phonon c

Cite this:J. Mater. Chem. C,2016, 4, 3281

Predicting high thermoelectric performance of ABX ternary compounds NaMgX (X = P, Sb, As) with weak electron–phonon coupling and strong bonding anharmonicity†

A. J. Hong,^aJ. J. Gong,^aL. Li,^aZ. B. Yan,^aZ. F. Ren^band J.-M. Liu*^a

The ABX ternary compounds have a variety of attractive physical properties. For example, some of the half-Heusler compounds as a class of ABX compounds exhibit high thermal power factors and thus promising thermoelectric performances. This motivates a search for potential thermoelectric candidates from the ABX compound family different from the half-Heusler structures. In this study, we investigate the electronic and phononic structures of tetragonal NaMgX (X = P, Sb, As) compounds, which belong to the ABX family using first-principles calculations based on density functional theory and density functional perturbation theory. The whole set of thermoelectric properties are thus evaluated from the semi-classical Boltzmann transport calculations. It is revealed that weak electron–phonon coupling and strong bonding anharmonicity are the critical physical ingredients for promising thermoelectric per- formance, giving rise to the optimized figure of merit factorZTof up to 1.38, 2.39, and 3.54 for NaMgX (X = P, Sb, As) compounds in the intermediate temperature range.

I. Introduction

Human being at present is facing two major problems: energy crisis and environment pollution. However, these problems are propelling explorations of new and clean energy sources.^1–11 Thermoelectric (TE) devices, which can directly convert heat to electricity and vice versa, are considered to be one of the potential ways of addressing these problems. For a good TE material, a high figure of merit (FOM) factorZT=S²sT/k_totis required, where k_tot = k_L + k_e, and S, T, s, k_tot, k_L, and k_e represent, respectively, the Seebeck coefficient, absolute tempera- ture, electrical conductivity, total thermal conductivity, and the lattice and electronic components ofk_tot. The electrical performance of a TE material is determined by the power factor (PF =S²s).

Conceptually, in order to possess a highZT, the PF must be large, while the total thermal conductivity must be minimized. However, the challenge exists of how to simultaneously obtain a large PF and a low k_tot due to the complex relationships between the physical parameters (S, s,k_L, and k_e), which have their respective electronic and phonon origins.

It is known thatk_eis related with s via the Wiedemann–

Franz law, suggesting that k_e may not be easily modulated without much modification ofs. One has to search for other possible opportunities to optimize the PF and ZT. In fact, substantial investigations on the relationship betweenZTand details of the electronic and phonon structures proposed that a highZTvalue can be obtained if the following TE dimension- less quality factorB_ZTis maximized:¹²

B_ZT ¼T2k_B²h 3p

c_lN_v;c mcl²kL

; (1)

where kB is the Boltzmann constant, h is the reduced Plank constant,clis the average longitudinal elastic constant,lis the deformation potential constant, mc* is the inertial effective mass for conduction electrons (holes), andNv,cis the number of degenerate hole or electron pockets. This equation suggests two schemes in terms of enhancing theZTby properly modulating the electronic and phonon structures. Obviously, a smallland a small mc* in addition to a lowk_Lbenefit the enhancement of quality factorB_ZT. On one hand, parameterldescribes the perturbation of electronic band energy against an elastic deformation and thus describes the electron–phonon coupling. Therefore, a small l represents a weak electron–phonon coupling, allowing a partially separated modulation of the electronic bands and phononic ones. On the other hand, strong bonding anharmonicity as an intrinsic character of some specific compounds has also been demonstrated to be effective in suppressingk_Ldue to the

aLaboratory of Solid State Microstructure and Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China.

E-mail: liujm@nju.edu.cn; Tel:+86 2583596595

bDepartment of Physics and TcSUH, University of Houston, Houston, TX 77204, USA

†Electronic supplementary information (ESI) available. See DOI: 10.1039/c6tc00461j Received 30th January 2016,

Accepted 10th March 2016 DOI: 10.1039/c6tc00461j

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strong phonon scattering, paving a road to manipulate the TE performance in parallel with the well-documented approaches viananostructuring of a TE material.

Along these two lines, a number of trials from both theore- tical and experimental explorations have been performed¹²and indeed, some seminal progress has been achieved. For exam- ple, some ABX ternary compounds¹³ with cubic LiAlSi-type structure (also called the half-Heusler structure) are considered to be ideal TE candidates. The n-type XNiSn (X = Zr, Hf) compounds,^14–23possessing a large S and high s, were opti- mized to acquire the reproducibly highestZT ofB1.0 at T= 1000 K. The p-type FeNbSb-based half-Heusler compounds were reported to have ZT values reaching up to B1.1 atT = 1100 K.²⁴ These materials are favored for their large PF and high thermal stability.

