Nernst-Ettingshausen effect at the trivial-nontrivial band ordering in topological crystalline insulator Pb_1−xSn_xSe

Narrow gap semiconductors have been for many years subject of intense experimental and theoretical studies in view of their interesting properties and important applications [1–3]. 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 Bi₂Se₃, Bi₂Te₃, bulk alloys Bi_1−xSb_x [5], two-dimensional quantum wells of HgTe/Hg_1−xCd_xTe [4] and, most recently, in bulk Pb_1−xSn_xSe alloys [6], bulk SnTe [7] and bulk Pb_1−xSn_xTe 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 [9–11]. 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−xSn_xSe 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 [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−xSn_xSe 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_0.77Sn_0.23Se [15–17].

In recent years the thermomagnetic Nernst-Ettingshausen (N-E) 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−xSn_xSe 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−xSn_xSe samples that, as the gap goes to zero, the N-E effect reaches a maximum. This contradicts the claims made in the literature, see [18]. All in all, our results confirm the conclusions of [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−xSn_xSe near the trivial-nontrivial transition of the band ordering will help further investigations of topological crystalline insulators and other topological materials.

Single crystals of Pb_1−xSn_xSe (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 cm³). Crystal compositions were determined by energy dispersive x-ray spectroscopy offered by a Scanning Electron Microscope Hitachi SU-70. Molar fractions were taken as averages from the scan covering 1 × 1.5 mm² 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 a razor blade along (001) planes in the form of rectangular parallelepipeds, with dimensions 1.5 × 3 × 10 mm³. All samples exhibited metallic behaviour of resistivity with almost temperature independent carrier concentration. The Hall effect measured electron density in PbSe to be 7 · 10¹⁸ cm⁻³ and the mobility 35000 cm²/Vs at liquid helium temperature. The corresponding values for ternary compounds Pb_1−xSn_xSe are included in table 1. For thermoelectric measurements, the samples were thermally anchored with silver epoxy to copper cold finger of continuous flow helium cryostat. 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:

$$\begin{eqnarray}&&{{\boldsymbol{E}}}_{{\rm{N}}-{\rm{E}}}={P}_{{\rm{N}}-{\rm{E}}}\cdot {\boldsymbol{\nabla }}T\times {\boldsymbol{B}},\end{eqnarray} \tag{ 1 }$$

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 a 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.

The conduction band of PbSe and Pb_1−xSn_xSe 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-of-states mass ${m}_{d}^{*}={({m}_{l}^{*}{{m}_{t}^{*}}^{2})}^{1/3}$ and in the description of mobility with the conductivity mass ${m}_{c}^{*}=3/({{m}_{l}^{*}}^{-1}+2{{m}_{t}^{*}}^{-1})$. In PbSe and Pb_1−xSn_xSe these masses differ little from each other, so we use an approximate spherical value of ${m}_{0}^{*}=0.048{m}_{0}$ at 0 K at the band edge (m₀ 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)

$$\begin{eqnarray}&&{ \mathcal E }=\displaystyle \frac{-{E}_{g}}{2}+{\left[{\left(\displaystyle \frac{{E}_{g}}{2}\right)}^{2}+{E}_{g}\displaystyle \frac{{{\hslash }}^{2}{k}^{2}}{2{m}_{0}^{*}}\right]}^{1/2},\end{eqnarray} \tag{ 2 }$$

This gives the energy-dependent mass, see [26]

$$\begin{eqnarray}&&{m}^{*}({ \mathcal E })={m}_{0}^{*}\left(1+2\displaystyle \frac{{ \mathcal E }}{{E}_{g}}\right),\end{eqnarray} \tag{ 3 }$$

It is seen from equations (2) and (3) that, as the gap vanishes, we deal with the linear energy dispersion E(k) i.e. with, so called, massless Dirac fermions. In principle the k·p theory is valid at T = 0, but here it is known to work also for higher temperatures. Using experimental data on the composition dependence of the band gap of Pb_1−xSn_xSe at various temperatures, see [1, 12, 27], we establish the following phenomenological dependence of E_g (meV) on x and T

$$\begin{eqnarray}&&{E}_{g}(T,x)=126-802\cdot x+{[400+0.258\cdot {T}^{2}]}^{1/2},\end{eqnarray} \tag{ 4 }$$

