Accurate high-throughput screening of I-II-V 8-electron Half-Heusler compounds forrenewable-energy applications

Bhawna Sahni, Vikram, Jiban Kangsabanik and Aftab Alam∗

Department of Physics, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, India(Dated: October 9, 2019)

Renewable energy resources have emerged as the best alternatives to fossil fuel energy whichare rapidly declining with time. Choice of correct materials plays a key role in designing deviceswith enhanched efficiency. Half-Heusler alloys, with various valence balanced compositions, havebeen phenomenal in showcasing a variety of rich properties, and thus provide a general plateformto search for candidates suitable for renewable energy applications. Here, eight valence-electroncount Half-Heusler(HH) alloys have been studied using reliable first principles calculations in thesearch of potential candidates for thermoelectric (TE), solar harvesting, topological insulator (TI)and transparent conductor (TC) applications. The initial screening parameters used for our studyare chemical and thermal stability, band gap, nature of bandgap and band inversion strength.The screening is further followed by application specific descriptor calculations to predict possiblecompounds for energy applications. We have performed quasistatic G0W0 calculation starting fromHSE groundstate wavefunction to predict the most accurate estimation of bandgap for these classof compounds. A total of 960 compounds were simulated. 121 out of 960 compounds were foundto be thermally and chemically stable. 31 compounds with bandgap less than 1.5 eV were studiedfor thermoelectric application out of which 13 compounds were found to show thermoelectric figureof merit ZT > 0.7 for both p-type and n-type conduction. 30 compounds with band gap 1-1.8eV were studied for optoelectronic application out of which 13 compounds were found to showSpectroscopic Limited Maximum Efficiency (SLME) more than 20%, comparable to existing stateof the art materials. 21 compounds were found to show band inversion at ambient conditionswhich is a necessary condition for topological insulators. The surface band structure calculationsfor one of the promising candidate was done to check robustness of the topological behaviour.29 compounds were found to have bandgap more than 2 eV which are promoted for transparentconductor applications with further band engineering. We strongly believe that our calculations willgive useful insights to experimentalists for synthesizing and investigating proposed compounds fordifferent energy applications.

I. INTRODUCTION

The demand for alternative energy resources has beenever increasing due to rapid decline of fossil fuel energyresources. Renewable energy resources are the best alter-natives for solving this problem and have indeed emergedas a unique solution to circumvent this. Among others,the technologies based on photovoltaics, thermoelectrics,topological electronics etc. remain at the forefront inproviding state of the art solutions to renewable energies.One of the key requirement, for devices based on all thesetechnologies, is the correct choice of materials. Hunt fornovel materials, which can enhance the efficiency of thesedevices, has always been an active area of research.

Historically, Half Heusler compounds show a plethoraof interesting properties, thus pave a general platformto search for candidates suitable for renewable energyapplications.1−3 They have a general formula unit XYZ,where X and Y are in, general, transition metals andZ is a p-block element. They are easy to synthesizewith easily tunable band gaps and can have numerouspossible combinations due to many choices for the threesites. Some examples of half heusler compounds havingexpceptionally good properties are p-type TaFeSb-basedHH compounds4 which show a record high thermoelec-tric performance (ZT) of ∼ 1.52 at 973 K and henceare highly promising for thermoelectric power genera-

tion. On the other hand, TaIrGe5 is found to be p-typetransparent conductor having a strong optical absorptionpeak at 3.36 eV with a phenomenal high hole mobilityof 2,730 cm2V −1s−1 at room temperature. LnPtBi (Ln= La, Y) HH compound6 was the first member found toshow non-trivial topological electronic structure makingthese class of materials also promising for the search ofnew topological quantum phases. HH compounds canbe classified on the basis of sum of the valence electronsof X, Y and Z elements. The 18 valence-electron sys-tems of HH alloys have been widely studied.3,7−10 But,8 valence-electron systems are less explored and hence areviable candidates for high throughput computation pre-dictions. High throughput computation has emerged asa great tool for the discovery of new compounds for vari-ous applications.11−13 Such studies can save tremendousamount of time/resources that can go in the correspond-ing experimental studies, yet provide useful insights toexperimentalists.

An interesting class of 8 valence-electron HH systemsare I-II-V class of compounds. There has been veryfew theoretical and experimental studies on these com-pounds. Although a few studies based on the optoelec-tronic properties have been reported14−16 but, a reli-able/accurate prediction of bandgap still remains a con-cern. Yadav et al.17 have studied the thermoelectricproperties of Li-based half-Heusler alloys using DFT in

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trl-

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which they studied the thermoelectric properties of LiYZ( Y = Be, Mg, Zn, Cd and Z = N, P, As, Sb, Bi).Jaiganesh et al.18 have studied the electronic and struc-tural properties of NaZnX ( X = P, As, Sb) compounds.Some of these class of compounds have been experimen-tally synthesized.19,20 Bacewiz et al.21 have preparedand characterized LiZnP, LiCdP and LiZnAs and foundthem to be p-type semiconductors with the help of op-tical and electrical measurements. The thermoelectricperformance of LiZnSb was measured experimentally22

motivated by the theoretical calculations23 on a bunchof antimony containing compounds. This experimentalsynthesis22 yields a p-type LiZnSb in hexagonal phase.Later, Miles A. White24 synthesized cubic LiZnSb andreported both n-type and p-type forms of this materialto exhibit high ZT values of 1.64 and 1.43 respectively at600 K with light doping. However, they assumed a con-stant value of lattice thermal conductivity which givespoor predictability of the figure of merit (ZT). Althoughthere are few thermoelectric studies on the I-II-V classof Heusler compounds, a thorough search for promisingcandidates with a more reliable high throughput in-depthstudy is lacking. Furthermore, there are still plenty ofroom for identifying reliable photo-voltaic materials fromthis class of compounds. Topological quantum materialsare other potential set of materials, many of which areexpected to find its home in this class, but are hardlyexplored.25,26

