Crystal Structure

Crystal Properties of Semiconductors

S

d fication of a polycrystalline solid from the melt. (a)Nucleation. (b) row h. (c) The solidified polycrystalline solid. Forsimplicity, cubes represent atoms.

(b)

Grain boundaries cause scattering of the electron andtherefore add to the resistivity by Matthiessen's rule. For a verygrainy solid, the electron is scattered from grain boundary to grainboundary and the mean free path is approximately equal to themean grain diameter.

Grain 1

Grain 2

GrainBoundary

(a)

Crystal Properties of Solid

Polycrystalline

Strained bond

Broken bond (danglingbond)

Grain boundary

Void, vacancySelf-interstitial type atomForeign impurity

The grain boundaries have broken bonds, voids, vacancies,strained bonds and "interstitial" type atoms. The structure of the grainboundary is disordered and the atoms in the grain boundaries have higherenergies than those within the grains.

Crystal Properties of Solid

Polycrystalline

Examples of Crystals

Snow Quartz Copper oxide

Salt (NaCl) crystal Gold (Au) crystals at 1000 C

Salt (NaCl) crystal

Examples of Crystals

Examples of Crystals

Fullerene

TEM image ofCarbon Nanotube

CarbonNanotube

CarbonNanofiber

Examples of Crystals

Single crystal Diamonds.

Single crystal Silicon.

Atomic Resolution Images of Solid Surfaces

Silicon (Si) surface Iron silicide surface

STM (Scanning Tunneling Microscope) images of solid surface

Silicon (Si) surface

3D-STM (Scanning Tunneling Microscope) images of solid surface

Hydrogen bonds on a Silicon surface.

Atomic Resolution Images of Solid Surfaces

TEM (Tunneling Electron Microscope) images of solid surface

High resolution image of a quasiperiodical

grain boundary in gold.

Atomic Resolution Images of Solid Surfaces

Lattice : The periodic arrangement of the atoms.

Unit Cell: Representative of the entire lattice and is regularly repeated throughout the crystal.

Primitive Cell: Smallest unit cell which can be repeated to form the lattices.

Crystal Structures and Definitions

Primitive CellUnit Cell

Each crystal built up of a repetitive stacking of unit cells each identical in size, shape, and orientation with every other one.

a/2a

Coordinates of position in the unit cell

x, y, z expressed in terms of the unit cell edges.

Example

reached by moving along the axis a distance of 3x the length of the vector , the parallel to , a distance 2 , and finally parallel to , a distance equal to the length of .  

czbyaxr xyz

r321

a b

b

c c

Crystal Structures and Definitions

Crystal Lattice Group

Triclinic abc 90 K2CrO7

Monoclinic abc ==90 -S, CaSO42H2OOrthorhombic abc ===90 -S, Ga, Fe3CTetragonal a=bc ===90 -Sn, TiO2

Cubic a=b=c ===90 Cu, Ag, Zn, NaClHexagonal a1=a2=a3c ==90, =120 Zn, CdRhombohedral a=b=c ==90 As, Sb, Bi

Bravais lattices

Length and Angle c

x

y

c

b

b

aa

O

Unit Cell Geometry

z

                   Monoclinic

a≠b≠c, ==90° ≠90°

                   Monoclinic

a≠b≠c, ==90° ≠90°

                   Orthorhombic

a≠b≠c, ===90°

                   Orthorhombic

a≠b≠c, ===90°

                   Orthorhombic

a≠b≠c, ===90°

Triclinic a≠b≠c, ≠≠≠90°

Crystal (Bravais) Lattice Group (I)

                   Orthorhombic

a≠b≠c, ===90°

                     Hexagonal

a1=a2=a3≠c, ==90° =120°

                 Rhombohedral

a=b=c, ==≠90°

Crystal (Bravais) Lattice Group (II)

                         Tetragonal

a=b≠c, ===90°

                         Tetragonal

a=b≠c, ===90°

                          Cubic

a=b=c, ===90°

                          Cubic

a=b=c, ===90°

                          Cubic

a=b=c, ===90°

Crystal (Bravais) Lattice Group (III)

