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Introduction to Crystal Structure and Bonding
Prof.P. Ravindran,
Department of Physics, Central University of Tamil
Nadu, India
http://folk.uio.no/ravi/semi2013
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Fundamental Properties of matter
Matter: - Has mass, occupies space
Mass – measure of inertia - from Newton’s first law of
motion. It is one of the fundamental physical properties.
States of Matter
1. Solids – Definite volume, definite shape.
2. Liquids – Definite volume, no fixed shape. Flows.
3. Gases – No definite volume, no definite shape. Takes the
volume and shape of its container.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Element – one type of atoms
Compound – Two or more different atoms chemically joined.
Constituent atoms (fixed ratios) can be separated only by chemical
means.
Mixture - Two or more different atoms combined. Constituent
atoms (variable ratios) can be separated by physical means.
Solid-State Physics – branch of physics dealing with solids.
Now replaced by a more general terminology - Condensed Matter
Physics. To include fluids which in many cases share same
concepts and analytical techniques.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Classification Of Solids
 Can be classified under several criteria based on
atomic arrangements, electrical properties, thermal
properties, chemical bonds etc.
 Using electrical criterion: Conductors, Insulators,
Semiconductors
 Using atomic arrangements: Amorphous,
Polycrystalline, Crystalline.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Atoms And Bonding
•
•
•
•
•
The periodic table
Ionic bonding
Covalent bonding
Metallic bonding
van der Waals bonding
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Atoms and bonding






In order to understand the physics of semiconductor
devices, we should first learn how atoms bond together to
form the solids.
Atom is composed of a nucleus which contains protons and
neutrons; surrounding the nucleus are the electrons.
Atoms can combine with themselves or other atoms. The
valence electrons, i.e. the outermost shell electrons govern
the chemistry of atoms.
Atoms come together and form gases, liquids or solids
depending on the strength of the attractive forces between
them.
The atomic bonding can be classified as ionic, covalent,
metallic, van der Waals,etc.
In all types of bonding the electrostatic force acts between
charged particles.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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The Periodic Table
8A
1A
2A
Li
Be
3A
4A
5A
6A
7A
He
Na
Mg
B
C
N
O
F
Ne
K
Ca
2B
Al
Si
P
S
Cl
Ar
Rb
Sr
Zn
Ga
Ge
As
Se
Br
Kr
Cs
Ba
Cd
In
Sn
Sb
Te
I
Xe
Fr
Rd
Hg
Ti
Pb
Bi
Po
At
Rn
Groups 3B,4B,5B,6B
7B,8B,1B lie in here
A section of the periodic table
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
The Periodic Table

Ionic solids
Group 1A (alkali metals) contains lithium (Li), sodium (Na),
potassium (K),..and these combine easily with group 7A (halogens) of
fluorine (F), chlorine (Cl), bromine (Br),.. and produce ionic solids of
NaCl, KCl, KBr, etc.

Rare (noble) gases
Group 8A elements of noble gases of helium(He), neon (Ne), argon
(Ar),… have a full complement of valence electrons and so do not
combine easily with other elements.

Elemental semiconductors
Silicon(Si) and germanium (Ge) belong to group 4A.

Compound semiconductors
1) III-V compound s/c’s; GaP, InAs, AlGaAs (group 3A-5A)
2) II-VI compound s/c’s; ZnS, CdS, etc. (group 2B-6A)
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Covalent bonding
 Elemental semiconductors of Si, Ge and diamond
are bonded by this mechanism and these are purely
covalent.
 The bonding is due to the sharing of electrons.
 Covalently bonded solids are hard, high melting
points, and insoluble in all ordinary solids.
 Compound semiconductors exhibit a mixture of
both ionic and covalent bonding.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Ionic bonding

Ionic bonding is due to the electrostatic force of attraction
between positively and negatively charged ions (between 1Å
and 7Å).

This process leads to electron transfer and formation of
charged ions; a positively charged ion for the atom that has
lost the electron and a negatively charged ion for the atom
that has gained an electron.

All ionic compounds are crystalline solids at room
temperature.

NaCl and CsCl are typical examples of ionic bonding.

