Thermoelectric Properties of PbTe, SnTe, and GeTe at High Pressure: an Ab Initio Study

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1 Journal of ELECTRONIC MATERIALS, Vol., No. 5, DOI:.7/s66--9-y Ó TMS Thermoelectric Properties of PbTe, SnTe, and GeTe at High Pressure: an Ab Initio Study LANQING XU,, HUI-QIONG WANG, and JIN-CHENG ZHENG,. Department of Physics, Institute of Theoretical Physics and Astrophysics, Fujian Key Laboratory of Semiconductor Materials and Applications, Xiamen University, Xiamen 65, People s Republic of China.. Key Laboratory of OptoElectronic Science and Technology for Medicine of Ministry of Education, Fujian Normal University, Fuzhou 57, People s Republic of China.. jczheng@xmu.edu.cn In this work we present an ab initio study of the transport properties of PbTe, SnTe, and GeTe crystals in the B structure under zero and high pressure and analyze the possibility of pressure-induced thermoelectric performance enhancement. GeTe displays higher thermoelectric coefficients in both the p- and n-doping cases at zero pressure, but with applied pressure they drop quickly. n-type SnTe has a higher Seebeck coefficient and figure of merit () than SnTe at ambient conditions. With increased pressure its thermoelectric performance is improved initially and degrades later. The highest appears at about 5 GPa. p-type PbTe possesses attractive thermoelectric properties at zero pressure. With pressure applied, the of this material undergoes a decline climb decline variation, and the optimal occurs at 8 GPa to GPa. Thermoelectric properties of PbTe degrade slightly with increasing pressure and improve later; the improvement can be observed for pressures up to GPa. These results suggest possible enhancement of thermoelectric properties for SnTe under intermediate pressure and PbTe under high pressure. Key words: Thermoelectrics, pressure, PbTe, SnTe, GeTe, ab initio, firstprinciples calculations, density-functional theory, transport property INTRODUCTION As a solid-state energy converter, thermoelectric (TE) materials have been extensively studied over the past several decades due to their promising performance in cooling and heating applications. The efficiency of TE materials is evaluated in terms of a dimensionless figure of merit =(S rt/j), where T is the absolute temperature, S is the thermopower, namely the Seebeck coefficient, r is the electrical conductivity, and j is the thermal conductivity. Researchers are devoting efforts to finding materials with high power factor S r and low thermal conductivity which can enable larger and better TE performance. Various pathways (Received May 8, ; accepted December 9, ; published online January, ) have been proposed to improve for energy conversion applications, including synthesis of new materials and use of materials with complex superlattice to utilize quantum effects, with induced disorder to decrease thermal conductivity, or with rattling ions and nanostructured systems.,,5 Te-based alloys (i.e., lead telluride, tin telluride, and germanium telluride) are well-known candidates for medium-temperature TE devices. Germanium telluride exhibits a higher value but is less temperature stable. 6 Within the working temperature range, these compounds possess attractive TE properties, and when treated, can have even higher values. In, Harman et al. reported a value of.6 at T = K for PbTe/PbSe nanodot superlattices. 7 In 5, Caylor et al. reported a PbTe-based superlattice structure which is believed to enhance the TE performance by reducing the 6

