Crossover from Boltzmann to Wigner thermal transport in thermoelectric skutterudites

Skutterudites are crystals with a cagelike structure that can be augmented with filler atoms (“rattlers”), usually leading to a reduction in thermal conductivity that can be exploited for thermoelectric applications. Here, we leverage the recently introduced Wigner formulation of thermal transport to elucidate the microscopic physics underlying heat conduction in skutterudites, showing that filler atoms can drive a crossover from the Boltzmann to the Wigner regimes of thermal transport, i.e., from particlelike conduction to wavelike tunneling. At temperatures where the thermoelectric efficiency of skutterudites is largest, wavelike tunneling can become comparable to particlelike propagation. We define a Boltzmann deviation descriptor able to differentiate the two regimes and relate the competition between the two mechanisms to the materials' chemistry, providing a design strategy to select rattlers and identify optimal compositions for thermoelectric applications.

Figure 1

Calculated ${\kappa }_{\mathrm{tot}}$ (solid), ${\kappa }_{\mathrm{P}}$ (dotted), and ${\kappa }_{\mathrm{C}}$ (dashed) for unfilled ${\mathrm{FeSb}}_3$, ${\mathrm{CoSb}}_3$, and ${\mathrm{IrSb}}_3$ and for their related filled compounds ${\mathrm{RFe}}_4{\mathrm{Sb}}_{12}$ (R = Ba, Ca, Nd, Yb), ${\mathrm{RCo}}_4{\mathrm{Sb}}_{12}$ (R = I, In, La, Yb), and ${\mathrm{RIr}}_4{\mathrm{Sb}}_{12}$ (R = Yb), between 100 and 800 K. Experimental ${\kappa }_{\mathrm{tot}}$ are taken from Ref. [34] for ${\mathrm{RFe}}_4{\mathrm{Sb}}_{12}$ (R = Ba, Ca, Nd, Yb), Refs. [35, 36] for ${\mathrm{CoSb}}_3$, Refs. [37, 38] for ${\mathrm{ICo}}_4{\mathrm{Sb}}_{12}$, Ref. [39] for ${\mathrm{InCo}}_4{\mathrm{Sb}}_{12}$, Refs. [40, 41] for ${\mathrm{LaCo}}_4{\mathrm{Sb}}_{12}$, Refs. [42, 43] for ${\mathrm{YbCo}}_4{\mathrm{Sb}}_{12}$, and Refs. [44, 45] for ${\mathrm{IrSb}}_3$. Symbols represents different experimental measurements, also color-coded according to the material. Experimental data referring to partial concentration of the filler are specified adjacent to the symbols. The universal $T^{-1}$ trend of ${\kappa }_{\mathrm{P}}$ [16, 46] is also shown; ${\kappa }_{\mathrm{tot}}$ displays a milder decay than ${\kappa }_{\mathrm{P}}$.

Figure 2

Distribution of phonon lifetimes $\tau {\left(\mathit{q}\right)}_s=\mathrm{\Gamma }{\left(\mathit{q}\right)}_s^{-1}$ as a function of energy $\hslash \omega {\left(\mathit{q}\right)}_s$ for unfilled (upper panels) and Yb-filled (lower panels) skutterudites at 800 K. The area of each scatter point is proportional to the contribution to the total lattice thermal conductivity and colored according to the origin of the contribution: $c=[\overline{{\kappa }_{\mathrm{P}}}{\left(\mathit{q}\right)}_s-\overline{{\kappa }_{\mathrm{C}}}{\left(\mathit{q}\right)}_s]/[\overline{{\kappa }_{\mathrm{P}}}{\left(\mathit{q}\right)}_s+\overline{{\kappa }_{\mathrm{C}}}{\left(\mathit{q}\right)}_s]$, where particlelike is green ($\mathrm{c}=1$), wavelike is blue ($\mathrm{c}=-1$) and intermediate mechanisms have intermediate colors, with red corresponding to 50% of particlelike and 50% of wavelike contributions [7]. The Wigner limit in time (dashed-black line) corresponds to a phonon lifetime equal to the inverse of the average interband spacing (${\tau }^{\omega }=\mathrm{\Delta }{\omega }_{\mathrm{avg}}^{-1}$). The dashed-purple hyperbola shows the Ioffe-Regel limit in time [7, 28] (${\tau }^{\mathrm{IR}}={\omega }^{-1}$), below which phonons are no longer well-defined quasi-particles. The pie charts have an area proportional to the total lattice thermal conductivity, and the slices resolve the particlelike conductivity (green) and the wavelike conductivity (blue). The black symbols connected by black lines are the points whose coordinates are the average energies and lifetimes projected on atoms [see Eqs. (A3) and (A4)]. Open (closed) symbols refer to unfilled (filled) skutterudites. The projections on the filler, transition-metal and antimony atoms are given by crosses, triangles and squares, respectively. The phonon lifetimes distribution for the remaining filled skutterudites are given in Appendix pp6-s2.

