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Quantum transport at nano-scale

Quantum transport at nano-scale. Zarand, Chung, Simon, Vojta, PRL 97 166802 (2006) Chung, Hofstetter, PRB 76 045329 (2007), selected by Virtual Journal of Nanoscience and Technology Aug. 6 2007

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Quantum transport at nano-scale

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  1. Quantum transport at nano-scale Zarand, Chung, Simon, Vojta, PRL 97 166802 (2006) Chung, Hofstetter, PRB 76 045329 (2007), selected by Virtual Journal of Nanoscience and Technology Aug. 6 2007 Chung, Zarand, Woelfle, PRB 77, 035120 (2008), selected by Virtual Journal of Nanoscience and Technology Jan. 8 2008 Chung, Glossop, Fritz, Kircan, Ingersent,Vojta, PRB 76, 235103 (2007) Chung, Le Hur, Vojta, Woelfle nonequilibrium transport near the quantum phase transition (arXiv:0811.1230) Chung-Hou Chung 仲崇厚 Electrophysics Dept. National Chiao-Tung University Hsin-Chu, Taiwan Collaborators: Matthias Vojta (Koeln), Gergely Zarand (Budapest), Walter Hofstetter (Frankfurt U.) Pascal Simon (CNRS, Grenoble), Lars Fritz (Harvard), Marijana Kircan (Max Planck, Stuttgart), Matthew Glossop (Rice U.) , Kevin Ingersent (U. Florida) Peter Woelfle (Karlsruhe), Karyn Le Hur (Yale U.)

  2. Outline • Introduction • Quantum criticality in a double-quantum-dot system • Quantum phase transition in a dissipative quantum dot • Nonequilibrium transport in a noisy quantum dot • Conclusions

  3. ed+U Coulomb blockade ed Vg VSD Quantum dot---A single-Electron-Transistor (SET) Single quantum dot Goldhaber-Gorden et al. nature 391 156 (1998) Coulomb Blockade

  4. Goldhaber-Gorden et al. nature 391 156 (1998) Quantum dot---charge quantization

  5. ed+U Coulomb blockade ed Vg VSD odd even Kondo effect in quantum dot Kondo effect conductance anomalies Glazman et al. Physics world 2001 L.Kouwenhoven et al. science 289, 2105 (2000)

  6. Kondo effect in metals with magnetic impurities logT (Kondo, 1964) electron-impurity spin-flip scattering (Glazman et al. Physics world 2001) For T<Tk (Kondo Temperature), spin-flip scattering off impurities enhances Ground state is Resistance increases as T is lowered

  7. Kondo effect in quantum dot (J. von Delft)

  8. Kondo effect in quantum dot

  9. Kondo effect in quantum dot AndersonModel New energy scale: Tk ≈ Dexp(-pU/G) For T < Tk : Impurity spin is screened (Kondo screening) Spin-singlet ground state Local density of states developesKondoresonance d ∝ Vg local energy level : charging energy : level width : All tunable! U Γ=2πV 2ρd

  10. P-H symmetry = p/2 Kondo Resonance of a single quantum dot Spectral density at T=0 Universal scaling of T/Tk M. Sindel L. Kouwenhoven et al. science 2000 particle-hole symmetry phase shift Fredel sum rule

  11. Numerical Renormalization Group (NRG) K.G. Wilson, Rev. Mod. Phys. 47, 773 (1975) W. Hofstetter, Advances in solid state physics 41, 27 (2001) • Non-perturbative numerical method by Wilson to treat quantum impurity problem • Logarithmic discretization of the conduction band • Anderson impurity model is mapped onto a linear chain of fermions • Iteratively diagonalize the chain and keep low energy levels

  12. Perturbative Renormalization Group (RG) approach: Anderson's poor man scaling and Tk HAnderson • Reducing bandwidth by integrating out high energy modes Anderson 1964 J J J J • Obtaining equivalent model with effective couplings • Scaling equation w < Tk, J diverges, Kondo screening J 0

  13. T g g c Quantum phase transitions Non-analyticity in ground state properties as a function of some control parameter g Avoided level crossing which becomes sharp in the infinite volume limit: Second-order transition True level crossing: Usually a first-order transition Sachdev, quantum phase transitions, Cambridge Univ. press, 1999 • Critical point is a novel state of matter • Critical excitations control dynamics in the wide quantum-critical region at non-zero temperatures • Quantum critical region exhibits universal power-law behaviors

  14. I. Quantum phase transition in coupled double-quantum-dot system

  15. Recent experiments on coupled quantum dots (I). C.M. Macrus et al. Science, 304, 565 (2004) • Two quantum dots coupled through an open conducting region which mediates an antiferromagnetic spin-spin coupling • For odd number of electrons on both dots, splitting of zero bias Kondo resonance is observed for strong spin exchange coupling.

