Nernst-Ettingshausen effect at the trivial-nontrivial band ordering in topological crystalline insulator Pb1-xSnxSe

The transverse Nernst Ettingshausen (N-E) effect and electron mobility in Pb$_{1-x}$Sn$_x$Se alloys are studied experimentally and theoretically as functions of temperature and chemical composition in the vicinity of vanishing energy gap $E_g$. The study is motivated by the recent discovery that, by lowering the temperature, one can change the band ordering from trivial to nontrivial one in which the topological crystalline insulator states appear at the surface. Our work presents several new aspects. It is shown experimentally and theoretically that the bulk N-E effect has a maximum when the energy gap $E_g$ of the mixed crystal goes through zero value. This result contradicts the claim made in the literature that the N-E effect changes sign when the gap vanishes. We successfully describe $dc$ transport effects in the situation of extreme band's nonparabolicity which, to the best of our knowledge, has never been tried before. A situation is reached in which both two-dimensional bands (topological surface states) and three-dimensional bands are linear in electron \textbf{k} vector. Various scattering modes and their contribution to transport phenomena in Pb$_{1-x}$Sn$_x$Se are analyzed. As the energy gap goes through zero, some transport integrals have a singular (nonphysical) behaviour and we demonstrate how to deal with this problem by introducing damping.


I. INTRODUCTION
Narrow gap semiconductors have been for many years subject of intense experimental and theoretical studies in view of their interesting properties and important applications [1][2][3]. In recent years they have become once again a source of excitement due to the discovery of topological insulators [4]. The topological boundary states have been observed in bulk compounds Bi 2 Se 3 , Bi 2 Te 3 , bulk alloys Bi 1−x Sb x [5], two-dimensional quantum wells of HgTe/Hg 1−x Cd x Te [4] and, most recently, in bulk Pb 1−x Sn x Se alloys [6], bulk SnTe [7] and bulk Pb 1−x Sn x Te alloys [8]. The latter were called topological crystalline insulators (TCI), because, in contrast to canonical topological insulators, specific crystalline symmetries warrant the topological protection of their metallic surface states [9][10][11]. Since the IV-VI lead chalcogenides are characterized by strong temperature dependence of their band structures, it is possible to reach proper band ordering for a suitably chosen chemical composition x by varying the temperature. For the Pb 1−x Sn x Se system at high temperatures the band ordering is called trivial (L − 6 band above L + 6 band) and no TCI state occurs. As the temperature is lowered for the properly chosen x, one can reach the vanishing band gap and then arrive at the inverted nontrivial ordering (L + 6 band above L − 6 band) in which TCI state can occur. Such transition was demonstrated with the use of angle-resolved photoemission spectroscopy in Ref. [6]. Generally speaking, the character of topological insulators is manifested in both their surface and bulk properties.
It has been recognized from the sixties that in the alloys Pb 1−x Sn x Se one can reach vanishing energy gap by changing the temperature. An early demonstration of this possibility was provided by Strauss [12] who used for this purpose the optical transmission measuring the gap on both sides of the band ordering. Also laser emission proved useful in this respect [13]. It was shown that the electric resistivity and the Hall coefficient depend on the band ordering and can be used to obtain information on the transition temperature even if the Fermi energy is quite high in either the valence or conduction band [14]. Quite recently, optical response with appropriate analysis was used to determine the band inversion temperature for Pb 0.77 Sn 0.23 Se [15][16][17].
In recent years the thermomagnetic Nernst-Ettingshausen effect experiences a real revival in investigations of semiconductors [18,19], graphene [20,21] and high-T c superconductors [22,23]. In the present work we undertake a bulk transport study of PbSe and Pb 1−x Sn x Se system for 0.25 ≤ x < 0.39. The above ternary alloys can reach the vanishing band gap in the available temperature range. We investigate experimentally and theoretically the conduction electron density, mobility and the N − E effect. Since our study concentrates on small gap values, particular features of the band structure are crucial and it turns out that the description of transport effects for the vanishing gap poses nontrivial theoretical problems. In fact, although we describe bulk properties, as the gap goes to zero we deal with linear energy bands which is a completely new situation for the transport theory. We demonstrate for four Pb 1−x Sn x Se samples that, as the gap goes to zero, the N − E effect reaches a maximum. This contradicts the claims made in the literature, see Ref. [18]. All in all, our results confirm the conclusions of Ref. [6] concerning the transition from trivial to nontrivial band ordering as a function of temperature. Our analysis of scattering modes and their relative importance in Pb 1−x Sn x Se near the trivial-nontrivial transition of the band ordering will help further investigations of topological crystalline insulators and other topological materials.

