resonant tunneling in carbon nanotube quantum dots
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RESONANT TUNNELING IN CARBON NANOTUBE QUANTUM DOTS
MILENA GRIFONI
M. THORWARTR. EGGERG. CUNIBERTI
H. POSTMAC. DEKKER
200 nmafter Postma et al. Science (2001)
25 nm
Discussions: Y. Nazarov
QUANTUM DOTS
eVTkB ,
addition energy
EC
eNN 2
)()1(2
dotdot
dotsourceV Vg
Cg drain
e e
001dot ),1( NNg EEVN
Coulomb blockade
)(dot N
a)
L R
single electron tunneling
)(dot N
b) ...1 NNN
L R
ORTHODOX SET THEORYC
ondu
ctan
ce
Gate voltage
hold woud)a(SWNTfor 8.0end
1MAX
end TGLuttinger leads SETs
Furusaki, Nagaosa PRB (1993)
1MAX
TG semiconducting dot + Fermi leads
Beenakker PRB (1993)
Sequential tunneling
Gate voltage C
ondu
ctan
ce T2 > T1(a)
TkE B
NANOTUBE DOT IS A SETPostma, Teepen, Yao, Grifoni, Dekker, Science 293 (2001)
Coulomb blockade in quantum regime
Ec = 41 meV, E = 38 meV > kBT up to 440 K
1MAX
TG1
MAXend TG
dI/dV d2I/dV2
Gate voltage (V)
Bia
s vo
ltage
(V)
30 K
unconventional
PUZZLE
Why nanotube SET not ? 1
MAXend TG
1MAX
endend TG endendend 2
Correlated sequential tunneling Gate voltage
T2 > T1(b)
Con
duct
ance
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)
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
METALLIC SWNT MOLECULES
metallic 1D conductor with 2 linear bands
k
EF
Energy
LUTTINGER FEATURES
Let us focus on spinless LL case,
generalization to SWNT case later
DOUBLE-BUCKLED SWNT´s
after Rochefort et al. 1998
buckles act as tunneling barriers
Luttinger liquid with two impurities
50 x 50 nm2
WHAT IS A LUTTINGER LIQUID ?
Fk k
R
linear spectrum
bosonization identity )(/ ~ iLR e
charge density
xFkx
)(
example: spinless electrons in 1D
L
Fk
)(kE
L L
R Rq~0
)'()'()('1])()([2v
'22F
0 xxxVxdxdxxxdxH xxx
+ forward scattering )(ˆ qV
FE
LUTTINGER HAMILTONIAN
])(1)([2
v 220 x
gxgdxH x
2.0ln
v81
1
v1
12
0
RReV
gS
FF
)(rangedshort )( 0 xVdxVxV captures interactioneffects
nanotubes
g/vv F
TRANSPORT
extimptot HHHH 0
gCgVV
Luttinger liquids
localized impurities voltage sources
constkUxUxdxH impimpimp )(ˆ),()( backscattering
constxVxVxdxeH extextext )(),()( forward scattering
TRANSPORT
2/)()(2/)( LR xxtN
)()()( LR xxtn charge transferred across the dot
charge on the island
Brownian`particles´ n, Nin tilted washboard potentialCn
CCeV
NeVHtot
ggext
2
)cos()cos( nNUH impimp
)(2
lim tNeI t
continuity equation
) 0, , ; , , ( lim2,
i i f fn N
f tn Nt n N P N e
f f
reduced densitymatrix
11
nnNNe1 e2
nN
2
CURRENTExact trace over bosonic modes
L Rx x x x, ), (
]' , [ ]' , [ ]' [ ] [ ' ) 0, , ; , , (*
n n F N N F N A N A ND ND n Nt n N Pn N i i f f
bare actionext impH H bulk modes reduced
density matrix
) , (n N N
mass gap for ncharging energy
LINEAR TRANSPORTdynamics states dynamics states2
nonlocal in time coupling )' (' ), (t N t N
1 , 1 n n
n n,
1 , 1 n n
N N,
1 , 1 N N
1 , 1 N N
CORRELATIONS
tiTk
tJdtWB
sincoth)cos1()()(
02
W
,
0 /
) range finite for only ( 0 / 1
SWNT , 2. 0
g
g
correlations involving different/same barriers /WN N,
1 , 1 N N
1 , 1 N N
0 , ) ( g
J
gx
E F
0
v
) ( J
)()()()( 32212321 SSSSW = S+iR
dipole
dipole-dipole
FINITE RANGE?
