近藤絶縁体－多体効果と磁場効果－

応用物理学会学術講演会
神奈川大学，2003年3月29日
12半導体B，12.1探索的材料物性
29p-ZD-1
近藤半導体とは何か
埼玉大学理学部物理学科佐宗哲郎
近藤半導体は普通の半導体と何が違う
か？
近藤半導体は役に立つか？
1.
近藤絶縁体とは
2.
近藤絶縁体のエネルギー・ギャップの形成機構
3.
近藤絶縁体における多体効果
4.
近藤絶縁体の強磁場誘起絶縁体-金属転移
5.
近藤絶縁体の熱電効果
1
4f and 5f electrons in rare earth and actinide ions under
CEF and spin-orbit interaction
L=3, S=1/2  J=5/2, 7/2 due to spin-orbit interaction
Ce:
J=7/2
4f
~7000K
~10~100K
J=5/2
spinorbit
7
8
CEF
Yb: J=7/2 is lower than J=5/2
8 states with J=7/2 under cubic CEF:
8
(1)
8
( 2)
 7 3
5 1 
5 0
  
Y3 
Y3    
Y3  
12
28
21


 3 2
 1
3 2 
15 1 
  
Y3 
Y3     
Y33 
Y3   
14
14
28
28




2
電子相関の役割

金属の自由電子模型
電子比熱

C/T
Cel  T ,  
2
3
T2
D( F )k B , D( F ) 
2
 Pauli  B 2 D( F )
スピン帯磁率
T
1  2m 


2 2   2 
3/ 2

常磁性!
自由を奪うもの：散乱
緩和時間
 k 1 
電気抵抗
0 
2
| V (k  k ' ) |2  D( F )

m
ne 2

0
T
自由を奪うもの：相互作用
遍歴性と局在性
s電子
遷移金属
d電子
アクチナイド金属
5f電子
希土類金属
4f電子
>
アルカリ金属
>
>
磁気モーメントの発生（磁性は電子相関から生じる)
U=0の時
EF
U>0
Ed+U
EF
Ed
Ed
3
4
近藤効果
希薄磁性合金における
電気抵抗極小の現象
R(T )  R0  AT 5  cJ log(T / TK )
TK:近藤温度, c:不純物濃度
T
s-d模型
H s d   k ck ck  J

k
S  s ' ck ck ' '

kk ' '
J. Kondo
(1964)
3次摂動によるスピン反転
散乱過程
TK  De1/ J
伝導電子
局在スピン
Ed+U
近藤模型とAnderson模型の等価性
H A    k c  k ck  Ed  nd  Und  nd 
k

  (Vk ck d  V * k d  ck )

EF
Ed
8V 2
 H s d ( J 
)
U
k
希土類不純物による近藤効果
Ce: 4f1, L=3, S=1/2, J=5/2, Jz=-5/2,・・・,5/2, N=6重縮退
Yb: 4f13, L=3, S=1/2, J=7/2, Jz=-7/2,・・・,7/2, N=8重縮退
5
Universal Behaviors in Kondo Effect
帯
磁
率
(T
)
 (T ) 
T(T)
電
気
抵
抗
C
T  TK
TK
R(T)
C/T
C(T)=T+AT3
比
熱
熱
電
能
Cel(T)

T
TK
S(T)
log T
0.01TK
TK
100TK
-log(T/TK)
6
T/TK
近藤絶縁体とは・・
電子相関の強いバンド絶縁体である。
・帯磁率
χ
YbB12
高温でCurie-Weiss則
低温で非磁性
FeSi
・比熱
T
C/T
Egap<T<TKで
C=T+AT3, 0

