Thermopower and resistivity of the topological insulator Bi2Te3 in the amorphous and crystalline phase

We have, in-situ, prepared and measured the temperature dependence of thermopower S(T) and resistance R(T) of Bi2Te3 topological insulator (TI) thin films in the amorphous and crystalline phase. Samples were prepared by sequential flash-evaporation at liquid 4He temperature. The S(T) in the amorphous phase is negative and much larger compared to other known amorphous materials, while in the crystalline phase it is also negative and behaves linearly with the temperature. The resistivity ρ(T) in the amorphous phase shows a semiconducting like behavior that changes to a linear metallic behavior after crystallization. S(T) an ρ(T) results in the crystalline phase are in good agreement with results obtained both in bulk and thin films reported in the literature. Linear behavior of the ρ(T) for T > 15 K indicates the typical metallic contribution from the surface states as observed in other TI novel materials. The low temperature conductivity T < 10 K exhibits logarithmic temperature dependent positive slope κ ≈ 0.21, indicating the dominance of electron-electron interaction (EEI) over the quantum interference effect, with a clear two dimensional nature of the contribution. Raman spectroscopy showed that the sample has crystallized in the trigonal R3―m space group. Energy-dispersive x-ray spectroscopy reveales high homogeneity in the concentration and no magnetic impurities introduced during preparation or growth.


Introduction
Topological insulators (TI) are a new class of materials, which were theoretically predicted to exist in different Bi, Sb and Te based compounds [1][2][3].Angle-resolved photoemission spectroscopy (ARPES) has been used to verify the topological surface states of such materials [4][5][6].TI have, due to strong spin-orbit coupling, an insulating bulk energy gap and generate conducting topologically protected gapless electronic surface states, which are robust against disorder and magnetic impurities [7].These new materials have very interesting properties from the basic physics point of view and for future applications, for example, the observation of the quantum spin Hall effect or the realization of Majorana fermions for the application in topological quantum computation [8,9].The alloy Bi 2 Te 3 was proposed as a three-dimensional (3D) TI, and later it was experimentally confirmed by ARPES [6] that this system possesses a single Dirac cone on the surface.Transport properties of different TIs, including Bi 2 Te 3 , were extensively studied and it had been observed that these materials exhibit a wide variety of exotic electric transport properties like extremely large MR, high carrier mobility µ, light effective mass, nontrivial Berry phase, and the anomalous Hall effect [10].
Since the 1950s [11], it is well known that those materials, today called TIs, are the best thermoelectrics (TE) at room temperature, with the most special being Bi 2 Te 3 in its bulk form and known as one of the best TE materials for application at room temperature and above (270-400 K) with the dimensionless high figure of merit Z = S 2 σ/(κ e + κ ph ) ≈ 1 [12], where σ is the electric conductivity, κ = κ e + κ ph is the thermal conductivity, consisting of a contribution from electrons κ e and phonons κ ph , and S is the thermopower.So, in order to improve the performance of TE it is necessary to increase the power factor defined as PF = S 2 σ, which is mainly associated to the electronic part of materials [13].
The influence of TI boundary states has long been neglected in early thermoelectric research due to the lack of information of its existence.In recent years this neglected issue has attracted intensive research efforts from the theoretical and experimental point of view to explore the contribution of TI boundary states to thermoelectricity.For example, Dirac fermions in topological edge and surface states are robust against backscattering with defects in the crystal structure and magnetic and non magnetic impurities [14].This opens the possibility of creating an ideal thermoelectric material that combines the properties of a phonon-glass electron-crystal proposed by Slack [15].Summarizing the idea, creating a large number of defects in a TI, e.g. using ion irradiation, it is possible to considerably reduce its lattice thermal conductivity κ ph while retaining the high electrical conductivity of topological surface states.First theoretical predictions of topological phases of matter are said to be ubiquitous in crystals, but less is known about their existence in amorphous systems that lack long-range order but possess well-defined local order that involves, up to, neighbors larger than the fifth order [16,17].But in recent years new theoretical studies have appeared stating that there is the possibility of realizing topological phases in amorphous systems [18,19].Very recently, the observation of topological states in a sample of amorphous Bi 2 Se 3 has been experimentally demonstrated.The authors have shown that applying a magnetic field to amorphous Bi 2 Se 3 , a typical weak anti-localization was observed in the magnetoresistance at small magnetic fields.Additionally, ARPES supports the topological state interpretation with data showing consistency with a dispersive two-dimensional surface state that crosses the bulk gap [20].
In the present research, for the first time in the literature, we are presenting a detailed study of the thermopower S(T) and the resistivity ρ(T) of a Bi 2 Te 3 thin film in the amorphous and crystalline phase, this was possible only because of the preparation method and the low temperature cryostat used.All results from the amorphous phase are unique and new, while the crystalline phase results are very similar to those from the literature, where samples were prepared with more expensive and sophisticated methods.The results of the crystallized sample evidently show contribution in the S(T) and ρ(T) of the 2D topological protected surface, while we cannot state that the results of the amorphous phase have a topological character, due to the limitations of magnetic fields during the measurements and the lack of theory that supports the results of thermopower.

