The impact of lone-pair electrons on the lattice thermal conductivity of the thermoelectric compound CuSbS 2

The discovery and design of compounds with intrinsically low thermal conductivity, especially compounds with a special bonding nature and stable crystal structure, is a new direction to broaden the scope of potential thermoelectric (TE) materials. This study revealed unambiguously the origin of the impact of the lone pair electrons on lattice thermal conductivity in Cu – Sb – S compounds by correlating the special bonding on the Sb site with the phonon dispersion spectrum and density of states. By substitution of Sb with the transition metal Fe and group III A element Ga without s 2 electrons, lone-pair electrons on some of the Sb sites were removed, which created a scenario with opposite in ﬂ uences on lattice thermal conductivity from the loss of lone-pair electrons and gain of alloy scattering. We investigated the competition between the alloy phonon scattering and the extra phonon scattering mechanism linked to lone-pair electrons on trivalent Sb 3+ sites in chalcostibite CuSbS 2 , which is a model system for benchmarking and quantifying the impact of lone-pair electrons on the lattice thermal conductivity of Cu – Sb – S compounds. A signi ﬁ cant deviation from the classic alloy model was observed. Along with the impact of the lone-pair electrons on the bonding arrangement and crystal structure, the role of lone-pair electrons in the phonon transport of the TE compound CuSbS 2 was well demonstrated and quanti ﬁ ed. Two Sb-related quasi-single-frequency vibration modes behaving like localised Einstein harmonic oscillators were discovered and correlated with the bonding circumstance around Sb sites. These results give unequivocal evidence that the trivalent V A atom creates special bonding and vibration modes because of its nonbonding 5s lone-pair electrons.


Introduction
The ability to discover and design materials with low lattice thermal conductivity is technologically important for thermoelectric (TE) generators and Peltier coolers, 1,2 which demand TE materials with crystal-like electrical transport properties and glass-like thermal conductivity, an ideal concept (phonon glasselectron crystal, PGEC) coined by Slack. 3 Several strategies have been explored over the last decade to minimize the thermal conductivity of TE materials while delicately avoiding any detrimental effects on electrical properties.Beneting from the difference between the phonon and charge carrier mean-free paths, 4,5 introducing a high density of grain boundaries by embedding nanoscale 'guest domains' or nanopores in a 'host matrix' is an effective way to block/scatter the movement of phonons without serious degradation of electrical conductivity.A series of record high TE gure-of-merits, zT, were achieved in AgPb m SbTe 2+m , 6 BiSbTe, 7,8 nano-microporous AgSbTe 2 (ref.9), skutterudites 10 and other systems by this method. 11,12Another strategy is to use the 'crystal complexity' to decrease the thermal conductivity and enhance the electrical properties. 13Some quantitative discussion has been carried out for the complex disordered system YbB 44 Si 2 (ref.14).For layered materials, it is an effective way to optimise the TE properties by decoupling the interrelated thermal and electrical transport parameters by designing a complex crystal structure with one block exhibiting excellent electrical properties, and another block acting as a phonon scatterer to minimize the thermal transport.Layered materials, including Na-Co-O, 15 TiS 2 (ref.16) and Bi-Cu-O-Se 17 are representative examples of this design strategy.Inserting/ lling rattling atoms into oversized cages/voids is another proven promising method to obtain the ideal phonon-glass thermal conductivity, while still maintaining electron-crystal electrical transport properties along the crystalline frame sublattice or cages. 18,19The vibration of the rattler strongly couples to the frame/cage vibration modes, which lowers the velocity of the phonons and results in low thermal conductivity. 20,21This strategy was well demonstrated both in skutterudites and clathrates. 22,23ecently, Li et al. 24 discovered the link between lattice vibration anharmonicity and electronic orbitals in SnSe 25 and revived the idea of designing materials with large Grüneisen parameters by anharmonicity engineering as proposed by Händel. 26Heremans 27 pointed out that high anharmonicity mostly occurs under conditions that are very close to a collapse of the crystal structure itself, which is veried by the unstable electrical structure and the ferroelectric-like lattice instability aroused by orbital interactions in SnSe. 24Another extreme example supporting Heremans's theory is copper selenide where Se atoms form a rigid cubic lattice with superionic copper ions with liquid-like mobility around them. 28The extraordinary 'liquid-like' nature of copper ions decreases the number of phonon modes and results in an intrinsically very low thermal conductivity in Cu 2Àx Se (around or less than 1 Wm À1 K À1 for Cu 2 Se). 28,29So, the discovery and design of compounds with intrinsically low thermal conductivity is a new direction to broaden the scope of potential TE materials, especially compounds with a special bonding nature and stable crystal structure.
