A Review of Nanocrystalline Film Thermoelectrics on Lead Chalcogenide Semiconductors: Progress and Application

Abstract

Submicron-structured films of thermoelectric materials, exhibiting an improved thermoelectric figure of merit, are reviewed, including methods of fabrication and characterization. The review emphasizes the beneficial role of the grain boundaries in polycrystalline films. The enhanced Seebeck coefficient of lead chalcogenide films is attributed to a potential relief that is built along the grain boundaries. It scatters charge carriers with low energy and does not affect carriers with higher energy. The model that accounts for the thermoelectric properties of the films is described and assessed experimentally. The application of a flexible thermoelectric device (module) based on the nanocrystalline film thermoelectric semiconductors as high sensitivity radiation detectors is suggested.

1. Introduction

Thermoelectric materials are used for solid-state refrigeration and energy conversion in bulk form. In recent years, nano-thermoelectrics have been investigated widely, with the goal of the efficiency optimization of the efficiency of TE devices [1,2,3,4,5]. Various methods including decomposition at solid-state synthesis [6] and preparation of nanopowders [7] are used for the preparation of nanostructured materials.

The enhanced performance (ZT > 1) of thermoelectrics with low-dimensional structures was reported by several groups, as recently reviewed by [1]. These improvements were mainly obtained by reduced phonon thermal conductivity. Another possibility that thin films may offer is the quantum confinement of electronic carriers. The present review discusses such a mechanism revealed in lead chalcogenide (PbTe, PbTe, PbS) semiconductor films.

The performance of the thermoelectric materials depends on a figure of merit Z or of dimensionless ZT, which is defined as [8]:

ZT = (S²σ/κ)T,

(1)

where S is the Seebeck coefficient of thermoelectric material (n- or p-type conductivity), σ and κ are their electrical and thermal conductivities, respectively, and T is the absolute temperature.

For the charge (electrons and holes) scattering on phonons, S is determined as [8]:

$$S=\frac{k_{\mathrm{B}}}{e}\left[2\frac{F^1\left(\mathrm{\mu }\ast \right)}{F^0\left(\mathrm{\mu }\ast \right)}-\mathrm{\mu }\ast \right],$$

(2)

where k_B is the Boltzmann constant, e is the charge of electrons, F_i is the Fermi integrals, µ* is the reduced Fermi level.

The electrical conductivity σ of a semiconductor is given by the formula [8]:

σ = en(p)µ_n(µ_p),

(3)

Here n(p) is the electron (hole) carrier concentration, and µ_n(µ_p) is the charge carrier mobility.

The electron carrier concentration is given by:

$$n=4\pi \frac{{\left(2{m\ast }_n{k}_{B}T\right)}^{3/2}}{h^3}F1/2\left(\mathrm{\mu }\ast \right),$$

(4)

Here m*_e is the density of state (DOS) effective mass.

In the case of many (N) ellipsoids band-structure model [8]:

$$m_e^{\ast }=N^{2/3}{\left(m_1{\mathrm{m}}_2{m}_3\right)}^{1/3},$$

(5)

where m₁, m₂, m₃ are the effective mass components along the main ellipsoid axis.

The hole concentration is given by

$$p=4\pi \frac{{\left(2{m\ast }_p{k}_{B}T\right)}^{\frac{3}{2}}}{h^3}F1/2\left(\mathrm{\mu }\ast \right),$$

(6)

where m*_p is the density of state effective (DOS) mass of holes.

The charge mobility in a semiconductor with a simple parabolic band is given by [8]:

$$\mu =e\tau /m_c^{\ast },$$

(7)

where τ is the relaxation time and m*_c is the effective mass of carrier mobility [8].

For semiconductor (thermoelectric) materials, the thermal conductivity к has three partial components [8]:

$$k=k_L+k_e+k_a,$$

(8)

where k_L is the lattice thermal conductivity, k_e is the electronic thermal conductivity, and k_a is the ambipolar thermal conductivity, which only appears at the intrinsic conduction area of semiconductor due to the diffusion of electron–hole pairs.

The lattice thermal conductivity k_L is given by [8]:

$$k_L=\frac{1}{3}{C}_{\mathrm{v}}{\nu }^3,$$

(9)

where C_v is the specific heat at constant volume, and ν is the phonon velocity.

