G = G0 + G0VG Electronic structure of thermoelectric systems

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


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


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


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


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 ?


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

−−=

∂∂


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


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 ]


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


Thermoelectric materials

κ

ZT

J. Snyder, JPL-NASA

Moduł TE

Skutterudites

Chevrel phases

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


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


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

Resistivity

Semiconductor from alloyed metals

Density of states at EF

M SC M

JT, et al., PRB (2001)

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


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


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)

Semiconductor-metal transition Ti1-xScxNiSnReal part of electronic energy bands: Re E(k)

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)

Metal-semiconductor transition

Stadnyk, …, J.T., JALCOM (2005)

Thermopower

100.651.136.510.22.1

94.460.533.416.77.1

30.315.310.03.10.6

29.812.64.64.0-0.4

Ti0.9Sc0.1NiSnTi0.8Sc0.2NiSnTi0.7Sc0.3NiSnTi0.6Sc0.4NiSnTi0.4Sc0.6NiSn

KKR-CPAExper.KKR-CPAExper.

T=300 KT=90 KKoncentracja

G = G0 + G0VG Electronic structure of thermoelectric systems

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