Electron Spectroscopies

In this section we describe various electron spectroscopies which have been used to study the electronic structure of carbon nanotubes. The techniques are illustrated schematically in Fig. 1. For this illustration we use the electronic structure of a carbon-based n-electron system. In these compounds the valence electrons of carbon are predominantly in the sp2 configuration, i.e., one s-electron and two p-electrons form the sp2-hybrid which has trigonally directed a bonds in a plane. In the solid, these a orbitals form strong covalent bonds with the a orbitals from neighboring carbon atoms. Therefore, occupied a and unoccupied a* bonds are formed. The third C 2p electron is in a 2pz-orbital perpendicular to the plane and forms a weaker n bond with the 2pz-orbitals of neighboring C atoms. The electrons from the C 2pz orbitals in this configuration are usually called n-electrons. Due to the weaker bonding, the splitting between the occupied n-bonds and the unoccupied n*-bands is weaker. The electronic structure of the valence and conduction bands in such systems together with the C 1 s core level are shown in Fig. 1. In addition, transitions used in the various techniques are shown in Fig. 1.

In Photoelectron Spectroscopy (PES) [1], photoelectrons are ejected from the solid by a photon with the energy hv. The kinetic energy, Ekin, and the intensity of the photoelectrons are measured. The binding energy of the ejected electron is then given by the Einstein relation EB = hv — Ekin — where <P is the work function. In a first approximation, the intensity of the photoelec-trons as a function of EB yields the density of occupied states. Detecting the angle of the ejected photoelectrons relative to the surface normal gives information on the wave vector of the electrons in the solid. Therefore, using this Angular Resolved PhotoEmission Spectroscopy (ARPES) the band structure of the occupied bands can be probed. Measuring the binding energy, EB,

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IPES

EELS (XAS)

Fig. 1. Illustration of various electron spectroscopies. The electronic structure of carbon-based ^-electron systems is sketched. PES: Photoemission, IPES: Inverse Photoemission, XES: X-ray Emission Spectroscopy, EELS: Electron Energy-Loss Spectroscopy, XAS: X-ray Absorption Spectroscopy relative to the Fermi level of the core electrons, provides information on the chemical bonding of the C atoms. The so-called chemical shift of EB is determined by the charge on the excited atom and the Madelung potential from the surrounding atoms. Finally, one should remark that the mean free path of the photoelectron is only several A. Therefore, this technique is extremely surface sensitive.

In the Inverse PhotoElectron Spectroscopy (IPES) [1], electrons are injected into the solid, and the intensity and the energy of the photons from the Bremsstrahlung is recorded. From these measurements, information on the density of unoccupied states can be derived. Similar to ARPES, in Angular-Resolved Inverse PhotoEmission Spectroscopy (ARIPES), the angle of the incoming electrons is varied and the band structure of the unoccupied bands can be measured. Since the mean free path of the injected electrons is again only several A IPES and ARIPES are also surface sensitive techniques.

In X-ray Emission Spectroscopy (XES) [2], a core hole, (e.g. in the C 1s shell) is created by electron bombardment or by X-ray irradiation. The fluorescent decay of the core hole by transitions from the occupied valence bands to the unoccupied core state is monitored. The intensity as a function of the energy of the fluorescent radiation gives information on the density of occupied states. Since a dipole transition to a localized core orbital is involved, only states with the appropriate symmetry character that are localized in the vicinity of the core orbital will contribute to the spectra. Consequently, this technique provides information on the local partial density of states. Starting from the core hole in the C 1s level, the contributions of C 2p states to the occupied density of states are measured.

In Inelastic soft X-ray Scattering (RIXS) [3,4], which is a technique closely related to XES, additional information can be obtained on the wave vector of the occupied states, similar to ARPES. Here, a core hole is created by selectively promoting the core electron into an unoccupied state of a chosen energy. Synchrotron radiation is used to induce excitations from the core level into various parts of the conduction bands. Under certain conditions, valence-band electrons with the same wave vector as the excited electron in the conduction band will then contribute predominantly in the consecutive X-ray emission process. As a result part of the band structure of the occupied bands can be probed selectively by variation of the excitation energy.

In Electron Energy-Loss Spectroscopy (EELS) [5], transitions from the core level into unoccupied states can be performed using the inelastic scattering of high-energy electrons. At small scattering angles, only dipole excitations are allowed. Therefore, starting from the C 1s level, the local partial density of unoccupied states having C 2p character is probed. However, since there is a core hole in the final state, excitonic effects have to be taken into account. When these effects are strong, the spectral weight at the bottom of the bands is enhanced at the expense of the spectral weight at higher energy. Similar information can be obtained in X-ray Absorption Spectroscopy (XAS) [6]. There, using synchrotron radiation, the absorption coefficient is measured near the threshold of a core excitation. In carbon-based materials, again, the local partial density of unoccupied states with C 2p character is probed.

In the low-energy range of the EELS spectra, excitations from occupied to unoccupied states are probed. In contrast to optical spectroscopy, the measured loss function is not dominated by absorption maxima but by the collective excitations of electrons, i.e. the plasmons [7]. The concept of plas-mons can be introduced in a continuous-medium approximation, at least in the long-wavelength limit. Neglecting retardation effects, the electric field, E, and displacement vector, D, generated by a perturbation of the polarization of the medium, satisfy the Maxwell equations V x E = 0 and V • D = 0. In an infinite, homogeneous material, solutions can be sought in the form of plane waves, with frequency w and wave vector q. The above equations then become q x E = 0 and q • D = 0, with D = eoe(w, q)E, where e is the dielectric function of the medium. The two possible plane-wave solutions are e(w, q) = 0 and q x E = 0 (1)

The first solution corresponds to longitudinal modes, the plasmons probed in EELS. The second modes are transverse ones and these can be excited by electromagnetic waves.

The probability per length unit that the electron loses the energy hw and momentum hq in a single-scattering event is proportional to the so-called loss function [8], the imaginary part of the inverse frequency- and momentum-dependent dielectric function,

For an anisotropic material such as graphite, e denotes q • e ■ q with q a unit vector in the direction of the wave-vector transfer, and e the dielectric tensor. The loss function has maxima when the real part Re[e] = ei is zero and Im[e] = e2 is small. That condition immediately shows that the longitudinal modes (1) are the ones that can be excited by a traveling electron. These modes generate an electric field in the medium that interacts with the electron through the Coulomb force eE, and this force slows the particle down. The energy losses come from that interaction.

EELS has the advantage to provide information on the dielectric function of the sample over a broad frequency interval. Kramers-Kronig analysis can then be used, which allows one to determine Re[- 1/e(w, q)], the dielectric function ei + ie2, and the optical constants from the measured loss function. Furthermore, performing angular resolved measurements, the wave vector and consequently the wavelength of the excitations can be varied. In terms of optical spectroscopy, not only vertical but also non-vertical transitions can be excited. Because EELS is performed with high-energy electrons (E ~ 200 keV) in transmission through free-standing samples with a thickness of ~ 0.1 ^m, this technique, like optical absorption spectroscopy, is not surface sensitive.

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