SOLID STATE PHYSICS PART II Optical Properties of Solids M. S. Dresselhaus 1 Contents 1 Review of Fundamental Relations for Optical Phenomena 1 1.1 Introductory Remarks on Optical Probes . 1 1.2 The Complex dielectric function and the complex optical conductivity . 2 1.3 Relation of Complex Dielectric Function to Observables . 4 1.4 Units for Frequency Measurements . 7 2 Drude Theory{Free Carrier Contribution to the Optical Properties 8 2.1 The Free Carrier Contribution . 8 2.2 Low Frequency Response: !¿ 1 . 10 ¿ 2.3 High Frequency Response; !¿ 1 . 11 À 2.4 The Plasma Frequency . 11 3 Interband Transitions 15 3.1 The Interband Transition Process . 15 3.1.1 Insulators . 19 3.1.2 Semiconductors . 19 3.1.3 Metals . 19 3.2 Form of the Hamiltonian in an Electromagnetic Field . 20 3.3 Relation between Momentum Matrix Elements and the E®ective Mass . 21 3.4 Spin-Orbit Interaction in Solids . 23 4 The Joint Density of States and Critical Points 27 4.1 The Joint Density of States . 27 4.2 Critical Points . 30 5 Absorption of Light in Solids 36 5.1 The Absorption Coe±cient . 36 5.2 Free Carrier Absorption in Semiconductors . 37 5.3 Free Carrier Absorption in Metals . 38 5.4 Direct Interband Transitions . 41 5.4.1 Temperature Dependence of Eg . 46 5.4.2 Dependence of Absorption Edge on Fermi Energy . 46 5.4.3 Dependence of Absorption Edge on Applied Electric Field . 47 5.5 Conservation of Crystal Momentum in Direct Optical Transitions . 47 5.6 Indirect Interband Transitions . 50 2 6 Optical Properties of Solids Over a Wide Frequency Range 57 6.1 Kramers{Kronig Relations . 57 6.2 Optical Properties and Band Structure . 62 6.3 Modulated Reflectivity Experiments . 64 6.4 Ellipsometry and Measurement of Optical Constants . 71 7 Impurities and Excitons 73 7.1 Impurity Level Spectroscopy . 73 7.2 Shallow Impurity Levels . 73 7.3 Departures from the Hydrogenic Model . 77 7.4 Vacancies, Color Centers and Interstitials . 79 7.5 Spectroscopy of Excitons . 83 7.6 Classi¯cation of Excitons . 87 7.7 Optical Transitions in Quantum Well Structures . 91 8 Luminescence and Photoconductivity 97 8.1 Classi¯cation of Luminescence Processes . 97 8.2 Emission and Absorption . 98 8.3 Photoconductivity . 104 10 Optical Study of Lattice Vibrations 108 10.1 Lattice Vibrations in Semiconductors . 108 10.1.1 General Considerations . 108 10.2 Dielectric Constant and Polarizability . 110 10.3 Polariton Dispersion Relations . 112 10.4 Light Scattering . 121 10.5 Feynman Diagrams for Light Scattering . 126 10.6 Raman Spectra in Quantum Wells and Superlattices . 128 11 Non-Linear Optics 132 11.1 Introductory Comments . 132 11.2 Second Harmonic Generation . 134 11.2.1 Parametric Oscillation . 135 11.2.2 Frequency Conversion . 136 12 Electron Spectroscopy and Surface Science 137 12.1 Photoemission Electron Spectroscopy . 137 12.1.1 Introduction . 137 12.1.2 Energy Distribution Curves . 141 12.1.3 Angle Resolved Photoelectron Spectroscopy . 144 12.1.4 Synchrotron Radiation Sources . 144 12.2 Surface Science . 146 12.2.1 Introduction . 146 12.2.2 Electron Di®raction . 147 12.2.3 Electron Energy Loss Spectroscopy, EELS . 152 12.2.4 Auger Electron Spectroscopy (AES) . 153 12.2.5 EXAFS . 154 3 12.2.6 Scanning Tunneling Microscopy . 156 13 Amorphous Semiconductors 165 13.1 Introduction . 165 13.1.1 Structure of Amorphous Semiconductors . 166 13.1.2 Electronic States . 167 13.1.3 Optical Properties . 173 13.1.4 Transport Properties . 175 13.1.5 Applications of Amorphous Semiconductors . 175 13.2 Amorphous Semiconductor Superlattices . 176 A Time Dependent Perturbation Theory 179 A.1 General Formulation . 179 A.2 Fermi Golden Rule . 183 A.3 Time Dependent 2nd Order Perturbation Theory . 184 B Harmonic Oscillators, Phonons, and the Electron-Phonon Interaction 186 B.1 Harmonic Oscillators . 186 B.2 Phonons . 188 B.3 Phonons in 3D Crystals . 189 B.4 Electron-Phonon Interaction . 