Role of Deep Defects on the Transport Properties of Polycrystalline Thermoelectric Alloys and Composites

Abstract

Deep traps at grain boundaries, or within grains, of polycrystalline alloys optimized for large-scale thermoelectric applications affect their transport properties. For an accurate description of these properties, defects require appropriate statistical treatment at the basic level, so that the chemical potential can be obtained as a function of temperature. This work reports on polycrystalline p- and n-type bulk hot-extruded ternary alloys and composites containing Bi, Sb, Te and Se which have advantageous mechanical properties and thermoelectric performance below 500 K. Calculations of the electrical transport properties of heavily doped ternary alloys taking into account their polycrystalline nature provide excellent accord to Hall effect transport measurements in the plane perpendicular to the extrusion axis in the range from 10 K to 440 K. To do so, a statistical treatment of deep donor traps at grain boundaries in p-type composites is required. Ignoring their role may lead to incorrect interpretation of transport measurements.

Introduction

Conventional zone-melted solidification of ternary p-type (Bi_1-xSb_x)₂Te₃ and n-type Bi ₂(Te_1-ySe_y)₃ single crystals have been recognized for their valuable properties at least since the late 1950s. ¹ They are among the highest performing bulk materials for thermoelectric applications below 500 K. There is a recently renewed interest in their development in polycrystalline form in order to improve their performance ^2,3,4,5,6,7 for industrial applications. They have been synthesized in various forms and are frequently compacted by spark plasma sintering for both cooling and energy conversion applications near room temperature. There have also been some studies of their transport properties.^8,9

The transport properties of polycrystalline materials are more difficult to understand because they must be modelled taking into account, at a minimum, two scales of length: within and external to crystal grains. Successful modelling of these properties requires knowledge of the structure within the grain at the atomic level, as well as transport across grain boundaries (GB), at a much larger scale. Furthermore, deep trap states in GB or within grains affect equilibrium properties such as carrier concentrations in these alloys. As a consequence, very few attempts can be found in the literature to quantitatively explain these basic properties in thermoelectric bulk polycrystalline compound and alloys. As an example, none of the most recent works on the subject show measurements, and modelization of the carrier concentration as a function of temperature. In addition, the complexity of the electronic band structures of crystals of the bismuth telluride family near the band edges, which do not admit a simplified parabolic description, renders the task more difficult. In many of the earlier characterizations of single crystals there was a need to simplify the approach using parabolic or quasi-parabolic approximations of the electronic dispersion relations, which often leads to erroneous conclusions, as was the case with Bi₂Te₃.¹⁰

Mechanical alloying combined with hot extrusion are well-established processes leading up to meter-long rods of ternary and quaternary alloys which can be optimized for industrial large-scale applications.^{11,12,13,14,15} Hot extrusion of mechanically alloyed grains containing Bi, Sb, Te and Se can produce highly textured materials, known as hot-extruded bulk polycrystalline (HEBP) alloys and composites. When combined with MoS₂ nanoparticles these composites acquire superior mechanical properties,^13,14 while their powders show a complex nanostructure.¹⁵

Compounds such as Bi₂Te₃ and Sb₂Te₃ have anisotropic crystal structures that can be described on a hexagonal unit cell^16,17 which translates in anisotropic transport properties enhanced along the plane perpendicular to the c-axis compared to those along the c-axis. Because of the severe crystal shearing when powders go through the hot extrusion process, HEBP alloys are highly textured such that planes perpendicular to the c-axis align along the extrusion axis. Thus HEBP (Bi_1-xSb_x)₂(Te_1-ySe_y)₃ alloys inherit anisotropic electrical transport properties, which are often enhanced along the extrusion axis.

In a previous work we provided a two-scale model of the electron transport properties in n-type HEBP Bi₂Te₃ in a temperature range from 200 to 500 K.¹⁰ As it will later be shown, one also must take account of the impact of deep defects in grain boundaries (GBs) on transport, which is important for p-type alloys, and this is one of the contributions of this work. In this work, we will extend the model to analyze the Hall effect measurements of one p-type composite of (Bi_1-xSb_x)₂Te₃ and an n-type conventional (random) Bi₂(Te_1-ySe_y)₃ ternary alloy, as examples of the impact of traps within GBs or within grains. As with our previous work,¹⁰ we will model charge carrier transport in the plane perpendicular to the extrusion axis, where first-order approximations to non-parabolic dispersion relations seem justified from ab initio calculations and render the modelling easier.¹⁶ In order to extend the model to a wide temperature range we found it necessary to incorporate two limiting scattering mechanism: (i) ionized impurity scattering mostly dominant at low temperatures (10 to ~240 K) and (ii) acoustic phonon scattering at higher temperatures (up to ~ 440K).

