Rattling-Induced Ultra-low Thermal Conductivity Leading to

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Rattling-Induced Ultra-low Thermal Conductivity Leading to Exceptional Thermoelectric Performance in AgInS 5

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Rinkle Juneja, and Abhishek K. Singh ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.9b10006 • Publication Date (Web): 27 Aug 2019 Downloaded from pubs.acs.org on August 28, 2019

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ACS Applied Materials & Interfaces

Rattling-Induced Ultra-low Thermal Conductivity Leading to Exceptional Thermoelectric Performance in AgIn5S8 Rinkle Juneja and Abhishek K. Singh∗ Materials Research Centre, Indian Institute of Science, Bangalore 560012, India E-mail: [email protected]

Abstract

Keywords

Rattling has emerged as one of the most significant phenomenon for notably reducing the thermal conductivity in complex crystal systems. In this work using first-principles density functional theory, we found that rattlers can be hosted in simpler crystal system such as AgIn5 S8 and CuIn5 S8 . Rattlers Ag and Cu exhibit weak and anisotropic bonding with the neighbouring In and S and reside in a very shallow anharmonic potential wells. The phonon spectra of these compounds have multiple avoided crossing of optical and acoustic modes, which are signature of rattling motion. This leads to ultralow thermal conductivity, which is inversely proportional to mass and frequency span of rattling modes. Even though Ag atoms contribute to the valence band states, the rattler modes of Ag do not scatter carriers significantly, leaving the electronic transport virtually unaffected. Moreover, AgIn5 S8 possesses a combination of heavy and light valence bands resulting in a very high power factor. A combination of favourable thermal and electronic transport results in a very high figure of merit of 2.2 in p-AgIn5 S8 at 1000 K. The proposed idea of having rattlers in simpler systems can be extended to wider class of materials, which would accelerate the development of thermoelectric modules for waste energy harvesting.

Anisotropic bonding, Rattling, thermal transport, electronic transport

∗

INTRODUCTION Thermoelectric materials are expected to play a major role in creating sustainable energy ecosystem. They convert waste heat to useful energy, 1 with the efficiency given by dimensionless figure 2σ of merit ZT, defined as ZT = κSe +κ T , where S, l σ, κe , κl , and T are Seebeck coefficient, electrical conductivity, electronic thermal conductivity, lattice thermal conductivity, and absolute temperature, respectively. A good thermoelectric material requires simultaneously high power factor (S2 σ) and low thermal conductivity (κ). However, the complex interdependence among S, σ, and κe makes the enhancement of ZT extremely challenging. Although there are various approaches to decouple the inverse relation among the transport parameters, 2–4 significant efforts have also been devoted to lower the κl . Most of these approaches, either focus on engineering the phonon scattering via nanostructuring/alloying, pressure tuning, or finding materials with inherent substantial anharmonicity. 5–8 Intrinsically low κl in various materials has been attributed to presence of lone pairs, mixed bonding character, soft phonon modes, resonant bonding, and presence of rattling atoms. 9–14 Among these, rattlers offer a rich playground for tuning the thermal conductivity by utilising the

To whom correspondence should be addressed

ACS Paragon Plus Environment

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ACS Applied Materials & Interfaces 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

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METHODOLOGY

vast chemical space. Rattlers are loosely bound guest atoms in a crystalline open framework systems. These rattlers lie in very shallow anharmonic potential wells. Their frequent vibrations produce multiple anti-crossing points between the acoustic and rattler modes in the phonon spectra. 15 This leads to increase in the phonon-phonon scattering resulting into very low κl . Prominent examples of reduction in κl due to rattling motion, include skutterudites, 16–20 AOs4 Sb12 (A = Sr, Ba, La, Ce), 21 sodium cobaltate, 22 LaOBiS2−x Sex , 23 TlInTe2 . 24 Most of these materials possess very complex crystal structures. For some of them dopants act as a rattlers, hence, the choice and concentration of dopants become critical factors. In order to carry forward the advantages of rattlers to simpler crystal classes, it is essential to find materials wherein one or some of the atoms bind weakly with others. In this work, we found simple crystalline cubic compounds AgIn5 S8 and CuIn5 S8 , where Ag and Cu atoms, respectively show very weak and anisotropic bonding with their neighbouring In and S atoms. Due to this unique combination of anisotropic bonding and simple crystal system, these two compounds are selected for this study. The Ag and Cu atoms reside in a very shallow anharmonic potential wells and act as rattlers. Consequently, there are multiple anti-crossing points in the phonon spectra, which leads to ultra-low κl of 0.29 W/mK and 0.54 W/mK at 1000 K for AgIn5 S8 and CuIn5 S8 , respectively. The apparent differences in the κl originate from its inverse dependence on the mass and the frequency span of the rattler atoms. Furthermore, AgIn5 S8 also exhibits combination of flat and dispersive valence bands, and hence possesses very high power factor. Inspite of the contribution of Ag rattling atoms in the valence bands, they do not significantly affect the electronic transport. The favourable electronic and thermal transport result in an unprecedented figure of merit of 2.2 for p-doped AgIn5 S8 at 1000 K. Our work highlights the importance of anisotropic bonding in simpler crystalline systems to obtain ultra-low thermal conductivity materials via rattling mechanism.

