Engineering Physics

Engineering Physics Notes

Dr. Arun Khalkar - Textbook on Engineering Physics

• Based on the PCCOE 2024 curriculum.
• Course Code: BSH21BS03 / BSH22BS03
• For F.Y.B.Tech students at Pimpri Chinchwad College of Engineering (PCCOE), Nigdi, Pune.
• Author: Dr. Arun Khalkar, Ph.D. (Sungkyunkwan University, SKKU, Korea), Assistant Professor, Department of Applied Sciences and Humanities, PCCOE.
• Dr. Khalkar has over 15 years of teaching experience, more than 10 research papers, and has presented at over 15 national and international conferences, including IEEE PVSC, SKKU Solar Forum, and Photovoltaic Science and Engineering Conference in China.
• He has mentored over 50 students in Project-Based Learning (PBL), Mini Projects, and Basics of Innovation (MPBI).

Preface

• Engineering Physics serves as a bridge between physics principles and engineering applications.
• The book aims to provide a balanced understanding of foundational concepts and practical applications.
• It is divided into four units:
  • Semiconductor Physics
    • Behavior of semiconductors and conductors.
    • Band Theory of Solids and Electrical Conductivity.
    • Hall Effect derivation.
    • Fermi-Dirac Probability Distribution.
    • Fermi Level.
    • Solar cells, I-V characteristics.
    • Sensors: characteristics, types, and applications.
  • Quantum Mechanics
    • Wave-particle duality.
    • De Broglie hypothesis.
    • Heisenberg Uncertainty Principle.
    • Wave function, probability interpretation.
    • Schrödinger’s Time-Independent Wave Equation.
    • Particle in a rigid box, tunneling effect, Scanning Tunneling Microscopy (STM).
  • Magnetism and Superconductivity
    • Magnetic Hysteresis.
    • Giant Magnetoresistance (GMR).
    • Magnetocaloric Effect.
    • Superconductivity: zero electrical resistance, Meissner Effect, BCS Theory
    • Type I and II Superconductors, Josephson Effect, SQUIDs, superconducting magnets, maglev trains.
  • Introduction to Nanoscience
    • Quantum confinement.
    • Surface-to-volume ratio.
    • Unique properties of nanomaterials.
    • Methods for preparation of nanomaterials: high-energy ball milling, physical vapor deposition, colloidal synthesis.
    • Applications of nanomaterials in medicine, energy, and defense.
    • Introduction to quantum computing.
• The book includes derivations, numerical problems, and practical applications, diagrams and examples.

Program and Course Information

• Program: B. Tech. (All Branches)
• Semester: I / II
• Course: Engineering Physics
• Code: BSH21BS03 / BSH22BS03
• Credits: 4
• Teaching Scheme: Lecture 2 hrs/week, Practical 2 hrs/week.
• Evaluation Scheme: FA (FA1 10 marks, FA2 10 marks), SA 30 marks, Total 50 marks.
• Prior knowledge required: Atom, molecule and nuclei, current, electricity and magnetism, and electromagnetic induction.
• Course Objectives:
  • Build conceptual understanding of Semiconductor and Quantum Physics.
  • Explore advances in Physics with Nanotechnology and Superconductivity.
  • Provide awareness of Physics principles in engineering applications.
• Course Outcomes: Students should be able to:
  • Apply semiconductor physics to explain charge carrier behavior.
  • Distinguish wave behavior of matter at quantum scale.
  • Interpret superconductor properties and applications.
  • Summarize nanomaterial properties, preparation, and applications.

