Previous Lecture - - - - Next Lecture

Physics of Thin Films

PES 449 / PHYS 549

Structure of Solids: ideal solids

Ohring: Chapter 1, Section 1.2

Classify solids

  • crystalline
    • atoms show short and long range order
  • amorphous (non-crystalline)
    • atoms show short range order only


We will study crystalline solids first.

Crystal structure

crystals have two parts
  1. lattice - regular periodic array of points in space
  2. basis - a group of atoms located at each point in the lattice


in three dimensions: only 14 unique lattices (Bravais lattices)

examples: (NOTE: the spheres are just points in space - NOT atoms !)

cubic structure models

lattice parameter = ao = length of a cube side

see Figure 1-1 for other lattices


number and arrangement of atoms we put at each lattice point


  • Cr
    • bcc lattice
    • one Cr atom basis
    • ao = 2.88 Angstroms
  • Cu
    • fcc lattice
    • one Cu atom basis
    • ao = 3.615 Angstroms
  • Si
    • fcc lattice
    • two Si atom basis
      • one Si atom at (0,0,0)
      • other Si atom at (1/4, 1/4, 1/4)a
    • ao = 5.43 Angstroms


For thin films, we are interested in surfaces.

=> cut crystals in different ways.

Describe surfaces by Miller indices.

General Procedure
Specific Example

1. Cut crystal along some plane.

model cutting along a cube face

cut along a face of a cube

2. Determine x, y, z, intercepts

(xo, yo, zo)
(1, infinity, infinity)

3. Take reciprocals and reduce to smallest integer.

equation of general Miller indices
(100) plane

[if we had used the (200) plane, reduce to (100)]

See Figure 1-3 for other examples.

Other cubic surfaces:

planes in cubic lattices

What would these surfaces look like ?

surfaces in cubic crystals

Note how open some surfaces are packed and how dense others are packed.

Different surfaces have different arrangements of atoms => different properties.

[crystal models demonstration]

Close packed planes:

Two surfaces produce the maximum possible density for packing spheres. The surface layers and second layers of atoms are identical, but the positions of atoms in the third planes are different.

  • hcp (0001) = base plane of the hexagonal close packed structure
    • ABABAB repeat pattern
  • fcc (111) = diagonal cut through the face centered cubic sructure (shown above)
    • ABCABCABC repeat pattern

[see handout figures]

Crystal Directions

define a vector [h k l] between two lattice points

when the direction is negative, instead of a minus sign, use a bar over the number.

angle (a) between any two vectors [h1 k1 l1] and [h2 k2 l2] :

equation for angle between vectors

for cubic lattices:

  • [h k l] is normal vector to (h k l) plane
  • spacing (d) between (hkl) planes is given by:

    distance between cubic planes

    where ao is the length of one side of the cube face

    example: spacing between (100) planes in a simple cubic lattice:

    equation for distance between 100 planes

Non-crytalline (amorphous) solids


not stable state for most pure metals

can be formed by very rapid cooling (106 K/sec)

readily formed from many metal alloys, semiconductors, oxides - especially at low temperatures

generally less dense than crystalline materials

no crystalline defects since no crystal structure

Previous Lecture Previous Lecture - - - INDEX OF LECTURES - - - Next Lecture Next lecture