solid state physics 02-crystal structure

Solid State PhysicsUNIST, Jungwoo Yoo

1. What holds atoms together - interatomic forces (Ch. 1.6)2. Arrangement of atoms in solid - crystal structure (Ch. 1.1-4) - Elementary crystallography - Typical crystal structures - X-ray Crystallography3. Atomic vibration in solid - lattice vibration (Ch. 2) - Sound waves - Lattice vibrations - Heat capacity from lattice vibration - Thermal conductivity4. Free electron gas - an early look at metals (Ch. 3) - The free electron model, Transport properties of the conduction electrons---------------------------------------------------------------------------------------------------------(Midterm I) 5. Free electron in crystal - the effect of periodic potential (Ch. 4) - Nearly free electron theory - Block's theorem (Ch. 11.3) - The tight binding approach - Insulator, semiconductor, or metal - Band structure and optical properties6. Waves in crystal (Ch. 11) - Elastic scattering of waves by a crystal - Wavelike normal modes - Block's theorem - Normal modes, reciprocal lattice, brillouin zone7. Semiconductors (Ch. 5) - Electrons and holes - Methods of providing electrons and holes - Transport properties - Non-equilibrium carrier densities8. Semiconductor devices (Ch. 6) - The p-n junction - Other devices based on p-n junction - Metal-oxide-semiconductor field-effect transistor (MOSFET)---------------------------------------------------------------------------------------------------------------(Final)

All about atoms

backstage

All about electrons

Main character

Main applications


Types of Bonding

Ionic Bonding

Van Der Waals Bonding

Metallic Bonding

Covalent Bonding

Hydrogen Bonding

High Melting Point

Hard and Brittle

Non conducting

solid

NaCl, CsCl, ZnS

Low Melting Points

Soft and Brittle

Non-Conducting

Ne, Ar, Kr and Xe

Variable Melting

Point

Variable

Hardness

Conducting

Fe, Cu, Ag

Very High Melting

Point

Very Hard

Usually not

Conducting

Diamond, Graphite

Low Melting Points

Soft and Brittle

Usually

Non-Conducting

İce,

organic solids


Crystal Structure

1. Crystal lattice, basis, unit cell, crystal planes and direction

2. Closed packed structure (cubic, hexagonal)

3. The body-centered cubic structure

4. Structure of ionic solids

5. The diamond and zincblende structures

6. X-ray crystallography

A solid is said to be crystal if atoms are arranged in such a way that their positions are exactly periodic.


Solid Mateirals

Crys-talline

Polycrys-talline

Amor-phous

Single Crystal

http://www.alaskanessences.com/gembig/Pyrite.jpg


5

Single Crystal

Single Pyrite Crystal

AmorphousSolid

Single crystal has an atomic structure that repeats periodically across its whole volume.

Crystalline solid

Even at infinite length scales, each atom is related to every other equivalent atom in the structure by translational symmetry

http://weblog.burningbird.net/fires/001936.htm


6

PolycrystallinePyrite form

(Grain)

Polycrystalline solid

Polycrystal is a material made up of an aggregate of many small single crystals (also called crys-tallites or grains).

Polycrystalline material have a high degree of order over many atomic or molecular dimensions.

These ordered regions, or single crytal regions, vary in size and orientation wrt one another.

These regions are called as grains (domain) and are separated from one another by grain boundaries. The atomic order can vary from one domain to the next.

The grains are usually 100 nm - 100 microns in diameter. Polycrystals with grains that are <10 nm in diameter are called nanocrystalline

http://www.alaskanessences.com/gembig/Pyrite.jpg


Amorphous solid

Amorphous (Non-crystalline) Solid is composed of randomly orientated atoms, ions, or molecules that do not form defined patterns or lattice structures.

Amorphous materials have order only within a few atomic or molecular dimensions.

Amorphous materials do not have any long-range order, but they have varying degrees of short-range order.

Examples to amorphous materials include amorphous silicon, plastics, and glasses

Amorphous silicon can be used in solar cells and thin film transistors.