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−xSn_xSe is

$$\begin{eqnarray}&&\displaystyle \frac{{m}_{0}^{*}}{{m}_{0}}=\displaystyle \frac{{E}_{g}{\rm{(eV)}}}{3.0546+1.8626\cdot {10}^{-3}\cdot T}.\end{eqnarray} \tag{ 5 }$$

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

$$\begin{eqnarray}&&\langle A\rangle ={\displaystyle \int }_{0}^{\infty }\left(-\displaystyle \frac{\partial {f}_{0}({ \mathcal E })}{\partial { \mathcal E }}\right)A({ \mathcal E }){k}^{3}{\rm{d}}{ \mathcal E },\end{eqnarray} \tag{ 6 }$$

where f₀ is the Fermi–Dirac distribution function depending on the Fermi level E_f. Since the derivative $\partial {f}_{0}({ \mathcal E })/\partial { \mathcal E }$ 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 $\partial {f}_{0}({ \mathcal E })/\partial { \mathcal E }\quad =\quad -\delta ({ \mathcal E }-{E}_{f})$ and the integrals (6) are equal to the integrands taken at the Fermi energy. The free electron density in the band is

$$\begin{eqnarray}&&N=\displaystyle \frac{{N}_{v}}{3{\pi }^{2}}{\displaystyle \int }_{0}^{\infty }\left(-\displaystyle \frac{\partial {f}_{0}({ \mathcal E })}{\partial { \mathcal E }}\right){k}^{3}{\rm{d}}{ \mathcal E }=\displaystyle \frac{{N}_{v}}{3{\pi }^{2}}\langle 1\rangle ,\end{eqnarray} \tag{ 7 }$$

where N_v = 4 is the number of equivalent ellipsoids. The electron mobility is $\mu ({ \mathcal E })=e\tau ({ \mathcal E })/{m}^{*}({ \mathcal E })$, in which τ is the relaxation time and ${m}^{*}({ \mathcal E })$ is given by equation (3). The electric conductivity is $\sigma ={eN}\bar{\mu }$, where the average electron mobility is

$$\begin{eqnarray}&&\bar{\mu }=\displaystyle \frac{\langle \mu \rangle }{\langle 1\rangle }.\end{eqnarray} \tag{ 8 }$$

An experimental determination of free electron density is nontrivial for our samples since it involves the Hall scattering factor

$$\begin{eqnarray}&&{A}_{r}=\displaystyle \frac{\lt {\mu }^{2}\gt \langle 1\rangle }{\lt \mu {\gt }^{2}}=\displaystyle \frac{\bar{{\mu }^{2}}}{{(\bar{\mu })}^{2}}\end{eqnarray} \tag{ 9 }$$

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 [26]

$$\begin{eqnarray}&&{P}_{{\rm{N}}-{\rm{E}}}=-\displaystyle \frac{{k}_{B}}{| e| c}{A}_{r}\bar{\mu }\left(\displaystyle \frac{\lt z{\mu }^{2}\gt }{\lt {\mu }^{2}\gt }-\displaystyle \frac{\lt z\mu \gt }{\langle \mu \rangle }\right),\end{eqnarray} \tag{ 10 }$$

where: k_B—Boltzmann constant and $z={ \mathcal E }/{k}_{B}T$. 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 ${\hslash }{\bf{k}}$ and energy, see [28]. Because of the narrow energy gap and fairly strong spin–orbit interaction the wave functions are in general linear combinations of conduction and valence periodic states as well as mixtures of spin-up and spin-down components. The wave functions for the conduction band in the spherical approximation are, see [28]

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{c}^{+}=\left[\sqrt{1-L}{bZ}+\sqrt{L}\displaystyle \frac{{k}_{z}}{k}{\rm{i}}{R}\right]\uparrow -\left[\sqrt{1-L}{{dX}}_{+}+\sqrt{L}\displaystyle \frac{{k}_{-}}{k}{\rm{i}}{R}\right]\downarrow ,\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{c}^{-}=\left[\sqrt{1-L}{bZ}+\sqrt{L}\displaystyle \frac{{k}_{z}}{k}{\rm{i}}{R}\right]\downarrow +\left[\sqrt{1-L}{{dX}}_{-}+\sqrt{L}\displaystyle \frac{{k}_{+}}{k}{\rm{i}}{R}\right]\uparrow ,\end{eqnarray} \tag{ 12 }$$

where $L={ \mathcal E }/(2{ \mathcal E }+{E}_{g})$ and ${k}_{\pm }={k}_{x}\pm {{\rm{i}}{k}}_{y}$, while Z, X_± = (X ± iY)/$\sqrt{2}$ and R denote the periodic amplitudes of Luttinger-Kohn functions taken at the L points of the Brillouin zone. The normalization coefficients satisfy the condition ${b}^{2}+{d}^{2}$ = 1. Plus and minus signs in the overscripts of the wave functions symbolize the effective spins up and down, respectively, while the arrows on the RHS symbolize spin-up and spin-down functions quantized on the z direction. The electron energies ${ \mathcal E }$ are counted from the conduction band edge.

