13 Optical Properties of Materials 13.1.Interaction of Light with Matter The most apparent properties of metals,their luster and their color,have been known to mankind since materials were known. Because of these properties,metals were already used in antiq- uity for mirrors and jewelry.The color was utilized 4000 years ago by the ancient Chinese as a guide to determine the compo- sition of the melt of copper alloys:the hue of a preliminary cast indicated whether the melt,from which bells or mirrors were to be made,already had the right tin content. The German poet Goethe was probably the first one who ex- plicitly spelled out 200 years ago in his Treatise on Color that color is not an absolute property of matter(such as the resistiv- ity),but requires a living being for its perception and descrip- tion.Goethe realized that the perceived color of a region in the visual field depends not only on the properties of light coming from that region,but also on the light coming from the rest of the visual field.Applying Goethe's findings,it was possible to ex- plain qualitatively the color of,say,gold in simple terms.Goethe wrote:"If the color blue is removed from the spectrum,then blue, violet,and green are missing and red and yellow remain."Thin gold films are bluish-green when viewed in transmission.These colors are missing in reflection.Consequently,gold appears reddish-yellow.On the other hand,Newton stated quite correctly in his "Opticks"that light rays are not colored.The nature of color remained,however,unclear. This chapter treats the optical properties from a completely different point of view.Measurable quantities such as the index of refraction or the reflectivity and their spectral variations are
13 The most apparent properties of metals, their luster and their color, have been known to mankind since materials were known. Because of these properties, metals were already used in antiquity for mirrors and jewelry. The color was utilized 4000 years ago by the ancient Chinese as a guide to determine the composition of the melt of copper alloys: the hue of a preliminary cast indicated whether the melt, from which bells or mirrors were to be made, already had the right tin content. The German poet Goethe was probably the first one who explicitly spelled out 200 years ago in his Treatise on Color that color is not an absolute property of matter (such as the resistivity), but requires a living being for its perception and description. Goethe realized that the perceived color of a region in the visual field depends not only on the properties of light coming from that region, but also on the light coming from the rest of the visual field. Applying Goethe’s findings, it was possible to explain qualitatively the color of, say, gold in simple terms. Goethe wrote: “If the color blue is removed from the spectrum, then blue, violet, and green are missing and red and yellow remain.” Thin gold films are bluish-green when viewed in transmission. These colors are missing in reflection. Consequently, gold appears reddish-yellow. On the other hand, Newton stated quite correctly in his “Opticks” that light rays are not colored. The nature of color remained, however, unclear. This chapter treats the optical properties from a completely different point of view. Measurable quantities such as the index of refraction or the reflectivity and their spectral variations are Optical Properties of Materials 13.1 • Interaction of Light with Matter
246 13.Optical Properties of Materials used to characterize materials.In doing so,the term "color"will almost completely disappear from our vocabulary.Instead,it will be postulated that the interactions of light with the electrons of a material are responsible for the optical properties. At the beginning of the 20th century,the study of the interac- tions of light with matter (black-body radiation,etc.)laid the foundations for quantum theory.Today,optical methods are among the most important tools for elucidating the electron structure of matter.Most recently,a number of optical devices such as lasers,photodetectors,waveguides,etc.,have gained con- siderable technological importance.They are used in telecom- munication,fiber optics,CD players,laser printers,medical di- agnostics,night viewing,solar applications,optical computing, and for optoelectronic purposes.Traditional utilizations of opti- cal materials for windows,antireflection coatings,lenses,mir- rors,etc.,should be likewise mentioned. We perceive light intuitively as a wave(specifically,an elec- tromagnetic wave)that travels in undulations from a given source to a point of observation.The color of the light is related to its wavelength.Many crucial experiments,such as diffraction,in- terference,and dispersion,clearly confirm the wavelike nature of light.Nevertheless,at least since the discovery of the photo- electric effect in 1887 by Hertz,and its interpretation in 1905 by Einstein,do we know that light also has a particle nature.