High Thermoelectric Figure of Merit via Tunable Valley Convergence

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C: Energy Conversion and Storage; Energy and Charge Transport

High Thermoelectric Figure of Merit via Tunable Valley Convergence Coupled Low Thermal Conductivity in AB C Chalcopyrites II

IV

V2

Madhubanti Mukherjee, George Yumnam, and Abhishek K. Singh J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b10564 • Publication Date (Web): 30 Nov 2018 Downloaded from http://pubs.acs.org on December 9, 2018

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The Journal of Physical Chemistry

High Thermoelectric Figure of Merit via Tunable Valley Convergence Coupled Low Thermal Conductivity in AII BIV CV2 Chalcopyrites Madhubanti Mukherjee, George Yumnam, and Abhishek K. Singh∗ Materials Research Centre, Indian Institute of Science, Bangalore 560012, India E-mail: [email protected]

Abstract

Introduction

Development of efficient thermoelectric materials requires a designing approach that leads to excellent electronic and phononic transport properties. Using first principles density functional theory and semiclassical Boltzmann transport theory, we report unprecedented enhancement in electronic transport properties of AII BIV CV2 (group II = Be, Mg, Zn and Cd; group IV = Si, Ge and Sn and group V = P and As) chalcopyrites via isoelectronic substitution. Multiple valleys in conduction bands, present in these compounds are tuned to converge by substitution of group IV dopant. Additionally, this substitution improves the convergence of valence bands, which is found to have a direct correlation with tetragonal distortion of these chalcopyrites. Furthermore, several chalcopyrite compounds with heavy elements such as Zn, Cd, As possess low phonon group velocities and large Gr¨ uneisen parameters that lead to low lattice thermal conductivity. Combination of optimized electronic transport properties as well as low thermal conductivity results in maximum ZT of 1.67 in CdGeAs2 at moderate n-type doping. The approach developed here to enhance the thermoelectric efficiency can be generalized to other class of materials.

A large amount of energy, approximating to 66%, produced by different sources such as combustion of fossil fuels, chemical reactions and nuclear decays, is wasted in the form of heat. Thermoelectric materials harvest this waste heat by converting it directly into useful electrical energy. The efficiency of a thermoelectric material is characterized by the figure of merit, ZT= S2 σT/(κe +κl ), where S, σ, κe , κl and T are the thermopower, electrical conductivity, electronic thermal conductivity, lattice thermal conductivity and the operating temperature, respectively. 1–4 Numerous efforts to enhance the efficiency have been focused towards simultaneously maximizing the powerfactor (S 2 σ) and minimizing thermal conductivity (κl ). 5–10 However, the biggest hurdle lies in inverse coupled dependence of different variables such as the Seebeck coefficient (S), electrical conductivity (σ) and electronic thermal conductivity (κe ). 11,12 Partially this challenge can be addressed by nanostructuring, alloying, band engineering, defect engineering, application of pressure and lattice strain. 13–21 Another approach of achieving high efficiency is to rationally design less expensive and earth abundant materials, which have simultaneously optimized electronic and thermal transport properties. Recently, II-IV-V2 tetragonal chalcopyrites have attracted significant attention in optoelec-

∗

To whom correspondence should be addressed

ACS Paragon Plus Environment

1

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mented in WIEN2k code. 31 The product of the interstitial plane-wave cutoff (kmax ) and smallest LAPW sphere radius (R) was set to 9.0, which ensures convergence of basis set. The Brillouin zone sampling was done with 5000 kpoints. The structural optimization was done using conjugate gradient scheme, until each component of forces on every atom were reduced to ≤ 0.01 mRyd/Bohr. Since band gaps are underestimated in LDA-GGA methods, 32,33 the Tran-Blaha modified functional of Becke Johnson (TB-mBJ) 34,35 was employed. Calculated mBJ band gaps are in good agreement with the experimental values (Table S1). All the calculations include spin orbit coupling (SOC). Transport properties were calculated using Boltzmann transport theory (BTE), 36,37 within constant scattering time approximation (CSTA). 36,38 The electrical conductivity is calculated as a multiplicative constant of relaxation time (τ ) in CSTA. BTE was solved numerically using the BoltzTraP code, using 35000 k-points, in the irreducible Brillouin zone. The relaxation time (τ ) was calculated using Bardeen’s deformation potential theory. 39 Different phonon-electron scattering mechanisms play an important role in the determination of τ . Here, we consider scattering of electrons with accoustic and optical phonons within a temperature range of 100-1000 K. The relaxation time for electron-accoustic phonon scattering is given by, 40

