CHAPTER3. Crystalline Structure Perfection

CHAPTER 3 Crystalline Structure–Perfection The transmission electron microscope (Section 4.7) can be used to image the regular arrangement of atoms ...
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CHAPTER

3

Crystalline Structure–Perfection The transmission electron microscope (Section 4.7) can be used to image the regular arrangement of atoms in a crystalline structure. This atomic-resolution view is along individual columns of gallium and nitrogen atoms in gallium nitride. The distance marker is 113 picometers or 0.113 nm. (Courtesy of C. Kisielowski, C. Song, and E. C. Nelson, National Center for Electron Microscopy, Berkeley, California.)

Unit cell Figure 3-1 Various structural units that describe the schematic crystalline structure. The simplest structural unit is the unit cell.

c

b a Figure 3-2 Geometry of a general unit cell.

Figure 3-3 The simple cubic lattice becomes the simple cubic crystal structure when an atom is placed on each lattice point.

=

(i)

(ii)

(iv)

(iii)

(v)

(a)

(b)

(c) Structure: body-centered cubic (bcc) Bravais lattice: bcc 1 Atoms/unit cell: 1 + 8 × 8 = 2 Typical metals: α-Fe, V, Cr, Mo, and W Figure 3-4 Body-centered cubic (bcc) structure for metals showing (a) the arrangement of lattice points for a unit cell; (b) the actual packing of atoms (represented as hard spheres) within the unit cell; and (c) the repeating bcc structure, equivalent to many adjacent unit cells (Part (c) courtesy of Molecular Simulations, Inc.).

(a)

(b)

(c) Structure: face-centered cubic (fcc) Bravais lattice: fcc 1 1 Atoms/unit cell: 6 × 2 + 8 × 8 = 4 Typical metals: γ-Fe, Al, Ni, Cu, Ag, Pt, and Au Figure 3-5 Face-centered cubic (fcc) structure for metals showing (a) the arrangement of lattice points for a unit cell; (b) the actual packing of atoms within the unit cell; and (c) the repeating fcc structure, equivalent to many adjacent unit cells (Part (c) courtesy of Molecular Simulations, Inc.).

Atom in midplane Atom centered in adjacent unit cell

Atom in midplane

2 atoms per lattice point One-twelfth of an atom

(a)

(b)

One sixth of an atom

(c) Structure: hexagonal close-packed (hcp) Bravais lattice: hexagonal 1 1 Atoms/unit cell: 1 + 4 × 6 + 4 × 12 = 2 Typical metals: Be, Mg, α-Ti, Zn, and Zr

Figure 3-6 Hexagonal close packed (hcp) structure for metals showing (a) the arrangement of atom centers relative to lattice points for a unit cell. There are two atoms per lattice point (note the outlined example). (b) The actual packing of atoms within the unit cell. Note that the atom in the midplane extends beyond the unit cell boundaries. (c) The repeating hcp structure, equivalent to many adjacent unit cells (Part (c) courtesy of Molecular Simulations, Inc.).

A A A A

B C A

C A B

A

A B C

B

A A B

B

A

A

(b) Stacking of close-packed planes

A

Normal to close-packed planes

A

(a) Stacking of close-packed planes Close-packed planes Normal to close-packed planes

A

C

B

A Close-packed planes B

A A

(c) Face-centered cubic

(d) Hexagonal close-packed

Figure 3-7 Comparison of the fcc and the hcp structures. They are each efficient stackings of close-packed planes. The difference between the two structures is the different stacking sequences. (After B. D. Cullity, Elements of X-Ray Diffraction, 2nd ed., AddisonWesley Publishing Co., Inc., Reading, Mass., 1978.)

Center of unit cell 2 ions per lattice point

Cs+ Cl–

(a)

(b) Structure: CsCl-type Bravais lattice: simple cubic Ions/unit cell: 1Cs+ + 1Cl–

Figure 3-8 Cesium chloride (CsCl) unit cell showing (a) ion positions and the 2 ions per lattice point, and (b) full-size ions. Note that the Cs + − Cl − pair associated with a given lattice point is not a molecule because the ionic bonding is nondirectional and a given Cs + is equally bonded to eight adjacent Cl − , and vice versa. (Part (b) courtesy of Molecular Simulations, Inc.)

