J Mater InfJournal of Materials InformaticsOAE Publishing Inc.10.20517/jmi.2021.03Research ArticlePrediction of the atomic structure and thermoelectric performance for semiconducting Ge_{1}Sb_{6}Te_{10} from DFT calculationsGanYu^{1}ZhouJian^{1}SunZhimei^{1}^{2}^{1}School of Materials Science and Engineering, Beihang University, Beijing 100191, China.^{2}Center for Integrated Computational Materials Engineering, International Research Institute for Multidisciplinary Science, Beihang University, Beijing 100191, China.Correspondence Address: Prof. Zhimei Sun, School of Materials Science and Engineering, Beihang University, No. 37 Xueyuan Road, Haidian District, Beijing 100191, China. Email: zmsun@buaa.edu.cn
Received: 6 Jul 2021  First Decision: 10 Aug 2021  Revised: 17 Aug 2021  Accepted: 24 Aug 2021  Available online: 28 Aug 2021
Academic Editor: Xingjun Liu  Copy Editor: XiJun Chen  Production Editor: XiJun Chen
Pseudobinary alloys (GeTe)_{m}(Sb_{2}Te_{3})_{n} (GST), known as the most popular phase change materials for datastorage applications, also exhibit great potential as thermoelectric (TE) materials due to their intrinsically low lattice thermal conductivity (κ_{l}) and high electrical conductivity. Among the GST compounds, the Sb_{2}Te_{3}rich Ge_{1}Sb_{6}Te_{10} (m = 1 and n = 3) crystallizes into a complex trigonal structure with a 51layer long period stacked along the caxis, which may generate various possible atomic arrangements, thereby affecting the electronic and transport properties. Here, using ab initio calculations, we demonstrate that, besides the two experimentally known atomic sequences (GSTI and GSTIII), Ge_{1}Sb_{6}Te_{10} has two novel stable stacking configurations (GSTII and GSTIV). GSTIV exhibits semimetallic behavior, whereas GSTI and GSTII are semiconductors. Both semiconducting stackings have low κ_{l} of 0.86 and 0.78 Wm^{1}K^{1} at 300 K, owing to their small phonon group velocities and short phonon lifetimes. Moreover, they show a combination of high ntype Seebeck coefficient and electrical conductivity due to the steep slope of conduction band density of states near bandgap, multiple conduction pocket electrons, and multiband conduction. The maximum ZT values of 2.23 and 1.91 are achieved in ntype stackings GSTI and GSTII at 710 K. Our work sheds light on the great potential of Ge_{1}Sb_{6}Te_{10} with different atomic stackings for TE applications and will stimulate further experimental study. More importantly, from the perspective of materials informatics, this study provides significant insights that crystal systems with multilayered structures may open a viable route for creating new functional materials.
The pseudobinary chalcogenides (GeTe)_{m}(Sb_{2}Te_{3})_{n} (GST) with m, n = integer are wellknown as excellent phase change materials for nonvolatile memory devices due to their fast and reversible phase transition between crystalline and amorphous states^{[1,2]}. Besides the applications in information storage, GST compounds are also expected to be potential thermoelectric (TE) materials because of their intrinsically low lattice thermal conductivity (κ_{l}) and relatively large electrical conductivity (σ)^{[35]}. Specifically, these two properties are beneficial to enhance the dimensionless figure of merit ZT = S^{2}σT/(κ_{l} + κ_{e}) (where S is the Seebeck coefficient and κ_{e} is the electronic thermal conductivity), which characterizes the performance of a TE material. In recent years, many studies on GST alloys have indeed focused on their TE performance^{[612]}. For example, the ZT value of bulk GeTerich GST compound is as high as 1.3 at 723 K^{[7]}. More recently, it has been reported that maximum ZT (ZT_{max}) values of 0.40.6 were achieved at 750 K for three undoped quasitwodimensional GST systems (Ge_{2}Sb_{2}Te_{5}, Ge_{1}Sb_{2}Te_{4}, and Ge_{1}Sb_{4}Te_{7})^{[9]}. The Insubstituted hexagonal Ge_{2}Sb_{2}Te_{5} (i.e., Ge_{1.85}In_{0.15}Sb_{2}Te_{5}) gives the ZT_{max} of 0.78 at 700 K, a ~2fold improvement compared with the undoped system^{[8]}.