The results obtained when incorporating these physical ingredients in a model for transport in hot-extruded p and n-type ternary alloys explains the following features: (1) it accounts for the Hall transport data in a wide temperature range up to 440 K as essentially limited by the two single scattering mechanisms described above, (2) it reveals the roles of deep defects on transport, and more important, (3) it highlights the importance of being able to correctly predict the temperature dependence of the equilibrium chemical potentials in polycrystalline alloys in order to obtain reliable information from transport measurements. This may appear as a rather trivial reminder, however judging from the recent literature it looks to be a necessary one.

Methods

Model for Electrical Transport

The electronic band structure near the band edges of the fundamental gap is a necessary ingredient to model transport properties but it is not sufficient for polycrystalline materials. For such inhomogeneous materials, often riddled with defects,^18,19 it is also necessary to accurately describe the statistics of the charge carriers. In what follows we summarily review a two-scale model, already presented in ¹⁰ to analyze n-type HEBP Bi₂Te₃. The modification of carrier statistics, which is necessary because of deep traps in GBs in p-type alloys will be treated in Sect. "Results and Discussion", as well as details of the calculations involving more than one scattering mechanism.

Transport in HEBP alloys can be accurately described by recognizing two of their basic physical properties: (1) its transport anisotropy which allows us to describe separately properties perpendicular to the extrusion direction $\left( { \bot_{ex} \equiv \bot } \right)$, on which we concentrate for Hall effect data, and (2) a simplified band structure that is based on first-principles calculation results for Bi₂Te₃.¹⁶ We extend this second assumption to ternary (Bi_1-xSb_x)₂Te₃ and Bi₂(Te_1-ySe_y)₃ alloys, and write an energy dispersion relation for holes $E_{v} \left( {\user2{x,K}} \right)$ near the valence band edge $E_{v0}$ at ${\varvec{K}}_{{0i_{v} }}$ ($i_{v} = 1,2...N_{V}$), where N_V is the number of equivalent valleys in the valence band, as:

$$ E_{v} \left( {\user2{x},\user2{K}} \right) = E_{v} \left( {\varvec{x},\varvec{K}_{{0i_{v} }} } \right) - \varepsilon _{p} \left( {\varvec{K} - \varvec{K}_{{0i_{v} }} } \right) = E_{v} \left( {\varvec{x},\varvec{K}_{{0i_{v} }} } \right) - \varepsilon _{p} \left( \varvec{k} \right) $$

(1)

where $E_{v} \left( {\user2{x,K}_{{0i_{v} }} } \right) = E_{v0} - \left| e \right|\psi \left( {\varvec{x}} \right)$ and $\psi \left( {\varvec{x}} \right)$ is the electric potential. $\varepsilon_{p} \left( {\varvec{k}} \right)$ is non-parabolic and of the form:

$$ {{\hbar^{2} {\varvec{k}}_{{}}^{2} } \mathord{\left/ {\vphantom {{\hbar^{2} {\varvec{k}}_{{}}^{2} } {2m_{h}^{*} }}} \right. \kern-\nulldelimiterspace} {2m_{h}^{*} }} = \varepsilon_{p} \left( {1 + \lambda \varepsilon_{p} } \right) $$

(2)

The parameter $\lambda \left( T \right)$ is in general a function of the temperature T and is taken as the inverse of the energy gap $E_{g} \left( T \right)$ of the material. A corresponding expression is proposed for the dispersion relation of electrons in the conduction band.¹⁰

We refer to Ref. 10, Sect. II, for the presentation of the transport model. The novel statistical treatment of deep defects and the updated description of the scattering mechanisms involved in the temperature range from 10 to 440 K will be shown in Sect. "Results and Discussion", before comparison to experimental results.

Material Synthesis and Characterization

Composite p-type ternary materials were synthesized with the total molar ratio of bismuth to antimony maintained constant at 1:4. Different alloys in powder form that ranged in composition from somewhat similar to the conventional (random) ternary alloy, to the extreme of two binary compounds of Bi₂Te₃ and Sb₂Te₃ were then compacted by hot extrusion. Their structural, mechanical and elastic properties were previously characterized.^12,13,14

