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Dresselhaus M.S. Solid State Physics 2.. Optical Properties Of Solids (lecture notes)(198s)_PS_.pdf |
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For the wave propagating in vacuum (ε = 1, µ = 1, σ = 0), Eq. 1.17 reduces to a simple plane wave solution, while if the wave is propagating in a medium of finite electrical conductivity, the amplitude of the wave exponentially decays over a characteristic distance δ given by δ= c c = ˜ ˜ ω N2 (ω ) ω k (ω ) (1.18)...
where R, A, and T are, respectively, the fraction of the power that is reflected, absorbed, and transmitted as illustrated in Fig. 1.1. At high temperatures, the most common observable is the emissivity, which is equal to the absorbed power for a black body or is equal to 1 − R assuming T =0. As a homework exercise, it is instructive to derive expressions for R and T when we have relaxed the restriction of no reflection from the back surface. Multiple reflections are encountered in thin films. The discussion thus far has been directed toward relating the complex dielectric function or the complex conductivity to physical observables. If we know the optical constants, then we can find the reflectivity. We now want to ask the opposite question. Suppose we know the reflectivity, can we find the optical constants? Since there are two optical constants, ˜ n and k , we need to make two independent measurements, such as the reflectivity at two ˜ different angles of incidence. Nevertheless, even if we limit ourselves to normal incidence reflectivity measurements, ˜ we can still obtain both n and k provided that we make these reflectivity measurements ˜ for all frequencies. This is possible because the real and imaginary parts of a complex ˜ physical function are not independent. Because of causality, n(ω ) and k (ω ) are related ˜ through the Kramers–Kronig relation, which we will discuss in Chapter 6. Since normal incidence measurements are easier to carry out in practice, it is quite possible to study the optical properties of solids with just normal incidence measurements, and then do a Kramers–Kronig analysis of the reflectivity data to obtain the frequency–dependent dielectric functions ε1 (ω ) and ε2 (ω ) or the frequency–dependent optical constants n(ω ) and ˜ ˜ k (ω ). In treating a solid, we will need to consider contributions to the optical properties from various electronic energy band processes. To begin with, there are intraband pro cesses which correspond to the electronic conduction by free carriers, and hence are more important in conducting materials such as metals, semimetals and degenerate semiconductors. These intraband processes can be understood in their simplest terms by the classical Drude theory, or in more detail by the classical Boltzmann equation or the quantum mechanical density matrix technique. In addition to the intraband (free carrier) processes, there are interband 6...
In narrow gap semiconductors, mαβ is itself a function of energy. If this is the case, the Drude formula is valid when mαβ is evaluated at the Fermi level and n is the total carrier density. Suppose now that the only conduction mechanism that we are treating in detail is the free carrier mechanism. Then we would consider all other contributions in terms of the core dielectric constant εcore to obtain for the total complex dielectric function ε(ω ) = εcore (ω ) + 4π iσ /ω so that σ (ω ) =
n...
in which the sum in Eq. 3.2 is over all valence and conduction band states labelled by i and j . Structure in the optical conductivity arises through a singularity in the resonant denominator of Eq. 3.2 [−iω + 1/τ + (i/¯ )(Ei − Ej )] discussed above under properties (1) h and (5). The appearance of the Fermi functions f (Ei ) − f (Ej ) follows from the Pauli principle in property (4). The dependence of the conductivity on the momentum matrix elements accounts for the tensorial properties of σαβ (interband) and relates to properties (2) and (3). In semiconductors, interband transitions usually occur at frequencies above which free carrier contributions are important. If we now want to consider the total complex dielectric constant, we would write ε = εcore + 4π i [σDrude + σinterband ] . ω (3.3)...
pendent of temperature. Although the carrier densities are low, the high carrier mobilities nevertheless guarantee a large contribution of the free carriers to the optical conductivity....
Assume that we know the solution to Eq. 3.25 about a special point k0 in the Brillouin zone which could be a band extremum, such as k0 = 0. Then the perturbation formulae Eqs. 3.22– 3.25 allow us to find the energy and wave function for states near k0 . For simplicity, we carry out the expansion about the center of the Brillouin zone k = 0, which is the most important case in practice; the extension of this argument to an energy extremum at arbitrary k0 is immediate. Perturbation theory then gives: En (k ) = En (0) + (un,0 |H |un,0 ) +
n...
Figure 3.6: Energy bands of Ge: (a) without and (b) with spin–orbit interaction. The listing above gives the Γ point splittings. The spin-orbit splittings are k -dependent and at the L-point are typically about 2/3 of the Γ point value. The one-electron Hamiltonian for a solid including spin-orbit interaction is from Eq. 3.36 H= p2 1 + V (r) − ( V × p) · S . 2m 2m2 c2 (3.44)...
where dkn is an element of a wave vector normal to S , as shown in Fig. 4.1. By definition of the gradient, we have | kE|dkn = dE so that for surfaces with energy difference Ec − Ev we write: | k(Ec − Ev )|dkn = d(Ec − Ev ). Therefore d3 k = dkn dS = dS so that ρcv (¯ ω ) = h 2 8π 3
d...
Figure 4.2: E (k ) for a few high symmetry directions in germanium, neglecting the spin-orbit interaction....
Figure 4.3: Frequency dependence of the real (ε1 ) and imaginary (ε2 ) parts of the dielectric function for germanium. The solid curves are obtained from an analysis of experimental normal-incidence reflectivity data while the dots are calculated from an energy band model....
˜ where the index of refraction n(ω ) is large and the extinction coefficient k (ω ) is small. For ˜ the imaginary part of the dielectric function, we have √˜ ˜ ˜ ε2 (ω ) ≡ 2n(ω )k (ω ) ≈ 2 ε0 k (ω ) = which is small, since ωp
2 ε 0 ωp τ 2 ω3 τ 3...
If we wish to consider the absorption process at finite temperature, we also need to include the Fermi functions to represent the occupation of the states at finite temperature f (Ev )[1 − f (Ec )] − f (Ec )[1 − f (Ev )] (5.27)...
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