solid state physics cee.pdf

7/24/2019 Solid State Physics CEE.pdf

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PHY 107C (General Physics) for CEE Major

Reference Books:

1. Structure of Matter

(a) Elementary Solid State Physics: Principle and Application - M. Ali Omar

(b) Introduction to Solid State Physics - Charles Kittel

(c) Perspectives of Modern Physics - Arthur Beiser

2. Sound Waves and Fluid Mechanics

(a) University Physics Sears, Zemnaski and Young

(b) Fundamental of Vibrations and Waves S. P. Puri

(c) Elements of Properties of Matter D. S. Mathur

Section 1: Structure of Matter

Matter consists of one or more elements or their chemical compounds.

Examples- Na, Cu, Fe, NaCl, CeP, CeSb, CaCO3, etc.

Matter exists in nature in the solid, liquid and gaseous states.

Examples- Solid state--NaCl, Cu, Fe, Al

Liquid state--H2O, CH3OH

Gaseous state--He, Ne, Ar


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1.1 Classification of Solids

The atoms and molecules in a solid are attached to one another with strong

attractive forces. The solids may be broadly classified as (i) crystalline solids

and (ii) non-crystalline solids.

1.1.1 Crystalline Solids

The crystalline state of solids is characterized by regular (or periodic)

arrangement of atoms or molecules. Fig. 1.1 shows a crystalline solid. Filled

circles in this figure represent the positions of atoms/molecules in the solid.

Here, all the atoms/molecules are arranged periodically.

Fig. 1.1: A crystalline solid.

The energy released during the formation of an ordered structure (i.e. crystalline

structure) is higher than that released during the formation of a disordered

structure (i.e. non-crystalline or amorphus structure). This means that the

Solids

Crystalline solids Non-crystalline (amorphous) solids

Single crystals Polycrystals


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crystalline state is a low-energy state. So, the crystalline state is preferred by

most of the solids.

The crystalline solids may be subdivided into (a) single crystal and (b)polycrystal.

(a) Single Crystal

A crystal in which the periodicity of atoms/molecules extends throughout the

whole crystal is called a single crystal. Examples of single crystal are NaCl,

diamond, quartz(SiO2), etc..

(b) Polycrystal

In a polycrystal the periodicity of atoms or molecules is found locally in the

material (say up to 100 atomic layers), and the same periodicity is observed in

other parts of the material, but with different orientations of the atomic planes.

A polycrystal can be imagined in the way as shown in Fig. 1.2.

Gaseous or Liquid state

Crystalline state

Amorphous state

Energy released

Ecrystalcrystal

, to form

crystalline solid

Ecrystal

> Eamorphous

amorphous solid

Eamorphous , to formEnergy released

Lower energy state

Higher energy state


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Fig. 1.2: A polycrystalline solid.

A polycrystalline material is composed of a large number of small single crystal

pieces (also called gains), which join together to form one solid. In other words,

a polycrystalline material is an aggregate (i.e., amount) of a large number of

small crystallites (or grains) with random orientations (i.e., like powder made

from a large single crystal) separated by well-defined grain boundaries.

Most of the metals (e.g., Fe, Cu, Al) and ceramics (e.g., metal oxide like Al 2O3)

exhibit polycrystalline structure.

1.1.2 Non-crystalline or Amorphous Solids

If the arrangement of atoms or molecules in the solid is completely random,

then the solid is called a non-crystalline or amorphous solid. The periodicity, ifat all present, extends up to a distance of a few atomic diameters only (say

24=8). In other words, these solids exhibit short range order.

Window glass is the most familiar example of an amorphous material. The basic

ingredient for most glasses is sand (Silica, SiO2). Ordinary window glass is

usually made from a mixture of SiO2, limestone (CaCO3) and Soda ash

(Na2CO3). Quartz glass is made from pure SiO2.


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1.2 Similarity between a liquid and an amorphous solid

A liquid has an amorphous structure. So the atoms in it appear to have random

distribution. As time passes, the atoms in the liquid drift from one region to

another, but their random distribution persists. This suggests a strong similarity

between a liquid and an amorphous solid, although the atoms in the latter case

(amorphous) are fixed in space and do not drift as they do in a liquid. This is

why amorphous solids, such as glass, are sometimes called as supercooled

liquids.

