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

Outline good contact to electrodes Theory and experimental results 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

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) High geometric aspect ration. Can have length of several mm but diameter of only a few nm.

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

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

Electron transport through metallic single-walled carbon nanotubes bad contacts to tube (tunneling regime): Mention CB at low T in QDs, two charge states degenerate-> transport, otherwise CB valley. Above CB, power-law scaling in Vds. 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

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

Electron transport through SWNT in good contact to reservoirs Gate SiO2 Drain Source Vg Vds 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) length of tube: 200-600 nm. Will refer to specific sample with L=360nm Metal contacts are :Ti/Au, Ti only, Pd (Palladium); SiO2 layer of 0.5 um, Heavily doped Si substrate as gate. including e-e interactions 2-channel Luttinger liquid with spin For reflectionless (ohmic) contacts : non-interacting value (Landauer Formula applies)

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 Forward scattering where e stay in same branch. Amplitude for backscattering is down by 1/N, N is # atoms around circumference. Use bosonization description of one-dimensional electrons where the electron creation and annihilation operators are written as an exponential of bosonic fields \phi and \theta. bosonization dictionary for right (R) and left (L) moving electrons: Cut-off length due to finite bandwidth 

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 Mention that charge mode has velocity vF/g whereas other modes have v_F Luttinger liquid parameter  strong correlations can be expected

Physical meaning of the phase-fields : Using: It follows immediately that : total charge density total current density total spin density n=number operator, j=current operator total spin current density It also holds that : which follows from the continuity eq. for charge : or

backscattering and modeling of contacts inhomogeneous Luttinger-liquid model: Safi and Schulz ’95 Maslov and Stone ’95 Ponomarenko ‘95 Combination of all flavor fields in the exponent, not only the charge field. This leads to interference in transport involving correlation of all flavors. 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

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

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 Action for the system without barrier source field 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

correlation function : 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)

Conductance In leading order backscattering where with without barriers backscattered current In leading order backscattering [see also Peca et al., PRB 68, 205423 (2003)] Conductance cannot be calculated analytically in general for g<1 and we have to refer to numerical calculation of integral. Still we can get inside into the physics by looking at the ret. and correlation functions to be discussed next. 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

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

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

for => exponential suppression of backscattering for

conductance plots bias difference between minimas (or maximas) Only at special Vg values we see the charge mode. Oscillation is probably hard to extract since oscillation amplitude is small (tuned by gate voltage) main effect of interaction: power-law renormalization

Differential conductance: Theory versus Experiment 0.38 0.36 0.34 0.32 0.30 dI/dV ds -20 -10 10 20 V (mV) measurement @ 4K 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

Current noise In terms of the generating functional: symmetric noise: We think that noise is much better suited than conductance to see interaction effects, since in the weak backsacttering regime, the conductance is dominated by the the larger value of transmission and the backscattered correction is modified bu g. But noise is shown to be directly proportional to the backscatterd component.

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)

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 Say that shot noise is directly proportional to the weak backscattered component of current which contains the information about g. Therefore, shot noise is well suited to extract g through power-law in the weak backscattering regime.

Experimental Setup and Procedures Gv ( ) 2 Lock-In Resonant Circuit RPD>>RCNT Signal DC VG RCNT -20V LED Vdc Vac CNT + Cparasitic * # Parallel circuit of two noise sources. LED/PD pair which is weakly coupled (PD) ~0.4 % => serves as an ideal full shot noise source: S=2eI. Resonance circuit filters frequency \omega~15-20 MHZ. Measures voltage noise via full modulation technique (modulation @ 22 Hz frequency)->get rid of thermal noise and enhances signal to noise ratio. Fano factor: F=S(I)/S_PD(I). This technique gets rid of noise from circuit which is assumed to be the same for both noise measurements. 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 Key point :

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 Here we display noise and Fano factor in a log-log Plot. We determine the power-law and compare it with the theoretical exponent, which leads to g. ( 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.

Device : 13A2426 Vg = - 7.9V Blue: Exp Yellow: g = 1 Red: g = 0.25 T = 4 K Fano factor in linear scale at temperature 4K with thermal noise subtracted. We use the same backscattering strengths U1 and U2 as we have used for the fitting of conductance. Fano factor can be fitted quantitatively in the range of the validity of the weak backscattering theory.

Finite frequency impurity noise Incoherent part frequency dependent conductivity of clean wire dominant at large voltages In addition to the usual beat note of shot noise we also see an additional modulation with frequency containing only the charge mode described by the conductivity of the clean wire. At large voltage, the incoherent part is dominant due to the monotonic increase with voltage and will be strongly modified by sigma^2. In addition we see that also the interference oscillation amplitudes will be modified with frequency, therefore the strength of Fabry-Perot oscillations is also modified by frequency dependent quantities. coherent part depends on point of measurement

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

Finite frequency excess noise for the non-interacting system 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 is zero for large frequencies in agreement with LB-transport. Further we see only one frequency scale. Oscillations in Omega and V are factor 2 different since in frequency epsilon+omega interferes with epsilon-omega Excess noise as a function of at =35 for Excess noise as a function of at

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 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

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

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