What is crystallography?

The branch of science that deals with the geometric description of crystals and their internal arrangement.

Crystallography is essential for solid state physics

• Symmetry of a crystal can have a profound influence on its properties.• Any crystal structure should be specified completely, concisely and unambiguously• Structures should be classified into different types according to the symmetries they

possess.

Crystallography


Crystal Lat-tice

CB ED

O AG

F

Graphene


Crystal Lat-tice

B

O A

• An infinite array of points in space,

• Each point has identical surroundings

to all others.

• Arrays are arranged exactly

in a periodic manner.

Lattice ?

a

b

Lattice vector: ba

,

bvauT

vu, are integer

Translational invariance

Bravais Lattice

Lattice is completely defined by lal and lbl and g

For graphene a=b=2.46 Å, g = 120º


Crystal Lat-tice

CB ED

O AG

F

Graphene


Crystal Lat-tice


Basis

a

b

O

A group of atoms associated with each lattice point to represent crystal structure

g

Lattice vector: ba

,

Basis vector: bar

3

1

3

2

Basis: C(0,0), C(2/3, 1/3)

Crystal structure = Lattice Basis

Basis gives identity to the lattice to from crystal


Crystal structure = Lattice Basis


5 Bravais lattice in 2D


Lattice in 3D

cwbvauT

Translational invariance

Lattice vector: cba

,,

wvu ,, are integer

Lattice vector


S

a

b

S

S

S

S

S

S

S

S

S

S

S

S

S

S

Unit Cell in 2D

The smallest component of the crystal (group of atoms, ions or molecules), which when stacked together with pure translational repetition reproduces the whole crystal.

2D crystal


Unit Cell in 2D

The smallest component of the crystal (group of atoms, ions or molecules), which when stacked together with pure translational repetition reproduces the whole crystal.

The choice of unit cell

is not unique.

2D crystal

S

S

a

Sb

S


We define lattice points ; these are points with identical environments

2D Unit Cell example – (NaCl)


Choice of origin is arbitrary - lattice points need not be atoms - but unit cell size should always be the same.


This is also a unit cell - it doesn’t matter if you start from Na or Cl


- or if you don’t start from an atom


This is not a unit cell


Are they unit cells ?


Unit Cell in 3D


simple cubic(sc)

body-centered cubic(bcc)

face-centered cubic(fcc)

3 Common Unit Cell in 3D


Unit Cell

Primi-tive

Conventional & Non-PrimitiveSingle lattice point per cell

Smallest area in 2D, orSmallest volume in 3D

More than one lattice point per cellIntegral multiples of the area of primitive cell

Simple cubic(sc)Conventional = Primitive cell

Body centered cubic(bcc)Conventional ≠ Primitive cell


Primitive and conventional cells of FCC

primitive unit cellconventional unit cell

Primitive vectors

.ˆˆ2

,ˆˆ2

,ˆˆ2

3

2

1

xza

a

zya

a

yxa

a

Q: How many lattice point ?


Primitive and conventional cells of bcc

primitive unit cell conventional unit cell

Primitive vectors

.ˆˆˆ2

,ˆˆˆ2

,ˆˆˆ2

3

2

1

zyxa

a

zyxa

a

zyxa

a

Q: How many lattice point ?


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


32Crystal Structure

• A primitive unit cell is made of primitive translation vectors a1 ,a2, and a3 such that there is no cell of smaller volume that can be used as a building block for crystal structures.

• A primitive unit cell will fill space by repetition of suitable crystal translation vectors. This defined by the parallelpiped a1, a2 and a3. The volume of a primitive unit cell can be found by

• V = a1.(a2 x a3) (vector products)

Cubic cell volume = a3

Primitive Unit Cell and Vectors


P = Primitive Unit CellNP = Non-Primitive Unit Cell

Primitive Unit Cell

The primitive unit cell must have only one lattice point.

There can be different choices for lattice vectors, but the volumes of these primitive cells are all the same


A simply way to find the primitive cell

Wigner-Seitz cell can be done as follows;

1. Choose a lattice point.2. Draw lines to connect these lattice point to its

neighbours.3. At the mid-point and normal to these lines

draw new lines.