Quaternary n-type (Bi_1-xSb_x)₂(Te_1-ySe_y)₃ alloys were also synthesized in a variety of compositions within the intervals $0 \le x,y \le 0.07$, of which we analyze here the case x=0, Bi₂(Te_1-ySe_y)₃ with y = 0.1. For both types of alloys and composites a mechanical alloying process carried out under an inert argon atmosphere starts from pure elements pellets to produce powders of alloys and/or phases of composites. High-purity pellets (99.99% or above from 5NPlus Inc.) of Bi, Sb, Se and Te are the starting materials for synthesis of different alloys (or phases of composites) by mechanical alloying for a minimum of 10 h. X-ray diffraction (XRD) analyses verify composition of powders as well as mixture homogeneity in the case of composites. The powders (or mixtures of them in the case of composites) were then introduced into the extrusion cylinder for hot extrusion at a fixed temperature in the range of 360°C to 460˚C, depending on material load and applied pressure. All processing steps were carried out under an argon atmosphere. The final alloy or composite products are in the shape of rods with 2.54 cm diameter and ~40 cm length. Adjustment of the p-type (n-type) doping level can be done by addition of Sb (SbI₃).

Scanning electron microscopy (SEM) of bulk samples whose surfaces were mechanically and chemically polished to reveal the grains allows characterizing grain size and size distribution. Grain size is in the range of a few µm (see, for example, Fig.S.1 of the Supplementary material).

Van der Pauw-Hall measurements of (Bi_1-xSb_x)₂Te₃ (Bi₂(Te_1-ySe_y)₃) have been performed from 10 to ~ 360 K (430 K) under a sub-Tesla magnetic field. The Hall effect samples are cut with a diamond blade in ~ 500 µm-thick slices which are perpendicular to the extrusion direction and at the center of the rods. This assures that the Hall carrier concentrations and mobilities are measured in a direction orthogonal to the extrusion direction. The relative errors in these measurements are estimated to be ~ 10%.

Results and Discussion

We first discuss the treatment of impurities in heavily doped polycrystalline materials. If there are positively charged ionized deep donors placed within the GBs of the material at an energy level $E_{DD}^{GB}$, an electric field surrounds the GB region, attracting electrons within the grains to screen these positive charges. This causes band bending and the formation of an energy barrier $\Delta E_{GB}$ for majority carriers within the grains surrounding the GB. An estimate of the value of this barrier as a function of temperature is obtained in Sect. S.I (see Fig. S.2) of the Supplementary material, for a p-type alloy. The GB can be modelled as a very narrow sheet of alloy (of width $\delta$) containing a high density of defects, among them dislocations, impurities and point defects.¹⁹

The problem of transport in polycrystalline semiconductors has been extensively discussed in the literature.^20,21,22 It was first treated by Volger²⁰ who concluded that the measured Hall constant $\overline{R}_{H}$ in these inhomogeneous solids can be written as $\overline{R}_{H} = R_{H}^{G} + a\left( {{\delta \mathord{\left/ {\vphantom {\delta d}} \right. \kern-\nulldelimiterspace} d}} \right)^{2} R_{H}^{{{\text{GB}}}}$, where $d > > \delta$ is a typical size of grains, $R_{H}^{G}$ and $R_{H}^{GB}$ are the Hall constants of the alloy within the grain (G) and the GB (of larger resistivity $\rho_{GB}$), respectively, and $a$ is a constant of order unity. If one defines the following ratios $\ell = {\delta \mathord{\left/ {\vphantom {\delta d}} \right. \kern-\nulldelimiterspace} d},$$m = {{\mu_{GB} } \mathord{\left/ {\vphantom {{\mu_{GB} } {\mu_{G} }}} \right. \kern-\nulldelimiterspace} {\mu_{G} }},$$\, r = {{\rho_{G} } \mathord{\left/ {\vphantom {{\rho_{G} } {\rho_{{{\text{GB}}}} }}} \right. \kern-\nulldelimiterspace} {\rho_{{{\text{GB}}}} }}$, all smaller than one, then it can be shown that $\overline{R}_{H} = R_{H}^{G} \left( {1 + \ell m} \right) \approx R_{H}^{G}$.²⁰ For example, for a dominant p-type majority carrier concentration one measures $\overline{p} \approx p$ the carrier density within the grains, while the measured Hall mobility obeys $\overline{\mu }_{H} = \mu_{H}^{G} \left( {1 + \ell /r} \right)^{ - 1}$.²¹ The measured Hall mobility is in general smaller than that of the alloy within the grain. However, in our case it turns out that the ratio $\ell /r$ is still much smaller than one since r is not as small as $\ell = {\delta \mathord{\left/ {\vphantom {\delta d}} \right. \kern-\nulldelimiterspace} d} < < 1$. It is worthwhile to notice that a prevalent model for conduction in polycrystalline semiconductors of moderate doping leads to a temperature activated mobility (see Eq. (18) of Ref. 22). This corresponds to the assumption of a sufficiently high carrier mobility within the grains (hence, carrier mean free path comparable to or larger than grain size), which may not be the case for most of the heavily doped thermoelectric semiconductors. Because of heavy doping, the energy barriers at GB are easily traversed either by field emission or by thermoelectronic (often referred to as thermionic) field emission for higher energies.

