Transcript Slide 1
Capri Spring School on Transport in Nanostructures, 2-9 April 2006
Current Noise
in Non-chiral Luttinger Liquids:
Appearance of Fractional Charge
F. Dolcini (Scuola Normale Superiore, Pisa)
in collaboration with
I. Safi (Lab. Phys. Solides, CNRS, Orsay (Paris))
B. Trauzettel (Universiteit Leiden)
H. Grabert (Universität Freiburg)
Outline
Quantum wires: experimental realizations and
theoretical modelling
Role of e- - e- interaction (Luttinger Liquid)
Results:
Shot Noise
Finite Frequency Noise
Discussion and Conclusions
Physical system
quantum wire (1D)
L
metallic lead
impurity
Edge States
FQHE
systems
(chiral LL)
SWNT
(Single
carbon
NanoTubes)
Cleaved
edgeinWall
overgrowth
on
GaAs-AlGaAs
metallic lead
Transport in ballistic wires
Impurity (barrier)
ballistic conductor
R
ni= 1
nR= R
T=1-R
nT= T
(nT)2 = T (1-T)
Average current:
I j(t )
Fluctuations:
(Noise)
S(ω) dt ei t j(t)j( 0 )
j = j -
j
Shot Noise: S( V)
1) Non-interacting electrons
Scattering Matrix Formalism (M.Büttiker’s lectures)
Impurity (barrier)
ballistic conductor
T=1-R
R
ni= 1
nT= T
nR= R
(nT)2 = T (1-T)
for a weak impurity T1
Blanter, Büttiker, Phys. Rep. 336,1 (00)
e2
S(ω V) 2 eV T(1 T)
h
S 2 e IBS
Noise due to incoherent back-scattering of electrons
-
-
2) 1-D wires: The effect of e - e interaction
in D=3
m m*
FERMI LIQUID:
Single-particle picture
is not destroyed !!
εk ε0k
τk = finite life
quasi-particles
in D=1
Fermi-Liquid theory breaks-down !
In D=1 there’s no geometrical room for
single particle excitations!
Only collective excitations
(Luttinger Liquid Theory)
R , L electron operators
H ivF dx : R x R L x L :
interaction
U
dx : R2 ( x) : : L2 ( x) : 2 R ( x) L ( x)
2
quartic in
R / L ( x) : R/ L R / L :
Luttinger Liquid Theory: Pass to a new basis
(x) e
i ΦR (x) 4π
(x) e
i ΦL (x) 4π
R
L
Plasmonic excitations
re-write the Hamiltonian:
Φ Φ
H vF
2
x
R
2
x
L
dx
H0
Hint
U
2
2
xΦR xΦL 2 xΦR xΦL dx
2π
Advantage: everything is quadratic now !
diagonalize with a rotation!
introduce the fields
~
1
ΦR (1 g) ΦR 1 gΦL
2
~
1
ΦL 1 gΦR 1 gΦL
2
~
H v
~
~ 2
xΦR xΦL
dx
2
where:
g
1
U
1
vF
1
g→1
no interaction
g→0
interaction → ∞
ΦR , ΦL
(coupled)
R (x)
(x) e
L
(decoupled)
new modes:
electron operators
R (x) e i Φ
~ ~
ΦR , ΦL
4π
i ΦL (x) 4π
~
R (x) e
i ΦR (x) 4π
L (x) e
i ΦL (x) 4π
~
{ R/L(x) , R/L(y) } = 0
R(x) R(y) = R(x) R(y) ei g sgn(x-y)
[R/L(x) , (y) ] = e R/L (x) (x-y)
[ R/L(x) , (y) ] = ge
(x) (x-y)
R/L
Backscattering: R L
ge
interaction
Is the fractional charge e*=ge observable ?
Indirectly: measures of the interaction strength g
Temperature dependence of the linear Conductance:
G T
Bockrath et al. Nature 397, 598 (99); Yao et al. Nature 402, 273 (99) ;
Postma et al. Science 293, 76 (01) ; Ausländer, Phys. Rev. Lett. 84, 1764 (00) ;
Displacement currents induced in the gate
Blanter, Hekking and Büttiker, Phys. Rev. Lett. 81, 1925 (98)
Oscillations in the I-V Characteristics
Peça and Balents, Phys. Rev. B 68, 205423 (03)
Dolcini et al. Phys. Rev. Lett. 91, 266402 (03)
Is it possible to observe directly the fractional
charge of the collective excitations ?
How is the relation
by the interaction ?