tronic and photovoltaic applications. 22 Most of these compounds are experimentally synthesized. 23–25 These chalcopyrites are diamond like compounds and exhibit unusual band structures combining heavy and light features. 26,27 Complex electronic structures of these class of materials offer an opportunity of achieving high electrical conductivity as well as high thermopower. In addition, different bond angles and bond lengths in these materials may influence the anharmonicity. This class of materials also allows isoelectronic doping, thus could provide new fundamental electronic properties. Herein, we estimate thermoelectric properties of II-IV-V2 chalcopyrites by performing first principles density functional theory in combination with Boltzmann transport theory and lattice dynamics. We report that the conflicting requirements of achieving simultaneously high S and high σ are reached by tuning crystal and electronic structure parameters through isoelectronic substitution. The substitution modify the electronic band structures in such a manner that the existing multiple valleys in conduction bands become nearly converged for Ge and Si doped AII BIV P2 and AII BIV As2 compounds, respectively. These degenerate carrier pockets near the band edges lead to an increase in thermopower S, 15,28 without reducing carrier mobility. Isoelectronic substitution with Sn, tunes the splitted valence bands to converge at Γ point and display peak powerfactor for ptype carriers in AII SnCV2 . This convergence is found to be influenced by tetragonal distortion parameter η. The calculated large Gr¨ uneisen parameters and low phonon group velocities for MgIVAs2 , ZnIVAs2 , CdIVP2 and CdIVAs2 result into kl less than 5 W/m-K. Excellent transport properties and low thermal conductivity lead to maximum figure of merit of 1.67 and 0.80 for n-type CdGeAs2 and ZnSnAs2 , respectively, at 1000 K.

(τ (E)ac )−1 =

2 (2m∗ )3/2 Dac kB T 1/2 E 4 2 2π¯h ρvLA

(1)

where ρ, m* and vLA are ion mass density, effective mass of carriers and longitudinal sound velocity, respectively. Dac is the deformation potential for the scattering of acoustic phonons by electrons. In general, chalcopyrites are polar, therefore we have considered polar optical phonons for electron-optical phonon scattering. The relaxation time for electron-polar optical

Methodology The first principles calculations were carried out using the linearized augmented plane-wave (LAPW) method with local orbitals, 29,30 imple-


2

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phonons scattering is characterized by, 40 !√ " 2 e ωLO m∗ 1 1 √ (τ (E)op )−1 = √ − (nq +1) 4 2πǫ0 h ¯ k∞ k0 E

×

r

+(nq )

E ¯ ωLO h ¯ ωLO h + sinh−1 −1 1− E E h ¯ ωLO

r

Born effective charges were obtained by taking into account the long range electrostatic interactions, using density functional perturbation theory (DFPT). The linearized PBTE was solved numerically for the calculation of lattice thermal conductivity, using the ShengBTE code. 46–48

!1/2 !

h ¯ ωLO h E ¯ ωLO 1+ − sinh−1 E E h ¯ ωLO

Results and discussion

!1/2 !#

where ωLO , k0 and k∞ are longitudinal optical phonon frequency, static dielectric constant and relative permittivity, respectively. The volume deformation potential for both n and ptype carriers was obtained from the positions of valence band maxima, conduction band minima and band gap as a function of volumetric strain. Relative permittivity was obtained from the Kramers-Kronig relation. The static dielectric constant was estimated using Lyddane-SachsTeller relation. 41 The total relaxation time is calculated using Mathiessen’s rule, 1 X1 = τ τi i

(2)

where τi is relaxation time for different scattering mechanisms. Using semiclassical phonon Boltzmann transport equation (PBTE), 36 lattice thermal conductivity (κlatt ) was estimated. Solving PBTE requires second order harmonic and third order anharmonic interatomic force constants (IFCs). Harmonic force constants were calculated using Phonopy code. 42 For the anharmonic IFCs, the interactions up to the third nearest neighbors were considered. Well converged supercell sizes of 4 × 4 × 2 and 2 × 2 × 1 were used for the harmonic and anharmonic IFCs calculations, respectively. The induced forces were calculated using Hellmann-Feynman theorem, as implemented in Vienna Ab Initio Simulation Package (VASP). 43–45 To obtain accurate values of forces, a k-grid of 4 × 4 × 2 and energy cutoff of 520 eV with a strict energy convergence criterion of 10−6 eV were used. The

