Faculty of Physics and Applied Computer Science, AGH University of Science and Technology,
Krakow, Polandemail: [email protected]
Janusz Tobola
G = G0 + G0VG
Electronic structure of thermoelectric systems
lecture I
OUTLINE
Goals of ab initio computations of thermoelectric materials - fundamental - computations of electron transport from „first principles”- practical - search for optimal thermoelectrics with maximum of ZT
Introduction to basic electron transport properties in solidsThermoelectric „tetragon” - Peltier, Seebeck, Ohm and Fourier effectsOnsager coefficients.
Electronic structure calculationsProblems with exact solutions for many-electrons systemsDensity Functional Theory (short introduction)Computational techniques (FLAPW, KKR, …)Femi surface and electron transport parameters
Electron transport properties Boltzmann transport equation (relaxation time approximation),Electron transport coefficients (electrical conductivity, thermopower, Hall coefficient, Lorentz factor).
Electronic structure and electron scattering (in practise)ordered systems (rigid band approach + constant relax. time) disordered alloys (KKR-CPA, complex bands, electron kinetic parameters).
GDR-Thermoelectricite, 5-9 may 2008, Carcans-Maubuisson
Investigations of electronic states near the Fermi surface E(k)=EF
*2)(
)(2222 m kkkkE zyx ++
=
5th Euroschool in Materials Science, Ljubljana, 24-29 May, 2010
Thermoelectric properties Resitivity
Thermopower
Thermal conductivity
ZTFigure of
merit
ρ
S
κ
ZT
INS SC M
A. Joffe
5th Euroschool in Materials Science, Ljubljana, 24-29 May, 2010
Thermoelectric „tetragon”
LEE LET
LTE LTT
E
-∇T
Π = S T (Kelvin-Onsager) LET=LTE/ T
κ/σ ≈ L0 T (Wiedemann-Franz, L0 Lorentz number) κ ≈ -LTT
∇−
=
T
ELL
LL
qj
TT
ET
TE
EE
Electrical current
Heat current
Electrical field
Temperature gradient
Ohm, 1826
Fourier, 1822
Seebeck, 1821Peltier, 1834
Volta (1800), Ampere (1820) , Faraday (1831), Gauss (1832), …5th Euroschool in Materials Science, Ljubljana, 24-29 May, 2010
Thermoelectric properties -search for optimum
COOLING ELEMENTSCOP =(TH-TC)(γ−1)(TC+ γTH)−1
POWER GENERATORSη = (γTC-TH)[(TH -TC+ (γ+1)]−1
γ = (1+ΖΤ)1/2
eLcalculatedLSTSZTκκκ
σ
+==
1
1²²
σκTL e≡
Improvement of figure of merit
Geometry of the devices
Physical properties of the system
Lorentz factor
Thermal conductivity(phonons /electrons)
Carnot limit
5th Euroschool in Materials Science, Ljubljana, 24-29 May, 2010
Seebeck effect (1821)
∇−
=
T
ELL
LL
qj
TT
ET
TE
EE
Temperature gradientElectric field E= S ∇ T
thermopower
1770 Tallin1854 Berlin
S = LEE-1LET
Explanation : thermomagnetism - „magnetic” polarisation of metals and alloys due to the difference of temperature !!
Vivid personality of the Romaticism
temperature gradient causes changes of magnetic field of Earth !!,
Oersted’s experiments (1820) „blind” scientists.
KVµ
5th Euroschool in Materials Science, Ljubljana, 24-29 May, 2010
Peltier effect (1834)
∇−
=
T
ELL
LL
qj
TT
ET
TE
EE
Electrical density currentHeat current q= Π j
Peltier coefficient 1785 Ham1845 Paris
Π = LTELEE-1
“Reverse” process to Seebeck effectThomson effect (1834)
Q = j2/σ +/- µ j dT/dx Joule Thomson
µ = T dS/dTΠ = S T (Thomson)
Heat generation in the presence of electricalcurrent j and temperature gradient dT/dx
LET=LTE/ T5th Euroschool in Materials Science, Ljubljana, 24-29 May, 2010
Nernst-(Ettingshausen) effect (1886)
Thermo-magneto-electric effect
1864 Wąbrzeźno1941 Niwica
“Reverse” process to Nernst effect = Ettingshausen effect
Phenomenon observed when a sample conducting electrical current is subjected to a magnetic field B and a temperature gradient dT/dx perpendicular to each other.
dxdT BEN ZY//=
EY is the y-component of the electric field that results from the magnetic field's z-component BZ and the temperature gradient dT/dx.
