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Lead telluride based thermoelectrics: approaches for higher efficiency
P. K. Rawat1, B. Paul2, and P. Banerji1
1
Materials Science Centre, Indian Institute of Technology, Kharagpur 721302, India
The Department of Physics, Chemistry, and Biology, Linköping University, Linköping 58183, Sweden
(All the authors having equal contribution in writing this chapter)
2
The current research scenario for alternative energy sources is primarily focused on the reduction of dependency on fossil
fuels, so that the harmful effects of green house gases can be minimised. Thermoelectricity is one such niche area of
research by which the waste heat can be utilized for electric power generation. Apart from the modern infrared
optoelectronic applications, lead chalcogenides gained a great interest worldwide in the field of thermoelectricity because
of its promising electrical and thermal properties. Among the other lead chalcogens, lead telluride (PbTe) is the most
studied thermoelectric material and holds the best performance in the mid-temperature range (500 – 900 K) power
generation. The focus of this chapter is to discuss the various approaches adopted to tune the carrier transport phenomena
for further enhancement of the performance of PbTe based materials for practical realization.
Keywords: thermoelectric; lead telluride; figure of merit
1. Introduction
The efficiency of any thermoelectric material is defined by a dimensionless parameter known as the figure of merit (zT)
[1],
zT =
S 2σ
T
k
(1)
where S is the Seebeck coefficient (also termed as thermopower); σ is the electrical conductivity; k is the thermal
conductivity, which consists of two factors, viz. (i) the lattice thermal conductivity (kl), and (ii) the electronic thermal
conductivity (ke) and T is the absolute temperature. So, for high value of zT, the numerator part of the above expression
known as power factor (PF=S2σ ) must be enhanced with simultaneous reduction in thermal conductivity. However, as
all the three parameters S, σ and k are coupled with each other, the optimization of one parameter without deteriorating
the others is quite a challenging task. Seebeck coefficient is inversely related with the carrier concentration (n) and for
any degenerate semiconductors they are coupled by the relation [1],
8π 2 k B2 ∗  π 
m T 
S=
3eh 2
 3n 
2/3
(2)
where kB is the Boltzmann constant, h is Planck’s constant, e is the electronic charge, and m* is density of state effective
mass of charge carriers. The electrical conductivity is related to the carrier concentration as
σ = neμ
(3)
where µ is the carrier mobility. In solids, although unipolar conduction mechanism and low carrier concentration are
found to favor the high Seebeck coefficient, however, at the same time it results in low electrical conductivity. So, for
higher power factor the proper tuning of carrier concentration is required. The most conventional way of optimizing PF
is the optimal doping. Further, electrical conductivity is coupled with the electronic contribution to thermal conductivity
by Wiedemann-Franz relation k e = LσT , where L is the Lorenz number. So, increase in electrical conductivity results
in increase in the value of ke leading to the reduction in zT. In the following sections, we shall discuss the various
strategies to decouple one parameter with other so that one can tune them quasi-independently.
2. Lead telluride
The compounds of lead with sulphur, selenium, and tellurium are known as lead chalcogenides. This class of materials
come under the category of narrow band-gap semiconductors as their band gap (Eg) at room temperature typically lies
in the range of 0.3 - 0.4 eV. The temperature coefficient of band gap of such materials are positive, i.e., the forbidden
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gap increases with the increase in temperature. Lead chalcogenides are well known for their excellent optical and
thermoelectric performance. Among all the three lead chalcogens, lead telluride (PbTe) shows the best thermoelectric
properties and used in the most modern thermoelectric generators, particularly for mid-temperature range (500 - 900 K)
power generation applications. The crystal structure of PbTe is m3m symmetry NaCl-rock salt type with coordination
number 6 and predominantly exhibits ionic type of bonding. The X-ray diffraction studies show that lattice constant of
PbTe is 6.46 Å and the material density is found to be 8.25 g/cm3. The melting temperature is 923 ºC for the intrinsic
PbTe. Due to the naturally occurring Pb vacancies, as-grown intrinsic PbTe is always p-type.
3. Approaches for achieving high zT in PbTe based systems
3.1 Nanostructured PbTe system
In recent years, it has been shown experimentally that the nanostructuring within the bulk matrix can effectively
enhance zT. Nanostructuring is found to produce huge number of interfaces, which, through predominant scattering of
phonons as compared to the charge carriers, increases the magnitude of ( μ / k ) [2]. Apart from the phonon scattering
such potential energy barriers at the interfaces are reported to casue filtering of low energy charge carrires leading to the
enhancement of Seebeck coefficient [3-5]. However, for this mechanism to come into play, the potential energy barriers
at the interfaces must have a certain value so that the transport of high energy charge carriers is not restricted. We
introduce such aspects in detail, below, showing how due to these nanostructures, the thermoelectrical properties of
PbTe is affected.