Ionic crystals are hard, high melting point, brittle and can be
dissolved in ordinary liquids.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Ionic bonding
The metallic elements have only up to the
valence electrons in their outer shell will lose their
electrons and become positive ions, whereas
electronegative
elements
tend
to
acquire
additional electrons to complete their octed and
become negative ions, or anions.
Na
Cl
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Comparison of Ionic and Covalent Bonding
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Potential energy diagram for molecules


This typical curve has a
minimum at equilibrium
distance R0
R > R0 ;
– the potential increases
gradually, approaching 0 as
R∞
– the force is attractive
V(R)
Repulsive
0
R0
R
Attractive

R < R0;
– the potential increases very
rapidly, approaching ∞ at
small radius.
– the force is repulsive
r
R
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Metallic bonding
 Valance electrons are relatively bound to the nucleus
and therefore they move freely through the metal and they
are spread out among the atoms in the form of a lowdensity electron cloud.
 A metallic bond result from the
sharing of a variable number of
electrons by a variable number of
atoms. A metal may be described
as a cloud of free electrons.
 Therefore, metals have high
electrical
and
thermal
conductivity.
+
+
+
+
+
+
+
+
+
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Metallic bonding





All valence electrons in a metal combine to form a “sea” of electrons
that move freely between the atom cores. The more electrons, the
stronger the attraction. This means the melting and boiling points are
higher, and the metal is stronger and harder.
The positively charged cores are held together by these negatively
charged electrons.
The free electrons act as the bond (or as a “glue”) between the
positively charged ions.
This type of bonding is nondirectional and is rather insensitive to
structure.
As a result we have a high ductility of metals - the “bonds” do not
“break” when atoms are rearranged – metals can experience a
significant degree of plastic deformation.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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van der Waals bonding

It is the weakest bonding mechanism.

It occurs between neutral atoms and molecules.


The explanation of these weak forces of attraction
is that there are natural fluctuation in the electron
density of all molecules and these cause small
temporary dipoles within the molecules. It is these
temporary dipoles that attract one molecule to
another. They are as called van der Waals' forces.
Such a weak bonding results low melting and
boiling points and little mechanical strength.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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van der Waals bonding
The dipoles can be formed as a result of unbalanced
distribution of electrons in asymmetrical molecules. This is
caused by the instantaneous location of a few more electrons
on one side of the nucleus than on the other.
symmetric asymmetric
Therefore atoms or molecules containing dipoles are attracted
to each other by electrostatic forces.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Classification of solids
SOLID MATERIALS
CRYSTALLINE
POLYCRYSTALLINE
AMORPHOUS
(Non-crystalline)
Single Crystal
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Crystalline Solids
Atoms arranged in a 3-D long range order. “Single crystals”
emphasizes one type of crystal order that exists as
opposed to polycrystals.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
Crystalline Solid

Crystalline Solid is the solid form of a substance in
which the atoms or molecules are arranged in a
definite, repeating pattern in three dimension.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
Crystalline Solid
•
Single crystal has an atomic structure that repeats periodically
across its whole volume. Even at infinite length scales, each
atom is related to every other equivalent atom in the structure by
translational symmetry
Single Pyrite
Crystal
Amorphous
Solid
Single Crystal
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Polycrystalline Solids
 Atomic order present in sections (grains) of the solid.
 Different order of arrangement from grain to grain.
Grain sizes = hundreds of m.
 An aggregate of a large number of small crystals or grains
in which the structure is regular, but the crystals or grains
are arranged in a random fashion.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Polycrystalline Solids
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
Polycrystalline Solid
•
•
Polycrystal is a material made up of an
aggregate of many small single crystals (also
called crystallites or grains).
The grains are usually 100 nm - 100 microns in
diameter. Polycrystals with grains that are <10
nm in diameter are called nanocrystalline
Polycrystalline
Pyrite form
(Grain)
Polycrystal
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Amorphous Solids
 No regular long range order of arrangement in the atoms.
Eg. Polymers, cotton candy, common window glass, ceramic.
 Can be prepared by rapidly cooling molten material.
 Rapid – minimizes time for atoms to pack into a more
thermodynamically favorable crystalline state.
 Two sub-states of amorphous solids: Rubbery and Glassy
states. Glass transition temperature Tg = temperature
above which the solid transforms from glassy to rubbery
state, becoming more viscous.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Amorphous Solids
Continuous random network structure of atoms in an
amorphous solid
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
Amorphous Solid