2 6 Xu, Wang, and Zheng lattice thermal conductivity to a minimum of 5 mw=cm K. 8 More recently, the LAST-m compounds AgPb m SbTe m+ have been shown to have values well above unity. 9 Tin telluride was claimed to be a semiconducting, nonstoichiometric compound with high cation vacancies and hole concentrations., Pseudoternary compounds Sn x Ge x Te and Pb x Ge x Te were also designed to ensure high electrical conductivities and low thermal conductivities simultaneously, to improve TE performance. Historically, pressure-induced phase transitions and TE enhancement have been reported in semiconductors. HgTe undergoes a phase transition which is accompanied by changes in thermopower when pressure is applied. 5 Li et al. studied the influence of pressure on the band structure of AuX (X = Al, Ga, and In) and found that their transport properties were sensitive to the pressure-tuned band structure close to the Fermi surface. 6 A colossal improvement of the TE efficiency of lead chalcogenide compounds was found by Ovsyannikov et al. with P GPa to GPa. 7 They also measured the thermomagnetic effect for Te and Se samples at the semiconductor metal phase boundary and found significant Nernst Ettingshausen effects. 8 A combination of high-temperature and high-pressure synthesis techniques has been proposed to greatly improve the TE properties of PbTe. 9 With further high-temperature and highpressure treatment, conventionally synthesized PbTe samples were found to undergo a reversible change of TE properties. Therefore, temperature and pressure seem to play an important role in improving TE properties. However, until recently few calculations have been performed to analyze the TE properties of Te-based materials under pressure. In this work we perform ab initio calculations using the full potential linearized augmented plane wave method (FP-LAPW) to study the electronic band structures of PbTe, SnTe, and GeTe (XTe) compounds in the B structure. Thermoelectric properties under pressure are derived using Boltzmann transport theory. COMPUTATIONAL METHOD We carry out first-principles calculations using the all-electron method, as implemented in the WIENK code within the framework of densityfunctional theory (DFT)., The muffin-tin radii are chosen to be.5 a.u. for all atoms. The planewave cutoff is defined by R MT K max ¼, which is sufficient to achieve good convergence. Spin orbit (SO) interaction is also included here to account for relativistic effects. During the self-consistent calculation (SCF) cycle a set of k points in the whole Brillouin zone (BZ) [88 in the irreducible Brillouin zone (IBZ)] is used. In the transport calculations, eigenenergies are calculated using a nonshifted mesh with 56, k points (9 in the IBZ). The exchange-correlation functional has been treated using the Perdew Burke Ernzerhof (PBE) generalized gradient approximation (GGA). 5 It is well known that most DFT-based functionals yield poor band gaps. The Engel Vosko (EV) GGA 6 is known to obtain band gaps in good agreement with the experimental values in several cases. 7 Since thermoelectric properties are sensitive to band structures near the Fermi surface, we used the values of band gap predicted by EV-GGA in the transport calculations. The Boltzmann transport equation and the rigid band approach are used to calculate transport properties as implemented in the BoltzTraP code, which relies on a well-tested smoothed Fourier interpolation to obtain an analytical expression of the bands. This ab initio approach has been successfully used in rationalizing and predicting the optimal doping level of known compounds and recently has been used to screen potential candidates for TE materials. The transport coefficients can be calculated from the transport distribution function. With the rigid band approximation 7 one can derive carrier concentrations from the chemical potential l and simulate the p- or n-doped environment by shifting the Fermi level down or up. As long as the shift is not large, the bands can be left fixed. Thus, only one band structure calculation is needed for each compound. To calculate the transport coefficients the relaxation time s is needed, which can be treated as a constant as shown in many previous studies. 