The first-principles calculations were performed using the linearised augmented plane-wave (LAPW) method with local orbitals, 25,26 as implemented in the WIEN2k code. 27 The Radius Muffin-Tin (RMT) for Ag, In, and S was set to 2.5, 2.4, and 2.23, respectively. Energy to separate core states from the valence states was set to -6.0 Ry. The product of the smallest LAPW sphere radius (R) and the interstitial plane wave cutoff (kmax ) was set to 9.0. The electronic structure was calculated using the Tran-Blaha modified functional of Becke-Johnson (TB-mBJ), 28,29 with inclusion of spin-orbit coupling (SOC). The harmonic interatomic force constants were calculated using PHONOPY 30 with 2×2×2 supercell and k-grid 5 × 5 × 5. The forces were calculated using Hellmann-Feynman theorem, as implemented in Vienna Ab initio Simulation Package. 31–33 The Perdew-Burke-Ernzerhof generalized gradient approximation was used for electronic exchange and correlation potential. 34 The projector augmented wave (PAW) potentials were used to represent the ion-electron interactions. 31 The valence electrons for Ag, In, and S were 11, 13, and 6, respectively. For the calculation of the forces and phonon frequencies, an energy cut off of 500 eV and energy convergence criterion of 10−8 eV were used. The lattice thermal conductivity was calculated by solving the phonon Boltzmann transport equation, as implemented in PHONO3PY. 35,36 The third order anharmonic force constants were calculated with 2×2×2 supercell and k-grid 1 × 1 × 1 by considering the default cut-off pair distances. The converged lattice thermal conductivity was obtained from the direct solution of linearised Boltzmann transport equation using the sampling mesh 11 × 11 × 11. The electronic transport coefficients were calculated using BoltzTraP 37,38 by considering constant scattering time approximation. 37,39 The Brillouin zone was sampled by taking 32000 k-points. The relaxation time τ was calculated by considering electron-acoustic phonon and electron-polar-optical phonon scattering. 40,41 For electron-acoustic phonon scattering, τ


2

(τap (E))−1

2 (2m∗ )3/2 Dac kB T 1/2 = E 2 4 2πvLA ℏ ρ

(b)

(a)

is determined by deformation potential theory, 40,41

1.0 S

0.8

(1)

∗

where m , Dac , vLA , and ρ are effective mass of carriers, volume deformation potential, group velocity of longitudinal acoustic mode, and ion mass density, respectively. Dac was estimated by the change in the positions of valence band maxima and conduction band minima as a function of volumetric strain. The deformation potentials were aligned with respect to the vacuum potential. For electron and polar-optical phonon scattering, τ is given as 41

0.4 0.2

c

(c)

√ [ e2 ωLO 1 1 m∗ (τop (E)) = √ ( − )√ (nq + 1) 4 2ε0 ℏ κ∞ κ0 E (√ ( )1 ) 2 ℏωLO ℏωLO E 1− + sinh−1 −1 E E ℏωLO (√ ( )1 ) ] 2 ℏωLO E ℏωLO −1 1+ − sinh +(nq ) E E ℏωLO −1

In

In S

S

In

In

0.0

b

a

S Ag

0.6

Ag/Cu In S

Potential energy (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


(d)

20

Ag In S

15 10 5 0

-1.0 -0.5 0 0.5 1.0 Displacement (Å)

Potential energy (eV)

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20 15

Cu In S

10 5 0

-1.0 -0.5 0 0.5 1.0 Displacement (Å)

Figure 1: (a) Unit cell of AgIn5 S8 /CuIn5 S8 , (b) electron localisation function of AgIn5 S8 along (110) plane, (c) and (d) calculated potential energy curves as a function of displacement of atoms from equilibrium configuration for AgIn5 S8 and CuIn5 S8 , respectively. The dotted black lines represent the perfect harmonic behaviour.