(The photoelectric effect describes the emission of electrons from a metallic surface after it has been illuminated by light of appro- priately high energy,e.g.,by blue light.)Interestingly enough, Newton,about 300 years ago,was a strong proponent of the par- ticle concept of light.His original ideas,however,were in need of some refinement,which was eventually provided in 1901 by quantum theory.We know today (based on Planck's famous hy- pothesis)that a certain minimal energy of light,that is,at least one light quantum,called a photon,with the energy: E=vh oh (13.1) needs to impinge on a metal in order that a negatively charged electron may overcome its binding energy to its positively charged nucleus,and can escape into free space.(This is true regardless of the intensity of the light.)In Eq.(13.1),h is the Planck con- stant whose numerical value is given in Appendix II and v is the frequency of light given as the number of vibrations(cycles)per second or hertz (Hz).Frequently,the reduced Planck constant: 方 2T (13.2)
used to characterize materials. In doing so, the term “color” will almost completely disappear from our vocabulary. Instead, it will be postulated that the interactions of light with the electrons of a material are responsible for the optical properties. At the beginning of the 20th century, the study of the interactions of light with matter (black-body radiation, etc.) laid the foundations for quantum theory. Today, optical methods are among the most important tools for elucidating the electron structure of matter. Most recently, a number of optical devices such as lasers, photodetectors, waveguides, etc., have gained considerable technological importance. They are used in telecommunication, fiber optics, CD players, laser printers, medical diagnostics, night viewing, solar applications, optical computing, and for optoelectronic purposes. Traditional utilizations of optical materials for windows, antireflection coatings, lenses, mirrors, etc., should be likewise mentioned. We perceive light intuitively as a wave (specifically, an electromagnetic wave) that travels in undulations from a given source to a point of observation. The color of the light is related to its wavelength. Many crucial experiments, such as diffraction, interference, and dispersion, clearly confirm the wavelike nature of light. Nevertheless, at least since the discovery of the photoelectric effect in 1887 by Hertz, and its interpretation in 1905 by Einstein, do we know that light also has a particle nature. (The photoelectric effect describes the emission of electrons from a metallic surface after it has been illuminated by light of appropriately high energy, e.g., by blue light.) Interestingly enough, Newton, about 300 years ago, was a strong proponent of the particle concept of light. His original ideas, however, were in need of some refinement, which was eventually provided in 1901 by quantum theory. We know today (based on Planck’s famous hypothesis) that a certain minimal energy of light, that is, at least one light quantum, called a photon, with the energy: E h &' (13.1) needs to impinge on a metal in order that a negatively charged electron may overcome its binding energy to its positively charged nucleus, and can escape into free space. (This is true regardless of the intensity of the light.) In Eq. (13.1), h is the Planck constant whose numerical value is given in Appendix II and is the frequency of light given as the number of vibrations (cycles) per second or hertz (Hz). Frequently, the reduced Planck constant: ' 2 h (13.2) 246 13 • Optical Properties of Materials