In fact, these half-Heusler compounds often have large mc*,^10,17and the bonding anharmonicity is not very significant due to the relatively high structure symmetry. The high TE performance is mainly ascribed to the weak electron–phonon coupling. Nevertheless, the ABX ternary compound family does have other members that have lower structure symmetry than of the half-Heuser compounds, allowing enhanced bonding anharmonicity. Tetragonal MgAgSb is one example of a com- pound havingZTB1.0 at room temperature andB1.4 atT= 475 K.¹⁵ Pure MgAgSb has ak_Llower than 1.0 W K¹ m¹.¹⁵ These results suggest that there are additional possibilities of finding some ABX ternary compounds with low-symmetry structures and thus strong bonding anharmonicity. One set of candidates are NaMgX (X = P, As, Sb) as ABX compounds that have tetragonal structure. In addition, it is known that layered materials often have weak electron–phonon coupling such as graphene.²⁵ The NaMgX compounds have a similar layered structure to that of BiCuSeO and LaFeAsO, and the only difference is that the layered structure of BiCuSeO and LaFeAsO is parallel to the (0 0 1) plane, while that of NaMgX is parallel to the (1 1 0) plane (see Fig. 1). Therefore, the tetragonal NaMgX (X = P, As, Sb) compounds are expected to have both weak electron–phonon coupling and strong bonding anharmonicity.

The only unknown physical factor left is the carrier effective massmc*, which should also be related to the electron–phonon coupling.

Herein, we use first-principles calculations based on density functional theory (DFT) and density functional perturbation

theory (DFPT)²⁶to obtain the electronic and phonon structures of NaMgX (X = P, As, Sb) compounds. Subsequently, the whole set of TE parameters of these compounds, given appropriate carrier doping, will be evaluated by means of a computational package based on the semi-classical Boltzmann transport theory and phonon transport approaches.²⁷We shall show that these compounds have weak electron–phonon coupling and strong bonding anharmonicity. The n-type NaMgX (X = P, As, Sb) compounds may obtain promising TE performances, such as the optimizedZTvalues of 1.38 for X = P atT= 1195 K, 2.39 for X = As atT= 965 K, and 3.54 for X = Sb atT= 1195 K.

The similar ZT values for the p-type NaMgX compounds, i.e.0.71 for X = P atT= 1195 K, 1.54 for X = As atT= 1195 K, and 1.59 for X = Sb atT= 1195 K are also predicted.

II. Approach and theory

In this study, we employ the DFT scheme with a plane-wave pseudopotential formulation to compute the electronic struc- tures and physical parameters using the VASP code.^28,29 The exchange-correlated potential is treated using the Perdew–

Burke–Ernzerhof (PBE) within the generalized gradient approxi- mation (GGA).³⁰The plane wave cutoff energy is set to 500 eV, and the first Brillouin zone is sampled with 30 30 18 Monkhorst–Packkmeshes. For the phonon calculation, we use the DFPT scheme as implemented in the VASP code under the GGA-PBE functional. A 553 mesh for the first Brillouin- zone sampling and 500 eV for cutoff of the plane-wave basis set are used.

According to the semi-classical Boltzmann transport equa- tion (BTE),³¹the electrical conductivity tensorss_aband electro- nic thermal conductivity tensorsk⁰_abat non-zero electric current are given as follows:

s_abðT;E_fÞ ¼ 1 O ð

s_abðeÞ @f

@e

de; (2)

k⁰_abðT;E_fÞ ¼ 1 e²TO

ð

s_abðeÞðeE_fÞ² @f

@e

de; (3) wheree,O,e, andfrepresent the unit electron charge (e), unit cell volume (O), carrier energy (e), and Fermi distribution function (f), respectively, and Ef is the Fermi energy level.

