Origin of different thermoelectric properties between Zintl compounds Ba3Al3P5 and Ba3Ga3P5: A first-principles study

Journal of Alloys and Compounds 636 (2015) 387–394

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Origin of different thermoelectric properties between Zintl compounds Ba3Al3P5 and Ba3Ga3P5: A first-principles study Zhenzhen Feng a, Jueming Yang a, Yuanxu Wang a,b,⇑, Yuli Yan a, Gui Yang a, Xiaojing Zhang a a b

Institute for Computational Materials Science, School of Physics and Electronics, Henan University, Kaifeng 475004, People’s Republic of China Guizhou Provincial Key Laboratory of Computational Nano-Material Science, Institute of Applied Physics, Guizhou Normal College, Guiyang 550018, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 7 December 2014 Received in revised form 15 February 2015 Accepted 17 February 2015 Available online 26 February 2015 Keywords: Zintl Ba3Al3P5 Ba3Ga3P5 Thermoelectric properties

a b s t r a c t The electronic structure and the thermoelectric properties of Zintl compounds Ba3M3P5 (M = Al, Ga) were investigated by the density functional theory (DFT) combined with the semiclassical Boltzmann transport theory. It is found that the transport properties of p-type Ba3M3P5 are better than that of n-type one at optimum carrier concentration. By p-type doping, the maximum ZT of Ba3Al3P5 and p-type Ba3Ga3P5 can reach 0.49 at 500 K and 0.65 at 800 K, corresponding to the carrier concentration of 7.1  1019 holes per cm3 and 1.3  1020 holes per cm3, respectively. The higher thermoelectric performance of p-type Ba3M3P5 than n-type one is mainly due to the large valence band dispersion near the Fermi level. For Ba3Ga3P5, the multiple extrema on the top of valence bands will increase its electrical conductivity. The calculated partial charge density near the Fermi level of Ba3M3P5 shows that there is little charge density around the P1 atoms in Ba3Al3P5. On the contrary, the high charge density appears around all P atoms in Ba3Ga3P5, which may be the reason why Ba3Ga3P5 has multiple extrema on its top of valence bands. Meanwhile, the minimum lattice thermal conductivities of Ba3Al3P5 and Ba3Ga3P5, are small and are comparable to those of Ca5Al2Sb6 and Ca5Ga2Sb6. Compared with p-type Ba3Al3P5 , p-type Ba3Ga3P5 shows better thermoelectric properties, which is mainly due to the multiple extrema on its top of the valence bands and its small band gap. Moreover, p-type Ba3Ga3P5 shows nearly isotropic transport behavior. Hence, good thermoelectric performance for p-type Ba3Ga3P5 can be predicted. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction Thermoelectric materials, which can convert waste heat into electricity without any moving parts or be used as solid-state Peltier coolers [1], could play an important role in a global sustainable energy solutions. Thermoelectric performance, as characterized by the dimensionless figure of merit, ZT, is a property of matter that has attracted much interest. The expression for ZT is ZT = S2 rT=j, where S is the Seebeck coefficient (also known as the thermopower), r is the electrical conductivity, T is the absolute temperature, and j is the thermal conductivity, typically written as the sum of electric (electrons and holes transporting heat) and lattice (phonon traveling trough the lattice) contributions,

j ¼ je þ jl . The term S2 r is called powerfactor, while not same as ZT, and it has often been optimized in the initial stages of materials development as a substitute for ZT. Obtaining high ZT is a fundamentally scientific challenge. Specifically, one requires ⇑ Corresponding author at: Institute for Computational Materials Science, School of Physics and Electronics, Henan University, Kaifeng 475004, People’s Republic of China. E-mail address: [email protected] (Y. Wang). http://dx.doi.org/10.1016/j.jallcom.2015.02.133 0925-8388/Ó 2015 Elsevier B.V. All rights reserved.

a large Seebeck coefficient, a high electrical conductivity, and a low thermal conductivity. Because S and r have an opposite dependence on carrier concentration, a balance between S and r must be achieved through chemical doping [2,3]. To achieve a low thermal conductivity, the method lies in scattering phonons without causing a corresponding reduction in the electronic mobility [4,5]. Thus, thermoelectric application need a rather type of unusual materials: ‘‘phonon-glass electron-crystal’’ [6–8]. Zintl phase compounds provide a good balance between high powerfactor and low thermal conductivity, and consequently they are prime candidates for applying this concept to obtain high ZT thermoelectric materials. In Zintl compounds, the electropositive cations (alkali, alkali-earth or rare earth) that donates their valence electrons to the anionic cluster which in turn forms covalent bonds to satisfy valence requirement. The need to satisfy valence often leads to a complex structure. Such a complex crystal structure can scatter phonons and consequently results in a low lattice thermal conductivity, making many Zintl compounds have ‘‘phonon-glass’’ behavior [9–11]. At the same time, the complex, covalent structures can potentially be harnessed for electronic conduction, leading to the desired ‘‘electron-crystal’’ behavior [12,13]. In addition, good thermoelectric performance is generally found in

