Solid State Physics Phys(471)
What is a
Solid ?
Solid is not a continuous rigid body, instead it is composed of discrete basic
units (ATOMS).
Those atoms may be arranged in a regular, geometric pattern
(crystalline solids) or irregularly (amorphous solids).
Atoms are
arranged in a
definite, repeating
pattern
randomly
orientated atoms
have high degree
of order
throughout the
entire volume
of the material.
have order only
within a few
atomic or
molecular
dimensions
Exp:
Metals, Semi
conductors, ceramics
& Diamond.
Exp:
amorphous
silicon, plastics,
and glasses
Solid State
Physics:
It is the study of the properties of solid materials. It is the main
branch of condensed matter physics (which also includes
liquids).
In this course, we mainly concerned with crystalline materials.
1- It describes the world around us:
Why do we study • why are metals shiny and opaque?
• why is glass transparent?
Solid State ?
• why is rubber soft and stretchy?
2- It is an Integral part of physics. In fact, it is the best
laboratory we have for studying quantum physics and statistical
physics.
3- It has the most important industrial applications, e.g.
electronics, solid state devices, smart materials.
The Perfect
Crystal:
A perfect crystal maintains the periodicity of atoms from () to (-).
But.... Strictly speaking, one cannot prepare a perfect crystal.
Why?
1- actual crystal always contains
some
foreign
atoms,
i.e.,
impurities. These impurities
spoils
the
perfect
crystal
structure.
2- the thermal vibrations of the
atoms around their equilibrium
positions for any temperature
T>0°K.
3- the surface of a crystal is a kind of
imperfection because the periodicity
is interrupted there.
The Lattice:
It is a geometrical pattern for the atomic arrangement in a crystal.
Bravais Lattice:
Non-Bravais Lattice:
all the lattice points are
equivalent. (I.e., appear
exactly the same when viewed
from any one of the points).
Basis
some of the lattice points are
not equivalent.
The Basis:
It is a set of atoms located at each site of a Bravais lattice.
Bravais Lattice + basis = Crystal structure
In the simplest case, basis is only 1 atom.
Odd Example: a- Mn & b-Mn (29 & 20 atoms per basis).
Bravais lattice is the set of points upon which a crystal
structure may be built by placing an identical basis, in the same
orientation, on each of the lattice points.
The lattice itself consists of nothing other than a grid of points.
Basis Vectors: A set of vectors in term of which the positions of all lattice points
can be expressed.
R
b
a
It is important to point out that:
 -the choice of basis vectors is not unique.
- A lattice is defined as integer sums of a set of basis vectors.
R = n1a + n2b
(2D)
The Unit Cell: In (2D), It is the area of the parallelogram whose sides are the basis
vectors.
There are two types of unit cells
Primitive Unit Cell:
which contains only
one lattice point
Nonprimitive Unit Cell:
which contains more
than one lattice point.
Useful Remarks:
 The same lattice may
have more than one unit
cell depending on the
chosen basis vectors.
 All primitive unit cells – in this lattice- have the same area.
 The area of nonprimitive unit cell is an integral multiple of the
primitive cell.
 In (3D), The same definitions BUT..
a
c
b
a
b
Conventional
Unit Cell
The lattice
constant
Co-ordination
number
Packing
fraction
The conventional unit cell is chosen to be larger than the
primitive cell, but with the more elements of symmetry of the
Bravais lattice.
Simple cubic(sc)
Conventional & Primitive cell
Body centered cubic(bcc)
Conventional But Not Primitive cell
It is the side length of the conventional unit cell (a).
It is the number of nearest neighbors(z).
It is the ratio of the volume of atoms to available space in a unit cell.
Each of the unit cells of the 14 Bravais lattices has one or more types
of symmetry properties
SYMMETRY
INVERSION
REFLECTION
Point
Plane
ROTATION
Around
Axis
Inversion center:
If there is a point at which transformation (r  -r) can be preformed
and the cell remains invariant.
Reflection plane
If a mirror reflection is performed on a plane and the cell remains
invariant.
Rotation axis If the cell rotated an angle () around an axis and remains
invariant. This axis is called n -fold axis of rotation
Bravais Lattices
There are only seven different shapes of unit cell which can be
stacked together to completely fill all space (in 3 dimensions)
without overlapping. This gives the seven crystal systems, with
14 structures in which all crystals can be classified.
Cubic Crystal System (SC, BCC,FCC)
Hexagonal Crystal System (S)
Triclinic Crystal System (S)
Monoclinic Crystal System (S, Base-C)
Orthorhombic Crystal System (S, Base-C, BC, FC)
Tetragonal Crystal System (S, BC)
Trigonal (Rhombohedral) Crystal System (S)
a
b
g
http://www.doitpoms.ac.uk/tlplib/crystallography3/systems.php
Body Centered Cubic (bcc)
Non-primitive unit cell, it has 2 atoms at 000 and ½½½.
Has eight nearest neighbors (z = 8). Each atom is in contact
with its neighbors only along the body-diagonal directions.
Examples: Na, Li, K .
Primitive and conventional cells of BCC
Primitive Translation Vectors:
1
a1  ( xˆ  yˆ  zˆ )
2
1
a2  ( xˆ  yˆ  zˆ )
2
1
a3  ( xˆ  yˆ  zˆ )
2
Face Centered Cubic (fcc)
Non-primitive unit cell, it has 4 atoms at 000, ½ ½ 0, ½ 0 ½
and 0 ½ ½.
Has twelve nearest neighbors (z = 12). Each atom is in contact
with its neighbors only along the face-diagonal directions.
