This document provides an introduction and syllabus for a module on electrical properties and free electron theory. It discusses key concepts from classical and quantum free electron theory, including postulates, failures of the classical theory, the importance of Fermi energy, and advantages of the quantum theory. Example questions are also provided to test comprehension of topics like thermal velocity, root mean square velocity, and the effects of temperature on conductivity.

1. FREE ELECTRON
THEORY

2. Introduction – Course Outcome and Syllabus of Module 1
 Intention of the module is to enable students to: Compare conductors,
semiconductors, dielectric and superconducting materials.
 Electrical Properties: Failures of classical free electron theory (CFT), postulates of
quantum free electron theory (QFT), merits of QFT, Fermi energy.
 Semiconductors: Fundamentals and types of semiconductors, p-n junction
formation, solar cells - working and efficiency of solar cells.
 Dielectric materials: Introduction, types of polarization in dielectrics (qualitative),
frequency and temperature dependence of polarization.
 Superconductivity: Introduction, Meissner effect, BCS theory, critical field, types of
superconductors.
 Total Duration: 12 Hours
 Blooms level selected: Comprehension
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3. Importance of Electrical Properties to a budding engineer
 Copper, silicon and diamond. They are all crystalline but they have different electrical
conductivities.
 Why do they exhibit different properties? The answer will help us modify, produce, design
or engineer new materials which are special and exhibit exact properties that we require.
 This unit deals with some fundamental concepts of electrical conductivity.
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Module 1: Electrical Properties – Classical Free Electron Theory

4. Classical Free Electron Theory
 Classical – the electron obey classical laws of Physics
 They are treated a gas molecules obeying the kinetic theory of gases.
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Module 1: Electrical Properties – Classical Free Electron Theory

5. Postulates of Classical Free Electron Theory – 1
 Valence electrons of atoms are free to move through out the volume of
the metal, like the molecules of a perfect gas.
 The attraction between the free electrons and the lattice ions, and the
repulsion between the electrons themselves are ignored.
 The free electrons travel in a constant potential inside the metal but stay
confined within its boundaries.
 The movements of free electrons obey the laws of the classical kinetic
theory of gases, Therefore their average kinetic energy, ½ mvth
2 = 3/2 k T,
where vth is the thermal velocity, k, Boltzmann’s constant and T is
temperature.
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Module 1: Electrical Properties – Classical Free Electron Theory

6. Electron movement in conductors
 Under the influence of electric field the randomly moving free
electrons drift in a direction opposite to the applied electric field and
acquire an additional velocity.
 The average velocity acquired by the free electron in the presence of
an applied electric field is known as Drift Velocity (νd)
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Module 1: Electrical Properties – Classical Free Electron Theory
Fig 1 (a) Random motion of electron without external electric field
(b) Motion of electron with drift caused by external electric field

7. Some important quantities
 The average time taken by the free electron to reach steady state
velocity from zero velocity is called Relaxation Time (τr)
 The average distance travelled between two successive collisions, in
absence of an external electric field, is known as Mean Free Path (λ)
 The average time taken by a free electron between two successive
collisions is called as mean free time or Collision Time (τc)
 The average velocity of free electrons is called Root Mean Square
Velocity (νrms) or the thermal velocity of electrons.
 In this model, it is assumed that collision time (τc) is equal to
relaxation time (τr) and the thermal velocity (νrms) is equal to drift
velocity (νd).
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Module 1: Electrical Properties – Classical Free Electron Theory

8. Ohm’s Law– microscopic form
 If electrons are free to move, what do they collide against?
 V= IR ohms law in macroscopic conditions
 𝑅 = ρ
𝐿
𝐴
 ρ is resistivity. Its inverse σ is conductivity
 Electric field developed across a metal is 𝑉 = 𝐸𝐿
 Current density 𝐽 =
𝐼
𝐴
 hence 𝐽 = σ𝐸 microscopic conditions
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Module 1: Electrical Properties – Classical Free Electron Theory

9. Mobility and Mean free path
 Mobility is defined as the steady-state drift velocity per unit electric
field
 Mobility of an electron μ = 𝑣𝑑 / 𝐸
 μ = −𝑒 τ
𝑚
(𝑚2𝑉−1𝑠−1)
 σ =
𝑛𝑒2τ
𝑚
 Mean free Path (λ) is the average distance travelled by electrons
between collisions.
 λ = velocity*time = 𝑣𝑟𝑚𝑠 ∗ τ
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Module 1: Electrical Properties – Classical Free Electron Theory

