CHAPTER 9 FUNDAMENTAL OPTICAL PROPERTIES OF SOLIDS Alan Miller Department of Physics and Astronomy Uni ersity of St. Andrews St. Andrews, Scotland and Center for Research and Education in Optics and Lasers ( CREOL ) Uni ersity of Central Florida Orlando, Florida 9. 1 GLOSSARY A vector field B magnetic induction c speed of light D displacement field d M penetration depth in a metal d T I R evanescent field depth for total internal reflection E electric field g band gap energy E l o c local electric field e electronic charge e i components of unit vectors of incident polarization e s components of unit vectors of scattered polarization f j oscillator strength G reciprocal lattice vector H hamiltonian H magnetic field I irradiance I unit tensor J c conduction current density dielectric constant or relative permittivity K 9.1
9.2 QUANTUM OPTICS k electron wave vector M induced magnetization m electron mass m * ef fective electron mass N charge density N L refractive index for left circularly polarized light N R refractive index for right circularly polarized light n real part of refractive index P Kane momentum matrix element P induced dipole moment per unit volume P D dielectric amplitude reflection polarization ratio P j j momentum matrix elements P M metallic amplitude reflection polarization ratio P p, ij elasto-optic coef ficients p electron momentum q complex photon wave vector qˆ unit photon wave vector R reflectance R e x exciton Rydberg R, p Raman scattering coef ficient r p field reflection amplitude for p-polarization r s field reflection amplitude for s-polarization S Poynting vector s i j components of lattice strain S R Raman scattering ef ficiency s j phonon oscillator strength T transmittance t p field transmission amplitude for p-polarization t s field transmission amplitude for s-polarization V volume of unit cell V ( r ) periodic lattice potential v phase velocity W ( ) transition rate Z number of electrons per atom absorption coef ficient two-photon absorption coef ficient damping constant D dielectric reflection polarization phase dif ference M metallic reflection polarization phase dif ference s specific rotary power
FUNDAMENTAL OPTICAL PROPERTIES OF SOLIDS 9.3 s o spin-orbit energy splitting energy walk-of f angle» electrical permittivity» complex permittivity» ( ) high frequency permittivity» (0) static permittivity» 1 real part of permittivity» 2 imaginary part of permittivity» 0 electrical permittivity of vacuum complex refractive index θ principal angle for metallic reflection θ O A angle between optic axis and principal axis θ B Brewster s angle θ c total internal reflection critical angle θ T I R phase change for total internal reflection imaginary part of refractive index magnetic permeability 0 magnetic permeability of vacuum Hooke s Law constant ˆ unit polarization vector f free charge density p band or polarization charge density p real field amplitude for p-polarization s real field amplitude for s-polarization t total charge density complex conductivity electrical conductivity 1 real part of conductivity 2 imaginary part of conductivity τ scattering or relaxation time scalar field i ground state wave function χ complex susceptibility χ real part of susceptibility χ imaginary part of susceptibility χ electrical susceptibility e χ m magnetic susceptibility k Bloch solution wave function photon frequency j resonant oscillation frequency L O longitudinal optical phonon frequency
9.4 QUANTUM OPTICS p plasma frequency s surface plasmon frequency T O transverse optical phonon frequency 9. 2 INTRODUCTION This chapter describes the fundamental interactions of light with solids. The discussion is restricted to intrinsic optical properties. Extrinsic optical properties of solids are described in Chap. 8 Optical Spectroscopy and Spectroscopic Lineshapes. Basic formulas, definitions, and concepts are listed for reference and as a foundation for subsequent chapters of this volume of the Handbook. More detailed accounts of specific optical properties and particular types of solid are given in later chapters, i. e., Chap. 42 Optical Properties of Films and Coatings, and in Vol. II, Chaps. 23 Crystals and Glasses, 35 Metals, 36 Semiconductors, and 13 Electro-Optic Modulators. The reader is referred to the many texts which provide more elaborate discussions of the optical properties of solids. 1 12 Electrical measurements distinguish three general types of solids by their conductivities, i. e., dielectrics, semiconductors, and metals. Optically, these three groups are characterized by dif ferent fundamental band gap energies, g. Although the boundaries are not sharply defined, an approximate distinction is given by, metals g 0, semiconductors, 0 g 3 ev, and dielectrics, g 3 ev. Solids may be found in single crystal, polycrystalline, and amorphous forms. Rudimentary theories of the optical properties of condensed matter are based on light interactions with perfect crystal lattices characterized by extended (nonlocal) electronic and vibrational energy states. These eigenstates are determined by the periodicity and symmetry of the lattice and the form of the Coulomb potential which arises from the interatomic bonding. The principal absorption bands in condensed matter occur at photon energies corresponding to the frequencies of the lattice vibrations (phonons) in the infrared, and electronic transitions in the near infrared, visible, or ultraviolet. A quantum mechanical approach is generally required to describe electronic interactions, but classical models often suf fice for lattice vibrations. Although the mechanical properties of solids can vary enormously between single crystal and polycrystalline forms, the fundamental optical properties are similar, even if the crystallite size is smaller than a wavelength, since the optical interaction is microscopic. However, electronic energy levels and hence optical properties are fundamentally altered when one or more dimensions of a solid are reduced to the scale of the debroglie wavelength of the electrons. Modern crystal growth techniques allow fabrication of atomic precision epitaxial layers of dif ferent solid materials. Ultrathin layers, with dimensions comparable with or smaller than the debroglie wavelength of an electron, may form quantum wells, quantum wires, and quantum dots in which electronic energy levels are quantized. Amorphous solids have random atomic or molecular orientation on the distance scale of several nearest neighbors, but generally have well-defined bonding and local atomic order which determine the overall optical response. 9. 3 PROPAGATION OF LIGHT IN SOLIDS Dielectrics and semiconductors provide transparent spectral regions at radiation frequencies between the phonon vibration bands and the fundamental (electronic) absorption edge. Maxwell s equations successfully describe the propagation, reflection, refraction, and scattering of harmonic electromagnetic waves.