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heavily-doped semiconductors with carrier concentrations on the order of 1019 to 1021 carriers per cm3, which can be easily realized because the rich solid-state chemistry of Zintl compounds. The complex structures of Zintl compounds provide numerous opportunities to modify their fundamental transport parameters to enhance thermoelectric performance. Previous studies have shown that Zintl compounds, for instance, Ca3AlSb3 [14], Sr3GaSb3 [15,16], Ca5Al2Sb6 [17], and Ca5Ga2As6 [18], are good thermoelectric materials. In the current study, we explore the thermoelectric properties of Ba3Al3P5 and Ba3Ga3P5 that were recently synthesized and that adopted a novel  [19]. By rhombohedral structure type with the space group of R3c studying the transport properties, we find that the transport properties of p-type Ba3M3P5 (M = Al, Ga) are most likely better than that of n-type ones at the optimum carrier concentration. For p-type Ba3Al3P5, the maximum ZT can reach 0.49 at 500 K, þ corresponding to the carrier concentration of 7.1  1019 h cm3. For p-type Ba3Ga3P5, the maximum ZT can reach 0.65 at 800 K, þ corresponding to the carrier concentration of 1.3  1020 h cm3. Compared with p-type Ba3Al3P5 ; p-type Ba3Ga3P5 shows better transport properties, which is mainly due to the multiple extrema on its top of valence bands and its smaller band gap.

respectively. For Ba3Ga3P5, the muffin-tin radii were chosen to be 2.5 and 2.22 for Ba and Ga or P, respectively. We set the cutoff parameter Rmt  K max to be 7 (K max is the magnitude of the largest k vector). The self-consistent calculations were done with 1000 k-points in the irreducible Brillouin zone and the total energy was converged to within 0.0001 Ry. The Seebeck coefficient, as well as the electrical conductivity relative to relaxation time were calculated by using the semiclassical Boltzmann theory [32,33] within the constant scattering time approximation as implemented in the BoltzTrap code [34]. This approximation, which has commonly been applied to many thermoelectric materials, including degenerately doped semiconductors, Zintl-type phases, and oxides [35,36], were based on the assumption that the scattering time determining the electrical conductivity does not vary strongly with energy on the scale of kT. The constant scattering time approximation has been used to calculate the transport coefficients of some known thermoelectric materials, and a very good agreement with experiment results was found [16,33,37–43].

2. Computational detail

Fig. 1 shows the optimized structures of Ba3M3P5 (M = Al, Ga). They are isotypic, crystallizing with a rhombohedral triangular  (No. 167), and there structure type with the space group of R3c are eighteen Ba atoms, eighteen M atoms, and thirty P atoms in each unit cell. The structure contains eight crystallographically independent atoms: three Ba atoms, two M atoms, and three P atoms. The three nonequivalent Ba sites are referred to as ‘‘Ba1’’ (12 atoms), ‘‘Ba2’’ (4 atoms) and ‘‘Ba3’’ (2 atoms), and the two nonequivalent M sites are referred to as ‘‘M1’’ (12 atoms) and ‘‘M2’’ (6 atoms), and the three nonequivalent P sites are referred to as ‘‘P1’’ (12 atoms), ‘‘P2’’ (12 atoms), and ‘‘P3’’ (6 atoms). Within the Zintl formalism, the structure of Ba3M3P5 (M = Al, Ga) constitutes with Ba2+ and [M3P5]6 polyanions based on a complex pattern of corner and edge-shared MP4 tetrahedra. Ba ions are situated between the tetrahedra section and provide overall charge balance. For Ba3Al3P5, the calculated lattice parameters of its primitive cell are a = 12.9216 Å, a = 69:41 , corresponding to the lattice constants of a = 14.7138 Å, c = 29.2099 Å in its unit cell. For Ba3Ga3P5, the calculated lattice parameters are a = 12.9442 Å, a ¼ 69:68 , corresponding to the lattice constants of a = 14.7897 Å, c = 29.185 Å in its unit cell. Those lattice parameters of unit cells for Ba3Al3P5 and Ba3Ga3P5 are comparable to the experimental