(2)

where κ∞ , κ0 and ωLO are relative permittivity, static dielectric constant and longitudinal optical phonon frequency, respectively. κ∞ is calculated from the Kramers-Kronig relation, 39 and κ0 was estimated using Lyddane-Sachs-Teller relation, 42 κ0 = κ∞ ( ωωTLOO )2 , where ωT O is transverse optical phonon frequency. The total relaxation rate is calculated using Mathiessen’s rule 1 ∑1 = τ τi i

1a. The optimised lattice parameter for AgIn5 S8 and CuIn5 S8 are 10.99 and 10.86 Å, respectively, which agree well with experimental values. 46–48 The melting temperature for AgIn5 S8 and CuIn5 S8 are 1348 and 1358 K, respectively. 49,50 Hence, both the compounds will be stable at very high temperatures. The bonding characteristics in these systems is analysed by calculating the the electron localisation function (ELF). ELF gives insights into the bonding character by measuring the probability of finding a same spin electron near to another electron. 51,52 It is quantified in the range 0 to 1, where 0 corresponds to perfect localisation and 1 corresponds to very less charge density. ELF for AgIn5 S8 along (110) plane is shown in Figure 1b. The system is comprised of huge anisotropic bonding between the constituent elements. There is no bonding between the Ag and neighbouring In atoms (ELF ∼ 0), whereas there is weaker bonding between Ag and the neighbouring S atoms (ELF ∼ 0.2). On the other hand, S and neighbouring In atoms have comparatively stronger bonding (ELF ∼ 0.4). Similar behaviour

(3)

where τi corresponds to relaxation time corresponding to electron-acoustic and electron-polaroptical phonon scattering.

RESULTS AND DISCUSSION The ternary compounds AgIn5 S8 and CuIn5 S8 from group I-III-VI crystallise in cubic spinel symmetry with space group F ¯43m. 43,44 The crystal structure contains 4 Ag/Cu, 20 In, and 32 S atoms occupying Wyckoff sites with multiplicity 4c, 4a16e, 16e-16e, respectively, 45 as shown in Figure


3


is observed in the ELF of CuIn5 S8 with comparatively weaker and stronger bonding between Cu and S (ELF ∼ 0), and In and S (ELF ∼ 0.5), respectively, as shown in Supporting Information Figure S1. Hence, Ag/Cu may act as rattlers due to weak and anisotropic bonding. The strength of bonding can also be correlated intuitively to the bond distances. Along (110) plane, the bond distance between Ag/Cu and neighbouring S atoms is around 2.5 Å, whereas between Ag/Cu and neighbouring In atoms, the bond distance is around 4.7 Å. The difference in bond distances explain the anisotropy of bonding as observed in the ELF plot. In order to study whether this prominent anisotropic bonding cause rattling motion, we calculated the change in potential energy as a function of displacement of atoms from their equilibrium positions. These potential energy curves are shown for AgIn5 S8 and CuIn5 S8 in Figure 1c and 1d, respectively. While In and S lie in very deep potential wells in both the systems, Ag and Cu lie in very shallow potential wells. Furthermore, these shallow potential wells deviate from the perfect harmonic behaviour, indicating a strong possibility of Ag and Cu being rattlers. (b) 3 A2

2

A1

A3 A4 LA

1 0 Γ XWK Γ LUWLKUX

(c)

A1

A2

A3

ω (THz)