13.2.The Optical Constants 247 Wavelength Energy Frequency (m) (ev) (H2) 10-14 108 1022 400nm 10-2 106 violet -----Y-Rays 1020 blue X-Rays--1-- 10-10 104 - nm 1018 500nm UVI 10-8 102 1016 green ----7 10-6 m yellow 100 104 600nm orange Infrared (heat) -不--1-- 10-4 10-2 1012 mm red/ Microwaves 10-2 10-4 1010 700nm 100 -m 106 GHz 108 Visible Radio, spectrum TV 102 10-8 106 -MHz km 104 10-0 104 106 10-2 kHz 102 FIGURE 13.1.The spec- is utilized in conjunction with the angular frequency,@=2mv. trum of electromag- In short,the wave-particle duality of light (or,more generally,of netic radiation.Note electromagnetic radiation)had been firmly established at about the small segment of 1924.The speed of light,c,and the frequency are connected by this spectrum that is the equation: visible to human eyes. C=入， (13.3) where A is the wavelength of the light. Light comprises only an extremely small segment of the entire electromagnetic spectrum,which ranges from radio waves via microwaves,infrared,visible,ultraviolet,X-rays,to y rays,as de- picted in Figure 13.1.Many of the considerations which will be advanced in this chapter are therefore also valid for other wave- length ranges,i.e.,for radio waves or X-rays. 13.2.The Optical Constants When light passes from an optically "thin"medium (e.g.,vac- uum,air)into an optically dense medium one observes that in the dense medium,the angle of refraction B(i.e.,the angle be- tween the refracted light beam and a line perpendicular to the surface)is smaller than the angle of incidence,a.This well-known
is utilized in conjunction with the angular frequency, & 2. In short, the wave-particle duality of light (or, more generally, of electromagnetic radiation) had been firmly established at about 1924. The speed of light, c, and the frequency are connected by the equation: c , (13.3) where is the wavelength of the light. Light comprises only an extremely small segment of the entire electromagnetic spectrum, which ranges from radio waves via microwaves, infrared, visible, ultraviolet, X-rays, to rays, as depicted in Figure 13.1. Many of the considerations which will be advanced in this chapter are therefore also valid for other wavelength ranges, i.e., for radio waves or X-rays. When light passes from an optically “thin” medium (e.g., vacuum, air) into an optically dense medium one observes that in the dense medium, the angle of refraction (i.e., the angle between the refracted light beam and a line perpendicular to the surface) is smaller than the angle of incidence, . This well-known FIGURE 13.1. The spectrum of electromagnetic radiation. Note the small segment of this spectrum that is visible to human eyes. 13.2 • The Optical Constants 247 10–14 10–12 10–10 10–8 10–6 10–4 10–2 100 102 104 106 108 106 104 102 100 10–2 10–4 10–6 10–8 10–10 10–12 1022 1020 1018 1016 1014 1012 1010 108 106 104 102 kHz MHz GHz nm mm m km m 400 nm 500 nm 600 nm 700 nm violet blue green yellow orange red Visible spectrum UV Wavelength (m) Energy (eV) Frequency (Hz) -Rays Microwaves Infrared (heat) Radio, TV X-Rays 13.2 • The Optical Constants
248 13.Optical Properties of Materials phenomenon is used for the definition of the refractive power of a material and is called the Snell law: sin a=nmed=n. (13.4) sin B nvac Commonly,the index of refraction for vacuum nvac is arbitrar- ily set to be unity.The refraction is caused by the different ve- locities,c,of the light in the two media: sin a Cvac (13.5) sin B Cmed Thus,if light passes from vacuum into a medium,we find: n=Cvac=c (13.6) Cmed v' where v cmed is the velocity of light in the material.The mag- nitude of the refractive index depends on the wavelength of the incident light.This property is called dispersion.In metals,the index of refraction varies also with the angle of incidence.This is particularly true when n is small. The index of refraction is generally a complex number,desig- nated as n,which is comprised of a real and an imaginary part ni and n2,respectively,i.e., n=ni-in2. (13.7) In the literature,the imaginary part of ni is often denoted by k. Equation (13.7)is then written as: n=n-ik. (13.8) We will call n2 or k the damping constant.(In some books,n2 and k are named absorption constant,attenuation index,or ex- tinction coefficient.We will not follow this practice because of its potential to be misleading.)The square of the (complex)index of refraction is equal to the (complex)dielectric constant(Sec- tion11.8): 2=e=e1-ie2, (13.9) which yields,with Eq.(13.8), n2=n2-k2-2nki=e1-i e2 (13.10) Equating individually the real and imaginary parts in Eq.(13.10) yields: e1=n2-k2 (13.11)
