Lattice dynamics and thermoelectric properties of diamondoid materials

The diamondoid compounds are a large family of important semiconductors, which possess various unique transport properties and had been widely investigated in the fields of photoelectricity and nonlinear optics. For a significantly long period of time, diamondoid materials were not given much attention in the field of thermoelectricity. However, this changed when a series of diamondoid compounds showed a thermoelectric figure of merit (ZT) greater than 1.0. This discovery sparked considerable interest in further exploring the thermoelectric properties of diamondoid materials. This review aims to provide a comprehensive view of our current understanding of thermal and electronic transport in diamondoid materials and stimulate their development in thermoelectric applications. We present a collection of recent discoveries concerning the lattice dynamics and electronic structure of diamondoid materials. We review the underlying physics responsible for their unique electrical and phonon transport behaviors. Moreover, we provide insights into the advancements made in the field of thermoelectricity for diamondoid materials and the corresponding strategies employed to optimize their performance. Lastly, we emphasize the challenges that lie ahead and outline potential avenues for future research in the domain of diamondoid thermoelectric materials.

[103][104][105][106] These reports have sparked intense interest in exploring the thermoelectric properties of diamondoid materials.Figure 1A shows the crystal structures of diamonds and some common diamondoid compounds.One typical diamondoid compound is zinc blende (F-43m), which is constituted by two elements and each anion is surrounded by four identical cations.By stacking two zinc blende cells along the c-axis, and evenly filling the cation site with two different elements, a ternary diamondoid compound (chalcopyrites, I-42d) is obtained.Similarly, different ternary (I-42m) and quaternary diamondoid structures can be obtained by altering the type of cation.
This article aims to provide an overview of the distinctive phonon and electronic transport properties exhibited by diamondoid compounds.We explore the links between lattice dynamics, thermal transport, and electronic band structure.Additionally, the paper will summarize the recent advancements in thermoelectricity for diamondoid materials and the corresponding strategies employed to optimize their performance.Because of the close-packed tetrahedral coordination, the diamond structure typically exhibits high thermal conductivity.For example, the room temperature thermal conductivity of diamond is as high as 2000 Wm −1 K −1 , [155] whereas Si has a thermal conductivity of 150 Wm −1 K −1 . [156]However, many ternary and quaternary diamondoid compounds possess an intrinsic low thermal conductivity, [157] which is essential for high thermoelectric performance.This unexpectedly low thermal conductivity originates from the fact that in ternary and quaternary diamondoid structures, each anion has two different types of cations as nearest neighbors.Due to the size and charge differences between the adjacent cations, the anions would acquire an equilibrium position closer to one pair of cations than the other, leading to distorted tetrahedral coordination and complex chemical bonding. [157,158]his results in intrinsically strong phonon scattering and low thermal conductivity.In this part, we will discuss the correlation between local structure, lattice dynamics, and thermal conductivity of diamondoid compounds.
Below, we first focus on the intriguing behavior of the classical Ag-based diamondoid compound AgGaTe 2 .Researchers discovered that Ag atoms are actually offcenter from the Te 4 tetrahedron center due to weak sd 3 orbital hybridization. [157]This weak chemical bonding allows Ag atoms to deviate from the tetrahedral geometry and shift along certain directions.[161] The induced local off-centering of Ag also affects neighboring Te atoms, giving rise to extremely low-frequency optical phonons and an avoided crossing with acoustic modes, indicating a strong coupling between acoustic and optical phonons.The resulting strong acoustic-optical phonon scattering drastically suppresses the thermal conductivity in AgGaTe 2 .Importantly, this emphanitic local distortion of the Ag coordination, rooted in electronic origin, is found to exist in other silver diamondoid compounds and is responsible for their low thermal conductivity. [157]he relationship between crystallographic distortion parameter (η) and chemical bonding strength provides a useful approach to evaluating the thermal conductivity of diamondoid compounds.
In the subsequent section, we explore another mechanism to introduce off-centering in diamondoid compounds through the presence of lone pair electrons.The example of In-doped CuFeS 2 showcases the different behaviors of In 3+ and In + when substituting Fe and Cu sites, respectively.The presence of lone pair electrons in In + induces a heavily distorted local tetrahedral structure, leading to reduced phonon velocity and enhanced phonon scattering, effectively suppressing thermal conductivity.A similar lone pair electron-induced structure distortion has been observed in the Cu-Sb-Se series, [162] further highlighting the impact of these electronic effects on thermal properties.
Lastly, we discuss interstitial atoms and intrinsic vacancies and the effective methods to reduce thermal conductivity in diamondoid compounds.Adding extra atoms in the interstitial sites or the presence of intrinsic vacancies enhances phonon scattering, thus lowering the thermal conductivity.Such strategies have been successfully demonstrated in various thermoelectric materials, offering valuable insights for designing high-performance thermoelectric materials.

