For each of s, p, d, and f subshells, there exist, respectively, one, three, five, and ... The extent of splitting depends on interatomic separation and begins with ... (Fig.5-a) one outermost band is only partially filled with electrons. The energy.
Diamond Optical Properties and Colorations M1 – TPA Project 2014-2015
By : Ali Fakih Supervised by : Professor Andreas Honecker
Diamond Optical Properties and Colorations I - Introduction In the framework of studying crystalline solids, some are interested in studying the material properties of the solid; such as solid structure, hardness, and toughness... Others are interested in the elastic properties, electric properties, thermal properties, or optical properties. In the following report I discuss the optical properties of diamond. I present the reasons why some diamonds are colorless and transparent and why some others are colored. 1. WHY Study the Optical Properties of Materials? When materials are exposed to electromagnetic radiation, it is sometimes important to be able to predict and alter their responses. This is possible when we are familiar with their optical properties and understand the mechanisms responsible for their optical behaviors. By “optical property” is meant a material’s response to exposure to electromagnetic radiation, and in particular, visible light.
Before talking about diamond, it is important to know very well the Energy Band-Gap theory and EM interaction with solids.
Fig.1, dispersion of light by a prism.
2.0 Energy Band-Gap Theory (Electronic Approach) For each individual atom there exist discrete energy levels that may be occupied by electrons, arranged into shells and subshells. Shells are designated by integers (1, 2, 3, etc.), and subshells by letters (s, p, d, and f ). For each of s, p, d, and f subshells, there exist, respectively, one, three, five, and seven states. The electrons in most atoms Fig.2, Energy shells and subshells. fill only the states having the lowest energies, two electrons of opposite spin per state, in accordance with the Pauli Exclusion Principle.
A solid may be thought of as consisting of a large number, say N, of atoms initially separated from one another, which are subsequently brought together and bonded to form the ordered atomic arrangement found in the crystalline material. At relatively large separation distances, each atom is independent of all the others and will have the atomic energy levels and electron configuration as if isolated. However, as the atoms come within close proximity of one another, electrons are acted upon, or perturbed, by the electrons and nuclei of adjacent atoms. This influence is such that each distinct atomic state may split into a series of closely spaced electron states in the solid, to form what is termed an electron energy band. The extent of splitting depends on interatomic separation and begins with the outermost electron shells, since they are the first to be perturbed as the atoms coalesce. Furthermore, gaps may exist between adjacent bands, normally; energies lying within these band gaps are not available for electron occupancy. The conventional way of representing electron band structures in solids is shown in Figure 2-b. We also note that the energies, in one energy band, are not continuous, but quantized. However, the numerous numbers of levels and small differences in the energies make it appear so. A clear picture is viewed in fig.4.
Fig.3, (a) The conventional representation of the electron energy band structure for a solid material at the equilibrium interatomic separation. (b) Electron energy versus interatomic separation for an aggregate of atoms, illustrating how the energy band structure at the equilibrium separation in (a) is generated.
Fig.4, The band–gap pattern of energy levels for an idealized crystalline solid. As the magnified view suggests, each band consists of a very large number of very closely spaced energy levels. (In many solids, adjacent bands may overlap; for clarity, we have not shown this condition.)
The number of states within each band will equal the total of all states contributed by the N atoms. For example, an s band will consist of N states, and a p band of 3N states. With regard to occupancy, each energy state may accommodate two electrons, which must have oppositely directed spins. Note that bands of lower energy are narrower than those of higher energy. This occurs because electrons that occupy the lower energy bands spend most of their time deep within the atom’s electron cloud. The wave functions of these core electrons do not overlap as much as the wave functions of the outer electrons do. Hence the splitting of the lower energy levels (core electrons) is less than that of the higher energy levels (outer electrons). 2.1
Types of Energy Band-Gaps
Four different types of band structures are possible at 0 K. In the first figure (Fig.5-a) one outermost band is only partially filled with electrons. The energy corresponding to the highest filled state at 0 K is called the Fermi energy as indicated. This energy band structure is typified by some metals, in particular those that have a single s valence electron (e.g., copper). Each copper atom has one 4s electron; however, for a solid comprised of N atoms, the 4s band is capable of accommodating 2N electrons. Thus only half the available electron positions within this 4s band are filled. For the second band structure, also found in metals (Fig.5-b), there is an overlap of an empty band and a filled band. Magnesium has this band structure. Each isolated Mg atom has two 3s electrons. However, when a solid is formed, the 3s and 3p bands overlap. In this instance and at 0 K, the Fermi energy is taken as that energy below which, for N atoms, N states are filled, two electrons per state. The final two band structures are similar; one band (the valence band) that is completely filled with electrons is separated from an empty conduction band, and an energy band gap lies between them. For very pure materials, electrons may not have energies within this gap. The difference between the two band structures lies in the magnitude of the energy gap; for materials that are insulators, the band gap is relatively wide (Fig.5-c), whereas for semiconductors it is narrow (Fig.5-d).