II. EXPERIMENT
Single crystals of Pb 1−x Sn x Se (0≤ x ≤ 0.39) were grown by self-selecting vapour growth technique [24,25]. Owing to the peculiarity of the method, where near equilibrium thermodynamic conditions are kept, we have obtained high quality compositionally uniform large monocrystals with natural (001) facets (typical dimensions: 1×1×1 cm 3 ). Crystal compositions were determined by Energy Dispersive X-ray Spectroscopy offered by Scanning Electron Microscope Hitachi SU-70. Molar fractions were taken as averages from the scan covering 1× 1.5 mm 2 of the sample surface. The accuracy of determination of the chemical composition is better than 0.005 molar fraction.
Samples for measurements were cleaved with razor blade along (001) planes in the form of rectangular parallelepipeds, with dimensions 1.5×3×10 mm 3 . All samples exhibited metallic behaviour of resistivity with almost temperature independent carrier concentration. The Hall effect measured electron density in PbSe to be 7·10 18 cm −3 and the mobility 35000 cm 2 /Vs at liquid helium temperature. The corresponding values for ternary compounds Table 1. For thermoelectric measurements, the samples were thermally anchored with silver epoxy to copper cold finger of continuous flow helium cryo-stat. A small SMD resistor was glued to the free end of the sample for use as a heater.
Copper potential leads were attached across the sample with silver paint. Temperature and temperature gradient were measured with two calibrated subminiature GaAlAs diodes by Lake Shore glued to the sample. For all temperatures, the temperature gradient was kept within 2-5 percent of the average temperature. An external magnetic field at given temperature, with proper temperature gradient steady developed, was swept from -0.5 to 0.5 Tesla using standard resistive magnet. The magnetic field was directed perpendicularly to the sample and the temperature gradient. In this configuration the transverse N − E electric field develops across the sample: where P N−E is the N − E coefficient, T is the temperature and B is the magnetic field.
The transverse N − E effect is a thermoelectric analogue of the Hall effect. The potential difference V N−E =w·E N−E (w is sample's width) was measured by Keithley nanovoltmeter 2182a. The potential was linearly dependent on the magnetic field for all applied fields.
Main uncertainties in the absolute P N−E data are systematic and arise mainly from finite size of the thermometers contacts to the sample. We estimate their contribution to the uncertainty of the temperature gradient to be less than 15 percent. All other possible errors are small in comparison.

