Avoided crossing of rattler modes in thermoelectric materials

M. Christensen, A. B. Abrahamsen, N. B. Christensen, F. Juranyi, N. H. Anderson, K. Lefmann, J. Anderson, C. H. Bahl,

and B. B. IversonNature Materials, 7, 811 (2008)

June 11th 2009 Seminar, T. Mori

Topics

• Neutron TAS on BGG• Simple spring-model→Avoided crossing• Avoided crossing (flat dispersion)

relates low thermal conductivity ?

referencesM. Christensen, A. B. Abrahamsen, N. B. Christensen, F. Juranyi,N. H. Anderson, K. Lefmann, J. Anderson, C. H. Bahl, and B. B. IversonNature Materials, 7, 811 (2008)

C. H. Lee, I. Hase, H. Sugawara, H. Yoshizawa, and H. SatoJournal of the Physical Society of Japan, 75, 123602 (2006)

C. H. Lee, H. Yoshizawa, M. A. Avila, I. Hase, K. Kihou, and T. TakabatakeJournal of Physics, 92, 012169 (2007)

κσ2TSzT =Dimensionless figure of merit Lcc κκκ ,↔

Wiedemann-Franz law (empirical, 1853) links κcc to σ

2

2

2

2

2

2

3212

3

31

ne

VdTdEmv

necmv

mne

cv

v

vcc

=

=

=τ

τ

σκ

vcc cv

mne

τκ

τσ

2

2

31

=

=

A reduction of κ is obtained only by reducing κL

Drude (1900)

Tek

ne

TkdTdTnk

nedTd

VNTk

B

BB

B

2

2

2

23

323

232

3232

⎟⎠⎞

⎜⎝⎛=

⎟⎠⎞

⎜⎝⎛

=

=

ε

l is assumed to be limited by the separation of the guest atoms,l = a/2 = 5.39 Ǻ

This assumption results τm=l/vm=0.18ps.Much shorter than observed in the present study.(τm=2-3ps, vm=3,046ms-1)

We find that the reduction of the thermal conductivity due to the guest atoms is caused by the flat modes of the avoided crossingrather than the guest atoms acting as scattering centres.

mmmL CvlCv τκ 2

31

31

==

sample

narrower narrower

monitor

2nd collimator

PG

monochromator

3rd collimator

4th collimator

analyzer

1st collimatorreactor

Neutron triple-axis spectroscopy(TAS)

Ref. Bizen thesis

κN = kf – ki= G + q

κN around G=(222)

kfki

κN around G=(004) κN around G=(330)

Phonon dispersion map

n-type Ba8Ga16Ge30Large 13 g single-crystal sample

κN around G=(004)

G=(004)

q=(hh0)

G=(330)

q=(hh0)κN around G=(330)

( )2Nκe •∝I e = cLeL + cTeT

TN // eκ

LN // eκ

G=(222)

q=(hh0)

κN around G=(222)

( )2Nκe •∝I e = cLeL + cTeT

Phonon dispersion map

←E1=4.6meV←E2=5.8meV

E3～7.5meV→

←E1

←E2

←E2

Simple spring model: LA & guest modes

Simple sine-function dispersion: TA mode

[ ] [ ]

[ ])(2)()()(

)(2)()()(2)()()(

122

2

121112

2

tvtutuKtvdtdm

tutvtvKtututuKtudtdM

jjjj

jjjjjjj

−+=

−++−+=

+

−+−

[ ] [ ]

[ ])(2)()()(

)(2)()()(2)()()(

122

2

121112

2

tvtutuKtvdtdm

tutvtvKtututuKtudtdM

jjjj

jjjjjjj

−+=

−++−+=

+

−+−

K2<<K1→one-atomic chain

K1<<K2→two-atomic linear chain

m(guest)<<M with finite K2→ m(guest) behaves as an independent oscillator(with a constant q-independent frequency).

