Hvar 2008 Sept. 21

Thermoelectrics ofK d S i d t dKondo Semiconductors and

Intermetallic Clathrates

T. TAKABATAKEDept. of Quantum Matter and

Inst. for Advanced Materials ResearchHiroshima University

Outline)(

2S=ZFigure of merit

Two strategies for high performance thermoelectric materials

)( elL κκρ +⋅

1. G.D. Mahan and J.O. Sofo (1996), a sharp singularity in thedensity of states very near EF enhance power factor S2/ρdensity of states very near EF enhance power factor S /ρ

2. G. A. Slack (1995), “phonon glass- electron crystal“

rattling of guest atoms reduce κ without increasing ρrattling of guest atoms reduce κ without increasing ρ1. Ce- and Yb-based Kondo semiconductors

Formation of a pseudogap in the renormalized band enhances the S while maintaining the ρ metallic. high power factor.

2. Intermetallic clathrates Type-I Ba8Ga16Sn30 is the best example of phonon glass due to off-center rattling of guest ions.

Thermoelectric Effects and Devices

Heat input Heat absorbedHeat input Heat absorbed

electronhole

Neuromagnetometer using V

Directly convert

g gSuperconducting Quantum Interference DevicesSeebeck effect Peltier effect

Quiet refrigerator without chloroflurocarbonsDirectly convertheat to electricityRecover vehicle waste heat

Quiet refrigerator without chloroflurocarbonsSmall-scale cooling of laser diodes, radiation detectorsTmin=150 KGoal: below 77K cool superconducting deviceswaste heat Goal: below 77K, cool superconducting devices

down to the operating temperature

Low efficiency <10%

Temp difference

Low efficiency <10%

Temp. difference∆Tmax = 2

2

1cZT

Figure of meritZ = S2/ ρ κ

2

Bi-TeZ S / ρ κS: thermopower; high>100 µV/Kρ: electrical resistivity; low<1 mΩcm G J S d d E S T bρ: electrical resistivity; low 1 mΩcmκ: thermal conductivity; low<2 WK/m

G.J. Snyder and E.S. Toberer,Nature materials, 7(2008)105.

High S and low ρ hardly coexist in conventional semimetals and semiconductors, but do ina Kondo semiconductor with a pseudogap.

Giant thermopower in 4f electron Kondo compoundsD Jaccard and J Sierro in Valence Instabilities (1982)D. Jaccard and J. Sierro, in Valence Instabilities (1982)

Mott’s equation

⎤⎡22Yb 4f 13+δ

80CePd3

0

µεεσπ

=⎥⎦⎤

⎢⎣⎡

∂∂−

=ln

3)(

22

eTkTS

Ce 4f 1-δ

Yb 4f 13+δ60

40

V / K

)

40

-20 YbSiTK =300 K

µεεετπ

=⎥⎦⎤

⎢⎣⎡

∂∂

+∂

∂−= cc N

eTk lnln

3

2220

0

S ( µ

V

CeCu2Si2 -60

-40

T 15 KThe lifetime is determined by the

-20

3002001000

-80

3002001000

YbAl3

TK =15 K

Kondo scattering.

)(21 2π NVJ=5/2 2J+1= 6 J=7/2 2J+1= 8

3002001000T ( K )

3002001000T ( K )

)()(

εετ fcf

c

NVh

=0/)( >∂∂ εεfNS >0 p-type

S <0 n-type

TSmax= TKH / 2

Dimension-less figure of merit ZTf C Pd YbAl d F Sifor CePd3, YbAl3 and FeSi

G Mahan B Sales J SharpG. Mahan, B. Sales, J. SharpPhysics Today 50 (1997) No.3, p.42.

Z S 2/ Z = S 2/ρ κ= 0.9 for Bi-Te

f 20 W/K @300Kκ of 20 W/Km @300Kis ten times of Bi-Te

In a good metal like CePd3, conduction electrons have a large contribution to κa large contribution to κ.κ = κel +κph

Ternary Ce valence-fluctuating compounds witha complex structure and a low carrier densitya complex structure and a low carrier density

Quasi Kagome latticeCeRhSn, CeRhIn TK=250 K

Ce momentsCe momentsare frustrated

M. S. Kim et al., PRB 68(2003) 054416.

No enhancement of Scompared to CePd3

κ is reduced to a level smaller than half Z is comparable

Enhance SCeTX honeycomb structure

Enhance Sat T < 100 K

Gap openingin f bands

KondoKondo semiconductors

CeNiSn CeRhSbCeNiSn, CeRhSb,CeRhAs, C

V(A3) Eg/kB (K)Ortho.

