Solid State Physics, Electronics & Digital Systems

Solid State Physics

Free Electron Theory (Drude Model)

The free electron theory, proposed by Paul Drude in 1900, treats valence electrons in a metal as a "gas" of free particles that can move through the crystal lattice.

Diffusion Current

When there is a concentration gradient of charge carriers (more electrons in one region than another), carriers diffuse from high to low concentration, creating a diffusion current:

$$J_{diff,n} = q D_n\frac{dn}{dx} \quad \text{(for electrons)}$$ $$J_{diff,p} = -q D_p\frac{dp}{dx} \quad \text{(for holes)}$$

where $D_n, D_p$ are diffusion coefficients. The Einstein relation connects diffusion and mobility: $D = \frac{kT}{q}\mu$.

Drift Current

Without an external field, electrons move randomly with zero net displacement. When an electric field $\vec{E}$ is applied, electrons acquire a small drift velocity $v_d$ superimposed on their random motion.

The drift current density $J$ is the current per unit cross-sectional area:

$$J_{drift} = \frac{I}{A} \quad \text{(A/cm²)}$$

Since current is the flow of charge carriers, we can write:

$$J = n \cdot q \cdot v_d = \rho_v \cdot v_d$$

where $n$ = carrier concentration (number/cm³), $q$ = charge of carrier, $v_d$ = drift velocity (cm/s), and $\rho_v = n \cdot q$ is the volume charge density (C/cm³).

Mobility ($\mu$)

The drift velocity is proportional to the applied electric field:

$$\boxed{v_d = \mu \cdot E}$$

where $\mu$ is the mobility of the charge carrier (cm²/V·s). Mobility is a measure of how easily charge carriers move under the influence of an applied electric field — it determines how mobile the charge carriers are.

Substituting $v_d = \mu E$ into the current density equation:

$$\boxed{J = n \cdot q \cdot \mu \cdot E}$$

Conductivity of a Semiconductor ($\sigma$)

In a semiconductor, both electrons and holes contribute to drift current. We write the current density for each carrier separately:

Hole drift current:

$$J_p = p \cdot q \cdot \mu_p \cdot E$$

Electron drift current:

$$J_n = n \cdot q \cdot \mu_n \cdot E$$

The total drift current density is the sum of both:

$$J = J_p + J_n = p q \mu_p E + n q \mu_n E$$ $$\boxed{J = (p\mu_p + n\mu_n) \cdot q \cdot E = \sigma \cdot E}$$

where the conductivity of the semiconductor is:

$$\boxed{\sigma = (p\mu_p + n\mu_n) \cdot q} \quad \text{(Ω·cm)}^{-1}$$

Fermi Energy & Fermi-Dirac Distribution

How to Count the Number of Carriers?

To determine how many electrons participate in conduction, we need two things:

1. Density of States $g(E)$ — tells us how many energy states exist at a given energy $E$.
2. Fermi-Dirac Distribution Function $f(E)$ — tells us the probability that an available state at energy $E$ is occupied by an electron.

The total carrier concentration is then:

$$n = \int g(E) \cdot f(E) \, dE$$

In other words: Number of carriers = (Number of available states) × (Probability of occupation).

The Fermi-Dirac Distribution Function

The Fermi energy $E_F$ is the highest energy level occupied by electrons at absolute zero (0 K). It represents the "top of the filled electron sea." It is the energy of the highest occupied quantum state.

At any temperature $T$, the probability that a quantum state at energy $E$ is occupied by an electron is given by the Fermi-Dirac distribution function:

$$f(E) = \frac{1}{1 + e^{(E - E_F)/kT}}$$

where $E_F$ = Fermi energy, $k$ = Boltzmann constant ($1.38 \times 10^{-23}$ J/K = $8.6 \times 10^{-5}$ eV/K), and $T$ = absolute temperature in Kelvin.