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

$$\begin{eqnarray}&&{\tau }^{-1}=\displaystyle \sum _{i}{\tau }_{i}^{-1},\end{eqnarray} \tag{ 13 }$$

where τ_i describe relaxation times for specific (independent) modes. Below we enumerate the relevant scattering modes, quote the involved material parameters and mention importance of various modes for the electron mobility and N-E effect in Pb_1−xSn_xSe for the chemical compositions 0.25 ≤ x ≤ 0.39 of our interest. The description includes band's nonparabolicity in the energies and wave functions, as well as screening of long-range interactions by the electron gas.

(1) Polar scattering by optic phonons (OP). This mode is important at higher temperatures. It is determined by the static and high-frequency dielectric constants ε₀ = 234 and ${\varepsilon }_{\infty }$ = 28.7 at T = 0. Their temperature variations can be found in [1, 29]. The mode is characterized by nonelastic scattering processes which complicates its description in relaxation time approximation. This mode contains no adjustable parameters. Taking into account the screening of the interaction by the electron gas one obtains for the relaxation time, cf [28]

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\tau }_{{\rm{pol}}}}=\displaystyle \frac{2{k}_{B}{{Te}}^{2}}{{\hslash }}\left(\displaystyle \frac{1}{{\varepsilon }_{\infty }}-\displaystyle \frac{1}{{\varepsilon }_{0}}\right)\displaystyle \frac{{\rm{d}}{k}}{{\rm{d}}{ \mathcal E }}{F}_{{\rm{pol}}},\end{eqnarray} \tag{ 14 }$$

where

$$\begin{eqnarray}&&{F}_{{\rm{pol}}}=\left[1-\displaystyle \frac{{\rm{ln}}(1+{\rho }_{\infty })}{{\rho }_{\infty }}\right]-2L(1-L)\left[1-\displaystyle \frac{2}{{\rho }_{\infty }}+2\displaystyle \frac{{\rm{ln}}(1+{\rho }_{\infty })}{{{\rho }_{\infty }}^{2}}\right],\end{eqnarray} \tag{ 15 }$$

in which ${\rho }_{\infty }=4{k}^{2}{{\lambda }_{\infty }}^{2}$, where ${\lambda }_{\infty }$ is the screening length for ${\varepsilon }_{\infty }$.

(2) Nonpolar scattering by optic phonons (NOP). This mode is determined by electron-optic-phonon deformation potential interaction. Due to the interband k·p mixing it involves both the conduction and valence deformation potential constants, E_np^c and E_np^v, which are treated as adjustable parameters, see [30]. We assumed that E_np^c and E_np^v are equal. A correction due to the nonelasticity is included, see [31]. The relaxation time for this mode is

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\tau }_{{\rm{np}}}}={({E}_{{\rm{np}}}^{c})}^{2}\displaystyle \frac{{k}_{B}T{\hslash }}{\pi {({\hslash }{\omega }_{{\rm{op}}})}^{2}{a}_{0}^{2}}\displaystyle \frac{\partial k}{\partial { \mathcal E }}{k}^{2}{T}_{{\rm{cor}}}{F}_{{\rm{np}}},\end{eqnarray} \tag{ 16 }$$

where

$$\begin{eqnarray}&&{F}_{{\rm{np}}}={\left[1-L\left(1-\displaystyle \frac{{E}_{{\rm{np}}}^{v}}{{E}_{{\rm{np}}}^{c}}\right)\right]}^{2}-L(1-L)\displaystyle \frac{8}{3}\displaystyle \frac{{E}_{{\rm{np}}}^{v}}{{E}_{{\rm{np}}}^{c}}.\end{eqnarray} \tag{ 17 }$$

Here ${\hslash }{\omega }_{{\rm{op}}}$ is the energy of the optical phonon and a₀ is the lattice constant. The correcting term ${T}_{{\rm{cor}}}={({\rm{exp}}z-1)}^{2}/({z}^{2}{\rm{exp}}\;z)$, where $z={ \mathcal E }/{k}_{B}T$.

