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Page 12 of 17 Kumar et al. Energy Mater. 2025, 5, 500109 https://dx.doi.org/10.20517/energymater.2025.22

By subtracting the κ from κ , the temperature-dependent lattice and bipolar thermal conductivities
total
e
κ +κ (T) of the hot-pressed BST+HEA samples for the Pa- and Pe-directions are obtained, as
L
b
x
shown Figure 6C. The κ +κ (T) shows the conventional 1/T behavior near room temperature and the bipolar
L
b
effect above 350 K. The 1/T behavior of the thermal conductivity is primarily caused by the Umklapp
processes of acoustic phonon . The thermal conductivity of narrow bandgap materials increases with
[39]
temperature due to the bipolar effect, which is associated with thermally excited minority carriers at high
temperatures. The bipolar effect is associated with the narrow bandgap (~0.1 eV) of bismuth tellurides at
high temperatures . Therefore, the κ dominates near room temperature in the κ +κ (T) of the BST+HEA x
[40]
L
b
L
samples.
Figure 6D shows the lattice thermal conductivities κ of the BST+HEA samples for the Pa- and
x
L
Pe-directions at 300 K. While the Pe-direction κ are slightly reduced, the κ of the Pa-direction are clearly
L
L
decreased by the HEA nanoparticles below 0.5 vol% but noticeably increase at 1.0 vol%. The increased
Pa-direction κ in the BST+HEA composites is attributed to enhanced grain connectivity, consistent with
L
the results for λ . Interestingly, the Pa-direction κ of pure BST is lower than that of the Pe-direction due to
L
e
the preferred orientation of the BST layers. However, in the BST+HEA composite with 1.0 vol% HEA, the
Pa-direction κ becomes higher than the Pe-direction κ . This unconventional increase in Pa-direction κ L
L
L
supports the idea that HEA nanoparticles enhance grain connectivity along the Pa-direction in the BST
matrix.
The lattice thermal conductivities κ can be described by the Debye-Callaway model using
L
(7)
where Θ is the Debye temperature, τ is the total scattering time rate, and ν is the sound velocity. The
C
scattering time τ , which is associated with boundary scattering, point defect scattering, and Umklapp
C
scattering, can be expressed as follows
(8)

where L is the average diameter of grain boundary, ω is phonon frequency, V is the lattice volume, Γ is the
scattering parameter of a point defect, A is the free parameter of Umklapp scattering, γ is the Grüneisen
N
parameter, and M is the average atomic mass. [43]

The thermal conductivity due to the bipolar contribution can be expressed as follows

(9)

where B is a fitting parameter .
[44]
Based on the theoretical equations for the κ and κ along with the parameters (in Supplementary Table 1)
L
b
such as the average group velocity of a phonon (ν = 2147 /s), acoustic Debye temperature (Θ = 94 K),
and Grüneisen parameter (γ = 2.3) , the lattice and bipolar thermal conductivities κ +κ (T) were
[43]
L
b
calculated using the fitting method. The fitting results are presented in Supplementary Figure 4A and B for
the Pa- and Pe-directions, respectively. The calculated κ and κ are shown separately in
L
b
Supplementary Figure 4C and D. The obtained parameters are listed in Supplementary Table 1.