(a) 3 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

2

shown in Figure S2a and S2d, respectively, implying their dynamical stability. The phonon density of states (PDOS) of AgIn5 S8 shows the dominant contribution of heavy atoms In and Ag in the low frequency regime, whereas the lighter S atoms contribute significantly above 5 THz, as shown in Figure S2b. While the PDOS of CuIn5 S8 is similar to AgIn5 S8 , the Cu atoms contribute at slightly higher frequency than Ag owing to the mass difference between Cu and Ag (Figure S2 e). The contribution of these atoms in the PDOS is also consistent with the bonding character observed in ELF plot. The weaker bonding between the atoms leads to delocalization in the ELF implying vanishingly low charge density between the atoms, which causes the softening of optical phonon modes. 14 Hence, the weaker bonding of Ag and Cu with their surrounding atoms leads to their low-lying states. Although there is a stronger bonding between In and neighbouring S atoms, the larger and smaller mass of In and S, respectively, push their states to low and high frequency. Additionally, in the phonon dispersion, we can identify the signature of the rattlers near the frequency regime of Ag/Cu atoms, where multiple avoided crossing points between the acoustic and optical phonon modes are present. The simpler intuitive picture of these avoided crossings between acoustic and rattler modes can be understood from the model spring system describing the interaction between host material and rattler atom. 15 The solution of this model system illustrates the avoided crossing in the dispersion. This can also be understood from symmetry of lattice vibrations, where two modes of same symmetry can not have crossing dispersion curves resulting in anticrossing. 53 The avoided crossings are shown in Figure 2a and 2b, for AgIn5 S8 and CuIn5 S8 , respectively with the corresponding zoomed version in Figure 2c. Hence, the anti-crossing behaviour confirms the rattling motion, which may result in the low lattice thermal conductivity in these compounds. The calculated lattice thermal conductivity of AgIn5 S8 and CuIn5 S8 are shown in Figure 3a. AgIn5 S8 has ultra-low κl of 0.29 W/mK at 1000 K. Even at room temperature, AgIn5 S8 has κl of 0.97 W/mK. This is close to the theoretical minimum of lattice thermal conductivity value 0.60 W/mK

C1 C2 LA

1 0 Γ XWK Γ LUWLKUX A4

C1

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C2

Figure 2: The phonon dispersion shown up to 3 THz for (a) AgIn5 S8 and (b) CuIn5 S8 . The red colour optical band shows avoided crossing behaviour with the longitudinal acoustic mode as highlighted by grey circles. (c) Corresponding zoomed region of avoided crossing points marked as A1, A2, A3, A4, C1, and C2. A weaker bonding may also lead to dynamical instability. In order to rule out this possibility, we calculated the phonon dispersion. There are no imaginary frequencies in the phonon dispersion of both the compounds AgIn5 S8 and CuIn5 S8 , as


4

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obtained using Cahill’s model 54 2 3 π 1 κmin = ( ) 3 kB vV − 3 2 6

atoms, whereas valance bands have major contributions from Ag and S atoms. Similar behaviour is observed in the density of states of CuIn5 S8 , however Cu contribution to the valence bands is more as compared to Ag. Further, we calculated the band decomposed charge density for the top most valence band, as shown in Figure S3b and S3d for AgIn5 S8 and CuIn5 S8 , respectively. The contribution to the charge density corresponding to valence band comes from 3 S atoms and 1 Ag atom for AgIn5 S8 , whereas in CuIn5 S8 , it is contributed only by Cu atom. This is consistent with the ELF analysis, and may lead to significant differences in the transport properties of these two compounds. Since the rattlers are contributing to the valence band states near the Fermi level, they can affect the p-type electrical conductivity. To check the possible effect of rattler on the electronic transport, we calculated the electronic transport properties for AgIn5 S8 and CuIn5 S8 by solving Boltzmann transport equation within constant relaxation time approximation. The calculated Seebeck coefficient (S), and electrical conductivity divided by relaxation time ( στ ), as a function of carrier concentration at three different temperatures are shown in Figure 4a (Figure S4a) and 4b (S4b), respectively for AgIn5 S8 (CuIn5 S8 ). The dotted and solid curves correspond to p- and n-type, respectively. As seen earlier, in the electronic band structures, the valence bands and conduction bands have very different dispersion, hence, the transport properties for p- and n-type will also be different. Although, the Seebeck coefficient as a function of carrier concentration varies in similar fashion for both p- and n-type AgIn5 S8 and CuIn5 S8 , it is always higher for p-type than for n-type over a wide carrier concentration range at all temperatures. This difference is attributed to the presence of flat valence bands, which results in high density of states effective mass leading to a very high Seebeck coefficient. On the other hand, the calculated σ is higher for n-type AgIn5 S8 and CuIn5 S8 than τ p-type, due to highly dispersive conduction bands. In addition, the presence of states originating from the rattlers in the valence band may also reduce p-type στ . This is evident from the significantly lower στ in CuIn5 S8 than AgIn5 S8 . This is due to the prominent contribution of Cu to the va-