| Local symmetry breaking and off-centering behavior
Despite having an identical crystal structure, most Agbased diamondoid compounds show significantly lower intrinsic lattice thermal conductivity than their Cu-based counterparts.Although one might expect the heavier Ag atom to result in a similar compound with lower thermal conductivity, the observed conductivity is much lower than anticipated. [163,164]This peculiar behavior has been a source of confusion for researchers for a considerable period.However, a recent breakthrough has shed light on this anomaly.With synchrotron X-ray measurements, the true nature of the local structures and heat transport behaviors in these diamondoid compounds has been revealed. [157]hrough an investigation of the classical Ag-based diamondoid compound AgGaTe 2 using the pair distribution function (PDF) technique, researchers made a significant discovery.It was found that the Ag atom actually shifts away from the Te 4 tetrahedron center by 0.1 Å at room temperature.The observed behavior was attributed to the substantial energy difference between the 5s and 4d electronic orbitals, as shown in Figure 2A.This disparity results in weak sd 3 orbital hybridization of tetrahedral Ag.Consequently, the chemical bonding becomes distorted and weaker (detected by the calculated interatomic force constant), allowing the Ag atom to deviate from the tetrahedral geometry and shift along the <100> and <010> directions. [157]As the temperature rises, this local distortion steadily increases, leading AgGaTe 2 to transition from an undistorted ground state to a locally distorted state, as depicted in Figure 2B,C. [159]he off-centering of Ag locally also causes distortion in the neighboring Te atoms, resulting in the emergence of several extremely low-frequency optical phonons in AgGaTe 2 .These optical modes appear at the Γ point and exhibit lower frequency values compared to the compound's acoustic modes.An interesting observation is the occurrence of an avoided crossing between these low-frequency optical modes and the acoustic modes in AgGaTe 2 , as shown in Figure 2D.This phenomenon indicates a strong coupling between the acoustic and optical phonons in the material, contributing to its unique vibrational properties.This type of phonon coupling [165] increases the phonon scattering at the intermediate frequency (Figure 2E), thereby dramatically suppressing the thermal conductivity in AgGaTe 2 .
More importantly, since the emphanitic local distortion of the Ag coordination is electronic origin, this offcentering behavior actually exists in other silver diamondoid compounds, and is the root cause of their low thermal conductivity. [157]This local distorted coordination results in negative c-axis thermal expansion and leads to a large tetragonal crystallographic distortion value, which can be represented as η = |2−c/a|, where c and a are the length of the c-axis and a-axis, respectively.As a result, the chemical bonding strength and off-centering behavior are closely linked with the crystallographic distortion parameter (η).This allows η to be used for evaluating the thermal conductivity of diamondoid compounds, resulting in a relationship of κ • T ~(1/m • V) • (1/η); (V is crystal volume and m is formula weight), [157,158] as shown in Figure 2G.The relationship between η and the thermal conductivity for different diamondoid compounds, the y-axis is the normalized thermal conductivity, the dashed line shows the 1/η relationship.Reproduced with permission. [157]Copyright 2022, Wiley-VCH.

| Lone pair electrons and discordant atoms
In addition to the intrinsic off-centering of Ag, another way to introduce the off-centering effect in diamondoid compounds is through a dopant atom with a lone pair of ns 2 electrons.An excellent example of this is In-doped CuFeS 2 . [166]By carefully designing the cations composition, researchers found that the In atom can substitute both the Fe 3+ and Cu + sites in CuFeS 2 .When the In atom replaces Fe (CuFe 1−x In x S 2 ), it contributes three electrons and performs as trivalent indium. [167]In this case, the In 3+ sits at a perfect tetrahedral coordination of sulfur atoms and suppresses thermal conductivity via the classical alloying scattering effect.However, when In substituting on the Cu site (Cu 1−x In x FeS 2 ), only 40% of the In atoms exist as In 3+ , and the remaining 60% actually behave as In + . [166]Thus, for the part of In 3+ that occupies Cu + sites, the trivalent indium also adopts the prefect S 4 tetrahedral coordination, similar to the CuFe 1−x In x S 2 case.However, for the In + part, the 5s 2 lone pair electrons of monovalent indium need their own space to stereochemically express itself, thus pushing the In + ion displace from the coordination center, see Figure 3A,B.The system energy of Cu 1−x In x FeS 2 in Figure 3C shows that, when the In + is approximately 0.12 Å away from the tetrahedral center and shifted towards one face of the tetrahedron, the structure is further stabilized, this in true suggests the discordant In + induces a heavily distorted local tetrahedral structure.This off-centering behavior of In + distorts and softens the In-S chemical bonding.Since the sound velocity (v) is determined by the stiffness coefficient (K s ) and the density (ρ) of the material, which expresses as v = (K s /ρ) 1/2 . [158]The weaker In-S chemical bonds lead to a smaller stiffness coefficient, which reduces the phonon velocity and enhances the phonon scattering, thus dramatically suppressing the heat conductivity.Figure 3D illustrates that, at the same doping level, the discordant In + doping (In Cu ) is more effective in reducing the thermal conductivity of CuFeS 2 than the conventional case (In Fe ).Introducing only 3% of the discordant atom can result in a 62% suppression of thermal conductivity.
Another case of the lone pair of electrons-induced low thermal conductivity has been observed in the Cu-Sb-Se series, [162] as shown in Figure 3E.The Cu 3 SbSe 4 possesses the diamondoid structure with the space group of I-42m.In this structure, Sb is surrounded by four Se atoms, and all Sb valence electrons form bonds, constructing a tetrahedral coordination structure with ideal Se-Sb-Se bond angles of 109.5°.This perfect SbSe 4 tetrahedral coordination and strong bonding results in high thermal conductivity (Figure 3F).When changing one Cu with Sb in Cu 3 SbSe 4 , the orthorhombic (Pnma) CuSbSe 2 is obtained.In this case, the Sb is coordinated with three Se atoms, and only the Sb 5p electrons form chemical bonds with Se, leaving the 5s electrons of Sb as lone pairs.Thus, Sb is surrounded by Se atoms in a trigonal pyramidal configuration.Although the CuSbSe 2 is not in a diamondoid structure, if the spread space of lone pairs is included, the SbSe 3 coordination can also be considered as a form of imperfect tetrahedral coordination.This lone pair effect results in a distorted local structure and a lower thermal conductivity in CuSbSe 2 , which is only half of that in Cu 3 SbSe 4 .The Cu 3 SbSe 3 has a similar SbSe 3 configuration to CuSbSe 2 , but the lone pair of electrons is farther removed from the Sb nucleus and is more dispersed.The dispersed lone pair electron induces a nonlinear repulsive electrostatic force, resulting in high anharmonicity in the lattice (in addition to the high anharmonicity exhibited often by Cu atoms) and a remarkably lower thermal conductivity, which is less than 10% of that in Cu 3 SbSe 4 .

