Types of Crystals

Physical Electronics, Spring 2017 18

1.8 The Crystalline State

Types of Crystals

•Crystalline ; periodicity, long-range order

•Amorphous

•Polycrystalline

“Solids” based on the arrangement of atoms.

Lattice

Basis

Crystal

Unitcell Unitcell

aa

(a) (b) (c)

90°

x

y

Basis placement in unit cell(0,0)

(1/2,1/2)

(d)

(a) A simple square lattice. The unit cell is a square with a side a. (b)Basis has two atoms. (c) Crystal = Lattice + Basis. The unit cell is asimple square with two atoms. (d) Placement of basis atoms in thecrystal unit cell.


2R

a

a

a

a

(c)(b)

FCCUnit Cell(a)

(a) The crystal structure of copper is Face Centered Cubic (FCC). Theatoms are positioned at well defined sites arranged periodically and there isa long range order in the crystal.(b) An FCC unit cell with closed packed spheres.(c) Reduced sphere representation of the FCC unit cell. Examples: Ag, Al,Au, Ca, Cu, γ-Fe (>912°C), Ni, Pd, Pt, Rh

Classification based on the patterns of the Unit Cell

Face-Centered Cubic (FCC)

“Three dimensional repetition of Unit Cell generates the whole crystal structure”

• Four atoms in the unit cell• The volume of the FCC unit cell is 74% full of atoms.• Cu is known as a close-packed crystal structure because the Cu atoms are tend to be packed as closely as possible.•The packing density of 74% is the maximum value possible with identical spheres.



a

a

b

Examples: Alkali metals (Li, Na, K, Rb), Cr,Mo, W, Mn, α-Fe (< 912 °C), β-Ti (> 882 °C).

Body centered cubic (BCC) crystal structure. (a) A BCC unit cellwith closely packed hard spheres representing the Fe atoms. (b) Areduced-sphere unit cell.


Body-Centered Cubic (BCC) Hexagonal Close-Packed (HCP)

•Two atoms in the unit cell•The volume of the BCC unit cell is 68% full of atoms

The Hexagonal Close Packed (HCP) Crystal Structure.

(a) The Hexagonal Close Packed (HCP) Structure. A collection of many Zn atoms. Color difference distinguishes layers (stacks).(b) The stacking sequence of closely packed layers is ABAB (c) A unit cell with reduced spheres (d) The smallest unit cell with reduced spheres.



Diamond Structure•Covalently bonded solids; Si, Ge, diamond, etc.•Eight atoms in the unit cell.

Zinc-Blende Structure

•III-IV compound Semiconductor :AlAs, GaAs, GaP, InP, InSb, ZnS, etc



A silicon ingot is a single crystal of Si. Within the bulk of the crystal, the atoms are arranged on a well-defined periodical lattice. The crystal structure is that of diamond. |Courtesy of MEMC, Electronic Materials Inc.



1¢

Ratio of radii = 1

Nearest neighbors = 6

25¢

1¢

Ratio of radii = 0.75

Nearest neighbors = 4

Unit cell

25¢

1¢

A two-dimensional crystal of pennies and quarters

Packing of coins on a table top to build a two dimensional crystal.

Ionic Solids

“…In ionic solids, the cations(e.g., Na+) and the anions (e.g., Cl-) attract each othernondirectionally. The crystalstructure depends on howclosely the opposite ions canbe brought together and howthe same ions can best avoideach other while maintaininglong-range order…Thesedepend on the relative chargeand size per ion. ”



Unit Cell of NaCl Crystal

A possible reduced sphere unit cell for the NaCl (rock salt) crystal. An alternative unit cell may have Na+ and Cl - interchanged.Examples: AgCl, CaO, CsF, LiF, LiCl, NaF, NaCl, KF, KCl, MgO

CsCl Structure

“When the cations and anions have equal charges and are about the same size, as in the CsCl crystal, the unit cell is called the CsCl structure.”Examples: CsCl, CsBr, CsI, TlCl, TlBr, TlI




Example 1.13 : Cu Crystal (FCC)

(a) # of atoms per unit cell?

1/8 of atom

2R

Half of atomR

a

R a

a

1 18 6 4 atoms per unit cell8 2

× + × =

(b) If R is the radius of the Cu atom, lattice parameter a ?

