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Page 10 of 17 Kumar et al. Energy Mater. 2025, 5, 500109 https://dx.doi.org/10.20517/energymater.2025.22

Table 1. Mean free paths of the carrier λ , Fermi energies E , the phonon mean free path λ , and average phonon scattering time τ of
Ph
ph
e
F
the BST+HEA samples
x
Sample E (eV) λ (nm) λ (nm) τ (s 10 )
-12
e
ph
f
Ph
Pa - 0 32.7 10.4 5.3 2.88
Pa - 0.1 33.1 12.6 4.9 2.88
Pa - 0.5 33.3 13.6 5.6 2.94
Pa - 1.0 33.3 12.3 6.1 2.95
Pe - 0 33.9 14.5 6.1 2.49
Pe - 0.1 33.4 14.4 5.8 2.16
Pe - 0.5 34.4 13.6 6.3 2.51
Pe - 1.0 33.7 14.7 6.6 3.08
HEA: High entropy alloy; BST: Bi Sb Te .
0.4 1.6 3
to the electrical conductivity,

(3)

where Δσ = σ(T ) - σ(T ) is the characteristic change in the electrical conductivity for the BST+HEA
high
low
samples. In this work, the T = 300 K and T = 500 K are used for the σ(T) of the pristine BST and
high
low
BST+HEA samples.
The tendencies of the S σ of the BST+HEA samples with the HEA additions for the Pa- and Pe-directions
2
x
2
are shown in Figure 5C. The trends in the S σ align well with the G conn. as a function of the HEA
concentrations for the Pa- and Pe-directions, as shown in Figure 5D. The Pa-direction G is enhanced by
conn.
the HEA nanoparticle, whereas the Pe-direction G conn. is not significantly affected. These trends are
consistent with the results for the λ . The enhanced G and λ clearly indicate that the HEA nanoparticles
e
e
conn
are more effective in increasing the electrical grain connectivity of BST in the Pa-direction, which originally
exhibits lower electrical conductivity compared to the Pe-direction. Therefore, the power factor can be
enhanced through increased electrical conductivity without a change in carrier concentration due to the
improved G induced by the addition of HEA nanoparticles in the BST matrix.
conn.
The temperature-dependent κ [κ (T)] of the hot-pressed BST+HEA (x = 0, 0.1, 0.5, and 1.0 vol%)
total
x
total
samples for the Pa- and Pe-directions is shown in Figure 6A. The thermal conductivity of the sintered HEA
(TaNb HfZrTi) sample is κ = 10.25 W m K at 300 K. The κ (T) of the BST+HEA samples decreases with
-1
-1
x
total
2
increasing temperature near room temperature. The κ (T) is increased with an increasing temperature
total
above 350 K due to the bipolar effect, consistent with the previous reports [9,11] . The lower Pa-direction
κ (T) compared to the Pe-direction κ (T) is attributed to the anisotropic alignment of the BST layers at
total
total
low HEA concentration (below 0.5 vol%). Also, the κ (T) values for the Pa- and Pe-directions are similar at
total
the higher HEA concentration (1.0 vol%).
The κ of the bismuth telluride is primarily affected by the phonons κ , electrons κ , and bipolar diffusion
L
total
e
κ . Using the Wiedemann-Franz law (κ = LσT, where L, σ, T are the Lorenz number, electrical conductivity
b
e
and absolute temperature, respectively.), the electronic thermal conductivity κ can be calculated. The
e
Lorenz number L = (π /3)(k /e) = 2.45 × 10 W Ω K can be assumed for a simple metal; however, it should
2
2
-8
2
B
0
be modified in case of correlated metals or degenerated semiconductors. Figure 6B presents the calculated
Lorenz numbers L(T) for correlated metals and degenerated semiconductors, as calculated using [38,39]