Heat and Nanotechnologies: Focus on Thermoelectricity Sebastian Volz
THERMOELECTRICITY: AN INTRODUCTION - · PDF fileTHERMOELECTRICITY: AN INTRODUCTION José...
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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.
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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.