3.1.1 Grain induced carrier energy filtering in PbTe nanocomposites
Filtering of low energy charge carriers increases the average energy of the carriers taking part in the transport process
and concomitantly enhances the Seebeck coefficient. This phenomenon of energy filterning is found to play an
important role in the enhancement of the Seebeck coefficient of the nanocomposites [3]. For a comperative study, three
specimens, produced by hot pressing of chemically sinthesized PbTe nanopowder (average size of 30 nm) at different
pressure and temperatures, are taken. The samples were prepared at a pressure of 125 MPa at temperature 773 K
(Specimen – I), at 200 MPa at temperature 803 K (Specimen – II), and at a pressure of 250 MPa at 873 K (Specimen –
III). SEM micrographs of the hot-pressed samples, as shown in fig. 1, reveal that those prepared at higher pressure and
temperature results in very sharp interface of the grains.
Fig. 1 Fracture surface SEM micrograph of (a) specimen-I, (b) specimen-II, and (iii) specimen-III.
The TEM micrograph also reveals that the specimen-III is considerably dense with fine grains and sharp interfaces
(shown in fig. 2(a)). High resolution transmission electron microscope image, as shown in fig. 2(b), indicates the
presence of self-formed nano regions (~ 2 – 3 nm) inside the grains with twisted boundaries relative to the surrounding
matrix, which acts as scattering centers for short wavelength phonons.
Energy dispersive spectroscopic (EDS) analyses show all the specimens are Pb-rich along with 2 – 3 at % of oxygen
impurity. It is observed that Pb-rich PbTe shows n-type conductivity. However, in the present case, the acceptor states
originated due to the adsorption of oxygen atoms at the grain boundaries, as confirmed by the EDS analyses, results in
p-type conductivity in all the specimens. Furthermore, as shown in fig. 3(a), the trapped charges at the grain boundaries
create energy barriers (Eb) and resist the conduction of charge carriers across the grains. The charge carriers with energy
lower than Eb (green solid circle in fig. 3 (b)) get trapped along the grain interfaces, whereas, the charge carriers with
energy higher than Eb (red solid circle in fig. 3 (b)) move freely across the grains.
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Fig. 2 (a) TEM image of specimen-III showing the grains having smooth interface. (b) High-resolution TEM image of selected
portion of specimen-III showing ~ 3 nm nano regions embedded inside the grains.
The mobility of the charge carriers depends on the number of interfaces acting as scattering centre. Further, with the
decrease in crystallite size (L), the interface density (interfacial area per unit volume) increases and thus the effective
mobility depends on crystallite size (L) as
1
μ eff

2
 Eb 
1

= Le 
 exp −
∗
 k BT 
 2πm k BT 
(4)
This implies that the mobility of the nanocomposite specimens increases with the increase in crystallite size, which
means the electric conductivity increases with crystallite size. Carrier concentration and crystallite size remaining
constant, equation (4) can be written as
σ ∝T
−
1
2
 E 
exp − b 
 k BT 
(5)
So, as the temperature increases, more and more charge carriers gain energy to surmount the energy barrier resulting
in an increase in electrical conductivity. Fig. 3 (b) schematically depicts the phenomenon of energy filtering of charge
carriers by the potential barrier at the grain interfaces where W is the width and L is the separation of the potential
barrier, which is equal to the grain size of the nanocomposite. Such filtration of charge carriers through grain boundary
potential barrier also enhances the Seebeck coefficient of the nanocomposites.
Fig. 3 (a) Scheme shows the trapping of low energy charge carriers (green solid circle) at the grain boundaries of the nanocomposite,
red solid circle represents higher energy charge carriers capable of overcome the grain boundary potential barrier, and the dark solid
circles depicts the self-formed nano-regions within the grains. (b) Schematic representation of the mechanisms of charge transport
across the grain interfaces of the nanocomposite specimen.
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Fig. 4 Temperature dependent (a) electrical conductivity, (b) thermopower, and (c) power factor.
Fig. 4 shows the electrical transport properties of all the samples. Temperature dependent electrical conductivity
shows semiconducting behavior in all three specimens, i.e. conductivity increases with temperature, a trend opposite of
which is found in case of melt grown bulk PbTe. The Seebeck coefficient decreases as the grain size increases.