・電気抵抗 ρ
Egap
T2
～ e Egap/T
低温で絶縁体
T
•低温で，狭いギャップ（数十～数百meV）の絶縁体。
•フェルミ準位に電子がないにもかかわらず近藤効果
的ふるまい(?)を示す。
•電子数偶数個で立方晶の4f ,5f化合物 (YbB12,
Ce3Bi4Pt3など)に見られる。
•d電子系でも類似 (FeSiなど)。ただし,χ(T=0)→0
•CeNiSnは斜方晶で擬ギャップ
7
•TmSeは電子数奇数個で，Mott絶縁体，TN=5K
Heavy Fermions with Energy Gap (Kondo Insulators)
 high
(mJ/K
2mole)
C/(0)
(K)
SmS
総
電子
数
78
SmB6
92
80
YbB12
130
80
TmSe
103
Ce3Bi4Pt3
740
Ce3Sb4Pt3
612
CeRhSb
154
CeRhAs
146
60
Tmax(K)
TK(K)
81
320
80
46
320
70
small
23
320
35
113
TN~3K
4
144
CeFe4P12
CeNiSn
Eg (K)
1500
136
190
92
12
2~5
U3Ni3Sb4
230
U3Pd3Sb4
260
U3Pt3Sb4
170
UFe4P12
pseudogap
Tc=3.15K
UNiSn
170
800
TN=45K
ThNiSn
168
800
no order
Sm3Se4
322
790
Sm3Te4
394
600
1600
TN=0.7K
Heavy Fermions with low carrier density
電子数
 high
C/(0)
Tmax
TK
nc/R
E
CeN
CeP
CeAs
TN=8K
20
CeBi
Yb4As3
205
92
no peak
0.05
TN=6K
0.05
TN=25K
0.001
T0~300K
8
近藤絶縁体を記述する模型
•近藤格子模型（U=∞)
•周期アンダーソン模型（U=0 → ∞)と
その一般形（2-バンド模型など）
○バンド計算でギャップが開いていて，そこに多体効
果が加わっている。
○複雑な結晶構造や軌道縮退のために，ギャップが
開かないことも有る。
○多体効果とスピンのゆらぎの効果は，連続したも
のである。
○近藤効果がきれいに見えるとは限らない。
k
Ef
k
Ef
9
No gap!
近藤絶縁体を記述する模型
•近藤格子模型（U=∞)
•周期アンダーソン模型（U=0 → ∞)と
その一般形（2-バンド模型など）
○バンド計算でギャップが開いていて，そこに多体効
果が加わっている。
○複雑な結晶構造や軌道縮退のために，ギャップが
開かないことも有る。
○多体効果とスピンのゆらぎの効果は，連続したも
のである。
○近藤効果がきれいに見えるとは限らない。
k
Ef
k
Ef
10
No gap!
Electronic structure of the Kondo Insulator YbB12
Harima and Yanase (1992)
4f
J=7/2
J=5/2
4f
J=7/2
J=5/2
11
LDA+U
→ Eg~0.01eV
4f
J=7/2
8
7,6
J=5/2
2002/03/15 15:07:02 fcc2.dat
T. Saso and H. Harima(2003)
Nearly-Free-Electron model?
Empty Band for fcc lattice (a=5.65A) cannot reproduce the band.
Energy (Ryd)
2
12 Saso
(C)T.
0

K
X
W
L

X
Tight-binding band for YbB12
Yb: 5d t2g band
5d (xy, yz, zx)
Effective d-d hopping through
B12 clusters
＋
－
＋
＋
－
－
2px±2py
B12
Yb 5dxy
13
Construction of d-f matrix elements
CEF and s.o.
8 states with J=7/2 under cubic CEF:
8
(1)
8
( 2)
 7 3
5 1 
5 0
  
Y3 
Y3    
Y3  
28 
21
 12
 3 2
 1
3 2 
15 1 
  
Y3 
Y3     
Y33 
Y3   
14
14
28
28




Slater-Koster integrals by K. Takegahara, et
al., J. Phys. C13(1980)583.---- CEF but no s.o.
Conversion of Ylm into l=3 cubic harmonics yields:

