THERMOELECTRICITY:

AN INTRODUCTION

José A. Flores-Livas

Laboratoire de Physique de la Matière Condensée et Nanostructures

Université Claude Bernard Lyon 118/01/2011

I. INTRODUCTIONThermal conduction in solids.

II. THERMOELECTRICMATERIALS.Seebeck effect.

III. FIGUREOFMERIT.

IV. CURRENTMETHODS (BTE).Phonon-Boltzmann Transport Equation.

V. STATEOF THEART.

Outline

2

L

SZ

I. THERMAL CONDUCTION IN SOLIDS

• Fluid particles

• Photons (classical electromagnetic waves treated as qsp)

• Electrons;

• Phonons; (lattice vibration treated as qsp)

Insulators Semiconductors

Phonons Phonons + Electrons

/

1

1P B

o

p E k Tf

e

( ) /

1

1e B

o

e E k Tf

e

Fermi-Dirac Distribution

Bose-Einstein Distribution

LT

Metals

Wiedemann-Franz law

Thermal carriers.

3

ΔV

ΔTS

“A temperature difference between two points in a conductor or semiconductor

results in a voltage difference between these 2 points”[1].

Seebeck coefficient

The sign of S:

If electrons diffuse from hot to cold end; then the cold side is negative.

4

Seebeck effect.II. THERMOELECTRIC MATERIALS.

HOT

0COLD

T

T

ΔV= SdT

[1] “Thermoelectric effects in metals: Thermocouples” S.O. Kasap (2001).

II. THERMOELECTRIC MATERIALS.

- ( ) ( )av ave V E T T E T

2 2

0

-2 F

k T Te V

E

2 2

02 F

V k T

T eE

2 2

03 F

k TS x

eE

VS

T

Assumptions;

• “Free” electron theory of metals.

• Effective mass constant (e-).

• Energy independent.

5

Seebeck effect.

Mott-Jones expression-dependent E.

“Imagine an electron diffuses from hot to cold”

It has to do work against the

potential difference.

Substituting E_average and expanding

the expression of T, one obtain:

Since: , the coefficient Seebeck

is:

2

0

0

3 51

5 12av F

F

kTE E

E

[1] “Thermoelectric effects in metals: Thermocouples” S.O. Kasap (2001).

III. FIGUREOFMERIT

eL e p

2

e

L

SZT T

Electrical

conductivity

Thermal

conductivity

ZT=1 are good, 3–4 essential for

thermoelectric. The best reported ZT values

have been in the 2–3 range. [2]

[2] R. Venkatasubramanian, et al., Nature 413, 597 2001

Thermo power

“State of the art”

Theoretically &

experimentally

6

σ : ELECTRICAL CONDUCTIVITY

DRUDE’S model

[3] Advanced Semiconductor Lab. University of Korea

Temperature dependence of resistivity. [3]

Δq

AΔtJ

e

xv

1 2 3

1[ ... ] x

dx x x x xN i

e

eEv v v v v t t

N m

d

e

e

m

dx d xv E

x d x dJ e E en

t

1

SS N

2

1 1

C

a T T

2

1 1eT

d T

m T

en e nC

= DV

We define a drift mobility:

Mean scattering

time, often called

relaxation time.

DV increase linearly with E;

The conductivity term, then is;

7

0 50 100 150 200 250 300

8x101

102

1.2x102

1.4x102

1.6x102

1.8x102

2x102

2.2x102

2.4x102

2.6x102

Pressure 4500 bar

BG_FIT- Parameters

= 565 K

= 71.42 cm

f = 0.69300698 cm

m = 1.73612619

[

cm

] | L

og

10

Temperature [K]

Matthienssen’s and Nordheinm’s rule

1 1 1

T i

1 1 1

d T ien en en

0( ) ( ) ( )ph magT T T

/

00

( ) ( 1)( 1)(1 )

m mT

f z z

T zT m dz

e e

2 2

6

2 (0)

B trf

F

k

e N v

2BaSi

Figure. Cooper resistivity (NSC)

Blöch Grüneisen FIT of BaSi2_Trigonal (SC) [4]

Compounds, (more complicated)

Blöch-Grüneisen [4]

8

Phonon conductivity

Non-equilibrium (NEMD)

Equilibrium (EMD) or

Green-Kubo

Boltzmann transport equation (BTE’s)

The so-called RTA’s

Varational principles approach

Exactly solve of the

liearized phonon BTE.

The continuum transport

theoryThe atomistic technique

Montecarlo, Molecular dynamics, etc

Etc…

IV. CURRENT METHODS (BTE). κ : THERMAL CONDUCTIVITY

9[5] Baoling Huang, University of Michigan (2008) Thesis

Phonon conductivity in

Diamond.