Ce3Bi4Pt3CeOs4Sb12,

gCePtSn 276.9 ---CeNiSn 264.3 14C RhSb 269 1 28

Cubic

4 12YbB12

CeRhSb 269.1 28CeRhAs 239.1 280

High thermopower at low T

150

100CeRhAs

CeOs4Sb12

0

50

V / K

)

CeRhSb

CeNiSnSmax >100 µV/Karises from gap formation

-50

0

S ( µ

V a ses o gap o at o

Ts: incoherent Kondo

-100

50

YbB12

T

s

-1501 10 100

TS

T ( K )

Opening of a narrow gap on cooling

Hybridization gap modelMagnetic susceptibility y g p

10

C PtSSteep slope of DOS

Conductionband8

u / m

ol ) CePtSn

　局在磁気モーメント有S eep s ope o OSenhances S

local 4f momentZ=S2/ρ κ

Gap Eg

6

4

10-3

em

u

CeNiSnB//a

Residual carriers retainhigh electrical conduction

4

2化率

　χ (

CeRhSbχ

Density of states0

磁化

3002001000

CeRhAs　　局在磁気モーメント無4f electron band

3002001000T ( K )

A pseudogap in the 4f electronic states of CeRhSb

Band structure

p g p

Photoemission t calculation

K. Shimada et al., F. Ishii and T. Oguchi, i

spectroscopy

PRB 66, 155202 (2002).

JPSJ 73, 145 (2004).semi-conductor

Electron dente

nsity

semimetal

nsity of statesaliz

ed In

t

metal

s / eV atomN

orm

a

Binding Energy (meV)Jz=±3/2

Kondo Metal Semimetal SemiconductorCePtSn CeNiSn, CeRhSb CeRhAs

Thermopower Thermal conductivityElectrical resistivity

4 I II b

S (µV/K) κ (W/Km)ρ (µΩcm) 100

80 CeRhAs CePd3

103

104

CeRhAsI II b 80

60

40

CeRhAs

CeRhSbCePd3 101

CeRhAs

3

102

10CeRhSbCeNiSn

40

20

0CeNiSn

CeNiSnCeRhSb

T *

101

10

CePtSnCePd3

0

-20

40

CePtSn

∆T II b

100 CePtSn∆T II b

T

1 10 100T ( K )

-401 10 100

T ( K )1 10 100

T ( K )S is enhanced at low Tby the gap formation

κph at high T is reduced by valence fluctuations

ρ remains low in the pseudo-gapped state

100

m) CeRhAsI II a

II bA i t i TE ti

1

10

ρ ( m

Ωcm

II b

II cAnisotropic TE properties

in CeRhAs

200

150

100V /

K ) ∆T II a

S//a of 186 µV/K is among th hi h t k f

12

100

50

0

S (

µV II c

II b

the highest known forCe-based compounds.

Fi f it 12

8

4(W /

K·m

)

∆T II b

II a

II c

Figure of meritZ = S2 / ρ κfor //c 1x10-3 /K at 115 K

4

0κ

( II a

1.00 81 )

c

cf. Z = 2x10-3 /K at 100 K 0.80.60.40.2Z

( 10-3

K-

a b

for CsBi4Te6the best p-type materialD Y Ch t l 0.0

1 10 100T ( K )

D.-Y. Chung et al., Science 287, 1024 (2000).

Ce3Bi4Pt3 cubic Y Sb Au typecubic Y3Sb4Au3-type

structure

M. F. Hundley et al., PRB42 (1990) 6842, Physica B171(1991)254.

K. Katoh and T. Takabatake,

Tetragonal point symmetry for Ce

,J. Alloys Compd. 268(1998)22.

CeT4Sb12 (T = Ru, Os)f

Issue: Does rattling of Ce scatters phonons?cubic filled skutterudite

120

100

Narazu et al., JPSJ 77 Suppl. A (2008).

80

60

40

S ( µ

V / K

)

CeOs4Sb12 II <001>

Ru Os

10

20

0

S

1

ρ ( m

Ωcm

)

Semimetallic for T = RuTakeda et al

6

)

Ce “rattles” in T8Sb12 cage

Takeda et al., Physica B 259-261 (1999) 92. Semiconducting for T = OsE.D. Bauer et al.,

4

2κ ( W

/ K

·m

Crystalline peakNo “rattling” effect Weak peak

E.D. Bauer et al., J. Phys.: Cond. Matt. 13 (2001) 4495.

