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12 CHAPTER 1. CLASSICAL MODELS FOR ELECTRONS IN SOLIDS |b> |b> |a> ħω T atomic levels broaden |a> Absorption line Absorption peak ω T ω ω T ω Figure 1.3: Absorption spectra produced by electronic transitions in atoms (discrete levels) and solids (levels are broadened). In solids, the sharp absorption peaks found in atomic spectra broaden out into resonances with a finite width. The resulting absorption spectra have a similar, Lorentzian form to that expected from the dipole oscillator model Im(ε) Permittivity ε Re(ε) ε static 1 0 0 ω T1 ω T2 ω T3 Frequency Figure 1.4: The electromagnetic response of an insulator can very generally be modelled by superimposing dipole oscillator responses with different natural frequencies, each scaled by a suitable oscillator strength. The low frequency, static permittivity then includes contributions from the low frequency tails of all the individual oscillator responses.
1.2. DRUDE MODEL 13 Im(ε) Permittivity ε 0 Re(ε) 0 ω p * Frequency Figure 1.5: Real and imaginary part of the dielectric permittivity in the Drude model. The peak at ω T1 illustrates the possibility of additional resonances due to the bound, core electrons. 1.2 Drude model The Lorentz, or dipole oscillator model extends naturally to metals, if we imagine that some of the electrons are no longer bound to the ions. The remaining inner, or core electrons are still closely bound and will continue to contribute to the permittivity according to the Lorentz oscillator model. The outer, or conduction electrons, however, have been cut loose from the ions and are now free to roam around the entire piece of metal. This corresponds to the spring constant in their linearised force law going to zero, and hence the natural frequency ω T vanishes. To model their contribution to ɛ ω we can then simply use our earlier expressions for the frequency dependent permittivity, but drop ω T : the resonance peak now occurs at zero frequency. This picture, in which some of the electrons are cut free from the ionic cores, is called the Drude model. ω T1 1.2.1 Optical properties of metals in the Drude model Setting ω T in the dipole oscillator response (1.2) to zero and inserting a background permittivity ɛ ∞ to take account of the polarisability of the bound core electrons, leads us to the Drude response ɛ ω = ɛ ∞ − N V q 2 mɛ 0 (ω 2 + iωγ) = ɛ ∞ − ω 2 p ω 2 + iωγ , (1.5) where ωp 2 = ne2 mɛ 0 (defining n = N/V , the number density of mobile electrons). ω p is called the plasma frequency. We can draw three immediate conclusions from the above form of the permittivitiy (Fig. 1.2): 1. |ɛ ω | diverges for ω → 0 =⇒ metals are highly reflecting at low frequency. 2. The imaginary part of ɛ ω peaks at ω = 0, giving rise to enhanced absorption at low frequency, the ’Drude peak’. 3. ɛ ω crosses through zero at high frequency ωP ∗ =⇒ metals become transparent in the ultraviolet.
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