2][33] They proposed that the electrostatic repulsion between the 5s 2 orbital lone-pair electrons on the trivalent Sb atoms and neighbouring chalcogenide ions results in so phonon modes and strong vibrational anharmonicity, which in turn arouse the ultralow thermal conductivity of Cu 3 -SbSe 3 compound. 30,346][37][38] The compounds always show lattice instabilities or different structures to those of the starting rocksalt structure, which result in strong phonon-phonon interactions and ultralow thermal conductivity approaching the amorphous limit. 35In fact, seminal work in 2008 reported I B -V A -(VI A ) 2 compounds with abnormally high Grüneisen parameters and so frequency lattice vibration modes, for example AgSbTe 2 and AgBiSe 2 (ref.34).The phonon mean free path is restricted to the interatomic distance by intrinsic normal and Umklapp phonon-phonon scattering processes alone because of the strong anharmonicity related to the special bonding arrangement.This has motivated research interest in the effect of bonding around trivalent group V A atoms on lattice thermal conductivity in related I B -V A -VI A compounds, containing group I B (Cu, Ag), group V A (P, As, Sb, Bi), and group VI A (S, Se, Te) elements. 39,40Most of the research focused on elucidating the relationship between the bonding arrangement and the low thermal conductivity by comparison of materials incorporating nominally trivalent VI A elements with materials incorporating only V A or III A elements. 30,34,41ctually, there is another way to assess the impact of the lonepair electrons on thermal conductivity by partly substituting the V A atoms by III A atoms without lone-pair electrons.First, this operation would introduce extra point defect phonon scattering and reduce the thermal conductivity based on the alloying model.Simultaneously, the substitution would break the special bonding arrangement related to the lone-pair electrons on some of the trivalent V A atom sites, which would likely restore the thermal conductivity to a typical value for a I B -III A -VI A compound.Since both effects happen on the atomic scale, it is reasonable to evaluate the role of the lone-pair electrons by analysing the doping concentration dependence of thermal conductivity as long as a system is free from the inuences of any other factors.
AgSbTe 2 is a representative example of I B -V A -(VI A ) 2 group compounds, and is a well-studied system with intrinsic minimum thermal conductivity (<0.6-0.7Wm2,43 However, its 'cubic' crystal structure is still a controversial issue. 36Also, the spontaneously generated nanostructure produced by a natural formation of nanoscale domains with different ordering on the cation sublattice plays some role in scattering phonons and achieving a low thermal conductivity. 44ismuth copper oxychalcogenides BiOCuQ (Q ¼ Se, Te) are another hot topic related to the impact of lone-pair electrons on thermal conductivity. 45,46Recently, rst-principles calculations and in situ neutron diffraction analysis have suggested that the low thermal conductivity of those materials may be attributed to the weak bonding of copper atoms within the structure, rather than to the Bi 3+ lone pairs. 47There is a sulphide counterpart (Cu-Sb-S, CAS system) of the CASe system. 395][56] Fig. 1 shows the temperature dependence of the lattice thermal conductivity of all of the four compounds. 41,57Famatinite has the highest value in nearly the whole temperature range, while tetrahedrite and skinnerite exhibit abnormally low and nearly temperatureindependent lattice thermal conductivity.Fig. 2 shows the crystal structures of the members of the CAS system.The Sb atoms in famatinite are tetrahedrally coordinated with four S atoms by sp 3 hybridisation. 58No non-bonded Sb 5s 2 lone-pair elections exist, which results in a relatively high lattice thermal conductivity.In contrast, the other three compounds Fig. 1 Temperature dependence of the lattice thermal conductivity of copper antimony sulphide, including famatinite, tetrahedrite, skinnerite, and chalcostibite. 