1. THE KONDO EFFECT IN CARBON NANOTUBES
Milena Grifoni
University of Regensburg
700 nm
Frauenchiemsee, October 5, 2015

2. TRANSPORT IN CARBON NANOTUBES: FROM FABRY-PEROT TO KONDO
Milena Grifoni
University of Regensburg Vg ms md
700 nm
Frauenchiemsee, October 5, 2015

3. TRANSPORT IN CARBON NANOTUBES
large transparencies
one electron
Kondo

4. TRANSPORT IN CARBON NANOTUBES
Fabry-Pérot
Coulomb blockade
Kondo
Courtesy of C. Strunk, A. Hüttel

5. TRANSPORT IN CARBON NANOTUBES
Strong coupling
Landauer-Büttiker
+ Green´s functions
DM-NRG,
Keldysh effective action
Perturbative master
equation for the reduced
density matrix

6. KONDO EFFECT IN CNTs ?

7. CARBON NANOTUBES Quantization of : boundary condition k║ = k κ║l ^ p 2 =
boundary condition
Ch·k κ║ κ K K’
τ = +1, -1
K´ = -K

8. Z-class valley preserving
FINITE CNTs
Reflection off-the boundaries  standing wave pattern, longitudinal quantization
Zig-zag like
Armchair like
ΔKK' = D+-
Z-class valley preserving
A-class valley mixing
Marganska et al., PRB 92 (2015)

9. Z-class valley preserving
FINITE CNTs
Zig-zag like
Armchair like
ΔKK' = D+-
Z-class valley preserving
A-class valley mixing
With spin: SU(4) Z-class, broken SU(4) A-class but E+(D)=E _ (-D)

10. Z-class valley preserving
FINITE CNTs
Zig-zag like
Armchair like
ΔKK' = D+-
Z-class valley preserving
A-class valley mixing
With spin-orbit: SU(4) broken for all classes but E1,2 (D)=E3,4 (-D)

11. CONJUGATION RELATIONS
Valid also for finite B !!! T T C P C T
Schmid et al., PRB 91, (2015)
T, P antiunitary; C=PT -1

12. SU(4) KONDO PHENOMENA IN CNTs
four-fold degeneracy of CNT shells allows for SU(4) Kondo when TK>>D
parallel magnetic field
lifts SU(4) symmetry
Jarillo-Herrero et al. Nature 434, 484 (2005)

13. SU(4) KONDO PHENOMENA IN CNTs
four-fold degeneracy of CNT shells allows for SU(4) Kondo when TK>>D ?
Jarillo-Herrero et al. Nature 434, 484 (2005)

14. KONDO EFFECT WITH BROKEN SU(4)
Nel 1 5 10 15 20 25 30 35 40
Nel 17 18 19 20 21 22 23

15. KONDO EFFECT WITH BROKEN SU(4)
Nel = 21
2√∆SO2 + ∆KK'2
Vg (V)
Vsd (mV)

16. MAGNETIC FIELD EVOLUTION
No splitting of satellite peaks  P-transition is missing !
Schmid et al. PRB 91, (2015)

17. MAGNETIC FIELD EVOLUTION
No splitting of satellite peaks  P-transition is missing !
Impose that conjugation relations are satisfied also at the level of the Keldysh effective action and hence at the level of the tunneling density of states
Schmid et al. PRB 91, (2015)

18. MAGNETIC FIELD EVOLUTION
No splitting of satellite peaks  P-transition is missing !

19. INELASTIC COTUNNELING
… but P-line present in the weak coupling limit ?
Jespersen et al., Nature Phys. 7, 348 (2011)

20. STRONG vs. WEAK COUPLING
With Jean-Pierre Grenoble

21. STRONG vs. WEAK COUPLING

22. STRONG vs. WEAK COUPLING

23. STRONG vs. WEAK COUPLING

24. ANGULAR DEPENDENCE
Ground state
is a spin singlet!

25. AND THANKS TO GRK 1570 SFB 689 Andreas Hüttel Davide Mantellli
Jean-Pierre Cleuziou (Grenoble)
and the Regensburg team
Alois Dirnaichner
Magdalena Margańska
Sergey Smirnov
Daniel Schmid
Christoph Strunk

26. CONJUGATION RELATIONS
Valid also at finite B :
E1,3(B)=E2,4(-B) T-conjugation
E1,2(D(B))=E4,3(-D(B)) P-conjugation
E3,4(D(-B))=E4,3(-D(-B)) C-conjugation
T, P antiunitary; C=PT -1
Schmid et al., PRB 91, (2015)

27. CONJUGATION RELATIONS

28. + PARALLEL MAGNETIC FIELD
Aharonov-Bohm effect:
changes the periodic boundary condition: Ch B 
Aharonov-Bohm
Ajiki, Ando J. Phys. Soc. Jpn 62 (1993)
metallic
semiconducting
Zeeman effect:
Zeeman term