Fig.5, The various possible electron band structures in solids at 0 K. (a) The electron band structure found in metals such as copper, in which there are available electron states above and adjacent to filled states, in the same band. (b) The electron band structure of metals such as magnesium, wherein there is an overlap of filled and empty outer bands. (c) The electron band structure characteristic of insulators; the filled valence band is separated from the empty conduction band by a relatively large band gap. (d) The electron band structure found in the semiconductors, which is the same as for insulators except that the band gap is relatively narrow.
As we shall see below, a diamond structures including impurities serves as a well semiconductor. In this case it would be beneficial to explain the band-gap of semiconductors. 2.2.1 Semiconductors The electronic properties of semiconductors are completely determined by the comparatively small numbers of electrons excited into the conduction band and holes left behind in the valence band. The electrons will be found near the conduction band minima, while the holes will be confined to the neighborhood of the valence band maxima. Therefore the energy vs. wave vector relations for the carriers can generally be approximated by the quadratic forms they assume in the neighborhood of such extrema.
Where is the energy at the bottom of the conduction band, is the energy at is the “effective mass tensor”. the top of the valence band and
1 % %
1 %&! % "
where the sign is - or + according to whether k us near a band maximum (holes) or minimum (electrons).The mass tensor plays an important role in determining the dynamics of holes located about anisotropic maxima(or electrons located about anisotropic minim). Since is real and symmetric, one can find a set of orthogonal principal axes for each such point, in terms of which energies have the diagonal forms.
The effective masses discussed above can be measured by the technique of cyclotron resonance. (This technique is discussed in Ashcroft Solid State Physics. see ch.28, p 570) 2.2.2 Number of carriers in thermal Equilibrium The most important property of any semiconductor at temperature T is the number of electrons per unit volume in the conduction band, nc, and the number of holes per unit volume in the valence band, pv. The determination of these as a function of temperature is done using Fermi-Dirac statistics. The values of nc(T) and pv(T) depend critically, on the presence of impurities although there are certain general relations that hold regardless of the purity of the sample and we'll consider those relations first. Suppose the density of levels is gc(ε) in the conduction band and gv(ε) in the valence band. The effect of impurities as we shall see below is to introduce
additional levels at energies between the top of the valence band, εv, and the bottom of the conduction band ,εc, without, however, appreciably alternating the form of gc(ε) and gv(ε). Since conduction is entirely due to electrons on conduction band levels or holes in the valence band levels, regardless of concentration of impurities the number of carriers present at temperature T will be given by : ) : number of electrons per unit volume in the conduction band, and * ) ∶ number of holes per unit volume in the valence band.
, -. 2 .
, -. 2 . 71
, -. 2 . 7
Impurities affect the determination of nc and pv only through the value of the chemical potential μ . To determine μ, one must know something about the impurity levels. However, one can extract some useful information from the above two equations which is independent of the precise value of the chemical potential, provided only that it satisfies the conditions: .
9 ≫ ;)
. ≫ ;)
We will assume that the above is valid, which makes the semiconductor nondegenerate, and substitute it in ) < - * ) . 0
Thereby we find :
) * )
A ) B )
0 45 0 45 01
. > .
. < .
