first-principles prediction of large thermoelectric ... · structural relaxations, phonon...

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2020, 22, 878

First-principles prediction of large thermoelectricefficiency in superionic Li2SnX3 (X = S, Se)†

Enamul Haque,a Claudio Cazorla *b and M. Anwar Hossaina

Thermoelectric materials create an electric potential when subjected to a temperature gradient and vice

versa; hence they can be used to harvest waste heat into electricity and in thermal management

applications. However, finding highly efficient thermoelectrics with high figures of merit, zT Z 1, is very

challenging because the combination of a high power factor and low thermal conductivity is rare

in materials. Here, we use first-principles methods to analyze the thermoelectric properties of Li2SnX3

(X = S, and Se), a recently synthesized class of lithium fast-ion conductors presenting high thermal

stability. In p-type Li2SnX3, we estimate highly flat electronic valence bands that produce high Seebeck

coefficients exceeding 400 mV K�1 at 700 K. In n-type Li2SnX3, the electronic conduction bands are

slightly dispersive; however, the accompanying electron–acoustic phonon scattering is weak, which

induces high electrical conductivity. The combination of a high Seebeck coefficient and electrical

conductivity gives rise to high power factors, reaching a maximum of B4.5 mW m�1 K�2 at 300 K in

both n-type Li2SnS3 and Li2SnSe3. Likewise, the thermal conductivity in Li2SnX3 is low as compared

to conventional thermoelectric materials, 1.35–4.65 W m�1 K�1 at room temperature. As a result, we

estimate a maximum zT of 1.1 in n-type Li2SnS3 at 700 K and of 2.1 (1.1) in n-type Li2SnSe3 at the same

temperature (300 K). Our findings of large zT in Li2SnX3 suggest that lithium fast-ion conductors, typically

employed as electrolytes in solid-state batteries, hold exceptional promise as thermoelectric materials.

1. Introduction

Thermoelectric (TE) materials represent an alternative to traditionalpower generation sources as they can transform waste heat directlyinto electricity.1,2 TE materials also can be employed in solid-staterefrigeration applications based on the Peltier effect, in which heatis removed via the application of electric fields.3,4 The efficiency ofTE materials is measured by a dimensionless parameter called thefigure of merit, which is defined as zT = S2sk�1T,5,6 whereS represents the Seebeck coefficient, s the electrical conductivity,k the thermal conductivity, and T the operation temperature. Themeasured efficiency of the best TE materials known to date typicallyis zT o 2.6;7–11 for instance, SnSe exhibits a maximum zT of 2.6 atT = 923 K8 and at room temperature Bi2Te3 and other related alloyspresent record zT’s of 0.8–1.2.12–16 However, in order to developnew and scalable energy conversion and thermal managementapplications, there is a pressing need to find highly efficient TEmaterials with larger zT’s.2

Highly efficient TE materials should display high powerfactors (PFs), defined as the product of the Seebeck coefficientand electrical conductivity, and low thermal conductivities.However, it is quite unusual to find these two qualities simulta-neously in the same material. Various strategies for theenhancement of TE efficiency have focused on reducing kwhile keeping PF almost unchanged.17–21 For example, somecompounds containing heavy atoms like YbFe4Sb3,22 CoSb3,23

Mg2Si1�xSnx,24 and Sb2Te325 show good thermoelectric properties

primarily due to low lattice thermal conductivity, kl. Other zTenhancement strategies have focused on optimizing the PFwhile keeping k almost unchanged.26 For example, a distortionin the electronic density of p-type PbTe caused by the intro-duction of Ti impurities enhances significantly its PF leadingto exceptionally high zT values of 1.5 at 773 K.26,27 However,S and s normally change in opposite ways (e.g., S increases withtemperature whereas the electrical conductivity is reduced, andS is low at high carrier concentrations whereas s is high); hencesystematic improvement of TE materials through PF enhance-ment is usually difficult.