The rhombohedral triangular structure of Ba3M3P5 (M = Al, Ga) [19] were taken as the initial bulk model to relax to find the minimum energy structure. The structures of Ba3M3P5 (M = Al, Ga) were optimized by the Vienna ab initio simulation package (VASP) [20–25] based on the density functional theory (DFT). We used the projector augmented wave (PAW) method of Blöchl [26] in the implementation of Kresse and Joubert. The Perdew–Burke– Ernzerhof generalized-gradient approximation (PBE-GGA) was used for the exchange correlation potential. The plane-wave cutoff energy was 500 eV. For the Brillouin zone integrations of Ba3M3P5 (M = Al, Ga), the k-point sampling of 9  9  9 grid mesh was chosen. Considering the positions of atoms and lattice constants, the Hellmann–Feynman forces on each ion were less than 0.005 eV/Å. The electronic structures of Ba3M3P5 (M = Al, Ga) were then calculated by the full-potential linearized augmented plane waves method [27], as implemented in the WIEN2k [28–30]. The Engel– Vosko with generalized-gradient approximation (EV-GGA) [31] was used for the exchange correlation potential. For Ba3Al3P5, the muffin-tin radii were chosen to be 2.5 and 2.2 for Ba and Al or P,

3. Results and discussions 3.1. Lattice structure, stability, and chemical bonding feature

Fig. 1. (a) Crystal structure of Ba3Al3P5. (b) Crystal structure of Ba3Ga3P5. (Green and purple spheres represent Ba and P atoms, respectively; blue and orange spheres represent Al and Ga atoms, respectively.) (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Z. Feng et al. / Journal of Alloys and Compounds 636 (2015) 387–394 Table 1 Calculated atomic coordinates of Ba3Al3P5 and Ba3Ga3P5.

Ba1 Ba2 Ba3 M1 M2 P1 P2 P3

Ba3Al3P5

Ba3Ga3P5

x

y

z

x

y

z

0.1462 0.3989 0.2500 0.0789 0.0207 0.1094 0.2032 0.3849

0.2232 0.3989 0.2500 0.5725 0.4793 0.4503 0.4716 0.1151

0.6517 0.3989 0.2500 0.4937 0.7500 0.3748 0.61497 0.7500

0.1403 0.3987 0.2500 0.0829 0.0207 0.1097 0.2056 0.3756

0.6498 0.3987 0.2500 0.4891 0.7500 0.3771 0.6144 0.7500

0.2261 0.3987 0.2500 0.5752 0.4793 0.4466 0.4713 0.1244

Frequency (THz)

Atomic type

(a)

(b)

4

4

2

2

0

Γ

Y

F

X

Γ

0 Z

R

F

Γ

Y

F

X

Γ

Z

R

F

Fig. 2. Phonon dispersion curves of (a) Ba3Al3P5; (b) Ba3Ga3P5.

Table 2 Calculated and experimental (brackets) bond distances of Ba3Al3P5 and Ba3Ga3P5 (in Å). Ba3Al3P5 Ba1–P1 Ba1–P2 Ba1–P2 Ba1–P2 Ba1–P3 Ba1–P3 Ba2–P1 Ba2–P2 Ba3–P1 Al1–P1 Al1–P1 Al1–P2 Al1–P3 Al2–P1 Al2–P2

3.448(3.415) 3.175(3.150) 3.223(3.199) 3.379(3.349) 3.400(3.361) 3.453(3.436) 3.669(3.632) 3.205(3.181) 3.189(3.154) 2.418(2.401) 2.478(2.474) 2.365(2.346) 2.341(2.320) 2.461(2.446) 2.383(2.364)

Ba3Ga3P5 Ba1-P1 Ba1–P2 Ba1–P2 Ba1–P2 Ba1–P3 Ba1–P3 Ba2–P1 Ba2–P2 Ba3–P1 Ga1–P1 Ga1–P1 Ga1–P2 Ga1–P3 Ga2–P1 Ga2–P2

3.544(3.482) 3.184(3.153) 3.240(3.199) 3.408(3.363) 3.392(3.344) 3.454(3.421) 3.655(3.623) 3.210(3.171) 3.189(3.142) 2.433(2.399) 2.540(2.524) 2.399(2.367) 2.362(2.328) 2.487(2.460) 2.422(2.388) Fig. 3. Calculated electron localization function of Ba3Ga3P5 with the isosurface value is 0.75. The Fermi level is at zero.

values [19]. The optimized lattice parameter and bond length of Ba3M3P5 (M = Al, Ga) determined by our simulations are shown in Tables 1 and 2. As seen in Table 2, the Ba–P bonds and the Al–P bonds in Ba3Al3P5 have relative shorter length than the Ba–P bonds and the Ga–P bonds in Ba3Ga3P5. Therefore, the M–P covalent interaction and Ba–P ionic interaction in Ba3Al3P5 should be stronger than those in Ba3Ga3P5, which is consistent with the calculated formation energies. The stability of Ba3M3P5 (M = Al, Ga) can be judged from their phonon frequency and formation energy. Dynamic stability is important for existing of a new structure because the appearance of soft phonon modes will lead to its distortion. The phonon dispersion curves of Ba3M3P5 (M = Al, Ga) were calculated and are plotted in Fig. 2. As seen in this figure, no imaginary phonon frequency appears in their whole Brillouin zone, which shows that they are dynamically stable. Moreover, the soft acoustic and optic vibration modes in Ba3M3P5 (M = Al, Ga) may result in their low lattice thermal conductivity. The formation energy can also determine the stability of a material, which is calculated by:

