SOLID STATE PHYSICS

Dr. M.J.M. JafeenDept. of Physical Sciences

South Eastern University of Sri Lanka

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.

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.

STATE OF Matters

GASES LIQUIDS SOLIDS

• 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.

Gases

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.

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.

SOLID MATERIALS

CRYSTALLINE

SINGLE CRYSTALLINE POLYCRYSTALLINE

AMORPHOUS(Non-crystalline)

An example for a crystalline structur

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.

• Single crystals, ideally have a high degree of order, or regular geometric periodicity, throughout the entire volume of the material.

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.

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

Amorphous Solids

Elementary Crystallography

EC is a study of geomentrical forms of crystalline solid.

Crystal LatticeCL 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.

α

a

bCB ED

O A

y

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.

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.

Honey comb

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

BasisA 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 one atom.

Basis for non-Bravais lattice more than one atom. Lattice + Basis = crystal

By appropriate selection of the basis, non-Bravais lattice may be set to Bravais lattice

Translational vectors

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.

A B

C

DE

P P

P

PP

Gernerally the lattice is defined by TVs (two basic vectors).

P

Point D(n1, n2) = (0,2)

Point F (n1, n2) = (0,-1)

Rn = n1 a + n2 b

A

B

rr'

T

O

ab

Primitive Translational vectors

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.

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,

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

An ideal 3-D crystal is described by 3 TVs(a, b and c).

Translational Vectors – 3D

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.

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.

S

a

b

S

S

S

S

S

S

S

S

S

S

S

S

S

S

2D Lattice Type

A B

C

DE

P P

P

PP

Possible Unit cells in 2 D lattice space

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.

The area enclosed by unit cell = a b

Five Bravais Lattices in 2D

In 2 –D , it is possible to have 4 crystal systems and 5 different lattice types (UNIT cell) .

• There are only seven different shapes of unit cell which can be stacked together to completely fill all space (in 3 dimensions) without overlapping.

3D Lattice Type and Unit Cells

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.

The volume enclosed by unit cell = a b . C

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)

THE SEVEN CRYSTAL SYSTEMS AND14 BRAVAIS LATTICES TYPES (UNIT CELL)

UNIT CELL

Primitive

§ Single lattice point per cell§ Smallest area in 2D, or §Smallest volume in 3D

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 cellSimple cubic(sc) unit cell

Conventional &Primitive cell

CLASSIFICATION OF UNIT CELLS IN 3D

Three common Unit Cell type in 3D

1.CUBIC CRYSTAL SYSTEM

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.

(a).Simple Cubic (SC)

a

b c

(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.

(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.

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 =a ß = 90 , g

=120 a =b ¹ c

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)

a ¹ ß ¹ g ¹ 90 a ¹ b ¹ c

4.MONOCLINIC CRYSTAL SYSTEM

Monoclinic (Simple) a = g = 90o, ß ¹ 90o

a ¹ b ¹c

Monoclinic (Base Centered) a = g = 90o, ß ¹ 90o

a ¹ b ¹ c,

5 .ORTHORHOMBIC SYSTEM

Orthorhombic (Simple) a = ß = g = 90o

a ¹ b ¹ c

Orthorhombic (Base-centred)

a = ß = g = 90o

a ¹ b ¹ c

Orthorhombic (BC) a = ß = g = 90o

a ¹ b ¹ c

Orthorhombic (FC) a = ß = g = 90o

a ¹ b ¹ c

6. TETRAGONAL SYSTEM

Tetragonal (P) a = ß = g = 90o

a = b ¹ c

Tetragonal (BC) a = ß = g = 90o

a = b ¹ c

7 . Trigonal or Rhombohedral system

Rhombohedral (R) or Trigonal (S) a = b = c, a = ß = g ¹ 90o

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.

Lattice CNSC 6BCC 8FCC 12

Coordinatıon Numbers of common Lattices

Crystal Planes and Directions in LSIt 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.

Crystal Directions

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.

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 .

a

c

b

P

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.