Despite the wide investigations of GST materials, however, the detailed atomic structures of the stable phase for many GST compositions have not yet been clarified, owing to their complex multilayered structures. For many GST compositions (e.g., Ge_{2}Sb_{2}Te_{5}, Ge_{1}Sb_{2}Te_{4}, Ge_{1}Sb_{4}Te_{7}, and Ge_{3}Sb_{2}Te_{6}), there is more than one proposed atomic stacking sequence considering the atomic positions of Ge and Sb^{[1316]}, thereby affecting their performance. In particular, it has been proposed that one of the stable atomic stackings of hexagonal Ge_{2}Sb_{2}Te_{5} is a topological insulator while the other is not at ambient condition^{[17,18]}. Moreover, we previously demonstrated that the atomic arrangements have great effects on the electronic, mechanical, and vibrational properties of layered GST systems^{[16,19,20]}. Besides, their lattice thermal conductivity and electrical transport properties show a significant dependence on the atomic stacking, thus resulting in differences in thermoelectric performance^{[10,21]}. These facts suggest that understanding the structural arrangement of stable GST compounds is essential for their technological applications.
Among the GST alloys, crystalline Ge_{1}Sb_{6}Te_{10}, which contains rich Sb_{2}Te_{3} units (i.e., n = 3), adopts a very complicated multilayered trigonal symmetry (R̅3m, No. 166) with 51 atomic layers (within a unit cell) stacked along the caxis via van der Waals (vdW) forces between TeTe layers^{[22,23]}. Such a complex structure with a very long layer period renders Ge_{1}Sb_{6}Te_{10} a promising TE material^{[24,25]} and offers great possibilities to explore its atomic stackingpolymorphs. Therefore, to get insights into the atomic stacking structures of Ge_{1}Sb_{6}Te_{10} and the corresponding effects on electronic and transport properties, we start our investigations from seven possible atomic arrangements [the inset of Figure 1, and Supplementary Figure 1 and Supplementary Figure 2 of Supplementary Material (SM)]. Using ab initio density functional theory (DFT) calculations, we first examine their structural stability. In addition to the two experimentally documented atomic stackings (GSTI^{[22]} and GSTIII^{[23]}), two types of novel state configurations, GSTII and GSTIV, are successfully identified. Then, we study the electronic structures of stackings GSTI, GSTII, and GSTIV and find that they vary between semimetal (GSTIV) and semiconductor (GSTI and GSTII). Finally, we focus on the transport properties and thermoelectric performance of semiconducting GSTI and GSTII. The results show that the lattice thermal conductivities of both stackings are below 1 Wm^{1}K^{1} at room temperature, which is attributed to their low phonon group velocities and short phonon lifetimes. In addition, they exhibit high ntype power factors, resulting from the large Seebeck coefficients and high electrical conductivities. Owing to the favorable thermal and electrical transport properties, the ZT_{max} values of ntype GSTI and GSTII reach up to 2.23 and 1.91 at 710 K, suggesting their great promise as mediumtemperature TE materials. Our work sheds light on the great potential of Ge_{1}Sb_{6}Te_{10} with different atomic stackings for TE applications and will stimulate further experimental study.
Calculated total energy for the seven atomic stackings of Ge_{1}Sb_{6}Te_{10}. The inset corresponds to the basic stacking units with 17 atomic layers of each configuration.
Materials and methods
Ab initio calculations were performed within the framework of DFT using the Vienna ab initio simulation package (VASP)^{[26]}. All theoretical calculations in this work were carried out automatically and intelligently using our newly developed ALKEMIE platform^{[27]}, in which automated workflows for structural relaxation, electronic structure, and transport properties calculations have been implemented. It is a useful and powerful informatics toolkit for materials science. We employed the projector augmented wave potential^{[28]} combined with the generalized gradient approximations (GGA) of PerdewBurkeErnzerhof (PBE) exchangecorrelation functional^{[29]}, where the valence electronic configurations are 4s^{2}4p^{2}, 5s^{2}5p^{3}, and 5s^{2}5p^{4} for Ge, Sb, and Te atoms, respectively. A semiempirical DFTD2 method^{[30]} was used for describing the vdW interactions in the layered crystal structure. The kinetic energy cutoff for the planewave basis set was 400 eV and the energy convergence criterion was 1 × 10^{6} eV. The crystal structure was fully optimized with the force convergence criterion of 1 × 10^{2} eV/Å and a 9 × 9 × 1 Γcentered kpoint grid. To accurately calculate the electronic bandgap, we also applied the HeydScuseriaErnzerhof screened hybrid functional (HSE06)^{[31]}, with the mixing parameter α of the HartreeFock exchanges to 0.25. Ab initio molecular dynamics (AIMD) simulations were carried out using supercells containing 204 atoms, and a canonical NVT (constant number, volume, and temperature) ensemble with Nosé thermostat^{[32]} was chosen. All AIMD simulations run for 30 ps with a timestep of 3 fs. We employed the LOBSTER code to calculate the projected crystal orbital Hamilton populations (pCOHP)^{[33]}.
