Engineering Physics Unit 1 Quiz

Engineering Physics Course Notes

• Course Title: Engineering Physics
• University: Marwadi University, Rajkot
• Department: Physics
• Course Code: 01GS1101
• Instructor: Dr. Yogesh Jani

Unit 1: Electronic Materials

• Introduction: Free electron theory, Types of electronic materials (metals, semiconductors, insulators)
• Energy and band diagrams: Direct and indirect band gaps, Kronig and Penny model, E-K diagrams, Fermi level

Basic Terminologies

• Ohm's Law: Current through a conductor is directly proportional to voltage across it. (I α V), I = V / R, V = IR
• Resistance (R): Opposition a material offers to electric current flow. Factors affecting resistance:
  • Length of the wire/conductor
  • Cross-sectional area of the wire/conductor
  • Nature of the material
  • Temperature (Resistance increases with temperature)
• Resistivity (ρ): Constant of proportionality, relating resistance to geometry of a conductor, ρ = RA/l
• Conductivity (σ): Reciprocal of resistivity, σ=1/ρ, measured in Ω⁻¹m⁻¹ (or Siemens)
• Current (I): Flow of electric charge, I = q/t (Coulombs/second)
• Current Density (J): Current per unit area of cross-section, J = I/A (A/m²)

Electrical Conductivity (σ)

• Definition: Measure of a material's ability to carry electric current.
• Formula: V = IR, where R = ρ l/A and thus V= σl I/A
• Units: Ω⁻¹m⁻¹ (or Siemens)

Relation between Current Density (J), Drift Velocity (Vd), and Mobility (µ)

• Formula: J = n e Vd; where n is the electron density, e is the electron charge, Vd the drift velocity.
• Formula: µ = Vd / E; where µ is the electron mobility, Vd is the drift velocity, and E is the electric field strength.
• Formula: σ = n e µ; where σ is the conductivity.

Expression for Electrical Conductivity

• Formula: σ = ne²τ/m, where n is the electron density, e is the electron charge, τ is the relaxation time, and m is the electron mass.

Conductivity in terms of kBT

• Formula: σ = n e² τ / m
• Relation to kinetic theory of gases. τ = λ / Vd, where λ is the mean free path and Vd is drift velocity

Examples (Try Yourself)

• Several example problems involving calculation of current density, drift velocity and mobility given relevant data.

Electron Theory of Metals

• Classical free electron theory: Treating electrons in metals like molecules in a gas, with application of Maxwell-Boltzmann statistics.
• Quantum free electron theory: Sommerfeld theory, adding quantum theory to address the limitations of the classical free electron theory.
• Band theory of solids (zone theory): Developed by Bloch, considering the periodic arrangement of atoms in solids.

Classical Free Electron Theory

• Developed by Drude and Lorentz.
• Free electrons treated like gas molecules, obeying Maxwell-Boltzmann statistics, high temperature and low electron density.
• Formula for f(E): f(E)=1/[exp(E/KbT)-1]

Main Assumptions or Postulates of Classical Free Electron Theory

• Solid metal has a positively charged nucleus and negatively charged electrons.
• Electrons move freely in a uniform potential field, due to ions in the lattice.
• In the absence of an electrical field, electrons move randomly, collisions are elastic, and no energy loss is observed.
• When an electric field is applied, free electrons accelerate in a direction opposite to the field.
• Free electrons obey Maxwell-Boltzmann statistics.

Basic Terms Involved in Free Electron Theory

• Drift Velocity (VD): Average velocity of electrons due to applied electric field.
• Mobility (µ): Drift velocity of an electron per unit electric field.
• Relaxation time (τ): Time taken by an electron to reach equilibrium position from a disturbed position in an electric field.
• Mean collision time (τc): Average time between successive electron collisions.
• Mean free path (λ): Average distance traveled by an electron between successive collisions.

Success of Free Electron Theory

• Verifying Ohm's law
• Explaining thermal and electrical conductivities of metals.
• Deducing the Wiedemann-Franz law.
• Explaining optical properties of metals.

Drawback of Free Electron Theory

• Theoretical specific heat values don't match experimental values.
• Unable to explain electrical conductivity of semiconductors and insulators.
• Theoretical paramagnetic susceptibility values exceed experimental values.
• Inability to explain ferromagnetism.
• Different temperature dependencies of electrical and thermal conductivity.

Energy Band

• Conduction band: High energy levels for electrons.
• Valence band: Lower energy levels, where most electrons usually reside.
• Forbidden band gap: Energy difference between valence band and conduction band.
• Illustrative diagrams.
• Different types of energy bands exist (full band gap, overlapping, etc.)

Classification of Solids on the basis of Band Theory

• Conductors: Easy flow of current, overlapping valence and conduction bands.
• Semi-conductors: Resistive flow of electricity, small forbidden/ band gap.
• Insulators: No flow of current, large forbidden/ band gap.

Kronig Penney Model

• Simplified model for electrons in a one-dimensional periodic potential.
• Demonstrates formation of energy bands and band gaps.
• Illustrates calculation of allowed and forbidden energies using Schrodinger's equation.
• Helps understand effective mass and E-k diagrams.

Points that lead to Kronig-Penney model

• Solid assumed as a collection of free electrons inside a box, constant potential.
• However, in a real crystal, periodic arrangement of positively charged ions leads to a drastic change.
• Potential of electron at positive ions sites is zero and maximum in between the ions.
• Outside crystal, potential energy is infinite.
• Electron potential varies periodically with the same period as the lattice. Illustrative diagrams.

E-K Diagram

• Dispersion diagram showing relationship between energy and momentum of electrons
• Explains bandgap properly
• Illustrative diagrams include energy band diagrams and E-k diagrams.
• Relationship between bottom and top values of conduction and valence bands.
• Semiconductors have ordered atoms in the lattice.
• Momentum values are quantized.
• Kinetic energy proportional to the square of velocity.

Direct and Indirect Band Gap Semiconductors

• Direct: Maximum valence band energy level aligns with the conduction band minimum in terms of 'k'. Example: GaAs. Electron-hole recombination directly produces photons. Simple and efficient optical devices,
• Indirect: Maximum valence band energy level does not align with the conduction band minimum in terms of 'k'. Example: Si, Ge. Electron-hole recombination requires a phonon to transfer momentum, indirect process, less efficient compared to direct. Indirect semiconductors are typically used in electronic device applications.

Fermi Energy Level

• Highest energy level an electron can occupy at absolute zero.
• Fermi-Dirac probability function defines the fraction of states occupied at energy E
• Explains the completely filled states below EF (Fermi level) and empty states above it at T=0
• At non-zero temperatures, electrons may occupy energy states above the Fermi level, although it gradually reduces from 1 at E to 0 above the Fermi level. Transitions are gradual, not abrupt. Graphs illustrate these changes.

Test your knowledge on electronic materials, including free electron theory and types of materials such as metals, semiconductors, and insulators. This quiz also covers Ohm's Law and factors affecting resistance. Get ready to assess your understanding of key concepts in Engineering Physics!