Traditional high throughput computation based onab-initio calculations mostly relies on the use of sim-ple local density or generalized gradient (LDA or GGA)type exchange correlation functionals.27 These function-als, though computationally faster, have severe draw-backs specially for band gapped systems. The short-comings arise due to the underestimation of the bandgap, as illustrated in various earlier studies.28 Linear re-sponse GW calculations29,30 within the hybrid (HSE06)functional31 comes to a rescue here, with a much bet-ter band gap predictions. HSE-GW is computationallymuch more expensive than LDA/GGA, but are foundto provide much closer comparison with experimentalfindings.32 Appropriate design of descriptors for a givenapplication also plays an extremely crucial role in anyhigh throughput computational study. This aspect isalso loosely taken up in many previous studies.11,23,24 Inthe present manuscript, we have carefully taken up thesetwo aspects and efficiently design a number of reliabledescriptors to screen three types of functional materialsi.e. thermoelectric, photovoltaic and topological quan-tum materials. These descriptors are calculated basedon the extremely accurate band gap evaluated from theHSE-GW simulation.

Here, we have performed a systematic and detailedhigh throughput calculations, based on the first princi-ples simulation, to screen I-II-V class of 8-electron halfHeusler compounds which can be promising for thermo-electric, photovoltaic and topological insulator applica-tions. A detailed electronic structure, phonon, thermo-

electric, opto-electronic and topological insulating prop-erties are simulated. Figure 1 shows the screening crite-ria which we follow for the high throughput calcualtions.Initially the thermal and chemical stability of these com-pounds was examined. Later, the thermally and chemi-cally stable compounds were shortlisted for different en-ergy applications based on their HSE-GW band gap val-ues. The parameters band inversion strength (BIS), ZTand SLME used for topological, thermoelectric and so-lar cell applications will be explained in the forthcomingsections. Such theoretical study will definitely help in

Screening Parameters

Band gap ≥ 2 eV

Thermal and Chemical Stability

Band gap between

1 - 1.8 eV

Band gap between

0 - 1.5 eV

Semi-metal/slightly metallic

Topological Insulators

Thermoelectric Solar cell Transparent Conductors

BIS < 0(21 compounds)

ZT > 0.7(13 compounds)

SLME ≥ 20%(13 compounds)

29 compounds

FIG. 1. Flowchart explaining step-by-step screening proce-dures in shortlisting the compounds for different applications.HSE-GW band gap values were used to shortlist these com-pounds.

discovering new potential materials and hence motivateexperimentalists to synthesize them and verify the theo-retical findings.

II. COMPUTATIONAL METHODS

For the DFT33 calculations, we have used Vienna Ab-initio Simulation Package (VASP)34−36 with a projectedaugmented wave basis37 and the generalized gradientapproximation (GGA)38 exchange-correlation functionalof Perdew−Burke−Ernzerhof (PBE).39 Single-shot GW(G0W0) calculations are performed with hybrid func-tional HSE0631 to get more accurate estimate of thebandgap. A kinetic energy plane wave cut-off of 500 eVwas used. The Brillouin zone sampling was done by usinga Γ-centered k-mesh.40 For all the compounds, a k-meshof 10×10×10 (for ionic relaxations) and 20×20×20 (forself-consistent-field calculations) were used for PBE cal-culations. For hybrid (HSE06)31 and G0W0 calculations,

3

a 10×10×10 k-mesh was used. In order to check band in-version for topological insulating compounds, spin-orbitcoupling (SOC) was included. Cell volume, shape, andatomic positions for all the structural configurations werefully relaxed using conjugate gradient algorithm41 untilthe energy (forces) converges upto 10−6 eV (0.001 eV/A).The tetrahedron method with Blochl correction42 wasused for accurate electronic density of states calculations.

HH compounds have an interpenetrating face-centeredcubic structure with a general composition of XYZ. Xand Y are in, general, transition metals and Z is a p-blockelement. All the compounds (XYZ) were considered tocrystallize in the F43m (#216) space group with X-atomat 4a (0,0,0), Y at 4c (0.25, 0.25, 0.25), and Z at 4b(0.5, 0.5, 0.5) Wyckoff positions. The formation energy(4EF ) was calculated as:

4EF = E(XY Z)− [E(X) + E(Y ) + E(Z)] (1)

where E(XY Z) is the total energy of the correspondingcompound and E(X), E(Y ), E(Z) are the energies ofthe constituent elements in their equilibrium phases. Anegative (positive) value of4EF signifies stability (insta-bility) of formation of the given compound with respectto its constituent elements.