The electronic transport properties were calculated by solving the semiclassic Boltzmann transport equation (BTE) under the constant relaxationtime approximation (CRTA) and rigid band approach using BoltzTraP^{[34]}. A dense Γcentered kmesh of 36 × 36 × 2 was used in the calculations of electronic transport properties. Using the ShengBTE package^{[35]}, we calculated the phonon thermal transport properties and lattice thermal conductivity. The phonon dispersion curves and secondorder interatomic force constants (IFCs) were obtained using 2 × 2 × 1 supercells within the density functional perturbation theory (DFPT)^{[36]}. On the basis of the finitedifference supercell method, the third anharmonic IFCs were calculated by using the 2 × 2 × 1 supercell with Γ kpoint, and a cutoff interaction range of third nearest neighbors was selected. A 13 × 13 × 1 kpoint sampling and scalebroad of 0.1, which have been tested to reach good convergence [Supplementary Figure 3], were employed in the calculation of corresponding lattice thermal conductivity.
Results and discussionStructural stability
Two kinds of atomic stacking configurations of Ge_{1}Sb_{6}Te_{10} have already been proposed experimentally, which are denoted as GSTI^{[22]} and GSTIII^{[23]} here, respectively, (see Supplementary Figure 1 for crystal structures). The layer stacking of GSTI, which exhibit a basic structural unit (BSU) of 17 atomic layers along the c axis (the inset of Figure 1), comprises two kinds of NaCltype slabs [i.e., TeSbTeSbTe (S5) and TeSbTeGeTeSbTe (S7)] as elemental structural units; the unit cell consists of three stacks of the BSU described by S5S7S5 (see Supplementary Figure 2 for details). The subtle structural difference between the two experimentally known atomic arrangements is that Ge and Sb atoms of stacking GSTIII are mixed in the same layer while GSTI is perfectly ordered, in which Ge, Sb, and Te atoms locate in their own specific layers. Based on the common stacking features of GSTI and GSTII, therefore, we only investigated the possible atomic sequences in the BSU instead of the whole unit cell. In other words, the sequential stacking of the BSU is maintained. Consequently, three hypothetical structural configurations (i.e., GSTII, GSTIV, and GSTV) were constructed by interchanging the positions of Ge and Sb atoms of stacking GSTI, which are likely to be stable and present experimentally, as suggested for other layered GST compositions^{[1316]}. Although it has been suggested that the existence of homopolar bonds of GST compounds (i.e., GeGe, GeSb, SbSb, and TeTe) will reduce the structural stability^{[37]}, there is no direct evidence to prove it. Thus, we also considered two kinds of antisite disordered stackings as representatives: one Sb/Te exchange (GSTVI) and one Ge/Te exchange (GSTVII), generating TeTe homopolar bonds. As a result, we started our investigations from the seven kinds of atomic arrangements for multilayered Ge_{1}Sb_{6}Te_{10}. Undoubtedly, the first step is to examine the structural stability of these new atomic stackings. Figure 1 shows the calculated total energy of all stacking configurations. Obviously, GSTI and GSTIII exhibit the lowest energies and thus should be the most stable configurations of Ge_{1}Sb_{6}Te_{10}, which is consistent with the fact that these two atomic stackings have been observed experimentally. Remarkably, the largest energy difference, which appears between GSTI and GSTVII, is only about 0.03 eV/atom, very close to the difference (~0.02 eV/atom) between total energies of the Ge_{2}Sb_{2}Te_{5} KH and Petrov configurations^{[14]}. Such small energy variation, therefore, suggests that the five newly conceived configurations of Ge_{1}Sb_{6}Te_{10} are energetically stable and further stability tests should be performed.