3.1 Crystal structure, bonding and electronic structure AII BIV CV2 compounds crystallize in tetragonal structure with spacegroup I42d, as shown in Figure 1a. The corresponding Brillouin zone along high symmetry points in k space is shown in Figure 1b. These compounds are analogous to cubic zinc blende (ZB) structure with doubled unit cell along z direction. 49 Unlike ZB, which has a perfect tetrahedral coordination, these structures have distorted tetrahedra. 50 This distortion can be described in terms of two parameters, u and η. The parameter u corresponds to anion displacement from the regular position (1/4, 1/4, 1/4) in deformed tetrahedra and η (=c/2a), known as the tetragonal distortion parameter, represents compression (1) in the crystal lattice along the zaxis. 51,52 In AII BIV CV2 , each A and B atom is tetrahedrally coordinated by four C-atoms and each C-atom is tetrahedrally surrounded by two A and two B atoms. The tetrahedral coordination in these class of materials implies covalent bonding though there is some ionic character present. The cation sublattices in these tetragonal chalcopyrites have cubic or nearly cubic framework, which might introduce highly degenerate electronic bands. To understand the bonding character, we have calculated electron localization function (ELF), which measures the probability of having an electron in the vicinity of another electron with same spin. 53,54 Possible values of ELF lie in the range 0-1. ELF = 1, 0.5, and 0 represent perfect localization, metallic, and no bonding character, respectively. ELF for CdSiP2 and CdSiAs2 , projected along the (001) plane, are


3


(a)

(b)

b*

(c)

Σ1 N Σ

Z

Page 4 of 15

1.0

a*

Y1 Γ P

0.5

Y X

(d)

c* 0.0

II

IV

V

Figure 1: (a) Crystal structure of AII BIV CV2 with space group I¯42d, (b) corresponding Brillouin zone, (c) and (d) projection of electron localization function along (001) plane of AII BIV P2 and AII BIV As2 . Γ-Σ, Γ-Z and P-Y1 directions. These valleys are within an energy range of 30-200 meV of CBM. Multiple carriers pockets near Fermi level and heavy bands contribute collectively to enhance the density of states effective mass, which is an important parameter to determine thermoelectric performance. 12 We next analyse the contribution of different atoms in density of states. The total and projected density of states (PDOS) are shown in Figure 2b. Low lying valence bands are mainly composed of localized d states of group II atom and p states of group II, IV and V atoms. Conduction band near Fermi level for each of the compounds, is derived from s and p states of group IV atoms and p states of V atoms, respectively. Both VBM and CBM show large slope near Fermi level, which generally leads to large Seebeck coefficient. 55 The multiple valleys in band structures are also observed from calculated Fermi surfaces. The Fermi surfaces at an isosurface value of 0.03 eV above conduction band (below the valence band) are plotted in Figure 2c, d. The Fermisurface below the valence band edge shows three regular ellipsoids at Γ (green pockets) point, which corresponds to converged bands at zone center. Above conduction band edges, multiple carrier pockets are observed along various high symmetry directions (blue pockets). These

shown in Figure 1c, d. The blue region corresponding to ELF = 1.0 between group IV and V atoms, indicates covalent bonding. Mixed covalent bond is observed between group II and nearest group V atoms having ELF= 0.7, with a delocalization of electrons around group II atoms (ELF = 0.2). This indicates longer bond length and weaker bond strength between group II and V atoms than that between group IV and V atoms. Mixed bonding nature and weak bond strength present in these structures may lead to low thermal conductivities. The calculated band structures are shown in Figure 2a, b. The energy spectra of all considered chalcopyrites are nearly same in a small energy range around Fermi level. P based compounds have larger band gap as compared to As based ones and band gap starts to decrease with changing the group IV element. This is due to the increase in atomic radii. Most of these compounds are found to be direct and quasi direct band gap semiconductors except BeGeP2 , BeSnP2 , BeGeAs2 , BeSnAs2 . The valence band maxima (VBM) and conduction band minima (CBM) of these compounds are located at Γ and in between Γ-Z or at Γ. The band structures exhibit light bands in X-Y, N-P and P-Y1 directions. Heavy bands are present in Γ-X and Γ−Z directions near the band edges. The conduction bands exhibit multiple valleys in Γ-X,


4

Page 5 of 15

(a)

(b)

3 E (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0

ZnGeP2

(c)

