ENG2000 Chapter 3 Crystals - York University · 2002-09-17 · ENG2000: R.I. Hornsey Crystal: 2...

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 andplanes in crystals because the properties can bedifferent in each direction

• There’s no such thing as a ‘perfect’ crystal, so wewill look at how imperfections occur

• Later, we will build on these ideas when we lookat 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 theminimum overall bonding energy of the system

• Crystals have well-defined chemical, physical andelectronic 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 blockof atoms§ this is called the ‘unit cell’

§ for most materials the unit cell is aparallelepiped with three sets ofparallel faces

§ the entire crystal structure can beconstructed from repeatedtranslations in 3-D of the unit cell

• Several unit cells may bepossible for a given crystal§ the simplest and most symmetric is

usually used

§ with atoms at the corners of the cellCallister

hard spherereducedsphere


FCC• Many common metals display the face-centred

cubic (FCC) structure§ Cu, Al, Ag, Au

• In the hard sphere representation, the atom coreson 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 inany 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 itis bonded§ in FCC, this is 12

§ Si has a coordination of 4

§ this metric is especially useful when discussing mixtures ornon-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) sphererepresentation§ 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 arerelated


HCP• Hexagonal close-

packed (HCP) isfound in Mg, Ti,Zn§ coordination = 12

§ APF = 0.74

§ same as FCC

• Now there are 6atoms in the unitcell§ prove it!

the unit cell

a

c

http://www.usc.edu/dept/materials_science/MASC110L/hcp.jpg


Comparison of metals

1.371BCCtungsten

1.445HCPtitanium

1.445FCCsilver

1.246FCCnickel

1.750FCClead

1.241BCCiron

1.442FCCgold

1.278FCCcopper

1.249BCCchromium

1.431FCCaluminum

Atomic Radius (Å)*structuremetal

* 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 unitcells is that the density of atoms on a surface (and hencesurface properties) depends on how the surface cuts throughthe unit cell

§ we need to be able to describe these planes – coming soon

http://www.physics.monash.edu.au/~adamf/images/silicon.gif


• Seven crystalsystems can bedefined according totheir latticeparameters

x

y

z

a

b

c

αβ

γ

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. Vectorsshould§ pass through the origin (but can

be translated without change)

§ the length of the vector projectedonto the axes is determined interms of a, b, and c

§ these numbers are reduced tothe smallest integer values bymultiplying or dividing by acommon 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 directionmay 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 givesand index of 0)

§ these indices are reduced by multiplication or division by acommon 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 ofimperfections – are closest to the ideal mechanicalstrength

• Single crystals are fundamental to thesemiconductor 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 andthe final ingot is about 1.5m long and up to 300mmin diameter

§ the ingot is then trimmed to a circular cross-sectionand 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 subsequentcoalescence, of crystals growing from multiple starting points

http://mimp.mems.cmu.edu/~ordofmag/alumina.jpghttp://www.mse.nthu.edu.tw/jimages/Beuty/Steel1.jpg

Crystalgrains ofaluminumoxideceramic

Sheetsteel


Amorphous materials• In contrast to crystals, which

have perfect long-range order,amorphous materials have nolong-range order

• Locally, the Si atoms still bondto 4 neighbours§ but the bond lengths and angles

vary randomly about the idealvalues

§ so after >100 inter-atomicdistances, the order is lost

• Amorphous materials areeffectively ‘frozen liquids’§ obtained if a liquid is cooled too

rapidly to allow crystal formation

http://www.research.ibm.com/amorphous/figure1.gif

the continuous randomnetwork 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 othersubstrates (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 requireshigher 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 electricalproperties

§ 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 avacancy, k is Boltzmann’s constant (1.38 x 10-23 J/K) and Tis 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 thesolution§ 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 lowervalency that higher valency

• Copper/nickel is the example system where thereis 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 beforethe 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 havegone forward, you will not come back to the same atomwhere you have started

§ the Burgers vector pointsfrom start atom to theend atom of your journey”

§ for the edge dislocationhere, the Burgers vectoris perpendicular to thedislocation 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 thedislocation

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 andmechanical effects at the surface (similar to surface tensionin 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 wavelengthdoe this correspond to?

• Another possibility is the scanning tunnellingmicroscope …

ENG2000: R.I. Hornsey Crystal: 37http://jmaps.d.umn.edu/images/stm/stm1.gif

For a small insulating gap, acurrent can flow because ofthe electron probabilityfunction.

If the gap is small enough,there is a finite possibilitythat the electron istransmitted to the other sideof 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 Chapter 3 Crystals - York University · 2002-09-17 · ENG2000: R.I. Hornsey Crystal: 2...

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