2 4 2 2 2 2 0.128 0.362a R a R nm= → = = × =

(c)

( )

3 3

33

4 44 4volume of atoms in unit cell 3 3Atomic Packing Factor= 0.74volume of unit cell 2 2

R R

a R

π π× ×= = =

(d) Calculate the atomic concentration (number of atoms per unit volume) in Cu and the density of the crystal given that the atomic mass of Cu is 63.55 g/mol and the radius of the Cu atom is 0.128 nm.

( )22 3

33 7

# of atoms in unit cell 4 4= 8.43 10volume of unit cell 0.362 10

atn cma cm

−

−= = = ×

×

22 33 23

mass of atoms in unit cell 4 mass of a atom 63.55 /= 8.43 10 8.9 /volume of unit cell 6.022 10 /

g mol g cma mol

ρ ×= = × × =

×




c

x

y

c

b

b

a aO αβγ

Unit Cell Geometryz

[010]

[100]

[001]

[010]

[110]

[111]

[110]

-a-yax

y

[111]

[111] [111]

[111]

[111]

[111][111]

[111] Familyof <111>directions

(c) Directions in cubic crystal system

ab

c

z

x

yyoxo

Pzo [121]

Unitcell

(a) A parallelepiped is chosen to describe geometryof a unit cell. We line the x, y and z axes with theedges of the parallelepiped taking lower-left rearcorner as the origin

(b) Identification of a direction in a crystal

Crystal Directions and Planes

Lattice Parameters :

Cu: a=b=c, 90Zn: a=b c, 90 , 120

α β γ

α β γ

= = =

≠ = = =

“Many properties, forexample, the elasticmodulus, electricalresistivity, magneticsusceptibility, etc., aredirectional within thecrystal.”

Directions :

Family of Directions :

[ ] [ ] [ ]100 , 010 , 001 ,100

100 , 010 , 001

⇒



(100)

(001) (110)

(111)

-z

y

x

z

x

(110)z

-yy

(111)

y

z(010) (010) (010)(010)

x

(010)

x

z

y

(b) Various planes in the cubic lattice

Miller Indices (hk) :1 1

11∞

(210)12

z intercept at ∞

a

b

c

x

yx intercept at a/2

y intercept at bUnitcell

z

(a) Identification of a plane in a crystal

Labelling of crystal planes and typical examples in the cubic lattice

Family of Planes

Miller Indices of a plane

0 0 0

0 0 0

1Intercepts , , and are , 1 , and 2

1 1 1Reciprocals , , and are

1 1 1 , , and 2,1,01 12

x y z a b c

x y z

∞

=∞

( ) ( ) ( )( ) ( ) ( ) { }100 , 010 , 001 ,

100100 , 010 , 001

⇒



Example 1.13 : Miller Indices and Planar Concentration

( ) ( )100 22 2 9

2

14 1 2 240.362 10

15.3 /

na a m

atoms nm

−

× += = =

×

=

( )2

110 2 2

1 14 2 24 2 10.8 /2 2

n atoms nma a

× + ×= = =

z

y

y a= −1

2z a=

( )012

x

a

FCC Unit Cell

a

a

2a

(100) plane (110) plane

0 0 0

0 0 0

1Intercepts , , and are , 1 , and 21 1 1Reciprocals , , and are

1 1 1 , , and 0,1,211 2

x y z a a

x y z

∞ −

=∞ −



Allotropy and Carbon

(c) Buckminsterfullerence

(d) Carbon Nanotube

“Certain substances can have more than one crystal structure, iron being one of the best-known examples. This characteristic is termed ploymorphism or allotropy.”

-α-Fe, BCC (below 912 oC; transition temperature)-γ-Fe, FCC (between 912 oC and 1400 oC)-δ-Fe, BCC structure (above 1400 oC)

“Carbon has threecrystalline allotropes:diamond, graphite, andnewly discoveredbuckminsterfullerene(buckyball).”


1.9 Crystalline Defects and Their Significance

“Key mechanical and electrical properties are controlled by the defects involved in the crystalline structures: Point, Line, and Planar Defects”

Point Defects: Vacancies and Impurities

Thermodynamic defects : Vacancies exist as a requirement ofthermal equilibrium

Equilibrium Concentration of Vacancies

nv = N exp −Ev

kT

nv = vacancy concentration

N = number of atoms per unit volume

Ev = vacancy formation energy

(a) Perfect crystal withoutvacancies

(b) An energetic atom at the surface breaksbonds and jumps on to a new adjoining positionon the surface. This leaves behind a vacancy.