Meaning of Fermi Level

The Fermi level ($E_F$) is the energy level with exactly 50% probability of finding an electron, at any temperature $T > 0$ K. When $E = E_F$:

$$f(E_F) = \frac{1}{1 + e^0} = \frac{1}{2} = 50\%$$

Effect of Temperature on Fermi-Dirac Distribution

• At $T = 0$ K, the electrons have the lowest energy and occupy the lower energy states. The distribution is a sharp step function — all states below $E_F$ are completely filled, all above are completely empty.
• As the temperature increases, electrons gain thermal energy. Some electrons near $E_F$ get excited to higher energy states. The sharp step smooths out into an S-shaped curve. The higher the temperature, the more gradual the transition around $E_F$.
• The Fermi level itself is independent of temperature in metals (it remains the 50% probability point).

Total Current Density in a Semiconductor

In a semiconductor, current flows due to both drift (due to electric field) and diffusion (due to concentration gradient). The total current density combines both contributions from electrons and holes:

$$J_{total} = J_{drift} + J_{diffusion}$$ $$J_{total} = \underbrace{(ne\mu_e + pe\mu_h)E}_{\text{Drift}} + \underbrace{\left(eD_n\frac{dn}{dx} - eD_p\frac{dp}{dx}\right)}_{\text{Diffusion}}$$

where $n, p$ = electron and hole concentrations, $\mu_e, \mu_h$ = mobilities, and $D_n, D_p$ = diffusion coefficients.

Band Theory of Solids

When atoms come together to form a solid, their discrete energy levels split into bands of closely spaced energy levels due to the interaction between atomic orbitals. Instead of single energy levels, there will be bands of energy levels formed due to the intermixing of atoms. These sets of closely packed energy levels are called Energy Bands.

Formation of Energy Bands

Consider $N$ atoms of silicon brought together:

• Each atom has discrete energy levels (1s, 2s, 2p, 3s, 3p).
• As atoms approach, the Pauli exclusion principle forces the $N$ identical energy levels to split into $N$ closely spaced but distinct levels.
• These $N$ levels form a quasi-continuous band.

Valence Band

The electrons present in the outermost shell are called Valence Electrons. These valence electrons, containing a series of energy levels, form an energy band called the Valence Band. The valence band is the band having the highest occupied energy.

Conduction Band

Some valence electrons gain enough energy to leave their atoms and become free electrons, which can move toward neighboring atoms. These free electrons are the ones that conduct current in a conductor and are hence called Conduction Electrons. The band containing these conduction electrons is called the Conduction Band — it is the band having the lowest unoccupied energy.

Forbidden Energy Gap ($E_g$)

The gap between the valence band and the conduction band is called the Forbidden Energy Gap. No electron can exist in this energy range. Depending upon the size of this forbidden gap, materials are classified as Insulators, Semiconductors, or Conductors.

Classification of Solids Based on Band Gap

Property	Conductor	Semiconductor	Insulator
Band Gap $E_g$	0 eV (bands overlap)	0.1 – 3 eV	> 3 eV
Examples	Cu, Ag, Al	Si (1.1 eV), Ge (0.67 eV)	Diamond (5.5 eV), Glass
Resistivity	$\sim 10^{-8}$ Ω·m	$10^{-5}$ to $10^{6}$ Ω·m	$\gt 10^{7}$ Ω·m
Temperature Effect	Conductivity decreases	Conductivity increases	Remains insulator
Current Carriers	Electrons only	Electrons and Holes	None at room temp

Concept of Effective Mass

Inside a crystal, an electron does not behave as a free particle because it interacts with the periodic potential of the lattice. We account for this by assigning it an effective mass $m^*$:

$$m^* = \frac{\hbar^2}{\frac{d^2E}{dk^2}}$$

where $E$ is the electron energy and $k$ is the wave vector (from the E-k diagram).

• At the bottom of the conduction band, the band curves upward sharply → small $m^*$ → electrons are "light" and mobile.
• At the top of the valence band, the band curves downward → negative $m^*$ → this is mathematically treated as a positive particle called a hole.

Hall Effect (Complete Derivation)

The Hall effect, discovered by Edwin Hall in 1879, is used to determine the type (n or p), concentration, and mobility of charge carriers in a material.

Setup

Consider a rectangular conductor of width $w$, thickness $t$, carrying current $I$ along the x-direction. A magnetic field $\vec{B}$ is applied along the z-direction (perpendicular to the current).

