1. Signatures of Tomonaga-Luttinger liquid behavior in shot noise of a carbon nanotube Patrik Recher, Na Young Kim, and Yoshihisa Yamamoto Institute of Industrial Science, University of Tokyo, Japan E.L. Ginzton Lab, Stanford University, USA Capri Spring School, April 8, 2006

2. Outline Brief overview of single-walled carbon nanotubes (SWNTs) The transport problem: Keldysh functional approach Conclusion Conductance and low-frequency noise properties: Theory and experimental results Finite frequency noise (theory only) Luttinger-liquid model for a metallic carbon nanotube in good contact to electrodes

3. Overview of carbon nanotubes wrapped graphene sheets with diameter of only few nanometer Ideal (ballistic) one-dimensional conductor up to length scales of 1-10 and energies of ~1 eV exists as semiconductor or metal with depending on the wrapping condition Wildoer et al., Nature 391, 59 (1998)

4. Density of states a)Metallic SWNT: constant DOS around E=0, van Hove singularities at opening of new subbands b)Semiconducting tube: gap around E=0 Energy scale in SWNTs is about 1 eV, effective field theories valid for all relevant temperatures

5. Predicted Tomonaga-Luttinger liquid behavior in metallic tubes at energies : => crucial deviations from Fermi liquid - spin-charge separation (decoupled movements of charge and spin) and charge fractionalization - Power-law energy density of states (probed by tunneling) - Smearing of the Fermi surface Tomonaga-Luttinger liquid parameter quantifies strength of electron-electron interaction, for repulsive interaction

6. Electron transport through metallic single-walled carbon nanotubes bad contacts to tube (tunneling regime): Differential conductance as function of gate voltage : Crossover from CB behavior to metallic behavior with increasing Differential conductance as function of bias voltage at different temperatures Dashed line shows power-law ~ which gives averaged over gate voltage

7. Well-contacted tubes: Conductance as function of bias voltage and gate voltage at temperature 4K. Unlike in Coulomb blockade regime, here, wide high conductance peaks are separated by small valleys. The peak-to-peak spacing determined by and not by charging energy Liang et al., Nature 411, 665 (2001) tube lengths 530 nm (a) – 220 nm (b)

8. Electron transport through SWNT in good contact to reservoirs Effective low-energy physics (up to 1 eV) in metallic carbon nanotubes: two-bands (transverse channels) cross Fermi energy including e-e interactions 2-channel Luttinger liquid with spin non-interacting value ( Landauer Formula applies) Gate SiO 2 Drain Source Vg Vds For reflectionless (ohmic) contacts : C. Kane, L. Balents and M.P.A. Fisher, PRL 79, 5086 (1997) R. Egger and A. Gogolin, PRL 79, 5082 (1997) P. Recher, N.Y. Kim, and Y. Yamamoto, cond-mat/0604613

9. Theory of metallic carbon nanotubes Hamiltonian density for nanotube: band indices i=1,2 ; Interaction couples to the total charge density :  Only forward interactions are retained : good approximation for nanotubes if r large bosonization dictionary for right (R) and left (L) moving electrons:  Cut-off length due to finite bandwidth is long-wavelength component of Coulomb interaction

10. It is advantageous to introduce new fields (and similar for ) : Where we have introduced the total and relative spin fields:  4 new flavors In these new flavors : Free field theory with decoupled degrees of freedom Luttinger liquid parameter  strong correlations can be expected

11. Physical meaning of the phase-fields : Using: It follows immediately that : total charge density total current density total spin density total spin current density It also holds that : which follows from the continuity eq. for charge : or

12. backscattering and modeling of contacts m =1,2 denotes the two positions of the delta scatterers The contacts deposited at both sides of the nanotube are modeled by vanishing interaction ( g=1) in the reservoirs  finite size effect are the bare backscattering amplitudes inhomogeneous Luttinger-liquid model: Safi and Schulz ’95 Maslov and Stone ’95 Ponomarenko ‘95

13. Including a gate voltage In the simplest configuration, the electrons couple to a gate voltage (backgate) via the term : This term can be accounted for by making the linear shift in the backscattering term The electrostatic coupling to a gate voltage has the effect of shifting the energy of all electrons. It is equivalent of shifting the Fermi wave number

14. Keldysh contour : source field Action for the system without barrier Keldysh rotation Definition of current correlation function : retarded function : and similar for Keldysh generating functional Keldysh contour : source field Action for the system without barrier Keldysh rotation Definition of current correlation function : retarded function : and similar for Keldysh generating functional Keldysh contour : source field Action for the system without barrier Keldysh rotation Definition of current correlation function : retarded function : and similar for Action for the system without barrier Action for the system without barrier : and similar for Keldysh rotation: source field; Keldysh form of current :

15. Green’s function matrix is composed out of equilibrium correlators correlation function : Correlation function : Retarded Green’s function : these functions describe the clean system without barriers and in equilibrium ( =0)

16. Conductance wherewith without barriers backscattered current In leading order backscattering [see also Peca et al., PRB 68, 205423 (2003)] sum of 1 interacting (I) and 3 non-interacting (F) functions, and similar for describes the incoherent addition of two barriers describes the interference of two barriers voltage in dimension of non-interacting level spacing

