Electronic structure and transport properties (II)

Faculty of Physics and Applied Computer Science, AGH University of Science and Technology,

Krakow, Poland email: [email protected]

Janusz Tobola

Electronic structure and transport properties (II)

complex multi-atom systems with

metal-semiconductor transition

G = G0 + G0VG

GDR-Thermoelectricite, 5-9 may 2008, Carcans-Maubuisson

CollaborationT. Stopa, B. Wiendlocha, S. Kaprzyk Faculty of Physics and Applied Computer Science AGH, Kraków, Poland

D. Fruchart, E. K. HlilInstitut Neel CNRS, Grenoble, France

B. Malaman, G. VenturiniLCSM Université H. Poincaré, Nancy, France

L. Chaput, C. Candolfi, B. Lenoir, P. Pecheur, H. Scherrer Laboratoire de Physique des Materiaux, Ecole des Mines, Nancy, France

A. Bansil Northeastern University, Boston , USA

GDR-Thermoelectricite, 5-9 may 2008, Carcans-Maubuisson

PLANBasic thermoelectric effects in solids

Double goal of ab initio electronic structure investigations :- fundamental (computations of electron transport from „first principles”)- practical (search for optimal thermoelectrics with maximum of ZT)

View on TE materials vs. Fermi surface peculiarities Thermoelectric „tetragon” – Onsager coefficients.Boltzmann transport equation (relaxation time approximation),Formulas for transport coefficients (electrical conductivity, thermopower, Hall coefficient,

Lorentz factor).Used methods to study electronic structure and electron scattering in

ordered systems (FLAPW, rigid band app + constant relax. time) disordered alloys (KKR-CPA, complex bands, kinetic parameters of electrons).Illustrative examples of theoretical results vs. experimental measurements

Skutterudites (normal and partly filled),Heusler and half-Heusler alloys,Zintl phases (3-3-4 type)

Chevrel phases.

GDR-Thermoelectricite, 5-9 may 2008, Carcans-Maubuisson

Investigations of electronic states near the Fermi surface E(k)=EF

mkkk

kE zyx

2)(

)(2222 ++

=

Thermoelectric properties Resitivity

Thermopower

Thermal conductivity

ZTFigure of

merit

ρ

S

κ

ZT

INS SC M

A. Joffe

COOLING ELEMENTSη = (TH-TC)(γ−1)(TC+ γTH)−1

POWER GENERATORSη = (γTC-TH)[(TH -TC+ (γ+1)]−1

γ = (1+ΖΤ)1/2

Thermoelectric “tetragon”

LEE LET

LTE LTT

j

q

E

-∇T

Π = S T (Kelvin-Onsager) LET=LTE/ T

κ/σ ≈ L0 T (Wiedemann-Franz, L0 liczba Lorentza) κ ≈ -LTT

∇−

=

T

ELL

LL

qj

TT

ET

TE

EE

Electrical current

Heat current

Electric field

temperature gradient

Ohm, 1826

Fourier, 1822

Seebeck, 1821Peltier, 1834

Volta (1800) - battery, Ampere (1820) – two conductors with currents,Faraday (1831), Gauss (1832), …

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

- new theory of colours (with Goethe) opposite to the theory of Newton,

- temperature gradient causes changes of magnetic field of Earth !!,- Oersted’s experiments (1820) „blind” scientists.

Fourier relation (1822)

∇−

=

T

ELL

LL

qj

TT

ET

TE

EE

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 = 0

when qgen=0

Heat conducted(balance)

= Heat generated in system

- Heat accumulated

in system

1768 Auxerre1830 Paris

T k mn

23

B τ=κ

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 conductivity 1789 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. !!

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 = Τ S (Thomson)

Heat generation in the presence of electricalcurrent j and temperature gradient dT/dx

LET=LTE/ T

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

ωGroup velocity of electrons

( ) ** mkE

m k kv

2

22 =⇔=v(k) parralel 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

ftfff

dtd

dtdf

∂∂+

∂∂+∇⋅−∇⋅−= 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

Current density

In the presence of electric field E & temperature gradient ∇T (without B)Transport functions

Under magnetic field B (Hall effect)

Mean-free path

Transport function

Magnetic transport function

Thermopower

Hall resistivity

Electrical conductivity

Thermal conductivity

Transport coefficients

Hall concentration

Lorenz factor

when ∇T = 0

when j = 0

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 =2.45 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

