Circuitry with a Luttinger liquid

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Circuitry with a Luttinger liquid

• K.-V. Pham
• Laboratoire de Physique des Solides

Pascals Festschrifft Symposium
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• Some Background

• New playgrounds (lt 10 yrs) for LL at the
  Meso/Nano scale
• e.g. quantum wires, carbon nanotubes, cold atoms
  
• Finite-size ergo New Physics due to the
  boundaries
• IMO Two quite relevant things
• nature of the BOUNDARY CONDITION
• Periodic (finite-size corrections, numerics)
• Open (e.g. broken spin chains), twisted
• Boundary conformal field theory (e.g. single
  impurity as a boundary problem, cf Kondo)
• interaction with PROBES (are invasive) (e.g.
  transport)

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• Towards Nanoelectronics / nanospintronics
• But before some more basic questions
• What happens to a LL plugged into a (meso)
  electrical circuit?

i.e. LL as an electrical component Impact of
finite-size? Coupling to other electrical
components?
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How would an electrical engineer view a LL?
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How would an electrical engineer view a LL?

• Condensed Matter theorist
• Low-energy effective Field Theory (harmonic
  solid)

Density
LL phase fields
Current
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How would an electrical engineer view a LL?

• Electrical engineer

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How would an electrical engineer view a LL?

• Electrical engineer

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How would an electrical engineer view a LL?

• Electrical engineer

Capacitive energy !
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How would an electrical engineer view a LL?

• Electrical engineer

Capacitive energy !
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How would an electrical engineer view a LL?

• Electrical engineer

Capacitive energy !
Inductive energy !
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How would an electrical engineer view a LL?

• Electrical engineer

The LL is just a (lossless) Quantum Transmission
line
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How would an electrical engineer view a LL?

• Electrical engineer

The LL is just a (lossless) Quantum Transmission
line
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How would an electrical engineer view a LL?

• Electrical engineer

The LL is just a (lossless) Quantum Transmission
line

• Further Ref- Bockrath PhD Thesis 99, Burke IEEE
  02
• circuit theory (Nazarov, Blanter)
• K-V P., Eur Phys Journ B 2003

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• Excitations (from bosonization)
• Density oscillations i.e. Plasmons (neutral)

• Zero modes
• (charged but dispersionless)

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• Excitations (from bosonization)
• Density oscillations i.e. Plasmons (neutral)

• Zero modes
• (charged but dispersionless)

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• Excitations (from bosonization)
• Density oscillations i.e. Plasmons (neutral)

• Zero modes
• (charged but dispersionless)

Electrical Engineer? Transmission line
telegrapher equation
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• Excitations (from bosonization)
• Density oscillations i.e. Plasmons (neutral)

• Zero modes
• (charged but dispersionless)

Electrical Engineer? Transmission line
telegrapher equation
excitations are also plasma waves
Wave velocity
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• Excitations (from bosonization)
• Density oscillations i.e. Plasmons (neutral)

• Zero modes
• (charged but dispersionless)

Electrical Engineer? Transmission line
telegrapher equation
excitations are also plasma waves
Wave velocity

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DC Conductance of infinite LL
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DC Conductance of infinite LL
A little mystery LL Conductivity is actually
infinite ! Dissipation should be impossible!
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DC Conductance of infinite LL
A little mystery LL Conductivity is actually
infinite ! Dissipation should be impossible!
E.E. answer resistance is non-zero because its
not really a resistance but the characteristic
impedance of the transmission line !
(quantifies the energy transported by a traveling
wave)
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DC Conductance of infinite LL
A little mystery LL Conductivity is actually
infinite ! Dissipation should be impossible!
E.E. answer resistance is non-zero because its
not really a resistance but the characteristic
impedance of the transmission line !
(quantifies the energy transported by a traveling
wave)
half-infiniteTransmission line ltgt resistance
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DC Conductance of infinite LL
A little mystery LL Conductivity is actually
infinite ! Dissipation should be impossible!
E.E. answer resistance is non-zero because its
not really a resistance but the characteristic
impedance of the transmission line !
(quantifies the energy transported by a traveling
wave)
half-infiniteTransmission line ltgt resistance
InfiniteTransmission line 2 half-infinite TL
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DC Conductance of infinite LL
A little mystery LL Conductivity is actually
infinite ! Dissipation should be impossible!
E.E. answer resistance is non-zero because its
not really a resistance but the characteristic
impedance of the transmission line !
(quantifies the energy transported by a traveling
wave)
half-infiniteTransmission line ltgt resistance
InfiniteTransmission line 2 half-infinite TL
gt conductance G1/2Z0
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DC Conductance of infinite LL
A little mystery LL Conductivity is actually
infinite ! Dissipation should be impossible!
E.E. answer resistance is non-zero because its
not really a resistance but the characteristic
impedance of the transmission line !
(quantifies the energy transported by a traveling
wave)
half-infiniteTransmission line ltgt resistance
InfiniteTransmission line 2 half-infinite TL
gt conductance G1/2Z0
Since
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DC Conductance of infinite LL
A little mystery LL Conductivity is actually
infinite ! Dissipation should be impossible!
E.E. answer resistance is non-zero because its
not really a resistance but the characteristic
impedance of the transmission line !
(quantifies the energy transported by a traveling
wave)
half-infiniteTransmission line ltgt resistance
InfiniteTransmission line 2 half-infinite TL
gt conductance G1/2Z0
One recovers
Since