8 Within this assumption, the Seebeck coefficient S, the electrical conductivity relative to the relaxation time r/s, can be calculated without any fitting parameters. The electrical conductivity, r, and hence the power factor S r and values, can however only be resolved with respect to s. With a given s, can be evaluated by ¼ S rt j e þ j l ¼ S L þ j l rt ; () where L is the Lorentz number. Using the rigid band approach and constant relaxation-time assumption, one can investigate thermoelectric properties by exploring the band structure. These results are only valid based on the assumption that the charge carrier mobility does not decrease due to carrier carrier scattering. To obtain a direct comparison with experiment, one should further obtain the carrier concentration from the chemical potential for each T. In the transport calculations, a smoothed Fourier interpolation with a k-point mesh 5 times denser than that in SCF calculation is used to obtain the analytical expression of the bands. RESULTS AND DISCUSSION An accurate energy band structure is desired to obtain reliable transport coefficients. The calculated fully relativistic band structures for XTe compounds

3 Thermoelectric Properties of PbTe, SnTe, and GeTe at High Pressure: an Ab Initio Study 6 (a) (b) (c) Fig.. Calculated electron density of states for XTe (X = Pb, Sn, Ge) crystals with the B structure. (a) Total DOS for PbTe, SnTe, and GeTe crystals. (b) Projected DOS of s and p orbital components for the Pb atom. (c) Projected DOS of s and p orbital components for the Te atom (Color figure online). (not shown) reveal that all three compounds exhibit similar energy band structures with narrow, direct band gaps at the L point. Compared with results from nonrelativistic calculations (not shown), spin orbit (SO) coupling band splitting can be observed in all three compounds at conduction bands. The conduction band minimum (CBM) and the next higher conduction band state have the same L 6 state, and the mutual repulsion moves the CBM down, shrinking the gap. Therefore in these materials, SO interaction should be taken into account when an accurate band structure is demanded. The electron density of states (DOS) (Fig. a) shows that all three compounds have a sharp increase of the DOS near the valence band maximum (VBM) or CBM, which suggests possible enhance of the TE performance. PbTe has the highest DOS at the VBM, while SnTe has the highest DOS at the CBM. This would lead to higher Seebeck coefficients in PbTe and SnTe, whereas GeTe has a steep increase of the DOS at both sides and with doping the Fermi level will shift up or down, thus attractive TE properties could be expected for both p- and n-doping cases. Analysis of the projected DOS confirms that the peaks near the CBM are mainly attributed to the Pb/Sn/Ge s states (Fig. b), while the primary contributions to the features close to the VBM come from the Te p states (Fig. c). The calculated transport coefficients at K and zero pressure are illustrated in Fig.. p-type PbTe and SnTe have higher Seebeck coefficients S (Fig. a, b), which is consistent with the DOS behavior as described above (Fig. ). The Seebeck coefficient of SnTe drops quickly at the low electron concentration region, due to its narrow band gap (.57 ev from the PBE-GGA calculation and. ev from the EV-GGA calculation). Both p- and GeTe have superior σ/τ (Ωm s) - S σ/τ (W/(mK s)) 6 PbTe 5 (a) SnTe GeTe 9 8 Hole concentration (cm - ) (c) 8 Hole concentration (cm - ) 9 9 (e) 8 Hole concentration (cm - ) σ/τ (Ωm s) - S σ/τ (W/(mK s)) SnTe GeTe 8 9 Electron concentration (cm - ) 9 8 (d) Electron concentration (cm - ) (f) Electron concentration (cm - ) Fig.. Calculated TE properties of XTe (X = Pb, Sn, Ge) crystals at K and zero pressure, as a function of carrier concentration: (a) Seebeck coefficient, S; (b) Seebeck coefficient, S; (c) electrical conductivity with respect to relaxation time, r/s; (d) electrical conductivity with respect to relaxation time, r/s; (e) power factor with respect to relaxation time, S r/s; (f) power factor with respect to relaxation time, S r/s. The transport coefficients for PbTe, GeTe, and SnTe are labeled as red squares, blue circles, and green triangles, respectively. Legends for the three subfigures are the same for each column and therefore are only shown for (a) and (b) (Color figure online).

