Vacancy-induced dislocations within grains for high-performance PbSe thermoelectrics

To minimize the lattice thermal conductivity in thermoelectrics, strategies typically focus on the scattering of low-frequency phonons by interfaces and high-frequency phonons by point defects. In addition, scattering of mid-frequency phonons by dense dislocations, localized at the grain boundaries, has been shown to reduce the lattice thermal conductivity and improve the thermoelectric performance. Here we propose a vacancy engineering strategy to create dense dislocations in the grains. In Pb1−xSb2x/3Se solid solutions, cation vacancies are intentionally introduced, where after thermal annealing the vacancies can annihilate through a number of mechanisms creating the desired dislocations homogeneously distributed within the grains. This leads to a lattice thermal conductivity as low as 0.4 Wm−1 K−1 and a high thermoelectric figure of merit, which can be explained by a dislocation scattering model. The vacancy engineering strategy used here should be equally applicable for solid solution thermoelectrics and provides a strategy for improving zT.

Predicted mean-free path dependent accumulative lattice thermal conductivity for PbSe (a), and the predicted frequency dependent accumulative reduction in the lattice thermal conductivity for Pb 0.95 Sb 0.33 Se due to point defects and/or dislocations (b). The modeling is based on a Born-von Karman approximation and the predictions are for 300 K.

Supplementary Tables
Supplementary Table 1. Equations for phonon relaxation times (τ) associated with different types of scattering processes, where τ U , τ N , τ PD , τ DC and τ DS are the relaxation times due to the scattering of Umklapp processes, Normal processes, point defects, dislocation cores and dislocation strains, respectively.

Supplementary Discussion
To better understand the mean free path and the frequency dependent lattice thermal conductivity (κ L ) accumulation, it is believed to be more precise if taking the effect of reduced phonon group velocity at high phonon energies into account 9,10 . This leads the Born-von Karman 11 dispersion relationship to be more reliable than that of Debye model. This improvement has been adopted to understand the lattice thermal conductivity of bulk and low-dimensional thermoelectrics or metals including Si-Ge 12 and PbTe 13 , silver 14 and Al-Si etc. 15 . Using the same method, we modeled the phonon transport for PbSe, based on a Born-von-Karman type phonon dispersion of ω = 2vq/πSin(2q/q c π) rather than the Debye type assuming ω = vq, where ω is the phonon frequency, v is the sound velocity, q is the wave vector and q c is the cut-off wave vector. The predicted accumulative κ L due to Umklapp and Normal scattering in pure PbSe, helps us understand the important range of mean free path that contributes to heat conduction. Furthermore, the predicted phonons frequency dependent accumulative reduction in κ L distinguishes the effect of each scattering mechanism.
As shown in the Supplementary Fig. 4a, 50% of the heat in pure PbSe is carried by phonon of mean-free path up to 13 nm and the central 80% heat is carried by mean-free path (MFP) between 4 nm and 400 nm at 300 K. As compared with the available prediction by first-principles calculations 16 , which includes the contributions of optical phonons (with even shorter MFPs) and results in a higher lattice thermal conductivity, the current model prediction shows a very good agreement on the normalized κ L -accumulation within the overlapped range of MFP. The randomly distributed dense in-grain dislocations here roughly enable a range of mean-free path to be achieved for reducing the lattice thermal conductivity by 50% at the 300 K (Fig. 3a). It is shown that dense in-grain dislocations, indeed lead to an effective scattering of phonons with mid-frequencies and therefore a significantly reduced lattice thermal conductivity ( Supplementary Fig. 4b).
The mechanical property measurements were carried out by modified small punch (MSP) technique 17 , a method has been successfully used to characterize the mechanical strength of thermoelectric materials 18,19 . For the MSP measurements, the disk sample was supported by a die with a center hole of 3.93 mm in diameter, and was punched by a cylindrical pressure head of 2.35 mm in diameter with a speed of 0.05 mm min -1 . All of the specimens were fine polished and the load was monitored by a high-accuracy transducer. The MSP strength σ MSP can be calculated via: σ MSP = 3P max /(2πt 2 )[1-(1-ν 2 )/4×b 2 /a 2 +(1+ ν)ln(a/b)], where P max is the measured load at failure, t is the thickness of the sample, ν is the Poisson's ratio which is estimated as 0.243 2 for PbSe-based materials and as 0.218 2 for PbTe-based materials, a is radius of the center hole and b is the radius of pressure head, respectively.
The mechanical strength is obtained by averaging 3~5 samples for each composition, and the results for Pb 1-x Sb 2x/3 Se, PbSe and PbTe are shown in Supplementary Table 4. One may claim that the mechanical strength of Pb 1-x Sb 2x/3 Se decreases a little with increasing density of dislocations (increasing x), however, the strength for samples with dense dislocations is still comparable to that of PbTe without dislocations. Therefore, dense in-grain dislocations here do not degrade the mechanical strength to be unacceptable.