FUNDAMENTAL OPTICAL PROPERTIES OF SOLIDS 9.5 Maxwell s Equations Four equations relate the macroscopically averaged electric field E and magnetic induction B to the total charge density, t (sum of the bound or polarization charge, p, and the free charge, f ), the conduction current density, J c, the induced dipole moment per unit volume, P, and the induced magnetization of the medium, M (expressed in SI units) : E t» 0 E B t (2) B 0 (3) B 0» 0 (1) E P t t J c M (4) where» 0 8. 854 10 1 2 F / m is the permittivity of vacuum, 0 4 π 10 7 H / m is the permeability of vacuum and c the speed of light in vacuum, c 1 / 4» 0 0. By defining a displacement field, D, and magnetic field, H, to account for the response of a medium D P» 0 E (5) H B 0 M (6) and using the relation between polarization and bound charge density, P p (7) Equations (1) and (4) may also be written in the form D f (8) and H D t J c (9) Vector A and scalar fields may be defined by B A (10) E A t (11) A convenient choice of gauge for the optical properties of solids is the Coulomb (or transverse) gauge A 0 (12) which ensures that the vector potential A is transverse for plane electromagnetic waves while the scalar potential represents any longitudinal current and satisfies Poisson s equation 2 t» 0 (13)
9.6 QUANTUM OPTICS Three constitutive relations describe the response of conduction and bound electrons to the electric and magnetic fields : J c E (14) D P» 0 E» E (15) B 0 ( H M ) H (16) Here is the electrical conductivity,» is the electrical permittivity, and is the magnetic permeability of the medium and are in general tensor quantities which may depend on field strengths. An alternative relation often used to define a dielectric constant (or relative permittivity), K, is given by D» 0 K E. In isotropic media using the approximation of linear responses to electric and magnetic fields,,», and are constant scalar quantities. Electronic, χ e, and magnetic, χ m, susceptibilities may be defined to relate the induced dipole moment, P, and magnetism, M to the field strengths, E and H : Thus, and P» 0 χ e E (17) M» 0 χ m H (18)»» 0 ( I χ e ) (19) 0 ( I χ m ) (20) where I is the unit tensor. Wave Equations and Optical Constants The general wave equation derived from Maxwell s equations is 2 E 2 E ( E )» 0 0 t 2 0 2 P t J c 2 t M (21) t For dielectric (nonconducting) solids 2 E» 2 E t 2 (22) The harmonic plane wave solution of the wave equation for monochromatic light at frequency,, E 1 2 E 0 exp i ( q r t ) c. c. (23) in homogeneous (» 0), isotropic ( E 0, q E 0), nonmagnetic ( M 0) solids results in a complex wave vector, q, q c» i» 0» 0 A complex refractive index,, may be defined by (24) q qˆ (25) c
FUNDAMENTAL OPTICAL PROPERTIES OF SOLIDS 9.7 where qˆ is a unit vector and n i (26) Introducing complex notation for the permittivity,»» 1 i» 2, conductivity, 1 i 2, and susceptibility, χ e χ e i χ e, we may relate and Alternatively,» 1»» 0 (1 χ e )» 0 ( n 2 2 ) 2» 2» 0 χ e 2» 0 n 1 n 1 2» 0 1 2» 0 1/2 ( 4» 2 1» 2 2» 1 ) 1/2 ( 4» 2 1» 2 2» 1 ) (27) (28) (29) (30) The field will be modified locally by induced dipoles. If there is no free charge, f, the local field, E l o c, may be related to the external field, E i, in isotropic solids using the Clausius-Mossotti equation which leads to the relation E l o c n 2 2 E i (31) 3 Energy Flow The direction and rate of flow of electromagnetic energy is described by the Poynting vector S 1 0 E H (32) The average power per unit area (irradiance, I ), W / m 2, carried by a uniform plane wave is given by the time averaged magnitude of the Poynting vector I S cn E 0 2 2 (33) The plane wave field in Eq. (23) may be rewritten for absorbing media using Eqs. (25) and (26). E ( r, t ) 1 2 E 0 ( q, ) exp c qˆ r exp i n qˆ r t c. c. (34) c The decay of the propagating wave is characterized by the extinction coef ficient,. The attenuation of the wave may also be described by Beer s law I I 0 exp ( z ) (35)
9.8 QUANTUM OPTICS where is the absorption coef ficient describing the attenuation of the irradiance, I, with distance, z. Thus, 2 4 π c 1» 0 cn» 2» 0 cn χ e cn The power absorbed per unit volume is given by (36) P a b s I χ e 2 E 0 2 (37) The second exponential in Eq. (34) is oscillatory and represents the phase velocity of the wave, c / n. Anisotropic Crystals Only amorphous solids and crystals possessing cubic symmetry are optically isotropic. In general, the speed of propagation of an electromagnetic wave in a crystal depends both on the direction of propagation and on the polarization of the light wave. The linear electronic susceptibility and dielectric constant may be represented by tensors with components of χ e given by P i» 0 χ i j E j (38) where i and j refer to coordinate axes. In an anisotropic crystal, D B q and E H S, but E is not necessarily parallel to D and the direction of energy flow S is not necessarily in the same direction as the propagation direction q. From energy arguments it can be shown that the susceptibility tensor is symmetric and it therefore follows that there always exists a set of coordinate axes which diagonalize the tensor. This coordinate system defines the principal axes. The number of nonzero elements for the susceptibility (or dielectric constant) is thus reduced to a maximum of three (for any crystal system at a given wavelength). Thus, the dielectric