Trigonal Warping Effect Research Articles

Effects of strain on the resonant Raman profile of metallic

AbstractIn this paper we report the effects of strain on the electronic properties of single wall carbon nanotubes and its consequence on the resonant Raman cross section. A quantum interference effect has been predicted for the radial breathing mode spectra for metallic tubes. For metallic tubes, the lower and upper components of Eii resulting from the trigonal warping effect are affected differently and for low chiral angle they cross for some strain value. Near (at) the crossing point, the resonant Raman spectra profile exhibits a maximum (minimum) value due to a quantum interference in the Raman cross section. This Raman cross section interference effect was observed in Raman experiments carried out on isolated SWNTs. The Raman experiment performed on an isolated strained metallic SWNT supports our modeling predictions.

Effects of Trigonal Warping on Perfect Channel in Metallic Carbon Nanotubes

Numerical calculations of the conductance are performed to study effects of trigonal warping of bands on a perfectly conducting channel in metallic carbon nanotubes existing unless the potential range of scatterers is smaller than the lattice constant. The perfect channel can easily be destroyed in the presence of small amount of trigonal warping when there are several bands at the energy, while it is much more robust in the energy range of metallic linear bands.

Science and applications of single-nanotube Raman spectroscopy.

A review is presented of the resonance Raman spectra from individual isolated single-wall carbon nanotubes (SWNTs). A brief summary is given of how the measurements are made. Why the resonance Raman effect allows single-carbon nanotube spectra to be observed easily and under normal operating conditions is summarized. The important structural information that is provided by single-nanotube spectroscopy using one laser line is discussed, and what else can be learned from tunable laser experiments is reviewed. Particular attention is given to the determination of the nanotube diameter and of the energy of its van Hove singularities Eii. Applications of single-nanotube spectroscopy are emphasized, such as measurements of isolated SWNTs connected with circuit-based samples and of isolated SWNTs mounted on an atomic force microscope tip. A critical assessment of the opportunities and limitations of the resonance Raman method for structural (n, m) identification is presented. The trigonal warping effect, which is central to the (n, m) identification in resonance Raman spectroscopy, is discussed in simple terms, and the importance of this effect in nanotube science and applications is reviewed.

Phonon trigonal warping effect in graphite and carbon nanotubes.

The one-dimensional structure of carbon nanotubes leads to quantum confinement of the wave vectors for the electronic states, thus making the double resonance Raman process selective, not only of the magnitude, but also of the direction of the phonon wave vectors. This additional selectivity allows us to reconstruct the phonon dispersion relations of 2D graphite, by probing individual single wall carbon nanotubes of different chiralities by resonance Raman spectroscopy, and using different laser excitation energies. In particular, we are able to measure the anisotropy, or the trigonal warping effect, in the phonon dispersion relations around the hexagonal corner of the Brillouin zone of graphite.

Structure-assigned optical spectra of single-walled carbon nanotubes.

Spectrofluorimetric measurements on single-walled carbon nanotubes (SWNTs) isolated in aqueous surfactant suspensions have revealed distinct electronic absorption and emission transitions for more than 30 different semiconducting nanotube species. By combining these fluorimetric results with resonance Raman data, each optical transition has been mapped to a specific (n,m) nanotube structure. Optical spectroscopy can thereby be used to rapidly determine the detailed composition of bulk SWNT samples, providing distributions in both tube diameter and chiral angle. The measured transition frequencies differ substantially from simple theoretical predictions. These deviations may reflect combinations of trigonal warping and excitonic effects.

Probing the electronic trigonal warping effect in individual single-wall carbon nanotubes using phonon spectra

We have studied the trigonal warping effect in metallic carbon nanotubes using resonance Raman scattering and examining the G ′-band profile for individual metallic single-wall carbon nanotubes. We show that the observed splitting in the G ′-band phonon spectra is directly correlated with the splitting in the singularities of the joint density of electronic states, i.e., detailed information about the 1D electronic structure is extracted from a phonon measurement. By correlating the phonon data with the calculated electronic structure for several ( n, m) nanotubes, we were able to determine the G ′-band dispersion [( ∂ω G ′ / ∂E laser )=108±6 cm −1/ eV] by using a single laser energy.

Anisotropy in the Phonon Dispersion Relations of Graphite and Carbon Nanotubes Measured by Raman Spectroscopy

ABSTRACTThe possible semiconducting device use of single wall carbon nanotubes (SWNTs) requires a technique for the determination of the exact structure of the nanotubes assembled in the device configuration. Raman spectroscopy has been established as a precise and non-destructive tool for the characterization of graphitic nanostructures. Double resonance theory, which is used to explain the dispersive nature of the Raman bands, has attracted much attention for its potential use for the characterization of the electronic and phonon spectra of these nanostructures. Dispersive features in the Raman spectra of low dimensional graphitic materials, such as carbon nanotubes, can be used to measure directly the anisotropy, or the trigonal warping effect, in the phonon dispersion relations about the hexagonal corner of the Brillouin zone (BZ) of graphite.