Zn-s Zn-p Zn-d Ge-s Ge-p P-s P-p

ZnGeAs2

Zn-s Zn-p Zn-d Ge-s Ge-p As-s As-p

c* b*

a*

(d)

a* b* c*

−3

Γ X Σ Γ Z N P Z DOS/eV

Γ X Σ Γ Z N P Z DOS/eV

Figure 2: (a) Electronic band structures of ZnGeP2 and ZnGeAs2 , (b) corresponding total DOS, (c) and (d) Fermi surfaces of ZnSiP2 and ZnSiAs2 plotted at an isosurface value 0.03 eV below (above) valence (conduction) band edges. compounds with substituion of group IV atom, whereas for other group II atoms (Mg, Zn and Cd), it increases up to 3-5%. These three bands have both heavy and light features. Thus, convergence of these bands is beneficial to achieve an optimal powerfactor. Another interesting electronic structural feature in these chalcopyrites are presence of many charge carrying valleys in conduction band. Good electronic transport properties of a material are strongly associated with the nature of band dispersion. High mobility originates from the bands with light mass (µ ∝ 1/m∗b , mb is single band effective mass), which is detrimental for achieving high Seebeck coefficient. In comparison, convergence of many valleys result into a large m∗dos by a multiplicative factor of 2/3 Nv , which is given by

pockets are anisotropic as zone boundary has a lower symmetry than zone center. These type of complex isosurfaces render large number of carrier pockets, which favour good thermoelectric performance. In the compounds, where the first and the second energy conduction bands do not have the same location in the Brillouin zone, the effective mass as well as electronic transport consequently become anisotropic. A very special character of these chalcopyrites is splitting of the valence bands into doubly degenerate and one non-degenerate bands due to crystal field effect (Figure 3a (inset)), whereas ZB has converged valence bands. The energy difference between these degenerate and non degenerate band is defined by ∆CF . Thus, ∆CF could be considered as a deviation parameter of these chalcopyrites from cubic ZB structure. ∆CF , approaching 0 therefore indicates smaller tetragonal distortion. These bands are found to converge (∆CF =0) at Γ with isoelectronic substitution of group IV dopant. Convergence of bands, essentially ∆CF , is thus possibly influenced by either changing lattice parameter ratio or anionic displacement due to isoelectronic substituion. In order to gain insights, ∆CF is plotted with respect to lattice parameter ratio η in Figure 3a. It shows that ∆CF approaches zero with increasing η, and as η exceeds 1, energy of non degenerate band becomes higher. η remains practically same for Be based

m∗dos = m∗b Nv2/3 where Nv is number of converged valleys. Large m∗dos significantly increase Seebeck coefficients, without affecting the mobility (µ) as m∗b remains unchanged. Figure 3b represents a single conduction band containing many carrier pockets. Si containing compounds (red bands), with group V element as P, have a smaller energy difference between CBM and Γ−X pocket (∆EΓ−X−CBM ), compared to the energy difference between Γ − Σ to CBM (∆EΓ−Σ−CBM ), whereas for Sn containing compounds (blue


5


(b) 0.0

1.01

1.00

0.98 η (c/2a)

0.94

0.90

IVP 2

Be

0.57 0.75

0.66 0.78

P Z

0.34

0.32

s2

IVA

Be

0.48

0.54

0.55 0.36

0.45

IVP 2 ZnIVP 2 CdIVP 2

Mg

CdIVAS2

P Z ΓX Σ ΓZ

0.17

MgSnAs2 MgGeAs2 MgSiAs2

Si Ge Sn

0.20 0.32

ZnSnAs2 ZnSnP2

MgIVAs2

ZnIVAs2

0.04

−0.50

BeIVAs2

P Z ΓX Σ ΓZ

0.18

−0.25

∆X-Σ (eV)

E(eV)

0.00

0.37

(c)

0.28

∆cf > −0.20

0.46

∆cf∼ −0.07 ∆cf∼ −0.15

CdIVP2

P Z ΓX Σ ΓZ

0.81

∆cf∼0.00

ZnIVP2

ΓX Σ ΓZ

0.18

0.0

a ∆cf∼ 0.05

MgIVP2

1.5

ΓX Σ ΓZ P Z

0.25

0.0 3.0

Si Ge BeIVP 2 Sn

0.15 0.22

E(eV)

−1.0

E (eV)

1.5

−0.5

c

3.0

0.03

(a)