0300250200150100500

T ( K )

107

x = 0Yb Lu B

Yb-based Kondosemiconductor

Photoemission

B103

105

µ Ω

cm )

0.25

0.01

0.05

0 50

Yb1-xLuxB12semiconductorYbB12

B

10-1

101

ρ (µ

0.75

1

0.50

-40

0

V /

K )

0.25 0.05

0.751

0.500.875

10 1

-120

-80S

( µV

x = 0

0.01

30

20)

x = 0Yb 5dB122p

F Iga et al

Yb 4f

20

10

κ (W

/ K

m)

0 25 κ

0.05

F. Iga et al.,J. Magn. Magn. Mater.226-230, 85 (2001).

Y T k d t l PRB73

hν =15.8eV

0

1 10 100T ( K )

0.25 κel

Binding energy (meV)Y. Takeda et al., PRB73, 033202 (2006).

Power factor S2/ρ : maximum heat Figure of merit ρabsorbed by the Peltier effect

gZ＝ S2/ρκ

4

CsBi4Te6

60

50

CeRhSb

Bi2Te3

CePd3

3

K-1) Bi-Te

CeNiSn

50

40

W /

cmK2 )

CsBi4Te6

2

Z ( 1

0-3 K

CeRhSb30

20

子 P

F ( µ

W

CeRhAs

YbB12

1Ce0.9Fe3CoSb12

CeRhAs

YbB12

10

0電

力因

子CeNiSn

CeOs4Sb12

0300250200150100500

Eu8Ga16Ge30CePd3

0

300250200150100500

Eu8Ga16Ge30 Ce0.9Fe3CoSb12Zr0.5Hf0.5NiSn

T ( K ) T ( K )

Potential of Kondo semiconductors for

H b idi ti f 10 100 K i 4f l t t

TE cooling application

• Hybridization gap of 10-100 K in 4f electron systems large thermopower of 100 µV/K coexists withlow resistivity of 1 mΩcm high power factor at T <100 K

• How to reduce the thermal conductivityAlloying cage atoms in Ce(Ru1-xOsx)4Sb12 change rattling ofCe ions incoherent, scatters the acoustic phonons effectively

Superlattices; CeRhSb/CeRhAs→ boundaries scatter phonons

• Approach to a Peltier p-n junction search for Yb-basedsemiconductors compatible with Ce counterparts.p p

Cool superconducting devices below 80 K

Acknowledgements

T. Sasakawa, S. Narazu, D. Hirata, M.S. Kim,J Kitagawa K Umeo F IgaJ. Kitagawa, K. Umeo, F. Iga

Dept. Quantum Matter,ADSM Hiroshima UniversityADSM, Hiroshima University

Y. Takeda, Spring-8H Sugawara Fac Integr Arts & Sci Univ TokushimaH. Sugawara Fac. Integr. Arts & Sci., Univ. Tokushima

H. Okamura Dept. Physics, Kobe Univ.D T Adroja ISIS RAL UKD.T. Adroja ISIS, RAL, UK

Off-Center Rattling and Thermoelectric Propertiesof β-Ba8Ga16Sn30o β a8Ga16S 30

Phonon glass- Electron crystalYes Perhaps

M A A il t l PRB 74 125109(2006)M. A. Avila et al., PRB 74,125109(2006)APL. 92,041901(2008)

K Suekuni et al PRB 75 195210(2007)K. Suekuni et al., PRB 75,195210(2007)PRB 77, 235119(2008)

JPSJ 77, Suppl. A, 61 (2008)

Y. Takasu et al., PRB 74, 174303 (2006)PRL 100, 165503 (2008).

CollaboratorsHiroshima University

Crystal growth and TE measurements

Other Institutions

EXAFSCrystal growth and TE measurementsM.A. Avila UNICAMP – BrazilK. Suekuni, M. Yamamoto, K. Umeo

F. Bridges UCSD – Santa Cruz

XPSK Tanigaki Tohoku U Sendai

Single-crystal XRDH. Fukuoka, S. Yamanaka

K. Tanigaki Tohoku U. – Sendai

Inelastic neutron scatteringC. H. Lee AIST – Tsukuba

Raman scatteringY. Takasu, M. Udagawa

ESRP.G. Pagliuso, C. Rettori g

Ultrasound measurementsI. Ishii, T. Suzuki

G ag uso, C ettoUNICAMP - Brazil

I. Ishii, T. Suzuki

XPS K. Shimada, M. Taniguchi

Dimorphic structure ofDimorphic structure of AA88EE1616XX3030

type I Pm3n

Ba8Ga16Si30B G G

β

A(2)

Ba8Ga16Ge30Ba8Ga16Sn30Ba8In16Ge30

H.G. Schnering et al.,(1998).