Double resonance raman spectrain disordered graphite and singlewall carbon nanotubes

The Raman spectra of the disorder-induced D-band and some other non-zone-center phonon modes of graphite and single wall carbon nanotubes are discussed using double resonance Raman theory. For two-dimensional disordered graphite, two D-band peaks and one G′-band peak are predicted. The trigonal warping effect of the constant energy contours in graphite and the van Hove singularities in the electronic and phonon energy dispersion of carbon nanotubes are of importance for discussing the line width and possible numbers of Raman peaks which come from inequivalent Raman processes.

Double resonance Raman spectra in disordered graphite and single wall carbon nanotubes

The Raman spectra of the disorder-induced D-band and some other non-zone-center phonon modes of graphite and single wall carbon nanotubes are discussed using double resonance Raman theory. For two-dimensional disordered graphite, two D-band peaks and one G'-band peak are predicted. The trigonal warping effect of the constant energy contours in graphite and the van Hove singularities in the electronic and phonon energy dispersion of carbon nanotubes are of importance for discussing the line width and possible numbers of Raman peaks which come from inequivalent Raman processes.

Trigonal warping effect of carbon nanotubes

Chirality-dependent van Hove singularities (vHs) of the one-dimensional electronic density of states (DOS) are discussed in connection with resonant Raman spectroscopy. The effect of trigonal warping on the energy dispersion relations near the Fermi energy splits the peaks of the density of states for metallic nanotubes, and the magnitude of this effect depends on the chiral angle of the carbon nanotube. The width of the peak splitting has a maximum for metallic zigzag nanotubes, and no splitting is obtained for armchair nanotubes or semiconducting nanotubes. We also find an additional logarithmic singularity in the electronic DOS for carbon nanotubes that is related to a two-dimensional singularity, which does not depend on either the diameter or the chirality.

Berry's Phase and Absence of Back Scattering in Carbon Nanotubes

The absence of back scattering in carbon nanotubes is shown to be ascribed to Berry's phase which corresponds to a sign change of the wave function under a spin rotation of a neutrino-like particle in a two-dimensional graphite. Effects of trigonal warping of the bands appearing in a higher order k · p approximation are shown to give rise to a small probability of back scattering.

Low Temperature Anomalies in the Thermoelectric Power of Highly Oriented Graphite

The magnetic field dependence of the thermomagnetic tensor components β x x and β y x for a high quality kish graphite specimen is studied in the temperature range 1.5-15 K with application of the field up to 0.1 T. β is a more fundamental quantity than thermoelectric power. β x x shows a negative dip at 3.8 K and a sharp peak at 11 K in zero magnetic field. With application of magnetic field the temperatures for the dip and peak change in complicated ways. The peak disappears when a filed of 0.1 T is applied. β y x is negative and decreases with increasing temperature. At temperatures below 4 K β x x shows a positive peak at a field around 0.01 T in the field dependence. The previous theory for the anomaly of β x x around 4 K is extended to the case with magnetic field. The observed behavior for β x x in weak field region is ascribed to the trigonal band warping effect.

Scattering between composite particles composed of fermions in the one-dimensional delta-force model

We have reported the electronic band structure of monolayer graphene when the combined effects arising from the trigonal warp and highest zone-boundary phonons having A1g symmetry with Haldane interaction which induced photo-irradiation effect. On the basis of our model, we have introduced a diagonalization to solve the associated Fröhlich Hamiltonian. We have examined that, a trigonal warping effect is introduced on the K and K' points, leading to a dynamical band gap in the graphene electronic band spectrum due to the electron-A1g phonon interaction and Haldane mass interaction. Additionally, the bands exhibited an anisotropy at this point. It is also found that, photo-irradiation effect is quite smaller than the trigonal warp effects in the graphene electronic band spectrum. In spite of this, controllability of the photo induced effects by the Haldane mass will have extensive implications in the graphene.

Diamagnetism of Dilute Acceptor Compounds of Graphite

Diamagnetism study on graphite-acceptor compounds has been further extended over dilute nitrates and slightly neutron-irradiated graphite. An analysis was made on the basis of Sharma-Johnson-McClure's theory taking into account the trigonal warping of the Fermi surface. The susceptibility vs reciprocal termperature plots, including those for IC l compounds not completely interpreted in the previous study, have been theoretically reproduced by assuming a mixed phase structure of orderly intercalated and purely graphitic domains in each sample. The hole production rate (ionization efficiency) of acceptors was evaluated by using Ono-Sugihara's formulae which take into account the trigonal warping effect on the carrier density. It has been found that 5% of intercalated IC l is ionized in the doped regions irrespectively of the total content.