Step-by-Step Derivation

Step 1: Electrons moving with drift velocity $v_d$ in the x-direction experience a Lorentz force:

$$\vec{F} = -e(\vec{v_d} \times \vec{B})$$

This force pushes electrons toward one face of the conductor (say, the bottom face).

Step 2: Electrons accumulate on the bottom face, creating a transverse electric field $E_H$ (Hall field) pointing from top to bottom. This field exerts a force $eE_H$ on the electrons, opposing further accumulation.

Step 3: At equilibrium, the electric force balances the magnetic force:

$$eE_H = ev_dB$$ $$E_H = v_dB$$

Step 4: The Hall voltage across the width $w$ is:

$$V_H = E_H \cdot w = v_d B w$$

Step 5: Since current density $J = nev_d$ and $I = Jwt$, we get $v_d = \frac{I}{newt}$. Substituting:

$$\boxed{V_H = \frac{BI}{net}}$$

Step 6: The Hall coefficient is defined as:

$$\boxed{R_H = \frac{1}{ne}} \quad \text{(for n-type: } R_H \text{ is negative)}$$ $$\boxed{R_H = \frac{1}{pe}} \quad \text{(for p-type: } R_H \text{ is positive)}$$

Solved Example: Hall Effect

Semiconductors: Intrinsic & Extrinsic

Intrinsic Semiconductors

An intrinsic semiconductor is a pure semiconductor with no impurities added. Germanium (Ge) and Silicon (Si), both Group IV elements, are the best examples. They possess a diamond cubic crystalline structure where each atom forms 4 covalent bonds with its neighbors by sharing one valence electron each.

At 0 K, all valence band states are filled and the conduction band is empty — the material behaves as an insulator. At room temperature, thermal energy excites some electrons across the band gap, creating electron-hole pairs (EHP).

Carrier Concentration in Intrinsic Semiconductors

Using the Density of States $g(E)$ and the Fermi-Dirac distribution $f(E)$, we can derive the equilibrium carrier concentrations:

Electrons in the Conduction Band:

$$n = N_C \exp\left(-\frac{E_C - E_F}{kT}\right)$$

where $N_C = 2\left(\frac{2\pi m_e^* kT}{h^2}\right)^{3/2}$ is the effective density of states in the conduction band.

Holes in the Valence Band:

$$p = N_V \exp\left(-\frac{E_F - E_V}{kT}\right)$$

where $N_V = 2\left(\frac{2\pi m_h^* kT}{h^2}\right)^{3/2}$ is the effective density of states in the valence band.

Intrinsic Concentration ($n_i$)

Since $n = p = n_i$ in an intrinsic semiconductor, multiplying the two equations:

$$n \cdot p = n_i^2 = N_C N_V \exp\left(-\frac{E_g}{kT}\right)$$ $$\boxed{n_i = \sqrt{N_C N_V} \exp\left(-\frac{E_g}{2kT}\right)}$$

This shows that intrinsic concentration increases exponentially with temperature and decreases with larger band gap.

Fermi Level in Intrinsic Semiconductor

The Fermi level is the energy level at which the probability of finding an electron is equal to the probability of finding a hole. For an intrinsic semiconductor, setting $n = p$ and solving:

$$E_F = \frac{E_C + E_V}{2} + \frac{3}{4}kT \ln\left(\frac{m_h^*}{m_e^*}\right)$$

If $m_e^* \approx m_h^*$ (which is approximately true), then:

$$\boxed{E_F \approx \frac{E_C + E_V}{2} = E_i}$$

The Fermi level lies approximately at the middle of the band gap. This mid-gap energy is called the intrinsic Fermi level $E_i$.