Skutterudites: prototype – a mineral CoAs3

Im-3 space group No. 204 (bcc)

Co : 8c (1/4,1/4,1/4)Sb : 24g (0,u,v)void: 2a (0,0,0)

u=0.335, v=0.159

Filled skutterudites RT4X12 (e.g. LaFe4Sb12)

Fe : 8c (1/4,1/4,1/4)Sb : 24g (0,u,v)La: 2a (0,0,0)

Name from Skutterud (place in Norway), important resource of cobalt-The most extended investigations in the world (last 15 years) within the thermoelectric materials (USA, Japan, France, Germany, Poland, ...) Systems close to realize the concept of Slack (1995) :

PGEC =„phonon-glass electron crystal”GOOD electrical conductivity like in metals

(by doping/substitution), HIGH thermopower – position of the Fermi level near the energy gap on the DOS slope, LOW thermal conductivity like in amorphous – due to specific scattering of phonons „rattling”

Doping in skutterudite CoSb3

Im-3 space group No. 204 (bcc)

Co : 8c (1/4,1/4,1/4)Sb : 24g (0,u,v)void: 2a (0,0,0)

u=0.335, v=0.159

Filled skutterudites RT4X12 (e.g. LaFe4Sb12)

Fe : 8c (1/4,1/4,1/4)Sb : 24g (0,u,v)La: 2a (0,0,0)

n & p types: Co site (Fe,Ni) lub vacancy site (La, Nd)

CoSb3 (density of states)

- narrow energy gap at EF ,

- strong hybridisation between p-states of Sb and d-states of Co,

- the theoretical value of gap strongly depends on the structural parameters, mostly Sb positions, Sofo et al., PRB ’98

- present KKR results support this conclusion, Eg=0.17 eV (0.2 eV)(a=8.94 A, u=0.3328, v=0.1599)

Eg =60-70 meV ) this work(a=9.034 A, u=335, v=0.158

close to results of Akai et al. ICT ’97Wojciechowski, JT et al., JALCOM (2003)

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 resitivity curves, includes also lattice contribution) (τ ~ 10-14 s)

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

Chaput, … JT, PRB (2005)

C

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 (green)Eg=0.22 eV (red)

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 orbitale sp3, d

Half-Heusler phases (1903)Wide variety of physical behaviours

ÿ metals, semiconductors, semimetals ÿ strong and weak ferromagnets, antiferromagnetsi (high TC / TN)

ÿ Pauli paramagnets, Curie-Weiss paramagnets ÿ half-metallic ferromagnets (HFM) ÿ strong thermoelectrics

Half-metallic ferromagnetism

ÿ lack of FS for one spin direction,

ÿ integer magnetic moment value,

ÿ anomalous ρ(T) dependence,

ÿ giant Magneto-Optic Kerr effect,

ÿ Predictions of HM-AF (1993)

ÿ HFM : CrO2, (La-Sr)MnO3, spinele.

ÿ Spintronic materials

4.0 µB

Crystal structure relations

Zinc blende structure

„Half-Heusler” structure

Cd1-xMnxTe

Bansil, Kaprzyk, JT, MRS Symp. Proc. 253 (1992)

Mn

total

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

Poperties ”on request”

Phase transitions :FM-PM FM-HMF FM-SC PM-SCPM-SC-PMFM-SC-PM

df

Structural stability

experimentsyntesis of C1b

fails

0

5

10

15

20

25

16 17 18 19 20 21 22 23

Number

theoryin C1b structure

1.5 µB /atom V !!

Number of valence electrons VEC

M SC HFM

NiVSb, CoCrSb, ... VEC=20

1998-2007: synthesis of many new VEC=18VEC=18 compounds inspired by computations ! JT, Pierre, JALCOM (2000)

Half-Heusler systems (other example of rigid band treatment)

Crystal structurescubic - parent compoundstetragonal - „disordered” phase

Density of states

NiTiSn NiHfSn

Ni(Ti0.5Hf0.5)Sn

Dispersion curves E(k)

Chaput, JT, et al., PRB (2007)

FLAPW (ordered) KKR-CPA (disordered)

Validity of rigid band approximation in Ni(Ti,Hf)Sn

FLAPW (ordered)

Close agreement between different codes well justifies using a rigid shift of the Fermi level when simulating different size of doping.