| Interstitial atoms and intrinsic vacancies
In most cases, adding extra atoms to the interstitial sites of the lattice is effective in reducing thermal conductivity in crystal solids.[174] The primary method to decrease heat conductivity in these materials involves filling their intrinsic structural voids with additional atoms.This process induces a phenomenon known as rattling motion, which enhances phonon scattering.A similar strategy has also been employed in the chalcopyrite CuFeS 2 . [128]t room temperature, the intrinsic CuFeS 2 possesses a tetragonal (I-42d) diamondoid structure.However, when adding 10% extra cations in the lattice, a novel cubic (I-43m) diamondoid compound Cu 17.6 Fe 17.6 S 32 is obtained. [128]Figure 4A [166] Copyright 2019, American Chemical Society.(E) Schematic representation of the local atomic environment of Sb in Cu 3 SbSe 4 , Cu 3 SbSe 3 , and CuSbSe 2 .(F) Temperature dependence of the lattice thermal conductivity of Cu 3 SbSe 4 , CuSbSe 2 , and Cu 3 SbSe 3 .Reproduced with permission. [162]Copyright 2019, American Chemical Society.(D) Crystal structure of Cu 2 SnSe 4 .(E) lattice thermal conductivity for Cu 2 SnSe 4 and Cu 2 SnSe 3 (solid dot).Reproduced with permission. [117]Copyright 2016, American Chemical Society.
scattering.Additionally, the interstitial cations expand the lattice and lead to a random arrangement of Cu and Fe atoms.This long-range cation disorder effectively boosts phonon scattering.All these factors result in low thermal conductivity in Cu 17.6 Fe 17.6 S 32 , which is only 0.5 Wm −1 K −1 at 625 K, and is less than one-sixth of CuFeS 2 in the whole temperature range, see Figure 5B,C.
By filling interstitial atoms into the structural voids, we can enhance phonon scattering by introducing mass and strain fluctuations between the host and guest atoms. [176]Conversely, creating vacancies in the matrix by removing some atoms enables the import of mass and strain fluctuations between the host atoms and the voids.This, too, results in enhanced phonon scattering and the suppression of lattice thermal conductivity. [177]ntroducing the vacancy to reduce heat conductivity has been successfully demonstrated in some thermoelectric materials, such as SnTe [178,179] and In 2 Te 3 . [180]However, in contrast to the artificial vacancy strategy, a fascinating diamondoid compound Cu 2 SnSe 4 has been discovered with intrinsic vacancies, which exhibits a very low thermal conductivity. [117]The crystal structure of Cu 2 SnSe 4 is shown in Figure 4D.The structure can be considered a zinc blende type, where the anion sites are fully occupied by Se.However, its cation sites are only partially and randomly occupied by Cu (50%) and Sn (25%), and 25% of its cation sites are vacant.Therefore, the chemical formula of Cu 2 SnSe 4 can be written as Cu 2 SnMSe 4 , where M represents the intrinsic vacancy.In fact, the Cu 2 SnSe 4 (Cu 2 SnMSe 4 ) serves as a prototype for some quaternary diamondoid compounds, such as Cu 2 ZnSnSe 4 [138] and Cu 2 CdSnSe 4 , [107,181] where the vacancy is filled by Zn or Cd.The large number of intrinsic vacancies lead to the very low thermal conductivity in Cu 2 SnSe 4 , which is about 0.5 Wm −1 K −1 at 700 K, see Figure 4E.We also noticed that the copper atom should be divalent in Cu 2 SnSe 4 to balance the charge, however, Cu 2+ is rare in diamondoid compounds, and there is no direct evidence to prove the existence of Cu 2+ in Cu 2 SnSe 4 .Thus, the compound could either have an extremely high hole concentration, exceeding 10 22 cm −3 , or it may actually be Cu 2 SnSe 3 with a very large number of cation vacancies.Future studies should focus on clarifying this issue.Unraveling the true nature of this material warrants further investigations.

| BAND STRUCTURE AND ELECTRICAL PROPERTIES
Generally, good thermoelectric performance is more likely to be obtained in narrow band gap semiconductors, such as the classical thermoelectric materials: Bi 2 Te 3 , [182] PbTe, [183] GeTe [63] and CoSb 3 . [75]However, many wide F I G U R E 5 Pseudocubic approach to realize highly degenerate electronic bands and good electronic-transport properties in noncubic chalcopyrites.(A) Crystal structure and electronic bands of cubic zinc blende structure.(B) Crystal structure and electronic bands of ternary chalcopyrites.Γ 4v is a nondegenerate band and Γ 5v is a doubly degenerate band.Δ CF is the energy split at the top of Γ 4v and Γ 5v bands.(C) Electronic bands of Cu 0.875 Ag 0.125 InTe 2 .(D) Thermoelectric power factors of tetragonal chalcopyrites and their correlation with the crystal field splitting Δ CF .Reproduced with permission. [175]Copyright 2014, Wiley-VCH.
band gap (E g > 1.0) diamondoid compounds not only show low thermal conductivities, but also present decent electrical transport properties and the high ZT values. [106,110,121,123,184]This mainly comes from the unusually high Seebeck coefficient and strong temperature dependence of carrier concentration.
In this part, we first focus on the effect of crystal symmetry on band structure and electronic transport properties.Researchers found a nondegenerate light band constitutes the valence bands of the tetragonal diamondoid compound and a doubly degenerate heavy band, and the splitting energy between these valence bands is strongly related to the crystal symmetry.By adjusting the crystal structure of the tetragonal diamondoid compound into a pseudocubic structure, the splitting energy of their valence bands is reduced, and a triply degenerate valence band can be obtained, which leads to the high Seebeck coefficient and good thermoelectric performance.
We next explore how a nonparabolic band structure influences the Seebeck coefficient.Some diamondoid compounds feature this unique band structure, showing a parabolic energy dispersion near the Γ points that quickly transitions to a linearly dispersing band as the wave vector increases.This distinctive band structure is advantageous for achieving higher Seebeck coefficients and contributes to high power factors in these materials.
Finally, we explore how in-gap states affect the electronic transport in diamondoid compounds.Many such compounds inherently possess in-gap states, leading to a significant temperature-dependent carrier concentration.Fine-tuning the chemical composition allows for the regulation of the density of these in-gap states and their activation temperature.This offers a valuable strategy for engineering the material's high-temperature transport properties.On the other hand, for the CuFeS 2 , when the temperature falls below 50 K, all the carriers are captured by the in-gap states, and variable range hopping processes dominate the electronic transport.This leads to a rather unusual Seebeck behavior in the material. [185]