Then, we assessed their mechanical stability based on the Born’s criteria for the trigonal system^{[37]}
where c_{ij} is the elastic stiffness constant. The calculated results for these five unreported atomic stackings of Ge_{1}Sb_{6}Te_{10} are summarized in Supplementary Table 1. It turns out that both GSTII and GSTIV well satisfy the above criteria, confirming their good mechanical stability. Stackings GSTV, GSTVI, and GSTVII were considered to be mechanically unstable due to failing to meet the stability conditions and thus were eliminated from our investigations. Subsequently, we calculated the phonon dispersion curves of the two mechanically stable structural configurations to evaluate their lattice dynamic stability. Figure 2A and B clearly shows that there are no imaginary phonon frequencies in the first Brillouin zone, verifying that stackings GSTII [Figure 2A] and GSTIV [Figure 2B] are dynamically stable. Furthermore, to further study their thermal stability, AIMD simulations were performed at 300 K for 30 ps using 2 × 2 × 1 supercells, as plotted in Figure 2C and D (GSTII and GSTIV, respectively). The timedependent evolution of total energy and temperature for both stackings fluctuate in a very narrow window during the entire simulated time range, substantiating their appreciable room temperature thermal stability. By means of comprehensive stability evaluations, we can conclude that the two novel atomic configurations GSTII and GSTIV are energetically, mechanically, dynamically, and thermally stable. Therefore, it is reasonable to think that stackings GSTI, GSTII, GSTIII, and GSTIV are stacking polymorphs of layered Ge_{1}Sb_{6}Te_{10} and could present at different experimental conditions.
Phonon dispersion curves of stackings (A) GSTII and (B) GSTIV. Evolution of temperature (orange line) and total energy (red line) of atomic stackings (C) GSTII and (D) GSTIV as a function of time in AIMD simulations at 300 K.
Structural and electronic properties
Herein, we only investigated the atomic arrangements GSTI, GSTII, and GSTIV because the documented stacking GSTIII can be considered as a mixture of GSTI and GSTII, as suggested in Ge_{2}Sb_{2}Te_{5}^{[13]}, and thus its properties would obey Vegard’s law resembling the rules of mixtures^{[20,38,39]}. After fully optimized structure relaxation, we found that the crystal symmetry has a slight difference that stacking GSTI retains the space group of R̅3m (No. 166) while GSTII and GSTIV adopt R3m (No. 160). As shown in Table 1, the calculated lattice parameters of stacking GSTI are in good agreement with experiments^{[22]}, suggesting the reliability of our calculations. In addition, the lattice constant c of stacking GSTIV (101.207 Å) is relatively smaller than those of GSTI (103.050 Å) and GSTII (103.546 Å), which results from the shorter TeTe bond of the former, indicating that the atomic configurations have a certain effect on structural properties. Overall, the average length of the TeTe bond in all the atomic stackings is appreciably longer than those of GeTe and SbTe bonds and is much larger than the sum of their covalent radii of 2.760 Å^{[40]}, manifesting the weak bonding nature between adjacent TeTe layers.
Calculated lattice parameters (a and c) compared with available experiments (parentheses)^{[22]}; average GeTe, SbTe, and TeTe bond lengths; and electronic bandgaps (E_{g}) calculated using PBE functional and HSE06 functional (parentheses), respectively
Stacking
a (Å)
c (Å)
Bond length (Å)
E_{g} (eV)
GeTe
SbTe
TeTe
GSTI
4.236 (4.236)^{[22]}
103.050 (101.059)^{[22]}
2.959
3.069
3.814
0.20 (0.45)
GSTII
4.229
103.546
3.010
3.068
3.812
0.05 (0.16)
GSTIV
4.244
101.207
2.957
3.088
3.543

Note that we first calculated the band structure including the effect of spinorbital coupling (SOC), with stacking GSTI as an example [Supplementary Figure 4]. The resulting band curves with SOC are almost the same as those without SOC, indicating that the SOC has negligible influence on the electronic band structures and thus was ignored in our following calculations. Figure 3 illustrates the electronic band structures of stackings GSTI, GSTII, and GSTIV, where the orange and red lines denote the highest valence band (HVB) and the lowest conduction band (LCB), respectively. The corresponding calculations were performed on their primitive cell containing 17 atoms (i.e., Ge, 1; Sb, 6; Te, 10). Clearly, GSTI has a direct bandgap at the Z point [Figure 3A], whereas GSTII is an indirect semiconductor, as the valence band maximum (VBM) and the conduction band minimum (CBM) locate at different kpoints along the ZF line (the inset in Figure 3B). The estimated bandgaps using PBE functional are 0.20 and 0.05 eV for GSTI and GSTII [Table 1], respectively. Interestingly, GSTIV has no gap around the Fermi level (E_{f}) because the HVB marginally overlaps with the LCB around the G point [Figure 3C]. These results demonstrate that the atomic stackings have remarkable impacts on the electronic structures of Ge_{1}Sb_{6}Te_{10}, thus affecting its electrical transport properties (as discussed below).