N ~ 0 in metals
N large in semiconductors, superconductors, heavy-fermions,Dirac electrons in Bi, graphen, Landau levels cross Fermi level
magnetic field B
TKV µ
Fourier relation (1822)
Heat current Temperature gradientq= -κ ∇ T
Thermal conductivity κ = LTELEE-1LET - LTT
∇q = qgen- du/dtdu/dt =ρ c dT/dt
∇(-κ ∇ T)+ ∂T/ ∂t =qgen
∇2T+ (ρc/k) ∂T/∂t = 0when qgen=0
Heat conduction equation
Heat conducted(balance)
= Heat generated in system
- Heat accumulated
in system
1768 Auxerre1830 Paris
∇−
=
T
ELL
LL
qj
TT
ET
TE
EE
5th Euroschool in Materials Science, Ljubljana, 24-29 May, 2010
Ohm law (1826)
Ohm’s study inspired by works of Fourier and Seebeck
∇−
=
T
ELL
LL
qj
TT
ET
TE
EE
Electrical density current Electric fieldj = σ E
σ=LEE=neµ=neτ/mElectrical conductivity1789 Erlangen1854 Munchen
„The Galvanic Circuit Investigated Mathematically” (1827)
Metallic wire in cyllinder
*Declination of magnetic needle proportional to electric current I
* Seebeck thermocouple – a source of electrical potential V
V/I = R = constant when R=const. !!
Electron motion in solids (semi-classical)
In general v-vector is NOT parallel to k–vector (e.g. ellipsoid), but it is perpendicular to isoenergetic surface E(k)
( ) ( ) )(1 kvkkk
kv k =∇=
∂∂== EE
dd
g ω
Group velocity of electrons
( ) ** mkE
m k kv
2
22 =⇔=v(k) parallel to k only if Fermi surface is spherical
Acceleration of electrons( ) ( ) F
kkkk
kkkva kF k
k ∂∂∂=
∂∂∂==⇒= E
dtdE
dtd
dtd 2
2
2 11
jiij kk
Em∂∂
∂== −2
211- 1)( where)
Fm(ak
In general, tensor of effective mass is independent on electron velocity
DOS near E=EF can be detected in specific heat and magnetic susceptibility
measurements
( )kk
∂∂∝ E)E(n F
jiij kk
E)m∂∂
∂∝−2
1(
Effective masses can be detected in dH-vA or transport measurements
How to measure ?
5th Euroschool in Materials Science, Ljubljana, 24-29 May, 2010
Boltzmann equation
)(4
13 t,,f rk
πElectron system described by distribution function f in the (r, k) space.
Electron density current krkvrJ k dt,,fet, ∫= )(4
)( 3π
Transport equation.
collt
ftfffdt
ddtdf
∂∂+∂
∂+∇⋅−∇⋅−= rk vk
Stationary condition 0=∂∂
tf
colltf
∂∂Collision integral
Describes e-e scatterings/collisions , probability of exit outside the dkdr volume
Fermi-Dirac functionin equilibrium state
time-independent forces
Relaxation time approximation τ0ff
tf
coll
−−=
∂∂
5th Euroschool in Materials Science, Ljubljana, 24-29 May, 2010
After linearisation
)( t,,f rk
Electric current density
Heat density current
1-electron Boltzmann eq. in the presence of fields : E, B & ∇T
where Mean-free path
Onsager coefficients
5th Euroschool in Materials Science, Ljubljana, 24-29 May, 2010
Electrical current density
With applied E and ∇T (B=0)
Transport functions
Applying additional magnetic field B (Hall effect)
Mean free path of electrons
Transport function
Magnetic transport function
Chaput et al.(JT), PRB 2006 5th Euroschool in Materials Science, Ljubljana, 24-29 May, 2010
Thermopower
Hall coefficient
Electrical conductivity
Thermal conductivity
Transport coefficients
Hall concentration
Lorenz factor
when ∇T = 0
when j = 0
Chaput et al.(JT), PRB 2006 5th Euroschool in Materials Science, Ljubljana, 24-29 May, 2010
Kinetic theory of Ziman
∇
=
T
ELL
LL
qj
TT
ET
TE
EE
σ(T) = e2/3 ∫ dE N(E) v2(E) τ (E,T) [ -∂f(E)/∂E ]
Electrical conductivity
S(T) = e(3Tσ)-1 ∫ dE N(E) v2(E) E τ (E,T) [ -∂f(E) / ∂E ] =
(3eTσ)-1 ∫ dE σ(E,T) E [-∂f(E) / ∂E ]
Thermopower (Seebeck coefficient)
N(E) = (2π)-3 ∫ δ(E(k)-E) dkDOS (density of states)
Thermal conductivityκ/σ ≈ L0 T, L0 =const κ ≈ -LTT
Wiedemann-Franz law, L0 Lorentz number
Relaxation time in transport Boltzman equation
5th Euroschool in Materials Science, Ljubljana, 24-29 May, 2010
Electronic structure peculiarities Half-Heusler (VEC=18)Semiconductors/semimetals (CoTiSb, NiTiSn, FeVSb, ...)9 + 4 + 5=18 wide variety !!