| Pseudocubic structure and band convergence
As we know, the Seebeck coefficient is determined by the density-of-states effective mass, m*, while the carrier mobility is related to the single valley band mass, m b *.For multiple degenerate carrier valleys, the density-ofstates effective mass is given by m* = N v 2/3 m b *, where the N v is the orbital degeneracy. [5,42,186]Thus, a higher valley degeneracy N v increases the density-of-states effective mass and Seebeck coefficient without explicitly reducing the carrier mobility (Mobility is nominally unaffected by N v , but there may be some reduction caused by intervalley scattering).High band degeneracy numbers can be observed in high symmetry crystal structures when the Fermi surface exhibits isolated pockets at low symmetry Brillouin zone points.More generally, bands may be regarded as effectively converged when their energy separation is small; this would also lead to an effective increase in N v , even if the bands are not precisely degenerate.This band energy alignment effect is called band convergence. [34,42,186,187]oreover, when the convergence happens in both energies and momentum at multiple electronic bands, the synglisis effect obtained, [53,55] which is effective to maximize the power factor of the materials.Band convergence and synglisis are the most successful states in a materials to enhance thermoelectric performance and have been observed in many advanced thermoelectric materials, such as SnSe, [53][54][55] PbTe, [42,183] Mg 2 (Si, Sn), [186] AgSbTe 2 , [188,189] and half-Heusler. [190,191]or tetragonal diamondoid compounds, because of the symmetry breaking, they should usually possess single valence and conduction bands.However, researchers found band convergence situations can also be obtained in the noncubic tetragonal diamondoid compounds, which is realized by adjusting the lattice parameters and constructing the pseudocubic structure. [175]Figure 5A shows the crystal structure and electronic bands of the zinc blende structure.Because of the high-symmetry cubic structure, zinc blende has degenerate electron band edges (triply degenerate valence bands).In contrast to the zinc blende lattice, the cation sites in ternary diamondoid compounds are occupied by two different elements.This results in the noncubic tetragonal structures with distorted tetrahedral coordination.Thus, the crystal field splitting energy Δ CF makes the triply degenerate valence bands split into a nondegenerate light band Γ 4v and a doubly degenerate heavy band Γ 5v in tetragonal diamondoid compounds. [192]However, researchers found that the crystal field splitting energy is related to the crystallographic distortion: c/2a.By tuning the cation sublattice and regulating the lattice parameter, when c/2a ≈ 1, a periodic supercell with a cubic framework can be obtained, which is referred to as a pseudocubic structure.This high symmetry cubic supercell reduces the crystal field splitting energy, resulting in the cubic-like degenerate electron band edge at the Γ point, [175] see Figure 5B,C.Thus, this degenerate band structure leads to the high power factor S 2 σ. Figure 5D plots the correlation between the power factors and the crystal field splitting energy for some reported diamondoid compounds.The trend clearly shows that the peak power factors of diamondoid compounds are achieved when the crystal field splitting energy is close to zero.
The Cu 2 SnSe 3 is another interesting instance demonstrating that crystal symmetry regulation's strategy leads to band convergence.Usually, pure Cu 2 SnSe 3 possesses a monoclinic structure with the Cc space group, and it shows a single valence band maximum (VBM) at Γ point, with two split bands located at lower energy with the largest splitting energy of 84 meV. [147]However, the crystal symmetry of Cu 2 SnSe 3 is easy to modify with doping [147] or introducing some cation vacancies. [193]n fact, these chemical modications cause Cu 2 SnSe 3 to crystallize in a cubic diamondoid structure with the space group of F-43m.In this case, the high crystal symmetry erases the valence bands' energy splitting and leads to the triple degenerate VBM at Γ point which creates high thermoelectric performance.

| Nonparabolic band structure
For many thermoelectric materials, their transport properties can be investigated by the parabolic band model with an underlying assumption of a spherical Fermi surface.Within the domain of the parabolic band structure, the band energy follows the E ∼ k 2 relationship (k is the wave vector), and the density-of-states (DOS) and the carrier concentration (p) are related to the wave vector as DOS ∼ k 2 dk/dE and p ∼ k 3 , respectively. [194,195]hus, the Mott relation can be simplified, and the Seebeck coefficient is related to the carrier concentration as S ∼ p −2/3 . [196]owever, this law is invalid in some materials, such as CoSb 3 , [197][198][199] Ba 8 Ga 16 Ge 30 [173] and some diamondoid compounds like CuFeS 2 , [194] Cu 1−x Ag x GaTe 2 [123,184] and Cu 2 SnSe 3 . [200]These materials actually possess a 3 GaTe 2 ; Reproduced with permission. [123]Copyright 2022, American Chemical Society.(D) The room-temperature Seebeck coefficient as a function of carrier concentration for CuFeS 2 , the inset illustrates band dispersions for the parabolic and quasilinear band models.Reproduced with permission. [194]Copyright 2022, American Physical Society.
nonparabolic band structure.Take the Cu 1−x Ag x GaTe 2 as an example, [123] its valence band exhibits a parabolic energy dispersion in a small region near the Γ points, but it rapidly changes to a linearly dispersing band as the wave vector increases, as shown in Figure 6A.In the case of a linearly dispersing band structure, the band energy follows the E ∼ k relationship, leading to a different energy dependence of the density-of-states and carrier concentration.Thus, the Seebeck coefficient becomes proportional to p −1/3 .This is known as the quasilinear band model, proposed by Singh and Pickett, and has been successful in describing the transport properties in the binary skutterudites: IrSb 3 , CoAs 3 , and CoSb 3 . [197]igure 6B shows the temperature-normalized Seebeck coefficient as a function of carrier concentration for Cu 0.7 Ag 0.3 GaTe 2 , and Figure 6C displays the Seebeck coefficient of Cu 0.7 Ag 0.3 GaTe 2 in the high-carrier concentration region.Obviously, the experimental data are well fitted by the quasilinear S ∼ T•p −1/3 relationship rather than the parabolic band model (S ∼ T•p −2/3 ), indicating the quasilinear band model is more suitable to describe the transport behavior of Cu 0.7 Ag 0.3 GaTe 2 .[123] A similar case has been observed in the n-type CuFeS 2 , [194] as shown in Figure 6D.These results suggest that the nontrivial band structure of diamondoid compound is beneficial for obtaining a higher Seebeck coefficient at a given carrier concentration, which is important for achieving high power factors in these materials.Additionally, while Singh's model aligns well with the observed Seebeck data, such a fitting offers limited insights into the electronic band structure.To gain a more comprehensive understanding of the unique transport behavior in diamondoid compounds, direct experimental tests on their accurate band structure are necessary.