(c) An atom in the bulk diffusesto fill the vacancy therebydisplacing the vacancy towardsthe bulk.

(d) Atomic diffusions cause the vacancy todiffuse into the bulk.

Generation of a vacancy by the diffusion of an atom to the surfaceand the subsequent diffusion of the vacancy into the bulk.



(a) A vacancy in the crystal. (b) A substitutional impurity in the crystal. Theimpurity atom is larger than the host atom.

(c) A substitutional impurity inthe crystal. The impurity atomis smaller than the host atom.

(d) An interstitial impurity in the crystal. Itoccupies an empty space between host atoms.

Point defects in the crystal structure. The regions around the pointdefect become distorted; the lattice becomes strained.

Different typesof the point defects.

For example, in Sisemiconductor,some impurities(e.g., As, P, B,etc.) are addedinto the Si crystal to make n- or p-type Si semiconductormaterials.



Frenkel defect

(b) Two possible imperfections caused by ionizedsubstitutional impurity atoms in an ionic crystal.

(a) Schottky and Frenkel defects in an ionic crystal.

Schottky defect

Substitutional impurity.Doubly charged

Point defects in ionic crystals

Point Defects in Ionic Crystal Example 1.15 and 1.16

Vacancy concentrations of metal (Al) and semiconductor (Ge) at the temperature close to its melting point.

Aluminum at 660 ºC

Germanium at 938 ºC

1922 3

23

19 3

0.70 1.6 10 /6.0 10 exp1.38 10 / 933

1.0 10

eV J eVn cmJ K K

cm

υ

−−

−

−

× ×= × × − × × = ×

1922 3

23

13 3

2.2 1.6 10 /4.41 10 exp1.38 10 / 1211

3.1 10

eV J eVn cmJ K K

cm

υ

−−

−

−

× ×= × × − × × = ×



Edgedislocationline(a)Dislocationisalinedefect.Thedislocationshownrunsinto thepaper.

CompressionTension

(b)Aroundthedislocationthereisastrainfieldastheatomicbondshavebeencompressedaboveandstretchedbelowtheislocationline

Dislocation in a crystal is a line defect which is accompanied by latticedistortion and hence a lattice strain around it.

Line Defects: Edge and Screw Dislocations

Edge Dislocation :

Line defect formedbecause an atomicplane terminateswithin the crystalinstead of passingall the way to theend of the crystal.This type of defectnormally forms duringthe crystal growth.



A

D

B

C

Atoms in theupper portion.

Atoms in thelower portion.

Dislocationline

(b)Thescrewdislocationin(a)asviewedfromabove.

(a) Ascrewdislocation inacrystal.

AC

D

Dislocation line

A screw dislocation involves shearing one portion of a perfect crystalwith respect to another portion on one side of a line (AB).

Screw Dislocation :

This is essentially a shearingof one portion of the crystalwith respect to another byone atomic distance.

“Both edge and screwdislocations are generallycreated by stresses resultingfrom thermal andmechanical processing.”



New molecule

Screw dis loca tion a ids crys ta l growth because the newlyarriving a tom can a ttach to two or three a toms ins tead ofone a tom and thereby form more bonds .

A mixed dislocation

Dislocation line



Grain

(c)

Grain boundary

(b)

CrystalliteNuclei

Liquid

(a)

Solidification of a polycrystalline solid from the melt. (a)Nucleation. (b) Growth. (c) The solidified polycrystalline solid.For simplicity, cubes represent atoms.

Planar Defects: Grain Boundary

“When a liquid is cooled to below its freezing temperature, solidification does not occur at every point: rather, it occurs at certain sites called nuclei, which are small crystal-like structures containing perhaps 50 to 100 atoms.”



Strainedbond

Brokenbond(danglingbond)

Grainboundary

Void,vacancySelf-interstitialtypeatomForeignimpurity

The grain boundaries have broken bonds, voids, vacancies, strained bondsand "interstitial" type atoms. The structure of the grain boundary isdisordered and the atoms in the grain boundaries have higher energies thanthose within the grains.