ME ¼ EðBa3 M3 P5 Þ  3EðBaÞ  3EðMÞ  5EðPÞ ;

ð1Þ

where EðBa3 M3 P5 Þ is the total energy of the Ba3M3P5 at their most stable states; EðBaÞ , EðAlÞ , EðGaÞ , and EðPÞ are the total energies of per atom with Ba-hexagonal, Al-face-centered, Ga-tetragonal, and P-tetragonal solid, respectively. For Ba3Al3P5 and Ba3Ga3P5, the calculated formation energies are 9.1 eV and 8.6 eV, respectively, which indicates that they are energetically stable. Due to the isotypic structure of Ba3Al3P5 and Ba3Ga3P5, we only investigate the chemical bonding features of Ba3Ga3P5. To gain a more detail insight into the bonding character of Ba3Ga3P5, we calculated its electronic localization function (ELF), which offers a reliable measure of electron pairing and localization. ELF is based on the Hartree–Fock pair probability of parallel spin electrons

and is widely used to describe and visualize chemical bonding in molecules and solid [44]. ELF, which is scaled between 0 and 1, can clearly understand the localization bonding in real space. ELF = 1 corresponds to the perfect localization that is characterized of covalent bonds or lone pairs (filled core levels), whereas ELF = 0 is typical for a vacuum (no electron density) or areas between atomic orbitals. ELF = 0.5 for a homogeneous electron gas, with values of this order indicating regions with bonding of a metallic character. ELF is useful in distinguishing between covalent, ionic, and metallic bonding. As shown in Fig. 3, the zero ELFs between the Ga–Ga contacts and the P–P contacts illustrate no bonding between these atoms. The ELF is evenly distributed around Ba, which means that it donates its valence electrons to the [Ga3P5]6 cluster, and charge balance is maintained: (Ba2+)3(Ga3+)3 (P3)5. Meanwhile, ELF is negligible at the Ga sites, whereas the ELF between the Ga and P atoms attains local maximum value and is close to the P site, indicating the combination of the stronger covalent and weaker ionic interactions between the Ga and P atoms. The result about the chemical bonding of Ba3Ga3P5 is consistent with the previous results from the crystal orbital Hamilton populations (COHP) [19]. 3.2. Lattic thermal conductivity Thermal conductivity in a material includes both electronic and lattice contributions [45–48]. The electronic component is described by Wiedemann–Franz law, scaling linearly with the electric conductivity and temperature. The lattice thermal conductivity comes from lattice vibration (phonons). Above the Debye temperature (HD ), the lattice thermal conductivity decreases with the 1/T temperature dependence expected when scattering is limited by Umklapp phonon–phonon scattering effect. This dependence

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relationship is maintained until the minimum lattice conductivity (jmin ) is reached. The large number of atoms per unit cell in Ba3M3P5 (M = Al, Ga) (N = 66) leads to 3 acoustic modes and 65  3 optical modes. Such complexity in the dispersion will enhance Umklapp scattering, and will flatten low velocity optical vibrations (Fig. 2), which can decrease heat transport in Ba3M3P5 (M = Al, Ga). To estimate jmin , we have exploited the formula defined by Cahill for disordered crystals [48,49]. At high temperatures ðT > HD Þ; jmin can be approximately calculated using the following formula:

jmin ¼

  1 p1=3 kB ½V 2=3 ð2v s þ v l Þ; 2 6

ð2Þ

3.3. Electrical transport properties The electrical transport properties of Ba3M3P5 (M = Al, Ga) were evaluated based on the calculated electronic structure by using the semiclassical Boltzmann theory. Thermoelectric performance as a function of the doping level was calculated with the rigid-band approach. For metals or degenerate semiconductors, Seebeck coefficient is given by [53]: 2

S¼

8p2 kB 2

3eh

mDOS T

 p 23 3n

ð6Þ

;

the electrical conductivity (r) is related to carrier concentration (n) through the carrier mobility l by

r ¼ nel; where V is the average volume per atom, jB is the Boltzmann constant, v s and v l are the shear and longitudinal velocities, respectively. The average wave velocity v m in a material can be expressed by the following formula [50]:

vm ¼

  1=3 1 2 1 þ ; 3 v 3s v 3l

where

vs ¼ vl

ð3Þ

q

;

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðB þ 43 GÞ ¼ ;

q

where is the density-of-states effective mass, kB is the Boltzmann’s constant, e is the charge of one electron, and h is the Planck’s constant. The carrier mobility is strongly affected by the band mass of a single valley:

l/

1 ; mb 5=2

ð4Þ

mDOS ¼ C 2=3 mb ;