/45 06 /45
A ) = C0 -. 2 . B )
06 -. 2 /
0 01 /45
06 0 /45
Because the ranges of integration in the previous equations include points where the arguments of the exponentials vanish, Nc(T) and Pv(T) are relatively slowly varying functions of temperature, compared with the exponential factors they multiply in. This is their most important feature. The level densities can then be taken as stated in chapter 28 of “Ashcroft/Mermin Solid State Physics”: (/ ', E2F. . , F ( 2, . G The integrals then give A ) B )
1 2' ;) I J 4 G
1 2' ;) I J 4 G
We still cannot infer nc(T) and pv(T) until we know the value of the chemical potential μ. However, the μ dependence disappears from the product of the two densities: * KA B
01 06 /45
This result means (sometimes called the law of mass action) that at a given temperature it suffices to know the density of one carrier type to determine that of the other. How this determination is made depends on how important the impurities are as a source carriers. Intrinsic case of semiconductor crystals: If the crystal is so pure that the impurities contribute negligibly to the carrier densities, one speaks of "intrinsic semiconductor". In this case, conduction band electrons can only have come from formerly occupied valence band levels, leaving holes behind. The number of conduction band electrons is therefore equal to the number of valence band holes. ) * ) ≡ ! )
Since nc = pv , then we can write ni = (ncpv)1/2 So
( 1 2;) I J ' ' O LM / 45 4 G We may now establish in the intrinsic case the condition for the validity of the assumption on which our analysis has been based. Let μi be the value of chemical potential in the intrinsic case, by setting ni = (ncpv)1/2 we find μi :
1 P 2 Q
1 ;) 2
1 P 2 Q
3 ;) 4
This asserts that as T→ 0, the chemical potential lies precisely in the middle of the energy gap. At temperatures KT small compared with Eg, the chemical potential will be found far from boundaries of the forbidden region, εc and εv, and the condition for nondegeneracy will be satisfied.
Fig.6, In an intrinsic semiconductor with an energy gap Eg large compared with KT, the chemical potential μ lies within order KT of the center of the energy gap, and is therefore far compared with KT from both boundaries of the gap at εc and εv.
Extrinsic case : If impurities contribute a significant fraction of the conduction band electrons and/or valence band holes, one speaks of extrinsic semiconductor". Because these added sources of carriers the density of conduction band electrons need no longer to be equal to the density of valence band holes: )
Since the law of mass action holds regardless of impurities, we can use the definition of ni(T) to write quite generally : * ! we can express the carrier densities in the extrinsic case in terms of their intrinsic values ni and the deviation ∆ from intrinsic behavior:
which measures the importance of the impurtites as a source of
carriers can be written in a simpler expression as a function of chemical potential: We have: ] ^ ! And
2sinh c 9
We have noted that if the energy gap Eg is large compared with KT, then the intrinsic chemical potential μi will satisfy the assumption of nondegeneracy. Note also that when ∆ is large compared to ni, then the density of one carrier type is essentially equal to ∆ , while that of the other type is smaller by a factor \ of order ( ] )2. Thus when impurities do provide the major source of carriers, one ∆\
of the two carrier types will be dominant. An extrinsic semiconductor is called "n-type" or "p-type" depending whether the dominant carriers are electrons or holes.
Now after discussing the Band-Gap theory, and going through details in that of semiconductors, we proceed and see how light interacts with solids in order to be able to have better understanding of the optical properties of diamond.
3. Light Interaction with Solids When light proceeds from one medium into another (e.g., from air into a solid substance), some of the light radiation may be transmitted through the medium, some will be absorbed, and some will be reflected at the interface between the two media. The intensity I0 of the beam incident to the surface of the solid medium must equal the sum of the intensities of the transmitted, absorbed, and reflected beams, denoted as IT, IA and IR and respectively.
4. Transparency, Translucency, and Opaqueness
Materials that are capable of transmitting light with relatively little absorption and reflection are transparent—one can see through them. Translucent materials are those through which light is transmitted diffusely; that is, light is scattered within the interior, to the degree that objects are not clearly distinguishable when viewed through a specimen of the material. Materials that are impervious to the transmission of visible light are termed opaque.