| In-gap state and dynamic doping
To obtain the best power factor in a thermoelectric material, it is important to regulate the carrier concentration at an optimal level.In many cases, the optimized carrier concentration of a thermoelectric material tends to increase with rising temperature. [11]However, for the most heavily doped semiconductors, their carrier concentrations are almost independent with temperature, making carrier concentration optimization challenging over a wide temperature range.For many diamondoid compounds, a strong temperature-dependent carrier concentration has been observed. [110,121,184]Studies demonstrate that this unusual behavior can be modulated by adjusting the defect state of the material. [123,184]his provides the concept of dynamic doping to optimize the carrier concentration and electrical properties of diamondoid compounds over a wide temperature range.
Figure 7A explains the mechanism of dynamic doping in a p-type semiconductor.In materials with ingap acceptor states, an empty impurity level situated above the valence band, rising temperatures can activate electrons from the valence band to fill these empty states.This process simultaneously creates holes in the valence band, thereby increasing the carrier concentration in the material.The activation energy of this process is E a , and is related to the Fermi level (E F ), which represents the intrinsic carrier concentration of the material and the energy of the in-gap state (type and density of the defect).Thus, by controlling the chemical composition, the activation energy can be adjusted, which in turn can regulate the dynamic doping effect in the material.
The Cu-based diamondoid materials can easily generate intrinsic Cu vacancies, which produce in-gap acceptor states in the material. [121,123,184]Figure 7B shows the temperature dependence of the carrier concentration in Cu 1−x Ag x GaTe 2 .Because of the dynamic doping effect, these materials exhibit strong temperature-dependent behavior, with the carrier concentration increasing four orders of magnitude with rising temperature.Moreover, adjusting the Cu/Ag ratio and Cu vacancy can significantly modulate the carrier concentration and dynamic doping effect, which is helpful in optimizing their electrical conductivity, [123] as shown in Figure 7C.Additionally, Figure 7D shows the energies of the conduction band minimum (CBM), VBM, and impurity level for Cu 0.8 Ag 0.2 In 1−x Ga x Te 2 extracted from experimental measurements. [184]This suggests that the activation energy and band structure of impurity levels in diamondoid materials are adjustable.As a result, by using the dynamic doping effect, the temperature-dependent behavior of carrier concentration can be optimized, leading to high ZT values for these diamondoid materials, see Figure 7E,F.

| Low temperature hopping transport and high Seebeck coefficient
Based on the traditional theory, the Seebeck coefficient of a material is determined by the energy asymmetry of electron transport and can be divided into two parts. [185,201]ne is the density-of-state asymmetric Seebeck coefficient (S D ), which reflects the Seebeck effect caused by the asymmetry of the electron distribution near the Fermi level.The other one is the relaxation time asymmetric Seebeck coefficient (S τ ), which represents the Seebeck effect caused by the energy-dependent relaxation time of the charge carrier (a scattering mechanism).
Generally, the S D is the contribution to the usual Seebeck effect, and it increases with the increasing density-of-states. [11]In Figure 8A, it is demonstrated that a sharp increase in the density-of-state near the Fermi level results in a significant Seebeck coefficient.Numerous electronic band engineering strategies, such as band convergence [42] and resonance states, [40] are founded on this principle.Moreover, in Figure 8B, it is revealed that a material with charge carriers possessing a large energy-dependent relaxation time can also exhibit a high Seebeck effect.However, the part of the Seebeck coefficient attributed to the contribution of relaxation time (S τ ) has commonly been disregarded because carrier scattering usually exhibits weak energy dependence.
A few exceptions have been found in the low temperature electronic transport for some materials, such as MoTe 2 , [202] CoSb 3 , [203] CeCu 2 Si 2 , [204] and diamondoid CuFeS 2 . [185]By studying the low temperature (2-300 K) electronic transport properties of CuFeS 2 , researchers found that, CuFeS 2 in fact possesses some in-gap states, [185] see Figure 8C.When the temperature is below 50 K, these in-gap states are occupied by electrons, and the electronic transport of the material proceeds via variable range hopping.In this case, the carrier migration increases by several orders of magnitude as the temperature rises, exhibiting a strongly energy-dependent relaxation time.This leads to the large relaxation time asymmetric Seebeck effect, resulting in a sharp increase in Seebeck coefficient, [185] see Figure 8D.As the temperature rises (T > 50 K), carriers transition from the in-gap states to the conduction band, they lead to dominant band conduction in electronic transport.Consequently, the mobility becomes weakly dependent on temperature, and the Seebeck effect, due to relaxation time asymmetry, becomes insignificant.This intriguing process gives rise to a substantial Seebeck coefficient in CuFeS 2 at low temperatures. [185]

| THERMOELECTRIC PERFORMANCE OPTIMIZATION
Diamondoid compounds showcase distinct thermal and electronic transport properties, rendering them promising candidates for thermoelectric applications.Numerous endeavors have been undertaken to enhance their ZT values, including elemental doping, [205,206] forming solid solutions, [140,158] introducing Reproduced with permission. [184]Copyright 2021, American Chemical Society.nano-precipitates, [139,141,207,208] and constructing nanostructures. [150,209,210]In this we discuss the range of successful strategies for optimizing thermoelectric performance, along with an overview of the latest advancements in diamondoid thermoelectric compounds.

