ObjectivesBy the end of this section you should:• know how the Lennard-Jones [12,6] potential describes

the interaction between atoms• be able to calculate the van der Waals radius, the

distance between the atoms that minimises their energy• know about vacancies, interstitials and Frenkel defects• be able to calculate the energy of vacancy formation

from quenching data• know the difference between intrinsic and extrinsic

conduction, p- and n-type silicon and donor and acceptor doping

• be able to describe the different types of line defect and use the Burgers’ vector

Each atom exludes other from the space it occupies

Attraction?Electrons are moving so that, at some instant, distribution is uneven

Positively (electron-deficient) and negatively (electron-rich) charged regions electrical dipole

Dipole induces an opposing dipole in neighbouring atom attraction

Close Packing

Attractive force is known as:

• van der Waals interaction

• London interaction

• induced dipole-induced dipole interaction

U rrr

rr

( )

4 0

12

0

6

The total potential energy for two atoms a distance, r, apart can be written as:

This is called the Lennard-Jones (12,6) potential function

First term is repulsive, second term is attractive.

We want to find a minimum - so differentiate w.r.t. r

U rrr

rr

( )

4 0

12

0

6

0r

r6

r

r124

dr

dU7

60

13

120

7

60

13

120

r

r6

r

r12

60

6

120

60

7

13

r

r

r

r

r

r

6

12

06

1r2r

This is the van der Waals radius, the distance between the atoms that minimises their energy

U rrr

rr

( )

4 0

12

0

6

0

61

r2r

Substituting back in to the (12-6) potential gives the minimum energy:

6

61

12

61min

2

1

2

14U

2

1

4

14Umin

Energy has a minimum value of - at the van der Waals radius

Defects

Up to now we have considered perfect crystals, i.e. crystals with perfect periodic arrangements.

Most “good” crystals show very little departure from this idea, e.g. silicon single crystals can be grown without defects over a range of several mm

This sounds small but is about 10 million unit cells!

However, defects are very important in processing and for optical and electrical properties.

1. VacanciesA vacancy is the absence of an atom in the lattice.

In ionic crystals (e.g NaCl) vacancies occur in pairs (Na + Cl) so that charge balance is maintained.

Also called a Schottky Defect.

Vacancies allow diffusion through the crystal:

Vacancy : point defect - associated with a point in the crystal

Vacancies

Vacancies are not energetically favourable - the number of vacancies increases with temperature (i.e. putting energy into the system)

Mathematically, for a crystal containing N atoms, there is an equilibrium number of vacancies, n, at temperature T (in K) given by:

Tk

EexpNn

B

V

where EV is the energy of vacancy formation and kB is Boltzmann’s constant. Applies to pairs also.

Diffusion

Similary, the diffusion coefficient, D, is given by:

Tk

EexpDD

B

DO

where ED is the energy of diffusion and DO is a diffusion constant specific to the element.

Strictly this applies only to self-diffusion, that is diffusion in an elemental substance.

Quenching

Non-equilibrium concentrations of vacancies may be obtained by rapidly cooling (quenching) metals from high temperatures.

These defects can cause additional resistivity proportional to the number of defects:

Tk

EexpCNCnR

B

V

where C is a proportionality constant.

R is the relative increase in resistance at low temperature after quenching from the temperature T.

Uses

so:

Tk

ECNlnRln

B

V

y = c + mx

EV can be obtained from a graph of lnR against (1/T)

Tk

EexpCNCnR

B

V

Example - Gold

33 1036.1109.0

)84.22(73.17Gradient

K11100k

E

B

V

J10536.1E 19V

eV96.0

2. Interstitials

Previously we discussed small tetrahedral and octahedral interstitial atoms within the close packed structure.

If the interstitial atom is the same size as the close packed atoms, then considerable disruption to the structure occurs.

Again, this is a point defect and requires much energy

3. Frenkel Defects

Often a vacancy and interstitial occur together - an ion is displaces from its site into an interstitial position.

This is a Frenkel Defect (common in e.g. AgCl) and charge balance is maintained.

Frenkel defects can be induced by irradiation of a sample

4. Impurities

Preparing pure crystals is extremely difficult - often foreign atoms enter the structure and substitute for “native” atoms - often by contamination from container

This can have a large effect (either detrimental or beneficial) on the properties of the crystal. We can also add impurities (or dopants) deliberately.

An important example is that of silicon.

Silicon

Silicon is a group IV element and, like carbon, bonds to four nearest neighbours:

At elevated temperatures bonds are broken to produce a (positive) gap - known as a hole - and a conduction electron.

T

This is known as the intrinsic effect in semiconductors

Doped Silicon

If we take a group V element (e.g. As) and substitute (at low levels) for Si there is a spare electron for conduction and no positive hole:

This process is known as “doping”. Arsenic acts as an electron donor to Si, making it easier to conduct electricity.

Si doped with As is an extrinsic semiconductor and because the electron is negative this is an n-type semiconductor

Doped Silicon

If we take a group III element (e.g. B) and substitute (at low levels) for Si there is a positive hole and no conduction electron

Boron acts as an electron acceptor to Si.

Electrons can move by diffusion - “hopping” into the hole leaves behind a new hole.

Again this is an extrinsic semiconductor and because the hole is negative this is a p-type semiconductor

Line Defects - 1. Stacking Faults

We discussed h.c.p which has sequence ABABABA and c.c.p. which has sequence ABCABCA.

A stacking fault occurs when the sequence goes wrong, e.g. ABCBCABCABC (A missing) or ABCABACABC (extra A)

Often these do not extend right across the plane, e.g.

This is also known as a partial dislocation

Line Defects - 2. Edge dislocations

Originally proposed to account for mechanical strength in crystals.

Consists of an extra plane of atoms which terminates within the crystal. This distorts the local environment.

Burgers Vector

If the dislocation was not present, then atom at A would be at A’

We define a vector B which shows the displacement of A due to the dislocation.

B is known as the Burgers’ Vector.

For an edge dislocation, the Burgers’ vector is perpendicular to the dislocation

SlippingSuch defects are produced by part of the crystal “slipping” with respect to the rest.

Consider a close packed structure:

For the top layer to slip to the right, to another close packed position, it must pass through a non-equilibrium position

Line Defects - 3. Screw dislocations

Here there are no extra planes - the defect appears as though part of the crystal has been cut in two, then shifted down on one side of the cut.

Burgers’ Vector

In this case, A would have been at A’ had the dislocation not occurred.

The Burgers’ Vector B is hence parallel to the direction of the screw dislocation.

Screw dislocations are important in the growth of single crystals since they provide nucleation sites for the growth of a new layer

Line Defects - 4. Twinning

Crystals are often grown with a fault in which one region of the crystal is a mirror image of the other:

In c.p. structures, twins are produced by stacking faults

ABCABCBACBA

Here C is the twin plane

Polymorphic compounds (i.e. ones with more than one crystal structure) are prone to twinning, e.g. YBa2Cu3Od

SummarySummary the attraction/repulsion between two atoms of size, r,

can be adequately described by the Lennard-Jones [12,6] potential

the point of minimum energy in the LJ potential is the van der Waals radius

Most crystals contain defects

Extra vacancies can be produced by quenching; this can produce an increase in resistivity which can be calculated.

Defects can be used to advantage, e.g. doped silicon

Line defect formation can be described using the Burgers’ Vector, B