Specimen with the highest grain size shows remarkable enhancement of power factor compared to other two samples,
which is caused due to large increase in electrical conductivity. The maximum power factor of specimen-III is found to
be 18.72 × 10-4 Wm-1K- 2 at 500 K, which is higher than that of the reported value for p-type bulk PbTe. The room
temperature thermal conductivity of specimen-I, II and III is found to be 1.42, 1.58 and 1.69 Wm-1K-1, respectively. One
of the most advantages of the nanocomposite is that certain distribution in size of its grains can cause the scattering of
wide spectrum of phonon wavelength, which in turn helps to reduce thermal conductivity below alloy limit. The
nanostructures do not require perfect interface and exact geometry for the reduction in thermal conductivity and it
depends only on the interface density. The interface density in all the nanocomposite specimens being higher, their
thermal conductivity is lower than that of the bulk PbTe. Again, the interface density of specimen-III being relatively
lower than specimen-I and II, thermal conductivity of the former is higher than the later two specimens. The increase in
power factor and decrease in thermal conductivity results in a remarkable increase in zT of the nanocomposites as
compared to their melt grown bulk counterpart.
3.1.2 Enhanced scattering of carriers by metallic nano-inclusion embedded into the grains of nanocomposites
Incorporation of metalic nano-inclusion within the grains, acting as an additional scattering centers, can casue further
enhancement of thermoelectric performance of the nanocomposite systems. For a comparative study, different samples
such as melt grown undoped bulk PbTe, melt grown Ag-doped PbTe (Ag0.01Pb0.99Te), and hot-pressed Ag-doped PbTe
nanocomposite (Ag0.02Pb0.98Te) are taken. (Hereafter, we refer hot-pressed sample as nanocomposite.)
A typical TEM image of PbTe:Ag nanoparticles is shown in fig. 5(a). Scanning electron microscopic (SEM) image
(fig.5(b)) of PbTe:Ag nanocomposites shows densely compact grains with certain distribution in size.
Fig. 5 (a) A typical TEM image of PbTe:Ag nanoparticles. (b) SEM micrographs of the fracture surface of the PbTe:Ag
nanocomposites. (c) A typical TEM image of PbTe:Ag nanocomposites showing the formation of densely packed grains with
embedded Ag-rich nanodots. (Reprinted with permission from J. Appl. Phys. 2010; 108: 064322-064327. Copyright 2010, American
Institute of Physics. [4] )
Fig. 5(c) shows high-resolution transmission electron microscopic (HRTEM) image of PbTe:Ag nanocomposites,
reveals the existence of Ag-rich nanoscale features of dimensions ~5-15 nm embedded within the densely packed grains
of nanocomposites, which can act as scattering centres for low wavelength phonon.
The Hall measurements show p-type conductivity in all the samples with room temperature hole concentration of
1.492 × 1018, 1.893 × 1018, and 1.96× 1018 cm-3 for melt grown undoped PbTe, melt grown Ag-doped PbTe and
PbTe:Ag nanocomposites, respectively. The thermoelectric properties of all the samples are shown in fig. 6. It is found
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that although hole concentration in PbTe:Ag nanocomposite sample is higher than the other two melt grown samples, its
low temperature (<450 K) electrical conductivity is lower than the other two samples due to the fact that the potential
energy barriers at the grain interfaces of the nanocomposite sample obstruct low energy charge carriers to take part in
the transport process. On the other hand, at temperatures higher than 450 K, the electrical conductivity of the
nanocomposite samples becomes higher than the other two melt grown samples. This is because the enhanced scattering
of charge carriers by the crystal lattice gradually decreases the electrical conductivity of melt grown samples, whereas,
in nanocomposite sample, as already discussed in previous section, with the increase in temperature more and more
carriers gain energy to overcome the grain boundary energy barriers leading to the enhancement of its electrical
conductivity.
As the Seebeck coefficient inversely related to the carrier concentration, it is higher in melt grown undoped PbTe
than the Ag-doped PbTe. But in case of nanocomposite, the phenomenon of carrier energy filtering leads to the highest
Seebeck coefficient, even though its carrier concentration is higher than the other two samples. Further, the presence of
Ag-rich nanodots within the grains of PbTe:Ag nanocomposites is found to provide better thermoelectric performance
than the undoped PbTe nanocomposites. The Ag-rich nanodots act as additional scattering centers enhancing the
scattering parameter leading to the increase in Seebeck coefficient of the system. At room temperature, such additional
scattering in PbTe:Ag nanocomposite is found to cause almost 31 % enhancement of its Seebeck coefficient as
compared to theoretically calculated value, which is remarkably higher than the enhancement of 13 % in undoped
PbTe-nanocomposite sample prepared under similar conditions. The random distribution of metallic nanodots within
PbTe:Ag nanocomposite is found to cause the reduced carrier mobility as compared to undoped PbTe-nanocomposite.
