Figure 1-3. (a)coordinates of lattice sites; (b) Miller indices of planes; (c, d)Miller dices of planes and directions (Fig. 1-3a). If the center of the coordinate axes is taken as x=0, y=0, z=0, or(0, 0, 0), then the coordinates of other nearest equivalent cube corner points are(1, 0, 0)(0, 1, 0)(1, 0, 0), etc. In this framework the two Si atoms referred to earlier, situated at the center of the coordinate axes, would occupy the(0, 0, 0)and (1 /4, 1/4, 1/4) positions. Subsequent repetitions of this oriented pair of atoms at each FCC lattice point generate the diamond cubic structure in which each Si atom has four nearest neighbors arranged in a tetrahedral configuration. Similarly, substitution of the motif(0, 0, 0)Ga and (1/4. 1/4. 1/4)As for each point of the FCC lattice would result in the zinc blende GaAs crystal structure(Fig. 1-2d Specific crystal planes and directions are frequently noteworthy because phenomena such as crystal growth, chemical reactivity, defect incorporation deformation, and assorted properties are not isotropic or the same on all planes and in all directions. Therefore, it is important to be able to identify accurately and distinguish crystallographic planes and directions. A simple recipe for dentifying a given plane in the cubic system is the following 1. Determine the intercepts of the plane on the three crystal axes in number of unit cell dimensions 2. Take reciprocals of those numbers 3. Reduce these reciprocals to smallest integers by clearing fractions
1.2. Structure 5 X Y J X C. d Z Z Y X Figure 1-3. indices of planes and directions. (Fig. 1-3a). If the center of the coordinate axes is taken as x = 0, y = 0, z = 0, or (0, 0, O), then the coordinates of other nearest equivalent cube comer points are (1,0,0) (0, 1,O) (1,0,0), etc. In this framework the two Si atoms referred to earlier, situated at the center of the coordinate axes, would occupy the (0, 0,O) and (1/4,1/4,1/4) positions. Subsequent repetitions of this oriented pair of atoms at each FCC lattice point generate the diamond cubic structure in which each Si atom has four nearest neighbors arranged in a tetrahedral configuration. Similarly, substitution of the motif (0, 0,O) Ga and (1/4, 1/4, 1/4) As for each point of the FCC lattice would result in the zinc blende GaAs crystal structure (Fig. 1-2d). Specific crystal planes and directions are frequently noteworthy because phenomena such as crystal growth, chemical reactivity, defect incorporation, deformation, and assorted properties are not isotropic or the same on all planes and in all directions. Therefore, it is important to be able to identify accurately and distinguish crystallographic planes and directions. A simple recipe for identifying a given plane in the cubic system is the following: 1. Determine the intercepts of the plane on the three crystal axes in number of 2. Take reciprocals of those numbers. 3. Reduce these reciprocals to smallest integers by clearing fractions. (a) coordinates of lattice sites; (b) Miller indices of planes; (c, d) Miller unit cell dimensions
Revlew of Materials Science The result is a triad of numbers known as the miller indices for the plane in question, i.e,(h, k, I). Several planes with identifying Miller indices are indicated in Fig. 1-3. Note that a negative index is indicated above the integer Crystallographic directions shown in Fig. 1-3 are determined by the compo- nents of the vector connecting any two lattice points lying along the direction If the coordinates of these points are u,, UI, w, and u?, Uz, W2, then the components of the direction vector are u1-u2, U1-U2,W,?. When educed to smallest integer numbers, they are placed within brackets and are known as the Miller indices for the direction, i.e., [hkl. In this notation the direction cosines for the e given directions are h/vh+k+ k/vh?+k2+/,1/vh2+k2+/. Thus, the angle a between any two directions [h,, k1, I,I and [h2, k2, l2] is given by the vector dot product h1h2+k:k2+l12 cos a= (1-1) h+k2+√h+k2+ Two other useful relationships in the crystallography of cubic systems are given without proof. 1. The Miller indices of the direction normal to the(hkh) plane are [hkn 2. The spacing between individual(hkl) planes is a= a,/vh2+k2+ where ao is the lattice parameter As an illustrative example, we shall calculate the angle between any two neighboring tetrahedral bonds in the diamond cubic lattice. The bonds lie along [111l-rype directions that are specifically taken here to be [1 Il and [Ill] Therefore, by Eq. 1-1 (1)(-1)+1(-1)+(1)(1) 12+12+12√(-12+(-1)2 These two bond directions lie in a common(110)-type crystal plane. The precise indices of this plane must be(110)or(110). This can be seen by noting that the dot product between each bond vector and the vector normal to the plane in which they lie must vanish We close this brief discussion with some experimental evidence in support of the internal crystalline structure of solids. X-ray diffraction methods have very convincingly supplied this evidence by exploiting the fact that the spacing between atoms is comparable to the wavelength(A) of X-rays. This results in easily detected emitted beams of high intensity along certain directions when