The volume(or area) enclosed is called a Wigner-Seitz cell.

Wigner-Seitz Method


Wigner-Seitz Cell - 3D

fcc wigner-seitz cell bcc wigner-seitz cell


Crystal Directions

Any vector r which specify direction in a crystal can be written in terms of lattice vector

cwbvauT

This direction in a crystal is expressed as

uvw

If the numbers u v w have a common factor, this factor is removed. Ex. [111] rather than [222], or [100], rather than [400].

wvu ,, are smallest integer

negative directions are indicated as wvu ,,


X = 1 , Y = ½ , Z = 0

[1 ½ 0] [2 1 0]

[210]

Ex.

X = ½ , Y = ½ , Z = 1

[½ ½ 1] [1 1 2]

[112]


Ex.

We can move vector to the origin. X = -1 , Y = 1, Z = -1/6

[-1 1 -1/6] [6 6 1]


Crystal Planes

• Within a crystal lattice it is possible to identify sets of equally spaced parallel planes. These are called lattice planes.

• In the figure density of lattice points on each plane of a set is the same and all lattice points are contained on each set of planes.

The set of planes in 2D lattice.

b

a

b

a


Miller Indices

Miller Indices are a symbolic vector representation for the orientation of an atomic plane in a crystal lattice and are defined as the reciprocals of the fractional intercepts which the plane makes with the crystallographic axes.

To determine Miller indices of a plane, take the following steps;

1) Determine the intercepts of the plane along each of the three crystallographic directions2) Take the reciprocals of the intercepts3) If fractions result, multiply each by the denominator of the smallest fraction

hkl lkh ,, are smallest integer

(100)


Ex.

(100) (110) (111)

(200) (100)

(102) (102) (233)


)111(),111(),111(),111(),111(),111(),111(),111(}111{

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

Indices of a Family or Form

Indices {h,k,l} represent all the planes equivalent to the plane (hkl) through rotational symmetry.

Sometimes when the unit cell has rotational symmetry, several nonparallel planes may be equivalent by virtue of this symmetry, in which case it is convenient to lump all these planes in the same Miller Indices, but with curly brackets.


43Crystal Structure

Close packing

Close-Packed Structure

A A A A A

A A A A

AA A A A

A A A A

A A A

A A A

A A A

BB B B

B B B

BB B B

BB B BB

BB

B B BB B

A A A A A

A A A A

AA A A A

A A A A

C C C C

C C CC C

C C C C

CC C C C

Two sequence: ABABABABAB…… ABCABCABC ……..


44

A A A A A

A A A A

AA A A A

A A A A

BB B B

B B B

BB B B

BB B BB

BB

B B BB B

A A A A A

A A A A

AA A A A

A A A A

For ABABABAB

For ABABABAB

Hexagonal structureor hexagonal close-packed structure

Lattice vector:

a=b g=120, c=1.633a, basis : (0,0,0) (2/3a ,1/3a,1/2c)


A A A A A

A A A A

AA A A A

A A A A

BB B B

B B B

BB B B

BB B BB

BB

B B BB B

C C C C

C C CC C

C C C C

CC C C C

For ABCABCABC

For ABCABCABC

Cubic close packed structureor face centered cubic structure

[111] plane


How about square ?

A A A

A A A

A A A

B B

BB

A A A

A A A

A A A

Body centered cubic structure


Coordination 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.

For SC: BCC: FCC:6 8 12

Atomic packing factor

APF = Volume of atoms in the unit cell

Volume of unit cell

For SC: BCC: FCC:


Structure of Ionic Solids

Sodium Chloride

Both Cl- and Na+ represent fcc latticeand they are displaced by half unit in [100] direction

Can be viewed as a fcc lattice with a basis ofNa+(0,0,0), Cl-(½, ½, ½), or ?