Detailed View of Grain Boundaries

Atoms can diffuse more easily along a grain boundary

Atomic diffusion allows big grains to grow at elevated temperature.

Grain size of polycrystalline silicon is important in poly-Si thin film transistor



Bulk crystal

SurfaceSurface atoms

Reconstructedsurface

OH

AbsorbedOxygen

H2O

OH2

Dangling bond

At the surface of a hypothetical two dimensional crystal, the atomscannot fulfill their bonding requirements and therefore have broken, ordangling, bonds. Some of the surface atoms bond with each other; thesurface becomes reconstructed. The surface can have physisorbed andchemisorbed atoms.

Crystal Surfaces and Surface Properties

Dangling: unbondedReconstructed: neighboring surface atomsAbsorbed: chemisorptionAdsorbed: physical adsorption

At high temp., some of the absorbed foreign surface atoms can diffuse into the crystal volume to become bulk impurities.

Natural Oxide



(a) Stoichiometric ZnO crystal with equal number ofanions and cations and no free electrons.

(b) Non-Stoichiometric ZnO crystal with excess Zn ininterstitial sites as Zn2+ cations.

O2-

Zn2+

"Free" (or mobile)electron within thecrystal.

Stoichiometry and nonstoichiometry and the resulting defectstructure.

Stoichiometry and Nonstoichiometry

Stoichiometric Compounds are those that have an integer ratio of atoms, ex) CaF2, Si3N4, ZnO

0.074R nm→ =0.14R nm→ =

The nonstoichiometric ZnO with excess Zn has Zn2+ cations in interstitial sites and mobile electrons within the crystal, which can contribute to the conduction of electricity.


1.10 Single-Crystal Czochralski Growth

Schematic illustration of the growth of a single-crystal Si ingot by the Czochralski technique.

• Metallurgical-grade Si (98%) : Heating quartzite (pure SiO2) and carbon (coal, coke, and wood chips) at high temperature.

SiC(s) + SiO2(s) Si(s) + SiO(g) + CO(g)

• Electronic-grade Si : 99.99999999999%

Si(s) + 3HCl(g) SiHCl3(g) + H2(g)@ 300oC : form trichlorosilane

SiHCl3 is liquid @ RT Distillation Purification

SiHCl3(g) + H2(g) Si(s) + 3HCl(g): this reaction takes place in a reactor containing a resistance-heated silicon rod, and this is a kind of CVD process.

Single Crystal Silicon Ingot


1.10 Single-Crystal Czochralski Growth

Single crystal Si ingot (about 2m)

Flat(100) Plane

Cut wafer

Ground edge or flat [100] Direction

The crystallographic orientation of the silicon ingot is marked by grounding a flat. Wafers are cut using a rotating annular diamond saw. Typical wafer thickness is 0.6-0.7mm

The diameter of ingot is determined by the temperature gradients, heat losses, and the rate of pull.


1.11 Glasses and Amorphous Semiconductors

Silicon(orArsenic)atomOxygen(orSelenium)atom

(a)AcrystallinesolidreminiscenttocrystallineSiO2.(Density=2.6gcm-3)

(b)Anamorphoussolidreminiscenttovitreoussilica(SiO2)cooledfromthemelt (Density=2.2gcm-3)

Crystalline and amorphous structures illustrated schematically in twodimensions.

“Crystal structure: Periodicityand degree of symmetry, long-range order resulting from a well-defined bond length and relative bond angle”.

Glasses: amorphous solids formed by rapidly cooling or quenching the liquid to temperatures where the atomic motions are so sluggish that crystallization is virtually halted bend and twisted bonds

Glasses and Amorphous Solids



Inert gas pressure

Molten alloyHeater coil

Quartz tube

Rotatingcooledmetal drum

Jet of moltenmetal

It is possible to rapidly quench a molten metallic alloy, thereby bypassing crystallization, and forming a glassy metal commonly called a metallic glass. The process is called melt spinning.

Metallic Glass



Crystalline and Amorphous Silicon

H H

H

HH

H (c) Twodimensional schematic representationof the structureof hydrogenatedamorphoussilicon. Thenumber of hydrogenatoms shownis exaggerated.

Danglingbond

(b) Two dimensional schematicrepresentation of the structure of amorphoussilicon. The structure has voids and danglingbonds and there is no long range order.