Experimental details
Bi 2 Te 3 alloys were prepared by melting pure bismuth and tellurium (purity 99.9999%, Alfa Aesar) in proper atomic proportions in an evacuated quartz ampoule at a base pressure of approximately P ≈ 10 −7 mbar.After the alloy was removed from the ampoule and milled into pieces of 200-300 µm in diameter, it was introduced into a chamber for in-situ film preparation and measurement.The amorphous Bi 2 Te 3 thin film was prepared by rapid quenching at liquid 4 He temperature under high vacuum conditions (≈5 × 10 −7 mbar), i.e. the technique called flash evaporation, which was already used successfully to prepare TI films of Bi 2 Se 3 [21] and Bi 100−x Sb x [22].The absolute Seebek coefficient S(T) was measured between 5 and 325 K by applying a slowly alternating temperature gradient across the thin film as described in the literature [22][23][24].The temperature dependence of resistance R(T) was measured parallel to the S(T) between 1.2 and 345 K, for that the film was structured in an conventional 4-point shape.After the sample was characterized in S(T) and R(T), it was removed from the cryostat for additional ex-situ sample characterization at room temperature.At this point we want to comment that due to the in-situ experimental conditions used in this work, where simultaneous measurements of resistance and thermopower are being performed, the setup does not allow for additional spectroscopic measurements (such as XPS, Raman, XRD, LEED etc).For this reason we have only structurally characterized the crystallized sample.Elemental analysis to determine the atomic composition was performed using the energy-dispersive x-ray (EDX) spectrometer coupled with a FEI NanoLab XT 200 Dual-Beam microscope, film thickness was determined by AFM and finally Raman spectroscopy measurements were performed with a LabRAM HR Evolution spectrometer, using wavelength of λ = 532 nm.Temperature dependence of the resistivity ρ(T) in the amorphous and crystalline phase of the investigated Bi 2 Te 3 thin film are shown in figure 1(a).The ρ(T) in the amorphous phase was measured immediately after preparation, i.e. the data were obtained during annealing and are non irreversible results (indicated with −→ in figure 1(a), and after maximal annealing temperature was reached, the sample was again cooled and the ρ(T) measured by increasing the temperature.These results are reversible and are indicated with double arrow (←→) in figure 1(a).After preparation, the amorphous Bi 2 Te 3 has a large resistivity which decreases by increasing the temperature in a non activated temperature behavior, and approximately at T K ≈ 269 K the film crystallizes, this is evident by the rapid and huge decrease of the resistivity, a common method to identify crystallization temperature of amorphous materials [25][26][27][28].After crystallization, the ρ behaves linearly during annealing up to T = 345 K.We observe a remarkable behavior of the amorphous ρ(T) at low temperatures, as can be seen in the inset of figure 1(a), a drop in the ρ(T) is observed starting at T = 2.29 K and reaching ρ(T) = 0 at T 0 = 0.78 K (determined by linear extrapolation).This behavior we interpret as a sign of superconductivity, which was already observed in amorphous TI samples of Bi 100−x Sb x [24].The fact that we cannot measure zero resistance is because we could not cool down our sample lower than 1.2 K. Another argument supporting our statement is based on the observation of phonon-induced electron-electron interaction in disordered superconductors [29].It was established experimentally, as well theoretically, that the effects of electron-electron (e-e) interaction in 3D disordered superconductors cause a resistivity change at low temperatures (i.e. in the residual resistivity regime) given by [30]