1 5
 x(5 x 2  3r 2 )  i y (5 y 2  3r 2 )
2 21
8(1)   [



1 9
5
 x ( y 2  z 2 )  i y ( z 2  x 2 ) ]  
z (5 z 2  3r 2 )  
2 7
21
8( 2 )   [



1 5
 x(5 x 2  3r 2 )  i y (5 y 2  3r 2 )
2 7



1 3
3
 x ( y 2  z 2 )  i y ( z 2  x 2 ) ]  
z( x 2  y 2 )  
2 7
7
Slater-Koster integrals  d-f mixing matrix elements:
8(1) 
xy 
yz 
zx 
xy 
yz 
zx 
8(1) 
8(2) 
8( 2) 
0
0
it 2 (c x s y  is x c y ) 
 5it1 (cx s y  is x c y )


 4t1c y s z
it1c y s z
 3it 2 s y cz
2t 2 c y s z



 4it1s z cx
it1c z sx
3it 2 c z s x
 2it 2 s z c x 


0
 5it1 (cx s y  is x c y )  it2 (cx s y  is x c y )
0



it1s y c z
 4t1c y s z
2t 2 c y s z
 3it 2 s y c z 



it1cz s x
4it1s z cx
2it 2 s z c x
3it 2 cz s x


t1 
14
5
15
(df ), t2 
(df ), s  sin(k / 2), c  cos(k / 2),   x, y, z
56
56
0.9
d,d bands
x2-y2, 3z2-r2
Energy (Ryd)
(dds) only
xy, yz, zx
0

K
X
W

L
X
2002/08/10 14:17:26 dxy-band2.dat
d-only bands
Ek
( xy )
xy, yz, zx
Ek
( yz )
Ek
( zx )
•Three independent bands
•Two-dimensional bands
ky
kx
) cos( )
2
2
ky
k
 3(dd ) cos( ) cos( z )
2
2
k
k
 3(dd ) cos( z ) cos( x )
2
2
 3(dd ) cos(
E
•Conduction band at X 0.3
point is doubly degenerate.
0
4f 8
15
-0.3

K
X
W
L

X
(dd)-(df)-(ff) tight-binding band
(dd)=0.06, (df)=0.01, (df)=－0.005, (ff)=－0.002,
LDA+U shift DE=－0.005 (in Ryd)
Energy(Ryd)
1.2
2002/09/14 16:39:45 YbB12-dos.dat
Eg=0.003 Ryd
0.8

K
X
W
L

X
400
YbB12
DOS
非対称！
0
0.8
1

16
Mechanism of the gap opening
Besides the Kramers degeneracy,
No gap!
Gap opens!
or
Semimetal!
No gap!
17
Gap opens!
Semimetal!
d-d model:
xy
yz
 Ekxy




