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SOLID STATE PHYSICS-1

Muhammad RizwanDepartment of Physics

University of Wah


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STRUCTURE OF SOLIDS

Can be classified under several criteria based on atomicarrangements, electrical properties, thermal properties,chemical bonds etc.

Using electrical criterion: Conductors, Insulators,

SemiconductorsUsing atomic arrangements: Amorphous,

Polycrystalline, Crystalline .


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No regular long range order of arrangement in the atoms.Example: Polymers, cotton candy, common window glass,

ceramic.Can be prepared by rapidly cooling molten material.

Rapid minimizes time for atoms to pack into a morethermodynamically favorable crystalline state.

Amorphous Solids


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Continuous random network structure of atoms in an amorphous solid

Amorphous Solids


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Atomic order present in sections (grains) of the solid.Different order of arrangement from grain to grain.

Grain sizes = hundreds of m.An aggregate of a large number of small crystals or

grains in which the structure is regular, but the crystals orgrains are arranged in a random fashion .

Polycrystalline Solids


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Polycrystalline Solids


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Atoms arranged in a 3-D long range order.Single crystals emphasizes one type of

crystal order that exists as opposed topolycrystals.

Crystalline Solids


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The Basis(or basis set)

The set of atoms which, when placed at eachlattice point, generates the Crystal Structure.

Crystal Structure Primitive Lattice + BasisTranslate the basis through all possible lattice vectors

T = n 1a 1 + n 2a 2 + n 3a 3

to get the Crystal Structure


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The periodic lattice symmetry is such that the atomic arrangement looks the same from an arbitrary

vector position r as when viewed from the pointr' = r + T (1)

where T is the translation vector for the lattice:

T = n 1a 1 + n 2a 2 + n 3a 3 Mathematically, the lattice & the vectors a 1,a 2,a 3 are

Primitive if any 2 points r & r' always satisfy (1) with asuitable choice of integers n 1,n 2,n 3.


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Consider a 2-dimensional lattice (figure). Let the 2 Dimensional Translation Vector

be R n = n 1a + n 2b (Sorry for the notation change!!)

a & b are 2 d Primitive Lattice Vectors , n 1, n 2 are integers .

2 Dimensional Lattice Translation Vectors

Once a & b are specified by thelattice geometry & an origin ischosen, all symmetrically equivalent points in the lattice are determined bythe translation vector R n. That is, the

lattice has translational symmetry .Note that the choice of Primitive

Lattice vectors is not unique! So,one could equally well take vectors a

& b' as primitive lattice vectors.

Point D(n 1, n 2) = (0,2)

Point F(n 1, n 2) = (0,-1)


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2-Dimensional Unit Cells

Unit Cell The smallest component of the crystal (group of

atoms, ions or molecules), which, when stacked together withpure translational repetition, reproduces the whole crystal.

S

ab

S

S

S

S

S

S

S

S

S

S

S

S

S

S


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Unit Cell The smallest component of the crystal (group of atoms,

ions or molecules), which, when stacked together with puretranslational repetition, reproduces the whole crystal.The choice of unit cell is not unique!

S

S

S


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Primitive Unit Cells Note that, by definition, the Primitive Unit Cell must

contain ONLY ONE lattice point. There can be different choices for the Primitive Lattice

Vectors , but the Primitive Cell volume must be independentof that choice.

A 2 DimensionalExample!

P = Primitive Unit CellNP = Non-Primitive Unit Cell


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Unit Cells Types

Primitive Face-Centered

Body-Centered End-Centered

A unit cell is the smallest component of the crystal that reproduces the whole crystalwhen stacked together.

Primitive (P) unit cells contain only a single lattice point . Internal (I) unit cell contains an atom in the body center . Face (F) unit cell contains atoms in the all faces of the planes composing the cell. Centered (C) unit cell contains atoms centered on the sides of the unit cell.

Combining 7 Crystal Classes (cubic, tetragonal, orthorhombic, hexagonal, monoclinic, triclinic,trigonal) with 4 unit cell types (P, I, F, C) symmetry allows for only 14 types of 3-D lattice.


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Crystal Lattices

Bravais Lattices(BL) Non-Bravais Lattices (non-BL)

All atoms are the same kind All lattice points are equivalent

Atoms are of different kinds. Somelattice points are not equivalent.

Atoms are of different kinds.Some lattice points aren t equivalent.

A combination of 2 or more BL

2 d examples


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Conventional & Primitive Unit Cells

U nt Cell Types

Primitive

A single lattice point per cell The smallest area in 2 dimensions, orThe smallest volume in 3 dimensions

Simple Cubic (sc)Conventional Cell = Primitive cell

More than one lattice point per cell Volume (area) = integer multiple of

that for primitive cell

Conventional( Non-primitive )

Body Centered Cubic (bcc)Conventional Cell Primitive cell


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Conventional Unit Cells A Conventional Unit Cell just fills space when

translated through a subset of Bravais lattice vectors. The conventional unit cell is larger than the primitivecell, but with the full symmetry of the Bravais lattice.

The size of the conventional cell is given by the lattice constant a .