However, such negative effect of Ag-nanodots on carrier mobility is compensated by their positive effect on Seebeck
coefficient. Similarly, enhancement in Seebeck coefficient is found to occur in PbTe by the enhanced scattering of
Pb-nanoprecipitates [5].
Fig. 6 Temperature dependent thermoelectric properties of melt grown undoped PbTe, melt grown Ag doped PbTe, and PbTe:Ag
nanocomposite. (Reprinted with permission from J. Appl. Phys. 2010; 108: 064322-064327. Copyright 2010, American Institute of
Physics. [4] )
Although at lower temperature regime (< 450 K) the Seebeck coefficient of the nanocomposite is remarkably higher
than the melt grown samples yet due to its lower electrical conductivity in the same temperature range, the power factor
is lower than that of the other melt grown samples. However, at higher temperature (> 450 K), due to the higher
Seebeck coefficient and electrical conductivity of nanocomposite, its power factor is higher than that of the melt grown
samples. Certain distribution in size of the grains in nanocomposites causes scattering of a wide spectrum of phonon
energy and thus the reduction in thermal conductivity of nanocomposite below its bulk counterpart. The room
temperature thermal conductivity of nanocomposite (1.69 Wm-1K-1) is ~30% less than that of undoped PbTe (2.42 Wm1 -1
K ), and PbTe:Ag (2.51Wm-1K-1). Such increase in power factor with simultaneous decrease in thermal conductivity
make nanocomposites superior over bulk samples for thermoelectric applications.
3.1.3 Unaffacted mobility with reduced thermal conductivity by endotaxial nanostructures
As discussed in previous section, the incorporation of metallic nanodots within the host matrix can lead to remarkable
reduction in thermal conductivity of the system, however, at the same time, through scattering, the mobility thus the
electrical conductivity is reduced drastically. This is because the band offset at the interface of the two different phases
cause the formation of energy barrier, which restricts the free movement of charge carriers leading to the decrease in the
ratio (μ / k ) of mobility (μ) to the thermal conductivity (k). However, the perfect band and lattice alignment of the
nanodots with the host matrix can prevent such unwanted scattering of charge carriers with simultaneous transmission
of phonon leading to the manifold enhancement of (μ / k ) . In PbTe, endotaxially embedded nanocrystals of SrTe are
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found to inhibit the heat conduction in host material without deteriorating the hole mobility, which is attributed to the
crystallographic alignment of SrTe with PbTe as shown in fig. 7 (a) and without any offset in valence band between
them (fig. 7 (b)) [6]. The alignment of valence band allows charge carriers to move between the two phases without any
obstacle. Such decoupling of phonon and electron transport enables a significant enhancement in zT with a maximum
value of zTmax = 1.7 at 800 K as is found from fig. 7 (c).
Fig. 7 (a) Hign resolution transmission electron micrograph of SrTe endotaxial nanostructures in PbTe matrix. (b) Schematic
representation of the alignment of valence band between the two phases. (c) Temperature dependent thermoelectric figure of merit
(zT) of PbTe–SrTe samples doped with 1% Na2Te. (Reprinted by permission from Macmillan Publishers Ltd.: Nature Chemistry.
2011; 3: 160-166. Copyright 2011 [6] )
3.1.4 Combined effect of mass fluctuation with endotaxial nanostructures on zT
The endotaxially embedded PbTe/PbSe nanostructure, shown in fig. 8 (a), is found to enhance the magnitude of
(μ / k ) in PbSe0.5Te0.5 leading to an enhancement of zT [7]. Similar to the previous case, the complete alignment of
coduction band of nanodots and the host matrix leads no band-offset between the two phases, schematically shown in
fig. 8 (b). On the other hand, the mass fluctuation at Te-sites in PbSe0.5Te0.5 reduces its lattice thermal conductivity to a
great extent (≤ 1.6 Wm-1K-1 at 300 K) compared to PbTe (2 Wm-1K-1 at 300 K) and PbSe (1.9 Wm-1K-1 at 300 K),
without compromise much with the high mobility values of PbTe and PbSe.The high values of power factor (due to
high mobility) as shown in fig. 9 (a) together with reduced thermal conductivity enables high zT values (fig. 9 (b)).
Fig. 8 (a) High magnification HRTEM image of a selected portion of PbSe0.5Te0.5 with x = 0.5 mol% PbI2 with selected area
diffraction (SAD) pattern (shown in inset). (b) Schematic of transmission of charge carriers between the endotaxial phases.