Results and discussion
where e is the electronic charge, k B is the Boltzmann constant, h is the Planck constant divided by 2π, D is the diffusion constant, F is the electron screening parameter and T c the superconducting transition temperature [30].In order to use the equation ( 1) to fit our results, it is necessary to have the value of D, which was obtained in similar experiments by measuring the magnetoresistance [31,32], which was not possible in our case.For this reason, in order to have a qualitative description of our results, we have rewritten the equation as following, ∆ρ(T)/ρ 2 0 = −P 1 T 1/2 + P 2 T 1/2 / ln(T c /T) and added the contribution of the usual Boltzmann transport term P 3 T 5 + P 4 , similar as done in the literature [29,31,33], with the constants P i with i = 1, 2, 3, 4 and T c being the fit parameters.The experimental ∆ρ(T)/ρ 2 0 , with ρ 0 = ρ(T = 10 K), and the fit result are shown in the inset of figure 1(b).It can be seen that the fit describes the experimental data well, with the parameters P 1 = 5.16 × 10 −6 , P 2 = 3.6 × 10 −6 , P 3 = 2.99 × 10 −11 , P 4 = 1.9 × 10 −5 and T c = 1.02K, a value little higher that the one predicted when we extrapolate the experimental results to zero resistance.This result leads us to state that it is highly probable that the Bi 2 Te 3 in the amorphous phase is superconductor, similar as the TI Bi 100−x Sb x in the amorphous phase.It should be noted that by doping Bi 2 Te 3 with Pd Hor et al, have reported superconductivity with a T c = 5.5 K [34].
In the crystalline phase, in general, ρ(T) of the Bi 2 Te 3 film shows a linear metallic behavior as shown in figure 1(b), very similar as in the case of other TI Bi 2 Te 3 thin films prepared with different techniques, such as molecular beam epitaxy [35,36], metal organic chemical vapor deposition [37] and DC magnetron sputtering [38].The estimated resistivity of our Bi 2 Te 3 thin film is in the same order as in thin films [38,39] and bulk [34] samples of Bi 2 Te 3 reported in the literature.We have payed attention to the low temperature results (i.e.T < 10 K) and plotted in figure 2 the results of the resistivity vs temperature and the conductivity (in units of e 2 /π h) vs the temperature in logarithmic scale.It is evident that for T ⪅ 8 K the ρ(T) exhibits an upturn by decreasing the temperature.This behavior was already reported in films [40] and bulk [41] samples of the TI Bi 2 Te 3 and Bi 2 Se 3 [42].
According to many literature results, both theory [43] and experiments, it was established that in many TI materials (non magnetic doped) quantum interference effect (QIE), i.e. weak localization, weak antilocalization, and many-body disorderenhanced electron-electron Coulomb interaction (EEI) dominate the conductivity at low temperature and are responsible for the upturn in the resistivity.
Assuming that both QIE and EEI contribute to the conductivity at low temperatures (T < 8 K in our case), the total correction ∆σ(T) of conductivity can be expressed as: with ∆σ (T) ] ln and where σ 0 is the Drude conductivity (ne 2 τ /m e ), n is the electron density, m is the mass of electron, τ is the relaxation time, F is the dimensionless screened Coulomb potential between electrons averaged over the Fermi surface [30], and T EEI is the characteristic temperature below which logarithmicdependent conductivity due of EEI starts to dominate.The dimensionless parameter p depends on various sources of inelastic scattering that destroy phase coherence and in a 2D disordered electronic system p = 1 for EEI, α is a dimensionless coefficient in Hikami-Larkin-Nagaoka (HLN) theory and T QIE is the characteristic temperature, above which the conductivity corrections due to QIE vanish.
Following literature works [40,[44][45][46], in order to analyze our results, in our case we may assume without any loss of generality, that T EEI = T QIE = T max = 5.5 K, where the conductivity of the material reaches its maximum as the characteristic temperature below which these corrections is valid.So, finally the equation ( 2) in the unit of e 2 /π h can be written as following: where ∆σ(T) = [σ(T) − σ(T = T max )], κ is the slope of ∆σ(T) vs ln(T) curve and it can be expressed as κ = (1 + αp − 3F/4) = ∂∆σ ∂ ln(T) , with F the Coulomb screening factor that quantifies the screened Coulomb interaction [29,47].The obtained value of κ from the fits to ∆σ(T) shown in figure 2(b) is κ = 0.212 83 ± 0.000 46.It was determined that in TI materials, κ is always positive due to the dominance of the EEI effect [43,44] over the QIE.In figure 2(b) we have also plotted (as dashed lines) a correction of electrical conductivity ∆σ(T) because of the EEI in 3D materials at low temperatures, i.e. ∆σ(T) ∝ √ T, evidencing that rightly so we can discard at all the 3D nature of the correction.
Considering the values obtained in the literature for bulk and thin films of Bi 2 Te 3 , we assume p = 1 and α = −1/2, values obtained after fitting the conductance ∆σ(B) = σ(B) − σ(0) as a function of the perpendicular magnetic field within the framework of the HLN theory [40,[48][49][50].Thus, using the above mentioned values for p and α, we obtain F = 0.382.According to the theory within the Thomas-Fermi approximation it is predicted −1 ⩽ F ⩽ 1, F → 0 in the limit of no screening (or, bad conductors), while F → 1 in the limit of complete screening (or, good metals) [30,47].The obtained value is very similar to the one reported for thin films of Bi 2 Te 3 [40,51] and Bi 2 Se 3 [42], not only in absolute value but also in sign.Another scattering mechanism which produce an logarithmic upturn in the resistivity at low temperatures is the Kondo effect [52,53], due to magnetic impurities, probably introduced during the sample preparation.This contribution we discard, because its very well known that thermopower is very sensitive to magnetic impurities scattering, manifesting as a bump-shaped feature at low temperatures as shown in figure 3(b).This figure serves as an example of what was measured in Au containing ≈10 ppm of Fe.

