

From the theory point of view, a three-dimensional (3D) topological insulator (TI) possesses insulating bulk and conducting surfaces, the conduction channels at surfaces being spin-polarized [1,2,3,4]. Since the spin polarization can potentially be utilized in spintronic devices, topological insulators have attracted a lot of attention in the past years [5,6]. In practice, the real samples of 3D topological insulators often conduct not only on their surfaces, but also in the bulk. Considerable efforts have been made to understand and separate the properties of surface and bulk charge carriers. These properties can particularly be studied via different spectroscopic techniques, such as angle-resolved photoemission spectroscopy (ARPES) or optical and magneto-optical spectroscopy. The optical conductivity and cyclotron resonance (CR) of a number of 3D TI materials have been reported in the literature. Perhaps the most studied family of such TIs is the bismuth selenide: bismuth telluride series, Bi₂(Te_{1 − x}Se_x)₃, which also includes the undoped members, Bi₂Te₃ and Bi₂Se₃ [7,8,9]. In this study, we concentrate on Bi₂Te₃. Namely, we investigate experimentally the CR in this compound. Surprisingly, the published cyclotron-resonance measurements performed on this well-studied TI produce rather diverging results [10,11,12,13] with the absorption features being generally of rather complex shapes. One of the reasons for such diversity might be the sample-dependent variation between the surface and bulk contributions, which, in turn, greatly depend on the exact position of the Fermi level.

Unlike in the majority of previous reports [11,12,13], the CR absorption observed in our study can be well described by a single Lorentzian-shaped mode (which is rather consistent with the earliest study on this issue from 1999 [10]). We believe the absorption we detect is of bulk origin. Thus, our findings might be useful for the proper interpretations of the CR modes in doped Bi₂Te₃, where the balance between the surface and bulk-states contributions can be shifted towards the former, but the bulk still cannot be completely ignored.

2. Materials and Methods

We grew thin layers of Bi₂Te₃ on (111)-oriented BaF₂ substrates by molecular beam epitaxy [14]. For the growth, we used binary Bi₂Te₃ and elemental Te. This is different from the standard practice, when elemental (Bi and Te) sources are utilized with the typical flux ratio of Te/Bi being about 10 to 20. Using Bi₂Te₃ and Te allowed us to reduce this ratio to the values below 1 and, hence, to precisely control the stoichiometry of the growing layer. Along with the employed “ramp up” growth procedure [15], these two approaches successfully suppress twin formation in the growing films. X-ray diffraction (XRD) φ scans about the [0 0 1] axis on the asymmetric (1 0 10) reflection revealed only 120°-periodic peaks and confirmed that the films obtained by this method are either single-domain or have a very small twin volume fraction (3–7% for the films with 1 cm² area) with the c axis of Bi₂Te₃ being perpendicular to the substrate surface. To the best of our knowledge, this thin-film growth method is unique.

In order to prevent possible influence of atmospheric oxygen and water, we have developed a method to cover the TI films in situ with optically-friendly protecting layers of BaF₂ [16]. We have found that 30–50 nm of BaF₂ provide the optimal protection. Our measurements have shown that the BaF₂ cap layers affect neither crystal-structure parameters nor optical properties at the frequencies of interest.

The sample used in this study was thoroughly characterized by XRD, scanning electron microscopy (SEM), atomic force microscopy (AFM), and angle-resolved photoemission spectroscopy (ARPES). The results of these investigations, presented in the Supplementary Materials, confirm high structural and morphological quality of the film and show that the film possesses the topological surface electronic states as well as the states in the bulk conduction band.

For optical measurements, we utilized the infrared optical setup available at the High Field Magnet Laboratory in Nijmegen [17]. This setup consists of a commercial Fourier-transform infrared (FTIR) spectrometer (Bruker IFS113v) (Bruker Optik GmbH, Ettlingen, Germany) combined with a continuous-field 33-Tesla Bitter magnet. A detailed description of this setup could be found elsewhere [18]. The measurements were performed in the Faraday geometry [19] at 2 K. A mercury lamp was used as a radiation source. The far-infrared radiation was detected using a custom-made silicon bolometer operating at 1.4 K. The FTIR spectra were recorded in a number of magnetic fields from 0 to 30 T. The optical data were collected between 10 and 170 cm⁻¹ (300–5100 GHz), using a 200-μm Mylar beamsplitter and a scanning velocity of 50 kHz. At each field, at least 100 scans were averaged. As will be seen below, the data obtained in the fields above 10 T cannot be used in our analysis because of a low signal-to-noise ratio. Thus, in this study we concentrate on the measurements performed between 0 and 10 T.