Amorphous (non-crystalline) Solid
is composed of
randomly orientated atoms, ions, or molecules that
do not form defined patterns or lattice structures.
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Single- Vs Poly- Crystal
 Properties of single crystalline materials vary with
direction, ie anisotropic.
 Properties of polycrystalline materials may or may
not vary with direction.
 If the polycrystal grains are randomly oriented,
properties will not vary with direction i.e isotropic.
 If the polycrystal grains are textured, properties will
vary with direction i.e anisotropic
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Single- Vs Poly- Crystal
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Single- Vs Poly- Crystal
200 m
-Properties may/may not
vary with direction.
-If grains are randomly
oriented: isotropic.
(Epoly iron = 210 GPa)
-If grains are textured,
anisotropic.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Solid state devices employ semiconductor materials in all
of the above forms.
Examples:
 Amorphous silicon (a-Si) used to make thin film
transistors (TFTs) used as switching elements in
LCDs.
 Ploycrystalline Si – Gate materials in MOSFETS.
 Active regions of most solid state devices are made of
crystalline semiconductors.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Solids

Crystalline Solids:
– Short-range Order
– Long-range Order

Amorphous solids:
– ~Short-range Order
– No Long-range Order
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
Crystals


The periodic array of atoms, ions, or
molecules that form the solids is called
Crystal Structure
Crystal Structure = Space (Crystal) Lattice
+ Basis
– Space (Crystal) Lattice is a regular periodic
arrangement of points in space, and is purely
mathematical abstraction
– Crystal Structure is formed by “putting” the
identical atoms (group of atoms) in the points
of the space lattice
– This group of atoms is the Basis
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Hard Sphere Model of Crystals
 Assumes atoms are hard spheres with well defined
diameters that touch.
 Atoms are arranged on periodic array – or lattice
 Repetitive pattern – unit cell defined by lattice
parameters comprising lengths of the 3 sides (a, b,
c) and angles between the sides (, , ).
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Lattice Parameters
c


a
b

P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Atoms in a Crystal
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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The Unit Cell Concept
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



The simplest repeating unit in a crystal is called a unit cell.
Opposite faces of a unit cell are parallel.
The edge of the unit cell connects equivalent points.
Not unique. There can be several unit cells of a crystal.
The smallest possible unit cell is called primitive unit cell of
a particular crystal structure.
 A primitive unit cell whose symmetry matches the lattice
symmetry is called Wigner-Seitz cell.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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 Each unit cell is defined in terms of lattice points.
 Lattice point not necessarily at an atomic site.
 For each crystal structure, a conventional unit cell, is chosen
to make the lattice as symmetric as possible. However, the
conventional unit cell is not always the primitive unit cell.
 A crystal's structure and symmetry play a role in
determining many of its properties, such as cleavage
(tendency to split along certain planes with smooth surfaces),
electronic band structure and optical properties.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Unit cell
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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b
a
•Unit cell: Simplest portion of the structure which is repeated
and shows its full symmetry.
•Basis vectors a and b defines relationship between a unit cell
and (Bravais) lattice points of a crystal.
•Equivalent points of the lattice is defined by translation vector.
r = ha + kb where h and k are integers. This constructs the entire
lattice.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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 By repeated duplication, a unit cell should reproduce the
whole crystal.
 A Bravias lattice (unit cells) - a set of points constructed
by translating a single point in discrete steps by a set of
basis vectors.
 In 3-D, there are 14 unique Bravais lattices. All
crystalline materials fit in one of these arrangements.
In 3-D, the translation vector
is
r = ha + kb + lc
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Crystal System
•The crystal system: Set of rotation and reflection
symmetries which leave a lattice point fixed.
•There are seven unique crystal systems: the cubic
(isometric), hexagonal, tetragonal, rhombohedral (trigonal),
orthorhombic, monoclinic and triclinic.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Bravais Lattice and Crystal System
 Crystal structure: contains atoms at every lattice point.
 The symmetry of the crystal can be more complicated than
the symmetry of the lattice.
 Bravais lattice points do not necessarily correspond to real
atomic sites in a crystal. A Bravais lattice point may be
used to represent a group of many atoms of a real crystal.
This means more ways of arranging atoms in a crystal
lattice.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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1. Cubic (Isometric) System
3 Bravais lattices
Symmetry elements: Four 3-fold rotation axes along cube diagonals
a=b=c
o
 =  =  = 90
c


