科研費新学術研究領域「コンピューティクスによる物質デザイン：複合相関と非平衡ダイナミクス」
計画研究「第一原理分子動力学法による構造サンプリングと非平衡ダイナミクス」

I型クラスレート化合物におけるフォノン散乱機構の理論解析
Theoretical studies of the phonon scattering
mechanism in type-I clathrate compounds
東京大学大学院理学系研究科
只野央将、合田義弘、常行真司

Outline
1. Background
2. Objective
3. Method
4. Results
5. Summary

1.Background
Thermoelectric material
Thermoelectric (TE) devices
efficiently convert atemperature difference toan electric potential.
TE efficiency

S2T
ZT =
c + `J.-F. Li et al., NPG Asia Mater 2, 152 (2010).

n: Electrical conductivity
S : Seebeck coefficientc : Thermal conductivity (carriers)` : Thermal conductivity (phonons)n

ZT > 1 is necessary,
ZT > 2 is demanded
Low lattice thermal-conductivity
is necessary. ` ⇠ 1 W/mK

Clathrate compounds•

Framework atoms (Host) & Guest ions (rattler)•Rattler reduces the lattice thermal conductivity: ` ⇠ 1 W/mK•

High ZT with optimal doping: Ba8Ga16Ge30 ZT=1.35
A. Saramat et al., J. Appl. Phys. 99, 3708 (2006)

Type-I

Type-IIT. M. Tritt and M. A. Subramanian, MRS Bulletin 31, 188 (2006).

Two explanations of low-k
It is experimentally evident that the guest atom (rattler)
reduces the thermal conductivity of clathrate. But how ?

2k` µ Cv t
1. The rattler reduces the relaxation time of phonons.

- Resonant interaction of Xe and acoustic phonons in clathrate
hydrate. [J. S. Tse et al., EPL 54, 354 (2001)]

2. The rattler reduces the group velocity of phonons.
-INS measurement & simplified model[M. Christensen et al., Nature Mat. 7, 811 (2008)]

2. Objective
Clarify the effect of rattlers in type-I clathratecompounds by first-principles calculations.

1. d-Ge
2. Ge46
3. Ba8Ga16Ge30 (BGG)
4. Ba8Ga16Sn30 (BGS)3. Method
Boltzmann transport theoryµ,⌫` =1 Xµ ⌫cks vks vks ⌧ks⌦Nkk,sk : Wave vectorn Dynamical matrix
X1M↵ M mD↵µ⌫ (k) = pdiagonalizes : Mode numbern Relaxation timeµ,⌫0↵,meik·(rm r0 )phonon-boundary

1 X 1
111
1
==++++ ··· .
⌧⌧⌧⌧⌧
⌧
p i
p pp ep Ip bi

2!ks= e⇤ks D(k)eks

phonon-phononn Group velocity@!ks!k+ ks !kvks =⇡@k2 kphonon-electronksphonon-impurityp p⌧ks (!) ⇡ ⌧ks(!) =12 ks (!)

ks (!) : Imaginary part of phonon self-energy

Perturbation theory of phononH=

X~!ksk,s✓1†bks bks +n Phonon Green’s func.
Dks (i!` ) =10 (i! )] 1[Dks`⌃ks (i!` )n Self-energy up to O( 2 )
(a)

(b)

Imaginary part ofdiagram (a) is zero.

2◆+ H3 +2

H4 + · · ·n Imaginary part of diagram (b)⇡ X X (3)|Ṽ ks,k1 s1 ,k2 s2 |2ks (!) =2
k1 ,k2 s1 ,s2⇥⇥ (n1 + n2 + 1) (! !1 !2 )(n1+ (n1(3)n2 ) (!

!1 + !2 )n2 ) (! + !1⇤!2 ) .

Ṽk1 s1 ,k2 s2 ,k3 s3◆ 12✓ ◆ 32 ✓31~2(k1 + k2 + k3 )= Ncell
2! k1 s 1 ! k2 s 2 ! k 3 s 3
XX
1i(k2 ·rm2 +k3 ·rm3 )p⇥eM↵ 1 M↵ 2 M ↵ 3
m2 ,m3↵1 ,↵2 ,↵3
X
µ1
µ2
µ3
µ1 ,µ2 ,µ3eee⇥
0↵1 ,m2 ↵2 ,m3 ↵3 (k1 s1 )↵1 (k2 s2 )↵2 (k3 s3 )↵3µ1 ,µ2 ,µ3

Force constants from first-principles
To calculate phonon frequencies and relaxation times,harmonic and cubic force constants are necessary.
Xµ1=F0↵
1µ1 ,µ20↵1 ,m2 ↵2uµm22 ↵2

m2 ,↵2 ,µ21 X
2 m2 ,m3 ,µ1 ,µ2 ,µ30↵1 ,m2 ↵2 ,m3 ↵3↵2 ,↵3 ,µ2 ,µ3n We employed the direct-method

approachn Onsite and interatomic force constants

are estimated in a systematic manner.
2=XXim,↵FPkFm↵(ti )Fm↵ (ti )k2n Translational/rotational invariances are

imposed.

uµm22 ↵2 uµm33 ↵3