4 6 Xu, Wang, and Zheng Seebeck coefficients, which is a result of the relatively homogeneous DOS distribution of GeTe at both the VBM and CBM. The electrical conductivity with respect to relaxation time r/s increases logarithmically with carrier concentration, therefore the doping level selection is crucial for optimization of TE properties (Fig. c, d). GeTe-based alloys often contain nonstoichiometric defects resulting from doubly ionized metal vacancies, 6 and suitable doping is thus highly recommended to achieve their optimal TE properties. p-type PbTe and GeTe have comparable S r/s, larger than that of SnTe (Fig. e), while GeTe and SnTe have a much higher S r/s than that of PbTe when the electron concentration exceeds 9 cm (Fig. f). With given values of s and j, the figure of merit can be easily calculated from physical quantities such as S r/s. In principle, s and j l should not be the same for different materials; deviations may even exist for the same material when carrier concentration, temperature or pressure fluctuates. The purpose of our work is to provide a trend of TE performance under pressure for the three materials. For the convenience of comparison, the same s and j l are used for all three compounds across the whole doping range. In the following computations we use an average relaxation time (s =.8 9 s 9 ) and constant lattice thermal conductivity (j l =. W/m K at K and j l =. W/m K at 6 K ) for the three materials, although the values were adopted from the experimental reports for PbTe. Therefore the computational results of electrical conductivity and will be more accurate for PbTe than those for SnTe and GeTe. The figures of merit for the three compounds are presented in Fig.. The calculated for PbTe at K coincides well with the experimental report. p-type PbTe has the highest at a hole concentration range of : cm to : cm. The optimal for GeTe occurs at n h : 9 cm. In the situation, the optimal doping level of GeTe is also about : 9 cm. The maximum of PbTe and SnTe occur at around 5: 8 cm and 5: 9 cm, respectively. The relaxation time at 6 K is in fact less than that at K, which could partly explain the relatively high at 6 K. Some experimental reports 6,5 indicated that s for SnTe and GeTe could be lower than that of PbTe, however since there are much fewer experimental data available for GeTe and SnTe than for PbTe, the comparison of s cannot be justified. Therefore, the calculated results will only provide a trendline for these three materials, and the absolute values could be overestimated; for example, it has been reported that the j value of SnTe could be as high as 7.9 W/m K, 5 which is higher than the total thermal conductivity j e þ j l used in our calculations. To better compare with experimental results for each compound, further corrections of s and j l are needed. However, the overall trend could remain the same. 5 (a) K PbTe SnTe GeTe 9 8 Hole concentration (cm - ) (c) 6K The temperature dependence of the Seebeck coefficients at zero pressure and fixed carrier concentrations were investigated (Fig. ) as well as the temperature dependence of the corresponding values (Fig. 5). The calculated Seebeck coefficients are in good agreement with experimental reports. 6,5 9 The calculated values of PbTe with n e ¼ 6: 8 cm, n e ¼ : 9 cm are also consistent with the experimental values, 9, but the calculated of PbTe is much higher than the experimental report for n h ¼ : 9 cm. The discrepancy possibly originates from the constant relaxation time used in our work, since in the real case it should vary with temperature and carrier concentration. The difference of thermal conductivities (j l =.5 W/m K in Ref. and j l =. W/m K in our case) could also contribute to the deviation. In the high temperature region some data points are omitted due to large fluctuations. Generally speaking, PbTe has the highest Seebeck coefficients, SnTe has higher S than that of PbTe with higher electron concentration, and GeTe has attractive values in both p- and n-doping cases. In the low doping situation, the optimal S occurs at the low temperature range, and it shifts gradually to the high temperature range with increasing carrier concentration. The maximum depends on the combination of doping level and temperature. For PbTe, although the Seebeck coefficients at n ¼ 5:9 8 cm are quite high, the corresponding electrical conductivity is (b) PbTe K SnTe GeTe Expt.I 8 9 Electron concentration (cm - ) 5 (d) 6K Hole concentration (cm - ) Electron concentration (cm - ) Fig.. The calculated versus carrier concentration at K and 6 K (zero pressure) for PbTe, SnTe, and GeTe crystals. The relaxation time used here is.8 9 s for the whole computation. This parameter may be overestimated in some cases, therefore some of the values could be overestimated too. The lattice thermal conductivity j l is. W/m K and. W/m K for K and 6 K, respectively. The values for PbTe, GeTe, and SnTe are labeled as red squares, blue circles, and green triangles, respectively. The experimental data Expt.I for PbTe are collected from Ref. and shown as open symbols (Color figure online).