(4)

where kB , v, V are Boltzmann constant, average group velocity, and volume per atom, respectively. However, κl for CuIn5 S8 is 0.54 W/mK at 1000 K. This can be attributed to the differences in the atomic mass and frequency range at which rattling behaviour is observed for Ag and Cu. Furthermore, significant reduction in phonon lifetime is observed around the frequency span of these rattler atoms, as shown in Figure 3b and 3c for AgIn5 S8 and CuIn5 S8 , respectively. Lower the frequency span of rattler atom, more is the reduction in scattering time. Hence, in addition to their mass differences, the frequency range of rattling modes is also an important factors in reducing of lattice thermal conductivity, as shown in Table S1.

(a)

3

(b) AgIn5S8 CuIn5S8

τ (ps)

τ (ps)

4 2 2 (c) 0 8 1 4 0 0 0 2 4 6 8 10 200 400 600 8001000 ω (THz) T (K)

κ l (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


Figure 3: (a) Calculated lattice thermal conductivity for AgIn5 S8 and CuIn5 S8 (b), and (c) Phonon lifetime as a function of frequency for AgIn5 S8 and CuIn5 S8 at T = 1000 K, respectively. The ultra-low κl compounds can be promising for thermoelectrics provided they simultaneously possess superior electronic transport. Hence, we next analysed the electronic properties of AgIn5 S8 and CuIn5 S8 . The electronic band structure calculated using TB-mBJ functional with spin orbit coupling show both AgIn5 S8 and CuIn5 S8 as semiconductors with band gap of 0.74 and 0.38 eV, respectively, Figure S3a and c. Although the conduction band states have parabolic dispersion, the valence bands show a combination of flat and dispersive bands. The combination of such heavy and light bands could result in high power factor. 55–59 Electronic density of states of AgIn5 S8 show that the conduction bands are mainly comprised of In


5


scattering time, 60 which is given as

lence states near the Fermi level than Ag atoms, which is consistent with inferences drawn from ELF. Hence, the contribution of rattling atoms to the low energy electronic states turns out to be one of the crucial parameters for electronic transport.

S (µV/K)

400 200 0

-200 1020

1.0

1021 1022 n, p (cm-3)

ZT τ x 10−14(s)

4 3 2 1 0 1020

2 πDif Zf 1 = (Ni +0.5∓0.5)gcf (E±ℏωif −∆Ef i ) τ 2ρωif (5) where the subscripts i and f correspond to initial and final valleys, Dif , Zf , ρ, ωif , Ni , gcf , and ∆Ef i represent intervalley deformation potential, number of final valleys, mass density of the system, frequency of intervalley phonons, density of states of final valleys, and energy difference between final and initial valleys, respectively. As a first approximation, the scattering time is proportional to change in vibrational frequency of phonon modes τ ∝ ωif . The gain in relaxation time due to shift in phonon frequency will ensure scattering of a phonon by the rattlers modes, before it scatters the carriers. This leaves to a large extent the electronic transport unaffected. The frequency shift is significantly more in AgIn5 S8 , than for CuIn5 S8 , therefore, AgIn5 S8 will have superior electronic transport than CuIn5 S8 as evident from the calculated electronic transport of these two compounds. Hence, AgIn5 S8 will behave as electron glass and phonon crystal, where the rattling motion would cause reduction in lattice thermal conductivity without affecting significantly the electronic transport. On the other hand, CuIn5 S8 will have relatively poor thermal and electronic transport due to lighter mass and significant contribution from Cu electronic states to the VBM, respectively. The combined effect of Seebeck coefficient and electrical conductivity are estimated by plotting 2 the power factor divided by relaxation rate ( Sτ σ ), as shown in Figure 4c (Figure S4c) or AgIn5 S8 (CuIn5 S8 ). The high value of electrical conductivity of n-type AgIn5 S8 could not suffice to overcome the low value of Seebeck coefficient, thereby resulting in very low power factor for n-type. The maximum of power factor for p-type doping is roughly 4 times higher than the corresponding power factor for n-type. At 900 K, the peak value of the power factor is 3.78 × 1011 at the carrier concentration of 1.49 × 1021 cm−3 . On the other hand, the very low p-type electrical conductivity of CuIn5 S8 results in low p-type power factor. Although the n-type power factor for CuIn5 S8 is higher, it is still significantly lower than AgIn5 S8 .

