AB INITIO MODELLING OF THERMOELECTRIC MATERIALS USING

DENSITY FUNCTIONAL THEORYMatthew Lane, Professor J. Staunton.

DENSITY FUNCTIONAL THEORY A brief summary.

DFT• Hartree approximation – many electron wavefunction as a product of single electron orbitals.

DFT• Hartree approximation – many electron wavefunction as a product of single electron orbitals.

• Hartree-Fock approximation – introduction of indistinguishability and spin using slater determinant.

Coulomb

Exchange

2

DFT• Hartree approximation – many electron wavefunction as a product of single electron orbitals.

• Hartree-Fock approximation – introduction of indistinguishability and spin using slater determinant.

• Hohenberg-Kohn theorems – ground state electron density contains the same information as the wavefunction.

DFT• Hartree approximation – many electron wavefunction as a product of single electron orbitals.

• Hartree-Fock approximation – introduction of indistinguishability and spin using slater determinant.

• Hohenberg-Kohn theorems – ground state electron density contains the same information as the wavefunction.

• Kohn-Sham equations – write in terms of an auxiliary system of non-interacting particles.

DFT• Hartree approximation – many electron wavefunction as a product of single electron orbitals.

• Hartree-Fock approximation – introduction of indistinguishability and spin using slater determinant.

• Hohenberg-Kohn theorems – ground state electron density contains the same information as the wavefunction.

• Kohn-Sham equations – write in terms of an auxiliary system of non-interacting particles.

• Density Functional Theory (DFT) – use fictitious potential of auxiliary system and self consistent field approach to iteratively minimise.

DFTInitial

potential •Intuit a reasonable guess at the potential.

Generate density function

•Use initial potential with Kohn-Sham equations.

Wavefunction•Determine improved wavefunction. Update potential

function

until self consistent

MODELLING THERMO-POWER

Extracting information from the density of states and maximising the thermoelectric figure of merit.

[5]

THE SEEBECK COEFFICIENT • Free electrons will diffuse from warmer to colder.• From the Drude model:

• More realistically, using the energy dependant electrical conductivity:

Large slope in the DOS at the Fermi

energy.

FIGURE OF MERIT• A good thermoelectric material:

• High Seebeck coefficient S• High electrical conductivity σ• Low thermal conductivity κ

• Maximise the thermoelectric figure of merit.

• Seebeck coefficient dominant.

MAGNESIUM SILICIDE

RIGID BAND APPROXIMATION Modelling the effect of

doping.

-0.05 0.00 0.05 0.10-1

0

1

2

3

4

5

6

7

8

9

DO

S (p

er R

h)

E-Ef (Rh)

-0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

dN

dN

Shifted Ef (Rh)

-0.05 0.00 0.05 0.10 0.15-800

-600

-400

-200

0

200

400

600

800

See

beck

(V

/K)

Shifted Ef (Rh)

Seebeck

D 0.01 0.03 0.01 0.03Ag 68 50 -55 -19Al -41 -33 76 66Ga -36 -34 78 427In -29 -46 79 78P -31 -31 -31 -47Sb -29 -47 -30 -45

Mg site Si site

RUNNING SIMULATIONS Generating new data.

[5]

TIN ALLOY• Evidence that alloying with tin on the Silicon site improves Seebeck coefficient.

• Want to know how much tin, and then suggest doping.

TIN ALLOY

-0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06-300

-200

-100

0

100

200

300

400

500

pure Mg2Si Mg2Si0.9Sn0.1

Mg2Si0.8Sn0.2

Mg2Si0.7Sn0.3

Mg2Si0.6Sn0.4

Mg2Si0.5Sn0.5

Mg2Si0.4Sn0.6

Mg2Si0.3Sn0.7

Mg2Si0.2Sn0.8

Seeb

eck

coef

ficie

nt (

VK

-1)

Dopant states

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.54

0.55

0.56

0.57

0.58

0.59

0.60

0.61

0.62

0.54402

0.55356

0.5632

0.57179

0.580550.58857

0.59688

0.60525

0.61378

Ferm

i ene

rgy

(eV

)

Tin concentration on Silicon site

-0.05 0.00 0.05 0.10-1

0

1

2

3

4

5

6

7

8

9

DO

S (p

er R

h)

E-Ef (Rh)

-0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

dN

dN

Shifted Ef (Rh)

-0.05 0.00 0.05 0.10 0.15-800

-600

-400

-200

0

200

400

600

800

See

beck

(V

/K)

Shifted Ef (Rh)

Seebeck

-0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06-300

-200

-100

0

100

200

300

400

500

pure Mg2Si Mg2Si0.9Sn0.1

Mg2Si0.8Sn0.2

Mg2Si0.7Sn0.3

Mg2Si0.6Sn0.4

Mg2Si0.5Sn0.5

Mg2Si0.4Sn0.6

Mg2Si0.3Sn0.7

Mg2Si0.2Sn0.8

Seeb

eck

coef

ficie

nt (

VK

-1)

Dopant states

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-240

-220

-200

-180

-160

-140

-120

-100

-80

See

beck

coe

ffici

ent (

VK

-1)

Sn

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-40

-30

-20

-10

0

10

20

30

40

50 pure Mg2Si Mg2Si0.9Sn0.1

Mg2Si0.8Sn0.2

Mg2Si0.7Sn0.3

Mg2Si0.6Sn0.4

Mg2Si0.5Sn0.5

Mg2Si0.4Sn0.6

Mg2Si0.3Sn0.7

Mg2Si0.2Sn0.8

Seeb

eck

coef

ficie

nt (

VK

-1)

number of states

-0.020 -0.015 -0.010 -0.005 0.000 0.005 0.010 0.015 0.0200

-5

-10

-15

-20

-25

-30

-35

pure Mg2Si Mg2Si0.9Sn0.1

Mg2Si0.8Sn0.2

Mg2Si0.7Sn0.3

Mg2Si0.6Sn0.4

Mg2Si0.5Sn0.5

Mg2Si0.4Sn0.6

Mg2Si0.3Sn0.7

Mg2Si0.2Sn0.8

Seeb

eck

coef

ficie

nt (

VK

-1)

number of states

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

-30

-28

-26

-24

-22

-20

-18

-16

Ther

mop

ower

(V

K-1

)

Sn

0.10 0.15 0.20 0.25 0.30 0.35

-30

-29

-28

-27

-26

-25

-24

Ther

mop

ower

(V

K-1)

Sn

AB INITIO MODELLING OF THERMOELECTRIC MATERIALS USING

DENSITY FUNCTIONAL THEORYMatthew Lane, Professor J. Staunton.