0



0




0



 3t s x s y

xy
yz
zx
x2  y2
3z 2  r 2
0
V
k
V
V
Ef
V
0
0
1.2
V 


V 


0 


E f 
Energy (Ryd)
0
3z 2  r 2
3t s x s y 



3
3
Ekyz
0
t s x s y 
t s x s y 
2
2


2002/10/29 12:28:02
dd-band05.dat 3

3
0
Ekzx
t s x s y 
t s x s y 
2
2


3
3
3

t s x s y
t s x s y

ts x s y
0

2
2
2


3
3
3

t s x s y 
t s x s y
0

ts x s y 
2
2
2

0
Constant mixing model:
k


0


V



V
x2  y 2
zx
0.8

K
X
W
L

X
Separate mixing model:
k


0


V



0
0
V
k
0
0
Ef
V
0
0  k
 
 
V  V
 

0   0
 
 
E f   0
V
0
Ef
0
0
k
0
V
0 


0 


V 


E f 
18
Summary
•New LDA+U band calculation was performed for the most
typical Kondo insulator YbB12. The gap (Eg=0.0013Ryd) opened
after additional shift of 4f level by 0.3Ryd.
•The conduction bands can be expressed very well by the simple tightbinding model (t2g band with effective d-d hopping matrix (dd)
through B12 cluster). It is impossible to express it by the nearly
free-electron model.
•Introduction of the d-f hybridization (df) opened up the energy gap
if the filled band is pulled down slightly.
•Relationship of the gap-opening and the degeneracy was
discussed.
•The CEF ground state must be 8, since otherwise a gap does
not open!!
Ref. T. Saso and H. Harima, to appear in J. Phys. Soc. Jpn. 71
(2003) No.5
Future problems
•Explicit inclusion of B12 clusters may improve the dispersion curves.
•Explicit separation of the Kramers degeneracy may provide
alternative representation. (see Maehira, et al.)
•Based on the present model, we have to take into account the
correlation effect ( via e.g. DMFT?), and calculate the physical 19
quantities.
FeSi
B-20 structure
•Magnetic susceptibility
(Jaccarino, et al., 1967)
vanishes at low T due to the
gap, and exhibits Curie law at
high T with a peak at 500K
•Magnetic specific heat
(Kanomata, et al., 2000)
has a peak at 220K.
•Optical conductivity
(Damascelli, et al., 1997)
has a gap of 570 cm-1
(810K) at low T, but it is
filled up at 300K
20
•Band calculation (Kulatov, et
al.) gives an indirect gap of
0. 04eV and direct gap of
0.16eV.
•Band calculation (Ohta, et al.,
1994) gives an indirect gap of
0. 02eV.

 ( )   d  ( )  (   ) f ( )[1  f (   )] +Drude term

Correct formula:

 ( )   d  ( )  (   )

f ( )  f (21
)

Spin-Fluctuation (SCR) theory
by Takahashi and Moriya (1979)
Sum rule for the dynamical susceptibility:
3



0
 0 (q,  )

2
d coth( ) Im
 Si 
2T q
1  U 0 (q,  )

 0 (0,0) 

 1  U 0 (0,0) 
 (T )  
is treated as a parameter to be
determined at each T.
Takahashi (1997)
3
2


0
d[1  2n( )] Im 
q
2 0 (q,  )
3
 (n  2  n n )
1  U 0 (q,  ) 4
For correlated metal:
  Si  (U  )
For an insulator (Eg>0):

0(q,) is calculated
from the T-independent
band model with a
constant gap.
•This theory can explain
(T) and C(T)
qualitatively, but not
(,T).
2
3
n (T  E g , n  1)
4

22
FLEX (Fluctuation Exchange)
•single-particle self-consistent theory
•weak coupling theory
1
G(k , i ) 
i   k  (k , i )
1
U
(k , i )  Un  T  G (k  q, i  i )
2
2  q
 3U (q, i )

U (q, i )


 2 (q, i )
1  U (q, i ) 1  U (q, i )

(q, i )  T  G(k , i )G(k  q, i  i )

k
U→0
1
(k , i )  Un  U 2T  (q, i )G(k  q, i  i )
2
 q
This is SC-SOPT self-energy.
The lower and upper Hubbard bands are not
reproduced! Thus, FLEX cannot be used for strong
correlation!
23
バンド計算の状態密度を用いた
2-band Hubbard 模型
H   (tij1 ci1 c j1  tij2 ci2 c j 2 )
ij
 U  (ni1 ni1  ni 2 ni 2 )
i
 U 2  (ni1 ni 2  ni 2 ni1 )
バンド内
バンド間
(反平行スピン)
i
 U 3  (ni1 ni 2  ni1 ni 2 )
i
 J  (ci1 ci1 ci2 ci 2  1  2)
バンド間
(平行スピン)
交換相互作用
i
(1,2 はband index)
Parameters: U=0.5eV，J=0.35U, U2=U-J, U3=U-2J
Band calculation at T=0 by Yamada et al., J. Phys.:
Condens. Matter 11 (1999) L309, but the gap is 24
enlarged by 16%.
(Self-Consistent Second-Order
Perturbation Theory in d→∞)
SC-SOPT
1
11 (i )  Un  U 2T 2  G22 (i ' )G22 (i 'i )G11 (i  i )  
2
 ',
G11 (i )  
k
1
i   k  11 (i )
2
( 2)
Σ11
 ( ) 

