| Home / lib / P_Physics / PS_Solid state / | ||
Dresselhaus M.S. Solid State Physics 2.. Optical Properties Of Solids (lecture notes)(198s)_PS_.pdf |
|
Size 1.7Mb Date Dec 9, 2005 |
Now that we have defined the complex dielectric function εcomplex and the complex conductivity σcomplex , we will relate these quantities in two ways: 1. to observables such as the reflectivity which we measure in the laboratory, 2. to properties of the solid such as the carrier density, relaxation time, effective masses, energy band gaps, etc. After substitution for K in Eq. 1.10, the solution Eq. 1.11 to the wave equation (Eq. 1.8) yields a plane wave E (z , t) = E0 e−iωt exp i ω z √εµ c
1...
With E in the x direction, the second relation between E0 , E1 , and E2 follows from the continuity condition for tangential Hy across the boundary of the solid. From Maxwell’s equation (Eq. 1.2) we have µ ∂H iµω H ×E =− (1.30) = c ∂t c which results in ∂ Ex iµω = Hy . ∂z c (1.31)...
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....
thereby introducing the electrical conductivity σ . Substitution for the drift velocity v 0 yields v0 = eE0 (m/τ ) − imω 8 (2.5)...
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)...
Figure 3.5: Schematic diagram showing the splitting of the = 1 level by the spin–orbit interaction....
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...
Figure 4.2: E (k ) for a few high symmetry directions in germanium, neglecting the spin-orbit interaction....
where mr is the reduced mass for the valence and conduction bands, we can estimate the absorption coefficient αabs (ω ). At very low temperature, a semiconductor has an essentially filled valence band and an empty conduction band; that is f (Ev ) = 1 and f (Ec ) = 0. We can estimate | v|p|c |2 from the effective mass sum–rule (Eq. 3.33) | v|p|c |2 m
0 Eg...
Figure 5.7: Energy band diagram in an electric field showing the wavefunction overlap (a) without and (b) with absorption of a photon of energy ¯ ω . h write the momentum matrix elements coupling two bands (for example, the valence and conduction bands) as n Operating with
k...
where En is the energy above the conduction band minimum Ec , and mn in Eq. 5.51 is the effective mass of an electron near the conduction band minimum. Since the valence band extremum is at k = 0, then ¯ kp is the crystal momentum for the h hole that is created when the electron is excited, corresponding to the kinetic energy of the hole h2 ¯ 2 kp Ep = . (5.52) 2mp 51...
Figure 5.11: Schematic diagram showing the frequency dependence of the square root of the absorption coefficient for indirect interband transitions near the thresholds for the phonon emission and absorption processes. The curves are for four different temperatures. At the lowest temperature (T4 ) the phonon emission process dominates, while at the highest temperature (T1 ) the phonon absorption process is most important at low photon energies. The magnitude of twice the phonon energy is indicated....
Figure 6.3: (a) Frequency dependence of the reflectivity of Ge over a wide frequency range. (b) Plot of the real [ε1 (ω )] and imaginary [ε2 (ω )] parts of the dielectric functions for Ge obtained by a Kramers–Kronig analysis of (a)....
Figure 6.4: Reflectance and frequency modulated reflectance spectra for GaAs. (a) Room temperature reflectance spectrum and (b) the wavelength modulated spectrum (1/R)(dR/dE ) (the solid curve is experimental and the broken curve is calculated using a pseudopotential band structure model. Adapted from Yu and Cardona)....
Figure 6.11: (a) Electric field vectors resolved into p and s components, for light incident (i), reflected (r), and transmitted (t) at an interface between media of complex indices of ˜ ˜ refraction Na and Ns . The propagation vectors are labeled by ki , kr , and kt . (b) Schematic diagram of an ellipsometer, where P and S denote polarizations parallel and perpendicular to the plane of incidence, respectively. then be determined from the angle φ, the complex σr , and the dielectric function εa of the ambient environment using the relation εs = εa sin2 φ + εa sin2 φ tan2 φ
1...
Figure 7.1: Hydrogenic impurity levels in a semiconductor. This interaction is described by the Coulomb perturbation Hamiltonian, H (r) = − e2 ε0 r (7.1)...
Figure 7.4: Photo-thermal ionization spectrum of phosphorus-doped Si measured by modulation spectroscopy. The inset shows schematically the photo-thermal ionization process for a donor atom....
Using Stirling’s approximation for ln x! when x is large, we write ln x! ∼ x ln x − x. = Equilibrium is achieved when (∂ F /∂ n) = 0, so that at equilibrium we have N! Es = k B T ln ∂ n (N − n)!n!
∂ =...
Table 7.1: Exciton binding energy (E1 ) and Bohr radius (r1 ) in some direct bandgap semiconductors with the zinc-blende structure (from Yu and Cardona). Semiconductor GaAs InP CdTe ZnTe ZnSe ZnS E1 (meV) 4.9 5.1 11 13 19.9 29 E1 (theory) (meV) 4.4 5.14 10.71 11.21 22.87 38.02 r1 (˚) A 112 113 12.2 11.5 10.7 10.22...
Figure 7.10: Plot of the square root of the absorption coefficient vs. ¯ ω for Ge for various h temperatures showing the effect of the excitons. Features associated with both indirect and direct excitons are found....
Figure 7.12: (a) A spectrum of the optical density of KBr showing Frenkel excitons. The optical density is defined as log(1/T ) where T is the optical transmission. (b) The energy bands of KBr, as inferred from tight-binding calculations of the valence bands and the assignments of interband edges in optical experiments. We note that the spectrum is dominated by exciton effects and that direct band edge contributions are much less important and the binding energy is on the order of an electron volt....
| © 2007 eKnigu | ||