tensor defined by the direction of the electric field vector with respect to the principal axes has the form The principal indices of refraction are» 1 0 0 0» 2 0 0 0» 3 n i 4 1 χ i j 4» i (39) with the E -vector polarized along any principal axis, i. e. E D. This case is designated as an ordinary or o -ray in which the phase velocity is independent of propagation direction. An extraordinary or e -ray occurs when both E and q lie in a plane containing two principal axes with dif ferent n. An optic axis is defined by any propagation direction in which the phase velocity of the wave is independent of polarization. Crystalline solids fall into three classes : (1) optically isotropic, (2) uniaxial, or (3) biaxial (see Table 1). All choices of orthogonal axes are principal axes and» 1» 2» 3 in isotropic solids. For a uniaxial crystal,» 1» 2» 3, a single optic axis exists for propagation in direction 3. In this case, the ordinary refractive index, n 0 n 1 n 2, is independent of the direction of polarization in the 1-2 plane. Any two orthogonal directions in this plane can be chosen as principal axes. For any other propagation direction, the polarization can be divided into an o -ray component in the 1-2 plane and a perpendicular e -ray component
TABLE 1 Crystalographic Point Groups and Optical Properties FUNDAMENTAL OPTICAL PROPERTIES OF SOLIDS 9.9
9.10 QUANTUM OPTICS FIGURE 1 Illustration of directional dependence of refractive indices and optic axes in : ( a ) a uniaxial, positive birefringent crystal ; ( b ) a uniaxial, negative birefringent crystal ; and ( c ) a biaxial crystal. (see Fig. 1). The dependence of the e -ray refractive index with propagation direction is given by the ellipsoid n i n j n j ( θ i ) (40) ( n 2 i cos 2 θ i n 2 j sin 2 θ i ) 1 / 2 where θ i is defined with respect to optic axis, i 3, and j 1 or 2. θ i 90 gives the refractive index n e n 3 when the light is polarized along axis 3. The dif ference between n 0 and n e is the birefringence. Figure 1 a illustrates the case of positive birefringence, n e n 0, and Fig. 1 b negative birefringence, n e n 0. The energy walk-of f angle, (the angle between S and q or D and E ), is given by sin 2 θ (41) tan n 2 ( θ ) 1 1 2 n 2 3 n 2 1 In biaxial crystals, diagonalization of the dielectric tensor results in three independent coef ficients,» 1» 2» 3» 1. For orthorhombic crystals, a single set of orthogonal principal axes is fixed for all wavelengths. However, in monoclinic structures only one principal axis is fixed. The direction of the other two axes rotate in the plane
FUNDAMENTAL OPTICAL PROPERTIES OF SOLIDS 9.11 perpendicular to the fixed axis as the wavelength changes (retaining orthogonality). In triclinic crystals there are no fixed axes and orientation of the set of three principal axes varies with wavelength. Equation (40) provides the e -ray refractive index within planes containing two principal axes. Biaxial crystals possess two optic axes. Defining principal axes such that n 1 n 2 n 3, then both optic axes lie in the 1-3 plane, at an angle θ O A from axis 1, as illustrated in Fig. 1 c, where sin θ O A Ú n 1 n 2 n 2 2 n 2 3 n 2 1 n 2 3 Crystals with certain point group symmetries (see Table 1) also exhibit optical activity, i. e., the ability to rotate the plane of linearly polarized light. An origin for this phenomenon is the weak magnetic interaction M [see Eq. (21)], when it applies in a direction perpendicular to P (i. e., M P ). The specific rotary power s (angle of rotation of linearly polarized light per unit length) is given by (42) s π ( n L n R ) (43) where n L and n R are refractive indices for left and right circular polarization. Optical activity is often masked by birefringence, however, polarization rotation can be observed in optically active materials when the propagation is along the optic axis or when the birefringence is coincidentally zero in other directions. In the case of propagation along the optic axis of an optically active uniaxial crystal such as quartz, the susceptibility tensor may be written χ 11 i χ 12 0 i χ 1 2 χ 1 1 0 0 0 χ 33 and the rotary power is proportional to the imaginary part of the magnetic susceptibility, χ m χ 1 2 : s π χ 1 2 n (44) Crystals can exist in left- or right-handed versions. Other crystal symmetries, e. g., 4 2m, can be optically active for propagation along the 2 and 3 axes, but rotation of the polarization is normally masked by the typically larger birefringence, except at accidental degeneracies. Interfaces Applying boundary conditions at a plane interface between two media with dif ferent indices of refraction leads to the laws of reflection and refraction. Snell s law applies to all o -rays and relates the angle of incidence, θ A in medium A, and the angle of refraction, θ B in medium B, to the respective ordinary refractive indices n A and n B : n A sin θ A n B sin θ B (45) Extraordinary rays do not satisfy Snell s law. The propagation direction for the e -ray can be found graphically by equating the projections of the propagation vectors in the two media along the boundary plane. Double refraction of unpolarized light occurs in anisotropic crystals. The field amplitude ratios of reflected and transmitted rays to the incident ray (r and t) in isotropic solids (and o -rays in anisotropic crystals) are given by the Fresnel relations.