DFT-Ground state and its

derivatives trough DFPT

κ : some examples…

10

Linear response of crystal to

determine the harmonic and

third order IFC’s

Characteristics [6] :

• Strong bond stiffness.

• Light atoms.

• High phonon frequencies and

• High acoustical velocities.

The highest thermal

conductivity of any bulk

material RT- values of

3000 W/m-K.

[6] A. Ward et al., Phys. Rev. B 80, 125203 (2009)

Continuum transport

Theory and the exactly

solve the linearized phonon

(BTE).

11

TOTAL ENERGY DERIVATIVES

Many physical properties are derivatives of the total energy, with respect to the

external perturbations. [7]

Related derivatives of the total energy on order are…..

1st Order : Forces, stress, dipole moment…

2nd Order: Dynamical matrix, elastic constants, dielectric susceptibility, Born

effective charge tensors, piezoelectricity, internal strains, etc…

3rd Order: Non-linear dielectric susceptibility, Grüneisen parameters,

PHONON-PHONON interaction, etc…

Further properties con be obtained by integration over the phononic degrees of

freedom; (entropy, thermal expansion). [8]

[7] Tutorials (RF)of ABINIT http://www.abinit.org/

[8] CECAM Tutorial (Gian-Marco Rignanese) 2010.

( )ele ionE E

12

There are 4 different method to get the 1st order wave functions. [8]

• Solving the Sternheimer equation directly.

• Using the Green’s function, technique.

• Exploiting the sum over states expression.

•Minimizing the constrained functional for the 2nd order correction to the energies.

With the 1st Order wavefunctions, corrections to the energies can be obtained.

More generally, the Nth order WF give access to the 2nd and (2n+1) order energy theorem.

THESE DERIVATIVES CAN BE OBTAINED FROM; [8]

Perturbative approaches

Direct approaches:

• Finite differences (Frozen phonons)

• Molecular dynamics spectral analysis

methods

[7] Tutorials (RF)of ABINIT http://www.abinit.org/

[8] CECAM Tutorial (Gian-Marco Rignanese) 2010.

13

PHONONS

Computation of phonon frequencies and eigenvector as solution of the

GENERALIZED EIGENVALUE PROBLEM :

[6] A. Ward et al., Phys. Rev. B 80, 125203 (2009)

[7] CECAM Tutorial (Gian-Marco Rignanese) 2010.

[9] http://www.tddft.org/bmg/seminars.php (2010)

'

''

' 21D ( ,q)e (q)= e (q)

M M

Where is the reciprocal space dynamical matrix constructed from the real-space

harmonic IFC’s given by:

'( )) 0 ; ' ' l

'

' iq RD ( ,q ( l )e

D

Notation changed respect to the last

presentation of professor Miguel Marques.[9 ]

Vector positionIndicating the atom

with mass MkR-S harmonic IFC’s

[10] X. Gonze and J. Vigneron, Phys. Rev. B 39,13120 (1989)

[6] A. Ward et al., Phys. Rev. B 80, 125203 (2009)

Further references:

[11] G. Deinzer, Phys. Rev. B 67, 144304 (2003)

[12] S. Baroni, Solid State Commun. 91, 813 (1994)

The theorem provides an analytic expression for the

third-order order anharmonic IFC’s. in there sets of

14

2

*

'

1( ) ( ) ( )el ion Eq q q

N u u

The matrix reciprocal-space harmonic IFC’s is a combination

of electronic and ionic parts [6]:

N = # of cells. The ionic terms involves the second derivative of energy.

The third order anharmonic IFC’s are evaluated first in R-Space, where they are given by the third

order derivatives of the total energy respect to the Fourier transformed atomic displacements. [11-12]

3' ''

' ''

( , ', '')( ) ( ') ( '')

totEq q q

u q u q u q

Thanks to 2n+1 theorem [10] within DFPT Third-order response function is accessible.

, ,q

THE LINEARIZED BOLTZMANN EQUATION [6]

0

c

n nT

T t

''

' ''j j j(q) (q) (q )

' ''q q q K

Consider a small gradient of temperature:

That perturbs a phonon distribution:

The anharmonicity of the interatomic potential causes phonon scatter inelastically from one another.

Umklampp process correspond to

And normal process is

0 1n n n ( , )j q

T

Bose distribution function Non-equilibrium part, proportional to

the small

We solve the linearized

BTE;

Conserving momentum and energy, satisfied:

The c is the collition term that

describes the scattering into

and out of the state. [13]

0K 0K

[13] Iterative solution; M. Omini and A. Sparavigna, Phys Rev. B 53, 9064 (1996).

