Solid State Physics Phys(471)
What is a Solid ?
Solidis 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).
Exp:
amorphous
silicon, plastics, and glasses
randomly
orientated atoms
have order only within a few atomic or molecular dimensions
Exp:
Metals, Semi
conductors, ceramics
& Diamond.
Atoms are arranged in a definite, repeating pattern
have high degree of order
throughout the entire volume of the material.
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.
Solid State Physics:
Why do we study Solid State ?
1- It describes the world around us:
• why are metals shiny and opaque?
• why is glass transparent?
• why is rubber soft and stretchy?
3- It has the most important industrial applications, e.g.
electronics, solid state devices, smart materials.
2- It is an Integral part of physics. In fact, it is the best
laboratory we have for studying quantum physics and statistical physics.
The Perfect Crystal:
2- the thermal vibrations of the atoms around their equilibrium positions for any temperature T>0°K.
1- actual crystal always contains some foreign atoms, i.e., impurities. These impurities spoils the perfect crystal structure.
A perfect crystal maintains the periodicity of atoms from () to (-).
But....Strictly speaking, one cannot prepare a perfect crystal.
3- the surface of a crystal is a kind of imperfection because the periodicity is interrupted there.
Why?
It is a geometrical pattern for the atomic arrangement in a crystal.
The Lattice:
all the lattice points are equivalent. (I.e., appear exactly the same when viewed from any one of the points).
some of the lattice points are not equivalent.
Basis
Bravais Lattice: Non-Bravais Lattice:
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.
Bravais Lattice + basis = Crystal structure
It is a set of atoms located at each site of a Bravais lattice.
The Basis:
In the simplest case, basis is only 1 atom.
Odd Example: a- Mn & b-Mn (29 & 20 atoms per basis).
Basis Vectors:
It is important to point out that:
A set of vectors in term of which the positions of all lattice points can be expressed.
- A lattice is defined as integer sums of a set of basis vectors.
R = n1a + n2b (2D)
-the choice of basis vectors is not unique.
R
a b
In (2D), It is the area of the parallelogram whose sides are the basis vectors.
The Unit Cell:
which contains more than one lattice point.
There are two types of unit cells
Primitive Unit Cell: Nonprimitive Unit Cell:
which contains only one lattice point
a
b a
b c
The area of nonprimitive unit cell is an integral multiple of the primitive cell.
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.
In (3D), The same definitions BUT..
Conventional Unit Cell
The conventional unit cell is chosen to be larger than the primitive cell, but with the more elements of symmetry of the Bravais lattice.
Body centered cubic(bcc)
ConventionalBut Not Primitive cell Simple cubic(sc)
Conventional&Primitive cell
It is the ratio of the volume of atoms to available space in a unit cell.
Packing fraction
It is the number of nearest neighbors(z).
The lattice
constant It is the side length of the conventional unit cell (a).
Co-ordination number
SYMMETRY
INVERSION REFLECTION ROTATION
Point Plane Axis
Each of the unit cells of the 14 Bravais lattices has one or more types of symmetry properties
Around
If there is a point at which transformation (r -r) can be preformed and the cell remains invariant.
Inversion center:
Reflection plane
If a mirror reflection is performed on a plane and the cell remains invariant.
If the cell rotated an angle () around an axis and remains invariant. This axis is called n -fold axis of rotation
Rotation axis
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)
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.
a g b
http://www.doitpoms.ac.uk/tlplib/crystallography3/systems.php
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 .
Body Centered Cubic (bcc)
1
2
3
1 ˆ ˆ ˆ( ) 2
1( ˆ ˆ ˆ) 2
1 ˆ ˆ ˆ( ) 2
a x y z
a x y z
a x y z
Primitive and conventional cells of BCC
Primitive Translation Vectors:
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.
Face Centered Cubic (fcc)
Primitive and conventional cells of FCC
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 .
Hexagonal close-packed (hcp )Structure
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.
Diamond & Related Structures
Zinc Blende is the name given to the mineral ZnS.
S Zn
(group 1) & (group 2) different
Zincblende Structure:
GaAs, ZnS, InSb.
(group 1) & (group 2) same
Diamond Structure:
C, Si, Ge
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.
Ionic Structures
23
Non-Bravais lattice composed of two sc sublattices displaced relative to each other by 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.
2. CsCl structure:
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
•Find the components of a vector in that direction.
•Reduce them to the smallest integers.
•Write them into square brackets [ ].
To determine a direction:
] [n1n2n3
X = 1 , Y = ½ , Z = 0 [1 ½ 0] [210]
Examples
X = -1 , Y = -1 , Z = 0 [110]
[210]
Examples
X =-1 , Y = 1 , Z = -1/6 [-1 1 -1/6]
We can move vector to the origin.
661• 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
a
b
a
What are the lattice planes?
Miller Indices (hkl):
Miller Indices are a symbolic representation for the orientation of a planein a crystal lattice
They are defined as the reciprocals of the fractional intercepts which the plane makes with the crystallographic axes.
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 ( ).
To determine Miller indices of a plane :
a
3
2 2
c
b[233]
Axis X Y Z Intercept
points 1 ∞ ∞
Reciprocals 1/1 1/ ∞ 1/ ∞ Smallest
Ratio 1 0 0
Miller İndices (100)
Examples
(1,0,0)
(0,1,0)
Axis X Y Z
Intercept
points 1 1 ∞
Reciprocals 1/1 1/ 1 1/ ∞ Smallest
Ratio 1 1 0
Miller İndices (110)
Axis X Y Z Intercept
points 1 ∞ ½
Reciprocals 1/1 1/ ∞ 2 Smallest
Ratio 1 0 2
Miller İndices (102)
Examples
Axis X Y Z
Intercept
points -1 ∞ ½
Reciprocals 1/-1 1/ ∞ 2 Smallest
Ratio -1 0 2
Miller İndices (102)
Examples
Large indices indicate closer Planes Note that :
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.
) 00 1 ( ), 1 00 ( ), 0 1 0 ( ), 001 ( ), 010 ( ), 100 ( } 100
{
Indices of a Family
) 1 1 1 ( ), 1 1 1 ( ), 1 1 1 ( ), 1 1 1 ( ), 11 1 ( ), 1 1 1 ( ), 1 11 ( ), 111 ( } 111
{
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
Bravais-Miller indices
h + k + i = 0
h + k + i = 0
Note how the "h + k + i = 0" rule applies here!
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)
Example 1
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)