A(2) Sr8Ga16Si30Sr8Ga16Ge30Sr8(Al,Ga)16Si30 K. Kishimoto et al.,(2008).

type VIII I43mA(1)

8( )16 30

Eu8Ga16Ge30α S. Paschen et al., (2001).

Ba8Ga16Sn30 W. Carrillo-Cabrera et al.,8 16 30

Sr8(Al,Ga)16Si30

Eu Ga Ge

W. Carrillo Cabrera et al., (2002).

S P h t l (2001)

K. Kishimoto et al.,(2008).

Eu8Ga16Ge30 S. Paschen et al.,(2001).

Phonon glass and Electron crystalPhonon glass and Electron crystalB C Sales et al PRB 63 245113 (2001)B.C. Sales et al., PRB 63, 245113 (2001).

ρ < 0.8 mΩcmκL < 1 W / m K

typetype--II AA88GaGa1616GeGe3030

BGGBGG n-typeS(300 K) ~ -50µV/K

EGGEGG

S(300 K) ~ -50µV/K

SGGSGG

κκLL of of A88GaGa1616GeGe3030: : GuestGuest--size and size and structurestructure--type dependencetype dependence

Ba2+Ba

Sr2+

B. C. Sales et al., PRB 63,245113 (2001).

Eu2+

S. Paschen et al., PRB 64, 214404 (2001).

α

β

Tc

βTc Split sites

B.C. Chakoumakos et al., J. Alloys Compd. 322, 127 (2001).

0.4Å

Cage-size tuning in Sr8Ga16Si30-xGex

K. Suekuni et al. PRB 75,195210(2007)

X=0 X=30a =10.446 10.726 Å 3% up

quasi on-center off-centerquasi on center off center

41

35 = θE (K)“Crystal”

5956

46

“Gl ” 59“Glass”

Phonon dispersion curves observedby inelastic neutron scattering

C. H. Lee et al., J. Phys.: Conf. Seri.92 (2007) 012169, JPSJ 77 (2008) Suppl. A 260.M. Christensen et al., Nature Materials (2008).

Th ti b h i fl tt d b th k h b idi ti b t th t d tiThe acoustic branch is flattened by the weak hybridization between the guest and acoustic modes. Avoided crossing contradicts with incoherent guest vibrations.Force constant F(Sr2-G1) = 0.008 mdynn / Å = 0.1×F(Sr1-G2)

Strong dependence of κL of Ba8Ga16Ge30g p L 8 16 30on the type of charge carries

A Bentien et al PRB69 (2004) 045107 PRB71 (2005) 165206A. Bentien et al., PRB69 (2004) 045107, PRB71 (2005) 165206, PRB73 (2006) 094301.

I , III n-typeNo peak for VI p typeNo peak for VI p-type,as similar with κ(T) forVII SrGaGe n-typeypVIII EuGaGe n-type• Identical crystal structure

for n- and p-type crystals.for n and p type crystals.• glasslike behavior and κ ∝ T 1.5 are due to scatteringfrom charge carriers.

• A dip at 10-30 K is scatteringfrom bound / localized charge carrier states.

CarrierCarrier--type dependence of type dependence of κκLL of of α- BaBa88GaGa1616SnSn30 30 and Baand Ba88GaGa1616GeGe3030

M. A. Avila et al., PRB 74, 125109 (2006).

A8Ga16Ge30

M. Christensen et al., Physica B 385, 505 (2006).

8 16 30

Sr

p-Ba

n-Baα n-Ba

κL(n) > κL(p)

Issue: Which plays more important role in the glass-like κ L?

Phonon lifetime τ (n) > τ (p)

ssue c p ays o e po a o e e g ass e Lguest rattling or charge carriers

Compared with ferromagnetic EuGaGe nonmagnetic BaGaSn has anCompared with ferromagnetic EuGaGe, nonmagnetic BaGaSn has an advantage in studying low-energy excitations.

Carrier type tuning in single crystals of Carrier type tuning in single crystals of ββ--BaBa88GaGa1616SnSn3030

sample Crystal Latticename composition parameter

Carrier type can be tuned by adjusting the Ga/Sn ratio i th i iti l iti

a e co pos t o pa a eteBa Ga Sn a (Å)

n1 8.0(1) 15.9(2) 30.1(2) 11.685(1)

in the initial composition.