9.12 QUANTUM OPTICS FIGURE 2 The electric field reflection amplitudes ( a, b ), energy reflectance ( c, d ), and phase change ( e, f ) for s- (solid lines) and p- (dashed lines) polarized light for external ( a, c, e ) and internal ( b, d, f ) reflection in the case n A 1, n B 1. 5. θ B is the polarizing or Brewster s angle and θ c is the critical angle for total internal reflection. 2 For s- ( or TE) polarization ( E -vector perpendicular to the plane of incidence) (see Fig. 2 a ) and p- ( π or TM) polarization ( E -vector parallel to the plane of incidence) (see Fig. 2 b ) : r s E r s n A cos θ A 4 n 2 B n 2 A sin 2 θ A E i s n A cos θ A 4 n 2 B n 2 A sin 2 θ A r p E r p n 2 B cos θ A n A 4 n 2 B n 2 A sin 2 θ A E i p n 2 B cos θ A n A 4 n 2 B n 2 A sin 2 θ A t s E t s 2 n A cos θ A E i s n S cos θ A 4 n 2 B n 2 A sin 2 θ A t p E t p 2 n A n B cos θ A E i p n 2 B cos θ A n A 4 n 2 B n 2 A sin 2 θ A (46) (47) (48) (49)
FUNDAMENTAL OPTICAL PROPERTIES OF SOLIDS 9.13 At normal incidence, the energy reflectance, R (see Figs 2 c and d ), and transmittance, T, are R E 2 r n 2 B n A E i n B n A T E 2 t E i (50) 4 n 2 A (51) n A n B 2 The p -polarized reflectivity (see Eq. (47)] goes to zero at Brewster s angle under the condition θ B tan n 1 A (52) n B If n A n B, total internal reflection (TIR) occurs when the angle of incidence exceeds a critical angle : θ c sin n 1 A (53) n B This critical angle may be dif ferent for s- and p-polarizations in anisotropic crystals. Under conditions of TIR, the evanescent wave amplitude drops to e 1 in a distance d T I R c ( n 2 A sin 2 θ A n 2 B ) 1/2 (54) The phase changes on reflection for external and internal reflection from an interface with n A 1. 5, n B 1 are plotted in Figs. 2 e and f. Except under TIR conditions, the phase change is either 0 or π. The complex values predicted by Eqs. (46) and (47) for angles of incidence greater than the critical angle for TIR imply phase changes in the reflected light which are neither 0 nor π. The phase of the reflected light changes by π at Brewster s angle in p-polarization. The ratio of s to p reflectance, P D, is shown in Fig. 3 a and the phase dif ference, D p s, in Fig. 3 b. Under conditions of TIR, the phase change on reflection, T I R, is given by tan T I R 2 4 sin2 θ A sin 2 θ c cos θ A (55) 9. 4 DISPERSION RELATIONS For most purposes, a classical approach is found to provide a suf ficient description of dispersion of the refractive index within the transmission window of insulators, and for optical interactions with lattice vibrations and free electrons. However the details of interband transitions in semiconductors and insulators and the ef fect of d-levels in transition metals require a quantum model of dispersion close to these resonances.
9.14 QUANTUM OPTICS FIGURE 3 Typical polarization ratios, P, ( a, c ) and phase dif ferences,, ( b, d ) for s- and p-polarizations at dielectric, D, and metallic, M, surfaces. 1 Classical Model The Lorentz model for dispersion of the optical constants of solids assumes an optical interaction via the polarization produced by a set of damped harmonic oscillators. The polarization P induced by a displacement r of bound electrons of density N and charge e is P Ne r (56) Assuming the electrons to be elastically bound (Hooke s law) with a force constant,, the dif ferential equation of motion is e E l o c r (57) m d 2 r d r 2 m dt dt r e E l o c (58) where m is the electron mass and is a damping constant. Here the lattice is assumed to have infinite mass and the magnetic interaction has been neglected. Solving the equation of motion for fields of frequency gives a relation for the complex refractive index and dielectric constant : 2»» 1 Ne2 0 m» 0 j f j ( 2 j 2 i j ) (59)
FUNDAMENTAL OPTICAL PROPERTIES OF SOLIDS 9.15 We have given the more general result for a number of resonant frequencies j j m (60) where the f j represent the fraction of electrons which contribute to each oscillator with force constant j. f j represent oscillator strengths. A useful semi-empirical relation for refractive index in the transparency region of a crystal known as the Sellmeier formula follows directly from Eq. (59) under the assumption that, far from resonances, the damping constant terms j are negligible compared to ( 2 j 2 ) : n 2 1 j A j 2 2 2 j (61) Sum Rules The definition of oscillator strength results in the sum rule for electronic interactions j f j Z (62) where Z is the number of electrons per atom. The periodicity of the lattice in solids (see Sec. 9. 7) leads to the modification of this sum rule to m f m n m 2 2 n k 2 1 m m * n 1 (63) where m n * is an ef fective mass (see Energy Band Structures in Sec. 9. 7). Another sum rule for solids equivalent to Eq. (62) relates the imaginary part of the permittivity or dielectric constant and the plasma frequency, p,» 2 ( ) d 1 2 π 2 p (64) 0 where 2 p Ne 2 /» 0 m (see Drude Model in Sec. 9. 6). Dispersion relations are integral formulas which relate refractive properties to the absorptive process. Kramers-Kronig relations are commonly used dispersion integrals based on the condition of causality which may be related to sum rules. These relations can be expressed in alternative forms. For instance, the reflectivity of a solid is often measured at normal incidence and dispersion relations used to determine the optical properties. Writing the complex reflectivity amplitude as r ( ) r r ( ) e i θ ( ) (65) the phase shift, θ, can be determined by integrating the experimental measurement of the real amplitude, r r, θ ( ) 2 ln r r ( ) π 2 2 d (66) 0
