UEEP2024 Solid State Physics

Topic 1 Crystal Structure

Introduction

Solid state physics : ‒ The physics in condensed matter

"condensed matter" : ‒ a collection of atoms (or molecules) arranged

in a well defined lattice with long range order.

Definitions

• Crystal structure = lattice + basis– A lattice is a set of regular and periodic

geometrical points in space – A basis is a collection of atoms or molecules at a

lattice point

A crystal is a collection of atoms or molecules arranged at all the lattice points.

Crystal structure

Crystal structure

Basic

Example

Given that the grains in a polycrystalline metal are typically 50 m across and that metal ions have a radius of 0.15 nm, estimate the average number of ions in a grain and the proportion of these ions which are adjacent to a grain boundary. (Assume the grain is roughly cubic in shape)

Solution

Volume of grain is = (50.0×10-6 m)3 = 1.25×10-13m3.Volume of ion is = (0.30×10-9 m)3 = 2.70×10-29m3.Number of ions per gain isSurface area of grain≈(6)(50× 10-6 )2 = 1.5×10-8m2.Area corresponding to one ion ≈(0.3× 10-9 )2= 9×10-20m2.Number of ions adjacent to surface of grain is

Proportion of ions adjacent to grain boundary

.1063.41070.21025.1 15

329

313

mm

.1067.1109105.1 11

220

28

mm

.106.31063.41067.1 5

15

11

translation operation

• long range order – One symmetry that all lattices must have is the

translation symmetry. This means that if one moves along some axis by a certain distance, one reaches another lattice point which looks the same as the first point in all respects. This movement is known as the translation operation and is also the definition of long range order in a crystal

Translation vectorsMathematically, the crystal translation operation may be defined as:

r’ = r + l a1 + m a2 + n a3 (l, m, n are integers)

The quantities a1, a2 and a3 are the smallest vectors called the primitive translation vectors.

T = l a1 + m a2 + n a3

T is the translation vector and any two points are connected by a vector of this form

T = -a1 + 2 a2

rr

T

primitive and conventional cells• A lattice can be formed by repetition of a cell

and the cell can be either primitive or conventional

Note : The ways to define a primitive cell or conventional cell are not unique

Primitive Lattice cell

• A primitive cell is a minimum-volume cell.• There is always one lattice point per primitive

cell.• The volume of primitive cell with axes a1, a2

and a3 is

• The basis associated with a primitive cell is called a primitive basis.

321 aaa cV

Wigner-Seitz cell• A Wigner-Seitz cell is a primitive cell constructed by the following method:

– (i) draw lines to connect a given lattice point to all nearby lattice points;

– (ii) at the mid point and normal to these lines, draw new lines or planes;

– (iii) the smallest volume enclosed by these new lines or planes is the Wigner-Seitz cell.

3-D Lattice types• seven major groups of lattice types

• if non-primitive cells (or conventional cell) are allowed, the number will expand to 14

Bravais lattice

diamond

Diamond lattice Zincblende lattice

Si, Ge GaAs, GaP, AlAs, InSb

Lattice planes

• Lattice planes are flat parallel planes separated by equal distance. All the lattice points lye on these lattice planes

Miller indices• Orientation of the lattice planes is specified by the

Miller indices (hkl).• To determine the Miller indices:

1. Find the intercepts on the axes in terms of the lattice constants a1, a2, a3. (The axes may be those of a primitive or nonprimitive cell.)

2. Take the reciprocals of these numbers.3. Reduce the numbers to three smallest integers by

multiplying the number with the same integral multipliers.

4. The results, enclosed in parenthesis (hkl), are called the Miller indices.

Miller indices

This plane intercepts the a, b, c axes at 3a, 2b, 2c. The reciprocals of these numbers are 1/3,1/2, 1/2. The smallest three integers having the same ratio are 2, 3, 3, and thus the Miller indices of the plane are (233).

Cubic crystal system

Simple cubic Body-centered cubic

Face-centered cubic

Simple Cubic (SC)• Primitive translation

vectors

zacyab

xaa

ˆ''

ˆ'

Body-centered cubic (BCC)• Primitive translation

vectors

)ˆˆˆ(2

'

)ˆˆˆ(2

'

)ˆˆˆ(2

'

zyxac

zyxab

zyxaa

orthogonal vectors of unit length

Face-centered cubic (FCC)• Primitive translation

vectors

).ˆˆ(2

'

)(2

'

);ˆˆ(2

'

xzac

zyab

yxaa

Example

Determine the actual volume occupied by the spheres in a simple cubic structure as a percentage of the total volume.