5. Optical Properties of Non-Metals By virtue of their electron energy band structures, nonmetallic materials may be transparent to visible light. Therefore, in addition to reflection and absorption, refraction and transmission phenomena also need to be considered. 5.0 Index of Refraction ( n ) The index of refraction n of a material is defined as the ratio of the velocity in a vacuum c to the velocity v in the medium. X
The magnitude of n (or the degree of bending) will depend on the wavelength of the light. This effect is graphically demonstrated by the familiar dispersion or separation of a beam of white light into its component colors by a glass prism seen in fig.1 . The velocity of light v in a medium can be written as i
Where k and l are respectively the permittivity and permeability of the particular substance. Now we can write
√kl mkn ln
Since the substances of interest to us are only slightly magnetic, lo ≅ Then
X ≅ √ko
For more details on the index of refraction formula one can refer to the book “Optics” by Eugene Hecht, Chapter 3. Thus, for transparent materials, there is a relation between the index of refraction and the dielectric constant. Since the retardation of electromagnetic radiation in a medium results from electronic polarization, the size of the constituent atoms or ions has a considerable influence on the magnitude of this effect—generally, the larger an atom or ion, the greater will be the electronic polarization, the slower the velocity, and the greater the index of refraction. For crystalline ceramics that have cubic crystal structures, and for glasses, the index of refraction is independent of crystallographic direction (i.e., it is isotropic).
5.1.0 Refraction Light that is transmitted into the interior of transparent materials experiences a decrease in velocity, and, as a result, is bent at the interface; this phenomenon is termed refraction. The refracted light may be absorbed, transmitted of both.
5.1.1 Absorption Nonmetallic materials may be opaque or transparent to visible light; and, if transparent, they often appear colored. Absorption of a photon of light may occur by the promotion or excitation of an electron from the nearly filled valence band, across the band gap, and into an empty state within the conduction band, as demonstrated in Fig.7-a.
Fig.7.a, Absorption of a photon by valence electron. Fig.7.b, Emission of a photon by a direct electron transition across the band gap.
These excitations with the accompanying absorption can take place only if the photon energy is greater than that of the band gap Eg—that is, if
ql > rs The minimum wavelength for visible light (which ranges between 0.4μm and 0.7μm) , λ (min), is about 0.4 μm, and since c= 3x108m/s and h=4.13x10-15 eV-s, the maximum band gap energy Eg (max) for which absorption of visible light is possible is just rs tuv
qh w txX
So, no visible light is absorbed by nonmetallic materials having band gap energies greater than about 3.1 eV; these materials, if of high purity, will appear transparent and colorless. On the other hand, the maximum wavelength for visible light, λ (max), is about 0.7 μm; computation of the minimum band gap energy Eg (min) for which there is absorption of visible light is according to
qh w tuv
This result means that all visible light is absorbed by valence band-to-conduction band electron transitions for those semiconducting materials that have band gap energies less than about 1.8 eV; thus, these materials are opaque. Only a portion of the visible spectrum is absorbed by materials having band gap energies between 1.8 and 3.1 eV; consequently, these materials appear colored.
Every nonmetallic material becomes opaque at some wavelength, which depends on the magnitude of its Eg. For example, diamond, having a band gap of 5.5 eV, is opaque to radiation having wavelengths less than about 0.22 μm.
5.1.2 Transmission When light strikes the interface between two media of different indices of refraction, part of the light is refracted, this refracted light itself is either absorbed, transmitted, or both. The intensity of the transmitted light is dependent to the distance it travels within the medium it crosses. The intensity of transmittance has the following equation as given in chapter 21 of the book “Material Science and Engineering”. ~•
Where R is the reflectivity (it will be discussed below), β is the absorption coefficient which depends on the material, and ‚ is the thickness of the material the light travels. Note that for this expression it is assumed that the same medium exists outside both front and back faces, see fig.8.
Fig.8, The transmission of light through a transparent medium for which there is reflection at front and back faces, as well as absorption within the medium.
5.2 Reflection When light radiation passes from one medium into another having a different index of refraction, some of the light is scattered at the interface between the two media even if both are transparent. The reflectivity R represents the fraction of the incident light that is reflected at the interface. „…
where IR and I0 are the intensities of the incident and reflected beams, respectively. If the light is normal (or perpendicular) to the interface, then
where n1 and n2 are the indices of refraction of the two media. If the incident light is not normal to the interface, R will depend on the angle of incidence. Thus, the higher the index of refraction of the solid, the greater is the reflectivity. For typical silicate glasses, the reflectivity is approximately 0.05. Just as the index of refraction of a solid depends on the wavelength of the incident light, so also does the reflectivity vary with wavelength.