Thesab(e) is given by the following formula:

sabðeÞ ¼e²t Nk

X

i;k

nanb

d e e_i;k

de ; (4)

where t, Nk, and na (nb) are the carrier relaxation time, total number of sampled k-points, and carrier group velocity along thea(b) direction, respectively. Thenacan be calculated as follows:

v_aði;kÞ ¼1 h

@eði;kÞ

@ka

: (5)

The Seebeck coefficient tensorsSaband electronic thermal con- ductivity tensorskabat zero electric current are given as follows:

Sab(T,Ef) = (s¹)laZ_lb, (6) Fig. 1 Schematic of lattice structure of NaMgSb compound.

kab(T,Ef) =k⁰abTZam(s¹)nmZ_mb, (7) where the coefficient tensorZ_abare written as follows:

Z_abðT;E_fÞ ¼ 1 eTO

ð

s_abðeÞðeE_fÞ @f

@e

de; (8) andE_fis determined from

n Eð _f;TÞ ¼n₀ ð

fðe;E_f;TÞDðeÞde; (9) wheren,n0and D(e) are the carrier density, valence electron number, and total density of states (DOS) as a function ofe, respectively. The effective mass approximation based on Bardeen and Shockley’s deformation potential (DP) theory is used to evaluate the carrier mobilitymand relaxation timet. In bulk systems,mandtalong thebdirection can be expressed as follows:³²

m_b¼ 2 ffiffiffiffiffiffi p2p

eh⁴c_ii

3ðk_BTÞ³⁼²m_b⁵⁼²l_b²; (10) t_b¼mm_b

e ; (11)

wherec_iiis the lattice elastic constant (i= 1, 2, 3) andl_bis the DP constant defined as follows:

l_b ¼dE_edge dd_b

d_b¼aa₀ a₀

_b¼a;

(12)

where db is the uniaxial strain along the b direction. For a tetragonal structure such as NaMgX (X = P, As, Sb) compounds, there are six independent elastic constants,i.e.,c11,c33,c44,c66, c12, andc13, whose evaluation requires six independent homo- geneous distortion modes. According to Voigt’s and Reuss’s approximations,³³ the bulk modulus Band shear modulusG can be evaluated by the following formulations:

B_V ¼1

9½2ðc₁₁þc₁₂Þ þ4c₁₃þc₃₃ G_V¼ 1

30ðc_Mþ3c₁₁3c₁₂þ12c₄₄þ6c₆₆Þ

B_R¼c_N² cM

G_R¼15 18B_V cN2þ 6

c11c12

þ 6 c44

þ 3 c66

1

c_M¼c₁₁þc₁₂þ2c₃₃4c₁₃ c_N²¼ðc₁₁þc₁₂Þc₃₃2c₁₃²:

(13)

from which the values ofBandGare obtained by taking the mean arithmetic values ofB_VandB_R,G_VandG_R, respectively:

B= (BV+BR)/2

G= (GV+GR)/2. (14)

In eqn (10) and (11),mbis the effective carrier mass along theb direction at the conduction (valence) band edge:

m_abðeÞ ¼ P

i;k

m_abd e e_i;k P

i;k

d e e_i;k

m_a¼m₁₁ðeÞj_e¼E

edge

m_c¼m₃₃ðeÞj_e¼Eedge;

(15)

wheremab* is the energy-dependent effective mass tensor and tensormabis given by

m_ab 1

¼ @²e

h²@k_a@k_b; (16)

According to the Slack equation,³⁴parameterk_Lis governed by k_L¼AY_D³V_per¹⁼³m

g²n_tot²⁼³T (17) where A is a dimensionless collection of physical constants (A B 3.1 10⁶), Y_D is the Debye temperature, V_per is the volume per atom,n_totis the number of atoms in the primitive unit cell,m% is the average mass of all the atoms in the crystal, and gis the Gru¨neisen parameter^35,36 calculated by the DFPT combined with the quasi-harmonic approximation (QHA), which is defined as follows:³⁷

g ¼ 1 C_V

X

i;q

gði;qÞC_Vði;qÞ

gði;qÞ ¼ V₀ oði;qÞ

@oði;qÞ

@V ;

(18)

where i and q represent the band index and phonon wave vector,V₀is the equilibrium volume,C_Vis the isometric heat capacity, andC_V(i,q) andg(i,q) are the mode heat capacity and mode Gru¨neisen parameter, respectively. TheC_V(i,q) and then C_Vare calculated from the phonon dispersions

C_V ¼ X

i;q

C_Vði;qÞ

¼ X

i;q

kB

hoði;qÞ k_BT

2 exp½hoði;qÞ=kBT exp½hoði;qÞ=kBT1

ð Þ²;

(19)

and then parameterY_Dis evaluated from the formula forCV: CV¼9NkB

T Y_D 3ðYD=T

0

x⁴e^x e^x1

ð Þ²dx (20) where N is the number of atoms in the unit cell. It should be noted that we specify the volume changes in 3%, 2%, 1%, 0%,1%,2%,3% for the QHA.

