1 INTRODUCTION

Topologically nontrivial materials are very promising for various applications of terahertz (THz) photonics. Massless electrons, which are topologically protected from backscattering, may exist on the surface of materials referred to as topological insulators (TIs) due to strong spin–orbit coupling [1, 2]. As a result, the surface states in TIs occupy the entire band gap of a bulk material and remain metallic even in the presence of impurities and surface defects. In addition, surface carriers are very mobile [3]. Topological insulators can be used not only as THz detectors and generators [4–6] but also as promising high-efficiency converters of the THz radiation [7–10]. Previously, THz high harmonic generation was observed in some Dirac materials: graphene [11, 12] and three-dimensional Dirac semimetal Cd3As2 [13]. Terahertz third-harmonic generation in Bi2Se3 TIs was experimentally investigated for the first time in [7]. A comparison of the nonlinear THz response of samples with nontrivial (Bi2Se3) and trivial ((Bi0.9In0.1)2Se3) topologies showed that the THz third harmonic is generated in the presence of surface electronic states. Terahertz high harmonic generation by Dirac electrons can be described using the thermodynamic nonlinearity mechanism, which depends on ultrafast absorption modulation. This mechanism is provided by the effective heating and subsequent cooling of the electron system upon the interaction with strong THz fields [14]. Thus, the behavior of the THz high harmonics is related to the characteristic time scales of the dynamics of Dirac fermion carriers. When the cooling lasts for several picoseconds, the saturation effects become pronounced with an increase in the incident field strength, which has recently been observed in graphene under strong THz excitation [15]. It was shown in [8] that the ultrafast relaxation of surface states in TIs leads to no accumulation effects during the long THz excitation pulse, which results in the no-saturation mode, in contrast to graphene with a strong saturation of THz harmonic generation and the carrier relaxation time of about several picoseconds.

In this study, we measured the parameters of the THz third-harmonic generation in TIs based on bismuth and antimony chalcogenides with different chemical compositions with advance to higher frequencies (in comparison with previous works [8, 9]). The third-harmonic generation efficiency was analyzed depending on the position of the Fermi level in order to find the most promising chemical compound in the family of bismuth and antimony chalcogenides to be applied as a nonlinear THz frequency converter.

2 TOPOLOGICAL INSULATOR SAMPLES

2.1 Growth Technology

Samples of TIs based on bismuth and antimony chalcogenides, described by the general chemical formula Bi2 – xSbxTe3 – ySey, were investigated. The samples were grown in a horizontal quartz reactor in hydrogen at atmospheric pressure by metal-organic chemical-vapor deposition on 400-µm-thick sapphire substrates with the basic (0001) orientation with a buffer ZnTe layer 5–20 nm thick [16]. Trimethylbismuth (BiMe3), trimethylantimony (SbMe3), diethylzinc (ZnEt2), diethyltelluride (Et2Te), and diisopropyl selenide (iPro2Se) were used as sources for bismuth, antimony, zinc, tellurium, and selenium. The buffer ZnTe layers were grown in a single technological cycle with TI films at the same temperature (445°C). The total hydrogen flow rate was 1.0 L/min at the deposition of the buffer ZnTe layers and 0.5 L/min at the epitaxy of the TI films. The ratio of group-VI to group-V elements in the vapor phase was no less than 10, and the total BiMe3 + SbMe3 partial pressure was maintained close to 6 × 10–5 bar. Stoichiometry and composition of the films were determined according to the energy-dispersive X-ray analysis and Raman spectroscopy data. The X-ray diffraction method was used to confirm the epitaxial nature of the grown films. The elemental composition of the films was determined using an X-MaxN X-ray spectrometer connected to an electron microscope. As a result, the TI films of binary and ternary compositions with different thicknesses and conductivity types were grown and characterized (Table 1). The compositions of two samples (1 and 2) in the composition and structural diagram y(x) are above Ren’s curve [17]; therefore, they are characterized by dominant electron bulk conductivity. The two other samples (3 and 4) have mainly the p-type bulk conductivity with compositions below Ren’s curve.