E (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 15

As 2 IVAs 2 IVAs 2 gIV Cd Zn M

Figure 3: (a) Valence band splitting and effective convergence at Γ point for a critical value of η, electronic band structure of MgSnP2 (inset), (b) single conduction band of BeIVP2 , MgIVP2 , ZnIVP2 , CdIVP2 and BeIVAs2 , MgIVAs2 , ZnIVAs2 , CdIVAs2 , respectively, and (c) comparison of energy difference between multiple valleys in conduction band in the band structures of II-IV-P2 and II-IV-As2 compounds. tures for good thermoelectric performance, we next evaluate electronic transport properties. Transport properties are calculated using Boltzmann transport equations within the rigid band approximation. The thermopower and electrical conductivity of these compounds as a function of carrier concentration at various temperatures are shown in Figure 4a, b. The electrical conductivities ( στ ) for n-type carriers are an order of magnitude larger compared to that of p-type. This is due to presence of highly dispersive lighter bands in the band structures which result into a smaller conductivity effective mass for n-type carriers as compared to the p-type (SI Table2, 3). According to Mott formula, 4,56 thermopower can be written as, ! 2 π 2 kB T d log[σ(E)] S= (3) 3e dE

bands) ∆EΓ−Σ−CBM is reduced compared to ∆EΓ−X−CBM . Interestingly, compounds containing Ge (green bands), with group V element as P, have nearly same energy at Γ − X and Γ − Σ, thus hold a larger number of degenerate valleys. In contrast, energy difference between Γ − X and Γ − Σ pockets increases from Si to Ge to Sn, resulting into highest number of degenerate valleys in AII SiAs2 . Large number of converged charge carrying valleys could enhance the electronic performance of these compounds. In order to observe the above trends in effective valley degeneracy in all compounds, the energy difference between Γ − X and Γ − Σ carrier pockets (∆EΓ−X−Γ−Σ ) are plotted in Figure 3c. It shows that most of these compounds have a small energy difference within few kB T between the valleys, leading to large number of degenerate pockets. Such combination of heavy and light bands with many converged charge carrying valleys and sharp nature of DOS below and above Fermi level show a possibility of achieving excellent transport properties. 3.2 Transport properties As described above, band dispersions and DOS of these compounds display beneficial fea-

|E=EF

where EF is the Fermi energy. Deriving thermopower within CSTA approximation (σ = ne2 τ , τ ∝ E0 ) gives, m∗ S=


6

2 4π 2 kB T ∗ 4π 2/3 m T( ) 2 eh 3n

(4)

Page 7 of 15

S (µV/K)

(a)

(b)

400 0

ASiP2

AGeP2

ASnP2

ASiAs2

AGeAs2

ASnAs2

Be

−400

Mg

σ/τ (Ω−1m−1s−1)

Zn

106

Cd

4

10

ASiP2

102 18 10 1020

AGeP2

1018 1020 n, p (cm−3)

ASnP2

1018

1018 1020

ASiAs2

AGeAs2

ASnAs2

1020

1018 1020 1018 −3 n, p (cm )

1020

Figure 4: Thermopower (S) and electrical conductivity divided by relaxation time ( στ ) of (a) AII SiP2 , AII GeP2 and AII SnP2 and (b) AII SiAs2 , AII GeAs2 and AII SnAs2 , where A is group II element. thereby providing direct dependence of m∗dos on thermopower. Many carrier pockets present in Fermi surface of these compounds have large number of electronic states, hence a large density of states effective mass. As a result, these materials possess high thermopower over a broad carrier concentration range of 1019 to 1020 cm−3 . At a carrier concentration of 5 × 1019 cm−3 at 900 K, thermopower of Mg based compounds are larger than 200 and 250 µV/K for n-type and p-type carriers, respectively. For both n-doped and p-doped II-IVV2 compounds, thermopower varies in a similar way and also it is largely free from any bipolar effects for wide range of carrier concentrations. High thermopower of these compounds has its origin in previously described electronic band structures. For both n-type and p-type doping, the power factors are plotted in Figure 5a, b, c. It shows that the peak of power factor shifts toward higher carrier concentration, with an increase in temperature, for all considered chalcopyrites. The substituiton of P in the place of As (as group V ) improves the power factor for p-type carriers, while for n-type carriers this substituion lower the power factor. Maximum powerfactor for n-type carriers of II-IV-P2 are obtained with substitution of Ge at group IV atom position. Whereas, II-IV-As2 exhibit peak

(a) x1011 14 ASiP 2 10 Be 6 Mg 2

S2σ/τ (W/m-K2s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


14 ASiAs 10 Zn 2 6 Cd 2 1018 1020

(b)

(c) AGeP2

ASnP2

AGeAs2

ASnAs2

1018 1020 −3 1018 1020 n, p (cm )

Figure 5: Powerfactor of (a) AII SiP2 , II A SiAs2 , (b) AII GeP2 , AII GeAs2 and (c) AII SnP2 , AII SnAs2 , where A is group II element. powerfactor for n-type carriers with substitution of Si at group IV atom position. These substitutions converge the carrier pockets and enhance power factor as a consequence of enhanced Seebeck coefficient contributed by collective effective mass of all the band pockets. In order to gain further insights, we have compared the trends of calculated thermopower and electrical conductivities of all compounds. Thermopower of p-type carriers decrease with substitution of group IV element (Si-Ge-Sn), following the trend of anion displacement pa-


