Transcript Correlation-induced resonances and population switching in

Kondo effect in a quantum dot
without spin
Hyun-Woo Lee (Postech)
& Sejoong Kim (Postech  MIT)
References:
H.-W. Lee & S. Kim, cond-mat/0610496
P. Silvestrov & Y. Imry, cond-mat/0609355
V. Kashcheyevs, A. Schiller, A. Aharony, & O. Entin-Wohlman, cond-mat/0610194
Kondo effect
 Temperature
dependence of resistance
 (T )   0  aT 2  bT 5
 Resistance
minimum
Kondo effect
(continued)
[J. Kondo, Prog. Theor. Phys. ’64]
 Scattering
by magnetic impurities
Before
After
s-d model
After


 
H sd  J sd S  s (0)   J k ,k ' S  ck,ck ',  S ck,ck ',  S z ck,ck ',  ck,ck ',
k ,k '
 (T )   0  aT  bT  cm ln
2
5

T

(*) 엄종화
LogT dependence in R(T)
Scattering amplitude for a channel of k   k  
t k ,k   t k ,k 
t k(1),k   
(1)
 t k  , k 
J
Sz
2N
t k( 2), k    t k( 2), k  
z
( 2)
 
2
 t k( 2), k  

 J 
 02
 S z N (0) ln( kT / D)
2
N


where N : number of d-electrons, N(0) : density of state at EF
D : width of conduction electron distribution around EF
Jk,q = J
= 0
where –D < ek, eq < D
otherwise
2
2
2J
RKondo (T )  c
t k  , k   c Rm [ 1 
N (0) ln( kT / D)   ]

N
This lnT dependence combined with the phonon contribution (T5 dependence)
makes a resistance minimum in R(T).
Kondo effect (continued)
 High
T vs. low T
Kondo singlet
Cf. Asymptotic
freedom
Kondo effect
(*) 엄종화
TK << T 일 때,
 (T )  ln( T / TK )
When T << TK,  ~ 0 - cT2
: unitary limit
TK ~ T 일 때, Hamann expression (Phys. Rev. 1967)
1
2

1
S ( S  1) 2 
 (T )  0 (1  1 
)
2
2
 [ln( T / TK )] 
For TK > T, take (-) in the equation
TK < T, take (+) in the equation
(*) 엄종화
Kondo effect in AuFe(26ppm) wire
Hamann expression (Phys. Rev. 1967)
 (ncm)
4
1
 (T )  0 (1 
2
1
2

S ( S  1) 2 
)
1 
2
[ln(
T
/
T
)]
K


2
From fitting the Hamann expression
to (T), we obtain
0
2
3
4 5 6 78
2
1
3
4 5 6 78
2
10
S = 0.12, TK = 0.99 K.
T (K)
Slope of Kondo resistivity = 0.11 ncm / (ppm decade K)
Concentration of AuFe is estimated by the slope of 
=> 26 ppm in the above figure
Kondo effect in quantum dot
Quantum dot (QD)
 “Metallic”
limit
(en) 2
E ( n) 
 Vg en
2C
n
Vg
~e2/2C >> kT
Transport through a QD
 Orthodox

theory of Coulomb blockade
Transport due to charge fluctuations
Quantum confinement
 Single
particle energy quantization
E >> kT
H dot   e i d i, d i ,
i ,
e2 1

ni , ni ', '  Vg e ni ,

2C 2 (i , )(i ', ')
i ,
Even-odd effect
 Spin
singlet (S=0) vs doublet (S=1/2)
n
0
3
2
1
1/2
0
S=1/2
Vg
 QD
with odd n = magnetic impurity ???
Kondo effect in QD ?
 Hamiltonian
n
H  H dot  H Source  H Drain  H Tunneling
H Tunneling 
 t
i, j,
c.f.

c d j ,  h.c. 
Vg

i , j i ,

H sd   J k ,k ' S  ck, ck ',  S  ck, ck ',  S z ck, ck ',  ck, ck ',
k ,k '
 Spin
Before

flip via second order processes
After
Kondo effect in QD w/ odd n

Theories  Kondo suppression of R

T. K. Ng and P. A. Lee
• Phys. Rev. Lett. 61, 1768 (1988)

L. I. Glazman and M. E. Raikh
• JETP Lett. 47, 452 (1988)

Experiments

D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D.
Abush-Magder, U. Meirav, and M. A. Kaster
• Nature 391, 156 (1998); Phys. Rev. Lett. 81, 5225 (1998)

S. M. Cronenwett, T. H. Oostercamp, and L. P.
Kouwenhoven
• Science 281, 540 (1998)
(*) 엄종화
Unitary limit of the Kondo effect in SET
[W. G. van der Wiel et al., Science ’00]
온도대역: 15 mK – 800 mK
G(T)
G at Vgl = -413mV shows
logarithmic T dependence (inset),
and saturates below 90mK (unitary limit)
This experiment shows a unitary limit
= 2e2/h (GR=GL의 경우)
Kondo resonance peak
Vgl was fixed at -413mV.
VSD was biased between S and D.
FWHM
0.78G  3.52kT
 
e
(*) 엄종화
Kondo temperature: TK
In Anderson model,
; Costi et al., J. Phys.: Condense. Matter 6, 2519 (1994)
TK in Log scale
kTK 
GU
e 0 (e 0  U )
exp[
]
2
GU
An empirical function
; Goldhaber-Gordon et al., PRL 81, 5225 (1998)