T = 300 K T = 600 K T = 900 K

0.5 0

1020

(d)

(c) 2

(b) 1.5 σ/τ x 1020 (Ω-1m-1s-1)

(a)

S2σ/τ x 1011(W/m-K 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

1021 -3 1022 n, p (cm )

9 6 3 0 2 1 0

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1021 1022 n, p (cm-3) p-type n-type

200 400 600 8001000 T (K)

Figure 4: Calculated (a) Seebeck coefficient, (b) electrical conductivity divided by relaxation time, (c) power factor divided by relaxation time as a function of carrier concentration at three different temperatures for AgIn5 S8 . The solid and dotted curves correspond to p-type and n-type doping, respectively. (d) Total relaxation time and Figure of merit as a function of temperature for p-doped and n-doped AgIn5 S8 . Although the presence of electronic states of rattlers is a good indicator of overall electronic transport properties, yet it may affect the thermal conductivity via electron and rattler phonon mode coupling. To get an estimate of this, we apply the hydrostatic pressure in both AgIn5 S8 and CuIn5 S8 and analyse the changes in the phonon density of states with respect to equilibrium, as shown in Figure S2b (Figure S2e) and c (f) for AgIn5 S8 (CuIn5 S8 ). As expected, the frequency span of rattling atoms Ag/Cu is shifted under this deformation. However, the shift takes place towards the high frequency range and is more prominent for Ag than for Cu. This shift can be related to the electron phonon relaxation time. This relaxation time for the flat and dispersive valence bands near the Fermi level can be approximated by intervalley


6

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Table 1: Parameters used in the calculation of relaxation time for AgIn5 S8 . Parameter Ion Mass density Longitudinal acoustic group velocity Effective mass for n-type carriers Effective mass for p-type carriers Deformation potential for n-type carriers Deformation potential for p-type carriers Relative permittivity Static dielectric constant Therefore, we estimated the thermoelectric efficiency for AgIn5 S8 only. The simultaneous presence of high power factor and ultra-low thermal conductivity in AgIn5 S8 could lead to high figure of merit. To determine the figure of merit, we next calculate the relaxation time τ and κe for AgIn5 S8 . The relaxation time τ is calculated by taking into account the scattering of electrons with acoustic and optical phonons. Various quantities required for the estimation of τ are listed in Table 1. The total relaxation time for both p-type and n-type carriers decrease with increasing temperature, as shown in Figure 4d. It is always higher for n-type compared to p-type because of the inverse dependence of acoustic relaxation time on the effective mass. κe is calculated using Wiedemann-Franz law 39 κe = LT σ

Symbol ρ vLA m∗n m∗p Dacn Dacp κ∞ κ0

Value 4722.44 kg/m3 2974.17 m/s 0.68 me 2.97 me 0.76 eV 3.65 eV 8.35 8.83

0.72 and 2.22, respectively, implying that it can act as an efficient thermoelectric material in a wide temperature range. The thermoelectric efficiency of this compound outperforms many of the current state-of-the-art promising thermoelectric materials such as Bi2 S3 , 2 SnSe, 8,61 Si2 Te3 , 12 Bi2 Te3 , 62 Pb chalcogenides, 63 Chalcopyrites, 64 transition metal dichalcogenides, 65 p-type based Half Heusler compounds TaFeSb, ZrCoBi. 66,67 Given the multitude of experiments showing the feasibility of p-type doping 68–70 in AgIn5 S8 and higher melting point, this unprecedented figure of merit can be realised in laboratory.

CONCLUSION In summary, using first-principles density functional theory, we report the presence of rattlers in simple cubic spinel compounds AgIn5 S8 and CuIn5 S8 . The rattlers, Ag and Cu atoms are weakly bound to their neighbouring atoms and lie in very shallow and anharmonic potential wells. The phonon spectra of these compounds show multiple avoided crossing points between acoustic and optical modes, which leads to ultra-low κl of 0.29 W/mK and 0.54 W/mK at 1000 K for AgIn5 S8 and CuIn5 S8 , respectively. The thermal conductivity is inversely proportional to mass and frequency span of rattling modes. The difference in mass and frequency span of rattler atoms results into even lower κl in AgIn5 S8 compared to CuIn5 S8 . The Ag minimal contribution to valence band in comparison to Cu, therefore, the rattler modes of Ag do not scatter the carriers strongly leading to a superior electronic transport properties. In addition, AgIn5 S8 also possesses a com-

