Chathurangi Kumarasinghe and Neophytos NeophytouSchool of Engineering, University of Warwick, Coventry, U.K.

Band convergence of half-Heuslers for a high thermoelectric power factor

Direct conversion of temperaturedifferences to electric voltage andvice versa.

Thermoelectricity -basics

Thermoelectric figure of merit

2

e l

S TZT sk k

=+

Seebeckcoefficient

Lattice thermal conductivity

Electronic thermal conductivity

Electrical conductivity

HOT COLD

ee

heat

2

Reducing the thermal conductivity (!" + !$)Ø Hierarchical architecturesØ Phonon engineering

2

e l

S TZT sk k

=+

Methods of improving ZT

Vineis, Adv. Mat., 2010

Li-Dong et al, Energy Environ. Sci., 2014

Increase the power factor (σ&')Ø Bandstructure engineering

-Band aligning-Modify band masses-Resonant doping

Biswas et al, Nature, 2012.(p-type PbTe)

Poehler et al, Energy Environ. Sci., 2012 3

Outline

Ø Band convergenceØ Introduction to transport equationsØ Scattering mechanisms : parabolic band approximation

Ø Introduction to half-Heuslers

Ø Modeling half-Heuslers : non parabolic band approximationØ Non parabolic band approximationØ Band aligning – NbCoSn,TiCoSb

Ø Aligning bands of half-Heuslers with strainØ NbCoSnØ TiCoSbØ ZrCoSb

Ø Conclusions 4

Band aligning

Conductivity

+ ,, ./ =1

2,3 4 ,, ./∫ Ξ78 . (. − ./) −

<

<.)= (., ./, , >.

4 ,, ./ =1

3? Ξ . −

<

<.)= (., ./, , >.

Seebeck coefEicient

Ξ . =FG2HIG . JG

H . DOSG (.)5

Transport distribution function

Power Factor = σ+H

• Increases number of carrier available for transport.• Also increase the scattering.• Depends on the nature of the bands.

VB0

Intra valley

Inter valley

• Constant rate of scattering (most commonly used)

• Scattering ∝ DOS% (Intra-valley scattering)

• Scattering ∝ ∑DOS% (Inter- and intra-valley scattering)

Ξ() * ∝+%

*,%

Ξ() * ∝+%,%-. *

/.

Ξ() * ∝∑% ,%

-. *

/.

∑%,%/. *

-.

Scattering mechanisms

0(*) = Constant

0 * % ∝1

DOS%(*)

0 * = 1;DOS%(*)

6

Case study: Constant scattering rate

!"=1, !# = 0.5Ξ%& ' ∝)*!*"# '

+#

-0.2 -0.1 0 0.1 0.2EF-EV0(eV)

0

0.5

1

1.5

PF[m

Wm

-1K-2

] 025810

E (kBT)

-0.2 -0.1 0 0.1 0.2EF-EV0(eV)

0

0.5

1

1.5

PF[m

Wm

-1K-2

] 025810

E (kBT)increaseincrease

7

Heavier masses results in a larger transport distribution, larger conductivity and a better PF.

Aligning any band will improve the transport

distribution.

!"=1, !# = 0.5 !"=1, !# = 2

!"=1, !# = 2

VB0 VB0

Case study: Scattering ∝ DOS% (Intra-valley scattering only)

-0.2 -0.1 0 0.1 0.2EF-EV0(eV)

0

0.5

1

1.5

PF[m

Wm

-1K-2

] 025810

E (kBT)

Ξ'( ) ∝*%

)+%

-0.2 -0.1 0 0.1 0.2EF-EV0(eV)

0

0.5

1

1.5

PF[m

Wm

-1K-2

] 025810

E (kBT)increase increase

Lighter masses results in a a larger transport distribution, larger conductivity and a better PF.

Aligning any band will improve the transport

distribution.8

+,=1, +- = 0.5 +,=1, +- = 2

+,=1, +- = 0.5 +,=1, +- = 2

VB0VB0

Case study: Scattering ∝ ∑DOS& (Inter- and intra-valley scattering)

Ξ() * ∝∑& +&

,- *

.-

∑& +&.- *

,-

/0 >/2

32 +/3

32

/202 +/3

02

+,=1, +- = 0.5

+,=1, +- = 2

-0.2 -0.1 0 0.1 0.2EF-EV0(eV)

0

0.5

1

1.5

PF[m

Wm

-1K

-2] 0

25810

E (kBT)

-0.2 -0.1 0 0.1 0.2EF-EV0(eV)

0

0.5

1

1.5PF

[mW

m-1

K-2

] 025810

E (kBT)

increase

decrease

In the case of 2 bands,

Aligning any band will NOT improve the

transport distribution

/0 > /2

In the case of 3 bands,

9

VB0

VB0

Case study: Scattering ∝ ∑DOS& (Inter- and intra-valley scattering)

10

'(=1, ') = 0.5 '(=1, ') = 2VB0 VB0

Summary of conditions

Under a constant scattering rateØ Any band will improve the powerfactor.Ø Improvement is better with heavier masses.