The lattice parameter is largest among type I clathratesn2 7.9(1) 15.8(2) 30.2(2) 11.685(1)

p1 7.8(1) 15.8(2) 30.2(2) 11.707(1)2 8 0(1) 15 8(2) 30 2(2) 11 708(1)

0.2%among type-I clathrates.

p2 8.0(1) 15.8(2) 30.2(2) 11.708(1)

crystalcrystal

fluxflux

Thermopower Thermopower S S andand Electrical resistivity Electrical resistivity ρρ in in β-Ba8Ga16Sn30

200

300 p2

100 n1

0

100

200

V /

K )

p1

Ω c

m )

n1p2

n2

-200

-100

0

S (

µ

1

n210

ρ (

mΩ

p1

3002001000T ( K )

-300n1

13002001000

T ( K )T ( K ) T ( K )

T bl i t

n (1018/cm3)@290 K

Tunable carrier type

Carrier density n can be changedin a limited range.

n1 _

n2 10

p1 8.5in a limited range. p2 5.5

OffOff--center split sites of Ba(2) in center split sites of Ba(2) in β-Ba8Ga16Sn30

• 24k site(0.2437, 0.5, -0.03656) n1(0.2430, 0.5, -0.03685) p1

Ba(2)• doff

0.434 Å n10 439 Å p1

( ) p

cage center 6d (0.25, 0.5, 0) 16i

24k

( )

71 % n1, 68 % p1• Ga occupancy on 6c site

0.439 Å p1

( , , )6c (0.25, 0, 0.5)

24k

0 50)

(b)

• doff increases on cooling0.06

0 05β-phase(a)

0.50

0.45

acem

ent (

Å

β-Ba8Ga16Sn30p1n1

(b)0.05

0.04

0 03Å2 ) Ueq_Ba(2)

p1n1

0.40

0.35-c

ente

r dis

pla

Sr8Ga16Ge30

β-Eu8Ga16Ge300.03

0.02

0.01

Ueq

(

Ueq_Ba(1)

U 0.30off-

3002001000T ( K )

Sr8Ga16Ge30

03002001000

T ( K )

Ueq_cage

Guest vibration mode in Raman scattering spectraGuest vibration mode in Raman scattering spectra

guest(2) site

Y. Takasu et al.,PRL, 100, 165503 (2008).T2g modes

Sr8Ga16Ge30

λ = 514.5nm

Raman active@2K

. uni

ts)

8 16 30

@2K

nsity

(arb

.

Eu8Ga16Ge30

@5K

Inte

n

Ba8Ga16Sn30pGuest mode

3002502001501005001

n

[001]

[110]

• Ba(2) vibrational energy15 cm-1 ≅ 21 K = θR (Cp) Energy Shift (cm-1) [001]

• Ga/Sn cage mode energies are lower than that for Ga/Ge cage(θD=210 K) (θD=312 K)

Specific heat Specific heat CC：： low-energy off-center rattling

5

200

10-3

Ba8Ga16Sn30R. C. Zeller et al., Phys. Rev. B 4, 2029 (1971).

5

4

3g K

4 )x

10-6

glass

SiO2

mol

K4 )x

glassSiO2

β Insulator3

2

/ T 3 (

µJ

/ g

crystal

100

/ T3 ( m

J / n1

p1t l1

0

C /

1 10 100

crystal

0

C /

1 10 100

n2 p2

crystalα

T ( K )1 10 100T ( K )

peak@ 4 Kk @ 10 K i d d

α β• peak@ 4 K,CE+CD do not reproduce the data

• peak @ 10 K is reproducedby CE+CD+Cel

CE: Einstein θE = 50 K CD: Debye Cel: electronic

T - linear coefficient of specific heat

10

0-6

SiO

100

802 )

Ba8Ga16Sn30R. C. Zeller et al., Phys. Rev. B 4, 2029 (1971).

