Electrical properties of matter lecture slide

Materials lecture slide

1. Chapter 2: Electrical and Thermal
Conduction in Solids
1

2. Electrical Conduction
•Electrical Conduction: Motion of charges in a material under the influence of an
applied electric field
•Recap: In metals ‘Cloud/Sea of Electrons’ or ‘Electron Gas’
Valence electrons are not bound to individual atoms
Free to move around form a sea/cloud of electrons in which the metal ions
are immersed Conduction electrons
2

3. Electrical Conduction
DRUDE MODEL (Classical Theory) explains electrical conduction in solids.
Proposed by Paul Drudeto explain the transport properties of electrons in materials
(especially metals).
The model assumes the microscopic behavior of electrons in a solid classically, with
a sea of constantly jittering electrons bouncing and re-bouncing off heavier,
relatively immobile positive ions.
3

4. 2.1 Classical Theory: The DrudeModel
Random Motion of Conduction Electrons:
•Kinetic energy originates from the electrostatic interaction of electrons with
positive metal ions and also with each other.
•Conduction electrons move about randomly (with a mean speed �) being
frequently and randomly scattered by thermal vibrations of the atoms.
•In absence of an applied electric field, there is no net drift in any direction.
 NOnet flow of charge
4

5. Under Applied Electric Field
•Conduction electrons experience a force of �??????
??????in the opposite direction of
??????
??????.
•The electron accelerates along the �direction under the action of the force
�??????
??????, and then it suddenly collides with a vibrating atom and loses the
gained velocity there is an average velocity in the �direction.
Anet flow of charge occurs
5

6. •The electric current density: ??????=
∆??????
??????∆??????
∆??????is the net quantity of charge flowing through an area ??????in time ∆�
6
No Applied E field Efield is applied
No net charge flow ??????=0 Drift of electrons A net charge flow
??????
??????≠0

7. The Drift Velocity
•Drift Velocity: Velocity of electrons averaged over many electrons
•Let, �
�??????electrons drift velocity along�direction
•Assume �=�/??????number of electrons per unit volume
•In time ∆�, electrons move a distance, ∆�=�
�??????.∆�and the charge crossing
??????is, ∆??????=��??????∆�=��??????�
�??????∆�
•The current density in the �direction: ??????
??????=
∆??????
??????∆??????
=
��????????????
�??????∆??????
??????∆??????
=���
�??????
•Time-dependent current: ??????
??????(�)=���
�??????(�)
7
�
�??????=
1
�
[�
??????1+�
??????1+�
??????1+⋯+�
??????�]
�
????????????�direction velocity of the ??????-thelectron
�Number of conduction electrons

8. Drift Velocity
•We had: �
�??????electrons drift velocity along�direction
�
�??????=
1
�
[�
??????1+�
??????1+�
??????1+⋯+�
??????�]
�
????????????�direction drift velocity of the ??????-thelectron
�Number of conduction electrons
•Consider i-thelectron
Let, the last collision happened at time �
??????
x-directed velocity right after the collision (initial velocity) �
????????????
Acceleration of the electron =�??????
??????/�
�
∴Velocity �
????????????in the x-direction at time t, �
????????????=�
????????????+
�????????????
��
(�−�
??????)
For electrons 1, 2, 3, ….. :
8

9. �
�??????=
1
�
[�
??????1+�
??????1+�
??????1+⋯+�
??????�]
=
1
�

??????=1
�
�
????????????+
�????????????
��
1
�

??????=1
�
[�−�
??????]
Immediately after a collision with a vibrating ion, the electron may move in any random
direction. So, �
????????????averaged over many electrons = 0.

??????=1
�
�
????????????=0
∴�
�??????=
�????????????
��
1
�

??????=1
�
[�−�
??????]=
�????????????
��
(�−�
??????)
(�−�
??????)≡average free time between collisions≡??????(tau)
??????:Mean free time/mean time between collisions/mean scattering time
∴�
�??????=
�??????????????????
��
•??????is directly related to the microscopic processes that cause the scattering of the electrons in
metal, e.g. lattice vibrations, crystal imperfections, and impurities, to name a few.
•1/??????represents the mean frequency of collisionsor scattering events.
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10. Drift Mobility
�
�??????=??????
�??????
??????
??????
�≡drift mobility
??????
�=
�??????
�
�
•??????
�represents the ease of electron conduction under an electric field.
??????
�↑easier for electrons to move
??????
�↓more difficult for electrons to move
•If the electron is not highly scattered, then the mean free time between collisions
will be long, τ will be large, and ??????
�will also be large.
The electrons will be highly mobile and be able to ‘respond well’ to the field.
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11. Ohm’s Law and Conductivity
??????
??????=���
�??????
•Using the expression for the drift velocity, �
�??????:
??????
??????=��??????
�??????
??????
•Ohm’s law: ??????
??????=????????????
??????
??????=��??????
�Conductivity
•A large ??????
�does not necessarily imply high conductivity, because ??????also depends
on the concentration of conduction electrons �.
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12. Example Problem
Calculate the diftmobility and the mean scatteingtime of conduction electrons in
copper at room temperature, given that the conductivity of Cu is 5.9×10
5
Ω
−1
cm
−1
. The density of Cu is 8.96 g cm
−3
and its atomic mass is 63.5 g mole
−1
.
Sol.:
We know: ??????=��??????
�
if �is the atomic volume concentration
�=
��
????????????
�
??????
here, �
??????=6.02×10
23
atoms
�
????????????=63.5g mole
−1
�=����??????��=8.96gcm
−3
�=8.5×10
22
atoms cm
−3
Assume: each Cu atom donates 1 electrons to the cloud of electrons.
then: �
�=8.5×10
22
atoms cm
−3
12

13. 13
See other worked out examples: Examples 2.1, 2.3, 2.4