Solution

• Volume of cube is Vc = (2r)3 = 8r3.• There are eight spheres of radius r each of

volume 1/8 of the sphere.• Volume of sphere Vs =

• Percentage of volume occupied

3

43

4818

33 rr

%4.52%1008

34

%100 3

3

r

r

VV

c

s

Real Crystal Structures(NaCl)

• FCC• The basis consists

of one Na atom and one Cl atom

Na Cl

                        

    

Real Crystal Structures(NaCl)

A 3x3x3 lattice of NaCl

Real Crystal Structures(NaCl)

                           

    

                             

      

The (111) plane of NaCl The (100) plane of NaCl

Real Crystal Structures(CsCl)

• Simple cubic

                      

   

A single unit cell of CsCl

Real Crystal Structures(CsCl)

A 3x3x3 lattice of CsCl

Real Crystal Structures(CsCl)

                           

     

                            

     

The (111) plane of CsCl The (100) plane of CsCl

• Zincblende

Real Crystal Structures(GaAs)

                        

    

A single unit cell of GaAs

Real Crystal Structures(GaAs)

A 3x3x3 lattice of GaAs

                           

    

                          

     

Real Crystal Structures(GaAs)

The (111) plane of GaAs The (110) plane of GaAs

X-ray Diffraction

• X-ray diffraction is the most commonly used technique for studying the structure of crystal lattice.

X-ray Diffraction

Typical x-ray diffraction data

X-ray Diffraction(Bragg Law)

• For constructive interference md sin2

X-ray Diffraction(Bragg Law)

• the longest possible wavelength is for sinθ=1 and m = 1, λmax = 2d

• This means that no diffraction is possible if the wavelength is greater than this maximum. This explains why we cannot study crystal structure with visible or I.R. radiation.

X-ray Diffraction

For crystal

f(r) = f(r+T) T = l a1 + m a2 + n a3

• Any local physical property of the crystal is invariant under T, such as the electron number density, magnetic moment density and etc.

X-ray Diffraction

1-D system n(x) = n(x+pa) p : arbitraty integers

Expand n(x) in a Fourier series

pp apxinxn )/2exp()(

X-ray Diffraction

3-D system n(r) = n(r+T)

Expand n(r) in a Fourier series

cellcG

GG

rGirdVnV

n

rGinrn

)exp()(1

)exp()(

Reciprocal lattice vectorsIf a1, a2 and a3 are primitive vectors of crystal lattice, then

b1, b2 and b3 are primitive vectors of reciprocal lattice.

Reciprocal lattice vector G= v1 b1 + v2 b2 + v3 b3 (v1, v2, v3 are any integers)

b1 = 2π(a2xa3)/(a1•a2xa3)b2 = 2π(a3xa1)/(a1•a2xa3)b3 = 2π(a1xa2)/(a1•a2xa3)

primitive vectors of the reciprocal lattice

Reciprocal lattice vectors• Properties of Reciprocal lattice vector G . From

these equations we observe the following properties:

1. The vector b1 is normal to both a2 and a3 . This is particularly simple for a cubic system in which case we can see that the reciprocal lattice is also a cubic system.

• For any component, say b1 , we have the relations: 211 ab

01312 abab

X-ray Diffraction

• Every crystal structure has two lattices associated with it, the crystal lattice and the reciprocal lattice. A diffraction pattern of a crystal is a map of the reciprocal lattice of the crystal. A microscope image is a map of the crystal structure in real space.

Diffraction condition

• Theorem- The set of reciprocal lattice vectors G determines the possible x-ray reflections.

• The Diffraction condition is written as

where k is the wavevector of the beam.

2

2

2

02

G

orG

Gk

Gk

X-ray Diffraction

• diffraction pattern of salt crystal

X-ray Diffraction

• Diffraction pattern of crystallized enzyme.

Brillouin Zones• Brillouin zones is defined as a Wigner-Seitz primitive cell in

reciprocal lattice.• The central cell in the reciprocal lattice is of special

importance in theory of solids, and is called the first Brillouin zone.

• The first Brillouin zone is the smallest volume entirely enclosed by planes that are the perpendicular bisectors of the reciprocal lattice vectors drawn from the origin.

• Only waves whose wavevector k drawn from the origin terminates on a surface of the Brillouin zone can be difffracted by the crystal.