7


in ELF plot and weak bond strength between II-V atoms (Figure 1c,d). Delocalization causes many optical modes to lie in low frequency region and strongly hybridize with acoustic modes, which significantly reduce thermal conductivity. In addition, large anharmonicity arises from soft chemical bonding. 57 Thus substitution of group II atoms increase available phonon states (Figure 6), which enhances the scattering of phonons at higher temperature and reduce κl . For lighter mass element like Be, Mg available states in low frequency region come from group IV or group V atoms because of the substantial mass difference between II, IV and V atoms. This should limit the reduction of thermal conductivity due to strong chemical bonding between group IV and V atoms.

rameter u, which decreases from Si, Ge to Sn. On the other hand, electrical conductivity increases from Si to Ge to Sn, following the opposite trend. It is observed that, this trend is same as that of tetragonal distortion parameter η, which increases from Si to Ge to Sn. Recent study of Zhang et al. 50 found that ∆CF changes (0.18-0.49 eV) with variation in η and unity value of η (unity-η rule) results into ∆CF = 0. This enhances power factor due to Γ band convergence in VB. However, ∆CF ∼ 0 for these compounds is achieved at a value of η, which is different from unity (within a narrow range of 0.97-1.0). A significant increase in η for Mg-IVV2 and Cd-IV-V2 compounds result into ∆CF ∼ 0 upon Sn substitution. As a result of convergence, Mg and Cd containing compounds with group IV element as Sn, show an increase in electrical conductivity as well as powerfactor for p-type charge carriers. For Be and Zn based compounds, ∆CF changes slightly with variation in η via group IV element subtitution and hardly affect the electrical conductivities. Complex features in the electronic structures allow all these chalcopyrites to attend a peak powerfactor in the range of 5-10×1011 W/m-K2 s. For compounds with low thermal conductivity, ZT predominantly depends on electronic properties.Therefore, tuning the parameter η close to unity can be a rational strategy to converge the bands (∆CF = 0) such that high powerfactor consequently an improve ZT can be achieved. 3.3 Dynamical stability, anharmonicity and thermal conductivity In order to have high thermoelectric efficiency, it is important to have equally favorable thermal transport properties. The phonon dispersion of these compounds are shown in Figure 6a-d. Absense of imaginary frequencies in phonon spectra imply that these structures are dynamically stable. The phonon density of states plots (Figure S5) show that group II atoms (Zn and Cd) have dominant contribution to the phonon modes in a low frequency range, particularly below 5THz, whereas group IV (Ge and Sn) and V (P, As) atoms contribute mostly in the high frequency region. This also can be realized from the delocalization around group II atoms

(a)

ω (THz)

(b)

BeSiP2

20

MgGeP2

8

6

10

4

3

0 20

0

0

ZnSnP2

5

CdGeP2

5

3

3

0 0 Γ X Σ Γ Z N Y1

0 Σ Γ N X Y Γ Z 1

10

(c) 20

0 20

(d)

BeGeAs2

10 ω (THz)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 15

ZnSiAs2

MgSnAs2

5

4

3

2

0

0

5

CdGeAs2

4

10

3

2

0

0

0

Γ X Σ Γ Z N Y1

Γ X Σ Γ Z N Y1

Figure 6: Phonon dispersion and corresponding longitudinal acoustic phonon velocity (km/s) represented by blue-red color bar for (a) BeSiP2 , ZnSnP2 , (b) MgGeP2 , CdGeP2 , (c) BeGeAs2 , MgSnAs2 and (d) ZnSiAs2 , CdGeAs2 . To get an qualitative understanding into the behavior of lattice thermal conductivity,


8

Page 9 of 15

logκlatt(W/m-K)

(a)

100

BeSiP2 BeGeP2 BeSnP2

(b) 100

(c) MgSiP2 MgGeP2 MgSnP2

100

100

BeSiAs2 BeGeAs2 BeSnAs2

(f) 100

10

ZnSiP2 ZnGeP2 ZnSnP2

(d) 100

10

10

10 (e)

logκlatt(W/m-K)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


CdSiP2 CdGeP2 CdSnP2

10 MgSiAs2 MgGeAs2 MgSnAs2

(g) 100

10

ZnSiAs2 ZnGeAs2 ZnSnAs2

(h) 100

10

CdSiAs2 CdGeAs2 CdSnAs2

10 200 600 1000

200 600 1000 200 600 1000 Temperature (K)