Recently, Cu- and Ag-based fast-ion conductors (FICs) havebeen proposed as very promising TE materials.28–30 In super-ionic materials, specific ions start to diffuse through thecrystalline matrix above a critical temperature producing highionic conductivities comparable to those in liquids. Ionic

a Department of Physics, Mawlana Bhashani Science and Technology University

Santosh, Tangail-1902, Bangladesh. E-mail: [email protected] School of Materials Science and Engineering, University of New South Wales

Australia, Sydney, New South Wales 2052, Australia.

E-mail: [email protected]

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9cp05939c

Received 1st November 2019,Accepted 20th November 2019

DOI: 10.1039/c9cp05939c

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diffusion produces high structural disorder in the crystal, thusenhancing zT by lowering k. Interestingly, the intrinsic highanharmonicity in FIC seems to be enough to give rise to low k’seven at temperatures below the superionic transition point.31,32

Likewise, the power factor reported for the investigated FIC issignificantly large (e.g., B1 mW m�1 K�2 in Cu2�xSe at roomtemperature28). But, unfortunately, FICs are prone to sufferfrom thermo- and electro-migration issues that limit theirapplication as TE materials.33,34 Simple strategies based onchemical doping, however, have been shown to solve theseproblems by simultaneously suppressing cation migration andmaintaining low lattice thermal conductivity.35,36 Therefore, inview of the large zT’s measured in Cu- and Ag-based FICs andthe straightforward solutions proposed for addressing theirlikely stability issues, it is of great interest to continue exploringthe TE potential of superionic materials.

Here, we investigate the TE performance of Li2SnX3 (X = S,and Se), a recently synthesized family of lithium superioniccompounds that present high thermal stability,37–39 by usingfirst-principles computational methods. Our calculationspredict high zT’s of 1 at room temperature and of 2 at 700 K,due to the combined effect of a high PF and low k. To the bestof our knowledge, this is the first prediction of large zT in aLi-based FIC that has been synthesized previously in thelaboratory. Therefore, lithium superionic conductors, whichare well known for their use in electrochemical devices andcurrently are being investigated very actively, appear to beextremely promising TE materials.

2. Computational methods

Structural relaxations, phonon dispersions, and electron–phonon(e–ph) dynamical matrix calculations are all performed with theplane-wave pseudopotential method as implemented in theQuantum Espresso code.40,41 The adopted exchange–correlationscheme corresponds to the generalized gradient approximationdue to Perdew–Burke–Ernzerhof (PBE).42 In these calculations, theVanderbilt ultrasoft pseudopotentials43 are employed, along witha 42 Ry cutoff energy in the wavefunctions and a Methfessel–Paxton smearing44 of 0.03 Ry. In the phonon calculations, non-shifted k-point grids of 4 � 4 � 2 and 4 � 4 � 4 are used forLi2SnS3 and Li2SnSe3, respectively. The electron–phonon dynamicalmatrices are calculated by using 2 � 2 � 2 and 2 � 2 � 1 q-pointgrids for Li2SnS3 and Li2SnSe3, respectively, since the corres-ponding unit cells are quite large (48 and 24 atoms, respectively),and therefore coarse q-point grids are sufficient to generate e–phmatrices with reasonable accuracy.45 Electron–phonon averagedapproximation (EPA) calculations are performed with the EPAcode45 by using an energy grid spacing of 0.6 eV. Consistency testsare performed by repeating some calculations with the EPA-MLScode, which computes electron–phonon matrices via moving leastsquares and produces accurate results with coarse q-point grids.46

In particular, a 3 � 3 � 3 q-point grid is used for Li2SnSe3 andgood agreement between the two series of EPA and EPA-MLSresults is obtained (data not presented here).