Electronic band structures of: (A) GSTI; (B) GSTII; and (C) GSTIV. The orange line is the highest valence band (HVB) and the red line represents the lowest conduction band (LCB). The inset of (B) is the zoomed in picture around the conduction band minimum (CBM) and valence band maximum (VBM). The Fermi level (E_{f}) is set to 0 eV.
Aiming to further improve the understanding of electronic structures, Figure 4 shows the projected density of states (PDOS) and pCOHP of stackings GSTI, GSTII, and GSTIV. For all the stackings, the PDOS pictures indicate that the bottom of the conduction bands mainly consists of Sb 5p and Te 5p orbitals, and the top of valence bands are primarily dominated by 5p electrons of Te atoms. It is also noticed that there are Ge 4s and Sb 5s peaks near the VBM, which hybridize with Te 5p orbitals, forming Ge 4sTe 5p and Sb 5sTe 5p antibonding states due to the negative values of pCOHP. Moreover, Ge 4p and Sb 5p orbitals states are widely distributed in the energy range of 6 to 0 eV and strongly couple with Te 5p electrons, implying the covalent bonding characters of GeTe and SbTe bonds. Meanwhile, the pCOHP graphs show that Ge 4pTe 5p and Sb 5pTe 5p interactions within the same energy region exhibit positive pCOHP values, substantiating their bonding states. Furthermore, no antibonding states were observed at E_{f} for these three configurations, manifesting their stable chemical bonding interactions.
PDOS and pCOHP of stackings: (A) GSTI; (B) GSTII; and (C) GSTIV. The E_{f} is set to 0 eV.
Notably, the GGAPBE method usually underestimates the electronic bandgaps of crystals, while the precise bandgap plays a significant role in obtaining more realistic electronic transport properties, especially at elevated temperatures^{[41]}. Hence, the HSE06 functional was further employed to predict more accurate bandgaps, and the resulting bandgaps of stackings GSTI and GSTII increase to 0.45 and 0.16 eV [Table 1], respectively. However, GSTIV still has no bandgap, exhibiting semimetallic characteristics. Since highperformance TE materials are generally narrow bandgap semiconductors^{[42]}, we only focused on studying the transport properties and TE performance of GSTI and GSTII. To give a qualitative description of electrical transport properties, Figure 5A and B displays the total DOS around conduction band (CB) and valence band (VB) edges of GSTI and GSTII. It is well known that a rapid change in DOS with energy is essential for obtaining a large Seebeck coefficient^{[43]}. Clearly, the CB DOS for both stackings is considerably larger than their VB DOS throughout the energy range of 00.6 eV. In addition, both VB and CB DOS of stacking GSTI are much steeper than those of stacking GSTII. These results suggest that both stackings would exhibit higher ntype S than that of the ptype, and higher Seebeck coefficients of stacking GSTI under both n and ptype dopings can be expected compared with GSTII.
Total DOS of conduction bands (red line) and valence bands (orange line) around the respective CBMs and VBMs for stackings (A) GSTI and (B) GSTII. Electronic band structures of stackings (C) GSTI and (D) GSTII. The blue and red dashed lines indicate the Fermi level at the carrier concentrations of 2 × 10^{19} and 1 × 10^{20} cm^{3}, respectively. The positive and negative carrier concentrations represent the p and ntype dopings, respectively. CBM: Conduction band minimum; VBM: valence band maximum.
Further, it is observed in Figure 5C that the second valence band maxima of GSTI located at the Gpoint is 0.1 eV lower in energy than VBM. This suggests that the hole transport in stacking GSTI is mainly dominated by the G pocket, and the Z pocket would participate in electrical transport when heavily doped, such as n = 1 × 10^{20} cm^{3} (red dashed line in Figure 5C). However, the second conduction band valley located along the ZF line is only about 0.02 eV from the CBM. Meanwhile, the third (on the GL line) and fourth conduction band minima (on the GF line) are slightly higher (~0.03 eV). In addition to the multiconduction pocket electrons, multiband conduction is also observed, even at very low electron concentrations (e.g., n = 2 × 10^{19} cm^{3}, Figure 5C). Stacking GSTII has a similar band configuration [Figure 5D] to GSTI. As a result, the unique conduction band structures could greatly promote electrical conduction and thus are favorable for achieving high electrical conductivities and power factors.