41,57ave spare Sb 5s 2 electrons free to orient along the missing vertex of their associated polyhedron.Tetrahedrite, which does not have a counterpart in the CASe system, forms a cubic structure, where half of the copper atoms occupy three-fold coordinated CuS 3 trigonal sites.The Sb lone-pair electrons on both sides of the CuS 3 triangular plane weaken the bonding of the Cu atoms in the direction perpendicular to the CuS 3 plane, which leads to quasi-localised and anharmonic out-of-plane rattling modes.This is likely to be the origin of the low thermal conductivity of tetrahedrite. 30,59There are no less than three temperature-dependent polymorphs of Cu 3 SbS 3 (ref.53).All of the structures show mixed character with lone-pair electron bonding arrangement and CuS 3 trigonal bonding units or fractionally occupied Cu sites, including the room temperature polymorph skinnerite.For similar reasons, both tetrahedrite and skinnerite are more thermally insulating than famatinite at room temperature and have nearly constant thermal conductivity with increasing temperature.So, both tetrahedrite and Cu 3 SbS 3 are not ideal compounds to evaluate the effect of lonepair electrons on thermal conductivity because of the inuences of other factors.Chalcostibite forms a stable orthorhombic structure (space group Pnma) until its melting point. 60It has lower thermal conductivity than famatinite but exhibits a similar trend, with intrinsic Umklapp phonon scattering dominating its thermal conductivity behaviour.All the Cu atoms are tetrahedrally coordinated in CuS 4 units.The edgeshared square pyramidal SbS 5 units are separated by CuS 4 units and face one another, which directs the lone-pair electrons into the void separating the SbS 5 units.In contrast to the co-contributions from mixed factors in tetrahedrite and skinnerite, the lone-pair electrons are the solo possible factor accounting for the low thermal conductivity in chalcostibite.So, chalcostibite is an ideal system to assess the role of lone-pair electrons free from the inuences of other factors.
To clarify the importance of lone-pair electrons on trivalent Sb 3+ sites in CuSbS 2 compound, alloy phonon scattering was purposely introduced and set up as a benchmark to quantify the impact of lone-pair electrons on thermal conductivity.By substitution of Sb with the transition metal Fe and group III A element Ga, lone-pair electrons on a fraction of the Sb sites were removed.All of the outer shell electrons around the Fe (Ga) sites involve in the formation of sp 3 hybridisation due to the valence number difference between Sb and Fe (Ga).We expected that there would be a deviation from the alloy mode of thermal conductivity in a system with opposite inuences of the loss of lone-pair electrons and gain of point defects.In this study, the competing impacts of lone-pair electrons and point defects on thermal conductivity were analysed.The doping concentration dependent thermal conductivity and phonon spectrum were also studied and correlated.

Experimental details
Two alloy systems CuFe x Sb 1Àx S 2 and CuGa x Sb 1Àx S 2 (x ¼ 0, 0.01, 0.025, 0.05, 0.1, 0.2, 0.9, and 1) were designed to remove lonepair electrons from some of the Sb sites by incorporating the trivalent transition metal atom Fe and group III A atom Ga in chalcostibite.Both systems were synthesised using a mechanical-alloying spark-plasma-sintering synthesis route.Pure elements Cu (150 mesh, 99.5%), Sb (100 mesh, 99.5%), S (reagent grade, puried by sublimation, 100 mesh) and Fe (200 mesh, 99+%) were used as raw materials to synthesise CuFe x -Sb 1Àx S 2 , while Ga 2 S 3 (powder, 99.99%, metals basis) was used as the gallium source to synthesise CuGa x Sb 1Àx S 2 .The powders were weighed and then sealed in stainless steel milling jars in an Ar lled glovebox.The jars were mounted and milled in a planetary mill at a rotational speed of 450 rpm for 20 h.The samples with x# 0.2 were spark plasma sintered using a graphite die in vacuum at 400 C for 5 min.For samples with x ¼ 0.9 or 1, the sintering temperature was adjusted based on the melting point of CuFeS 2 (950 C) and CuGaS 2 (1236 C) to achieve dense pellets.For comparison, Cu 3 SbS 3 was synthesised using the same processing method and its thermal conductivity is listed in Fig. 1 with tetrahedrite, famatinite and chalcostibite.
