Opis zjawisk transportu elektronów

w materii skondensowanej

konwersja termoelektryczna

Pierwsza kwantowa teoria elektronów w metalach (Drude-Sommerfeld)

1838 Michael Faraday obserwuje przepływ prądu przez rozrzedzone powietrze, pomiędzy anodą i katodą wytwarza się łuk świetlny oprócz okolicy katody (tzw. ciemnia Faradaya) odkrycie promieni katodowych

1897 Joseph John Thomson odkrywa elektron w podobnym eksperymencie, ale z przyłożonym poprzecznym polem E, którym można kierować wiązkę „promieni katodowych”; wyznacza e/m.

1900 Paul Drude formułuje pierwszą elektronową teorię metali

1927 Arnold Sommerfeld używa statystyki Fermiego-Diraca do modelu Drudego i proponuje pierwszą kwantową teorię ruchu elektronów w metalach dając początek tzw. modelowi Drudego-Sommerfelda elektronów swobodnych.

j= -e n v

Współczynnik Halla wybranych pierwiastków w słabych i średnich polach B.

dobra zgodność

słaba zgodność

niezgodność

Eksperymentalne wartości przewodności cieplnych i liczb Lorenza dla wybranych metali

Pomiar doświadczalny ciepła właściwego w metalicznym potasie

Współczynniki elektronowe dla ciepła właściwego (pomiar/teoria)

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

5th Euroschool in Materials Science, Ljubljana, 24-29 May, 2010

E(k)=ℏ(kx

2+k y

2+kz

2)

2m

E k⃗ ¿=∑i=0

∞ f (i)(0)i !

x if (x)=f ( i)

(0)

i !x i

Thermoelectric properties Resitivity

Thermopower

Thermal conductivity

ZTFigure of

merit

S

ZT

INS SC M

A. Joffe

Thermoelectric „tetragon”

LEE LET

LTE LTT

E

-T

S = T (Kelvin-Onsager) LET=LTE/ T

L0 T (Wiedemann-Franz, L0 Lorentz number) -LTT

Electrical current

Heat current

Electrical field

Temperature gradient

Ohm, 1826

Fourier, 1822

Seebeck, 1821Peltier, 1834

Volta (1800), Ampere (1820) , Faraday (1831), Gauss (1832), …

Thermoelectric properties -search for optimum

COOLING ELEMENTSCOP =(TH-TC)(TC+ TH

POWER GENERATORSTC-TH)[(TH -TC+ (

ZT=S ²σκT=

S ²L⏟

calculated

1

1+κLκeL=

κe

σT

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)

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.

[ μVK ]

Peltier effect (1834)

Electrical density currentHeat current q= j

Peltier coefficient 1785 Ham1845 Paris

= LTELEE-1

“Reverse” process to Seebeck effect

Thomson effect (1834)

Q = j2/ +/- j dT/dx Joule Thomson

= T dS/dTS (Thomson)

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

LET=LTE/ T

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.

|N |=EY /B ZdT /dx

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

[ μVKT ]

Fourier relation (1822)

Heat current Temperature gradientq= - T

Thermal conductivity = LTELEE-1LET - LTT

q = qgen- du/dtdu/dt = c dT/dt

(- T)+ T/ t =qgen

2T+ (c/k) T/t = 0when qgen=0

Heat conduction equation

Heat conducted(balance)

= Heat generated in system

- Heat accumulated

in system

1768 Auxerre1830 Paris

Ohm law (1826)

Ohm’s study inspired by works of Fourier and Seebeck

Electrical density current Electric fieldj = E

=LEE=nene/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)v g=

dωdk

=1ℏ

∂ E ( k )

∂ k=∇ k E ( k )=v (k )

Group velocity of electrons

v=ℏ km

⇔ E ( k )=ℏ

2 k 2

2m

v(k) parallel to k only if Fermi surface is spherical

Acceleration of electrons

F=ℏdkdt

⇒ ak=dv kdt

=1ℏ

∂2E ( k )

∂ k ∂kdkdt

=1ℏ2

∂2E ( k )

∂k ∂ kF

a k=(m )-1F where (m ij)−1=

1ℏ 2

∂2 E

∂ k i∂ k j

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

n(EF )∝∂E ( k )

∂k(mij )

−1∝∂

2 E∂ k i∂ k j

Effective masses can be detected in dH-vA or transport measurements

How to measure ?

Boltzmann equation

1

4 π3f (k,r,t )

Electron system described by distribution function f in the (r, k) space.

Electron density current J (r,t )=e

4 π 3∫ vk f ( k,r,t )dk

Transport equationdfdt

=−dkdt

⋅∇k f−v⋅∇r f+∂ f∂ t

+( ∂ f∂ t ) coll .

Stationary condition ∂f∂t

=0

( ∂ f∂ t ) collCollision integral

Describes e-e scatterings/collisions , probability of exit outside the dkdr volume

Fermi-Dirac functionin equilibrium state

time-independent forces

Relaxation time approximation ( ∂ f∂ t ) coll=−

f−f 0

τ

After linearisation

f ( k,r,t )

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

Thermopower

Hall coefficient

Electrical conductivity

Thermal conductivity

Transport coefficients

Hall concentration

Lorenz factor

when T = 0

when j = 0

Kinetic theory of Ziman (T) = e2/3 ∫ dE N(E) v2(E) τ (E,T) [ -∂f(E)/∂E ]

Electrical conductivity

S(T) = e(3T)-1 ∫ dE N(E) v2(E) E τ (E,T) [ -∂f(E) / ∂E ] =

(3eT)-1 ∫ dE (E,T) E [-∂f(E) / ∂E ]

Thermopower (Seebeck coefficient)

N(E) = (2π)-3 ∫ δ(E(k)-E) dkDOS (density of states)

Thermal conductivity

L0 T, L0 =const -LTT

Wiedemann-Franz law, L0 Lorentz number

Relaxation time in transport Boltzman equation

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 properties KKR-CPA code (S. Kaprzyk, AGH Krakow+ NEU Boston)

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, …

Thermoelectric materials

ZT

J. Snyder, JPL-NASA

Moduł TE

Skutterudites

Chevrel phases

26

Concepts of ZT improvementSharp DOS (heavy fermions, QC, low dimension).