Face centered cubic

Sim ple cubic Body centered cubic

Sim plem onoclinic

Sim pletetragonal

Body centeredtetragonal

Sim pleorthorhom bic

Body centeredorthorhom bic

Base centeredorthorhom bic

Face centered orthorhom bic

Rhom bohedralHexagonal

Base centeredm onoclinic

Triclinic

UNIT CELL GEOMETRY

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

CUBIC SYSTEMa = b = c 90°

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

TETRAGONAL SYSTEMa = b c ===90°

In, Sn, Barium Titanate, TiO2

ORTHORHOMBIC SYSTEMa b c ===90°

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

HEXAGONAL SYSTEMa = b c = = 90° ; = 120°

Cadmium, Magnesium, Zinc,Graphite

RHOMBOHEDRAL SYSTEMa = b = c = = 90°

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

TRICLINIC SYSTEMa b c 90°

Potassium dicromate

MONOCLINIC SYSTEMa b c = = 90° ; 90°

Selenium, PhosphorusLithium SulfateTin Fluoride

Miller Convention Summary

Convention Interpretation

(hkl) Crystal Plane {hkl} Equivalent Planes [hkl] Crystal Direction <hkl> Equivalent Directions

plane {111}: (111) (-111) (1-11) (11-1) direction <111>: [111] [-111] [1-11] [11-1]

Examples

Identification of a plan in a crystal

Crystal Planes

Miller Indices (hkl)

11/2

11

1 (210)

z intercept at

a

b

c

x

y

x intercept at a/2

y intercept at bUnit cell

z

c

x

y

c

b

b

aa

O

Unit Cell Geometry

z

ab

c

z

yyoxo

Pzo [121]

Identification of a plane and direction in a crystal

Crystal Planes

Crystal Planes

Miller Index

Examples

Miller Index

Examples

Miller Index

Crystal Planes in the Cubic Lattice

y(111)

z

y

x

z

x

(110)z

y

y

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

x

(100)

(001) (110)

(010)

x

z

y

(111)

Various planes in cubic lattice

The value of d, the distance between adjacent planes in the set (hkl), may be found from the following equations

Cubic :

Tetragonal :

Hexagonal :

lkhad

222

2

cl

akh

d 2

2

2

22

21

2

2

2

22

2 341

cl

akhkh

d

Interplanar spacing

Crystal Planes

X-Ray Diffraction

Crystal Planes

Each set of planes has a specific interplanar distance and will give rise to a characteristic angle of diffracted X-rays.

The relationship between wavelength, atomic spacing (d) and angle was solved as the Bragg Equation.

nsind 2Bragg’s Law

Single between (h1 k1 l1) of sparing d, and the plane (h2 k2 l2), of spacing, may be found from the followings.

Cubic : cos =

Tetragonal : cos =

Hexagonal : cos =  

))(( 2

2

2

2

2

2

2

1

2

1

2

1

212121

lkhlkhllkkhh

))(( 2

2

2

2

2

2

22

2

12

2

1

2

1

221

22121

ch

akh

cl

akh

callkkhh

)22)((

)(21

2

22

22

2

2

2

2

12

2

112

1

2

1

212

2

1212121

43

43

43

1

lcakhlc

akh

ca

khkh

llkhkhkkhh

Crystal Planes

Interplanar Angles

[010]

[100]

[001]

[010]

[110]

[111]

[110]

a y

ax

y

[111]

[111] [111]

[111]

[111]

[111]

[111]

[111] F a m ily o f < 1 1 1 > d ir e c t io n s

Crystal Directions

Crystal Directions in Cubic Crystal System

SC (Simple Cubic) Atoms situated at the corners of the unit cell. Atoms touch along <100> and a = 2r (r = atomic radius)

BCC (Body-Centred Cubic) Atoms situated at the corners of the unit cell and at the centre. Atoms touch along <111> and a = 4r/3

FCC (Face-Centred Cubic) Atoms situated at the corners of the unit cell and at the centre of

all cubic faces. Atoms touch along <110> and a = 2r/2

Cubic Lattices

Tightest Way to Pack Spheres (I)

ABC stackingSequence

(FCC)

ABAB stackingSequence

(HCP)

other close packed structures, ABABCAB… etc.

Tightest Way to Pack Spheres (II)

ABC stackingSequence

(FCC)

ABAB stackingSequence

(HCP)

Cubic Structures

a : lattice constant

(a) Simple Cubic (b) Body-Centered CubicBCC

(C) Face-Centered CubicFCC

Cubic Lattices Atoms situated at the corners of the unit cell.