| Ag-alloying and domain structure
In general, Cu-based diamondoid compounds display moderate electrical conductivities but have high intrinsic lattice thermal conductivities.On the other hand, as explained above, Ag-based compounds typically exhibit very low thermal conductivities and inferior electronic properties.Due to their decent electrical conductivity and power factor, Cu-based diamondoid compounds are commonly chosen as the starting matrix.Consequently, reducing the lattice thermal conductivity becomes crucial in achieving high thermoelectric performance for diamondoid materials.Numerous efforts have been dedicated to suppressing the lattice thermal conductivity of Cu-based diamondoid materials, with alloying Ag on the Cu site proving be the most effective method. [158]lloying elements into a crystal solid introduces additional mass and strain fluctuation scattering of phonons, as seen in Figure 9A.The intrinsic offcentering behavior of Ag in tetrahedral coordination further influences thermal conductivity when Ag is alloyed into a diamondoid lattice.This makes Agalloying the most effective strategy to suppress thermal conductivity in diamondoid materials, as depicted in Figure 9B,C.By combining the alloy scattering and offcentering effect, extremely low lattice thermal conductivities of 0.47 and 0.44 Wm −1 K −1 can be achieved at 850 K in the Cu 1−x Ag x InTe 2 [121] and Cu 1−x Ag x GaTe 2 , [123] respectively.This results in a ZT of ~1.6 at 850 K in these materials.Furthermore, when mixing the Cu 1−x Ag x InTe 2 with Cu 1−x Ag x GaTe 2 , researchers found that the  [185] Copyright 2020, American Chemical Society.
Cu 1−x Ag x In 1−y Ga y Te 2 possesses a single diamondoid structure, while its chemical composition is [110] see Figure 9D.This leads to the spatial fluctuations of lattice parameters, thus resulting in a spatially inhomogenous strain field.Besides, this also leads to a hierarchical microstructure, where some small grains with nanometer size precipitate in the large grains, which substantially increases the grain boundaries.This domain structure in grain and inhomogeneous chemical composition contributes to an extremely low thermal conductivity of ~0.25 Wm −1 K −1 in Cu 1−x Ag x In 1−y Ga y Te 2 , and then results in a high ZT of 1.64 at 873 K, [110] as shown in Figure 9E,F.

| Introducing nano-precipitates
The p-type CuInTe 2 compound has gained significant attention due to its high Seebeck coefficient, prompting efforts to enhance its thermoelectric performance.One successful approach is incorporating nano secondary phases of ZnQ (Q = O, S, Se, and Te) into the lattice. [139,141,207]This has been achieved by directly mixing ZnS and ZnSe powders into the CuInTe 2 matrix, followed by hot press sintering. [184]This resulted in the dissolution of part of the ZnS/ZnSe into the matrix, creating Zn In point defects that substantially increased hole density and improved electrical conductivity.The room temperature experimental thermal conductivities and theoretical thermal conductivities of Cu 1−x Ag x GaTe 2 ; the solid line indicates the combined effect of alloy scattering and the off-centering effect; and the dash-dotted line is calculated considering only the effect of alloy scattering.(C) Temperature dependence of thermal conductivity of Cu 1−x Ag x GaTe 2 .Reproduced with permission. [158]Copyright 2023, American Chemical Society.Reproduced with permission. [110]Copyright 2019, Wiley-VCH.
Additionally, oversaturated ZnS/ZnSe formed nanoprecipitates (100-200 nm) that were uniformly distributed throughout the matrix, as depicted in Figure 10A,B.These nano-precipitates enhance scattering and effectively suppress the lattice thermal conductivity.Moreover, they purportedly introduce an energy filtering effect, which plays a significant role in scattering the minority carrier and enhancing the Seebeck coefficient.
Generally, thermally excited minority carriers transport simultaneously with the majority carriers, which results in a reduction of the net Seebeck coefficient.However, for the CuInTe 2 -ZnS/ZnSe materials, the large energy difference of the conduction band generates a minority carrier barrier at the CuInTe 2 -ZnS/ZnSe interface, as shown in Figure 10C.This strongly restricts the transport of electrons (minority carriers), while having less effect on the holes (majority carriers), thus increasing the Seebeck coefficient and the thermoelectric performance.By introducing the nanoscale ZnS/ZnSe heterostructure barrier, a high ZT value of 1.52 was achieved. [207]hen introducing ZnO into CuInTe 2 , it was discovered that ZnO reacts with CuInTe 2 during the hot press sintering process. [139]This reaction results in the in situ formation of In 2 O 3 nano-inclusions and leaves Zn In − defects in the matrix.The Zn In − defect significantly increases the hole concentration of the material, improving its electrical conductivity.At the same time, the In 2 O 3 nano-inclusions seem to enhance phonon scattering and suppress heat conductivity, thereby improving the thermoelectric performance.Moreover, it was observed that the Fermi levels of n-type In 2 O 3 and p-type CuInTe 2 move in opposite directions and enlarge the energy barrier between the interface with increasing temperature, as shown in Figure 10E,F.This leads to the higher carrier scattering at high temperature, resulting in a large Seebeck coefficient and power factor, thus contributing to a high ZT.

