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Martin-Gonzalez et al. Energy Mater. 2025, 5, 500121 https://dx.doi.org/10.20517/energymater.2025.32 Page 11 of 35

is realized, then many valleys (or transport carrier pockets) would provide increased carrier density and
conductivity. Another way to appreciate the importance of the number of valleys, is when examining the
material at a fixed carrier density. In this case, a large band degeneracy increases the overall density-of-states
effective mass (m ) and improves the Seebeck coefficient (the reason being that at a fixed carrier density
DOS
the Fermi level will be positioned lower if m is higher). On the other hand, a low conductivity effective
DOS
mass (m ) leads to high carrier mobility and conductivity. The favorable situation will be many light valleys
C
for large m and low m . An additional important aspect is the degree of inter-valley scattering between
C
DOS
these many valleys. This is desired to remain low, because the additional scattering could negate the increase
in carrier density and conductivity that large N brings [120,121] . In many TE materials, the dominant scattering
V
mechanisms are polar optical phonons and ionized impurity scattering (IIS), since TE materials are heavily
doped and many are polar. Fortunately, both scattering mechanisms are anisotropic and their strength
decays fast with the distance between the valleys in momentum space. Thus, ideally, the optimal TE material
electronic structure will consist of: (i) many bands and valleys per band packed closely in energy, (ii) but as
further apart as possible in the k-space of the Brillouin zone [118,122] (a definition of N as the average number
v
[118]
of valleys per band captures some of this ), (iii) with light conductivity effective masses, and (iv) large
dielectric constant values for enhanced screening and reduction of Coulomb and polar phonon scattering.
Such behavior is typically encountered, for example, in most of the materials that exhibit large PFs, such as
half-Heusler materials (primarily p-type), chalcogenides, some Zintl phases, silicides, and others. Note that
at high carrier densities exceeding 10 cm , screening effects are typically strong and reduce the strength of
20
-3
polar optical phonons. Typically, screening does not become strong enough to do the same for IIS, since IIS
also increases with the density of ionized dopants, which is equal to the carrier density. On the other hand,
these are anisotropic (small angle) scattering mechanisms, and at elevated densities and Fermi levels, where
the Fermi surfaces are large, they only scatter carriers locally in the Brillouin zone, according to the DOS in
the vicinity of the initial state, thus not proportionally to the entire DOS. This could make their relative
strength reduced compared to non-polar optical phonon scattering, for example, especially at higher
temperatures. In other materials such as PbTe with ultra-high dielectric constants, the IIS strength can be
diminished, since the scattering strength is inversely proportional to the square of the dielectric constant .
[123]
Another band structure-related feature that allows for large PFs in complex materials is the presence of
complex shaped, elongated energy surfaces in the electronic structure (see Figure 4). Compared to isotropic
spherical, or even ellipsoidal bands, elongated, non-parabolic and highly anisotropic features can deliver
higher PFs as they can provide simultaneously light effective masses for large electrical conductivity and
heavy masses for enhanced Seebeck coefficient. In the limiting case, largely elongated tube-like bands of
seemingly lower dimensionality can also provide possibilities for PF improvements and are actively
investigated.

Band alignment materials engineering
For materials with many valleys, but misaligned in energy, significant effort is devoted to properly aligning
these valleys using iso-electric alloying (referred to as band alignment or convergence), with very promising
[124]
results across materials . This is a major band engineering optimization direction currently undertaken
(see Figure 4). Many promising examples of materials whose PF benefits upon band alignment can be found
in the literature with 10%-50% PF improvements . An effective strategy to achieve band convergence at
[120]
proper compositions is to use compounds in solid solutions with different band ordering. One example is
the solid solution of Mg X, where X can be Si or Sn. These materials present a conduction band with a dual-
2
band electronic structure (with one heavy and another light band) with an inverted band order for Mg Si
2
and Mg Sn. A mixing of the two compounds, around composition Mg Si Sn , makes the band edges of
0.35
0.65
2
2
the light and heavy conduction bands coincide . It is very popular to use solid solutions in
[125]
thermoelectricity to reduce the lattice thermal conductivity, but equally important is to examine the band