U
U
2
2
2
+
2
+
U3

U2
U2

2

2
2

U3
+
J
J

Density of states:
 ( )  
1
ImG11 (  i )  G22 (  i )

Optical conductivity:
2 e 2 2
 ( ) 




(Assume breaking of momentum
conservation)
d  ( )  (   )
f ( )  f (   )

(Good for interband transition)
Density of states and (,T) become strongly
T-dependent.
25
Optical conductivity of FeSi by SCSOPT
and comparison with experiment
K．Urasaki and T.S.，JPSJ 68(1999)3477.
Exp. data taken from A. Damascelli, et al., Physica B 230-232 (1997) 787.
•The gap (~700K) is filled up at 300K.
•The gap is filled rather slowly at T<100K, but is filled
faster at T>150K.
•The gap edge shifts to low  when T rises.
•Rigid band model
cannot reproduce the
correct temperaturedependence.
26
電気伝導度の公式(d→∞)
d→∞では，バーテックス
補正は効かない。
2e 2
f
 (T ) 
d

L
(

)(

)
3 

厳密に言うと，d→∞では，
vk2～O(1/d)→0
2
1
2
L( )  k ImG(k ,  )
N k
G (k ,  ) 
1
1

   k  ( )    
k
i
2 ( )
d→∞では， (k,)
はkによらない。
1




2

(

)
2

L( )   d ' ( ' )  ( ' )
1 2

2
(



'
)

(
) 

2

(

)


  ( ) 2  ( ) ( )
よって，Boltzmann方程式と同じ形に書ける:
2e 2
f
2
 (T ) 
d


(

)

(

)

(

)(

)

3

2
注意: 周期系では
Umklappが必要
光学伝導度の公式(d→∞)
2e 2
f ( )  f (   )
 ( ) 
d

L( ,  )
3 

1
2
L( ,  )  k Im G (k ,  ) Im G (k ,    )
N k
  ( ) 2  ( )  d k Im G (k ,  ) Im G (k ,    )


1

  ( ) 2  ( ) Im
R
A
    (   )   ( ) 
0

  ( ) 2  ( ) ( )
27
Specific heat of FeSi
E
 1

d

f
(

)
Im



(

)

G( )




 2

2

 E 
CV (T )  

 T V
K. Urasaki and T. Saso, Correlation Effects in Multi-Band
Hubbard Model and Anomalous Properties of FeSi, ``New
Properties of Matter due to Ordering and Fluctuation of
Electron Orbitals--Comprehensive Study of f- and dElectrons--'' News Letter Vol.1 No.2 (2000) p.83.
•Peak at T=200～300K
agrees with experiments.
Kanomata,
et al. (2000)
28
Magnetic susceptibility of FeSi
The susceptibility can be obtained by
1. Solving the Bethe-Salpeter equation
2. Numerical derivative of the electron number.
n  n
h 0
h
  lim
•Peak position is too low compared to the
experiments.
•Curie behavior at high T is not well
reproduced.
•Inclusion of spin fluctuation is necessary.
29
YbB12
Sugiyama(1988)
Yb2.9+
30
Yanase-Harima (1992)
YbB12
Optical conductivity
Okamura
(1998)
Anisotropy of magnetization curves in YbB12
F. Iga, et al.
(1998)
•YbB12 has a cubic structure, so that the linear
susceptibility must be isotropic.
•Anisotropy may appear in non-linear regions.
31
動的分子場理論（d=∞理論）

自己エネルギーが局所的： ij () d
()ij
~
G ( ) 
1
G ( ) 1  ( )
を無摂動Green関数とする
１不純物問題と同じ
~
G( ) : 中心サイト以外の効果を取り入れ
た(cavity) Green関数
  
 U 
  

  
  