Conductivity of an intrinsic semiconductor: $\sigma = n_i e(\mu_e + \mu_h)$

Extrinsic Semiconductors

Extrinsic semiconductors are those in which impurities are deliberately added (doping) to dramatically increase conductivity. Based on the type of impurity, they are classified into two categories:

Property	N-type	P-type
Dopant	Pentavalent (P, As, Sb)	Trivalent (B, Al, Ga, In)
Impurity type	Donor	Acceptor
Majority carrier	Electrons	Holes
Minority carrier	Holes	Electrons
Energy level	Donor level near $E_C$	Acceptor level near $E_V$
Fermi level	Closer to conduction band	Closer to valence band
Current direction	Electron current	Hole current (opposite to electron flow)

Charge Neutrality & Charge Densities

In a semiconductor under thermal equilibrium, the total positive charge must equal the total negative charge. This gives the charge neutrality equation:

$$\boxed{n + N_A^- = p + N_D^+}$$

where $N_D^+$ = ionized donor concentration, $N_A^-$ = ionized acceptor concentration. At room temperature, all dopants are typically fully ionized, so $N_D^+ \approx N_D$ and $N_A^- \approx N_A$.

For an N-type semiconductor ($N_D \gg N_A$, $n \gg p$): $n \approx N_D$ and $p = n_i^2/N_D$

For a P-type semiconductor ($N_A \gg N_D$, $p \gg n$): $p \approx N_A$ and $n = n_i^2/N_A$

Fermi Level in Extrinsic Semiconductors

In N-type: The Fermi level shifts toward the conduction band. At low temperatures, it lies between the donor level and $E_C$. As temperature increases, more donor atoms get ionized, and eventually the material behaves intrinsically, with $E_F$ moving back toward the mid-gap $E_i$.

In P-type: The Fermi level shifts toward the valence band. At low temperatures, it lies between the acceptor level and $E_V$. With increasing temperature, $E_F$ gradually moves toward $E_i$.

Properties of Germanium & Silicon

Property	Germanium (Ge)	Silicon (Si)
Atomic Number	32	14
Band Gap $E_g$	0.67 eV	1.12 eV
Intrinsic concentration $n_i$ (at 300 K)	$2.4 \times 10^{13}$ /cm³	$1.5 \times 10^{10}$ /cm³
Crystal Structure	Diamond cubic	Diamond cubic
Electron mobility $\mu_e$	3900 cm²/V·s	1350 cm²/V·s
Hole mobility $\mu_h$	1900 cm²/V·s	480 cm²/V·s
Dielectric constant	16	11.8

Generation & Recombination

Generation is the process where electron-hole pairs are created by exciting an electron from the valence band to the conduction band. Recombination is the reverse process where electrons and holes recombine and are annihilated.

In thermal equilibrium, the generation and recombination processes are in dynamic equilibrium — the rate of generation equals the rate of recombination.

Carrier Lifetime

The carrier lifetime ($\tau$) is the average time a minority carrier exists before recombining. It determines how far carriers can diffuse before recombining, and is critical for device performance:

$$L = \sqrt{D \cdot \tau}$$

where $L$ is the diffusion length and $D$ is the diffusion coefficient. A longer carrier lifetime means carriers travel farther before recombining — essential for solar cells and transistors.

Direct vs Indirect Band Gap

Property	Direct Band Gap	Indirect Band Gap
E-k diagram	CB minimum and VB maximum at same $k$	CB minimum and VB maximum at different $k$
Transition	Electron can directly emit/absorb photon	Requires phonon (lattice vibration) assist
Examples	GaAs, InP, GaN	Si, Ge
Applications	LEDs, Laser diodes, solar cells	Transistors, ICs, CPUs

Solar Cells & Computational Applications

Solar Cell Basics

A solar cell is a PN junction operated under illumination. When photons with energy $h\nu \geq E_g$ strike the cell, electron-hole pairs are generated in the depletion region. The built-in electric field separates them — electrons go to n-side, holes to p-side — creating a photocurrent and photovoltage.

• Open-circuit voltage: $V_{OC} = \frac{kT}{q}\ln\left(\frac{I_L}{I_0} + 1\right)$
• Short-circuit current: $I_{SC} \approx I_L$ (photogenerated current)
• Efficiency: $\eta = \frac{P_{max}}{P_{incident}} = \frac{V_{OC} \cdot I_{SC} \cdot FF}{P_{in}}$ where $FF$ is the fill factor (~0.7–0.85).