Skutterudites (VEC=96) semiconductors/semimetals (CoSb3, RhSb3, IrSb3, CoP3 ...) 4 x 9 +12 x 5 = 96
Zintl phases (VEC=62) semiconductors/semimetals (Y3Cu3Sb4, Y3Au3Sb4, ...)
3 x 4 + 3 x 10 + 4 x 5 = 62
a
b
c
DOS vs. transport function σ(E)
Small substitution/doping 0.01-0.05 el. per Co4Sb12 → ∆EF≈1-2 mRy
Chaput, … JT, PRB (2005)
Doped CoSb3 : FS vs. Hall concentration (rigid band)Chaput, … JT, PRB (2005)
nH=f(n)
EF=0.5191 Ry, n=0.01 EF=0.5570 Ry, n=0.01 EF=0.5585 Ry, n=0.06
1.1
1.6
Valence bands Conduction bands
Doped CoSb3 : electrical resistivity
Constant relaxation time (only one free parameter selected in order to gain the best fits to experimental resistivity curves, includes also lattice contribution) (τ ~ 10-14 s), concentration of carriers taken from Hall measurements
(not from nominal composition !)
Experiment (literature) FLAPW calculations
Chaput, … JT, PRB (2005)
Constant relaxation time – NOT important, since Seebeck coefficient Does not depend on this parameter – Excellent test for theory !!
Doped CoSb3 : thermopower
Experiment (literature) FLAPW calculations
S(T) = e(3Tσ)-1 ∫ dE N(E) v2(E) E τ(E,T) [-∂f(E)/∂E ]
Doped CoSb3: Lorentz factor& effect of band gap on thermopower S
Chaput, … JT, PRB (2005)
ATTENTION: one must be careful when estimating lattice contribution to thermal conductivity using Wiedemann-Franz law & total thermal conductivity measurements – it can appear NEGATIVE !
FLAPW calculations
Band gap effect on S:Eg=0.05 eV Eg=0.22 eV
Heusler phases X2YZ, XYZ (1903)
Normal Heusler L21
Fm3m (type Cu2MnAl)X : (0,0,0), (1/2,1/2,1/2)Y : (3/4,3/4,3/4)Z: (1/4,1/4,1/4)
Half-Heusler C1b
F-43m (type AgMgAs)X : (0,0,0) 4aY : (3/4,3/4,3/4) 4dZ: (1/4,1/4,1/4) 4c
structure DO3
Fm3m (type Fe3Al)X : (0,0,0), (1/2,1/2,1/2)X : (3/4,3/4,3/4)Z: (1/4,1/4,1/4)
Crystal stability orbitals sp3, d
Electron phase diagram of half-Heusler systemsJT et al., JMMM (1996), J. Phys. CM (1998), JALCOM (2000), PRB (2001,2003)
CoTiSn27Co : 18Ar 4 s2 3 d7 (9)22Ti : 18Ar 4 s2 3 d2 (4)50Sn:[36Kr4d10] 5s25p2 (4)
VEC = 17
FeVSb26Fe: 18Ar 4 s2 3 d6 (8)23V : 18Ar 4 s2 3 d3 (5)51Sb:[36Kr4d10] 5s25p3 (5)
VEC = 18
NiMnSb28Ni: 18Ar 4 s2 3 d8 (10)25Mn: 18Ar 4 s2 3 d5 (7)51Sb:[36Kr4d10] 5s25p3 (5)
VEC = 22
Electrical resitivity
Metal–semiconductor-metal crossovershalf-Heusler
Seebeck coefficient Resistivity
FeTiSb (VEC=17)Curie-Weiss PM
(0.87µB)NiTiSb (VEC=19)
Pauli PM JT et al., PRB (2001)
1-hole system 1-electron system
p
n
Transport properties in disordered systems (KKR-CPA methodology)
The idea of Butler (1980-82) of complex energy bands applied to study metallic alloys as Ag-Pd, Ag-Cu, etc.