The remaining problem associated with band bending near the GBs is to establish how to quantitatively take into account the effect of the deep traps, which may be located within the GB, on the equilibrium carrier concentrations in the grain. Because of the long range of the Coulomb potential, even if screened, it is expected that these impurities will have an effect on the carriers within the grain, in particular when the thermal energy becomes comparable to the energy barrier near the GB. Understanding the effect on the carriers within the grain is necessary because the measurements yield a Hall constant which is related to the majority carrier density in the grain. This problem has been treated by Seto²² only for the case where the semiconductor is not heavily doped so that its mobility remains high, with a mean free path comparable to grain size. To this end, an average majority carrier density was calculated by taking into account the effect of the space charge regions within the grain close to the GB.²² To our knowledge, the case of heavily doped semiconductors, where the carrier’s mean free path becomes much smaller than the grain size, and the space-charge region may be negligible, has not yet been treated in the abundant literature on semiconductors. We consider here the case of p-type grains whose majority carriers are holes, which are surrounded by GBs containing ionized deep donors. In this example the bands will bend downwards presenting a positive electric potential barrier for holes which generates a majority carrier depletion region within the grain (see Fig. S.1 in Sect. S.I of the Supplementary material for more details). One can immediately note that as the thermal energy becomes comparable to the energy barrier $\Delta E_{{{\text{GB}}}}$, the majority carriers (holes) will be able to surmount the barrier and “spill” into the GB, where they either recombine with the electrons within the GB or “escape” to the next grain. This thermally activated process will tend to remove holes from the grain, thus decreasing the hole density as the temperature increases, at least at low temperatures close to 0 K. Eventually, as the temperature continues to increase, the thermal generation of electron-hole pairs will increase both hole and electron densities within the grains, as is expected in semiconductors, and as will later be shown. We aim to describe this physical situation by ignoring the GB, and concerning ourselves with an average carrier concentration $\overline{p}$ (which is what one actually measures) in an otherwise isolated grain. In our approximate description, this average carrier concentration $\overline{p}$ is found by introducing an (average) fictitious or virtual concentration of deep donors $\left\langle {N_{D}^{ + } } \right\rangle$ in the grain, which is related to the concentration $N_{DD}^{{}}$ of deep donors located at the GB. Thus, the effect of these deep donor traps is incorporated in the neutrality equation for carriers within the grain:

$$ \overline{p} - \overline{n} = N_{a}^{ - } - N_{d}^{ + } - \left\langle {N_{D}^{ + } } \right\rangle \approx N_{{\text{a,res}}} - \left\langle {N_{D}^{ + } } \right\rangle $$

(3)

where the shallow acceptors and donors constituting the residual carrier concentration are assumed to be ionized $N_{a,res} = N_{a}^{ - } - N_{d}^{ + } \approx N_{a}^{{}} - N_{d}^{{}} \approx N_{a}$ . This approximation is expected to hold for the very shallow donors in these narrow-gap semiconductors down to temperatures of ~10 K.

Next, we must determine $f_{DD}^{( + )}$, such that $\left\langle {N_{D}^{ + } } \right\rangle = f_{{{\text{DD}}}}^{ + } N_{{{\text{DD}}}}$. The concentration of ionized donors placed in the GB is usually written as $N_{{{\text{DD}}}}^{\left( + \right)} \left( {\text{in GB}} \right) = N_{{{\text{DD}}}} \left( {1 + 2\exp \left( {\frac{{F - E_{D} }}{{k_{B} T}}} \right)} \right)^{ - 1}$, where F is the electrochemical potential and the difference $F - E_{D} = \zeta - E_{c0} + E_{c0} - E_{D}$ where $E_{c0}$ denotes the minimum of the conduction band in the absence of an electric field; the contributions of the electrical potential $\left( { - \left| e \right|\psi \left( x \right)} \right)$ cancels out in the difference $F - E_{D}$. Within the GB the difference $F - E_{D}$ can be written as $\Delta_{c}^{{{\text{GB}}}} + \varepsilon_{D}$, where the quantity $\varepsilon_{D} = E_{c0}^{{{\text{GB}}}} - E_{D}^{{{\text{GB}}}}$, and $\Delta_{c}^{{{\text{GB}}}} = \zeta - E_{c0}^{{{\text{GB}}}}$ is the position of the Fermi level with respect to the conduction band minimum (Fig. S.1 of Supplementary material). We are looking for a quantity which will, on average, have the same effect on the carrier concentration within the grain. We then calculate this average and determine that

