First-Principles Study of the Thermoelectric Properties of the Zintl

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First-Principles Study of the Thermoelectric Properties of Zintl Compound KSnSb Shan Huang, Huijun Liu, Dengdong Fan, Peiheng Jiang, Jinghua Liang, Guohua Cao, Ruizhe Liang, and Jing Shi J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b00099 • Publication Date (Web): 08 Feb 2018 Downloaded from http://pubs.acs.org on February 9, 2018

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The Journal of Physical Chemistry

First-principles Study of the Thermoelectric Properties of Zintl Compound KSnSb S. Huang, H. J. Liu*, D. D. Fan, P. H. Jiang, J. H. Liang, G. H. Cao, R. Z. Liang, J. Shi Key Laboratory of Artificial Micro- and Nano-Structures of Ministry of Education and School of Physics and Technology, Wuhan University, Wuhan 430072, China

ABSTRACT: The unique structure of Zintl phases makes it an ideal system to realize the concept of phonon-glass and electron-crystal in the thermoelectric community. In this work, by combining first-principles calculations and Boltzmann transport theory for both electrons and phonons, the thermoelectric properties of the Zintl compound KSnSb are investigated, where the carrier relaxation time is accurately treated within the framework of electron-phonon coupling. It is demonstrated that the ZT value of KSnSb can reach ~2.2 at 800 K. Such extraordinary thermoelectric performance originates from the large Seebeck coefficient due to multi-valley band structures and particularly small lattice thermal conductivity caused by mixed-bond characteristics.

1. Introduction Energy is the backbone of modern civilization. Developing new and renewable energy has become an urgent issue. Among them, the thermoelectric materials can directly convert heat into electrical energy and vice versa, which is simple, durable and environmentally-friendly. The conversion efficiency of thermoelectric materials depends on the figure of merit (ZT) which can be expressed as1

ZT 

S 2 T , e   p

(1)

where S ,  , T ,  e and  p are the Seebeck coefficient, the electrical conductivity, the absolute temperature, the electronic thermal conductivity, and the lattice thermal conductivity, respectively. In order to obtain a higher ZT value, a *

Author to whom correspondence should be addressed. E-mail: [email protected] 1

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thermoelectric material should have larger Seebeck coefficient, higher electrical conductivity and lower thermal conductivity. Slack proposed the concept of an ideal thermoelectric material, namely, the phonon-glass and electron-crystal (PGEC) 1, which has very low thermal conductivity like phonons in glass, as well as high electrical conductivity like electrons in crystal. It is believed that Zintl phases are typical representatives of PGEC materials and generally exhibit pretty good thermoelectric performance.2 The complex structures of Zintl compounds, which are usually composed of the cations and the polyanions, contribute to the low thermal conductivity. In addition, the cations are often made up of group I and II elements which provide electrons to the polyanions, and the covalent bond in the polyanions can ensure relatively high carrier mobility. All these favorable characteristics could lead to good thermoelectric performance of Zintl compounds. In recent years, more and more Zintl compounds with higher thermoelectric performance have been found. For instance, Hu et al. reported that the Zintl material Yb14MgSb11 synthesized by annealing the mixture of elements can reach a high ZT value of 1.02 at 1075 K.3 Yamada et al. prepared the polycrystalline sample of the Zintl phase Na2+xGa2+xSn4–x via a series of procedures including pulverization, mixing, compaction, and heating. They found that the ZT value is 1.28 at 340 K when x = 0.19.4 Shuai et al. synthesized the Zintl compound Eu0.2Yb0.2Ca0.6Mg2Bi2 by ball milling and hot pressing, and the ZT value can be as high as 1.3 at 873 K.5 Xu et al. theoretically predicted that the ZT value of n-type Mg3Sb2 can reach 1.53 with a carrier concentration of 1.154  1020 cm−3 at 700 K. 6 On the other hand, it is interesting to note that many Sb-based compounds have been demonstrated to be good thermoelectric materials, such as Sb2Te37, β-Zn4Sb38, Yb14MnSb119 and so on. It is thus natural to ask whether the Sb-based Zintl compound KSnSb10 is a potential thermoelectric material. In fact, the β value, which is usually identified as a metric of thermoelectric performance, is calculated to be 79.611 for n-type KSnSb and larger than those of many good thermoelectric materials. These observations indicate that KSnSb is likely to have good thermoelectric performance, which warrants deep investigation. 2