9.16 QUANTUM OPTICS and the optical constants determined from the complex Fresnel relation, ( n 1 i ) r r ( ) e i θ ( n 1 i ) (67) Sum rules following from the Kramers-Kronig relations relate the refractive index n ( ) at a given frequency,, to the absorption coef ficient, ( ), integrated over all frequencies,, according to n ( ) 1 c ( ) d (68) 2 2 Similarly, the real and imaginary parts of the dielectric constant,» and», may be related via the integral relations,» 1 ( ) 1 2 π 0» 2 ( ) 2 π 0» 2 ( ) 2 2 d (69) 0» 1 ( ) 1 2 2 d (70) 9. 5 LATTICE INTERACTIONS The adiabatic approximation is the normal starting point for a consideration of the coupling of light with lattice vibrations, i. e., it is assumed that the response of the outer shell electrons of the atoms to an electric field is much faster than the response of the core together with its inner electron shells. Further, the harmonic approximation assumes, that for small displacements, the restoring force on the ions will be proportional to the displacement. The solution of the equations of motion for the atoms within a solid under these conditions give normal modes of vibration, whose frequency eigenvalues and displacement eigenvectors depend on the crystal symmetry, atomic separation, and the detailed form of the interatomic forces. The frequency of lattice vibrations in solids are typically in the 100 to 1000 cm 1 range (wavelengths between 10 and 100 m). Longitudinal and doubly degenerate transverse vibrational modes have dif ferent natural frequencies due to long range Coulomb interactions. Infrared or Raman activity can be determined for a given crystal symmetry by representing the modes of vibration as irreducible representations of the space group of the crystal lattice. Infrared Dipole Active Modes If the displacement of atoms in a normal mode of vibration produces an oscillating dipole moment, then the motion is dipole active. Thus, harmonic vibrations in ionic crystals contribute directly to the dielectric function, whereas higher order contributions are needed in nonpolar crystals. Since photons have small wavevectors compared to the size of the Brillouin zone in solids, only zone center lattice vibrations (i. e., long wavelength phonons) can couple to the radiation. This conservation of wavevector (or momentum) also implies that only optical phonons interact. In a dipole-active, long wavelength optical
FUNDAMENTAL OPTICAL PROPERTIES OF SOLIDS 9.17 mode, oppositely charged ions within each primitive cell undergo oppositely directed displacements giving rise to a nonvanishing polarization. Group theory shows that, within the harmonic approximation, the infrared active modes have irreducible representations with the same transformation properties as x, y, or z. The strength of the light-dipole coupling will depend on the degree of charge redistribution between the ions, i. e., the relative ionicity of the solid. Classical dispersion theory leads to a phenomenological model for the optical interaction with dipole active lattice modes. Because of the transverse nature of electromagnetic radiation, the electric field vector couples with the transverse optical (TO) phonons and the maximum absorption therefore occurs at this resonance. The resonance frequency, T O, is inserted into the solution of the equation of motion, Eq. (59). Since electronic transitions typically occur at frequencies 10 2 to 10 3 higher than the frequency of vibration of the ions, the atomic polarizability can be represented by a single high frequency permittivity,» ( ). The dispersion relation for a crystal with several zone center TO phonons may be written S j» ( )» ( ) (71) j ( 2 TOj 2 i j ) By defining a low frequency permittivity,» (0), the oscillator strength for a crystal possessing two atoms with opposite charge, Ze, per unit cell of volume, V, is S (» ( ) /» 0 2) 2 ( Ze ) 2 2 TO (» (0)» ( )) (72) 9 m r V where Ze represents the ef fective charge of the ions, m r is the reduced mass of the ions, and the local field has been included based on Eq. (31). Figure 4 shows the form of the (a) (b) (c) FIGURE 4 ( a ) Reflectance ; and ( b ) real and imaginary parts of the permittivity of a solid with a single infrared active mode. ( c ) polariton dispersion curves (real and imaginary parts) showing the frequencies of the longitudinal and transverse optical modes.
9.18 QUANTUM OPTICS real and imaginary parts of the dielectric constant, the reflectivity and the polariton dispersion curve. Observing that the real part of the dielectric constant is zero at longitudinal phonon frequencies, L O, the Lyddane-Sachs-Teller relation may be derived, which in its general form for a number of dipole active phonons, is given by» (0)» ( ) j 2 L j T j These relations [Eqs. (71) to (73)] give good fits to measured reflectivities in a wide range of ionically (or partially ionically) bonded solids. The LO-TO splitting and ef fective charge, Ze, depend on the ionicity of the solid ; however the magnitude of Ze determined from experiments does not necessarily quantify the ionicity since this rigid ion model does not account for the change of polarizability due to the distortion of the electron shells during the vibration. In uniaxial and biaxial crystals, the restoring forces on the ions are anisotropic, resulting in dif ferent natural frequencies depending on the direction of light propagation as well as the transverse or longitudinal nature of the vibration. Similar to the propagation of light, ordinary and extraordinary transverse phonons may be defined with respect to the principal axes. For instance, in a uniaxial crystal under the condition that the anisotropy in phonon frequency is smaller than the LO-TO frequency splitting, infrared radiation of frequency propagating at an angle θ to the optic axis will couple to TO phonons according to the relation (73) 2 T 2 T sin 2 θ 2 T cos 2 θ (74) where T is a TO phonon propagating with atomic displacements parallel to the optic axis, and T is a TO phonon propagating with atomic displacements perpendicular to the optic axis. The corresponding expression for LO modes is 2 L 2 L cos 2 θ 2 L sin 2 θ (75) In Table 2, the irreducible representations of the infrared normal modes for the dif ferent crystal symmetries are labeled x, y, or z. Brillouin and Raman Scattering Inelastic scattering of radiation by acoustic phonons is known as Brillouin scattering, while the term Raman scattering is normally reserved for inelastic scattering from optic phonons in solids. In the case of Brillouin scattering, long wavelength acoustic modes produce strain and thereby modulate the dielectric constant of the medium, thus producing a frequency shift in scattered light. In a Raman active mode, an incident electric field produces a dipole by polarizing the electron cloud of each atom. If this induced dipole is modulated by a lattice vibrational mode, coupling occurs between the light and the phonon and inelastic scattering results. Each Raman or Brillouin scattering event involves the destruction of an incident photon of frequency, i, the creation of a scattered photon, s, and the creation or destruction of a phonon of frequency, p. The frequency shift, i Ú s p, is typically 100 to 1000 cm 1 for Raman scattering, but only a few wavenumbers for Brillouin scattering.