Electronic structure calculations are performed with the full-potential linearized augmented plane wave method as implementedin the WIEN2k code,47 using the PBE and TB-mBJ (Tran–Blahamodified Becke–Johnson) exchange–correlation functionals.48

2600 irreducible k-points and a RmtKmax product of 8.0 (whereRmt is the muffin-tin sphere radius and Kmax the plane wavecut-off energy) are used to calculate the electronic band structureand density of states. 28 � 28 � 12 and 22 � 22 � 17 non-shiftedk-point grids are used for the self-consistent calculation of energyeigenvalues, which are required for the transport calculations inLi2SnS3 and Li2SnSe3, respectively. The calculation of transportcoefficients is performed with the modified BoltzTrap code,45,49

in which the carrier relaxation time is calculated through theequation45

t�1 e; m;Tð Þ ¼ 2pOgs�h

Xv

gv2ðe; eþ �ovÞ n �ov;Tð Þ þ f ðeþ �ov; m;TÞ½ �

�

� r eþ �ovð Þ þ gv2 e; e� �ovð Þ

� n �ov;Tð Þ þ 1� f ðe� �ov; m;TÞ½ �r e� �ovð Þg(1)

where O is the volume of the primitive unit cell, h� the reducedPlanck’s constant, v the phonon mode index, �ov the averagedphonon mode energy, gv

2 the averaged electron–phonon matrix,n(�ov,T) the Bose–Einstein distribution function, f (e + �ov, m, T )the Fermi–Dirac distribution function, gs = 2 the spin degen-eracy, e the electron energy, and r the density of states per unitenergy and unit volume. Further details of the calculation oftransport coefficients can be found in work.45

The elastic constants were calculated using the stress–strainrelations as implemented in the IRelast package of WIEN2k.50

In these calculations, all the parameters except for the k-pointgrids, which were increased to 6� 6� 3 for Li2SnS3 and 6� 6� 6for Li2SnSe3, were the same as explained above.

Harmonic (2nd order) and anharmonic (3rd order) forceconstant matrices were calculated using the Phono3py package.51

For these calculations, we employed 2� 2� 2 supercells (containing384 and 192 atoms for Li2SnS3 and Li2SnSe3, respectively), andgamma k-point sampling for calculation of the atomic forces.The lattice thermal conductivity was calculated by solving thephonon Boltzmann transport equation within the constantrelaxation time approximation and considering 4 � 4 � 4 and6 � 6 � 6 q-point grids.

3. Results and discussion3.1. Structural, electronic, and vibrational properties

Fig. 1 shows the crystal structure of Li2SnX3 (X = S and Se).Li2SnS3 crystallizes in a monoclinic phase with space groupC2/c (#15)37,38 and Li2SnSe3 in another monoclinic phase withspace group Cc (#9).39 The number of atoms per unit cell is 48 and24 for Li2SnS3 and Li2SnSe3, respectively. The corresponding fullyoptimized lattice parameters are listed in Table 1, which are ingood agreement with the available experimental data and previoustheoretical calculations reported by other authors.37–39,52

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We note that PBE-GGA overestimates the experimentallattice parameters by less than 1% while LDA underestimatesthem by B2%. In view of the superior agreement with experimentsprovided by PBE-GGA, we will use this functional in the rest of thecalculations if not stated otherwise. Further structural details ofLi2SnX3 (fractional atomic coordinates) can be found in theESI,† Table S1.

The electronic band structure and projected electronicdensity of states of Li2SnX3 are shown in Fig. 2. Both compoundsdisplay similar electronic band structure properties. In Li2SnS3,the maximum of the valence band (VBM) lies between the L and Gpoints while the minimum of the conduction band (CBM) sitsat L. An indirect band gap of 2.14 eV is estimated in Li2SnS3, inconsistent agreement with previous theoretical calculations andan experimental value of 2.38 eV.38 A direct band gap with almostthe same value exists at the L point; hence Li2SnS3 may behavealso as a direct band gap semiconductor. In Li2SnSe3, both CBMand VBM lie at the G-point and generate a direct band gap of1.95 eV. The energy bands in both compounds are highly flat,which, as we will explain in detail below, leads to high Seebeckcoefficients (according to Mott’s relation53) and high powerfactors. The projected density of states in both compounds areshown to the right of each band structure figure. The electronicorbitals Sn 5p and S 3p–Se 4p are strongly hybridized andcontribute the most to the VBM and CBM, which suggests thatdoping in the Sn and/or S–Se sites could be used to further tunethe band gap of Li2SnX3.

