ENG2000 Chapter 3
Crystals
ENG2000: R.I. Hornsey
Crystal: 1
Overview of chapter
• In this chapter we seek to understand the types of
crystal structures and their properties
• We also need to describe different directions and
planes in crystals because the properties can be
different in each direction
• There’s no such thing as a ‘perfect’ crystal, so we
will look at how imperfections occur
• Later, we will build on these ideas when we look
at material properties
 semiconductors, magnetism, optical properties etc.
ENG2000: R.I. Hornsey
Crystal: 2
Crystalline Solids
• A crystal is a material in which the atoms
possess perfect ‘long-range order’
 i.e. a repeating or periodic array of infinite dimension
 this array is three-dimensional
 for materials which crystallise, the crystal represents the
minimum overall bonding energy of the system
• Crystals have well-defined chemical, physical and
electronic properties
 theoretically simpler
 uniform and predictable properties
 some properties are unique to crystalline form
• Generally, metals have the simplest crystal forms
ENG2000: R.I. Hornsey
Crystal: 3
Unit cell
• All crystals comprise a
fundamental, repeating block
of atoms
hard sphere
reduced
sphere
 this is called the ‘unit cell’
 for most materials the unit cell is a
parallelepiped with three sets of
parallel faces
 the entire crystal structure can be
constructed from repeated
translations in 3-D of the unit cell
• Several unit cells may be
possible for a given crystal
 the simplest and most symmetric is
usually used
 with atoms at the corners of the cell
ENG2000: R.I. Hornsey
Callister
Crystal: 4
FCC
• Many common metals display the face-centred
cubic (FCC) structure
 Cu, Al, Ag, Au
• In the hard sphere representation, the atom cores
on each face touch each other
 hence the unit cell dimension, a, is given by a = 2R√2
 where R is the core diameter
a
ENG2000: R.I. Hornsey
Crystal: 5
How many atoms in a unit cell?
• We have to be careful not to count atoms more
than once
 in FCC the corner atoms are divided between 8
neighbouring unit cells, so only 1/8 of each corner atom is in
any one cell
 but the face atoms are shared between only 2 unit cells
• So the total number is
 (8 X 1/8) corners + (6 x 1/2) face = 4 atoms
ENG2000: R.I. Hornsey
Crystal: 6
Other metrics
• The coordination of an atom (or coordination
number) is the number of other atoms to which it
is bonded
 in FCC, this is 12
 Si has a coordination of 4
 this metric is especially useful when discussing mixtures or
non-crystalline materials
• The atomic packing factor (APF)
 APF = (total sphere volume)/(unit cell volume)
 for FCC, the APF is 0.74
 this is the largest possible for identical spherical atoms
ENG2000: R.I. Hornsey
Crystal: 7
BCC
• Body-centred cubic (BCC) is
found in materials such as W, Cr,
and Fe
• For the hard (touching) sphere
representation
 a = 4R/√3
 APF = 0.68
• For BCC materials
 the number of atoms in the unit cell is
(8 x 1/8) + 1 = 2
 and the coordination is 8
 note that APF and coordination are
related
ENG2000: R.I. Hornsey
Crystal: 8
HCP
• Hexagonal closepacked (HCP) is
found in Mg, Ti,
Zn
 coordination = 12
 APF = 0.74
 same as FCC
the unit cell
c
a
• Now there are 6
atoms in the unit
cell
 prove it!
http://www.usc.edu/dept/materials_science/MASC110L/hcp.jpg
ENG2000: R.I. Hornsey
Crystal: 9
Comparison of metals
metal
structure
Atomic Radius (Å)*
aluminum
FCC
1.431
chromium
BCC
1.249
copper
FCC
1.278
gold
FCC
1.442
iron
BCC
1.241
lead
FCC
1.750
nickel
FCC
1.246
silver
FCC
1.445
titanium
HCP
1.445
tungsten
BCC
1.371
* 1 angstrom (Å) - 10-10m = 0.1nm
ENG2000: R.I. Hornsey
Crystal: 10
Silicon Unit Cell
• Unit cells can contain even more atoms
 silicon has 8
 one especially important consequence of more complex unit
cells is that the density of atoms on a surface (and hence
surface properties) depends on how the surface cuts through
the unit cell
 we need to be able to describe these planes – coming soon
ENG2000: R.I. Hornsey
http://www.physics.monash.edu.au/~adamf/images/silicon.gif
Crystal: 11
• Seven crystal
systems can be
defined according to
their lattice
parameters
z
c



b
y
a
x
Callister
ENG2000: R.I. Hornsey
Crystal: 12
Point coordinates
• In order to describe the directions and planes in a
crystal, a set of coordinates has been developed
 the coordinates of P are qrs (no commas), where q, r, and s
are <1
z
c
P
qa
a
sc
b
y
rb
x
ENG2000: R.I. Hornsey
Crystal: 13
Crystallographic directionsz
• A direction is a vector
between two points. Vectors
should
 pass through the origin (but can
be translated without change)
 the length of the vector projected
onto the axes is determined in
terms of a, b, and c
 these numbers are reduced to
the smallest integer values by
multiplying or dividing by a
common factor (also in units of a,
b, c)
 these three values are given as
[uvw]
ENG2000: R.I. Hornsey
[111]
y
[100]
[???]
x
e.g. [???]:
vector is 0.5a1b0c
multiply through by 2 x (a, b, or c)
gives [120]
Crystal: 14
Other directions
• e.g.
