1. RESONANT TUNNELING IN CARBON NANOTUBE QUANTUM DOTS
200 nm
after Postma et al. Science (2001)
MILENA GRIFONI
M. THORWART
R. EGGER
G. CUNIBERTI
H. POSTMA
C. DEKKER
25 nm
Discussions: Y. Nazarov

2. single electron tunneling
QUANTUM DOTS
addition energy
dot
source V Vg Cg
drain e
Coulomb blockade a) mL mR
single electron tunneling b) mL mR

3. ORTHODOX SET THEORY (a) Luttinger leads SETs Conductance
Gate voltage
semiconducting dot + Fermi leads
Beenakker PRB (1993)
Sequential tunneling
Gate voltage
Conductance
T2 > T1
(a)
Luttinger leads SETs
Furusaki, Nagaosa PRB (1993)

4. NANOTUBE DOT IS A SET Ec = 41 meV, DE = 38 meV > kBT up to 440 K
Postma, Teepen, Yao, Grifoni, Dekker, Science 293 (2001)
Ec = 41 meV,
DE = 38 meV > kBT up to 440 K
dI/dV
d2I/dV2
Gate voltage (V)
Bias voltage (V)
30 K
unconventional
Coulomb blockade in quantum regime

5. Correlated sequential tunneling
PUZZLE
Why nanotube SET not ?
unscreened Coulomb interaction ? Maurey, Giamarchi, EPL (1997)
weak tunneling at metallic contacts ?
Kleimann et al., PRB (2002)
asymmetric barriers ?
Nazarov, Glazman, PRL (2003)
correlated tunneling ?
Postma et al., Science (2001), Thorwart et al. PRL (2002)
Hügle and Egger, EPL (2004)
Correlated sequential tunneling
Gate voltage
T2 > T1
(b)
Conductance

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 SWNT´s
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 L R
q~0
+ forward scattering

10. LUTTINGER HAMILTONIAN
captures interaction
effects
nanotubes

11. TRANSPORT Luttinger liquids voltage sources localized impurities
backscattering
forward scattering

12. TRANSPORT charge transferred across the dot charge on the island 2 1 e
Brownian`particles´ n, N
in tilted washboard potential
continuity equation
reduced density
matrix

13. CURRENT Exact trace over bosonic modes reduced bare action
bulk modes
reduced
density matrix
nonlocal in time
coupling
mass gap for n
charging energy
LINEAR TRANSPORT

14. CORRELATIONS dipole W = S+iR dipole-dipole
correlations involving different/same barriers

15. FINITE RANGE?

16. FINITE RANGE?
not needed

17. CORRELATIONS II W = S+iR zero range WD : purely oscillatory
WS : Ohmic + oscillations
<cosh LD> const,
<sinh LD>=0

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
Uncorrelated sequential tunneling:
only lowest order tunneling process
master equation for populations:
Ingold, Nazarov (1992) (gr = 1), Furusaki PRB (1997)
Gtot
golden rule rate
linear regime:
only n = 0,1 charges
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
Different view from path integral approach
generating function
exact series expression
contributions to the f/b current
of order D2m
Example: m = 2 (divergent!)
(c)
(a)
(b)
cotunneling

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 exact! Short cut notation: divergent l =0 Let us look
order by order:
m=2
Short cut notation:
cosh LD
sinh LD
divergent l =0

29. CST II
divergent
m=3
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 Systematic expansion in L summation over m 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 ï î í ì » G µ / T spinless LL: nanotubes:
Thorwart et al., PRL (2002) ï î í ì G -
end * 1 / a T
MAX
4 bosonic fields
Kane, Balents, Fisher PRL (1997), Egger, Gogolin, PRL (1997)
spinless LL:
nanotubes:

33. CONCLUSIONS & REMARKS dot leads REMARK LOW TEMPERATURES :
BREAKDOWN OF UST IN LINEAR REGIME
dot
leads
UNCONVENTIONAL COULOMB BLOCKADE
REMARK
NONINTERACTING ELECTRONS gr =1