a
b

P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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By convention, the edge of a unit cell always connects
equivalent points. Each of the eight corners of the unit cell
therefore must contain an identical particle.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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(1-a): Simple Cubic Structure (SC)
Rare due to poor packing (only Po has this structure)
• Close-packed directions are cube edges.
•
Coordination # = 6
(# nearest neighbors)
1 atom/unit cell
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Coordination Number = Number of
nearest neighbors
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One atom per unit cell
1/8 x 8 = 1
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Atomic Packing Factor
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(1-b): Face Centered Cubic Structure (FCC)
• Exhibited by Al, Cu, Au, Ag, Ni, Pt
• Close packed directions are face diagonals.
• Coordination number = 12
• 4 atoms/unit cell
All atoms are identical
Adapted from Fig. 3.1(a),
Callister 6e.
– Semiconductor
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FCC
Coordination number = 12
3 mutually perpendicular planes.
4 nearest neighbors on each of the three planes.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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How is a and R related for an FCC?
[a= unit cell dimension, R = atomic radius].
All atoms are identical
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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(1-c): Body Centered Cubic Structure (BCC)
• Exhibited by Cr, Fe, Mo, Ta, W
• Close packed directions are cube diagonals.
• Coordination number = 8
All atoms are identical
2 atoms/unit cell
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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How is a and R related for an BCC?
[a= unit cell dimension, R = atomic radius].
All atoms are identical
2 atoms/unit cell
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Which one has most packing ?
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Which one has most packing ?
For that reason, FCC is also referred to
as cubic closed packed (CCP)
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
2. Hexagonal System
Only one Bravais lattice
Symmetry element: One 6-fold rotation axis
a=bc
= 120o
 =  = 90o
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Hexagonal Closed Packed Structure (HCP)
2D Projection
• Exhibited by ….
• ABAB... Stacking Sequence
• Coordination # = 12
3D Projection
A sites
B sites
A sites
Adapted from Fig. 3.3,
Callister 6e.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
3. Tetragonal System
Two Bravais lattices
Symmetry element: One 4-fold rotation axis
a=bc
=  =  = 90o
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4. Trigonal (Rhombohedral) System
One Bravais lattice
Symmetry element: One 3-fold rotation axis
a=bc
= 120o
 =  = 90o
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
5. Orthorhombic System
Four Bravais lattices
Symmetry element: Three mutually perpendicular 2fold rotation axes
abc
 =  =  = 90o
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6. Monoclinic System
Two Bravais lattices
Symmetry element: One 2-fold rotation axis
abc
 =  = 90o,   90o
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7. Triclinic System
One Bravais lattice
Symmetry element: None
abc
      90o
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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•The crystal system: Set of
symmetries which leave a lattice point
fixed. There are seven unique crystal
systems.
• Some symmetries are identified by
special name such as zincblende,
wurtzite, zinc sulfide etc.
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Layer Stacking Sequence
A sites
HCP
B sites
= ABAB…
A sites
= ABCABC..
FCC
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FCC: Coordination number
FCC
Coordination number = 12
3 mutually perpendicular planes.
4 nearest neighbors on each of the three planes.
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Diamond Lattice Structure
 Exhibited by Carbon (C), Silicon (Si) and Germanium (Ge).
 Consists of two interpenetrating FCC lattices, displaced
along the body diagonal of the cubic cell by 1/4 the length of
the diagonal.