The full cube is theConventional Unit Cell

for the FCC Lattice

FCC Bravais Lattice


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Conventional & Primitive Unit CellsFace Centered Cubic Lattice

Primitive Lattice Vectors

a 1 = ()a(0,1,0)

a 2 = ()a(1,0,1)a 3 = ()a(1,1,0)

Note that the a is are

NOT MutuallyOrthogonal!

Conventional Unit Cell(Full Cube)

Primitive Unit Cell(Shaded)

LatticeConstant


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Conventional & Primitive Unit CellsBody Centered Cubic Lattice

Primitive Lattice Vectorsa 1 = ()a(1,1,-1)a 2 = ()a(-1,1,1)

a 3 = ()a(1,-1,1)Note that the a is are NOT mutually

orthogonal!

Primitive Unit Cell

Lattice

Constant

Conventional Unit Cell(Full Cube)


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It can be shown that, in 2 Dimensions , there are Five (5) & ONLY Five Bravais Lattices!


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21

This results in the fact that, in 3 dimensions, there are only 7 different shapes of unit cell which can be stacked together to completely fill all spacewithout overlapping . This gives the 7 crystal systems, in which all crystalstructures can be classified. These are:

The Cubic Crystal System (SC, BCC, FCC) The Hexagonal Crystal System (S) The Triclinic Crystal System (S) The Monoclinic Crystal System (S, Base-C) The Orthorhombic Crystal System (S, Base-C, BC, FC) The Tetragonal Crystal System (S, BC) The Trigonal (or Rhombohedral ) Crystal System (S)

Classification of Crystal Structures Crystallographers showed a long time ago that, in 3 Dimensions, there are

14 BRAVAIS LATTICES + 7 CRYSTAL SYSTEMS


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22


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Some concepts

Number of atoms per unit cell Corner atom = 1/8 per unit

cell Body centered atom = 1 Face centered atom = 1/2

Face diagonal=

Body diagonal= 3 a

2 a


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Crystal Structure 24

For a Bravais Lattice,

The Coordinaton Number The number of lattice points closest to a given point

(the number of nearest-neighbors of each point).

Because of lattice periodicity, all points have the same number of nearest neighbors or coordination number . (That is, the coordination number is intrinsic to the lattice.)

Examples

Simple Cubic (SC) coordination number = 6Body-Centered Cubic coordination number = 8Face-Centered Cubic coordination number = 12

Coordination Number


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3 Common Unit Cells with Cubic Symmetry


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The SC Lattice has one lattice point in its unit cell , so its

unit cell is a primitive cell. In the unit cell shown on the left, the atoms at the corners are cutbecause only a portion (in this case 1/8) belongs to that cell. Therest of the atom belongs to neighboring cells .

The Coordinatination Number of the SC Lattice = 6 .

The Simple Cubic (SC) Lattice

a

b c
http://www.kings.edu/~chemlab/vrml/simcubun.wrl


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1 atom/unit cell

Simple Cubic Structure (SC)


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Simple cubic(P)

Number of atomsper unit cell

1/8 X 8 = 1

Coordinationnumber

Atomic packing

factor


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Simple cubic

Number of atomsper unit cell 1/8 X 8 = 1

Coordinationnumber 6

Atomic packingfactor


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Simple cubicNumber of atoms

per unit cell1/8 X 8 = 1



0.52


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33

Simple Cubic (SC) Lattice Atomic Packing Factor

520. = V

V = PF

cellunit

atomsSC


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34

The BCC Lattice has two lattice points

per unit cell so the BCC unit cell is anon-primitive cell. Every BCC lattice point has 8 nearest-

neighbors. So (in the hard sphere model) each atom is in contact with itsneighbors only along the body-diagonaldirections.

Many metals (Fe,Li,Na..etc), includingthe alkalis and several transitionelements have the BCC structure.

a

b c

The Body Centered Cubic (BCC) Lattice


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Body centered cubic(I)

Number of atomsper unit cell 1/8 X 8 + 1 = 2

Coordinationnumber



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Body centered cubic





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Body centered cubic




0.68


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BCC Structure


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0.68 = V

V = APF

cellunit

atoms BCC

2 (0.433a)

Body Centered Cubic (BCC) Lattice Atomic Packing Factor


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Elements with the BCC Structure


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Face centered cubic(F)

Number of atoms per unit

cell1/8 X 8 + 1/2 X 6 = 4

Coordinationnumber



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Face centered cubic


cell1/8 X 8 + 1/2 X 6 = 4




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Face centered cubic


cell1/8 X 8 + 1/2 X 6 = 4



0.74*

*Highest packing possible in real structures


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FCC Structure


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0.68 = V

V = APF

cellunit

atoms BCC FCC 0.74

Face Centered Cubic (FCC) Lattice Atomic Packing Factor

APF =

4

3( 2 a /4 ) 34

atoms unit cell atom

volume

a 3

unit cell

volume

Unit cell contains:

6 x 1/2 + 8 x 1/8

= 4 atoms/unit cell


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Atomic Positions

X

Y

Z

(0,0,0)

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

(0,1,1) (1/2,1/2,1)

(1/2,0,1/2)

(0,0,1)


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49

Coordination # = 12

ABAB... Stacking Sequence

APF = 0.74

3D Projection 2D Projection

Hexagonal Close-Packed Structure(HCP)

6 atoms/unit cell

ex: Cd, Mg, Ti, Zn

c / a = 1.633

c

a

A sites

B sites

A sites Bottom layer

Middle layer

Top layer


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FCC & BCC: Conventional Cells With a Basis

Alternatively, the FCC lattice can be viewed interms of a conventional unit cell with a 4-pointbasis.

