If we want to learn something about a system, a general experimental approach is a scattering technique: we shoot some particles in a well-prepared state on the target and look at particles coming out of the target (which do not have to be the same). In surface science the most basic questions we want to solve with this approach are for example: Is the surface clean? Which elements are on the surface? And in which chemical compound? What is the exact geometric structure of the surface?
The most common particles to scatter from surfaces are electrons, ions, atoms and photons both as probe and response particles. An important issue is the surface sensitivity of an experiment. In general, it is high if we choose particles which have a small mean free path in the solid because this means that the detected particles must originate near the surface. The opposite is true, for example, when the scattering of light by a surface is investigated (reflectivity and change of polarization). The photons will penetrate relatively deeply into the crystal. The amount of photons scattered at or near the surface will be very small. Hence, light scattering is not a good tool to study surfaces. In some cases we can increase the surface sensitivity by choosing an experimental set-up where we use a very grazing angle of incidence or emission. In this way the particles travel a long way close to the surface, even if their mean free path is relatively long.
Very many surface science techniques are based on electrons as a probe. Electrons have very useful properties: they are, at certain energies, very surface sensitive. Electrons in this energy range carry also enough momentum to explore the whole surface Brilloin zone of a material (in contrast to light), they also carry a spin and they are easy to generate and to handle. The extensive use of electrons in surface sciences justifies a lecture explaining the physics of electron-solid interaction in some more detail. Along with this, we will start to learn about some electron-based analytical techniques.
A technique which is of particular interest in this lecture is
Electron Energy Loss Spectroscopy (EELS)
where a beam of monochromatic
electrons is scattered from the surface. A sketch of this experiment
is given in Fig. 4.1.
Figure 4.1: An EELS experiment. The momentum transfer parallel to the surface is determined by the electron energy and the scattering geometry.
One of the main reasons to use electrons in surface science is the mean free path of electrons in matter. This mean free path λ is determined by collisions
| λ (Ekin)= v(Ekin) τ = |
| τ, (4.1) |
where v is the velocity and τ is the collision time. In the
Drude model τ is the mean time between two scattering events. In
a quasiparticle-picture τ is given by the imaginary part of the
self-energy, i.e. by the lifetime of the quasi-particle.
We are interested in energies of the electrons between a few eV and
many hundred eV. The mean free part of the electrons
in this regime
is
plotted in Fig. 4.2. The dashed curve shows a
calculation
of the mean free path independent of the material and the points are
measured data from many elemental solids. The data points scatter
more or less around the calculation. The curve is therefore often
called a
universal curve.
The reason for this universality is
that the inelastic scattering of electrons in this energy range is
mostly involving excitations of conduction electrons, which have more
or less the same density in all elements. Note that at lower
energies other scattering mechanisms will be important, like the
scattering with phonons.
Figure 4.2: The mean free part of the electrons in solid. The dots are measurements the dashed curve is a calculation. After Ref. [13].
The mean free path curve has a broad (note the log-log scale) minimum around a kinetic energy of about 70 eV. There it is less than 10 Å. This means that if we observe an electron with this kinetic energy which has left the solid without suffering an inelastic scattering event it must originate from the first few layers. How do we know that the electron has not been scattered inelastically? Fortunately, the energy loss associated with a scattering from the valence electrons is rather large (as we shall see below). Therefore it is relatively easy to distinguish between inelastically scattered and non-scattered electrons.
One big advantage of using electrons is that they are relatively easy to
produce. The most common way is electron emission from a hot filament.
A filament is heated by passing a current through it. To “help” the
thermally excited electrons out of the metal one additionally puts
an anode in front of the filament. The electron beam is focused by
placing a so-called Wehnelt cylinder between the anode and the
filament. The Wehnelt cylinder is at a negative potential with
respect to the filament. The basic principle is shown in Fig 4.3.
The simple filament
has two disadvantages when one eventually wants to produce a
monochromatic
beam of electrons. The first is that the voltage drop over the
length of the filament (0.5 V) is also reflected in the kinetic energy of
the electrons. The second is the thermal broadening due to the
high temperature needed to emit the electrons.
A better design for emitting monochromatic electrons is an
indirectly heated crystal which has a low work function.
Figure 4.3: An electron gun.
Electrons can be detected using an electron multiplier, usually a so-called channeltron. Such a device is essentially a glass tub with a resistive coating on the inside. A high voltage is applied between the front and the end. An electron which enters the channeltron will be accelerated to the wall where it kicks out more electrons. In this way an electron avalanche is created which eventually leads to a measurable current pulse.
Electron monochromators are needed both for creating a mono-energetic
probe-beam and for analysing the energy distribution of scattered or
emitted electrons. Electrostatic monochromators are the most common
choice. Actual designs represent a trade-off between the need for
high
count rates and high angular / energy resolution. The so-called cylindrical
mirror analyser (CMA)
is mostly used for checking the chemical
composition of the surface. It consists of two co-axial cylinders in
front of the sample. The inner cylinder is held at a positive
potential and the outer cylinder at a negative potential. Only the
electrons with the right energy can pass through this set-up and are
detected at the end.
