transport a dimensions reduitesgdr-thermoelectricite.cnrs.fr/ecole2012/ermit2012... · 2012. 9....

Ecole CNRS, Ventron II 2012 p 1

Transport a dimensions reduitesJoseph P. Heremans

Department of Mechanical and Aerospace EngineeringDepartment of Physics

The Ohio State University, Columbus, Ohio 43210, [email protected]

Les notes sont en anglais, mais je parlerai francais (ni accents, ni cedilles)


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


FACTORS LIMITING zT

S2 = electronic properties of the solid ("Power factor")

Thermal conductivity = ELECTRONS + LATTICE LATTICE

Problem: counter-indicated properties: S, and are interrelated1. Increase n to increase => decrease S (Pisarenko S(n))

2. Add scattering to decrease LATTICE=> decrease

Low-Dimensionality add a new design parameter

Nanostructures reduce L more than

Nanostructures enhance S(n)

TSzT2

0

0.6

0 2n

S

S S2

n e

ne

eTnSeTnSzTL

22


Performance limits for zT

3

Glen Slack, CRC Handbook on Thermoelectrics, David Rowe, Ed., 1996

Hypothesis1. Amorphous limit (for PbTe)2. Mobility of Bi / single-crystal PbTe3. Effective mass of free electron4. Energy gap of PbTe


How do low-dimensional structures work ?

LnEm

TqvC

nSzT

LTvC

qTnSzT

nq

FV

e

VELECTRONICLATTICE

21*2

2

2

23

13

1

System of characteristic dimension dIfThen it is possible to choose d so that:and enhance • Very successful• Limit: minimal thermal conductivity

e

Quantum size effect

Concept: Hicks and Dresselhaus, PRB (1993)

Experiment: Heremans et al., PRL 88 216801 (2002)

ed

a


A little history1993: Theory / 2002 experiments– Hicks and Dresselhaus Theory (Phys.Rev.B 47

16631,1993)– Nanowires should give much greater ZT– Quantitative theory based on Bismuth nanowires– Enhanced Seebeck coefficient of bismuth nanowires

(Heremans, Phys. Rev. Lett. 88 216801, 2002)1996: Penn state Theory

Mahan and Sofo, PNAS 93 7436 (1996)2001: Venkatasubramanian (Nature 413 597, 2001)

– Bi2Te3/Sb2Te3 superlattices – Claim ZT=2.4 by reduction of thermal conductivity– Thin-films by MOCVD

2002: T. Harman et al., (Science 297 2229, 2002)– Claim ZT = 2 demonstrated on PbTe/PbSe Quantum Dots

• ½ due to reduction of thermal conductivity• ½ due to enhancement of Seebeck coefficient

2004: Kanatzidis: (Science 303 818 2004)• “LAST” nanoparticles of AgSbTe2 in PbTe: ZT= 2

2008: Chen/Ren/Dresselhaus: (Science 320 634 2008)• ZT= 1.4 on Ball-milled BiSbTe alloys


1. Thermal conductivity reductions

Thin films Bulk materials

MOCVD MBE ThermalMetallurgical processes

Ball-millingCompacting

Tetradymites(Bi1-xSbx)2(Te1-ySey)3

Lead salts(Pb1-xSnx)(Te1-ySey)

III-V compoundsIn1-xGaxAs + ErAs

Tetradymites(Bi1-xSbx)2(Te1-ySey)3

Lead salts(Pb1-xSnx)(Te1-ySey)(Pb1-xSnx)(Te1-ySy)(PbTe)(AgSbTe2)


Kanatzidis: LAST = (PbTe)1-x(AgSbTe2)xScience 303 818, 2004


Chen/Ren/Dresselhaus nano-Bi2 Te3Science 320 634, 2008


Chen/Ren/Dresselhaus nano-Bi2 Te3

Science 320 634, 2008

Direct measurement of zT:

zTTT CMAX 2


Silicon nanowiresMajumdar & Peydong Yang, Nature 451 163 (2008)

Also another paper but they invoke phonon-

drag at 200K!Heath, Nature 451 168 (2008)


Silicon nanowiresMajumdar & Peydong Yang, Nano Letters (2012) dx.doi.org/10.1021/nl3005868

Observations:1. Very strange T-dependence2. Strong correlation between surface roughness and reduction in thermal conductivity

I tend to believe it, but cannot explain it.


