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Dresselhaus M.S. Solid State Physics 2.. Optical Properties Of Solids (lecture notes)(198s)_PS_.pdf |
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where we have assumed that the charge density is zero. The constitutive equations are written as: D = εE B = µH j = σE (1.5) (1.6) (1.7)...
˜ where we note that ε1 , ε2 , n and k are all frequency dependent. ˜ Many measurements of the optical properties of solids involve the normal incidence reflectivity which is illustrated in Fig. 1.1. Inside the solid, the wave will be attenuated. We assume for the present discussion that the solid is thick enough so that reflections from the back surface can be neglected. We can then write the wave inside the solid for this one-dimensional propagation problem as Ex = E0 ei(K z −ωt) (1.27)...
processes which correspond to the absorption of electromagnetic radiation by an electron in an occupied state below the Fermi level, thereby inducing a transition to an unoccupied state in a higher band. This interband process is intrinsically a quantum mechanical process and must be discussed in terms of quantum mechanical concepts. In practice, we consider in detail the contribution of only a few energy bands to optical properties; in many cases we also restrict ourselves to detailed consideration of only a portion of the Brillouin zone where strong interband transitions occur. The intraband and interband contributions that are neglected are treated in an approximate way by introducing a core dielectric constant which is often taken to be independent of frequency and external parameters....
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...
The free carrier term makes a negative contribution to ε1 which tends to cancel the core contribution shown schematically in Fig. 2.1. We see in Fig. 2.1 that ε1 (ω ) vanishes at some frequency (ωp ) so that we can write ˆ ε1 (ωp ) = 0 = εcore − ˆ which yields ωp = ˆ2 4π ne2 τ 2 m(1 + ωp τ 2 ) ˆ2 (2.22)...
In a semiconductor at low frequencies, the principal electronic conduction mechanism is associated with free carriers. As the photon energy increases and becomes comparable to the energy gap, a new conduction process can occur. A photon can excite an electron from an occupied state in the valence band to an unoccupied state in the conduction band. This is called an interband transition and is represented schematically by the picture in Fig. 3.1. In this process the photon is absorbed, an excited electronic state is formed and a hole is left behind. This process is quantum mechanical in nature. We now discuss the factors that are important in these transitions. 1. We expect interband transitions to have a threshold energy at the energy gap. That is, we expect the frequency dependence of the real part of the conductivity σ1 (ω ) due to an interband transition to exhibit a threshold as shown in Fig. 3.2 for an allowed electronic transition. 2. The transitions are either direct (conserve crystal momentum k : Ev (k ) → Ec (k )) or indirect (a phonon is involved because the k vectors for the valence and conduction bands differ by the phonon wave vector q). Conservation of crystal momentum yields kvalence = kconduction ± qphonon . In discussing the direct transitions, one might wonder about conservation of crystal momentum with regard to the photon. The reason we need not be concerned with the momentum of the photon is that it is very small in comparison to Brillouin zone dimensions. For a typical optical wavelength of 6000 ˚, the wave vector for the photon K = 2π /λ ∼ 105 cm−1 , while a typical dimension A across the Brillouin zone is 108 cm−1 . Thus, typical direct optical interband processes excite an electron from a valence to a conduction band without a significant change in the wave vector. 3. The transitions depend on the coupling between the valence and conduction bands and this is measured by the magnitude of the momentum matrix elements coupling the valence band state v and the conduction band state c: | v|p|c |2 . This dependence results from Fermi’s “Golden Rule” (see Chapter A) and from the discussion on the perturbation interaction H for the electromagnetic field with electrons in the solid (which is discussed in §3.2). 15...
The term εcore contains the contributions from all processes that are not considered explicitly in Eq. 3.3; this would include both intraband and interband transitions that are not treated explicitly. We have now dealt with the two most important processes (intraband and interband) involved in studies of electronic properties of solids. If we think of the optical properties for various classes of materials, it is clear from Fig. 3.3 that ma jor differences will be found from one class of materials to another. 17...
The reason why interband transitions depend on the momentum matrix element can be understood from perturbation theory. At any instance of time, the Hamiltonian for an electron in a solid in the presence of an optical field is H= (p − e/cA)2 e2 A 2 p2 eA ·p+ + V (r) = + V (r) − 2m 2m mc 2mc2 (3.12)...
Assuming that the first order term in perturbation theory (Eq. 3.26) can be neglected by parity (even and oddness) arguments, we obtain for En (k ) about k = 0 En (k ) = En (0) + |(v |pα |c)(c|pβ |v )| ¯2 h kk 2αβ m Eg (3.30)...
A detailed discussion of this topic is found in any standard quantum mechanics text. This spin-orbit interaction gives rise to a spin-orbit splitting of the atomic levels corresponding to different values of the total angular momentum J J =L+S where L and S , respectively, denote the orbital and spin angular momentum. Thus J · J = (L + S ) · (L + S ) = L · L + S · S + (L · S + S · L) (3.39) (3.38)...
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)...
We now carry out the integral over d(Ec − Ev ) to obtain 2 ρcv (¯ ω ) = 3 h 8π S . | k(Ec − Ev )|Ec −Ev =¯ ω h
d...
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