(3) Scattering by acoustic phonons (AC). It involves conduction and valence acoustic deformation potentials E_ac^c and E_ac^v treated as adjustable parameters. We assumed that E_ac^c and E_ac^v are equal—a good approximation for mirror-like conduction band—valence band symmetry in IV-VI semiconductors. The relaxation time for this mode is

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\tau }_{{\rm{ac}}}}={({E}_{{\rm{ac}}}^{c})}^{2}\displaystyle \frac{{k}_{B}T}{\pi {\hslash }{v}_{{\rm{av}}}^{2}\varrho }\displaystyle \frac{\partial k}{\partial { \mathcal E }}{k}^{2}{F}_{{\rm{ac}}},\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{F}_{{ac}}={\left[1-L\left(1-\displaystyle \frac{{E}_{{ac}}^{v}}{{E}_{{ac}}^{c}}\right)\right]}^{2}-L(1-L)\displaystyle \frac{8}{3}\displaystyle \frac{{E}_{{ac}}^{v}}{{E}_{{ac}}^{c}},\end{eqnarray} \tag{ 19 }$$

where v_av is the averaged sound velocity and ϱ 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−xSn_xSe 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 ε₀. Thus, one is left with the short-range contribution to electron scattering. The relaxation time for this mode is, see [32]

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\tau }_{{\rm{sr}}}}=\displaystyle \frac{{A}^{2}{N}_{{\rm{d}}}{F}_{{\rm{sr}}}}{2\pi {\hslash }}\displaystyle \frac{{\rm{d}}{k}}{{\rm{d}}{ \mathcal E }}{k}^{2},\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&{F}_{{sr}}={\left[1-L\left(1-\displaystyle \frac{B}{A}\right)\right]}^{2}-L(1-L)\displaystyle \frac{8}{3}\displaystyle \frac{B}{A},\end{eqnarray} \tag{ 21 }$$

where N_d is the concentration of ionized defects, while A = $\lt R| {V}_{{\rm{sr}}}| R\gt$ and B = $\lt X| {V}_{{\rm{sr}}}| X\gt$ 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_Pb 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]

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\tau }_{{\rm{ad}}}}={({U}_{{\rm{ad}}}^{c})}^{2}\displaystyle \frac{4x(1-x)}{\pi {\hslash }{\rm{\Omega }}}\displaystyle \frac{\partial k}{\partial { \mathcal E }}{k}^{2}{F}_{{\rm{ad}}},\end{eqnarray} \tag{ 22 }$$

where U^c_ad is the matrix element of the potential difference ${V}_{{\rm{Pb}}}-{V}_{{\rm{Sn}}}$ for the conduction band and Ω is the volume of unit cell. Further

$$\begin{eqnarray}&&{F}_{{\rm{ad}}}={\left[1-L\left(1-\displaystyle \frac{{U}_{{\rm{ad}}}^{v}}{{U}_{{\rm{ad}}}^{c}}\right)\right]}^{2}-L(1-L)\displaystyle \frac{8}{3}\displaystyle \frac{{U}_{{\rm{ad}}}^{v}}{{U}_{{\rm{ad}}}^{c}},\end{eqnarray} \tag{ 23 }$$

where U_ad^v is the matrix element of the potential difference ${V}_{{\rm{Pb}}-{\rm{Sn}}}$ for the valence band. According to the theory, see [33], there exists a relation

$$\begin{eqnarray}&&{E}_{g}({\rm{SnSe}})={E}_{g}({\rm{PbSe}})+\displaystyle \frac{({U}_{{\rm{ad}}}^{c}-{U}_{{\rm{ad}}}^{v})}{{\rm{\Omega }}}.\end{eqnarray} \tag{ 24 }$$

Knowing the gaps of both materials and fitting the value of U_ad^c one automatically obtains the value of U_ad^v.

