Electrons in solids: elastic and inelastic scattering

Let us now consider the interaction of electrons with solids in some more detail. First consider the scattering of an electron beam from the surface of the solid. In an elastic scattering event the energy is (by definition) conserved, i.e.

 

where is the energy of the incoming electrons and that of the scattered electrons. The momentum parallel to the surface is also conserved apart from a surface reciprocal lattice vector

 

The crystal itself provides perpendicular momentum such that () and () can be fulfilled simultaneously. Observing the elastically scattered electrons provides information about the surface reciprocal lattice and the surface geometry. The technique concerned with this is called LEED and will be discussed later. Here we are more interested in the inelastic scattering since it determines the mean free path of the electrons and hence the surface sensitivity .  

The dielectric function

The dielectric function is a very useful concept because it describes the macroscopic absorption of both light and charged particles in solids and, at the same time, has a microscopic interpretation. Let us remind ourselves about some fundamental optical equations. Let the light vector be described by a plane wave which propagates in the x direction.

 

with

 

Between the complex index of refraction N and the dielectric function we have the Maxwell relation

 

The description of the optical properties in terms of N and is completely equivalent. The two parts of N and are not independent but can be transformed into each other using the Kramers-Kronig relations  

 

and

 

Note that technically spoken one quantity has to be known over the whole frequency spectrum if we wish to obtain the other. In similar ways both parts of N or can be obtained from just measuring the normal-incidence reflectivity over a large spectral range. The absorption of light in matter is given by Lambert's law  

 

The probability p for the electrons to suffer an inelastic scattering event is given by

 

This probability is exactly what we are concerned with here. When looking at the mean-free path, there seems to be a scattering probability which is very high for electrons with kinetic energies around 70 eV. In the following subsections we go quickly through the elementary excitations which are important contributions to the dielectric function, ordered by energy. These excitations provide a detailed microscopic picture for the dielectric function.  

Phonons

On its way through the solid and at the surface the electrons can be scattered inelastically by absorbing or creating phonons. The phonon energies are small (usually less than 100 meV) but the vector can be large. The phonon losses one observes in an EELS spectrum can be used to map the dispersion of the surface phonons or to learn something about the adsorbates by measuring their vibrational frequencies. We will come back to this in a later lecture. In our context here, phonon scattering is not very important because it only has to be considered at low energies.  

Excitons

Consider the case that an electron is excited from a bound state to a previously unoccupied state. In a metal, the screening is so strong that the electron and the hole will have very little interaction. In a semiconductor, however, electron and hole can remain loosely bound to form a so-called exciton. This exciton has a spectrum like a hydrogen atom but the Coulomb potential is screened by the dielectric function

 

The energy levels of this ``hydrogen atom'' lie just below the conduction band in an insulator or semiconductor. Ionizing the exciton means exciting the electron into the conduction band. The exciton is not bound to a particular site: the hole and the electron have some finite probability to hop to an adjacent site. This probability broadens the excitonic energy levels into bands. At the surface of a solid, the reduced coordination changes both the Coulomb potential for a single exciton and the hopping matrix elements between the excitons. This results in a so-called surface exciton which is shifted and has a different width.  

Interband transitions

Another loss mechanism is the creation of electron-hole pairs. In a metal electron-hole pairs can be created with infinitely small energies by lifting an electron from an energy level just below the Fermi energy to a level just above. Electron-hole creation does thus contribute to the dielectric function at all energies. For a semiconductor is the situation is different. There is a smallest energy for electron-hole pair creation, the energy of the fundamental gap. In semiconductors, a structure in the dielectric function can be found which corresponds to excitations over the gap. At slightly lower energy, the excitons are found. For both, metals and semiconductors so-called critical points   in the band structure give rise to strong features in the dielectric function. A critical point is, for example, a situation where the occupied bands and unoccupied bands are parallel in a larger region of k-space. Then the optical transitions from the region all have the same energy and contribute strongly to .

Bulk and surface plasmons

  In the Drude model of metals, the dielectric function is

 

where is the so-called plasma frequency

 

has a simple interpretation. It corresponds to a longitudinal collective vibration of the electron gas against the positively charged ions (see Fig. ). These excitations are called plasmons  .

  
Figure: A simple picture for a plasma oscillation

The plasma frequency is very important for the optical properties of a metal. We write equ. as

 

We can distinguish between two cases: if then is real and negative and () gives only exponentially damped solutions. This means that an electric field can not penetrate a metal, the metal is reflecting all the light. Above the plasma frequency () does permit propagating solutions of the electric field. For simple metals, there is a good agreement with the calculated plasma frequency , or plasmon energy , and the experimental values. There is also a plasmon mode which is localized at a metal surface and decays exponentially towards both metal and vacuum. It can be described as a longitudinal wave

 

Fig. shows the field and charge distribution for such a mode.

  
Figure: Charge and field distribution for a surface plasmon

The planar component of the field associated with this is continuos but the perpendicular component is not. Just above and below the surface it is

 

Now the field must be continuos. This gives us the condition for the existence of the surface plasmon

 

and hence

 

The energy loss of electrons due to plasmons and surface plasmons is illustrated in Fig. . It shows to energy distributions from an electron beam with approx. 2 keV kinetic energy which has been scattered from a surface of -Ga. Distinct losses are visible which can be ascribed to bulk and surface plasmons. Note that the energy difference between these losses is in good agreement with equ. . The difference between the spectra is due to the experimental geometry. This will be picked up in the exercises.

  
Figure: EELS spectra from the (010) surface of -Ga. The surface and bulk plasmon losses can be identified. The difference between the spectra is due to the experimental geometry (to be discussed in the exercises  ).

The excitation of plasmons and surface plasmons is the major reason for the inelastic scattering of the electrons in the energy regime we are interested in. When looking again at the universal curve we can see that the mean free path is long for lower energies because it is not possible to excite the plasmons. Above the edge for plasmon creation the mean free path drops drastically. At high energies it goes up again because the cross section for the plasmon creation diminishes.

Core levels

At much higher energies, several hundred eV or so, small structures in the dielectric function can be found which are due to the excitation of core electrons.

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