  
•局所電子相関がfullに考慮されている。
•電子系に対する最良の１サイト理論。
有効不純物問題の解法
厳密対角化法
数値繰り込み群
量子モンテカルロ法
NCA, slave boson
改良反復摂動論 (mIPT)
~
()を G ( ) を用いて，U2まで計算して繰り返す。
U→∞にも適用できる。(Mott転移)
32
7 300K
Model
6 150K
Yb3+ (J=7/2)
8
0K
Periodic Anderson model (N=2)
H   tij ci c j   E f f i f i
ij
i
  Vij (ci f j  H .c.)  U  n fi n fi
ij
i
E f  E f  h, h  g f B H |  | J z |  |
g f  gc → (T=0) > 0
  1
(We neglect gc since gf>gc.)
Methods
Dynamical Mean Field Theory
(exact in d→∞)
Solution of the impurity problem
•Exact diagonalization of small clusters
•Iterative perturbation theory (IPT)
Exact in zero conduction band width
E. Lange, cond-mat/9810208
Self-Consistent Second-Order Perturbation
Theory (SCSOPT)
33
Field-induced insulator-to-metal transition
(FIIMT) in Kondo insulators
Two effects of magnetic field on Kondo insulators
(1) Zeeman effect→ ↑ and ↓ bands are shifted
Gap decreases.
(2) Suppression of the Kondo effect
→ renormalization disappears.
Gap increases.
Which dominates?
Answer
Renormalization factor scarecely changes when
there is a gap.
Gap is closed mainly by the
Zeeman effect.
(Note that the DOS shape changes.)
34
M(H)/M()
Magnetization curves
1
0.8
SCSOPT
DMFT
0.6
EG(H).SMP 3-17-1999 19:54
0.4
U/D=2
U/D=4
0.2
0
0
0.2
0.4
0.6
0.8
1
H/TK
Gap closing
0.3
0.2
Eg
U=0
(Scaled at h=0)
Rigid Band
Eg(0)-2h
0.1
SCSOPT
(U=2)
0
0
0.1
h
0.2
35
H. Yamada, Phys. Rev. B47(1993)11211.
DF ( M ) 
1
1
1
aM 2  bM 4  cM 6
2
4
6
Stoner theory
+spin fluctuation
(T>0)
(T=0)

E (m)    d   ( )  Un  n


n   d  ( ),
E(h)  E(m)  mh
()
36
Seebeck Coefficient
S
Metals:
hole-like
 2 k B2 L' ( )
S (T )  
T
3 | e | L( )
T
L( )  vc ( )2 c ( ) c ( )
electron-like
S
Semiconductors:
S (T )  
hole-like
k B Ec,  
| e | k BT
T
electron-like
As a portable refrigerators:
Figure of Merit Z= S2/k
k: thermal
conductivity
Power factor P=S2
Ordinary semiconductors:
(T)=n(T)e
n(T)~exp[-(Ec-)/kBT]
P=S2
e.g. Bi2Te3
S2
Eg~0.15eV
T
ZT~1 at 300K
37
|S|max=2kB/e=173V/K
at kBT=(Ec-)/2
Kelvin’s relations
W. Thomson (Lord Kelvin), Proc. Roy. Soc. Edinburgh (1851) 91.
Thermodynamic arguments :
a unit charge is adiabatically moved
through ABCDA.
T+DT
T0
T
T0
metal b
A
metal a
C
B
metal a
D
 S ab DT   ab (T  DT )   ab (T )  ( b   a )DT  0,
 ab (T  DT )  ab (T )  b   a


DT  0
T  DT
T
T
Seebeck coefficient S(T), Peltier coefficient (T),
Thomson coefficient (T)
T
 (T ' )
0
T'
S (T )  
T
 (T ' )
0
T'
dT '
 (T )
T
: Kelvin’s relations
Contradictions:
S (T )  
dT ' , S (T ) 
S(T) should →0 at T=0,
but S(T)→1/T in semiconductors??
(T )  TS (T )  ( Ec,v   ) / e
Π=finite even at T=0 !?
One can remove finite
heat at T=0??
Contradictory to 3rd law of thermodynamics !?
38
Modification of Kelvin’s argument at T→0 for insulatos:
T+DT
T0
|e|
T
T0
C
Semicond. a D
Semicond. b
A Semicond. a B
E
e-
Conduction band
Ecｂ
T+DT
Eca
T
 S ab DT   ab (T  DT )   ab (T )  ( b   a )DT  W  0,
 ab (T  DT )  ab (T )  b   a