Detailed Syllabus

• Unit I: Semiconductor Physics (8 Hrs)
  • Band Theory of solids
  • Electrical conductivity of conductors (qualitative) and semiconductors (Derivation), Numerical.
  • Hall effect (with derivation), Numerical
  • Fermi Dirac probability distribution function
  • Fermi level (Definition)
  • Fermi level variation with i) temperature and ii) doping concentration
  • Solar cell I-V characteristics
  • Basics of sensors
  • Characteristics of sensors: range, sensitivity, resolution, accuracy, repeatability
  • Types of sensors: Active and Passive sensors
  • Applications of sensors.
• Unit II: Quantum Mechanics (7 Hrs)
  • Wave-particle duality of radiation and matter
  • De Broglie hypothesis
  • De Broglie wavelength in terms of kinetic and potential energy, Numerical concept of wave packet
  • Properties of matter waves
  • Heisenberg’s uncertainty principle
  • Wave function and probability interpretation, well behaved wave function
  • Schrödinger’s time independent wave equation
  • Applications of independent wave equation to the problem of (i) particle in rigid box (Derivation), Numerical.
  • Tunneling effect
  • Scanning tunneling microscope (STM).
• Unit III: Magnetism and Superconductivity (7 Hrs)
  • Magnetism: Magnetic hysteresis loop, giant magneto-resistance (GMR), magneto caloric effect (only statement), adiabatic demagnetization.
  • Superconductivity: Introduction, critical temperature, properties of superconductors - zero electrical resistance, persistent current, Meissner effect, critical magnetic field, Isotope effect, Numerical BCS theory, type I and II superconductors, low Tc and high Tc superconductors, Josephson effect, DC-SQUID- construction, working and applications, applications - superconducting magnets, maglev trains.
• Unit IV: Introduction to Nanoscience (8 Hrs)
  • Introduction.
  • Surface to volume ratio.
  • Quantum confinement
  • Properties of nanomaterials- optical, electrical, mechanical, magnetic
  • Methods of preparation of nanomaterials- bottom-up and top-down approaches
  • Physical methods- high energy ball milling, physical vapor deposition Chemical method - colloidal route for synthesis of gold nanoparticles.
  • Applications of nanomaterials in medical, energy, automobile, space, defense
  • Introduction to quantum computing.

Unit 1: Semiconductor Physics

• Semiconductor physics underlies modern technology, enabling advancements in communication, computation, energy, and sensing.
• Semiconductors fill the space between conductors and insulators; semiconductors offer versatile platforms for various applications.
• Key concepts include:
  • Band Theory of Solids: Explains electron distribution in energy bands, differentiating conductors, insulators, and semiconductors.
  • Electrical Conductivity: Semiconductor conductivity is tunable and changes with temperature or doping.
  • Hall Effect: Reveals charge carrier properties under magnetic fields - type, density and mobility.
  • Fermi Level and Fermi-Dirac Distribution: Central in understanding electron behavior in semiconductors, especially under varying conditions.
  • Solar Cells: Convert light to electrical energy; I-V characteristics show efficiency.
  • Sensors: Can be active ot passive, with parameters like sensitivity, range, resolution, accuracy, and repeatability; used in diverse fields.

Semiconductor Materials

• Elemental Semiconductors
  • Composed of a single element, primarily from Group IV (Silicon and Germanium).
  • Silicon (Si): Abundant, stable, band gap of $1.12 eV$, ideal for diodes and transistors.
  • Germanium (Ge): Smaller band gap ($0.69 eV$), used in infrared sensors.
• Compound Semiconductors
  • Formed by combining two or more elements, typically from Groups III-V or II-VI.
  • Gallium Arsenide (GaAs): High electron mobility, direct band gap of $1.43 eV$, ideal for high-frequency and optoelectronic applications (LEDs and lasers).
  • Indium Phosphide (InP): Used in high-speed communication systems and solar cells because of its direct band gap and high efficiency.
  • Cadmium Telluride (CdTe): Used in thin-film solar cells, ideal band gap of $1.45 eV$ for solar energy absorption.
• Alloy Semiconductors
  • Formed by mixing two or more semiconductors in varying proportions; properties tuned by composition.
  • Silicon-Germanium ($Si<em>xGe</em>{(1-x)}$): Band gap and lattice constants tailored by varying 'x' for high-speed integrated circuits.
  • Aluminum Gallium Arsenide ($Al<em>xGa</em>{(1-x)}As$): Used in optoelectronic devices like laser diodes, band gap adjusted by changing 'x'.