We note that recently some wide bandgap semiconductors(B1.5–3.5 eV) have been found to be promising materials forTE applications,54–61 both in experimental and theoreticalstudies. In fact, wide band gap materials may display high

thermoelectric power owing to the high effective mass of theircharge carriers. As it will be shown in the following sections,this turns out to be the case of superionic Li2SnX3.

Phonons characterize the dynamic stability of a compound;real frequencies in the phonon dispersion curves indicate thatthe crystal is vibrationally stable.62–66 Elastic stability, on theother hand, can be assessed from the elastic constants of thematerial. The phonon band structures calculated in Li2SnX3 areshown in Fig. 3. As can be appreciated in there, all latticephonon modes and frequencies are well-behaved (i.e., are real).The elastic constants computed in both compounds are listedin Table 2, along with the corresponding Pugh’s ratio,67 Poisson’sratio, and Debye temperature.

3.2. Lattice thermal conductivity

The elastic constants in Li2SnX3 fulfill the elastic stabilitycriteria corresponding to monoclinic crystals as reported inref. 68 and 69 (e.g., the cij matrix is positive definite70–73). Wenote that elastically stable monoclinic phases may presentnegative elastic constants,74 as is the case of Li2SnX3. Thevalue of the Pugh’s ratio is very close to the critical valueof 1.75 that differentiates ductile from brittle materials.The low value of the Debye temperature indicates low heattransport by phonons, that is, low lattice thermal conductivity, kl.In order to calculate kl, we use the longitudinal (vl) and transverse(vt) sound velocities to estimate the acoustic Gruneisen para-meter:75

g ¼9 vl

2 � 4vt2=3

� �2 vl2 þ 2vt2ð Þ (2)

which then can be used in the Slack equation to calculate thelattice thermal conductivity as:76,77

kl ¼ AMavyaco3dg2n2=3T

(3)

where Mav is the average atomic mass, d the cubic root of thevolume of the primitive cell per atom,76 yaco the acoustic Debyetemperature, n the number of atoms per unit cell, and A thecoefficient defined by the relation:78

A gð Þ ¼ 5:720� 107 � 0:849

2� 1� 0:514

g

� �þ 0:228

g2

� �� (4)

Eqn (3) provides an averaged isotropic value for kl; however,in most materials the lattice thermal conductivity depends on

Fig. 1 Ground state crystal structures of (a) Li2SnS3 (monoclinic symmetry, space group C2/c) and (b) Li2SnSe3 (monoclinic symmetry, space group Cc)considering different views. The black lines represent the corresponding unit cells and a, b, and c characteristic crystallographic directions.

Table 1 Fully relaxed lattice parameters in Li2SnX3 (X = S and Se). Availableexperimental data are shown for comparison. SC-XRD stands for singlecrystal X-ray diffraction and SXRPD for synchrotron X-ray powder diffraction.Further structural data can be found in the ESI

Compounds a (Å) b (Å) c (Å) b Ref.