The electronic thermal conductivity k_e is given by [8]:

$$k_{\mathrm{e}}=L_0\mathsf{\sigma }T,$$

(10)

where L₀ is a Lorenz number. For the acoustic phonons, the scattering Lorenz number is given by [8]:

$$L_{\mathrm{o}}=(k_o/e)\left[\right(3F_2\left(m^{\ast }\right)/F_o\left(m^{\ast }\right))-(2F_1\left(m^{\ast }\right)/F_2\left(m^{\ast }\right)\left)\right],$$

(11)

The Seebeck coefficient S decreases with the increase in charge carriers’ concentration n(p), and in a metal with charge carriers concentration, n~10²² cm⁻³ S is close to zero. At the same time, electrical conductivity σ increases with increasing the charge carriers concentration n(p). Therefore, the dependence of the product S²σ on Fermi level E_F is bell-shaped. The maximum of S²σ is achieved when E_F~E_C—the bottom of the conduction band in n-type semiconductor or E_F~E_V—the top of the valence band in the p-type semiconductor. In this case, the value of the Seebeck coefficient S_n ≈ −180 μV/K for the n-type and S_p ≈ 180 μV/K for the p-type, respectively, for the charge carriers’ concentration n(p)~3 × 10¹⁸ cm⁻³ [9]. For this carrier concentration, the thermal conductivity κ is determined mainly by lattice thermal conductivity κ_L [8,9]. For the optimal charge carrier concentration (reduced Fermi level μ* ≈ 0), the figure of merit Z is determined by only three parameters [8]:

$$Z\sim {m_n^{\ast }}^{3/2}(\mathrm{\mu }/k_L),$$

(12)

Nanostructured film materials have become a topic of intense interest in recent years, arising from their potential applications in the field of thermoelectric power conversion. The authors have systematized and generalized the methods and investigation results for nanostructured films of lead chalcogenide semiconductors for thermoelectric applications. Factors that influence the figure of merit Z include the barrier structure on grain boundaries in such films and the phonon scattering on nanograin boundaries. The practical application of the thermoelectric device as a thermal detector of radiation is suggested based on nanocrystalline-film materials.

2. Materials and Methods

Various methods including decomposition at solid-state synthesis [6] and preparation of nanopowders [7] are used for the preparation of nanostructured materials. In this work, the flash evaporation technology method, first developed by Prof. Z. Dashevsky for Bi₂Te₃ based compounds [10,11], was used for the preparation of the lead chalcogenides (PbTe, PbTe, PbS) semiconductor films [12,13,14,15,16,17,18,19,20,21,22,23,24,25]. The setup for the film preparation is presented in Figure 1 [11]. The flash evaporation method provides an opportunity to deposit the same composition of the evaporated PbTe, PbTe and PbS powder with a small dopant amount (for the n-type materials it is 0.1 to 1 at % of I or In, and for the p-type ones it is 0.1 to 1 at % of Na). The films were evaporated on a thin polyimide substrate with a thickness of ~10 µm. The benefits of the polyimide material are a low thermal conductivity (of ~3.5 × 10⁻³ Wcm⁻¹K⁻¹) and mechanical flexibility [26]. The temperature of the substrate during the film preparation ranged from T = 100 K (cooling of the substrate by liquid nitrogen during the evaporation) up to T_s = 573 K (by heating the substrate). The evaporation speed was v_e = 0.1 mm/min. The film thickness was determined by the deposition time and was measured using a micro interferometer MII-4 with an accuracy of 0.1 mm. After the evaporation process, certain films were annealed in the same chamber at T_t = 673 K for 0.5 h in an oxygen atmosphere at the pressure p = 0.9 atm. It is known [27,28,29,30,31] that under an oxygen atmosphere an oxide barrier forms which prevents nanocrystal agglomeration and growth, thus keeping a high grain boundary area, and this may affect the position of the Fermi level.

The method of X-ray structural analyses of films was described in [18]. SEM images were taken with a Quanta 200 environmental scanning electron microscope (HRSEM) equipped with energy dispersive X-ray spectroscopy.