Themopower in the amorphous and crystalline phase.
Thermopower results in the amorphous ad crystalline phase are plotted in figure 3. The S(T) in the amorphous phase is negative and increases its magnitude by increasing the temperature up to 200 K to later slowly change its slope and when the sample starts to crystallize, a sharp change in the S(T) is observed.The temperature at which the sample crystallizes visible by the S(T) results is the same as observed by the resistance measurements.Above T K the S(T) still shows a negative slope and increases in magnitude by increasing the temperature.The absolute value of the S(T) of the amorphous Bi 2 Te 3 is large in comparison to other amorphous metallic alloys, e.g. they reach a value of S(T = 200 K) < 2µ V K −1 [54], i.e. two orders less than in our investigated sample.After maximal annealing at T = 345 K the crystallized sample was cooled and again the S(T) measured, in order to make that more evident, we have plotted the results in the figure 3(b) which show negative thermopower over the entire range of temperatures.The negative S(T) in Bi 2 Te 3 samples is because its Fermi energy E F lies in the bulk conduction band due the n-type bulk carriers induced by Te vacancies, as confirmed by the Hall effect [36] and ARPES measurements [35].As it can be seen, the thermopower of the crystallized sample behaves almost linearly in all temperatures measured, but careful observation leads us to see that there is a fast change in the slope at T sl ≈ 25 K which is made clear by linear fits shown in the figure 3(b) and more evident if we plot the first derivative dS(T)/dT, as shown as inset in the same figure.From our knowledge, there is no theory discussing such behavior, although observed in some experimental works [36,40], pointing out the mentioned regions with linear behavior but not discussed [40] explicitly.But we do have an idea of where this is coming from.If we take into account diverse results in the literature, where magnetotransport properties are investigated, we can observe that one parameter which changes steadily with the temperature is the phase coherence length l ϕ , which signifies the length over which an electron can move without losing the phase of its eigenstate [55].Literature results of Bi 2 Te 3 [48,51] show that the l ϕ (T) tends to have a value close to zero at approximately T ≈ 25 K, which is similar to the temperature T sl .To remark, the steady decrease in l ϕ with temperature is expected because of the increased elastic and inelastic scattering of charge carriers by impurities and phonons [56], and we believe that it is the reason for the slope change in S(T), at the same range of temperature.The change of slope can not be related to some magnetic impurities as the effect of only a few ppm is already noticeable.An example is thermopower of gold containing ≈10 ppm of Fe plotted as inset in figure 3(b), the deep negative peak in S(T) is a consequence of the Kondo effect [57], which in this specific case even originated a change in the sign of S(T).The dotted lines indicate what we would observe without Fe impurities and the peak around 35 K is identified as consequence of the phonon drag effect [57].

