G = G0 + G0VG Electronic structure and transport properties (I)gdr-thermoelectricite.cnrs.fr/GDR2007-2010/ecole2008/... · 2008. 5. 13. · Total density of states DOS Component,

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 (I)

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


PLAN (I)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” – Onsager coefficients.

Electronic structureProblems with exact solutions for many-electrons systemsDensity Functional Theory (short introduction)Pseudopotential and tight-binding aproachesUsed 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 (FLAPW, rigid band app + constant relax. time) disordered alloys (KKR-CPA, complex bands, electron kinetic parameters).


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

mkkk

kE zyx2

)()(

2222 ++=


Thermoelectric properties Resitivity

Thermopower

Thermal conductivity

ZTFigure of

merit

ρ

S

κ

ZT

INS SC M

A. Joffe


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

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

γ = (1+ΖΤ)1/2

eL

calculatedLSTSZT

κκκ

σ+

==1

1²²

σκTL

e≡

Improvement of figure of merit

Geometry of the devices

Physical properties of the system

Lorentz factor

Thermal conductivity(electrons / phonons)


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 electric currents,Faraday (1831), Gauss (1832), … GDR-Thermoelectricite, 5-9 may 2008, Carcans-Maubuisson

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 cylinder

*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 =∇=∂

∂== EEdd

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 kk ∂∂

∂=∂∂

∂==⇒= EdtdE

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 ?


H : Hamiltonian operator

E : total energy of system (eigenvalue)

Ψ: wavefunction (eigenfunction)

• Time independent Schroedinger equation

Electronic s tructure (many-body problem)

• The complete Hamiltonian

• The Born-Oppenheimer approximation: - nuclei are much heavier than electrons, - nuclear kinetic energy is zero, - inter-nuclear potential energy constant.

( = m = c =1)


• ‘Electronic’ Hamiltonian

Electronic structure theory

• only normalized wavefunctions

=1

Expectation values = observable properties (what is measured)

kinetic E(e-n) E(e-e)


Electronic structure theory

• Foundation of approximate methods is necessary , Nat∼1026

• Exact ground state solution:

• Variation theorem

• E[Ψ] is the energy functional

• Hartree-Fock (HF) solution when minimizing E[ΨSD]• ΨSD is a Slater determinant

• Describes system as n one-electron systems with an effective potential


Electronic structure theoryDensity Functional Theory ( DFT ):P. Hohenberg and W. Kohn. Phys. Rev. B 76, 6062 (1964).W. Kohn and L. J. Sham. Phys. Rev. 140, A1133 (1965).

DFT theorems:1.Total energy functional of the electronic charge density

2.The true ground state density is the density that minimizes E [ρ]

We have replaced problem of solving one equation for N-electrons

by the fictious system of N non-interacting electrons(N x 1-electron equations)

Kohn-Sham one-electron equations in effective potential

Electronic s tructure theory

How to approximate EXC (LDA –local density approximation) ?

Exc[n] is a sum of contributions from each points, depending only on the density at each point, independent of other points.

exchange-correlation energy per electron

GGA –generalised gradient approximation

There are many analytical expressions for exchange-correlations part Exc-Barth-Hedin, Vosko-Nusair, Lunqvist, Perdew-Wang, …(1970-90), -Car-Parinello (QMC)

LSD –local spin density approximation (magnetic systems)


DFT - summary

• Density functional theory is an exact reformulation of many-body QM in terms of the probability density rather than the wave function.• The ground state energy can be obtained by minimisation of the energy functional E[n]. All we know about the functional that it exists, but its form is unknown.

• KS reformulation in terms of single-particle orbitals helps to develop of approximations (LDA, GGA, GW,… and is used in current DFT calculations today.


DFT computations in solids are good for ...

• Predict of ground state: Metal, Semiconductor, Insulator

• Fermi Surface Areas (transport, optical properties)

• Bulk vs. Surface properties

• Spectra: X-Ray, XPS

• Electron-Phonon Interaction (e.g. superconductivity, transport )

• Stoner Criterion for Ferromagnetism (magnetic moments,hyperfine interactions, magnetic structures)

• Band Gaps (?)

• Equilibrium Lattice Parameter and Crystal Structure stability

• Bulk modulus and Elastic Constants

• Phonon Frequencies


Tight-binding methods (LCAO, LMTO)

ψ = ∑ncn φn

Rc = [∑m∑n cm*cnHmn]/ [∑m∑n cm*cnSmn]

Hnm = ∫ φm*H φndr Snm = ∫φm* φndr

∑n(Hmn -E Smn) Cn = 0

Rc = [∫ ψ*Hψdr]/ [∫ψ* ψdr] Rc ≥ Eexact

Wave function in term of atomic orbitals

Coulomb integral

Minimisation of the Rayleigh quotient (normalisation)

Example of used atomic orbitals

more complex orbitals for more complex systems

Domain of quantum chemistry

Overlap integralResolve 'eigenvalue problem'to find coefficients

Tight-binding methods (LCAO, LMTO)

Snm = ∫φm* φndr

Rc ≥ Eexact

1D E(k) = α + βcos(ka)

2D, 3D E(k) = α + 2β{∑ Rcos(k.R)}

in solids overlap only between NN atoms

+ Bloch theorem leads to

Origin of energy gaps

Rc ≥ Eexact

Kronig-Penney 1D model

Using Bloch's theorem

-1 ≤ [ cos(αa)-P sin(αa) / (αa) ] ≤ 1Energy bands

Some energy ranges – forbidden

Brillouin zone = Wigner-Seitz cell in the reciprocal k-space

One needs to resolve geometrical problem

Theory of Brillouin zone and Fermi surfaces

For given Bravais lattice in direct space, the lattice in reciprocal k-space is known

1st BZ = the smallest volume enclosed entirely by the planes that are perpendicular bisectors of the reciprocallattice vector G, drawn from the origin.

2nd BZ and 3rd BZ etc. can be defined accordingly

More and more complex polyhedra = background for the Fermi surface


Lorenz factor

Mizutani


Lorenz factor

Electronic s tructure calculations (Linearized) Augmented Plane Waves (LAPW)

Green function Korringa-Kohn-Rostoker (KKR)

Linearized Muffin-Tin Orbitals (LMTO); tight-binding (TB)Pseudopotential Methods

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

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, …

Boltzmann equation)(

41

3 t,,f rkπElectron 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 τ

0fftf

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)

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 =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

PLAN (II)

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

Electronic structure and electron scattering in practiseordered systems (FLAPW, rigid band app + constant relax. time) disordered alloys (KKR-CPA, complex bands, electron kinetic

parameters)

Illustrative examples (theory vs. experiment)Skutterudites (normal and partly filled),Heusler and half-Heusler alloys,Zintl phases (3-3-4 type)Chevrel phases.


Documents

G = G0 + G0VG Electronic structure and transport properties (I)gdr-thermoelectricite.cnrs.fr/GDR2007-2010/ecole2008/... · 2008. 5. 13. · Total density of states DOS Component,