Dressel.Gruner.Electrodynamics of Solids.2003

Fig. B.6. Transmission through a plane parallel sapphire plate (thickness d3 = 0.41 mm) with a metallic NbN film of different thicknesses: bare substrate (curve 1); d2 = 20 A˚ (curve 2); d2 = 140 A˚ (curve 3); and d2 = 300 A˚ (curve 4). The spectra are taken at room temperature. The lines are fits according to Eq. (B.27) (from [Gor93a]).

interference pattern can be well described by Eq. (B.27) using n3 = 3.39 and k3 = 0.0003 as the refractive index of the substrate [Gor93a]. It is interesting to compare these results with those displayed in Fig. 11.17, where not the thickness of the film but the conductivity of the material changes; the transmission and the

In Fig. B.7a the transmission spectra of a 115 A˚ thick NbN film at two different temperatures above and below the superconducting transition are shown. The solid lines in the upper panel represent the theoretical fit of the multireflection in this two-layer system. In the normal state transmission spectra vary only slightly while the temperature decreases from 300 K down to transition temperature Tc ≈ 12 K. Below Tc we see that at small frequencies the transmission decreases by about

Fig. B.7. (a) Transmission spectra TF(ν) of an asymmetrical Fabry–Perot resonator, formed by a NbN film (d2 = 115 A)˚ on a sapphire substrate (0.43 mm), measured at two temperatures above and below critical temperature Tc ≈ 12 K. (b) Transmission spectra TF(ν) of a plane parallel sapphire plate (0.41 mm) covered with NbN films on both sides (d2 = 250 A˚ and d4 = 60 A),˚ again measured at 300 K and 5 K. The solid lines represent the theoretical fit. (c) For the latter case the temperature dependence of the optical conductivity σ1(T ) of NbN film is calculated for two frequencies, 8 cm−1 and 29 cm−1 (after [Gor93a, Gor93b]). Note the good agreement with the theoretical predictions shown in Fig. 7.4.

an order of magnitude, while at higher frequencies ω > 2 /h¯ it increases. If the substrate is covered on both sides, a three-layer system is obtained where the internal reflections in the substrate are significantly enhanced. A system with two metallic films on a dielectric substrate is essentially a symmetrical Fabry–Perot resonator with a much higher quality factor Q (or finesse F according to Eq. (11.3.1)) compared to the asymmetrical one. The sensitivity of the measurement is increased because the interaction of the wave with the material is multiplied. The transmission coefficients of such a Fabry–Perot resonator with NbN mirrors is shown in Fig. B.7b. The Q factors of the resonances increase strongly when the films become superconducting, which is connected with decrease of losses and increase of reflectivities of the mirrors. The material parameters of the substrate do not change significantly. Fig. B.7c shows temperature behavior of optical conductivity of the NbN film, obtained by least mean square fitting treatment of the transmission spectra. It is clearly seen that at the superconducting transition the conductivity drops for T → 0, but for low frequencies (ν = 8 cm−1) a peak develops right below Tc. The results are in good agreement with the theoretical predictions by the BCS theory as calculated by the Mattis–Bardeen formula (7.4.20) and displayed in Fig. 7.4.

References

[Bor99] M. Born and E. Wolf, Principles of Optics, 6th edition (Cambridge University Press, Cambridge, 1999)

[Gor93a] B.P. Gorshunov, G.V. Kozlov, A.A. Volkov, S.P. Lebedev, I.V. Fedorov, A.M. Prokhorov, V.I. Makhov, J. Schutzmann,¨ and K.F. Renk, Int. J. Infrared Millimeter Waves 14, 683 (1993)

[Gor93b] B.P. Gorshunov, I.V. Fedorov, G.V. Kozlov, A.A. Volkov, and A.D. Semenov,

Solid State Commun. 87, 17 (1993)

[Hea65] O.S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1963) [Hec98] E. Hecht, Optics, 3rd edition (Addison Wesley, Reading, MA, 1998)

[Ram94] S. Ramo, J.R. Whinnery, and T.v. Duzer, Fields and Waves in Communication Electronics, 3rd edition (John Wiley & Sons, New York, 1994)

[Sch75] H. Schunk, Stromverdrangung¨ (Huthig¨ Verlag, Heidelberg, 1975) [Sch94] G. Schaumburg and H.W. Helberg, J. Phys. III (France) 4, 917 (1994) [Wol34] W. Woltersdorff, Z. Phys. 91, 230 (1934)

Further reading

[Kle86] M.V. Klein, Optics, 2nd edition (John Wiley & Sons, New York, 1986)

Appendix C

k · p perturbation theory

The method of k·p perturbation theory was originally introduced by J. Bardeen and F. Seitz as a means of determining effective masses and crystal wavefunctions near high symmetry points of k [Sei40]. Considering small q as a perturbation, the k · p perturbation theory can be utilized to evaluate the transition between |k and |k states. The matrix element which appears in Eq. (4.3.20) can be evaluated using Bloch functions (Eq. (4.3.5)). Using the wavefunction ψkl = −1/2ukl exp{ik·r} = |kl the Schrodinger¨ equation Hkψkl = Ekl ψkl becomes

In order to find uk+q,l in the Schrodinger¨ equation Hk+quk+q,l = Ek+q,l uk+q,l , we assume that q is small, and consequently the Hamiltonian becomes

where the second term is considered a perturbation. The state uk+q,l can be found by performing a perturbation around ukl

where the integration over the squared terms (q · k and q2) in Eq. (C.2) is zero since they are not operators and the Bloch functions are orthogonal. Here the

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Appendix D

Sum rules

The interaction of an electronic system with the electromagnetic field lead to an absorption process. The total absorption is finite, and this is expressed by sum rules. These can be derived by examining the absorption process, but often the Kramers–Kronig relations are used as a starting point (together with some general assumptions about the absorption itself).

D.1 Atomic transitions

The interaction of a free atom with the electromagnetic field is treated in numerous textbooks (see for example [Mer70]), and therefore only the main results will be recalled here. For an atom initially in the ground state, the response to an electromagnetic field of the harmonic form E = E0 cos{ωt} leads to transitions between the ground and excited states. The relevant matrix element is related to the dipole moment

rl0 = l er 0 dr ,

where 0 and l are the wavefunctions of the ground state and excited states. The transition probability between the ground state and state l is described by the so-called oscillator strength (Eq. (6.1.7)):

where m is the electron mass and h¯ ωl0 = El − E0 is the energy difference between the ground state and the excited state in question; h¯ ωl0 is positive if the energy of the final state l is higher (upward transition) and negative for downward transitions. Here the transition matrix element is given in terms of the electric dipole moment,

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