RESONANT TUNNELING IN CARBON NANOTUBE QUANTUM DOTS

1. 200 nm after Postma et al. Science (2001) RESONANT TUNNELING INCARBON NANOTUBE QUANTUM DOTS MILENA GRIFONI M. THORWART R. EGGER G. CUNIBERTI H. POSTMA C. DEKKER 25 nm Discussions: Y. Nazarov

2. e e source dot drain Cg V Vg addition energy a) b) mL mR mL mR Coulomb blockade single electron tunneling QUANTUM DOTS

3. Conductance Gate voltage Sequential tunneling semiconducting dot + Fermi leads Beenakker PRB (1993) (a) T2>T1 Conductance Luttinger leads SETs Furusaki, Nagaosa PRB (1993) Gate voltage ORTHODOX SET THEORY

4. Bias voltage (V) dI/dV d2I/dV2 30 K Gate voltage (V) Ec = 41 meV, DE = 38 meV > kBT up to 440 K NANOTUBE DOT IS A SET Postma, Teepen, Yao, Grifoni,Dekker, Science 293 (2001) unconventional Coulomb blockade in quantum regime

5. Why nanotube SET not ? unscreened Coulomb interaction ? Maurey, Giamarchi, EPL (1997) weak tunneling at metallic contacts ? Kleimann et al., PRB (2002) asymmetric barriers ? Nazarov, Glazman, PRL (2003) correlated tunneling ? Postma et al., Science (2001), Thorwart et al. PRL (2002) Hügle and Egger, EPL (2004) (b) T2>T1 Conductance Correlated sequential tunneling Gate voltage PUZZLE

6. OVERVIEW • METALLIC SINGLE-WALL NANOTUBES (SWNT) • SWNT LUTTINGER LIQUIDS • SWNT WITH TWO BUCKLES • UNCOVENTIONAL RESONANT TUNNELING EXPONENT 1D DOT WITH LUTTINGER LEADS CORRELATED TUNNELING MECHANISM

7. METALLIC SWNT MOLECULES Energy EF metallic 1D conductor with2 linear bands k LUTTINGER FEATURES

8. DOUBLE-BUCKLED SWNT´s buckles act as tunneling barriers after Rochefort et al. 1998 50 x 50 nm2 Luttinger liquid with two impurities Let us focus on spinless LL case, generalization to SWNT case later

9. L L + forward scattering q~0 R R WHAT IS A LUTTINGER LIQUID ? example: spinless electrons in 1D linear spectrum bosonization identity charge density

10. LUTTINGER HAMILTONIAN captures interaction effects nanotubes

11. Luttinger liquids voltage sources localized impurities backscattering forward scattering TRANSPORT

12. charge transferred across the dot charge on the island continuity equation 2 1 e e Brownian`particles´ n, N in tilted washboard potential reduced density matrix TRANSPORT

13. Exact trace over bosonic modes reduced density matrix bare action bulk modes nonlocal in time coupling mass gap for n charging energy LINEAR TRANSPORT CURRENT

14. CORRELATIONS dipole W = S+iR dipole-dipole correlations involving different/same barriers

15. FINITE RANGE?

16. FINITE RANGE? not needed

17. CORRELATIONS II W = S+iR • zero range WD: purely oscillatory • WS : Ohmic + oscillations <cosh LD> const, <sinh LD>=0

18. EFFECT OF THE CORRELATIONS ? • FIRST CONSIDER UNCORRELATED TUNNELING • MASTER EQUATION APPROACH • Ingold, Nazarov (1992) (gr= 1), Furusaki PRB (1997) • GENERATING FUNCTION METHOD (FROM PI SOLUTION) • Grifoni, Thorwart, unpublished

19. Ingold, Nazarov (1992) (gr= 1), Furusaki PRB (1997) • Uncorrelated sequential tunneling: • only lowest order tunneling process • master equation for populations: linear regime: only n = 0,1 charges golden rule rate MASTER EQUATION FOR UST Gtot example

20. MASTER EQUATION FOR UST II Note: can also be obtained from the master eq. Is there a simple diagrammatic interpretation of Gf/b ?

21. Different view from path integral approach exact series expression generating function contributions to the f/b current of order D2m Example: m = 2 (divergent!) (a) (b) (c) cotunneling GENERATING FUNCTION METHOD

22. GENERATING FUNCTION METHOD FOR ST Sequential tunneling approximation: Consider only (butall)paths which are back to the diagonal after two steps (giustified for strong Coulomb interaction)

23. GENERATING FUNCTION METHOD FOR ST Sequential tunneling approximation: Consider only (butall)paths which are back to the diagonal after two steps (giustified for strong Coulomb interaction)

24. GENERATING FUNCTION METHOD FOR ST Sequential tunneling approximation: Consider only (butall)paths which are back to the diagonal after two steps (giustified for strong Coulomb interaction) L R non trivial cancellations among contribution of different paths

25. GENERATING FUNCTION METHOD FOR CST Sequential tunneling approximation: Consider only (butall)paths which are back to the diagonal after two steps (giustified for strong Coulomb interaction) L R non trivial cancellations among contribution of different paths Correlations!

26. GENERATING FUNCTION METHOD FOR UST Sequential tunneling approximation: Consider only (butall)paths which are back to the diagonal after two steps (giustified for strong Coulomb interaction) L R UST: only intra-dipole Correlations! again

27. GENERATING FUNCTION METHOD FOR UST II Interpretation: Higher order paths provide a finite life-time for intermediate dot state, which regularizes the divergent fourth-order paths L Gtot R

28. Let us look order by order: exact! cosh LD sinh LD Short cut notation: divergent l =0 CST m=2

29. CST II divergent m=3 As for UST, sum up higher order terms to get a finite result Approximations: • Consider only diverging diagrams • Linearize in dipole-dipole interaction LS/D; FS/D = 0

30. CST III Systematic expansion inL summation over m UST modified line width at resonance

31. MASTER EQUATION FOR CST transfer through 1 barrier (irreducibile contributions of second and higher order) transfer trough dot (irreducibile contributions at least of fourth order) Thorwart et al. unpublished finite life-time due to higher order paths found self consistently

32. ì a - µ G » 1 G / T T ï - end end MAX í ï a µ G » * G T î - end end 4 bosonic fields Kane, Balents, Fisher PRL (1997), Egger, Gogolin, PRL (1997) RESULTS GMAX Thorwart et al., PRL (2002) spinless LL: nanotubes:

33. LOW TEMPERATURES : BREAKDOWN OF UST IN LINEAR REGIME CONCLUSIONS & REMARKS dot leads • UNCONVENTIONAL COULOMB BLOCKADE REMARK NONINTERACTING ELECTRONS gr=1