Origin of Energy Bands in Solids

• Energy bands arise from the interaction of atomic orbitals when numerous identical atoms form a solid (crystal lattice).
• In isolated atoms, electrons occupy discrete energy levels within atomic orbitals.
• When atoms are closely packed in a solid, atomic orbitals overlap, causing discrete energy levels to split into N closely spaced energy levels for each atomic energy state.
• These closely spaced levels merge into continuous ranges of allowed energies, called energy bands.
• Regions between energy bands where no electron energy states exist are called forbidden bands or energy gaps.
• This band structure determines whether a material behaves as a conductor, semiconductor, or insulator.
• Atomic orbitals such as 1s, 2s, 2p transform into energy bands. The transformation of discrete energy levels into bands enables the understanding of electrical and optical properties of materials.

Energy Bands

• Energy bands fundamentally describe the allowed energy ranges for electrons in a solid.
  • Valence Band (VB): Highest range of electron energies fully or partially occupied in the ground state; electrons in the valence band are responsible for the bonding structure of the material.
  • Conduction Band (CB): Energy levels of free electrons sufficiently energetic to become mobile within the solid; free electrons in the conduction band contribute to electrical conductivity.
  • Forbidden Energy Gap: Region between the valence and conduction bands where no energy levels exist; the energy required to transition an electron from the valence to conduction band is the band gap energy ($E_g$).

Classification of Solids Based on Band Theory

• Metals / Conductors
  • Large numbers of free electrons move freely throughout the solid.
  • Valence and conduction bands overlap, zero band gap energy ($E_g = 0$).
  • High electrical conductivity, with resistivity ranging from $10^{-8} \Omega \cdot m$ to $10^{-6} \Omega \cdot m$, and conductivity around $10^7 mho/m$.
• Semiconductors
  • Narrow band gap between valence and conduction bands ($E_g \leq 2 eV$).
  • Moderate energy excites electrons from VB to CB.
  • Electrical properties are intermediate between conductors and insulators, with resistivity from $10^{-4} \Omega-m$ to $10 \Omega-m$ and conductivity ranging from $10^{-6} mho/m$ to $10^4 mho/m$.
  • Can be classified as intrinsic or extrinsic.
    • Intrinsic Semiconductors: Pure semiconductors, electrical conductivity depends on crystal structure and temperature. At absolute zero, they act as insulators.
    • Extrinsic Semiconductors: Doped with other elements to enhance conductivity.
      • N-Type: Doped with elements having extra valence electrons, where free electrons are majority charge carriers (e.g., silicon doped with phosphorus).
      • P-Type: Doped with elements having fewer valence electrons, creating holes as majority charge carriers (e.g., silicon doped with boron).
• Insulators
  • Very wide band gap between valence and conduction bands (E_g > 5 eV).
  • Electrons tightly bound to their atoms.
  • Resistivity is very high, ranging from $10^{12} \Omega-m$ to $10^{16} \Omega-m$, and conductivity is around $10^{-12} mho/m$.

Electrical Conductivity in Conductors (Qualitative)

• Ability to allow electric current to flow through them, determined by ease of electron movement.
• Conductors have free electrons, with an electric field pushing them to flow, creating current.
• Resistivity, denoted by $\rho$, measured in ohm-meters ($\Omega \cdot m$).
• Conductivity, denoted by $\sigma$, measured in Siemens per meter (S/m).
• Conductivity is the reciprocal of resistivity: $\sigma = \frac{1}{\rho}$
• Factors Affecting Conductivity:
  • Material:
    • silver and copper are excellent conductors due to high free-electron density.
    • Temperature:
      • increasing temperature increases resistance, reducing conductivity.
    • Impurities:
      • Adding impurities disrupts electron flow, lowering conductivity.

