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.

Drift Velocity and 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:

where $e$ = electron charge, $m_e$ = electron mass, $\tau$ = relaxation time.

The drift current density is:

where $n$ = number density of free electrons and $\sigma = \frac{ne^2\tau}{m_e}$ is the electrical conductivity.

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:

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

Fermi Energy & Fermi-Dirac Distribution

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."

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

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.

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.
• The lower filled band = Valence Band (VB).
• The upper empty band = Conduction Band (CB).
• The gap between VB and CB = Forbidden Energy Gap ($E_g$).

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 $m^*$:

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:

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:

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

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

Step 6: The Hall coefficient is defined as:

Solved Example: Hall Effect

Semiconductors: Intrinsic & Extrinsic

Intrinsic Semiconductors

A pure semiconductor (no impurities). At 0 K, all valence band states are filled and the conduction band is empty — it behaves as an insulator. At room temperature, thermal energy excites some electrons across the gap.

• Electron concentration = Hole concentration: $n_i = p_i$
• Fermi level lies approximately at the middle of the band gap: $E_F \approx \frac{E_C + E_V}{2}$
• Conductivity: $\sigma = n_i e(\mu_e + \mu_h)$

Extrinsic Semiconductors

Adding controlled impurities (doping) dramatically increases conductivity:

Direct vs Indirect Band Gap

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:

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:

Kirchhoff's Voltage Law (KVL)

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

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 $R_1$:

Voltage across $R_2$:

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 $R_1$:

(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

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

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 $\eta = 81.2\%$

Bipolar Junction Transistor (BJT)

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

Current Relations

Regions of Operation

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 $\theta_r = 90°$:

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

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

Number Systems & Logic Gates

Number System Conversions

A number $N$ in base $r$: $N_r = d_n r^n + d_{n-1} r^{n-1} + \dots + d_0 r^0 + d_{-1} r^{-1} + \dots$

Decimal to Binary (Division-by-2 Method)

45 ÷ 2 = 22  remainder 1  (LSB)
22 ÷ 2 = 11  remainder 0
11 ÷ 2 = 5   remainder 1
 5 ÷ 2 = 2   remainder 1
 2 ÷ 2 = 1   remainder 0
 1 ÷ 2 = 0   remainder 1  (MSB)
Read remainders bottom to top: (101101)₂

Binary ↔ Octal ↔ Hexadecimal

• Binary → Octal: Group bits in 3s from right. Example: $101\,101_2 = 55_8$
• Binary → Hex: Group bits in 4s from right. Example: $0010\,1101_2 = 2D_{16}$

Codes

BCD (Binary Coded Decimal)

Each decimal digit is represented by its 4-bit binary equivalent. Example: $93_{10} = 1001\,0011_{BCD}$.

Gray Code (Binary ↔ Gray Conversion)

Binary to Gray: MSB stays same. Each subsequent bit = XOR of adjacent binary bits.

Gray to Binary: MSB stays same. Each subsequent bit = XOR of previous binary bit with current gray bit.

Binary:  1  0  1  1
         ↓  ↓  ↓  ↓
Gray:    1  1  1  0
         │  ↑  ↑  ↑
         │  1⊕0 0⊕1 1⊕1
         MSB same

Excess-3 Code

Add 3 to each decimal digit, then convert to BCD. Example: $5_{10} → 5+3 = 8 → 1000_{Excess-3}$

Complements & Binary Arithmetic

1's Complement

Flip all bits: $1's(1010) = 0101$

2's Complement

Flip all bits + add 1: $2's(1010) = 0101 + 1 = 0110$

Subtraction Using 2's Complement

7 in binary:   0111
3 in binary:   0011
2's comp of 3: 1101 (flip: 1100, add 1: 1101)

   0111
+  1101
-------
 1|0100  ← Discard carry. Result = 0100 = 4 ✓

Logic Gates

Boolean Algebra

Important Laws

De Morgan's Theorems

Theorem 1: $\overline{A \cdot B} = \overline{A} + \overline{B}$ (Break the bar, change the sign)

Theorem 2: $\overline{A + B} = \overline{A} \cdot \overline{B}$

SOP and POS Forms

• SOP (Sum of Products): OR of AND terms = $\sum m(minterms)$. Example: $F = \overline{A}B + AB = \sum m(1,3)$
• POS (Product of Sums): AND of OR terms = $\prod M(maxterms)$. Example: $F = (A+B)(\overline{A}+B)$

Karnaugh Map (K-Map)

A visual method for simplifying Boolean expressions. Adjacent cells differ by only one variable.