For the investigation of the transport properties of thin films over a wide temperature range of 10 K–300 K, including the Seebeck coefficient S (6% accuracy), the electrical conductivity σ (6% accuracy), the Hall coefficient R_H (8% accuracy), and the Nernst coefficient Q_N (12% accuracy), a setup presented in Figure 2 was used [32]. The measurement of R_H was carried out in permanent magnetic fields up to 2 T, and the reported results are the average of measurements in two directions of the electrical and magnetic fields. The accuracy of the temperature measurement was 0.1–0.2 K, and of the magnetic field it was ±3%.

The thermal diffusivity of thin films at the 300–350 K temperature range was measured in a setup presented in [32,33]. It used the method of dynamic gratings (Figure 3). Thermal gratings were generated by a 20 ns pulsed laser radiation with a wavelength of 532 nm and recorded by a 635 nm continuous laser radiation. By analysis of the kinetics of the diffraction signal, the lifetime of the recorded thermal gratings was determined. The accuracy of thermal diffusivity measurements was 10%. The total thermal conductivity of thin films was calculated from the equation [34]:

$$\kappa =\mathsf{\alpha }\mathsf{\rho }{c}_p,$$

(13)

where α is the thermal diffusivity, ρ is the single crystal density, and c_p was calculated within the Dulong-Petit limit.

3. Results and Discussion

3.1. Structure Properties

Figure 4 presents an SEM image of as-grown PbTe films on an amorphous polyimide substrate at different substrate temperatures. The grain size is found to increase with the increase in temperature. The average grain size is 100 ± 20 nm at T_s = 100 K, 250 ± 40 nm at T_s = 300 K, and 400 ± 50 nm at T_s = 500 K. As-grown polycrystalline PbTe films on an amorphous substrate have a columnar structure (Figure 4e).

The XRD patterns of the grown PbTe films at different substrate temperatures T_s are presented in Figure 5. The main peak corresponding to (100) planes is similar to a single crystal with a cubic NaCl structure. Other XRD peaks with small intensities correspond to the (311) and (111) planes of PbTe.

3.2. The Features of Charge-Carriers Scattering in Nano (Submicron)-Grained Films of Lead Chalcogenides

The charge scattering mechanism was studied in the n-PbTe films on a polyimide substrate with a grain size of ~400 nm. σ, R_H, and the S, and Q_N coefficients were investigated over the 80 K–300 K temperature range. The electron concentration varied in the range of 10¹⁸ to 10²⁰ cm⁻³. A comparison of the properties, obtained for the films with a thickness of d~1 μm with data for the bulk n-type PbTe single crystals, was reported in [32,35,36,37]. The following features of the films were observed: a decrease in electrical conductivity (electron mobility) by about 0.6 times at 300 K and slightly more at 100 K for all carrier concentrations (Figure 6), and an increase in S that is greater the higher the carrier concentration (more than two times at the carrier concentration n > 10²⁰ cm⁻³). This implies a decrease by 1.7–2 times of an absolute value for the dimensionless parameter (e/k_B) (Q_N/|R_H|σ).

The calculation of the density of a state’s effective mass m*_d and scattering parameter r by measuring of the four transport coefficients σ, S, R_H, Q_N and using Equations (14) and (15) was made by the equations [32,38]:

m*_d = (3n/π)^2/3(eℏ ²/k_B²T) (S − Q_N/|R_H|σ),

(14)

r = (Q_N−/|R_H|σ/(S − Q_N/|R_H|σ),

(15)

where ℏ is the reduced Planck constant. The electron mean free path is l = l₀E^r, where l₀ is a constant, and E is the electron energy.