The semi-classical Boltzmann transport formalism asimplemented in BoltzTraP code43 was used to calculatethe TE properties (Seebeck coefficient (S), electrical con-ductivity (σ) and electronic contribution to the ther-mal conductivity (κe)) of selected candidates. Densityfunctional perturbation theory (DFPT) combined withphonopy44 was used to obtain relevant phonon proper-ties. Further details are provided in the supplementarymaterial (SM).45 Lattice thermal conductivity is calcu-lated using the Debye-Callaway (D-C) model.46 Withinthis model, the major contribution to the lattice thermalconductivity comes from the 3-phonon scattering process(normal & Umklapp phonon scattering) and is majorlydue to the 3 acoustic modes of vibrations. The thermalcontribution due to each of the acoustic vibrational modein D-C model46,47 is given as,

κi = CiT3/3

θi/T∫0

τ ic(x)x4ex

(ex − 1)2 dx+

[∫ θi/T0

τ ic(x)x

4ex

τ iN (ex−1)2 dx

]2∫ θi/T0

τ ic(x)x

4ex

τ iNτ

iU (ex−1)2 dx

(2)

where x = (~ω/kBT ) and the index i denotes LA, TAand TA′ modes of vibration. The constant Ci is givenby,

Ci =k4B

2π2~3νi(3)

The scattering rates corresponding to the normal andUmklapp scattering processes are,

1

τLAN (x)=k3Bγ

2LAV

M~2ν5LA

(kB~

)2

x2T 5 (4)

1

τTA/TA′

N (x)=k4Bγ

2TA/TA′V

M~3ν5TA/TA′

(kB~

)xT 5 (5)

1

τ iU (x)=

~γ2

Mν2i θi

(kB~

)2

x2T 3e−θi/3T (6)

where, kB is the Boltzmann’s constant, ~ is the reducedPlank’s constant, T is the temperature, θ is the Debyetemperature, V and M are the volume and mass per atomrespectively, γ is the mode gruneisen parameter, ν is ve-locity and ω is the angular frequency. In addition tolattice and electronic contribution to the thermal conduc-tivity, we have also calculated the bipolar components ofthermal conductivity (κb) which often becomes impor-tant at relatively higher temperature. Further detailsabout the formulation of κb is provided in SM.45

The optical properties were simulated from the lin-ear response calculations as implemented in VASP. Theoptical transition probabilities were calculated from thedipole-dipole transition matrix elements. The indepen-dent particle approximation was used to obtain frequencydependent dielectric function. All the other optical prop-erties are then obtained from real and imaginary part ofthe dielectric function. The band structure was obtainedby using PBE exchange correlation functional. Becausethe optical properties are extremely sensitive to the bandgap of the concerned material, a more accurate HSE-GW calculations were performed to estimate the same.SLME is the maximum solar power conversion efficiencywhich incorporates the nature of the bandgap and ab-sorption coefficient for a particular compound.48−50 De-tails of SLME formulation can be found in our earlierarticle.51 SLME51,52 was used as a figure of merit for thephotovoltaic applications.

III. RESULTS AND DISCUSSION

Figure 2 highlights the elements from I, II and V groupof periodic table chosen for our high throughput study.The elements from VB group were not taken since forXYZ alloys to form, Z should be a p-block element.

This gives us a total of 320 compounds to be studied.Furthermore, there are 3 unique possibilities for site pref-erence of X, Y and Z elements at the three Wyckoff posi-tions (4a, 4b and 4c) in a given half Heusler structure (i.e.XYZ, XZY and YZX). This makes a total of 960 (320×3)simulations to be carried out. As illustrated before, de-pending on the concerned applications, the screening pa-rameters used in the present study are formation energy(chemical stability), phonon dispersion (thermal stabil-ity), accurate bandgap values (PBE + HSE06 + GW),nature of bandgap (direct/indirect), thermoelectric figureof merit(ZT), band inversion strength (BIS) and SLME.

4

I Li, Na, K, Rb, Cs, Cu, Ag, Au

II Be, Mg, Ca, Sr, Ba, Zn, Cd, Hg

V N, P, As, Sb, Bi

FIG. 2. Periodic table of elements showing I-II-V class ofelements (highlighted) used in the present high throughputstudy. The figure (on the top) shows the primitive unit cellof Half-Heusler compounds with one possible configuration ofX, Y and Z elements.

A. Initial Screening Parameters

1. Stability

In this section, we have examined the chemical andthermal stability of all the compounds. For every com-pound, we simulated the total energy for the three dis-tinct structural configurations (XYZ, XZY and YZX),and chose the energetically most stable one. As such, weshortlisted 320 unique compounds out of 960.

Figure 3 shows the chemical and thermal stability ofthese 320 unique compounds. Here, negative (–) signindicates those compounds which are chemically unsta-ble (+ve formation energy), positive (+) sign indicatesthose which are chemically stable (-ve formation energy)but thermally unstable (imaginary phonon frequencies),while tick (X) sign indicates those which are both chem-ically as well as thermally stable. 192/320 compoundswere found to have negative formation energy. Phonondispersion for all of these 192 compounds are then sim-ulated. 71/192 compounds were found to show imagi-nary phonon frequencies. Hence, a total of 121/960 com-pounds were found to be chemically as well as thermallystable, which are then chosen for further screening. Thenumerical values of formation energies and the possibilityof the thermal stability for all these 192 compounds areshown in Table SI of SM.45

2. Band gap

Band gap was used as the next screening parameter tocategorize the shortlisted 121 compounds into specific ap-plications. Band gaps are initially evaluated using PBEexchange correlation functional. PBE functional, how-ever, is traditionally known to underestimate the band

gap values. As such, compounds having bandgap be-tween 0-1.5 eV or with negligibly small density of statesat Fermi level (within PBE) were further chosen for HSE-GW calculations to get an accurate estimate of the band-gap. 21 compounds were found to be highly metallicwithin PBE simulations, and are not expected to open agap under HSE-GW calculations, hence neglected.

