eng2000 chapter 3 crystals
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ENG2000: R.I. Hornsey Crystal: 2
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: 3
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
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Unit cell• All crystals comprise a
fundamental, repeating block of atoms 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
Callister
hard spherereducedsphere
ENG2000: R.I. Hornsey Crystal: 5
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: 6
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: 7
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: 8
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: 9
HCP• Hexagonal close-
packed (HCP) is found in Mg, Ti, Zn coordination = 12 APF = 0.74 same as FCC
• Now there are 6 atoms in the unit cell prove it!
the unit cell
a
c
http://www.usc.edu/dept/materials_science/MASC110L/hcp.jpg
ENG2000: R.I. Hornsey Crystal: 10
Comparison of metalsmetal 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: 11
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
http://www.physics.monash.edu.au/~adamf/images/silicon.gif
ENG2000: R.I. Hornsey Crystal: 12
• Seven crystal systems can be defined according to their lattice parameters
x
y
z
a
b
c
Callister
ENG2000: R.I. Hornsey Crystal: 13
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
x
y
z
a
b
c
P
qa
rb
sc
ENG2000: R.I. Hornsey Crystal: 14
Crystallographic directions• 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]
[111]
[???][100]
x
y
z
e.g. [???]:
vector is 0.5a1b0c
multiply through by 2 x (a, b, or c)
gives [120]
ENG2000: R.I. Hornsey Crystal: 15
Other directions• e.g.
• Negative directions are indicated also is in the opposite (antiparallel) direction to
• In a particular structure, more than one direction may have identical structures e.g. cubic crystals these is a family of directions, written as <100>
http://python.rice.edu/~arb/Courses/Images/360dot.gif
€
11 1[ ]
€
1 11 [ ]
€
11 1[ ]
€
100[ ], 1 00[ ], 010[ ], 01 0[ ], 001[ ], 001 [ ]
ENG2000: R.I. Hornsey Crystal: 16
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: 18
Si [111]
http://www.mse.nthu.edu.tw/jimages/Beuty/Si(111)-7x7%20.jpg
ENG2000: R.I. Hornsey Crystal: 19
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
http://www.csc.fi/elmer/examples/czmeltflow/growth.gifhttp://www.ami.bolton.ac.uk/courseware/mdesign/ch2/SingleCrystalSiliconIngot.jpg
ENG2000: R.I. Hornsey Crystal: 20
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
http://mimp.mems.cmu.edu/~ordofmag/alumina.jpghttp://www.mse.nthu.edu.tw/jimages/Beuty/Steel1.jpg
Crystal grains of aluminum oxide ceramic
Sheet steel
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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
http://www.research.ibm.com/amorphous/figure1.gif
the continuous random network of amorphous silicon
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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: 23
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: 24
vacancy
substitution
self-interstitial(low probability in metals)
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
interstitial
ENG2000: R.I. Hornsey Crystal: 26
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: 27
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: 28
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: 29
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
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Edge dislocations - linear defects• An edge dislocation occurs when there is an
extra crystal plane
http://pilot.mse.nthu.edu.tw/tem/gallery/Tem-11.JPG http://www.mse.nthu.edu.tw/jimages/Beuty/
copper sulphidecactus!
ENG2000: R.I. Hornsey Crystal: 32http://www.iap.tuwien.ac.at/www/surface/STM_Gallery/Burgers_circuit.jpg
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 dislocationhere, the Burgers vector is perpendicular to the dislocation line
ENG2000: R.I. Hornsey Crystal: 34
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
350Å
GaNhttp://www.iap.tuwien.ac.at/www/surface/STM_Gallery/screw_disl_schem.gifhttp://nano.phys.uwm.edu/li/new_pa4.jpg
Burgers vector
ENG2000: R.I. Hornsey Crystal: 35
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: 36
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: 37http://jmaps.d.umn.edu/images/stm/stm1.gif
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.
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