 Also regarded as an FCC lattice with two atoms per lattice
site: one centered on the lattice site, and the other at a
distance of a/4 along all axes, ie an FCC lattice with the twopoint basis.
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Diamond Lattice Structure
a = lattice constant
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Diamond Lattice Structure
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
Two merged FCC cells offset by a/4 in x, y and z.
Basic FCC Cell
Omit atoms outside Cell
Merged FCC Cells
Bonding of Atoms
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8 atoms at each corner, 6 atoms on each
face, 4 atoms entirely inside the cell
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Zinc Blende
 Similar to the diamond cubic
structure except that the two atoms at
each lattice site are different.
 Exhibited by many semiconductors
including ZnS, GaAs, ZnTe and CdTe.
 GaN and SiC can also crystallize in
this structure.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Zinc Blende
Each Zn bonded to 4 Sulfur
- tetrahedral
Equivalent if Zn and S are
reversed
Bonding often highly
covalent
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Zinc sulfide crystallizes in two different forms: Wurtzite and
Zinc Blende.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
GaAs
Green = Ga-atoms, Blue = As-atoms
•Equal numbers of Ga and As ions distributed on a diamond
lattice.
• Each atom has 4 of the opposite kind as nearest neighbors.
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Wurtzite (Hexagonal) Structure
 This is the hexagonal analog of the zinc-blende lattice.
 Can be considered as two interpenetrating close-packed
lattices with half of the tetrahedral sites occupied by
another kind of atoms.
 Four equidistant nearest neighbors, similar to a zincblende structure.
 Certain compound semiconductors (ZnS, CdS, SiC) can
crystallize in both zinc-blende (cubic) and wurtzite
(hexagonal) structure.
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WURTZITE
A sites
B sites
A sites
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Wurtzite Gallium Nitride (GaN)
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80
Miller Index For Cubic Structures
 Miller index is used to describe directions and planes in a
crystal.
 Directions - written as [u v w] where the integers u, v, w
represent coordinates of the vector in real space.
 A family of directions which are equivalent due to symmetry
operations is written as <u v w>
 Planes: Written as (h k l).
 Integers h, k, and l represent the intercept of the plane with x, y-, and z- axes, respectively.
 Equivalent planes represented by {h k l}.
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
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Miller Indices: Directions z
y
x
[1] Draw a vector and take components
[2] Reduce to simplest integers
[3] Enclose the number in square
brackets
x
0
0
y
2a
1
[0 1 1]
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
z
2a
1
82
Negative Directions
z
y
x
[1] Draw a vector and take components
[2] Reduce to simplest integers
[3] Enclose the number in square brackets
x
0
0
y
-a
-1
0 1 2
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
z
2a
2
83
Miller Indices: Equivalent Directions
Equivalent directions due to crystal symmetry:
z
1:
2:
3:
[100]
[010]
[001]
3
y
2
x
1
Notation <100> used to denote all directions equivalent to [100]
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Directions
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Miller Index
Step 1 : Identify the intercepts on the x- , y- and z- axes.
Step 2 : Specify the intercepts in fractional co-ordinates
Step 3 : Take the reciprocals of the fractional intercepts
(i) in some instances the Miller indices are best multiplied or divided
through by a common number in order to simplify them by, for
example, removing a common factor. This operation of multiplication
simply generates a parallel plane which is at a different distance from
the origin of the particular unit cell being considered.
e.g. (200) is transformed to (100) by dividing through by 2 .
(ii) if any of the intercepts are at negative values on the axes then the
negative sign will carry through into the Miller indices; in such cases
the negative sign is actually denoted by overstriking the relevant
number.
e.g. (00 -1) is instead denoted by 00 1


