Si2Ge: A New VII-Type Clathrate with Ultralow Thermal Conductivity and High Thermoelectric Property

Based on global particle-swarm optimization algorithm and density functional theory methods, we predicted an alloyed Si2Ge compond with body centered tetragonal type VII clathrate (space group I4/mmm) built by a truncated octahedron fromed by six quadrangles and eight hexagons ([4668]). Si2Ge clathrate is 0.06 eV/atom lower than VII Si clathrate and thermally stable up to 1000 K. It has an indirect band gap of 0.23 eV, high p-doping Seebeck coefficient and n-doping electrical conductivity. It owns a low lattice thermal conductivity of 0.28 W/mK at 300 K because of its weak bonding and strong anharmonic interaction of longitudinal acoustic and low-lying optical phonons. The moderate electronic transport properties together with low lattice thermal conductivity results in a high optimal thermoeletric performance value of 2.54 (1.49) at 800 (1000) K in n (p)-doped Si2Ge.


Methods
Structure prediction. We employ the efficient particle swarm optimization (CALYPSO) code 24 to search for low-energy 3D Si 2 Ge clathrate. The number of formula units per simulation cell is set to be 1~2. Unit cells containing total number atoms of 6 and 12 are considered. The structure relaxations are performed using Vienna ab initio simulation package (VASP) 25,26 . The projector-augmented plane wave (PAW) approach 27 is used to represent the ion-electron interaction. The generalized gradient approximation in the form of Perdew, Burke and Ernzerhof (PBE) is adopted 28 . The plane-wave cutoff energy for wave function is set to 600 eV. Monkhorst-Pack k-mesh of 5 × 5 × 5 is adopted to represent the first Brillouin zone. For structure optimization, the convergence thresholds are set to 10 −7 eV and 10 −3 eV/Å for total energy and force component, respectively.
Electronic and phonon structure. The Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional 29,30 are also used for the high accuracy of electronic structure calculations. The plane-wave cutoff energy for wave function is set to 400 eV. Monkhorst-Pack k-mesh of 7 × 7 × 7 is adopted to represent the first Brillouin zone. Ab initio molecular dynamics (AIMD) simulations at different temperatures are performed using the canonical ensemble (NVT) with the Nosé thermostat 31 to examine thermal stability. Simulations lasted for 10 ps with a time step of 1 fs at the temperature of 500, 1000, and 1200 K were carried out. Phonon spectrum calculation is carried out using the linear response method within density functional perturbation theory 32 implemented in the Phonopy code 33 . TE performance calculation. Based on the Boltzmann transport theory, the Seebeck coefficient, the ratio of electrical conductivity to electrical relaxation time and the electronic thermal conductivity are evaluated by using the semiclassical Boltzmann transport theory with the relaxation time approximation, which is implemented in the so-called BoltzTraP code 34 . Here it is assumed that the acoustic phonon is the main scattering mechanism, we calculated carrier mobility by the deformation potential (DP) theory 35 as following 36,37 ⁎ ⁎ e m e m m k E where μ is carrier mobility, ⁎ Ι m is inertial effective mass, m s is the density of states effective mass of a single band, ρ is the crystal mass density, υ is the average sound velocity from phonon dispersion listed in Table S1 (Supplementary Information). The term E 1 represents the deformation potential constant of the valence-band minimum (VBM) for hole or conduction-band maximum (CBM) for electron along the transport direction. The deformation potential constant (E 1 ) is calculated by the linear fitting of the CBM (VBM)-strain relation, the result is shown in Fig. S1 (Supplementary Information). With E 1 , and the effective mass is known, the carrier motilities are calculated by Eq. (1).
Lattice thermal conductivity. The first-principles lattice thermal conductivity κ L was calculated by solving Boltzmann transport equation for phonons. The interatomic force constants (IFCs) were calculated within a real-space supercell approach using the Phonopy package 33 for the two-order harmonic IFCs and the ShengBTE package 38 for the thirdorder anharmonic IFCs. The IFCs were calculated using a 3 × 3 × 3 supercell with a 19 × 19 × 19 q-mesh. The electron-phonon (e-p) coupling properties are obtained using the Quantum Espresso package 39 with ultrasoft pseudopotentials, energy cutoff of 40 Ry and a q-grid of 8 × 8 × 8.

