Kondo, Fano and Dicke effects
in side quantum dots
Pedro Orellana
UCN-Antofagasta
Collaborators
Gustavo Lara, Universidad de Antofagasta
Enrique V. Anda P. Universidade Católica de Rio de Janeiro
Outline
Kondo effect in quantum dots
 Fano effect in side attached quantum dots
 Model
 Results
 Conclusions

Kondo effect-Phenomenology:
Behavior of the resistance as a function
of the temperature in macroscopic systems.
Kondo Effect – More of 40 Years
after the Discovery
Jun Kondo's paper "Resistance Minimum in
Dilute Magnetic Alloys" was published in
Progress of Theoretical Physics 32 (1964)
37. Although more forty years have passed
since then, the importance of this work has
not diminished, but continues to increase.
Kondo solved the long-standing mystery of
resistance minimum phenomenon in his
study, thereby opening a door to fundamental
and universal physics; this is now known as
the
Kondo
effect.
Kondo effect in quantum dots
Conductance as a function of the
temperature in quantum dots
Kondo effect in quantum dots
• Ng, T. K. & Lee, P. A. On-site Coulomb repulsion and
resonant tunneling. Phys. Rev. Lett. 61, 1768–1771
(1988).
• Glazman, L. I. & Raikh, M. E. Resonant Kondo
transparency of a barrier with quasilocal impurity states.
JETP Lett. 47, 452–455 (1988).
Fano effect in a single quantum dot
The condition for the Fano resonance is the existence of two scattering
channels: a discrete level and a broad continuum band.
Noninteracting picture
Fano antiresonance
  1 

T ( ) 
2
2







1
2
Destructive interference between two paths
Kondo effect in a quantum wire with
a side coupled quantum-dot
Two side attached quantum dots
H  t   ci†, ci 1,  h.c. 
i

U



†
ˆ
ˆ

t
c
f

h
.
c
.



n
n


  l , 0, l ,
l ,   l , 
 l ,
2


l 1,2, 

We adopt the two-fold Anderson Hamiltonian. Each dot has a single level l
(with l = 1, 2), and intra-dot Coulomb repulsion U. The two side attached
dots are coupled to the QW with coupling t0 . We use the finite-U slave
boson mean-field approach in which all the boson operators are
replaced by their expectation value.
To fin the solution of this correlated fermions system, we use the finite-U slave boson
mean-field approach. We appeal to this semi analytical approach where, generalizing
the infinite-U slave-boson approximation the Hilbert space is enlarged at each site, to
contain in addition to the original fermions a set of four bosons represented by the
creation (annihilation) operators
 
ˆel†  eˆ l  , ˆpl†,  ˆpl ,  ,dˆ l† dˆ l
They act as projectors onto empty, single occupied (with spin up and down) and
doubly occupied electron states, respectively. Then, each creation (annihilation)
operator of an electron with spin  in the l-th QD, is substituted by
†
l ,
f Z
†
l ,
Z
l ,
fl ,

As we work at zero temperature, the bosons operators expectation values and the
Lagrande multipliers are determined by minimizing the energy <H> with respect to
these quantities. It is obtained in this way, a set of nonlinear equations for each
quantum dot, relating the expectation values for four bosonics operator, the three
Lagrange multipliers, and the electronic expectation values,
The operator Zi, ‘s are chosen to reproduce the correct limit when U tends to zero in
the mean field approximation,
To obtain the electronic expectation values <…>, the Hamiltonian, He is
diagonalized. Their stationary states can be written as
k 

a
j 
2
k
j
j  b l
l 1
k
l
where ajk and blk are the probabilities amplitudes to find
the electron at the site j and the l-th QD respectively,
with energy
  2t cos k
The amplitudes a’s and b’s obey the following linear difference
equations
 a  t  a
k
j
k
j 1
a
k
j 1
 a  t  a  a
k
0
k
1
k
1
t b
k
k



b


t
a

1 1
0 0
   2  b
k
2
 t a
,
k
0 0
k
0 1
 j  0
t b
k
0 2
 1  Vg   V
 2  Vg   V
CONTACTS
Transmission
For U sufficiently large the transmission can be written
approximately as a superposition of Fano and BrietWigner line shapes
  q

T    2
2

 2
2
 1   
2
where,   / 2, q  0,    V / 2
2
Density of states
This phenomenon is in analogy to the Dicke effect in optics, that take place in
the spontaneous emission of two closely lying atoms radiating a photon, with a
wavelength larger than the separation between them, in the same environment
The luminescence spectrum is characterized by a narrow and a broad peak,
associated with long-and-short lived states, respectively. The former state, coupled
weakly to the electromagnetic field, called subradiant, and latter, strongly coupled,
superradiant state
In the electronic case, however the level broadening are produced
by indirect coupling of the up-down QDs through the QW.


