Signatures of Tomonaga-Luttinger liquid behavior in shot noise of a carbon nanotube

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

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

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 • Metallic SWNT: constant DOS around E=0, van Hove singularities at opening of new subbands • 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. Differential conductance as function of gate voltage : Crossover from CB behavior to metallic behavior with increasing Electron transport through metallic single-walled carbon nanotubes bad contacts to tube (tunneling regime): 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: • tube lengths 530 nm (a) – 220 nm (b) Liang et al., Nature 411, 665 (2001) 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

8. Vds Source Drain SiO2 Gate Vg Electron transport through SWNT in good contact to reservoirs P. Recher, N.Y. Kim, and Y. Yamamoto, cond-mat/0604613 • two-bands (transverse channels) • cross Fermi energy • Effective low-energy physics (up to 1 eV) in metallic carbon nanotubes: C. Kane, L. Balents and M.P.A. Fisher, PRL 79, 5086 (1997) R. Egger and A. Gogolin, PRL 79, 5082 (1997) • including e-e interactions 2-channel Luttinger liquid with spin • For reflectionless (ohmic) contacts : non-interacting value (Landauer Formula applies)

9. Theory of metallic carbon nanotubes Hamiltonian density for nanotube: band indices i=1,2 ; is long-wavelength component of Coulomb interaction 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 

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 inhomogeneous Luttinger-liquid model: Safi and Schulz ’95 Maslov and Stone ’95 Ponomarenko ‘95 are the bare backscattering amplitudes 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

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

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

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

19. for => exponential suppression of backscattering for

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

21. 0.38 0.36 ds 0.34 dI/dV 0.32 0.30 • From the first valley-to-valley distance • around we extract -20 -10 0 10 20 V (mV) ds Differential conductance: Theory versus Experiment measurement @ 4K • damping of Fabry-Perot oscillation • amplitude at high bias voltage observed • clear gate voltage dependence of • FP-oscillation frequency

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

23. Low-frequency limit of noise: for renormalization of charge absent due to finite size effect of interaction * ! What kind of signatures of interaction can we still see ? Fano Factor: * 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)

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

25. * -20V + # Gv Lock-In ( ) 2 RPD>>RCNT LED Signal RCNT Resonant Circuit Cparasitic CNT + DC VG Vdc Vdc Vac Vac 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 Power-law scaling • 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 ( red dashed line) giving g~0.18 for this particular gate voltage. In average over many gate voltages we have g~0.22 with g~0.16 for particular gate voltage shown and g~0.25 if we average over many gate voltages.

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

28. Finite frequency impurity noise Incoherent part frequency dependent conductivity of clean wire dominant at large voltages coherent part • 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 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) independent of not true for real part and imaginary part of (in units of ) • oscillations are due to backscattering of partial charges arising from inhomogeneous

30. Finite frequency excess noise for the non-interacting system g=1 T=4K 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

31. Signatures of spin-charge separation in the interacting system g=0.23 Interacting levelspacing and non-interacting levelspacing clearly distinguished in excess noise ! from oscillation periods without any fitting parameter 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 ) charge roundtrip time Excess noise as a function of at =35 for Excess noise as a function of at

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

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