Computational Applications

Semiconductors: Intrinsic & Extrinsic

Intrinsic Semiconductors

An intrinsic semiconductor is a pure semiconductor with no impurities added. Germanium (Ge) and Silicon (Si), both Group IV elements, are the best examples. They possess a diamond cubic crystalline structure where each atom forms 4 covalent bonds with its neighbors by sharing one valence electron each.

At 0 K, all valence band states are filled and the conduction band is empty — the material behaves as an insulator. At room temperature, thermal energy excites some electrons across the band gap, creating electron-hole pairs (EHP).

Carrier Concentration in Intrinsic Semiconductors

Using the Density of States $g(E)$ and the Fermi-Dirac distribution $f(E)$, we can derive the equilibrium carrier concentrations:

Electrons in the Conduction Band:

$$n = N_C \exp\left(-\frac{E_C - E_F}{kT}\right)$$

where $N_C = 2\left(\frac{2\pi m_e^* kT}{h^2}\right)^{3/2}$ is the effective density of states in the conduction band.

Holes in the Valence Band:

$$p = N_V \exp\left(-\frac{E_F - E_V}{kT}\right)$$

where $N_V = 2\left(\frac{2\pi m_h^* kT}{h^2}\right)^{3/2}$ is the effective density of states in the valence band.

Intrinsic Concentration ($n_i$)

Since $n = p = n_i$ in an intrinsic semiconductor, multiplying the two equations:

$$n \cdot p = n_i^2 = N_C N_V \exp\left(-\frac{E_g}{kT}\right)$$ $$\boxed{n_i = \sqrt{N_C N_V} \exp\left(-\frac{E_g}{2kT}\right)}$$

This shows that intrinsic concentration increases exponentially with temperature and decreases with larger band gap.

Fermi Level in Intrinsic Semiconductor

The Fermi level is the energy level at which the probability of finding an electron is equal to the probability of finding a hole. For an intrinsic semiconductor, setting $n = p$ and solving:

$$E_F = \frac{E_C + E_V}{2} + \frac{3}{4}kT \ln\left(\frac{m_h^*}{m_e^*}\right)$$

If $m_e^* \approx m_h^*$ (which is approximately true), then:

$$\boxed{E_F \approx \frac{E_C + E_V}{2} = E_i}$$

The Fermi level lies approximately at the middle of the band gap. This mid-gap energy is called the intrinsic Fermi level $E_i$.

Conductivity of an intrinsic semiconductor: $\sigma = n_i e(\mu_e + \mu_h)$

Extrinsic Semiconductors

Extrinsic semiconductors are those in which impurities are deliberately added (doping) to dramatically increase conductivity. Based on the type of impurity, they are classified into two categories:

Property	N-type	P-type
Dopant	Pentavalent (P, As, Sb)	Trivalent (B, Al, Ga, In)
Impurity type	Donor	Acceptor
Majority carrier	Electrons	Holes
Minority carrier	Holes	Electrons
Energy level	Donor level near $E_C$	Acceptor level near $E_V$
Fermi level	Closer to conduction band	Closer to valence band
Current direction	Electron current	Hole current (opposite to electron flow)

Charge Neutrality & Charge Densities

In a semiconductor under thermal equilibrium, the total positive charge must equal the total negative charge. This gives the charge neutrality equation:

$$\boxed{n + N_A^- = p + N_D^+}$$

where $N_D^+$ = ionized donor concentration, $N_A^-$ = ionized acceptor concentration. At room temperature, all dopants are typically fully ionized, so $N_D^+ \approx N_D$ and $N_A^- \approx N_A$.

For an N-type semiconductor ($N_D \gg N_A$, $n \gg p$): $n \approx N_D$ and $p = n_i^2/N_D$

For a P-type semiconductor ($N_A \gg N_D$, $p \gg n$): $p \approx N_A$ and $n = n_i^2/N_A$

Fermi Level in Extrinsic Semiconductors

In N-type: The Fermi level shifts toward the conduction band. At low temperatures, it lies between the donor level and $E_C$. As temperature increases, more donor atoms get ionized, and eventually the material behaves intrinsically, with $E_F$ moving back toward the mid-gap $E_i$.