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The KSnSb compound was first synthesized in 1987 and its nonmetallic behavior was suggested.10 Subsequently, Schmidt et al. calculated the electronic band structure and density of states (DOS) by using density functional theory (DFT). They adopted the concept of the Zintl phase to understand the electronic properties of KSnSb, and found that the SnSb substructures have stable and saturated covalent bonds because the free valence electrons of K atoms are transferred to the polyanion.12 In a later work of Alemany et al., first-principles calculations on the Zintl phase KSnSb were carried out and they clearly pointed out that the system should be semiconducting.13 However, the thermoelectric properties of the Zintl compound KSnSb have been less studied before. Indeed, there is only few experimental and theoretical works 14-16 suggested that KSnSb could have good thermoelectric performance. In this work, by using first-principles calculations combined with Boltzmann transport theory, we carry out a detailed investigation of the electron, phonon, and thermoelectric transport properties of the KSnSb compound. We shall see that a high ZT value of ~2.2 can be realized at 800 K by optimizing the carrier concentration, which suggests the intriguing thermoelectric application potentials of such a Zintl phase in bulk form.

2. Computational methods The electronic properties of KSnSb are calculated by using the pseudopotential method within the framework of DFT, which are implemented in the so-called QUANTUM ESPRESSO (QE) package.17 The exchange and correlation energy is in the form of Perdew-Burke-Ernzerhof (PBE)18. To accurately predict the band gap, we also employ the hybrid density functional of Heyd-Scuseria-Ernzerhof (HSE06). 19-21 The kinetic energy cutoff of the plane-wave basis is set to be 80 Ry and an 11  11  4 k-mesh is used for the total energy calculations. Meanwhile, the phonon calculations are performed on an 11  11  4 q-mesh within density functional perturbation theory (DFPT)17. After obtaining the electron and phonon eigenvalues, the electron scattering rate can be calculated by combining the maximally localized Wannier functions with the electron-phonon coupling, which are implemented in the electron-phonon Wannier (EPW) package.22,23 In order to achieve converged scattering rate, we apply a dense 3



mesh of 88  88  32 k-point and 22  22  8 q-point in the Wannier interpolation. By inserting the energy-dependent relaxation time into the Boltzmann transport equation24 under the rigid band picture25, the electronic transport coefficients can be calculated. On the other hand, the lattice thermal conductivity can be obtained from the Boltzmann transport theory for phonons.26 A 6  6  2 supercell is adopted for the calculation of 2nd-order interatomic force constants (IFCs)27, and the cutoff is set to be the 8th nearest neighbor for the 3rd-order IFCs 28 . Based on the harmonic (2nd-order) and anharmonic (3rd-order) IFCs, the lattice thermal conductivity is calculated with a fine 24  24  8 q-grid.

3. Results and discussions 3.1 Electronic and transport properties The KSnSb compound has a typical layered structure which is made up of the strong electropositive cation K+ and the covalent bonding polyanion (SnSb)−. There are ionic bonds and van der Waals (vdW) interactions between the K and SnSb layers. Fig. 1 shows the crystal structure of KSnSb, which has the space group of P63mc.10 The primitive cell is hexagonal and contains 6 atoms, where the K atoms occupy the site (2/3, 1/3, 0) and (1/3, 2/3, 1/2), the Sn atoms are located at (0, 0, zSn) and (0, 0, 1/2 + zSn), and the Sb atoms are at (2/3, 1/3, zSb) and (1/3, 2/3, 1/2 + zSb). The internal coordinates zSn and zSb and the lattice constants of KSnSb are summarized in Table 1, where we see that the results with vdW corrections29 give better agreement with the experimental data10 than those without vdW interactions. Fig. 2 plots the energy band structure of KSnSb, where the calculations using PBE and HSE06 functionals are both shown for comparison. In each case, we see that the conduction band minimum (CBM) is located at the Γ point. Besides, there is a conduction band extreme (CBE) along the ΓM direction with energy slightly higher (~0.02 eV) than that of CBM. As for the valence band maximum (VBM), we observe two valleys with almost identical energy along the ΓM and ΓK directions. Such kind of multi-valley structures usually leads to large power factor (PF= S 2 ), 30 as

4


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previously found in the Bi2Te3, Sb2Te331 and Mg3Sb26 systems. We will return to this point later. The indirect band gap of KSnSb is calculated to be 0.38 eV using the PBE functional. It is well known that standard DFT tends to seriously underestimate the band gap. As an alternative, the HSE06 calculation predicts a larger gap of 0.76 eV which is exclusively used in the following calculations. It should be mentioned that we have actually done calculations with and without spin-orbit coupling, and found almost the same energy band structure for both of them. Within the framework of Boltzmann transport theory24, the electrical conductivity

 , the Seebeck coefficient S , and the electronic thermal conductivity  e can be obtained from the band structure S

ek B



 f    

 ( )   0  k BT

d ,

 f 0   d ,   

  e2  ( )  

(2)

(3)

2

e 

k B2T

 f 0       2  ( )      kBT  d  TS  .