1.3 The Crystal Lattice

In crystallography, only the geometrical properties of the crystal are of interest,

rather than those arising from the particular atoms constituting the crystal. The

crystal lattice is a regular periodic array of points in space, and these points are

basically located at the equilibrium positions of the atoms in the crystal. There

are two types of lattice: (i) Bravais lattice and (ii) non-Bravais lattice.

1.3.1 Bravais Lattice

A lattice in which all lattce points are equivalent, ie. all atoms in the crystal are

of the same kind, is called a Bravais lattice. Fig. 1.3 shows a Bravais lattice in

two dimensions. Here the lattice sites A, B, C are eqivalent to each other.

Liquid

(in amorphous state)

Very quickly cooling(i.e., supercooling)

Amorphous solid(example-glass)


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Fig.1.3: A Bravais lattice

1.3.2 Non-Bravais Lattice

A lattice in which some of the lattice points are non-equivalent is called a non-

Bravais lattice. Fig. 1.4 shows a non-Bravais lattice. Here the lattice sites A, B,

C are equivalent to each other, and so are the sites A/, B/, C/among themselves,

but the two sites A and A/are not equivalent to each other.

Fig. 1.4: A non-Bravais lattice

A non-Bravais lattice is sometimes referred to as a lattice with a basis. A non-

Bravais lattice may be regarded as a combination of two or more

interpenetrating Bravais lattices with fixed orientation relative to each other. So

the points A, B, C etc. form one Bravais lattice, while the points A /, B/, C/etc.

form another.


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1.4 Basis

A basis is a set of atoms stationed near each site of a Bravais lattice. In Fig. 1.4,

the basis is the two atoms A and A/, or any other equivalent set.

1.5 Unit Cell

The volume of the parallelepipe (opposite faces are parallel) whose sides are the

basis vectors a

, b

and c

is called a unit cell of the lattice. Fig. 1.5 shows a

unit cell of a lattice in three dimensions. The volume of the unit cell is

|| cbaV

. The number of lattice point per unit cell is one in this figure.

Fig. 1.5: A unit cell in three dimensions.

1.6 Primitive and Non-primitive Unit Cells

A unit cell having one lattice point is called a primitive unit cell. A primitive

unit cell is a minimum volume cell. In order to exhibit or to understand the

symmetry of the lattice more clearly, we sometimes deal with a unit cell which

is larger (non-primitive unit cell). The concept of primitive and non-primitive

unit cell can be obtained from the two-dimensional Bravis lattice as illustrated

in Fig. 1.6.

Fig. 1.6: A primitive (area S1) and a nonprimitive (area S2) unit cell.

a

bc

a

a

1a2

S2

S1

b


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The unit cell S1 formed by the basis vectors 1a

and 2a

is a primitive unit cell

and the cell S2 formed by the vectors a

and b

is a non-primitive unit cell. In

the former case the number of lattice point per unit cell is one, while in the latter

case, the number of lattice point per unit cell is two. Unit cells of triclinic (o90; cba ), simple monoclinic ( o90;cba ),

simple cubic ( o90; cba ), etc. are examples of primitive unit

cell. All nonsimple lattices like base-centered orthorhombic (

o90; cba ), face-centered cubic (NaCl structure), etc. are

examples of non-primitive unit cell.

There is no connection between a non-primitive cell and a non-Bravais lattice. A

primitive cell refers to the particular and somewhat arbitrary choice of basis

vectors a

,b

, c

in a Bravais lattice, while a non-Bravais lattice refers to the

physical fact of non equivalent sites.

1.7 Bravis Lattice in Two Dimensions

There are five Bravais lattices or nets in two dimensions. These are oblique,

square, hexagonal, simple rectangular, centered (body-centered) rectangularlattices. These lattices are summarized in the following table.


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Table 1.1: Five Bravais lattices in two dimensions.