III. Results and discussion

A. Electronic structures

NaMgAs and NaMgSb were reported in the Inorganic Crystal Structural Database (ICSD) with the ZrCuSiAs type structure

(P4/nmmspace group).³⁸Previous theoretical studies predicted that NaMgP has theP4/nmmspace group.³⁸It is noted that the layered TE oxychalcogenide BiCuSeO also has this space group, and itsZT was enhanced from 0.5 for pristine BiCuSeO to 0.67, 0.9, 0.76, and 1.1 for Mg, Ca, Sr, and Ba doped samples, respectively.^39–43The crystal structure of NaMgSb is illustrated in Fig. 1 as an example, and NaMgAs and NaMgP show similar structures with substitution at the X site. The Na and X atoms are located in the (1 1 0) planes, and Mg atoms are located in the (1 1 0) planes.

In our calculations, all the geometry optimizations were performed on a primitive cell of NaMgX containing six atoms (each element has two atoms) by the VASP code. All the Hellman–Feynman forces on every atom are converged to less than 1.0 meV Ź. The optimized lattice constants for each material are listed in Table 1. The calculated electronic disper- sion relations along the high symmetry lines and the density of states (DOS) are shown in Fig. 2. It is seen that NaMgP and NaMgAs both have a direct gap at theGpoint, and NaMgSb has an indirect gap. The calculated band gaps for NaMgP, NaMgAs and NaMgSb areD= 1.59 eV, 0.97 eV, and 1.04 eV, respectively.

The reduction in the band gap for X = As and Sb with respect to X = P is attributed to the fact that the conduction band minimum has strong Na-s states, which become less localized upon substitution from P to Sb (see ESI,†Fig. S1). Hence, the conduction bands shift towards the Fermi level from X = P to X = As and Sb. In fact, the reduced electronegativity at X = Sb compared to that of X = P results in the change of localization of

the Na-s states. It is expected that a big gapDshould not result in any bipolar conduction, and this bipolar effect is not favored for the improvement of TE properties. In fact, the empirical rule suggests that a TE material with a big gap performs well at high T. It is worth mentioning that NaMgP, NaMgAs, and NaMgSb exhibit two localized conduction band maximums (LCBMs), which have differences of 0.33 eV, 0.57 eV, and 0.14 eV, respectively. The first conduction band maximum (CBM) of NaMgSb is in the vicinity of pointM, different from the cases for NaMgP and NaMgAs, whose first CBM are both at the G point. It was reported that multi-valley bands due to a large number of degenerate holes or electron pockets are the electro- nic origin for high TE performance in good TE materials such as CoSb3 and SnSe.^44,45However, the n-type NaMgSb has five degenerate electron pockets, corresponding to one pocket around theGpoint and four pockets around theMpoint and its equivalent points. A mixture of heavy and light bands in the vicinity of the VBM emerges for NaMgX, which is usually favorable. One can see that there are non-parabolic bands at the CBM and VBM, while the non-parabolic band has important effects in the TE properties of semiconductors.⁴⁶

The DOS data of the three compounds are presented in Fig. 2(b), (d), and (f). The total DOS increases more rapidly near the VBM than near the CBM. The X atoms contribute more than Na and Mg to the DOS near the VBM. It is implied that an appropriate doping at Na and Mg sites with either an anion or cation will only change the carrier densitynwithout seriously distorting the topology of the electronic structure. However, Na, Mg and X atoms have almost the same contributions to the DOS near the CBM.

B. Elastic properties

Subsequently, we list the calculated elastic parameters, including elastic constants cii (nine components), bulk modulus B, and shear modulusG, in Table 2. All the elastic constants for NaMgP are larger than those of NaMgAs and NaMgSb, while NaMgSb has the smallest elastic constants. It is known that constantsc11,c22, andc33represent the stiffness against principal strains, whereas c44, c55, and c66 correspond to the resistance against shear deformations. Obviously, NaMgSb can easily undertake the shear elastic deformation, which may be roughly measured by theG/B ratio. Namely, a bigG/Bratio implies a fragile material, which is unfavorable for TE devices. TheG/Bratio for NaMgP, NaMgAs, and NaMgSb is 0.58, 0.61, and 0.59, respectively, significantly Table 1 Calculated lattice constants (aandc, in unit of Å) and band gapD

(in units of eV) for tetragonal NaMgX (X = P, As, Sb)

Compound a c D

NaMgP 4.3141 6.9491 1.5905

NaMgAs 4.4221 7.1796 0.9725

NaMgSb 4.6575 7.7109 1.0389

Fig. 2 The calculated band structure of NaMgP (a), NaMgAs (c) and NaMgSb (e), and the total DOS spectra for NaMgP (b), NaMgAs (d), and NaMgSb (f), respectively.