17. Retarded Green’s functions smeared step function : reflection coefficient of charge : cut-off parameter associated with bandwidth : The retarded functions are temperature independent non-interacting functions obtained with =1 I. Safi and H. Schulz, Phys. Rev. B, 52 17040 (1995) sum indicates the multiple reflection at inhomogeneity of

18. Correlation functions Relation to retarded functions via fluctuation dissipation theorem: correlation at finite temperature correction

19. for => exponential suppression of backscattering for

20. main effect of interaction: power-law renormalization (tuned by gate voltage) conductance plots bias difference between minimas (or maximas)

21. Differential conductance: Theory versus Experiment @ 4K measurement damping of Fabry-Perot oscillation amplitude at high bias voltage observed clear gate voltage dependence of FP-oscillation frequency From the first valley-to-valley distance around we extract

22. Current noise In terms of the generating functional: symmetric noise:

23. Low-frequency limit of noise: Fano Factor: renormalization of charge absent due to finite size effect of interaction * ! What kind of signatures of interaction can we still see ? * The same conclusion for single impurity in a spinless TLL: B. Trauzettel, R. Egger, and H. Grabert, Phys. Rev. Lett. 88, 116401 (2002) B. Trauzettel, I. Safi, F. Dolcini, and H. Grabert, Phys. Rev. Lett. 92, 226405 (2004) F. Dolcini, B. Trauzettel, I. Safi, and H. Grabert, Phys. Rev. B 71, 165309 (2005) for

24. Asymptotic form of backscattered current shot noise is well suited to extract power-laws in the weak backscattering regime I. Safi and H. Schulz ’95 reflection coefficient of charge : g=0.23

25. GvGv ( ) 2 Lock-In Resonant Circuit R PD >>R CNT Signal DC V G R CNT -20V LED Vdc Vac CNT Vdc Vac + + C parasitic * # Experimental Setup and Procedures Key point : Parallel circuit of 2 noise sources: LED/PD pair (exhibiting full shot noise S=2eI) and CNT. Resonant Circuit filters frequency ~15-20 MHZ. Voltage noise measured via full modulation technique (@ 22 Hz) -> get rid of thermal noise

26. Comparison with experiments on low frequency shot noise PD=Shot noise of a photo diode light emitting diode pair exhibiting full shot noise serving as a standard shot noise source. Experimental Fano factor F (blue) compared with theory for g~0.25 (red) and g~1(yellow). F is compared with power-law scaling with g~0.16 for particular gate voltage shown and g~0.25 if we average over many gate voltages. ( red dashed line) giving g~0.18 for this particular gate voltage. In average over many gate voltages we have g~0.22 Power-law scaling

27. Blue: Exp Yellow: g = 1 Red: g = 0.25 T = 4 K Device : 13A2426 Vg = - 7.9V

28. Incoherent part coherent part dominant at large voltages frequency dependent conductivity of clean wire Finite frequency impurity noise depends on point of measurement

29. Frequency dependent conductance of clean SWNT+reservoirs related to retarded function of total charge only ! is assumed to be in the right lead and (in units of ) independent of not true for real part and imaginary part of oscillations are due to backscattering of partial charges arising from inhomogeneous see also: B. Trauzettel, I. Safi, F. Dolcini, and H. Grabert, Phys. Rev. Lett. 92, 226405 (2004) F. Dolcini, B. Trauzettel, I. Safi, and H. Grabert, Phys. Rev. B 71, 165309 (2005)

30. 3D plot of excess noise in units of at T=4K for g=1 measured at barrier as function of bias (in units of ) and frequency (in units of ) Excess noise as a function of at =35 for Excess noise as a function of at g=1 Finite frequency excess noise for the non-interacting system T=4K

31. 3D plot of excess noise in units of at T=4K for g=0.23 measured at barrier 2 as function of bias (in units of ) and frequency (in units of ) Excess noise as a function of at =35 for Excess noise as a function of at g=0.23 charge roundtrip time Signatures of spin-charge separation in the interacting system Interacting levelspacing and non-interacting levelspacing clearly distinguished in excess noise ! from oscillation periods without any fitting parameter

32. Dependence of excess noise on measurement point g=0.23 g=1 T=4K d=0.14 d=0.3 d=0.6 =35

33. Conclusions conductance and shot noise have been investigated in the inhomogeneous Luttinger-liquid model appropriate for the carbon nanotube (SWNT) and in the weak backscattering regime conductance and low-frequency shot noise show power-law scaling and Fabry-Perot oscillation damping at high bias voltage or temperature. The power-law behavior is consistent with recent experiments. The oscillation frequency is dominated by the non-interacting modes due to subband degeneracy. finite-frequency excess noise shows clear additional features of partial charge reflection at boundaries between SWNT and contacts due to inhomogeneous g. Shot noise as a function of bias voltage and frequency therefore allows a clear distinction between the two frequencies of transport modes  g via oscillation frequencies and info about spin-charge separation