ð5Þ

where C includes orbital degeneracy imposed by the symmetry of the Brillouin zone. The band dispersion relationship determines the effective mass (m ), the effective mass is given by:

" 

m ¼ h where B, G, and q are the polcrystalline bulk modulus, the shear modulus, and the density, respectively. B and G were estimated using the Voigt–Reuss–Hill approximation [51] from the calculated elastic constants, shown in Table 3. The elastic constants were obtained by the stress–strain method. Table 3 shows the calculated elastic constants and jmin . For Ba3Al3P5 and Ba3Ga3P5, the calculated

jmin are 0.63 W m1 K1

1

and 0.61 W m1 K , respectively. These values are comparable to the

ð8Þ

where mb is the band mass of a single valley. The relation between the density-of-states effective mass and the band mass of a single valley is given by [54,55]:

v s and v p are obtained from equation as follows:

sffiffiffiffi G

ð7Þ

mDOS

jmin of Zintl compounds Ca5Al2Sb6 (0.53 W m1 K1 ) and

Ca5Ga2Sb6 (0.50 W m1 K1 ) [52]. Such low lattice thermal conductivities are helpful for increasing their ZT values. Moreover, we calculated the eigenvalues of elastic constants matrix of Ba3M3P5 and found that the eigenvalues are all positive, which prove that they are elastically stable. As expected, the larger atomic radius of Ga than Al results in the higher density, and the smaller sound velocities of Ba3Ga3P5 than those of Ba3Al3P5. With the increasing of atomic mass, the shear velocity v s and the longitudinal velocity v l will decrease, which may lead to a decrease in the jmin . To seek thermoelectric with low lattice thermal conductivities, structural complexity is a graceful method to achieve glass-like thermal transport across much of the phonon spectrum. To further decrease thermal conductivity, it is necessary to lowering the sound velocity through increasing density or decreasing stiffness.

2

2

d EðkÞ dk

ð9Þ

#1 ð10Þ

;

2 EðkÞ¼Ef

where k is the wave vector,  h is the Planck constant divided by 2p. The explicit forms of Eq. (6) suggests that the Seebeck coefficient is proportional to temperature and density-of-states effective mass, yet is inversely related to the carrier concentration. Eqs. (7) and (8) suggest that the electrical conductivity is proportional to the carrier concentration, and inversely proportional to the effective mass. A good thermoelectric material requires a large Seebeck coefficient and a high electrical conductivity. Hence, we have studied the thermoelectric properties of Ba3M3P5 (M = Al, Ga) as a function of carrier concentration to achieve a good balance between the Seebeck coefficient and the electrical conductivity. The electronic relaxation time, s decreases as temperature T increases at the whole range which shows a T 1 dependence. For the doping dependence, there is a standard electron–phonon interaction relaxation time, s / n1=3 [42]. Therefore, the s yield s = C T 1 n1=3 with s in s, T in K, and n in cm3. Previous work about Zintl compounds [16,17], showed that the order of magnitude for the constant C is in the range of 106  105 by comparing with the experimental values [15,52]. Further, we can estimate the electrical conductivity and the thermoelectric figure of merit tively, where

2

r

r ¼ rs  s and ZT = S jsminsT , respec-

s ¼ 2:3  105 T 1 n1=3 . Fig. 4 shows the calculated

Table 3 Calculated elastic constants (in GPa), theoretical density (q in g/cm3), bulk modulus (B in GPa), shear modulus (G in GPa), shear sound velocities (ms in m/s), longitudinal sound velocities (ml in m/s), and the minimum lattice thermal conductivity (jmin in W/mK) of Ba3Al3P5 and Ba3Ga3P5.

Ba3Al3P5 Ba3Ga3P5

C11

C12

C13

C33

C44

q

B

G

ms

ml

jmin

81 87

26 28

29 30

71 80

26 29

3.53 4.17

44 48

28 32

2810 3070

4800 4660

0.63 0.61

391

(a)

0 -200

σ (mΩ−1cm-1)

(e)

(b) p-type n-type

500 K 800 K 1100 K

0.1 0

0.1

p-type xx p-type yy p-type zz n-type xx n-type yy n-type zz

0.6

0.4

ZT

ZT

0.2

(b)

(e)

(c)

(f)

0

(f)

(c)

0.6

0.2 0 1019

(d)

200

-400

0.3 0.2

(a)

400

(d)

S (μV/K)

400 200 0 -200 -400 0.4

σ (mΩ−1cm-1)

S (μV/K)

Z. Feng et al. / Journal of Alloys and Compounds 636 (2015) 387–394

0.4 0.2

10211019

1020

Carrier concentration

1020

1021

(cm-3)

0 1019

1020

1021 1019

1020

1021

-3

Carrier concentration (cm )

Fig. 4. Transport coefficients of p-type and n-type Ba3Al3P5 (left) and Ba3Ga3P5 (right) as a function of carrier concentration at 500 K, 800 K, and 1100 K.