Silicon nanowires / Thermal conductivity of glassesMajumdar & Peydong Yang, Nature 451 163 (2008)

SiO2crystal

SiO2glass

Zeller & Pohl, 1971


Lattice thermal conductivity in reduced dimensions

R. Berman, Diamond Research 1976

R. Ruoff, Nat. Mater. 11 203 (2012)

Graphene

(10,10) C. nanotubes

Tomanek, PRL 20 4613 (2000)


Lattice thermal conductivity in reduced dimensions

R. Berman, Diamond Research 1976

B. Kelly, Physics of Graphite, ASP 1981

Diamond (3D) Graphite (2.5-D) Polydiacetylene (1D)

Single crystals T3

T2.5

T1

U UMorelli-Heremans, PRL 57 869 (1986)

T1


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


Spectral thermal conductivity

Asegun S. Henry, and Gang Chen Gang, Journal of Computational and Theoretical Nanoscience, 5, Number 2, pp. 141-152(12), (2008)


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


Starting point: AgSbTe2

2SzT el


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

→ Strong anharmonicity

→ Low

Tn

VMA atomL

322

3/13

Average Mass of AtomsDebye Temp

Volume per atom

# atoms in cellGrueneisen Parameter

Smattering of Constants


Material Systems: Cubic I-V-VI

[Ag,Na][Sb,Bi][Te,Se]2 systems– AgSbTe2:Single Phase Off-Stoichiometric Ag0.366Sb0.588Te

– AgSbSe2– AgBiSe2: cubic & hexagonal– Ag1-xNaxSbTe2: x= 0.2, 0.5– NaSbTe2– NaSbSe2– NaBiTe2


Origin of the very high Grueneisen parameter: lone pair electrons on Sb

]011[ // NTDISPLACEME Se]111[ // NTDISPLACEME Se

Polarizability density plot for s-electrons on Sb sites when Se atoms are displaced


XRD: Rock Salt Structure

Reference Peaks– ICDD PDF database

20 30 40 50 60 70

50

100

150

200

250

300300

10

AgSbTe2

OffStoich

65 2intref

7020 OS ref

Ag0.366Sb0.558TeAgSbTe2

0 20 40 60 80 100% Na (Ag1-xNaxSbTe2)

6.1

6.2

6.3

6.4

Latti

ce C

onst

ant (Å)

20 30 40 50 60 70Angle (2)

Inte

nsity

(A.U

.)

NaBiTe2

NaSbTe2

NaSbSe2

Ag0.73Sb1.12Te2

AgSbSe2

AgBiSe2 cubic

AgBiSe2 hex


Temperature Dependent XRD: Coefficient of Thermal Expansion

Linear Expansion Coefficient: α (ppm/K)

[AgSbTe] – 30.8 ± 0.9

AgSbSe2 – 24.5 ± 0.8

NaSbSe2 – 22.5 ± 1.0

NaSbTe2 – 27.7 ± 1

PbTe ~20

Angle (2)

Linear Expansion

y = 0.000187x + 6.063561R2 = 0.997944

y = 0.000137x + 5.954447R2 = 0.994554

y = 0.000142x + 5.780304R2 = 0.962200

y = 0.000173x + 6.311627R2 = 0.998309

5.7

5.8

5.9

6

6.1

6.2

6.3

6.4

6.5

0 100 200 300 400 500 600

Temperature (C)

Latti

ce P

aram

eter

(A)

AgSbTeNaSbSe2AgSbSe2NaSbTe2

Goal: Calculate Gruneisen Parameter:


Comparison: Theory and Experimental Grüneisen parameters

CBVcell)3(3

Linear coefficient of expansion

Bulk modulus

Unit Cell Volume

Specific heat

Grüneisen parameters fit theory much better than bulk moduli

Grüneisen parameter:

a/2 (nm) B (GPa) a/2 (nm) B (GPa)

NaSbSe2 0.2906 44 1.76 0.298 ‐ 1.5NaSbTe2 0.3103 35 1.57 0.316 ‐ 1.8NaBiTe2 0.3138 34 1.46 ‐

AgSbSe2 0.2819 78 3.54 0.29 129 3.71AgSbTe2 0.2972 67 2.34 0.304 45 2.11AgBiSe2 0.2893 74 2.5 0.29 ‐

calculated experimental


Measurement Low T Static Heater and Sink Method

1 10 100T (K)

1x10-5

1x10-4

1x10-3

1x10-2

1x10-1

1x100

C (J

/ gr

K)

AgBiSe2-TriAgSbSe2

Ag0.73Sb1.1Te2

AgBiSe2

NaSbTe2 NaSbSe2

NaBiTe2

1 10 100T (K)

0.1

1

(W

/m K

)

min

min


Measurement Low T Static Heater and Sink Method

Amorphous limit:Umklapp processes

i

T

x

x

iiB

i

edxexTnk

/

02

323/2

3/1

min16

1 10 100T (K)

0.1

1

(W

/m K

)

UmklappProcesses

resistive

NormalProcesses

not resistive

1/2 zone edgephonon temp.