(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

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\tau }_{{\rm{dis}}}}=\displaystyle \frac{{f}^{2}{e}^{4}{\lambda }_{0}^{4}{N}_{{\rm{dis}}}{m}_{0}^{*}(1+2{ \mathcal E }/{E}_{g})}{{{\hslash }}^{3}{\varepsilon }_{0}^{2}{(1+{4k}_{\perp }^{2}{\lambda }_{0}^{2})}^{3/2}{a}^{2}},\end{eqnarray} \tag{ 25 }$$

where f is the fraction of filled traps, a is the lattice constant, λ₀ is the screening length for ε₀ and N_dis is the dislocation density.

In table 1 we quote the employed values of material parameters. After the adjustment they are kept the same for all four samples. The density of dislocations N_dis and the damping constant G are adjusted for each sample separately.

Trying to compare experimental transport data with the theory one has to connect the measured quantities with the calculated ones. To work out this question we proceed in the following way. We begin with the experimentally determined Hall constant R_H for a given temperature T from which we determine the Hall electron density ${N}_{H}=1/{{eR}}_{H}$. The real free-electron density is $N={A}_{r}{N}_{H}$, where A_r is the Hall scattering factor. From the given value of N_H we compute the first value of the Fermi energy E_f and use E_f to compute the first value of mobility $\bar{\mu }$ using equation (8) with the above mentioned scattering modes, and finally the first value of A_r given by equation (9). Employing this value of A_r we determine the first value of N which concludes the first round of iteration. For this value of N we compute the second value of E_f, etc, all the way to the second value of N. We terminate the iteration when the (n + 1)th value of N is practically the same as the nth. This procedure determines the final values of N, E_f, A_r and $\bar{\mu }$ for the given temperature. These values can then be used to calculate the N−E coefficient given by equation (10). The adjustable parameters mentioned above are chosen to obtain an overall best agreement between the experiment and theory for $\bar{\mu }$ and P_N-E for our four samples at all available temperatures.

When employing the above scheme we have found that the computed quantities containing the integral 〈 μ² 〉 , that is A_r and P_N-E, (see equations (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 figure 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−xSn_xSe 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.

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Figure 1 shows our main experimental results on the N-E effect in PbSe and four Pb_1−xSn_xSe samples in which the gaps E_g in Pb_1−xSn_xSe alloys can go through zero as functions of temperature. Our absolute experimental values of P_N-E are in general agreement with those measured on other narrow-gap semiconducting materials [19, 36]. The vertical thin arrows indicate critical temperatures T_c which, according to the dependence (4), correspond to E_g = 0 for the indicated chemical compositions. It can be seen that the maximum of P_N-E for each sample corresponds to the critical temperature T_c. In other words, a maximum of P_N-E indicates that the gap goes through zero value. On the other hand, in PbSe there is no possibility of reaching vanishing gap and one observes no maximum. The observed values of T_c, are compared to the line drawn on the basis of E_g(T, x) given in equation (4) and presented in figure 2. The agreement between the two is very good.

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This result contradicts the claim made in [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 in our experimental set-up because it requires a very small temperature gradient.

In order to understand and appreciate contributions and relative importance of various scattering modes in Pb_0.75Sn_0.25Se, which are used in the description of N-E effect, we show in figure 3 an example of experimental mobility $\bar{\mu }$ 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 $\bar{\mu }=\mu (H)/{A}_{r}$, 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 = 10⁹ cm⁻² the corresponding partial mobility is too high to be seen in the figure.

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In figure 4 we plot theoretical behaviour of P_N-E for Pb_1−xSn_xSe having four different chemical compositions x. All calculations are carried for small density of dislocations N_dis = 1 · 10⁸ cm⁻², no damping (G = 0) and fixed values of other material parameters. In addition, the electron density N = 1 · 10¹⁸ cm⁻³ is kept the same for all samples. This somewhat hypothetical calculation is instructive for several reasons. It shows that for x = 0 and x = 0.15, when no E_g = 0 occurs, the behaviour of P_N-E is very flat. On the other hand, for higher values of x, where the gap vanishes as a function of T, the N-E effect has a maximum at the critical temperature T_c. It is seen that, when no damping is introduced, theoretical P_N-E coefficients have high and very sharp peaks not observed experimentally. This, as we explained previously, is a result of the 'explosive' behaviour of the < μ² > integral at E_g = 0, which appears in formula (10) for P_N-E.