PGEC – „phonon glass” + „electron crystal” (Slack, ‘95)

more complex structures + specific vibrations (rattling, phonon, magnon)

skutterudites clathrates B. Lenoir, GDR-Thermo, Bordeaux (2008)

27

Enhancement of TE efficiency in PbTe(distortion of electronic DOS)

Mott’s formula for thermopower

J. Heremans et al., Science 321 (2008) 544

S

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

Electron transport coefficients

Electrical conductivity

Seebeck coefficient (thermopower)

Electronic thermal conductivity

Onsager-related functions

Transport functions (in general tensors)

Wiedemman-Franz-Lorenz

SCTE2016, April 11-15, Zaragoza, Th. AK.2., Thermoelectrics

Boltzmann transport & KKR-CPA calculationsof complex energy bands and thermopower

Different approximations used(1) τ = const; (2) λ = const; (3) μ = const; (4) CPA (velocity + life-time);

K. Kutorasinski, Ph.D. Thesis (2014)

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 (E,T) [-∂f(E)/∂E ]

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

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

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

Half-metallic ferromagnetism

lack of FS for one spin direction,

integer magnetic moment value,

anomalous (T) dependence,

giant Magneto-Optic Kerr effect,

Predictions of HM-AF (1993)

HFM : CrO2, (La-Sr)MnO3, spinels.

Spintronic materials

4.0 B

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

Properties ”on request”

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

df

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

Crystal structurescubic - parent compoundstetragonal - „disordered” phase

Density of states

Calculated thermopower in Ni(Ti,Hf)Sn

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

Metal–semiconductor-metal crossovershalf-Heusler

Seebeck coefficient Resistivity

FeTiSb (VEC=17)Curie-Weiss PM

(0.87B)

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

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-x

NixHfSb

Res

isti

vity

(

.m)

Temperature (K) 100 150 200 250 300 3500

30

60

90

120

150

x=0.4

Fe1-xPtxZrSb

x=0.5

Res

isti

vit

y (

.m)

Temperature (K)

0.2 0.3 0.4 0.5 0.6 0.7 0.8-150

-100

-50

0

50

100

150

Se

ebec

k co

effi

cien

t (

V K

-1)

Concentration x

Fe1-xNixTiSb

Fe1-xPtxZrSb

Fe1-xNixHfSb

Experimental curves: resistivity, Seebeck coefficient (RT)

Fe2VGa i Fe2VGa semimetals

theory

experiments

Bansil,...JT, PRB (1999)

Lue el al., PRB (2002)

44

Convergence of conduction bands in Mg2X

Kutorasinski et al. (JT), Phys. Rev. B 87 (2013)195205.

Mg2Si Mg

2Ge

Mg2Sn

Conduction band degeneracy tends to increase thermopower in n-doped systems and is expected in Mg

2(Si-Sn) alloys !

45

Pb(Te-Se)

Mg2(Si-Sn)

„Engineering” of band degeneracy to improve zT

-- shrinking of band gap and mutual shift of conduction bands near X : not effect of Si/Sn disorder, rather due to important variation of lattice constant

-- similiar thermopower but higher electrical conductivity (more carriers) BUT p-type

Pei et al., Nature 473 (2011) 66

-Achievement of HIGH band degeneracy – general strategy to improve TE properties in bulk materials (n-type)

~Mg2Si0.4Sn0.6

Zwolenski et al.(JT), J. Electr. Mater. 40 (2011) 889.

46

More realistic treatment of “band gap problem”

more complex structures + specific vibrations (rattling, phonon-magnon)

J. Bourgois et a. (JT), Functional Mat. Lett. 6 (2013) 1340005.

Application of more sophisticatedexchange-correlation potentialimproves exp-theory agreement for band gap.

BUT the overall bands shape remains quite similiar -> important results for transport calculations based on LDA results

47

Complex energy Fermi surface

Mg2Si

0.4Sn

0.6

Kutorasinski et al. (JT), Phys. Rev. B 87 (2013)195205.

48

Electronic thermal conductivity

Seebeck coefficient

TEST for relaxationtime approach

Kutorasinski et al. (JT), Phys. Rev. B 87 (2013)195205.

49

ZT=S2σκe+κlat

T

Lorentz factor :deviation from WF law

Figure of merit

50

ZT vs. n/p & T

lattice thermal conductivity as parameter

Kutorasinski et al. (JT), Phys. Rev. B 87 (2013) 195205.

Relativistic effects on TE properties

Full form of Dirac equation including four components

More readable: non-relativistic approach of Dirac equation

SO splitting ~

Fig 1 Band structure of Mg2X

Effect of spin-orbit (S-O) coupling on bands

Kutorasinski et al. (JT), Phys. Rev. B 89, 115205 (2014)

Fig 1 Thermopower Mg2Si

Effect of S-O on electronic bands in Mg2X

effect of S-O: on conduction bands - negligibleon valence bands – HUGE !

Mg2Si Mg

2SnMg

2Ge

X=

S-O influence on transport function

III Conclusions

Net effect of S-O on Seebeck coefficient

Bands' contribution to hole effective mass