$$ \left\langle {N_{D}^{ + } } \right\rangle = f_{{{\text{DD}}}}^{( + )} N_{{{\text{DD}}}}^{{}} = \left( {\exp \left( {\frac{{\Delta E_{{{\text{GB}}}} }}{{k_{B} T}}} \right) + 2\exp \left( {\frac{{\Delta_{c} + \varepsilon_{D} + \Delta E_{{{\text{GB}}}} }}{{k_{B} T}}} \right)} \right)^{ - 1} N_{D} $$

(4)

where $\Delta E_{{{\text{GB}}}} = \left| e \right|\Phi_{{{\text{GB}}}} = \left| e \right|\psi \left( {x_{J} } \right) \ge 0$ ($x_{J}$ indicates the position of the junction between the grain and the GB) and the effective deep donor density $N_{D} \equiv N_{{{\text{DD}}}} \left( {{{V_{{{\text{GB}}}} } \mathord{\left/ {\vphantom {{V_{{{\text{GB}}}} } {V_{G} }}} \right. \kern-\nulldelimiterspace} {V_{G} }}} \right)_{{{\text{av}}}}$ incorporates the (average) effect of the difference in volume occupied by GB $\left( {V_{{{\text{GB}}}} } \right)$ to that occupied by grains $\left( {V_{G} } \right)$.

As already noted, in our approach, in contrast to other modelization of GB effects on transport properties,²² the measured mobility is not predicted (nor observed) to be thermally activated. This is simply a consequence of the high scattering rates due to impurities in these heavily doped samples, so that the charge carriers mean free path (~ 70 nm or less) turns out to be much smaller than the average grain size (diameter ~ 1 µm or larger). This fact places the problem outside the assumptions of the analysis in Ref. 22.

Intraband carrier scattering caused by absorption or emission of acoustic phonons is usually the limiting mechanism determining carrier mobility near room temperature. At lower temperatures, there are other well-known mechanisms that play a limiting role. The most frequent is due to ionized impurity scattering, but in ternary alloys compacted through hot extrusion there are at least two other mechanisms which could be active. They are alloy scattering, present as well in crystalline alloys, and space charge scattering due to possible pores in HEBP¹⁴ or other compacted materials.¹⁹ Our description of these rates $\tau_{ \bot ,\alpha }^{ - 1}$ will require at least one adjustable parameter for each mechanism (α), as we can only provide the (carrier) energy and temperature dependence of the relaxation rate. Expressions for these rates are well known in the literature when assuming parabolic dispersion relations. We have derived appropriate expressions for the scattering rate due to acoustic phonons within the non-parabolic approximation in Ref. 10. When comparing with experimental data of alloys and composites, we could determine that neither space-charge limited nor alloy scattering play a significant role in our samples, so we will only need expressions for acoustic scattering (ac, see expressions in Ref. 10):

$$ {1 \mathord{\left/ {\vphantom {1 {\tau_{ \bot }^{{{\text{ac}}}} }}} \right. \kern-\nulldelimiterspace} {\tau_{ \bot }^{{{\text{ac}}}} }} \propto T\left( {1 + 2\lambda \varepsilon } \right)^{ - 1} D\left( \varepsilon \right) $$

(5)

while for ionized impurity scattering, we obtain a relaxation time (see derivation in Sect. S.II, Supplementary material):

$$ \tau_{{{\text{ii}}}} \propto N_{{{\text{ii}}}}^{ - 1} \left( {\varepsilon + \lambda \varepsilon_{{}}^{2} } \right)\left( {1 + 2\lambda \varepsilon } \right)^{ - 4} D\left( \varepsilon \right) $$

(6)

In both expressions $D\left( \varepsilon \right)$ is the (non-parabolic) density of states (DOS) of the majority carriers whose energy scales are defined, as usual, by $\varepsilon \left( {\varvec{k}} \right) = E_{c} \left( {\varvec{k}} \right) - E_{c0}$ for electrons ($E_{c0}$ is the minimum of the conduction band) and for holes $\varepsilon_{p} \left( {\varvec{k}} \right) = - \left( {E_{v} \left( {\varvec{k}} \right) - E_{v0} } \right)$, $E_{v0}$ is the maximum of the valence band. The non-parabolicity parameter $\lambda = 1/E_{g}$ is taken as the inverse of the (temperature dependent) fundamental gap $E_{g}$. The parameter $N_{ii}^{{}}$ is the concentration of charged impurities. As is usual in the case of quasi-equilibrium transport, these quantities are defined in thermal equilibrium and are field-independent.

