ENG2000: R.I. Hornsey Crystal: 1 ENG2000 Chapter 3 Crystals.

ENG2000: R.I. Hornsey Crystal: 1

ENG2000 Chapter 3Crystals


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.


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


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


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


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


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


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


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


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


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


• Seven crystal systems can be defined according to their lattice parameters

x

y

z

a

b

c

Callister


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


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]


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 [ ]


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: 17Callister


Si [111]

http://www.mse.nthu.edu.tw/jimages/Beuty/Si(111)-7x7%20.jpg


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


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


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


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


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


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


Impurities


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


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


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


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


‘Mechanical’ defects


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


http://www.uet.edu.pk/dmems/EdgeDislocation.gif


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


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!


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.


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: 1 ENG2000 Chapter 3 Crystals.

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Transcript of ENG2000: R.I. Hornsey Crystal: 1 ENG2000 Chapter 3 Crystals.