Session U18: Focus Session: Carbon Nanotubes: Transport III

Electron Phonon Coupling Effects in Nanotubes

Electron-phonon coupling (EPC) is a key physical parameter in nanotubes. Here we discuss its effects on phonon dispersions, Raman spectra and electron transport. The main EPC effect on the phonon dispersions is the presence of Kohn anomalies. These are distinct features of the phonon dispersion in metallic systems, associated to the presence of a Fermi surface [1]. Graphite has two Kohn anomalies for the Gamma-E$_ {2g}$ and K-A'$_{1}$ optical modes [2]. Their strength is proportional to the EPC square [2]. Kohn anomalies are enhanced in metallic nanotubes due to their reduced dimensionality, but absent in semiconducting nanotubes [2,3]. At 0 K all metallic nanotubes are not stable and undergo a Peierls distortion. We show that the Peierls distortion temperature decreases exponentially with the tube diameter [3]. For nanotubes generally used in experiments, with diameters larger than 0.8 nm, we find that this temperature is smaller than 10$^{-8}$ K [3]. We then show that EPC is the major source of broadening for the Raman G and G$^{-}$ peaks in graphite and metallic nanotubes [3]. The EPC explains the difference in the Raman spectra of metallic and semiconducting nanotubes and their dependence on tube diameter [3]. We dismiss the common assignment of the G$^{-}$ peak in metallic nanotubes to a Fano resonance between phonons and plasmons. We assign the G$^ {+}$ and G$^{-}$ peaks to TO (circumferential) and LO (axial) modes, the opposite of what often done. We then present five independent approaches to directly measure the optical phonons EPC in graphite and nanotubes from their phonon dispersions and Raman spectra. This allows us to quantify the EPC effects on high field electron transport in nanotubes. High field measurements show that electron scattering by optical phonons breaks the ballistic behavior. From our EPCs we derive a simple formula for the electron mean free path for optical phonon scattering in high-field transport [4]. The comparison with the scattering lengths fitted from experimental I-V curves shows that hot phonons are created during high-bias transport [4]. Their effective temperature is thousands K and sets the ultimate limit of ballistic transport [4]. \begin{enumerate} \item W. Kohn, Phys. Rev. Lett. \textbf{2}, 393 (1959) \item S. Piscanec et al. Phys. Rev. Lett. \textbf{93}, 185503 (2004) \item M. Lazzeri et al. cond-mat/0508700; S. Piscanec et al. Phys. Rev. B submitted (2005) \item M. Lazzeri et al. Phys. Rev. Lett. \textbf{95}, 236802 (2005) \end{enumerate}   

Self-Heating and Non-Equilibrium Optical Phonons in Suspended Carbon Nanotubes

Understanding of current-limited transport in single-walled carbon nanotubes (SWNTs) is vital to many potential nanotube applications. In this talk I will discuss the high bias electrical transport characteristics of well-contacted suspended SWNTs in various environments. Negative differential conductance at low bias (below 0.4V) appears as a result of extreme self-heating and the formation of non-equilibrium optical phonons. Various gas and molecular solid environments lead to the reduction or elimination of the non-equilibrium phenomenon. Finally I will discuss the ways in which we can use the data to directly and indirectly measure the nanotube's intrinsic properties and temperature.   

Direct measurements of electron-phonon coupling of radial breathing modes in carbon nanotubes and their chirality dependence

A method for direct measurement of electron-phonon coupling matrix elements, M$_{e-ph}$, is proposed and demonstrated experimentally by correlating resonant Raman excitation profiles of the first and second harmonics of the radial breathing mode. M$_{e-ph}$ values are quantitatively determined for individual carbon nanotubes (CNT) excited in small ropes suspended in air. The results show that the matrix elements satisfy S. V. Goupalov and coworker’s empirical tight binding theory calculation$^{1}$ with quantitative values that show a smaller electron-phonon coupling than reported from ab initio calculations$^{2}$ for isolated carbon nanotubes. We find that resonant excitation profile broadening $\eta$ for CNTs in small ropes show a correlation with chiral angles that appears to be unchanged from isolated carbon nanotubes. 1 S. V. Goupalov, Satishkumar B. C., and S. K. Doorn, Pre- print (2005). 2 M. Machon, S. Reich, H. Telg et al., Phys. Rev. B 71 (3) (2005).   