[6] A. Ward et al., Phys. Rev. B 80, 125203 (2009).15

T

0 0 0

' '' 2

, ', '' , ', '' ' ''

' ''

1 1( 1)( )

2 2 ( )4

n n n

WN

Three-phonon scattering rates are computed from Fermi’s golden rule with the anaharmonic IFC’s

as input [6]:

Three-phonon matrix elements is given by:

' '

''

' ''' '

, ', ''

' ' '' '' ' ''

(0 , ' ', '' '') l liq R iq R

l l

e e el l e e

M M M

[6] A. Ward et al., Phys. Rev. B 80, 125203 (2009) 16

THE HARMONIC AND THIRD-ORDER ANHARMONIC IFC’S

Phonon frequencies

R-space anharmonic IFC’sPhonon eigenvectors.

17

1 0 0( 1)n n n F T

0F F F

SCATTERING RATES

The scattering events are used in iterative solution to the L-phonon BTE, using

the substitution [13];

The phonon BTE can be recast as:

, ,x y z

0 00 ( 1)n n

FTQ

, ', '' , ', '' , '

', '' '

1

2

impQ W W W

[13] Iterative solution; M. Omini and A. Sparavigna, Phys Rev. B 53, 9064 (1996).

Total scattering rate defined as [6]:

18

21zz z zC

V

/ zz zTF

LATTICE THERMAL CONDUCTIVITY [6]

Considering a temperature gradient along [001] (z) direction.

Starting from:

The phonon scattering time is related to as :

Then, the lattice thermal conductivity:

Where : is the specific heat per mode, and: 0 0 2( 1)( )BC k n n 1/( )Bk T

' '' 0z zF F

Correspond to 0th a single mode RTA.

[6] A. Ward et al., Phys. Rev. B 80, 125203 (2009)

19

THERMAL CONDUCTIVITY RESULTS [6]

[6] A. Ward et al., Phys. Rev. B 80, 125203 (2009)

RTA solution and full converged solution.

(Dashed line represent percent error.)

Results of Full p-BTZ calculation of Si and Ge, Solid

lines represent the isotopically enriched values.

Naturally occuring

Isotopically enriched

Experimental

ab initio (solid lines)

20

V. STATEOF THEART. “Mater ia l Challenges”

Desirable electronic characteristics. [14] ACS National meeting , talk of Mercouri Kanatzidis, April 2010.

[15] Tutorial of summer school Michigan State University, Kanatzidis’s group 2006.

[15]

[15]

[14]

21

P romis ing systems

Alloys (ZrNiSn)

Zn 4Sb3.

Skut terudites (CoSb3).

Yb14MnSb11

Bulk “nano” composites based PbTe.

Bulk “nano” composites based Si,Ge.

Clathrate s . (Slack’s proposal of PGEC.)

IN THERMOELECTRICITY…

• All compounds are strongly anisotropic.

• “There are many promising materials”.

• Nanostructures reduce the lattice thermal conductivity.

• Doping seems to be the key-gold”.

• “Nano” saves thermoelectricity?

Predict thermoelectric properties by calculations?

• Yes!

SUMMARY+CONCLUSIONS. (literature…)

BIBLIOGRAPHY

22

Articles.

• Nature 413, 597 2001

• Phys. Rev. B 80 125203 (2009) §

• PRL. 104, 208501 (2010)

• Nanoletters, Vol. 8 N. 11, 3750-3754, (2008)

• X. Gonze and J. Vigneron, Phys. Rev. B 39,13120 (1989)

• G. Deinzer, Phys. Rev. B 67, 144304 (2003)

• S. Baroni, Solid State Commun. 91, 813 (1994)

• Phys. Rev. B 77 12509 (2008)

• arXiv:cond-mat/0602203v1. (2006)

• Dalton Trans., 2010, 39, 978–992

Thesis.

• Al Thaddeus Avestruz. Massachusetts Institute of technology MIT. (1994).

• Baoling Huang, University of Michigan (2008).

Books, tutorial and presentations.

• Book: “Thermoelectric effects in metals: Thermocouples” S.O. Kasap (2001).

•Tutorials (RF)of ABINIT http://www.abinit.org/

• ACS National meeting , talk of Mercouri Kanatzidis, April 2010.

• Tutorial of summer school Michigan State University, Kanatzidis’s group 2006.

• CECAM Tutorial (Gian-Marco Rignanese) 2010.

• “A brief introduction to phonons” http://www.tddft.org/bmg/seminars.php

23

Migue l.

S ilvana .

Lauri.

David .

Woyten

and

Guilherme .

I want to Thanks: my group, and friends.