    2
 2
2
2
   
2
Tk1  , Tk 2     V / 2
2
2
Two Kondo
temperatures
Non equilibrium transport

1
k * k
fl c j 
f   k    Im bl a0

2 N   L , R ,k
†
J 4
e


 L , R , k

f   k    Im a a
k * k
0
1


Current(solid line, black) and differential conductance (dashed line,red) for
Vg=-3, U=6 for a) δV=0.1 and b) δV=0.5
Current (solid line, black) and differential conductance (dashed line) for Vg=-3,
on site energy, U=6, for a) δV=0.1 and b) δV=0.5
2e 
V 
 V 
J  V  2 arctan 
   arctan    ,
h 
 2 
  


dJ 2e2 
4 2
4 2 

 2
1  2

V
dV
h  V
2
2 

4


4




4
4

CONCLUSIONS





We have studied the transport through two single side coupled quantum
dots using the finite-U slave boson mean field approach at T=0.
We have found that the transmission spectrum shows a structure with two
antiresonances localized at the renormalized energies of the quantum dots.
The DOS of the system shows that when the Kondo correlations are
dominant there are two Kondo regimes each with its own Kondo
temperature.
The above behavior of the DOS is due to quantum interference in the
transmission through the two different resonance states of the quantum dots
coupled to common leads.
This result is analogous to the Dicke effect in optics.
Side attach double quantum dot
molecule
Transmission probability





2
       


T   
2
2
               2 

 




where the renormalized bonding and antibonding energies are,

1
  1   2
2
and
~
   t   0
2
0

1
2

1  2

2
2
 4t c .
Artificial Molecule
Coupled quantum dot
system
Series connection
Science 274 5291 (1996)
 
1   2
2

(1   2 )2  4tc2
2
Energetically the double quantum
dot molecule can be modeled as
two wells potential connected by
a barrier
Transmission spectrum
Transmission spectrum for 1=2=Vg=-3, tc = 0.5  (solid line) and tc
=  (dashed line), for on site energy, a) U = 2 ,b) U = 4 , c) U = 8 
and d) U =16 .
The figure allows to study the interplay between the Kondo effect and the inter-dot
anti-ferromagnetic correlation. Increasing U, a sharp feature develops close to the
Fermi energy, indicating the appearance of a Kondo resonance. We can see that
this process is more rapidly defined for the case where tc =0.5 than for tc =
This behavior is due to the anti-ferromagnetic interaction between the spins
of the dots, proportional to tc 2/U, that destroys the Kondo effect when it is
greater than the Kondo temperature creating a singlet ground state for the
quantum-dot molecule.
For U sufficiently large the transmission can be written approximately as
the superposition of two Fano-Kondo line shapes.
 ' q ' 

T   
2
2
 '' q '' 


2
 ' 1
 ''  1
where  '     2 / , q '   2    / ,


2




2
c
 ''      / , q ''  0, with   t / .
Density of states
of the double quantum dot molecule
   

   2



2
 
2

               2 

 

2

t


2
c
 
2

               2 

 

2
Density of states
In fact, the DOS can be written as the superposition of two
Lorentzian. We can observe the superposition of a narrow
and a broad Kondo peak
   
1



     
2

1



   2  
2
Density of states
The DOS is shown for various values of U, for the case tc =0.5, where the Kondo
spin correlation is dominant over the anti-ferromagnetic one.
This behavior is understood considering that the Kondo regime of quantum-dot 2 (QD2)
is a result of its weak coupling to the conduction electrons mediated by the intermediate
quantum-dot 1 (QD1) that is as well at the Kondo regime. This coupling creates a very
sharp Kondo peak at QD2 created by depleting, at the vicinity of the Fermi level,
the larger Kondo peak of QD1.
Existence of two Kondo temperature associated to each dot.
~
Tk 1  
~
Tk 2  t / 
2
c
Linear Conductance
2
dI
2e
G  lim

T 0
V  0 dV
h
  
      
2
2e
G
h
2


2


2
2
Linear conductance versus for Vg= 1=2, for tc = 0.5  (solid line) and tc =
 (dashed line), for on site energy, a) U = 2 ,b) U = 4 , c) U = 8  and d)
U =16 .
The linear conductance shows two Fano anti-resonances corresponding to the
bonding and anti-bonding energies of the quantum-dot molecule and one resonance
between them. The separation between the two antiresonance grows linearly with
U. The conductance vanishes when the bonding or the antibonding energies
coincide with the Fermi energy.
CONCLUSIONS




We have studied the transport through a side-coupled double
quantum-dot molecule using the finite-U slave boson mean
field approach.
The transmission spectrum shows a structure with two antiresonances localized at the bonding and antibonding
renormalized energies of the quantum-dot molecule, and one
resonance at the renormalized site energy of the outside
quantum-dot.
The LDOS at each dot shows that when the Kondo
correlations are dominant both dots are at the Kondo regime
each one with its own Kondo temperature.
Increasing the inter dot interaction, the anti-ferromagnetic
correlation becomes dominant destroying the Kondo effect and
the physics associated with it.