FUNDAMENTAL OPTICAL PROPERTIES OF SOLIDS 9.19 Atomic polarizability components have exactly the same transformation properties as the quadratic functions x 2, xy,..., z 2. The Raman activity of the modes of vibration of a crystal with a given point group symmetry can thus be deduced from the Raman tensors given in Table 2. The scattering ef ficiency, S R, for a mode corresponding to one of the irreducible representations listed is given by 2 S R A e i R, p e s, (76) where A is a constant of proportionality, R, is the Raman coef ficient of the representation, and e i and e s are components of the unit vectors of polarization of the incident, i, and scattered, s, radiation along the principal axes, where and x, y, and z. Not all optic modes of zero wavevector are Raman active. Raman activity is often complementary to infrared activity. For instance, since the optic mode in the diamond lattice has even parity, it is Raman active, but not infrared active, whereas the zone center mode in sodium chloride is infrared active, but not Raman active because the inversion center is on the atom site and so the phonon has odd parity. In piezoelectric crystals, which lack a center of inversion, some modes can be both Raman and infrared active. In this case the Raman scattering can be anomalous due to the long-range electrostatic forces associated with the polar lattice vibrations. The theory of Brillouin scattering is based on the elastic deformation produced in a crystal by a long wavelength acoustic phonon. The intensity of the scattering depends on the change in refractive index with the strain induced by the vibrational mode. A strain, s i j in the lattice produces a change in the component of permittivity,», given by»» p, ij» s i j (77), where p, ij is an elasto-optical coef ficient. 3 The velocity of the acoustic phonons and their anisotropy can be determined from Brillouin scattering measurements. 9. 6 FREE ELECTRON PROPERTIES Fundamental optical properties of metals and semiconductors with high densities of free carriers are well described using a classical model. Reflectivity is the primary property of interest because of the high absorption. Drude Model The Drude model for free electrons is a special condition of the classical Lorentz model (Sec. 9. 4) with the Hooke s law force constant, 0, so that the resonant frequency is zero. In this case» 1 n 2 2 1 2 p (78)» 0 2 τ 2
TABLE 2 Infrared and Raman-Active Vibrational Symmetries and Raman Tensors 1 3 9.20
9.21
9.22 QUANTUM OPTICS and» 2 2 n 2 p» 0 2 τ 2 where p is the plasma frequency, p Ne2» 0 m 0 c 2 τ 1 τ (79) and τ ( 1 / ) is the scattering or relaxation time for the electrons. In ideal metals ( 5 ), n. Figure 5 a shows the form of the dispersion in the real and imaginary parts (80) FIGURE 5 Dispersion of : ( a ) the real and imaginary parts of the dielectric constant ; ( b ) real and imaginary parts of the refractive index ; and ( c ) the reflectance according to the Drude model where p is the plasma frequency.
FUNDAMENTAL OPTICAL PROPERTIES OF SOLIDS 9.23 of the dielectric constant for free electrons, while the real and imaginary parts of the refractive index are illustrated in Fig. 5 b. The plasma frequency is defined by the point at which the real part changes sign. The reflectance is plotted in Fig. 5 c and shows a magnitude close to 100 percent below the plasma frequency, but falls rapidly to a small value above p. The plasma frequency determined solely by the free electron term is typically on the order of 10 ev in metals accounting for their high reflectivity in the visible. Interband Transitions in Metals Not all metals are highly reflective below the plasma frequency. The nobel metals possess optical properties which combine free electron intraband (Drude) and interband contributions. A typical metal has d -levels at energies a few electron volts below the electron Fermi level. Transitions can be optically induced from these d -states to empty states above the Fermi level. These transitions normally occur in the ultraviolet spectral region, but can have a significant influence on the optical properties of metals in the visible spectral region via the real part of the dielectric constant. Describing the interband ef fects,» b, within a classical (Lorentz) model, the combined ef fects on the dielectric constant may be added :» 1» b» f (81) The interband contribution to the real part of the dielectric constant is positive and shows a resonance near the transition frequency. On the other hand, the free electron contribution is negative below the plasma frequency. The interband contribution can cause a shift to shorter wavelengths of the zero crossover in» 1, thus causing a reduction of the reflectivity in the blue. For instance d -states in copper lie only 2 ev below the Fermi level, which results in copper s characteristic color. Reflectivity Absorption in metals is described by complex optical constants. The reflectivity is accompanied by a phase change and the penetration depth is d M c 2 4 π At normal incidence at an air-metal interface, the reflectance is given by R ( n 1)2 2 ( n 1) 2 1 2 2 1 2 2» 0» 2 (82) (83) By analogy with the law of refraction (Snell s Law) a complex refractive index can be defined by the refraction equation sin θ t 1 sin θ i (84) Since is complex, θ t is also complex and the phase change on reflection can take values other than 0 and π. For nonnormal incidence, it can be shown that the surfaces of constant amplitude inside the metal are parallel to the surface, while surfaces of constant phase are at an angle to the surface. The electromagnetic wave in a metal is thus inhomogeneous. The real and imaginary parts of the refractive index can be determined by measuring the amplitude and phase of the reflected light. Writing the s and p components of the complex reflected fields in the form E r p p e i p ; E r s s e i s (85)