The results of the calculation of m*_d from Equation (14) were only slightly different from those obtained for bulk single crystals of PbTe [9]. This result, as well as the results of measurements of the optical band gap in the same films [20], made it possible to exclude changes in the band parameters in PbTe films. The observed differences between S and Q_N of PbTe films and of the bulk single crystals can be related to the appearance of an additional scattering mechanism. A two- to three-fold decrease in electron mobility indicates that the relative contribution of this mechanism is approximately the same as the contribution of phonon scattering to the electron mobility. Phonon scattering dominates the bulk PbTe properties over the 100–300 K temperature range. The increase in |S| and a decrease in |Q| indicate a higher scattering parameter r due to the additional mechanism. Using the experimental values of σ, R_H, S, Q_N, the effective scattering parameter, characterizing the mixed scattering, r_ef was determined by the formula from [10]:

$$r_{ef}\left({\mathrm{\mu }}^{\ast }\right)=\left(\frac{3}{2}\right)\left(\frac{S}{S-Q_N}\right)-1+3\frac{d \mathrm{l}\mathrm{n} m_d^{\ast }}{d \mathrm{l}\mathrm{n} n},$$

(16)

After that, the parameter only characterizing the additional scattering, r₂ was calculated. This was carried out by assuming strongly degenerate electron gas,

$$\mathrm{\mu }={\mathrm{\mu }}_1{\mathrm{\mu }}_2/({\mathrm{\mu }}_1+{\mathrm{\mu }}_2),$$

(17)

$${\mathrm{r}}_{\mathrm{e}\mathrm{f}}=\left({\mathrm{\mu }}_2{r}_1+{\mathrm{\mu }}_1{r}_2\right)/\left({\mathrm{\mu }}_1+{\mathrm{\mu }}_2\right),$$

(18)

Here, μ is the electron mobility in the PbTe film, and μ₁ and r₁ are the electron mobility and scattering parameter, for the PbTe single crystal (the scattering occurs due to phonons [29]), respectively. The results of the calculation for temperature T = 120 K are listed in

Table 1. The values of electron mobility μ and scattering parameter r in PbTe films and single crystals at different carrier concentration n.

Carrier Concentration n × 10−18, сm−3	Fermi Level μ*, eВ	μ1, cm2/Vs	r1	ref	μ2, cm2/Vs	r2
6	0.075	8500	0.8	1.6	6900	2.2
20	0.13	4900	0.6	1.5	3000	2
100	0.3	1300	0.55	1.5	1100	2.3

.

The mobility μ₂ depends weakly on temperature, consistent with the assumption of scattering by defects. Despite the large values of r₂, indicating a strong increase in the relaxation time with carrier energy, the mobility μ₂ decreases significantly with an increase in the Fermi energy µ*. This is possible if the concentration of scattering defects increases with increasing electron concentration, or if the power of the scattering potential of each defect increases with increasing Fermi energy. Therefore, the additional scattering mechanism can be associated with intergrain boundaries. This assumption is supplemented by the following considerations. Electronic states can be localized at the boundaries of the grains, the filling of which charge the boundaries, creating potential barriers on them that scatter electrons. With increasing Fermi energy, the filling of these states, their charge, and the height of the potential barrier increase, which thus follows the Fermi level. If the barrier height is E_b » µ*, then such a barrier effectively dissipates carriers with energy less than the Fermi energy and weaker with higher energy, thereby creating a strong dispersion of the free path near the Fermi level. This gives rise to a large scattering parameter r₂ and r_ef.

A scattering model on two-dimensional potential barriers is of considerable interest from the point of view of increasing the thermoelectric figure of merit Z. Consider, for example, a metal—a material with high electrical conductivity and low Seebeck coefficient. In the presence of highly selective barriers, electrons with energy lower than the Fermi energy scatter and practically do not contribute to charge transfer, while electrons with higher energy retain the same free path as in a defect-free crystal. In this case, the electrical conductivity, roughly speaking, would decrease by half. At the same time, the Seebeck coefficient would increase to a value of the order of 100–200 μV/K. At the same time, the specific power P = S²σ would significantly exceed the value currently achieved by the best thermoelectric semiconductor materials. For metals with k~k_e and ZT » S²/L (L is the Lorentz number). At S » 100–200 μV/K and L~(2–3) (k_B/e)² thermoelectric figure of merit Z can be achieved, which is at the level of the best known semiconductors, or even higher than that. It is clear that such a model is idealized. Nevertheless, even stricter estimates have shown the reality of the increasing of Z, due to the scattering by two-dimensional potential barriers.

