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Density functional theory A brief summary.
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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.