Plasma Frequency

The induced electric polarisation in the material shown in equation 2.16 can also be written in terms of the displacement, D:

since the susceptibility, χ, is related to the dielectric function, ε, from equation 2.17.

When the number of fixed positive ions and free electrons are equal the solid can be treated like a plasma. In this system the free electrons are best described using a free electron gas model. For a displacement, x, the equation of motion of the electron is given by:

where x and E have a time dependence of the form where ω is the frequency of the light. Substituting this into equation 2.25 gives:

Therefore:

The single electron dipole moment is given by:

(2.24)

(2.25)

(2.26)

23

The polarisation can be defined as the dipole moment per unit volume which is written as:

where n is the number of electrons per unit volume.

Furthermore, equation 2.24 can be rearranged to give:

which, by substituting equation 2.29 into equation 2.30 gives:

This is the equation for the dielectric function of the free electron gas. If the plasma frequency is then defined as:

then equation 2.31 can be rewritten as:

The plasma frequency is a major determinant of the optical activity, absorbing or reflecting, of a material. Equation 2.20 shows how the complex refractive index, N, is related to the complex dielectric function, ε. Hence, when , , N will be imaginary whereas when , , N will be real and finally when

, . From equation 2.23 for the reflectivity, R, when , R will be unity.

R will decrease when and then approach zero when . Therefore, up (2.29) (2.30) (2.31) (2.32) (2.33) (2.28)

24 to the plasma frequency, the reflectivity of a gas of free electrons will be at a maximum whereas after the plasma frequency the gas will absorb incident light.

When the dielectric function equals zero there are longitudinal frequencies of oscillation, ωL caused by a displacement, u, of the entire electron gas with respect to the fixed positive ions. A schematic of this process is shown in figure 2.9. At position (1) the free electron gas is around the fixed positive ions. However, if the entire gas is displaced the positive ions will exert a restoring force to oppose the displacement (2). Position (3) shows the ‘overshoot’ of the electron gas in the opposite direction resulting in a longitudinal oscillation. These oscillations are called the plasma oscillations where quantised oscillations are known as plasmons.

Figure 2.9: Schematic diagram showing the plasma oscillation with displacement u.

A plasmon may be excited by an electron passing through a metallic thin film or reflecting an electron or photon from a film. Although the above has been concerned with metals, plasma oscillations can be excited in dielectrics such as silicon in the same way with the valence electron gas oscillating with respect to the positive ion cores.

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2.4 The Three Phase Model

Later, chapter three will describe the path of polarised light through the experimental RAS instrument. Here a three phase model is introduced to describe the changes in polarisation as the light is reflected from the surface. McIntyre and Aspnes [10] introduced a linear approximation that can be used for reflectivity calculations and developed the theory of the three layer model which was then progressed further by Cole et al [11]. This is shown schematically in figure 2.10.

Figure 2.10: Schematic diagram of the three phase model.

As shown in figure 2.10 the model is based on an anisotropic surface in between an isotropic semi-infinite bulk and an ambient/vacuum layer each with their own dielectric constants. The dielectric function of the anisotropic surface is split into two orthogonal directions, εsx and εsy. Due to the surface anisotropy these two values may not be equal so Δεs is referred to as the surface dielectric anisotropy (SDA). If it is assumed that the dielectric function of the ambient or vacuum is equal to one and the thin film approximation that d<<λ , the fractional anisotropy of the surface (Δr/r) can be given by equation 2.34 [10]. This will be a complex function as it is determined using the dielectric functions of the surface and bulk which are both complex quantities, as shown by the schematic diagram in figure 2.11.

26 However the bulk and surface dielectric functions are complex quantities as shown in figure 2.10 and by equations 2.35 and 2.36.

where and . So equation 2.34 can be written in

terms of its real and imaginary parts:

Although most studies using RAS have focused on the real part of the function, in this research some studies have been conducted using the imaginary part and the changes in the experimental set up will be discussed later.

Figure 2.11: Schematic diagram to show the complex dielectric functions.

(2.35) (2.36)

(2.38) (2.37)

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2.5 References

[1] Kittel, C., Introduction to Solid State Physics, Eighth Edition. (John Wiley & Sons Inc, Hoboken) 2005

[2] Hummel, R. E., Electronic Properties of Materials, Fourth Edition. (Springer, New York) 2011

[3] Myers, H. P., Introductory Solid State Physics, Second Edition. (Taylor & Francis, London) 1997

[4] Ibach, H. and Lüth, H., Solid State Physics, An Introduction to Principles of Materials Science, Fourth Edition. (Springer-Verlag, Berlin Heidelberg) 2009

[5] Alloul, H., Introduction to the Physics of Electrons in Solids. (Springer-Verlag, Berlin Heidelberg) 2011

[6] Ho, K. -M. and Bohnen, K. P., Phys. Rev. Lett. 59 (1987) 1833

[7] Kolasinski K. W., Surface Science: Foundations of Catalysis and Nanoscience, Third Edition. (John Wiley & Sons Inc, Chichester) 2012

[8] Zangwill, A., Physics at Surfaces. (Cambridge University Press, Cambridge) 1988

[9] Goldmann, A., Dose, V. and Borstel, G., Phys. Rev. B 32 (1985) 1971 [10] McIntyre, J. D. E. and Aspnes, D. E., Surf. Sci. 24 (1971) 417

[11] Cole, R. J., Frederick, B. G. and Weightman, P., J. Vac. Sci. Technol. A 16 (1998) 3088

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Chapter 3