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Figure 5 shows two calculations of the N-E coefficient for the Pb_0.75Sn_0.25Se sample, carried out with the same values of adjustable material parameters. The dashed line indicates a calculation without damping, i.e. with G = 0. It gives a sharp peak at the temperature T_c = 141 K at which E_g = 0. This peak is related to the integral $\bar{{\mu }^{2}}$ discussed above and it is not observed experimentally. For this reason we use the damping procedure mentioned in the preceding section with the adjusted value of G = 20 meV. This gives the result indicated by the solid line. It is seen that now there is no sharp peak and the theory describes quite well the experiment shown in figure 1 with the exception of low temperatures. Finally, we indicate a marked difference of theoretical results for P_N-E calculated for the sample x = 0.25 with the use of G = 0, as seen in figures 4 and 5. This difference is due to largely different dislocation densities assumed for both calculations: N_dis = 10⁸ cm⁻² in figure 4 and N_dis = 10⁹ cm⁻² in figure 5.

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Further check of our theoretical model is presented in figure 6. Here we plot the temperature dependence of P_N-E for three different densities of the electron gas corresponding to different Fermi energies. It is seen that, while low temperature slope changes with concentration, the position of the P_N-E remains unchanged. This fact strongly supports our interpretation of P_N-E maximum as a hallmark of the vanishing gap. We note that the absolute value of P_N-E depends on electron density. This dependence suggests the origin of discrepancy between our data and those presented in [18] since we employed the samples having one order of magnitude higher electron densities.

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Finally, figure 7 shows theoretical dependence of P_N-E versus temperature, calculated for all four Pb_1−xSn_xSe samples of our interest. In the description we use for all samples the same values of material parameters, excepting the density of defects, the density of dislocations, and the damping constant G. The latter are adjusted for each sample and their values are used also in the description of average mobilities, as illustrated for the sample with x = 0.25 in figure 3. In the same figure we reproduce for comparison the experimental data shown in figure 1 with the same vertical arrows indicating the critical temperatures T_c at which E_g = 0. The vertical arrows drawn at maxima of P_N-E denote the critical temperature T_c at which E_g = 0. The same arrows are transferred to figure 1 for comparison with experimental data. It is seen that the calculated and experimental maxima occur at the same temperatures within several percent accuracy. The general experimental and theoretical temperature behaviour near the maxima as well as the absolute values of P_N-E agree quite with each other. By comparing figures 4 and 5 it becomes clear that the maximum of N-E effect is directly related to the gap going through zero. This is a general feature of the theory which should find applications for other zero-gap semiconductor and semimetal systems. On the other hand, the low temperature behaviour of the samples with x = 0.325 and x = 0.39 remains unexplained according to the presented theory. It should be observed that the unexplained region corresponds to the nontrivial band ordering in which the topologically protected surface states appear, so we deal with a two-channel transport. According to the formulas for two-channel mixed conductivity, the N-E coefficient depends on weighted contributions of both conducting channels and difference of their thermoelectric powers [37]. Consequently, it is possible that the low temperature behaviour is modified by the topological surface states which are not accounted for in our theory. It should be mentioned that higher values of the damping constant G result in wider and lower theoretical maxima of P_N-E. 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].

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The main conclusion of our work is that a maximum of the electron Nernst-Ettingshausen effect, measured as a function of decreasing temperature in Pb_1−xSn_xSe alloys, indicates a critical temperature T_c corresponding to the zero energy gap and transition from trivial to nontrivial band ordering in terms of the topological crystalline insulators. The above property allows one to determine experimentally critical temperatures T_c by means of transport data. In particular, we checked theoretically that the maximum of the N-E effect does not depend on the carrier density in the sample. Electron mobility and N-E effect are measured and successfully described for four Pb_1−xSn_xSe samples with 0.25 ≤ x ≤ 0.39 in the vicinity of E_g = 0 , establishing the dominant scattering modes. It is demonstrated that the description of Hall scattering factor and N-E effect presents theoretical problems as the gap goes to zero and the bands become linear in the wave vector. The latter have been overcome introducing damping procedure. We emphasize that our conclusion concerning the maximum of the N-E effect is a result of the detailed theoretical treatment. Our analysis of electron transport phenomena in the proximity of vanishing energy gap should be of value for other temperature and composition driven topological crystalline insulators and three-dimensional Dirac semimetals.

We acknowledge support from Polish National Science Centre grants No. 2011/03/B/ST3/02659 and No. DEC-2012/07/B/ST3/03607, and from European Regional Development Fund through the Innovative Economy grant (POIG.01.01.02-00-108/09).