RESONANT TUNNELING IN CARBON NANOTUBE QUANTUM DOTS

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

Discussions Y. Nazarov
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QUANTUM DOTS
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ORTHODOX SET THEORY
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NANOTUBE DOT IS A SET
Postma, Teepen, Yao, Grifoni, Dekker, Science 293
(2001)
Coulomb blockade in quantum regime
unconventional
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PUZZLE
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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
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METALLIC SWNT MOLECULES
Energy
EF
metallic 1D conductor with 2 linear bands
k
LUTTINGER FEATURES
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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
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WHAT IS A LUTTINGER LIQUID ?
example spinless electrons in 1D
linear spectrum
bosonization identity
charge density
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LUTTINGER HAMILTONIAN
captures interaction effects
nanotubes
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TRANSPORT
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TRANSPORT
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CURRENT
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CORRELATIONS
dipole
W SiR
dipole-dipole
correlations involving different/same
barriers
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FINITE RANGE?
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FINITE RANGE? not needed
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CORRELATIONS II
W SiR

• zero range WD purely oscillatory
• WS Ohmic oscillations

ltcosh LDgt const,
ltsinh LDgt0
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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• Grifoni, Thorwart, unpublished

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MASTER EQUATION FOR UST
Gtot
example
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MASTER EQUATION FOR UST II
Note
can also be obtained from the master eq.
Is there a simple diagrammatic interpretation of
Gf/b ?
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GENERATING FUNCTION METHOD
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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)
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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)
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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
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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!
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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
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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
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CST
m2
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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

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CST III
summation over m
Systematic expansion in L
UST
modified line width
at resonance
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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
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RESULTS GMAX
Thorwart et al., PRL (2002)
spinless LL
nanotubes
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CONCLUSIONS REMARKS
dot
leads

• UNCONVENTIONAL COULOMB BLOCKADE

REMARK
NONINTERACTING ELECTRONS gr 1