Low Field Galvanomagnetic Properties of Graphite Acceptor Compounds and their Relation to Trigonal Warping

ABSTRACTIn first stage graphite intercalation compounds with cylindrical Fermi surfaces and isotropic relaxation times, no magnetoresistance occurs. Holzwarth pointed out that trigonal band warping provides a finite magnetoresistance in first stage compounds. A calculation of the galvanomagnetic coefficients of stage-2 acceptor compounds is performed including the effect of trigonal warping on the bands. In the present problem a perturbation calculation provides a good approximation. Without trigonal warping, the stage-2 compounds exhibit a finite magnetoresistance since they have two kinds of holes. Retaining terms up to order (γ3/γ0)2, the weak field magnetoresistance is calculated in limit H→0; Δρ/ρoH2 ≍ (e/h2c)2 (τ/EF)2 (3γ0b/2)4[1/4(γ1/EF)2+57/2(γ3/γ0)2] where b = 1.42 Å is the nearest C-C distance, EF the Fermi energy and τ the carrier relaxation time. The first term in curly brackets is due to the existence of the two kinds of holes. This formula indicates that the term due to trigonal warping is much larger than the other term.

Specific heat of pure graphite in the ultra-quantum limit region

We calculate the electronic specific heat of pure graphite in the ultraquantum limit region for fields between 60 and 200 kG, at very low temperatures, using the Slonczewski-Weiss band model with values of the energy-band parameters which are in agreement with recent magneto-reflection experiments. The effect of trigonal warping of the Fermi surfaces associated with the parameter γ 3 is neglected in the calculation. Our results show that, for most of the range of fields considered, the electronic specific heat C is very nearly proportional to both the magnetic field strength H and the temperature T, according to the relation C ≈ αHT with a coefficient α of about 0.091 μJ/g-at. K 2kG. The results also indicate that, at the upper end of the magnetic field range, the C( H) curves, at a given T, depart progressively, though slightly, from linearity with increasing H.

Specific heat of pure graphite in the ultra-quantum limit region

We calculate the electronic specific heat of pure graphite in the ultraquantum limit region for fields between 60 and 200 kG, at very low temperatures, using the Slonczewski-Weiss band model with values of the energy-band parameters which are in agreement with recent magneto-replection experiments. The effect of trigonal warping of the Fermi surfaces associated with the parameter γ 3 is neglected in the calculation. Our results show that, for most of the range of fields considered, the electronic specific heat C is very nearly proportional to both the magnetic field strength H and the temperature T, according to the relation C ≈ α HT with a coefficient α of about 0.091 μJ/g-at. K 2kG. The results also indicate that, at the upper end of the magnetic field range, the C( H) curves, at a given T, depart slightly and progressively from linearity with increasing H, essentially as a result of the variation of the Fermi energy with magnetic field.

Diamagnetism of graphite

A calculation of the diamagnetism of graphite including the effects of trigonal warping of the Fermi surfaces has been performed using Fukuyama's formulation of the diamagnetism of Bloch electrons. Inclusion of the trigonal warping increases the diamagnetism by about 13% at low temperatures and reduces it by about 1% at high temperatures. A paramagnetic constant is used to represent all effects not associated with the free carriers. The experimental diamagnetism can be fitted quite well using values of the energy band parameters which give agreement with the de Haas-van Alphen effect, optical absorption, and magnetoreflection experiments.

Diamagnetism of graphite

The temperature-dependent diamagnetism of graphite has been accounted for using band parameters agreeing with magnetoreflection, optical, and de Haas-van Alphen results. The agreement requires inclusion of the effects of trigonal warping and addition of a small paramagnetic constant.

Effect of Trigonal Warping on Landau Levels of Graphite

AbstractWe have devised a successive approximation scheme for taking account of the trigonal warping (γ3) terms in the calculation of the lower‐lying magnetic levels of graphite at fields ⩾ 20 kG. A two‐component effective Hamiltonian has been introduced for these levels. Starting from analytic expressions in the case γ;3 = 0, straightforward and simple iteration schemes have been devised (i) to take account of the higher levels and (ii) to obtain the γ3 contributions. It is found that the trigonal warping changes the levels only by a few per cent except for the n = 0 levels. The levels n = 0, 1, 2, 3 have been calculated for a range of fields from 20 to 200 kG.