DT  0
T  DT
T
T
Q+W
W
A work necessary to push
up a charge from Eca to Ecb.
W  (Ec  Ec ) / | e |
b
a
Modified Kelvin’s relations:
T2=T
Q
T1=0
(valid only at T→0)
 ab (T ) Ec  Ec
S ab (T ) 

,
T
| e|T
b
a
T
 a (T ' )   b (T ' )
0
T'
S ab (T )  
Ec  Ec
dT '
| e|T
b
a
Thus,
E  Ec
Sab (T )  c
| e|T
b
a
but Πab(T)→0,
T
 a (T ' )   b (T ' )
0
T'

dT '  0
Contradiction to the 3rd law of the thermodynamics is resolved!
39
•At low but finite T, nonequilibrium effect must be included!
Correct behaviors due to Modified Kelvin Relations:
S
S
T
T


T
T
Many-Body Effect or
Non-Stoichiometry
S
T
40
Heat current carried by the strongly correlated electrons
G. D. Mahan, Solid State Physics 51 (1998) 81.
Hubbard:
jQ   ( k   ) c c
k
jQ  
PAM:

k k k
S


k
i
( titj )R ij ci c j  Vt ij R ij (ci f j  H .c.)

2 ij 
ij
M. Jonson and G. D. Mahan,
PRB42(1990)9350.
Linear Response Theory
  e2 
ni   n j 
 

 Uk  ci c j
2
k

d  f 
  [ F (  i ,   i )  Re F (  i ,   i )],
   
e d  f 
  (   )[F (  i ,   i )  Re F (  i ,   i )],
T     
F ( , ' )  k ck ( )ck ( ' ) j (0)
k
For detailed analysis and application to High Tc’s, see:
H. Kontani, J. Phys. Soc. Jpn. 70 (2001) 2840; cond-mat/0206501,
cond-mat/0204193
Many-Body Effect in terms of DMFT for PAM
For Kondo insulators:
•Self-energy () ≠0 even in the gap when T>0.
•Quasi-particle DOS () ≠0 even in the gap at T>0 and is Tdependent due to ().
•Thus, Seebeck coefficient S(T)
∝T,
~1, at low T.
41
Kondo Insulators
Periodic Anderson Model (N=2) +DMFT
500
U=2
D=0.5
(T) [arb. unit]
100
50
Ef=-1
Ef=-0.8
Ef=-0.7
Ef=-0.6
Ef=-0.5
Ef=-0.3
Ef=0
10
5
1
0
0.2
0.1
T [K]
S(T) [V/K]
600
Ef=-1
Ef=-0.8
Ef=-0.7
Ef=-0.6
Ef=-0.5
Ef=-0.3
Ef=0
400
200
0
0
0.1
T
0.2
42
Kondo insulators: S(T) ∝T, ~1, because ()≠0 at T>0.
Yb1-xLuxB12
phonon
drag?
43
F. Iga, et al.
2-Band model
Using total DOS from the band calculation
1000
Density of States
H. Harima
500
YbB12
Total
f-part
4f 8
EF
0
0.86
0.87
0.88
0.89
 (Ryd)
0.9
0.91
Small carrier (electron) doping:
200
S(T) (V/K)
100
YbB12
U=0
Eg=63K
0
Exp.
n=0.01
n=0.005
n=0
-100
-200
100
200
T [K]
300
T. Saso (2001)
44
Temperature-dependence of the Kondo peak may be important.
Temperature-dependence of DOS of PAM with N=2
1.5
U=2, Ef=-1.2
f
()
1
c
0.5
0
T=0
T=1
-4
-2
0

