1. SOLID STATE PHYSICS
Dr. M.J.M. Jafeen
Dept. of Physical Sciences
South Eastern University of Sri Lanka
2. The structure of this course
•
•
•
•
•
Course Code: PHM 2021
Credit weight : 1
Lecture hours : 15
Tutorial and Discussion hours : 5 hrs
Minimum Cont. Assest. : 3
• Main References:
Charles Kittel - Introduction to Solid State Physics.
J.S. Blakemore - Solid State Physics.
3. Introduction
Solid state physics is largely concerned with crystals and electrons in the
crystals.
The study of solid state physics began in the early years of the 19th century
following the discovery of x-ray diffraction by crystals and the publications
of series of simple calculations and successful predictions of the properties
of the crystals.
It gives details about how the large-scale properties of solid materials
results from their atomic -scale properties.
It forms the theoretical basis of Material Science.
It has many direct applications.
5. Gases
• Gases contain either atoms or molecules that do
not bond each other in a range of
pressure, temperature and volume.
• These molecules do not have any particular order
and move freely within a container.
6. Liquids
• Similar to gases, liquids does not have
any
atomic/molecular order and takes the shape of the
containers.
• Applying low levels of thermal energy can easily break the
existing weak bonds.
7. Solids
• Solids consist of atoms or molecules which are
attached with one another with strong force.
• Solids (at a given temperature, pressure, and
volume) have stronger bonds between molecules
and atoms than liquids.
• Solids require more energy to break the bonds.
• Solids
can
take
the
form
of
crystalline, polycrstalline, or amorphous materials.
10. Crystalline Solids
It contains regular periodic arrangememnt
of atoms or
molecules.
i.e. The atoms or molecules are stacked in ordered manner.
Most of
solids are in crystalline state because of lower
energy state.
i.e. Energy released during the formation of crystalline
solid is larger than non crystalline solid.
Single Crystalline Solids
A periodicity extends throughout the materials.
E.g. Diamond.
Bond length and Angles are the same within the whole
crystal.
11. •Single crystals, ideally have a high degree of order, or
regular geometric periodicity, throughout the entire volume
of the material.
12. Polycrystalline Solids
Made up of an aggregate of many small single crystals (also
called crystallites or grains).
As a whole, discontinuity exists in the Periodicity.
E.g. Most of metals and ceramics.
The periodic extension is only for few Å- the locus of
discontinuity is called a grain boundery .
The grains are usually 100 nm - 100 microns in diameter.
Polycrystals with grains that are <10 nm in diameter are called
nanocrystalline.
13. Amorphous Solids
o Amorphous (Non-crystalline) Solid composed of randomly
orientated atoms , ions, or molecules that do not form
defined patterns or lattice structures.
o ASs have order only within a few atomic or molecular
dimensions.
o ASs do not have any long-range order, but they have varying
degrees of short-range order.
Eg: plastics, glasses ; amorphous silicon-used in solar cells
and transistors.
Crystal Structure
13
14. Elementary Crystallography
EC is a study of geomentrical forms of crystalline solid.
Crystal Lattice
CL is a regular peridic array of points in space, representing a
crystal.
In any crystal , atoms or molecules are arranged periodically in
3-dimentional space. When replacing these atoms by exact
’points’ , then the arrangement of points will have the same
geometrical symmetry of the crystals. This arrangment of points
in space is called crystal lattice.
•An array of imaginary points forms the framework on which the
actual crystal structure is based.
15. y
B
C
α
b
O
D
a
A
E
x
•Each point in CL has identical surroundings to all
other points- Bond length and Angles are the same.
•Crystals can form two types of lattice spaces.
16. Crystal Lattice
Bravais Lattice (BL)
Non-Bravais Lattice (non-BL)
§ All atoms are of the same kind
§ All lattice points are equivalent
§ Atoms can be of different kind
§ Some lattice points are not
equivalent
§A combination of two or more BL
17. Bravais lattice
An infinite array of discrete points with an
arrangement and orientation that appears exactly the
same, from whichever of the points the array is viewed.
E.g.: Copper crystal
In non – Bravais lattice , the lattice points are not identical.
NOTE: The vertices of a 2D honey comb does not form a bravais lattice
Not only the arrangement but
also the orientation must appear
exactly the same from every
point in a bravais lattice
Honey comb
18. Basis
A unit which may consist one atom or more.
A crystal can be generated by replacing this unit(basis) in each
point of the lattice set.
Basis for Bravais lattice
Basis for non-Bravais lattice
Lattice + Basis = crystal
one atom.
more than one atom.
20. Translational vectors
Gernerally the lattice is defined by TVs (two basic vectors).