Equations for Electrical Conductivity in Solids

• Consider a rectangular block with length L and cross-sectional area A, having n free electrons.
• Potential difference V creates electric field E: $E = \frac{V}{L}$
• Current I is the charge Q transported per unit time t: $I = \frac{Q}{t}$
• Ohm’s law: $I = \frac{V}{R}$
• Resistance R is given by: $R = \rho \frac{L}{A}$
• Conductivity is the reciprocal of resistivity: $\sigma = \frac{1}{\rho}$
• Total number of electrons: $N = nAL$
• Total charge: $Q = neAL$
• The current through the solid $I = \frac{Q}{t} = \frac{neAL}{t} = neAv<em>d$ where $v</em>d$ is average drift velocity of electrons.
• Current density J is: $J = \frac{I}{A} = nev_d$
• Microscopic form of Ohm's law $J = \sigma E$ and conductivity $\sigma = ne\mu$ where $\mu = \frac{v_d}{E}$ is the mobility of electrons.
• Mobility of Electrons is a measure of how easily electrons can move through a material when subjected to an electric field. It quantifies the efficiency of electron movement in response to an applied force. The mobility ( ) of electrons is defined as the drift velocity per unit electric field:
  $μ = \frac{v_d}{E}$

Electrical Conductivity of Semiconductors (Derivation)

• Based on contributions of both electrons and holes.
• Current Density:
  • $J = J<em>n + J</em>p$ where $J<em>n = qnv</em>{dn}$ and $J<em>p = qpv</em>{dp}$
  • $v<em>{dn} = \mu</em>n E$ and $v<em>{dp} = \mu</em>p E$
• Relating Current Density and Conductivity: By Ohm's law, the current density is related to the electric field through conductivity: $J = \sigma E$ so $\sigma = q(n\mu<em>n + p\mu</em>p)$
  • q: Elementary charge ($1.6 × 10^{-19} C$)
  • n: Concentration of electrons
  • $\mu_n$: Mobility of electrons
  • p: Concentration of holes
  • $\mu_p$: Mobility of holes
• Special Cases:
  • Intrinsic Semiconductors: n = p = ni $\sigma = qn<em>i(\mu</em>n + \mu_p)$
  • Extrinsic Semiconductors:
    • N-type (n >> p): majority are electrons $\sigma \approx qn\mu_n$
    • P-type (p >> n): majority are holes $\sigma \approx qp\mu_p$

Hall Effect and Hall Coefficient (with Derivation)

• Production of a voltage difference (Hall voltage) across a conductor or semiconductor when it carries an electric current and is placed in a magnetic field perpendicular to the current.
• Hall coefficient is the ratio of induced electric field to the product of current density and applied magnetic field.
• At equilibrium, the electric force equals the magnetic (Lorentz) force.
  • Lorentz Force: $F<em>B = q(\vec{v</em>d} \times \vec{B})$
  • Electric Force: $F<em>E = qE</em>H$
  • At equilibrium $qE<em>H = qv</em>dB$ and from $J = nqv<em>d$, we get $E</em>H = \frac{J}{nq}B$
  • Hall Coefficient: $R<em>H = \frac{E</em>H}{JB} = \frac{1}{nq}$
• Hall Voltage: $V_H = \frac{BI}{nq*d}$ Where d is thickness of the material.
• Hall Voltage:
• Applications:
  • Determining semiconductor type (P-type if $R<em>H$ is positive, N-type if $R</em>H$ is negative).
  • Determining carrier concentration: $n = \frac{1}{R_Hq}$
  • Determination of charge carrier mobility: $\mu = \sigma R_H$

Fermi Energy, Fermi Level, and Fermi-Dirac Probability Distribution Function

• Fermi Energy ($E_F$):
  • Energy of the highest occupied quantum state in a system of fermions at absolute zero (0 K).
• Fermi Level:
  • Energy level at which the probability of an electron being present is 50% at a given temperature
  • In intrinsic semiconductors: Fermi level lies approximately at the center of the energy gap.
  • In extrinsic semiconductors:
    • n-type: Fermi level shifts closer to the conduction band.
    • p-type: Fermi level shifts closer to the valence band.
• Fermi-Dirac Probability Distribution Function:
  • Describes the probability of an electron occupying a particular energy state E at a given temperature T.
  • The Formula: $f(E) = \frac{1}{e^{(E - E<em>F) / (k</em>BT)} + 1}$
    • where:
      • $E$: Energy of the electron state
      • $E_F$: Fermi energy
      • $k_B$: Boltzmann constant ($1.38 × 10^{-23} J/K$)
      • $T$: Absolute temperature in Kelvin
  • Behavior at Different Temperatures:
    • At Absolute Zero Temperature (T = 0 K):
      • E < EF, $f(E) = 1$ All states below $E</em>F$ are fully occupied.
      • E > EF, $f(E) = 0$ All states above $E</em>F$ are unoccupied.
    • At Non-Zero Temperatures (T > 0 K):
      Some electrons occupy states above $E<em>F$, and some states below $E</em>F$ become unoccupied.
      $E = E_F$, $f(E) = 0.5$. The probability is 50%.
  • Position of Fermi level in intrinsic semiconductor: $E<em>F = \frac{E</em>C + E_V}{2}$