Na

2 ions per lattice point

+

Cl–

(a)

(b)

(c) Structure: NaCl-type Bravais lattice: fcc Ions/unit cell: 4Na+ + 4Cl– Typical ceramics: MgO, CaO, FeO, and NiO

Figure 3-9 Sodium chloride (NaCl) structure showing (a) ion positions in a unit cell, (b) full-size ions, and (c) many adjacent unit cells. (Parts (b) and (c) courtesy of Molecular Simulations, Inc.)

F – ions located at corners of a cube(at one-quarter of the distance along the body diagonal)

Ca2+ F–

(a)

(b)

Structure: flourite (CaF2)-type Bravais lattice: fcc Ions/unit cell: 4Ca2+ + 8F– Typical ceramics: UO2, ThO2, and TeO2 Figure 3-10 Fluorite (CaF 2 ) unit cell showing (a) ion positions and (b) full-size ions. (Part (b) courtesy of Molecular Simulations, Inc.)

Interior Si4+ located at positions one-quarter of the distance along the body diagonal

Si4+ O2–

(a)

(b)

(c)

Structure: cristobalite (SiO2) -type Bravais lattice: fcc Ions/unit cell: 8Si4+ + 16O2–

Figure 3-11 The cristobalite (SiO 2 ) unit cell showing (a) ion positions, (b) full-size ions, and (c) the connectivity of SiO 44− tetrahedra. In the schematic, each tetrahedron has a Si 4+ at its center. In addition, an O 2− would be at each corner of each tetrahedron and is shared with an adjacent tetrahedron. (Part (c) courtesy of Molecular Simulations, Inc.)

Crystallographic Form

Bravice lattice

2000

1723 (melting point) 1500

T(˚C)

High cristobalite (shown in Fig. 3-11)

fcc

High tridymite

Hexagonal

High quartz

Hexagonal

Low quartz

Hexagonal

1470

1000 867

500

573

0

Figure 3-12 Many crystallographic forms of SiO 2 are stable as they are heated from room temperature to the melting temperature. Each form represents a different way to connect adjacent SiO 44− tetrahedra.

Top view

Side view

Al3+ O2–

Unit cell Structure: corundum (Al2O3) -type Bravais lattice: hexagonal (approx) Ions/unit cell: 12Al3+ + 18O2– Typical ceramics: Al2O3, Cr2O3, αFe2O3

Close-packed layer of O2– with octahedral sites filled with Al3+

2 3

Unit cell (6O2– layers high) of

Figure 3-13 The corundum (Al 2 O 3 ) unit cell is shown superimposed on the repeated stacking of layers of close-packed O 2− ions. The Al 3+ ions fill two-thirds of the small (octahedral) interstices between adjacent layers.

Ti4+: at the body center +

Ca2 : at corners – O2 : at face centers

(a)

(b)

Structure: perovskite (CaTiO3)-type Bravais lattice: simple cubic Ions/unit cell: 1Ca2+ + 1Ti4+ + 3O2− Typical ceramics: CaTiO3, BaTiO3

Figure 3-14 Perovskite (CaTiO 3 ) unit cell showing (a) ion positions and (b) full-size ions. (Part (b) courtesy of Molecular Simulations, Inc.)

Oxygen Octahedral positions Tetrahedral positions Figure 3-15 Ion positions in the spinel (MgAl 2 O 4 ) unit cell. The circles in color represent Mg 2+ ions (in tetrahedral or four-coordinated positions), and the black circles represent Al 3+ ions (in octahedral or six-coordinated positions). (From F. G. Brockman, Bull. Am. Ceram. Soc. 47, 186 (1967).)

b a

0.514nm

Octahedral coordination

Tetrahedral coordination

–12 602–

0.737nm

c +16 4Si4+

–10 0.893nm α = 91˚48′ β = 104˚30′ γ = 90˚0′ Anions O2–

4 O2– + 2 OH–

Cations

+12 4Al3+

OH– Al3+ Si4+

–6 6OH+

Figure 3-16 Exploded view of the kaolinite unit cell, 2(OH) 4 Al 2 Si 2 O 5 . (After F. H. Norton, Elements of Ceramics, 2nd ed., Addison-Wesley Publishing Co., Inc., Reading, Mass., 1974.)

