Ecole CNRS, Ventron II 2012 p 1 Transport a dimensions reduites Joseph P. Heremans Department of Mechanical and Aerospace Engineering Department of Physics The Ohio State University, Columbus, Ohio 43210, USA [email protected]Les notes sont en anglais, mais je parlerai francais (ni accents, ni cedilles) Ecole CNRS, Ventron II 2012 p 1 References: Heikes et Ure, Thermoelectricity: Science and Engineering, Interscience, New York 1961 Heremans, Thrush & Morelli, Phys. Rev. B 70 115334 (2004) Chasmar et Stratton, J. Electronics and Control 7 52-72 (1959) J. M. Ziman, Electrons and Phonons, Clarendon, Oxford (1960), reprint 1972
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Ecole CNRS, Ventron II 2012 p 1
Transport a dimensions reduitesJoseph P. Heremans
Department of Mechanical and Aerospace EngineeringDepartment of Physics
Les notes sont en anglais, mais je parlerai francais (ni accents, ni cedilles)
Ecole CNRS, Ventron II 2012 p 1
References: Heikes et Ure, Thermoelectricity: Science and Engineering, Interscience, New York 1961Heremans, Thrush & Morelli, Phys. Rev. B 70 115334 (2004)Chasmar et Stratton, J. Electronics and Control 7 52-72 (1959)J. M. Ziman, Electrons and Phonons, Clarendon, Oxford (1960), reprint 1972
Ecole CNRS, Ventron II 2012 p 22
FACTORS LIMITING zT
S2 = electronic properties of the solid ("Power factor")
Lattice thermal conductivity in reduced dimensionsTwo effects:
1. Phonon dispersion relation has branches that extend over 3, 2 or 1 D => specific heat goes as TD
2. Phase space argument for scattering:
k
k1 k2
E1
E2
E1+E2
k1+k2
in 1-D, no space point to scatter into
in 2 or 3-D,space points at different angles
Basically:1. at low T, 2. at high T, reduced Umklapp
DT
Ecole CNRS, Ventron II 2012 p 16
Spectral thermal conductivity
Asegun S. Henry, and Gang Chen Gang, Journal of Computational and Theoretical Nanoscience, 5, Number 2, pp. 141-152(12), (2008)
Ecole CNRS, Ventron II 2012 p 17
Another way to reduce but not : anharmonicity
Design the chemical bonds to limit lattice thermal conductivityCase study: I-V-VI2 Compounds and their Alloys
Summary: Lone pair electrons → large anharmonicity → strong phonon-phonon interactions → thermal conductivity at the amorphous limit
Ecole CNRS, Ventron II 2012 p 18
Starting point: AgSbTe2
2SzT el
Ecole CNRS, Ventron II 2012 p 19
Intrinsic Mechanisms for Low Lattice Thermal Conductivity
Nielsen et al., in preparation; Morelli et al. Phys. Rev. Lett., 101 035901, (2008); [i]S.-H. Wei and A. Zunger. Physical Review B (1997) vol. 55 (20) pp. 13605-13610
AgInTe2: In’s s2p1 valence electrons participate in forming sp3 hybridized bonds with Te.
AgSbTe2: Sb’s s2p3 electrons:
p electrons form the conduction/valence bands
s electrons form an isolated ("lone pair") band, which couples to and is repelled from the valence band
Summary• Intrinsically low thermal conductivity of cubic I-V-VI2 compounds
• Thermal conductivity limited by phonon-phonon interactions
• Lone pair electrons → Extra strong anharmonicity
• Works with I-V-VI2 compounds where – Group I includes noble metals, alkali metals and their alloys.– Group V = Bi, Sb, As, P– Group VI = Te, Se, S
• No effects from defects or alloy scattering → robust vis-à-vis variations in sample preparation / manufacturing processes
• Future Work:– Dope Na compounds to get high zT