phenomenon is used for the definition of the refractive power of a material and is called the Snell law: s s i i n n n n m va e c d n. (13.4) Commonly, the index of refraction for vacuum nvac is arbitrarily set to be unity. The refraction is caused by the different velocities, c, of the light in the two media: s s i i n n c c m va e c d . (13.5) Thus, if light passes from vacuum into a medium, we find: n c c m va e c d v c , (13.6) where v cmed is the velocity of light in the material. The magnitude of the refractive index depends on the wavelength of the incident light. This property is called dispersion. In metals, the index of refraction varies also with the angle of incidence. This is particularly true when n is small. The index of refraction is generally a complex number, designated as nˆ, which is comprised of a real and an imaginary part n1 and n2, respectively, i.e., nˆ n1 i n2. (13.7) In the literature, the imaginary part of nˆ is often denoted by k. Equation (13.7) is then written as: nˆ n i k. (13.8) We will call n2 or k the damping constant. (In some books, n2 and k are named absorption constant, attenuation index, or extinction coefficient. We will not follow this practice because of its potential to be misleading.) The square of the (complex) index of refraction is equal to the (complex) dielectric constant (Section 11.8): nˆ 2 ˆ 1 i 2, (13.9) which yields, with Eq. (13.8), nˆ 2 n2 k2 2nki 1 i 2. (13.10) Equating individually the real and imaginary parts in Eq. (13.10) yields: 1 n2 k2 (13.11) 248 13 • Optical Properties of Materials
13.2.The Optical Constants 249 and E2=2nk. (13.12) e is called polarization whereas e2 is known by the name ab- sorption.Values for n and k for some materials are given in Table 13.1.For insulators,k is nearly zero,which yields for dielectrics e1≈n2ande2→0. When electromagnetic radiation (e.g.,light)passes from vac- uum (or air)into an optically denser material,then the ampli- tude of the wave decreases exponentially with increasing damp- ing constant k and for increasing distance,z,from the surface, as shown in Figure 13.2.Specifically,the intensity,I,of the light (that is,the square of the electric field strength,)obeys the fol- lowing equation(which can be derived from the Maxwell equa- tions): 1=82=106 (13.13) TABLE 13.1.Optical constants for some materials (A 600 nm) 么 k R%ob Metals Copper 0.14 3.35 95.6 Silver 0.05 4.09 98.9 Gold 0.21 3.24 92.9 Aluminum 0.97 6.0 90.3 Ceramics Silica glass (Vycor) 1.46 3.50 Soda-lime glass 1.51 4.13 Dense flint glass 1.75 a 7.44 Quartz 1.55 4.65 Al203 1.76 7.58 Polymers Polyethylene 1.51 4.13 Polystyrene 1.60 5.32 Polytetrafluoroethylene 1.35 2.22 Semiconductors Silicon 3.94 0.025 35.42 GaAs 3.91 0.228 35.26 "The damping constant for dielectrics is about 10-7;see Table 13.2. bThe reflection is considered to have occurred on one reflecting surface only. See also Table 15.1
and 2 2nk. (13.12) 1 is called polarization whereas 2 is known by the name absorption. Values for n and k for some materials are given in Table 13.1. For insulators, k is nearly zero, which yields for dielectrics 1 n2 and 2 0. When electromagnetic radiation (e.g., light) passes from vacuum (or air) into an optically denser material, then the amplitude of the wave decreases exponentially with increasing damping constant k and for increasing distance, z, from the surface, as shown in Figure 13.2. Specifically, the intensity, I, of the light (that is, the square of the electric field strength, ) obeys the following equation (which can be derived from the Maxwell equations): I 2 I0 exp 4 c k z . (13.13) 13.2 • The Optical Constants 249 TABLE 13.1. Optical constants for some materials ( 600 nm) n kR %b Metals Copper 0.14 3.35 95.6 Silver 0.05 4.09 98.9 Gold 0.21 3.24 92.9 Aluminum 0.97 6.0 90.3 Ceramicsc Silica glass (Vycor) 1.46 a 3.50 Soda-lime glass 1.51 a 4.13 Dense flint glass 1.75 a 7.44 Quartz 1.55 a 4.65 Al2O3 1.76 a 7.58 Polymers Polyethylene 1.51 a 4.13 Polystyrene 1.60 a 5.32 Polytetrafluoroethylene 1.35 a 2.22 Semiconductors Silicon 3.94 0.025 35.42 GaAs 3.91 0.228 35.26 aThe damping constant for dielectrics is about 107; see Table 13.2. bThe reflection is considered to have occurred on one reflecting surface only. cSee also Table 15.1.