Electrical transport properties
Noticeably, the thermal stability of GSTI and GSTII at 800 K was verified by performing AIMD simulations [Supplementary Figure 5]. Hence, both stackings are thermally stable in a wide temperature range of 300800 K, and here we focus on the range from 300 to 710 K. In addition, the calculated temperaturedependent Seebeck coefficients of GSTI agree well with the experiments (see comparison details and Supplementary Figure 6A in the Supplementary Materials)^{[25]}, indicating the validity of our theoretical predictions. It is noted that layered GST compounds generally tend to show the ptype character. However, our previous work^{[44]} made an extensive analysis to unravel the defect physics in these materials and suggests that it is quite likely to make ntype GST semiconductors by tuning the atomic chemical environments. Therefore, Figure 6A and E shows the Seebeck coefficients of GSTI and GSTII as a function of carrier concentration at different temperatures for both n and ptypes. At the same temperature, the absolute S value of stacking GSTI basically decreases with increasing carrier concentration, and, for the same carrier concentration, S increases with the temperature. GSTII exhibits peak S values for both p and ntype dopings at higher temperatures (e.g., 520 and 710 K). It is also observed that GSTI has relatively larger Seebeck coefficients compared with GSTII. Meanwhile, the absolute ntype S values are significantly higher than those of the ptype for both stackings. For example, under the carrier concentration of 1 × 20 cm^{3} and T = 710 K, the S values of stacking GSTI for p and ntype dopings are 187 and 287 μV/K, respectively, while stacking GSTII has the values of 145 and 246 μV/K. Our calculated results are consistent with the above analysis of electronic properties.
Electronic transport properties of (AD) GSTI and (EH) GSTII as a function of carrier concentration for p and ntype dopings at different temperatures: (A, E) Seebeck coefficient S; (B, F) electrical conductivity σ; (C, G) electronic thermal conductivity κ_{e}; and (D, H) power factor PF.
In principle, using the BoltzTraP, the electrical conductivity can be estimated only if the electronic relaxation time (τ) is given. Here, τ is determined by the comparison between calculated σ/τ value and reported σ (see comparison details and Supplementary Figure 6B in the Supplementary Materials)^{[25]}, which has been widely employed in the evaluation of relaxation time for many materials^{[4547]}. The resulting τ values of stacking GSTI at different temperatures are shown in Supplementary Figure 6C, in which τ decreases with temperature (e.g., 20.1 fs at 310 K, 11.8 fs at 520 K, and 7.0 fs at 710 K). Given the fact that there are no experimental data for GSTII, the same relaxation time as stacking GSTI was used for better comparison. Figure 6B and F illustrates the changes of electrical conductivity with respect to carrier concentration at different temperatures. It is seen that electrical conductivity increases with carrier concentration at the same temperature, and, for the same doping level, σ decreases when increasing the temperature, exhibiting the metallike behavior. In addition, the ntype σ for both stackings are larger than that of ptype, owing to their complex conduction bands discussed above. Interestingly, by comparing with stacking GSTII, GSTI with higher ptype Seebeck coefficient also possesses larger electrical conductivity for hole doping, suggesting its higher power factors and better ptype TE performance.
The electronic thermal conductivity was further calculated based on the WiedemannFranz law^{[48]}: κ_{e} = LσT, where L is the Lorenz number. L of 1.5 × 10^{8} WΩK^{2} was selected as an empirical constant in our calculations, which has been used for many Tebased semiconductors^{[45,49,50]}. As clearly shown in Figure 6C and G, κ_{e} gradually increases with increasing carrier concentration and decreases with the increase of temperature. The ntype κ_{e} is larger than that of ptype, owing to the larger ntype doped electrical conductivity. Furthermore, the power factor (PF) is derived from the Seebeck coefficients and electrical conductivities [Figure 6D and H]. At 310 K, the maximum ntype PF values are 3.24 and 2.95 mWm^{1}K^{2} for GSTI and GSTII, respectively, while the ptype values are 1.88 and 0.63 mWm^{1}K^{2}. Apparently, both stackings exhibit significantly larger ntype PF than that of ptype due to the combination of higher ntype S and σ.
