Solid State Electronics EC 210 EC 211 Prof

Solid State Electronics EC 210 – EC 211 Prof. Dr. Iman Gamal Eldin Morsi 1

Miller Indices • It is often necessary to be able to specify certain directions and planes in crystals. • Many material properties and processes vary with direction in the crystal. • Directions and planes are described using three integers -Miller Indices. 2

Point coordinates • Point position specified in terms of its coordinates as fractional multiples of the unit cell edge lengths. ( 1 a, 1 b, 1 c) example: (1 a, 1 b, 1 c) ( 0 a, 0 b, 0 c ) a, b, c are called unit measurements a=b=c 1 c 1 a 1 b ( 0 a, 0. 5 b, 0 c) 3

General Rules for Lattice Directions, Planes & Miller Indices Miller indices used to express lattice planes and directions x, y, z are the axes. a, b, c are lattice parameters ( a = b = c for cubic unit cell ) h, k, l are the Miller indices for planes and directions expressed as planes: ( hkl ) and directions: [ hkl ]. • • Conventions for naming – There are NO COMMAS between numbers – Negative values are expressed with a bar over the number • • Example: -2 is expressed • • 4

Miller Indices for Directions • Find the name of direction which passes by the point Solution: Ø Point : Ø Drop the unit measurements Ø Ø Remove fractions by multiplying by the least common factor Ø LCF = 60. Ø Miller indices : 45 24 Ø Enclose in square brackets [ 45 24 ] 5

Miller Indices for Directions • How to sketch a given direction • For example [ 45 24 ] Solution: Ø Miller indices: 45 24 Ø Coordinates: 45 a -20 b 24 c [ 45 24 c Z a 24 ] -20 b ------------------45 a X 0 Y 6

Families of Directions • Equivalence of directions • <123> Family of directions • [123], [213], [312], [132], [231], [321] , ………etc only in a cubic crystal • In the cubic system directions having the same indices regardless of order or sign are equivalent. 7

Miller Indices for Planes • Find the name of the plane which intercepts the 3 axes x, y, z at 2 a , ⅓ b , -⅕c respectively. Solution: Ø intercepts : 2 a ⅓b -⅕c Ø Drop the unit measurements Ø : 2 ⅓ -⅕ Ø Reciprocals: ½ 3 -5 Ø Remove fractions by multiplying by the least common factor LCF = 2. Ø Miller indices : 1 6 Ø Enclose in parentheses: [ 1 6 ] 8

Miller Indices for Planes • How to sketch a given plane • For example ( 1 6 ) Solution: Ø Miller indices: 1 6 Ø Reciprocals: 1 a ⅙b c Ø intercepts: 1 a ⅙b c Z ⅙b 1 a Y X (1 6 ) 9

Cont’d 10

Families of planes • ( hkl ) name of plane • { hkl } Family of planes – e. g. (hkl), (lhk), (hlk) … etc. only in the cubic crystal • Planes having the same indices regardless of order or sign are equivalent 11

Note • Parallel planes have the same name (same (hkl) ). • In the cubic system, a plane and a direction with the same indices are orthogonal 12

Diamond cubic crystal structure and planes. Determine what portion of a blackcolored atom belongs to the plane that is hatched.

Distance between parallel planes • Parallel planes have the same name same ( hkl ). Distance between parallel planes can be Lattice obtained by the formula parameter 14

Angle between planes • Angle between planes can be obtained by the formula 15