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A simple Series circuit

• Ref Lederer, Piéchon, Imura K-V P., PRB 03

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• Rationale
• Phenomenological Model for mesoscopic electrodes
• The 2 Resistors modelize contact resistances.
• Implementation
• Are described in term of dissipative boundary
  conditions.
• Quantization not trivial (NO normal eigenmodes)
  but bosonization still holds (Ref K-V P, Progr
  Th Ph 07)

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• Some Straightforward Properties (at least for an
  E.E.) (ref K-V P, EPJB 03)
• DC resistance
• AC conductance is a 3 terminal measurement

Conductance is a 3x3 matrix.
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• Resonances for Gij (i,j1,2)

• Interpretation
• Infinite Transmission Line (TL) Traveling waves

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• Resonances for Gij (i,j1,2)

• Interpretation
• Infinite Transmission Line (TL) Traveling waves
• Open TL Standing waves (nodes perfect
  reflections of plasma wave at boundaries)

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• Resonances for Gij (i,j1,2)

• Interpretation
• Infinite Transmission Line (TL) Traveling waves
• Open TL Standing waves (nodes perfect
  reflections of plasma wave at boundaries)
• TLresistors Standing waves are leaking
  (imperfect reflections gt finite life-time)

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Reflection coefficients for a TL (classical and
quantum i.e. LL)
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Reflection coefficients for a TL (classical and
quantum i.e. LL)
Resonances
Reflections in a TL due to impedance mismatch
(cf Safi Schulz, inhomogeneous LL,
Fabry-Perot)
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• Impedance matching of a TL and implications.

Impedance mismatch leads to reflections gt novel
physics for Luttinger (E.E. not so new, standing
waves of a TL)
Match impedances to Z0 gt kills reflections
!
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• Impedance matching of a TL and implications.

Impedance mismatch leads to reflections gt novel
physics for Luttinger (E.E. not so new, standing
waves of a TL)
Match impedances to Z0 gt kills reflections
!
gt finite TL now behaves like infinite TL
Property still true for quantum TL (i.e.
Luttinger) ! (cf K-V P., Prog. Th. Ph. 07)
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• Impedance matching of a Luttinger Liquid
• Remedy to invasiveness of probes
• The finite LL exhibits the same properties as the
  usual infinite LL
• allows measurements of intrinsic properties of a
  LL in (and despite) a meso setup.

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• Impedance matching of a Luttinger Liquid
• Remedy to invasiveness of probes
• The finite LL exhibits the same properties as the
  usual infinite LL
• allows measurements of intrinsic properties of a
  LL in (and despite) a meso setup.

• Experimental realization

Rheostat??? Depends on type of measurement (DC or
AC)
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• Tuning of (contact) resistances at the mesoscopic
  level in quantum wires (Yacoby)

Two-terminal conductance of a quantum wire
Electron density in the wire
Ref Yacoby et al, Nature Physics 07
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In this setup, contact resistances (barriers at
electrodes) are equal
So that
Impedance matching if
(crossing of curves GG(nL) and Ke2/hf(nL) )
The two curves cross impedance matching realized
!
(unpublished courtesy A. Yacoby)
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Applications of impedance matching Shot noise
(detection of fractional excitations in the LL)

• Issue
• shot noise for infinite LL in various setups
  should exhibit anomalous charges (Kane, Fisher
  PRL 94 T. Martin et al 03)
• These charges are irrational in general and can
  be shown to correspond to exact eigenstates of
  the LL
• Description of LL spectrum in terms of fractional
  eigenstates (holons, spinons, 1D Laughlin qp, )
  K-V P, Gabay Lederer PRB 00
• But probes are invasive so that it is predicted
  that fractional charges can not be extracted from
  shot noise (Ponomarenko 99, TrauzettelSafi 04)

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Interferences by probes circumvented by impedance
matching
A promising setup (A. Yacoby expts) Two parallel
quantum wires

• Spin-charge separation observed in this setup
  (Auslaender et al, Science 05)
• Current asymetry incompatible with free electrons
  observed (predicted by Safi Ann Phys 97) can be
  ascribed to fractional excitations (K-V P, Gabay,
  Lederer PRB 00).
• (Consistent with fractional excitations but not
  definite proof more expts needed)

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Other interesting things but no time for
discussion

• Gate conductance G33, DC AC shot noise, bulk
  tunneling, charge relaxation resistance

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(Setup idea Burke 02)
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Conclusion Main message 1) The LL is a Quantum
Transmission Line 2) The Physics of classical
Transmission lines can bring many interesting
insights into the LL physics at the meso scale
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Conclusion Main message 1) The LL is a Quantum
Transmission Line 2) The Physics of classical
Transmission lines can bring many interesting
insights into the LL physics at the meso scale
Thank You
Thank you, Pascal , for many fruitful years of
Physics !!!
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Addenda Gate conductance
Here RC is the contact resistance
Rq is the charge relaxation resistance
NB Recover earlier results of Blanter et al as
special limit