(4)

Here e, k B ,  , f 0 and μ are the electron charge, Boltzmann’s constant, energy eigenvalue, Fermi distribution function and chemical potential, respectively.    vi k vi k i k is the transport distribution, where vi k is the group velocity and i ,k

 i k is the relaxation time of the i-th band with wavevector k. The last term TS 2 in Eq. (4) is usually ignored due to its small effect when the power factor and the temperature are low. 32 At elevated temperature, however, we should explicitly consider it. As indicated in Eqs. (2)-(4), the energy dependent relaxation time plays a very important role in determining the electronic transport coefficients. To accurately evaluate the relaxation time, one should know the detailed scattering mechanism such as electron-phonon coupling and impurity scattering. It was demonstrated33 that the impurity scattering plays a significant role in the low-temperature region, while the 5



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electron-phonon coupling dominates in the high-temperature region. Indeed, the electrical conductivity at the low temperature region can be largely reduced by impurity scattering and the ZT value decreases accordingly. However, as a high-temperature thermoelectric material, it is appropriate to consider only the electron-phonon scattering in the KSnSb compound, where the effect of impurity scattering becomes negligible with increasing temperature. Within the EPW framework,23 the relaxation time  i k is given by 1

ik





2  i' k' 2   

dk g j i ,v (k , q )   BZ  j ,v BZ

2

  nq v  f j k q         j k q   F    q v 

,

(5)



  nq v  1  f j k q         j k q   F    q v 

Here  i''k is the imaginary part of the electron self-energy at the i-th band with wavevector k in the Briliouin Zone (BZ), g ji ,v (k , q) is the electron-phonon matrix element with the electron wavevector k and phonon wavevector q,  BZ is the volume of the BZ, n is the Bose-Einstein distribution, and f is the electronic occupation. The other quantities  ,  and  F are the reduced Planck constant, angular frequency and Fermi energy, respectively. Fig. 3 depicts the energy dependent relaxation time around the Fermi energy at two typical temperatures of 300 K and 800 K. We do not consider higher temperature since it was previously found that the melting point of KSnSb should be lower than the temperature of synthesis reaction (803 K).10 At 300 K, we see that the relaxation time of KSnSb can reach as high as femtosecond (fs), which is similar to those of good thermoelectric materials such as Bi2Te3 (~22 fs)34 and Sb2Te3 (~20 fs)7 and is very beneficial for the thermoelectric performance. In addition, we find that the relaxation time of n-type carriers is obviously larger than that of p-type carriers, which suggests better transport coefficients of electrons. We will thus focus on it in the following discussions. Fig. 4 shows the calculated Seebeck coefficient, the electrical conductivity, and the power factor of n-type KSnSb as a function of carrier concentration at 300 K and 800 6


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K. As can be seen from Fig. 4(a), the absolute values of the Seebeck coefficient decrease with the carrier concentration, and are almost identical to each other along the x (in-plane) and z (out-of-plane) directions. At room temperature, the absolute values of Seebeck coefficients can reach ~250 μV/K at a carrier concentration of 1  1019 cm−3, which is higher than that of good thermoelectric material SnSe (~200 μV/K)35. When the temperature is increased to 800 K, the Seebeck coefficients become even larger. The origin of such a high Seebeck coefficient should be attributed to the multi-valley band structures mentioned before. As indicated in Fig. 2, there are two energy valleys around the CBM, which lead to a significant increase of the DOS around the Fermi level and an enhanced energy-dependence of the carrier density n(ε). According to the Mott relation,36 the Seebeck coefficient can be written as S

 2 k B2T  1 dn( ) 1 d c ( )    , 3q  n d  c d  

(6)

where q is the carrier charge and c is the carrier mobility. It is obvious that the enhanced

dn ( ) will results in higher Seebeck coefficient. In contrast, the electrical d

conductivity increases with carrier concentration and shows strong direction and temperature dependence, as shown in Fig. 4(b). In particular, the in-plane (x direction) electrical conductivity is obviously higher than that of out-of-plane (z direction), which is generally found in many layered thermoelectric materials such as Bi2Te334, Sb2Te37 and SnSe37. Meanwhile, the electronic thermal conductivity  e is related to the electrical conductivity  by e  L T according to the Wiedemann-Franz law38, where L is the Lorenz number. In principle, the variation of  e with carrier concentration is almost identical to that of  and is thus not shown here. Due to the well-known competitive behavior of the Seebeck coefficient S and the electrical conductivity  , one can only find the optimized power factor ( S 2 ) at appropriate carrier concentration, as shown in Fig. 4(c). For example, at 300 K, the power factor of KSnSb can reach 4.4  10−3 W/mK2 at a carrier concentration of 2.7  1019 cm−3. The large power factor of KSnSb comes mainly from the enhanced Seebeck coefficient 7



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caused by multi-valley band structures, as discussed above.

