w materii skondensowanej konwersja termoelektryczna
TRANSCRIPT
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