Does low-dimensional (pipe-like) Fermi-surface topology raise the net thermoelectric figure of merit of real bulk compounds under realistic scattering and thermal conductivity?

Parker, Chen and Singh [Phys. Rev. Lett. 110, 146601 (2013)] showed, within a parabolic-band Boltzmann model under the constant scattering time approximation (CSTA), that a bulk cubic semiconductor whose valence-band constant-energy surfaces form low-dimensional 'pipes' connecting the L-point pockets can roughly double the Seebeck coefficient and multiply the power factor by up to 2.5 relative to a sphere of equal radial mass, and suggested searching for new thermoelectrics among such compounds (naming SnTe). Because the model omitted thermal conductivity, energy-dependent scattering, and realistic non-parabolic multi-band structure, it remains open whether this low-dimensional band topology yields a genuine net increase of the thermoelectric figure of merit $ZT$ in real materials. The problem is to settle this question — by first-principles identification of bulk compounds with robust pipe-like (or other low-dimensional) constant-energy surfaces, beyond-CSTA transport calculations of $\sigma$, $S$, and $\kappa_{electronic}$ together with the lattice contribution $\kappa_{phonon}$.

Background

The thermoelectric figure of merit is $ZT = \sigma S^2 T / \kappa$, where $\sigma$ is the electrical conductivity, $S$ the Seebeck coefficient (thermopower), $T$ the absolute temperature, and $\kappa$ the total thermal conductivity (electronic plus lattice). Good thermoelectrics need large $S$, large $\sigma$, and small $\kappa$ simultaneously, and these requirements conflict: within Boltzmann transport theory — the standard semiclassical treatment in which the transport coefficients are energy integrals of the transport distribution $\sigma(E) \propto N(E) v^2(E) \tau(E)$ weighted by the derivative of the Fermi function — high conductivity favors large Fermi energy $E_F$, while Mott's formula $S \propto d\ln\sigma/dE|_{E_F}$ makes thermopower large near a band edge where $E_F$ is small. Hicks and Dresselhaus [Phys. Rev. B 47, 16631 (1993)] proposed quantum wells and wires as a resolution: in low dimensionality the density of states sets in faster with energy, raising $S$ at fixed carrier concentration. Superlattice implementations, however, suffer from parasitic heat conduction in barrier layers and anisotropic performance. Parker, Chen, and Singh [Phys. Rev. Lett. 110, 146601 (2013)] observed that a three-dimensional bulk crystal can host an effectively low-dimensional electronic structure without being low-dimensional itself: in the rocksalt lead and tin chalcogenides (PbTe, PbSe, PbS, SnTe), first-principles calculations show the valence-band L-point pockets (L being the zone-boundary point of the face-centered-cubic Brillouin zone along $\langle111\rangle$) connecting near, but not at, the valence-band maximum into tube-like ('pipe') constant-energy surfaces. In a cubic network of pipes along the $k_x$, $k_y$, $k_z$ directions, transport is isotropic yet 2D-like, since for a current along a given direction the pipes parallel to that direction behave like the stacking direction of a superlattice and do not contribute to the conductivity, while the pipes along the other directions contribute as in-plane. Comparing an idealized 12-fold-degenerate cylindrical Fermi surface against a sphere of equal radial mass within the constant scattering time approximation (CSTA — the assumption that the carrier relaxation time $\tau$ is energy-independent), they found the 2D-like case nearly doubles the thermopower at heavy doping (consistent with their analytic result $S_{2D}/S_{3D} = (\pi/p)^{1/3}$ in carrier concentration $p$ per unit cell) and raises the maximum power factor $S^2\sigma$ by roughly 2.5, at a conductivity cost of only one third. They explicitly framed this as an idealization (real pipes reconnect at the L-point pockets; real bands are non-parabolic), computed no thermal conductivity and hence no $ZT$, and suggested searching for new thermoelectric materials among compounds with low-dimensional electronic structures, naming SnTe as a candidate. Valley degeneracy — the number of symmetry-equivalent band extrema ('valleys') — is a distinct, established booster of thermoelectric performance, and disentangling topology from degeneracy is central to this problem, since in the chalcogenides the pipes appear together with high valley degeneracy (4 L-point and 12 off-axis $\Sigma$-line valleys).

Problem Statement

Determine whether low-dimensional Fermi-surface topology — pipe-like (approximately cylindrical) constant-energy surfaces embedded in a three-dimensional bulk crystal, as in the valence bands of the rocksalt chalcogenides where L-point pockets connect near the band maximum — genuinely enhances the net thermoelectric figure of merit $ZT = \sigma S^2 T / \kappa$ of real materials, or whether the power-factor enhancement established by Parker, Chen, and Singh within the constant scattering time approximation is an artifact of that idealization. Settling this question entails the following linked components: (1) identify, from first principles, bulk compounds whose band structures produce robust pipe-like or other low-dimensional constant-energy surfaces at practically relevant doping and temperature; (2) compute $\sigma$, $S$, and $\kappa_{electronic}$ for such a compound with realistic energy-dependent carrier scattering (electron–phonon and defect mechanisms) and full multi-band, non-parabolic band structure, going beyond the constant scattering time approximation; (3) include the lattice thermal conductivity $\kappa_{phonon}$, computed or measured for the same compound, so that the net $ZT$ — not merely the power factor — is established; and (4) compare against a control with matched valley degeneracy, effective mass, and carrier concentration but three-dimensional (spherical/ellipsoidal) Fermi-surface topology, so that any $ZT$ difference is attributable to topology itself. The problem is resolved by a demonstrated net-$ZT$ enhancement for at least one real bulk compound with a pipe-like band structure under this realistic treatment, not by model power factors alone; establishing the negative branch — that the CSTA power-factor enhancement is an artifact of the idealization — correspondingly requires controlled evidence that the enhancement fails to translate into a net $ZT$ gain for such compounds generally, rather than its absence in a single compound.

References

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