The constituent phases of the samples were characterized using powder X-ray diffraction (XRD, X'Pert PRO-PANalytical, CuKa) in the range 5-120 .RAMAN spectra were obtained from powders using a RENISHAW machine equipped with a He-Ne laser source with 633 nm wavelength and optical lens of 50Â.The error associated with RAMAN measurements was 1 cm À1 .The microstructure images of freshly fractured surfaces were observed using a scanning electron microscope (SEM, FEI Inspect TM-F) with energy dispersive X-ray spectroscopy (EDS).The temperature dependent electrical resistivity and Seebeck coefficient of CuFe x Sb 1Àx S 2 (x ¼ 0.9, and 1) were measured using a commercial instrument (ZEM-3, Ulvac, Inc.) in a He atmosphere.The error of resistivity and Seebeck coefficient measurements are less than 5%.Thermal conductivity k was determined using the equation k ¼ lC p d.The temperature dependent thermal diffusivity l was measured using a laser ash method (LFA-457, Netzsch) on a pellet (f 12.7 mm, height 1.5 mm).The machine was calibrated using a standard specimen within the same temperature range.The repeatability of the measurement was better than 2%, while the error of the thermal diffusivity was less than 5%.The specic heat C p was calculated using the Dulong-Petit law to avoid the large uncertainty in the routine differential scanning calorimetry method.The density d was obtained using the mass and volume of the sintered pellets with an error less than 1%.

Computational details
First principles calculations were performed using the Quantum-ESPRESSO package. 61We used the Garrity-Bennett-Rabe-Vanderbilt (GBRV) high-throughput pseudopotential library. 62The Perdew-Burke-Ernzerhof (PBE) function was used along with ultraso pseudopotentials for all the atoms.A plane wave basis with kinetic energy cut off of at 500 eV and a dense kpoint sampling of 7 Â 11 Â 3 were used to ensure the convergence in all of the calculations.The atomic positions were relaxed until all the force components on each atom were less than 10 À3 a.u., and the lattice constants were optimized until the stress was less than 0.5 kbar.The phonon dispersions and partial phonon density of states were calculated using the density perturbation functional theory (DPFT) and Quasi-Harmonic Approximation (QHA) package, both implemented in Quantum-ESPRESSO.

Results and discussion
According to the literature and the above structural analysis, the lone-pair electrons are probably the solo decisive factor in determining the ultralow thermal conductivity of single crystal chalcostibite.To discuss the competing impacts of the loss of lone-pair electrons and the gain of point defects in the substituted polycrystalline samples, we need to separate their effect from other inuences, such as phase structure and microstructure.

Phase structure
Fig. 3 shows the X-ray diffraction spectra of (a) CuGa x Sb 1Àx S 2 (x ¼ 0, 0.01, 0.025, 0.05, 0.1, 0.2, 0.9, and 1) and (b) CuFe x Sb 1Àx S 2 (x ¼ 0, 0.01, 0.025, 0.05, 0.1, 0.2, 0.9, and 1).The two bottom traces were generated based on the crystal structures of pure CuSbS 2 and CuGaS 2 /CuFeS 2 .For both the Fe and Ga substituted samples, the samples are divided into two groups based on the crystal structure.Samples with x up to 0.2 crystallise with the chalcostibite structure without any trace of impurity.Samples with x ¼ 0.9 and 1 are phase-pure materials with a chalcopyrite structure (cation ordered structure based on zinc blende) for both Ga and Fe substituted samples.This conrmed that it is possible to maintain the structure of chalcostibite even when 20% of the lone-pair electrons are removed from the trivalent Sb sites.Based on the principle of minimum energy, the Fe/Ga atoms should be randomly distributed on the Sb sites of chalcostibite, which in turn restrains the formation of a secondary phase.However, the chalcostibite structure collapses without the support from the lone-pair electrons in the voids separating the SbS 5 units in samples with a specic value of x between 0.2 and 0.9.The phases of CuFeS 2 and CuGaS 2 with the chalcopyrite structure then form with tetrahedral coordination geometry similar to famatinite.