Similarly, the BCC lattice can be viewed in terms of a conventional unit cell with a 2- point basis.

2 HEXAGONAL CRYSTAL SYSTEMS


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Crystal Structure

In a Hexagonal Crystal System , three equal coplanar axesintersect at an angle of 60 , and another axis is

perpendicular to the others and of a different length.

2- HEXAGONAL CRYSTAL SYSTEMS

The atoms are all the same.


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The HCP lattice is not a Bravais lattice, because orientation of theenvironment of a point varies from layer to layer along the c-axis.

Hexagonal Close Packed (HCP) Lattice


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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 pack

Sequence 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

Comments on Close Packing


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Miller index is used to describe directions and planes in acrystal.

Directions - written as [u v w] where u, v, w.Integers u, v, w represent coordinates of the vector in real

space.A family of directions which are equivalent due tosymmetry operations is written as

Planes: Written as ( h k l).

Integers h, k , and l represent the intercept of the plane with x-, y-, and z- axes, respectively.

Equivalent planes represented by {h k l}.

Miller Index For Cubic Structures

Mill I di Di i


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x y z

[1] Draw a vector and take components 0 2a 2a

[2] Reduce to simplest integers 0 1 1

[3] Enclose the number in square brackets [0 1 1]

z

y

x

Miller Indices: Directions

Negative Directions


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z

y

x

x y z[1] Draw a vector and take components 0 -a 2a

[2] Reduce to simplest integers 0 -1 2

[3] Enclose the number in square brackets 210

Negative Directions


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Miller Indices: Equivalent Directions

z

y

x1

2

3

1: [100]

2: [010]3: [001]

Equivalent directions due to crystal symmetry:

Notation used to denote all directions equivalent to [100]


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Directions

Th i f l l i h h i d fi d b f


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The intercepts of a crystal plane with the axis defined by a set of unit vectors are at 2a, -3b and 4c. Find the corresponding Millerindices of this and all other crystal planes parallel to this plane.

The Miller indices are obtained in the following three steps:1. Identify the intersections with the axis, namely 2, -3 and 4.2. Calculate the inverse of each of those intercepts, resulting in

1/2, -1/3 and 1/4.3. Find the smallest integers proportional to the inverse of the

intercepts. Multiplying each fraction with the product of each of the intercepts (24 = 2 x 3 x 4) does result in integers,

but not always the smallest integers.4. These are obtained in this case by multiplying each fraction

by 12.5. Resulting Miller indices is6. Negative index indicated by a bar on top .

346

zMiller Indices of Planes


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y

x

z=

y=

x=a

x y z

[1] Determine intercept of plane with each axis a [2] Invert the intercept values 1/a 1/ 1/ [3] Convert to the smallest integers 1 0 0

[4] Enclose the number in round brackets (1 0 0)

z


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z

y

x

x y z

[1] Determine intercept of plane with each axis 2a 2a 2a[2] Invert the intercept values 1/2a 1/2a 1/ 2a[3] Convert to the smallest integers 1 1 1

[4] Enclose the number in round brackets (1 1 1)

Miller Indices of Planes

zPlanes with Negative Indices


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y

x

Planes with Negative Indices

x y z

[1] Determine intercept of plane with each axis a -a a[2] Invert the intercept values 1/a -1/ a 1/ a[3] Convert to the smallest integers 1 -1 -1

[4] Enclose the number in round brackets 111

z(001) planeE i l Pl


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Planes (100), (010), (001), (100), (010), (001)are equivalent planes. Denoted by {1 0 0}.

Atomic density and arrangement as well aselectrical, optical, physical properties are

also equivalent.

yx

(100)plane

(010)plane

(001) planeEquivalent Planes


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The (111) surface Assignment

Intercepts : a , a , a Fractional intercepts : 1 , 1 , 1Miller Indices : (111)

The (210) surface Assignment Intercepts : a , a , Fractional intercepts : , 1 , Miller Indices : (210)


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S mmetr eq i lent s rf ces


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Symmetry-equivalent surfaces

the three highlightedsurfaces are related by thesymmetry elements of the

cubic crystal - they areentirely equivalent.In fact there are a total of 6 faces related by the symmetry elements and equivalent to the(100) surface - any surface belonging to this set of symmetry related surfaces may bedenoted by the more general notation {100} where the Miller indices of one of thesurfaces is instead enclosed in curly-brackets.

(001)(010),Family of Planes { hkl }

(100), (010),(001),Ex: {100} = (100),

Diamond Lattice Structure


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a = lattice constant

Diamond Lattice Structure


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Crystalline Structure of Diamond


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