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

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

Outline Single wall carbon nanotubes (SWCNT)

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

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

Outline Single wall carbon nanotubes (SWCNT) Quantization effects

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

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

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

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

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

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

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

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

Kondo effect causes increased resistivity at low temperatures

Kondo Effect in bulk Kondo cloud at low temperature

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

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

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

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

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]

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

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)

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

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

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

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

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