[010]

[100]

[001]

[110]

[101]

[011]

[110]X

Y

Z

Important directions in 3D cubic lattice

[111]

Body diagonal

Face diagonal

Crystal Planes in LS

A

B

C

Op a

q c

r b

a

b

c

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

This (hkl) are known as Miller indices of the plane ABC.

h:k:l =1/p : 1/q : 1/r

Example 1

• 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) a

b

C1.0

0.50.75

Planes in Lattices and Miller Indices

Planes in Lattices and Miller Indices

Planes in Lattices and Miller Indices

Planes in Lattices and Miller Indices

The Miller indices (hkl) represents a set of all parallel planes in a LS.

The distance between two parallel crystal planes

a

b

c

A

B

C

pa

qb

rc

The expression for an interplaner spacing mainly depends on the type of crystal system.

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)

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.

Since the minimum value for n is 1,

Interplanar spacing (dhkl) in cubic lattice

2 2 2

cubic latticehkl

ad

h k l

010 2 2 20 1 0

cubic lattice ad a

020 2 2 2 20 2 0

cubic lattice a ad

010020 2

dd

E.g.: Estimate the seperations between (100) planes and (111) planes in a cubic crystal.

111 / 3 3 / 3d a a

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

Related to ‘k’ index

Related to ‘h’ index

Related to ‘i’ index

Related to ‘l’ index

Hexagonal crystals → Miller Indices

(h k i l)i = (h + k)

a1

a2

a3 Intercepts → 1 1 - ½ Plane → (1 12 0)

The use of the 4 index notation is to bring out the equivalence between

crystallographically equivalent planes and directions

a1

a2

a3

Intercepts → 1 -1 Miller → (0 1 0)

Miller-Bravais → (0 11 0)

Intercepts → 1 -1 Miller → (1 1 0 )

Miller-Bravais → (1 1 0 0 )

Examples to show the utility of the 4 index notation

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)

a1

a2

a3

Intercepts → 1 1 - ½ Plane → (1 12 0)

Intercepts → 1 -2 -2 Plane → (2 11 0 )

Examples to show the utility of the 4 index notation

Intercepts → 1 1 - ½ 1

Plane → (1 12 1)

Intercepts → 1 1 1

Plane → (1 01 1)

Atomic Packing Fraction

• It is defined as the volume of atoms within the unit cell divided by the volume of the unit cell.

Atomic Packing Fraction of Simple Cubic (sc)

0.68 = V

V = APF 3

R 4 = a

cell unit

atomsBCC

2 (0,433a)

Atomic Packing Factor of BCC

4 (0,353a)

0.68 = V

V = APF 3

R 4 = a

cell unit

atomsBCCFCC

0,74

Atomic Packing Fraction of FCC

Note: FCC lattice has a larger packing fraction among all lattice types.

Characteristics of cubic lattices –chrles kitle 12

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.

Some Common Crystal Structures

• 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) ;

)(2

zyxa

• The unit cell consists of 4 NaCl molecules.• LiH,MgO,MnO,etc• The lattice constants are in the order of 4-7 Å.

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

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.

• 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 structureElement Cubic sizeC(diamond) 3.57 A˚Si 5.43 A˚Ga 5.66 A˚

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.

Crystal Structure 76

Bravais Lattice : Hexagonal LatticeHe, Be, Mg, Hf, Re (Group II elements)

Basis : (0,0,0) (2/3a ,1/3a,1/2c)

The packing fraction of HCP = F.C.C = 0.74 , maximum possible denser packing.

a1= a2 = awith in cluded angle of 1200 . c= 1.633 a for ideal hcp

Crystal Structure 77

A A

AA

AA

A

AAA

AA

AAA

AAA

B B

B

B

B B

B

B

B

BB

C C C

CC

C

C

C C C

Sequence ABABAB..-hexagonal close packSequence ABCABCAB..

-face centered cubic close pack

Close pack

B

AA

AA

A

A

A

A A

B

B B

Sequence AAAA…- simple cubic

Sequence ABAB…- body centered cubic

Packing