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approaches such point). However, improving the zT further requires improvements to the PF as well, on top
of retaining the nanostructured material geometry, for which targeting the roadblock of the adverse
interdependence of the electrical conductivity and Seebeck coefficient is encountered. In this regard,
directions such as energy filtering and band engineering are shown to be very powerful tools, enabled by
developments in the control of nanostructure and alloy synthesis. This was also driven by developments in
theoretical and computational tools that enhanced our understanding of transport in complex materials and
how chemical bonding affects electronic structure features [13,56] .

Importantly, in some of the zT record-breaking works mentioned above, care was taken such that the
insertion of nanoinclusions does not degrade the PF significantly. Typically, nanostructuring reduces the
electrical conductivity and increases the Seebeck coefficient slightly, but the reduction in conductivity has a
[57]
stronger influence on the PF , which is also reduced, albeit at a smaller degree compared to the thermal
conductivity. Some of the ways this PF reduction is mitigated are by aligning the band edges of the
constituent material phases such that electronic transport is not noticeably interrupted [17,38,39] alloying using
iso-valent substitutions in the lattice (which minimize electron-alloy scattering) using atoms with different
[17]
atomic weight, like Sr instead of Pb in lead tellurides , or Zr and Hf instead of Ti in TiNiSn half-Heusler
compounds [58,59] ; introducing energy filtering through potential barriers which improve the Seebeck
[60]
coefficient ; through charge transfer from doped islands (modulation doping) , or doped interstitial
[57]
[61]
voids in the crystal matrix, which improves conductivity without degrading the mobility, but still
reducing the thermal conductivity, etc.
USING NANOSTRUCTURING TO IMPROVE THE POWER FACTOR
Developments in energy filtering
From the above strategies to mitigate the reduction by nanostructuring on the PF, energy filtering is a
common underlying cause when PF improvements are encountered. It is typically achieved by the
introduction of potential barriers that form at the interfaces between different materials phases, at the grain
boundaries and material discontinuities, in the presence of nanoscale defects, etc. These barriers allow high-
energy electrons to propagate more easily, while blocking low-energy electrons, thus directly increasing the
Seebeck coefficient (see Figure 3). Of course, potential barriers degrade the electrical conductivity. However,
design "sweet spots" can be found for which the PF is improved, but even so not significantly, only at the
order of 30% [57,62-65] . Nevertheless, a surge in efforts to use energy filtering and design the grain/grain-
boundary system efficiently in a variety of materials has recently emerged [66-69] .

A few experimental designs, however, backed by theoretical calculations, have demonstrated extraordinarily
high PFs in Si-based nanocrystalline materials. Specifically, a simultaneous improvement in both the
electrical conductivity and the Seebeck coefficient in p-type Si layers was achieved, which led to PF
[68-73]
improvements in various instances from 2×-10× . For this to be realized, certain design “ingredients”
needed to be fit in place: (i) the presence of energy filtering barriers between the nanograins, (ii) the size of
the nanograins had to be a few 10 s of nanometers, (iii) ultra-high doping up to the levels of solid solubility
had to be used, and finally, and (iv) the materials had to be annealed at high temperatures to allow
precipitation of the dopants at the grain boundaries. Compared to bulk Si, whose maximum room
-1
-1
-2
temperature PF is approximately ~4-5 mW·m ·K for p-type and ~5 mW·m ·K for n-type , values of
[74]
-2
~6.5 mW·m ·K were reported for n-type films , ~15 mW·m ·K for ultra-highly doped p-type
-1
[72]
-2
-2
-1
-1
nanocrystalline Si films with grain sizes ~30 nm , ~22 mW·m ·K in the latter after the additional
-2
[70]
inclusion of nanopores , and up to ~33 mW·m ·K in such ultra-highly doped nanocrystalline films after
[71]
-2
-1
they had undergone a dehydrogenated process .
[73]