Two Band Model Interpretation of the p to n Transition in Ternary Tetradymite Topological Insulators

The requirement for large bulk resistivity in topological insulators has led to the design of complex ternary and quaternary phases with balanced donor and acceptor levels. A common feature of the optimized phases is that they lie close to the p to n transition. The tetradymite Bi2Te3_xSex system exhibits minimum bulk conductance at the ordered composition Bi2Te2Se. By combining local and integral measurements of the density of states, we find that the point of minimum electrical conductivity at x=1.0 where carriers change from hole-like to electron-like is characterized by conductivity of the mixed type. Our experimental findings, which are interpreted within the framework of a two band model for the different carrier types, indicate that the mixed state originates from different type of native defects that strongly compensate at the crossover point.

Unfortunately, the presence of significant bulk conduction poses a challenge for the observation of surface states, leading to significant efforts aimed at suppressing the bulk conductance.
Appropriate doping or alloying can lead to bulk insulating behavior. Large bulk resistivities have been reported for Bi 2 Te 2 Se [5,6], Bi 1.5 Sb 0.5 Te 1.3 Se 1.7 [7] and Bi 1.4 Sb 0.6 Te 1.8 S 1.2 [8]. All of these compositions are located close to a p-to n-crossover of conductivity. The crossover in the conductivity type is the result of a counteracting effect between different types of defects that arise upon changing the stoichiometry. A crossover point can be attained in multiple ways. For the Bi 2 Te 3-x Se x system a p-to n-transition is expected close to x=1.0 [9]. For Sb x Bi 2-x Te 3-y Se y the p-to n-transition may be obtained by tuning either x or y [10]. For the Bi 2 Te 3-x S x system the crossover is located at x~0.12 [11]. Alloying Bi 2 Te 2 S with Sb shifts the system closer to the crossover [8].
In this report, we examine the p-to n-transition of the Bi 2 Te 3-x Se x system more closely by combining results from electrical transport and nuclear magnetic resonance (NMR) measurements. The Seebeck coefficient is positive and increases up to x=0.6, due to a reduction in the hole density, which is reflected in a drop in electrical conductivity. For x ≥ 1.4 the Seebeck coefficient is negative and decreases, reflecting an increase in the electron density, leading to higher electrical conductivity. The compositions with x<0.8 and x ≥ 1.4 are nearly extrinsic and the transport properties are described on the assumption of a single carrier type. The chemical potential of holes was found close to the valence band edge. On the other hand, the chemical potential of electrons was found within the conduction band revealing the metallic behavior of the Se-rich compositions. For 0.8<x<1.0 the Seebeck coefficient changes sign, the electrical conductivity displays a minimum and the optical energy gap is maximized. The behavior is qualitatively explained within the framework of a two-band model allowing for the overlap of two types of carriers with opposite sign. Based on this two-band model the total Seebeck coefficient changes sign for chemical potentials located within the gap yielding intrinsic conduction. At the same time the electrical conductivity displays a minimum due to minimum carrier density. The combination of the Seebeck and NMR analysis unveils a mixed conduction state across the entire compositional range as well as a decrease of the Fermi level density of states at the ordered composition Bi 2 Te 2 Se. The model captures the general features of the p-to n-transition, explaining the large bulk resistivity obtained for properly alloyed ternary and quaternary TIs.