Name of

lattice

Conventiona

l unit cell

Restriction on

conventional

unit cell axes &

angles

Illustration of unit cell

Oblique Parallelogram o90|,||| ba

(arbitrary)

Square Square o90|,||| ba

Hexagonal o60 rhombus o120|,||| ba

Simplerectangular

Rectangle o90|,||| ba

Centeredrectangular

Rectangle o90|,||| ba

a

b

b

aa

a

b=120

o

b

aa

b

aa

b

aa

Centered rectangularunit cell (non-primitive)

Primitive un it cell


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1.8 The Fourteen Bravais Lattices and the Seven Crystal Systems

There are only fourteen different Bravais latteces in three dimensions due to the

consequence of the translational-symmetry condition demanded of a lattice. To

understand the fact of reeducing the number of Bravais lattices, let us consider

the two-dimensional case of a regular pentagon. It can be drawn as an isolated

figure (like ), but one cannot place many such pentagons side by side so that

they fit tightly to fill the area of a plane with a connected array of

pentagons.(Fig. 1.7)

Fig. 1.7: A regular array of pentagons. The area of the plane cannot be filledcompletely by the pentagons.

Many similar trials exhibit that the requirement of translational symmetry in

two dimensions restricts the number of possible lattices to only five. The

number of non-Bravais lattices in three dimentions is 230.

The fourteen Bravais lattices are grouped into seven crystal systems according

to the type of cells as listed in Table 1.2.

The cell of each system is a parallelepiped whose sides are the bases a

,b

&c

.

The opposite angles are called , and . The cells in Table 1.2 are conventional

unit cells.


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Table 1.2: Fourteen Bravais lattices grouped into seven crystal systems.

System Bravais lattice

(symbol)

Restriction on

conventional

unit cell axes &

angles

Illustration of conventional

unit cell

1. Triclinic Simple(P) |||||| cba

o90

=NOM, =NOL, =MOL

2. Monoclinic Simple(P)

Base-centered (C)

|||||| cba

o90

3. Orthorhombic Simple(P)

Base-centered (C)

Body-centered(I)

Face-centered(F)

|||||| cba

o90

4. Tetragonal Simple(P)

Body-centered(I)

|||||| cba

o90

O L

N

M

a

c

b

b

c

a

(P) (C)

b

c

a

(P) (C)

b

c

a

(I) (F)

bc

a

(P) (I)


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System Bravais lattice

(symbol)

Restriction on

conventional

unit cell axes &

angles

Illustration of conventional

unit cell

5. Cubic Simple(P)

Body-centered(I)

Face-centered(F)

|||||| cba

o90

6. Trigonal(Rhombohedral)

Simple(P) |||||| cba

,120o o90

7. Hexagonal Simple(P) |||||| cba

o120

o90

Important notes-

A simple lattice has points only at the corners of the unit cell. A body-centered lattice has one additional point at the center of the unit

cell. A face-centered lattice has six additional points, one on each face of the

unit cell. In all the nonsimple lattices the unit cells are nonprimitive.

bc

a

(P) (I)

(F)

a

c

b

a

b

c

(P) shaded cell


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1.9 Index System for Crystal Planes

The orientation of a crystal plane can be determined by three points in the plane,

provided they are not collinear. If these points lay on the respective crystal

axes(X, Y, Z axes), the plane can be specified by giving the coordinates of the

points in terms of the lattice constantsa,b, c.

However, it is more useful for structure analysis to specify the orientation of a

plane by giving its Miller indices determined by the following rules.

(a) To determine the indices for the plane P as shown in Fig. 1.8, let us find its

intercepts with the axes along the basis vectors a

,b

and c

.Let these intercepts

be x, y and z (x, y and z may be integer or fractional multiple of a, b and c,

respectively).

Fig. 1.8: Plane P intercepting X, Y and Z axes at czbyax 1and5.1,2 ,respectively.

(b) We then form the triplet cz

by

ax ,,

and invert it to obtain the triplet

z

c

y

b

x

a,,

(c) Then we reduce this set to a similar one having the smallest integers by

multiplying by a common factor (say, n) like

z

cn

y

bn

x

an,,

This set is called the Miller indices of the plane and is indicated by )( lkh .

Plane P

Z

X

Y

a

c

b


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Example

Suppose the intercepts are czbyax 1,2

3,2 (Fig. 1.8).

Form the set

1,2

3,2

1,

2

3,

2,,

c

c

b

b

a

a

c

z

b

y

a

xand inverting it we find

1,

32,

21 .

Multiplying the set

1,3

2,

2

1by the common dinominator of 6 to obtain the set

(3, 4, 6), which is written as (3 4 6) and is called the Miller indices of the plane.

Important Notes:

The Miller indices are so defined that all equivalent, parallel planes are

represented by the same set of indices.

