1. Introduction
The thermoelectric (TE) effect offers the direct conversion of a temperature gradient into electrical energy, and vice-versa. Among the many TE materials, (Bi,Sb)
2(Te,Se)
3-based alloys (p-type (Bi,Sb)
2Te
3 and n-type Bi
2(Te,Se)
3) have been intensively investigated because they exhibit the highest TE performance near room temperature [
1,
2]. However, current TE applications are limited due to low conversion efficiency, which is evaluated from a dimensionless TE figure of merit,
zT =
S2·
σ·
T/
κ (
S: Seebeck coefficient,
σ: electrical conductivity, and
κ: thermal conductivity) at a given absolute temperature (
T). The
zT value can be improved by reducing the
κ and enhancing the
S while maintaining the
σ. Low
κ maintains the temperature difference between the hot and cold sides of a material, and high
S associates the high voltages generated at a certain temperature gradient.
zT can be enhanced by reducing the lattice thermal conductivity (
κlatt) caused by interface phonon scattering in nanostructured materials, which effectively scatter low-frequency phonons. For example, Poudel et al. [
1] reported a peak
zT of 1.4 at 100 °C in nanograin composite Bi
0.5Sb
1.5Te
3. Additionally, it has been recently found that dense dislocations formed at grain boundaries can intensify phonon scattering by the additional scattering of mid-frequency phonons [
3]. Substitutional doping is another effective approach to reducing the
κlatt by introducing point defects for phonon scattering, which target high-frequency phonons. It has been experimentally found that certain elements, such as Al, Ga, In, Cu, Ag, and Fe, reduce
κlatt effectively [
4,
5,
6,
7,
8]. However, the substitutional doping approach is inherently accompanied with the modification of electronic structure, resulting in the variation of power factor (
σ·
S2) and bipolar conduction.
We have recently demonstrated that the substitutional doping of Pb in Bi
0.48Sb
1.52Te
3 can reduce
κlatt and increase power factor (
σ·
S2) [
9]. In this study, the TE transport properties, including
S,
σ, and
κlatt of Pb-doped Bi
0.48Sb
1.52Te
3 polycrystalline samples, were further analyzed by using the single parabolic band (SPB) model [
10] and Callaway model [
11], and compared with a Bi
0.48Sb
1.52Te
3 polycrystalline sample in order to closely examine the effect of Pb substitution on bipolar conduction and
κlatt reduction. The
S and
σ were fitted to experimental values by adjusting the deformation potentials and the effective masses of both the conduction and valence bands by the two-parabolic band model based on the SPB model. Through this model, the bipolar conduction (
κbp) was estimated, while the
κlatt was analyzed using the Callaway model.
2. Results and Discussion
The temperature dependence of
σ and
S of the Pb-doped samples (Bi
0.48-xPb
xSb
1.52Te
3,
x = 0.0025, 0.005, 0.01, 0.015, and 0.02) and undoped Bi
0.48Sb
1.52Te
3 (
x = 0) is shown in
Figure 1. The value of
σ of Bi
0.48Sb
1.52Te
3 at 300 K is about 640 S/cm, which increased significantly to 2700 S/cm (
x = 0.02) with an increase in Pb content. The hole concentrations (
Np) at 300 K were 2.47, 3.54, 4.96, 6.60, 8.76, and 11.96 × 10
19/cm
3 for
x = 0, 0.0025, 0.005, 0.01, 0.015, and 0.02, respectively. Therefore, the substitution of Pb for Sb induces significant hole carriers by the introduction of acceptor defects. The value of
S decreased from 215 μV/K to 117 μV/K, revealing a clear trade-off relationship with the
σ value. It is noteworthy that while the maximum value of
S is obtained at 360 K for
x = 0, it shifts to a higher temperature of 480 K for
x = 0.02; this implies that the charge compensation from bipolar conduction is reduced due to the significant increase in
Np as Pb is added.
Figure 2 shows the temperature dependence of
κtot and (
κtot-
κelec) of the Pb-doped samples (Bi
0.48-xPb
xSb
1.52Te
3,
x = 0.0025, 0.005, 0.01, 0.015, and 0.02) and undoped Bi
0.48Sb
1.52Te
3 (Bi
0.48-xPb
xSb
1.52Te
3,
x = 0), where
κtot is the measured total thermal conductivity and
κelec is the electronic thermal conductivity without considering the bipolar conduction. Thus, (
κtot-
κelec) is the thermal conductivity excluding the contribution of the increased
Np. The
κelec was estimated using the Wiedemann–Franz law, and the Lorenz number was calculated using Equation (1) [
12] which is deduced under the assumption of a single parabolic band (plus acoustic phonon scattering),
Indeed, the
κtot increased with Pb content due to a significantly enhanced
Np value (
Figure 2a). As seen in
Figure 2b, the (
κtot-
κelec) value decreased by over 14% to 0.55 W/mK for
x = 0.02, as compared to 0.64 W/mK of the undoped sample at 300 K. Furthermore, the amount of reduction in the (
κtot-
κelec) value increased with increasing temperature and reduced up to 22% for
x = 0.02 at 480 K. For
x = 0.02, which is only a 1% substitution for the cation site, a significant reduction of 14–22% in (
κtot-
κelec) was achieved.
