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Electronic transport in one-dimensional wires
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Electronic transportin one-dimensional wires
Akira Furusaki (RIKEN)
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Aug 14, 2003 Electronic transport in 1D wires 2
Outline Tomonaga-Luttinger (TL) liquid Bosonization Single impurity in a TL liquid Two impurities in a TL liquid
linear conductance G
Random-matrix approach to transport in disordered wires
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Aug 14, 2003 Electronic transport in 1D wires 3
1D metals= Tomanaga-Luttinger liquid No single-particle excitations Collective bosonic excitations
spin-charge separation charge density fluctuations spin density fluctuations
Power-law decay of correlation functions (T=0)
tunneling density of states
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Aug 14, 2003 Electronic transport in 1D wires 4
TL liquids are realized in: Very narrow (single-channel) quantum wires edge states of fractional quantum Hall liquids Carbon nanotubes
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Aug 14, 2003 Electronic transport in 1D wires 5
Interacting spinless fermions Simplified continuum model
kinetic energy
short-range repulsive interaction (forward scattering)
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Aug 14, 2003 Electronic transport in 1D wires 6
Abelian Bosonization Fermions = Bosons in 1D
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Aug 14, 2003 Electronic transport in 1D wires 7
Electron density
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Aug 14, 2003 Electronic transport in 1D wires 8
Kinetic energy
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Aug 14, 2003 Electronic transport in 1D wires 9
Bosonized Hamiltonian
TL liquid parameter g
g < 1: repulsive interaction FQHE edgeg = 1: non-interacting case g > 1: attractive interaction
Interacting fermions = free bosons
12
1
m
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Aug 14, 2003 Electronic transport in 1D wires 10
Correlation functions ( T=0 )
Scaling dimension of is
4a
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Aug 14, 2003 Electronic transport in 1D wires 11
Single impurity
Non-interacting case (free spinless fermions)
transmission probability
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Aug 14, 2003 Electronic transport in 1D wires 12
Current
Conductance G changes continuously. no temperature dependence. is a marginal perturbation
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Aug 14, 2003 Electronic transport in 1D wires 13
Interacting spinless fermions reflection at the barrier potential
Hamiltonian
free boson + = pinning of charge density wave
electric current
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Aug 14, 2003 Electronic transport in 1D wires 14
Partition function (path integral)
effective action for
linear: dissipation due to gapless excitations in TL liquid
(Caldeira-Leggett: Macroscopic Quantum Coherence)
a particle (with coordinate ) moving in a cosine potential with friction
0x
|| n
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Aug 14, 2003 Electronic transport in 1D wires 15
Renormalization-group analysis Weak-potential limit weak perturbation:
scaling equation (lowest order): renormalized potential:
conductance
4cos0V
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Aug 14, 2003 Electronic transport in 1D wires 16
Strong-potential limit (weak-tunneling limit) duality transformation [A. Schmid (’83); compact QED by A.M. Polyakov]
“ dilute instanton (=tunneling) gas”
t: tunneling matrix element (fugacity)
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Aug 14, 2003 Electronic transport in 1D wires 17
scaling equation:
renormalized tunneling matrix element:
conductance
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Aug 14, 2003 Electronic transport in 1D wires 18
Flow diagram for transmission probability (Kane & Fisher, 1992)
g<1 (repulsive int.) perfect reflection at T=0
g=1 (free fermions) marginal
g>1 (attractive int.) perfect transmission at T=0
1
01
Trans.Prob.
g
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Aug 14, 2003 Electronic transport in 1D wires 19
Exact results “Toulouse limit” g=1/2
introduce new fields
refermionization
quadratic Hamiltonian
cf. 2-channel Kondo problem (Emery-Kivelson, 1992)
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Aug 14, 2003 Electronic transport in 1D wires 20
Conductance at g=1/2
General gThe boundary sine-Gordon theory is exactly solvable (Ghoshal & Zamolodchikov, 1994)
Bethe ansatz elastic single-quasiparticle S-matrix (Fendley, Ludwig & Saleur, 1995)
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Aug 14, 2003 Electronic transport in 1D wires 21
Spinful case (electrons)(Furusaki & Nagaosa, 1993; Kane & Fisher, 1992)
charge boson: spin boson:
Hamiltonian
: non-interacting electrons
: repulsive interactions
: if spin sector has SU(2) symmetry
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Aug 14, 2003 Electronic transport in 1D wires 22
Weak-potential limit
Strong-potential limit (weak-tunneling limit)single-electron tunneling: t
RG flow diagram
critical surface
at intermediate
coupling
tKKdl
dt
11
2
11
00
2
11 VKK
dl
dV
1K 21 K
211
2
KKTtG
0
1
0
1
1
Trans.Prob.
Trans.Prob.