9.24 QUANTUM OPTICS and defining the real amplitude ratio and phase dif ference as P M tan s ; M p s (86) p then the real and imaginary parts of the refractive index are given by n sin θ i tan θ i cos 2 sin 1 sin 2 cos θ i tan θ i cos 2 (87) M tan 2 sin M tan 2 (88) θ i is the principal angle which occurs at the maximum in P M at the condition M π / 2 (see Fig. 3 c and d ), which is equivalent to Brewster s angle at an interface between two nonabsorbing dielectrics (see Fig. 3 a and b ). Plasmons Plasmons are oscillations of fluctuations in charge density. The condition for these oscillations to occur is the same as the condition for the onset of electromagnetic propagation at the plasma frequency. Volume plasmons are not excited by light at normal incidence since they are purely longitudinal. Oscillations cannot be produced by transverse electromagnetic radiation with zero divergence. However, at the surface of a solid, an oscillation in surface charge density is possible. At an interface between a metal with permittivity,» m, and a dielectric with permittivity,» d, the condition» m» d such that (neglecting damping and assuming a free electron metal) a surface plasmon can be created with frequency s (89) (» d /» 0 1) 1 / 2 By altering the angle of incidence, the component of the electromagnetic wavevector can be made to match the surface plasmon mode. p 9. 7 BAND STRUCTURES AND INTERBAND TRANSITIONS Advances in semiconductors for electronic and optoelectronic applications have encouraged the development of highly sophisticated theories of interband absorption in semiconductors. In addition, the development of low-dimensional structures (quantum wells, quantum wires, and quantum dots) have provided the means of engineering the optical properties of solids. The approach here has been to outline the basic quantum mechanical development for interband transitions in solids. Quantum Mechanical Model The quantum theory of absorption considers the probability of an electron being excited from a lower energy level to a higher level. For instance, an isolated atom has a characteristic set of electron levels with associated wavefunctions and energy eigenvalues. The absorption spectrum of the atom thus consists of a series of lines whose frequencies are given by f i f i ( f i ) (90)
FUNDAMENTAL OPTICAL PROPERTIES OF SOLIDS 9.25 where f and i are a pair of energy eigenvalues. We also know that the spontaneous lifetime, τ, for transitions from any excited state to a lower state sets a natural linewidth of order / τ based on the uncertainty principle. The Schrö dinger equation for the ground state with wavefunction, i, in the unperturbed system, 0 i i i (91) is represented by the time-independent hamiltonian, 0. The optical interaction can be treated by first order perturbation theory. By introducing a perturbation term based on the classical oscillator e E r (92) this leads to a similar expression to the Lorentz model, Eq. (59), where 2» 1 Ne2 f f i» 0 m» 0 m ( 2 fi 2 i f i ) f j j 2 p j j 2 m j j (93) (94) and p j j are momentum matrix elements defined by p j j j p j * j ( i ) j d r (95) Perturbation theory to first order gives the probability per unit time that a perturbation of the form ( t ) p exp ( i t ) induces a transition from the initial to final state : W f i 2 π f p i 2 ( f i ) (96) This is known as Fermi s Golden Rule. Energy Band Structures If we imagine N similar atoms brought together to form a crystal, each degenerate energy level of the atoms will spread into a band of N levels. If N is large, these levels can be treated as a continuum of energy states. The wavefunctions and electron energies of these energy bands can be calculated by various approximate methods ranging from nearly free electron to tight-binding models. The choice of approach depends on the type of bonding between the atoms. Within the one electron and adiabatic assumptions, each electron moves in the periodic potential, V ( r ), of the lattice leading to the Schrö dinger equation for a single particle wavefunction p 2 2 m V ( r ) ( r ) ( r ) (97)
9.26 QUANTUM OPTICS where the momentum operator is given by p i. The simple free electron solution of the Schrö dinger equation (i. e., for V ( r ) 0) is a parabolic relationship between energy and wavevector. The solution including a periodic potential, V ( r ) has the form k ( r ) exp ( i k r ) u k ( r ) (98) where k is the electron wavevector and u k ( r ) has the periodicity of the crystal lattice. This is known as the Bloch solution. The allowed values of k are separated by 2 π / L where L is the length of the crystal. The wavevector is not uniquely defined by the wavefunction, but the energy eigenvalues are a periodic function of k. For an arbitrarily weak periodic potential 2 2 m k G 2 (99) where G is a reciprocal lattice vector (in one dimension G 2 π n / a where a is the lattice spacing and n is an θ integer). Thus we need only consider solutions which are restricted to a reduced zone, refered to as the first Brillouin zone, in reciprocal space (between k π / a and π / a in one dimension). Higher energy states are folded into the first zone consistent with Eq. (99) to form a series of energy bands. Figure 6 shows the first Brillouin zones for face centered cubic (fcc) crystal lattices and energy levels for a weak lattice potential. A finite periodic potential, V ( r ), alters the shape of the free electron bands. The FIGURE 6 Free electron band structure in the reduced Brillouin zone for face-centered-cubic lattices. The insert shows the first Brillouin zone with principal symmetry points labeled. This applies to crystals such as Al, Cu, Ag, Si, Ge, and GaAs. 5
FUNDAMENTAL OPTICAL PROPERTIES OF SOLIDS 9.27 curvature of the bands are described by m *, an ef fective mass, which is defined by the slope of the dispersion curve at a given k : 1 1 d m k * 2 k dk (100) At the zone center, k 0, this reduces to the parabolic relationship 2 k 2 2 m * 0 (101) Ef fective masses can be related to interband momentum matrix elements and energy gaps using perturbation theory. Substituting the Bloch function [see Eq. (98)] into the Schrö dinger equation [see Eq. (97)] and identifying each band by an index, j, gives 2 p 2 m m k p 2 k 2 2 m V ( r ) u j k ( r ) j k ( r ) u j k ( r ) (102) The k p term can be treated as a perturbation about a specific point in k -space. For any k, the set of all u j k ( r ) (corresponding to the N energy levels) forms a complete set, i. e., the wavefunction at any value of k can be expressed as a linear combination of all wavefunctions at another k. Second order perturbation theory then predicts an ef fective mass given by 1 m * 1 m 2 p j j 2 m j (103) j ( k ) j ( k ) In principle the summation in Eq. (103) is over all bands, however, this can usually be reduced to a few nearest bands because of the resonant denominator. For example, in diamond and zinc blende structured semiconductors, the Kane momentum matrix element, P, defined by P i m S C O p x X V O ) (104) successfully characterizes the band structure and optical properties close to zone center. Here, S C O is a spherically symmetric s-like atomic wavefunction representing the lowest zone center conduction band state and X V O is a p-like function with x symmetry from the upper valence bands. In this case, including only the three highest valence bands and the lowest conduction band in the summation of Eq. (91), the conduction band ef fective mass is given by 1 m * CO 1 m 2 3 2 2 P 2 P 2 (105) g g S O where g is the band gap energy and S O is the spin-orbit splitting. By inverting this expression, the momentum matrix element may be determined from measurements of ef fective mass and the band gaps. P is found to have similar magnitudes for a large number of semiconductors. Equation (105) illustrates the general rule that the ef fective mass of the conduction band is approximately proportional to the band gap energy.