Table 2 The calculated elastic constants and modules for NaMgX (X = P, As, Sb) compounds (in units of GPa)

Elastic parameters NaMgP NaMgAs NaMgSb

c11 86.53 77.70 66.60

c₃₃ 71.97 62.68 50.97

c44 22.72 17.76 9.95

c₆₆ 16.17 15.61 13.97

c₁₂ 6.70 5.29 4.12

c₁₃ 22.58 18.72 14.71

B 38.75 33.72 27.89

G 22.65 20.48 16.38

lower than 1.06 for the reference fragile material a-SiO2,⁴⁷ suggesting that the NaMgX materials possess low resistance against shear deformation and good machinability.

Microscopically, moduliB andG reflect the magnitude of force between the atoms. The empirical rule also tells us that largeBandGgive rise to high melting point and high Debye temperature, leading to high lattice thermal conductivity. In comparison with the high-Thalf-Heusler NbSbFe compound, which has B= 163.62 GPa and G = 63.04 GPa,²⁷ the NaMgX compounds are lower in the two moduli, implying lower melting points and lower thermal conductivities.

C. Electro-transport behaviors

The carrier effective mass reflects the effect of a periodic crystal potential field on the carrier transport, and its values near the VBM and CBM are important for electrical transport. According to eqn (15), thee-dependent effective massmcan be calculated, and the data near the VBM and CBM along the different directions are listed in Table 3. In general, for the present compounds, the effective mass at the VBM along thea-axis is larger than that along thec-axis, but the opposite situations are observed at the CBM.¹⁰

It is known that the deformation potential constantlscales the degree to which the carriers interact with phonons, namely, a low or high |l| corresponds to weak or strong electron–

phonon coupling. It represents one of several key parameters for band engineering in the search for high performance TE materials. Considering that the carriers near the VBM and CBM are important for electrical transport, we calculate the deformation potential constant of the VBM for holes and CBM for electrons along the transport direction. The calculated l andtare also summarized in Table 3. In fact, for n-type poly- crystalline PbSe, the |l| is lower than 25 eV, characterizing the weak electron–phonon coupling and thus weak carrier scatter- ing. It is noted that most half-Heusler materials have high TE performance mainly due to weak electron–phonon coupling.

For example, HfNiSn has a |l| value of B5.0 eV,¹⁰ which compensates the effect from the large carrier effective mass, beneficial to obtaining a high PF. As seen in Table 3, the deformation potential constant, effective mass, mobility, and relaxation time all indicate much weaker carrier scattering of

the p-type compounds than the n-type compounds. The p-type NaMgAs has a very low |l| ofB4.5 eV along thec-axis.

An understanding of these low carrier scattering effects in these compounds is challenging, considering that the electron–

phonon coupling in a solid is ubiquitous. The origin for the weak electron–phonon coupling can be complicated. It is shown that these compounds have the Mg layer on the ab-plane, benefiting carrier transport in a manner like that in metals.

This may be the primary cause for the weak electron–phonon coupling, considering the fact that the electron–phonon coupling in metals is weak.

We then look at the relaxation timet. For the p-type carriers, the time along thea-axis is shorter than that along thec-axis. In contrast, for the n-type carrier, the time along the a-axis is longer than that along the c-axis. Along all the directions, the p-type carriers have much shortertvalues than those for the n-type. For instance, in NaMgAs with p-type carriers, the time along thea-axis is as short as 6.81 fs, but for the n-type carriers, it becomes 1112.15 fs.

Based on the whole set of data on the electronic structures, the electrical transport properties, including parameterss,S, andk_e, in the (T,n) plane for the p-type and n-type NaMgSb compounds as examples are plotted in Fig. 3. The data for the p-type and n-type NaMgP and NaMgAs compounds are presented in the ESI†(Fig. S2 and S3). In the general sense, the large |S| corresponds to the lowsand lowk_e, which appears in the region of lownand intermediateT. The small |S|, highs, and high k_e appear in the region of high n and low T.

