Lone Pair Rotation and Bond Heterogeneity Leading to Ultralow Thermal Conductivity in Aikinite

Understanding the relationship between the crystal structure, chemical bonding, and lattice dynamics is crucial for the design of materials with low thermal conductivities, which are essential in fields as diverse as thermoelectrics, thermal barrier coatings, and optoelectronics. The bismuthinite-aikinite series, Cu1–x□xPb1–xBi1+xS3 (0 ≤ x ≤ 1, where □ represents a vacancy), has recently emerged as a family of n-type semiconductors with exceptionally low lattice thermal conductivities. We present a detailed investigation of the structure, electronic properties, and the vibrational spectrum of aikinite, CuPbBiS3 (x = 0), in order to elucidate the origin of its ultralow thermal conductivity (0.48 W m–1 K–1 at 573 K), which is close to the calculated minimum for amorphous and disordered materials, despite its polycrystalline nature. Inelastic neutron scattering data reveal an anharmonic optical phonon mode at ca. 30 cm–1, attributed mainly to the motion of Pb2+ cations. Analysis of neutron diffraction data, together with ab-initio molecular dynamics simulations, shows that the Pb2+ lone pairs are rotating and that, with increasing temperature, Cu+ and Pb2+ cations, which are separated at distances of ca. 3.3 Å, exhibit significantly larger displacements from their equilibrium positions than Bi3+ cations. In addition to bond heterogeneity, a temperature-dependent interaction between Cu+ and the rotating Pb2+ lone pair is a key contributor to the scattering effects that lower the thermal conductivity in aikinite. This work demonstrates that coupling of rotating lone pairs and the vibrational motion is an effective mechanism to achieve ultralow thermal conductivity in crystalline materials.


Polyhedral distortions:
The bond angle variance (σ 2 ) is calculated using the relation, 2 , the bond length distortion (distortion index) as defined by Baur 1 as, and the quadratic elongation as < >= 1 ∑ ( ) =1 , which gives a quantitative measure of polyhedral distortion.
The subscript "poly" refers to the polyhedron type (octahedron or tetrahedron), with angles θpoly as 90° and 109.47° respectively. θi refers to the ith bond angle; the summation is taken over the m angles of the polyhedron or the n vertexes of the polyhedron, li is the distance from the central atom to the ith coordinating atom, lpoly is the centre-to-vertex distance of a regular polyhedron of same volume, and lav is the average bond length. θpoly in the bond angle variance and lpoly in the quadratic elongation calculation could be defined only for a regular polyhedron and hence could not be calculated for the PbS7 capped octahedron. For an undistorted polyhedron, the bond angle variance and distortion index would be 0 whereas the quadratic elongation would be 1.
The number of atoms coordinated to a central atom in a coordination polyhedron is defined as the effective coordination number (ECoN) and is ideally 4, 6 and 7 for a tetrahedron, octahedron and capped octahedron respectively. ECoN was obtained using the formulation of Robinson et.al. 2 and Hoppe 3 as implemented in VESTA where the surrounding atoms to a central atom in a coordination polyhedron are given a weighting scheme with numbers between 0 and 1; as the distance from the central atom to a surrounding atom increases, this number gets closer to zero.

Sound velocity measurements:
The following expressions were used to extract elastic parameters, the Debye temperature and the Grüneisen parameter from the sound velocity measurements.
Average velocity Poisson ratio

Grüneisen parameter
Young's modulus Where N is the number of atoms in the unit cell (N = 24), V is the volume of the unit cell as obtained from refinement (534.250(4) Å 3 ), and ρ is the Archimedes' density 6.921 g cm -3 ; .
Within the quasi-harmonic approximation, we compute the mode resolved Grüneisen parameter for the wave vector q and the phonon branch j and its contribution by each atomic species by taking the derivative of the dynamical matrix with respect to the volume as explained by Siloi et al. 4 . In order to have a complete description of the system's equilibrium, thermodynamical potential functions such as Helmholtz free energy F, internal energy E, entropy S and specific heat Cv at zero pressure are obtained using the calculated phonon density of states employing the quasi-harmonic approximation. The following equations have been used to calculate F; E; S; and Cv: 5,6 where is the Boltzmann constant, n is the number of atoms per cell, N is the number of cells, is the phonon frequency, is the cut-off phonon frequency, and ( ) is the normalized phonon density of states.

Calculation of the minimum thermal conductivity:
Considering the Cahill-Watson-Pohl (CWP) model, where the transport of thermal energy within a material takes place via a random walk to the nearest neighbour of a localized Einstein oscillator, the minimum thermal conductivity at high temperature can be approximated as, Using this relation, the minimum thermal conductivity is ca. 0.414 W m -1 K -1 .
Based on the Allen-Feldman theory, 7,8 the minimum thermal conductivity (diffusive thermal conductivity) at high temperature can be approximated as, Using this relation, the minimum thermal conductivity is ca. 0.260 W m -1 K -1 . Figure S1. Computed Helmholtz free energy F, internal energy E, entropy S and specific heat Cv at zero pressure. Refined parameters from Rietveld refinement of powder neutron diffraction: