Quaternary compounds Ag2XYSe4 (X = Ba, Sr

1. Introduction

During the last few decades, with the increasing problems of energy exhaustion and environment pollution, thermoelectric (TE) materials have already aroused people’s widespread concern [1, 2], due to their special properties of directly con­

verting heat to electricity and vice versa. One can use a dimen­

sionless figure  of merit, namely ZT = S²σT/κ, to assess the TE materials’ conversion efficiency from heat to electricity [3, 4]. Herein, S and σ represent the Seebeck coefficient and electrical conductivity, T and κ are the temperature (in unit of Kelvin) and total thermal conductivity composed of elec­

tronic κe and lattice κL contributions. Thus, a good TE mat­

erial requires high S and σ and low κ. However, this is hard to achieve due to the complex interdependences between S, σ and κ [5]. Recently, it was reported that achieving large band

degeneracy [6] or localized resonant states [7] via energy band engineering do improve the TE transport properties in few existing TE materials [8]. Nevertheless, this strategy has great blindness because one is difficult to decide which dopant can induce beneficial change to electronic structure. Another common strategy is to reduce lattice thermal conductivity by low­dimension [9] or solid solution strategies [10, 11] that usually require very rigorous preparation conditions. Thus, seeking new TE materials with medium TE performance and then improving ZT value by tuning the carrier density remains a simple and feasible method [12, 13].

How to screen good TE materials through electronic struc­

tures has always been of concern to researchers. For example, Liu et al [14] propose the generalized material parameter B^*_∝ U^*Eg/κL as a screening criterion where the weighted mobility U^*, the total thermal conductivity κL, and the band gap E_g.

Good TE material should have B^* as high as possible. Thus, theoretically, wide bandgap materials should possess better TE

Journal of Physics D: Applied Physics

Quaternary compounds Ag ₂ XYSe ₄

(X = Ba, Sr; Y = Sn, Ge) as novel potential thermoelectric materials

A J Hong^1,3 , C L Yuan^1,3 and J-M Liu²

1 Jiangxi Key Laboratory of Nanomaterials and Sensors, School of Physics, Communication and Electronics, Jiangxi Normal University, Nanchang 330022, People’s Republic of China

2 Laboratory of Solid State Microstructures and Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, People’s Republic of China

E­mail: [email protected] and [email protected] Received 13 August 2019, revised 20 October 2019 Accepted for publication 10 December 2019 Published 3 January 2020

Abstract

Experimental results have shown that the quaternary compound Cu2ZnSnSe4 is an excellent thermoelectric (TE) material. This inspires us to seek the other quaternary compounds with similar chemical formula to Cu2ZnSnSe4 as TE materials. In this paper, we use the first­

principle method to systematically explore the electronic and phonon structures, mechanical, thermal and TE properties of p­ and n­type Ag2XYSe4 (X = Ba, Sr; Y = Sn, Ge). It is found that the ZT maximum for n­type Ag2SrGeSe4 can reach up to 1.22 at 900 K, and those for p­type Ag2SrSnSe4, Ag2SrGeSe4 and Ag2BaSnSe4 can reach up to 1.20, 1.13 and 1.12, respectively. Our work not only shows that Ag2XYSe4 (X = Ba, Sr; Y = Sn, Ge) are a kind of potential TE material, but also can inspire more theoretical and experimental research on the TE properties of quaternary compounds.

Keyword: thermoelectric materials, electrical conductivity, thermal conductivity (Some figures may appear in colour only in the online journal)

A J Hong et al

Printed in the UK 115302

JPAPBE

© 2020 IOP Publishing Ltd 53

J. Phys. D: Appl. Phys.

JPD

10.1088/1361-6463/ab6056

Paper

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Journal of Physics D: Applied Physics IOP

3 Authors to whom any correspondence should be addressed.

2020

1361-6463

https://doi.org/10.1088/1361-6463/ab6056 J. Phys. D: Appl. Phys. 53 (2020) 115302 (14pp)

and ~0.35 eV [26], respectively.

Recently, quaternary compounds Cu2ZnSn(S/Se)4 (CZTSSe) are considered as a promising photovoltaic absorber material due to high efficiency of about 12.6% [15]. It was reported that the band gaps of pure Cu2ZnSnSe4 (CZTSe) and Cu2ZnSnS4

(CZTS) were about 1.0 [16–18] and 1.5 eV [19, 20] close to ideal values of 1.3–1.4 eV for solar cell materials. By judging from band gaps, instead of CZTS, compound CZTSe seems to be a promising TE material due to smaller band gap suit­

able for carrier density regulation. Indeed, previous works have indicated that the ZT of ln­doped CZTSe reached up to 0.95 at 850 K [21] and that of modified CZTS was only 0.36 at 700 K [22]. Therefore, we tried to replace Cu, Zn and Sn atoms of compound Cu2ZnSnSe4 with Ag, Ba/Sr and Sn/Ge atoms, con­

structing four new compounds Ag2XYSe4 (X = Ba, Sr; Y = Sn, Ge). The structures of Ag2BaGeSe4 and Ag2BaSnSe4 have been experimentally verified to be space group I222 (No. 23) [23] and the other two were theor etically predicted to be space group I222 [24], also. Obviously, Ag2XYSe4 structures pos­

sess lower symmetry than kesterite and stannite Cu2ZnSnSe4

belonging to space groups I4 (No. 82) and I42m (No. 121) [25], respectively. Low symmetry can favorite the strength­

ening anharmonic vibration of phonon, and thus the low sym­

metry materials such as SnSe, Zn4Sb3 and MgAgSb usually possess low lattice thermal conductivity [26]. Furthermore, it is expected Ag2XYSe4 have high electrical conductivity due to containing two Ag atoms in each primitive cell.

To the best of our knowledge, there have been few theor­

etical reports on the TE properties of Ag2XYSe4 (X = Ba, Sr; Y = Sn, Ge). Hence, in this work, we used density func­

tional theory (DFT) with semi­classical Boltzmann equa­

tion to explore the TE property of p­ and n­type Ag2XYSe4

(X = Ba, Sr; Y = Sn, Ge). It was showed that band gaps of Ag2BaGeSe4, Ag2BaSnSe4, Ag2SrGeSe4 and Ag2SrSnSe4

are 0.909 eV, 0.832 eV, 0.708 eV and 0.729 eV, respectively.

In addition, their lattice thermal conductivities are 2.22 W m⁻¹ K⁻¹, 1.95 W m⁻¹ K⁻¹, 2.26 W m⁻¹ K⁻¹ and 2.00 W m⁻¹ K⁻¹ at 300 K that are obviously lower than 3.2 W m⁻¹ K⁻¹ of Cu2ZnSnSe4 [21]. Lattice thermal conductivity at high temper ature for each compound is below 1.0 W m⁻¹ K⁻¹. It is predicted that the n­type Ag2SrGeSe4, having ZT maximum of 1.22 (at 900 K), is the most excellent in the compounds Ag2XYSe4 (X = Ba, Sr; Y = Sn, Ge), and its p­type counter­

part has a ZT maximum of 1.13. The ZT maximums of both p­type Ag2BaSnSe4 and Ag2SrSnSe4 are above unity. This paper reveals Ag2XYSe4 are a new family of promising can­

didates for TE materials and may pave a way for seeking TE materials with high the ZT values.

Firstly, structure optimization was performed using the generalized gradient approximation of Perdew, Burke, and Ernzerhof (GGA­PBE) [27] as the electronic exchange­corre­

lation functional in VASP code. The plane­wave energy cutoff and Monckhorst–Pack k­point mesh were set to 500 eV and 10 × 10 × 10, and the total­energy and force convergences were 10⁻⁶ eV and 10⁻³ eV Å⁻¹.

Secondly, we used the full potential method with GGA­PBE modified Becke and Johnson potential scheme (mBJ) [28] to perform atomic coordinate optimization and then calculate electronic structures for all the compounds in Wien2k code [29]. In the whole calculation procedure, the total­energy and charge convergences were taken as 0.0001 eV and 0.0001 e.

3000 k­points in the whole first Brillouin zone were sampled for calculations of such as band structure and density of states.

Taking a very dense mesh of 20 000 k­points, the electrical transport parameters such as the Seebeck coefficient, elec­

trical conductivity and electronic thermal conductivity, were calculated using the semi­classical Boltzmann theory. In this theoretical framework, the electronic transport parameters can described as follows [30]:

σαβ(T,EF) = 1 Ω

ˆ

σαβ(ε) ï

−∂f

∂ε ò

dε,

(1) Sαβ(T,EF) = (σ⁻¹)_lαηlβ,

(2) καβ(T,EF) =κ⁰_αβ−Tηαm(σ⁻¹)_nmηmβ,

(3) where T, Ef , Ω, f and ε are the absolute temperature, Fermi energy, unit cell volume, Fermi–Dirac distribution function and energy for electronic state. As noted, καβ is the electronic thermal conductivity without the external electric field. The energy projected conductivity tensors, σαβ, can be expressed as the following form

σαβ(ε) =e²τ N

i,k

νανβ

δ(ε−ε_i,k) dε .

(4) Herein, e is the charge of an electron, τ is the carrier relax­

ation time, N is the total number of sampled k­points, εi,k is the energy for the kth point at the ith band. The other physical parameters are given as follows:

ηαβ(T,Ef) = 1 eTΩ

ˆ

σαβ(ε) (ε−Ef) ï

−∂f

∂ε ò

(5)dε,

κ⁰_αβ(T,Ef) = 1 e²TΩ

ˆ

σαβ(ε) (ε−Ef)² ï

−∂f

∂ε ò

(6)dε.

Subsequently, we used Slack’s equation  (see equation  (7)) to calculate the lattice thermal conductivity for all systems [31].

κL=AΘ³_DVper^1/3m¯

γ_a²n^2/3tot T . (7) Herein, A is a collection of physical constants with the value of about 3.04 × 10⁻⁶, ΘD is the Debye temperature, Vper is the volume per atom, m is the average atom mass in the whole unit cell, ntot is the total number of all atoms in the primitive cell. γa is the Grüneisen parameter for only acoustic pho­

nons. Previous works [32–34] show lattice thermal conduc­

tivity attained by Slack’s equation is in good agreement with experimental data. For the purpose of comparison, we also used compressive sensing lattice dynamics (CSLD) method [35] combined with Boltzmann transport equation  (BTE) [36] to compute the lattice thermal conductivity. In addi­

tion, we used the Phonopy code [37] to calculate the phonon spectrum.

3. Results and discussion

3.1. Crystal structures and electronic structures

Crystal structures of compounds Ag2XYSe4 have been veri­

fied experimentally or theoretically to belong to group I222 (No. 23). There are 16 and 8 atoms in the conventional and primitive cells, respectively (see figure 1). The lattice con­

stants of conventional cells in this work and other literatures are summarized in table 1. It is seen that our results for Sr compounds are in good agreement with the other theoretical data. However, for Ba compounds, there is a little differ­

ence compared to the experimental data. Experimental and theoretical lattice constants are determined at room and zero temperatures respectively, thus the difference is perfectly acceptable. The cell volume of Ag₂SrGeSe₄ is smaller than that of Ag₂SrSnSe₄ due to the difference in the atomic radii of Ge and Sn, which is the same as Ba compounds. Similarly, due to the Ba radius being smaller than the Sr radius, the cell

Figure 1. The (a) conventional (b) and primitive cells of Ag2XYSe4.

Table 1. Experimental and calculated lattice constants (in Å) and calculated band gaps (in eV) of Ag₂XYSe₄ (X = Ba, Sr; Y = Sn, Ge) using different exchange­correlation functionals (PBE, mBJ and HSE06).

Ag₂BaGeSe₄ Ag₂BaSnSe₄ Ag₂SrGeSe₄ Ag₂SrSnSe₄

This work Expt.^a This work Expt.^b This work Theo.^c This work Theo.^c

a (Å) 7.053 7.058 7.141 7.116 7.110 7.115 7.183 7.193

b (Å) 7.501 7.263 7.713 7.499 7.413 7.389 7.673 7.657

c (Å) 8.481 8.263 8.582 8.337 8.061 7.951 8.135 8.034

PBE bandgap (eV) 0.284 0.195 0.000 0.041

mBJ bandgap (eV) 0.909 0.832 0.708 0.729

HSE06 bandgap (eV)^c 0.85 0.77 0.68 0.66

a From Tampier and Johrendt [23].

b From Assoud et al [38].

c From Zhu et al [24].

volumes of Ag2BaGeSe4 and Ag2BaSnSe4 are smaller than those of Ag2SrGeSe4 and Ag2SrSnSe4, respectively.

Calculated band structures and corresponding density of states (DOS) of the four compounds are showed in figure 2. Sn compounds have indirect band gaps of 0.765 eV (Ag2BaSnSe4)

and 0.694 eV (Ag2SrSnSe4), of which valence band maximum (VBM) and conduction band minimum (CBM) are at the Z and Γ points, respectively. However, the Ge compounds have direct band gaps of 0.870 eV (Ag2BaGeSe4) and 0.667 eV (Ag2SrGeSe4), with both CBM and VBM appearing at Z

Figure 2. Calculated band structures along high symmetry points shown in (a0) and DOS for Ag₂XYSe₄ (X = Ba, Sr; Y = Sn, Ge).

point. It is worth mentioning the difference between indirect and direct band gaps for each compound is very small. We have also noticed that other work [24] points Ag2BaGeSe4

and Ag2SrGeSe4 possess indirect band gaps. Perhaps, the full potential method with mBJ scheme was adopted in this work, which is different from the pseudopotential method, leading to this difference. The PBE, mBJ and HSE06­hybrid (from [24]) band gaps for the four compounds are also listed in table 1.

The mBJ band gaps, close to the HSE06­hybrid band gaps, are obviously larger than the PBE band gaps. The normal PBE method usually underestimates the band gap, but the mBJ method, comparable with GW, usually gets band gap in good agreement with the experimental data.

The DOS diagrams show Se–Ag hybridization is the major ingredient of valence band (VB) in all the compounds, which implies Se–Ag covalent bonds are in the four compounds.

The conduction band (CB) displays significant density of Se and Ge/Sn atoms. The Ba/Sr atoms have negligible contrib­

utions to both VB and CB in each compound. It is implied that doping vacancy or heterogeneous atom at the site of Ba/

Sr atoms can result in little effect on electronic transport prop­

erties. The valence bandwidth (VBW) is ~2 eV greater than the conduction bandwidth (CBW) of less than 1 eV in each

compound. This possibly implies that n­type carrier has larger effective mass than p­type carrier does.

Calculated effective mass along the a, b and c axes as a function of energy is plotted in figure 3. It is expected that the electron effective mass is larger than the hole effective mass along the a and b axes in each compound, although it is in the opposite case along the c axis. In addition, the hole effec­

tive mass decrease with the increasing energy, which is the opposite of the electron effective mass. For each compound, the hole effective mass along the c axis is larger than along the a and b axes in each compound, and the electron effective mass along the b axis is larger than along the other axes. For instance, the effective mass for p­type Ag2SrSnSe4 along the c axis can reach up to 1.33 me, and the effective mass for n­type Ag2BaSnSe4 is as large as 9.82 me (not shown in figure 3) at

~0.01 Ry above the CBM.

3.2. Structure stabilities

In order to verify structural stability, phonon spectrum with corresponding phonon DOS for the four compounds was cal­

culated and plotted in figure 4. There are eight atoms in one primitive cell and one atom can yield three phonon branches.

Figure 3. The effective mass of p­ and n­type carriers for Ag2XYSe4 (X = Ba, Sr; Y = Sn, Ge) as a function of energy. Both VBM and CBM are set to 0 eV.

Thus, there are twenty four phonon branches composed of sixteen longitudinal branches and eight transverse branches, including one longitudinal acoustic (LA), two transverse acoustic (TA) modes.

Phonon DOS in the range of low frequency mainly comes from the contribution of Ag atom. The phonon thermal trans­

port property is primarily decided by phonon modes of low frequency. This implies Ag atom play important role and then Sn atom has negligible effect in lattice thermal transport. As is known, for a system with dynamical stability, the frequency of each phonon mode should be not imaginary. Our result shows no obvious imaginary frequencies in the full Brillouin zone, thus ensuring structure stability.

Elastic constants were obtained by second­order­deriva­

tive of polynomial fit of energy as a function of strain at zero strain, which called energy approach. For the orthorhombic Ag2XYSe4, there are nine independent elastic constants (c11,

c22, c33, c44, c55, c66, c12, c13 and c23). The elastic constants c11, c22 and c33 are indicative of the stiffness against prin­

cipal strains that are along the a, b and c axes, respectively.

The other elastic constants represent resistance against shear deformations, which are obviously smaller than the c_ii (i = 1, 2, and 3). Table 2 shows the c11, c22 and c33 follow the order of c11 > c22 > c33. Interestingly, the lattice constants follow the opposite order of a < b < c. The elastic constants can be used to assess structural mechanical stability. The criteria of mechanical stability for orthogonal crystals are expressed by [39]

c₁₁ >0,c₂₂>0,c₃₃ >0,c₄₄>0,c₅₅ >0,c₆₆>0, (8)

c11+c22+c33+2c12+2c13+2c23>0,

(9) c₁₁+c₂₂−2c₁₂ >0,

(10)

Figure 4. The phonon spectrums along high symmetry points for Ag₂XYSe₄ (X = Ba, Sr; Y = Sn, Ge) and phonon DOS.

c11+c13−2c13 >0,

(11) c₂₂+c₃₃−2c₂₃>0.

(12) It is clear that the four compounds are of mechanical stability.

Some mechanical and thermal parameters can be calculated by the elastic constants [40]. For examples, using the Voigt– Reuss–Hill approximations bulk and shear moduli B and G can be calculated (summarized in table 2). B for each com­

pound is larger than G. The G/B ratio is usually used to eval­

uate the brittleness of materials. High G/B ratio is indicative of fragile material. For example, the fragile material α­SiO2

has a high G/B ratio of 1.06 [41]. The G/B ratios for the four compound are all less than 0.5, showing they are unbreakable.

Using B and G Poisson’s ratio can be computed:

ν= 3B−2G 6B+2G.

(13) In macroscopic view, Poisson’s ratio can estimate the corre­

sponding lateral strain to applied axial strain. Microcosmically, it represents the degree of covalent bond. The low value of Poisson’s ratio is indicative of covalent bond; high value rep­

resents strong ionic or metallic bonds [43]. Poisson’s ratios of Ag2XYSe4 are lower than 0.367 of silver bulk. This indicates Ag atom in Ag2XYSe4 participates in covalent bonds, which agrees with results from electronic band structures.

Debye temperature Θ and acoustic Grüneisen parameter γa

are attained using the following formulas:

γa= 9v²_l −12v²t

2v²_l +4v²t ,

(14)

ΘD= h kB

Å3ntotNAρ 4πM

ã1/3

vm,

(15) where physical constants h, NA are Plank’s constant and Avogadro’s number, M and ρ are the total atomic mass in primitive unit cell and the volume density of materials, vm is the average sound velocity given by the velocities for trans­

verse and longitude waves velocities vl and vs. The three velocities can be given by the following formulas:

vs=

Å3B+4G 3ρ

ã1/2

, vp= ÅG

ρ ã1/2

,

(16)

vm= ñ1

3 Ç2

v³_s + 1 v³_p

åô−1/3

(17).

Due to similarities of structure and composition, the four com­

pounds have close values of both Θ and γa. The largest differ­

ence of Θ is 12.36 K between Ag2BaGeSe4 and Ag2SrSnSe4. The Grüneisen parameter scales the degree of anharmonic vibrations. The Grüneisen parameters γa of the four compounds are about 1.8, the Ag2SrSnSe4 has the largest γa value of 1.85, indicating strong anharmonic vibrations of acoustic wave. As is known the thermal resistivity (reciprocal thermal conduc­

tivity) origins from anharmonic vibrations of phonon and espe­

cially acoustic phonon. It is noted that recent experiential work shows the average Grüneisen parameter of Ag2BaSnSe4 is very

small value of 0.65 [42] that is less than the acoustic Grüneisen parameter ~1.8 in this work. This implies that anharmonic vibrations of optical phonon are very weak.

3.3. TE properties

3.3.1. Carrier relaxation time. It is well known that the pre­

cise calculation for the carrier relaxation time is difficult to be implemented due to complex carrier­scattering mechanism. In this work, we calculated the relaxation time based on con­

sidering the carrier­phonon scattering using the Fermi­golden rule and the deformation potential (DP) theory. According to the Fermi­golden rule, the scattering rate Pk at the electronic state k can be described by

P_kk =

k

Mk,k²δε(k)−ε(k)±∆E, (18) where the matrix element M(k, k′) represents the scattering from k state to k′ state, which is known as the transition matrix element, ΔE is the phonon energy. In this work, we taken the phonon energy as 3kBT. Based on solid state physics theory, the reciprocal of the carrier relaxation time can expressed as:

1 τ(k) =

k

Pkk(1−cosθ)

=2π

k

Mk,k²δε(k)−ε(k)±∆E

(1−cosθ). (19) Herein, the angle θ is between the k state to the k’ state. In the scattering process, the carrier absorbs or emits a phonon.

Thus, by the DP theory, the effective transition matrix ele­

ment around the valence band maximum (VBM) or conduc­

tion band maximum (CBM) can be given by [43]

Table 2. Calculated elastic constants cij in GPa, bulk and shear moduli B and G in GPa, velocities of transverse and longitudinal waves vt and vl in m s⁻¹, Poisson’s ratios ν, Debye temperatures Θ in K and acoustic Grüneisen parameters γa for Ag₂XYSe₄ (X = Ba, Sr; Y = Sn, Ge).

Ag2BaGeSe4 Ag2BaSnSe4 Ag2SrGeSe4 Ag2SrSnSe4

c11 90.1953 90.8454 98.4429 99.8721

c₂₂ 56.0873 64.6186 66.9605 63.6739

c₃₃ 52.2170 52.8386 62.3014 56.6002

c44 24.5931 22.7369 24.7166 23.2939

c55 26.0964 21.4068 25.2265 23.4543

c₆₆ 33.7824 31.6366 29.2304 30.8721

c12 48.8688 53.5752 54.4242 56.4909

c13 33.0205 29.9881 35.1455 32.7599

c₂₃ 9.3571 30.3142 29.9960 30.7913

B 44.207 45.954 49.460 48.150

G 20.685 19.939 21.969 20.606

vt 1944.30 1900.12 2016.75 1943.25

vl 3622.08 3624.24 3818.37 3722.74

va 2171.08 2125.24 2254.55 2174.10

ν 0.297 0.310 0.306 0.312

Θ 212.535 204.537 224.897 212.985

γa 1.75 1.83 1.81 1.85

Figure 5. The relaxation time of p­ and n­type carrier as a function of temperature for Ag₂XYSe₄ (X = Ba, Sr; Y = Sn, Ge).

Figure 6. Calculated Seebeck coefficient with respect to temperature T and carrier density n for p­ and n­type Ag₂XYSe₄ (X = Ba, Sr;

Y = Sn, Ge).

|M|²=kBTE_β² Ωcβ

,

(20) where Eβ and cβ are the deformation potential constant and elastic constant along the β direction. It is noted that the anisotropy is neglected and the matrix element M is indepen­

dent of the θ in the DP theory, and thus the above equation can be described as

1

τ(k) = 2π

k

Mk,k²δε(k)−ε(k)±∆E.

(21) It is worth mentioning that the anisotropy is not completely ignored in the DP theory, and it arises from the anisotropic deformation potential constants and anisotropic elastic con­

stants, which has small difference in the different directions.

The carrier relaxation time was showed in figure 5. It is found that the τ for each compound reduces with increasing T and is of the order of magnitude of 10⁻¹⁴–10⁻¹⁵. The τ for p­type Ag2BaSnSe4 and Ag2SrSnSe4 is obviously higher than their n­type. However, for Ag2BaGeSe4 and Ag2SrGeSe4, p­

and n­type carriers have close values of relaxation time. It is worth mentioning that the scatters from kinds of defects were not considered. Thus, the actual value should be lower that the calculated.

3.3.2. Electronic transport properties. According to Boltzman theory, Seebeck coefficient S, electrical conductivity σ and electronic thermal conductivity κe are all related to temper­

ature and chemical potential. The chemical potential corre­

sponds to carrier density. Thus, calculated S with respect to

Figure 7. Calculated electrical conductivity with respect to temperature T and carrier density n for p­ and n­type Ag₂XYSe₄ (X = Ba, Sr;

Y = Sn, Ge).

temperature T and carrier density n for p­and n­type Ag2XYSe4

(X = Ba, Sr; Y = Sn, Ge) were calculated and then plotted in figure 6. The largest |S| of n­type system is higher than that of the p­type system. For example, the largest |S| of n­ and p­type Ag2BaSnSe4 are 775 and 668 µeV respectively. The largest

|S| for each n­type systems is above 700 µeV. |S| follows the following rules: 1. The |S| reduces with increasing carrier den­

sity at low temperatures, and reduces and then rises at high temperature. 2. The |S| rises and then reduces with increasing temperature at low carrier density. The |S| rises with increas­

ing temperature at high carrier density.

Figure 7 shows that the electric conductivity is more sen­

sitive to carrier density than to temperature. In the case of p­type Ag2BaSnSe4, σ at 300 K changes from 4.12 S cm⁻¹ to 2.71 × 10⁴ S cm⁻¹ with the carrier density increasing from

10¹⁷ to 10²¹ cm⁻³. σ at 300 K reduces to 1.46 S cm⁻¹ at 620 K and then rises to 7.33 S cm⁻¹ at 900 K. Figure 8 shows that electronic thermal conductivity at low carrier densities <10¹⁹ cm⁻³ has negligible contribution to the total thermal conduc­

tivity. At 300 K, κe of p­type Ag2BaGeSe4 is 0.004 W m⁻¹ K⁻¹ at 10¹⁷ cm⁻³, which can reach up to 0.37 W m⁻¹ K⁻¹ at 10¹⁹ cm⁻³. κe is not sensitive to the change of temperature like σ. At 10¹⁹ cm⁻³, when temperature increases 300 K to 900 K, κe reduces to 0.06 W m⁻¹ K⁻¹.

3.3.3. Phonon transport properties. Figure 9 shows the calculated specific heat at constant volume Cv of the four compounds and experimental specific heat at constant press Cp of Ag2BaSnSe4. Below 550 K, Cv and Cp of Ag2BaSnSe4

agree with each other. Above 550 K, Cp is higher than Cv. The

Figure 8. Calculated electronic thermal conductivity with respect to temperature T and carrier density n for p­ and n­type Ag₂XYSe₄ (X = Ba, Sr; Y = Sn, Ge).

difference between Cv and Cp above 550 K origins from the thermal expansion coefficient and isothermal compressibility, which is reasonable and acceptable. Θ was also obtained by fitting the calculated specific heat Cv according to the follow­

ing equation:

CV =9NkB

Å T ΘD

ã3ˆ ΘD/T 0

x⁴e^x (e^x−1)²dx.

(22) Herein, N represents the total number of atoms in the primitive cell. In figure 9(b), Θ at 300 follows the order of Ag2SrGeSe4 (255.0K) > Ag2BaGeSe4 (249.9) > Ag2BaSnSe4

(228.3) > Ag2SrSnSe4 (228.0), which indicates the values are slightly higher than these by equation (15). Experimental Θ of Ag2BaSnSe4 is about 160 K [42] that is obviously lower than the calculated value of 244.9 K. We consider the different fitting equations lead to the difference of Debye temperatures between in this work and [42].

We used Slack’s equation  and BTE to calculate the lat­

tice thermal conductivity of Ag2XYSe4, respectively. Lattice thermal conductivities calculated by two equations  both reduce with rising temperatures (see figures 10(a) and (b)). κL

obtained by Slack’s equation is higher than that by BTE. κL

of Ag2BaSnSe4 by BTE is closer to the experimental data (see figure 10(c)) from [42] but that by Slack’s equation  seems more reasonable. The calculated κL is without the effects of grain boundary, vacancy and doping, and only is average of anisotropic property, which should be higher than exper­

imental data of materials containing all kinds of defects. Thus, we used κL by Slack’s equation to calculated ZT.

3.3.4. Figure of merit. ZT with respect to T and n in a suf­

ficiently broad region (n, T) were plotted in figure 11, where the red, green and blue areas denote the high, moderate, and low ZT. N­type Ag2BaSnSe4 and Ag2SrSnSe4 possess respec­

tively low ZT maximums 0.72 and 0.92, although they have larger red closed region than the other systems have the red open region. ZT maximums for p­ and n­type Ag2SrGeSe4

reach up to 1.22 and 1.13, respectively. However, these for p­ and n­type Ag2BaGeSe4 are below unit (0.91 and 0.97). ZT maximums for P­type Ag2BaSnSe4 and Ag2SrSnSe4 are 1.20 and 1.12, respectively. For all the p­type systems and n­type Ag2BaGeSe4 and Ag2SrGeSe4, ZT maximums appear at 900 K and in the range of 10¹⁹–10²⁰ cm⁻³. However, these for n­type Ag2BaSnSe4 and Ag2SrSnSe4 appear in the range of 10²⁰–10²¹ cm⁻³ and respectively at 670 K and 810 K.

Figure 9. (a) Calculate experimental heat capacity at constant volume for Ag₂XYSe₄ (X = Ba, Sr; Y = Sn, Ge) and experimental heat capacity for Ag₂BaSnSe₄, (b) calculate Debye temperature for Ag₂XYSe₄ (X = Ba, Sr; Y = Sn, Ge).

Figure 10. Calculated lattice thermal conductivity obtained by (a) BTE and (b) Slack’s equation for for Ag₂XYSe₄ (X = Ba, Sr; Y = Sn, Ge), (c) comparisons of the calculated and experimental lattice thermal conductivities for Ag₂BaSnSe₄.

In order to study TE properties in detail, we extract ZT as a function of carrier density n at 300 K, 600 K and 900 K from figure 11 and show them in figure 12. For all the systems, ZT rises to the peak value and then reduces with increasing carrier density. At most carrier densities, ZT rises with increasing tem­

perature. In the range of 10¹⁸–10¹⁹ cm⁻³, ZT is sensitive to both carrier density and temperature. At high temperature, ZT is more sensitive to carrier density. In the case of p­type Ag2BaSnSe4, The ZT at 900 K changes from 0.06 to 1.20 with the carrier den­

sity increasing from 10¹⁷ cm⁻³ to 3.3 × 10¹⁹ cm⁻³.

In this work, we defined the optimal carrier density nopt

and the optimal temperature Topt at which the system has the ZT maximum. The nopt and Topt of the four compounds

with p­ and n­type carrier were summaried in table 3. The nopt for most systems are ~10¹⁹ cm⁻³, but nopt for n­type Ag2BaSnSe4 and Ag2SrSnSe4 reach up to 2.5 × 10²⁰ cm⁻³ and 2.7 × 10²⁰ cm⁻³, respectively. The optimal carrier density is very useful for promoting ZT by experimental doping. Theoretically, one can easily calculate the optimal carrier density, but the optimal carrier density and the optimal temperature are difficult to achieve experimentally.

For example, when temperature rises to 900 K, material may undergo phase transition or melt. These are the issues to be considered and solved in the experiment, but we still consider Ag2XYSe4 (X = Ba, Sr; Y = Sn, Ge) as a class of potential TE materials.

Figure 11. Calculated ZT with respect to temperature T and carrier density n for p­ and n­type Ag₂XYSe₄ (X = Ba, Sr; Y = Sn, Ge).

4. Conclusions

In summary, we use the first­principles method to systemati­

cally explore electronic structures, mechanical and thermal and TE properties of p­ and n­type Ag₂XYSe₄ (X = Ba, Sr;

Y = Sn, Ge). The results show that low crystal symmetry and suitable band gap mainly contribute to weak thermal transport and medium electrical transport performances. The largest ZT in the p­type Ag₂XYSe₄ comes from Ag₂BaSnSe₄ compound reaching up to 1.20 at 900 K, and the largest ZT in the n­type is 1.22 of Ag₂SrGeSe₄ at 900 K. Although the values for ZT

is not very excellent, we believe the ZT for those compounds can be prompted by such as doping, low­dimensional strategy.

Acknowledgments

The authors gratefully acknowledge great support from the National Natural Science Foundation of China (Grant No.

11804132).

ORCID iDs

A J Hong https://orcid.org/0000­0002­5328­8760 C L Yuan https://orcid.org/0000­0002­8088­0313 References

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