[ ] [ ]

[ ])(2)()()(

)(2)()()(2)()()(

122

2

121112

2

tvtutuKtvdtdm

tutvtvKtututuKtudtdM

jjjj

jjjjjjj

−+=

−++−+=

+

−+−

))21(exp()()exp()(

tidjiqAtvtiiqjdAtu

qj

qj

ω

ωα

−+=

−=

])2/cos(1[2

]2)2/cos(1[2)]cos(1[222

2

122

21

2

αωω

αβωωω

qd

qdqd

q

q

−=

−−+−= −

)2/cos(2)2/(cos4)])cos(1[1()]cos(1[1 22

qdqdqdqd βγβγβ

α+−−−±−−−

=

22

21

222

121

///

/

ωωγ

βω

ω

=

==

=

MmmK

MK

])2/cos(1[222

2

αωω

qdq −=

)2/cos(2)2/(cos4)])cos(1[1()]cos(1[1 22

qdqdqdqd βγβγβ

α+−−−±−−−

=

MKmK

MmmKMK

21

22

21

222

121

//

///

==

==

=

ωωγ

βω

ω

β=1/3, γ=2 β=1/3, γ=1.25 β=1/12, γ=1.25

β=1/3～8mBa/(16MGa+30MGe)

β=1/12～mBa/(16MGe+8MGa)

γ=1.25=βK1/K2=1/12*K1/K2 K1/K2=12*1.25=15

Phonon dispersion map

vT(222) = 2,795ms-1vL(222)

= 4,096ms-1

vm=3,046ms-13vm-3=vL

-3+vT-3

θD=ħkBvmn=301K n=(6π2Na-3)1/3

N=46, a=10.78 Ǻ

τL=2.6ps τrattler=2.0ps

τL=1.3ps

τrattler=1.6ps

τL～2ps>>τm=0.18ps

∑=j

LL TjCTjjvV .

2 ),,(),,(),(31

q

qqq τκ

Finite phonon lifetime result fromphonon-phonon scattering.:Normal process:Umklapp process

Normal process

Umklapp process

K1+K2=K3

K1 K2

K3

K1+K2=K3+G

C. H. Lee et al. JPSJ, 2006

θD=ħkBvmn=301K～25meV

Umklapp processes are frozen out qD/2(=150K).But, energy at half the BZ is only 2.5-3.0meV.Umklapp processes cannot be frozen out until below 30K.

xTQ

T

x ∂∂

−=

∇−=

κ

κQ

( )∑

∑

∑

∑

−=

=

=

×=

jx

jx

jxx

xx

vnnV

vnV

jvjnjV

Q

AtvnV

AtQ

.

0

.

.

00

1

1

),(),(),(1

1

q

q

q

qqq

ω

ω

ω

ω

η

η

η

η

decaydifftn

tn

dtnd

∂∂

+∂∂

=

00 =→=dtnd

dtdT

decaydifftn

tn

∂∂

+∂∂

=0

xx q

v∂∂

=ω

vxt0

A

τ

0nntn

decay

−−=

∂∂

x

x

x

t

x

tdiff

vxT

TxTn

vxxn

t

xntvxxn

xn

txntvxn

tn

∂∂

∂∂

−=

∂∂

−=

Δ

−+Δ−∂

∂+

=

Δ−Δ−

=∂∂

→Δ

→Δ

))((

)(

)()()(

)(lim

)()(lim

0

0

Λ

( )∑ −=j

xx vnnV

Q.

01q

ωη

decaydifftn

tn

∂∂

+∂∂

=0

xT

Tn

vnn

x ∂∂

∂∂

−=−τ

0

( )

∑

∑

∑

∑

∑

=

∂∂

=

∂∂

=

∂∂

=

∂∂

=

jL

j

j

j

jL

jCjljvV

TTvl

V

Tn

vlV

Tn

vlV

Tn

vV

.

.

.

.

.

2

),(),(),(31

),(31

31

31

31

q

q

q

q

q

qqq

ωε

ω

ω

ωτκ

η

η

η

∑

∑

∑

∂∂

∂∂

−=

∂∂

∂∂

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

−=

j

jx

jxxx

xT

Tn

vV

xT

Tn

vV

vxT

Tn

vV

Q

.

2

.

2

.

31

1

1

q

q

q

ωτ

ωτ

τω

η

η

η

xT

Tn

vnn

x ∂∂

∂∂

−=−τ

0

( )∑ −=j

xx vnnV

Q.

01q

ωη

C. H. Lee et al. J. Phys.2008

β-BGS