250 13.Optical Properties of Materials X ←-Vacuum Material ep(e】 FIGURE 13.2.Exponential decrease of the amplitude of electromagnetic radiation in optically dense materials such as metals. We define a characteristic penetration depth,W,as that dis- tance at which the intensity of the light wave,which travels through a material,has decreased to 1/e or 37%of its original value,i.e.,when: L=1=e1 (13.14) Io e This definition yields,in conjunction with Eq.(13.13), &=w=k (13.15) Table 13.2 presents experimental values for k and W for some materials obtained by using sodium vapor light (A=589.3 nm). The inverse of W is sometimes called the (exponential)atten- uation or the absorbance,a,which is,by making use of Eq. (13.15)and(13.12), a=Amk=Amk =2me2 (13.16) A c An It is measured,for example,in cm-1.The energy loss per unit length(given,for example,in decibels,dB,per centimeter)is ob- tained by multiplying the absorbance,a,with 4.34,see Problem 13.7.(1dB=101ogI1o). TABLE 13.2.Characteristic penetration depth,W,and damping con- stant,k,for some materials (A=589.3 nm) Material Water Flint glass Graphite Gold W(cm) 32 29 6×10-6 1.5×10-6 k 1.4×10-7 1.5×10-7 0.8 3.2
We define a characteristic penetration depth, W, as that distance at which the intensity of the light wave, which travels through a material, has decreased to 1/e or 37% of its original value, i.e., when: I I 0 1 e e1. (13.14) This definition yields, in conjunction with Eq. (13.13), z W 4 c k 4 k . (13.15) Table 13.2 presents experimental values for k and W for some materials obtained by using sodium vapor light ( 589.3 nm). The inverse of W is sometimes called the (exponential) attenuation or the absorbance, , which is, by making use of Eq. (13.15) and (13.12), 4 k 4 c k 2 n 2 . (13.16) It is measured, for example, in cm1. The energy loss per unit length (given, for example, in decibels, dB, per centimeter) is obtained by multiplying the absorbance, , with 4.34, see Problem 13.7. (1 dB 10 log I/I0). FIGURE 13.2. Exponential decrease of the amplitude of electromagnetic radiation in optically dense materials such as metals. 250 13 • Optical Properties of Materials x Vacuum Material exp – &kz c z TABLE 13.2. Characteristic penetration depth, W, and damping constant, k, for some materials ( 589.3 nm) Material Water Flint glass Graphite Gold W (cm) 32 29 6 106 1.5 106 k 1.4 107 1.5 107 0.8 3.2
13.2.The Optical Constants 。 251 The ratio between the reflected intensity IR and the incom- ing intensity lo of the light is the reflectivity: R=会 (13.17) Quite similarly,one defines the ratio between the transmitted in- tensity,Ir,and the impinging light intensity as the transmissiv- ity: (13.18) The reflectivity is connected with n and k(assuming normal in- cidence)through: R=n-12+k2 (n+1)2+k2 (13.19) (Beer equation).The reflectivity is a unitless material constant and is often given in percent of the incoming light (see Table 13.1).R is,like the index of refraction,a function of the wave- length of the light.For insulators (k=0)one finds that R de- pends solely on the index of refraction: R=-1)2 (13.20) (n+1)2 Metals are characterized by a large reflectivity.This stems from the fact that light penetrates metals only a short distance,as shown in Figure 13.2 and Table 13.2.Thus,only a small part of the impinging energy is converted into heat.The major part of the energy is reflected (in some cases as much as 99%,see Table 13.1).In contrast to this,visible light penetrates into glass (and many other dielectrics)much farther than into metals,that is, approximately seven orders of magnitude more;see Table 13.2. As a consequence,very little light is reflected by glass.Never- theless,a piece of glass about 1 or 2 m thick eventually dissipates a substantial part of the impinging light into heat.(In practical applications,one does not observe this large reduction in light intensity because windows are,as a rule,only a few millimeters thick.)It should be noted that window panes,lenses,etc.,reflect the light on the front as well as on the back side. An energy conservation law requires that the intensity of the light impinging on a material,Io,must be equal to the reflected inten- sity,IR,plus the transmitted intensity,I7,plus that intensity which has been extinct,IE,for example,transferred into heat,that is, Io IR IT IE. (13.21)