The numerical solution of the neutrality Eq. 3 as a function of temperature gives the chemical potential $\zeta \left( T \right)$, or equivalently, the difference $\Delta_{c} \left( T \right) = \zeta \left( T \right) - E_{c0}$ when electrons are the majority carriers (or $\Delta_{v} \left( T \right) = E_{v0} - \zeta \left( T \right)$ for p-type alloys). Once the chemical potential as a function of temperature has been determined one obtains the majority and minority carrier concentrations within the grains as a function of temperature, which can be directly compared to Hall effect measurements (as will be shown below). Once these values are known, we can obtain the temperature dependence of the transport properties perpendicular to the extrusion axis through the following expressions for the hole mobility:

$$ \mu_{n/p, \bot } = \left\{ {\left( {\mu_{n/p, \bot }^{{{\text{ac}}}} } \right)^{ - 1} + \left( {\mu_{n/p, \bot }^{{{\text{ii}}}} } \right)^{ - 1} } \right\}^{ - 1} $$

(7)

where, for example, for p-type alloys:

$$ \mu_{p \bot }^{{{\text{ac}}}} = \mu_{p \bot }^{{{\text{ac}}(0)}} \left( \frac{300K}{T} \right)^{3/2} \frac{{L_{ - 1}^{0, \, 1} \left( {\eta_{\nu } ,\beta } \right)}}{{L_{0}^{0, \, 3/2} \left( {\eta_{\nu } ,\beta } \right)}} $$

(8a)

$$ \mu_{p \bot }^{{{\text{ii}}}} = \mu_{p \bot }^{ii\left( 0 \right)} \left( {\frac{{N_{{\text{a,res}}} }}{{N_{{{\text{ii}}}} }}} \right)\left( \frac{T}{300} \right)^{3/2} \frac{{L_{ - 4}^{0,3} \left( {\eta_{\nu } ,\beta } \right)}}{{L_{0}^{0,3/2} \left( {\eta_{\nu } ,\beta } \right)}} $$

(8b)

and similar expressions for n-type carriers. The functions $L_{h}^{n,m} \left( {\eta ,\beta } \right)$ are the triple-index generalized Fermi-Dirac (GFD) integrals defined in Ref. 10, originally introduced by Zawadsky²³, whose arguments are $\eta_{\nu /c} \equiv {{\Delta_{\nu /c} } \mathord{\left/ {\vphantom {{\Delta_{\nu /c} } {k_{B} T}}} \right. \kern-\nulldelimiterspace} {k_{B} T}}$), $\beta \equiv {{k_{B} T} \mathord{\left/ {\vphantom {{k_{B} T} {E_{g} }}} \right. \kern-\nulldelimiterspace} {E_{g} }}$. The parameters with dimension of mobility $\mu_{p \bot }^{\left( 0 \right)}$ are specific to each sample. The expression of the electrical conductivity in the plane perpendicular to the extrusion axis is:

$$ \sigma_{ \bot } = \left| e \right|\left( {n\mu_{n \bot } + p\mu_{p \bot } } \right) = \sigma_{n \bot } + \sigma_{p \bot } $$

(9)

Please note that the temperature dependence of measured quantities given by Eqs. 7–9 depend on the temperature dependence of the chemical potential.

We now compare the predictions of this model with Hall effect experiments carried out on a p-type ternary composite alloy of (Bi_1-xSb_x)₂Te₃ produced as described in Material synthesis and characterization Section and Ref. 12. For completeness, we also examine an n-type Bi₂(Te_1-ySe_y)₃ conventional (random) alloy.¹¹ The experimental procedure to characterize HEBP compounds and alloys has been briefly reviewed at the end of Material synthesis and characterization Section.

