Electron–phonon coupling and a resonant-like optical observation of a band inversion in topological crystal insulator Pb1−x Sn x Se

The optical reflectivity of n-type Pb0.865Sn0.135Se and Pb0.75Sn0.25Se solid solutions was measured in the THz spectral region energetically corresponding to optical phonon excitations and in the temperature range from 40 K to 280 K. The first solid solution exhibits an open energy gap with trivial band ordering at all temperatures, while for the second one the transition from trivial insulator to topological crystal insulator phase is expected. The analysis of Pb0.75Sn0.25Se data performed within the dynamic dielectric function formalism revealed an anomaly of resonance-like character in the temperature dependence of LO phonon frequency. The formula for LO phonon energy renormalization reproduced observed anomaly for energy gap equal to zero, the phase transition occurred at T 0 = (172 ± 2) K. This effect was absent for Pb0.865Sn0.135Se. Present results show that reflectivity measurements in the THz range in the vicinity of LO phonon frequency can be valuable experimental method for precise determining of band inversion temperature in narrow-gap topological materials.


Introduction
The studies of topological properties of semiconducting and semimetallic materials and related topological phase transitions attract a lot of attention in recent years being one of the hot topics of contemporary condensed matter physics.The canonical topological materials, like bismuth or antimony chalcogenides, are narrow-gap semiconductors with an inverted band structure, for which electronic Dirac-like metallic states protected by the time-reversal symmetry occur on the surface of insulating bulk crystals [1,2].Another class of Dirac topological materials are topological crystal insulators (TCI) for which crystal symmetry guarantees the existence of surface Dirac states [3].It is expected that due to their particular electronic properties topological insulators could serve as a new type of electrical conductors with almost no energy dissipation in nanoscale interconnects, improve presently considered devices in the area of thermoelectricity and infrared or THz optoelectronics as well as constitute materials platform for quantum computation [1,2].
While considering possible applications or devices taking advantage of the unique electronic properties of TCI materials the crucial point is the determination of topological phase diagram, in particular temperature, pressure and chemical composition corresponding to topological transitions.Several experimental methods were applied in the past for this purpose.The bulk band gap closing at the topological phase transitions followed by the opening afterward is accompanied by the inversion of the symmetry character of the bottom of the conduction band and the top of the valence band.It was the reason that a big part of experiments applied so far for studies of solid solutions which could exhibit the trivial insulator-TCI phase transition are focused on demonstration of the presence of inverted bulk band structure.However, mentioned above important modifications of the band structure usually have a minor impact on a number of electrical and optical properties of investigated material and a clear demonstration of the bulk band inversion is a challenging task.
The Pb 1−x Sn x Se is a TCI material as it was demonstrated a few years ago using the angle-resolved photoemission spectroscopy (ARPES) [4].This solid solution is one of the best-known materials exhibiting the topological transition from a trivial insulator to TCI phase when modifying chemical composition, temperature, or pressure [5][6][7].The Pb 1−x Sn x Se solid solutions belong to the well-known, IV-VI based, narrow-gap semiconductors family.For the chemical composition 0 ⩽ x ⩽ 0.37 they crystallize in the rock salt structure and the conduction and valence band edges are located at the L points of the Brillouin zone.Below some critical composition x at all temperatures the conduction band is predominantly derived from the cation 6p (Pb) or 5p (Sn) orbitals (L 6 − symmetry), whereas the highest valence band is in majority formed from the anion 4p (Se) orbitals (L 6 + symmetry).With an increasing Sn content at given temperature depending on x (where 0.18 ⩽ x ⩽ 0.37) the system undergoes energy band inversion and the symmetries of these bands above the critical x value change respectively.
The dependence of Pb 1−x Sn x Se energy gap E g (in meV) on temperature T and chemical composition x of the solid solution can be described by the expression [8]: The temperature dependence of the energy gap for Pb 0.75 Sn 0.25 Se crystal resulting from equation (1) shown in figure 1 predicts the zero gap and band inversion at T 0 = 168 K.The gap is always open for Pb 0.185 Sn 0.135 Se crystal-our topologically trivial reference material [4].
The first experimental method to estimate the temperature of band inversion for Pb 1−x Sn x Se solid solution was based on the observation of a broad minimum in the temperature dependence of resistivity [9] and T 0 = 150 K for x = 0.23 was determined.The authors did not estimate possible experimental error for this value, but it probably could be as high as ± 20 K.The possibility of determination of pressure corresponding to the band inversion for Pb 1−x Sn x Se solid solution was demonstrated by pressure modification of the laser diode emission energy [10] and by pressure-dependent minimum of plasma frequency observed in the reflectivity spectrum [11].