In alloys (solid state systems containing chemical disorder) the electronic energy bands are „fat-fuzzy” related to finite life-time of excited carriers.
Mattheissen rules (contributions to total resistivity coming from different relaxation/scattering mechanisms are additive quantities:
At low temperature electrical conductivity is in general dominated by various types of disorder.
Information on the Fermi surface (with complex energy) from the KKR-CPA computations allows for finding kinetic parameters of electrons (velocities and life-times depending on k-vector). It permits next determining transport coefficients (residual resitivity, thermopower)
...ef
+++=321
1111ττττ
KKR-CPA method & complex bandsKKR-CPA method & complex bands
CPA “trick”CPA condition
CPA crystal - restored periodicity • price: complex potential
• Disordered alloys: periodic - Coherent Potential Approximation (CPA):
In multi-atomic systems more imagination needed !
Present KKR-CPA code allows for treatment of more than 2 atoms on disordered site, but within muffin-tin potential (problem with CPA condition), CPA is also solved self-consistently; imaginary part of E(k) related to electron life-time due to disorder !
CPA much better than virtual crystal approx. VVCA
5th Euroschool in Materials Science, Ljubljana, 24-29 May, 2010
Lloyd formula Kaprzyk et al. Phys. Rev. B (1990)
CPA
Densityof states
Full GF
Fermi energy N(EF)=Z
Korringa-Kohn-Rostoker with coherent potential approximation
Bansil, Kaprzyk, Mijnarends, (JT), Phys. Rev. B (1999)) conventional KKR
Stopa, Kaprzyk, (JT), J.Phys.CM (2004)novel formulation of KKR
KKR-CPA method for disordered alloys
Ground state properies KKR-CPA code (S. Kaprzyk, Krakow)
Total density of states DOS
Component, partial DOS
Total magnetic moment
Spin and charge densities
Local magnetic moments
Fermi contact hyperfine field
Bands E(k), total energy, electron-phonon coupling, magnetic structures, transport properties, magnetocaloric, photoemission spectra, Compton profiles, superconductivity, … 5th Euroschool in Materials Science, Ljubljana, 24-29 May, 2010
Complex energy bands & Fermi surface
Stopa, JT, Kaprzyk, European Conference on Thermoelectrics 2004
semiconductorx =0.50
FS pecularities
Group velocity
Life-time
)(ERev k kk ∇=1
)Eh
k k( Im=τ
Influence of disorder on residual resistivity & thermopower
Stopa, JT, Kaprzyk, European Conference on Thermoelectrics 2004
Electrical conductivity
Thermopower(Mott formula)
Theory vs. experiment
FE
B
EE
eTkS
∂∂−= )(ln
3
22 σπ
kkk3
2
232 τ
πσ vdS
)(e)E(
)E(∫
Σ
=
Semiconductors from metals
50 100 150 200 250 300 3500
50
100
150
200
250
x=0.6
x=0.4
x=0.5
Fe1-xNixHfSb
Res
istiv
ity (
µΩ.m
)
Temperature (K) 100 150 200 250 300 3500
30
60
90
120
150
x=0.4
Fe1-xPtxZrSb
x=0.5
Res
istiv
ity (µ
Ω.m
)
Temperature (K)
0.2 0.3 0.4 0.5 0.6 0.7 0.8-150
-100
-50
0
50
100
150
See
beck
coe
ffici
ent (
µV K
-1)
Concentration x
Fe1-xNixTiSb Fe1-xPtxZrSb Fe1-xNixHfSb
Experimental curves: resistivity, Seebeck coefficient (RT)
JT et al., JALCOM (2004)
Semiconductor-metal transition(Ti-Sc)NiSn
Electrical resistivity
KKR-CPA
Thermopower
Stopa, JT, Kaprzyk, ECT 2005, Nancy
Experiment
SRT =(S/T)0*300 K Residual resitivity
Horyn et al., JALCOM (2004)
Velocities and life-times on FS Ti0.7Sc0.3NiSn
1 FS sheet
2 FS sheet
3 FS sheet
Stopa, JT et al.., J. Phys. CM (2006)