The calculated data in the low-n region do not support the

Table 3 Calculated DP constant l, carrier effective mass m at zero temperature, carrier mobilitym(in units of cm²V¹s¹), and relaxation timet(in units of fs) atT= 300 K for NaMgX (X = P, As, Sb) compounds.

The subscript n or p denotes the carrier type

NaMgX X = P X = As X = Sb

Axis a c a c a c

lp(eV) 6.67 4.97 6.48 4.50 7.70 5.12

l_n(eV) 10.59 11.10 9.50 10.09 7.46 8.01

mp(me) 5.53 2.34 4.28 0.90 2.30 1.54 m_n(m_e) 0.16 0.37 0.09 0.16 0.46 0.46

m_p 1.65 21.31 2.67 246.73 8.58 40.4

m_n 4615.25 429.66 21701.40 3682.70 510.77 339.06

t_p 5.22 28.39 6.81 126.43 11.23 35.48

t_n 420.45 90.52 1112.25 335.49 133.78 88.80

Fig. 3 The calculatedS(top row), logs(middle row), and logk_e(bottom row) in the (T,n) plane for the carrier-doped NaMgSb compound. Left panel: p-Type and right panel: n-type. Herein, log is the natural logarithm.

Wiedemann–Franz lawk_e=LsTwhereL= (1.49–2.45)10⁸W K². Remarkable dependence of Lon both Tandnis seen if this law is applied.

D. Phonon structure and lattice thermal transport

Our major attention goes to the thermal transport properties.

For evaluating the lattice thermal conductivityk_L, one needs to obtain the phonon spectrum. The calculated phonon disper- sions for the three compounds along several representative symmetry lines within the first Brillouin zone of the primitive cell are plotted in Fig. 4(a), (c), and (e). Since each atom gives rise to three phonon branches containing one longitudinal branch and two transverse branches, there are in total eighteen branches for six atoms in this cell. They are one longitudinal acoustic (LA), two transverse acoustic (TA), five longitudinal optical (LO), and ten transverse optical (TO) modes. The high- est optical frequencies for NaMgP, NaMgAs, and NaMgSb are 11.20 THz, 9.28 THz, and 8.28 THz, respectively, while the highest acoustic frequencies are only 4.08 THz, 3.16 THz, and 2.38 THz.

The lattice thermal conductivity k_L is mainly from the acoustic branches. In this sense, NaMgX should possess lowk_L. The two TA modes along the highest symmetry lines are two-fold degenerate. Moreover, a small phononic band gap for NaMgSb is noted, and the phononic DOS data for these compounds are presented in Fig. 4(b), (d), and (f). Considering that the main contribution to k_L is from the low-frequency branches, we focus our discussion on the phononic DOS only in the range of low frequency. For NaMgP, as shown in the phonon DOS (see Fig. 4(b)), the low-frequency branches up to 4.2 THz are mainly from Na vibration, while the Mg atoms and P atoms have relatively weak contributions to the DOS.

For NaMgSb, the low-frequency branches up to 2.1 THz are mainly from Sb atomic vibrations, while the frequency branches between 2.1 THz and 3.8 THz are mainly from Sb and Na atomic vibrations. Based on the electronic structure analysis, it is noted that the carrier doping at the Na site for NaMgX can not only reduce the lattice thermal conductivity but also change the carrier densityn. However, a carrier doping at the Mg site changes the carrier density n, while the lattice thermal conductivity may only be weakly affected.

Subsequently, we calculate parameters g and Y_D, which are related to the lattice thermal conductivity, as shown in Fig. 5(a)–(i). The remarkableT-dependences of both parameters apply mainly to the low-T range, and they become nearly T-independent in the high-Trange. The values ofgfor NaMgP, NaMgAs, and NaMgSb at T= 300 K are 1.65, 1.64, and 1.61, respectively, but increase up to 1.71, 2.03, and 2.14 atT= 1195 K.