Figure 3-17 Transmission electron micrograph (see Section 4.7) of the structure of clay platelets. This microscopic-scale structure is a manifestation of the layered crystal structure shown in Figure 3–16. (Courtesy of I. A. Aksay)

1C 0.67nm 2C 0.25nm C

1C R = 0.08nm (a)

(b)

Figure 3-18 (a) An exploded view of the graphite (C) unit cell. (From F. H. Norton, Elements of Ceramics, 2nd ed., Addison-Wesley Publishing Co., Inc., Reading, Mass., 1974.) (b) A schematic of the nature of graphite’s layered structure. (From W. D. Kingery, H. K. Bowen, and D. R. Uhlmann, Introduction to Ceramics, 2nd ed., John Wiley & Sons, Inc., New York, 1976.)

(a)

(b) Figure 3-19 (a) C 60 molecule, or buckyball. (b) Cylindrical array of hexagonal rings of carbon atoms, or buckytube. (Courtesy of Molecular Simulations, Inc.)

Figure 3-20 Arrangement of polymeric chains in the unit cell of polyethylene. The dark spheres are carbon atoms, and the light spheres are hydrogen atoms. The unit cell dimensions are 0.255 nm × 0.494 nm × 0.741 nm. (Courtesy of Molecular Simulations, Inc.)

Top view of fold plane

b a Orientation of unit cell Side view of fold plane

Figure 3-21 Weaving-like pattern of folded polymeric chains that occurs in thin crystal platelets of polyethylene. (From D. J. Williams, Polymer Science and Engineering, Prentice Hall, Inc., Englewood Cliffs, N.J., 1971.)

3 4 2 1

CH2

O

CH2

NH

O

NH O

C

NH

NH CH2 CH2 CH2

CH2 NH

CH2 NH CH2

O

NH O C CH2

O NH b

CH2 a

Figure 3-22 Unit cell of the α -form of polyhexamethylene adipamide or nylon 66. (From C. W. Bunn and E. V. Garner, “Packing of nylon 66 molecules in the triclinic unit cell: α form, Proc. Roy. Soc. Lond. 189A, 39 (1947).)

Courtesy of SEAMATECH.

Pull Rotate

Seed crystal Container Crystal Crucible

Heating elements

Melt

Schematic of growth of single crystals using the Czochralski technique. (After J. W. Mayer and S. S. Lau, Electronic Materials Science: For Integrated Circuits in Si and GaAs, Macmillan Publishing Company, New York, 1990.)

Interior atoms located at positions one-quarter of the distance along the body diagonal

2 atoms per lattice point

(a)

(b) Structure: diamond cubic Bravais lattice: fcc 1 1 Atoms/unit cell: 4 + 6 × 2 + 8 × 8 = 8 Typical semiconductors: Si, Ge, and gray Sn

Figure 3-23 Diamond cubic unit cell showing (a) atom positions. There are two atoms per lattice point (note the outlined example). Each atom is tetrahedrally coordinated. (b) The actual packing of full-size atoms associated with the unit cell. (Part (b) courtesy of Molecular Simulations, Inc.)

Zn2+

S2– Two ions per lattice point

(a)

(b) Structure: Zinc blende (ZnS)-type Bravais lattice: fcc Ions/unit cell: 4Zn2+ + 4S2– Typical semiconductors: GaAs, AlP, InSb (III-V compounds), ZnS, ZnSe, CdS,HgTe (II-VI compounds)

Figure 3-24 Zinc blende (ZnS) unit cell showing (a) ion positions. There are two ions per lattice point (note the outlined example). Compare this with the diamond cubic structure (Figure 3–23a). (b) The actual packing of full-size ions associated with the unit cell. (Part (b) courtesy of Molecular Simulations, Inc.)

S2– ion 3 at of 8 height of unit cell

S2– ion 7 at of 8 height of unit cell Zn2+ ion in midplane (tetrahedrally coordinated by S2–)

Zn2+ S2–

(a)

(b)

Structure: wurtzite (ZnS)-type Bravais lattice: hexagonal Ions/unit cell: 2Zn2+ + 2S2– Typical semiconductors: ZnS, CdS, and ZnO.

