Suppressed Lone Pair Electrons Explain Unconventional Rise of Lattice Thermal Conductivity in Defective Crystalline Solids

Abstract Manipulating thermal properties of materials can be interpreted as the control of how vibrations of atoms (known as phonons) scatter in a crystal lattice. Compared to a perfect crystal, crystalline solids with defects are expected to have shorter phonon mean free paths caused by point defect scattering, leading to lower lattice thermal conductivities than those without defects. While this is true in many cases, alloying can increase the phonon mean free path in the Cd‐doped AgSnSbSe3 system to increase the lattice thermal conductivity from 0.65 to 1.05 W m−1 K−1 by replacing 18% of the Sb sites with Cd. It is found that the presence of lone pair electrons leads to the off‐centering of cations from the centrosymmetric position of a cubic lattice. X‐ray pair distribution function analysis reveals that this structural distortion is relieved when the electronic configuration of the dopant element cannot produce lone pair electrons. Furthermore, a decrease in the Grüneisen parameter with doping is experimentally confirmed, establishing a relationship between the stereochemical activity of lone pair electrons and the lattice anharmonicity. The observed “harmonic” behavior with doping suggests that lone pair electrons must be preserved to effectively suppress phonon transport in these systems.


Calculation of theoretical lattice thermal conductivity
The theoretical lattice thermal conductivity is calculated based on the modified Debye-Callaway model: ,where  is the average sound velocity and   is Debye temperature of AgSbSnSe3.For determination of relaxation time, Umklapp process, normal process, and grain boundary scattering is considered in the calculation.The relaxation time by Umklapp and normal process is given as follows: Sound velocities were experimentally determined from ultrasound measurements.Experimental Grüneisen parameter measured in this work was used for the calculation.Debye temperature was estimated from elastic properties by the equation , where ℎ is Planck constant,   is Boltzmann constant,  is the number of atoms in the unit cell,  is Avogadro's number, ρ is the bulk density,  is the molecular weight, and  is the average sound velocity (J.Phys.Chem.Solids 24, pp.909-917 (1963)).The relaxation time by grain boundary and point defect scattering is given as follows: The scattering parameter for point defect is calculated according to the previously reported method [Appl.Phys. Lett. 94, 153101 (2009)].However, we noticed that including point defect scattering process significantly underestimate the theoretical lattice thermal conductivity of Cd-doped AgSbSnSe3.

Estimation of lattice thermal conductivity of alloys using Klemens model
,where  0 is the lattice thermal conductivity of the pure solid without defects and  is the disorder parameter.The disorder parameter is determined from the elastic properties of the pure material and point defect scattering parameter as follows: , where  is the average volume per atom,   is the Boltzmann constant,   is the average sound velocity (1686 m/s), and  is the point defect scattering parameter. is estimated from the two principal components: mass fluctuation (  ) and strain field fluctuation (  ), which is given as follows: , where 〈 ̅ 〉 is the average atomic mass, 〈∆ ̅̅̅̅̅ 〉 is the average atomic mass variation, 〈 ̅ 〉 is the average atomic radius of the site, 〈∆ ̅̅̅̅ 〉 is the average atomic radius variation, and  is the phenomenological parameter for the strain fluctuation that typically varies from 1 to 500 to fit the experimental data.

Notes on the effect of intrinsic point defects (VAg and AgSb)
It can be seen that some intrinsic defects (e.g., VAg or AgSb) are observed that have lower formation energy than that of CdSb.However, given that these intrinsic point defects are present regardless of the Cd content, their effect on the lattice thermal conductivity should be present regardless of Cd doping, while the effect of VAg and AgSb is not immediately clear.Supplementary Table 4|

Supplementary
Figure 9| Temperature-dependent thermoelectric properties.Temperaturedependent (a) Seebeck coefficient, (b) electrical conducticity, (c) total thermal conductivity, (d) lattice thermal conductivity of AgSb1-xCdxSnSe3.Measured properties of pristine AgSbSnSe3 are consistent with those of the reference.The thermally activated charge transport observed in (b) near 450 K may originate from weak dielectric screening at the grain boundaries.The positive slope of electrical conductivity at 600 K is presumed to be the signature of cation disordering observed in ABX2-SnSe thermoelectrics.Supplementary Figure 10| Temperature-dependent XRD spectra for AgSb1-xCdxSnSe3.The temperature ranges from 298 K to 498 K with step interval of 25 K.

Furthermore
, suppose VAg formation becomes more pronounced with Cd doping.As we already discussed in the main text, the formation of VAg would not affect the LPE concentration considering the electron configuration of Ag.However, the increased point defect scattering due to mass fluctuation (zero mass versus matrix) would reduce the lattice thermal conductivity, which cannot explain the experimentally observed rise of the lattice thermal conductivity.We might exclude the possibility of the formation of AgSb as well to see the decrease in the lattice parameter.The ionic radius of Ag + and Sb 3+ is 115 pm and 76 pm, respectively.If the antisite defect is dominating in AgSnSbSe3, we should observe an increase in the lattice parameter.Therefore, neither VAg nor AgSb are significantly affecting the properties of Cd-doped AgSbSnSe3.

Table 2| Extrinsic chemical potentials for different equilibrium phases.
Extrinsicchemical potentials (Δμi) of each element in all phase regions in the Ag-Sb-Se-Cd chemical space where AgSbSe2 is stable.