It is known that a largeg indicates strong anharmonic vibra- tions. The PbTe compound has agB1.45 at 300 K, implying a strong anharmonic phonon scattering.⁴⁸Such largegvalues for NaMgX compounds also imply that these compounds may exhibit strong anharmonic phonon scattering. In general, it is approved that good TE materials often havegvalues larger than 1.0. For example, PbSe and PbS havegB1.65 andB2.0 at room temperature, respectively.⁴⁸The layered BiCuSeO withgB2.5 at room temperature has a lowk_LbelowB1.0 W K¹m¹.⁴⁹The gvalue for AgSbTe2isB2.05, while itsk_Lat room temperature is onlyB0.68 W K¹m¹.⁵⁰In particular, thegvalues of SnSe along the three axes areB4.1,B2.1, andB2.3, noting that itsk_L is lower thanB1.0 W K¹m¹.¹

Microscopically, it is noted that the X atom in NaMgX compounds has some non-bonded s-valence electrons and p-valence electrons. These non-bonded valence electrons can form a shell of relatively large radius, which is the reason for strong bond anharmonicity in a compound of low lattice thermal conductivity. In lattice-driven thermal transport, atoms approach one another, which leads to an overlapping of the

Fig. 4 The calculated phonon spectra of NaMgP (a), NaMgAs (c), and NaMgSb (e), and the total phononic DOS spectra for NaMgP (b), NaMgAs (d), and NaMgSb (f). The corresponding spectra and phononic DOS are plotted in (b), (d), and (f), respectively.

Fig. 5 The calculated Gru¨neisen parameterg, Debye temperatureY_D, and intrinsic lattice thermal conductivityk_Las a function ofTfor NaMgP (a–c), NaMgP (d–f), and NaMgSb (h–j), respectively.

wave functions of these non-bonded electrons and offers a repulsive force or in other words an anharmonicity. This effect applies to the NaMgX compounds concerned here.

The Debye temperatureY_Dreflects the magnitude of sound velocity determined by elastic moduli BandG. Its values for NaMgP, NaMgAs, and NaMgSb compounds in the high-Trange are relatively low,e.g.405 K, 338 K, and 292 K, respectively, at TB 1195 K. The relatively low Y_D and largeg for all these NaMgX compounds benefit the reduction ofk_L. The evaluated intrinsick_Lfor these compounds are presented in Fig. 5(c), (f), and (j), reflecting the monotonous decreasing with increasingT, e.g.from 4.2 W K¹m¹, 4.6 W K¹m¹, and 4.4 W K¹m¹at T= 300 K to 1.1 W K¹m¹, 0.8 W K¹m¹, and 0.7 W K¹m¹ atT= 1195 K for NaMgP, NaMgAs, and NaMgSb, respectively.

Such low lattice thermal conductivities, due to the intrinsic weak bonding force and thus the small elastic constants, are impress- ive and less reported for available TE materials.

E. FOM factorZT

Finally, one can evaluate the FOM factorZTfor these NaMgX compounds, and the data in the (n,T) plane for the p-type and n-type compounds are plotted in Fig. 6(a)–(f), where the red, green and blue colors represent the high, moderate, and low ZT factors, respectively. Several prominent features deserve mention here. First, for all the three types of doped com- pounds, rather high ZT values in the optimized states are predicted. At some optimized doping levels, the ZT values may reach up to 2.0–3.0 at a relatively high T. Second, the n-type doped compounds have higher ZT factors than the corresponding p-type doped ones due to the specific electronic

and phonon structures. Third, one finds that the ZT in the optimized region is not so sensitive to variation of eithernorT, suggesting that a practical control of the carrier doping in these compounds for a highZTvalue is not so challenging; moreover, high ZTcan be obtained in a relatively broadT-range. These issues are usually inevitable for many TE materials, leading to additional complexity in TE device design for broad-Tapplications, but they are not so critical for the present NaMgX compounds.

For details, the optimized ZT values and corresponding carrier densitynfor NaMgP, NaMgAs, and NaMgSb compounds are summarized in Table 4. For NaMgP, theZT values at the optimized p- and n-type doping levels, corresponding ton_optB 9.1210²⁰cm³and 7.2410¹⁸cm³, reach 0.71 and 1.38 at TB1195 K and 1155 K but rapidly decrease to 0.27 and 0.68 atTB 600 K, corresponding tonoptB4.1710²⁰cm³and 3.16 10¹⁸ cm³. In comparison, the doped NaMgAs and NaMgSb have better TE performance than the doped NaMgP.

For the doped NaMgAs, theZTvalues at the optimized p-type and n-type doping levels, corresponding tonoptB4.7910²⁰cm³ and 2.7610¹⁸cm³are 1.54 and 2.39 atT= 1195 and 965 K. It is noted that the optimizedZTfor the n-type NaMgSb can reach as high as 3.54, while the p-type NaMgSb hasZT= 1.59 as the highest value, lower than the n-type doping cases.