FINITE RANGE?not needed
CORRELATIONS II
tiTk
tJdtWB
sincoth)cos1()()(
02
W
,
N N,
1 , 1 N N
1 , 1 N N
)()()()( 32212321 SSSS
W = S+iR
• zero range W : purely oscillatory WOhmic + oscillations
<cosh const, <sinh
0321 ,, Ohm
EFFECT OF THE CORRELATIONS ?
FIRST CONSIDER UNCORRELATED TUNNELING
• MASTER EQUATION APPROACH Ingold, Nazarov (1992) (g = 1), Furusaki PRB (1997)
• GENERATING FUNCTION METHOD (FROM PI SOLUTION) Grifoni, Thorwart, unpublished
MASTER EQUATION FOR UST
Rb Lb Rf Lf
tot
Rb Lb Rf Lfe
n N
N tt n P N e I,
) , ( lim2
linear regime: only n = 0,1 charges
end Lf
f LT e e dn iE W L
~ Re20
/ ) 1 ( ) (2
N N,
1 , 1 N N
1 , 1 N N
W
golden rule rateexample
Uncorrelated sequential tunneling:• only lowest order tunneling process • master equation for populations:
2/R L
Ingold, Nazarov (1992) (g = 1), Furusaki PRB (1997)
) , ( ) , 1 ( ) , 1 (
) , ( ) , ( ) ( ) , (
1 1 1
1
t n P t n P t n P
t n P t n P t n P
N Lb N Lf N Rb
N Rf N Rb Rf Lb Lf N
tot
MASTER EQUATION FOR UST II
) (b ftot
Rb Lb Rf Lfe e I
) , ( ) , ( ) , ( ) ( ) , (2 2t n P t n P t n P t n PN b N f N b f N
b
f
Note:
can also be obtained from the master eq.
Is there a simple diagrammatic interpretation of f/b ?
GENERATING FUNCTION METHOD
Different view from path integral approach
n N
Nn N
N tn P N t n P N I,
20
,
) , ( ˆ lim ) , ( lim
0
20) , ( lim
F
n N
NN
n P e F,
) , ( ˆ ) , (
generating function
exact series expression
2
) ( ) (0) ( ˆ ) ( ˆ lim
m
mb
mfI I I
contributions to the f/b current of order mExample: m = 2 (divergent!)
(c)2)N,2(N
N),(NN),(N
2)N,2(N
N),(N
2)N,2(N (a) (b)
W
cotunneling
GENERATING FUNCTION METHOD FOR ST
N N,
1 , 1 N N
1 , 1 N N Sequential tunneling approximation:
Consider only (but all) paths which are back tothe diagonal after two steps(giustified for strong Coulomb interaction)
)(ˆlim2
)(0
m
mII
GENERATING FUNCTION METHOD FOR ST
N N,
1 , 1 N N
1 , 1 N N Sequential tunneling approximation:
Consider only (but all) paths which are back tothe diagonal after two steps(giustified for strong Coulomb interaction)
)(ˆlim2
)(0
m
mII
GENERATING FUNCTION METHOD FOR ST
Sequential tunneling approximation:
Consider only (but all) paths which are back tothe diagonal after two steps(giustified for strong Coulomb interaction)
)(ˆlim2
)(0
m
mII
bm
bfm
fm LTRRTLI 22)( )()()(ˆ
L
R
non trivial cancellationsamong contributionof different paths
GENERATING FUNCTION METHOD FOR CST
Sequential tunneling approximation:
Consider only (but all) paths which are back tothe diagonal after two steps(giustified for strong Coulomb interaction)
)(ˆlim2
)(0
m
mII
bm
bfm
fm LTRRTLI 22)( )()()(ˆ
L
R
non trivial cancellationsamong contributionof different paths
Correlations!