Studies of n- and p-type nano- and submicron-grained PbSe have shown the effect of a strong increase in the Seebeck coefficient S with a slight decrease in electron mobilityμ. Figure 7 presents an increase in Seebeck coefficient S for PbSe films compared to bulk single crystals. The m*_d values determined from S, R_H, σ and Q_N indicate the identity of the band parameters of the bulk single crystals and PbSe films. The value of the additional scattering parameter r₂ calculated from (16) is greater than two. In such films, the crystalline grains grow from the substrate to the surface of the film (Figure 4e), i.e., the height of the crystalline grain is equal to the thickness of the film d. Films with a thickness d up to 2 mm were studied. The dimension size of the crystalline grains was 500 nm, as with n-type PbTe films.

In such a spatially anisotropic barrier structure, there should be a significant anisotropy of the coefficients describing the transport phenomena in the directions perpendicular and parallel to the plane of the film. However, due to the small thickness of the sample, it was difficult to experimentally measure the electric field, current, and temperature gradient in the direction perpendicular to the plane of the film. Therefore, the magnetoresistance was measured at two orientations of the magnetic field (perpendicular and parallel to the surface of the film). PbSe was chosen among the lead chalcogenides, since its iso-energy ellipsoids are weakly elongated [29], so that the anisotropy associated with the multipole spectrum is weakly expressed and should not obscure the spatial-structural anisotropy.

The mobility in the films, determined by the dependence of the electrical conductivity on the magnetic field, turned out to be approximately three times less than in bulk PbSe samples. When measuring the magnetoresistance, the current was directed in the plane of the film, the magnetic field was perpendicular to the electrical current, and it had two orientations: B || (111)—(∆ρ_||), and B $\perp$ (111)—(∆ρ_⊥) (Figure 8). The following features should be noted: (1) the magnetoresistance assigned to the square of the Hall angle (mB)² are several times higher in the orientation in || (111) than in n-type PbSe bulk single crystals; (2) the anisotropy of the magnetoresistance, characterized by the ratio K = ∆ρ_||||/∆ρ_⊥, increases with the increasing concentration to values of 2 and 4 respectively, at 300 and 80 K; (3) for some samples with electron concentration n » 2 × 10¹⁹ cm⁻³, the K value reaches tens of units. In bulk n-type PbSe single crystals, K is slightly less than one [29]. The observed effect with K >> 1 for PbSe films is undoubtedly related to the scattering anisotropy.

A new confirmation of the proposed model was obtained by studying the effect of IR illumination with a wavelength of about 1 µm on the kinetic effects in PbTe and PbSe films. The kinetics of photoconductivity for two p-type PbTe films is presented in Figure 9. The slow processes of increase and decrease of photoconductivity, as well as residual conductivity after switching off the light, was observed. Changes in time of other coefficients (R_H, S and Q_N) have the same character as changes in σ. At the same time, the degree of influence of the light on different effects depends on the level of doping. At the hole concentration p < 1 × 10¹⁸ cm⁻³ (T = 77 K), the ratio of the electrical conductivities with and without illumination σ_light/σ_dark reaches values of 2–4, while R_Hdark/R_Hlight < 1.15–1.20. In films with p > 1 × 10¹⁹ cm⁻³, the relative changes in σ do not exceed 20%, and R_H remains unchanged within the measurement error (about 5%) but the Seebeck coefficient S and Nernst coefficient Q_N are the most sensitive to illumination.

The features of the photovoltaic properties of these films can be explained in the framework of a model connected with a creation of a potential relief on grain boundaries. In the presence of such a relief, nonequilibrium electrons and holes that arise during illumination are spatially separated, which makes it difficult to recombine them. The recombination rate is limited by the need to overcome the energy barrier, the height of which is close to the maximum fluctuations of the electrostatic potential (bending energy bands). The accumulation of nonequilibrium holes in the region of the humps of the potential relief and electrons in its dips smooth the relief, having a significant effect on the conductivity of inhomogeneous samples, the kinetic effects of which are described in the framework of the flow theory. At large nonequilibrium concentrations both drift and recombination barriers are greatly reduced, which leads to an increase in conductivity and a drop in lifetime by several orders of magnitude. The kinetics of the photo- response decay after the light is turned off is described by the equation [10]:

$$t={|dln\Delta \sigma /dlnt|}^{-1}~{\tau }_o+t,$$

(19)

Similar kinetics that are characteristic of the initial part of the photoconductivity decay and other kinetic effects in nano (sub-micron) PbTe and PbSe films on amorphous (polyimide) substrate was observed. At T = 80 K in the time range t~10⁻²–10⁻³ s

$$t\gg ({{\tau }_o+t)}^{\beta },$$

(20)

where τ_o~10⁻²–10⁻¹ s and β ≈ 0.8–1.