Power factor in the amorphous and crystalline phase
The temperature-dependent power factors PF(T) in the amorphous and crystalline phase for the studied Bi 2 Te 3 sample have been obtained from ρ(T) and S(T) results and are shown in figure 4. Generally, in both phases, the PF(T) increases with T, without displaying any maximum up to the maximum temperature used in this investigation.The PF in the amorphous phase is almost negligible for temperature T < 100 K for later increase up to the temperature where the sample crystallizes.During the crystallization the PF decreases suddenly for after increase by increasing the temperature (shown as star symbols in figure 4).When we compare the PF of both phases we can observe that for temperature 230 K < T < 270 K, the PF corresponding to the amorphous phase is evidently larger than that from the crystalline phase.If we take into the consideration that the thermal conductivity κ of the amorphous phase is less than in the crystalline phase, mainly due to phonon contribution κ ph , we can state that the figure of merit Z of the amorphous phase is larger than in the crystalline phase, probably in a larger range of temperature than the one mentioned above.From the PF results in the crystalline phase we can observe that it behaves almost quadratic with the temperature, as shown by the fit done and plotted as line in the same figure.Using the extrapolated data we have compared our PF with those from the literature for higher temperatures as shown in the inset of figure 4, showing that our results are in good agreement with the literature where samples were prepared with more complex methods and annealed to 650 K [58] and 423 K [59], respectively.

Raman spectroscopy
Raman spectroscopy is used in the last years as powerful method to provide useful structural information, crystalline phase, composition, and stoichiometry of different TI samples [60,61].We have performed Raman measurements on our sample at ambient conditions after the sample was annealed at T = 345 K and the results are shown in figure 5.After analyzing the Raman results (figure 5) by Gaussian fitting, the frequencies of the peaks and their assignment are listed in the table 1, together with the previously reported data from the literature in bulk and similar film thickness.In figure 5 we can see that to obtain a good fit of the most relevant peaks of the Raman spectrum, we have used two Gaussians in the same position but with different intensities and widths.We interpret this as necessary since our sample is polycrystalline and there must be regions of the sample with crystalline structure of nanometric size (or perhaps also grain boundaries).This interpretation is supported by the Raman results of the TI Sb 2 Te 3 [62], where the Raman spectral evolution through amorphous to crystalline transitions was studied, and shown that Raman peaks immediately after crystallization are wide and not very intense.The use of Gaussian peaks that are not indexed can be attributed to surface oxidation, also observed in other thin films [63,64] but not in the case of in-situ Raman measurements of a ≈40 nm thick sample [65].
Considering the experimental resolution, one can see that the peak positions of the most characteristic expected optical phonon modes are in good agreement with the previously reported data of Bi 2 Te 3 in the literature [61,[64][65][66].The Raman results shown in figure 5 and the corresponding assignment, indicate that our preparation method (after annealing) leads to obtain well ordered crystalline films of Bi 2 Te 3 , with the corresponding crystal structure belonging to the space group R3m(D 5 3d ) [67].