2
Kondo peak
disappears.
4
U=2, Ef=-1.5
1.5
()
f
1
T=0
c
0.5
0
-4
-2
T=1
0

2
Kondo peak
disappears.
4
45
Comparison with the rigid-band model
(DOS is fixed to DOS at T=0)
100
S(T) [V/K]
0
-100
rigid band
-200
-300
Due to temperature-dependence of DOS
PAM, U=2, Ef=-1.2
-400
-500
0
0.1
X
0.2
S(T) [V/K]
0
rigid band
-100
Due to temperature-dependence of DOS
PAM, U=2, Ef=-1.5
-200
0
0.1
0.2
0.3
T
0.4
0.546
FeSi
Sales, et al.
PRB50(1994)
8207
Jarlborg (1999)
Jarlborg (1995)
47
Band DOS (H. Yamada) + SCSOPT(d=∞)
FeSi
FeSi
EF
4000
DOS
(T) (Wcm)
6000
2000
-1000
0
0
100
0  1000
200
300
T
600
-4
1x10 holes (U=0)
-4
1x10 holes (U=0.5 eV)
Exp.
S [V/K]
400
200
FeSi
0
-200
0
100
200
T [K]
300
48
Summary
•Kelvin’s relations must be modified at low temperature limit of
insulators and semiconductors. The contradiction to the 3rd law of the
thermodynamics is resolved by the new formula.
•In reality, the effect will be smeared by nonstoichiometry and manybody interaction.
•The energy bands of the most typical Kondo insulator YbB12 can be
expressed very well by the simple tight-binding model (t2g band with
effective d-d hopping matrix (dd) through B12 cluster, and (df)
mixing). It is impossible to express it by the nearly free-electron
model.
•The gap was reproduced by a realistic band model for the first
time (after LDA+U correction). Relationship of the gapopening and the degeneracy was discussed.
•The CEF ground state must be 8, since otherwise a gap does
not open!!
•The temperature-dependence of the quasi-particle density of states is
indispensable to understanding the thermopower. It cannot be
understood by the rigid band model.
Future problems
•Based on the present model, we have to take into account the
correlation effect, and recalculate the thermopower, etc..
•Band anisotropy may be important for a comparison with experiment
49
on the thermopower.
Summary
•近藤絶縁体は，相関の強いバンド絶縁体である。
•電子相関とスピンのゆらぎの効果とは，連続的につな
がったものである。 dとfも連続している。
•状態密度とギャップは，多体効果のため，強く温度依存
する。
•相互作用のバンド間成分の大きさで，温度依存性が異
なり得る。
•光学伝導度のギャップは，さらに強く温度依存する。
(K.Urasaki & TS, JPSJ 68(1999)3477)
•σ(,T)の記述には，自己無撞着な計算が必要。（SCSOPTなど。）
•バンド計算の状態密度を反映した，正しいσ(,T)が必
要。
•χ(T)にはスピンの揺らぎが大事。
•S(T)にはdopingが大事。
•動的分子場理論とスピンの揺らぎの理論の統合理論
が可能。（TS，JPSJ 68 (1999) 3941; 69 (2000) No.12;
J.Phys. (2001))
•問題点：バーテックス補正
50
まとめ
•近藤半導体は，多体効果の強いバンド絶縁体(半導
体)である。
•例：YbB12，FeSi，CeRhSn，etc.
•多体効果が強いために，バンドギャップが近藤温度
程度の小さな値になる場合がある。
•多体効果が強いために，バンドギャップ（と状態密度
の形）が温度変化する。
•ギャップの成因は，基本的には半導体と同じである
ので，バンド計算で決まる。ただし，多体効果が強い
場合，バンド計算に問題が生じる。また，希土類化合
物では，スピン-軌道相互作用と結晶場分裂を正しく
扱うことが必要である。
•近藤半導体は役に立つか? 低温で性能のよい熱電
素子となる可能性は十分にある。
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