P
E
D
P
A
P
P
C
P
B
Consider a 2D lattice space.
•By selecting any arbitary point as
origin , any position of a point (P) in the
lattice space can be written as a linear
combination of two vectors.
Rn = n1 a + n2 b,
where- a and b are called translational
vectors and n1 and n 2 are real integers.
• For a particular lattice it is possible to have a set of translational
vectors to represent each lattice points in the LS. Moreover, the
choice of translational vectors ( a and b) is not unique, arbitary.
21. Rn = n1 a + n2 b
P
Point D(n1, n2) = (0,2)
Point F (n1, n2) = (0,-1)
22. Primitive
Translational vectors
Even though the choice of TVs is arbitary, they do not form the
translational invariance in the LS- the same appearnce when viewed
from all the position in the LS. But, PTVs have this property.
Consider a 2D LS with a pair of T V of a and b,
b
a
B
T
A
r'
r
O
The mathematical Conditions:
If a and b are said to be primitive
translational vectors of the lattice
space, then it
satisfies the following
condition,
T = r’-r = 3a + b
i.e.) All the positions of the lattice points in the LS are defined by r’ = T+ r and
are identical.
In other words, a pair of PT Vs form the smallest cell that can serve as a building
block for the crystal structure.
23. Translational Vectors – 3D
An ideal 3-D crystal is described by 3 TVs(a, b and c).
If there is a lattice point represented by the position vector r, then there is
also a lattice point represented by the position vector r’ which satisfies the
following conditions.
r’ = r + u a + v b + w c , , where u, v and w are arbitrary integers.
Then a, b and c are termed as PT Vs of the 3D LS.
24. Unit Cell
The smallest unit of the lattice which, on continous
repeatition, generate the complete lattice.
To generate a LS by translational symmetry , a set of lattice
vectors may be sellected. Each parrallelogram generated by
a pair of lattice vectors is formed unit cell.
The unit cells produced by primitive translatinal vectors are
the least building blocks of a LS and called as a primitive unit
cells. Others are termed as non-primitive unit cells.
The primitive unit cell of a LS is always the smallest building
block and it has only one lattice point for the LS.
26. Possible Unit cells in 2 D lattice space
P
E
The area enclosed
by unit cell = a b
P
D
P
C
P
P
A
B
All unit cells from A to D are Primitive unit cells except the E
because E does not have the least volume and a lattice point.
L.P . of A = L.P . of B =L.P . of C = L.P . of D = ¼4 = 1.
L.P . of E = ¼4 + ½ 2 = 2
Non Primitive unit cell.
27. Five Bravais Lattices in 2D
In 2 –D , it is possible to have 4 crystal systems and
5 different lattice types (UNIT cell) .
28. 3D Lattice Type and Unit Cells
The volume enclosed
by unit cell = a b . C
Only 7 combination of a ,b and C and
, and are allowed where , and
are called axial angle.
The crystal can be devided into 7
system which are known as 7 crystal
system.
•
There are only seven different shapes of unit cell which
can be stacked together to completely fill all space (in 3
dimensions) without overlapping.
29. THE SEVEN CRYSTAL SYSTEMS AND
14 BRAVAIS LATTICES TYPES (UNIT CELL)
1. Cubic Crystal System (SC, BCC,FCC)
2. Hexagonal Crystal System (S)
3. Monoclinic Crystal System (S, Base-C)
4. Orthorhombic Crystal System (S, Base-C, BC, FC)
5. Tetragonal Crystal System (S, BC)
6. Trigonal (Rhombohedral) Crystal System (S)
7. Triclinic Crystal System (S)
30.
31. CLASSIFICATION OF UNIT CELLS IN 3D
UNIT CELL
Primitive
§ Single lattice point per cell
§ Smallest area in 2D, or
§Smallest volume in 3D
Simple cubic(sc) unit cell
Conventional &Primitive cell
Non-primitive
§ More than one lattice point per cell
§ Integral multiples of the area of
primitive cell
Body centered cubic(bcc)
Conventional but non- Primitive cell
33. 1.CUBIC CRYSTAL SYSTEM
(a).Simple Cubic (SC)
Simple Cubic has one lattice point
primitive cell.
In the unit cell on the left, the atoms at the corners are cut
because only a portion (in this case 1/8) belongs to that cell.
The rest of the atom belongs to neighboring cells.
Coordinatination number of simple cubic is 6.
b
c
a
34. (b)Body Centered Cubic (BCC)
BCC has two lattice points
a non-primitive cell.
BCC has eight nearest neighbors. Each atom is in contact with
its neighbors only along the body-diagonal directions.
Many metals (Fe,Li,Na..etc), including the alkalis and several
transition elements choose the BCC structure.