Figure 3-25 Wurtzite (ZnS) unit cell showing (a) ion positions and (b) fullsize ions.

112

001 c

111 1 1 1 2 2 2

0

000

100

a

0

1 1 2 2

0

1 2

–1

1 2

0

b

010

110

Figure 3-26 Notation for lattice positions.

121

2×c c 1×b b a Figure 3-27 Lattice translations connect structurally equivalent positions (e.g., the body center) in various unit cells.

[112] [111]

[111] [111] c

1 1 1 2 2 2

111

[111]

b a

[111] Figure 3-28 Notation for lattice directions. Note that parallel [uvw] directions (e.g., [111]) share the same notation because only the origin is shifted.

[111] [111]

[111]

[111]

a3 a2

a1

[111]

[111]

[111]

[111]

Figure 3-29 Family of directions, h 111 i , representing all body diagonals for adjacent unit cells in the cubic system.

Intercept at ∞

Miller indices (hkl): 1 1/2

c

Intercept at

1 2

a

1 ∞

(210)

b

a

1 1

Intercept at b (a)

c

c

c

b a

b

b

a (010)

c

a (020)

(111)

b

a (111)

(b)

Figure 3-30 Notation for lattice planes. (a) The (210) plane illustrates Miller indices (hkl) . (b) Additional examples.

c

120˚

a1

a3

a2

120˚

Miller-Bravais indices (hkil): Note: h + k = −i

1 1 1 1 , , , ∞ 1 –1 ∞

→ (0110)

Figure 3-31 Miller-Bravais indices, (hkil) , for the hexagonal system.

(100) on back face

(010) on side face

(001) a3 (100)

(010) a2

a1

(001) on bottom face {100} Figure 3-32 Family of planes, {100}, representing all faces of unit cells in the cubic system.

c

000

1 1 2 2

0 b 110

a

[110]

Anions

Cations

Glass plate d a b

Incident rays

Figure 3-33 Diffraction grating for visible light. Scratch lines in the glass plate serve as light-scattering centers. (After D. Halliday and R. Resnick, Physics, John Wiley & Sons, Inc., New York, 1962.)

X-radiation

γ-radiation

10–6

10–3

Visible light Microwaves

UV

1

ir

103

Radio waves

106

109

1012

Wavelengh (nm) Figure 3-34 Electromagnetic radiation spectrum. X-radiation represents that portion with wavelengths around 0.1 nm.

Incident X-ray beam (in phase) λ

Diffracted beam (in phase)

θ

θ

A

θ θ B

C d

ABC = nλ (for constructive interference) AB = BC = d sin θ Therefore nλ = 2d sin θ

Figure 3-35 Geometry for diffraction of x-radiation. The crystal structure is a three-dimensional diffraction grating. Bragg’s law ( nλ = 2d sin θ ) describes the diffraction condition.

X-ray source

X-ray detector

Sample

Figure 3-36 Relationship of the Bragg angle ( θ ) and the experimentally measured diffraction angle (2 θ ).

Figure 3-37 Diffraction pattern of a single crystal of MgO (with the NaCl structure of Figure 3–9). Each spot on the film represents diffraction of the x-ray beam from a crystal plane ( hkl ).

Collimator

180 – 2θ Sample

X-ray source (a)

Holder (b)

Figure 3-38 (a) Single-crystal diffraction camera (or Laue camera). (Courtesy of Blake Industries, Inc.) (b) Schematic of the experiment.

(111)

100

Intensity (arbitrary units)

λ = 0.1542nm (CuKα-radiation) 80 60

(200)

40 (220) 20 0 20

(311) (222)

30

40

50

60 70 80 2θ (degrees)

(400) 90

100

(331)(420) 110

120

Figure 3-39 Diffraction pattern of aluminum powder. Each peak (in the plot of x-ray intensity versus diffraction angle, 2 θ ) represents diffraction of the x-ray beam by a set of parallel crystal planes ( hkl ) in various powder particles.

(a) Scan directions

Collimator

Collimator Detector

X-ray source 2θ Sample

Computer display (b)

Figure 3-40 (a) An x-ray diffractometer. (Courtesy of Scintag, Inc.) (b) A schematic of the experiment.

Film 2θ 1cm X-ray beam

θ ϕ θ 3cm

Sample