AgSbSe2

Ag0.73Sb1.1Te2

Working Hypothesis:

1. Phonon-phonon interactions extremely intense at all temperatures

2. T>30 K: Umklapp processes limit thermal conductivity to amorphous limit

3. T<30 K: Excess conductivity: switch from Umklapp processes to Normal Processes


Measurement Results: RT and Above Flash Diffusivity

Material Kmin[W/m*K]

1/3*C*v*l[W/m*K]

AgSbTe2 0.44 0.73

300 400 500 600Temperature (K)

0.2

0.4

0.6

0.8

Ther

mal

Con

duct

ivity

(W/m

K) Ag0.366Sb0.558Te

Ag0.8Na0.2SbTe2

Ag0.5Na0.5SbTe2

NaSbTe2

*Cahill, Solid State Communications, v70:10, p972-930 (1989)

CL 31

i

T

x

x

iiB

i

edxexTnk

/

02

323/2

3/1

min16

*


Alloy Thermal Conductivity @ ~373KAg1-xNaxSbTe2

0 20 40 60 80 100y (% Se)

0.5

1

1.5

2

LW

/m K

)

PbTe PbSe

0 25 50 75 100Na Concentration (%)

0.3

0.4

0.5

0.6

0.7

Ther

mal

Con

duct

ivity

(W/m

K)

Conclusion:Alloy scattering ineffectivePhonon scattering mechanism: INTRINSIC

A composition(%) B

Generic AB alloy


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)


2. Power factor enhancements


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


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



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



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


Density of states g(E) depends on dimensionality

E


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


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


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


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


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)


Bismuth Quantum Wires

Bulk Bi is a semimetal

Brillouin zoneElectron Fermi surfacesHole Fermi surfaces

Eg

Bismuth quantum wires:Anisotropic, assume wire // [0112]• d > 49 nm : semimetals• d < 49 nm: semiconductors• Gap opens d -2

Lin, Sun & Dresselhaus, Phys. Rev. B 62 4610 (2000)


Experiments: grow Bi nanowires in porous host materials

1. Porous anodic alumina: pores 200 – 7 nm

2. Porous SiO2 (silica-gel): pores 15 nm

3. Porous Al2O3 (substrates for automotive gas sensors): pores 9 nm

4. Porous SiO2 : pores 8 nm (Batelle-PNNL labs)

5. Porous Vycor glass (Corning glass): pores 4 nm


Vapor-phase growth technique

Bi vapor

Vacuum chamber

Pump

Liquid Bi

Top plate

Clip

Crucible Heater

Crucible

Spacer plate

PorousAluminaHostPlate

Heremans & al., Phys. Rev. B 61 2921 (2000),Thrush & Heremans, US Patent Number 6,159,831 (2000)

1 st stage:heat crucible to evaporate Bi through the pores

Cleans the porous plate Bi vapor escapes into

vacuum

2nd stage: cool crucible down slowly

Top plate is always cooler than crucible

Nanowires grow into the pores from top to bottom

Top plate heater


Scanning Electron Microscope Images

200 nm Bi wires in anodic Al2O3 15 nm Bi wires in SiO2


TRANSPORT MEASUREMENTS

Hex BN holder

Sample

T

V

Ag-paint or Wood’s metal

Heater

Sample

Hex BN holder

Heater TV


1 10 100T (K)

0.0

0.5

1.0

1.5

2.0

R [

T ] /

R [

T=30

0 K

]

200 nm

70 nm

36 nmWire diameter

Bulk Bi

.

50 nm

28 nm

Blue: semimetalRed: semiconductors

7 to 10 nm

Host: anodic Al2O3

LE

HE

70 nm d 200 nm: metalsBrandt &al. (Sov. Phys. JETP 65 515, 1987)

d 70 nm: semiconductorsZ. Zhang & al. APL 73 1589, 1998

Heremans et al, Phys. Rev. B 61 2921 (2000)

ELECTRICAL RESISTANCE Bi-NANOWIRES


ELECTRICAL RESISTANCE Bi-NANOWIRES

0 100 200 300T (K)

1

10

100

1000

R (T

) / R

(300

K)

0.1

1

10

100

9nm, Al2O3

sample 1

9nm, Al2O3

sample 2

15nm, SiO2

sample 1

15 nm, SiO2

sample 2

Eg=0.18eV

Eg = 0.19 eV

Eg = 0.39 eV

Eg = 0.29 eV

4 nm Bi/Vycor

Thermally activated conductivity

= n e = n0 eEg/2kT e

So we fit: R=R0 e-Eg/2kT

Host: SiO2 Al2O3 Vycor glass

dw 15nm 9nm 4nm

Mechanism Semi-Cond.

Semi-Cond.

Locali-zation

Eg (eV) 0.18 –0.19

0.29 –0.39

-

J. P. Heremans & al., Phys. Rev. Lett. 88 216801 (2002)


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


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


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)


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)


Other ways to create distortions of the DOS in bulk materials


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)


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)


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


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


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


Tl: PbTe => thermopower enhancement

e

V

mm

bEEaEg

5.1

)(*

Shape of excess DOS(E) nearly free-electron like => excellent thermopower


Ti: PbTe => Fermi level pinning


Thallium in PbTe: correction

Resonant level

UVB

LVB

This not this

EF = 60 meV


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

quantum wires on the DOS

LnEm

TqvC

nSzT

FV

e

21

2

2

23

1

*

transport a dimensions reduitesgdr-thermoelectricite.cnrs.fr/ecole2012/ermit2012... · 2012. 9....

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