1. Kondo Effects in Carbon Nanotubes
Master Colloquium by
Jeppe Vilstrup Holm
Supervisor: Poul Erik Lindelof

2. Outline Single wall carbon nanotubes (SWCNT) Quantization effects
SWCNT quantum dots
The Kondo effect in bulk
Kondo effects in carbon nanotubes
Equilibrium
In equilibrium
Conclusion

3. Outline
Single wall carbon nanotubes (SWCNT)

4. Single Wall Carbon Nanotubes
Graphene
Carbon atom
~1 nm a2 a1

5. One dimensional conductor. Metallic or Semiconducting
Metallic if: 2n+m/3=q
Metallic
Metallic
Semiconducting

6. Outline
Single wall carbon nanotubes (SWCNT)
Quantization effects

7. Conductance quantization: Gmax= 4 e2/h
Carbon nanotube
L~300nm
Electrode
Electrode
E+Uc
Back gate
SiO2
(insulator) E
SWCNT
Conductance quantization: Gmax= 4 e2/h
SWCNT (1nm): Four channels (Rmin= 26 k)
Size quantization: E~1/L~ 1 meV
(Wave nature of electron)
Separated energy levels
Charge quantization: Uc~e2/C~5 meV
(Particle nature of electron)
Single electron transport

8. Outline Single wall carbon nanotubes (SWCNT) Quantization effects
SWCNT quantum dots

9. Device Fabrication (c) (a) (b) (e) (d)
(a) Alignment marks made using lithography and metal evaporation
(b) Catalyst islands made using lithography and material spinning
(c) Grow SWCNT by chemical vapour deposition
(d) Electrodes made using electron beam lithography and metal evaporation
(e) Bonding pads made using optical lithography and metal evaporation

10. Cryogenics – measurements at low temperature
Low temperature is needed to observe these quantum phenomena
4K: Helium bath (4He)
300mK Heliox: Closed liquid/gas system (3He)
30mK Kelvinox: 3He/4He dilution refrigerator

11. Measurement setup Vsd Vgate Back gate Two voltage knobs:
SWCNT
Back gate
Vgate
Two voltage knobs:
Source-drain voltage Vsd
Gate voltage

12. Coulomb blockade and single electron tunneling
G[e2/h] 1 2 3 4 5
Vg[V] ΔE 2x Uc μs μd
Single electron tunneling
Coulomb Blockade

13. Vsd[mV] ΔE Uc μs μd ΔE Uc Uc Uc/(eα) 1 2 3 4 5 eVsd
White: High dI/dV (increase in current)
Black: Low dI/dV

14. Outline Single wall carbon nanotubes (SWCNT) Quantization effects
SWCNT quantum dots
The Kondo effect in bulk

15. Kondo effect causes increased resistivity at low temperatures

16. Kondo Effect in bulk Kondo cloud at low temperature

17. Outline Single wall carbon nanotubes (SWCNT) Quantization effects
SWCNT quantum dots
The Kondo effect in bulk
Kondo effects in carbon nanotubes
Equilibrium (Zero bias)

18. Kondo effect with odd occupancy on tube
Spin on Tube
Kondo Cloud in Electrodes

19. Spin flipping proces (a) (b) (c) Uc Odd Odd Odd Odd x x x x Even Even
Δt ~ h/Uc
(c) Uc

20. Extra peak in density of states
N is even
N is odd Γs Γd Γs Γd μs μd μs μd
~2kBTK ε0 ε0 Γ Γ
Γ= Γs + Γd

21. G[e2/h] Vsd[mV] Vg[V] (a) Even Even Even Odd Odd Odd Odd Even (b) 70mK
-8,0
-7,9
-7,8
-7,7
-7,6
-7,5
0,0
0,3
0,6
70mK
181mK
445mK
987mK
(a)
Even
Even
Even
Odd
Odd
Odd
Odd
Even
G[e2/h]
(b)
Vsd[mV]
Vg[V]

22. (c)
G[e2/h]
T[K]

23. Outline Single wall carbon nanotubes (SWCNT) Quantization effects
SWCNT quantum dots
The Kondo effect in bulk
Kondo effects in carbon nanotubes
Equilibrium
In equilibrium (finite bias)

24. Kondo effect with even occupancy on tube
Odd
Even
Odd
Even
Odd
Odd
Even
Odd
Gateswitch
Even
Odd
eV = Δ eV Δ

25. Extra peak in density of states
eV < Δ,δ
eV ≥ Δ,δ
Odd
Even eV eV Γ Γ

26. Cotunneling I
dI/dV
|V|
|V|
Kondo Effect I
dI/dV
|V|
|V|

27. I
|V|
dI/dV
Combined
(e)
(d)

28. Conclusions SWCNT quantum dots devices Coulomb blockade Kondo effects
Equilibrium
In equilibrium