The evidence of the band inversion in Pb 1−x Sn x Se bulk crystals was also obtained from thermoelectric measurements of the Seebeck and the Nernst-Ettingshausen effects [12,13].The theoretical analysis of thermoelectric effects in temperature driven band inversion regime pointed out that the Nernst-Ettingshausen effect is particularly strongly affected by the band inversion and changes sign or reaches maximum when the energy gap vanishes.The T 0 value equal to 180 K [12] and to (150 ± 10) K [13] for Pb 0.75 Sn 0.25 Se solid solution, respectively, was obtained in these papers.Interestingly, from the analysis of infrared reflectivity data (in the plasma edge region) for different samples with the same solid solution composition (x = 0.23) a quite different values of the temperature corresponding to the phase transition, T 0 = 100 K and (160 ± 15) K were proposed in [14,15], respectively.The present method has an additional advantage of better accuracy for temperature determination over the Nernst-Ettingshausen effect used in [13].
The direct evidence of the presence of TCI electronic states for Pb 1−x Sn x Se solid solution obtained by ARPES directly probing the surface electronic structure suggested T 0 value close to 100 K [4].Later on this technique was successfully applied by a few research groups [8,[16][17][18].The T 0 equal to (130 ± 15) K and (175 ± 15) K for solid solutions with x = 0.23 and 0.27, respectively, was obtained in [8].The ARPES-data based full composition-temperature (x-T) topological phase diagram of Pb 1−x Sn x Se crystals was presented in [8,18].The TCI states were also detected in scanning tunneling microscopy and spectroscopy studies locally probing the density of electronic states at the surface [7,[19][20][21].The ultrahigh mobility surface states observed in Pb 1−x Sn x Se (where x = 0.23-0.25)by infrared reflectivity in high magnetic fields were reported in [22].Finally, an important difference in surface phonon dispersion in trivial insulator and TCI phases was also recently demonstrated for Pb 0.7 Sn 0.3 Se solid solution using inelastic He atom scattering measurements [23].The possible additional problem in determination of the trivial insulator-TCI phase transition parameters for a solid solution was pointed out by the density functional theory (DFT) analysis suggesting that this transition is broadened by local chemical (crystal field) disorder with the band inversion transition in a substitutional alloy involving zero-gap states with several cation and anion band crossings (degenerated only for ideal rock salt symmetry) [24].However, this DFT approach concerns zero temperature limit and neglects the electron-phonon coupling.
An influence of the phonon system on the electron band structure of solids resulting from the electron-phonon coupling was investigated for many years and today is a well-established phenomenon.The opposite effect, i.e. possible influence of the electron system and, in particular, of the band structure on the phonon system was much less explored in the past.The simple Drude-Lorentz model satisfactorily describing previous reflectivity and transmission spectra in the infrared assumed independent damping constants of free-carrier plasma and phonon excitations [10,13,14,25].However, a deviation from this model can be expected when the light frequency approaches the optical phonon frequency.A strong mixing of the electron plasma and polar phonon modes resulting in the collective motion of electrons and ions causes a significant modification of the effective scattering of electrons by impurities [26].The influence of the electron system on the phonon system resulting in a correction to the TO phonon frequency was theoretically investigated for Pb 1−x Sn x Te, material closely related to the Pb 1−x Sn x Se, providing the equation for renormalized TO phonon energy [27].A step in reflectivity spectra resulting from an influence of plasmon-phonon coupling on the electron-impurity interaction in the free carrier absorption and corresponding to the optical phonon mode frequency can also be expected in this case.This effect was previously demonstrated for LO phonon in a narrow-gap semiconductor with an inverted band structure, HgSe [28].The optical phonon softening accompanied by an anomaly in the phonon linewidth resulting from electron-phonon interaction observed in Raman spectra for Sb 2 Se 3 bulk crystal were explained by an electronic topological transition occurring under pressure for this compound [29].Possible evidence of the electronic band inversion by an analysis of the bulk optical phonon linewidth was also demonstrated in [30].A modification of the formula given in [27], predicting two discontinuities in the temperature dependence of phonon frequency for E g = 0 and E g = hω TO was shown to be valid for II-VI semiconductor with zero-energy gap [31].The direct experimental evidences of this effect were demonstrated for Hg −x Cd x Te [31] and Hg 1−x Zn x Te [32] solid solutions.
In summary, depending on an experimental method, electron conductivity type and free-carrier concentration, published values of determined or estimated topological phase transition temperature for Pb 0.77 Sn 0.23 Se solid solution are scattered throughout almost 100 K. Recently, using state of the art Landau level spectroscopy, this temperature was estimated as being close to 150 K in epitaxial layers with the same chemical composition [33].Although reflectivity measurements and Kramers-Krönig (KK) analysis are known for years, the temperature-dependent reflectivity studies of topological insulators are still very rare.In this work, we show that topological phase transition temperature can be determined in bulk typical TCI from reflectivity measurements and KK analysis in the THz spectral range corresponding to LO phonon frequency.