Raman spectroscopy analysis
To clarify the effect of Ga doping on the structure, the Ramanactive modes were investigated at ambient temperature for CuSbS 2 , CuGa 0.025 Sb 0.975 S 2 and CuGaS 2 (Fig. 4).The CuSbS 2 sample has three pronounced peaks at about 187, 250 and 329 cm À1 , along with a much weaker peak at about 450 cm À1 , which is in agreement with data reported in the literature. 63The broad peak at 329 cm À1 includes two adjoining modes with A g symmetry, which are assigned to the vibration modes from the Fig. 3 X-ray diffraction spectra of (a) CuGa x Sb 1Àx S 2 (x ¼ 0, 0.01, 0.025, 0.05, 0.1, 0.2, 0.9, and 1) and (b) CuFe x Sb 1Àx S 2 (x ¼ 0, 0.01, 0.025, 0.05, 0.1, 0.2, 0.9, and 1).The peaks related to the [0 0 4] plane are labeled in (a).The two bottom traces were generated for pure CuSbS 2 and CuGaS 2 / CuFeS 2 .The samples are categorized into two groups based on the crystal structure in both Ga and Fe doped systems.
Sb-S bonds.Peaks at 250 and 450 cm À1 are attributed to the vibration of Cu-S and S-S bonds, respectively.Compared to CuSbS 2 the peak positions of CuGa 0.025 Sb 0.975 S 2 are not shied, but there is a signicant change in the relative peak intensities.This probably means that 2.5 mol% of Ga doping shis the orientation of the Sb-S bonds, which also manifested itself in the XRD pattern.As shown in Fig. 3(a), the intensity of the [0 0 4] peak at 24.5 (2q) decreases with increasing Ga substitution.In pristine CuSbS 2 , two S atoms of the bottom face of one pyramidal CuS 5 and the top S atom of the opposite CuS 5 unit form the [0 0 4] atomic plane.Any doping on the Sb sites with atoms without lone-pair electrons causes a delicate reconstruction of the CuS 5 unit, which leaves a ragged [0 0 4] lattice plane, which in turn decreases the XRD intensity from the plane.In moving from CuSbS 2 to CuGaS 2 , the peak at 329 cm À1 disappears completely and an intense peak at 307 cm À1 emerges due to the A 1 vibrational mode in the chalcopyrite structure.The above discussion showed that the special bonding circumstance on Sb sites in chalcostibite plays an important role in determining its crystal structure.

Microstructure
Fig. 5 shows the typical SEM images of the Cu(Fe/Ga) x Sb 1Àx S 2 samples.All samples are free of pores and have relative densities between 98% and 99%.Due to the different crystalline structures and sintering temperatures, samples with x up to 0.2 have very ne grain size ranging from 100 to 300 nm as shown in Fig. 5(a), while samples with the chalcopyrite structure (x ¼ 0.9 and 1) have coarse grains between 1 and 4 mm as shown in Fig. 5(b).According to the phase structure and microstructure, we separated the samples into two categories.One is the samples with x up to 0.2 sharing the chalcostibite structure and ne average grain size, the other is the samples with x ¼ 0.9 and 1 with tetrahedral coordination geometry and coarse grains.
This classication made it possible to put aside the inuences of phase and microstructure on thermal conductivity in the following discussion.