Cesium Chloride

Both Cl- and Cs+ represent sc latticeand they are displaced by half unit in [111] direction

Can be viewed as a sc lattice with a basis ofCs+(0,0,0), Cl-(½, ½, ½)


Structure of Covalent Bond Solids

Height of atoms

Diamonds

Q: what type of hybridzation? sp3

Diamonds structure consists of two Interpenetrating fcc lattice

Structure is far from being close packed structure

Why so ? Nature of covalent bond restrict # of bondingi.e. nearest neighbor

Nature of covalent bond very different from metallic, ionic, VDW

Q: What other crystalline materials have diamond structure?

Si, Ge

Made of Carbon

Can be viewed as a fcc lattice with a basis ofC(0,0,0), C(1/4, 1/4, 1/4)


Structure of Covalent Bond Solids

For group III- group V

GaAs (gallium arsenide)InSb (indium antimonide)

Q: what type of bonding? covalent

The structure of these materials closely related to ZnS (zincblende)

Height of atoms

Can be viewed as a fcc lattice with a basis ofZn(0,0,0), S(1/4, 1/4, 1/4)


X-ray Crystallography

X-rays were discovered in 1895 by the German physicist Wilhelm Conrad Rönt-gen

X ray, invisible, highly penetrating electromagnetic radiation of much shorter wavelength (higher frequency) than visible light. The wavelength range for X rays is from about 10-8 m to about 10-11 m, the corresponding frequency range is from about 3 × 1016 Hz to about 3 × 1019 Hz.

/hcE

For l = 1Å,

X-ray

E ~ 104 eV

h = 6.626ⅹ10-34 J∙s = 4.136ⅹ10-15 eV


Production of x-ray

Evacuated glass bulb

Anode

Cathode

X rays can be produced in a highly evacuated glass bulb, called an X-ray tube, that contains essentially two electrodes—an anode made of platinum, tung-sten, or another heavy metal of high melting point, and a cathode. When a high voltage is applied be-tween the electrodes, streams of electrons (cathode rays) are accelerated from the cathode to the anode and produce X rays as they strike the anode

K, L, M lines for emitted photon from electron Transition to n = 1, 2, 3 orbitals


Interference of wave

Constructive interference is the result of synchronized light waves that add to-gether to increase the light intensity.

Destructive İnterference results when two out-of-phase light waves cancel each other out, resulting in darkness.

Young’s double slit exp. Constructive interference

Path difference: Dx =

Destractive interference

Path difference: Dx =

nl

n = 0, 1, 2, 3 …

nl + /2ln = 0, 1, 2, 3 …

To see interference: d ~ l

d

L

s

Principle of superposition


X-ray crystallography

Consider the solid as a diffraction grating with a spacing of ~1Å

/hcE

For l = 1Å, E ~ 104 eV

Similar to diffraction grating, measurement of the x-ray diffraction maxima from a crystal al-lows us to determine the size of the unit cell

Lattice point is the analogue of the line on an optical diffraction grating Basis represents the structure of the line


A B

CD

2

Bragg’s law

W.L. Bragg considered crystals to be made up of parallel planes of atoms. Incident waves are reflected specularly from parallel planes of atoms in the crystal, with each plane is reflecting only a very small fraction of the radiation, like a lightly silvered mirror.

In mirrorlike reflection the angle of incidence is equal to the angle of reflection.Therefore AC = DB for any angle. No interference pattern!The diffracted wave looks as if it has been reflected from the plane.

Coherent scattering from a single plane is not sufficient to obtain a diffraction maximum. It is also necessary that successive planes should scatter in phase


Path difference: sin2d

For constructive interference sin2d n

Consider successive planes

There is path difference between the light scattered from successive planes. And it varies with incident angle

Bragg’s law

DE + EF =


A beam corresponding to a value of n>1 could be identified by a statement such as ‘the nth-order reflections from the (hkl) planes’.(nh nk nl) reflection

Third-order reflection from (111) plane a (333) reflection

Bragg’s law

Rewriting the Bragg law

which makes n-th order diffraction off (hkl) planes of spacing ‘d’ look like first-order diffraction off planes of spacing d/n.