K K
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Aug 14, 2003 Electronic transport in 1D wires 23
External leads (Fermi-liquid reservoir) (Maslov & Stone, 1994)Tomonaga-Luttinger liquid:
Fermi-liquid leads:
Action
Current I vs Electric field E
dc conductance is not renormalized by the e-e interaction
if the wire is connected to Fermi-liquid reservoirs
Lx 0Lxx ,0
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Aug 14, 2003 Electronic transport in 1D wires 24
Weak e-e interactions (Matveev, Yue & Glazman, 1993)
small parameter:
V(q): Fourier transform of interaction potential
scaling equation for the transmission probability
lowest order in
but exact in
conductance
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Aug 14, 2003 Electronic transport in 1D wires 25
Coulomb interactions (Nagaosa & Furusaki, 1994; Fabrizio, Gogolin & Scheidel, 1994) : width of a quantum wire
scaling equation for tunneling
conductance
stronger suppression than power law
W
)/1log()( qWqV
|)|/log(11nF WvrK
trldl
dt 11
2
1
Fv
er
2
2/32/1 log
3
2exp
WT
vrG F
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Aug 14, 2003 Electronic transport in 1D wires 26
Experiments on tunneling Edge states in FQHE
(Chang, Pfeiffer & West, 1996)
tunneling between a Fermi liquid and edge state
[Fig. 1 & Fig. 2 of PRL 77, 2538 (1996) were shown in the lecture]
3/1
1
VI
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Aug 14, 2003 Electronic transport in 1D wires 27
Single-wall carbon nanotubes Yao, Postma, Balents & Dekker, Nature 402, 273 (1999)
[Fig. 1 and Fig. 3 were shown in the lecture.]
Segment I & II: bulk tunneling
Across the kink: end-to-end tunneling
exp:
TG
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Aug 14, 2003 Electronic transport in 1D wires 28
Resonant Tunneling (Double barriers) Non-interacting case
transmission amplitude: t
has maximum when resonance
(symmetric barrier)
symmetric case backscattering is irrelevant
asymmetric case backscattering is marginal = single impurity
0 dx
L R
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Aug 14, 2003 Electronic transport in 1D wires 29
When
life time of discrete levels
Conductance
if coherent tunneling
if
incoherent
sequential tunneling
peak width
1||,|| 22 RL tt
22
2
41
||
RL
RLt
2,, ||
2 RLF
RL td
v
22
2
41
RL
RL
EdE
dfdE
h
eG
RLT ,
TRL ,
22
2
41
RL
RL
h
eG
d
df
h
eG
RL
RL )(22
1
max
111
RLT
G
T
RL 2
1
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Aug 14, 2003 Electronic transport in 1D wires 30
Resonant tunneling in TL liquidsSpinless fermions
Hamiltonian
gate voltage
Current
Excess charge in [0, d ]
is massive
dt
deI
02
1 d 0~ d
~e
Q
~
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Aug 14, 2003 Electronic transport in 1D wires 31
Weak-potential limit (Kane & Fisher, 1992)
effective action for
single-barrier problem
scaling equation
if (symmetric) and (on resonance)
dvn /||
,01 V RL VV 1)cos(
g g1 1/4
1V 2V
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Aug 14, 2003 Electronic transport in 1D wires 32
Resonance line shape symmetric
¼ < g < 1 is the only relevant operator, on resonance
universal line shape peak width
not Lorentzian
RL VV
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Aug 14, 2003 Electronic transport in 1D wires 33
Weak-tunneling limit (Furusaki & Nagaosa, 1993; Furusaki,1998)
Off resonance
process is not allowed at low T
virtual tunneling
On resonancesequential tunneling
life time due to tunneling through a barrier
peak width
1t
e
2t
21
2
tt
TG
on :
11
21
gTt
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Aug 14, 2003 Electronic transport in 1D wires 34
Phase diagram at T=0 Symmetric barriers
Asymmetric barriers
g<1 g=1 g>1
G
g0
1
11/21/4
Transmissionprobability
GG
G
h
e2
11 )1( Vgdl
dV
22 1
1 tgdl
dt
h
e2G G G
0 1
1
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Aug 14, 2003 Electronic transport in 1D wires 35
T > 0 Weak potential
Weak tunneling
sequential tunneling
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Aug 14, 2003 Electronic transport in 1D wires 36
Experiments on resonant tunneling in TL liquids
Auslaender et al., Phys. Rev. Lett. 84, 1764 (2000)
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Aug 14, 2003 Electronic transport in 1D wires 37
Carbon nanotubes
Postma et al., Science 293, 76 (2001)
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Aug 14, 2003 Electronic transport in 1D wires 38
Summary In 1D e-e interaction is crucial
Tomonaga-Luttinger liquid Repulsive e-e interaction
backward potential scattering is relevant power-law suppression of tunnel density of states
Problems nontrivial fixed points at intermediate coupling Resonant-tunneling experiment?