5 Thermoelectric Properties of PbTe, SnTe, and GeTe at High Pressure: an Ab Initio Study Expt.I (e) GeTe 5 n h n h n h n hp Ex t.iii Expt.IV (d) SnTe n e n e n e n e Expt.II (f) GeTe Fig.. Temperature dependence of Seebeck coefficient S at zero pressure for (a) PbTe, (b) PbTe, (c) SnTe, (d) SnTe, (e) GeTe, and (f) GeTe. Legends for the three subfigures are the same for each column and therefore only presented for (e) and (f). Carrier concentrations are: n h ¼ 5:9 8 cm, n h ¼ : 9 cm, n h ¼ 6:5 cm, n h ¼ 5: cm, n e ¼ 6: 8 cm, n e ¼ : 9 cm, n e ¼ n h, and n e ¼ n h. Experimental data for Expt.I (n = n h ), Expt.II (n = n e ), Expt.III (n unknown), and Expt.IV (n = n h, rhombohedral structure) are collected from Refs. 6 8, Ref. 9, Ref. 5, and Ref. 6, respectively. They are shown as open symbols. Some curves do not extend to the high temperature region. Because the carrier concentration here is rather high, the computational results fluctuate and become unreliable. GeTe and SnTe have higher carrier concentrations than PbTe under the same situation, therefore more data points are omitted (Color figure online). E xpt.i n h n h n h n h (e) GeTe Ex pt.ii Ex pt.iii (d) SnTe n e n e n e n e (f) GeTe Fig. 5. Temperature dependence of values at zero pressure for (a) PbTe, (b) PbTe, (c) SnTe, (d) SnTe, (e) GeTe, and (f) GeTe. Legends for the three subfigures are the same for each column and therefore only presented once for (e) and (f). Carrier concentrations are the same as in Fig.. Some data points are omitted for the same reason described in Fig.. The lattice thermal conductivity used to calculate is set to j l =. W/m K, and the relaxation time used here is.8 9 s. This parameter may be overestimated in some situations, therefore the could be overestimated, too. The experimental data are shown as open symbols, in which Expt.I (n = n h ), Expt.II (n = n e ), and Expt.III (n = n e ) are collected from Ref., Ref., and Ref. 9, respectively. The lattice thermal conductivity used in Expt.I is j l =.5 W/m K, which is larger than the parameter we used here (Color figure online). rather small, resulting in poor values. For example, our calculated results (not shown) confirm that r/s with n ¼ : cm is several magnitudes larger than that with n ¼ 5:9 8 cm in the K to 5 K temperature range, and the value increases further with temperature. The optimal working temperature is found to be 5 K for the low doping level and more than 7 K for the high doping level. For SnTe, a mid doping level is favored to achieve high. The SnTe has apparently attractive TE properties, with the maximum of at about 55 K for n e ¼ : 9 cm. With a higher doping level this peak shifts to the higher temperature domain. The experimental of this compound is low because of its high carrier concentration. As for GeTe, both p- and materials have high values in the high temperature domain. Its optimal carrier concentration is about 9 cm, and the optimal working temperature ranges from 6 K to 9 K, depending on the doping range. With pressure applied, the TE properties vary, as depicted in Figs. 6 and 7, where the pressure is obtained through the fitting of the energy volume curve to the Murnaghan equation of state. It can be seen that, theoretically, as the pressure increases, the Seebeck coefficients of PbTe have an

6 66 Xu, Wang, and Zheng (e) GeTe n n n n anomaly at the pressure range of 5 GPa to 8 GPa (Fig. 6a, b), which is consistent with the experimental report. Since thermopower or Seebeck coefficient is sensitive to the microstructural state, the microstructure displacements induced by a phase transition may result in a sharp increase or decrease of its TE performance. The pressure range where the abnormality appears matches the critical pressure needed for the BðNaClÞ!Pnma transition. Another interesting phenomenon is that the maximum value of PbTe occurs at a pressure of 8 GPa, while that for PbTe occurs at a pressure of GPa or even higher. Therefore, high-pressure