| Hierarchical and ordered microstructures
Constructing the hierarchical microstructure has been demonstrated as a successful method to suppress the lattice thermal conductivity, [43] which has been applied to improve the thermoelectric performance in many materials, including diamondoid compounds.One successful case is the well-organized two-phase structure observed in the CuFeS 2 /Cu 1.1 Fe 1.1 S 2 mixture formed by using the so-called combustion synthesis approach. [209]I G U R E 10 Introducing nano-precipitates to optimize thermoelectric performance.TEM images show the nano-precipitates of (A) ZnS and (B) ZnSe in CuInTe 2 ; Reproduced with permission. [207]Copyright 2017, Elsevier.(C) The schematic diagram of the hole and electron transport in the energy band of a p-type semiconductor with a minority blocking barrier.(D) TEM image shows the nano-precipitates of In 2 O 3 in CuInTe 2 .Reproduced with permission. [141]Copyright 2015, Elsevier.Schematic illustration of hole transport mechanism in the heterojunction interface of CuInTe 2 and In 2 O 3 at (E) low and (F) high temperatures.
The combustion synthesis process is to press the wellmixed raw elemental powders into a pellet, and heat the pellet to its "ignition" temperature to trigger the reaction.[213] Because both the CuFeS 2 and Cu 1.1 Fe 1.1 S 2 can be synthesized by the combustion synthesis process, designing the starting composition of the mixed powders and controlling the combustion synthesis process, a stripe-type two-phase stacking structure can be obtained in the CuFeS 2 /Cu 1.1 Fe 1.1 S 2 composite, [209] see Figure 11A.This composite structure exhibits suppressed thermal conductivity.Moreover, since the Cu Reproduced with permission. [209]opyright 2017, Springer Nature.(C) Microstructure of CuFeS 2 /ZnS composite.Reproduced with permission. [208]Copyright 2016, Wiley-VCH.(D) Microstructure of 3D flower-like hierarchical Cu 3 SbSe 4 microspheres.Reproduced with permission. [210]Copyright 2018, Elsevier.
from Cu 1.1 Fe 1.1 S 2 to CuFeS 2 , which raises the carrier concentration of CuFeS 2 (main phase), as depicted in Figure 11B.This high carrier concentration is akin to the modulation doping, enhancing the overall electrical conductivity of CuFeS 2 /Cu 1.1 Fe 1.1 S 2 composite.
In the Zn-doped CuFeS 2 compound, a unique hierarchical microstructure has been identified. [208]The maximum amount of Zn that can dissolve in CuFeS 2 is less than 3%.When excess Zn is added and the mixture is heated to 1323 K, Zn initially dissolves in the CuFeS 2 melt.However, as the temperature decreases, the solubility limit for Zn is reduced, resulting in the spontaneous formation of ZnS nanoparticles.A eutectic reaction then occurs between ZnS and CuFeS 2 , resulting in the layered distribution of ZnS nanoparticles, [208] Figure 11C.These ZnS nanoparticles enhance phonon scattering and effectively suppress the lattice thermal conductivity of CuFeS 2 .
For the Cu 3 SbSe 4 diamondoid compound, the microwave-assisted solvothermal synthesis method has been employed to synthesize self-assembled flower-like hierarchical Cu 3 SbSe 4 microspheres, [150,210] as shown in Figure 11D.By controlling the synthetic process, the flower-like hierarchical structure of Cu 3 SbSe 4 can be tailored.This unique hierarchical structure can strengthen phonon scatterings in different wavelengths (Figure 11E), leading to a low thermal conductivity of 0.38 Wm −1 K −1 at 623 K in Cu 3 SbSe 4 . [210]| OUTLOOK In summary, the diamondoid materials' phonon and electronic transport properties are strongly related to the lattice dynamics and crystal symmetry.The diamond structure should normally exhibit high thermal conductivity because of the close-packed tetrahedral coordination.However, the hidden local symmetry breaking and offcentering behavior leads to the distorted tetrahedral coordination and the extremely low thermal conductivity in Ag-based diamondoid compound.Moreover, introducing the lone pair electron or interstitial atoms could also effectively modify the coordination structure, thus could be the valid method to regulate the thermal conductivity for other diamondoid compounds.Conversely, crystal symmetry has a significant impact on the band structure.Finetuning the crystal structure can lead to band convergence within the material.Additionally, many diamondoid compounds exhibit adjustable in-gap states, which can enable dynamic doping effects to enhance thermoelectric performance.These intriguing phenomena have been utilized to develop high-performance diamondoid thermoelectric materials.As a result, a series of diamondoid compounds have shown impressive ZT values of 1.6-1.8.
Furthermore, diamondoid materials have also found significant applications in photoelectricity and nonlinear optics, and the aforementioned discoveries are expected to further advance their use in these fields as well.Overall, the exploration of diamondoid materials has opened up exciting possibilities in both thermoelectricity and other areas of technology and science.
Despite the promising thermoelectric properties exhibited by diamondoid materials, there are several challenges that need to be addressed for their practical applications.One major concern is the high cost associated with some of the advanced diamondoid thermoelectric materials due to the inclusion of rare and expensive elements like Indium and Tellurium.Additionally, while these materials may demonstrate high ZT values at high temperatures (T > 700 K), their thermoelectric performance in the lower temperature range (300-700 K) is relatively lower.Moreover, another limitation is the scarcity of n-type diamondoid materials with favorable thermoelectric performance.The existing n-type diamondoid materials often possess inferior thermoelectric properties, which hampers the development of thermoelectric devices based on diamondoid materials.To overcome these challenges and unlock the full potential of diamondoid thermoelectric materials, future research should focus on fundamental understanding of their complex chemistry and physics as well as finding cost-effective alternatives to the expensive elements currently used.Additionally, efforts should be directed toward improving the thermoelectric performance at lower temperatures and exploring novel high-performance n-type diamondoid materials.Besides, revealing and improving the mechanical properties of diamondoid thermoelectric materials [214][215][216] are also important to their applications, and further study should be conducted.Since the diversity and full miscibility of diamondoid compounds can offer a vast space for optimizing thermoelectric performance and discovering novel candidates for thermoelectric applications, we believe by leveraging the high throughput calculations and machine learning techniques, more highperformance diamondoid thermoelectric materials could be identified, ultimately leading to more practical and sustainable thermoelectric solutions.

F
I G U R E 1 The development of diamondoid thermoelectric materials.(A) Structure evolution of diamondoid materials.(B) Current state-of-the-art bulk diamondoid thermoelectric materials, the figure-of-merit ZT as a function of temperature and year illustrating important milestones, the red rectangular bars represent the n-type materials, while green bars represent the p-type ones. [101,102,107-128]THERMAL CONDUCTIVITY This section delves into the fascinating realm of diamondoid compounds, where recent breakthroughs have seen the attainment of ZT values beyond 1.6 in ternary I−III−VI 2 diamondoid compounds.This significant progress has been primarily achieved by reducing the lattice thermal conductivity through Ag alloying.Strikingly, despite possessing identical crystal structures, Ag-based diamondoid compounds exhibit considerably lower intrinsic lattice thermal conductivity compared to their Cu-based counterparts.This phenomenon has puzzled researchers for some time.However, recent advances in synchrotron X-ray measurements have shed light on the true nature of local structures and heat transport behaviors in these diamondoid compounds.