8

6

µJ /

g K

2 )x

10

glass

SiO2

60

mJ

/ mol

K 2

β -phase n1

β-phaseSiO2

Tunneling4

2C /

T 3 (

µ

crystal40

20C /

T (

m

α -phase p1 n2

α-phase crystal

TunnelingC = DT

0543210

T 2 ( K )0

1.00.80.60.40.20.0T 2 ( K2 )

p2

T ( K )

A ( J/ l K2)

2/ BTATC +=Free electron model

β phase 29 >>

A (mJ/mol K2)α-phase 8

electronic γ carrier density n5 mJ/mol K2 ⇐ 4x1019 /cm3

3 mJ/mol K2 ⇐ 1x1019 /cm3

θD = 207 K

β-phase 29 >>Tunneling DT ＞＞ Electronic γ T

3 mJ/mol K ⇐ 1x10 /cmθD = 218 K

from sound velocity

Thermopower Thermopower S S andand Electrical resistivity Electrical resistivity ρρ in in β-Ba8Ga16Sn30

200

300 p2

100 n1

0

100

200

V /

K )

p1

Ω c

m )

n1p2

n2

-200

-100

0

S (

µ

1

n210

ρ (

mΩ

p1

3002001000T ( K )

-300n1

13002001000

T ( K )T ( K ) T ( K )

T bl i t

n (1018/cm3)@290 K

Tunable carrier type

Carrier density n can be changedin a limited range.

n1 _

n2 10

p1 8.5in a limited range. p2 5.5

OffOff--center split sites of Ba(2) incenter split sites of Ba(2) in β-Ba8Ga16Sn30

• 24k site(0.2437, 0.5, -0.03656) n1(0.2430, 0.5, -0.03685) p1

Ba(2)• doff

0.434 Å n10 439 Å p1

( ) p

cage center 6d (0.25, 0.5, 0) 16i

24k

( )

71 % n1, 68 % p1• Ga occupancy on 6c site

0.439 Å p1

( , , )6c (0.25, 0, 0.5)

24k

0 50)

(b)

• doff increases on cooling0.06

0 05β-phase(a)

0.50

0.45

acem

ent (

Å

β-Ba8Ga16Sn30p1n1

(b)0.05

0.04

0 03Å2 ) Ueq_Ba(2)

p1n1

0.40

0.35-c

ente

r dis

pla

Sr8Ga16Ge30

β-Eu8Ga16Ge300.03

0.02

0.01

Ueq

(

Ueq_Ba(1)

U 0.30off-

3002001000T ( K )

Sr8Ga16Ge30

03002001000

T ( K )

Ueq_cage

Guest vibration mode in Raman scattering spectraGuest vibration mode in Raman scattering spectra

guest(2) site

Y. Takasu et al.,PRL, 100, 165503 (2008).T2g modes

Sr8Ga16Ge30

λ = 514.5nm

Raman active@2K

. uni

ts)

8 16 30

@2K

nsity

(arb

.

Eu8Ga16Ge30

@5K

Inte

n

Ba8Ga16Sn30pGuest mode

3002502001501005001

n

[001]

[110]

• Ba(2) vibrational energy15 cm-1 ≅ 21 K = θR (Cp) Energy Shift (cm-1) [001]

• Ga/Sn cage mode energies are lower than that for Ga/Ge cage(θD=210 K) (θD=312 K)

Specific heat Specific heat CC：： low-energy off-center rattling

5

200

10-3

Ba8Ga16Sn30R. C. Zeller et al., Phys. Rev. B 4, 2029 (1971).

5

4

3g K

4 )x

10-6

glass

SiO2

mol

K4 )x

glassSiO2

β Insulator3

2

/ T 3 (

µJ

/ g

crystal

100

/ T3 ( m

J / n1

p1t l1

0

C /

1 10 100

crystal

0

C /

1 10 100

n2 p2

crystalα

T ( K )1 10 100T ( K )

peak@ 4 Kk @ 10 K i d d

α β• peak@ 4 K,CE+CD do not reproduce the data

• peak @ 10 K is reproducedby CE+CD+Cel

CE: Einstein θE = 50 K CD: Debye Cel: electronic

T - linear coefficient of specific heat

10

0-6

SiO

100

802 )

Ba8Ga16Sn30R. C. Zeller et al., Phys. Rev. B 4, 2029 (1971).

8

6

µJ /

g K

2 )x

10

glass

SiO2

60

mJ

/ mol

K 2

β -phase n1

β-phaseSiO2

Tunneling4

2C /

T 3 (

µ

crystal40

20C /

T (

m

α -phase p1 n2

α-phase crystal

TunnelingC = DT

0543210

T 2 ( K )0

1.00.80.60.40.20.0T 2 ( K2 )

p2

T ( K )

A ( J/ l K2)

2/ BTATC +=Free electron model

β phase 29 >>

A (mJ/mol K2)α-phase 8

electronic γ carrier density n5 mJ/mol K2 ⇐ 4x1019 /cm3

3 mJ/mol K2 ⇐ 1x1019 /cm3

θD = 207 K

β-phase 29 >>Tunneling DT ＞＞ Electronic γ T

3 mJ/mol K ⇐ 1x10 /cmθD = 218 K

from sound velocity

Soft Potential Model for CC((TT) of Glasses) of GlassesM. A. Ramos and U. Buchenau, 1998