(6)

where L is the Lorenz number. The Lorenz number is calculated using Boltzmann transport equation by considering the temperature dependence. Since στ for n-type was much higher than p-type, the contribution of κe to the total thermal conductivity for n-type is much higher as compared to the p-type, which will further reduce its figure of merit. Using these quantities, we estimate the figure of merit ZT for AgIn5 S8 as a function of temperature for both p- and n-type at the optimized carrier concentration of 1.49 × 1021 cm−3 , Figure 4d. As expected, due to poor electrical transport, ZT for n-type AgIn5 S8 is very low. However, the high power factor for ptype AgIn5 S8 results in exceptionally high ZT. At 300 and 1000 K, ZT for p-type AgIn5 S8 is


7


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Pressure. J. Mater. Chem. C 2016, 4, 1979– 1987.

bination of heavy and light valence bands resulting in a very high power factor. The simultaneous presence of ultra-low κl and high power factor result in very high figure of merit of 2.22 for pAgIn5 S8 at 1000 K. The proposed idea of having rattlers, which can provide major resistance to heat transport without affecting the electronic transport, in simple crystalline systems, provides rich playground for designing efficient thermoelectrics.

(3) Tan, G.; Hao, S.; Cai, S.; Bailey, T. P.; Luo, Z.; Hadar, I.; Uher, C.; Dravid, V. P.; Wolverton, C.; Kanatzidis, M. G. All-scale Hierarchically Structured p-type PbSe Alloys with High Thermoelectric Performance Enabled by Improved Band Degeneracy. J. Am. Chem. Soc. 2019, 141, 4480–4486. (4) Tan, G.; Zhang, X.; Hao, S.; Chi, H.; Bailey, T. P.; Su, X.; Uher, C.; Dravid, V. P.; Wolverton, C.; Kanatzidis, M. G. Enhanced Density-of-States Effective Mass and Strained Endotaxial Nanostructures in SbDoped Pb0.97 Cd0.03 Te Thermoelectric Alloys. ACS Appl. Mater. Interfaces 2019, 11, 9197–9204.

ASSOCIATED CONTENT Supporting Information Available: The electron localisation function of CuIn5 S8 along (110) plane, phonon dispersion and phonon density of states of AgIn5 S8 and CuIn5 S8 , electronic band structure and density of states of AgIn5 S8 and CuIn5 S8 , charge density of top most valence band of AgIn5 S8 and CuIn5 S8 , electronic transport properties of CuIn5 S8 . This material is available free of charge via the Internet at http://pubs.acs.org/.

(5) Joshi, G.; Lee, H.; Lan, Y.; Wang, X.; Zhu, G.; Wang, D.; Gould, R. W.; Cuff, D. C.; Tang, M. Y.; Dresselhaus, M. S.; Chen, G.; Ren, Z. Enhanced Thermoelectric Figure-of-Merit in Nanostructured ptype Silicon Germanium Bulk Alloys. Nano Lett. 2008, 8, 4670–4674.

AUTHOR INFORMATION Corresponding Author ∗ Email: [email protected]

(6) Bux, S. K.; Blair, R. G.; Gogna, P. K.; Lee, H.; Chen, G.; Dresselhaus, M. S.; Kaner, R. B.; Fleurial, J.-P. Nanostructured Bulk Silicon as an Effective Thermoelectric Material. Adv. Funct. Mater. 2009, 19, 2445– 2452.

Notes The authors declare no competing financial interests. Acknowledgement RJ thanks DST for INSPIRE fellowship (IF150848). The authors thank the Materials Research Centre, Thematic Unit of Excellence, and Supercomputer Education and Research Centre, Indian Institute of Science, for providing computing facilities. AKS and RJ thank the support from DST Nano Mission.

(7) Farahi, N.; Prabhudev, S.; Botton, G. A.; Salvador, J. R.; Kleinke, H. Nano-and Microstructure Engineering: An Effective Method for Creating High Efficiency Magnesium Silicide Based Thermoelectrics. ACS Appl. Mater. Interfaces 2016, 8, 34431– 34437.

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Graphical TOC Entry 2

In

In S

S

In

In

kl

P. E.

S Ag

1 0

Displacement

T (K)

2 ZT

S

Efficient TE

Rattling

Anisotropic Bonding

Frequency

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


k-path


13

1 0

T (K)

Rattling-Induced Ultra-low Thermal Conductivity Leading to

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