200 600 1000

Figure 7: Lattice thermal conductivities (log scale) of AII BIV P2 and AII BIV As2 compounds, where (a) BeIVP2 , (b) MgIVP2 , (c) ZnIVP2 , (d) CdIVP2 , (e) BeIVAs2 , (f) MgIVAs2 , (g) ZnIVAs2 and (h) CdIVAs2 . To uphold the above mentioned phononic features, that suggest lower thermal conductivities for these chalcopyrites, we next calculate the lattice thermal conductivity solving BTE iteratively. The calculated κl with a logarithmic scale as a function of temperature for all of the compounds are shown in Figure 7a-h. κlatt follows a 1/T dependence and becomes below 5 W/m−K at high temperatures for Mg, Zn and Cd based compounds. Lattice thermal conductivity trend is consistent with earlier discussed observations of phonon group velocity and γi (k). Lowest values of κlatt arise for Cd based compounds as a result of low-lying optical modes, low group velocities, and large negative Gr¨ uneisen parameters. Be based compounds exhibit much high thermal conductivities due to lighter mass of atoms and presence of large phononic gap. The interesting opposite trend of increasing lattice thermal conductivity with increasing atomic mass for ZnIVP2 , CdIVP2 and CdIVAs2 compounds is attributed to the differences in the ratio of atomic masses of constituent atoms (r). 59,60 Atomic mass ratio (AMR) of cationic to anionic character of the constituent atoms is greater than 1 for ZnP, Cd-(P/As) based chalcopyrites (SI Table 4).

phonon group velocities and mode Gr¨ uneisen parameters were calculated. Phonon group velocities are plotted in Figure 6. The calculated mode resolved speed of sound in tetragonal chalcopyrite compounds are in the range of 1700-5000 ms−1 for the two transverse acoustic modes (TA1/TA2) and 3100-8000 ms−1 for one longitudinal acoustic mode (LA). Among all the compounds, CdGeAs2 possesses lowest phonon group velocity (3019 ms−1 for LA), which is comparable to that of PbTe 58 (∼ 3500 ms−1 ). The mode Gr¨ uneisen parameter γ, a measure of the anharmonicity in a material, is defined as negative logarithmic derivitive of phonon frequency with respect to changing volume of the cell and can be written as, γi (k) = −

V δωi (k) ωi (k) δV

(5)

where ωi (k) is the frequency of a normal mode of ith branch at wave vector k in first Brillouin zone and V is volume of the cell. The calculated Gr¨ uneisen parameters have large negative values (∼ −3 for accoustic modes at 300 K) for compounds containing heavier elements (Figure S6, 7). This implies large anharmonicity in the crystal structures.


9


T = 1000 K) for n-type carriers. Apart from that, ZnSnAs2 , ZnGeAs2 and CdSnP2 exhibit a maximum ZT of 0.80, 0.55 and 0.52 for n type doping, respectively at high temperatures. CdGeAs2 also shows a ZT of 0.56 for p-type doping. (a) 2

(b) 20

TA1 TA2

κtot (W/m-K)

AMR > 1 indicates higher cation (Zn, Cd) contribution to acoustic branches, whereas AMR < 1 indicates higher anion (P, As) contribution. Furthermore, increasing the ratio greater than 1 results in unprecedented hybridization of phonon density of states (Figure S5, S8) from different atoms in low frequency region. Such hybridization of acoustic phonons enhances the phonon group velocity via resonant mode vibrations, which results in higher thermal conductivity for ZnSnP2 , CdSnP2 and CdSnAs2 . Electronic transport properties are observed to have high values in moderate carrier concentration range, which indicates presence of large number of charge carriers. Therefore, the contribution of electronic part of thermal conductivity κe to the total thermal conductivity might not be neglected. κe is calculated using WiedemannFranz law, κe = LTσ, where L is the Lorenz number. The contribution of κe to the total thermal conductivity is only 11% for n type carriers at high temperatures. Whereas κe has larger contribution to κtotal for p-type carriers, thereby affects the figure of merit. 3.4 Figure of merit The next step is to estimate the dimensionless figure of merit, ZT= S2 σT/(κl + κe ). The accurate prediction of thermoelectric efficiency can be obtained by calculating relaxation time (τ ). Within all range of doping and temperature conditions, τ range from 1×10−14 s to 8×10−15 s. For some extreme cases τ reaches 1×10−16 s at higher temperatures and carrier concentrations (Figure S10). Calculated ZT has been plotted with respect to temperature. We have achieved reasonable ZT for several compounds within a range of 0.35-1.67. In particular, highest ZT has been achieved for CdGeAs2 (1.67), owing to its much lower thermal conductivity (1.7 W/mK), shown in Figure 8b, d. The calculated ZT of CdGeAs2 is higher than/comparable to state of the art thermoelectric materials such as, PbTe, Bi2 Te3 , Si2 Te3 , SnSe. 61–64 Figure 8a, c represents Gr¨ uneisen parameter and calculated electronic relaxation time for CdGeAs2 . The total relaxation time for CdGeAs2 varies from 2.74×10−14 s (at T = 100 K) to 2.52×10−15 s (at T = 1000 K) for p-type carriers and from 4.16×10−14 s (at T = 100 K) to 4.39×10−14 s (at