• Negative directions are indicated 11 1
 also 1 11  is in the opposite (antiparallel) direction to 11 1
• In a particular structure, more than one direction
may have identical structures
 e.g. cubic crystals 100, 1 00 , 010 , 01 0, 001, 00 1
 these is a family of directions, written as <100>
ENG2000: R.I. Hornsey
http://python.rice.edu/~arb/Courses/Images/360dot.gif
Crystal: 15
Crystallographic planes
• Lastly, we can describe planes in a similar
fashion
 using (hkl), also called the Miller indices
• The procedure is as follows
 the plane should not pass through the origin; if it does, either
translate the plane or chose a new origin
 the plane now either intercepts or is parallel with all the axes;
the length of the intercept is determined in multiples of abc
 the reciprocal of these multiples is taken (no intercept gives
and index of 0)
 these indices are reduced by multiplication or division by a
common parameter (in units of abc) to their lowest integers
 these are written (hkl)
ENG2000: R.I. Hornsey
Crystal: 16
Callister
ENG2000: R.I. Hornsey
Crystal: 17
Si [111]
http://www.mse.nthu.edu.tw/jimages/Beuty/Si(111)-7x7%20.jpg
ENG2000: R.I. Hornsey
Crystal: 18
Single crystals
• Perfect single crystals are hard to form
 because impurities or defects are tough to prevent
 single crystal metals – because of the lack of
imperfections – are closest to the ideal mechanical
strength
• Single crystals are fundamental to the
semiconductor industry
 they are drawn from a crucible of molten Si using a
‘seed’ crystal as a template
 the growth rate is typically 1-10µm per second and
the final ingot is about 1.5m long and up to 300mm
in diameter
 the ingot is then trimmed to a circular cross-section
and sliced into wafers, which are then polished
 imperfections in the crystal are measured per cm2,
a typical value being about 10 defects/cm2
ENG2000: R.I. Hornsey
http://www.csc.fi/elmer/examples/czmeltflow/growth.gif
http://www.ami.bolton.ac.uk/courseware/mdesign/ch2/SingleCrystalSiliconIngot.jpg
Crystal: 19
Polycrystalline materials
• As they solidify naturally from the molten state,
materials tend to become polycrystalline
 consisting of many crystal grains, each with a random
orientation, joining at grain boundaries
 this results from the simultaneous growth, and subsequent
coalescence, of crystals growing from multiple starting points
Crystal
grains of
aluminum
oxide
ceramic
Sheet
steel
ENG2000: R.I. Hornsey
http://mimp.mems.cmu.edu/~ordofmag/alumina.jpg
http://www.mse.nthu.edu.tw/jimages/Beuty/Steel1.jpg
Crystal: 20
Amorphous materials
• In contrast to crystals, which
have perfect long-range order,
amorphous materials have no
long-range order
• Locally, the Si atoms still bond
to 4 neighbours
 but the bond lengths and angles
vary randomly about the ideal
values
 so after >100 inter-atomic
distances, the order is lost
• Amorphous materials are
effectively ‘frozen liquids’
 obtained if a liquid is cooled too
rapidly to allow crystal formation
the continuous random
network of amorphous silicon
http://www.research.ibm.com/amorphous/figure1.gif
ENG2000: R.I. Hornsey
Crystal: 21
Applications
• Both amorphous and polycrystalline
semiconductors find applications in electronics
• Single crystals must be formed from a single
‘seed’, and so cannot be formed on other
substrates (e.g. glass)
 so large-area devices, such as active matrix LCD displays,
must be constructed from amorphous silicon
 polycrystalline Si has better electrical properties but requires
higher temperatures to form which distorts the glass
ENG2000: R.I. Hornsey
Crystal: 22
Imperfections in solids
• As we mentioned before, effectively all crystals
include imperfections
 these can dominate the properties of the material in both
desirable and undesirable ways
• The addition of impurities (i.e. other substances)
is vital in metallurgy and microelectronics
 e.g. sterling silver = 92.5% silver + 7.5% copper is much
harder than pure silver
 addition of B or P to Si drastically alters the Si electrical
properties
 we will get to these later in the course
• For the moment, we will discuss physical defects
ENG2000: R.I. Hornsey
Crystal: 23
Point defects
• Vacancies – missing atoms – are present in all
materials
 the number is given by NV = N exp (-QV/kT)
 where N is a constant, QV is the energy required to create a
vacancy, k is Boltzmann’s constant (1.38 x 10-23 J/K) and T is
the absolute temperature
 for a metal just below melting, there is 1 vacancy for every
~104 atoms
self-interstitial
(low probability in metals)
vacancy
interstitial
ENG2000: R.I. Hornsey
substitution
Crystal: 24
Impurities
ENG2000: R.I. Hornsey
Crystal: 25
Impurities
• Because of the relatively ‘free-and-easy’ bonding
