Transcript Kondo, Fano and Dicke effects in side quantum dots
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 the Kondo
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 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
T
1 1 2 2 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
i
c c i
1,
l
1,2,
t l
,
c
† 0,
f l
,
l
,
U
2
n l
,
n
ˆ
l
, 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
t 0
. 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
l †
l
ˆ
† l ,
,d ˆ
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
† f Z l , † l ,
Z l ,
f l ,
As we work at zero temperature, the bosons operators expectation values and the Lagrande multipliers are determined by minimizing the energy
The operator
Z i,
‘s are chosen to reproduce the correct limit when the mean field approximation,
U
tends to zero in
To obtain the electronic expectation values <…>, the Hamiltonian, H e diagonalized. Their stationary states can be written as is
k
j
a
k j
j
l
2 1
k
b l
l
where
a j k
and
b l k
are the probabilities amplitudes to find the electron at the site j and the l-th QD respectively, with energy
2 cos
k
The amplitudes
a
’s and
b
’s obey the following linear difference equations
a k j
a
0
k
t a
1
k j k
1
a k
1
a
j k
1 ,
t b
0 1
k
t b
0 2
k
1 2
b
1
k
b
2
k
t a
0 0
k
t a
0 0
k j
0
1 2
V g
V g
V
V
CONTACTS
Transmission
For
U
sufficiently large the transmission can be written approximately as a superposition of Fano and Briet Wigner line shapes 2
q T
where,
1
2
2
2
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.
T k
1
2
,
T k
2
2
2
2
V
2
2
2
/ 2
Two Kondo temperatures
Non equilibrium transport
J
†
f c l j
1 2
N
4
e
f
k
Im
b l k
*
a
0
k
f
k
Im
a
0
k
*
a
1
k
Current(solid line, black) and differential conductance (dashed line,red) for
V g =-3, U=6 for a) δV=0.1 and b) δV=0.5
Current (solid line, black) and differential conductance (dashed line) for V g =-3, on site energy, U=6, for
a
) δV=0.1 and b) δV=0.5
J dJ dV
2
e h
V
2
e
2
h
1
V
2 4 4 2
V
2 arctan 2
V
4 2 4 2 2 ,
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
T
2 2 2 2 ~ where the renormalized bonding and antibonding energies are, and 1 2 1 2 1 2 1 2 2
t
2
t
0
Artificial Molecule Coupled quantum dot system Series connection
Science
274
5291 (1996)
1 2 2 ( 1 2 ) 2 4
t c
2 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 =V g =-3 , t c = and d) U =16 .
= 0.5 (solid line) and t c (dashed line), for on site energy, a) U = 2 ,b) U = 4 , c) U = 8 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 t c =0.5
than for t c =
This behavior is due to the anti-ferromagnetic interaction between the spins of the dots, proportional to t c 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.
T
' ' 2
q
' 1
2
'' '' 2
q
'' 1
2 / 2 '' 2 0, with
t
2
c
/ ,
Density of states of the double quantum dot molecule
2 2 2
t c
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 t c =0.5
spin correlation is dominant over the anti-ferromagnetic one. , where the Kondo
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.
T k
1 ~
T k
2
t c
2 / ~
Linear Conductance
G
lim
V
0
dI dV G
2
e
2
h
2 2
e h
2
T
2 2
Linear conductance versus for V U =16 .
g = 1 = 2 , for t c = 0.5 (solid line) and t c (dashed line), for on site energy, a) U = 2 ,b) U = 4 , c) U = 8 = and d) 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 anti resonances 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.