Results and Discussion
The stable structure of Si 2 Ge obtained from global structure search is shown in Fig. 1. The optimized Si 2 Ge crystallizes in the Tetragonal space group, I4/mmm (no. 139), with a = b = 6.759 Å, c = 6.868 Å (Fig. 1). The lattice strain is mostly induced by the distorted tetrahedral coordination of SiGe alloy, or, alternatively, by the 90.2° (∠GeSiGe) and 89.7° (∠SiGeSi) of 4-membered (Si 2 Ge 2 ) rings along c direction. The 3D framework is composed of a 24-atom tetrakaidecahedra (Si 8 Ge 4 )2 formed by four-fold coordination of Si at 8j and Ge at 4d sites (Fig. 1b,d). The clathrate-forming polyhedron is a truncated octahedron, so-called clathrate-VII pattern 40 , formed by six quadrangles and eight hexagons ([4 6 6 8 ]). The Si 2 Ge-VII clathrate is 0.11 and 0.06 eV/atom lower in energy than Si and Ge-VII clathrates, but higher than those well-known Si-II and Si-VIII clathrates (0.10 and 0.07 eV/atom) because of containing a large number of four-membered rings resulting strained in comparison to type II frameworks 41 34-x Ge x alloy clathrate 6,18 . Generally, a longer bond length corresponds to weaker bond interactions, and weak bond interaction decrease the speed of the sound, which conversely drop the thermal conductivity of the lattice 43 . Therefore Si 2 Ge-VII clathrate shows relatively stable and weak covalent bonds which is responsible for the low lattice thermal conductivity.
Also, we simulate the thermal stability of Si 2 Ge clathrate. A 3 × 3 × 3 supercell is used in the simulations at temperatures of 500, 1000 and 1200 K by performing ab initio molecular dynamics (AIMD) simulations. The snapshots of the geometries at the end simulations show that Si 2 Ge clathrate can maintain its original configuration at temperature up to 1000 K ( Fig. 2). At 1200 K, some bonds begin to break and lead to cage structure distorted. The radical distribution functions (RDF, Fig. S2, Supplementary Information) at 500 K and 1000 K have also shown the typical feature of VII-type clathrate. When the temperature reaches 1200 K, RDF exhibit a few feature of liquid. This indicates that Si 2 Ge has a melting/decomposition temperature close to that of Si and Ge-based clathrates. For instance, Ba 8  www.nature.com/scientificreports www.nature.com/scientificreports/ The well-preserved geometry of Si 2 Ge at such high temperature suggests the thermal stability of Si 2 Ge clathrate and its possible utilization at a high temperature. Figure 3a and Table 1 show the band structure, effective mass and carrier mobility for Si 2 Ge. It is shown that an indirect band gap of 0.23 eV for Si 2 Ge from Fig. 3a. The valence band maximum (VBM) is located at the Z point with 3-degeneracy, are named by VB1, VB2 and VB3 in Fig. 3a. The conduction band minimum (CBM) is along the Z-Γ line of 2-degeneracy imposed by the symmetry of the Brillouin zone which is shown in the inset of Fig. 3a. It is obvious that p-type doped will display slightly higher degeneracy of carrier pockets than that of n-type doped Si 2 Ge. It is well known the Seebeck coefficient is proportional to the density of state effective mass 2,46 , given by m d  The optimized Si 2 Ge was used as the initial structure. The temperature of the system was controlled at 500 K, 1000 K and 1200 K. The estimated melting temperature is around 1200 K. Yellow apex: Si atom; green apex: Ge atom. The dotted red lines represent the broken bonds.
www.nature.com/scientificreports www.nature.com/scientificreports/ m I * of the conduction band is beneficial to increase µ and then enhance ZT performance. Therefore, it is clear that the light mass plays a crucial role in carrier transport and TE performance 48 .