The ratio between the reflected intensity IR and the incoming intensity I0 of the light is the reflectivity: R I I R 0 . (13.17) Quite similarly, one defines the ratio between the transmitted intensity, IT, and the impinging light intensity as the transmissivity: T I I T 0 . (13.18) The reflectivity is connected with n and k (assuming normal incidence) through: R (13.19) (Beer equation). The reflectivity is a unitless material constant and is often given in percent of the incoming light (see Table 13.1). R is, like the index of refraction, a function of the wavelength of the light. For insulators (k 0) one finds that R depends solely on the index of refraction: R . (13.20) Metals are characterized by a large reflectivity. This stems from the fact that light penetrates metals only a short distance, as shown in Figure 13.2 and Table 13.2. Thus, only a small part of the impinging energy is converted into heat. The major part of the energy is reflected (in some cases as much as 99%, see Table 13.1). In contrast to this, visible light penetrates into glass (and many other dielectrics) much farther than into metals, that is, approximately seven orders of magnitude more; see Table 13.2. As a consequence, very little light is reflected by glass. Nevertheless, a piece of glass about 1 or 2 m thick eventually dissipates a substantial part of the impinging light into heat. (In practical applications, one does not observe this large reduction in light intensity because windows are, as a rule, only a few millimeters thick.) It should be noted that window panes, lenses, etc., reflect the light on the front as well as on the back side. An energy conservation law requires that the intensity of the light impinging on a material, I0, must be equal to the reflected intensity, IR, plus the transmitted intensity, IT, plus that intensity which has been extinct, IE, for example, transferred into heat, that is, I0 IR IT IE. (13.21) (n 1) 2 (n 1)2 (n 1)2 k 2 (n 1)2 k2 13.2 • The Optical Constants 251
252 13.Optical Properties of Materials Dividing Eq.(13.21)by Io and making use of Eq.(13.17)and (13.18)yields: R+T+E=1. (13.22) (It has been assumed for these considerations that the light which has been scattered inside the material may be transmitted through the sides and is therefore contained in Ir and IE.) The reflection losses encountered in optical instruments such as lenses can be significantly reduced by coating the surfaces with a thin layer of a dielectric material such as magnesium flu- oride.This results in the well-known blue hue on lenses for cam- eras. Metals are generally opaque in the visible spectral region be- cause of their comparatively high damping constant and thus high reflectivity.Still,very thin metal films (up to about 50 nm thickness)may allow some light to be transmitted.Dielectric ma- terials,on the other hand,are often transparent.Occasionally, however,some opacifiers are inherently or artificially added to dielectrics which cause the light to be internally deflected by mul- tiple scattering.Finally,if the diffuse scattering is not very se- vere,dielectrics might appear translucent,that is,objects viewed through them are vaguely seen,but not clearly distinguishable. Scattering of light may occur,for example,due to residual poros- ity in ceramic materials,or on grain boundaries(which have a small variation in refractive index compared to the matrix),or on finely dispersed particles,or on boundaries between crys- talline and amorphous regions in polymers,to mention only a few mechanisms. There exists an important equation which relates the reflec- tivity of light at low frequencies (infrared spectral region)with the direct-current conductivity,o: R=1-4,T0- (13.23) This relation,which was experimentally found at the end of the 19th century by Hagen and Rubens,states that materials having a large electrical conductivity (such as metals)also possess es- sentially a large reflectivity (and vice versa). 13.3.Absorption of Light If light impinges on a material,it is either re-emitted in one form or another (reflection,transmission)or its energy is extinct,for example,transformed into heat.In any of these cases,some in-
Dividing Eq. (13.21) by I0 and making use of Eq. (13.17) and (13.18) yields: R T E 1. (13.22) (It has been assumed for these considerations that the light which has been scattered inside the material may be transmitted through the sides and is therefore contained in IT and IE.) The reflection losses encountered in optical instruments such as lenses can be significantly reduced by coating the surfaces with a thin layer of a dielectric material such as magnesium fluoride. This results in the well-known blue hue on lenses for cameras. Metals are generally opaque in the visible spectral region because of their comparatively high damping constant and thus high reflectivity. Still, very thin metal films (up to about 50 nm thickness) may allow some light to be transmitted. Dielectric materials, on the other hand, are often transparent. Occasionally, however, some opacifiers are inherently or artificially added to dielectrics which cause the light to be internally deflected by multiple scattering. Finally, if the diffuse scattering is not very severe, dielectrics might appear translucent, that is, objects viewed