LA

0

κ

tot

γ

10

−2 0

(c) 10−11

0

4 8 12 ω(ΤΗz)

10−13

200

600 T(K)

1000

(d) 2.0

τtotal τap τop

1.5 ΖΤ

−4

τ(s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 15

n-type p-type

1.0 0.5

−15

10

200

600 T(K)

1000

0.0

200

600 T(K)

1000

Figure 8: (a) Gr¨ uneisen parameter (γ), (b) total thermal conductivity and (c) relaxation time (τ ) and (d) figure of merit ZT of CdGeAs2 . Very low ZT of Be based compounds can be attributed to their high lattice thermal conductivity compared to Mg, Zn and Cd-based compounds. These compounds are easy to dope, which would open up a possibility for all these compounds to be used in thermoelectric devices.

Conclusion In summary, we explored the thermoelectric properties of AII BIV CV2 chalcopyrites. We demonstrate tuning of multiple charge carrying valleys present in conduction band of these compounds to converge, via isoelectronic substituion of group IV elements. Si and Ge in group IV atom position for AII BIV P2 and AII BIV As2 repectively, turns out to be the best dopant to provide large number of converged valleys, which offer enhanced powerfactors for


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her useful comments and discussions. Madhubanti Mukherjee acknowledges support from DST through INSPIRE Fellowship.

n type carriers. Convergence of valence bands at Γ is also achieved through isoelectronic substituion, which changes lattice parameter ratio c/2a or η. Complete convergence of the bands is obtained with Sn substitution with η ∼ 0.99, indicating a smaller distortion in the compounds. Enhanced powerfactors are thus achieved as a result of this band convergence, arising from smaller tetragonal distortion. Combination of optimized transport properties and low thermal conductivities give maximum ZTs of 1.67, 0.80, 0.55, 0.52 and 0.56 for n-doped CdGeAs2 , ZnSnAs2 , ZnGeAs2 and CdSnP2 and p-doped CdGeAs2 at high temperature. Convergence of bands or many charge carrying valleys via isoelectronic substituion does not invlove doping, alloying or any kind of structural engineering, thus can be realized readily and can be an useful strategy of designing new thermoelectric materials.

References (1) Ioffe, A. F.; Stil’Bans, L.; Iordanishvili, E.; Stavitskaya, T.; Gelbtuch, A.; Vineyard, G. Semiconductor Thermoelements and Thermoelectric Cooling. Phys. Today 1959, 12, 42. (2) Wood, C. Materials for Thermoelectric Energy Conversion. Rep. Prog. Phys 1988, 51, 459. (3) Bell, L. E. Cooling, Heating, Generating Power, and Recovering Waste Heat with Thermoelectric Systems. Science 2008, 321, 1457–1461. (4) Snyder, G. J.; Toberer, E. S. Materials for Sustainable Energy: A Collection of Peer-Reviewed Research and Review Articles from Nature Publishing Group; World Scientific, 2011; pp 101–110.

Supporting Information Calculated and experimental band gaps, calculated lattice parameters, electronic band structures, effective masses, valence band convergence and enhanced powerfactor, phonon density of states, Gr¨ uneisen parameters, average relaxation time

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Author Information Corresponding Author ∗ Email: [email protected]

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Notes The authors declare no competing financial interests

Acknowledgement This work was financially supported by DST Nanomission. The authors thank Materials Research Centre(MRC) and Supercomputer Educational and Research Centre (SERC), Indian Institute of Science, Bangalore for providing the required computational facilities. The authors are thankful to Ms. Rinkle Juneja for

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Graphical TOC Entry

EX−Σ < 0

Sn

EX−Σ = 0 E > 0 X−Σ

Ge

ZT=1.67

n-type p-type

Si

∆CF = 0 ∆CF > 0


15

High Thermoelectric Figure of Merit via Tunable Valley Convergence

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