Li2SnS3 6.4542 11.1853 12.4891 99.884 Present work6.3964 11.0864 12.405 99.867 Expt. (SC-XRD)38

6.3961 11.089 12.416 99.860 Expt. (SC-XRD)37

6.4004 11.0854 12.4222 99.883 Expt. (SXRPD)38

6.30 10.91 12.15 99.94 LDA52

6.28 10.87 12.08 99.82 LDA38

Li2SnSe3 12.7363 7.3198 7.9506 121.14 Present work12.522 7.2137 7.7692 120.96 Expt. (SC-XRD)39

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the direction of heat propagation. Eqn (3) can be modified asfollows to reproduce the likely anisotropic effects in the latticethermal conductivity:

kli ¼ AMavyi3dg2n2=3T

i ¼ a; b; cð Þ (5)

where yi is the anisotropic Debye temperature, which can becalculated with the equation:79

yi ¼h

kB

3n

4pNArM

� �vmi; i ¼ a; b; c: (6)

In eqn (6), vmi represents the average speed of sound along thecrystallographic directions a, b, and c (Fig. 1). By using the elastic

constants and density of the material, vmi can be calculated via theChristoffel eigenvalue equation:80–82

Xijkl

rv2ðnÞdij � cijklnknl ¼ 0 (7)

where r is the density of the compound, v2(n) is the speed ofsound along with the unit vector n, cijkl is the fourth rank elastictensor, and nk and nl are unit vector components representing thedirections of propagation (one longitudinal and two transverse).83

The anisotropic Debye temperatures and wave velocitiesestimated along the corresponding crystallographic directionsin Li2SnX3 are shown in Table 3. The average of the anisotropicDebye temperatures is in reasonable agreement with theisotropic Debye temperature (see Table 2), which supports thevalidity of the computational method employed in this study

Fig. 3 Phonon dispersion relations of (a) Li2SnS3 and (b) Li2SnSe3.

Fig. 2 Electronic band structure and projected density of states of (a) Li2SnS3 and (b) Li2SnSe3. The dashed blue line at zero energy represents theFermi level. The k-point paths for the calculation of the electronic band structures of Li2SnS3 and Li2SnSe3 involve the high-symmetry points G(0,0,0),Y(1/2,1/2,0), A(0,0,1/2), M(1/2,1/2,1/2), V(1/2,0,0), and L(1/2,0,1/2).

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for the estimation of kl. Meanwhile, the calculated Gruneisenparameters are large, which indicates high anharmonicity inthe crystals (as generally expected in fast-ion conductors84), andhence intense phonon scattering. By using eqn (5), we estimatethe lattice thermal conductivity along the three crystallographicdirections and represent them in Fig. 4. Since Li2SnS3 undergoesa structural phase transition around 750 K,38 it is reasonableto calculate kl just within the temperature interval of 200–700 K.In both compounds, non-negligible anisotropies in thermal trans-port are observed and kl is the lowest along the c-axis.

Li2SnS3 presents larger kl than Li2SnSe3, essentially due tothe higher anisotropic Debye temperatures (Table 3). Thisoutcome is consistent with the phonon lifetimes shown inFig. 4(c) and (d). The shorter phonon lifetimes in Li2SnSe3

indicate more intense phonon scattering than that in Li2SnS3.At 300 K, the lattice thermal conductivity along the b-axis isequal to 4.65 W m�1 K�1 in Li2SnS3 and 1.35 W m�1 K�1 inLi2SnSe3. We also note that the highest kl in Li2SnS3 is obtainedalong the b-axis, whereas in Li2SnSe3 it is obtained along thea-axis. This lattice thermal conductivity behavior is similar tothe thermal transport trends observed in SnS and SnSe.85 Forexample, the average kl calculated in Li2SnSe3 is very close tothat in SnSe (B1 W m�1 K�1 at 300 K).85 It is worth noting thatthe kl’s estimated in Li2SnX3 are low in comparison to the onesreported for other typical thermoelectric materials like CoSb3

(11.5 W m�1 K�1 at 300 K).23

The lattice thermal conductivities obtained within thephonon Boltzmann transport equation (BTE) formalism agree

fairly well with the values calculated using the modified Slack’smodel (mSM). The slight kl deviations observed in Fig. 4(a)and (b) result from using the constant Debye temperaturesand Gruneisen parameters in the mSM, since both quantitiesdepend on temperature. Moreover, in the mSM only scatteringcaused by acoustic phonons is considered and hence otherpossible scattering mechanisms are neglected. Nevertheless,the simple mSM is shown to provide an overall good descriptionof the thermal transport in complex materials like Li2SnX3. Foraccuracy purposes, in all subsequent calculations we will employthe average of the kl values obtained with both the mSM andBTE approaches.