6 A Review of Materials Science The result is a triad of numbers known as the Miller indices for the plane in question, i.e., (h, k, l). Several planes with identifying Miller indices are indicated in Fig. 1-3. Note that a negative index is indicated above the integer with a minus sign. Crystallographic directions shown in Fig. 1-3 are determined by the components of the vector connecting any two lattice points lying along the direction. If the coordinates of these points are u1 , ul, w1 and u,, u,, w2, then the components of the direction vector are u1 - u2, u1 - u,, w1 - w,. When reduced to smallest integer numbers, they are placed within brackets and are known as the Miller indices for the direction, Le., [hkl]. In this notation the direction cosines for the given directions are h/dh2 + k2 + 12, k/dh2 + k2 + 12, l/dh2 + k2 + 1'. Thus, the angle a between any two directions [ h, , k, , 11] and [ h, , k, , /2] is given by the vector dot product h,h, + k,k2 + /]12 Jh: + k: + 1: dh; + ki + /; cos Q = (1-1) Two other useful relationships in the crystallography of cubic systems are given without proof. 1. The Miller indices of the direction normal to the (hkl) plane are [ hkl]. 2. The spacing between individual (hkl) planes is a = a,/dh2 + k2 + 12, where a, is the lattice parameter. As an illustrative example, we shall calculate the angle between any two neighboring tetrahedral bonds in the diamond cubic lattice. The bonds lie along [ 11 11-type directions that are specifically taken here to be [i 'i 11 and [l 1 11. Therefore, by Eq. 1-1, 1 3 and (I)(- 1) + I(- 1) + (1)(1) cos a = = -- d12 + 1, + l2 J( - 1)2 + (- 1)2 + l2 a = 109.5". These two bond directions lie in a common (110)-type crystal plane. The precise indices of this plane must be 010) or (1iO). This can be seen by noting that the dot product between each bond vector and the vector normal to the plane in which they lie must vanish. We close this brief discussion with some experimental evidence in support of the internal crystalline structure of solids. X-ray diffraction methods have very convincingly supplied this evidence by exploiting the fact that the spacing between atoms is comparable to the wavelength (A) of X-rays. This results in easily detected emitted beams of high intensity along certain directions when
1.2. Structure incident X-rays impinge at critical diffraction angles(0). Under these condi tions the well-known Bragg relation n入=2asin0 (1-2 lolds, where n is an integer. In bulk solids large diffraction effects occur at many values of In thin films, however, very few atoms are present to scatter X-rays into the diffracted beam when 8 is large. For this reason the intensities of the diffraction lines or spots will be unacceptably small unless the incident beam strikes the film surface at a near-glancing angle. This, in effect, makes the film look thicker Such X-ray techniques for examination of thin films have been developed and will be discussed in Chapter 6. a drawback of thin films relative to bulk solids is the long counting times required to generate enough signal for suitable diffraction patterns. This thickness limitation in thin films is turned into great advantage, however, in the transmission electron microscope. Here electrons must penetrate through the material under observation, and this can occur only in thin films or specially thinned specimens. The short wavelength of the electrons employed enables high-resolution imaging of the lattice structure as well as diffraction effects to be observed. As an example, consider the remarkable electron micrograph of Fig. 1-4, showing atom positions in a thin CoS interface 192A Figure 1-4. High resolution lattice image of epitaxial CoSi, film on(111)Si((112) projection).( Courtesy J. M. Gibson, AT& T Bell Laboratories)