When Scattering ∝ DOS% (Intra-valley scattering only)Ø Any band will improve the powerfactor.Ø Improvement is better with lighter masses.

When Scattering ∝ DOS% (Intra- and inter- valley scattering)

Ø Only specific masses improve the powerfactor.Ø In the case 2 bands, aligning mass has to be

lighter than the existing one.11

1

1

Outline

Ø Band convergenceØ Introduction to transport equationsØ Scattering mechanisms : parabolic band approximation

Ø Introduction to half-Heuslers

Ø Modeling half-Heuslers : non parabolic band approximationØ Non parabolic band approximationØ Band aligning – NbCoSn,TiCoSb

Ø Aligning bands of half-Heuslers with strainØ NbCoSnØ TiCoSbØ ZrCoSb

Ø Conclusions 12

Why half-Heuslers?

• Complex band structure offering a high band degeneracy, multiple valleys contributing to conduction

• XYZ form, where X and Y are transition metals and Z is in the p-block.

• Many combinations of X,Y and Z

Relatively high thermoelectric performance combined with Ø relatively inexpensive elemental composition Ø robust mechanical properties.Ø high temperature stability

Ti Co Sb

TiCoSb Unit cell

Ti Co Sb

Ti Co Sb

Ti Co Sb

13

Much work focuses on the reduction of the thermal conductivity• nanocomposites • grain size• point defects• alloying for large mass contrast

Graf et.al, Prog. Solid State Chem. (2011).Fu et.al, Nature Comm. (2015).

Half –Heuslers : Current state of research

14

Co based half-Heuslers :TiCoSb, NbCoSn, ZrCoSb

ZrCoSbNbCoSn

Potential for improving power factor through band aligning

Band aligning techniquesØ StrainØ AlloyingØ Second phasing

TiCoSb

15

Outline

Ø Scattering mechanisms : parabolic band approximationØ Introduction to transport equationsØ Case study with two bands

Ø Introduction to half-Heuslers

Ø Modeling half-Heuslers : non parabolic band approximationØ Non parabolic band approximationØ Band aligning – NbCoSn,TiCoSb

Ø Aligning bands of half-Heuslers with strainØ NbCoSnØ TiCoSbØ ZrCoSb

Ø Conclusions 16

Non parabolic band approximation

! 1 + $! = ℏ'('2*

+,- = *.'

/'ℏ. 0 2! 1 + $! 1 + 2$!

1 = 2! 1 + $! − !0*

11 + 2$!

Ξ ! =567'86 ! 16' ! DOS6 (!)

17Fermi surface below 0.1eV of valance band edge of NbCoSn

Bandstructure of of NbCoSn

Ø Align valleys at X point with valence band edge

Ø 2 bands with similar masses

Δ" = 12.24k)Tbands 4,5

band 2band 3

band 1

NbCoSn :Non parabolic band approximation (NPBA)

18

Numerical calculation(Boltztrap)

Non parabolicapproximation

NbCoSn – Aligning X valley

19

TiCoSb – Aligning L valley

Reasonable match between NPBA and full

band coalculations.

Aligning 2 bands with two different masses.

20

TiCoSb

21

TiCoSb – Aligning L valley

Outline

Ø Scattering mechanisms : parabolic band approximationØ Thermoelectric Transport theoryØ Case study with two bands

Ø Introduction to half-Heuslers

Ø Modeling half-Heuslers : non parabolic band approximationØ Non parabolic band approximationØ Band aligning – NbCoSn,TiCoSb

Ø Aligning bands of half-Heuslers with strainØ NbCoSnØ TiCoSbØ ZrCoSb

Ø Conclusions 22

NbCoSn bandstructure with strain

Compression can align the bands at the X point. Masses reduce with compression.

23

NbCoSn thermoelectric performance with strain

24

TiCoSb – Strain bandstructure

Compression can align the bands at the L point.Masses reduce with compression.

25

TiCoSb thermoelectric performance with strain

ZrCoSb – Strain bandstructure

Expansion can align the bands at the G point. Masses increase with expansion.

27

ZrCoSb thermoelectric performance with strain

29

Summary of strain analysis

Outline

Ø Band convergenceØ Introduction to transport equationsØ Scattering mechanisms : parabolic band approximation

Ø Modeling half-Heuslers : non parabolic band approximationØ Non parabolic band approximationØ Band aligning – NbCoSn, TiCoSb

Ø Aligning bands of half-Heuslers with strainØ NbCoSnØ TiCoSbØ ZrCoSb

Ø Conclusions

30

Ø Band aligning outcome depend on the electron scattering rate

Ø Constant : any band will improve PF, higher masses are better

Ø Intra-valley only : any band will improve the PF, lower masses are better.

Ø Intra and inter-valley: only certain masses will improve PF

Ø Non parabolic approximation can model NbCoSn ,TiCoSb

Ø Strain can be used for aligning bands in NbCoSn ,TiCoSb and ZrCoSb.

Conclusion

31

Ø Other complex crystalline structure material

Ø Multi-phase material

Ø Material screening using machine learning techniques

Ø Study larger systems MC,MD and NEGF codes.

Future work

32

33

Thank you!