Fig. 5. The anisotropy of thermoelectric coefficients of Ba3Al3P5 (left) and Ba3Ga3P5 (right) as a function of carrier concentration at 800 K.

thermoelectric properties of Seebeck coefficient, S, electrical conductivity, r, and figure of merit, ZT of p-type and n-type Ba3M3P5 (M = Al, Ga) as a function of carrier concentration at 500 K, 800 K, and 1100 K. The combined Seebeck coefficient is given by [56]

p-type Ba3Ga3P5 is 0.65 at 800 K with the carrier concentration of 1.3  1020 holes per cm3. Compared with Ba3Al3P5, at same temperature, the peak values of ZT of p-type Ba3Ga3P5 are larger than that of p-type Ba3Al3P5, which is mainly caused by the large number of band valleys on the top of valence bands for Ba3Ga3P5. The increasing of band valleys near band edge may increase electrical conductivity with little detrimental effect on Seebeck effect [54]. This is confirmed by the higher electrical conductivity of Ba3Ga3P5 than Ba3Al3P5, as shown in Fig. 3. Moreover, the smaller band gap of Ba3Ga3P5 is helpful for increasing its carrier concentration, and consequently produces the larger ZT of Ba3Ga3P5 than that of Ba3Al3P5 by increasing its electrical conductivity. Thus, p-type Ba3Ga3P5 may be a promising thermoelectric compound. To investigate the most favorable direction for thermoelectric application in Ba3Al3P5 and Ba3Ga3P5, we calculated their transport properties along the x-, y-, and z- directions as a function of carrier concentration from 1019 cm3 to 1021 cm3 at 800 K, shown in Fig. 5. Fig. 5(a) and (d) shows that the anisotropy of Seebeck coefficient of Ba3M3P5 (M = Al, Ga) is little affected by the change of carrier concentration. n-type Ba3M3P5 (M = Al, Ga) have larger absolute values of Seebeck coefficient than p-type one at the same carrier concentration, owing to their larger band degeneracy at the bottom of conduction bands than that at the top valence bands, as shown in the calculated band structure (Fig. 6). Fig. 5(b) and (e) show the anisotropy of electrical conductivity. As seen in Fig. 5(b), for Ba3Al3P5, the anisotropy of r of p-type doping is increased with the increasing of carrier concentration, and the electrical conductivity along the z-direction is much higher than those along the x- and y-direction, mainly due to its larger band dispersion along the F–R direction. For n-type Ba3Al3P5, the electrical conductivity along the x-direction is much higher than those along y- and z-direction. On the contrary, the r for p-type Ba3Ga3P5 shows a fair degree isotropy shown in Fig. 5(e), which is mainly due to its isotropy valence band dispersion along the three directions. For n-type Ba3Ga3P5, the electrical conductivity along the z-direction are higher than those along x- and ydirection. The anisotropy of ZT is shown in Fig. 5(c) and (f). Obviously, for n-type Ba3Al3P5, the ZT along the x-direction is larger than those along the y-direction and z-direction, which mainly comes from its larger r along the x-direction. For p-type Ba3Al3P5, the ZT along z-direction is larger than those along x- and y-directions. For Ba3Ga3P5, the anisotropy of ZT for n-type doping is larger than that

Se re þ Sh rh ; re þrh   kB Nv þ 2:5  c ; Sh ¼ ln e np     kB Nc þ 2:5  c ; Se ¼  ln e nn

S¼

ð11Þ ð12Þ ð13Þ

where c is the scattering mechanism parameter, Se (Sh ) is the Seebeck coefficient of electrons (holes), rh (re ) is the electrical conductivity of holes (electrons), Nv (Nc ) is the density of states in the valence band (conduction band), np is the number of holes, and nn is the number of electrons. As can be seen from Fig. 4(a) and (d), the Seebeck coefficients of p-type Ba3M3P5 (M = Al, Ga) are positive, while those of n-type are negative, which are in agreement with Eqs. (12) and (13), respectively. Moreover, the absolute values of S increase with the increasing of temperature at the same carrier concentration, and decrease with the increasing of carrier concentration at the same temperature, which corresponds to Eq. (6). Somewhat higher absolute S values for n-type doping than that for p-type doping are mainly due to the larger band degeneracy at the bottom of the conduction bands, as shown in the calculated band structure (Fig. 6). As seen in Fig. 4(b) and (e), r increases with the increasing of carrier concentration, which is consistent with electrical conductivity being proportional to carrier concentration (Eq. (7)). However, the electrical conductivity decreases from 500 K to 800 K, and eventually to 1100 K, which is mainly due to the motion of the carriers is impeded by the increased lattice vibration with the increasing of temperature. The r of p-type Ba3M3P5 (M = Al, Ga) are larger than that of the n-type ones, due to the large valence band dispersion near the Fermi level for Ba3M3P5. Moreover, as shown in Fig. 6, the multiple extrema on the top of valence band for Ba3Ga3P5 will increase its electrical conductivity by increasing its carrier concentration. Consequently, for Ba3M3P5 (M = Al, Ga), p-type doping may induce a higher ZT than n-type doping at same carrier concentration, as shown in Fig. 4. As seen in Fig. 4, the maximum of ZT of p-type Ba3Al3P5 is 0.49 at 500 K corresponding to the carrier concentration of 7.1  1019 holes per cm3, and that of