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The intercepts of a crystal plane with the axis defined by a set of
unit vectors are at 2a, -3b and 4c. Find the corresponding Miller
indices of this and all other crystal planes parallel to this plane.
The Miller indices are obtained in the following three steps:
1. Identify the intersections with the axis, namely 2, -3 and 4.
2. Calculate the inverse of each of those intercepts, resulting in
1/2, -1/3 and 1/4.
3. Find the smallest integers proportional to the inverse of the
intercepts. Multiplying each fraction with the product of
each of the intercepts (24 = 2 x 3 x 4) does result in integers,
but not always the smallest integers.
4. These are obtained in this case by multiplying each fraction
by 12.
5. Resulting Miller indices is 6 4 3
6. Negative index indicated by a bar on top.


P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
z
Miller Indices of Planes
87
z=
y
x=a
y=
x
x
y
z
∞
0
[2] Invert the intercept values
1/a
∞
1/∞
[3] Convert to the smallest integers
1
0
[1] Determine intercept of plane with each axis a
[4] Enclose the number in round brackets
(1 0 0)
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
1/∞
88
Miller Indices of Planes
z
y
x
x
[1] Determine intercept of plane with each axis 2a
[2] Invert the intercept values
1/2a
[3] Convert to the smallest integers
1
[4] Enclose the number in round brackets
y
z
2a
1/2a
2a
1/2a
1
1
(1 1 1)
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z
Planes with Negative Indices
y
x
x
[1] Determine intercept of plane with each axis a
[2] Invert the intercept values
1/a
[3] Convert to the smallest integers
1
[4] Enclose the number in round brackets
y
-a
-1/a
-1
111
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
z
a
1/a
1
z
Equivalent Planes
90
(001) plane
(010) plane
(100) plane
y
x
• Planes (100), (010), (001), (100), (010), (001) are
equivalent planes. Denoted by {1 0 0}.
• Atomic density and arrangement as well as
electrical, optical, physical properties are also
equivalent.
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In the cubic system the (hkl) plane and the
vector [hkl] are normal to one another.
This characteristic is unique to the cubic
crystal system and does not apply to crystal
systems of lower symmetry.
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The (110) surface
Assignment
Intercepts : a , a , 
Fractional intercepts : 1 , 1 , 
Miller Indices : (110)
The (100), (110) and (111) surfaces considered are the so-called low
index surfaces of a cubic crystal system (the "low" refers to the Miller
indices being small numbers - 0 or 1 in this case).
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Crystallographic Planes
Miller Indices (hkl)
reciprocals
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The (111) surface
Assignment
Intercepts : a , a , a
Fractional intercepts : 1 , 1 , 1
Miller Indices : (111)
The (210) surface
Assignment
Intercepts : ½ a , a , 
Fractional intercepts : ½ , 1 , 
Miller Indices : (210)
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
Symmetry-equivalent surfaces
The three highlighted surfaces
are related by the symmetry
elements of the cubic crystal they are entirely equivalent.
In fact there are a total of 6 faces related by the symmetry
elements and equivalent to the (100) surface - any surface
belonging to this set of symmetry related surfaces may be
denoted by the more general notation {100} where the Miller
indices of one of the surfaces is instead enclosed in curlybrackets.
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97
Angle Between Crystal Directions
Angle () between directions [h1 k1 l1] and [h2 k2
l2] of a cubic crystal is:
cos( ) 
h1h2  k1k 2  l1l2
(h1  k1  l1 )(h2  k 2  l2 )
2
2
2
2
2
2
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Miller Index for Hexagonal Crystal System
 Four principal axes used, leading to four Miller
Indices:
 Directions [h k i l]; Planes (h k i l), e.g. (0001)
surface.
 First three axes/indices are related: h + k + i = 0
or i = -h-k.
 Indices h, k and l are identical to the Miller
index.
 Rhombohedral crystal system can also be
identified with four indices.
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Miller Index for Hexagonal System
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
•
10
0
Miller indices
Referring to the origin of the reciprocal lattice’s definition, i.e, Bragg refraction, a reciprocal lattice vector G
actually represents a plane in the real space
z
{001}
y
x
(100)
(200)
Easier way to get the indices:
Reciprocals of the intercepts
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding
Wigner-Seitz primitive unit cell and first Brillouin zone
The Wigner–Seitz cell around a lattice point is defined as the locus of points in space that are
closer to that lattice point than to any of the other lattice points.
The cell may be chosen by first picking a lattice point. Then, lines are drawn to all nearby
(closest) lattice points. At the midpoint of each line, another line (or a plane, in 3D) is drawn
normal to each of the first set of lines.
1D case
Important
3D case: BCC
2D case
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101
102
The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice
1D
2D
Real space
Reciprocal space
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10
3
3D:
Recall that the reciprocal lattice of FCC is BCC.
4
4
4
 X = ???
4/a
Why is FCC so important?
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Why is FCC so important?
It’s the lattice of Si and many III-V semiconductors.
Si: diamond, a = 5.4 Å
GaAs: zincblende
Crystal structure = lattice + basis
Modern VLSI technology uses the (100) surface of Si.
Which plane is (100)? Which is (111)?
P.Ravindran, PHY01E – Semiconductor Physics, Autum 2013 18 December : Introduction to Structure and Bonding