S 2 e IBS modified
Fractional charge in chiral Luttinger Liquids
Lectures by C.Glattli
Edge States in FQHE device
= filling factor = 1/3
g = = 1/3
weak back-scattering limit:
Theory (Kane & Fisher PRL 94)
Experiments
(de Picciotto et al. Nature 97,
Saminadayar et al. PRL 97 )
S (V ) = 2 e* IBS
e* = g e
fractional charge
OPEN QUESTION: and for non-chiral LL (e.g. SWNT) ?
chiral
pair of chiral LL
vs
non-chiral
non-chiral LL
R
R
L
L
e2
G g
h
S ( V) = 2 e g IBS
2 tuned separately
R , L cannotebe
G
h
sensitive to presence
of boundaries
S (finite
V)length
= ? effects
Model
for an interacting (non–chiral) quantum wire
1
g
1
U0
π vF
x
g→1
no interaction
g→0
interaction → ∞
Inhomogeneous
Luttinger Liquid
Safi & Schulz, PRB (95)
Maslov & Stone, PRB (‘95)
Ponomarenko, PRB (95)
L
Effect of the finite size of the wire
L
ħ L = ħ vF/L ~ meV
Ballistic frequency
2.0
T=0
g=0.25
IBS
1.5
I(V) oscillates with period ħ L
1.0
0.5
0.0
0
5
10
15
eV / L
20
25
30
Dolcini, Grabert, Safi, Trauzettel,
Phys. Rev. Lett. 91, 266402 (2003)
shot noise ( V) for non-chiral LL
Sshot = S( eV) IBS · 2 e F( / L)
F
F (ω ωL ) → 1
2
1.0
e* = e
0.5
0.0
F ω = g
/ L
Frequency-averaging gives the fractional charge
Trauzettel, Safi, Dolcini, Grabert, Phys. Rev. Lett. 92, 226405 (2004)
e* = g e
Discussion
LEAD
WIRE
In the regime ω ωL
Lω L
one always probes the
physics of the leads
e* = e
LEAD
Finite frequency noise ( V)
S( x;) S0 ( x;) Simp ( x;)
does not depend on V !
(the same as in equilibrium)
Excess
noise
depends on V
SEX ( x;) S( x; ) S( x;) |V 0
SEX ( x;) Simp ( x;) Simp ( x;) |V 0
For a non-interacting wire (g=1)
SEX =
0
if | | > eV
2 (eV -| |) T (1-T)
if | | < eV
1.0
0.8
0.6
Diagonal shape
0.4
0.2
0.0
0.0
0.2
0.4
0.6
eV
0.8
1.0
SEX ( x;) S( x; ) S( x;) |V 0
in an interacting wire
Dolcini, et al. PRB 71, 165309 (2005)
a)
30
12
b)
30
16
9
25
20
20
8
15
4
12
3
0
15
/L
6
Y Axis
g=0.25
25
-3
10
10
-6
5
5
g=0.50
0
-4
-8
-9
0
5
10
15
20
25
c)
0
30
0
30
30
25
25
0
5
10
15
20
25
30
d)
70
20
60
15
20
10
5
15
0
-5
-10
20
/L
g=0.75
50
40
30
15
20
10
10
5
5
0
30
0
10
0
-10
-15
0
5
10
15
20
eV / L
25
0
5
10
15
20
eV / L
25
30
g=0.95
Summary and Conclusions
Interacting 1D quantum wire (Luttinger Liquid), attached to
metallic leads, and with an impurity
Shot Noise: Fano factor depends on / L
o
L
physics of non-interacting wire
o
L
interaction effects are observable
Fractional charge e* = eg observable
through frequency averaging
Excess Noise: V- diagram: diagonal
interaction
horizontal striped
Experimental realizability
ħ L = ħ vF/gL ~ meV
Linearity of the spectrum fulfilled
SIS on-chip detector:
2; Al 100 GHz; Nb 1 THz
THz frequencies realistic
DeBlock et al. Science 301, 203 (03)
1.0
0.8
F () 1 (1 g
2
) L / 2vF
2
g=0.75
F()
Low-frequency analysis
0.6
0.4
O( )
g=0.5
3
0.2
g=0.25
0.0
0.0
0.5
1.0
/ 2L
1.5
2.0
Andreev-type reflections at the contacts
WIRE
LEAD
(Interacting: 0 < g <1 )
(Non Interacting: g=1)
t
x
x
1+
-
F(ω /ωL ) fn ei n/
L
h.c.
n 0
x
n = # of Andreev reflections
n = 0
fn=0 = <F>
bulk