(Reprinted from Nanotechnology 2013; 24: 215401-215408. Copyright 2013, IOP Publishing Ltd. [7])
Fig. 9 (a) Temperature dependent power factor. (b) Themperature dependent figure of merit. (Reprinted from Nanotechnology 2013;
24: 215401-215408. Copyright 2013, IOP Publishing Ltd. [7])
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3.2 Modification of electronic density of states
For a significant enhancement of zT, only the enhancement of (μ/k) is not sufficient; drastic enhancement of Seebeck
coefficient without deteriorating the electrical conductivity is required. However, changing Seebeck coefficient and
electrical conductivity independently is quite a challenging task as they are strongly coupled with each other as
mentioned earlier. One interesting strategy can be the local distortion of band near to the Fermi level. Mahan and Sofo
[8] theoretically predicted that steep increase in density of states (DOS) can drastically enhance the Seebeck coefficient.
The phenomenon can be explained by the Mott relation given by,
S=
 1 dn(E ) 1 dμ (E ) 
k BT 
+

3 e
μ dE  E = EF
 n dE
π 2 kB
(6)
where n(E) is the density of charge carriers and can be expressed as n( E ) = g ( E ) f ( E ) , where g (E ) and f (E )
stands for the electronic density of states and the Fermi function. The above equation shows that an increase
in dn( E ) / dE , which means a local increase in g(E), should lead to an enhancement in Seebeck coefficeint. The most
outstanding feature of this strategy is that such enhancement of Seebeck coefficient is not related to the reduction in
electrical conductivity. Heremans et al. [9] first implemented such strategy experimentally in thallium (Tl) doped PbTe.
Thallium is found to introduce impurity levels, which resonates with the valance band of PbTe enhancing locally the
density of states near Fermi level leading to the double degree enhancement of zT of about 1.5 at 773 K. Sodium (Na) is
used as an acceptor dopant to tune the Fermi level (EF) within the distorted band. Fig. 10 shows the schematic
representation of enhanced density of states by resonant impurity.
Fig. 10 Schematic representation of enhanced density of states by resonant impurity.
Titanium (Ti) is also found to modify the DOS in the conduction band of PbTe, which resides approximately
52 meV above the conduction band bottom. However, because of Fermi level pinning it does not show any
enhancement of thermopower [10]. Indium (In) also creates resonant levels inside the conduction band, but due to the
increase in the band gap of PbTe it shifts into the band gap at higher temperatures and thus does not show any
enhancement in thermopower [11]. Theoretically, it was predicted that cadmium (Cd) can also form its resonant state in
the conduction band, but experimentally no evidence has been found [12]. Optical measurements showed that aluminum
(Al) makes its resonant states very deep inside the conduction band of PbTe (300 meV), but it could not be utilized for
the enhancement of thermopower because of the solubility limitation of Al in PbTe [13]. Chromium (Cr) is also found
to introduce resonant levels within the conduction band of PbTe. In the next section we will discuss the effect of Cr
resonant states on thermoelectric properties in PbTe.
3.2.1 Effect of chromium (Cr) resonance states on thermoelectric properties of PbTe
Chromium is found to create its impurity levels 100 meV above the conduction band bottom of PbTe. The energy
separation between the resonant states and middle of the band gap remains unaffected with the temperature change. In
the present case, four specimens with different Cr content, viz. specimen-I (0.70 at. % Cr), specimen-II (1.31 at. % Cr),
specimen-III (1.45 at. % Cr), and specimen-IV (2.08 at. % Cr) were taken. From x-ray diffraction studies of specimen-I
a homogeneous solid solution was observed, however, for specimen-II, -III, and -IV the presence of few impurity peaks
of elemental Pb along with Cr2Te3 and Cr3Te4 hexagonal phases were found. Resonant states of Cr are found to play an
important role in redistribution of electrons and thus tune the nature of temperature dependency of Hall coefficient as
shown in fig. 11 (a). The negative sign of Hall coefficient reveals n-type conduction in all the samples. Due to the
trapping of thermally excited electrons by the resonant states of Cr, as shown in fig. 11 (b), the Hall coefficient of all of
the samples increases in magnitude with temperature and attains the highest value at a particular temperature, Topt and
beyond which the magnitude of Hall coefficient begins to decrease with temperature. The electrical resistivity of all the
specimens increases with the increase in temperature (fig. 11 (c)), showing semimetallic nature of the specimens.
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Fig. 11(d) shows temperature dependent Seebeck coefficient in all the specimens. The room temperature values of
Seebeck coefficient of specimen-I, -II, -III, and -IV are found to fall near theoretically calculated value confirming no
enhancement in Seebeck coefficient. However, to understand the effect of Cr induced local enhancement of the density
of states on Seebeck coefficient, tuning of Fermi level is required, though in the present case the carrier concentration is
not high enough to tune the Fermi level near to the distorted band.