First Brillouin Zones

Second Brillouin Zones

Third Brillouin Zones

Higher Brillouin Zones

Reciprocal lattice to SC lattice

• The primitive translation vectors of a simple cubic lattice may be taken as

• are orthogonal vectors of unit length.• The volume of the cell is • The primitive translation vectors of reciprocal

lattice are

. , , 321 zayaxa aaa

zyx , ,

.3321 a aaa

.2 ,2 ,2321 zbybxb

aaa

Reciprocal lattice to SC lattice

• The reciprocal lattice is itself a simple cubic lattice of lattice constant 2/a.

• The boundaries of the first Brillouin zones are planes normal to the six reciprocal lattice vectors b1, b2, b3 at their midpoints:

• The six planes bound a cube of edge 2/a and of volume (2/a)3. This cube is the first Brillouin zone of the sc crystal lattice.

.21 ,

21 ,

21

321 zbybxb

aaa

Reciprocal lattice to bcc lattice

• Primitive translation vectors of bcc lattice are

where a is the side of the conventional cube.• The volume of primitive cell is• The primitive translations of the reciprocal

lattice are

);(21 );(

21 );(

21

321 zyxazyxazyxa aaa

.21 3

321 aV aaa

.2 ;2 ;2321 yxbzxbzyb

aaa

Reciprocal lattice to fcc lattice

• Primitive translation vectors of fcc lattice are

where a is the side of the conventional cube.• The volume of primitive cell is• The primitive translations of the reciprocal

lattice are

);(21 );(

21 );(

21

321 yxazxazya aaa

.41 3

321 aV aaa

.2 ;2 ;2321 zyxbzyxbzyxb

aaa

Crystal binding

• What holds a crystal together?– Electrons and electrostatic forces play an important role in

binding atoms together to form a solid (crystal).• Common types of crystal bindings:

– (i) Ionic bonding– (ii) Covalent bonding– (iii) Metallic bonding– (iv) Hydrogen bonding– (v) Van der Waals interaction

Crystal binding

• Cohesive energy (u)– the energy required to disassemble the solid into its

constituent part (e.g. atoms of the chemical elements out of which the solid is composed)

• For a stable, the cohesive energy has an attractive term when the inter atomic distance is large (so that the crystal can be formed), and a repulsive term when the inter atom distance is short (so the crystal will not collapse).

Crystal binding

The equilibrium distance between two atoms is given by

Ionic bonding

• When the difference in electronegativity between two different types of atom is large, electrons will be transferred from the low electronegative atom to the high electronegative atom. The low electronegative atom will become a positive ion and the high electronegative atom will become a negative ion (e.g. Na + Cl → Na+ + Cl-). These ions will attract each other by electrostatic force to form a solid.

• The repulsive force is due to the Pauli exclusion principle – this prevents the crystal from collapsing.

• The attractive force is due to the Coulomb attraction between the ions.

electronegativity• Electronegativity is the average of the first ionization energy

and the electron afinity. It is the measure of the ability of an atom or molecule to attract electrons in the context of a chemical bond.

Covalent bonds

• When the electronegativiy between two atoms is small, the two atoms can form covalent bond by sharing a pair of electrons (one from each atom).

• Most atoms can form more than one covalent bond. For example, C has four outer electrons and hence it can form 4 covalent bonds.

• A crystal can be formed with one atom forming covalent bonds with several other atoms.

Metallic bonding

• Atoms bounded by “free electrons”. Good example is alkali metals (Li, K, Na, etc.)

Van der Waals interaction• Coulomb attraction can occur between two neutral spheres,

as long as they have some “internal charges” so that the neutral spheres can be polarized.

• The repulsive force is due to the Pauli exclusion principle.• The attractive force is due to the Coulomb attraction

Van der Waals interaction

• Larger molecule stronger Van der ⇒ Waals force ⇒ higher melting point.For example: He Ne Ar Kr Xe Rn Increasing melting point

This is also true for many organic molecules.

Hydrogen bonding• First ionization energy of atomic hydrogen is very high (13.6

eV). It is highly unlikely for hydrogen to form ionic bonding.• The complete shell of hydrogen atom is 2 electrons and a

hydrogen atom has only one electron. It can form only one covalent bond and it does not have sufficient bond to bind the whole crystal together with covalent bond.

• However, the covalent bond between hydrogen and the other atom (e.g. oxygen) can often be polarized,

Hydrogen bonding• These polarized molecules will “stick” to each other

by Coulomb attraction. This is possible because the hydrogen size is very small. For example, for water:

Hydrogen bonding