Figure 1, scale on the right, shows the calculated chemical potential, solution of the neutrality Eq. 3 for a p-type ternary composite with the parameters in Table I (3rd column). The majority and minority carrier concentrations as well as the (average) virtual ionized deep donor concentration are shown, scale on the left, as a function of temperature from 10 K to 440 K, as calculated once the chemical potential is known. The figure also compares the result to our Hall effect measurement for the Hall carrier density $p_{H} \approx \left( {eR_{H} } \right)^{ - 1}$ for this (Bi_1-xSb_x)₂Te₃ composite. Note that when measuring p-type samples there is always a concern that the electron-to-hole mobility ratio $b \equiv {{\mu_{n} } \mathord{\left/ {\vphantom {{\mu_{n} } {\mu_{p} }}} \right. \kern-\nulldelimiterspace} {\mu_{p} }}$ could be very large, as is often the case in direct band gap III-V compounds. In such cases $R_{H} \cdot \left| e \right| = \frac{{A_{p} \left( {p - nb_{{}}^{2} } \right)}}{{\left( {p + nb} \right)^{2} }}$ may need a careful analysis to obtain the majority (p) carrier concentration. This is not a concern with the p-type samples treated here, as the (scarce) literature data on (Bi_1-xSb_x)₂Te₃ n-type doped ternary alloys indicate b mobility ratio values less than 0.4. As it is clear from the calculated values of electron density n, shown in Fig. 1 for this composite sample, the approximation $R_{H} \cdot \left| e \right| \approx {{A_{p} } \mathord{\left/ {\vphantom {{A_{p} } p}} \right. \kern-\nulldelimiterspace} p}$ is valid. We then take the (unknown) Hall factor $A_{p} \approx 1$ which is assumed as a reasonable approximation given the relative uncertainty of ~ 10% in these measurements.

Fig. 1

The chemical potential (dash-dot curve) with respect to the maximum of the valence band $E_{v,0} - \zeta$ (scale to the right) calculated as a function of temperature for a (Bi_0.2Sb_0.8)₂Te₃ composite alloy with a residual concentration $N_{a} - N_{d} = 2.1 \times 10^{19} {\text{cm}}^{ - 3}$ following the neutrality Eq.(3). Calculated majority (black dots) and minority (blue dashed) carrier concentrations, and virtual deep ionized donors (green crosses), scale on the left. The measured Hall concentration p_H (empty black squares) for this HEPBC p-type ternary composite is also shown for comparison (Color figure online).

Table I Parameters required to describe the transport properties of one HEBP p-type (Bi_1-xSb_x)₂Te₃ composite and an n-type Bi₂(Te_1-ySe_y)₃ random alloy

Notice that considering the estimated error in the measurements (~ 10%) the fit to the carrier concentration data using the parameter values on the 3rd column of Table I and our assumption of a deep donor trap in the GB is quite satisfactory. In particular, if the energy barrier at the boundary were zero, or equivalently, if the deep donor resided within the grain, then the noticeable drop of the hole density for temperatures below about 200 K could not be explained by conventional statistics, as the carrier density in a homogeneous semiconductor can only increase with temperature. Also notice the marked sensitivity of the fit reflected by the maximum allowed variations in the values of the fitting parameter (figures in parenthesis in the third column of Table I).

In contrast, the results for the carrier concentration vs. temperature of the n-type Bi₂(Te_1-ySe_y)₃ sample, show an increase in the majority carrier concentration as the temperature increases, starting at a low temperature (~ 100 K) way before thermal generation of electron-hole pairs (which is evidenced by the increase of the minority carrier concentration). This is shown in Fig. 2, where we see that in this case a deep donor trap within the grain is able to provide this additional source of mobile electrons.

Fig. 2

Measured (red squares) and calculated carrier concentration (black dots, LHS scale) as a function of temperature for a Bi₂(Te_1-ySe_y)₃ alloy with y=0.1. The chemical potential $\zeta - E_{c,0}$ with respect to the minimum of the conduction band is shown as a function of temperature on the RHS scale. The residual shallow donor concentration is $N_{d} - N_{a} = 1.5 \times 10^{19} {\text{cm}}^{ - 3}$ and a deep donor within the grain (no energy barrier) whose concentration is $N_{D}^{{}} = 2.8 \times 10^{19} {\text{cm}}^{ - 3}$, whose ionized concentration $N_{D}^{\left( + \right)} \left( T \right)$ is shown by green crosses. The calculations are carried out with the parameters shown in Table I, 4th column (Color figure online).

Notice that, according to the neutrality Eq. 3 and the data shown in Fig. 1 (p-type sample) and Fig. 2 (n-type sample), there is no need to consider the contribution of deep acceptors, as in both cases they would provide the opposite of the observed effect with temperature. For example, for the p-type sample, the effect of considering a deep acceptor in place of a deep donor would increase the hole concentration at low temperatures, not decrease it, as is shown in Fig. 1. We also note that the thermal generation of electron-hole pairs in these narrow band gap alloys is quite large, as is expected. However, due to the heavy doping, characteristic of most thermoelectric alloys, the thermal generation begins to be noticeable in Figs. 1 and 2 only ~ 350 K.