1.2. Structure 7 incident X-rays impinge at critical diffraction angles (8). Under these conditions the well-known Bragg relation nX = 2asin8 (1-2) holds, where n is an integer. In bulk solids large diffraction effects occur at many values of 8. In thin films, however, very few atoms are present to scatter X-rays into the diffracted beam when 8 is large. For this reason the intensities of the diffraction lines or spots will be unacceptably small unless the incident beam strikes the film surface at a near-glancing angle. This, in effect, makes the film look thicker. Such X-ray techniques for examination of thin films have been developed and will be discussed in Chapter 6. A drawback of thin films relative to bulk solids is the long counting times required to generate enough signal for suitable diffraction patterns. This thickness limitation in thin films is turned into great advantage, however, in the transmission electron microscope. Here electrons must penetrate through the material under observation, and this can occur only in thin films or specially thinned specimens. The short wavelength of the electrons employed enables high-resolution imaging of the lattice structure as well as diffraction effects to be observed. As an example, consider the remarkable electron micrograph of Fig. 1-4, showing atom positions in a thin Figure 1-4. High resolution lattice image of epitaxial CoSi, film on (111) Si ((112) projection). (Courtesy J. M. Gibson, AT & T Bell Laboratories)
of Ma m of cobalt silicide grown with perfect crystalline registry (epitaxially)on a silicon wafer. The silicide film-substrate was mechanically and chemically thinned normal to the original film plane to make the cross section visible. Such evidence should leave no doubt as to the intemal crystalline nature of solid 1.2. 2. Amorphous Solids In some materials the predictable long-range geometric order characteristic of crystalline solids breaks down. Such materials are the noncrystalline amor- phous or glassy solids exemplified by silica glass, inorganic oxide mixtures and polymers. When such bulk materials are cooled from the melt even at low rates, the more random atomic positions that we associate with a liquid are frozen in place within the solid. On the other hand, while most metals cannot be amorphized, certain alloys composed of transition metal and metalloid combinations (e. g, Fe-B) can be made in glassy form but only through extremely rapid quenching of melts. The required cooling rates are about 10 C/sec, and therefore heat transfer considerations limit bulk glassy metals to foil, ribbon, or powder shapes typically -0.05 mm in thickness or dimen- sion. In general, amorphous solids can retain their structureless character practically indefinitely at low temperatures even though thermodynamics gests greater stability for crystalline forms. Crystallization will, however, proceed with release of energy when these materials are heated to appropriate elevated temperatures. The atoms then have the required mobility to seek out Thin films of amorphous metal alloys, semiconductors, oxides, and chalco- genide glasses have been readily prepared by common physical vapor deposi tion(evaporation and sputtering) as well as chemical vapor deposition(CVD) methods. Vapor quenching onto cryogenically cooled glassy substrates has made it possible to make alloys and even pure metals-the most difficult of all materials to amorphize-glassy. In such cases, the surface mobility of deposit ing atoms is severely restricted, and a disordered atomic configuration has a greater probability of being frozen in Our present notions of the structure of amorphous inorganic solids are extensions of models first established for silica glass. These depict amorphous SiO, to be a random three-dimensional network consisting of tetrahedra joined at the comers but sharing no edges or faces. Each tetrahedron contains a central Si atom bonded to four vertex oxygen atoms, i. e,(SiO4). The oxygens are, in turn, shared by two Si atoms and are thus positioned as the pivotal links between neighboring tetrahedra. In crystalline quartz the tetrahe-
8 A Review of Materials Science film of cobalt silicide grown with perfect crystalline registry (epitaxially) on a silicon wafer. The silicide film- substrate was mechanically and chemically thinned normal to the original film plane to make the cross section visible. Such evidence should leave no doubt as to the internal crystalline nature of solids. 