(a) Two dimensional schematicrepresentation of a silicon crystal

Silicon can be grown as a semiconductor crystal or as an amorphoussemiconductor film. Each line represents an electron in a bond. A fullcovalent bond has two lines and a broken bond has one line.

Amorphous silicon, a-Si, can be prepared by an electron beam evaporation of silicon. Silicon has a high melting temperature so that an energetic electron beam is used to melt the crystal in the crucible locally and thereby vaporize Si atoms. Si atoms condense on a substrate placed above the crucible to form a film of a-Si.


Hydrogenated amorphous silicon, a-Si:H, is generally prepared by the decomposition of silane molecules in a radio frequency (RF) plasma discharge. Si and H atoms condense on a substrate to form a film of a-Si:H



1.12 Solid Solutions

(a)DisorderedSubstitutionalSolidSolution. Example: Cu-Nialloys ({100}planes)

(c) Interstitial Solid Solution.Example: Small number of Catoms in FCC Fe (austenite).({100} planes)

(b) Ordered Substitutional SolidSolution. Example: Cu-Zn alloyof composition 50%Cu-50%Zn.({110} planes).

Solid solutions can be disordered substitutional, ordered substitutionaland interstitial substitutional. There is only one phase within the alloywhich has the same composition, structure and properties everywhere.

Solid Solubility

Phase : same composition, structure, and property. (cocktail and water/oil )

Solvent and Solute : majority and minority.

HEATDirection of travel

ImpureSolid

Purifiedregion

AB

Melt

C0

ImpureSolidA'B' C0

(b) As the torch travels towards theright, the refrozen solid at B' has CB'where CB < CB' < C0. The impurityconcentration in the melt is now evengreater than CL'

x

C0

CB

CB'

Zone refined region

Impu

rity

conc

entra

tion

(a) Heat is applied locally starting at oneend. The impurity concentration in the re-frozen solid at B is CB < C0. The impurityconcentration in the melt is CL' > C0.

(c) The impurity concentration profile in therefrozen solid after one pass.

x

C0

Zone refined regionC0

106Impu

rity

conc

entra

tion

(d) Typical impurity concentration profileafter many passes.

Zone refining

Impurity content

TemperatureLIQUID

L + S

1412°C CLB

TB CL'B'

TB'

SOLID

CB CB' C0

SOLIDUS

LIQUIDUS

The phase diagram of Si with impurities near the lowconcentration region.


1.13 Bravais Lattice

Face centeredcubic

Simple cubic Body centeredcubic

Simpletetragonal

Body centeredtetragonal

Simpleorthorhombic

Body centeredorthorhombic

Base centeredorthorhombic

Face centeredorthorhombic

RhombohedralHexagonal

RHOMBOHEDRAL SYSTEMa = b = c α = β = γ ≠ 90°

Arsenic, Boron, Bismuth, Antimony, Mercury(<-39°C)

Simplemonoclinic

Base centeredmonoclinic

Triclinic

TRICLINIC SYSTEMa ≠ b ≠ c α ≠ β ≠ γ ≠ 90°

Potassium dicromate

UNIT CELLGEOMETRY

The seven crystal systems (unit cell geometries) and fourteen Bravaislattices.

CUBIC SYSTEMa = b = c α = β = γ = 90°

Many metals, Al, Cu, Fe, Pb. Many ceramics andsemiconductors, NaCl, CsCl, LiF, Si, GaAs

TETRAGONAL SYSTEMa = b ≠ c α = β = γ = 90°

In, Sn, Barium Titanate, TiO2

ORTHORHOMBIC SYSTEMa ≠ b ≠ c α = β = γ = 90°

S, U, Pl, Ga (<30°C), Iodine,Cementite (Fe3C), Sodium Sulfate

HEXAGONAL SYSTEMa = b ≠ c α = β = 90° ; γ =120°

Cadmium, Magnesium, Zinc,Graphite

MONOCLINIC SYSTEMa ≠ b ≠ c α = β = 90° ; γ ≠90°

α−Selenium, PhosphorusLithium SulfateTin Fluoride

Basis

Unit cell

Crystal

a

Lattice

Unit cell

a

Basis placement in unit cell(0,0)

(1/2,1/2)

x

y


Chapter 1. Homework

3, 7, 22, 24, 25, 27, 28

Due day: 1장수업끝나고일주일후수업시작전.

Types of Crystals

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