F
I G U R E 2 Origin of off-centering behavior and its effect on thermal conductivity.(A) The molecular orbital diagram of Ag-Te bonding.(B) Local Ag environment (left), local distortion (middle), and coupling of local distortions on neighboring AgTe 4 tetrahedra (right) in AgGaTe 2 .(C) Temperature evolution of Ag and Te local displacements in AgGaTe 2 .(D) Phonon dispersions and phonon density of states for AgGaTe 2 , the insert shows the distorted AgTe 4 tetrahedra.(E) The spectral lattice thermal conductivity κ(ω) for AgGaTe 2 .(F) Crystal structure of ZnTe and diamondoid materials, the η is the tetragonal distortion parameter.(G)

F
I G U R E 3 The effect of lone-pair electron on thermal conductivity.(A) and (B) The bonding environment of In + and In 3+ in CuFeS 2 , the discordant In + caused the local distorted structure and the distorted weak chemical bonding.(C) The energy profile of In + substitute of Cu in CuFeS 2 at various positions.(D) Room temperature lattice thermal conductivity dependence on the alloying content for the discordant case In Cu and the accordant In Fe .Reproduced with permission.

F
I G U R E 4 The effect of cation disorder, interstitial atoms and intrinsic vacancy on thermal conductivity.(A) Crystal structures of chalcopyrite CuFeS 2 , quadruple chalcopyrite Cu 16 Fe 16 S 32 , and talnakhite Cu 17.6 Fe 17.6 S 32 .(B) Lattice thermal conductivity of CuFeS 2 and Cu 17.6 Fe 17.6 S 32 as a function of temperature, in comparison to other typical thermoelectric materials.(C) Temperature dependence of the lattice thermal conductivity of CuFeS 2 and Cu 17.6 Fe 17.6 S 32 ; Reproduced with permission.

F I G U R E 6
Nontrivial Band Structure in diamondoid materials.(A) Band structure of Cu 0.5 Ag 0.5 GaTe 2 , the insert shows the enlarged view of the valence band near the Γ point.The horizontal dashed lines indicate the Fermi levels with different hole concentrations.(B) Temperature-normalized Seebeck coefficient as a function of the carrier concentration for Cu 0.7 Ag 0.3 GaTe 2 .The dashed curves show the parabolic band model corresponding to different effective masses.The solid curve shows the S ∼ p −1/3 relationship based on the quasilinear band model.(C) Seebeck coefficient as a function of carrier concentration for Cu 0.7 Ag 0.
In-gap states and the temperature dependent carrier concentration.(A) Schematic diagram of the in-gap acceptor levels.(B) Temperature dependence of the carrier concentration for Cu 1−x Ag x GaTe 2 .(C) Temperature dependence of the electrical conductivity for Cu 1−x Ag x GaTe 2 ; Reproduced with permission. [123]Copyright 2022, American Chemical Society.(D) Energies of the conduction band minimum (CBM), valence band maximum (VBM), and impurity level for Cu 0.8 Ag 0.2 In 1−x Ga x Te 2 .(E) Temperature dependence of carrier concentration for Cu 0.8−y Ag 0.2 In 0.2 Ga 0.8 Te 2 compounds.(F) The figure of merit ZT for CuInTe 2 , Cu 0.8 Ag 0.2 InTe 2 , Cu 0.8 Ag 0.2 In 0.2 Ga 0.8 Te 2 , and Cu 0.78 Ag 0.2 In 0.2 Ga 0.8 Te 2 .

F I G U R E 8
Relaxation time modification and Seebeck coefficient enhancement.Schematic diagrams showing (A) an increase in the density-of-states (DOS) and (B) the relaxation time modification leading to an increase of the term −∂f 0 /∂ε and an enhancement in the Seebeck coefficient.(C) Scanning tunneling spectrum and a topographic image obtained using STM on a CuFeS 2 crystal, the dashed lines indicate the energy of the conduction band minimum (CBM) and valence band maximum (VBM).(D) Temperature dependence (7−300 K) of the Seebeck coefficient, Hall mobility, and the calculated values of dμ H /dT for CuFeS 2 , the dμ H /dT value indicates the Seebeck coefficient derived from the steep mobility gradient.Reproduced with permission.

F I G U E 9
The role of Ag off-centering and domain structure in thermal conductivity.(A) Schematic diagrams of the role of mass fluctuation scattering, strain fluctuation scattering, and off-centering effect in phonon transport.(B) (D) Scanning electron microscopy (SEM) image and electron backscattered diffraction (EBSD) microstructures for Cu 0.7 Ag 0.3 Ga 0.4 In 0.6 Te 2 .(E) Schematic image of the transport process for holes and phonons in the domain structure.(F) Temperature dependence of ZT values for Cu 1−x Ag x Ga 0.4 In 0.6 Te 2 .

1 . 1
Fe 1.1 S 2 compound has a relatively high carrier concentration, the Fermi level imbalance in the interface with CuFeS 2 prompts electron diffusion F I G U R E 11 Constructing well-organized nano-structure to improve thermoelectric performance.(A) Microstructure of CuFeS 2 /Cu 1.1 Fe 1.1 S 2 composite.(B) Schematic description of the electron redistribution at the CuFeS 2 /Cu 1.1 Fe 1.1 S 2 phase boundary.
shows the crystal structure of CuFeS 2 and Cu 17.6 Fe 17.6 S 32 .Obviously, the crystal structure of Cu 17.6 Fe 17.6 S 32 is closely related and possesses similar atomic sites with the quadruple supercell of CuFeS 2 (Cu 16 Fe 16 S 32 ).The Cu 16 Fe 16 S 32 structure has eight cation positions that are not occupied, but in Cu 17.6 Fe 17.6 S 32 , these positions are partially occupied by the extra Cu and Fe atoms (marked as red spheres in the crystal structure), making the Cu 17.6 Fe 17.6 S 32 be a "stuffed" CuFeS 2 structure.The presence of interstitial cations adds several low-frequency optical vibration modes.These modes interact with heat-carrying acoustic phonons, resulting in significant acoustic-optical phonon