1.2.2. Amorphous Solids In some materials the predictable long-range geometric order characteristic of crystalline solids breaks down. Such materials are the noncrystalline amorphous or glassy solids exemplified by silica glass, inorganic oxide mixtures, and polymers. When such bulk materials are cooled from the melt even at low rates, the more random atomic positions that we associate with a liquid are frozen in place within the solid. On the other hand, while most metals cannot be amorphized, certain alloys composed of transition metal and metalloid combinations (e.g., Fe-B) can be made in glassy form but only through extremely rapid quenching of melts. The required cooling rates are about lo6 "C/sec, and therefore heat transfer considerations limit bulk glassy metals to foil, ribbon, or powder shapes typically - 0.05 mm in thickness or dimension. In general, amorphous solids can retain their structureless character practically indefinitely at low temperatures even though thermodynamics suggests greater stability for crystalline forms. Crystallization will, however, proceed with release of energy when these materials are heated to appropriate elevated temperatures. The atoms then have the required mobility to seek out equilibrium lattice sites. Thin films of amorphous metal alloys, semiconductors, oxides, and chalcogenide glasses have been readily prepared by common physical vapor deposition (evaporation and sputtering) as well as chemical vapor deposition (CVD) methods. Vapor quenching onto cryogenically cooled glassy substrates has made it possible to make alloys and even pure metals-the most difficult of all materials to amorphize-glassy. In such cases, the surface mobility of depositing atoms is severely restricted, and a disordered atomic configuration has a greater probability of being frozen in. Our present notions of the structure of amorphous inorganic solids are extensions of models first established for silica glass. These depict amorphous SiO, to be a random three-dimensional network consisting of tetrahedra joined at the comers but sharing no edges or faces. Each tetrahedron contains a central Si atom bonded to four vertex oxygen atoms, Le., (Si04r4. The oxygens are, in turn, shared by two Si atoms and are thus positioned as the pivotal links between neighboring tetrahedra. In crystalline quartz the tetrahe-
Figure 1-5. Schematic representation of (a) crystalline quartz;(b) random network (amorphous);(c)mixture of crystalline and amorphous regions.( Reprinted with permis sion from John Wiley and Sons, E. H. Nicollian and J. R. Brews, MOS Physics and Technology, Copyright@ 1983, John Wiley and Sons dra cluster in an ordered six-sided ring pattern, shown schematically in Fig 1-5a, should be contrasted with the completely random network depicted in Fig. 1-5b. In actuality, the glassy solid structure is most probably a compro- mise between the two extremes consisting of a considerable amount of short range order and microscopic regions (i. e, less than 100 A in size)of crystallinity (Fig. 1-5c). The loose disordered network structure allows for a considerable amount of " holes''or "vacancies to exist, and it. therefore omes as no surprise that the density of glasses will be less than that of their whereas in silica glass it is 2.2 g/cm. Amorphous silicon, which has /cm i crystalline counterparts. In quartz, for example, the density is 2.65 commercial use in thin-film solar cells, is, like silica, tetrahedrally bonded and believed to possess a similar structure. We return later to discuss further structural aspects and properties of amorphous films in various contexts throughout the book
1.2. Structure a. 9 b. C. Figure 1-5. Schematic representation of (a) crystalline quartz; (b) random network (amorphous); (c) mixture of crystalline and amovhous regions. (Reprinted with permission from John Wiley and Sons, E. H. Nicollian and J. R. Brews, MOS Physics and Technology, Copyright 0 1983, John Wiley and Sons). dra cluster in an ordered six-sided ring pattern, shown schematically in Fig. 1-5a, should be contrasted with the completely random network depicted in Fig. 1-5b. In actuality, the glassy solid structure is most probably a compromise between the two extremes consisting of a considerable amount of shortrange order and microscopic regions (i.e., less than 100 A in size) of crystallinity (Fig. 1-5c). The loose disordered network structure allows for a considerable amount of “holes” or “vacancies” to exist, and it, therefore, comes as no surprise that the density of glasses will be less than that of their crystalline counterparts. In quartz, for example, the density is 2.65 g/cm3, whereas in silica glass it is 2.2 g/cm3. Amorphous silicon, which has found commercial use in thin-film solar cells, is, like silica, tetrahedrally bonded and believed to possess a similar structure. We return later to discuss further structural aspects and properties of amorphous films in various contexts throughout the book