Solid State Physics 460- Lecture 2a Structure of Crystals (Kittel - PowerPoint PPT Presentation

Solid State Physics 460- Lecture 2a Structure of Crystals (Kittel Ch. 1) See many great sites like “Bob’s rock shop” with pictures and crystallography info: http://www.rockhounds.com/rockshop/xtal/index.html Physics 460 F 2006 Lect 2a 1

Crystals From Last Time • A crystal is a repeated array of atoms ¤ • Crystal Lattice + Basis Lattice of points Crystal (Bravais Lattice) Basis of atoms • Crystals can be classified into a small number of types – See text for more details Physics 460 F 2006 Lect 2a 2

From Last Time Two Dimensional Crystals a 2 φ a 1 Basis Lattice • Infinite number of possible lattices and crystals • Finite number of possible lattice types and crystal types Physics 460 F 2006 Lect 2a 3

From Last Time Possible Two Dimensional Lattices a 2 a 2 φ a 1 a 1 Hexagonal Φ = 60, a 1 = a 2 General oblique 6-fold rotation , reflections a 2 a 2 a 2 a 1 a 1 a 1 Rectangular Centered Rectangular Square 2-fold rot., reflect. 2-fold rot., reflect. 4-fold rot., reflect. • These are the only possible special crystal types in two dimensions Physics 460 F 2006 Lect 2a 4

Three Dimensional Lattices z a 3 y a 1 a 2 X • Every point on the Bravais lattice is a multiple of 3 primitive lattice vectors T(n 1 ,n 2 ,n 3 ) = n 1 a 1 + n 2 a 2 + n 3 a 3 where the n’s are integers Physics 460 F 2006 Lect 2a 5

Three Dimensional Lattices Simplest examples a 3 a 3 a 1 a 2 a 1 a 2 Simple Orthorhombic Bravais Lattice Hexagonal Bravais Lattice • Orthorhombic: angles 90 degrees, 3 lengths different Tetragonal: 2 lengths same Cubic: 3 lengths same • Hexagonal: a 3 different from a 1 , a 2 by symmetry Physics 460 F 2006 Lect 2a 6

Cubic Lattices Length of each side - a a 3 a a 2 a a 1 a Simple Cubic Body Centered Cubic Primitive lattice vectors (BCC) a 1 = (1,0,0) a a 2 = (0,1,0) a Conventional Cell with 2 atoms at positions a 3 = (0,0,1) a (000), (1/2,1/2,1/2) a One atom per cell at position (0,0,0) Physics 460 F 2006 Lect 2a 7

Cubic Lattices Length of each side - a a 3 a 3 a 2 a 2 a 1 a 1 Face Centered Cubic Simple Cubic (FCC) Primitive lattice vectors a 1 = (1,0,0) a Conventional Cell with 4 atoms at positions a 2 = (0,1,0) a (000 ), (0,1/2,1/2), (1/2,0,1/2), (1/2,1/2,0)a a 3 = (0,0,1) a One atom per cell at position (0,0,0) Physics 460 F 2006 Lect 2a 8

Face Centered Cubic Two views - Conventional Cubic Cell z z y y X X Conventional Cell of Face Centered Cubic Lattice 4 times the volume of a primitive cell Physics 460 F 2006 Lect 2a 9

Face Centered Cubic (fcc) Also called cubic closed packed (ccp) z z y y X X Each atom has 12 equal neighbors We will see later that this is a “close packed” lattice Physics 460 F 2006 Lect 2a 10

Face Centered Cubic z y a 2 a 3 X a 1 One Primitive Cell Wigner-Seitz Cell Primitive lattice vectors a 1 = (1/2,1/2,0) a a 2 = (1/2,0,1/2) a a 3 = (0,1/2,1/2) a One atom per cell at position (0,0,0) Physics 460 F 2006 Lect 2a 11

Body Centered Cubic z a 3 y a 1 a 2 X One Primitive Cell Wigner-Seitz Cell Primitive lattice vectors a 1 = (1/2,1/2,-1/2) a a 2 = (1/2, -1/2,1/2) a a 3 = (-1/2,1/2,1/2) a One atom per cell at position (0,0,0) Physics 460 F 2006 Lect 2a 12

Lattice Planes - Index System s 3 a 3 s 1 a 1 s 2 a 2 Plane through the points s 1 a 1 , s 2 a 2 , s 3 a 3 Each s can be an integer or a rational fractions • Define the plane by the reciprocals 1/s 1 , 1/s 2 , 1/s 3 • Reduce to three integers with same ratio h,k,l • Plane is defined by (h,k,l) Physics 460 F 2006 Lect 2a 13

Schematic illustrations of lattice planes Lines in 2d crystals Lattice (01) a 2 φ (02) a 1 (14) Basis (22) (11) • Infinite number of possible planes • Can be through lattice points or between lattice points Physics 460 F 2006 Lect 2a 14

Schematic illustrations of lattice planes Lines in 2d crystals Lattice (01) a 2 (02) φ a 1 (14) Basis • Equivalent parallel planes • Low index planes: more dense, more widely spaced • High index planes: less dense, more closely spaced Physics 460 F 2006 Lect 2a 15

Schematic illustrations of lattice planes Lines in 2d crystals (01) a 2 (02) a 1 (14) • Planes “slice through” the basis of physical atoms Physics 460 F 2006 Lect 2a 16