Thermal conductivity and TE properties of the CuFe x Sb 1Àx S 2 system Fig. 6 shows the temperature dependence of thermal conductivity k for CuFe x Sb 1Àx S 2 (x ¼ 0, 0.01, 0.025, 0.05, 0.1, 0.2, 0.9, and 1).The sample category with x up to 0.2 possesses much lower thermal conductivity than the other category with x ¼ 0.9 and 1.The thermal conductivities decrease with increasing temperature for all of the samples, which indicates that Umklapp phonon scattering prevails in the testing temperature range.No apparent ambipolar transport phenomenon was observed.The overall thermal conductivity is the sum of two nearly independent terms k ¼ k latt + k carr , where k latt and k carr are the lattice and carrier contributions, respectively.The carrier term is related to the electrical resistivity r via the Wiedemann-Franz law, k carr ¼ LT/r, where L is the Lorentz constant, which is 2.45 Â 10 À8 V 2 K À2 for a fully degenerate semiconductor. 64The high resistivity of the samples with x up to 0.2 means that their carrier thermal conductivity accounts for less than 0.1% of their overall thermal conductivity.However, the carrier contributions  in CuFe 0.9 Sb 0.1 S 2 and CuFeS 2 are not negligible, and were calculated using the Wiedemann-Franz law.Fig. 7 displays the resistivity, Seebeck coefficient, lattice/carrier thermal conductivity and gure-of-merit of the CuFe 0.9 Sb 0.1 S 2 and CuFeS 2 samples.Both samples have rather high Seebeck coefficient and moderate resistivity and thermal conductivity.A gure-of-merit zT of 0.19 was obtained for the CuFe 0.9 Sb 0.1 S 2 sample.This value is comparable to values reported in the literature, 65,66 which demonstrates that n-type CuFeS 2 is a good candidate to work with the p-type members of the CAS system in TE generators in the intermediate temperature range.

Lattice thermal conductivity
Fig. 8 displays the temperature dependence of lattice thermal conductivity for (a) CuFe x Sb 1Àx S 2 (x ¼ 0, 0.01, 0.025, 0.05, 0.1, 0.2, 0.9, and 1) and (b) CuGa x Sb 1Àx S 2 (x ¼ 0, 0.01, 0.025, 0.05, 0.1, 0.2, 0.9 and 1).The carrier thermal conductivity for all of the Ga substituted samples is negligible, including the CuGaS 2 sample due to their high resistivity. 67So, we regarded the total thermal conductivity of all the Ga substituted samples as their lattice thermal conductivity in the following discussion.The CuGa x Sb 1Àx S 2 system is also divided into two categories based on their crystal structure and microstructure, as was done for the CuFe x Sb 1Àx S 2 system.Excluding the x ¼ 0.9 and 1 samples with different crystal structures, the thermal conductivities initially increase slightly, reach a maximum, and then decrease with the increasing substitution in both systems.This trend apparently contradicts with the alloying model.In conventional solid solutions or alloys, point defect phonon scattering is one of the dominant factors in reducing the lattice thermal conductivity because of mass contrast, charge uctuation, local strain and other accompanying changes around the foreign atoms.Historically, alloying is the cornerstone for the design of the most commercial room temperature TE materials (bismuth chalcogenide solid solutions) 68 and high temperature materials (SiGe). 69To illustrate the results more clearly, the lattice thermal conductivities are re-plotted against substitution concentration at chosen temperatures, 300, 373 and 600 K in Fig. 9.A 5% error bar was used for each point.In both Fe and Ga substituted systems and at all temperatures, the lattice thermal conductivity increases in the lightly doped samples.Taking into account the special bonding arrangements and the electron density distribution around the trivalent Sb sites in chalcostibite, the abnormal lattice thermal conductivity increment in lightly doped samples is naturally connected to the 5s 2 lone-pair electrons.To quantify the impact of the lone-pair electrons on lattice thermal conductivity, the modelled lattice thermal conductivities based on an alloy model are also plotted in Fig. 9.According to Klemens and Abeles's model, 70,71 the lattice thermal conductivity k of a disordered alloy can be calculated from the k p value in the absence of point defects (pure CuSbS 2 in this work), where a is the ratio of the relaxation times of three-phonon normal and Umklapp process, and U is calculated by the equation Fig. 7 Electrical, thermal transport properties, and figure-of-merit zT of CuFe 0.9 Sb 0.1 S 2 and CuFeS 2 samples.