1

G(T )  G0 
 1  (21/ S  1) T 2 / T 2 
K 

S
: universal functional form of T/TK
s is a fit parameter, but is almost constant
~0.2 in the Kondo regime.
Kondo effect in QD w/o spin?
Two level QD
 QD
w/ two single-particle level
2
H dot   e j d j d j  U (n1  1 / 2)( n2  1 / 2)
j 1
 Source
& Drain
H lead  t 


(
c
 m,l cm1,l  h.c.)
l  L , R m 0
H T   (t lj c0,l d j  h.c.)
 Tunneling
j ,l
t
 “Spin”
?
t 1L
t1R
t2L
t1R
t
Pseudospin for e1=e2(= e)
 Unitary
transformations
Pseudospin up
Pseudospin
Pseudospin down
Schrieffer-Wolf transformation:
QD system (Anderson model)  s-d model
 Fock
space decomposition
   n0  n1  n2
 Full
Hamiltonian
 H 00

 H10
H
 20
H 01
H11
H 21
 Projection
H 02  0 
 0 
 
 
H12  1   E  1 
 
H 22  2 
 2
to n=1 Fock space


1
1
H 21  H10
H 01  1  E 1
 H11  H12
E  H 22
E  H 00


H s d
1
1
 H11  H12
H 21  H10
H 01
Eg  H 22
Eg  H 00
0
1
2
Effective Hamiltonian Hs-d
 Total
Hamiltonian

H s d  H ex  H lead  H B

H ex   J kk ' S  ck ck '  J kk ' S  ck ck '  J kkz ' S z ck ck   ck ck  
k ,k '
H B  S z B
eff
z

Bzeff  Bz   J kkz , 2' ck ck '  ck ck ' 
k ,k '
Anisotropic antiferro-exchange J kkz '  J kk '  J kk '  0
• U(1) instead of SU(2)

Pseudomagnetic field Bzeff

(*) For =
• SU(2): Jz=J+=J• Bzeff=0
Pseudomagnetic field Bzeff
 Expectation
hz   B  
eff
z
value
G  G

U / 2e
ln
U / 2 e
G /   0V2/ 

hz
0
1
e=+U/2 0 -U/2
For G  >G 
•  Population switching from  level to  level with
decreasing e
2
Population switching (PS)
[Silvestrov & Imry, PRL’00]
 Energy


renormalization
eeff= e bare+ e (hopping)
e : gate voltage dependent
Charge
10
0
1
2
e=+U/2 0 -U/2
Charge
U
12
U
Poor man’s scaling
 [1]
Fock space decomposition
D
   el ,top (D)  g  hole,bottom(D)
 [2]
 [3]

Full Hamiltonian
 H ee

 H ge
H
 he
D
H eg
H gg
H hg
 e 
H eh  e 
 
 
H gh  g   E  g 
 
H hh  h 
 g
Projection to “g” sector of Fock space
New Hamiltonian w/ reduced D
 [4]
D
Back to [1]
Scaling equations
 Exchange
J’s
dJ 
 2  0 J  J z
d ln D


dJ z
 2  0 J  J 
d ln D
( J 0 )2  ( J z )2  ( J  )2
Scaling invariant
Integration: Characteristic energy scale
(Kondo temperature)

1
J z  J0 

k BTK  U exp  
ln z
 4 0 J 0 J  J 0 
 Pseudomagnetic

field Bzeff
Integration: Bzeff  hz
Anisotropic s-d model
 Approximation
  Anisotropic

H B  S z B
eff
z
~
 H B   S z hz
s-d model
Exact solution (via Bethe ansatz) available !!!
• Tsvelick & Wiegmann, Adv. Phys. 32, 453 (1983)
Conductance at T=0

and  scattering states

  ( x)    e  e
 Friedel
ikx
2i ikx
e



( x)    e  e
ikx
2i ikx
e

sum rule
    n 
 Landauer-Buttiker
    n 
formula
G  Gmax sin 2   n    n  
Gmax
L R* 2
2 2
4t t t t
e

h  t L 2  t L 2  t R 2  t R 2   4 t L t R*  t L t R* 2
2
1
2
1 1
2 2
 1

2
L R*
1 1
Anisotropic s-d model
[Tsvelick and Wiegmann, Adv. Phys. (1983)]
 Sz=(n-n)/2
G
vs. hz
vs. e
hz   B  
eff
z
G  G

ln
U / 2e
U / 2 e
Cf. Conventional spin Kondo
 Conventional
spin Kondo
0
1
e=+U/2 0 -U/2
 Kondo

w/o spin
Correlation-induced resonance
[Meden & Marquardt, PRL 96, 146801 (2006)]
2
w/o degeneracy
e1-e2 0
 Same
Unitary transformation
 Additional
pseudomagnetic field h
hz  2 (|  |2  |  |2 )

hx  2 Re( * )
Parallel to z
• Shift of CIRs

Perpendicular to z
• Asymmetry in CIRs (Fano-like)
hy  2 Im( * )
Summary

Kondo effect in QD w/o spin


Distinct conductance pattern (cf. spin Kondo in QD)
Future directions



w/o degeneracy
Temperature dependence
Pseudospin & real spin
• Real Spin [SU(2)]
• Pseudospin [Not SU(2) invariant]
• Connection w/ anomalous transmission phase problem ?