Chaput, JT, et al., PRB (2007)

Thermopower computations in Ni(Ti,Hf)SnS – nH correlation

Chaput, JT, ...., PRB (2007)

Calculated thermopower in Ni(Ti,Hf)Sn

Different approximations: - constant relaxation time,- constant mean-path,- free electron.

Chaput, JT, ...., PRB (2007)

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

Multiple Scattering Theory (1) Dyson equations

Crystal potential

Free-electron Green function

Path-scattering operator (for many potentials)

Scattering operator t for single potential v

Multiple Scattering Theory (2)

KKR-constants matrix

Expression of full GF

Korringa-Kohn-Rostoker method

and

Free-electron part

Regular part

In spherical muffin-tin model matrices are site-diagonal

KKR-CPA method (code S. Kaprzyk)

Lloyd formula Kaprzyk et al. Phys. Rev. B (1990)

CPA

Densityof states

Green function

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

Novel formulation of the KKR equations

- monotonic dependence of eigenvalues with energy,- very important in the case of many bands – huge, complex systems !! - numerically easy to find bands E(k),

Adequate normalisation of Schrodinger eq. solutions

diagonalisation of full Green function

Ground state properies KKR-CPA code (S. Kaprzyk)

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, …

Metal–semiconductor-metal crossovers

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)

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

=

New half-Heusler phases close to VEC=18

6.221(4)6.231(5)6.242(2)

6.063(5)6.060(4)6.048(6)

5.917(4)5.910(3)5.906(4)

6.260(5)5.884(2)

Fe1-xPtxZrSbx=0.4x=0.5x=0.6Fe1-xNixHfSbx=0.4x=0.5x=0.6Fe1-xNixTiSbx=0.4x=0.5x=0.6

RhZrSbCoTiSb

a (Å)Compound corresponding mixtures were first molten in an induction furnace in a water cooled copper crucible;

resulting ingots were sealed in silica tube under argon and annealed during one week at 1073K;

purity of the sample was checked by powder X-ray diffraction (Guinier camera Co Kα);

cell parameters have been refined by a least square procedure from powder X-ray diffraction data recorded with high purity silicon (a = 5.43082 Å ) as internal standard.

New alloys with M-SC-M transitionstheoretical predictions (1)

JT et al., JALCOM (2004)

JT et al., JALCOM (2004)

New alloys with M-SC-M transitionstheoretical predictions (II)

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

Role of point defects FeVSb – “dirty” semiconductor

Jodin, J.T., …, PRB (2004)

KKR-CPA – total energy

Thermopower

Resistivity

Defects should be accounted : to interpret metallic electron conductivity + large Seebeck coefficient (FeVSb - rather semiconductor)

Defects in Heusler alloysKKR-CPA density of statesupon inclusion Fe/Sb vac/Fe defects

Vacancy on Fe-site & Sb on Fe-site behaves as a HOLE donorEPMA data

Doping of n and p types

Jodin, JT, … PRB (2004)

Fe2VGa i Fe2VGa semimetals

Oporność elektryczna

theory

experiments

Bansil,...JT, PRB (1999)

B

Lue el al., PRB (2002)

Zintl phases: Y3Au3Sb4 structure “filled “Th3Sb4 Structural relations with Heusler phases (tetraedral coordinations of Co, etc.)

Sportouch, Kanatzidis, J. Solid State Chem. (2001)

I-43d space group No. 220 (bcc)

R : 12a (0, 3/4, 1/8 )Cu : 12b (1/8 , 1/2, 1/4 ) Sb : 16c (-x , ½ - x, x), x 0.08

3-3-4 Zintl systems crystallize

“alternatively” with Heusler alloys

Systems R3T3X4 (R- rare earth,T=Au, Cu, Pt, Co, X=Sb, Bi)(I-43d space group, Y3Au3Sb4 –type)

Dwight, Acta Cryst. 1977- mixed valence properties e.g. Ce3Pt3Sb4

- Kondo insulator, e.g. Pr3Pt3Sb4 , - heavy fermion systems,- superconductors (e.g. Th3Ni3Sb4),- FM & AFM magnets, R=Sm, Gd,- narrow band semiconductors R3Cu3Sb4

-large thermopower, -very low thermal conductivity

Critical in optimisation of ZT !