b is nearly a constant within a given covalent crystal system.G is a function of the strain parameter 3 and other known basic alloy parameters.d 3 is the atomic volume derived from Vegard's law.So, there are three adjustable parameters, the ratio of normal and Umklapp scattering rate a, the anharmonicity parameter g, and the strain parameter 3, to estimate the thermal conductivities of alloys.As shown in Fig. 9, the discrepancy between the model values and the experimental data highlights the signicance of the lone-pair electrons in chalcostibite.In alloy or solid solution, the lattice thermal conductivity is quite sensitive to the point defects due to its effective scattering of short and medium-wavelength heat-carrying phonons.However, the lattice thermal conductivity witnesses an increase rather than a decrease in lightly Fe and Ga doped chalcostibite.This suggests that the inuence of the loss of lone-pair electrons outweighs the impact of point defects in lightly doped samples.With further increase in doping, the effect of phonon scattering by point defects becomes more dominant in the trend of the lattice thermal conductivity.This trend also manifested itself at 373 K and 600 K, which indicates that lone-pair elections play an important role in thermal transport properties both at low and high temperature.Taking into account the origins of point defects scattering and lone-pair electrons, both mechanisms are closely related to the bonding properties at the atomic scale.So both phonon scattering mechanisms should have a similar working temperature range and compete with each other in the whole temperature range.Actually, the lattice thermal conductivity is controlled by the competition between the opposite effects of point defects and the loss of lone-pair electrons on Sb sites.This is the rst direct demonstration of the role of lonepair electrons in comparison with another well-known scattering mechanism.It demonstrates that lone-pair electrons provide a strong mechanism to transform phonon transport and may eclipse point defects in lightly doped solid solution/ alloys.This suggests that compounds with lone-pair electrons (trivalent V A atoms) provide a new direction to design low thermal conductivity materials, and can be used to screen for prospective TE materials.

Phonon band structure and density of states
To illustrate the mechanism of the effect of lone-pair elections on thermal transport properties, the phonon spectra of CuSbS 2 and CuGaS 2 were calculated (Fig. 10).Although the chalcostibite structure has been reported for CuSbS 2 at room temperature, the negative phonon frequencies indicate that this structure is unstable at 0 K, at which the rst principles calculations were performed.The eigen-displacements of the negative modes (so modes) will lead to phase transition at low temperatures, known as 'so mode hardening'.In the following discussions, the phonon dispersion of the CuSbS 2 chalcostibite structure will be used because the discussions focus on the lattice thermal conductivity above room temperature.Besides, most of the optical modes and the three acoustic modes are unaffected by the so modes and have positive values.
Due to the high symmetry of the chalcopyrite structure, CuGaS 2 has much fewer optical modes than CuSbS 2 .Most of the optical modes exhibit frequencies higher than 125 cm À1 , except two modes at 92 and 94 cm À1 .Compared to its Ga counterpart, the most obvious feature of CuSbS 2 is that the frequencies of the optical modes are very low, although an underestimation of the frequencies may be possible because of the band-gap problem in generalized gradient approximation (GGA).In fact, most of the optical modes are located in a narrow low frequency range of 20-80 cm À1 , which is similar to AgSbTe 2 .Ye et al. linked the soness of those modes to the heavy atomic masses, the relatively weak bonds between Te and Sb, and the large anioncation distances. 