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p-type doping, which mainly comes from the larger anisotropy of r for n-type doping. The smaller anisotropy of ZT for p-type Ba3Ga3P5 mainly comes from its isotropic Seebeck coefficient and electrical conductivity. The maximum of ZT for p-type Ba3Al3P5 appears in the z-direction at 800 K, corresponding to the carrier concentration þ of 1.02  1020 h cm3. For p-type Ba3Ga3P5, the maximum of ZT appears in the x-direction at the same temperature, with the carrier þ concentration of 1.2  1020 h cm3. Hence, good thermoelectric performance for p-type Ba3Ga3P5 along the x-direction is predicted. 3.4. Electronic structure The transport properties of materials can be analyzed from their electronic structure. The relation between electronic structure and thermoelectric figure of merit ZT was defined by Hicks and Dresselahaus [57]. In an anisotropic three-dimensional single-band circumstance, ZT increases with an inherent parameter B when the thermal and electrical currents travel in a certain direction, the inherent parameter B and the maximum attainable figure of merit (Z max ) are defined by:

 3=2 2 1 2kB T    1=2 kB T l ðm m m Þ ; x y z 3p 2 ekl h2 qffiffiffiffiffiffiffiffi ffi   m m T 3=2 sz mx  y z Z max / Nv eðrþ1=2Þ ; kl

B¼

ð14Þ

ð15Þ

where N v is the degeneracy of band extrema, sz is the relaxation time of carriers moving along the transport (z) direction, r is the scattering parameter. mi (i = x; y; z) is the effective mass of carriers (electrons or holes) in the ith direction, l is the carrier mobility along the transport direction, and kl is the lattice thermal conductivity. Eqs. (14) and (15) suggest that a high ZT value is benefited from a large effective mass, a high carrier mobility, a low lattice thermal conductivity, and high degeneracy of band extrema. Band structure calculations can give us important informations of these properties directly. Because transport properties are closely related to the electronic states near the valence band maximum (VBM) and the conduction band minimum (CBM), we only focus on the electronic states near the Fermi level. Fig. 6 shows the band structure of

(a)

Ba3M3P5 (M = Al, Ga). As seen in Fig. 6, the band structures are characterized by an indirect band gap with the VBM at F and CBM at C for Ba3Al3P5, and the VBM at F–C and CBM between Z– R for Ba3Ga3P5. For Ba3Al3P5 and Ba3Ga3P5, the calculated band gap are 1.5 eV and 1.25 eV, respectively. During optimizing thermoelectric performance, the effective mass of carriers makes an opposite contribution to Seebeck coefficient and electrical conductivity. The large effective mass of carriers is helpful to increasing Seebeck coefficient. However, the high carrier mobility needs a small band mass, corresponding light bands near band edge. mDOS can be appropriated as    ðm1 m2 m3 Þ1=3 N 2=3 v , where m1 ; m2 , and m3 are the effective mass components along three perpendicular directions, and N v is the band degeneracy. Therefore, multiple degenerate valleys are helpful for producing large mDOS without explicitly decreasing in mobility. We can see from Fig. 6 that the band degeneracy of Ba3M3P5 at the bottom of conduction band equals 2, while that on the top of valence band only equals 1, which is conducive to obtaining large Seebeck coefficients for n-type Ba3M3P5. Fig. 6(a) shows that the top of the valence band has a stronger dispersion than the bottom of the conduction bands. Such stronger dispersion of the valence bands is conducive to increasing their carrier mobility. Hence, p-type Ba3Al3P5 may have a higher electrical conductivity than n-type one. As seen in Fig. 6(b), multiple extrema appears on the top of the valence bands of Ba3Ga3P5, which will produce a large carrier concentration in Ba3Ga3P5. Due to more band valleys, more intrinsic carriers can be activated across the band gap in Ba3Ga3P5. Thus, the multiple band extrema and the small band gap of Ba3Ga3P5 are helpful to achieving a higher carrier concentration in Ba3Ga3P5 than Ba3Al3P5 at same temperature. High carrier concentration is favorable for increasing electrical conductivity. The high thermopower and electrical conductivity will lead to a good thermoelectric performance of Ba3Ga3P5. Fig. 7 shows the density of states (DOS) per atom in Ba3M3P5 (M = Al, Ga). Fig. 7 reveals that the P states dominate the valence band edge for Ba3M3P5 (M = Al, Ga), which indicates that the P atoms play an important role for a high DOS near the Fermi level. The hybridized DOS of the M and P atoms indicates a covalent M–P bonding character, which is beneficial to electrical transport. The Ba1 states also make contribution on DOS at the VBM. Thus, if