Table 1. Samples under study
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2.2 Characterization of the Samples

The transmission of the TI samples in the THz range was studied using the conventional scheme of THz time-domain spectroscopy [18]. The measurements were carried out with a THz photoconductive InGaAs antenna as a THz source. The field strength of THz pulses passed through the TI samples was measured by the electro-optical detection method in a ZnTe crystal. The transmission function T(f) of a thin film is directly related to its complex conductivity G(f) and, in accordance with the Tinkham formula [19, 20], can be written as

$$T(f) = \frac{{1 + n}}{{1 + n + {{Z}_{0}}G(f)}}{{e}^{{i\Phi (f)}}}.$$
(1)

Here, n is the refractive index of the Al2O3 substrate, Φ( f ) is the phase delay due to the passage through the thin film, and Z0 = 377 Ω is the vacuum impedance. The total conductivity can be represented as the sum of contributions from the bulk and surface carriers. Since the surface contribution to the conductivity is much smaller than the bulk one, below we can consider and analyze only the bulk part of the conductivity:

$$G(f) \approx {{G}^{{{\text{bulk}}}}} = {{\sigma }^{{{\text{bulk}}}}}(f)d,$$
(2)

where \({{\sigma }^{{{\text{bulk}}}}}(f)\) is the bulk conductivity and d is the film thickness. The results of measuring \({{\sigma }^{{{\text{bulk}}}}}(f)\) are presented in Fig. 1. The resonance near 1.6 THz is due to the well-known phonon, which is active in IR absorption. It can be seen that the static conductivity is maximum in samples 3 and 4, which is related to the high charge carrier densities because the Fermi level is located in the valence band. In contrast, the bulk conductivity of sample 2, the chemical composition of which in the configuration and structural diagram y(x) is characterized by a point near Ren’s curve, is suppressed most significantly, in complete agreement with the results of [17, 21]. The fitting function providing correspondence with the experimental dependences in Fig. 1 can be represented as the sum of three terms: Drude contribution for free carriers, Drude–Lorentz contribution for bound electrons, and contribution from all high-frequency resonances (for example, interband transitions), which decreases linearly with increasing frequency:

$$\begin{gathered} {{\sigma }^{{{\text{bulk}}}}}(f) \propto \frac{{f_{{pD}}^{2}}}{{{{\gamma }_{D}}{\text{/}}2\pi - if}} \\ + \;\frac{{f_{{pL}}^{2}f}}{{f{{\gamma }_{L}}{\text{/}}2\pi + i(f_{{0L}}^{2} - {{f}^{2}})}} - if({{\varepsilon }_{\infty }} - 1), \\ \end{gathered} $$
(3)

where \({{f}_{{pD}}}\)(\({{f}_{{pL}}}\)) and \({{\gamma }_{D}}\)(\({{\gamma }_{L}}\)) are the plasma frequency and decay constant of free (bound) electrons, respectively; \({{f}_{{0L}}}\) is the phonon frequency; and \({{\varepsilon }_{\infty }}\) and \({{\varepsilon }_{0}}\) are the high-frequency and static permittivities, respectively. The calculated spectra of the bulk conductivity are shown by a dashed line in Fig. 1; good agreement between the simulation result by Eq. (3) and the experimental data can be observed. For the subsequent characterization of the TI samples, it is important to know the Fermi level position relative to the bulk band gap. To this end, we calculate the free-carrier density related to the static conductivity. Then, in the approximation of highly degenerate semiconductors, the following expressions for the energy difference between the bottom of the conduction band and the Fermi level ΔE = Ec – EF can be obtained for the n- and p-type conductivities:

$$\Delta E = \left\{ {\begin{array}{*{20}{l}} { - \frac{{{{h}^{2}}}}{{2m_{n}^{*}}}\sqrt[3]{{{{{\left( {\frac{3}{{8\pi }}\frac{{{\text{Re}}\sigma _{0}^{{{\text{bulk}}}}}}{{e{{\mu }_{n}}}}} \right)}}^{2}}}},\quad n{\text{-type}},} \\ {{{E}_{{\text{g}}}} + \frac{{{{h}^{2}}}}{{2m_{h}^{*}}}\sqrt[3]{{{{{\left( {\frac{3}{{8\pi }}\frac{{{\text{Re}}\sigma _{0}^{{{\text{bulk}}}}}}{{e{{\mu }_{h}}}}} \right)}}^{2}}}},\quad p{\text{-type}},} \end{array}} \right.$$
(4)

where \(\sigma _{0}^{{{\text{bulk}}}}\) is the static bulk conductivity, \(m_{{n,h}}^{*}\) are the effective electron and hole masses, \({{\mu }_{{n,h}}}\) are the electron and hole mobilities, and Eg is the band gap. In contrast, in the case of a low static conductivity (low free-carrier density), we can apply the nondegenerate semiconductor approximation and use the expressions