(valence band structure has high DOS; with theory input from UCLA)– Find and exploit other compounds with lone pair electrons– Explore relations with IV-VI compounds (with Northwestern U.)– Translate to Cu compounds (see Michigan State work)
Ecole CNRS, Ventron II 2012 p 30
2. Power factor enhancements
Ecole CNRS, Ventron II 2012 p 31
Quantum Size Effects
)()(
)(
)(
g
z
z
y
y
x
x
EEEE
EE
mk
mk
mkE
1
222
222222
iy
y
x
x
z
i
y
y
x
x
Emk
mk
dma
mk
mkE
22
2222222
2
2222222
)(
ijz
x
z
j
y
i
x
x
Emk
dmb
dma
mkE
2
22222
2
222
2
22222
)(
3-D solid Quantum well Quantum wire
ijkEE )(Quantum dots:Standing waves
Ecole CNRS, Ventron II 2012 p 32
Energy Dependence of the Density of states g(E) (3D)
32
23
32
23233
3
3
22
32
34
81
3
4
EmdE
dEg
EmERR
R
/
//
)(
)()(
EmkkkR
kkkm
E
zyx
zyx
22
222
2222
)k(Energy
DefineRadius
How many “lattice points” lie at a fixed distance from the origin in this velocity space?Number of allowed k-values per unit volume of k-space is
Volume of sphere of phase space of radius R:
381
Ecole CNRS, Ventron II 2012 p 33
Energy Dependence of the Density of states g(E) (2D)
i EE
EE
iii
i
i
mdE
dEg
EEmERR
R
22
222
2
2
2
42
4
)(
)()(
mEEkkR
Ekkm
E
iyx
iyx
)(
)k(
22
22
222
Energy
DefineRadius
How many “lattice points” lie at a fixed distance from the origin in this velocity space?Number of allowed k-values per unit area of k-space is
Area of circle of phase space of radius R:
kx
ky
iEE ,241
Ecole CNRS, Ventron II 2012 p 34
Energy Dependence of the Density of states g(E) (1D)
ijEEij
EE
ij
ij
x
ij
ij
EEm
dEdEg
EEmE
kR
122
2
)(
)(
mEEkR
Ekm
E
ijx
ijx
22
2
22
)k(Energy
DefineLength
How many “lattice points” lie at a fixed distance from the origin in this velocity space?Number of allowed k-values per unit volume of k-space is
Length of line of phase space:
kx
21
Ecole CNRS, Ventron II 2012 p 35
Density of states g(E) depends on dimensionality
E
Ecole CNRS, Ventron II 2012 p 36
Variations to the Mott Relation (3D, 2D)
1. For metals and heavily-doped semiconductors:
0
0 dEEfE)(
0
01 dEEf
TkEEE
ekS
B
FB )(
FEEE
)(
FF EEB
B
EEB
B
EETk
ek
EETk
ekS
)(ln)(
31
3
22
25
TkEE
ekS
B
FB
2. For semiconductors with band conduction, very dilute electron concentrations
Ecole CNRS, Ventron II 2012 p 37
1D: Landauer-Büttiker formalism (Butcher, 1990)
0
2
2
2
2222
0
2
2
2
F
F
B F
B
B F
B
E
g E
g E
g E
eG dEh E
k E EeGS dEh e E k
g E
G EeLT
f
fT
k E EeS G dET h e E k T
E
S Gf LT
,
2n m E
nmnmg E t
nmt transmission coefficient
total transmission probability n
m
f E Fermi-Dirac function
Variation of Mott formula
Ecole CNRS, Ventron II 2012 p 38
Mahan-Sofo theory
Transport function:
DOS
• Optimal DOS = delta function
• Fermi energy at 2.4 kBT from it
• As little background DOS as possible
Degradation of zTwith increase in background DOS
Ecole CNRS, Ventron II 2012 p 39
1. d THERMOPOWER ENHANCEMENT
ENERGY DIFFERENCEThermal => Electrical
0
1
f(E)
3D
-D
OS
(E)
N(E
)D
OS
(E)
Electron Energy E
N(E
)k k
COLD HOTFermi surfaces:
- -- -
-
- -
---
T
- -- --
V Blue = cold endRed = hot endof the sample
Seebeck coefficient or Thermoelectric power:
Mott N. F., Conduction in Non-crystalline Materials, Oxford, 1987
Maximize
Delta-function-like densities of states
1. f – levelsG. Mahan, Proc. Natl. Acad. Sci. USA
93 7436 (1996)
2. Quantum wiresHicks and Dresselhaus,
Phys. Rev. B 46 16631 (1993)
EE
)(
EE
ETk
ekS B
B
)()(
13
2
Ecole CNRS, Ventron II 2012 p 40
Theory: Enhanced zT in Bi Quantum Wires
0 4 8 12 16 20Energy (a.u.)
