crystal structures

Crystal structures

1 Md Imran Momtaz, Dept. of EEE, BUET

The Crystalline State

• Lattice• Basis

• Unit Cell

2 Md Imran Momtaz, Dept. of EEE, BUET

Crystal = Lattice + Basis at each lattice pointCrystal = Lattice + Basis at each lattice point

Molecules and general bonding principles

3 Md Imran Momtaz, Dept. of EEE, BUET

(a) Force vs. interatomic separation(b) Energy vs. interatomic separation

Molecules and general bonding principles

4 Md Imran Momtaz, Dept. of EEE, BUET

• In previous slide, only electron scenario has been shown. Many electron case can be analyzed in the same manner.

Crystal structure of some well known material

5 Md Imran Momtaz, Dept. of EEE, BUET

• Copper, Cu:

FCC unit cellCrystal structure of Cu An FCC unit cell Reduced sphere representation

4 atoms per unit cell4 atoms per unit cell2 2

Ra=

3 2APF

π=

Crystal structure of some well known material

6 Md Imran Momtaz, Dept. of EEE, BUET

• Iron, Fe:

BCC structure of Fe Reduced sphere representation

2 atoms per unit cell2 atoms per unit cell3

4R

a= 3

8APF

π=

Crystal structure of some well known material

7 Md Imran Momtaz, Dept. of EEE, BUET

• Zinc, Zn:

HCP Crystal structure of Zn Unit cellSmallest unit cell

(Hexagonal unit cell)

74%APF =

Crystal structure of some well known material

8 Md Imran Momtaz, Dept. of EEE, BUET

• Silicon, Si & Germanium, Ge:

Diamond Cubic Crystal Structure of Si (Same for Ge as well)

8 atoms per unit cell8 atoms per unit cell

Note: A diamond cubic crystal structure can be formed by superimposing two FCC structures by (a/4, a/4, a/4) distance

Crystal structure of some well known material

9 Md Imran Momtaz, Dept. of EEE, BUET

• ZnS & GaAs (and some other well known III-V semiconductor as well):

Zinc Blend Cubic Crystal Structure of Si (Same for Ge as well)

8 atoms per unit cell8 atoms per unit cell

Note: A Zinc Blend cubic crystal structure can be formed by superimposing two FCC structures by (a/4, a/4, a/4) distance

Crystal structure of some well known material

10 Md Imran Momtaz, Dept. of EEE, BUET

• NaCl & CsCl:

FCC Crystal of NaCl BCC Crystal of CsCl

Numerical Problem

11 Md Imran Momtaz, Dept. of EEE, BUET

• Consider FCC structure of Cu crystal.

a) How many atoms are there per unit cell?

b) Prove that,

c) Determine APF.

d) Determine atomic concentration and density of crystal. Given, atomic mass of Cu is 63.55 gm mol-1 and Rcu = 0.128 nm.

2 2R a=

Crystal Direction & Plane

12 Md Imran Momtaz, Dept. of EEE, BUET

a, b, c, α, β and γ are known as lattice parameters.

For Cubic Crystal structure, a = b = c andα = β = γ = 90°

For Cubic Crystal structure, a = b = c andα = β = γ = 90°

For Hexagonal Close structure, a = b ≠ c andα = β = 90°, γ = 120°

For Hexagonal Close structure, a = b ≠ c andα = β = 90°, γ = 120°

Crystal Direction

13 Md Imran Momtaz, Dept. of EEE, BUET

1. Take the x, y and z axis intercepts. a/2, b, c/2

Result

2. Normalize them w.r.t. lattice constants. 1/2, 1, 1/2

3. Put them within [] without comma. [½ 1 ½]

4. Multiply with suitable number, if needed. [1 2 1]

a/3, b/2, c

Another example

1/3, 1/2, 1

[1/3 ½ 1]

[2 3 6]

Crystal Direction

14 Md Imran Momtaz, Dept. of EEE, BUET

Note: If the intercept is negative, place a bar over the number.

-a, b, c

Another example

-1, 1, 1

[-1 1 1]

[111]

Note: If the direction does not start from origin, shift that direction to some unit cell so that its initial point starts from origin.

Family of planes:It is represented by <> .

<100> means [100],[010],[001],[100],[010] [001]and

<110> means

[110],[011],[101],[110],[110] .etc

Crystal Planes: Miller Indices

15 Md Imran Momtaz, Dept. of EEE, BUET

1. Take the x, y and z axis intercepts. a/2, b, ∞

2. Normalize them w.r.t. lattice constants. 1/2, 1, ∞

3. Invert them, put them within () without comma.

(2 1 0)

4. Multiply with suitable number, if needed. (2 1 0)

Result

This is Miller Indices.

Crystal Planes: Miller Indices

16 Md Imran Momtaz, Dept. of EEE, BUET

Crystal Plane

17 Md Imran Momtaz, Dept. of EEE, BUET

Note: If the intercept is negative, place a bar over the number.

-a, b, c

Another example

-1, 1, 1

(-1 1 1)

(111)

Note: If the direction starts from origin, shift that direction to a unit cell.

Family of planes:It is represented by {} .

{100} means (100), (010), (001), (100), (010) (001)and

{110} means

(110), (011), (101), (110), (110) .etc

Important Point

18 Md Imran Momtaz, Dept. of EEE, BUET

[hkl] direction is perpendicular to (hkl) plane. This is applicable for cubic crystal

system ONLY.

[hkl] direction is perpendicular to (hkl) plane. This is applicable for cubic crystal

system ONLY.

Planer Concentration

19 Md Imran Momtaz, Dept. of EEE, BUET

• Significance:

It can justify different properties of the materials.

e.g. It can justify the oxide growth probability. It also can justify the required amount of force required to penetrate the layer.

•Planer Concentrationatoms ina certain plane

areaof that plane=

Numerical Problem

20 Md Imran Momtaz, Dept. of EEE, BUET

• Consider FCC structure of Cu crystal.

a) Determine planer concentration at (100) plane

b) Determine planer concentration at (110) plane