Generally, the deformation potential (DP) theory overestimates the mobility due to the neglect of scatterings from other phonon modes 49 . The calculated average e-p coupling constant (λ) is to be about 0.082 from the dominated three acoustic branches using Quantum Espresso package. Such weak e-p coupling indicates that the low carrier scattering rates from e-p coupling and large carrier relaxation time of e-p coupling. The detail e-p coupling constants vs. frequency is shown in Fig. S3 (Supplementary Information). Seen from Fig. S3, low frequency phonons, especially those less than 2 THz, have greater e-p coupling than that of high frequency phonons and have strong carrier scattering rates. The phonons in this region are mainly derived from the acoustic branches. Therefore, for Si 2 Ge, deformation potential method can give a reasonable carrier relaxation time. Fig. 3b shows the calculated phonon structure of Si 2 Ge clathrate. The low frequency vibrations, <4 THz, are strongly contributed from Ge atoms. Three extremely anomalous low-lying optical (LLO) phonons are overlapped with the longitudinal acoustic (LA) phonons. The boundary frequency of LLO1 branch at the Γ point is about 1.2 THz (43 cm −1 ), is similar to most of the LLO phonons in other low κ L PGEC compounds, for example, Yb filled skutterudites (42 cm −1 ) 50 and Ba 8 Ga 16 Ge 30 (44 cm −1 ) 51 . LLO branches have such large phonon dispersion slope near the Γ point, which means high phonon velocity and strong anharmonic behaviour and may be provided essential scattering channels for heat-carrying phonons, similar to that of PbTe 52-54 . More importantly, the "avoided crossing" interaction between LLO and longitudinal acoustic (LA) branches has been observed in Fig. 4a along Z-Γ line at 1.5 THz. There is a small gap at avoided crossing point indicates strength of coupling between LA and LLO modes seen from the inset of Fig. 4a. It leads to enhance the phonon scattering rates and reduce acoustic mode velocities, and then result the low κ L . Figure 4b shows the Grüneisen parameter (γ) for Si 2 Ge as a function of the phonon frequency. The γ shows similar features as the Si-VII 55 , where negative γ are spread out at low frequency values. TA and LLO branches possess high absolute γ, typically, the minimum γ is extraordinarily low ~−14.16. The average Grüneisen parameter calculated from ShengBTE is 3.19 at 300 K. This value is a little larger than that of AgSbSe 2 (3.05, a low thermal conductivity material, 0.48 W/mK at 300 K) 56 . The acoustic and LLO modes have much larger absolute γ and play an important role in lattice thermal resistance of Si 2 Ge.
The phonon scattering rates (SC) related to phonon-phonon interactions (PPI) and electron-phonon (EPI) are shown in Fig. 5a. The phonon-phonon SC from acoustic phonons is as low as the order of 0.006 ps −1 , while the low lying optical phonons is in the range of 0.06~8 ps −1 and are 1-2 orders of magnitude higher than acoustic branches with frequencies above ~5 THz for Si 2 Ge clathrate. High SC around 5 THz from flat optical phonons.    66 , m 0 is the free electron mass), the density of states effective mass of a single band (m s ), inertial effective mass (m I * ), density of state effective mass (m d * ), carrier mobility (μ) and predicted relaxation time (τ) for hole and electron of Si 2 Ge at room temperature (300 K), Number of degenerate carrier pocket (N v ), deformation potential constant (E 1 ).
One can see the electron-phonon SC due to EPI is much smaller than the phonon-phonon scattering. Si 2 Ge has stronger lattice anharmonicity, as a consequence, electron-phonon scattering nearly has no contributions to the lattice thermal transport.
