Structure of Solids: defects in solids

Classification of Defects in Crystalline solids

polycrystalline solid - contains several crystalline regions which are randomly oriented
Most crystalline materials have many defects.
Some (like Silicon) can be grown with relatively few defects.

Three types of defects:

1. Planar defects - grain boundaries

interfaces between two single crystal regions of different orientation
atoms at grain boundaries tend to be loosely bound
=> more reactive (corrosion) and accelerated diffusion along grain boundaries
typical grain sizes: 0.01 mm - 100 mm (micron)
How many atoms in a solid are at grain boundaries ?

rough model:
assume grains are all cubes with sides of length l
l = grain size
a = atomic lattice parameter
n = number of atoms in one row of the grain
then, l = na

What would be the result for spherical grains of diameter, l ?

A somewhat better argument using truncated octahedra leads to the fraction being about 3.35 (a/l). (See Christian "Theory of Transformations in Metals and Alloys", p. 332).
example:

for l = 0.1 micron (1000 Angstroms) and a = 3 Angstroms about 10 atoms out of 1000 are at grain boundaries (1 %)
Number of grain boundaries in film (grain size) depends on deposition rate and substrate temperature.

generally:
lower T => smaller grains => many boundaries

hight T => larger grains => fewer boundaries grain size is often proportional to film thicknes
(thinner films tend to have smaller grains)

2. line defects - dislocations

example: edge dislocation - from inserting an extra row of atoms distorts lattice => stresses (compression and tension) very common: often 10¹⁰ - 10¹² dislocations/ cm² in films form from:

film growth process

dislocations in substrate continuing into film

contamination on substrate

3. point defects

self interstitial - extra atom

vacancy - missing atom

substitutional impurity - impurity atom in lattice

interstitial impurity - impurity atom not in regular lattice site

in principle you can eliminate all of these except vacancies
vacancies arise from thermodynamics (entropy)

fraction of vacancies (f)

$equation for fraction of vacancies$ where k_B = Boltzmann's constant = 1.381 x 10^-23 J/K
typically E_f is about 1 eV
at room temperature, f is about 10^-17

point defects often arise from

fast deposition

low substrate temperatures => no time for atoms to move to crystal lattice sites