3.2 Phonon transport properties We now move to the discussion of phonon transport in KSnSb. The corresponding thermal conductivity  p can be calculated according to the phonon Boltzmann theory26

p 

1 N 0 k B T 2

 n0  n0  1 v 

2



 2 .

(7)

where N0 ,  , n0 , v ,   and   are the number of q-points in the first Brillouin zone, the volume of the primitive cell, the Bose-Einstein distribution function, the group velocity, the relaxation time, and the angular frequency of phonon mode  , respectively. As shown in Fig. 5, the  p decreases monotonously with temperature increasing from 300 to 800 K. Moreover, the  p along the z direction is obviously lower than that along the x direction, which is also found in layered thermoelectric materials.7,34,37 It is interesting to find that the lattice thermal conductivity of KSnSb is rather small. For example, the x and z components at 300 K are 1.6 and 0.9 W/mK, respectively. To understand the origin of such intrinsically lower thermal conductivity, we first investigate the phonon dispersion relations of KSnSb, as shown in Fig. 6(a). We see there is no imaginary frequency which indicates that the structure is dynamically stable. Note that the maximum phonon frequency of KSnSb is about 5.3 THz. Such a lower value is comparable with those of Bi 2Te3 (4.6 THz)34, Sb2Te3 (5.2 THz)39, and SnSe (5.6 THz)35. Besides, there are two large phonon gaps around 2 and 4 THz, which is a sign of low lattice thermal conductivity.40 The analysis of phonon density of states (PDOS) indicates that the phonon branches in the lower and higher frequency region are mainly contributed by the Sn and Sb atoms, while the K atoms dominate the intermediate frequency region. In Fig. 6(b), we show the lattice thermal conductivity contribution as a function of frequency along the x and z directions at 300 K. It can be found that more than 90% of 8


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 p is contributed by low-frequency phonons (< 2 THz), or caused by the Sn and Sb atoms as discussed above. In addition, we find from Fig. 6(c) that the average group velocity v below 2 THz is only about 2.0 km/s, which is similar to that of good thermoelectric material PbTe41. Such a small value again suggests the low lattice thermal conductivity of KSnSb. The normalized cumulative lattice thermal conductivity of KSnSb with respect to phonon mean free paths (MFP) at 300 K is shown in Fig. 6(d). It is clear that  p is dominated by phonons with MFP ranging from 1 to 200 nm. Besides, the MFP values for 50%  p accumulation are 23 and 19 nm for the x and z directions, respectively. Such lower MFP values at half  p are close to that found in PbTe (~20 nm)42. Meanwhile, the average phonon scattering rate of KSnSb we obtained is 0.2 ps−1 which is similar to that of SnSe37. The calculated total Grüneisen parameter  of KSnSb is about 1.0, which is ~28% lower than that of Bi2Te3 (   1.4 )34 and ~59% higher than that of SnSe (   0.63 )37. The anharmonic phase space volume P3 of KSnSb is calculated to be 1.3  10−2 eV, which is a little lower than that of SnSe (1.9  10−2 eV)37 and much higher than that of Si (0.35  10−2 eV)43. All these findings confirm that the layered KSnSb indeed has intrinsically lower lattice thermal conductivity, and is in principle governed by the co-existence of K-K metallic bonds, K-Sn ionic bonds, Sn-Sb covalent bonds, as well as the interlayer vdW interactions. 44

3.3 Figure of merit After all the transport coefficients have been calculated, we can predict the thermoelectric performance of KSnSb. Fig. 7 shows the n-type ZT values as a function of carrier concentrations at two typical temperature of 300 and 800 K, and the corresponding transport coefficients are summarized in Table 2. We see that at the optimized carrier concentration of 2.8  1019 cm−3, the maximum ZT values of 2.2 and 2.0 can be reached at 800 K along the x and z directions, respectively. Such 9



extraordinary thermoelectric performance of KSnSb exceeds those of most bulk thermoelectric materials. Moreover, we see that the in-plane ZT value is comparable to that of out-of-plane, which suggests that the polycrystalline KSnSb may exhibit almost the same electronic transport properties as single crystal. Consider the fact that polycrystalline could have smaller lattice thermal conductivity, it is expected that polycrystalline KSnSb may exhibit even better thermoelectric performance.