A total of 31 compounds with HSE-GW bandgap lessthan 1.5 eV were chosen for thermoelectric studies. 30compounds were found to show band gap between 1.0-1.8 eV which are suitable for optical absorption andhence were shortlisted for performing further simula-tion for photovoltaic application. Figure 4(a) showsthe simulated band gaps and the optimized lattice pa-rameters for 30 compounds with band gap between 1.0-1.8 eV, found suitable for solar cell application. Thisalso includes 19 common compounds with band gap 1-1.5 eV that are suitable for both solar and thermo-electric application. Figure 4(b) shows the simulatedband gaps and the optimized lattice parameters for thecompounds suitable for only thermoelectric application(HSE-GW band gap< 1.0 eV). One can easily notice thatPBE functional significantly underestimates the bandgap. In addition, 22 compounds were found to be semi-metallic/slightly/weakly metallic, which may serve as po-tential candidates for topological insulator and hence fur-ther investigated.

29 compounds were found to have bandgap above 2eV (within PBE) which is a suitable band gap range fordiscovering transparent conductors. Although, we havenot performed detailed calculations for compounds hav-ing bandgap greater than 2 eV, these candidates can bepromoted for transparent conductor applications withfurther band gap engineering. The list of these com-pounds are given in Table SIV of SM.45 Some of theseclass of compounds have already been synthesized exper-imentally in cubic phase. Table SII and SIII of SM45

show a comparison of the experimental lattice parame-ter (aexp) and band gap with our theoretically optimizedvalues for those compounds which are synthesized exper-imentally. Our simulated values matches fairly well withthe experimentally reported ones. This sets a benchmarkand signifies the accuracy of our calculations.

After categorizing the pool of compounds for three dif-ferent applications, we then performed detailed calcula-tions for each of them. We will now show the simulatedresults for each applications and hence screen the mostpromising candidates out of them.

B. Thermoelectric application

Thermoelectric properties of 31 compounds with bandgap < 1.5 eV were simulated. D-C model was usedto calculate the lattice thermal conductivity, which isa reasonable approximation for κL in high temperaturerange. The accuracy of D-C model and cross-validationwith experimental results can be found in our previous

5

Li Be

Li Mg

Li Ca

Li Sr

Li Ba

Li Zn

Li Cd

Li Hg

Na Be

Na Mg

Na Ca

Na Sr

Na Ba

Na Zn

Na Cd

Na Hg

N P As Sb Bi N P As Sb Bi

K Be

K Mg

K Cd

K Sr

K Ba

K Zn

K Ca

K Hg

Rb Be

Rb Mg

Rb Ca

Rb Sr

Rb Ba

Rb Zn

Rb Cd

Rb Hg

Cs Be

Cs Mg

Cs Ca

Cs Sr

Cs Ba

Cs Zn

Cs Cd

Ag Be

Cu Be

Cu Mg

Cu Ca

Cu Sr

Cu Ba

Cu Zn

Cu Cd

Cu Hg

N P As Sb Bi N P As Sb Bi

Cs Hg

Ag Mg

Ag Ca

Ag Sr

Ag Ba

Ag Zn

Ag Cd

Ag Hg

Au Be

Au Mg

Au Ca

Au Sr

Au Ba

Au Zn

Au Cd

Au Hg

I-II-V

FIG. 3. Chemical and thermal stability of all possible I-II-V class of 8 electron half-Heusler compounds XY Z obtained bytaking the most stable configuration out of the three possibilities (XY Z, Y XZ, and XZY for every compounds). Negative (–)sign indicate those compounds which are chemically unstable, positive (+) sign shows compounds which are chemically stablebut thermally unstable (imaginary phonon frequencies), while tick (X) sign indices compounds which are chemically as well asthermally stable.

report.47 13 compounds were found to have figure ofmerit, ZT > 0.7 for both n-type and p-type conduction.Table I represent the Seebeck coefficient (S), power fac-tor (S2σ), ZT, and optimal carrier concentration (N) forthe compounds having ZT > 0.7 for both n- and p-typeconduction. The optimal concentration (N) is the carrierconcentration at which ZT is maximum in the given tem-perature range. Similar data for all the 31 compoundsare shown in Table SV of SM.45

In the next section, we shall choose one of the promis-ing candidate material AgPMg and discuss its TE prop-erties in some details.

1. Electronic Structure:

AgPMg is found to be both chemically (4EF=-368.5meV/atom) as well as thermally stable with an optimized

cubic lattice constant 6.14 A. The GW band gap is ∼ 0.59eV, which is further used in TE calculations.

Figure 5 shows the atom projected electronic bandstructure and the density of states for AgPMg. The va-lence band edges are mostly composed of Ag while con-duction band has contribution from all the three atoms.The valence band edges are relatively more flat than theconduction bands, pointing towards higher hole effectivemass. This, in turn, yields a higher Seebeck coefficientsfor p-type conduction as compared to n-type for a givencarrier concentration.