EDX spectroscopy
These experiments were done at room temperature after the transport measurements were performed.It is here important to note that up to this point the samples were exposed to air for more than six months.The main aim of this experiment was to confirm the nominal composition and if it was maintained in comparison to the starting one, as well as to determine if some ferromagnetic contaminants were present in the sample.One example of the result is shown in figure 6, where peaks corresponding to Bi and Te are evidently observable, as well as the presence of peaks related to the oxygen.As we mentioned, before EDX measurements, the sample was exposed to air and its well known that a oxidation process occurs.The oxidation was confirmed by XPS measurements in the literature [68,69].So, without losing generality, we have quantified the results without considering the oxygen contribution and calculated the sample composition from the EDX results obtained at three different parts of the sample.The results are presented in the table 2.Besides confirming the absence of ferromagnetic impurities within the detection limits of this technique, EDX analysis allowed to verify that the actual composition of the samples agrees with the nominal value with an error of of ±1.3%, confirming that our synthesis method (sequential flash evaporation) is very accurate in producing the desired composition and in a good agreement with previous results of other TIs prepared by the same technique like Bi 60 Se 40 [21] and Bi 100−x Sb x [22,24].We emphasize the EDX results regarding   the ferromagnetic impurities such as Fe, Co and Ni, because this supports the interpretation that the upturn of the R(T) at low temperatures is due to the EEI interaction and not because of some ferromagnetic centers scattering process.

Conclusions
We have prepared Bi 2 Te 3 thin films samples by the sequential flash evaporation method in the amorphous and crystalline phase and measured in-situ the thermopower and resistance.
The amorphous film shows a huge and semiconductor-like type resistivity, with a peculiar rapid decreasing for temperatures T < 2.3 K which we interpret as a sign of a probable superconducting transition as observed in amorphous TI Bi 100−x Sb x , while the thermopower also shows huge negative values up to when crystallization occurs.In the crystalline phase, the ρ(T) is metallic in all ranges of temperature measured, except for T < 10 K, where we have described the upturn in R(T) because of the EEI.Thermopower in the crystalline phase shows linear behavior in two regions, one down to T ≈ 25 K and above another, our S(T) results are in agreement with literature results, where linearity was observed in similar samples and other TI alloys.Raman results are in good agreement with the expected peaks for a well ordered Bi 2 Te 3 , corresponding to the crystal structure belonging to the space group R3m(D 5 3d ), and finally EDX results confirm that our preparation method retained the initial concentration and that during the film production process no magnetic impurities were introduced.We want to conclude by mentioning that our preparation method not only leads us to obtain high quality films, but there will also be the opportunity to prepare thin films in the amorphous phase to stepwise investigate the changes in their electronic (as well as magnetic) properties in its evolution towards crystalline transformation.In addition, our preparation method is compatible with processes that include lithography, giving the opportunity to study bilayers between TIs or with other non-magnetic and magnetic materials, and exploration of new properties for future applications.

Figure 1 .
Figure 1.Temperature dependence of the resistivity ρ(T).In (a) results of the amorphous and crystalline phase.In (b) results of the crystalline phase.The inset in (a) shows the lowest temperature of the resistivity in the amorphous phase.The inset in (b) shows the fit using the modified equation (1).

Figure 2 .
Figure 2. Low temperature results of the crystalline phase, in (a) the resistivity ρ(T) and in (b) the corresponding conductivity normalized in units of e 2 /π h.

Figure 3 .
Figure 3. Temperature dependence of the thermopower S(T).In (a) results of the amorphous phase, during crystallization and crystalline phase.In (b) results of the crystalline phase.The down inset in (b) are results from a crystalline Au sample and upper one is the derivative of the thermopower of our crystallized sample.

Figure 4 .
Figure 4. Temperature dependence of the power factor PF(T) of the amorphous and crystallized phase of Bi 2 Te 3 thin film.The symbols are results during and after crystallization as explained in the text.The inset is our extrapolated data compared with results from the literature.

Figure 5 .
Figure 5. Raman spectra of the crystallized phase of Bi 2 Te 3 thin film.

Figure 6 .
Figure 6.EDX spectrum obtained for sample Bi 40 Te 60 at the position P o1 .The inset shows a scanning electron microscope (SEM) image of the investigated sample at position P o1 .

Table 1 .
Identified Raman peaks and widths (FWHM) in units of cm −1 .The first row are data of this work while the other rows are from the literature.The samples thickness is d.

Table 2 .
Elemental composition of the Bi 40 Te 60 sample obtained at two (P o1 , P o2 ) different positions in the sample.