through them are vaguely seen, but not clearly distinguishable. Scattering of light may occur, for example, due to residual porosity in ceramic materials, or on grain boundaries (which have a small variation in refractive index compared to the matrix), or on finely dispersed particles, or on boundaries between crystalline and amorphous regions in polymers, to mention only a few mechanisms. There exists an important equation which relates the reflectivity of light at low frequencies (infrared spectral region) with the direct-current conductivity, : R 1 4 0 . (13.23) This relation, which was experimentally found at the end of the 19th century by Hagen and Rubens, states that materials having a large electrical conductivity (such as metals) also possess essentially a large reflectivity (and vice versa). If light impinges on a material, it is either re-emitted in one form or another (reflection, transmission) or its energy is extinct, for example, transformed into heat. In any of these cases, some in- 252 13 • Optical Properties of Materials 13.3 • Absorption of Light
13.3.Absorption of Light 253 teraction between light and matter will take place,as was ex- plained in the preceding section.One of the major mechanisms by which this interaction occurs is called absorption of light. The classical description of absorption and reemission of light was developed at the turn of the 20th century by P.Drude,a Ger- man physicist.His concepts were described in Chapter 11.1 when we discussed electrical conduction in metals.As explained there, Drude postulated that some electrons in a metal (essentially the valence electrons)can be considered to be free,that is,they can be separated from their respective nuclei.He further assumed that the free electrons within the crystal can be accelerated by an external electric field.This preliminary Drude model was re- fined by considering that the moving electrons on their path col- lide with certain metal atoms in a nonideal lattice.If an alter- nating electric field (as through interaction with light)is envolved,then the free electrons are thought to perform oscil- lating motions.These vibrations are restrained by the above-men- tioned interactions of the electrons with the atoms of a nonideal lattice.Thus,a friction force is introduced which takes this in- teraction into consideration.The calculation of the frequency de- pendence of the optical constants is accomplished by using the classical equations for vibrations whereby the interactions of electrons with atoms are taken into account by a damping term which is assumed to be proportional to the velocity of the elec- trons.The Newtonian-type equation (Force mass times accel- eration)is essentially identical to that of Eq.(11.9)except that the direct-current excitation force e is now replaced by a peri- odic (i.e.,sinusoidal)excitation force: F=e 80 sin(2Tvt), (13.24) where v is the frequency of the light,t is the time,and o is the maximal field strength of the light wave.In short,the equation describing the motion of free electrons which are excited to per- form forced,periodic vibrations under the influence of light can be written as: m- dv+yv=e &o sin (2mvt) (13.25) t where y is the damping strength which takes the damping of the electron motion into account. The solution of this equation,which shall not be attempted here,yields the frequency dependence (or dispersion)of the op- tical constants. The free electron theory describes,to a certain degree,the dis- persion of the optical constants quite well.This is schematically
teraction between light and matter will take place, as was explained in the preceding section. One of the major mechanisms by which this interaction occurs is called absorption of light. The classical description of absorption and reemission of light was developed at the turn of the 20th century by P. Drude, a German physicist. His concepts were described in Chapter 11.1 when we discussed electrical conduction in metals. As explained there, Drude postulated that some electrons in a metal (essentially the valence electrons) can be considered to be free, that is, they can be separated from their respective nuclei. He further assumed that the free electrons within the crystal can be accelerated by an external electric field. This preliminary Drude model was refined by considering that the moving electrons on their path collide with certain metal atoms in a nonideal lattice. If an alternating electric field (as through interaction with light) is envolved, then the free electrons are thought to perform oscillating motions. These vibrations are restrained by the above-mentioned interactions of the electrons with the atoms of a nonideal lattice. Thus, a friction force is introduced which takes this interaction into consideration. The calculation of