Materials and methods
The Pb 0.75 Sn 0.25 Se and Pb 0.865 Sn 0.135 Se solid solution single crystals used in this study were grown by the self-selecting vapor-growth method [34,35].Their chemical composition and homogeneity of the crystal was determined by energy dispersive x-ray spectroscopy (for details, see [4,13]).The 2 mm thick crystal plates with the dimensions 4 × 3 mm 2 selected for optical studies were cleaved by the razor blade along (001) natural cleavage plane.The electron transport measurements carried out previously for this Pb 1−x Sn x Se solid solution demonstrated an n-type conductivity, for the sample with x = 0.25 the electron concentration equal to 3.1 • 10 18 cm −3 and 2.4 • 10 18 cm −3 at T = 4.2 K and T = 295 K, respectively [13].
The reflectance measurements were performed with Bruker FTIR vacuum spectrometer Vertex 80 v operating in a rapid-scan mode [36].As a light source the mercury lamp with mylar multilayer beamsplitter was used.The Si bolometer with 700 cm −1 cut-off low-pass filter employed as a detector spectrally covered the range of THz frequencies.The measurements were performed from 50 cm −1 -450 cm −1 for oblique incidence of 12 degrees utilizing the transmission/reflection unit and temperature changes (40 K-280 K) provided by the optical cryostat equipped with polyethylene window.The spectral resolution was equal to 1 cm −1 .
From the measured reflectivity coefficient a real and an imaginary part of dynamic dielectric function (DDF) was calculated using the standard KK relations, the previously described procedure valid for solid solutions was applied [37].
The effect demonstrated in [31,32] concerning TO phonon frequency discontinuity in II-VI narrow-gap semiconductors occurred in the frequency range from 120 to 150 cm −1 , i.e. far enough from the end of the spectral range accessible for reflectivity measurements using a standard, far-IR spectrometer.Moreover, the authors used a synchrotron beam with a high photon flux as a source of radiation.In our case TO-phonon frequency (about 70 cm −1 [38]) is very close to this end where the measured signal is noisy and of very low intensity because we used globar as a source of radiation (a black body-type of intensity spectral distribution).Such experimental conditions exclude a possibility to study the expected small effect which requires a high precision for TO phonon and we focused our efforts on investigation of LO phonon frequency.This frequency equal to about 144 cm −1 for PbSe [39] is expected to be almost the same for Pb 0.75 Sn 0.25 Se.