3. Results and Discussion

In Figure 1, we show raw transmission data measured though the 115-nm-thick Bi₂Te₃ film on a 0.49-mm-thick BaF₂ substrate (cf. S2 in the Supplemental Material) in magnetic fields B up to 10 T. We note that all the measurements reported in this study are performed on a single sample. As seen from Figure 2, the substrate has no detectable field dependence. Hence, all the field-inducted changes come from the film. We note that the BaF₂ substrate has intense phonon modes at roughly 50 and 140 cm⁻¹ [20]. Thus, accurate measurements around these frequencies are impossible. The spectra of Figure 1 are dominated by a single broad mode, which position shifts to higher frequencies in increasing field. The spectra can be fitted by a single Lorentzian, as exemplified in Figure 3 for 2, 4, and 6 T. In zero field, the Lorentzian central frequency is zero, i.e., the observed absorption mode is due to free carriers (Drude conductivity). The field evolution of the mode can be traced in Figure 1: with increasing field, the mode shifts upwards and eventually goes above 100 cm⁻¹, i.e., in the range, where the signal-to-noise ratio is worsened by the spectrometer noise and the phonons in the substrate (this prevents a meaningful spectra analysis in higher fields). Still, the shift of the mode in the applied magnetic field is apparent and can straightforwardly be interpreted as a magnetic-field-induced free-carrier localization or, in other words, a cyclotron-resonance absorption.

The Lorentzian-fit results for this CR absorption mode in the fields from 0 to 10 T are shown in in Figure 4. One can see that the central frequency of the absorption line is linear in field (left panel). This immediately signals that the electronic band(s) responsible for the observed absorption have a quadratic dispersion relation. For linear electronic bands, the field dependence of the CR lines is supposed to have a square root dependence on applied field [21]. Thus, following the Occam’s razor principle, we conclude that the mode is due to bulk (i.e., not linear, not topological) electronic bands. This conclusion is in full agreement, e.g., with ARPES [9] and quantum-oscillations [22] measurements, which show that the Fermi level in undoped Bi₂Te₃ crosses the bulk conduction band and hence there exists a large bulk Fermi surface.

Figure 3. Examples of Lorentzian fits of the transmission spectra from Figure 1 for a few magnetic-field strengths as indicated. The raw experimental data are smoothed, using a Savitzki–Golay method [23]. Note that the spectra for 4 and 6 T are shifted upwards for clarity.

Figure 3. Examples of Lorentzian fits of the transmission spectra from Figure 1 for a few magnetic-field strengths as indicated. The raw experimental data are smoothed, using a Savitzki–Golay method [23]. Note that the spectra for 4 and 6 T are shifted upwards for clarity.

We note that weak modes due to the surface conduction channels may exist on top of the dominating bulk abortion, but within our accuracy they cannot be resolved.

The linear field dependence of the central CR frequency, ω₀, can be fitted with the standard parabolic-band expression, connecting the slope of ω₀(B) and the carrier cyclotron mass, m*, ω₀ = eB/m*c (CGS units are used, e is the elementary charge, c is the speed of light). This fit is shown in the left panel of Figure 4 as a straight line and provides m* = 0.1m_e (m_e is the free-electron mass). This value is in very good agreement with the available literature data on the bulk effective mass in Bi₂Te₃ [24]: m* = 0.109m_e for the response perpendicular to the c-axis, which we do probe in our transmission experiment with unpolarized light. This match provides another confirmation for the correctness of our interpretation. We would like to note here that the complete agreement between the calculated electronic band structure of Bi₂Te₃ and the entire body of the available experimental work is still to be achieved, as emphasized in a recent review [25].

Finally, we turn to the width of the absorption band. As one can see from the right panel of Figure 4, the full width at half maximum (FWHM) of the band demonstrates a non-monotonous field dependence: in low fields, it decreases with increasing B and then, starting at approximately 5 T, the FWHM starts growing with the applied field. The initial decrease of FWHM can be naturally explained by the decreasing cyclotron-orbit radius with increasing B and the consequent decrease of impurity scattering. The reason for the CR mode broadening in B > 5 T is not entirely clear. We propose that this could be due to the increased electron-phonon scattering. In higher fields, the CR mode approaches the frequencies, where phonon density grows (roughly, above 40 cm⁻¹; cf. the left panel of Figure 4 and [26], where the phonon density for Bi₂Te₃ was calculated) and hence the rate of the electron-phonon scattering starts to increase, leading to the observed total broadening of the CR line according to the Matthiessen rule.

4. Conclusions

We have investigated the magneto-optical response of undoped Bi₂Te₃ films at terahertz frequencies and in magnetic fields of up to 10 T. We observed an intense CR line, which can be fitted with a single Lorentz oscillator. The central frequency of the CR increases linearly with applied field, signaling the bulk origin of this resonance. In addition, we found the “in-plane” cyclotron mass, m* = 0.1m_e, which matches well the literature data for bulk Bi₂Te₃. The width of the CR mode demonstrates a behavior non-monotonous in field. We propose that the CR width is defined by two competing factors: impurity scattering, which rate decreases in increasing field, and electron-phonon scattering, which rate demonstrates the opposite behavior. We believe our findings can be exploited in future measurements of the surface-states CR in Bi₂Te₃ to disentangle the bulk and surface contributions.