  16. Von der Zant et al. (PRL, 2005) • A quantum dot coupled to magnetic impurities in the leads • Antiferromagnetic spin coupling between impurity and dot suppresses Kondo effect • Kondo peak restored at finite temperatures and magnetic fields

  17. T Non-fermi liquid Kondo Spin-singlet K Kc Quantum phase transition in coupled double-quantum-dot system G. Zarand, C.H. C, P. Simon, M. Vojta, PRL, 97, 166802 (2006) C.H. C and W. Hofstetter, PRB 76 045329 (2007) R1 L1 K L2 R2 • Critical point is a novel state of matter • Critical excitations control dynamics in the wide quantum-critical region at non-zero temperatures • Quantum critical region exhibits universal power-law behaviors

  18. 2-impurity Kondo problem Affleck et al. PRB 52, 9528 (1995) Jones and Varma, PRL 58, 843 (1989) Sakai et al. J. Phys. Soc. Japan 61, 7, 2333 (1992);ibdb. 61, 7, 2348 (1992) 1 2 K X Heavy fermions -R/2 R/2 H = H0+ Himp H0 =

  19. Quantum criticality of 2-impurity Kondo problem • Particle-hole asymmetry T Non-fermi liquid 1 2 even Kondo Spin-singlet K odd Kc Quantum phase transition as K is tuned Kc = 2.2 Tk Jump of phase shift at Kc K < Kc, d = p/2 ; K >KC , d = 0 • Particle-hole symmetry V=0 H  H’ = H under Affleck et al. PRB 52, 9528 (1995) Jones and Varma, PRL 58, 843 (1989) Jones and Varma, PRB 40, 324 (1989) Kc is smeared out, crossover Misleading common belief ! We have corrected it!

  20. Quantum Phase Transition in Double Quantum dots: P-H Symmetry triplet states L1 R1 Izumida and Sakai PRL 87, 216803 (2001) Vavilov and Glazman PRL 94, 086805 (2005) K Simon et al. cond-mat/0404540 Hofstetter and Schoeller, PRL 88, 061803 (2002) L2 singlet state R2 • Two quantum dots (1 and 2) couple to two-channel leads • Antiferrimagnetic exchange interaction K, Magnetic field B • 2-channel Kondo physics, complete Kondo screening for B = K = 0 K

  21. Transport properties • Current through the quantum dots: • Transmission coefficient: • Linearconductance:

  22. n k-kc Crossover energy scale T* NRG Flow of the lowest energy Phase shift d d Kondo K<KC JC Kondo p/2 K>KC Spin-singlet Spin-singlet 0 Kc K Two stable fixed points (Kondo and spin-singlet phases ) Jump of phase shift in both channels at Kc One unstable fixed point (critical fixed point) Kc, controlling the quantum phase transition

  23. J < Jc, transport properties reach unitary limit: • T( = 0) 2, G(T = 0) 2G0 where G0 = 2e2/h. • J > Jc spins of two dots form singlet ground state, • T( = 0) 0, G(T = 0) 0; and Kondo peak splits up. • Quantum phase transition between Kondo (small J) and spin singlet (large J) phase. Quantum phase transition of a double-quantum-dot system J=RKKY=K Chung, Hofstetter, PRB 76 045329 (2007)

  24. Restoring of Kondo resonance in coupled quantum dots Singlet-triplet crossover at finite temperatures T NRG Result Experiment by von der Zant et al. T=0.003 T=0.004 • At T= 0, Kondo peak splits up due to large J. • Low energy spectral density increases as temperature increases • Kondo resonance reappears when T is of order of J • Kondo peak decreases again when T is increased further.

  25. J=-0.005, Tk=0.0025 B in Step of 0.001 Singlet-triplet crossover at finite magnetic fields J=0.007, Jc=0.005, Tk=0.0025, T=0.00001, in step of 400 B NRG: P-h symmetry EXP: P-h asymmetry Ferromagnetic J<0 Antiferromagnetic J>0 J close to Jc, smooth crossover J >> Jc, sharper crossover

  26. _ _ G G 1 2 K even 2 (L2+R2) even 1 (L1+R1) T Non-fermi liquid Kondo Spin-singlet K Kc Quantum criticality in a double-quantum –dot system: P-H Asymmetry G. Zarand, C.H. C, P. Simon, M. Vojta, PRL, 97, 166802 (2006) V1 ,V2 break P-H sym and parity sym.  QCP still survives as long as no direct hoping t=0

  27. _ _ G G 1 2 K Quantum criticality in a double-quantum –dot system No direct hoping, t = 0 Asymmetric limit: T1=Tk1, T2= Tk2 QCP occurs when 2 channel Kondo System Goldhaber-Gordon et. al. PRL 90 136602 (2003) QC state in DQDs identical to 2CKondo state Particle-hole and parity symmetry are not required Critical point is destroyed by charge transfer btw channel 1 and 2

  28. Transport of double-quantum-dot near QCP (only K, no t term) K K NRG on DQDs without t, P-H and parity symmetry At K=Kc Affleck andLudwig PRB 48 7279 (1993)