structure of metals, mixtures of elements – alloys
– are straightforward
 the maximum purity achievable is ~99.9999%, or 1 in a
million atoms is foreign
• An alloy is effectively a solid solution
 the solvent is the species with the highest concentration
 the solute is the lower concentration element
• There are two possibilities for forming the
solution
 substituting one atom for another
 the solute atom fits in the interstitial site
ENG2000: R.I. Hornsey
Crystal: 26
Substitution
• Solubility for substitutional impurities depends
on a number of factors
 relative sizes of atoms - typically limited to ±15% for high
solubility
 crystal structure – should be similar for high solubility
 electronegativity – should be similar for high solubility
 valence – a metal dissolves easier in another metal of lower
valency that higher valency
• Copper/nickel is the example system where there
is excellent solubility




RNi = 1.25Å, RCu = 1.28Å
both are FCC
electronegativities are almost equal
valence for Ni is +2, for Cu is +1
ENG2000: R.I. Hornsey
Crystal: 27
Interstitial
• Since the metal packing densities are relatively
high, the interstices are small
 so only small atoms can dissolve in this way
 even then, typically only 10% impurities can be added before
the strain induced is too high
• Carbon is an interstitial impurity in iron
 up to about 2%
 RFe = 1.24Å, RC = 0.71Å
ENG2000: R.I. Hornsey
Crystal: 28
Specification of composition
• The composition of an allow can be specified in
two principal ways
• Weight percent (wt%)
 wt%1 in 2 = m1/(m1 + m2) x 100
 where m is the mass of each element
• Atom percent (at%)
 at% is the number of moles of one element as a fraction of
the total number of moles
 the number of moles of material 1 is nm1 = m’1/A1, where m’
is the mass (in g) and A1 is the atomic weight for material 1
 at%1 = nm1/(nm1 + nm2) x 100
ENG2000: R.I. Hornsey
Crystal: 29
‘Mechanical’ defects
ENG2000: R.I. Hornsey
Crystal: 30
Edge dislocations - linear defects
• An edge dislocation occurs when there is an
extra crystal plane
copper sulphide
cactus!
ENG2000: R.I. Hornsey
http://pilot.mse.nthu.edu.tw/tem/gallery/Tem-11.JPG h
ttp://www.mse.nthu.edu.tw/jimages/Beuty/
Crystal: 31
Burgers vector
• The direction and magnitude of a dislocation is
expressed in terms of the ‘burgers vector’
 “If you imagine going around the dislocation line, and exactly
going back as many atoms in each direction as you have
gone forward, you will not come back to the same atom
where you have started
 the Burgers vector points
from start atom to the
end atom of your journey”
 for the edge dislocation
here, the Burgers vector
is perpendicular to the
dislocation line
ENG2000: R.I. Hornsey
http://www.iap.tuwien.ac.at/www/surface/STM_Gallery/Burgers_circuit.jpg
Crystal: 32
http://www.uet.edu.pk/dmems/EdgeDislocation.gif
ENG2000: R.I. Hornsey
Crystal: 33
Screw dislocation
• In screw dislocations, the atom planes look like
they have been ‘sheared’
• The Burgers vector is parallel to the line of the
dislocation
Burgers vector
350Å
http://www.iap.tuwien.ac.at/www/surface/STM_Gallery/screw_disl_schem.gif
http://nano.phys.uwm.edu/li/new_pa4.jpg
ENG2000: R.I. Hornsey
GaN
Crystal: 34
Interfacial defects
• It is worth noting that any surface or interface is
an imperfection
 surface – dangling bonds that would otherwise have been
occupied with other atoms lead to non-bulk electronic and
mechanical effects at the surface (similar to surface tension
in liquids)
 grain boundaries
 atomic vibrations – only a perfect crystal at 0 kelvin!
ENG2000: R.I. Hornsey
Crystal: 35
How do we ‘see’ atoms?
• One cannot observe anything smaller than the
wavelength of the illumination
 ~500nm for visible light
• So how do we see atoms of size ~ 0.1nm?
 not with light – this is in the x-ray part of the EM spectrum
• One possibility is an electron microscope
 electrons have a wavelength that is inversely proportional to
their energy, which depends on the acceleration voltage
 energies in the range MeV are possible – what wavelength
doe this correspond to?
• Another possibility is the scanning tunnelling
microscope …
ENG2000: R.I. Hornsey
Crystal: 36
For a small insulating gap, a
current can flow because of
the electron probability
function.
If the gap is small enough,
there is a finite possibility
that the electron is
transmitted to the other side
of the gap.
This is called tunnelling.
http://jmaps.d.umn.edu/images/stm/stm1.gif
ENG2000: R.I. Hornsey
Crystal: 37
Summary
• Unit cells
 FCC, BCC, HCP
• Coordinates, directions and planes
 Miller indices
• Polycrystalline and amorphous materials
• Impurities
 solid solutions
 substitutions, interstitial
• Dislocations
 edge, screw, Burgers vector
ENG2000: R.I. Hornsey
Crystal: 38