When a unit cell has rotational symmetry, several non-parallel planes

may be equivalent by virtue of this symmetry. The indices within a curly

bracket like {hkl} represent all the planes equivalent to the plane (hkl)

through rotational symmetry.

For an intercept (of a plane with any axis) at infinite, the corresponding

miller index is zero.

Example-In the cubic system the indices {100} refer to the six planes (100),

(010), (001), ( 1 00), (0 1 0) and (00 1 ).


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Some important planes in a cubic crystal are shown below in Fig. 1.9.

Fig. 1.9: Some important planes in a cubic crystal.

1.10 Spacing between Planes of the Same Miller Indices

The interplanar distance between planes labelled by the same Miller indices(hkl) is denoted by dhkl. The actual formula of dhkl depends on the crystal

structure. Let us consider the case of an orthorhombic lattice ( |||||| cba

,

o90 ). The dhkl can be calculated by referring to the Fig. 1.10,

imagining another plane parallel to the one shown and passing through the

origin. The distance between these planes, dhkl, is simply the length of the

normal line drawn from the origin to the plane shown.

Z

X

Y

(100)(010)

Z

X

Y

(110)

Z

X

Y

(200)

Z

X

Y

(111)


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Fig. 1.10: (a) (hkl) plane and (b) interplanar distance dhkl.

We define the angles which the normal line makes with the axes as , and

(Fig. 1.10), and the intercepts of the plane (hkl) with the axes asx,yandz. Then

we can write,

dhkl=xcos (using Fig. 1.10(b))

=ycos

=z

cos ..............................................(1.1)The relation among the directional cosines cos, cos& cosis

cos2+cos2+cos2= 1 .......................(1.2)

or, 1222

z

d

y

d

x

d hklhklhkl

or,222

2

1111

zyx

dhkl

or,222

111

1

zyx

dhkl .............(1.3)

( ) plane

Z

X

Y

O hkl

Normal

ML

N

P

(a) (b)

O

L

P

hkld =x cos

Normal

x

sinxx y

z


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x,yandzare related to the Miller indices h, kand lby the following relations

x

anh ,

y

bnk ,

z

cnl

or,h

anx ,

k

bny ,

l

cnz .................(1.4)

wherenis the common factor used to reduce the indices to the smallest integers.

Substituting Eq. (1.4) into Eq. (1.3) we get

21

2

2

2

2

2

2

)()()(

1

nc

l

nb

k

na

h

dhkl

=2

1

2

2

2

2

2

2

c

l

b

k

a

h

n.................(1.5)

For the cubic lattice we have a=b=c, so the above expression reduces to

222lkh

na

dhkl .................(1.6)

1.11 Close-packed Structure

A crystal whose constituent atoms are so arranged as to occupy the maximum

possible volume of the unit cell by the atoms is said to have a close-packed

structure. Close-packed structures occur when the bonding forces are

spherically symmetric, as in the inert gases, or very nearly so, as in many

metals.

1.12 Formation of fccand hcpStructures

There are two ways of arranging equivalent spheres to minimize the interstitial

volume. One way leads to a structure with cubic symmetry and it is the face-

centered cubic and the other with hexagonal symmetry and is called thehexagonal close-packed structure.


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Spheres may be arranged in a single close packed layer by placing each sphere

in contact with six others (Fig. 1.11). A second similar layer may be placed on

top of this by placing each sphere in contact with three spheres of the bottom

layer (with centers of spheres over the points marked B). A third layer can be

added in two ways; in the cubic (fcc) structure, the spheres in the third layer areplaced over the holes in the first layer not occupied by second layer (over points

C); in the hexagonal close-packed structure, the spheres in the third layer are

places directly over the spheres of the first layer (over points A) and the packing

sequence is ABABAB.....

Fig. 1.11: Close-packed stacking of spheres.

1.13 Packing Fraction

Packing fraction is defined as the fraction of the total volume of the unit cell

filled by the atom(s).

Packing fraction,f=cellunittheofVolume

atom(s)byoccupiedVolume.

1.13.1 Packing Fraction of hcpStructure

LMNO is the base of hexagonal close-packed unit cell (Fig. 1.12) withLO=LM=a=2R(a=bc). The sphere in the next layer has its center F vertically

. A . A . A . A . A . A

. . A . A . A .