F. Discussion

Up to this stage, one understands that the predicted electrical and thermal transport behaviors favored for good TE perfor- mance are substantially attributed to the weak electron–

phonon coupling and strong bonding anharmonicity of these compounds. These compounds represent typical examples in which the control of the electrical and thermal transports can be partially separated so that the TE performance can be consider- ably optimized. Indeed, very large FOM factor ZT values are obtained for all three compounds upon proper carrier doping.

It is also understood that a theoretically predicted highZT may not be reached experimentally due to some practical challenge or unforeseen factors of the theoretical calculations.⁵¹ First, these compounds are difficult to synthesize and may be unstable and easily oxidized due to the presence of sodium.

Second, carrier doping may not be easy for many cases.

Fig. 6 The calculatedZTmap as a function ofTand carrier densitynfor NaMgP (a and b), NaMgP (c and d), and NaMgSb (e and f), respectively.

Table 4 Optimized ZT values (ZTopt) and the corresponding optimal carrier density (nopt) for NaMgX (X = P, As, Sb) compounds. The evaluated PF values at (noptandZTopt) are also listed, which may not be optimized

NaMgX Carrier n_opt(cm³) PF (mW m¹K²) ZT_opt X = P Hole 9.1210²⁰ 0.98 (1195 K) 0.71 (1195 K)

4.1710²⁰ 1.27 (600 K) 0.27 (600 K) Electron 7.2410¹⁸ 3.58 (1155 K) 1.38 (1155 K)

3.1610¹⁸ 3.83 (600 K) 0.68 (600 K) X = As Hole 4.7910²⁰ 1.75 (1195 K) 1.54 (1195 K)

2.8810²⁰ 2.57 (600 K) 0.57 (600 K) Electron 2.7610¹⁸ 7.28 (965 K) 2.39 (965 K) 1.7410¹⁸ 8.54 (600 K) 1.48 (600 K) X = Sb Hole 3.3110²⁰ 1.48 (1195 K) 1.59 (1195 K)

2.1910²⁰ 2.39 (600 K) 0.58 (600 K) Electron 6.3110¹⁹ 6.90 (1195 K) 3.54 (1195 K)

4.1710¹⁹ 9.33 (K) 1.67 (600 K)

A favored carrier doping should impose an impact as weak as possible on the electronic and phononic band structures although an increase of k_e is inevitable. For the three com- pounds, one may propose a doping at the Na site with some transition metals, such as Ti, V, Nb, Ta, Fe, Co, Ni, Cu, and Zn, in order to obtain the n-type systems, given a consideration of atomic radius and valence. A challenge exists for the p-type doping, and it may be feasible to optimize the electron density by introducing a Na or Mg vacancy by C or Si doping at the Sb site. In these two cases, the impact on the electronic structure may be remarkable.

On the other hand, the high operating temperature may not be reachable due to the degraded mechanical properties of these NaMgSb compounds in such high temperatures. In addition, the charge polarity for a specific compound may not be easily modulated when some materials prefer one polarity and the other becomes difficult. However, the experimental testing of our calculation predictions is appealing.

IV. Summary

In summary, we investigated in detail the electronic, phonon, and elastic properties of ABX ternary NaMgX (X = P, As, Sb) compounds and predicted the whole set of TE parameters using first-principles calculations. First, we used the DFT combined with semi-classical BTE and DP theories to calculate the elec- trical parameters. Second, we calculated the lattice thermal conductivity using the DFPT combined with the QHA scheme and found that the lattice thermal conductivity for NaMgX at B600 K is lower than 2.0 W m¹K¹, which is ascribed to the strong bonding anharmonicity. For all three compounds, a doping at the Na site with either an anion or cation will not only reduce the lattice thermal conductivity but also enhance the carrier densityn. Finally, we also predicted that the optimizedZTvalues for the p-type NaMgX (X = P, As, Sb) compounds can reach up to 0.71, 1.54, and 1.59, respectively, while theZTvalues for the n-type doping can be as high as 1.38, 2.39, and 3.54. The main physics underlying such large ZT values originates from the weak electron–phonon coupling and strong bonding anharmonicity. The present study not only approves the two physical ingredients as effective schemes for enhancing the TE performance of a TE material but also suggests the possibility of more ABX ternary compounds with promising TE properties.

Acknowledgements

This study is supported by the National 973 Projects of China (Grant No. 2015CB654602), the National Natural Science Foundation of China (Grant No. 51431006), and the ‘‘Solid State Solar Thermal Energy Conversion Center (S3TEC)’’, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Science under award number DE-SC0001299.

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