10. Success of Classical Free Electron Theory
 Explains electrical and thermal conductivities of most metals
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Module 1: Electrical Properties – Classical Free Electron Theory

11. Failures of Classical Free Electron Theory-1
 CFET could not account for specific heat of metals
(The value of electronic specific heat is equal to 3R/2. while the actual
value is about 0.01 R only)
 Temperature dependence of σ=1/T and
 The dependence of electrical conductivity on free electron
concentration 𝝈 =
𝒏𝒆𝟐𝝉
𝒎
 n(Zn)= 13.1×1028/m3 n(Cu)= 8.45×1028/m3
σ(Zn)=1.09×107 /Ωm σ(Cu)= 5.88x 107 /Ωm
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Module 1: Electrical Properties – Classical Free Electron Theory

12. Central Ideas of Quantum Free Electron Theory
 Quantum – available energy values are not continuous but quantized
 The distribution of electrons in the various allowed energy levels
occurs as per Pauli Exclusion Principle.
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Module 1: Electrical Properties – Quantum Free Electron Theory

13. Postulates of Quantum Free Electron Theory
 Valence electrons of atoms are free to move through out the volume
of the metal, like the molecules of a perfect gas.
 The attraction between the free electrons and the lattice ions, and the
repulsion between the electrons themselves are ignored.
 The free electrons travel in a constant potential inside the metal but
stay confined within its boundaries.
 The energy values of the conduction electrons are quantized.
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Module 1: Electrical Properties – Quantum Free Electron Theory

14. Fermi Energy and its importance
Fermi energy: It is the highest energy of the free electron at 0K.
Fermi energy EF level is the maximum energy level up to which electrons
can be filled at 0K.
Fermi level acts as a reference level that separates the vacant and filled
states at 0 K.
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Module 1: Electrical Properties – Quantum Free Electron Theory

15. Advantages of Quantum Free Electron Theory
 It is successfully explains the electrical and thermal conductivity of
metals.
 Temperature dependence of conductivity of metals can be explained
by this theory.
 It explains the specific heat of metals.
 Phenomenon of thermionic emission can be explained by this theory.
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Module 1: Electrical Properties – Quantum Free Electron Theory

16. Multiple Choice Questions - Recollection
 In classical free electron theory
Thermal velocity is equal to
a) Drift Velocity
b) Root mean square velocity
c) Zero velocity
d) Absolute velocity
 The velocity of electrons in a
copper wire is called as
a) Drift Velocity
b) Root mean square velocity
c) Zero velocity
d) Absolute velocity
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Module 1: Electrical Properties – Example Questions
 With respect to the classical free electron
theory choose the right statement
a) energy values of the free electrons are
discontinuous
b) free electrons obey the Pauli’s exclusion principle
c) free electrons obey the laws of kinetic theory of
gases
d) Electrons can jump to the conduction band

17. Multiple Choice Questions - Recollection
 Thermal velocity can be expressed in terms of
a) Moles
b) microns/milliseconds
c) eV
d) m/s-1K-1
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Module 1: Electrical Properties – Example Questions

18. Multiple Choice Questions - Comprehension
 What causes the random motion of electrons in quantum free electron
theory?
a) Applied electric field
b) Electric field produced by the neighbouring ions
c) Repulsion or collision by other electrons
d) Thermal energy
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Module 1: Electrical Properties – Example Questions

19. Multiple Choice Questions - Comprehension
 With increase in temperature, the root mean square velocity of
electrons in classical free electron theory
a) Increases
b) Decreases slightly
c) Remains same
d) Decreases significantly
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Module 1: Electrical Properties – Example Questions

20. Multiple Choice Questions - Comprehension
 Keeping temperature constant, electrons with lower degrees of freedom will have
root mean square velocities that are
a) Higher than electrons with higher degrees of freedom
b) Lower than electrons with higher degrees of freedom
c) Similar to electrons with higher degrees of freedom
d) Independent of the degrees of freedom
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Module 1: Electrical Properties – Example Questions