+ B + B + B + B + B

+ B + B + B + B + B+ B

. A . A . A . A . A . A

. A. A

* C * C * C * C

* C* C* C* C

* C* C


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above E and it touches the three spheres whose centers are at L, M and O (Fig.

1.13).

Fig. 1.12: Close-packed stacking of spheres.

Fig. 1.13: Positions of spheres in the first and second layers.

Now LE is 2/3 of median LG, as in LMO, LM=MO=OL=a.

So, LE= 2/3 LG .............................................................(1.7)

We can write

(LM)2 =(LG)2 +(MG)2

or, 2a = 2(LG) +2

2

a

ab

c

(2/3, 1/3, 1/2)

L M

NO

F

L M

NO

E

G

R R

L E

F

c/2


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or, LG=4

22 a

a = a4

3= a

2

3

LE= a2

3

3

2 =

3

a

From Fig. 4.13 we can write

EF= aa

aa

ac

3

2

332

22

22

or, 3

22

a

c1.633 ........................................................(1.8)

Fig. 1.14: Area of the base of hcpunit cell.

From Fig. 1.14, the area of the base of hcpunit cell is

Xa= aa

a

22

2= 2

2

3a .

Packing fraction = Fraction of total volume filled

=cellunittheofVolume

atom(s)byoccupiedVolume=

cXa

R

)(

3

42 3

=

aa

a

3

22

2

3

23

42

2

3

=0.74

Examples of hcp structures are Be, Mg, Zn, Cd, etc..

L M

NO

a

a /2

X

a


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1.13.2 Packing Fraction of Simple Cubic (sc) Structure

The unit cell of a simple-cubic crystal is shown in Fig. 1.15. Each atom has six

neighbouring atoms in this structure. The upper plane of the real structure of the

unit cell is shown in Fig. 1.16. Let the radius of each atom isR.

Fig. 1.15: Unit cell of simple cubic structure.

Fig. 1.16: The upper plane of the unit cell unit cell of scstructure.

So, a=R+R=2R

The volume of the unit cell is

V=a3=(2R)3=8R3

The number of atoms in the unit cell is (1/8) 8=1

Packing fraction=cellunittheofVolume

atom(s)byoccupiedVolume= 52.0

8

3

41

3

3

R

R

.

R R


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1.13.3 Packing Fraction of Body-centered Cubic (bcc) Structure

All metallic elements (e.g., Li, Na, K, Cr, Mo, W, etc.) crystallize in the body-

centered cubic structure. The unit cell of bcc structure is shown in Fig. 1.17.

The structure is not a close-packed one because each atom has only eight

neighbours.

Fig. 1.17: Unit cell of bccstructure.

In the bccstructure, the number of atoms in the unit cell is (1/8) 8+1=2.

Let us denote the unit cell edge by a.

From Fig. 1.17,

AB2 =AD 2 +DB2=a2 +a2

or, AB= 2a

IfRis the radius of each atom, so we can writeAC=R+2R+R=4Rfor closer packing.

Now, AC2 =BC 2 +AB2

(4R)2 =a 2 +( 2a)2

16R2 =3a2

or,

3

4Ra

a

aaA D

B

C

A B

C

a2

a


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So the volume of the unit cell is3

3

3

4

Ra

Packing fraction =cellunittheofVolume

atom(s)byoccupiedVolume=

3

3

34

3

42

R

R

=8

3=0.68

1.13.4 Packing Fraction of Face-centered Cubic (fcc) Structure

The unit cell of face-centered cubic structure is shown in Fig. 1.18. The face-

centered cubic structure is a close-packed structure, each atom has 12 nearest

neighbour atoms. In the real structure, the atoms are in contact along the

diagonal of the faces. Let Rbe the radius of sphere (atom). So the diagonal AB

of the face can be written as AB=R+2R+R=4R (Fig. 1.19).

Fig. 1.18: Unit cell offccstructure.

Fig. 1.19: A face offccstructure.

A

B

a

a

C


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Now, the number of atoms in one unit cell is

488

16

2

1

Again, AB

2

= AC

2

+BC

2

= a2

+a2

= 2a2

Or, (4R)2 = 2a2

Or, a=2

4R

Hence, the volume of the unit cell is3

3

2

4

Ra

Packing fraction =3

3

2

4R

3

44

R=

6

2=0.74

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