2. Phonon Properties:

Figure 6(a) shows the phonon dispersion for AgPMgalong high symmetry directions. The acoustic and op-tical modes of vibration are shown by green and black

6

0

0.2

0.4

0.6

0.8

1

Ba

nd

ga

p (

eV

)

PBEHSEGW

5.6

5.8

6

6.2

6.4

6.6

6.8

7

7.2

7.4L

att

ice

co

ns

tan

t (Å

)

Lattice Constant

Ba

LiB

i

LiA

sC

d

Ag

As

Ca

Ca

Ag

Sb

Ag

PM

g

Ca

Cu

Sb

Na

PC

d

Na

As

Cd

Ag

As

Mg

Na

As

Zn

Na

Sb

Cd

Na

BiM

g

0

0.5

1

1.5

2B

an

d g

ap

(e

V)

PBEHSEGW

5

6

7

8

9

La

ttic

e c

on

sta

nt

(Å)

Lattice Constant B

aL

iSb

CsP

Ba

Cu

PM

g

KS

bM

gK

BiC

aL

iAsZ

nL

iBeB

iL

iBiM

g

KA

sB

a

LiP

Cd

LiS

bC

dL

iZn

Sb

NaA

sB

aN

aB

iBa

NaB

iCa

NaP

Zn

Rb

AsB

aR

bA

sC

aR

bB

iBa

Rb

BiC

aR

bP

Ba

Rb

Sb

Ba

KB

iBa

KP

Ba

CaC

uA

sC

sB

iCa

Ag

PC

aN

aZ

nS

bC

sB

iBa

Cu

PB

e

(a) Solar + Thermoelectric

(b) Thermoelectric

FIG. 4. Band gap data calculated at the level of PBE(square; red), HSE06 (triangle down; magenta) and HSE06+ GW (circle; green) functionals. The absence of data forany given compound corresponding to a particular exchangecorrelation functional implies its metallic behaviour. Opti-mized lattice constants calculated at the PBE level are shownby star (×) symbol. Top (bottom) panel shows compoundswith HSE06 + GW band gaps in the range 1.0-1.8 eV (<1 eV). 30 compounds are found with band gap in the range1.0-1.8 eV, suitable for solar application, out of which 19 com-pounds with band gap 1-1.5 eV are suitable for both solar andthermoelectric application. 12 compounds with band gap lessthan 1 eV are also chosen for purely thermoelectric studies.

lines, respectively. Since, the primitive unit cell has threeatoms, there are a total of 9 phonon branches (3 acousticsand 6 optical). The acoustic modes are further classifiedinto one longitudinal (LA) and two transverse acousticmodes (TA and TA′). The thermal conduction is mainlydue to the acoustic modes of vibration. Notably, the twoTA and TA′ bands are degenerate along Γ-X and X-W di-

S.No. Compound Doping S S2σ ZTmax N T

1 BaLiBi n 373.3 2.49 0.70 2.4 900p 386.0 0.96 1.01 0.9 900

2 CaAgSb n 323.3 3.85 0.99 5.2 900p 330.5 1.49 0.56 1.3 900

3 AgPMg n 135.7 4.79 0.76 26.5 900p 337.1 3.88 1.56 5.4 900

4 AgAsMg n 174.2 5.43 0.70 13.3 900p 197.2 2.74 0.93 7.6 900

5 NaSbCd n 147.3 3.81 0.78 12.2 900p 306.1 4.39 1.38 8.2 900

6 BaLiSb n 384.6 2.09 1.01 1.9 900p 425.6 0.83 0.74 0.7 900

7 LiSbCd n 177.2 5.11 0.91 12.8 900p 308.9 3.77 1.15 4.6 900

8 KPBa n 202.27 6.13 0.74 44.94 900p 364.46 3.34 1.03 19.14 660

9 RbPBa n 183.1 5.67 0.73 47.9 900p 390.1 2.45 1.37 10.8 520

10 CuPMg n 246.4 8.01 0.88 18.3 900p 392.2 1.75 1.04 1.9 620

11 CuPBe n 409.5 4.39 0.93 5.0 900p 442.9 2.64 0.83 1.8 900

12 NaZnSb n 207.1 4.41 0.77 5.2 900p 400.1 2.09 1.04 1.7 880

13 CaCuSb n 317.3 3.77 0.92 5.2 900p 287.9 1.85 0.59 1.9 900

TABLE I. Selected candidate materials whose maximum ZTvalue is >0.7 for both n- and p-type conduction. Listed arethe maximum ZT values, and the corresponding Seebeck coef-ficient (S, µVK−1 ), Power factor (S2σ, mWm−1K−2), carrierconcentration (N, ×1019cm−3) and the temperature (T, K).