With long observation times, the decline becomes so slow that it is difficult to accurately estimate t, but approximate estimates show that the lifetime here exceeds 10^4–10⁵ s.

The effects of illumination on such films and the presence of long-term relaxation can naturally be explained by assuming that charge carriers excited by light are localized near the grain boundaries, reducing their electric charge, the height of potential barriers, and the scattering efficiency on them. It is essential that the values of the carrier mobility μ_n (μ_p), Seebeck S, and Nernst Q_N coefficients with increasing light intensity or duration of illumination asymptotically approach the values characteristic of bulk single crystals with the same concentration of charge carriers, as observed in Figure 9.

The analysis of the t value and its temperature dependence shows that recombination is associated with overcoming an energy barrier of 0.1 eV. This means that the maximum height of the barriers at the grain boundaries exceeds the Fermi energy, which in the studied samples is at the conduction band for the n-type and at the valence band for the p-type. At the same time, consideration of the features of the carrier mobility, Seebeck coefficient, and magnetoresistance led to the conclusion that the height of the barrier responsible for the scattering of charge carriers is close to the Fermi energy [39]. To explain this contradiction, we have to assume that the barrier is inhomogeneous in height, i.e., it has the form of a fence. Estimates made using the parameters of lead chalcogenides (deformation potentials, effective masses) showed that the localization radii of these states are very small (about 0.5 nm), and ionization energies can reach values of 0.1 eV. In the area of the interblock boundary, the potential barrier has the form of a mountain ridge (chain) with alternating peaks (near the defects) and passes (between defects) (Figure 10). The shape of the barrier here is complicated by the influence of the deformations. If, when partially filled, the energy of the bound states is close to the Fermi energy (at temperature T = 0 K it coincides with Fermi energy μ*), then the edge of the energy band in the same region of space is above μ* by an amount of E_i, and the recombination barrier may be equal to 0.1 eV. The estimation of the drift barrier, responsible for the scattering of charge carriers, is more complicated. The drift barrier should be lower than the recombination barrier, and both should decrease under illumination due to a decrease in the charge of the interblock boundaries.

4. Thermoelectric Properties of Submicron Pb_1−xIn_xTe Films on Polyimide Substrate

As presented in [34,36], a reliable method to control the electron concentration in the PbTe semiconductor is doping with indium—Pb_1−xIn_xTe. Indium creates a resonance level in the conduction band over a wide temperature range [40,41].

Figure 11 demonstrates a temperature dependence of Hall coefficient R_H from 10 to 400 K for Pb_1−xIn_xTe films (x = 0.01, 0.002 and 0.03 at. %). All films have negative values of the Hall coefficient over the entire temperature range, which is related to the electronic conductivity (n- type) in PbTe films. The Hall concentration n_H of the prepared Pb_1−xIn_xTe films was determined using Equation (17). For all film specimens up to heavily doped Pb_0.97In_0.03Te film, the Hall electron concentration does not exceed in the order 10¹⁸ cm⁻³. A nonmonotonic temperature dependence of the Hall coefficient is observed, which is unusual for n- type PbTe films doped with donor iodine impurity [32].

The Fermi level E_F was calculated from the electron concentration n_H Equation (5) using a value of $m_n^{\ast }$ for bulk PbTe specimens ($m_n^{\ast }$ ≈ 0.27 m_o) [9]. E_F is 0.06–0.08 eV above the bottom of the conduction band at a temperature of T ≤ 80 K. As the temperature increases, E_F rapidly decreases, crossing the bottom of the conduction band at T = 250–350 K.