Examples: Cu, Au, Ag, Pb.
Primitive and conventional cells of FCC
Hexagonal close-packed (hcp )Structure
The unit cell is a simple hexagonal cell with a basis of 2 atoms one at
000 and the other at 2/3 1/3 ½ .
 6 atoms per unit cell.
 z = 12.
The ratio c/a =1.633.
Examples: He, Be, Mg, Zn .
Diamond & Related Structures
The unit cell is an fcc cell with a basis of two atoms (none- Bravais lattice).
z = 4.
 It has 8 atoms, where (group1) atoms located at 000, ½½0, ½0½, 0½½ ,
and (group2) atoms located at ¼¼¼, ¼¾¾, ¾¼¾, ¾¾¼.
Si, Ge and C crystallizes in diamond structure.
Zn
S
(group 1) & (group 2) same
 Diamond Structure:
C, Si, Ge
(group 1) & (group 2) different
 Zincblende Structure:
GaAs, ZnS, InSb.
Zinc Blende is the name given to the mineral ZnS.
Ionic Structures
1. NaCl structure:
Non-Bravais lattice composed of two fcc sublattices displaced relative
to each other by ½ a.
One made up of 4 Na atoms (at 000, ½½0, ½0½, 0½½;)
and the other of 4 Cl atoms (at ½00, 0½0, 00½, ½½½).
z = 6.
Examples: LiF, NaBr, KCl.
2. CsCl structure:
Non-Bravais lattice composed of two sc sublattices displaced relative to
each other by 23 a.
One made up of 1 Cs atom (at 000) and the other of 1 Cl atom (at ½½½).
z = 8.
Examples: CsBr,CsF, CsI.
 We choose one lattice point on the line as an origin, say the point O.
Choice of origin is completely arbitrary, since every lattice point is
identical.
 Then we choose the lattice vector joining O to any point on the line,
say point T. This vector can be written as;
R = n1 a + n2 b + n3c
 To distinguish a lattice direction from a lattice point, the triple is
enclosed in square brackets [ ...] is used.
 [n1n2n3] is the smallest integer of the same relative ratios.
 negative directions can be written as [n1n2n3 ]
To determine a direction:
•Find the components of a vector in that direction.
•Reduce them to the smallest integers.
•Write them into square brackets [ ].
Examples
[210]
X=1,Y=½,Z=0
[1 ½ 0]
[210]
X = -1 , Y = -1 , Z = 0
[110]
Examples
We can move vector to the origin.
X =-1 , Y = 1 , Z = -1/6
[-1 1 -1/6]
66 1 
What are the lattice planes?
•
•
Within a crystal lattice it is possible to identify sets of equally
spaced parallel planes. These are called lattice planes.
In the figure density of lattice points on each plane of a set is
the same and all lattice points are contained on each set of
planes.
b
b
a
a
Miller Indices (hkl):

Miller Indices are a symbolic representation for the
orientation of a plane in a crystal lattice

They are defined as the reciprocals of the fractional
intercepts which the plane makes with the crystallographic axes.
To determine Miller indices of a plane :
1) Determine the intercepts of the plane
along each of the three crystallographic
directions
2) Take the reciprocals of the intercepts
3) If fractions result, multiply each by the
denominator of the smallest fraction
4) Write the set into round brackets ( ).
3
[233]
2
c
a
b
2
Examples
(0,1,0)
(1,0,0)
Axis
X
Y
Z
Axis
X
Y
Z
Intercept
points
1
∞
∞
Intercept
points
1
1
∞
Reciprocals
1/1
1/ ∞
1/ ∞
Reciprocals
1/1
1/ 1
1/ ∞
Smallest
Ratio
1
0
0
Smallest
Ratio
1
1
0
Miller İndices
(100)
Miller İndices
(110)
Examples
Axis
X
Y
Z
Axis
X
Y
Z
Intercept
points
1
∞
½
Intercept
points
-1
∞
½
Reciprocals
1/1
1/ ∞
2
Reciprocals
1/-1
1/ ∞
2
Smallest
Ratio
1
0
2
Smallest
Ratio
-1
0
2
Miller İndices
(102)
Miller İndices
(102)
Examples
Note that :
Large indices indicate closer Planes
Indices of a Family
Sometimes when the unit cell has rotational symmetry, several
nonparallel planes may be equivalent.
Indices {hkl} represent all the planes equivalent to the plane (hkl)
through rotational symmetry.
{100}  (100), (010), (001), (0 1 0), (00 1), (1 00)
{111}  (111), (11 1), (1 1 1), (1 11), (1 1 1), (1 1 1), (1 1 1), (1 1 1)
Bravais-Miller indices
The Bravais-Miller indices are used in the case of hexagonal
lattices.
In that case, one uses four axes, a1, a2, a3, c and four Miller
indices, (hkil), where h, k, i, l are prime integers inversely
proportional to the intercepts OP, OQ, OS, OR of a plane of the
family with the four axes. The indices h, k, i are related by
h+k+i=0
Example 1
h+k+i=0
Axis
a1
a2
a3
c
Intercept
points
1
∞
-1
∞
Reciprocals
1/1
1/ ∞
1/-1
1/ ∞
Smallest
Ratio
1
0
-1
0
Miller İndices
(1010)
Note how the "h + k + i = 0" rule applies here!
Example 2
h+k+i=0
Axis
a1
a2
a3
c
Intercept
points
1
1
-1/2
∞
Reciprocals
1/1
1/ 1
-2
1/ ∞
Smallest
Ratio
1
1
-2
0
Miller İndices
(1020)