Results and discussion
When analyzing the experimental data a well pronounced step on the reflectivity curve was found at about 145 cm −1 (figure 2). Figure 3 presents the imaginary part of DDF resulting from the KK analysis of this step.The full width at half maximum of the structure is of the order of 3 cm −1 , the amplitude passes by a maximum below about 200 K.Its frequency is fixed within 1% but also exhibits a non-monotonous evolution, shown in detail in figure 4. The error of determined LO phonon frequency does not exceed 0.12 cm −1 , the dashed line is a guide for the eye.The clear discontinuity in temperature evolution of LO phonon frequency is seen in figure 4 at temperature T = (172 ± 2) K.
In order to explain this effect we carried out for LO phonon the calculations similar to these presented in [27] using the same rough approximations apart from negligence of the phonon energy and assuming possible transition from trivial insulator to TCI phase.The obtained, renormalized LO phonon energy for 2E F > hω LO is given by the formula: where Ξ cv is the optical deformation potential matrix element between valence and conduction band, M is the reduced mass of two different ions, a is the lattice parameter, W is the sum of the width of the conduction and the valence band, E F is the Fermi energy calculated from the band edge, and E g is the energy gap.For smallest E F this formula takes slightly modified form.The equation ( 2) predicts the largest correction to LO phonon frequency in the vicinity of E g = 0 and only a single anomaly, contrary to more complex formula proposed in [32].This anomaly can correspond to resonance-like discontinuity in this frequency dependence on E g at the zero energy gap.Not all parameters required for the calculations analogous to those presented in [32] are known and possible second resonance for E g = hω LO is not clearly observed in our case so a simplified formula given by equation ( 2) was used for the calculations.The best-known parameter values for Pb 0.75 Sn 0.25 Se are the lattice parameter a = 6.12 Å and the reduced ion mass M, which could be estimated as 9.49•10 −26 kg (both values correspond to PbSe [40]).The valence and conduction band width are of the order of 7-10 eV [41] so the W value can be taken as 18 eV.Due to both nonparabolicity and anisotropy of energy bands the value E F can be only roughly estimated from known electron concentration followed by the approximate formula given in [13].The obtained Fermi energy is equal to about 20 meV, but it could also be smaller due to possible modification of the form of the band edge.The hybridization-driven modification of the energy gap and some flattening of the energy dispersion in Pb 1−x Sn x Se were indeed previously suggested [24].
A few proposed in the literature values of optical deformation potential Ξ cv are scattered.According to widespread opinion this parameter is of the order of a few eV e.g.2.5 ± 0.8 eV [42] or 5.3 eV [43].
We were able to reproduce qualitatively the effect shown in figure 4 using equation ( 2) and the following parameters Ξ cv = 0.5 eV and E F = 9 meV apart from the well-known a, M, and W values.The solid curve plotted in figure 4 was the result of these calculations.The first vertical line indicated temperature corresponding to energy gap value E g = 0, so observed anomaly can be considered as a new signature of temperature-induced band inversion in the bulk corresponding to the trivial insulator-TCI phase transition.The second one is shifted from this position towards higher temperatures and marked   temperature for which in this case expected E g value is equal to hω LO .In comparison to the literature data value of Ξcv used for calculations is significantly reduced and E F probably is too small which demonstrates that the effect predicted by present model is clearly quantitatively overestimated.The reflectivity spectra similar to those determined for the Pb 0.75 Sn 0.25 Se sample were measured for the comparison for Pb 0.865 Sn 0.135 Se single crystal (figure 5).According to the equation ( 1) the solid solution composition for x = 0.135 corresponds to the trivial band ordering at all temperatures.One cannot expect an anomaly in the temperature dependence of LO phonon frequency for this sample.Nevertheless, the reflectivity spectra were measured at temperatures T < 130 K every 10 K.This experimentally determined dependence is shown in figure 5.As previously, the dashed line is a guide for the eye.According to the prediction the observed smooth temperature dependence of LO phonon frequency at low temperatures excludes possible change of the band ordering for the Pb 1−x Sn x Se solid solution crystal with x = 0.135.
All proposed up to date experimental methods to determine parameters of possible phase transition from a trivial insulator to a TCI phase have some disadvantages or at least their applications are limited to some extent.The ARPES measurements [4,8,[16][17][18] can be performed on a non-oxidized, clean crystal surface only.In the case of bulk samples this technique requires single crystals of a big size (several mm long) because of a necessity of their cleavage under very high vacuum conditions.It could be difficult to satisfy such condition for many materials.The experimental methods taking advantage of electron transport techniques e.g [12,13] require first a careful preparation of high-quality electric contacts what could depend on the electron concentration and could be different for n-type and p-type samples.In the case of new or less-known materials it could be a real difficulty hampering an application of electron transport techniques.The published optical method based on a modification of absorption edge slope results from the analysis of reflectivity in the mid-infrared spectral range versus temperature [14,15].Another possible experimental method could be based on direct measurements performed for a few micrometer thin plane-parallel slab.A sensitivity of the first method is not very high due to very limited influence of this absorption mechanism on the reflectivity coefficient.A careful analysis of the absorption edge requires transmission measurements with the use of very thin samples and not all materials can be grown in the form of such thin plates.In principle, a preparation of required for transmission measurements sample starting from a bulk crystal should be possible [44].However, it is really a challenging experiment not applicable for a practical use.Importantly, none of the experimental methods previously applied has a resonance-like character.In most of cases the temperature of the trivial insulator-TCI phase transition was previously estimated from a modification of the slope of a smooth curve (methods based on the resistivity measurements or on the analysis of a reflectivity).A part of methods based on the maximum or the minimum value of selected parameter (the Fermi energy, the plasma frequency, the electron effective mass) requires complex interpretation of experimental data with several additional assumptions or approximations.Probably the best up to date experimental determination of the phase transition temperature based on a sophisticated analysis of the high field magnetospectroscopy data [25] is too complicated and time-consuming method for a practical, 'every-day' use.