  29. charge transfer between two channels of the leads Relevant operator Generate smooth crossover at energy scale The only relevant operator at QCP: direct hoping term t dim[ ] = 1/2 (wr.t.QCP) RG most dangerous operators: off-diagonal J12 typical quantum dot At scale Tk, may spoil the observation of QCP

  30. << How to suppress hoping effect and observe QCP in double-QDs assume effective spin coupling between 1 and 2 off-diagonal Kondo coupling more likely to observe QCP of DQDs in experiments

  31. The single quantum dot can get Kondo screened via 2 different channels: At low temperatures, blue channel finite conductance; red channel zero conductance At the 2CK fixed point, Conductance g(Vds) scales as The single quantum dot can get Kondo screened via 2 different channels: At low temperatures, blue channel finite conductance; red channel zero conductance The 2CK fixed point observed in recent Exp. by Goldhaber-Gorden et al. Goldhaber-Gorden et al, Nature 446, 167 ( 2007) At the 2CK fixed point, Conductance g(Vds) scales as

  32. 1 2 V J even Side-coupled double quantum dots Chung, Zarand, Woelfle, PRB 77, 035120 (2008), • Two coupled quantum dots, only dot 1 couples to single-channel leads • Antiferrimagnetic exchange interaction J • 1-channel Kondo physics, dot 2 is Kondo screened for any J > 0. • Kosterlitz-Thouless transition, Jc = 0

  33. Jk 4V2/U J: AF coupling btw dot 1 and 2 Tk D rc 1/G 2 stage Kondo effect 1st stage Kondo screening Jk: Kondo coupling 2nd stage Kondo screening dip in DOS of dot 1

  34. Log (T*) 1/J J 0 8 Kondo spin-singlet NRG:Spectral density of Model (II) U=1 ed=-0.5 G=0.1 Tk=0.006 L=2 Kosterlitz-Thouless quantum transition No 3rd unstable fixed point corresponding to the critical point Crossover energy scale T* exponentially depends on |J-Jc|

  35. Dip in DOS of dot 1: Perturbation theory 1 2 when w Dip in DOS of dot 1 J = 0 d1 wn< Tk J > 0 but weak self-energy vertex sum over leading logarithmic corrections

  36. Dip in DOS: perturbation theory U=1, ed=-0.5, G= 0.1, L=2,J=0.0005, Tk=0.006, T*=8.2x10-10 • Excellence agreement between Perturbation theory (PT) and NRG for T* << w << Tk • PT breaks down for w T* • Deviation at larger w > O(Tk)due to interaction U

  37. L1 R1 R2 L2 n J x J-Jc T* 8 J 8 0 Jc 0 Kondo spin-singlet Kondo spin-singlet Summary I • Coupled quantum dots in Kondo regime exhibit quantum phase transition quantum critical point • The QCP of DQDs is identical to that of a 2-channel Kondo system • correct common misleading belief: The QCP is robust against particle-hole and parity asymmetries • The QCP is destroyed by charge transfer between two channels • The effect of charge transfer can be reduced by inserting additional even number of dots, making it possible to be observe QCP in experiments K-T transition

  38. II. Quantum phase transition in a dissipative quantum dot

  39. ed+U Coulomb blockade ed Vg VSD Quantum dot as charge qubit--quantum two-level system charge qubit-

  40. Quantum dot as artificial spin S=1/2 system Quantum 2-level system

  41. Dissipation driven quantum phase transition in a noisy quantum dot Noise = charge fluctuation of gate voltage Vg K. Le Hur et al, PRL 2004, 2005, PRB (2005), Noise ~ SHO of LC transmission line Impedence H = Hc + Ht + HHO N=1/2 Q=0 and Q=1 degenerate Caldeira-Leggett Model

  42. Spin Boson model

  43. / Delocalized-Localized transition delocalized localized h ~ N -1/2 K. Le Hur et al, PRL 2004,

  44. Charge Kondo effect in a quantum dot with Ohmic dissipation Hdissipative dot non-interacting lead N=1/2 Q=0 and Q=1degenerate Anisotropic Kondo model de-localized Jz = -1/2 R localized g=J Kosterlitz-Thouless transition

  45. Generalized dissipative boson bath (sub-ohmic noise) Ohmic { Sub-Ohmic

  46. Generalized fermionic leads: Power-law DOS Anderson model Quantum phase transition in the pseudogap Anderson/Kondo model d-wave superconductors and graphene: r =1 Fradkin et al. PRL 1990 Local moment (LM) J X Kondo Jc 0

  47. Delocalized-Localized transition in Pseudogap Fermi-Bose Anderson model C.H.Chung et al., PRB 76, 235103 (2007) Pseudogap Fermionic bath Sub-ohmic bosonic bath

  48. Phase diagram Field-theoretical RG

  49. Critical propertiesvia perturbative RG exact

  50. Critical properties via NRG

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