Characteristics of Cubic Lattices

Simple BCC FCC

Volume of cubic cell a3 a3 a3

Volume of primitive cell a3 1/2a3 1/4a3

Type of primitive cell SC rhombohedral rhombohedral

Lattice points per cubic cell 1 2 4 Lattice points per unit cell 1/a3 2/a3 4/a3

Nearest neighbour distance a 1/23a 1/22a # of nearest neighbours 6 8 12 Next nearest neighbour distance 2 a a a # of next nearest neighbours 12 6 6

Crystal Structure Model

Crystal Structure Model

(a) Simple CubicSC

(b) Body-Centered CubicBCC

(C) Face Centered CubicFCC

Hard Sphere Model Assume that the atoms are considered as hard spheres

a

a

a

(c )

2R

(b )

a

FC C U nit C ell(a )

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

Crystal Structure Model

FCC Lattices

Crystal Structure Model

BCC Lattices

a

Body centered cubic (BCC) crystal structure. (a)unit cell with closely packed hard spheres representing the Featoms. (b) A reduced-sphere unit cell.

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

(a) (b)

Atomic Packing Factor

Simple Cubic and FCC Lattices

Volume of unit cell

Volume of atoms

Volume of atoms

Volume of unit cell

Number of atoms

Number of atoms

Atomic Packing Factor

Four Cubic Lattices

Of the 18 atoms shown in the figure, only 8 belong to the volume ao3.

Because the 8 corner atoms are each shared by 8 cubes, they contribute a total of 1 atom; the 6 face atoms are each shared by 2 cubes and thus contribute 3 atoms, and there are 4 atoms inside the cube. The atomic density is therefore 8/ao

3, which corresponds to 17.7, 5.00, and 4.43 X 1022 cm-3, respectively.

Semiconductor Lattice Structures

Diamond Lattices

The diamond-crystal lattice characterized by four covalently bonded atoms. The lattice constant, denoted by ao, is 0.356, 0.543 and 0.565 nm for diamond, silicon, and germanium, respectively. Nearest neighbors are spaced ( ) units apart.4/3 oa

(After W. Shockley: Electrons and Holes in Semiconductors, Van Nostrand, Princeton, N.J., 1950.)

Semiconductor Lattice Structures

Diamond Lattices

Semiconductor Lattice Structures

Diamond and Zincblende Lattices

Diamond latticeSi, Ge

Zincblende latticeGaAs, InP, ZnSe

Diamond lattice can be though of as an FCC structures with an extra atoms placed at a/4+b/4+c/4 from each of the FCC atoms

The Zincblende lattice consist of a face centered cubic Bravais point lattice which contains two different atoms per lattice point. The distance between the two atoms equals one quarter of the body diagonal of the cube.

Semiconductor Lattice Structures

Diamond and Zincblende Lattices

Diamond latticeSi, Ge

Zincblende latticeGaAs, InP, ZnSe

Arrangement of atoms on various crystal surfaces.

Crystal Surfaces and Atomic Arrangement

Low Miller Index Planes of Cubic Lattice

BCC

FCC

(100) (110)

(111)

(110)

(100)

(111)

Low Miller Index Planes Diamond Lattice

Diamond Lattice Structures Number of atoms per unit cell : 8Atomic packing factor : 0.34maximum packing density is 34 %.

Arrangement of atoms in Diamond lattice structures on various crystal directions.

Crystal Directions and Atomic Arrangement

Moving through Lattice.mov

Actual Crystal Surfaces Observed by Scanning Tunneling Microscope

Silicon (111) surfaceSilicon (100) surface

Common Crystal Structures of Semiconductor

IV Compounds SiC, SiGe

III-V Binary CompoundsAlP, AlAs, AlSb, GaN, GaP, GaAs, GaSb, InP, InAs, InSb

III-V Ternary CompoundsAlGaAs, InGaAs, AlGaP

III-V Quternary CompoundsAlGaAsP, InGaAsP

II-VI Binary CompoundsZnS, ZnSe, ZnTe, CdS, CdSe, CdTe

II-VI Ternary CompoundsHgCdTe

Semiconductor Materials

Semiconductor Materials