)()( 42 xxDxDWxV ++=

Anharmonic potential

W : characteristic energy of the potential)()( 21 xxDxDWxV ++= Ps : density of the soft modesA : distribution broadness of the soft modes

• SM: soft mode term

CL(T) = CSM + CTS + CDebye

νν deexkgC x

x

BSM 2

2

0 )1()( −

−∞

−= ∫

( ) ( )∫ −−⎠⎞

⎜⎝⎛=

1

0

22264

]12/exp[81)( ttWhAdt

Wh

WPhg s ννν

texp, τmin: constantSoft vibrational density of states

( ) ( )∫⎠⎝ 08 WW

• TLS: two level system (tunneling) term⎞⎛⎞⎛

3/12

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

)(ln

91

6 min

exp3/123/12

Tkt

TkWPC

BB

STS τ

π CTLS ∝ T

Analysis of C by the soft potential modelAnalysis of C by the soft potential model200

x10-3

( TH

z-3 )

nits

)

Vibrationaldensity

/ mol

K4 )x

Ba8Ga16Sn30

g ( ν

) / ν

2 (

2 01 51 00 50

(arb

. un y

100C

/ T3 ( m

J /

Sr8Ga16Ge30p

nEu8Ga16Ge30

2.01.51.00.50ν ( THz ) guest(2)

0

C

TLSSM 0

1 10 100T ( K )

SMDebye

A W/k P θ h t i ti fA W/kB Ps θR characteristic energy of (K) (mol-1) soft modes (off-center rattling)

SGG 0.002 4.4 2.0×1021 0.7 THz = 35 K21β-BGS 0.017 3.5 6.4×1021 0.4 THz = 20 K

Interacting dipole model by Nakayama and Kaneshita, Cond mat, 0809.1070v1

Phonon glass behavior in Phonon glass behavior in κκ ofof β-Ba8Ga16Sn30

t l

R. C. Zeller et al., RRB 4, 2029 (1971).

1α-phase n1 p2α-phase

crystal

SiO2∝T3

1

m ) p1

n1

n2

Sr8Ga16Ge30

crystalline

glass

0.1

W /

K m

β h

n1p1v - SiO2

β-phaseT2 1 glass

0.01κ L

( β-phaseβ-phase

glass-like

∝T2.1

∝T2

• lowest among type-I clathrates0 001

∝T1.9∝T1.7

• lowest among type-I clathrates • κL(n) ≅ κL(p)

0.0010.1 1 10 100

T ( K )

off-center rattling is most responsible for the glass-like κ L

TRR model forfor GlassGlass--like like κκLL((TT))

( ) )(1)( TlvTCdT D ωωωκω

∫=

• Debye approximationJ. L. Cohn et al.,RRL 82, 779 (1999).

• Average phonon mean free path1 1

( ) ),(,3

)(0

TlvTCdT DebyeL ωωωκ ∫=

( ) min1111

1 1llll

lRresTS +++

= −−−−

−

• Phonon scattering process

Tunneling states

Phonon scattering process

1

31 11

22tanh

−−

⎠⎞

⎜⎝⎛ ++

⎠

⎞⎜⎜⎝

⎛

⎠

⎞⎜⎜⎝

⎛=

TBkA

TkkAl B

TSωω

h

hh

BA

A

k θResonant scattering by guest vibration

∑ TC 22ωC

322 ⎠⎜⎝⎠

⎜⎝⎠

⎜⎝ TBTkk BB

TS ωh B

Rayleigh scattering

hRB

Rik θωω ==( )∑

Γ+−=−

i iii

ires

TCl22222

1

ωωωωωC

A/B ∝ the density of tunnel states coupled to phonons Rayleigh scattering

C : resonant scattering strengthD : Rayleigh scattering strength

41

⎟⎟⎠

⎞⎜⎜⎝

⎛=−

BR k

Dl ωhD

Analysis of Analysis of κκLL((TT) by) by TTRRRR modelmodel1

Eu8Ga16Ge30: A. Bentien et al., PRB 71, 165206(2005).