Not many semiconductors of R3T3X4 comparing to the Heusler XYZ phases. different unit cells

12 atoms in MgAgAs-type 40 atoms in Y3Cu3Sb4-type→ bigger problem for electronic structure computations

Cava et al., APL (2001)

Electronic structure of Y3Cu3Sb4

zoom near Fermi level

Y3Cu3Sb4Partial density of states

Energy gap (or pseudo-gap ?) appears near the Fermi level,due to strong hybridisation p-states of Sb and d-states of R,d-Cu states form a large peak well below EF ( plays a role of electron „reservoir” filling bonds completing VEC=62).

Electronic structure of La3Cu3Sb4

zoom near Fermi level

La3Cu3Sb4partial density of states

In comparison with Y3Cu3Sb4, bands are narrower in La3Cu3Sb4, but interestingly, states near EF suggests rather pseudogap than the true energy gap.

Dispersion curves E(k)

Eg ~ 0.1-0.2 eVKKRThe value of computed band gap strongly depends on applied positional parameters of Sb

Young et al., APL (1999)FLAPW Eg=0.23 eV (Y3Cu3Sb4)Eg=0.61 eV (Y3Au3Sb4)

Y3Cu3Sb4 similarly in La3Cu3Sb4

Experimental dataElectrical resitivityThermopower

Thermal conductivitySkolozdra et al.. Acta Phys. Pol. (2002)

+28.4+56.8+85.2

+38 +20+72 +39+100 +63

100 K200 K300 K

TheoryExper.Temperature

S(T) = AT + BT3 + …

S/T (µV/K2) = 0.2877 x 10-2 d lnNtot(E)/dE Ntot(E) (states/Ry) at E=EF

σ(E) ~Ntot(E) μ(E)

Mott formula

La3Cu3Sb4 Simplified estimation of thermopower from DOS

Residual electrical conductivity

Seebeck coefficient

S/T (µV/K2)=0.28 (±0.05 !!)Very sensitive on EF position !!

Half-Heusler compounds: RCuSb (hypot.)

LaCuSbLarge peak at EF

tentatively explainsPreference of ZintlstructureBut it is less evident in the case of YCuSb

More detailed analysis of total energy necessary to give answer – a difficult task40 atoms in unit cell !!

Chevrel phases

R-3 (rhomboedral) Mo (6f) : (x,y,z)Se (6f) : (x,y,z)Se (2e) : (x,x,x)M (1a) : (0,0,0) big cations (Sn, Pb)M (6f) : (x,y,z) small cations (Cu, Zn)

P-3 triclinicboth open structures

MX(2)

X(1)Mo

Crystal parameters of lattice structure

Search for Chevrel phases as thermoelectrics Band structure calculations:

Roche et al. (1998) TB-LMTO; Roche et al. (1999) KKR

TiMo6Se8

Cd2Mo6Se8

Resistivity experiments

CrMo6Se8

TiMo6Se8

Zn2Mo6Se8

Search for Chevrel phases as thermoelectrics (2) Band structure calculations:

Mo6Se8

Zn2Mo6Se8 (narrow gap Eg<0.1 eV)

Singh et al. (1999) FP-LAPW

LixMo6Se8

jiij kk

EM∂∂

∂=−2

1)(

Effective masses

me= 0.2 mo

mh = 1.2 mo

JT et al. (1999) KKR

Thermoelectrical properties of Chevrel phases Thermopower Resistivity

Thermal conductivityCaillat et al., Solid State Sciences (1999)

The best ZT for TixMo6Se8

ZT = 0.6 at T=1100 K

Very low thermal conductivity ! 0.1 W/(K m)

Chevrel phases with defects (2)

MSe(2)X

Mo

Real samples: TixMo6Se8-y, SnxMo6Se8-y

defects not only in M network but also in Se sublattice

DOS in ordered Chevrel compounds

total

Se

Mo

Ti

s-states4 holes

2 holes

Semiconductor or semimetal

Ti

total

TiMo6Se8 – a very narrow gap semiconductor

KKR calculations

FLAPW calculations

KKR-CPA DOS in Mo6Se8-y with defectstotal Mo Se

JT, Pecheur, ....,J. Phys. CM (2003)

KKR-CPA DOS in TiMo6Se8 with defectstotal Mo Se Ti

JT, Pecheur, ....,J. Phys. CM (2003)