Phonon thermal transport properties
Lattice thermal conductivity is one of the critical factors that determine the TE performance of TE materials. Figure 7A presents temperaturedependent κ_{l} of GSTI and GSTII calculated by iteratively solving the phonon BTE. The κ_{l} of both stackings gradually decreases with increasing temperature, suggesting that the phononphonon scattering is primarily dominated by Umklapp processes. Moreover, they have similar κ_{l} values in the whole studied temperature range, although the former is marginally larger, indicating that stacking arrangement has no significant impact on phonon thermal transport performance. At 300 K, the predicted κ_{l} values are 0.86 (GSTI) and 0.78 Wm^{1}K^{1} (GSTII) and decrease to 0.37 and 0.33 Wm^{1}K^{1} at 710 K, respectively. To unravel the origin of the low κ_{l}, we calculated the group velocity of each phonon mode for stackings GSTI and GSTII. Figure 7B shows that the group velocities of most vibrations are in the range of 02500 m/s, indicating the slow phonon propagation in the crystal lattice. For a better understanding of the slow phonon transport, the longitudinal (v_{L}), transverse (v_{T}), and mean sound velocities (v_{m}) were extracted from the elastic moduli (see the Supplementary Materials and Supplementary Table 2 for details) and bulk density ρ^{[51]}:
(A) Temperature dependence of lattice thermal conductivity; (B) group velocity; (C) room temperature phonon lifetime; and (D) weighted phase space of GSTI and GSTII.
As shown in Table 2, bulk and shear moduli of GSTI (B, 39.91 GPa; G, 25.66 GPa) are relatively higher than those of GSTII (B, 38.48 GPa; G, 24.25 GPa), leading to higher v_{L} and v_{T} of the former, thereby higher lattice thermal conductivity. Moreover, the v_{m} values of GSTI and GSTII are 2.21 and 2.15 km/s, respectively, comparable to Bi_{2}Te_{3}based alloys (~2.15 km/s)^{[52]}, further substantiating their slow phonon propagation.
Calculated bulk modulus B (GPa), shear modulus G (GPa), bulk density ρ (g/cm^{3}), and sound velocities v_{L}, v_{T}, and v_{m} (km/s) for stackings GSTI and GSTII, respectively
Stacking
B
G
ρ
v_{L}
v_{T}
v_{m}
GSTI
39.91
25.66
6.47
3.39
1.99
2.21
GSTII
38.48
24.25
6.46
3.31
1.94
2.15
The phonon lifetime (τ), another essential parameter in evaluating the κ_{l} of crystals, was further calculated [Figure 7C]. Obviously, both stackings exhibit lower τ values at 300 K, which mainly stay within the range of 110 ps, similar to the region of τ for cubic methylammonium lead iodide MAPbI_{3} (κ_{l} < 1 Wm^{1}K^{1} at room temperature)^{[53]}. To better describe the τ, we averaged the phonon times of all vibration modes. By comparing with the average τ (3.22 ps) of stacking GSTI, GSTII gives a bit lower value of 2.98 ps, thereby leading to the slightly lower κ_{l}. In addition, the magnitude of the Grüneisen parameter (γ) is considered a good measure of lattice anharmonicity and able to reflect the strength of anharmonic phononphonon scattering^{[54]}. Generally, a crystal with larger γ value indicates that it has higher degree of anharmonicity and stronger anharmonic scattering rate, giving rise to shorter phonon lifetime and lower lattice thermal conductivity. Therefore, the total Grüneisen parameter γ_{total} at 300 K was determined from a weighted sum of the mode contributions. It turns out that the γ_{total} is 1.40 for stacking GSTI and 1.15 for stacking GSTII. These γ_{total} values are comparable to, although slightly lower than, the value of PbTe (~1.45)^{[55]}, manifesting the appreciable phonon anharmonicity in both stackings of Ge_{1}Sb_{6}Te_{10} and thus short phonon lifetimes. Meanwhile, it should be expected that the larger γ_{total} of stacking GSTI gives a lower τ value than that of stacking GSTII. Nonetheless, our calculation results above demonstrate the opposite situation. To explain this, we calculated the weighted scattering phase space (W) [Figure 7D], which measures the number of threephonon scattering channels for each vibration mode^{[56]}. It shows that the W values of stacking GSTII are overall larger than those of GSTI, indicating that there are more threephonon scattering processes of the former, which results in shorter phonon lifetimes. Collectively, it can be concluded that the low lattice thermal conductivity for both stackings of Ge_{1}Sb_{6}Te_{10} is well understood from the small phonon group velocity and short phonon lifetime.