3.3. Seebeck coefficients and power factors

In Boltzmann transport theory, the constant relaxation timeapproximation can be used to calculate electrical conductivitiesand the electronic contribution to the thermal conductivity.Previous theoretical calculations, however, have shown thatsuch an approximation may lead to some inaccuracies,45 sincethe carrier relaxation time, t, depends strongly on the carrierdensity and composition of the material. Here, we employ amore sophisticated method to estimate carrier relaxation timesthat is able to take into account electron–phonon interactionswith reasonable accuracy (eqn (1) in the Computational methodssection).45 Fig. 5 shows the t results obtained for Li2SnX3 at 300 Kexpressed as a function of energy.

Both compounds present practically isotropic relaxationtimes. In Li2SnS3, the relaxation time of the electrons in theconduction band is almost the same as that of the holes in thevalence band near the band edge. In Li2SnSe3, the electronsin the conduction band near the Fermi level have longerrelaxation times than the holes in the valence band. Theslightly dispersive nature of the conduction band, as comparedto the valence band near the Fermi level, is responsible for theparticular t behavior observed in Li2SnSe3. In both compounds,t decreases sharply near the band edges. This effect can berationalized with the help of the relationship:45

t�1 B g2(e)r(e) (8)

which points out that the relaxation time varies inversely withthe carrier density of states per unit energy and volume, r,when the electron–phonon matrix elements, g, depend onlyweakly on the energy.

Above the Debye temperature, electron–phonon (e–ph)scattering is dominant86,87 and the e–ph coupling matrix alonecan be used to describe, with reasonable accuracy, the electricalconductivity and electronic contribution to the thermal conduc-tivity. Here, we are interested in the temperature interval of300–700 K, which is above yi (Table 3); hence we take intoconsideration only the e–ph coupling matrix for the calculationof electronic transport coefficients. Fig. 6 shows the Seebeckcoefficients calculated in Li2SnX3 as a function of carrierconcentration at temperatures T = 300, 500, and 700 K. Bothcompounds show practically isotropic Seebeck coefficients thatare larger than 400 mV K�1 at 700 K and carrier concentrationsof B1019 cm�3 for p-type systems. The Seebeck coefficients of

Table 2 Calculated elastic constants (cij) in GPa, Pugh’s ratio (B/G) (whereB and G are both in GPa), Poisson’s ratio (n), longitudinal and transversesound velocities (vl & vt) in m s�1, and Debye temperature (yD) in K

Property Li2SnS3 Li2SnSe3

c11 90.24 34.89c12 29.36 15.08c13 31.06 13.10c16 �4.29 0.14c22 86.64 46.39c23 16.40 13.02c26 3.78 1.45c33 89.28 38.42c36 8.16 �1.92c44 24.75 11.81c45 2.39 �0.10c55 33.75 13.57c66 22.28 13.48B/G 1.66 1.73n 0.249 0.257vl 4933.76 3195.14vt 2850.49 1825.55vm 3164.39 2028.54yD 356.18 202.74

Table 3 Calculated anisotropic sound velocity (vmi), anisotropic Debyetemperature (yi) and Gruneisen parameter (g) of Li2SnX3

Compoundvma

(m s�1)vmb

(m s�1)vmc

(m s�1) ya (K) yb (K) yc (K) g

Li2SnS3 2606.4 2838.8 2344.9 279.4 303.9 251.04 1.495Li2SnSe3 1778.0 1754.04 1617.5 177.8 175.4 161.7 1.534

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p-type Li2SnX3 are much larger than those of n-type compoundsdue to the relatively flatter valence bands calculated inthe former cases. We also note that the Seebeck coefficientcalculated in Li2SnS3 is much larger than that in Li2SnSe3

due to the smaller band gap estimated in the latter case.The Seebeck coefficients in p-type Li2SnS3 and Li2SnSe3 arevery similar to those in p-type SnS and SnSe, respectively;85

however, the Seebeck coefficients in n-type Li2SnX3 are slightlysmaller.

