Download QUANTUM TRANSPORT IN THE TUNNELING AND

QUANTUM TRANSPORT IN THE
TUNNELING AND COTUNNELING
REGIMENS
Javier F. Nossa M.
Goal
•I
want to show you an animation
that I have been working with!!!
•Why?
•Because I like animations!!!
•Quantum transport.
25 Nov 2010
Quantum transport in the Tunneling and Cotunneling regimens
J.F. Nossa
Sequential Regimen
System
Fermi Level of Source is
lower than first unoccupied
level of dot
L
unoccupied
energy levels
R
E
EF
source
D
occupied
energy levels
quantized
energy levels
gate
drain
E  k T
B
Energy spacing must be greater
E   ( N  1)   ( N )
than thermal smearing
25 Nov 2010
Quantum transport in the Tunneling and Cotunneling regimens
addition energy
J.F. Nossa
Sequential Regimen
Applying a potential, the dot energy levels are shifted.
An electron can now
occupy the dot
Fermi energy of source is higher
than first dot unoccupied energy
level
Coulomb blockade
prevents others.
 L electrons
R
EF
source
25 Nov 2010
gate
drain
Quantum transport in the Tunneling and Cotunneling regimens
J.F. Nossa
Sequential Regimen
Ground state energy levels
Constant interaction model:
U ( N )  EC N 2  eVg N
U ( N  1)  EC ( N  1) 2  eVg ( N  1)
EC Charging energy
 ( N  1)  U ( N  2)  U ( N  1)
L
tunneling in
R
tunneling out
This cycle is known as
single electron tunneling
 ( N )  U ( N  1)  U ( N )
source
gate
drain
We fix VG and vary VDS  VL  VR
25 Nov 2010
Quantum transport in the Tunneling and Cotunneling regimens
J.F. Nossa
Sequential Regimen
 ( N  1)
Current
 (N )
Current Step
25 Nov 2010
Quantum transport in the Tunneling and Cotunneling regimens
J.F. Nossa
Sequential Regimen
Energy levels
Constant interaction model:
U ( N )  EC N 2  eVg N
U ( N  1)  EC ( N  1) 2  eVg ( N  1)
L
EC Charging energy
 ( N  1)  U ( N  2)  U ( N  1)
R
 ( N )  U ( N  1)  U ( N )
Source§
gate
drain
We fix VDS  VL  VR and vary VG
25 Nov 2010
Quantum transport in the Tunneling and Cotunneling regimens
J.F. Nossa
Sequential Regimen
25 Nov 2010
Quantum transport in the Tunneling and Cotunneling regimens
J.F. Nossa
Sequential Regimen
Energy levels
Constant interaction model:
U ( N )  EC N 2  eVg N
U ( N  1)  EC ( N  1) 2  eVg ( N  1)
EC Charging energy
 ( N  1)  U ( N  2)  U ( N  1)
L
R
 ( N )  U ( N  1)  U ( N )
Source
gate
drain
We change both VDS  VL  VR and VG
25 Nov 2010
Quantum transport in the Tunneling and Cotunneling regimens
J.F. Nossa
Sequential Regimen
Coulomb diamonds, differential conductance
Current Steps
Current Oscillations
25 Nov 2010
Quantum transport in the Tunneling and Cotunneling regimens
J.F. Nossa
Anderson Model
H
  n  Un n
Two level system
 ,
Coulomb diamond
E (0)  0
E ()   0
E ()   0
E (2)  2 0  U
25 Nov 2010
N=0
Quantum transport in the Tunneling and Cotunneling regimens
N=1
N=2
J.F. Nossa
Sequential Regimen
So far we have taken into account only ground states...
... what if we included excited states?
25 Nov 2010
Quantum transport in the Tunneling and Cotunneling regimens
J.F. Nossa
Sequential Regimen
N
N+1
If a line terminates at the N-electron Coulomb blockade region,
the transition necessarily involves an N-electron excited state
25 Nov 2010
Quantum transport in the Tunneling and Cotunneling regimens
J.F. Nossa
Anderson Model
 
H     n  Un n  B  S
Two level system
 ,
Coulomb diamond
E (0)  0
E ()   0  B
N=0
E ()   0  B
N=1
N=2
E (2)  2 0  U
25 Nov 2010
Quantum transport in the Tunneling and Cotunneling regimens
J.F. Nossa
Cotunneling Regimen
So far, only single-particle processes have been considered.
First order processes in the tunneling Hamiltonian.
Two-particle processes.
Higher order tunneling process
Cotunneling
One electron
occupies the dot
while, at the
same time,
another electron
leaves the dot
L
 ( N  1)  U ( N  2)  U ( N  1)
R
 ( N )  U ( N  1)  U ( N )
Source§
Because EIS  Ei
25 Nov 2010
Intermediated state
gate
drain
Virtual process
Quantum transport in the Tunneling and Cotunneling regimens
J.F. Nossa
Cotunneling Regimen
Elastic Cotunneling
D (N )  0
 D ( N  1)  0
L  R  0
Initial state
Intermediate
state
Final state
Spin Effects in Single-Electron Transistors, Ghislain Granger, MIT 2005
25 Nov 2010
Quantum transport in the Tunneling and Cotunneling regimens
J.F. Nossa
Cotunneling Regimen
Inelastic Cotunneling
E f  Ei
D (N )  0
 D ( N  1)  0
L  0
R  0
 L  
Initial state Intermediate
state
Final state
The dot is left in
an excited state
 L  
E f  Ei
25 Nov 2010
Quantum transport in the Tunneling and Cotunneling regimens
J.F. Nossa
Cotunneling Regimen
Cotunneling in the differential conductance
dI / dV
V
Sequential tunneling: current inside
the diamond is not allowed
25 Nov 2010
Two regimes: one consisting of elastic
processes only, and one including both
elastic and inelastic contributions
S. De Franceschi et.al., Phys. Rev. Lett. 86, 878
Quantum transport in the Tunneling and Cotunneling regimens
J.F. Nossa
Anderson Model
 
H     n  Un n  B  S
Two level system
 ,
Coulomb diamond
E (0)  0
E ()   0  B
N=0
E ()   0  B
N=1
N=2
E (2)  2 0  U
E ()  E ()   0  B  ( 0  B)  2 B
25 Nov 2010
Quantum transport in the Tunneling and Cotunneling regimens
J.F. Nossa
Anderson Model
 
H     n  Un n  B  S
Two level system
 ,
Coulomb diamond
E (0)  0
E ()   0  B
N=0
E ()   0  B
N=1
N=2
E (2)  2 0  U
E ()  E ()   0  B  ( 0  B)  2 B
25 Nov 2010
Quantum transport in the Tunneling and Cotunneling regimens
J.F. Nossa
Anderson Model
dI / dV
Step instead
of peak
N=0
E (N )
25 Nov 2010
N=1
N=2
V
Quantum transport in the Tunneling and Cotunneling regimens
J.F. Nossa
TUNNELING AND COTUNNELING
PHENOMENA IN SET
THANK
YOU
!!!
Javier F. Nossa M.