In P-type: The Fermi level shifts toward the valence band. At low temperatures, it lies between the acceptor level and $E_V$. With increasing temperature, $E_F$ gradually moves toward $E_i$.

Properties of Germanium & Silicon

Property	Germanium (Ge)	Silicon (Si)
Atomic Number	32	14
Band Gap $E_g$	0.67 eV	1.12 eV
Intrinsic concentration $n_i$ (at 300 K)	$2.4 \times 10^{13}$ /cm³	$1.5 \times 10^{10}$ /cm³
Crystal Structure	Diamond cubic	Diamond cubic
Electron mobility $\mu_e$	3900 cm²/V·s	1350 cm²/V·s
Hole mobility $\mu_h$	1900 cm²/V·s	480 cm²/V·s
Dielectric constant	16	11.8

Generation & Recombination

Generation is the process where electron-hole pairs are created by exciting an electron from the valence band to the conduction band. Recombination is the reverse process where electrons and holes recombine and are annihilated.

In thermal equilibrium, the generation and recombination processes are in dynamic equilibrium — the rate of generation equals the rate of recombination.

Carrier Lifetime

The carrier lifetime ($\tau$) is the average time a minority carrier exists before recombining. It determines how far carriers can diffuse before recombining, and is critical for device performance:

$$L = \sqrt{D \cdot \tau}$$

where $L$ is the diffusion length and $D$ is the diffusion coefficient. A longer carrier lifetime means carriers travel farther before recombining — essential for solar cells and transistors.

Direct vs Indirect Band Gap

Property	Direct Band Gap	Indirect Band Gap
E-k diagram	CB minimum and VB maximum at same $k$	CB minimum and VB maximum at different $k$
Transition	Electron can directly emit/absorb photon	Requires phonon (lattice vibration) assist
Examples	GaAs, InP, GaN	Si, Ge
Applications	LEDs, Laser diodes, solar cells	Transistors, ICs, CPUs

Solar Cells & Computational Applications

Solar Cell Basics

A solar cell is a PN junction operated under illumination. When photons with energy $h\nu \geq E_g$ strike the cell, electron-hole pairs are generated in the depletion region. The built-in electric field separates them — electrons go to n-side, holes to p-side — creating a photocurrent and photovoltage.

• Open-circuit voltage: $V_{OC} = \frac{kT}{q}\ln\left(\frac{I_L}{I_0} + 1\right)$
• Short-circuit current: $I_{SC} \approx I_L$ (photogenerated current)
• Efficiency: $\eta = \frac{P_{max}}{P_{incident}} = \frac{V_{OC} \cdot I_{SC} \cdot FF}{P_{in}}$ where $FF$ is the fill factor (~0.7–0.85).

Computational Applications

MCQ Assessment — Unit I

Fundamentals of Electricity & Devices

Fundamental Electrical Laws

Ohm's Law

The current $I$ through a conductor is directly proportional to the voltage $V$ across it:

$$V = IR$$

Resistance depends on material and geometry: $R = \rho \frac{L}{A}$, where $\rho$ = resistivity, $L$ = length, $A$ = cross-sectional area.

Kirchhoff's Current Law (KCL)

The algebraic sum of all currents at any node is zero:

$$\sum I_{in} = \sum I_{out} \quad \text{or} \quad \sum I = 0$$

Kirchhoff's Voltage Law (KVL)

The algebraic sum of all voltages around any closed loop is zero:

$$\sum V = 0$$

Voltage & Current Division Rules

Voltage Division Rule (VDR) — Derivation

For two resistors $R_1$ and $R_2$ in series connected to a source $V_s$:

Since series current is the same: $I = \frac{V_s}{R_1 + R_2}$

Voltage across $R_1$:

$$\boxed{V_1 = V_s \times \frac{R_1}{R_1 + R_2}}$$

Voltage across $R_2$:

$$V_2 = V_s \times \frac{R_2}{R_1 + R_2}$$

Solved Example — VDR

Current Division Rule (CDR) — Derivation

For two resistors $R_1$ and $R_2$ in parallel connected to a current source $I_s$:

Voltage across parallel combination: $V = I_s \times \frac{R_1 R_2}{R_1 + R_2}$

Current through $R_1$:

$$\boxed{I_1 = I_s \times \frac{R_2}{R_1 + R_2}}$$

(Note: current through $R_1$ depends on the other resistor $R_2$.)