GENERATING FUNCTION METHOD FOR UST
Sequential tunneling approximation:
Consider only (but all) paths which are back tothe diagonal after two steps(giustified for strong Coulomb interaction)
)(ˆlim2
)(0
m
mII
])()([)(ˆ 221
)(Lb
mtotRbRf
mtotLfm
m eI
L
R
b f e IRftot tot
Lf
... 1 1 lim
2
0
tot
Rb Lb Rf Lfe
0 lim
again
UST: onlyintra-dipoleCorrelations!
GENERATING FUNCTION METHOD FOR UST II
Interpretation:
Higher order paths provide a finite life-timefor intermediate dot state, which regularizes the divergent fourth-order paths
)(ˆlim2
)(0
m
mII
tot
LbRb
tot
RfLfbf ee )(
)](cos[)](cos[4 1111
)()(32
01
22231
REREeeeddd RfLfSSRL
ftot
)()()( iRStW
L
R
tot
2,lim 0
totecc RfLff
CST)()(22)( ˆˆ)()()(ˆ m
bm
fbm
bfm
fm IILTRRTLI
Let us lookorder by order:
)()()(32
01
22
0)2( 31321
4lim
SSRLf eeeddd
133132113 coscos),,(cosh RERE RfLf
133132113 sinsin),,(sinh RERE RfLf
m=2
/// RiSW
Short cut notation: RfLfRfLff sscceI )/()(ˆ )2(
cosh
sinh
divergent =0
exact!
CST II
)()(22)( ˆˆ)()(ˆ mb
mfb
mbf
mf
m IILTRRTLI
m=3
As for UST, sum up higher order terms to get a finite result
divergent
RfLfRfLfRfLfRfLff scsssccssccceI )/(ˆ )3(
RfLfRfLfRfLfRfLf ccccssccsssc
• Consider only diverging diagrams• Linearize in dipole-dipole interaction = 0
RbLbRfLf ccccc
Approximations:
CST III
summation over m Systematic expansion in
RfLfRfLfRfLfRfLff csscssccsscceI )/()(ˆ
tote
RfLff cc0lim
modified line width
end
end
TT
tot
RfLff const.
~~2
UST
at resonance
ss
tote
MASTER EQUATION FOR CST
2 / /,a b f L R n N
N tt n P N e I,
) , ( lim2
) , ( ) , (
) , ( ) , 1 ( ) , 1 (
) , ( ) , ( ) ( ) , (
2 2 2 2
1 1 1
1 2 2 1
t n P t n P
t n P t n P t n P
t n P t n P t n P
N f N b
N Lb N Lf N Rb
N Rf N f b tot N
transferthrough 1 barrier(irreducibile contributions ofsecond and higher order)
transfer trough dot (irreducibilecontributions at least of fourth order)
Thorwart et al. unpublished finite life-time due to higher orderpaths found self consistently
RESULTS GMAX
nanotubes:
spinless LL:
31
41, 1
effeff
gggg
112endend
g
endend
endend
*
1/
TG
TTGMAX
23.0g
4 bosonic fields
spin2 band2
Kane, Balents, Fisher PRL (1997), Egger, Gogolin, PRL (1997)
Thorwart et al., PRL (2002)
CONCLUSIONS & REMARKS
0endendend 1MAX )1( TgG
leads
dot
• UNCONVENTIONAL COULOMB BLOCKADE
0F /v xTkL BT LOW TEMPERATURES : BREAKDOWN OF UST IN LINEAR REGIME
NONINTERACTING ELECTRONS g =1REMARK
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