Since metals are highly reflective (reflectivity of most metals is between 0.90 and 0.95), they are opaque, the perceived color is determined by the wavelength distribution of the radiation that is reflected and not absorbed. A bright silvery appearance when exposed to white light indicates that the metal is highly reflective over the entire range of the visible spectrum. In other words, for the reflected beam, the composition of these reemitted photons, in terms of frequency and number, is approximately the same as for the incident beam. Aluminum and silver are two metals that exhibit this reflective behavior. Copper and gold appear red-orange and yellow, respectively, because some of the energy associated with light photons having short wavelengths is not reemitted as visible light.
Fig.9, different types and colors of diamond.
Fig.10, Transmission spectrum for a Type IIa diamond window of 1 mm thickness.
Carbon crystallizes in very different forms (diamond being one of them) with very different physical properties. Diamonds may be found in nature or manufactured in the laboratory with different colorations. The picture in fig. 9 shows different colors of diamond. For visible light range (between 0.4μm and 0.7μm), type IIa diamond of 1mm thickness transmits with a high percentage (about 65%) the visible light. If it were to be pure, the transmittance would have been higher; however, the presence of the impurity within makes allows less transmitted light.
6.1 Transparency From what has been discussed above, it is now clear why diamond appears colorless and transparent. In fact, this is the case for pure diamond in which no impurities lie within the diamond lattice. To make it clear, diamond appears transparent to visible light because the maximum energy within this spectrum (3.1 eV) is less than the Gap between the energy bands in diamond, thus the energy is not enough to allow the transition of an electron from the valence band to the conduction band. Therefore the will be no absorption of the photons and light will continue its transmission to leave to the exterior of diamond.
6.2 Fluorescence Fluorescence is an optical phenomenon in which a diamond's molecules absorb high-energy photons, re-emitting them as lower-energy, or longer-wavelength photons.
Fig.11-a, Long Wave/Short Wave UV Cabinet
Fig.11-b, Diamond Fluorescence Under UV Light
Diamond types that exhibit the phenomenon of fluorescence radiate or glow in a variety of colors when exposed to long wave ultra-violet light, and give off a bluish-white, greenish or yellow fluorescence when exposed to the X-ray wavelength.
6.2 Types of Diamond and Colorations Diamonds can occur in a wide variety of colors: colorless or white, blue, steel grey, pink, orange, red, brown, green, yellow, and black. All colored diamonds contain certain specific impurities and/or structural defects that cause their coloration, while chemically "pure" diamonds are basically transparent, and therefore colorless.
Diamonds are scientifically classed into two main types and several subtypes, according to the nature of defects present and how they affect light absorption: Type I : As many as 99% of all natural diamonds are classified as Type I, and contain nitrogen atoms as an impurity, which replace some carbon atoms within the crystal lattice structure. These Nitrogen impurities found in Type I diamonds are evenly dispersed throughout the gemstone, absorbing some of the blue spectrum, and thereby making the diamond appear yellow. There are also two subcategories (a and b) within each diamond 'type' (either Type I or Type II) that are based on a stone's electrical conductivity. All Type 1 diamonds have nitrogen atoms as their main impurity. If the nitrogen atoms are grouped in clusters they do not necessarily affect the diamond's overall color, and they are classified as Type 1-A. The title picture of the project shows a Type 1-A diamond. If the nitrogen atoms are dispersed evenly throughout the crystal, they can give the stone a yellow tint, and are classified as Type 1-B. Typically, a natural diamond may contain both Type 1-A and Type 1-B material. Man-made synthetic diamonds containing nitrogen are classified as Type 1-B.