Materials and methods
Bi 2 Te 3-x Se x ingots were synthesized by mixing the appropriate ratios of high purity (99.999%) elemental Bi, Te, and Se metals. Starting materials were sealed under high vacuum (~10 -4 Torr) in quartz tubes, heated to 800 °C for 15 hours, kept at 800 °C for 10 hours, and cooled to room temperature for at least 15 hours. Powder X-ray diffraction (PXRD) measurements were performed using a CPS120 INEL powder X-ray diffractometer (Cu Kα, 1.54056A°). Transmission Electron Microscopy (TEM) images were collected using a JEOL 2100F transmission electron microscope operating at 200 kV. TEM samples were prepared by cryogenic crushing to minimize sample preparation artifacts. Scanning Electron Microscopy (SEM) imaging and Energy Dispersive Spectroscopy (EDS) were performed with a Hitachi S-3400 scanning electron microscope equipped with a PGT energy-dispersive X-ray analyzer. Data were acquired with an accelerating voltage of 20 kV. The room temperature electrical conductivity and Seebeck coefficient were measured for rectangular shaped samples in helium atmosphere on a ULVAC-RIKO ZEM-3 instrument. The measurement direction coincides with the in-plane direction of the hexagonal structure. Room temperature diffuse reflectance measurements of finely ground powders were collected using a Nicolet 6700 FT-IR spectrometer. The direct energy gaps were obtained by extrapolating to zero energy the squared Kubelka -Munk function (α/s) 2 relating the diffuse reflectance with the absorption coefficient α and the scattering coefficient s [12]. 125 Te NMR spin-lattice relaxation times (T 1 ) and spectra were acquired using a Bruker DSX-300 spectrometer. A standard Bruker X-nucleus probe with 5 mm solenoid coil was used.
To avoid skin depth issues of the RF transmission power, ingots were ground by mortar and pestle. Spectral data were acquired using a spin-echo sequence. T 1 data were acquired with a saturation-recovery technique. The 125 Te chemical shift scale was calibrated using the unified Ξ scale [13,14]. Figure 1a displays the PXRD patterns of selected compositions. All reflection peaks are assigned to the hexagonal crystal structure. A high magnification TEM image at a grain boundary of the Bi 2 Te 2 Se composition is shown in Fig. 1b. The corresponding selected area electron diffraction pattern confirms that we have a single-phase material with hexagonal crystal structure. The layered crystal structure is further supported by the SEM image of cleaved Bi 2 Te 2 Se (Fig. 1c). EDS results obtained from different locations of the rectangular shaped x=1 sample are summarized in Table I  the Bi-Te (2) bond is weaker than the Bi-Te (1) bond and hence determines the energy gap. The substitution of the Te (2) sites by selenium atoms increases the bond strength and hence the energy gap until x~1 [15]. The observed increase of the direct energy gap up to x=1 may also be explained by a possible crossover of different valleys within the conduction or valence bands [16]. Optical studies of the Bi 2 Te 3-x Se x system have shown that for x=1.0 a value of ~0.3 eV is expected for the direct energy gap [15,17]. The slightly lower value in our case may be due either to incomplete ordering or the formation of impurity bands caused by native defects [5,6].