Conclusions
The presently proposed method of determination of the band inversion and phase transition temperature for a solid solution is based on a new effect resulting from electron-phonon interaction, modifying LO phonon frequency at E g = 0.It is free from several limitations listed above and seems to be 'a user friendly' method.The reflectivity measurements do not require a big size of analyzed bulk samples.A typical surface of investigated single crystal plate could be small because the light is focused approximately into a 2 mm diameter spot on the sample surface.This condition can be easily satisfied for a variety of materials.The proposed method is not destructive and the sample surface does not need to meet particular requirements.A natural, cleaved in air, mechanically polished or chemically etched crystal surface is suitable for such kind of measurements.The interpretation of reflectivity curves requires standard, well established KK relations.Last but not least, the anomaly in the temperature dependence of LO phonon mode frequency can have a resonant-like character and its energy position can be determined relatively easy.The Fourier spectrometer dedicated to optical measurements in the infrared is a typical laboratory equipment and its experimental possibilities can be easily extended to the reflectivity measurements in the THz spectral range.Considering all circumstances mentioned above the proposed method can be an excellent tool to study semiconductors considered as possible candidates for topological materials.

Figure 2 .
Figure 2. The part of reflectivity curves determined at several temperatures for Pb0.75Sn0.25Sesolid solution.The selected spectral range close to LO phonon frequency is shown.

Figure 3 .
Figure 3.The imaginary part of DDF calculated from the experimental spectra shown in figure 2 according to the procedure described in the text.

Figure 4 .
Figure 4. Temperature evolution of LO phonon frequency resulting from data presented in figures 2 and 3 (details in text).

Figure 5 .
Figure 5. Temperature evolution of LO phonon frequency for Pb0.865Sn0.135Sesolid solution crystal determined by the same procedure as that applied for Pb0.75Sn0.25Sesample.