PN Junction Diode

Formation of PN Junction

When p-type and n-type semiconductors are joined:

1. Electrons from n-side diffuse into p-side; holes from p-side diffuse into n-side.
2. This creates a region depleted of free carriers: the depletion region.
3. Immobile ions in the depletion region create a built-in potential $V_{bi}$ (≈0.7V for Si, 0.3V for Ge).
4. The built-in field opposes further diffusion, establishing equilibrium.

Shockley Diode Equation

$$I = I_s\left(e^{qV/nkT} - 1\right)$$

where $I_s$ ≈ 10⁻¹² A (reverse saturation current), $n$ = ideality factor (1–2), $V_T = kT/q$ ≈ 26 mV at 300K.

V-I Characteristics

Region	Condition	Current
Forward bias	$V > 0$ (p positive, n negative)	Exponentially increasing after $V_{bi}$
Reverse bias	$V < 0$	Small constant $-I_s$ (leakage)
Breakdown	$V < -V_{BR}$	Sudden large reverse current

Rectifier Applications

Half-Wave Rectifier: Uses one diode. Only positive half-cycle passes.

• $V_{dc} = \frac{V_m}{\pi} \approx 0.318 V_m$
• Ripple factor $\gamma = 1.21$ (very high)
• Efficiency $\eta = 40.6\%$

Full-Wave Rectifier (Bridge): Uses 4 diodes. Both half-cycles are rectified.

• $V_{dc} = \frac{2V_m}{\pi} \approx 0.636 V_m$
• Ripple factor $\gamma = 0.48$ (much better)
• Efficiency $\eta = 81.2\%$

Parameter	Half-Wave	Full-Wave (Bridge)
No. of diodes	1	4
$V_{dc}$	$V_m/\pi$	$2V_m/\pi$
Ripple factor	1.21	0.48
Efficiency	40.6%	81.2%
Transformer	Not needed	Not needed

Bipolar Junction Transistor (BJT)

A BJT has three regions: Emitter (heavily doped), Base (thin, lightly doped), Collector (moderately doped, large area).

Current Relations

$$I_E = I_B + I_C$$ $$\alpha = \frac{I_C}{I_E} \quad (\text{common-base current gain, } \approx 0.95\text{–}0.99)$$ $$\beta = \frac{I_C}{I_B} \quad (\text{common-emitter current gain, } \approx 50\text{–}300)$$ $$\beta = \frac{\alpha}{1 - \alpha} \quad \text{and} \quad \alpha = \frac{\beta}{\beta + 1}$$

Regions of Operation

Region	BE Junction	BC Junction	Use
Active	Forward biased	Reverse biased	Amplification
Saturation	Forward biased	Forward biased	Switch ON
Cutoff	Reverse biased	Reverse biased	Switch OFF

Solved Example — BJT

Optical Fiber Communication

Optical fibers transmit data as light pulses via Total Internal Reflection (TIR).

Critical angle (from Snell's law): When $\theta_r = 90°$:

$$n_1 \sin\theta_c = n_2 \sin 90° \implies \boxed{\theta_c = \sin^{-1}\left(\frac{n_2}{n_1}\right)}$$

Numerical Aperture: $NA = \sqrt{n_1^2 - n_2^2}$ determines the light-gathering ability.

Type	Single-Mode	Multi-Mode
Core diameter	8–10 μm	50–62.5 μm
Distance	Up to 100 km	Up to 2 km
Bandwidth	Very high	Moderate
Use	Long-haul telecom	LAN, data centers

Wireless Communication — Basics

Wireless signals use electromagnetic waves modulated to carry information:

• AM (Amplitude Modulation): Signal varies the amplitude of the carrier wave. Used in AM radio.
• FM (Frequency Modulation): Signal varies the frequency. Better noise immunity. Used in FM radio.
• Digital Modulation: ASK, FSK, PSK — used in WiFi, 4G, 5G.

MCQ Assessment — Unit II