P-N Junction

• A fundamental structure in semiconductor physics, formed by joining P-type and N-type semiconductor materials.
• P-type Semiconductor:
  • Doped with acceptor impurities (e.g., boron or gallium).
  • Abundance of holes (positive charge carriers).
• N-type Semiconductor:
  • Doped with donor impurities (e.g., phosphorus or arsenic).
  • Abundance of electrons (negative charge carriers).
• Formation of the PN Junction:
  • Electrons from N-side diffuse into P-side, and holes from P-side diffuse into N-side; recombination occurs near the junction.
  • A depletion region forms with immobile positive ions in the N-region and negative ions in the P-region.
  • An electric field generates a built-in potential, opposing further diffusion and maintaining equilibrium.

A PN Junction Diode

• A two-terminal semiconductor device allowing current flow in one direction.
• Electrons and holes diffuse and recombine at the junction, creating a depletion region with an internal electric field.
• Zero Bias (Unbiased PN Junction):
  • No external voltage applied is applied.
  • Electrons diffuse from N to P, holes diffuse from P to N, leading to recombination.
  • A built-in potential establishes equilibrium.
  • No net current flow across the junction.
• Forward Bias:
  • Positive voltage applied to P-side, negative to N-side.
  • Reduces the depletion region width, allowing charge carriers to move freely across the junction.
  • Current increases exponentially as voltage increases.
• Reverse Bias:
  • Positive voltage applied to N-side, negative to P-side.
  • Increases the depletion region width, blocking current flow.
  • A minimal leakage current may flow.
• Applications of PN Junctions:
  • Diodes, transistors, solar cells, LEDs.

Solar Cell

• A semiconductor device converting light energy into electrical energy through the photovoltaic effect.
• Basic principle involves light exciting electrons, creating electron-hole pairs, separated by the p-n junction’s electric field, generating electric current.
• Working Mechanism:
• * Absorption of Photons:
  Sunlight strikes the surface with photons absorbed by material.
• * Generation of Electron-Hole Pairs:
  absorbed photons cause electrons to move to new postitions.
• Key Parameters of a Solar Cell:
  • $I_{sc}$ (Short-Circuit Current)
  • $V_{oc}$ (Open-Circuit Voltage)
  • FF (Fill Factor): $FF = \frac{I<em>{max} \times V</em>{max}}{I<em>{sc} \times V</em>{oc}}$
  • $\eta$ (Efficiency) : $\eta$

Basics of Sensors

• Specialized devices detecting environmental changes, translating them into electrical signals.
• Components:
  • Sensor: Detects physical parameters.
  • Signal Conditioning: Amplifies and filters the signal.
  • Analog-to-Digital Converter (ADC): Converts analogue signal into digital signal.
  • Microcontroller: Processes the digital signal and makes decisions.
  • Output/Display: Shows the processed data.
• Basic Functionality:
  • Detection, Conversion, Transmission, Processing, Output
• Types of Sensors:
  • Temperature, position, light, sound, proximity, pressure, ultrasonic, touch, humidity, color, chemical, seismic, magnetic.
  • Active and Passive Sensors
• Piezoelectric Sensor: Generates electric charge under mechanical stress.
  • Piezoelectric Effect and Inverse Piezoelectric Effect
  • Internal resistance, inductance, and capacitance are primary factors.
  • Used for vibration sensing, pressure measurement, and medical devices.
• Photodiode Sensor: Converts light energy into electrical energy.
  • Operation through Photovoltaic Effect
  • Consists of P-type and N-type semiconductor layers, with the junction exposed to light.
  • Applications involve light sensing, fiber optic communication, solar cells, and medical devices.
• Characteristics of Sensors:
  • Range, Sensitivity, Resolution, Accuracy, Repeatability
• Applications of Sensors:
  • Environmental monitoring, industrial automation, medical and healthcare, the automotive industry, consumer electronics, agriculture.