III. THEORY
The conduction band of PbSe and Pb 1−x Sn x Se alloys consists of four ellipsoids of revolution with minima at the four L points of the Brillouin zone. As a consequence, one deals with the longitudinal m * l and transverse m * t effective masses, the anisotropy in PbSe at 0 K is m * l /m * t = 1.7. In the description of electron scattering one deals with the density-ofstates mass m * d = (m * l m * t 2 ) 1/3 and in the description of mobility with the conductivity mass . In PbSe and Pb 1−x Sn x Se these masses differ little from each other, so we use an approximate spherical value of m * 0 = 0.048m 0 at 0 K at the band edge (m 0 is free electron's mass). This means that we use the standard dispersion relation for the two-band Kane model (the zero of energy is chosen at the band edge) This gives the energy-dependent mass, see [26] m * (E) = m * 0 1 + 2 It is seen from Eqs. (2) and (3) In the two-band k·p model the effective mass is proportional to the gap and inversely proportional to the matrix element of momentum squared. The temperature dependence of the mass at the band edge of Pb 1−x Sn x Se is In order to consistently describe transport effects we use the formalism developed for nonparabolic energy bands, see [26]. According to this scheme, statistical and transport quantities are given by the integrals where f 0 is the Fermi-Dirac distribution function depending on the Fermi level E f . Since the derivative ∂f 0 (E)/∂E does not vanish only in the limited range of energies around the Fermi level, the integrals (6) are not difficult to compute. If the electron gas is strongly degenerate, there is ∂f 0 (E)/∂E = −δ(E − E f ) and the integrals (6) are equal to the integrands taken at the Fermi energy. The free electron density in the band is where N v = 4 is the number of equivalent ellipsoids. The electron mobility is µ(E) = eτ (E)/m * (E) , in which τ is the relaxation time and m * (E) is given by Eq. (3). The electric conductivity is σ = eN µ, where the average electron mobility is An experimental determination of free electron density is nontrivial for our samples since it involves the Hall scattering factor which for materials with vanishing energy gap should be carefully evaluated, see below. The N − E effect, which is the main subject of our interest, is described in Ref. [26] where: k B -Boltzmann constant and z = E/k B T . For the complete degeneracy of electron gas the difference in the parenthesis is zero and the N − E effect vanishes.
In order to describe electron scattering mechanisms one needs to know the electron wave functions for the conduction band. These are taken in the form of true Bloch states with the periodic amplitudes depending on the pseudo-momentum k and energy, see [28].
where L = E/(2E + E g ) and k ± = k x ± ik y , while Z, X ± = (X± iY)/ The scattering probabilities have been calculated for various scattering modes by computing matrix elements of corresponding perturbing potentials. The total scattering probability is a sum of separate probabilities, which amounts to calculating the total relaxation time according the well known formula where τ i describe relaxation times for specific (independent) modes. Below we enumerate the where 2) Nonpolar scattering by optic phonons (NOP). This mode is determined by electronoptic-phonon deformation potential interaction. Due to the interband k·p mixing it involves both the conduction and valence deformation potential constants, E c np and E v np , which are treated as adjustable parameters, cf. [30]. We assumed that E c np and E v np are equal. A correction due to the nonelasticity is included, see [31]. The relaxation time for this mode is where Here ω op is the energy of the optical phonon and a 0 is the lattice constant. The correcting

3) Scattering by acoustic phonons (AC). It involves conduction and valence acoustic
deformation potentials E c ac and E v ac treated as adjustable parameters. We assumed that E c ac and E v ac are equal -a good approximation for mirror-like conduction band -valence band symmetry in IV-VI semiconductors. The relaxation time for this mode is where v av is the averaged sound velocity and is the crystal density.

4) Scattering by ionized defects is due to electrostatic interaction between electrons and
charged defects in the crystal. In Pb 1−x Sn x Se each native defect furnishes two free electrons (Se vacancies) or two free holes (metal vacancies) [1]. In general, the defect potential is of the form V=V C + V sr , where V C is the Coulomb interaction and V sr symbolizes the short range interaction related to size of the defect. The Coulomb interaction is negligible in lead chalcogenides due to very high value of the static dielectric constant ε 0 . Thus, one is left with the short-range contribution to electron scattering. The relaxation time for this mode is, see [32] 1 where N d is the concentration of ionized defects, while A = < R|V sr |R > and B=< X|V sr |X > are the matrix elements of the short-range potential V sr for the conduction and valence bands, respectively. The elements A and B are treated as adjustable parameters and were assumed to be equal.

5)
Alloy disorder scattering (AD), that appears only in the ternary alloys, is due to the fact that the V P b and V Sn atomic potentials are not the same. This results in perturbations of crystal periodicity and, consequently, in electron scattering, see [33,34]. The disorder scattering is important at high values of x and low temperatures. The relaxation time for this mode is, see [34] 1 where U c ad is the matrix element of the potential difference V P b − V Sn for the conduction band and Ω is the volume of unit cell. Further where U v ad is the matrix element of the potential difference V P b−Sn for the valence band. According to the theory, see [33], there exists a relation Knowing the gaps of both materials and fitting the value of U c ad one automatically obtains the value of U v ad .