9.28 QUANTUM OPTICS Direct Interband Absorption In the case of a solid, the first order perturbation of the single electron hamiltonian by electromagnetic radiation is more appropriately described by ( t ) e A p (106) mc rather than Eq. (92). A is the vector potential, A ( r, t ) A 0 ˆ exp [ i ( q r t )] c. c. (107) q is the wavevector, and ˆ is the unit polarization vector of the electric field. (Note that this perturbation is of a similar form to the k p perturbation described above. ) Using Fermi s Golden Rule, the transition probability per unit time between a pair of bands is given by W f i 2 π ea 2 0 f ˆ p i 2 ( f ( k ) i ( k ) ) (108) mc Conservation of momentum requires a change of electron momentum after the transition, however, the photon momentum is very small, so that vertical transitions in k -space can be assumed in most cases (the electric dipole approximation). The total transition rate per unit volume, W T ( ), is obtained by integrating over all possible vertical transitions in the first Brillouin zone taking account of all contributing bands : W T ( ) 2 π 2 π e 2 I ncm 2 2 f d k (2 π ) 3 ˆ p f i ( k ) 2 ( f ( k ) i ( k ) ) (109) Here the vector potential has been replaced with the irradiance, I, of the radiation through the relation A 0 2 π c n 2 I (110) Note that the momentum matrix element as defined in Eq. (95) determines the oscillator strength for the absorption. p f i can often be assumed slowly varying in k so that the zone center matrix element can be employed for interband transitions and the frequency dependence of the absorption coef ficient is dominated by the density of states. Joint Density of States The delta function in the integration of Eq. (109) represents energy conservation for the transitions between any two bands. If the momentum matrix element can be assumed slowly varying in k, then the integral can be rewritten in the form J f i ( ) 1 (2 π ) 3 d k ( f ( k ) i ( k ) ) 1 (2 π ) 3 d S k f i ( k ) (111) where d S is a surface element on the equal energy surface in k -space defined by f i ( k ) f ( k ) i ( k ). Written in this way, J ( ) is the joint density of states between the two bands (note the factor of two for spin is excluded in this definition). Points in k -space for which the condition k f i ( k ) 0 (112)
FUNDAMENTAL OPTICAL PROPERTIES OF SOLIDS 9.29 hold form critical points called van Hove singularities which lead to prominent features in the optical constants. In the neighborhood of a critical point at k c, a constant energy surface may be described by the Taylor series 3 f i ( k ) c ( k c ) k 2 (113) 1 where represents directional coordinates. Minimum, maximum, and saddle points arise depending on the relative signs of the coef ficients,. Table 3 gives the frequency dependence of the joint density of states in three dimensional (3-D), two-dimensional (2-D), TABLE 3 Density of States in 3, 2, 1, and 0 Dimensions
9.30 QUANTUM OPTICS FIGURE 7 Illustration of absorption edge of crystals with : ( a ) direct allowed transitions ; and ( b ) direct forbidden transitions, based on density of state (dashed lines) and excitonic enhanced absorption models (solid lines). one-dimensional (1-D), and zero-dimensional (0-D) solids. The absorption coef ficient,, defined by Beer s Law may now be related to the transition rate by 1 di ( ) I dz I W T (114) Thus, the minimum fundamental absorption edge of semiconductors and insulators (in the absence of excitonic ef fects) has the general form (see Fig. 7 a ) ( ) C 0 ( c ) 1/2 (115) Selection Rules and Forbidden Transitions Direct interband absorption is allowed when the integral in Eq. (95) is nonzero. This occurs when the wavefunctions of the optically coupled states have opposite parity for single photon transitions. Transitions may be forbidden for other wavefunction symmetries. Although the precise form of the wavefunction may not be known, the selection
FUNDAMENTAL OPTICAL PROPERTIES OF SOLIDS 9.31 rules can be determined by group theory from a knowledge of the space group of the crystal and symmetry of the energy band. Commonly, a single photon transition, which is not allowed at the zone center because two bands have like parity, will be allowed at finite k because wavefunction mixing will give mixed parity states. In this case, the momentum matrix element may have the form p f i ( k ) ( k k 0 ) k [p f i ( k )] k k 0 (116) i. e., the matrix element is proportional to k. For interband transitions at an M 0 critical point, the frequency dependence of the absorption coef ficient can be shown to be (see Fig. 7 b ) ( ) C 0 ( c ) 3 / 2 (117) Indirect Transitions Interband transitions may also take place with the assistance of a phonon. The typical situation is a semiconductor or insulator which has a lowest conduction band minimum near a Brillouin zone boundary. The phonon provides the required momentum to move the electron to this location, but supplies little energy. The phonon may be treated as an additional perturbation and therefore second order perturbation theory is needed in the analysis of this two step process. Theory predicts a frequency dependence for the absorption of the form ( ) ( Ú p h c ) 2 (118) where p h is the phonon frequency, absorption or emission being possible. For forbidden indirect transitions, this relationship becomes ( ) ( Ú p h c ) 3 (119) Multiphoton Absorption Multiphoton absorption can be treated by higher order perturbation theory. For instance, second order perturbation theory gives a transition rate between two bands W (2) fi 2 π ea 4 0 mc t 2 f ˆ p t t ˆ p i ( f ( k ) i ( k ) 2 ) (120) t i where the summation spans all intermediate states, t. The interaction can be regarded as two successive steps. An electron first makes a transition from the initial state to an intermediate level of the system, t, by absorption of one photon. Energy is not conserved at this stage (momentum is) so that the absorption of a second photon must take the electron to its final state in a time determined by the energy mismatch and the uncertainty principle. In multiphoton absorption, one of the transitions may be an intraband self-transition. Since the probability depends on the arrival rate of the second photon, multiphoton absorption is intensity dependent. The total transition rate is given by W (2) T ( ) 2 π 4 π 2 e 4 I 2 n 2 c 2 m 4 4 f d k (2 π ) 3 t ˆ p f t ( k ) ˆ p t i ( k ) 2 ( f ( k ) i ( k ) 2 ) (121) t i