Once the chemical potential is known as a function of temperature, transport properties can be calculated according to Eqs. 7–9. Figure 3 shows a log-log plot of the hole mobility calculated in the plane perpendicular to extrusion (red solid line), compared to our Hall data (black empty squares with error bars) for both ternary alloys considered above. For illustration purposes we have also included the calculated values of the mobilities when the results of each one of the two scattering mechanism are calculated in the absence of the other. The (blue) dotted curves correspond to acoustic phonons as the sole limiting scattering mechanism, while the dashed dot (green) curves correspond to ionized impurity scattering only. These curves show that the scattering by ionized impurities is the limiting mechanism at low temperatures (< 100 K), while acoustic phonons dominate transport above ~300 K for holes in the (Bi_0.2Sb_0.8)₂Te₃ composite, and above ~ 200 K for electrons in the Bi₂(Te_0.9Se_0.1)₃ alloy. Note that these calculations are consistent with the features of the model already discussed in the introduction to “Results and Discussion”. Since the mobility values do not exceed 0.2 m²/Vs, one can estimate a mean free path for these carriers which is always $d_{{{\text{mfp}}}} < 70{\text{nm}}$, so that most scattering events occur within grains. The effects of deep traps are included in the calculations shown in Fig. 3 through their impact on the chemical potential (see, for example, expressions in Eqs. 8a and 8b). It is also worth noting (see section S.III of the of the Supplementary material) that the model can also provide the electrical conductivity, both perpendicular and parallel to the extrusion axis, in a temperature range where there is only one limiting scattering mechanism (see Fig. S.3 of the Supplementary material).

Fig. 3

(a) Hole mobilities as a function of temperature for the HEBP composite (Bi_0.2Sb_0.8)₂Te₃ composite alloy sample of Fig. 1, with a residual $N_{a} - N_{d} = 2.1 \times 10^{19} {\text{cm}}^{ - 3}$ (data originally published in Ref. 12). (b) Corresponding figure for the Bi₂(Te_1-ySe_y)₃ alloy with y=0.1, and a residual $N_{d} - N_{a} = 1.5 \times 10^{19} {\text{cm}}^{ - 3}$. The concentrations $N_{ii}$ were approximated by the corresponding ionized residual impurity concentration. The measured Hall mobilities are shown as empty squares with the corresponding error bars (Color figure online).

The success of non-parabolic bands to model the behavior of the mobility with temperature in these ternary alloys indicates that both acoustic phonon and ionized impurity scattering do not express themselves merely as power laws of temperature as would be the case for parabolic bands. These power laws have to be modified by the non-parabolicity corrections as indicated by the GFD corrections in Eq. (8).

Speculations on the nature of the deep defects that may be responsible for the behavior observed remains outside the scope of this article.

Conclusion

In this work we have stressed the importance of correctly modelling the carrier concentration, as measured by the Hall effect, as a starting point for understanding the transport properties of heavily doped polycrystalline thermoelectric alloys. We have analyzed two ternary alloys based on Bi₂Te₃, where the role of deep donors has a significant impact on the determination of the chemical potential, the carrier concentration and the mobility. In particular, we have shown how deep traps in grain boundaries can be statistically modelled in the case of heavily doped, narrow band alloys useful for large scale thermoelectric applications up to 500 K. This work complements what is known about electron transport in polycrystalline alloys, which was previously limited to moderate doping. Work remains to be done for transport in polycrystalline material between moderate and heavy doping. In order to explain the transport properties of ternary alloys obtained by hot extrusion we extended our previous model for HEBP Bi₂Te₃ of Ref. 10. We have shown how the model is capable to satisfactorily reproduce the transport properties of a p-type (Bi_0.2Sb_0.8)₂Te₃ composite and an n-type (Bi_0.2Sb_0.8)₂Te₃ conventional (random) alloy. We found that two scattering mechanisms are necessary, and sufficient, to reproduce the measured majority carrier mobilities for both alloys in a wide temperature range (10 to ~ 400 K). A perusal of the most recent literature on the transport properties of TE polycrystalline alloys, shows few actual measurements of the carrier concentration as a function of temperature on which the modelization of the transport properties has been based. A key ingredient of any model of near equilibrium transport heavily relies on the determination of the chemical potential as a function of temperature, and the most basic property needed to corroborate the accuracy of its determination is the carrier concentration vs. temperature. Future work will study the Seebeck coefficient and the extraction of the lattice contribution to the thermal conductivity in HEBP ternary and quaternary alloys and composites. This will hopefully provide needed insight to further improve them for refrigeration at lower temperatures and waste heat recovery at higher temperatures.