6) Scattering by charged dislocations (DIS)
. This mode is due to repulsive interaction of electrons with dislocation lines which, forming acceptor centers, attract conduction electrons and become negatively charged, see [35]. The mode depends strongly on dislocation density (which is an adjustable parameter) and rather weakly on the temperature. The relaxation time for this mode is where f is the fraction of filled traps, a is the lattice constant, λ 0 is the screening length for ε 0 and N dis is the dislocation density.
In Table 1  When employing the above scheme we have found that the computed quantity containing the integral < µ 2 >, that is A r and P N−E , (see Eqs. (9) and (10)), have a strong and narrow peak at the temperature T c for which E g = 0. Such peak has no physical meaning and, as it can be seen in Fig. 1 for P N−E , it is not observed experimentally. Also, there is no reason to expect a sharp peak of the scattering factor A r because it would lead to a sharp peak of the free electron density N without a physical reason. However, since our Pb 1−x Sn x Se samples are certainly not homogeneous, i.e. they have somewhat different chemical compositions x at various parts, one can not expect to have E g = 0 in the whole sample at one temperature T . We simulate this nonhomogeneity by using the well known mathematical measure to avoid singularities in resonances. Thus, we introduce damping by replacing in all formulas the value of E g by E g +iG and take real values of the resulting expressions. By adjusting the damping constant G we can bring both the scattering factor A r and the N − E coefficient into a reasonable and experimentally observed behaviour, see below. given in Eq. (4) and presented in Fig. 2. The agreement between the two is very good.

IV. RESULTS AND DISCUSSION
This result contradicts the claim made in Ref. [18] that the temperature at which the band gap vanishes corresponds to the change of sign of P N−E . In particular, in the samples with x = 0.25 and x = 0.277 the coefficient P N−E does not change sign at all but the gap goes through zero and the maxima are well observed. In principle, the N − E effect should go to zero as the temperature goes to zero, but this is experimentally difficult to achieve because it requires a very small temperature gradient.
In order to understand and appreciate contributions and relative importance of various scattering modes in Pb 0.75 Sn 0.25 Se, which are used in the description of N −E effect, we show in Fig. 3 an example of experimental mobility µ for this sample, compared with the theory for the total mobility and partial mobilities related to single modes. The values of adjusted material parameters are given in Table 1. As indicated above, there is µ = µ(H)/A r , so that we have to go here through the calculation of A r as well. As the temperature goes from 0 to 300 K the calculated value of A r goes smoothly from 0.85 to 1.55. It can be seen that, with the indicated values of material parameters for various scattering modes, the overall description of mobility is quite satisfactory. It follows from the figure that the dominant scattering mechanism is due to alloy disorder. At higher temperatures the polar and nonpolar optical, as well as acoustical phonon modes become important. For the assumed low density of linear dislocations N dis = 10 9 cm −2 the corresponding partial mobility is too high to be seen in the figure.
In Fig. 4  This, as we explained previously, is a result of the "explosive" behaviour of the < µ 2 > integral at E g = 0, which appears in formula (10) for P N−E .  Fig. 4 and N dis = 10 9 cm −2 in Fig. 5.
Finally, Fig. 6  All in all, we achieve a good description of the N − E effect and electron mobility for four investigated samples in the critical range of small forbidden gaps in which the transition between trivial-nontrivial band ordering takes place. Our analysis has general significance for zero-gap systems and, in particular, it should be useful for a description of electron transport in topological crystalline insulators and in recently discovered three-dimensional topological Dirac semimetals -bulk analogues of graphene [38,39].    Table 1.  Table   1, open circles -experiment. Thin vertical arrows are drawn at the critical temperatures T c for which the corresponding gaps vanish (the same arrows are also drawn in Fig. 1). For temperatures near maxima the theory describes very well the experimental data both in terms of shape and absolute values.