Figure 3a shows the temperature dependence of power factor, which shows ~40% increase to 4.11 mW/mK
2 for
x = 0.01 from 2.97 mW/mK
2 for
x = 0. The
Np value of 4.96 × 10
19/cm
3 at
x = 0.01 seems to be the optimum value for the highest power factor. For
x = 0.015 and 0.02, the power factor decreased due to a significant reduction in the
S value.
Figure 3b,c exhibit
zT values of the samples. The
zT values at temperatures above 400 K increased as the substitution increased to
x = 0.01, as shown in the inset of
Figure 3b. For
x = 0.02, the
zT value reduced up to 440 K due to a significant increase in the
σ value, which results in a significant rise in
κtot. The optimal
σ range is 700–1200 S/cm for a high
zT in this case.
Pb substitutional doping significantly increases σ due to an increase in Np and reduces the (κtot-κelec) value, which includes the bipolar thermal conductivity (κbp) and lattice thermal conductivity (κlatt). In order to further understand the origin of (κtot-κelec) reduction, we first estimated the κbp from a two parabolic band model, which is an extension of the SPB model that considers acoustic phonon scattering. Thereafter, the contribution of additional point defect scattering to κlatt was closely analyzed using the Debye–Callaway model and compared with the point defect contribution from its native cation disorder. The Pb substitutes are additional point defects in the native cation disorder of (Bi0.48Sb1.52) in their mother compound Bi0.48Sb1.52Te3.
In the two-band model, the thermoelectric parameters of valence and conduction bands computed from the Boltzmann transport equations [
10] can be substituted into Equations (2)–(5),
Here,
σi,
Si, and
RHi in Equations (2)–(5) are the electrical conductivity, Seebeck coefficient, and Hall coefficient of an individual band, respectively. The parameters with subscript “
total” include contributions from all the participating bands (these variables can be measured directly). Equation (5) describes the contribution of
κelec,
κlatt, and
κbp to the
κtot. In order to estimate
κbp, the calculated
σtotal and
Stotal of one sample were fitted to the experimental
σ and
S of the sample by adjusting the deformation potentials and density-of-states (DOS) effective masses (
m*) of its valence and conduction bands (
Table 1). Since it is nontrivial to extract a single band’s contribution to TE transport (band parameters) experimentally, we estimated each band’s contribution via modelling. For this purpose, we referred to the values of
m* and the mobility of Bi
2Te
3 reported in literature. For example, the
m* of holes (or electrons) in Bi
2Te
3 reported in the literature can elucidate the band structure of Bi
0.48Sb
1.52Te
3 when two bands (valence and conduction) are assumed to participate in the transport. Due to the high crystal symmetry of Bi
xSb
2-xTe
3, more than one pocket of Fermi surface contributes to
m* as
m* =
NV2/3mb*, where
NV and
mb* are the valley (pocket) degeneracy and band mass of a single valley, respectively. The
NV of the highest valence band of Bi
xSb
2-xTe
3 is 6, while that of the lowest conduction band is 2, as listed in
Table 1 [
13]. Similarly, the mobility of holes and electrons in Bi
2Te
3 reported in the literature along with
m* can help us estimate reasonable deformation potentials. Moreover,
RHi was calculated from the measured Hall carrier concentration (=1/(
e RHtotal)) and other band parameters computed above (deformation potential and
m*). The Fermi level of each band obtained from
RHi was used to crosscheck that it calculated from
Si. Because the band gap between the valence and conduction bands are given, we only needed to calculate the Fermi level of either band.
The estimated
κbp is plotted in
Figure 4a,b. In
Figure 4b, the plot of ln(
κbp) vs. (1/
T) reveals that the bipolar conduction increases exponentially with temperature. It is clearly seen that the estimated
κbp decreases systematically with Pb doping, as inferred from
Figure 1a and
Figure 2. The inset in
Figure 4b shows that the
κbp decreases by ~30% from 0.47 W/mK to 0.33 W/mK at 480 K for
x = 0.02. The Pb doping of Bi
0.48Sb
1.52Te
3 effectively suppresses the
κbp by increasing the concentration of the major carrier (holes) and reducing the concentration of the minor carrier (electrons), which partly accounts for the reduced
κlatt values at high temperatures in Pb-doped Bi
0.48Sb
1.52Te
3.
The estimated
κbp for each sample was added to the theoretical
κlatt, which is calculated below, to compare with the experimentally determined
κlatt. Heat flow (
q) is defined as a product of thermal conductivity (
κ) and temperature gradient (Δ
T). The negative sign of the product (
q = −
κ ΔT) indicates that heat flows from hot to cold.