Figure 3

(Upper panel) Relation between the relative strength of particlelike and wavelike conduction, $\frac{\overline{{\kappa }_{\mathrm{P}}}}{\overline{{\kappa }_{\mathrm{C}}}}$, at 800 K and $B$ as given in Eq. (3); unfilled (filled) symbols represent unfilled (filled) skutterudites. Circles, triangles and squares represent Fe-, Co- and Ir-based skutterudites, respectively. The black dashed line correspond to the $\frac{\overline{{\kappa }_{\mathrm{P}}}}{\overline{{\kappa }_{\mathrm{C}}}}=B^{-1}$ power law interpolating the data [see Eq. (B7)]. The shaded regions show a smooth crossover from a dominant particlelike heat conduction in green, to a competing particle- and wavelike mechanism in red. The burgundy cross represents the correlation for ${\mathrm{YbFe}}_4{\mathrm{Sb}}_{12}$ at 300 K. (Lower panel) Relation between the relative strength of particlelike and wavelike conduction ($\frac{\overline{{\kappa }_{\mathrm{P}}}}{\overline{{\kappa }_{\mathrm{C}}}}$) at 800 K and the descriptor $\eta$ given in Eq. (4). The linear correlation between $\eta$ and $B$ shows how skutterudites' chemistry determines the degree of Wigner thermal transport in these materials. The color scale goes from red (Wigner regime) to green (Boltzmann regime).

Figure 4

PDOS of ${\mathrm{YbFe}}_4{\mathrm{Sb}}_{12}$.

Figure 5

Convergence of the lattice thermal conductivity of ${\mathrm{YbFe}}_4{\mathrm{Sb}}_{12}$ with respect to the number of input atomic configurations and the third- and second-order cutoffs of interaction used for the hiPhive calculation and with respect to the third-order cutoff of interaction used for the SHENGBTE calculation. The first column of the legend refers to the calculations performed with hiPhive, in fact both the number of input structures (atomic positions configurations randomly generated) and the cutoffs of interactions are specified. The second column instead contains the calculations carried out with SHENGBTE and the experimental reference data. Note that all the calculations were carried out using the second-order force constants obtained through Quantum ESPRESSO DFPT calculations. For hiPhive calculations the first value of the cutoffs is the one of the second order of interaction while the second is the one of the third order of interaction. Instead, for SHENGBTE calculations, the only cutoff value that appears is clearly the one of the third order of interaction. The cutoff values are measured in angstroms (Å). The experimental values are taken from Ref. [34].

Figure 6

Convergence of the lattice thermal conductivity of a generic skutterudite with respect to the number of input configurations and the cutoff values of the second and third order of interaction.

Figure 7

Comparison of D3Q with smearing method and Phono3py with tetrahedron method calculations in ${\mathrm{YbFe}}_4{\mathrm{Sb}}_{12}$. The used smearing value used is measured in ${\mathrm{Wm}}^{-1}{\mathrm{K}}^{-1}$ and the $k$-points grid has also been specified. The experimental values are taken from Ref. [34].

Figure 8

Total lattice thermal conductivity (solid), particlelike contribution (dotted), and wavelike contribution (dashed) of ${\mathrm{YbFe}}_4{\mathrm{Sb}}_{12}$ obtained from DFT calculations on top of hiPhive displaced atomic position configurations (green) for both second- and third-order interatomic force constants, PHonon DFPT calculations for second-order force constants, and DFT calculations on top of hiPhive displaced atomic position configurations for third-order force constants (blue) and $\mathrm{DFT}+\mathrm{U}$ calculations on top of hiPhive displaced atomic position configurations (red) for both second- and third-order interatomic force constants. Here “2” and “3” superscripts refer to the order of the interatomic force constants. A Hubbard parameter U = 2.5 eV was used for the $\mathrm{DFT}+\mathrm{U}$ calculations. Note that the $\mathrm{PBE}+\mathrm{U}+$ hiPhive calculation derives from the related antiferromagnetic electronic state with lowest energy, while the $\mathrm{PBE}+$ Quantum ESPRESSO DFPT and $\mathrm{PBE}+$ hiPhive ones derive from the related nonmagnetic electronic state with lowest energy.

Figure 9

Phonon dispersions of nonmagnetic ${\mathrm{YbFe}}_4{\mathrm{Sb}}_{12}$ obtained using $\mathrm{PBE}+$Quantum ESPRESSO DFPT (blue), antiferromagnetic ${\mathrm{YbFe}}_4{\mathrm{Sb}}_{12}$ with $\mathrm{PBE}+\mathrm{U}+$hiPhive (red) and nonmagnetic ${\mathrm{YbFe}}_4{\mathrm{Sb}}_{12}$ with $\mathrm{PBE}+$hiPhive (green).

Figure 10

Total lattice thermal conductivity of ${\mathrm{BaFe}}_4{\mathrm{Sb}}_{12}$ obtained by substituting the atomic weight of Ba with that of Yb (solid green), by substituting the atomic weight of Ba with that of Ca (dotted red) and by using the actual atomic weight of Ba (dashed blue).

Figure 11

Phonon dispersions of the skutterudites studied. Note that the phonon dispersion of ${\mathrm{FeSb}}_3$ comes from DFPT calculations carried out on top of the antiferromagnetic electronic ground state, while for the other materials the nonmagnetic ground state was considered.

Figure 12

Phonon lifetimes distribution $\tau {\left(\mathit{q}\right)}_s=\frac{1}{\mathrm{\Gamma }{\left(\mathit{q}\right)}_s}$ as a function of the energy $\hslash \omega {\left(\mathit{q}\right)}_s$ for the skutterudites studied at 800 K.

Figure 13

Distribution of phonon lifetimes $\tau {\left(\mathit{q}\right)}_s=\mathrm{\Gamma }{\left(\mathit{q}\right)}_s^{-1}$ as a function of energy $\hslash \omega {\left(\mathit{q}\right)}_s$ for ${\mathrm{YbFe}}_4{\mathrm{Sb}}_{12}$ at 300 K (left), 800 K (center), and 1200 K (right). We see that the wavelike contribution reaches a plateau at 800 K as we do not have further appreciable increase of ${\kappa }_{\mathrm{C}}$ up to 1200 K.