Planes of this reduced spacing would have Miller indices (nh nk nl)

sin2

n

d


Details of the structure can also be deduced from diffraction pattern

Ex. NaCl, KCl

Both shows diffraction peak for (100)

NaCl shows weak but clear (111) peaksBut, absence of (111) peaks for KCl

K+ and Cl- both have argon electron shell a nearly same strength of x-ray scatteringBut, Na+ and Cl- have different shell structure

(111) plane: only Cl-

(222) plane: only Na+

For KCl: reflection from (222) plane induces destructive interference


X-ray Diffraction Methods

Laue Rotating Crystal

Powder

Lattice parametersPolycrystal (powdered)Monochromatic beamVariable angle

Lattice constantSingle crystalMonochromatic beamVariable angle

OrientationSingle crystalPolychromatic beamFixed angle


Laue methods

The Laue method is mainly used to determine the orientation of large single crystals while radiation is reflected from, or transmitted through a fixed crystal.

The diffracted beams form arrays of spots, that lie on curves on the film

The Bragg angle is fixed for every set of planes in the crystal. Each set of planes picks out and diffracts the particular wavelength from the white radiation that satisfies the Bragg law for the val-ues of d and θ involved

The symmetry of the spot pattern reflects the symmetry of the crystal when viewed along the direction of the incident beam. Laue method is often used to determine the orientation of single crystals by means of illuminating the crystal with a continuos spectrum of X-rays

http://jcrystal.com/steffenweber/JAVA/jlaue/jlaue.html


X-RayFilmSingle

Crystal X-Ray Film

SingleCrystal

Transmission laue method

Back reflection laue method


Rotating crystal method

In the rotating crystal method, a single crystal is mounted with an axis normal to a monochromatic x-ray beam. A cylindrical film is placed around it and the crystal is rotated about the chosen axis

As the crystal rotates, sets of lattice planes will at some point make the correct Bragg angle for the monochromatic incident beam, and at that point a diffracted beam will be formed

The reflected beams are located on the surface of imaginary cones. By recording the dif-fraction patterns (both angles and intensities) for various crystal orientations, one can de-termine the shape and size of unit cell as well as arrangement of atoms inside the cell.


The powder methods

If a powdered specimen is used, instead of a single crystal, then there is no need to rotate the specimen, because there will always be some crystals at an orientation for which diffraction is permitted. Here a monochromatic X-ray beam is incident on a powdered or polycrystalline sample. This method is useful for samples that are difficult to obtain in single crystal form.

The powder method is used to determine the value of the lattice pa-rameters accurately. Lattice parameters are the magnitudes of the unit vectors a, b and c which define the unit cell for the crystal.

For every set of crystal planes, by chance, one or more crystals will be in the correct orientation to give the correct Bragg angle to satisfy Bragg's equation. Every crystal plane is thus capable of diffraction. Each diffraction line is made up of a large number of small spots, each from a separate crystal. Each spot is so small as to give the appear-ance of a continuous line.


If the sample consists of some tens of randomly orientated single crystals, the diffracted beams are seen to lie on the surface of several cones. The cones may emerge in all directions, forwards and backwards.

A sample of some hundreds of crystals (i.e. a powdered sample) show that the diffracted beams form continuous cones. A circle of film is used to record the diffraction pattern as shown. Each cone intersects the film giv-ing diffraction lines. The lines are seen as arcs on the film.

66

• If a monochromatic x-ray beam is directed at a single crystal, then only one or two diffracted beams may result.

The powder methods


The specimen is placed in the Debye Scherrer camera and is accurately aligned to be in the centre of the camera. X-rays enter the camera through a collimator.

The powder diffracts the x-rays in accordance with Braggs law to produce cones of diffracted beams. These cones intersect a strip of pho-tographic film located in the cylindrical camera to produce a characteristic set of arcs on the film.

solid state physics 02-crystal structure

Science

solid crystal structure

crystal lattice

crystal planes

jungwoo yoo crystal

solid state physics

fixed crystal

solid nacl

solid lattice vibration