-2

0

2

4

6

8

E -

EF (

eV

)

0 1 2 3DOS (states/eV)

-2

0

2

4

6

8

Ag

PMg

W L Γ X W

Eg = 0.59 eV

FIG. 5. Atom projected electronic band structure and densityof states (Fermi level at 0 eV) for AgPMg.

rections. The velocity of each acoustic mode (ν) is givenby the slope of the band corresponding to the vibrationalmode at the Γ point. Debye temperature (θ) can be ob-tained from the maximum frequency corresponding tothe vibrational mode. In addition to ν and θ, the D-Cmodel for the lattice thermal conductivity also requires

7

W L K Γ X0

2

4

6

8

Freq

uen

cy,

ν (

TH

z)

LA

TA’

TA

W

0 2 4 6 8Frequency, ν (THz)

0

1

2

3

4

Gru

neis

en

para

mete

r, γ

TA’TALA

300 400 500 600 700 800 900T (K)

0

0.5

1

1.5

2

κ (

Wm

-1K

-1)

n typep type

κL

κb

κL + κ

b

(b)

(c)

(a)

FIG. 6. For AgPMg, (a) phonon dispersion,(b) variation ofmode Gruneisen parameter γi with phonon frequency, (c)temperature dependence of lattice and bipolar thermal con-ductivity (for n and p type carriers).

the mode Gruneisen parameters (γi). Figure 6(b) showsthe phonon frequency dependence of Gruneisen parame-ters. The maximum value of γi for each vibrational modeis taken for the calculation of lattice thermal conductiv-ity in the D-C model. The numerical values of variousparameters used to calculate (kL) for AgPMg are; νLA =5170.02 m/s, νTA = 2586.92 m/s, νTA′ = 2296.99 m/s,γLA = 3.35, γTA = 2.38,γTA′ = 2.16, γ=3.35, θLA =110.73 K, θTA = 96.44 K, θTA′ = 94.06 K, V = 19.24 ×10−30 m3, and M = 54.38 × 10−27 kg.

The simulated values of kL vary between 1.12 to 0.18W m−1K−1 in the temperature range 300-900 K asshown in fig. 6(c). At sufficiently high temperatures,the effect of minority charge carriers become significantwhich generates electron-hole pairs and the excitation ofelectrons across the band gap. The thermal transport dueto such a process introduces an additional component toκ, called the bipolar thermal conductivity(κb).

53 Figure6(c) shows the temperature dependence of κb for bothn-type and p-type carriers. κb plays an important rolein the thermal conductivity beyond ∼ 480 K, as evidentfrom Fig. 6(c).

3. Thermoelectric Properties:

Figure 7 shows the chemical potential (µ) and tem-perature dependence of the simulated Seebeck coefficient(S), power factor (S2σ), and ZT for both n- and p-typeconduction for AgPMg. The temperature dependent TEproperties were calculated for a carrier concentration atwhich ZT attains its maxmum value. Thes value ofthese optimized concentrations are 2.65 × 1020 cm−3

and 5.42 × 1019 cm−3 for n- and p- type conduction,respectively. At these carrier concentrations, the cal-culated Seebeck coefficient and power factor for n-typeconduction are 135.69 µVK−1 and 4.79 mWm−1K−2,whereas for p-type conduction are 337.10 µVK−1 and3.88 mWm−1K−2, respectively. This gives a significantlyhigh value of ZT, 0.76 and 1.56 for n-type and p-type

-500

0

500

1000

0

100

200

300

400

S (

µV

K-1

)

0

2

4

6

8

0

1

2

3

4

5

S2σ

(m

Wm

-1K

-2)

n type

p type

-1 -0.5 0 0.5 1µ (eV)

0

0.5

1

1.5

300 K

500 K

700 K

900 K

300 400 500 600 700 800 900T (K)

0

0.5

1

1.5

ZT

FIG. 7. Variation of important TE properties (S, S2σ andZT) with chemical potential (µ) (left) and temperature (right)for AgPMg. The temperature dependent TE properties werecalculated at a carrier concentration of 2.65 × 1020 cm−3 and5.42 × 1019 cm−3 for n- and p- type carrier, respectively.

300 400 500 600 700 800 900T (K)

0

0.5

1

1.5

2κ (

Wm

-1K

-1)

κL

κe

κb

κe+κ

L

κe+κ

L+κ

b

AgPMg (p-type)

kL

ke

ke + k

L

kb

ke + k

L + k

b

FIG. 8. Temperature dependence of lattice (κL), electronic(κe) and bipolar (κb) thermal conductivity for p-type AgPMg

conduction, respectively at 900 K.To illustrate further, we show, in Figure 8, various

components of the thermal conductivity with tempera-ture for p-type conduction. The thermoelectric figure ofmerit is defined as:

ZT =S2σ

(κe + κL + κb)T (7)

Since the total thermal conductivity (κ = κe+κL+κb) isquite small (2.28 W m−1K−1) at 900 K (as evident fromFig. 8), the ZT value is reasonably high (1.56) for p-typeconduction. ZrNiSn has been shown to be a state of theart material in the HH family for thermoelectric applica-tions. With some selective doping, Ti0.5Zr0.25Hf0.25NiSn,this compound has been proved to be highly promising,with a ZT value of 1.5 at 820 K.54 In comparison tothis state-of-art material, our proposed compound when

8

1 1.5 2 2.5 3 3.5 4 4.5 5Photon Energy (eV)

103

104

105

Ab

so

rpti

on

co

eff

icie

nt,

α (

cm

-1)

MAPbI3CaCuAsCsPBaCuPMg

KBiCaKSbMg

LiBiMg

LiPCdLiSbCdLiZnSbNaBiCaNaPZnAgPCa

NaZnSb

0 1 2 3 4 5Thickness (µm)

0

10

20

30

SL

ME

, η

(%

)

MAPbI3

CsPBa

CuPMg

KSbMg

KBiCa

LiBiMg

LiPCd

LiSbCd

LiZnSb

NaBiCa

NaPZn

CaCuAs

NaZnSb

AgPCa

(b.)