36In chalcostibite, the Sb5s 2 electrons orienting toward the voids between the SbS 5 units are nonbonding, which may lead to a soening of the optical modes.The small gap between the optical modes and acoustic modes suggests that the energy transfer between those modes is very likely and easy.As a result, the soening optic modes must have a strong scattering effect on the heat-carrying acoustic modes, which may be the mechanism of the ultralow lattice thermal conductivity in chalcostibite.Fig. 11 shows the calculated total and partial phonon density of states (DOS) of (a) CuSbS 2 and (b) CuGaS 2 .The peaks of the DOS shi toward higher frequency in CuGaS 2 because of the lower mass of Ga compared with Sb.In fact, most of the modes with frequency higher than 160 cm À1 in CuSbS 2 (275 cm À1 in CuGaS 2 ) are only connected to the S and Cu bonds, while the Sb bonds contribute to the peaks below 153 cm À1 in chalcostibite (Ga below 236 cm À1 in CuGaS 2 ).To elaborate the difference between the two materials, the details of the phonon DOS between 20 and 150 cm À1 are highlighted in Fig. 11(c) and (d).In CuGaS 2 , there is only one Ga-related broad peak at around 90 cm À1 .There are no signicant contributions from Ga out of the range of 70-105 cm À1 .In sharp contrast to CuGaS 2 , there are three apparent Sb related peaks at 82, 111 and 138 cm À1 for CuSbS 2 .Except for the peak at around 111 cm À1 showing a broad shoulder similar to that observed in Ga substituted CuGaS 2 , the other two peaks are very sharp and show nearly no shoulders on the high frequency side.This is the characteristic of quasi-single-frequency Einstein modes created by the individual 'rattling' of the guest atoms in skutterudites and clathrates. 21,72In CuSbS 2 , the phonon structure is well dened and can be understood in terms of the ideal crystal.No glass-like behaviour was observed.However, the two Sb-related quasi-single-frequency vibration modes behave like localised Einstein harmonic oscillators and qualitatively modied the whole vibration spectrum. 73These results give unequivocal evidence that the trivalent V A atom creates special bonding and vibration modes because of its nonbonding 5s lone-pair electrons.Combined with the crystal structure shown in Fig. 2(d), we conjecture that quasi-single-frequency modes may be related to the Sb vibration in a direction perpendicular to the SbS 4 plane of the pyramidal SbS 5 unit.Due to the asymmetric bonding, Sb atoms are free to undergo large displacement or shi toward the voids separating the SbS 5 units, while the movement toward the vertex is restrained by the opposing S atoms.Moreover, the non-bonding 5s 2 electrons are expected to form a shell of relatively large radius, especially in the direction toward the voids.During the thermal vibration, the wave functions of lone-pair electrons overlap with each other, which leads to an additional repulsive force to the restoring force that in turn leads to extreme anharmonicity of the lattice vibrational spectrum. 34This is likely to be the origin of the ultralow lattice thermal conductivity in systems with lone-pair electrons.

Conclusions
To clarify the importance of lone-pair electrons on trivalent Sb 3+ sites in Cu-Sb-S systems, we purposely introduced alloy phonon scattering in the carefully chosen chalcostibite CuSbS 2 and used it as a benchmark to quantify the impact of lone-pair electrons on thermal transport properties.The thermal conductivity measurements show an apparent deviation from the conventional alloy model.Most impressively, the role of lone-pair electrons eclipses point defects in lightly doped solid solutions, which gives the rst direct demonstration of the importance of the lone-pair electrons by comparison with another well-known scattering mechanism.Phonon dispersion calculations disclosed two Sb related quasi-single-frequency vibration modes behaving like localised Einstein harmonic oscillators, similar to the modes created by the individual 'rattling' of the guest atoms in skutterudites and clathrates.Combined with the crystal structure evolution with increasing substitution, we conjecture that quasi-single-frequency modes may be related to the Sb vibration in a direction perpendicular to the SbS 4 plane of the pyramidal SbS 5 unit.Due to the asymmetric bonding, the wave functions of lone-pair electrons in the voids overlap with each other during thermal vibration, which leads to an additional repulsive force on the restoring force, which leads to extreme anharmonicity of the lattice vibrational spectrum and ultralow lattice thermal conductivity in chalcostibite.This suggests that compounds with lone-pair electrons (trivalent V A atoms) are a new direction to design low thermal conductivity materials.

Fig. 5 Fig. 6
Fig. 5 Typical SEM images of bulk Cu(Fe/Ga) x Sb 1Àx S 2 samples with (a) x up to 0.2 and (b) x ¼ 0.9 or 1.