0
2
4
6
8
10
g(E)
EF
n(EF)
1S(E) [ + ]
E 2.(E – Enm)1/2
= scattering exponent : = -(1/2) for acoustic phonon = 0 for neutral impurity = 3/2 for ionized impurity
1Z (EF – En,m)1/2 . [ + ]
EF 2.(EF – Enm)
Lin, Sun & Dresselhaus, Phys. Rev. B 62 4610 (2000)
J. P. Heremans & al., Phys. Rev. Lett. 88 216801 (2002)
Ecole CNRS, Ventron II 2012 p 48
OPTICAL SPECTROSCOPY Bi-NANOWIRES
0 0.1 0.2 0.3 0.4 0.5E (eV)
1
2
3
4
5
6
Ref
lect
ion
(%)
Eg=0.4 eV
Infrared spectrumOf 8 nm diameterBi wires in SiO2Mark Myers
Optical energy gap:
Eg = 0.4 eV
J. P. Heremans, p. 324-329, Proc. 22nd Int.Conf. Thermoelectrics, IEEE, 2003
Ecole CNRS, Ventron II 2012 p 4949
Eg
COMPARISON with THEORYTheory Lin, Sun & Dresselhaus, Phys. Rev. B 62 4610 (2000)
Energy levels for each band are:
gLz
z
gLwyx
nmgLgLznm Em
kEdm
EEkE
22
2
2 281
22)(
5 10 15dw (nm)
0
500
1000
E g (m
eV)
Theory2 differentcrystallographic directions
Experiment:resistance vs temperature
Experiment:Infrared absorption
Ecole CNRS, Ventron II 2012 p 5050
THERMOPOWER 200 nm Bi-NANOWIRES
0 100 200 300T (K)
-40
-30
-20
-10
0
( V/
K)Bi nanowires200 nm diameter
Slope =-0.10 V/K 2
-0.5 V/K 2
cooldown
warmup
Pure Bimountedwith Ag-paint
Pure Bimountedwith Wood's
Te-doped Bi
Experiment:• Pure Bi nanowire• Bi nanowires doped n=type with
5x1018cm-3 Te
Theory:• Shubnikov-deHaas oscillations in magnetoresistance =>• electron and hole densities =>• Fermi energies =>• Partial electron and hole Seebeck =>• Seebeck coefficient with no adjustable parameters
J. Heremans & al., Phys. Rev. B 59, 12579-12583 (1999)
Ecole CNRS, Ventron II 2012 p 51
THERMOPOWER 4-15 nm Bi-NANOWIRES
Bi 200 nm diameter wires
Bulk Bi
15nm SiO2 sample 1
0 100 200 300T(K)
1x100
1x101
1x102
1x103
1x104
1x105
1x106
|S| (V
/K)
9 nm Bi/Al2O3 sample 1
9 nm Bi/Al2O3 sample 2
15nm SiO2 sample 2
4 nm BiVycor glass
• Data on bulk nanowire composites
Thermopower 9 nm wires = 1000Thermopower bulk Bi
• S(9 nm) T-1 @ T>100 K
• S(4nm) < S(9nm)• There is an optimum at 9 nm• Why? Localization !
J. Heremans & al., Phys. Rev. Lett. 88 216801 (2002)
Ecole CNRS, Ventron II 2012 p 52
Other ways to create distortions of the DOS in bulk materials
Ecole CNRS, Ventron II 2012 p 53
Resonant energy levels: definition
k
E
Donor level: "hydrogenoid" model, ED=R*
Resonant level:ED is IN the band
ED
ED
g(E)
E
Dispersion Density of states
Concept comes from atomic physics (1930’s)
Adapted to metals by Korringa & Gerritsen (1953), and Friedel (1956)
Ecole CNRS, Ventron II 2012 p 54
Origin of resonant levels
Position in space
Ene
rgy
Conduction band
Valence band
GapResonantstate
V
hyper-deepstate a L
Resonantimpurity
kk k ||0 EH
OEOH | | 00
kkk 3
0
|)(||
||)(|
daOc
EVHH
)(||2
2)(
||
2
22
0
22
EgVO
EE
VOc
k
k
Extended state (plane wave)
Localized state (atom-like wave)
Mixed state (resonant level)
Ecole CNRS, Ventron II 2012 p 55
Localized or delocalized? 1. Localized (atom-like) and extended (plane-wave-like) states cannot coexist at the same energy for a given configuration (Morell Cohen)
2. Concept of Wigner delay timeE
EOW
)(2
Bound state real space
Resonant state
Dephasing angle of conduction electron (d-character)
Resonance when the phase shift O(E) changes from 0 to over the energy interval
The value of ED is the energy where O (ED)=/2.
Localized state: W = ; Extended state: w = 0Resonant state: localized W , extended rest
Ecole CNRS, Ventron II 2012 p 56
Optimal Wigner delay time in Tl: PbTe
Contributions of the constituent atoms to this hump at EF are:Tl 12%, Pb 15%, and Te 50%
Low Tl contribution => Tl atoms as acting like a catalystTl allows the formation of the excess DOS of PbTe, but contributes minimally.
C.M Jaworski, B. Wiendlocha, V. Jovovic and J.P. Heremans, Energy Environ. Sci., 2011, 4, 4155
Ecole CNRS, Ventron II 2012 p 57
Too high Wigner delay time in Ti :PbTe
Contributions of the constituent atoms to this hump at EF are: Ti > 50%
Ti state becomes too localized
Jan D. König, Michele D. Nielsen, Yi-Bin Gao, Markus Winkler, Alexandre Jacquot, Harald Böttner, and Joseph P. Heremans, Phys. Rev. B, accepted, 2011
Ecole CNRS, Ventron II 2012 p 58
Tl: PbTe => thermopower enhancement
e
V
mm
bEEaEg
5.1
)(*
Shape of excess DOS(E) nearly free-electron like => excellent thermopower
Ecole CNRS, Ventron II 2012 p 59
Ti: PbTe => Fermi level pinning
Ecole CNRS, Ventron II 2012 p 60
Thallium in PbTe: correction
Resonant level
UVB
LVB
This not this
EF = 60 meV
Ecole CNRS, Ventron II 2012 p 61
Summary
• Low dimensions add a design parameter that breaks the two counter-indicated effects in zT
• They are not the only solutions possible– High anharmonicity mimics the benefits of low-dimensional
phonon scattering– Kondo, heavy-fermion and resonant levels mimic the effects of