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RESONANT TUNNELING IN CARBON NANOTUBE QUANTUM DOTS

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Postma, Teepen, Yao, Grifoni, Dekker, Science 293 (2001) ... in tilted washboard potential. continuity equation. reduced density. matrix. e. 1. e. 2. CURRENT ... – PowerPoint PPT presentation

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Title: RESONANT TUNNELING IN CARBON NANOTUBE QUANTUM DOTS


1
RESONANT TUNNELING IN CARBON NANOTUBE QUANTUM
DOTS
MILENA GRIFONI
M. THORWART R. EGGER G. CUNIBERTI H. POSTMA C.
DEKKER
25 nm

Discussions Y. Nazarov
2
QUANTUM DOTS
3
ORTHODOX SET THEORY
4
NANOTUBE DOT IS A SET
Postma, Teepen, Yao, Grifoni, Dekker, Science 293
(2001)
Coulomb blockade in quantum regime
unconventional
5
PUZZLE
6
OVERVIEW
  • METALLIC SINGLE-WALL NANOTUBES (SWNT)
  • SWNT LUTTINGER LIQUIDS
  • SWNT WITH TWO BUCKLES
  • UNCOVENTIONAL RESONANT TUNNELING EXPONENT

1D DOT WITH LUTTINGER LEADS
CORRELATED TUNNELING MECHANISM
7
METALLIC SWNT MOLECULES
Energy
EF
metallic 1D conductor with 2 linear bands
k
LUTTINGER FEATURES
8
DOUBLE-BUCKLED SWNTs
buckles act as tunneling barriers
after Rochefort et al. 1998
50 x 50 nm2
Luttinger liquid with two impurities
Let us focus on spinless LL case, generalization
to SWNT case later
9
WHAT IS A LUTTINGER LIQUID ?
example spinless electrons in 1D
linear spectrum
bosonization identity
charge density
10
LUTTINGER HAMILTONIAN
captures interaction effects
nanotubes
11
TRANSPORT
12
TRANSPORT
13
CURRENT
14
CORRELATIONS
dipole
W SiR
dipole-dipole
correlations involving different/same
barriers
15
FINITE RANGE?
16
FINITE RANGE? not needed
17
CORRELATIONS II
W SiR
  • zero range WD purely oscillatory
  • WS Ohmic oscillations

ltcosh LDgt const,
ltsinh LDgt0
18
EFFECT OF THE CORRELATIONS ?
  • FIRST CONSIDER UNCORRELATED TUNNELING
  • MASTER EQUATION APPROACH
  • Ingold, Nazarov (1992) (gr 1), Furusaki PRB
    (1997)
  • GENERATING FUNCTION METHOD (FROM PI SOLUTION)
  • Grifoni, Thorwart, unpublished

19
MASTER EQUATION FOR UST
Gtot
example
20
MASTER EQUATION FOR UST II
Note
can also be obtained from the master eq.
Is there a simple diagrammatic interpretation of
Gf/b ?
21
GENERATING FUNCTION METHOD
22
GENERATING FUNCTION METHOD FOR ST
Sequential tunneling approximation Consider
only (but all) paths which are back to the
diagonal after two steps (giustified for strong
Coulomb interaction)
23
GENERATING FUNCTION METHOD FOR ST
Sequential tunneling approximation Consider
only (but all) paths which are back to the
diagonal after two steps (giustified for strong
Coulomb interaction)
24
GENERATING FUNCTION METHOD FOR ST
Sequential tunneling approximation Consider
only (but all) paths which are back to the
diagonal after two steps (giustified for strong
Coulomb interaction)
L
R
non trivial cancellations among contribution of
different paths
25
GENERATING FUNCTION METHOD FOR CST
Sequential tunneling approximation Consider
only (but all) paths which are back to the
diagonal after two steps (giustified for strong
Coulomb interaction)
L
R
non trivial cancellations among contribution of
different paths
Correlations!
26
GENERATING FUNCTION METHOD FOR UST
Sequential tunneling approximation Consider
only (but all) paths which are back to the
diagonal after two steps (giustified for strong
Coulomb interaction)
L
R
UST only intra-dipole Correlations!
again
27
GENERATING FUNCTION METHOD FOR UST II
Interpretation Higher order paths provide a
finite life-time for intermediate dot state,
which regularizes the divergent fourth-order
paths
L
Gtot
R
28
CST
m2
29
CST II
divergent
m3
As for UST, sum up higher order terms to get a
finite result
Approximations
  • Consider only diverging diagrams
  • Linearize in dipole-dipole interaction LS/D
    FS/D 0

30
CST III
summation over m
Systematic expansion in L
UST
modified line width
at resonance
31
MASTER EQUATION FOR CST
transfer through 1 barrier (irreducibile
contributions of second and higher order)
transfer trough dot (irreducibile contributions
at least of fourth order)
Thorwart et al. unpublished
finite life-time due to higher order paths found
self consistently
32
RESULTS GMAX
Thorwart et al., PRL (2002)
spinless LL
nanotubes
33
CONCLUSIONS REMARKS
dot
leads
  • UNCONVENTIONAL COULOMB BLOCKADE

REMARK
NONINTERACTING ELECTRONS gr 1
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