(a.)

FIG. 9. (a) Absorption coefficient vs incident photon en-ergy and (b) ”Spectroscopic Limited Maximum Efficiency(SLME)” vs film thickness for compounds having SLMEgreater than 20%.

lightly doped, exhibits a high thermoelectric figure ofmerit (ZT∼ 2), making them even better for TE appli-cations. Electronic band structure, phonon dispersion,Gruneisen parameters and temperature dependence ofTE properties (S, S2σ and ZT) for the rest of 30 com-pounds are shown in SM45(Fig SI to Fig SXXX).

C. Solar cell application

The compounds having band gap in the range 1-1.8eV were investigated for solar cell application. First, wesimulated the optical transition probabilities for direct

-4

-2

0

2

4

6

8

En

erg

y (

eV

)

NaZnSb

0

100

200

p2 (

a.u

.)

ΓLW X WWave vector (k)

1 1.5 2 2.5 3 3.5 4 4.5 5Photon Energy (eV)

103

104

105

Ab

so

rp

tio

n c

oeff

icie

nt,

α (

cm

-1)

MAPbI3NaZnSb

0 1 2 3 4 5Thickness (µm)

10

20

30

SL

ME

η (

%)

MAPbI3NaZnSb

(c)(b)

(a)

FIG. 10. For NaZnSb, (a) Band structure(top) and tran-sition probability (bottom) for NaZnSb. Here green circlesshow contribution from Na atoms, while red and blue squaresrepresent contribution from Zn and Sb atoms respectively(b) Absorption coefficient vs incident photon energy and (c)”Spectroscopic Limited Maximum Efficiency (SLME)” vs filmthickness.

transitions from valence band maxima (VBM) to conduc-tion band minima (CBM) for all these compounds. Thetransition probabilities are estimated by calculating thesquare of dipole transition matrix elements. Using this,the allowed direct bandgap for all these compounds wascalculated. The compounds having a difference less than0.3 eV between optically allowed direct band gap (Edag )and the electronic band gap (Eg) were finally shortlistedfor further calculations. Out of 30 compounds with elec-tronic band gap between 1-1.8 eV, 18 compounds werefound to have a difference, (Edag -Eg) < 0.3 eV. We thencalculated the next relevant properties, namely the ab-sorption coefficient (α) and SLME50−52 for these 18 com-pounds. The absorption coefficient for a material showsthe depth to which a photon with a particular wavelengthcan penetrate into the material before it gets absorbed.For a given material, calculation of SLME needs specificinputs like the nature/magnitude of bandgap, frequencydependent optical absorption spectra etc.

SLME and GW band gap data for all the 18 com-pounds is listed in Table SVI of SM.45 13 compoundswere found to have SLME greater than 20%. Figure9(a) shows the frequency dependent absorption coeffi-cient for all the compounds having SLME >20%. WhileFig. 9(b) shows our calculated SLME vs film thicknesscompared with the state of the art material CH3NH3PbI3(MAPbI3).

Interestingly, NaZnSb turns out to be the most promis-ing candidate for solar cell application with SLME ∼ 31%at 2µm film thickness. Figure 10(a) shows the atom pro-jected band structure and transition probability (p2) forNaZnSb for which HSE - GW band gap and the opti-mized lattice constant are 1.25 eV and 6.59 A respec-tively. An accurate prediction of band gap is extremelyimportant for the calculation of SLME. As no significantchange was observed in the band topology with respect tothe exchange correlation functional, we have used PBEband structure with bandgap value shifted to GW val-ues for the calculation of absorption coefficients. Figure10(a) (Figure 10(b)) shows a comparison of the opticalabsorption coefficient (SLME) for NaZnSb and MAPbI3.

9

One can notice that, absorption coefficient of NaZnSb isoverall larger compared to those of MAPbI3. SLME forthe proposed compound is also much higher, approach-ing the ShockleyQueisser limit, as compared to MAPbI3.Also, this value is comparable to the efficiency potentialof the GaAs/Si dual-junction solar cell which has been re-ported to be 31.7% in previous study.55 Electronic bandstructure, transition probability and phonon dispersionfor the rest of 12 compounds are shown in SM45(see Fig.SXXX to SXLIV).

D. Topological Insulators

5.7 6.0 6.3 6.6 6.9 7.2 7.5aopt (Å)

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

BIS

(E

Γ6 -

EΓ

8) (e

V) NaNCd

MgCuAsLiPHg

NaPHg

AuAsCaCaCuBi

CaAuSb

MgAgSb

LiAsHg

LiZnBi

NaAsHg

LiBiCdCaAgBi

NaBiCd

RbMgBi

BaAgBi

CaAuBi

KZnBiKCdBi

BaAuBi

LiHgSb

FIG. 11. The band inversion strength, BIS (EΓ6 -EΓ8) vs op-timized lattice constants (aopt) for all the topologically non-trivial compounds in the 8-electron HH family.