UNIT 2: Quantum Mechanics

• Quantum physics explores matter and energy at atomic/subatomic scales, with unintuitive concepts such as wave paticle duality, quantum superpostions and entanglement. It forms foundation for semiconductors, lasers, quantum computing, and nanotechnology.
• Engineers are enables to design transistors and sensors for medicine, defense, and industry.
• Quantum mechanics is a revolutionary franework to describe behavior of particles at atomic and subatomic levels.

Invalidity of Classical Mechanics and Need for Quantum Mechanics

• Classical mechanics fails to describe objects moving at light speed and at atomic/subatomic scale. In 1905 Einstin introduced Special Theory of Relativity, and explained the behavior of electrons in atoms which predicted that electrons should spiral into nuecleus as they moved around atom but contradicts oberserved atomic stabilies.

Limitations of Classical Physics

*Classical mechanics is not fit for atomic stability, blackbody radiation, photoelectric effect, spectrum lines of atoms, dual nature of lightmatter, and fails in microscopic system.

Wave-Particle Duality

• Light and matter exhibit both wave-like and particle-like behavior.
• Einstein’s explanation of the photoelectric effect, Compton effect, blackbody radiation, and emission and absorption is about particle nature. Macrocopic and optical phenomena is about the wave nature..

De Broglie Hypothesis of Matter Waves

• Proposed by Louis de Broglie in 1924; all moving particles have an associated wave-like nature.
• $\lambda = \frac{h}{p} = \frac{h}{mv}$ (p is momentum, h is Planck's constant).
  *Also $E = hν = mc^2$
  $\lambda = \frac{h}{\sqrt{2m⋅K.E.}}$
  For particles accelerated through a potential difference, $\lambda = \frac{h}{\sqrt{2m⋅qV}}$

Properties of Matter Waves

• Inverse relationship between wavelength and mass/velocity: the smaller the velocity of the particle and lighter mass, the longer the wavelength.
• Matter waves are produced by motion and are independent of charge.
• Travel through vacuum and exhibit diffraction.

Concept of Wave Packet

*In quantum mechanics, a wave packet is a superposition of several waves with different wavelengths.
Phase Velocity $v<em>p = \frac{ω}{k}$ Group velocity $v</em>g = \frac{dω}{dk}$

Velocity of wave packet (energy) propagations is group velocity.
It is a measure of how easily electrons can move through a material when subjected to a electric field. It quantifies the efficiency of electron movement in response to an applied force and is applicable to electron/holes.

Heisenberg Uncertainty Principle

• States that it is impossible to precisely measure certain pairs of physical properties like position and momentum simultaneously: $\Delta x \cdot \Delta p \geq \frac{\hbar}{2}$
• The uncertainty principle also extends to the energy-time uncertainty relation, which states that it is impossible to measure the energy of a system exactly over a finite time period: $\Delta E ⋅ \Delta t ≥ \frac{\hbar}{2}$

Schrodinger’s Time-Independent Equation

It's a key mathematical formulation in quantum mechanics that describes the quantum state of a system in a stationary energy state and is used to determine the allowed energy levels and the corresponding wave functions for a particle in a given potential field i.e. :
$\nabla^2\psi + \frac{2m}{\hbar^2}(E - V ) \psi = 0$

Particle in Infinite Potential Well (Rigid Box)

• Energy values are discrete and quantized: $E_n = \frac{n^2h^2}{8mL^2}$
• Wave Function: $\Psi_n(x) = \sqrt{\frac{2}{L}} sin(\frac{n\pi x}{L})$ for $n = 1,2,3,…$

Quantum Tunneling

• The wavelike properties of matter permit particles to penetrate potential barriers.
  Also that the process of adiabatic demagnetization relies on fact that reducing magnetic field in an isolated system causes internal energy redistribution, lowering the temperature.
  Scanning Tunneling Microscope (STM)