The high Seebeck coefficients calculated in Li2SnX3 lead tohigh power factors (PFs), as shown in Fig. 7. The estimated PFdisplays non-isotropic behavior due to the anisotropies foundin the electrical conductivity (ESI†). The energy band gaps inboth Li-based compounds are almost two times larger than

Fig. 4 Anisotropic lattice thermal conductivities of (a) Li2SnS3 and (b) Li2SnSe3. Solid (dashed) lines represent the lattice thermal conductivity obtainedfrom the modified Slack’s model (phonon Boltzmann transport equation). Symbols a, b, and c denote the characteristic crystallographic directions (Fig. 1).The bottom panel shows the phonon lifetimes of (c) Li2SnS3 and (d) Li2SnSe3.

Fig. 5 Computed carrier lifetimes (at 300 K) of (a) Li2SnS3 and (b) Li2SnSe3. The dotted blue line at zero energy represents the Fermi level. Labels a, band c denote the characteristic crystallographic directions (Fig. 1).

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those found in SnS and SnSe; however, their electrical con-ductivities are much higher as well. In particular, Li–S/Se ionshave low effective masses (me* = 0.13m0, and mh* = 0.14m0 andme* = 0.04m0, and mh* = 0.09m0 in Li2SnS3 and Li2SnSe3, respec-tively), as compared to SnS (me* = 0.20m0, and mh* = 0.28m0

88) and

SnSe (me* = 0.12m0, and mh* = 0.13m089), thus inducing higher

electrical conductivity in Li2SnX3. The electrical conductivity inn-type Li2SnX3 is higher than those in the analogous p-typesystems due to the lower effective mass of the electrons ascompared to that of the holes. We note that the PF calculated

Fig. 6 Calculated absolute value of Seebeck coefficients as a function of carrier concentrations of Li2SnS3 (top panel) and (b) Li2SnSe3 (bottom panel).The symbols a, b, and c denote the crystallographic directions (Fig. 1), while p and n indicate the type of the carrier.

Fig. 7 Computed power factor (PF = S2s) as a function of carrier concentrations of Li2SnS3 (top panel) and Li2SnSe3 (bottom panel).

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for n-type Li2SnX3 along the c-axis can reach large values ofB4.5 mW m�1 K�2 at room temperature for a carrier concentration

of around 1019 cm�3 (Fig. 7). The PFs estimated at roomtemperature are high and comparable to the values reported

Fig. 8 Total thermal conductivities (k = ke + kl) of Li2SnS3 (top panel) and Li2SnSe3 (bottom panel) as a function of carrier concentration andcrystallographic direction.

Fig. 9 Dimensionless thermoelectric figures of merit (zT) of Li2SnS3 (top panel) and Li2SnSe3 (bottom panel) as a function of carrier concentration andcrystallographic direction.

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for other promising thermoelectric materials (e.g., a PF of10 mW m�1 K�2 measured in p-type half-Heusler NbFeSb alloysat 500 K is the current experimental record90).