synthesis of PbTe is suggested. For SnTe, increases initially with pressure and decreases after passing a maximum point, for any carrier concentration. The thermoelectric enhancement is observed at the intermediate pressure range, i.e., GPa to 7 GPa and GPa to 5 GPa for Expt. 5 5 (d) SnTe 5 5 (f) GeTe n n n n 5 5 Fig. 6. Pressure dependence of Seebeck coefficient at K for (a) PbTe, (b) PbTe, (c) SnTe, (d) SnTe, (e) GeTe, and (f) GeTe. Carrier concentrations are: n ¼ : 8 cm, n ¼ : 9 cm, n ¼ : 9 cm, and n ¼ : cm. Experimental data collected from Ref. are shown as open symbols, where the electron concentration for the sample is unknown. Anomalies in S can be observed for PbTe as the pressure is applied, which may indicate a phase transition (Color figure online) (e) GeTe n n n n 5 5 p- and SnTe, respectively (Fig. 7c, d). For GeTe, a phase transition can easily occur at high temperature or high pressure. Its TE properties at the high pressures in the B phase are not satisfied. In the pressure range of GPa to GPa, S decreases with pressure (Fig. 6e, f). A similar behavior is observed for its power factor with respect to relaxation time S r/s (not shown), resulting in reduced values in the high pressure region. Above 5 GPa, its Seebeck coefficient S decreases to nearly zero, resulting in zero at the high pressure domain. These results confirm that GeTe in the B phase is not applicable under high pressure with reduced thermoelectric coefficients. CONCLUSIONS The electronic band structures of PbTe, SnTe, and GeTe crystals in the B phase are investigated by first-principles calculations and their transport properties are studied by exploring the energy band (d) SnTe 5 5 (f) GeTe n n n n 5 5 Fig. 7. Pressure dependence of values at K for (a) PbTe, (b) PbTe, (c) SnTe, (d) SnTe, (e) GeTe, and (f) GeTe. Carrier concentrations are: n ¼ : 8 cm, n ¼ : 9 cm, n ¼ : 9 cm, and n ¼ : cm. s and j l are the same as in Fig. 5 (Color figure online).

7 Thermoelectric Properties of PbTe, SnTe, and GeTe at High Pressure: an Ab Initio Study 67 using Boltzmann transport theory. It is found that both p- and GeTe have high TE transport coefficients at zero pressure, and these transport coefficients could be optimized at carrier concentration of about 9 cm and a temperature range of 6 K to 9 K, depending on the doping level. As the pressure increases, its transport coefficients including S, S r/s, and all decrease quickly, indicating the unsuitability of B phase GeTe for application at high pressure. n-type SnTe has TE properties superior to its case. Its optimal doping level is about n e 5 9 cm to 9 cm, depending on the working temperature required. With pressure applied, its TE performance improves initially and reduces afterwards with a maximum at about 5 GPa. Therefore for SnTe, slight pressure would be favorable in thermoelectric applications. The PbTe has larger S and compared with its case. Its optimal working temperature is 5 K for the low doping level and higher for the high doping level. Under pressure this material exhibits an unusual decline climb decline variation trend. The optimal working pressure is located at about 8 GPa for the p-doping case and above GPa for the n-doping situation. These findings suggest that improved TE performance would be expected at high pressure for PbTe in the B structure. ACKNOWLEDGEMENTS This work is supported by the Minjiang Scholar Distinguished Professorship Program through Xiamen University of China, Specialized Research Fund for the Doctoral Program of Higher Education (Grant Nos. 98, and 6), National Natural Science Foundation of China (Grant No. ), Natural Science Foundation of Fujian Province, China (Grant Nos. 9J5, and J58), Program for New Century Excellent Talents in University (NCET) (Grant No. NCET-9-68), and the 9 Project for Scientific and Technical Development of Xiamen (No. 5Z 997). REFERENCES. J.C. Zheng, Front. 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