Fig. 11 (a) Temperature-dependent Hall coefficient. (b) Schematic representation of electron transition from the conduction band to
the impurity state. Where Cr+3, Eg, EF, and ΔE represents the Cr-impurity states, the band gap, The Fermi level, and the gap between
the impurity levels and middle of the band gap, respectively. (c) Temperature dependent electrical resistivity. (d) Temperature
dependent thermopower. (e) Temperature dependent power factor. (Reprinted with permission from J. Appl. Phys. 2011; 109:
103710- 103717. Copyright 2011, American Institute of Physics. [14] )
The power factor (PF) in almost all thermoelectric materials investigated so far is found to achieve its highest value
at a particular temperature; beyond which it falls drastically, and thus it does not allow the materials to provide its
higher energy conversion efficiency beyond a certain limited temperature range. However, the power factor of
specimen-III and -IV shows nearly flat response (fig. 11 (e)) throughout the investigated temperature range. Although
specimen-III and IV provides higher PF the excess Cr incorporation in those samples is found to cause random
distribution of Pb nanodots of size 2-13 nm as can be found from fig. 12 within the PbTe matrix, which is caused due to
the nano-precipitation of excess Pb substituted by Cr. Such nano-precipitation is considered as unfavorable for high
carrier mobility. In the present case, the most interesting thing is that the electron mobility in those samples is found to
be as high as that of single crystalline n-type PbTe. Such high mobility of electrons in those samples is attributed to the
perfect crystallographic alignment of nanodots with the surrounding PbTe matrix. The selected area diffraction pattern
of Pb-nanodots and PbTe matrix does not show any splitting in Bragg spots indicating the complete alignment between
them.
Fig. 12 (a) Typical TEM image showing random distribution of nearly spherical nanodots within the PbTe matrix (taken from
specimen-III). (b) HRTEM image of a nanodot showing the coherency of the lattice parameter of the nanodot with the host PbTe
matrix. Inset shows the electron diffraction pattern of the region including the nanodots and the host PbTe matrix. (Reprinted with
permission from J. Appl. Phys. 2011; 109: 103710- 103717. Copyright 2011, American Institute of Physics. [14] )
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3.3 Optimization of carrier concentration
The most well-known and straight-forward strategy for the enhancement of thermoelectric performance is the
optimization of carrier concentration through optimal doping. As Seebeck coefficient is inversely related to the carrier
concentration ( n ), the optimization of carrier concentration is an essential requirement for higher power factor ( S σ ).
However, it is worth to be noted that optimal carrier concentration (n*) is not independent of temperature and varies
2
*
*
3/ 2
with temperature as n ∝ ( m T ) , where m is density of states effective mass of charge carriers. So, at low
temperature carrier concentration (n) in optimally doped thermoelectric materials is typically found to be much higher
*
*
than the required n value, whereas, at high temperature the former is found to be much lower than the later leading to
the poor thermoelectric performance [15]. The overall performance throughout the temperature range can be fully
*
realized only when the dopant is functional in such a way that it would maintain the value of n close to n . Some new
*
interesting and alternative strategies, which can be functional to achieve the carrier concentration near to n throughout
the operational temperature range, are discussed below.
*
3.3.1 Tuning temperature-dependent density of states effective mass ( m )
As optimal carrier concentration depends on density of states effective mass such that
n* ∝ (m*T ) 3 / 2 , the value of n*
*
can be tuned by tuning m . On the other hand the density of states effective mass depends on degeneracy of valleys
∗
2/3
∗
∗
(Nv= number of valleys taking part in transport process) through the relation m = N v mb , where mb is is average
*
band mass for each valley. So, a novel approach for tuning m could be the carrier pocket engineering. In this approach
the effort is to increase the number of degeneracy, Nv, by tailoring electronic band structure of the material system. In
PbTe the number of valleys at L point in the Brillouin zone is 4, i.e., degeneracy Nv = 4. In p-type PbTe below 450 K
only those valleys at L point can take part in transport process as other valleys along the Σ - line of the Brillouin zone is
found to reside below the L point. At low temperature, the energy separation ΔEL-Σ between those bands is found to be
∼0.2 eV. Mg-doping is found to widen the band gap of PbTe only at L point keeping unaltered the position of Σ band
with respect to conduction band bottom at L point resulting in the reduction in the value of ΔEL-Σ , as shown in fig.