Based on ShengBTE, Si 2 Ge actually possess a low lattice thermal conductivity seen from Fig. 5b. With the temperature rising the lattice thermal conductivity decreases monotonically. At 300 K, lattice thermal conductivity is 0.28 W/mK, which is lower than majority of clathrates, such as Sr 8 Ga 16 Ge 30 (0.9 W/mK) 45 , Sn-based clathrates (~1 W/mK) 57 , and comparable to the unconventional transition metal-phosphorus clathrates with ordered superstructures and heavy elements, such as Ba 8 Cu 16 P 30 (~0.3 W/mK) 58 and Ba 8 Au 16 P 30 (~0.2 W/mK) 59 . At 1000 K the lattice thermal conductivity decreases dramatically to ~0.12 W/mK, which is lower than that measured for SnSe single crystals at 973 K (0.23 ± 0.03 W/mK) 60 . The inset of Fig. 5b shows the cumulative lattice thermal conductivity vs. phonon frequnency of Si 2 Ge. We found that the lattice thermal conductivity increases quickly with ω in the low-frequency region. By setting a cutoff of 4 THz, the accumulated thermal conductivity is found to be as high as ~73%, which means low frequency (<4 THz) phonons may make an importance role on κ L due to low scattering rates because of large group velocity of acoustic modes which are mainly from vibration of Ge discussed in the previous description (see Fig. 3b). The high-frequency optical phonons have SC of 1 ps −1 , which are less contribution on heat current. The cumulative lattice thermal conductivity divided by total lattice thermal conductivity of Si 2 Ge with respect to phonon mean free path (MFP) at 300, 500 and 1000 K, are plotted in in Fig. 5c. As the MFP increases, the normalized κ L integration increases, and then approaches 1. It is found that the thermal conductivities are dominated by phonons with MFPs ranging from 0.1 to 5 µm at room temperature. At width about 70 nm, the lattice thermal conductivity drops about 50%. At high temperatures, the phonon MFPs become even shorter, the MFP corresponding to the median κ L accumulation in Si 2 Ge reduces from 33 nm at 500 K to 19 nm at 1000 K. The phonon MFPs in Si 2 Ge are notably longer than those in other clathrate (around 10 nm at 300 K for Type-I Si clathrate) [61][62][63] , which means κ L of Si 2 Ge is more sensitive to size effects.
The electronic thermal conductivity (κ e ) was evaluated via Wiedemann-Franz law: κ e = L 0 σT with L 0 = 2.44 × 10 −8 W·Ω/K 2 . The Seebeck coefficient S, electrical conductivity σ, and TE power factor S 2 σ (PF) as a function of carrier concentration at 300 K have been shown in Fig. 6. Clearly, p-doped Si 2 Ge has the higher  Seebeck coefficient than n-dope ones over the full carrier concertation range (0.01~10 × 10 20 cm −3 ), while the higher conductivity values of electrons than that of holes. This consistent with the discussed above. Since S decreases as carrier concentration where σ increase, the maximum power factor is 0.63 mW/mK 2 at the hole concentration of 1.91 × 10 20 cm −3 , while 2.81 mW/mK 2 at the electron concentration of 4.31 × 10 19 cm −3 . From  Fig. 6c, at 300 K, the n-type power factor is much higher than p-type, which further confirmed that the low effective mass contributes to the enhancement of the TE performance.
ZT at different temperature vs. carrier concentration is plotted in Fig. S4 (see Supplementary Information). The ZT value is peaked at a specific carrier concentration at the different temperature. For electrons at room temperature, the peaked ZT value is predicted to be 0.41 at 5.41 × 10 18 cm −3 and that for holes is 1.09 at 5.41 × 10 19 cm −3 . This peaked ZT value is named the maximum ZT (ZT max ). ZT max as a function of temperature is plotted in Fig. 7, which demonstrates a linear increase below 800 K and then decrease for n-doped, while a linear increase with temperature for p-doped. The highest ZT max achieved at 800 K is 2.54 for n-doped Si 2 Ge clathrate and 1.49 for p-doped at 1000 K. These values are superior those realized in K 8 Ba 16 Ga 40 Sn 96 (n-type, 1.12 at 637 K) 64 , and type-I Ba 8 Ga 16 Ge 30 , (p-type, 1.10 at 823 K) 65 .
To summarize, we extend a new clathrate materials, namely Si 2 Ge-VII clathrate on basis of global structure search and density functional theory. This clathrate has a tetrakaidecahedral lattice similar to sodalite and exhibits excellent thermal and dynamical stabilities. Si 2 Ge clathrate has an indirect band gap of 0.23 eV, with higher p-doping Seebeck coefficient owing to higher hole density-of-sates mass and higher n-doping electrical conductivity thanks to lower electron effective mass. Interestingly, it owns a low lattice thermal conductivity due to its weak bonding interaction and strong anharmonic LA-LLO coupling results in avoided-crossing. The fascinating electronic properties together with the low lattice thermal conductivity make Si 2 Ge clathrate a promising TE