(b)

20

Energy (eV)

1

Ef

0

Density of states (states/eV)

2

Ba1 Ba2 Ba3 Al1 Al2 P1 P2 P3

10

0 20

Ba1 Ba2 Ba3 Ga1 Ga2 P1 P2 P3

10

-1

Γ

Y

F

X

Γ

Z

R

F

Γ

Y

F

X

Γ

Z

R

F

(a)

0 -2

-1

0

1

(b)

2

Energy (eV) Fig. 6. Band structures of (a) Ba3Al3P5; (b) Ba3Ga3P5. Top of the valence band is set to zero. The special k points C, Y, F, X, Z, and R in the figure represent the points (0, 0, 0), (0, 0.5, 0), (0.5, 0.5, 0), (0.5, 0, 0), (0, 0, 0.5), and (0.5, 0.5, 0.5), respectively.

Fig. 7. Calculated partial density of states for (a) Ba3Al3P5; (b) Ba3Ga3P5. The Fermi level is at zero.

Z. Feng et al. / Journal of Alloys and Compounds 636 (2015) 387–394

393

Fig. 8. Calculated band decomposed charge density of (a) Ba3Al3P5 and (b) Ba3Ga3P5 for valence bands from 0.09 eV to the Fermi level, with the isosurface value of 0.0015. The unit of charge density is e/Å3.

we need to adjust the hole carrier concentrations and does not alter the band shape to enhance the Seebeck coefficient of p-type Ba3M3P5 (M = Al, Ga), an effective way is adjusting the amount of the M atoms. If we need to adjust the hole carrier concentrations and alter the band shape, we can investigate the thermoelectric properties of p-type Ba3M3P5 (M = Al, Ga) by replacing the P sites. The conduction band edge is dominated primarily by Ba states for Ba3Al3P5 and Ga states for Ba3Ga3P5, respectively. To deeply understand the states near the band edge, we calculated the partial charge densities near the Fermi level of Ba3Al3P5 and Ba3Ga3P5 with VASP. The calculated charge densities with the energy range from 0.09 eV to the Fermi level are shown in Fig. 8(a) and (b). Because there is little charge density around Ba and M atoms, we only display the charge density distribution around the P atoms. From Fig. 8, we can see that the states on the top of valence bands mainly comes from the P p orbital. For Ba3Al3P5, there are little charge around the P1 atoms. On the contrary, there are large charge density around the P1 atoms in Ba3Ga3P5. Compared with Ba3Al3P5, more electrons of the P atoms in Ba3Ga3P5 can be activated from its valence bands to its conduction bands. This result indicates that the carrier concentration of p-type Ba3Ga3P5 should be larger than that of p-type Ba3Al3P5 at same temperature. High carrier concentration is helpful to increase the electrical conductivity of Ba3Ga3P5. 4. Conclusion The electronic structures and the transport properties of Ba3M3P5 (M = Al, Ga) have been investigated by the first-principles calculations and the semiclassical Boltzmann theory. The calculated phonon frequency and formation energy confirm that Ba3M3P5 (M = Al, Ga) are stable. The calculated minimum lattice thermal conductivity jmin of Ba3M3P5 (M = Al, Ga) are comparable to those of Ca5Al2Sb6 and Ca5Ga2Sb6. The transport properties of p-type Ba3M3P5 (M = Al, Ga) are most likely better than those of n-type ones at the optimum carrier concentration due to the large electrical conductivity for p-type Ba3M3P5 (M = Al, Ga). The smaller valence-band effective mass induces the high electrical conductivity of p-type Ba3Al3P5. Moreover, the multiple extrema on the top of valence bands in Ba3Ga3P5 will increase its electrical conductivity by increasing its carrier concentration. For p-type Ba3Al3P5, the maximum ZT can reach 0.49 at 500 K, corresponding þ to the carrier concentration of 7.1  1019 h cm3. Moreover, for p-type Ba3Ga3P5, the maximum ZT can reach 0.65 at 800 K, þ corresponding to the carrier concentration of 1.3  1020 h cm3. The calculated band structure shows that multiple extrema appears on the top of the valence bands of Ba3Ga3P5, which may

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