The Seebeck coefficient S for Pb_1−xIn_xTe films also has negative values over the 80–400 K temperature range, typical to semiconductors with an electron type of conductivity (Figure 12). The value of S for Pb_0.97In_0.03Te achieves ≈−450 μV/K at T = 300 K. Figure 13 presents electrical conductivity σ as a function of temperature. The behavior of σ(T) is different from the temperature dependence of S(T). An increase in the content of the indium dopant in PbTe leads to a decrease in the electrical conductivity σ in the entire operation temperature range. Such unusual behavior of σ can be explained by a strong decrease in electron mobility μ_H with a growth of indium concentration in PbTe films.

Table 2. Thermoelectric properties of n- type Pb_1−xIn_xTe (1–3), PbTe_1−yI_y films, and Pb_1−xIn_xTe_1−yI_y bulk specimens at the temperature T = 300 K.

Composition	S, μV/K	σ, Ω−1cm−1	κ, W/mK	κL, W/m K	Z × 103, K−1	Reference
Pb0.99In0.01Te	−350	110	1.5	1.4	0.9	-
Pb10.98In0.02Te	−400	70	1.4	1.4	0.8	-
Pb0.97In0.03Te	−440	50	1.3	1.3	0.75	-
PbTe0.999I0.0001	−330	100	1.8	1.7	0.6	[32]
Pb0.999In0.001Te	−205	600	2.2	1.9	1.1	[34]
Pb0.999In0.001Te0.999I0.001	−155	1100	2.5	2.0	1.05	[42]

presents the thermoelectric properties at T = 300 K for polycrystalline n-PbTe_1−xIn_xTe and PbTe_0.999I_0.001Te films on a thin polyimide substrate with a thickness of 1 ± 0.1 μm. The thermoelectric parameters of bulk n-type Pb_1−xIn_xTe_1−yTe specimens are also shown in Table 2. The calculated thermoelectric figure of merit Z for the n-Pb_0.99In_0.01Te film with the concentration n = 3 × 10¹⁸ cm⁻³ achieves the value Z ≈ 1.0 × 10⁻³ K⁻¹ at 300 K, which is close to the Z value for n-PbTe bulk crystals doped with In [34] or codoped with In and I [42]. The lattice thermal conductivity κ_L was calculated according to Equations (8) and (10), and is also presented in Table 2. A significant decrease in the value of κ_L in polycrystalline Pb_1-xIn_xTe films in comparison with bulk crystals [34,42] may be attributed to the scattering of long wave phonons by the grain boundaries [43].

The parameter r for Pb_1−xIn_xTe films has a value of 1.5–2 in the 200–300 K temperature range. The additional scattering in polycrystalline Pb_1−xIn_xTe films can be related to the creation of energy barriers for electrons on the grain boundaries. The schematic view of the energy band diagram for Pb_1−xIn_xTe films is presented in Figure 14. Such barriers allow fast electrons with energy of E > E_F to pass through and delay the slow electrons E < E_F.

5. Application

5.1. Film Thermoelectric Module

The challenge in thermoelectricity is the development of film thermoelectric batteries (modules) for miniature autonomic generators. Such devices were presented in [44,45,46,47,48].

The high performance a film thermoelectric module (FTEM) on a thin polyimide substrate using effective Bi₂Te₃ thermoelectrics was developed and presented in [49]. The fabrication of FTEM is described as follows (Figure 15):

• Preparation of submicron p-type Bi_0.5Sb_1.5Te₃ layers [11] (the thickness ~5 μm) on two sides of the flexible substrate (the thickness ~10 μm);
• Preparation of submicron n-type Pb_0.99In_0.01Te layers (the thickness ~5 μm) on two sides of the flexible substrate;
• Perforation between p- and n-type layers. The width of the cut ~0.2 mm, and a length between the cuts ~1 mm;
• Preparation of the electrical contacts (two layers: the first Cr—with thickness ~0.1 μm and the second Au—with thickness ~0.5 μm) on the two sides of flexible substrate between the p- and n-type layers;
• Preparation of a polymeric cover (the thickness ~0.5 mm) on two sides of the film thermoelectric module [50].

The characteristics of the FTEM (number of thermocouples N = 100) are presented in

Table 3. Output parameters of the FTEM (T_h = 400 K, T_c = 300 K) with the number of thermocouples N = 100, film thickness d_f = 10 µm, length l = 1 mm, and width b = 4 mm.