3.4. Thermoelectric figures of merit

Fig. 8 shows the total thermal conductivity, equal to the sum ofthe lattice and electronic contributions, calculated for Li2SnX3

at different temperatures and expressed as a function of carrierconcentration. As in SnS and SnSe, the thermal conductivitiesin both compounds show anisotropic behavior. At low tem-perature and low carrier concentration the lattice contributionto the thermal conductivity is dominant in both compounds,whereas at high carrier concentration (above B1020 cm�3) theelectronic contribution to the thermal conductivity is dominant(further details of the electronic part of the thermal conductivitycan be found in the ESI,† Fig. S2). The k values estimated for thesecompounds are lower than those for other typical semiconductors(e.g., 11.5 W m�1 K�1 in CoSb3 at 300 K23), although higher thanthose in SnS and SnSe due to larger lattice thermal contributions.Such estimated low thermal conductivities can potentially lead tohigh thermoelectric figures of merit.

Fig. 9 shows the zT results obtained for Li2SnX3 expressed asa function of carrier concentration and temperature. At roomtemperature, the zT estimated for Li2SnS3 is relatively low,0.1–0.4, due to the relatively high thermal conductivity obtainedat low temperatures (Fig. 8). Nevertheless, zT increases signifi-cantly with increasing temperature; for instance, at 500 K the zTfor n-type Li2SnS3 along the c-axis amounts to 0.78 at a carrierconcentration of 2.1� 1019 cm�3. Yet, this value is lower than thatfound, for instance, in n-type SnS (zT = 1.5 at 750 K and a carrierconcentration of B8 � 1019 cm�3 ).85 The maximum zT estimatedin n-type Li2SnS3 is 1.07, which corresponds to the c-axis at 700 Kand a carrier concentration of 4.2 � 1019 cm�3.

Meanwhile, the combination of the low thermal conductivityand high power factor leads to large zT’s in n-type Li2SnSe3. At300 K, our calculations predict a maximum zT of B1 along thec-axis and for a carrier concentration of 2.9 � 1018 cm�3.Therefore, Li2SnSe3 represents a promising room-temperaturethermoelectric material. Remarkably, the maximum zT calcu-lated in n-type Li2SnSe3 amounts to 2.05 at 700 K (along thec-axis and for a carrier concentration of 8.8 � 1018 cm�3), whichrepresents an unusually large TE figure of merit. It is worthnoting that Li, Sn, and Se are all abundant and cost-effectiveelements, hence thermoelectric compounds containing themare particularly attractive from an application point of view.

4. Conclusions

In summary, the thermoelectric properties of superionicLi2SnX3 (X = S, and Se) have been studied extensively by usingaccurate first-principles methods. Highly flat valence bandsin p-type Li2SnX3 lead to high Seebeck coefficients, which canexceed 400 mV K�1 at 700 K. Slightly dispersive conductionbands induce comparatively lower Seebeck coefficients inn-type compounds and longer electron lifetimes near the band

edges. Such conduction bands lead to weak electron–phononinteractions that produce high electrical conductivity in n-typeLi2SnX3. The combination of a high Seebeck coefficient andelectrical conductivity gives rise to high power factors, which atroom temperature can reach B4.5 mW m�1 K�2 in both n-typeLi2SnS3 and Li2SnSe3. Likewise, the thermal conductivitiesestimated in Li2SnX3 are low, for instance, 1.35–4.65 W m�1 K�1

at 300 K. Based on these values, we predict that n-type Li2SnS3

is a potentially good TE material at high temperatures (zT = 1.1at 700 K) whereas n-type Li2SnSe3 is already superb at lowtemperatures (zT = 1.1 at 300 K and 2.1 at 700 K). Such large zT’scould be further increased in practice by tuning the corres-ponding material bandgaps with doping. The reported largefigures of merit suggest that lithium-based fast-ion conductorshold tremendous promise for energy conversion and thermalmanagement applications, thus the field of thermoelectricitycould benefit immensely from the intensive research alreadyundertaken on electrochemical energy devices. We hope thatthe theoretical results presented in this study will motivateexperimentalists to investigate the TE performance of Li2SnX3

(X = S and Se) and other similar Li-based superionic materials.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors thank Professor Boris Kozinsky and Daehyun Weefor sharing their personal codes and Dr K. C. Bhamu for helpfuldiscussions about EPA calculations.

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