$$E = \left\{ {\begin{array}{*{20}{l}} {{{k}_{{\text{B}}}}T\ln \left[ {\frac{{2{{\mu }_{n}}e}}{{{\text{Re}}\sigma _{0}^{{{\text{bulk}}}}}}\sqrt[3]{{{{{\left( {\frac{{2\pi m_{n}^{*}{{k}_{{\text{B}}}}T}}{{{{h}^{2}}}}} \right)}}^{2}}}}} \right],\quad n{\text{-type}},} \\ {{{E}_{{\text{g}}}} - {{k}_{{\text{B}}}}T\ln \left[ {\frac{{2{{\mu }_{h}}e}}{{{\text{Re}}\sigma _{0}^{{{\text{bulk}}}}}}\sqrt[3]{{{{{\left( {\frac{{2\pi m_{h}^{*}{{k}_{{\text{B}}}}T}}{{{{h}^{2}}}}} \right)}}^{2}}}}} \right],\quad p{\text{-type}}.} \end{array}} \right.$$
(5)
Fig. 1.
figure 1

(Color online) Dispersions of the (a) real and (b) imaginary parts of the bulk conductivity of the topological insulator sa-mples.

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Under the assumption that the effective electron mass for samples 1 and 2 is \(m_{n}^{*} = 0.21{{m}_{e}}\) [22], the effective hole mass for samples 3 and 4 is \(m_{h}^{*} = 0.42{{m}_{e}}\) [23], and \({{E}_{{\text{g}}}} = 0.203\) eV [24], we determined the shifts of the Fermi energy (Table 2). As a result, the Fermi level in Bi2Se3 (sample 1) is higher than the bottom of the conduction band Ec by 0.1 eV \(({{E}_{{\text{c}}}} - {{E}_{{\text{F}}}} < 0)\), whereas in Bi2Te3 (sample 3) it is below the top of the valence band by 0.04 eV. The Fermi level in the Sb2Te3 sample 4 takes the lowest position, which completely correlates with the highest conductivity and, correspondingly, significant absorption in the THz spectral range.

Table 2. Position of the Fermi level
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3 TERAHERTZ THIRD-HARMONIC GENERATION

3.1 Experiment

The source of a strong THz field for subsequent frequency conversion was provided by the method of optical rectification of pulses with a tilted leading edge, generated by a Ti:sapphire laser, in a lithium niobate crystal using the system described in detail in [8]. A bandpass filter was mounted behind the lithium niobate crystal to select the THz pump radiation at a fundamental frequency of 0.7 THz. The peak strength of the THz field was about 60 kV/cm. Another bandpass filter was placed behind the TI sample providing multiplication of the THz pump frequency, which suppressed THz radiation at the fundamental frequency and selected only the third harmonic at a frequency of 2.1 THz. The amplitudes of the THz pump pulse and the third harmonic were measured by the electro-optical method on a 2-mm-thick ZnTe crystal using 100-fs probe laser pulses. In this study, we analyzed the time dependences of the THz third harmonic obtained in TIs with different thicknesses and compositions (see Table 1). This time dependence for sample 2 is presented in Fig. 2a. Figure 2b shows the Fourier spectrum of the signal. It can be seen that, despite the presence of the narrowband THz filter selecting radiation at the third-harmonic frequency, a part of the THz pump radiation at the fundamental frequency passes through the filter. Below, we will consider only the Fourier amplitude of the third-harmonic signal at a frequency of 2.1 THz.

Fig. 2.
figure 2

(Color online) Example of the time dependence of the field strength of the THz third-harmonic pulse and its spectrum in sample 2 (Bi2Te2Se).

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3.2 Analysis of the Experimental Results

The measured third-harmonic field strength at the output of the samples under study \(E_{{3\omega }}^{{{\text{out}}}}\) is given in Table 3. The magnitudes of \(E_{{3\omega }}^{{{\text{out}}}}\) were obtained proceeding from calibration with the third-harmonic field strength in graphene with a known cubic susceptibility [8, 11], in which the third-harmonic field conversion efficiency was 0.5%. However, these data are insufficient to determine the frequency conversion efficiency owing to the dynamics of surface carriers in the TI, because the reflection loss at the interface and the radiation absorption in the TI bulk must be taken into account. The field \(E_{{3\omega }}^{{{\text{in}}}}\) generated in the sample bulk was determined using the results of Subsection 2.2, where the conductivity was calculated for all samples under study based on their field transmission in the THz range. Thus, the third-harmonic field strength \(E_{{3\omega }}^{{{\text{in}}}} = E_{{3\omega }}^{{{\text{out}}}}{\text{/}}{{T}_{{3\omega }}}\) and the radiation conversion efficiency \(\eta = E_{{3\omega }}^{{{\text{in}}}}{\text{/}}{{E}_{\omega }}\) were calculated with allowance for loss (Table 3). Figure 3 shows the dependence of the conversion efficiency of THz radiation to the third harmonic on the energy difference between the bottom of the conduction band Ec and the Fermi level EF. It can be seen that this efficiency increases monotonically with a decrease in the Fermi level. According to the kinetic theory of third-harmonic generation by free carriers on the TI surface [9], the third harmonic of the photocurrent in a TI is inversely proportional to the Fermi energy EF, which is the main parameter characterizing the TI samples under study. It is extremely important for the THz high harmonic generation, because it determines what part of the Dirac cone is not occupied by charge carriers. The nonlinear current depends on many factors, including the carrier relaxation times. We believe that the relaxation times differ only slightly for related materials; therefore, the main factor affecting the third-harmonic amplitude is the Fermi energy. The linear character of the electron dispersion relation is most important for the third-harmonic generation; it leads to a nonlinear response to an external THz field, as follows from both the experiment and the theory [9]. For example, the current at the third-harmonic frequency is zero for parabolic bands in two-dimensional systems. The strict relationship between the spin and pulse is undoubtedly important for TIs; however, this circumstance is disregarded in the model under consideration [9]. Apparently, the difference in the conversion efficiencies of the TI and graphene is mainly due to different relaxation times, which was demonstrated in the pump–probe experiments [8]; this factor also leads to the earlier saturation of the third harmonic with an increase in the pump field strength. The strength of the field formed by the nonlinear current is also inversely proportional to EF and can be approximated by the expression

