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

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

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

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

Correlated sequential tunneling PUZZLE 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) Correlated sequential tunneling Gate voltage T2 > T1 (b) Conductance

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

METALLIC SWNT MOLECULES Energy EF metallic 1D conductor with 2 linear bands k LUTTINGER FEATURES

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

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

LUTTINGER HAMILTONIAN captures interaction effects nanotubes

TRANSPORT Luttinger liquids voltage sources localized impurities backscattering forward scattering

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

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

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

FINITE RANGE?

FINITE RANGE? not needed

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

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

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

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

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

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

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

GENERATING FUNCTION METHOD FOR ST Sequential tunneling approximation: Consider only (but all) 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

GENERATING FUNCTION METHOD FOR CST Sequential tunneling approximation: Consider only (but all) 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!

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

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

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

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

CST III Systematic expansion in L summation over m UST modified line width at resonance

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

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

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