Results and discussion
The saturation of the energy gap for x>1.0 ( Fig. 1d) is the result of the Burstein effect [15,18]. For x>1.4 the Fermi level is located within the conduction band. Under these conditions where E g is the direct gap of the undoped material [18]. In general, a reduction of the energy gap is expected for x>1.0, assuming that as selenium atoms go to the Te (1) sites they tend to attract charge along the Bi-Te x Se 1-x axis, making the Bi-Se (2) bond less ionic [15,19], although the effect of the complex band structure cannot be excluded [16]. This reduction is not seen in our case due to the Burstein effect.
The variation of the room temperature Seebeck coefficient versus Se content is presented in Fig. 2a Fig. 2b. One can see that the electrical conductivity initially decreases, reaching a minimum at x~1.0 and increases for higher Se amounts.
Assuming a single type of carriers and based on the semi-classical Mott−Jones formula the Seebeck coefficient is inversely proportional to the carrier density [20]. This means that for x≤0.6 the increase of the Seebeck coefficient (Fig. 2a) captures the decrease in the hole density, i.e. the reduction of the electrical conductivity (Fig. 2b). On the other hand, the reduction of the Seebeck coefficient for x>1.4 demonstrates the increase of the electron density followed by a subsequent increase in the electrical conductivity.
Accounting for acoustic phonon scattering and band parabolicity, the Seebeck coefficient as a function of the reduced chemical potential where F is the Fermi integral which, in the general case with index r, is given where f is the carrier distribution function. For acoustic phonon scattering r = 0 [21].We note that the assumption of pure acoustic phonon scattering may not fully hold, especially for Bi 2 Se 3 -rich compositions. Transport studies on Bi 2 Se 3 have shown mixed scattering by both acoustic phonons and ionized impurities. The ionized impurities are due to Se vacancies [22].
The reduced chemical potential of holes is related to the reduced chemical potential of electrons where E g is the energy gap between the two bands. The total Seebeck values and the belonging reduced Fermi level depend sensitively on the gap width and on the A parameter [23].
The direct application of a two-band model in our case is not straightforward due to the relatively large number of parameters. For n-type doped Bi 2 Te 3-x Se x , studies on the transport properties have shown that the electron effective mass decreases with x [24,25], while the electron mobility was proposed to remain nearly unchanged up to x~1 [25]. The mobility of holes is expected to decrease upon alloying with Se. For both Bi 2 Te 3 and Bi 2 Se 3 the valence band structure is complex [26,27]. Different valleys in both the conduction and the valence bands may contribute to conduction upon alloying [16]. All the above make the estimation of the A parameter challenging. Owing to this, this analysis of mixed conduction can be at most qualitative. antisites, while the n-type character is due to Te vacancies or Te Bi antisites [28]. Since in our case the x=1.0 material is not fully ordered Se vacancies are also a source of n-type conduction The EDS results (Table 1)  On the contrary, LaChance et al. reported Seebeck values close to zero and intrinsic conduction [31]. The similarity of the experimental and theoretical behaviors of Fig. 2 suggests that a two-band model may describe the whole compositional dependence of the transport properties. For a given Fermi level, band width and conductivity ratio one expects the Seebeck coefficient for each x value to be described by the theoretical curve of Fig. 2c.
The aforementioned interaction of the n-type defects with the p-type ones leads to the inaccurate determination of carrier concentration derived from volume-averaged (i.e. integral) techniques such as Seebeck coefficient measurements [15,[29][30][31]. In contrast to electrical transport methods, the NMR lineshape and T 1 measurements allow for a site-specific characterization of the electronic band structure [14] due to their proportionality to the effective mass and the carrier density. We have measured 125 Te NMR spectra and T 1 measurements as function of Se content to extract microscopic information related to changes in the density of states at the Fermi-level. We extracted frequency shift and linewidth of each spectrum as a function of Se content. There is a linear dispersion between the 125 Te shift and conductivity, indicating that the Bi 2 Te 3-x Se x system follows a mixed conduction mechanism provided by ptype and n-type carriers (Fig.3a). Interestingly, this composition dependence of the NMR shift is more pronounced in x=1, which exhibits a minimum with a shift difference of 300 ppm from the case of x=0.2 and 100 ppm from the x=1. 8. In addition, the 125 Te NMR linewidth narrows with increasing Se content. Close to x=1, the linewidth achieves its lowest value, which we attribute to a minimum in electrical conductivity (Fig. 2b).
The 125 Te magnetization recovery data in Bi 2 Te 3-x Se x series were fitted to a stretched exponential function, as a stretched exponential model describes well materials dominated by structural defects and electronic inhomogeneities [14,32,33]. The presence of structural defects within the composition range is consistent with the observed value of the Kohlrausch exponent (β) [32]. For Bi 2 Te 3-x Se x β was found to be 0.7 [14]. In case of semiconductors such as Bi 2 Te 3- x Se x , the spin-lattice relaxation rate is proportional to ( * ) The charge carrier concentration due to the mixed conductive state of Bi 2 Te 3-x Se x originates from both electrons and holes, and ℎ that change differently by a thermally activated T-dependence [37]. NMR in PbSe [38] and PbTe [39] has probed similar behavior. At ambient temperature, the interplay between and ℎ is constant as expressed by the conductivity but strongly varies across the Bi 2 Te 3-x Se x series. In Fig. 3b, we monitored this interplay between electrons and holes by plotting the T 1 -1 versus the electrical conductivity. The linear dispersion between the relaxation rate and conductivity indicates that the entire system follows a mixed conduction mechanism analogous to the aforementioned 125 Te shift behavior.
Hence counteracting effects between the hole-rich, and electron-rich, conductivity varies systematically. To obtain a clear view of the electronic changes in the vicinity of the p-to-n transition, we plotted the spin-lattice relaxation time, the Seebeck coefficient and the carrierdensity product 1/(T 1 .T 1/2 ) versus Se content (Fig. 3c). The gray region indicates an n-type phase, whereas the white region shows a p-type phase. The larger T 1 value at the p-to n-crossover is consistent with a decreased net carrier concentration in the x=1.0 composition. In addition, the quantity 1/(T 1 .T 1/2 ) which is proportional to the charge carrier concentration (N) [14,[35][36][37], achieves a minimum value at the x=1.0 composition. All the above results suggest that counteracting processes between defects modify the electronic characteristics. This countereffect could be the driving force of the gradual transition from the p-type to n-type conductivity in Bi 2 Te 3-x Se x .

Conclusion
The p-to n-transition of the Bi 2 Te 3-x Se x series was monitored via room temperature This product is proportional to the charge carrier density in the vicinity of the p-to n-crossover.
Therefore, the two-band model captures the general features of the p-to n-transition and explains the large bulk resistivity obtained for properly alloyed ternaries and quaternaries TIs.