$$E_{{3\omega }}^{{{\text{in}}}}(f) \propto \frac{1}{{{{E}_{{\text{F}}}}}} = \frac{1}{{{{E}_{{\text{c}}}} - ({{E}_{{\text{c}}}} - {{E}_{{\text{F}}}})}} \equiv \frac{1}{{{{E}_{{\text{c}}}} - \Delta E}},$$
(6)

where \(\Delta E \equiv {{E}_{{\text{c}}}} - {{E}_{{\text{F}}}}\) and Ec is the energy of the bottom of the conduction band. The approximation of the experimental data by Eq. (6) indicates that the theory and the experiment are in satisfactory agreement. The Fermi energy of surface electrons in TIs based on bismuth and antimony chalcogenides (such as Bi2Se3), may be fairly high. According to Eq. (6), the third-harmonic generation efficiency at a specified amplitude of the pump electric field decreases in two-dimensional Dirac systems with a high Fermi energy EF. One can also see that the conversion efficiency of the pump radiation to the third harmonic in sample 4 (Sb2Te3) is much higher than that in the other samples. However, this sample in the THz range has the highest bulk conductivity (Fig. 1) and, correspondingly, the absorption coefficient, which reduces significantly the final output strength of the third-harmonic field \(E_{{3\omega }}^{{{\text{out}}}}\). The absorption can be avoided using samples with a much smaller thickness (on the order of few elementary five-layer blocks). Such a block has the thickness of about 1 nm. As is known [24], TIs lose their properties when the thickness decreases below a certain value; at the thickness equal to eight elementary blocks, the Dirac cone is transformed into a pair of electronic bands separated by the band gap. Since the transition from the linear to parabolic electronic spectrum in two-dimensional systems leads to disappearance of the nonlinear current at the third-harmonic frequency, we believe it reasonable (for achieving the maximum field strength) to use antimony telluride samples with a minimum possible thickness bounded below with a value of ten elementary five-layer blocks (10 nm) to exclude such undesirable effects. It is also noteworthy that possible deviations from a linear dispersion relation should undoubtedly affect the third-harmonic conversion efficiency, with both an increase and a decrease in the Fermi energy. As follows from the experiment, these changes are insignificant in the range of chemical compounds in the configuration and structural diagram (x, y) from Bi2Se3 to Sb2Te3. However, beyond this range, one might expect that the dependence of the third-harmonic field amplitude deviates from the dependence \( \sim {\kern 1pt} 1{\text{/}}{{E}_{{\text{F}}}}\).

Table 3. Field strength and the third-harmonic generation efficiency
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Fig. 3.
figure 3

(Color online) Terahertz third-harmonic conversion efficiency versus the position of the Fermi level in the topological insulator samples.

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4 CONCLUSIONS

Terahertz third-harmonic generation in topological insulators based on bismuth and antimony chalcogenides (Bi2–xSbxTe3–ySey) with different bulk conductivities has been investigated. It has been shown that the third-harmonic generation efficiency, calculated from the experimental data disregarding the influence of loss during propagation in the samples, is inversely proportional to the Fermi energy. It has been found that antimony telluride has the highest conversion efficiency. It has been proposed to use antimony telluride samples with a thickness of about ten elementary five-layer blocks to obtain the maximum efficiency of the conversion of the pump radiation to the output THz third-harmonic radiation.