1Sr8Ga16Ge30Eu8Ga16Ge30

ResonanceRayleigh

0.1

( W /

K m

)

pBa8Ga16Sn30

Tunneling states

0.01κ L

nBa8Ga16Sn30

∝ T2

Tunneling states0.001

0.1 1 10 100T ( K )

parameter Unit SGG EGG n-BGS

A/B 105 K/m 4 5 22

C1 1030 /(m s2 K2) 1.3 1.1 2.0Tunneling state density

θ 1K 103 82 78

C2 1030 /(m s2 K2) 3.9 3.3 6.0

θ K 35 25 20 Rayleigh scattering center:

Resonant scattering strength

θ 2K 35 25 20

D 1 /(m K4) 1.8 8 10

θ DK 312(BGG) 312(BGG) 210

Rayleigh scattering center:The positional disorder of A(2) among off-center sites

Structural parameter relevant to low κL

2 0 Ba8Zn8Ge38

Ba8Ni6Ge40 type-1l h2.0

/ K m

)

Ba8Ga16Ge30 Cs8Zn4Sn42

8 8 38Ba8Al16Si30 Ba8Cu6Ge40

Cs Sn44

clathrates24k6d

1.5

0K) (

W / Ba8Ga16Si30

Rb8Zn4Sn42

Rb8Ga8Sn38

Cs8Sn44

6d 24k (cage)1.0

κ L(1

50Eu Ga Ge

Ba8In16Ge30

Sr8Ga16Si30

β Ba Ga Sn

Sr8Ga16Ge30

Cage radius Cage radius = RR6d 6d -- 24k24k0.5

4.84.64.44.24.0

Eu8Ga16Ge30 β − Ba8Ga16Sn30 Ba8In16Ge28Sn2

κL (150K) depends on the cage radius ( Å )RR6d 6d -- 24k 24k (A8E16X30)

= R6d-24k(Sr8Ga16Ge30) (S G G )a (A8E16X30)×

guest ion radius, but is not scaledby the cage radius.

6d 24k( 8 16 30) a (Sr8Ga16Ge30)

Scaling ofScaling of κκL L with the guest free spacewith the guest free space

2.0)

type-1clathrates

Ba8Zn8Ge38

B Al Si

Ba8Ni6Ge40

Ba Cu GeHostr (Å) = 1 11 (Si)

Ionic radius for high coordinationR G (Å) = 1.35 (Eu2+)

2

1 5W /

K m

)

Ba8Ga16Ge30

Ba Ga Si

Cs8Zn4Sn42 clathratesBa8Al16Si30Ba8Cu6Ge40

Cs8Sn44

r H(Å) = 1.11 (Si)1.22 (Ge)1.41 (Sn)

1.44 (Sr2+)1.61 (Ba2+) 1.83 (Rb1+)1.88 (Cs1+)

1.5

150K

) ( W

Ba8Ga16Si30

Sr8Ga16Si30

Rb8Zn4Sn42 Rb8Ga8Sn38

24k6d

1.0

κ L( 1 Sr8Ga16Ge30

Eu Ga Ge

Ba8In16Ge30

β−Ba8Ga16Sn30Ba8In16Ge28Sn2n1

10.5

1.71.61.51.41.31.2guest free space ( Å )

Eu8Ga16Ge30 β 8 16 30 p1

6d 24k guest free space ( Å )increase of rgfs suppression of κL

Guest free spaceGuest free space rr

6d 24k

Guideline for low κ LGuest free space Guest free space rrgfsgfs= R R 6d 6d -- 24k24k－r r guestguest－r r hosthost

Summary on TE clathratesSummary on TE clathrates

• Both p-type and n-type crystals are obtained.

Type-I and type-VIII Ba8Ga16Sn30

p yp yp y• The type-Ι BaGaSn is the best example of “phonon glass”

among intermetallic clathrates. However, more carriersare necessary to realize “electron crystal”.

• Off-center rattling scatters effectively the acoustic phonons, g y pwhich mechanism remains to be further studied.

Universal relation between κ and the guest free spaceUniversal relation between κ L and the guest free spaceprovides a useful guideline for caged TE materials with lower κ L.with lower κ L.

Searching for Rattling Kondo SemiconductorsHow to satisfy two requisites1. A sharp pseudogap at EF

2S=Z1. A sharp pseudogap at EF

in the renormalized band2. Off-center rattling in a caged structure

)( elL κκρ +⋅Z

2. Off center rattling in a caged structureComplex caged compoundswith multiple sites for Ce Sm and Yb ionswith multiple sites for Ce, Sm, and Yb ions one site: Kondo effect / hybridization gap other sites: off-center rattlingother sites: off-center rattlingApproach to greentech applicationefficiency of p-n coupleefficiency of p-n couple, phase stability against heat cycles,costs of materials and so oncosts of materials, and so on.