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Circuitry with a Luttinger liquid

Circuitry with a Luttinger liquid. K.-V. Pham Laboratoire de Physique des Solides. Pascal’s Festschrifft Symposium. New playgrounds (< 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

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Circuitry with a Luttinger liquid

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  1. Circuitry with a Luttinger liquid K.-V. Pham Laboratoire de Physique des Solides Pascal’s Festschrifft Symposium

  2. New playgrounds (< 10 yrs) for LL at the Meso/Nano scale: • e.g. quantum wires, carbon nanotubes, cold atoms • Finite-size: ergo NewPhysics due to the boundaries • IMO Two quite relevant things: • nature of theBOUNDARY 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) • Some Background:

  3. 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?

  4. How would an electrical engineer view a LL?

  5. How would an electrical engineer view a LL? • Condensed Matter theorist: • Low-energy effective Field Theory (harmonic solid) Density: LL phase fields Current:

  6. How would an electrical engineer view a LL? • Electrical engineer:

  7. How would an electrical engineer view a LL? • Electrical engineer:

  8. How would an electrical engineer view a LL? • Electrical engineer: Capacitive energy !

  9. How would an electrical engineer view a LL? • Electrical engineer: Capacitive energy !

  10. How would an electrical engineer view a LL? • Electrical engineer: Capacitive energy ! Inductive energy !

  11. How would an electrical engineer view a LL? • Electrical engineer: The LL is just a (lossless) Quantum Transmission line

  12. How would an electrical engineer view a LL? • Electrical engineer: The LL is just a (lossless) Quantum Transmission line

  13. 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

  14. Excitations (from bosonization): • Density oscillations i.e. Plasmons (neutral) • Zero modes • (charged but dispersionless)

  15. Excitations (from bosonization): • Density oscillations i.e. Plasmons (neutral) • Zero modes • (charged but dispersionless)

  16. Excitations (from bosonization): • Density oscillations i.e. Plasmons (neutral) • Zero modes • (charged but dispersionless) Electrical Engineer? Transmission line: telegrapher equation

  17. 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

  18. 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

  19. DC Conductance of infinite LL:

  20. DC Conductance of infinite LL: A little mystery: LL Conductivity is actually infinite ! Dissipation should be impossible!

  21. 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 it’s not really a resistance but the characteristic impedance of the transmission line ! (quantifies the energy transported by a traveling wave)

  22. 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 it’s not really a resistance but the characteristic impedance of the transmission line ! (quantifies the energy transported by a traveling wave) half-infiniteTransmission line <=> resistance =

  23. 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 it’s not really a resistance but the characteristic impedance of the transmission line ! (quantifies the energy transported by a traveling wave) half-infiniteTransmission line <=> resistance = InfiniteTransmission line = 2 half-infinite TL

  24. 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 it’s not really a resistance but the characteristic impedance of the transmission line ! (quantifies the energy transported by a traveling wave) half-infiniteTransmission line <=> resistance = InfiniteTransmission line = 2 half-infinite TL => conductance: G=1/2Z0

  25. 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 it’s not really a resistance but the characteristic impedance of the transmission line ! (quantifies the energy transported by a traveling wave) half-infiniteTransmission line <=> resistance = InfiniteTransmission line = 2 half-infinite TL => conductance: G=1/2Z0 Since:

  26. 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 it’s not really a resistance but the characteristic impedance of the transmission line ! (quantifies the energy transported by a traveling wave) half-infiniteTransmission line <=> resistance = InfiniteTransmission line = 2 half-infinite TL => conductance: G=1/2Z0 One recovers: Since:

  27. A simple Series circuit • Ref: Lederer, Piéchon, Imura + K-V P., PRB 03

  28. 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)

  29. 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.

  30. Resonances for Gij (i,j=1,2): • Interpretation: • Infinite Transmission Line (TL): Traveling waves

  31. Resonances for Gij (i,j=1,2): • Interpretation: • Infinite Transmission Line (TL): Traveling waves • Open TL: Standing waves (nodes: perfect reflections of plasma wave at boundaries)

  32. Resonances for Gij (i,j=1,2): • Interpretation: • Infinite Transmission Line (TL): Traveling waves • Open TL: Standing waves (nodes: perfect reflections of plasma wave at boundaries) • TL+resistors: Standing waves are leaking (imperfect reflections => finite life-time)

  33. Reflection coefficients for a TL (classical and quantum i.e. LL):

  34. 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)

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

  36. Impedance matching of a TL and implications. Impedance mismatch leads to reflections => novel physics for Luttinger (E.E. :not so new, standing waves of a TL) Match impedances to Z0 => kills reflections ! => finite TL now behaves like infinite TL Property still true for quantum TL (i.e. Luttinger) ! (cf K-V P., Prog. Th. Ph. 07)

  37. 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.

  38. 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)

  39. 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

  40. In this setup, contact resistances (barriers at electrodes) are equal: So that: Impedance matching if: (crossing of curves G=G(nL) and Ke2/h=f(nL) ) The two curves cross: impedance matching realized ! (unpublished; courtesy A. Yacoby)

  41. 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, Trauzettel+Safi ’04)

  42. 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)

  43. Other interesting things but no time for discussion… • Gate conductance G33, DC & AC shot noise, bulk tunneling, charge relaxation resistance

  44. (Setup idea: Burke ’02)

  45. 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

  46. 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 !!!

  47. 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:

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