The count rates are high but the
resolution (both in energy and angle) is poor. A hemispherical
analyser
is often used for applications where higher resolution is
needed. It consists of two con-centric hemispheres held a different
potentials. The electrons enter and leave through slits.
Again, only the electrons with the right kinetic energy, the
so-called pass energy Ep can pass the analyser. An electrostatic
lens-system can be placed in front of the hemispheres in order to
focus the electrons into the analyser and to change the angular
acceptance. Such an analyser is shown in Fig. 4.4.
Figure 4.4: A hemispherical electron analyser with a lens system.
In the EELS experiment mentioned above two electron monochromators are needed: one to produce a monochromatic beam and one to analyse the scattered electrons. In a typical apparatus one of these monochromators is movable in order to change the scattering geometry and the momentum transfer (see Fig. 10.3).
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.
| Es=E0, (4.2) |
where E0 is the energy of the incoming electrons and Es that of the scattered electrons. The momentum parallel to the surface is also conserved apart from a surface reciprocal lattice vector g
| k∥ s=k∥ 0+g (4.3) |
The crystal itself provides perpendicular momentum such that (4.2) and (4.3) 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 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 E vector be described by a plane wave which propagates in the x direction.
| E=E0ei(kx−ω t), (4.4) |
with
| k= |
| and N=n+i κ (4.5) |
Between the complex index of refraction N and the dielectric function є we have the Maxwell relation
| N= | √ |
| = | √ |
| (4.6) |
The description of the optical properties in terms of N and є is completely equivalent. The two parts of N and є are not independent but can be transformed into each other using the Kramers-Kronig relations
| єr(ω)=єr(∞)+ |
| ∫ |
|
| dω′ (4.7) |
and
| єi(ω)= |
| ∫ |
|
| dω′ (4.8) |
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 є 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
| I=I0e−α x and α= |
| (4.9) |
The probability p for the electrons to suffer an inelastic scattering event is given by
| p(ω) ∝ ℑ( |
| ). (4.10) |
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.
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 q 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.
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
| VCoul(r,R)=− |
| (4.11) |
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.
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 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 є.
In the Drude model of metals, the dielectric function is
| є (ω) = 1− |
| , (4.12) |
where ωP is the so-called plasma frequency
| ωP2= |
| . (4.13) |
ωP has a simple interpretation. It corresponds to a longitudinal collective vibration of the electron gas against the positively charged ions (see Fig. 4.5).
These excitations are called plasmons .
Figure 4.5: A simple picture for a plasma oscillation
The plasma frequency is very important for the optical properties of a metal. We write equ. 4.4 as
| E=E0e |
| . (4.14) |
We can distinguish between two cases: if ω < ωP then є is real and negative and (4.14) 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 (4.14) does permit propagating solutions of the electric field.
For simple metals, there is a good agreement with the calculated plasma frequency ωP, or plasmon energy ℏ ωP, 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
| Φ(r)=Φ0eiq∥r∥e−q∥|z|. (4.15) |
Fig. 4.6 shows the field and charge distribution for
such a mode.
Figure 4.6: Charge and field distribution for a surface plasmon
The planar component of the E field associated with this is continuos but the perpendicular component is not. Just above and below the surface it is
| E(z+0)= Φ0q∥eiq∥r∥ and E(z−0)= −Φ0q∥eiq∥r∥ (4.16) |
Now the D=є E field must be continuos. This gives us the condition for the existence of the surface plasmon
| є (ωsp) =−1 (4.17) |
and hence
| ωsp=ωP / | √ |
| (4.18) |
The energy loss of electrons due to plasmons and surface plasmons is illustrated in Fig. 4.7. It shows two energy distributions from an electron beam with approx. 2 keV kinetic energy which has been scattered from a surface of α-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. 4.18. The difference between the spectra is due to the experimental geometry. This will be picked up 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.
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.
We have seen that the minimum in the electron mean free path in Fig. 4.2 is mainly determined by the excitations of plasmons. Below the edge for plasmon excitation, the most important loss mechanism is interband transitions. This raises the question about how universal the curve actually is. If we take, for example, a wide band semiconductor, then interband transitions below the gap energy will not be possible and this will increase the mean free path considerably.
The fact that plasmon creation is the most important mechanism for inelastic scattering makes it clear why it is simple, in most cases, to distinguish elastically scattered (emitted) electrons from electrons which have been scattered by plasmons. The reason is that the plasma energies are rather large so that the loss peaks are far away from the elastic peak (see Fig. 4.7).
The most common way to measure the mean free path of electrons in a material is to evaporate a thin film of the material on a substrate. Then the intensity of characteristic Auger transitions in the substrate is measured as a function of film thickness.
The basic principles of electron spectroscopy are discussed in most surface science books [3, 2, 1, 4]. A good section about analysers and electron optics can be found in [4]. The elementary excitations which contribute to the dielectric function are discussed nicely in [3]. The basic physics of the Drude model and the plasmons is discussed in [11]. Surface plasmons are described in [3].