thermoelectrics : the search for better materials

Thermoelectrics:The search for better materials

Jorge O. SofoDepartment of Physics,

Department of Materials Science and Engineering,and

Materials Research InstitutePenn State

The basics

Abram F. Ioffe

The devices

2SZ

The performanceT1

T2

21 2 1

22 1

( ) / 2( )

T T TQW T

I I RRI

SSI T

1 2 1

MAX

2 1

1 /

( ) 1 1

T T T T

T

Z

T ZT

1

2 1( )Z

TT T

The materials

n-type p-typeJ.-P. Fleurial, DESIGN AND DISCOVERY OF HIGHLY EFFICIENT THERMOELECTRIC MATERIALS Download Design and Discovery, Jet Propulsion Laboratory/California Institute of Technology, 1993.

Conductivity 101

mne2Drude et al.

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q

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ky

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,

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k kk

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k k

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f fdkdt tk

1dk dp eEdt dt

1dk dp eEdt dt

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kk k

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k k

k k

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1dk dp eEdt dt

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kk k

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coll k

f fft

k k

coll

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1dk dp eEdt dt

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kk k

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k k

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f fft

k k

coll

f fdkdt tk

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k

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1dk dp eEdt dt

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kk k

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k k

0k kk

coll k

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k k

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k

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k

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Q

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2 20k k

k k

fe v

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k Bk

fekS vk T

2

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k Bk

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2

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SZ

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f fe v e d

20 0( )kB Bk k

k B Bk

f fek ekS v dk T k T

2 2

2 2 20 0( )kel B Bk k

k B Bk

f fk v k dk T k T

2( ) ( )k k kk

v

Transport distribution

[ ] [ ]S

[ ]el

max [ ] [ ]best bestZ Z

)()( 0 Cbest TkB4.20

2

0 [ ]el ph

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2

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fk d

k T

“The best thermoelectric,” G. D. Mahan and J. O. SofoProc. Nat. Acad. Sci. USA, 93, 7436 (1996)

The “Best” Thermoelectric

)()( kkk

kk vv

)()()( 2 vN

)()( 0 Cbest TkB4.20

k

dSN

1)(

kv )(

Limitations of the Boltzman Equation Method

• Also known as the Kinetic Method because of the relation with classical kinetic theory

• According to Kubo, Toda, and Hashitsume(1) cannot be applied when the mean free path is too short (e.g., amorphous semiconductors) or the frequency of the applied fields is too high.

• However, it is very powerful and can be applied to non linear problems.

(1) R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II: Non-equilibrium Statistical Mechanics (Springer-Verlag, Berlin, 1991) p. 197

k k k k

coll

f f f fdr dkt dt r dt tk

,k k k k kk

coll

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Using Boltzman with ab-initio

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0

2kv

kk

1

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1 kpk

mˆ1

k

C. Ambrosch-Draxl and J. O. SofoLinear optical properties of solids within the full-potential linearized augmented planewave methodComp. Phys. Commun. 175, 1-14 (2006)

First Born Approximation• Defect scattering

• Crystal defects• Impurities

• Neutral• Ionized

• Alloy• Carrier-carrier scattering• Lattice scattering

• Intravalley• Acoustic

• Deformation potential

• Piezoelectric• Optic

• Non-polar• Polar

• Intervalley• Acoustic• Optic

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T. J. Scheidemantel, C. Ambrosch-Draxl, T. Thonhauser, J. V. Badding, and J. O. Sofo. “Transport Coefficients from First-principles Calculations.” Phys. Rev. B 68, 125210 (2003)

Bi2Te3

Georg Madsen’s

Relaxation time from e-p interaction

2

3

2 12

1

1 1

qe p

q p p q q q p p q q

q p p q q q p p q q

f p dq M f p f p qt

N N

f p q f p N N

2

0

23

0

1 22

1

q

q q q qk k q k k q

t

dq Mk

N f k q N f k

q kk

q

22 2

2qq

qM D

Deformation Potential Calculations

Van de Walle, Chris G. “Band Lineups and Deformation Potentials in the Model-solid Theory.” Phys. Rev. B 39, 1871–1883 (1989).

ln

qdD q

d V

Bardeen, J., and W. Shockley. “Deformation Potentials and Mobilities in Non-Polar Crystals.” Phys. Rev. 80, 72–80 (1950).

Wagner, J.-M., and F. Bechstedt. “Electronic and Phonon Deformation Potentials of GaN and AlN: Ab Initio Calculations Versus Experiment.” Phys. Status Solidi (b) 234, 965–969 (2002)

Lazzeri, Michele, Claudio Attaccalite, Ludger Wirtz, and Francesco Mauri. “Impact of the Electron-electron Correlation on Phonon Dispersion: Failure of LDA and GGA DFT Functionals in Graphene and Graphite.” Physical Review B 78, no. 8 (August 26, 2008): 081406.

Careful…

• Doping: rigid band• Gap problem• Temperature dependence of the electronic

structure.• Alloys. Single site approximations do not work.• Many k-points• Correlated materials?• Connection with magnetism and topology?

Linear Response Theory (Kubo)• Valid only close to equilibrium

• However– Does not need well defined energy “bands”– It is easy to incorporate most low energy excitations of the solid– Amenable to diagrammatic expansions and controlled approximations– Equivalent to the Boltzmann equation when both are valid.

20

†

0

, , 0

1, , ,nin

n eiq qm

q i d e T j q j qV

Summary• Tool to explore new compounds, pressure, “negative” pressure.• Prediction of a new compound by G. Madsen.• Easy to expand adding new Scattering Mechanisms• Limited to applications on “non-correlated” semiconductors.

Questions• Should we start the program of calculating all parameters from

ab-initio?• What about an implementation based on the Kubo formula?• Where the “stochastization” will come from in a small periodic

system? Remember that there should be an average somewhere to get irreversibility…