κ can be expressed as arising from the heat capacity (
CV), velocity (
v), and distance between the collisions (
l) according to the kinetic gas model presented in Equation (6).
The
l in Equation (6) is a product of
v and the relaxation time,
τ. Here,
κ can also be described in terms of
CV,
v, and
τ (Equation (6)). By applying Equation (6) to the phonons in solids instead of gas particles, we obtain the Callaway equation for
κlatt (Equation (7)).
Because each parameter is dependent on frequency (
ω), the product of each parameter is integrated over the frequency. Here,
vg refers to the phonon group velocity (
vg =
dω/
dk). By treating the
vg and phase velocity (
vp =
ω/
k) as equivalent to the speed of sound,
v, Equation (7) can be written as Equation (8) (Debye–Callaway model).
where
kB,
ћ,
θ, and
z are the Boltzmann constant, reduced Planck’s constant, Debye temperature
θa, and
ћω/
kBT, respectively [
11]. The values of Debye temperature
θa (94 K) [
14] and average phonon velocity (2147 m/s) [
15] were obtained from experimental literature data. The
κlatt of a material can be calculated using Equation (8) once its
τtotal (
z) is determined from the individual relaxation times (
τi) for different scattering processes, according to Matthiessen’s rule (Equation (9)),
The parameter for relaxation times is associated with Umklapp scattering (
τU), boundary scattering (
τB), and point-defect scattering (
τPD). Here, point-defect scattering arises from atomic disorders in alloys, and is described in terms of a scattering parameter (
Г) within the
τPD formula as:
and
In Equations (10) and (11), P is the fitting parameter and
Г is a scattering parameter related to the difference in mass (Δ
M) and lattice constant (Δ
a) between two constituents of an alloy. The parameters
f, Δ
M,
G,
, r, and Δ
a are the fractional concentration of either constituents, difference in mass, parameter representing a ratio of the fractional change of bulk modulus to that of local bond length, Grüneisen parameter, Poisson ratio, and the difference in lattice constant, respectively. The parameter (
G) represents material dependent (Δ
K/K) (
R/Δ
R), where Δ
K and Δ
R are the changes in bulk modulus and local bond length, respectively. We used the same parameters used in reference [
16] for estimating the relaxation times for Umklapp scattering (
τU) and boundary scattering (
τB).
For estimating the relaxation time for point-defect scattering (
τPD), we regard
P and
f (or
) in Equation (10) as adjustable parameters in the calculation; these parameters were fitted to experimental
κlatt varying with
f for the (Bi
1-fSb
f)
2Te
3 alloy by Stordeur and Sobotta [
17] to model the
κlatt of the undoped sample in
Figure 2b with
f = 0.48/2 = 0.24 (Bi fraction in the cation site). The P and
were fitted to 28.19 × 10
−41 s
3 and 5.142 × 10
−41 s
3 and the result is represented as a black solid line in
Figure 5. For the Pb-doped samples, the additional scattering due to Pb point defects was estimated with separated relaxation time for Pb disorder,
τPD(Pb), using the fractional concentration of Pb in the cation site, which is
fPb =
x/2 (0.0013, 0.0025, 0.005, 0.0075, and 0.001 for
x = 0.0025, 0.005, 0.01, 0.015, and 0.02, respectively). The fitting parameter for the scattering of Pb defects, P
Pb, is 208.8 × 10
−41 s
3, and thus, the
values are 0.229, 0.521, 1.039, 1.553 and 2.067 × 10
−41 s
3 for
x = 0.0025, 0.005, 0.01, 0.015, and 0.02, respectively (
Table 2). The solid lines in
Figure 5 exhibit the results of the calculation to which the estimated
κbp was added. It is noteworthy that
PPb is ~7.4 times higher than
P for its native cation disorder. The effect of Pb substitution on
κlatt reduction appears to be much greater than that of the native cation disorder between Bi and Sb. It seems that the interaction of disorders within the matrix, for example, bonding nature and structure, is important to the inducing of strong point-defect scattering, thus influencing the local vibration mode around the point defects.
Although no meaningful enhancement in the maximum zT was observed since Pb doping induced excessive hole carriers, it was evident that Pb doping effectively suppressed both κlatt and κbp simultaneously. The bipolar contribution to the thermal conductivity decreased by ~30% from 0.47 W/mK to 0.33 W/mK at 480 K for x = 0.02, while the bipolar contribution to the reduction in total thermal conductivity increased at higher temperatures. At 480 K, the contribution of bipolar conduction suppression to thermal conductivity reduction increased up to 0.13 W/mK, which is 70% of the total reduction of 0.19 W/mK, due to Pb doping (x = 0.02). Therefore, the substitutional doping approach should be considered not only for the introduction of additional point defects for phonon scattering, but also for the suppression of bipolar conduction in BixSb1-xTe alloys, especially at high temperatures.