Fig.12, Type Ib diamonds
Fig.13,Cloud Inclusion Under UV AGS Labs
Fig.14, Type I UV Fluorescence AGS Labs
Type II : Type II diamonds do not contain any detectable nitrogen, thereby allowing the passage of short-wave ultra-violet (SWUV) light through the stone. Natural blue Type II diamonds containing scattered boron impurities within their crystal matrix are good conductors of electricity, classifying them as Type IIb diamonds. The absorption spectrum of boron causes these gems to absorb red, orange, and yellow light, lending Type IIb diamonds a light blue or grey color. Type II diamonds that lack boron impurities are classified as Type IIa. These diamonds are almost or entirely devoid of impurities, and consequently are usually colorless and have the highest thermal conductivity. They are very transparent in ultraviolet, down to 230 nm. Occasionally, while Type IIa diamonds are being extruded towards the surface of the Earth, the pressure and tension can cause structural anomalies arising through plastic deformation1 during the growth of the crystal structure, leading to imperfections. These imperfections can confer a yellow, brown, orange, pink, red, or purple color to the gem. Some Type IIa diamonds can be found with pink, red, or brown coloration, due primarily to certain structural anomalies arising from "plastic deformation" which occurred during their formation.
Fig.15, The hope diamond (Type IIb)
In physics and materials science, plasticity describes the deformation of a material undergoing non-reversible changes of shape in response to applied forces.
Consider nitrogen impurity atoms substituting for carbons in a diamond crystal (Eg about 5 1 /2 eV). Since N has one more electron than C, each nitrogen donates one extra electron above the Fermi surface and these "donor" electrons form an impurity level within the band-gap, as shown in Figure 16-A. The nitrogen impurity level is still 4eV below the conduction band (a "deep" donor) and is, in fact, somewhat broad. It can only absorb a little violet at 3eV, thus giving a yellow color. A typical deep-yellow diamond may contain one nitrogen atom for every 100,000 carbon atoms. The reduction in the bandgap is insufficient to permit electrical conductivity at room temperature.
Fig. 16, Band diagrams for (A) yellow N-containing diamond and (B) blue B-containing diamond.
Boron, having one electron less than carbon, acts in a similar way to produce "acceptor levels" in the band-gap. These are "hole" levels which can accept electrons from the valence band and are located only 0.4eV above the top of the valence band, as shown in Figure 16-B. Here again the impurity band is not just a single level, but has a complex structure. resulting in a blue color. A1, which also has one less electron than C and is present in significant amounts in some blue diamonds, had been previously thought to provide the acceptor; however, laboratory synthesis showed that B additions gave blue and that A1 did not (Chrenko, 1973). Since the boron acceptor level is "shallow," ambient thermal excitation can raise electrons from the valence band into the acceptor level, and the resulting holes in the valence band can then conduct electricity. Type IIB blue conducting diamonds such as the "Hope" contain typically one boron atom per million carbon atoms.
The presence of Boron as an impurity in the Type IIb diamond crystal makes it a good semi-conductor, and there are plenty of factories that are manufacturing synthetic diamond with the properties they demand. The question is will diamond replace silicon in our electronic devices in the following years?
Appendix Information References : Books : ●Solid State Physics Neil W. Ashcroft, N. David Mermin Harcourt College Publishers 1976
● Fundamentals of Physics 10th Edition Halliday & Resnick
● Intrinsic OpVcal ProperVes of Diamond Richard P. Mildren
●Materials Science and Engineering An IntroducVon William D. Callister, Jr. 7th Edition
Sites : ● http://www.allaboutgemstones.com ● https://www.wikipedia.org
Subtitle Picture https://en.wikipedia.org/wiki/Diamond_cubic Title Picture http://www.prinsandprins.com/blog/diamonds-blog/
Fig.1 Materials Science and Engineering An Introduction th 7 Edition ,chapter 21 (Chapter introduction photo)
Fig.3 Materials Science and Engineering An Introduction th 7 Edition ,chapter 18
Fig.4 Materials Science and Engineering An Introduction th 7 Edition ,chapter 18
Fig.5 Materials Science and Engineering An Introduction th 7 Edition ,chapter 18 Fig.6 Solid State Physics (Ashcroft/Mermin) chapter 28
Fig.7 Materials Science and Engineering An Introduction th 7 Edition ,chapter 18
Fig.8 Materials Science and Engineering An Introduction th 7 Edition ,chapter 21
Fig.10 Intrinsic Optical Properties of Diamond