Nonequilibrium phonon occupation in carbon nanotube quantum dots

We present a formalism for electron transport through a coulomb blockaded quantum dot strongly coupled with vibrations and weakly with leads and the thermal environment. By calculating the joint electron-phonon probability distribution, we show that recently observed anomalous conductivity through single-walled carbon nanotube (SWCNT) quantum dots arises from `hot' phonons that are generated by the current at a faster rate than their extraction rate by the surrounding. We explain semi-quantitative details of the experiment and predict a nontrivial temperature dependence of the phonon population arising from a subtle interplay between phonon emission and absorption rates at specific bias voltage values.   

Phonon scattering in Carbon Nanotube Field Effect Transistors -- an NEGF Treatment.

We examine the influence of phonon scattering on DC current transport in carbon nanotube field-effect transistors using the non-equilibrium Green's function (NEGF) formalism. Both optical and acoustic phonon modes are considered, and electron-phonon interaction is modeled through a scattering self-energy. Intra-valley scattering due to longitudinal optical (LO) and radial breathing mode (RBM) phonons is examined. Zone-boundary phonon eigenmodes that mediate inter-valley scattering are found to be a mixture of fundamental polarizations such as LO/TA and to couple strongest to the electrons. The effect of phonon scattering on the current vs. voltage characteristic of a filed-effect transistor is found to have distinct gate voltage (Vg) dependence. High-energy optical phonons can significantly degrade the ON-current (large Vg) while their effect is negligible in the OFF-state (low Vg). On the other hand, low-energy phonons (acoustic and RBM) can considerably affect the current transport for all gate biases. Their influence is enhanced at low Vg due to the one-dimensional density of states.   

Transport properties of suspended carbon nanotubes

The study of suspended doubly clamped carbon nanotubes allows for the observation of many novel phenomena due to the intimate coupling of the mechanical and electrical degree of freedom, e.g. high frequency quantum limited displacement sensing, phonon adsorption and emission spectroscopy and quantized frequency tuning. We use a high frequency mixing technique originally develloped by Sazonova et al. to monitor the high frequency properties of suspended carbon nanotubes. Our setup allows for measurements from DC up to 4 GHz from room temperature down to 300 mK.   

Electrical Switching in Metallic Carbon Nanotubes

We present first-principles calculations of quantum transport which show that the resistance of metallic carbon nanotubes can be changed dramatically with homogeneous transverse electric fields if the nanotubes have impurities or defects. The change of the resistance is predicted to range over more than two orders of magnitude with experimentally attainable electric fields. This novel property has its origin that backscattering of conduction electrons by impurities or defects in the nanotubes is strongly dependent on the strength and/or direction of the applied electric fields. We expect that this property will open a path to new device applications of metallic carbon nanotubes. Ref.) Young-Woo Son {\em et al.}, Phys. Rev. Lett. {\bf 95}, 216602 (2005).   

High Temperature Conductivity and Reactivity of Carbon Nanotube Electronic Circuits

At sufficiently high temperatures, carbon nanotubes (CNTs) begin to react with their immediate environment. For example, adsorbates first desorb, then the carbon may react with connective electrodes, and ultimately Stone-Wales defects become mobile and can be annealed. These reactions are conventionally studied by thermogravimetric analysis (TGA), but they can also profoundly effect the conductance of the nanotubes. We have measured the four probe impedance and transimpedance of individual metallic and semiconducting nanotube devices from room temperature to 1200 K in ultra-high vacuum. When the nanotubes are initially heated from room temperature, the conductance increases as adsorbates are desorbed. On subsequent heating, the device resistance is linearly dependant on temperature over the range 300 to 900 K. Above 900 K the temperature dependence becomes more complex as chemical reactions change the nanotube and as optical phonon modes become thermally populated. This electronic characterization agrees with and complements TGA of bulk, purified CNTs.   