35. (c). Face Centered Cubic (FCC)
•
Atoms are located at the corners of the unit cell and at the
centre of each face.
• Face centred cubic has 4 atoms
non primitive cell.
• Many of common metals (Cu, Ni, Pb..etc) crystallize in FCC
structure.
36.
37. 2 . HEXAGONAL SYSTEM
A crystal system in which three equal coplanar axes
intersect at an angle of 60 , and a perpendicular to the
others, is of a different length.
Hexagonal
=ß = 90 , =120
a =b c
38. 3 TRICLINIC
• Triclinic minerals are the least symmetrical. Their three axes are
all different lengths and none of them are perpendicular to each
other. These minerals are the most difficult to recognize.
Triclinic (Simple)
ß 90
abc
42. 7 . Trigonal or Rhombohedral system
Rhombohedral (R) or Trigonal (S)
a = b = c, = ß = 90o
43. Coordinatıon Number
• Coordinatıon Number (CN) : The Bravais lattice points closest
to a given point are the nearest neighbours.
• Because the Bravais lattice is periodic, all points have the
same number of nearest neighbours or coordination number.
It is a property of the lattice.
Coordinatıon Numbers of common Lattices
Lattice
SC
CN
6
BCC
FCC
8
12
44. Crystal Planes and Directions in LS
It is possible to identify set of equally spaced parallel planes
within a LS. So, it is often necessary to describe a particular
crystallographic plane or a particular direction within a real 3D
crystal.
c
Crystal Directions
The position vector of a lattice pont P , r = u a + v b + w c
Then, the direction of the lattice vector r is given by
[ u*,v*,w*], where u*,v* and w* are possible integers so
that u*: v*: w* = u : v : w .
P
a
b
OR: The direction of a line in a LS is determined by finding out the
projections of the vector drawn from the origin to that point on the
crystellographic axes.
45. E.g.: Consider a lattice vector R= ½ a + ½ b +½ c
The direction of the lattice vector is given by [1,1,1].
Similarly, The direction of the lattice vector R= a + b + c is
also given by [1,1,1].
[ u*,v*,w*] represents a set of parallel lines in this direction.
46. Important directions in 3D cubic lattice
[011]
[001]
Z
[101]
Y [010]
[100]
X
[110]
[110]
[111]
Face diagonal
Body diagonal
47. Crystal Planes in LS
Consider a crystal plane ABC,
Let the pa, qb and rc are the intercepts
of the plane ABC on the crystal axis a, b
and c respectively.
Then the crystal plane ABC is
represented by (hkl), where h,k and l
are smallest possible integers so that
b
B
rb
O
A
a
qc
h:k:l =1/p : 1/q : 1/r
C
This (hkl) are known as Miller
indices of the plane ABC.
pa
c
48. Example 1
C
• The intercepts of the
plane are at
0.5a, 0.75b, and 1.0c
• Take the reciprocals to
get (2, 4/3, 1)
• Reduce common factors
to get Miller Index of
(643)
1.0
0.5
a
0.75
b
53. The Miller indices (hkl) represents a set of all parallel planes in a LS.
54. The distance between two parallel crystal planes
c
The expression for an interplaner spacing mainly
depends on the type of crystal system.
b
Consider CLs of orthogonal axes.
If , and are the direction cosine of the normal
, then
dhkl = pa cos
= qb cos
= rc cos .
But, cos2 + cos2 + cos2 = 1 (orthogonal axes)
C
B
qb
rc
pa A
If (hkl) are the miller indices of the plane ABC, then
h= n/p; k= n/q ; l= n/r, where n is the common factor.
a
55. Since the minimum value for n is 1,
Interplanar spacing (dhkl) in cubic lattice
d
cubic lattice
hkl
=
a
h k l
2
2
2
56. cubic
d 010 lattice =
cubic
d 020 lattice =
a
0 1 0
2
2
2
a
02 22 02
=a
d 020 =
=
a
2
d 010
2
E.g.: Estimate the seperations between (100) planes and (111) planes in a cubic
crystal.
d111 = a / 3 = a 3 / 3
57. Miller Indices for Hexagonal crystals
Directions and planes in hexagonal crystals are designated by the
4-index Miller notation
In the four index notation:
the first three indices are a symmetrically related set on the basal plane
the third index is a redundant one (which can be derived from the first two)
and is introduced to make sure that members of a family of directions or planes
have a set of numbers which are identical
this is because in 2D two indices suffice to describe a lattice (or crystal)
the fourth index represents the ‘c’ axis ( to the basal plane)
Hence the first three indices in a hexagonal lattice can be permuted to get the
different members of a family; while, the fourth index is kept separate.
