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Kautsar et al. Energy Mater. 2025, 5, 500129 https://dx.doi.org/10.20517/energymater.2025.26 Page 7 of 14

Figure 4A summarizes the σ of the studied magnets, showing a slight increase after HD and subsequent
xx
GBDP. No notable differences in σ were observed among the RE-Cu (RE = Dy-Nd, Nd, Pr) GBDP
xx
magnets. Figure 4B presents the κ and the lattice contribution to thermal conductivity (κ ) of the studied
lat
magnets. The κ is extracted by subtracting the electronic contribution (κ ) from the total κ : = κ - κ . The κ e
e
lat
e
lat
is estimated using the Wiedemann-Franz Law κ = Lσ T, where L represents the Lorenz number
e
xx
(2.44 × 10 WΩK ) and T is the absolute temperature . Figure 4B shows that the κ exhibits a slight
-8
-2
[30]
lat
increase after HD but rises significantly following GBDP. This contributes to the increase in κ in the HD
and GBDP magnets compared to the HP magnet. The κ values of the studied magnets are presented in
e
Supplementary Figure 3; κ shows only a slight increase after HD and GBDP.
e
The observed increase in σ and κ following HD and GBDP can be attributed to the grain growth [53,54] .
xx
However, the substantial rise in κ after GBDP suggests the involvement of an additional mechanism. To
lat
investigate this, microstructural analyses comparing the IGP of HP, HD and GBDP magnets-represented by
the Dy-Nd-Cu GBDP magnet-were conducted, as illustrated in Figure 5A-C. The HP magnet exhibits a thin
amorphous IGP [Figure 5A], which is retained in the HD magnet [Figure 5B]. However, this phase
transforms into a thick crystalline IGP after GBDP [Figure 5C and D]. This crystallization of the IGP after
GBDP is likely a key factor driving the pronounced increase in κ , as crystalline IGPs typically exhibit
lat
higher phonon mean free paths and reduced phonon scattering compared to their amorphous counterparts,
[30,55,56]
thereby improving κ . Additionally, the prolonged heat treatment during GBDP may also contribute to
the increase in κ by reducing point defect scattering. The reduced content of ferromagnetic elements
lat
(Fe + Co) in the IGP after GBDP contributes to the enhanced coercivity [Figure 1B] by reducing the M of
the IGP [39-41,57] . However, the impact of this compositional change on σ and κ still requires further
xx
investigation. The mechanism of IGP thickening and crystallization can be described as follows. During
GBDP, the diffusion source, consisting of eutectic alloys, melts and infiltrates the magnet through the grain
boundaries. This process increases the thickness and volume fraction of the RE-rich phase in the IGP and
modifies its composition. The thickening of the IGP during GBDP, combined with prolonged annealing,
likely explains the observed crystallization.
Here we show the transverse thermoelectric conversion properties of the studied magnets. Figure 6A
presents the A and ϕ images of the studied magnets at f = 1.0 Hz and J = 1.0 A, measured in the M state
c
r
odd
odd
under zero H. Uniform current-induced temperature modulation is clearly observed across the entire
surface of the magnet slabs. To quantitatively estimate the anomalous Ettingshausen coefficient
Π (=S T), the A per unit charge current density j , i.e., A /j , was measured at different f ranging from
AEE
odd c
odd
ANE
c
1.0 Hz to 10.0 Hz, as shown in Figure 6B. The A values at each f were obtained by averaging the A odd
odd
values over the marked rectangular area (1.2 × 4.5 mm ) in Figure 6A. A clear decrease in A /j with
2
odd c
increasing f was observed, which is well replicated by considering thermal diffusion in the sample using the
one-dimensional heat diffusion equation in the frequency domain (solid lines in Figure 6B) . Finally, the
[27]
steady-state value of A /j , corresponding to f → 0 Hz (A , /j ), was calculated from fitting the curve in
odd 0Hz c
odd c
Figure 6B. The signal with ϕ approximately 180º indicates that the bottom surface of the sample
odd
(-y direction) is being heated, as illustrated in Figure 6C.
The Π and S values were calculated using Π = S T = , where L is the sample thickness and
AEE
AEE
ANE
ANE
∆T represents the temperature difference between the top and bottom surfaces of the sample induced by
AEE
[45]
AEE. This temperature difference is determined as ∆T = 2A , (M /M ) . An M /M correction is
s
AEE
r
s
r
odd 0Hz
applied to address the incomplete saturation of the studied magnets’ M during the AEE measurements. To
ensure accurate determination of the M /M values for the LIT sample slabs, their magnetic hysteresis loops
s
r
were measured using a pulse B-H tracer [Supplementary Figure 4], and the results are summarized in