13(a). At room temperature, such widening of band gap is found to result in the coincidence of L and Σ bands leading
to the enhancement of the value of Nv of MgxPb1-xTe with concomitant increase in n*, whereas, at elevated temperatures,
the temperature dependent widening is found to cause the misalignment of the valleys resulting in the decrease in the
value of Nv with simultaneous decrease in n*. Such optimization of carrier concentration without changing the actual
doping level leads to a significant improvement of overall efficiency in MgxPb1-xTe by 40% as shown in fig. 13(b).
Fig. 13 (a) Schematic band structure of MgxPb1-xTe alloys at 300 K., (b) Thermoelectric figure of merit for MgxPb1-xTe:Na.
(Reprinted with permission from Adv. Mater. 2011; 23: 5674-5678. Copyright 2011, John Wiley and Sons.[15] )
3.3.2 Optimization of carrier concentration through temperature-dependent dopant
The carrier concentration of any thermoelectric material can also be tuned by an interesting strategy like thermally
induced enhanced solubility of excess dopant element residing in the interstitial site of the host material. This
phenomenon is found to play an important role in tuning the electron concentration of Ag-doped PbTe/Ag2Te
composites throughout the temperature range of operation [16]. The excess Ag is found to reside in the interstitial
position of PbTe/Ag2Te and also to form separate nano-phase as shown by the black spot in the back scattering image in
fig. 14(a). The increased solubility of this excess Ag with the increase in temperature is found to contribute additional
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extrinsic carriers favouring almost 50 % enhancement of average zT than optimally La-doped PbTe [17]. Such increase
in carrier concentration is found to decrease the magnitude of Hall coefficient of PbTe/Ag2Te as a function of
temperature as shown in fig. 14(b).
Fig. 14 (a) shows back scattering image of Ag doped PbTe/Ag2Te. (b) Temperature dependent Hall coefficient. (Reprinted with
permission from Adv. Energy Mater. 2011; 1: 291-296. Copyright 2011, John Wiley and Sons. [16] )
3.3.3 Carrier concentration optimization by the temperature-dependent redistribution of carriers
The optimization of carrier concentration can also be achieved by the incorporation of certain impurity in the vicinity of
band edge with an aim to compensate the excess donor or acceptor activity in as-grown materials by the temperaturedependent redistribution of carriers between the impurity levels and the energy bands, as investigated in Pb1-x-ySnxYbyTe
[18]. The alloy of PbTe and SnTe is considered to be favourable for the thermoelectric applications due to its reduced
thermal conductivity originated from the mass fluctuation at Pb-sites. However, due to the point defects at Pb-sites the
hole concentration in as-grown Pb1-xSnxTe samples is found to be much higher than the optimal carrier concentration;
thus full realization of thermoelectric performance from these class of materials becomes elusive. We discuss here a
new approach to compensate such defect-induced excess hole in as-grown Pb1-xSnxTe for achieving optimal carrier
concentration, which can be implemented in other material systems. For the investigation we incorporated ytterbium
(Yb) in Pb1-xSnxTe, which produces its own impurity levels in the band gap above the valence band edge. The
investigation was done on five different samples of Pb1-x-ySnxYbyTe having different x and y values. The composition
and other related parameters of the samples are tabulated in table 1 below.
Table 1 Room temperature values of various electrical parameters with elemental composition of Pb1-x-ySnxYbyTe. (Reprinted with
permission from Phys. Status Solidi RRL. 2012; 6: 481-483. Copyright 2012, John Wiley and Sons. [18] )
Sample
x
y
RH (cm3/C)
p (cm-3)
EF (meV)
ρ (ohm-cm)
S (µV/K)
µ (cm2/V-s)
1
0.11 0.004
+30.64
0.2 × 1018
5.57
48.16 × 10-4
+142.80
6362
2
0.14 0.002
+9.11
0.7 × 1018
13.42
31.90 × 10-4
+118.37
2856
3
0.18 0.001
+1.06
5.9 × 1018
51.80
19.40 × 10-4
+117.91
546
4
0.24 0.001
+0.67
9.3 × 1018
73.84
16.82 × 10-4
+60.73
398
5
0.18 0.000
+0.56
11.2 × 1018
73.34
10.90 × 10-4
+89.23
514
Yb levels are found to take part in redistribution of charge carriers providing electrons to the empty states of the
valence band of Pb1-xSnxTe following the conversion of charge state from Yb2+ to Yb3+ resulting in an increase in Hall
coefficient of sample-1, 2, 3, and 4, as shown in fig. 15. Due to the partial compensation of acceptor activity of native
point defects by the donor activity of Yb-impurity levels hole concentration of sample-1, 2, 3, and 4 is found to have
remarkably reduced compared to the reported values (depicted by the solid line in inset of fig. 15) for Pb1-xSnxTe with
different Sn-content.