192 4 Chapter 1 Review of Fundamental Relations for Optical Phenomena References: G. Beke¯ and A.H. Barrett, Electromagnetic Vibrations Waves and Radiation, MIT ² Press, Cambridge, MA J.D. Jackson, Classical Electrodynamics, Wiley, New York, 1975 ² Bassani and Pastori{Parravicini, Electronic States and Optical Transitions in Solids, ² Pergamon Press, NY (1975). Yu and Cardona, Fundamentals of Semiconductors, Springer Verlag (1996) ² 1.1 Introductory Remarks on Optical Probes The optical properties of solids provide an important tool for studying energy band struc- ture, impurity levels, excitons, localized defects, lattice vibrations, and certain magnetic excitations. In such experiments, we measure some observable, such as reflectivity, trans- mission, absorption, ellipsometry or light scattering; from these measurements we deduce the dielectric function "(!), the optical conductivity σ(!), or the fundamental excitation frequencies. It is the frequency-dependent complex dielectric function "(!) or the complex conductivity σ(!), which is directly related to the energy band structure of solids. The central question is the relationship between experimental observations and the electronic energy levels (energy bands) of the solid. In the infrared photon energy region, information on the phonon branches is obtained. These issues are the major concern of Part II of this course. 1 1.2 The Complex dielectric function and the complex optical conductivity The complex dielectric function and complex optical conductivity are introduced through Maxwell's equations (c.g.s. units) 1 @D~ 4¼ H~ = ~j (1.1) r £ ¡ c @t c 1 @B~ E~ + = 0 (1.2) r £ c @t D~ = 0 (1.3) r ¢ B~ = 0 (1.4) r ¢ where we have assumed that the charge density is zero. The constitutive equations are written as: D~ = "E~ (1.5) B~ = ¹H~ (1.6) ~j = σE~ (1.7) Equation 1.5 de¯nes the quantity " from which the concept of the complex dielectric func- tion will be developed. When we discuss non{linear optics (see Chapter 11), these linear constitutive equations (Eqs. 1.5{1.7) must be generalized to include higher order terms in E~ E~ and E~ E~ E~ . From Maxwell's equations and the constitutive equations, we obtain a wave equation for the ¯eld variables E~ and H~ : "¹ @2E~ 4¼σ¹ @E~ 2E~ = + (1.8) r c2 @t2 c2 @t and "¹ @2H~ 4¼σ¹ @H~ 2H~ = + : (1.9) r c2 @t2 c2 @t For optical ¯elds, we must look for a sinusoidal solution to Eqs. 1.8 and 1.9 i(K~ ¢~r¡!t) E~ = E~0e (1.10) where K~ is a complex propagation constant and ! is the frequency of the light. A solution similar to Eq. 1.10 is obtained for the H~ ¯eld. The real part of K~ can be identi¯ed as a wave vector, while the imaginary part of K~ accounts for attenuation of the wave inside the solid. Substitution of the plane wave solution Eq. 1.10 into the wave equation Eq. 1.8 yields the following relation for K: "¹!2 4¼iσ¹! K2 = : (1.11) ¡ ¡ c2 ¡ c2 If there were no losses (or attenuation), K would be equal to ! K = p"¹ (1.12) 0 c 2 and would be real, but since there are losses we write ! K = p" ¹ (1.13) c complex where we have de¯ned the complex dielectric function as 4¼iσ " = " + = " + i" : (1.14) complex ! 1 2 As shown in Eq. 1.14 it is customary to write "1 and "2 for the real and imaginary parts of "complex. From the de¯nition in Eq. 1.14 it also follows that 4¼i "! 4¼i " = σ + = σ ; (1.15) complex ! 4¼i ! complex · ¸ where we de¯ne the complex conductivity σcomplex as: "! σ = σ + (1.16) complex 4¼i Now that we have de¯ned the complex dielectric function "complex 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, e®ective 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 ¡i!t !z 4¼iσ E~ (z; t) = E~0e exp i p"¹ 1 + : (1.17) 0 c s "! 1 @ A For the wave propagating in vacuum (" = 1, ¹ = 1, σ = 0), Eq. 1.17 reduces to a simple plane wave solution, while if the wave is propagating in a medium of ¯nite electrical conductivity, the amplitude of the wave exponentially decays over a characteristic distance ± given by c c ± = = (1.18) !N~2(!) !k~(!) where ± is called the optical skin depth, and k~ is the imaginary part of the complex index of refraction (also called the extinction coe±cient) 4¼iσ N~(!) = p¹"complex = "¹ 1 + = n~(!) + ik~(!): (1.19) "! s µ ¶ This means that the intensity of the electric ¯eld, E 2, falls o® to 1=e of its value at the j j surface in a distance 1 c = (1.20) ®abs 2!k~(!) 3 where ®abs(!) is the absorption coe±cient for the solid at frequency !. Since light is described by a transverse wave, there are two possible orthogonal direc- tions for the E~ vector in a plane normal to the propagation direction and these directions determine the polarization of the light. For cubic materials, the index of refraction is the same along the two transverse directions. However, for anisotropic media, the indices of refraction may be di®erent for the two polarization directions, as is further discussed in 2.1. x 1.3 Relation of Complex Dielectric Function to Observables In relating "complex and σcomplex to the observables, it is convenient to introduce a complex index of refraction N~complex N~complex = p¹"complex (1.21) where ! K = N~ (1.22) c complex and where N~complex is usually written in terms of its real and imaginary parts (see Eq.