Topological quantum materials are the other class ofpotential materials which often find home in half Heusleralloys. These materials are usually zero (small) bandgap systems with surface protected bulk band inversion.While screening the 8-electron half Heusler compounds,we found several of them having semi-metallic and/orweakly metallic behavior where the conduction bandsget mixed with fewer valence bands (within the PBEfunctional). Topological insulators have non-trivial bandtopology induced by spin-orbit coupling which requiresinverted band order between s-like (Γ6 band) and p-like (Γ8 band) orbitals at the Γ point. Band Inversionstrength (BIS) is a measure of the non triviality whichis defined as the difference between the energy of s-like(Γ6 band) and p-like (Γ8 band) bands at the Γ point.The negative value of BIS implies the non-trivial bandorder. Because the band topology and the magnitude ofBIS is extremely sensitive to the computational methodused in a given study, we have performed HSE-SOC bandstructure calculations for all those compounds which aresemimetallic or weakly metallic at the PBE level. Figure11 shows the BIS and optimized lattice parameters of21 compounds which display non-trivial band topology

at ambient conditions. It should also be noted that, forcompounds having small positive BIS values (e.g. Mg-CuSb having BIS value 0.37), the band inversion can beinduced by applying small perturbation such as pressure,strain, or doping.

-1.5

-1

-0.5

0

0.5

1

1.5

E(k

)-E

F (

eV

)

s-orbitalp-orbital

W L Γ X W

NaNCd

FIG. 12. HSE06+SOC bulk bandstructure of NaNCd at am-bient lattice parameter (a = 5.68 A). Red circles and greensquares represent the s and p orbital character respectively.

-0.8

-0.4

0

0.4

0.8

E(k

)-E

F (

eV

)

Γ M K Γ

Na

N

Cd

FIG. 13. (Left)Surface bands for NaNCd (red circles andgreen squares represent the contribution of surface atoms fromthe bottom three (X-Y-Z) and top three (Y-Z-X) atomic lay-ers, respectively). (Right) unit cell for (111) surface with Natermination at the top and Cd at the bottom.

For TIs, surface states play a crucial role in further con-firming the topological behavior of the material. Here, wechose a representative system NaNCd with BIS value -0.64 to further study the surface properties. Figure 12shows the bulk band structure of NaNCd at optimizedlattice constant (a = 5.68 A) where red circles and greensquares represent s and p-orbital contributions, respec-tively. As seen from the figure, p-like bands have four-folddegeneracy and s-like states lie below them. The bandinversion is just the necessary condition whereas the ro-bustness of conducting states is a sufficient condition for

10

an acceptable TI.56,57

HH compounds have a layered structure along the [111]direction. Thus, the most naturally cleavable surfaceis along the [111] plane. Hence, for investigating sur-face bands, a surface slab along [111] direction was con-structed for NaNCd at the optimized lattice parameter.A 36 atomic-layer thick slab (38.02A), with Na atom ter-mination at top and Cd termination at the bottom sur-face, is shown in Fig. 13. A vacuum of 15 A was used tomake sure there is no interaction between the repeatingsurface layers due to the periodic boundary conditions.Five atomic layers on each of the top and bottom sides ofthe slab was fully relaxed, keeping the remaining middlelayers fixed, such that force (energy) is converged up to0.001 eV/A (10−6 eV).

Slabs Surface energy(eV/A2)

Na (top surface) 0.028and Cd(bottom surface) terminated

N (top surface) 0.037and Cd(bottom surface) terminated

Both side Cd terminated 0.1352

TABLE II. Surface energies of various terminated surfaces forsurface slab of NaNCd along [111] direction.

Table II lists the surface energies of the slab with dif-ferent termination layers. The slab with Na and Cd ter-mination at the top and bottom surfaces respectively isfound to be energetically the most favorable surface andhence our choice for further simulations. The surfaceband structure for this slab is shown in Fig. 13. The sizeof red circles and green squares represent the weightedcontributions of surface atoms from bottom three andtop three consecutive atomic layers, respectively. Clearlythe surface bands show metallic character and hence con-

ducting in nature. HSE-SOC structures showing theband inversion, for the rest of 20 compounds are shownin SM45 (see Fig. SXLV to SLIV).

IV. CONCLUSION

To summarize, our work is focused on high through-put calculation of I-II-V class of compounds for searchingpromising materials for renewable energy applications.121 out of 960 compounds were found to be both chem-ically and thermally stable. An accurate HSE-GW levelband gap calculation lists out 31 compounds for thermo-electric application having band gap in the range 0-1.5eV. 30 compounds having band gap in the range 1-1.8 eVwere selected to study solar harvesting problem. 22 com-pounds which were metals/semi-metals were checked forband inversion in presence of spin-orbit coupling for iden-tifying potential topological insulators. 13 compoundswere found to have thermoelectric figure of merit (ZTvalue) greater than equal to 0.7. 13 compounds showexcellent optoelectronic properties with SLME greaterthan 20%. 21 compounds were found to show band inver-sion for which robustness of surface conducting states waschecked to make them qualify as strong topological insu-lators. The properties of proposed compounds from ourcalculations are found to be comparable to the state-of-art materials. We believe our detailed high throughputinvestigation and the accurate shortlisting of potentialmaterials will be of timely interest to the readers andshould be helpful in recent pursuit of novel functionalmaterials for renewable energy applications. We stronglysuggest experimentalists to synthesize these shortlistedcompounds for corresponding applications and verify ourclaim.

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ACKNOWLEDGMENT

Bhawna acknowledges financial support from IndianInstitute of Technology, Bombay in the form of teach-ing assistantship. AA acknowledges National Centre forPhotovoltaic Research and Education (NCPRE) (finan-cially supported by Ministry of new renewable energy(MNRE), Government of India) to support this research.