4. Summary In summary, we demonstrate by first-principles calculations that the Zintl compound KSnSb is semiconducting with an indirect band gap of 0.76 eV predicted from hybrid functional. The multi-valley band structures around the Fermi level lead to significantly enhanced Seebeck coefficient and power factor, as derived from the Boltzmann transport theory with carrier relaxation time rigorously obtained from the electron-phonon coupling. By solving the phonon Boltzmann transport equation, we find that the KSnSb compound exhibits very small lattice thermal conductivity, as characterized by its large values of phonon gaps, Grüneisen parameter, and phase space volume, as well as small values of group velocity and phonon mean free path. In principle, the intrinsically lower lattice thermal conductivity is traced back to the mixture of various bonding in the KSnSb compound, including the K-K metallic bonds, the K-Sn ionic bonds, the Sn-Sb covalent bonds, and the interlayer vdW interactions. All these findings make n-type KSnSb a very promising candidate for thermoelectric application in the high temperature region, with a high ZT value of 2.2 at optimized electron concentration of 2.8  1019 cm−3. It should be noted that KSnSb is known to be highly air sensitive like other Zintl compounds. In order to realize the promising thermoelectric application of KSnSb, one needs to take special care by synthesizing it in a low-oxygen or vacuum environment. We also want to mention that our theoretical predictions are based on the assumption that KSnSb can be n-type doped. Although currently the experimental report of such kind of doping is still lacking, we believe that by using techniques similar to those done for n-type Mg3Sb2,45,46 another Sb-based Zintl compound, one can realize n-type KSnSb with 10


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desired carrier concentration.

Acknowledgments We thank financial support from the National Natural Science Foundation (Grant No. 11574236 and 51772220) and the “973 Program” of China (Grant No. 2013CB632502). The numerical calculations in this paper have been done on the supercomputing system in the Supercomputing Center of Wuhan University.

11



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Table 1 The calculated structural parameters of the Zintl compound KSnSb with and without consideration of vdW interactions. The experimental data10 are also shown for comparison. This work

Exp.

w/o vdW

with vdW

a (Å)

4.359

4.45

4.40

c (Å)

13.15

13.3

13.0

zSn

0.2010

0.203

0.200

zSb

0.3115

0.312

0.312

Table 2 The optimized ZT values of n-type KSnSb along the x and z directions at 300 K and 800 K. The corresponding carrier concentration and transport coefficients are also listed.

x

z

T

n

S



S 2

e

p

(K)

(1019 cm−3)

(μV/K)

(S/cm)

(10−3 W/mK2)

(W/mK)

(W/mK)

300

1.5

−212

934

4.19

0.43

1.61

0.6

800

2.8

−276

352

2.67

0.35

0.60

2.2

300

1.5

−206

461

1.96

0.22

0.92

0.5

800

2.8

−273

168

1.26

0.16

0.34

2.0

12


ZT

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Figure Captions

Fig. 1 The crystal structure of KSnSb: (a) side-view and (b) top-view.

Fig. 2 The energy band structures of KSnSb, calculated with PBE and HSE06 functionals. The Fermi level is at 0 eV.

Fig. 3 The energy-dependent relaxation time of KSnSb. The Fermi level is at 0 eV.

Fig. 4 The calculated electronic transport coefficients as a function of carrier concentration along the x (in-plane) and z (out-of-plane) directions for n-type KSnSb at 300 K and 800 K: (a) the Seebeck coefficient S , (b) the electrical conductivity  , and (c) the power factor S 2 .

Fig. 5 The lattice thermal conductivity as a function of temperature for KSnSb along the x and z directions.

Fig. 6 (a) The phonon dispersion relations and density of states for KSnSb, (b) the lattice thermal conductivity contribution as a function of phonon frequency along the x and z directions at 300 K, (c) the group velocity as a function of phonon frequency, and (d) the normalized cumulative lattice thermal conductivity as a function of phonon mean free path.

Fig. 7 The calculated ZT values as a function of carrier concentration along the x and z directions for n-type KSnSb at 300 and 800 K.

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Figure 1 396x215mm (96 x 96 DPI)



Figure 2 272x208mm (150 x 150 DPI)


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Figure 3 242x189mm (150 x 150 DPI)





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Figure 5 272x208mm (150 x 150 DPI)





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Figure 7 272x208mm (150 x 150 DPI)


First-Principles Study of the Thermoelectric Properties of the Zintl

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