Aim

To study the conductivity of Fe2O3

with different particle sizes using

Impedance Spectroscopy and find

its activation energy.

Impedance

Is the measure of ability of a circuit to

resist the flow of electrical current.

It introduces the concept of resistance to

AC circuits because of the two mechanisms

i. The induction of voltages in conductors

self induced by magnetic fields of the

current (Inductance)

ii. The electrostatic storage of charge induced

by voltages between conductors

(Capacitor)

The excitation signal ,expresses as a function of

time,has the form

ET = EOsin(wt), ET is the potential at time t, Eo is the

amplitude of the signal and w is the frequency given

by

w=2∏f , f is frequency in hertz.

In linear system , the response signal IT is shifted in

phase Ø and has amplitude Io.

IT = Io sin(wt+Ø) , then using Ohm’s Law,

Z=ET/IT , Z is the impedance of the system

= EO sin(wt)/Io sin(wt+Ø)

= Zo sin(wt)/sin(wt+Ø)

If we plot the applied sinusoidal signal E (t) on X-axis

and sinusoidal response I(t) on Y-axis, we get an oval

figure called “Lissajous Figure “,analysed using

oscilloscope for impedance measurement .

With Euler’s relationship

exp(jØ)= cosØ+ jsinØ ,

the potential is described as

ET = Eo exp(jwt)

and current response as,

IT = Ioexp(jwt+Ø) ,

then impedance is represented as a complex no. given

by :

Z(w)=E/I=Zo exp(jØ)= Zo(cosØ+jsinØ)

Impedance Spectroscopy

It measures the electrical response of a material of

interest and its subsequent analysis of the properties

of the system, analysis generally being carried out in

the frequency domain .

We make AC dependent measurement to study the

transporting property of charge carrier in materials

which is the basic requirement for manufacturing

electronic devices.

Density of states

Density of states : It is defined as the no. of

quantum states available between energy levels E

and E+dE per unit volume in real space.

Bulk: Is defined as the system/material with

dimensions larger than the de-broglie wavelength of

the charge carriers so that the quantum effects do

not come into picture.

Density of states is proportional to E1/2 and is a

continuous function of energy.

Classification of

nanostructure materials

I-D confinement : electron movement confined along one dimension (2-D structure)E.g. Quantum well, disc, platelets, films etc.

Density of states given by :

D2D (E)=Σ(all states) (me / pi*h2 ) * Ɵ(E-Enz ) ,

so there is a minimum energy E>Enz below which electrons cannot occupy the states as Ɵ function becomes zero. So the continuous energy levels become discrete and the band gap increases.

2-D confinement : electron movement confined

along two dimensions(1-D structure).E.g. Quantum

wire, nano tubes, nano rods etc.

Density of states is given by :

D(1D) (E) =Σ(all states) √( 2me ) /(pi*h2) * Ɵ(E-Eny,nz) *

(1/√( E-Enx,ny)

The density of states is a step funtion of energy

levels and exists only for E>Enx,ny.

3-D confinement : electron movement restricted

along all the three dimensions (0-D structure).

E.g. quantum dot, nano particle etc.

Density of states is given by :

D(0D)(E) = ∑(all states)δ(E-Enx,ny,nz ) .

The density of states exist only when E=Enx,ny,nz

and no states available below this. So we obtain

sharp energy levels.

With decrease in the dimensions the band gap

increases.

Activation Energy

It is defined as minimum energy required for

charge carrier to participate in conduction

process.

It is given by Arrhenius Equation,:

σ=σo*exp(−Ea/kT),where σo is the temperature independent quantity, Ea is the activation

energy, k is the boltzmann constant, T is the temperature (in Kelvin)

and σ is the temperature dependent quantity (in general) .

Taking log both sides we get,

ln(σ)= -(Ea/k)*(1/T)+ ln(σo),

which is straight line equation( y=mx+c ).Hence slope of graph b/w (ln σ)

and (1/T) gives the activation energy for that temperature range.

Ball milling

Principle:A ball mill works on the principle of

impact: size reduction is done by impact as the

balls drop from near the top of the shell.

Ball mills rotate around a horizontal axis, partially

filled with the material to be refined plus the

grinding medium (zirconium balls). An internal

cascading effect reduces the material to a fine

powder.

Procedure

1).Take the pellets of different sizes (ball milled for

different time periods), measure their diameter and

thickness using vernier caliper and one by one at

room temperature place them in the sample holder

attached to the main instrument. Set the desired

apparatus specifications and start the experiment.

2). Now consider one of the pellets (say ball milled

for 48 Hrs) and take the conductivity reading for

varying frequency at different temperatures starting

from RT.

3).Plot the graphs and analyse the result .

Graphs ,Observations and

Discussion

( at RT )

6Hr

Original

24Hr

48 Hr

(Hr)

Why the decrease in

conductivity?

When we reduce the particle size, the quantum

confinement come into play, the no. of density

states available for electrons to occupy decrease as

the energy levels become discrete and the band gap

increases, so the conductivity decreases.

Why the increase in

conductivity for even more

reduced particle size ?

The increase in conductivity after a limit may be

attributed to the strain in the system or other

interstitial defects.

This strain energy may supersede the quantum

confinement effect and the iron metal which

experiences compressive strain may start

compressing so the hopping distance b/w the

density of states for the electrons reduces, their

transport becomes easier and the conductivity

increases.

Variation in Conductivity With

Temperature

In general the increase in conductivity with

increase in temperature is attributed to the theory

of semiconductors i.e. with increase in

temperature even though mobility decreases but

the no. of charge carriers increases exponentially

hence conductivity too increases.

DC and AC components

The measured conductivity is composed of two terms:

σ=σdc(T) +σac(w,T)

First term represents the temperature dependent DC

component of conductivity that is related to the drift

mobility of electric charge carriers and dominates in low

frequency case where it shows an independent behavior

or high temperature case when sufficient time and

energy and provided for charge carrier to jump from

valence to conduction band i.e. why we see a linear

(frequency independent) or dc conductivity at low

frequency .

Second term represents the frequency and

temperature dependent AC component and is

attributed to the Hopping Conduction caused by

localized electric charge carriers and obeys the power

law form which in the graph is represented by the

increasing curve .

Plot of Log(σDC) Vs (1000/T) shows linear

dependence for given Fe2O3 (48 hour ball milled)

sample in a certain temperature range.

Hence, given plot is Arrhenius Plot.

Slope of this plot will be given by: Ea/k

Slope obtained from the graph is 6893.6 K.

Hence, activation energy of a given Fe2O3 sample is

= Slope × k.

=6893.6(K) × 8.617E-5(eV/K)= .59402(eV)

RESULT: Ea =0.594 eV

Thank you