On Current Carrying Capacity of Single Wall Semiconducting Carbon Nanotubes

The current carrying capacity of single wall semiconducting carbon nanotubes (CNTs) in the presence of phonon scattering and band-to-band tunneling is studied by self-consistently solving Poisson and Schr\"{o}dinger equation using the non-equilibrium Green's function formalism. We show that the current delivery capacity of semiconducting CNTs strongly depends on the bias regime and is drastically different from metallic CNTs. A long metallic single wall CNT carries a saturation current of $\sim $25$\mu $A due to optical phonon (OP) scattering. In contrast, a semiconducting CNT can deliver a current well above 25$\mu $A in ambipolar transport regime even when the channel length is much longer than the OP scattering mean free path (mfp). When the channel length is short (comparable to the OP scattering mfp), a semiconducting CNT biased in unipolar transport regime can deliver a current larger than 25$\mu $A. Biasing the CNT in ambipolar transport regime, however, further doubles the current. The different current carrying capacity in the ambipolar transport regime is due to nearly uncoupled dissipative transport through both the lowest conduction and valence subbands in the channel.   

A comparison of measured electron-phonon and electron-photon coupling strengths in isolated and small ropes of single wall carbon nanotubes

Resonant Raman scattering excitation profiles and photoluminescence (PL) are measured for isolated carbon nanotubes (CNT) and small ropes suspended in air. Most of the measured CNTs do not exhibit PL and are believed to be in small ropes. The radial breathing mode electron-phonon coupling, M$_ {e-ph}$, are measured, and values for the isolated CNT are in good quantitative agreement with ab initio calculations. The matrix elements M$_{e-ph}$ and electron-photon coupling, M$_{e- op}$, for a CNT in a small rope are 1.7 times and 1.4-2.7 times weaker than in an isolated CNT. The reduced e-phonon coupling in small ropes is correlated with a smaller RRSE broadening $\eta$, compared to the value (45meV) obtained from an isolated CNT. Despite the reduced values of M$_{e-ph}$ and $\eta$, M$_{e- ph}$ in small ropes still display the same chiral dependence predicted for isolated CNTs.   

Anharmonic decay of the Radial Breathing Mode in Suspended Single-walled Nanotubes

The growth of isolated single-walled nanotubes (SWNTs) suspended over trenches in Si substrates makes it possible to study the Raman lineshapes of individual tubes. High-resolution room temperature resonant micro-Raman spectra from a number of suspended SWNTs exhibit very narrow radial breathing modes (RBMs), with full-width at half maximum (FWHM) values ranging from 1.3-2.5 cm$^{-1}$. These values are much smaller than previously reported in the literature. The observed FWHM is \textit{not} a smooth function of the tube's radius. We note that the two-phonon density of states (2DOS) for the anharmonic decay of the RBM phonon shows many singularities whose energies depend both on the tube's radius \textit{and} chirality. Therefore, tubes with very similar RBM frequencies, and similar radii, could have different linewidths because of a different 2DOS. The observed linewidths increase with increasing incident laser power, as expected if the origin of the linewidth is anharmonic. We analyze the RBM linewidth in terms of down-conversion and up-conversion third-order anharmonic contributions. A comparison of the temperature dependence of both FWHM and peak frequency suggest that up-conversion processes are important, as found previously for low-frequency optical phonons in semiconductors.   

Phonon anomalies in the resonance Raman spectra of graphite and single-wall carbon nanotubes

Phonon dispersion relations for a graphene sheet and single-wall carbon nanotubes (SWNTs) are calculated within the extended tight-binding model that has recently been shown to accurately predict the optical transition energies in small-diameter SWNTs. Anomalies in the dispersion relations are found at certain high- symmetry points of the reciprocal lattice and these anomalies are attributed to the strong electron-phonon coupling. These anomalies are very sensitive to changes to electron and lattice temperatures, electron doping, mechanical stress, SWNT diameter, and SWNT metallicity. Resonance Raman measurements of doping and strain induced shifts of the phonon frequencies in SWNTs are in qualitative agreement with the present calculations. The MIT authors acknowledge support under NSF Grant DMR 04-05538.   

Phonon stiffening in semiconducting single-walled carbon nanotubes under n-type doping

The doping dependence of the high-frequency Raman-active modes in single-walled semiconducting carbon nanotubes is studied by density functional theory. We find that the $A_{1g}$ longitudinal mode in $(3*n+1,0)$ zigzag tubes shows a small anomalous upshift, followed by a large downshift under electron doping. This doping-induced stiffening of the $A_{1g}$ mode is related to the large anharmonicity of the mode. Connections are made to recent experiments in the group of P. C. Eklund.