Note: In this first three Miller indices must add up to zero
58. Related to ‘l’ index
Related to ‘i’ index
Related to ‘k’ index
Related to ‘h’ index
59. Hexagonal crystals → Miller Indices
a3
Intercepts → 1 1 - ½
Plane → (1 12 0)
(h k i l)
i = (h + k)
a2
a1
The use of the 4 index notation is to bring out the equivalence between
crystallographically equivalent planes and directions
60. Examples to show the utility of the 4 index notation
a3
a2
a1
Obviously the ‘green’ and
‘blue’ planes belong to the
same family and first three
indices have the same set of
numbers (as brought out by the
Miller-Bravais system)
Intercepts → 1 -1
Intercepts → 1 -1
Miller → (1 1 0 )
Miller → (0 1 0)
Miller-Bravais → (1 1 0 0 )
Miller-Bravais → (0 11 0)
61. Examples to show the utility of the 4 index notation
a3
a2
Intercepts → 1 -2 -2
Plane → (2 11 0 )
a1
Intercepts → 1 1 - ½
Plane → (1 12 0)
65. Atomic Packing Factor of BCC
4R
a=
3
APF BCC
V atoms
=
= 0.68
V unit cell
2
(0,433a)
66. Atomic Packing Fraction of FCC
4R
V atoms
0,74
a=
= 0.68
APF BCC =
FCC
3
V unit cell
4
(0,353a)
Note: FCC lattice has a larger packing fraction among all lattice types.
68. Some Common Crystal Structures
1. Sodium Chloride Structure
• Sodium chloride also crystallizes in a
cubic lattice, but with a different unit
cell.
• Sodium chloride structure consists of
equal numbers of sodium and
chlorine ions placed at alternate
points of a simple cubic lattice.
• Each ion has six of the other kind of
ions as its nearest neighbours.
69. •
The space lattice of CsCl is cubic
lattice but non-Bravise.
•
This structure can be considered
as a face-centered-cubic Bravais lattice
with a basis consisting of a sodium ion
at (0,0,0) and a chlorine ion at the
center of the conventional cell(½,½,½) ;
a
2
•
•
•
( x y z )
The unit cell consists of 4 NaCl molecules.
LiH,MgO,MnO,etc
The lattice constants are in the order of 4-7 Å.
70.
71. 2. Cesium Choloride structure
•The space lattice of CsCl is cubic lattice but non-Bravise.
•This structure can be considered as a Base-centered-cubic Bravais lattice
with a basis consisting of a Cesium ion at (0,0,0) and a chlorine ion at the
center of the conventional cell (½,½,½).
•CsBr,CsI,CuZn,BeCu,etc
72. 2.Diamond Structure
• The space lattice of the diamond is a cubic lattice but non-Bravais .
• The diamond lattice can be viewed as two interpenetrating F.C.C.
Lattices displaced from each other by one quarter of the cube diagonal
distance. So, there are two same type of atoms in the basis at (0,0,0)
and (¼,¼,¼) to make F.C.C. Bravais lattice.
• The conventional unit cell consists of eight atoms. There is no way to
choose the primitive unit cell in Diamond.
• Each atom bonds covalently to 4 others equally spread about atom in 3d.
73. • Each atom has 4 nearest neighbours and 12 next
nearest neighbours .
• The packing fraction of Diamond is only 34 %- very low.
• Elements with diamond crystal structure
Element
Cubic size
C(diamond)
3.57 A˚
Si
5.43 A˚
Ga
5.66 A˚
74. 4.Hexagonal Close-Packed Structure (hcp).
• This is another structure that is
common, particularly in metals. In
addition to the two layers of atoms
which form the base and the upper
face of the hexagon, there is also an
intervening layer of atoms arranged
such that each of these atoms rest
over a depression between three
atoms in the base.
75.
76. The packing fraction of HCP = F.C.C = 0.74 , maximum possible denser
packing.
Bravais Lattice : Hexagonal Lattice
He, Be, Mg, Hf, Re (Group II elements) Basis : (0,0,0) (2/3a ,1/3a,1/2c)
a1= a2 = awith in cluded angle of 1200 .
c= 1.633 a for ideal hcp
Crystal Structure
76
77. Packing
Close pack
A
A
A
A
A
A
B
BA BA BA B A
C
C
C
BA BA B A
A
C
C
C
C
A
A
B
A
B
A
A
A
A
B
A
A
BA BA B A B A
C
C
C
Sequence AAAA…
- simple cubic
Sequence ABABAB..
-hexagonal close pack
Sequence ABCABCAB..
-face centered cubic close pack
Crystal Structure
Sequence ABAB…
- body centered cubic
77