Dimensionless Figure of Merit (ZT)av	Efficiency η, %	Resistance R, Ω	Electric Current IL, mA	Output Voltage VL, V	Electric Power PL, mW
0.37	2.2	70	29	2	58

.

5.2. Film Thermoelectric Detector

Thermal detectors are heated by incident IR radiation. Their temperature increase may be measured by any temperature-dependent mechanism, such as thermoelectric voltage [51]. The design of a film thermoelectric detector (FTED) is presented in Figure 16. The flexible thin substrate with a thickness d ~10–12 μm and perforation cuts between the p- and n-film layers is bended along the perforations. The film thermoelectric detector (FTED) consists from an absorption layer, film thermoelectric module (FTEM), and a heat sink. The temperature difference between the hot and cold sides of FTEM creates an electrical voltage that is directly proportional to the power of the incoming radiation.

The main characteristics of a thermoelectric detector are the response time τ_r and sensitivity S_s [51]. The calculation of these parameters for a concrete design of FTED is given in [50]. In the case where the heat transfer is mainly determined by the process of heat conduction, the response time of such FTED (Figure 16) is determined by [50]:

$${\tau }_r\approx \frac{4}{{\pi }^2}\frac{c\gamma H^2}{\kappa },$$

(21)

where c is the heat capacity of the thermoelectric films, γ is their density, H is the height of thermocouples, and κ is the thermal conductivity of the thermoelectric films.

The response time as a function of the thickness of the thermoelectric film d_f and the height (length) of thermocouple H is presented in

Table 4. Response time τ_r and sensitivity S_s as a function of the thickness of thermoelectric film d_f and the height of thermocouple H for a Film Thermoelectric Detector (FTED).

H, mm	Response Time τr, s			Sensitivity Ss, V/W
	df = 1 μm	df = 5 μm	df = 10 μm	df = 1 μm	df = 5 μm	df = 10 μm
1	1.5	0.9	0.7	85	35	15
2	5.1	2.5	2.0	110	58	35
3	12	6.5	5.0	120	100	90

. The sensitivity S_s is defined by [51]:

$$S_{\mathrm{s}}=V/P,$$

(22)

where V is the voltage, and P is the power of the incident radiation. V is determined for a thermoelectric detector by

$$V=(S_{\mathrm{n}}+S_{\mathrm{p}})m,$$

(23)

where S_n and S_p are Seebeck coefficients of the n- and p- film thermoelectric legs, and m is the number of thermocouples. The sensitivity S_s as a function of the thickness of thermoelectric film d_f and the height of thermocouple H for FTEB is listed in Table 4.

6. Conclusions

The transport properties of the Hall coefficient, Seebeck coefficient, electrical conductivity, and Nernst-Ettinshausen coefficient over the temperature range of 80–300 K were investigated for the nano(submicron) grained films of lead chalcogenide semiconductors (n- and p-type PbTe and n- and p-type PbSe) on an amorphous (polyimide) substrate.

The experimental measurements show a decrease in the electric conductivity and carrier mobility that is accompanied with an increase in the Seebeck coefficient of submicron-grained films of the lead chalcogenides. These properties result in an increase of the figure of merit Z for the submicron-grained Pb_1−xIn_xTe films on the polyimide substrate. A model which may account for the beneficial properties of the films was presented and discussed in some detail. It assumes that a mountain ridge-like with alternating peaks potential relief is built along the grain boundaries. It scatters electrons with energy lower than the potential relief, which behaves as an energy barrier for them, and it does not affect electrons with higher energy. The model was assessed by magnetoresistance and photoconductivity experiments.

A Flexible Film Thermoelectric Module (FTEM) based on the p-type Bi_2−xSb_xTe₃ and n-type, In-doped, PbTe materials on a thin polyimide substrate was developed. The dimensionless figure of merit ZT was ~0.4 and the TE efficiency η_max ≈ 2.2% at temperature difference ΔT = 100 K (T_c = 300 K, T_h = 400 K). The original design of a parallel-serial connection sharply increased the reliability of the FTEM, even if the number of thermoelectric unicouples was large (N ≥ 100). FTEM may be used as a radiation detector for short laser pulse diagnostics applications.