the frequency dependence of the optical constants is accomplished by using the classical equations for vibrations whereby the interactions of electrons with atoms are taken into account by a damping term which is assumed to be proportional to the velocity of the electrons. The Newtonian-type equation (Force mass times acceleration) is essentially identical to that of Eq. (11.9) except that the direct-current excitation force e is now replaced by a periodic (i.e., sinusoidal) excitation force: F e 0 sin (2t), (13.24) where is the frequency of the light, t is the time, and 0 is the maximal field strength of the light wave. In short, the equation describing the motion of free electrons which are excited to perform forced, periodic vibrations under the influence of light can be written as: m d d v t v e 0 sin (2t), (13.25) where is the damping strength which takes the damping of the electron motion into account. The solution of this equation, which shall not be attempted here, yields the frequency dependence (or dispersion) of the optical constants. The free electron theory describes, to a certain degree, the dispersion of the optical constants quite well. This is schematically 13.3 • Absorption of Light 253
254 13.Optical Properties of Materials shown in Figure 13.3,in which the spectral dependence of the reflectivity is plotted for a specific case.The Hagen-Rubens re- lation (13.23)reproduces the experimental findings only up to aboutv=1013 s-1.In contrast to this,the Drude theory correctly reproduces the frequency dependence of R even in the visible spectrum.Proceeding to yet higher frequencies,however,the ex- perimentally found reflectivity eventually rises and then de- creases again.Such an absorption band cannot be explained by the free electron theory.For its interpretation,a different con- cept needs to be applied;see below.By making use of the Drude theory,one can obtain the number of free electrons per unit vol- ume,Nf,from optical measurements: N=1-n2+km4m20 e2 (13.26) where m is the mass of the electrons,e their charge,and eo is the permittivity of empty space;see Appendix II and Section 11.8. The number of free electrons is a parameter which is of great in- terest because it is contained in several nonoptical equations (Hall effect,electromigration,superconductivity,etc.). The Drude theory also provides the plasma frequency which is that frequency at which all electrons perform collective,fluid- like oscillations: e2Nf (13.27) 4r2e011m A selection of plasma frequencies is given in Table 13.3 and a value for v is shown in Figure 13.3. Hagen-Rubens FIGURE 13.3.Schematic frequency dependence of the reflectivity of metals,experimentally Experimental (solid line)and calcu- lated according to Drude I three models.The Lorentz spectral dependence of the reflectivity is often 1013 1014y1 %1015 Frequency (1/s) quite similar to that of Absorption band the absorption,e2.The plasma frequency is Classical IR absorption red violet marked by v. Visible spectrum
shown in Figure 13.3, in which the spectral dependence of the reflectivity is plotted for a specific case. The Hagen–Rubens relation (13.23) reproduces the experimental findings only up to about 1013 s1. In contrast to this, the Drude theory correctly reproduces the frequency dependence of R even in the visible spectrum. Proceeding to yet higher frequencies, however, the experimentally found reflectivity eventually rises and then decreases again. Such an absorption band cannot be explained by the free electron theory. For its interpretation, a different concept needs to be applied; see below. By making use of the Drude theory, one can obtain the number of free electrons per unit volume, Nf, from optical measurements: Nf (13.26) where m is the mass of the electrons, e their charge, and 0 is the permittivity of empty space; see Appendix II and Section 11.8. The number of free electrons is a parameter which is of great interest because it is contained in several nonoptical equations (Hall effect, electromigration, superconductivity, etc.). The Drude theory also provides the plasma frequency which is that frequency at which all electrons perform collective, fluidlike oscillations: 1 4 e 2 2 N 0m f . (13.27) A selection of plasma frequencies is given in Table 13.3 and a value for 1 is shown in Figure 13.3. (1 n2 k2) 2m 42 0 e2 FIGURE 13.3. Schematic frequency dependence of the reflectivity of metals, experimentally (solid line) and calculated according to three models. The spectral dependence of the reflectivity is often quite similar to that of the absorption, 2. The plasma frequency is marked by 1. 254 13 • Optical Properties of Materials 1013 1014 1015 0 Frequency (1/s) Reflectivity Hagen-Rubens Experimental Drude Lorentz Absorption band Classical IR absorption red violet Visible spectrum 1