Thermoelectric figure of merit
Using the obtained power factor and thermal conductivity, we estimated the ZT values of GSTI and GSTII, as shown in Figure 8. Apparently, the calculated ZT values of stacking GSTI are in excellent agreement with the experiments (Figure 8A, top)^{[25]}. For both p and ntype dopings, it is seen that the ZT_{max} of stackings GSTI and GSTII all rise with temperature, mainly due to the decrease in κ_{e} and κ_{l}. By modulating the carrier concentration to the optimal value of 5.3 × 10^{19} cm^{3}, the ZT_{max} of ptype GSTI reaches up to 1.36 at 710 K, exhibiting excellent ptype TE performance. This means that the low ZT values reported by experiments are mainly ascribed to the high hole concentration, and the TE performance can be appreciably improved through the reduction of carrier density using appropriate dopant. The PDOS above [Figure 4] demonstrates that the tops of valence bands are primarily dominated by 5p electrons of Te atoms and the Ge/Sb p orbitals have slight contributions. Therefore, it is possible to modulate the hole carrier concentration via Ge/Sb defect engineering without considerably influencing the valence band structure, thereby increasing the ZT value. For example, it is found that the homologous compound Ge_{2}Sb_{2}Te_{5} gives the maximal ZT value of 0.78 at 700 K by substituting the Ge sites with In, an about twofold improvement compared with the undoped material system because the introduction of indium as a potentially donorlike dopant lowers the hole carriers density^{[8]}. In addition, substituting with Cd leads to a considerable increase of the Seebeck coefficient of Ge_{1}Sb_{2}Te_{4} due to the significant reduction of hole concentration, and the power factor of Cd_{0.2}Ge_{0.8}Sb_{2}Te_{4} exceeds that of undoped Ge_{1}Sb_{2}Te_{4} by a factor of about 2.5^{[12]}. The ZT_{max} of ptype GSTII is 0.60 (n = 9.3 × 10^{19} cm^{3}) at 710 K [Figure 8B], overtly lower than that of stacking GSTI, which is attributed to its smaller Seebeck coefficient and electrical conductivity. It is very exciting that the peak ZT values for ntype GSTI and GSTII are as high as 2.23 and 1.91 at 710 K, respectively, with corresponding optimal carrier concentrations of 1.10 × 10^{20} and 1.36 × 10^{20} cm^{3}. Such high ZT_{max} values are comparable to and even slightly larger than those of GeTebased TE material Ge_{0.87}Pb_{0.13}Te + 3% Bi_{2}Te_{3} (ZT = 1.9 at 773 K)^{[57]}. Therefore, our results indicate that the layered Ge_{1}Sb_{6}Te_{10} will be of immense promise for mediumtemperature TE applications.
Dimensionless figure of merit ZT of stackings (A) GSTI and (B) GSTII as a function of carrier concentration for p and ntype dopings at different temperatures. The dots in (A) represent the experimental results^{[25]} at the concentration of 7.3 × 10^{20} cm^{3}.
Conclusions
By means of firstprinciple calculations combined with the Boltzmann transport theory, we carried out systematic investigations on the atomic stacking configurations, electronic structures, transport properties, and thermoelectric performance of layered Ge_{1}Sb_{6}Te_{10}. Two novel atomic arrangements (GSTII and GSTIV) were predicted to be energetically, mechanically, dynamically, and thermally stable. GSTII and the experimentally known stacking GSTI were found to be semiconductors with narrow bandgaps of 0.45 and 0.16 eV, respectively, while GSTIV is a semimetal. Importantly, for the two semiconducting stacking configurations, high DOS of conduction band edge, multiple conduction pocket electrons, and multiband conduction give rise to large ntype Seebeck coefficients and electrical conductivities, thereby resulting in large power factors and better ntype TE performance. Moreover, both GSTI and GSTII exhibit low lattice thermal conductivities, 0.86 and 0.78 Wm^{1}K^{1} at 300 K, and as low as 0.37 and 0.33 Wm^{1}K^{1} at 710 K, owing to short phonon lifetimes and small phonon group velocities arising from the low elastic moduli. Consequently, the combination of high power factors and ultralow lattice thermal conductivities lead to the maximum ZT values of ntype GSTI and GSTII reaching up to 2.23 and 1.91 at 710 K, respectively, indicating the great promise of stacking polymorphic Ge_{1}Sb_{6}Te_{10} for TE application. We believe that our results provide great possibilities to discover novel functional materials from layered crystals and will stimulate further experimental research in the future.
DeclarationsAuthors’ contributions
Contributed to conception and design of the study and performed data analysis and interpretation: Gan Y
Provided professional guidance: Zhou J, Sun Z
Availability of data and materials
Supplementary materials are available from the Journal of Materials Informatics or from the authors.
Financial support and sponsorship
This work is supported by the National Key Research and Development Program of China (2017YFB0701700), the National Natural Science Foundation of China (51872017) and the high performance computing (HPC) resources at Beihang University.
Conflicts of interest
All authors declared that there are no conflicts of interest.
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