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Materials
and processes for energy: communicating current research and technological developments (A. Méndez-Vilas, Ed.)
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Fig. 15 Temperature dependent Hall coefficient. Inset shows variation of hole concentration (symbols) of the samples as a function
of x at 300 K. Solid line in inset shows the mean curve obtained by Orihashi et al. in Pb1-xSnxTe. (Reprinted with permission from
Phys. Status Solidi RRL. 2012; 6: 481-483. Copyright 2012, John Wiley and Sons. [18] )
Other function of Yb-impurity levels is observed as the negation of bipolar effect. At high temperatures because of
bipolar effect the Seebeck coefficient of many thermoelectric materials is found to be reduced. The conventional way to
negate such bipolar effect is to heavily dope the material. However, heavy doping can deteriorate low temperature
thermoelectric performance leading to operational temperature range small. Such impurity levels through their acceptor
activity can negate the bipolar effect to a great extent preventing the degradation of high temperature thermoelectric
performance. Due to the same bipolar effect a transition from extrinsic to intrinsic conduction is reported to occur in
Pb0.9Sn0.1Te resulting in change of sign of Seebeck coefficient from positive to negative [19]. However, no such
transition of Seebeck coefficient is observed in sample-1 with x = 0.11 and y = 0.004, which is attributed to the trapping
of thermally excited electrons from valence band by the Yb-impurity levels through transformation of Yb ions from
Yb3+ to Yb2+ charge state. In other words, doping of Yb in Pb1-xSnxTe can eliminate the intrinsic carrier conduction at
high temperature without any heavy doping. Fig. 16(a) shows the energy level diagram of Pb1-x-ySnxYbyTe at 300 K.
The dotted line shows the estimated position of Yb impurity levels, which lies above EF (diamond symbols) in all the
Yb-doped samples. Fig. 16(b) schematically shows the charge redistribution mechanism in Pb1-x-ySnxYbyTe, where the
vertically up arrows (red colour) indicates the charge trapping of thermally excited electrons by Yb impurity levels. As
shown in fig. 17 (a), the temperature dependent electrical resistivity shows degenerate conduction of all the samples.
Seebeck coefficient as a function of temperature of all the samples is shown in fig. 17(b). Nearly 24 % enhancement of
thermopower of sample-3 (315 μV/K at 500 K) is observed as compared to sample-5 (254 μV/K at 500 K) (shown in
the inset of fig. 17(b)) and is attributed to the reduction in hole concentration due to the compensation of acceptor
activity of the native point defects in Pb1-x-ySnxYbyTe by the donor action of Yb-impurity levels. Fig. 17(c) shows the
temperature-dependent power factor of all the samples. Sample-3 provides the highest power factor of 16.07 × 10-4 Wm1 -2
K at 450 K, which is nearly 29 % higher than the maximum power factor of sample-5 (12.45 × 10-4 Wm-1K-2 at
450 K). Such enhancement of power factor of sample-3 can be attributed to ~24 % enhancement of its Seebeck
coefficient as compared to sample-5. Overall, such optimization of carrier concentration by the incorporation of Ybimpurity levels near valence band edge is observed to be fruitful for the remarkable enhancement of power factor in
Pb1-xSnxTe systems in a wide temperature range.
Fig.16 (a) Energy level diagram of Pb1-x-ySnxYbyTe at 300 K. (b) Schematic shows the donor activity of Yb impurity levels
(downward arrows) and thermally excited electron trapping by Yb levels (upward arrows). (Reprinted with permission from Phys.
Status Solidi RRL. 2012; 6: 481-483. Copyright 2012, John Wiley and Sons. [18] )
850
©FORMATEX 2013
Materials
and processes for energy: communicating current research and technological developments (A. Méndez-Vilas, Ed.)
____________________________________________________________________________________________________
Fig. 17 Temperature dependent (a) electrical resistivity, (b) thermopower, and (c) power factor in Pb1-x-ySnxYbyTe. Inset shows 24 %
enhancement in thermopower in Sample-3 as compared to Sample-5 at 500 K. (Reprinted with permission from Phys. Status Solidi
RRL. 2012; 6: 481-483. Copyright 2012, John Wiley and Sons. [18] )
4. Summary
In this chapter, various strategies such as carrier energy filtering, nanostructuring, distortion of density of states, and
optimization of carrier concentration were discussed to explicitly demonstrate how the carrier transport phenomenon
can be tuned for further enhancement of thermoelectric efficiency in the existing high performance materials. The
quasi-independence of the interdependent thermoelectric parameters by such strategies is found to enhance the
performance of PbTe-based material system significantly.
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