Lattice planes in cubic crystals z z y y X X (100) and (110) planes in a cubic lattice (illustrated for the fcc lattice) Physics 460 F 2006 Lect 2a 17

(111) lattice planes in cubic crystals z y X Face Centered Cubic Lattice Lattice planes perpendicular to [111] direction Each plane is hexagonal close packed array of points Physics 460 F 2006 Lect 2a 18

Stacking hexagonal 2d layers to make close packed 3-d crystal A B C • Each sphere has 12 equal neighbors • 6 in plane, 3 above, 3 below • Close packing for spheres • Can stack each layer in one of two ways, B or C above A Physics 460 F 2006 Lect 2a 19 • Also see figure in Kittel

Stacking hexagonal 2d layers to make hexagonal close packed (hcp) 3-d crystal • Each sphere has 12 equal neighbors • Close packing for spheres • See figure in Kittel for stacking sequence • HCP is ABABAB….. Stacking • Basis of 2 atoms Physics 460 F 2006 Lect 2a 20

21 Physics 460 F 2006 Lect 2a Hexagonal close packed Hexagonal Bravais Lattice Two atoms per cell a 1 a 2 a 3 A B A

Stacking hexagonal 2d layers to make cubic close packed (ccp) 3-d crystal Cube ABC stacking • Each sphere has 12 equal neighbors • Close packing for spheres • See figure in Kittel for stacking sequence • CCP is ABCABCABC….. Stacking • Basis of 1 atom Physics 460 F 2006 Lect 2a 22

Stacking hexagonal 2d layers to make cubic close packed (ccp) 3-d crystal Cube [111] direction in cube Physics 460 F 2006 Lect 2a 23

Recall from Face Centered Cubic (fcc) before Also called cubic closed packed (ccp) [111] z direction in cube y X Each atom has 12 equal neighbors The figure at the right shows the face centered character Physics 460 F 2006 Lect 2a 24

(111) planes in an fcc crystal z [111] y direction in cube X C B [111] A direction C in cube B A ABCABC... stacking of hexagonal planes fi fcc crystal fcc is a close packed crtsal – cubic close packed - ccp Physics 460 F 2006 Lect 2a 25

More on stacking hexagonal 2d layers A B C B A B A B A C A C B B B A A A CCP Other polytype HCP • Infinite number of ways to stack planes • Polytypes occur in some metals, some compounds like silicon carbide (SiC) Physics 460 F 2006 Lect 2a 26

Cubic crystals with a basis z y X ZnS Structure with Face Centered Cubic Bravais Lattice NaCl Structure with C, Si, Ge form diamond structure with Face Centered Cubic Bravais Lattice only one type of atom Physics 460 F 2006 Lect 2a 27

NaCl Structure z y X NaCl Structure with Face Centered Cubic Bravais Lattice Physics 460 F 2006 Lect 2a 28

CsCl Structure z y X a 3 a 2 a 1 CsCl Structure Simple Cubic Bravais Lattice From http://www.ilpi.com/inorganic/structures/cscl/index.html Physics 460 F 2006 Lect 2a 29

Atomic planes in NaCl and ZnS crystals z y X (110) plane in ZnS crystal (110) planes in NaCl crystal zig-zag Zn-S chains of atoms rows of the Na and Cl atoms Physics 460 F 2006 Lect 2a 30

(110) plane in diamond structure crystal z y X (100) plane in ZnS crystal Calculated valence electron density zig-zag Zn-S chains of atoms in a (110) plane in a Si crystal (diamond if the two atoms are the same) (Cover of Physics Today, 1970) Physics 460 F 2006 Lect 2a 31

(111) planes in ZnS crystals z [111] y direction Zn C in cube S X Zn B S Zn A S Zn C S B Zn S Zn A S CCP (111) planes in cubic ZnS crystal Note: ABAB... stacking gives hexagonal ZnS Physics 460 F 2006 Lect 2a 32

Perovskite Structure ABO 3 B B B B A O B B B B A atoms have 12 O neighbors Simple Cubic Bravais Lattice B atoms have 6 closer O neighbors Many compounds form the perovskite structure, SrTiO 3 , BaTiO 3 , LaMnO 3 , . . . Physics 460 F 2006 Lect 2a 33

Symmetries of crystals in 3 dimensions • All Crystals can be classified by: • 7 Crystal systems (triclinic, monoclinic, orthorhombic, tetragonal, cubic, hexagonal, trigonal) • 14 Bravais Lattices (primitive, face-centered or body-centered for each system – 14 of the 7x3 possibilities describe all Bravais lattices ) • 32 Points groups (rotations, inversion, reflection) • See references in Kittel Ch 1, G. Burns, “Solid State Physics” Physics 460 F 2006 Lect 2a 34

Is a crystal really different from a liquid? Crystal Liquid Yes – the crystal has “order” – different directions are different Other crucial differences? Yes – dislocations Example of a dislocation - a crystal with an extra plane of atoms on the left Crystal with a “dislocation” - The dislocation can move but it cannot disappear! Important for deformations, … See Kittel Ch. 20 Physics 460 F 2006 Lect 2a 35

36 Physics 460 F 2006 Lect 2a Next Time • Diffraction from crystals Reciprocal lattice • Read Kittel Ch 2 •