29. STRONG vs. WEAK COUPLING
Absence of P-transition is independent of direction of magnetic field !

30. FROM GRAPHENE TO CNTs
a = 2.4 A° K K’
Dirac cones

31. Reflection off-the boundaries  standing wave pattern
FINITE CNTs
Reflection off-the boundaries  standing wave pattern
Valley preserving
Valley swapping

32. SYMMETRIES
(8,2)

33. BUILDING A NANOTUBE: HELICAL CONSTRUCTION
(8,2)

34. + CURVATURE curvature spin-orbit coupling other terms in κ┴:
determined by the boundary conditions
T. Ando, J. Phys. Soc. Jpn 67 (2000)

35. UNIVERSALITY plot G(Vsd /TK) for different TK
universality around Vsd = 0

36. TEMPERATURE DEPENDENCE
experiment
theory
both main peak and satellites are washed out if T becomes comparable to the Kondo temperature

37. KELDYSH EFFECTIVE ACTION THEORY
slave-boson technique + Keldysh field integral:

38. THANK YOU

39. PEAKS EVOLUTION B^ Nel = 21 B|| Nel = 17 one free parameter DSO/DKK'
P-transitions
T-transitions

40. TRANSMISSION IN PARALLEL B
(6,3)
(8,2)
ΔSO Δ
ΔKK‘
ΔKK' ≠ 0
ΔKK' = 0
Marganska et al., PRB 92, (2015)

41. SINGLE ELECTRON SPECTROSCOPY∆∥
√∆SO2 + ∆KK'2
∆KK'
B sweep
Experimental determination of

42. A-CLASS vs Z-CLASS A-class m=0 Z-class m=1, m=-1 Z-class
m: quasi-angular momentum, distinguishes valley in Z-class only

43. A-CLASS vs Z-CLASS A-class m=0 Z-class m=1, m=-1 Z-class
A-class: use odd/even eigenstates under U

44. Z-class valley preserving
FINITE CNTs
Z-class valley preserving
A-class valley mixing
Valley swapping

45. FINITE CNTs Dso=0 Zig-zag like Armchair like Valley preserving
Valley swapping

46. (classical electrostatic)
QUANTUM DOTS Vb e
eVb=ms-md Dm
source
drain
N-1
gate VG
dot electrochemical potential
addition energy
charging energy
(classical electrostatic)
mean level spacing
(quantum confinement)
1nm x 100nm dot:
kBT273°K = 23 meV

47. QUANTUM DOTS b
Transport condition: mN
mN-1
blockade

48. QUANTUM DOTS b Transport condition in linear regime: mN mN mN-1
resonant tunneling mN
mN-1
mN-1
kT >> G
blockade

49. KONDO RESONANCE b Transport condition: mN+1 e N odd Kondo mN resonance
kT < G, width ~ TK
TK ~ Wexp(-pe/G), large EC

50. KONDO RESONANCE b Transport condition: mN+1 e N odd Kondo mN resonance
kT < G, width ~ TK
TK ~ Wexp(-pe/G), large EC

51. KONDO RESONANCE b Transport condition: mN+1 kT < G, width ~ TK
Van der Wiel et al., Science 289, 2105 (2000)
mN+1
kT < G, width ~ TK
TK ~ Wexp(-pe/G), large EC

52. KONDO RESONANCE b Transport condition: kT < G, width ~ TK
Van der Wiel et al., Science 289, 2105 (2000)
kT < G, width ~ TK
TK ~ Wexp(-pe/G), large EC

53. INFLUENCE OF MAGNETIC FIELDBc
B = 0
B = Bc
see e.g. Kretinin et al., PRB 85, (2012)
spin degeneracy of state on the QD lifted by Zeeman energy
central resonance peak splits above Bc
linear split in B at large fields

54. KONDO EFFECT s=1/2 LEVEL Low T, V High T ,V Fermi liquid behavior
(Nozieres, Yosida and Yamada)
High T ,V
Perturbative regime
(Anderson, Hamann) G
universal cV/cT TK
T,V
Keldysh effective action theory
 analytic tunneling DOS in the whole regime of parameters
Smirnov and Grifoni, PRB 87, R (2013)

55. KONDO EFFECT s=1/2 LEVEL Keldysh effective action theory
 analytic tunneling DOS in the whole regime of parameters
Smirnov and Grifoni, PRB 87, R (2013)

56. FINITE CNTs
With spin: SU(4) Z-class, broken SU(4) A-class but E+(D)=E _ (-D)
Marganska et al., PRB 92, (2015)

57. FINITE CNTs
With spin-orbit: SU(4) broken for all classes class but E3,4(D)=E1,2 (-D)
Marganska et al., PRB 92, (2015)

58. KONDO EFFECT IN QUANTUM DOTS
N even
quantum
dot
N odd
Van der Wiel et al., Science 289, 2105 (2000)

59. FROM GRAPHENE TO CNTs

60. BUILDING A NANOTUBE quantized (yellow lines) zigzag (n,0) θ E θ =0 k||kx ky θ
zigzag (n,0)
metal
k|| E
θ =0
armchair (n,n)
θ =π/6 l ^ p 2 =
boundary condition
Ch·k
k|| E l ^ p 2 =
boundary condition
Ch·k
semiconducting