Grouping Rules

1. Groups must contain $2^n$ cells (1, 2, 4, 8, 16)
2. Groups must be rectangular
3. Groups can wrap around edges
4. Every 1 must be covered by at least one group
5. Make groups as large as possible to eliminate more variables

4-Variable K-Map Solved Example

MCQ Assessment — Unit III

Combinational Logic Circuits

Adders & Subtractors

Half Adder

$Sum = A \oplus B$    $Carry = A \cdot B$

Full Adder

A Full Adder = Two Half Adders + OR gate.

Half Subtractor

$D = A \oplus B$    $B_{out} = \overline{A} \cdot B$

Full Subtractor

Multiplexer (MUX)

A MUX selects one of $2^n$ inputs and routes it to a single output using $n$ select lines.

4:1 MUX

Implementing Boolean Functions with MUX

De-Multiplexer (DEMUX)

A DEMUX routes a single input to one of $2^n$ outputs using $n$ select lines. It is the reverse of a MUX.

1:4 DEMUX: Input $D$, select lines $S_1, S_0$:

$Y_0 = D \cdot \overline{S_1} \cdot \overline{S_0}$,   $Y_1 = D \cdot \overline{S_1} \cdot S_0$,   $Y_2 = D \cdot S_1 \cdot \overline{S_0}$,   $Y_3 = D \cdot S_1 \cdot S_0$

Decoder & Encoder

2:4 Decoder

Each output represents one minterm. A decoder with an enable input can function as a DEMUX.

Priority Encoder (4:2)

If multiple inputs are active, the highest priority input is encoded.

2-Bit Magnitude Comparator

Compares two 2-bit numbers $A = A_1A_0$ and $B = B_1B_0$. Three outputs: $A>B$, $A=B$, $A

Equality: $A = B$ when $A_1=B_1$ AND $A_0=B_0$:

where $\odot$ is XNOR.

Greater than: $A > B$ when $A_1>B_1$, or $A_1=B_1$ and $A_0>B_0$:

Sequential Logic Circuits

Latches

Key Difference: A latch is level-sensitive (transparent when enable is high). A flip-flop is edge-triggered (changes only on clock edge).

SR Latch (using NOR gates)

D Latch

Eliminates the invalid state by using a single data input $D$. Internally: $S = D$, $R = \overline{D}$.

Characteristic equation: $Q_{next} = D$ (when enabled)

Flip-Flops

SR Flip-Flop

Characteristic equation: $Q_{next} = S + \overline{R} \cdot Q$ with constraint $SR = 0$

Excitation Table (given desired transition, find required inputs):

JK Flip-Flop

Solves the invalid state problem of SR. When J=K=1, the output toggles.

Characteristic equation: $\boxed{Q_{next} = J\overline{Q} + \overline{K}Q}$

Excitation Table:

D Flip-Flop

Characteristic equation: $Q_{next} = D$. Most widely used in registers and memory.

Excitation Table: $Q \rightarrow Q_{next}$: D = $Q_{next}$ (always!)

T Flip-Flop (Toggle)

Characteristic equation: $Q_{next} = T \oplus Q = T\overline{Q} + \overline{T}Q$

T flip-flops are ideal for counters because each stage divides the frequency by 2.

Master-Slave JK Flip-Flop

The basic JK flip-flop has a race condition: if J=K=1 and the clock pulse is wide, the output may toggle multiple times during a single clock pulse.

Solution: Master-Slave configuration:

1. Master (first FF): Captures input on the positive edge (clock = 1)
2. Slave (second FF): Transfers master's output on the negative edge (clock = 0)
3. Since master and slave never operate simultaneously, race condition is eliminated.

Flip-Flop Conversion

General Procedure:

1. Write the characteristic table of the desired flip-flop.
2. Write the excitation table of the available flip-flop.
3. For each row, determine the required inputs of the available FF.
4. Use K-maps to derive the expressions connecting desired inputs to available inputs.

Solved Example: Convert JK → D Flip-Flop

Convert D → T Flip-Flop

T FF: $Q_{next} = T \oplus Q$. D FF excitation: $D = Q_{next}$.

So: $D = T \oplus Q$. Circuit: XOR gate with T and Q feeding into D input.

Shift Registers

SISO Operation Example

Loading data 1011 into a 4-bit SISO register (MSB first):

After 4 clock pulses, data appears serially at $Q_0$ output.

Asynchronous (Ripple) Counters

3-Bit UP Counter

Uses 3 T flip-flops with T=1 (always toggle). Clock of each FF is connected to the Q output of the previous FF.

Counts from 0 to $2^N - 1$. For 3 bits: 0 to 7. DOWN counter: Connect clock to $\overline{Q}$ instead of $Q$.

Mod-N Counter

A Mod-N counter counts from 0 to N-1 and then resets. Design: Use enough flip-flops ($\lceil \log_2 N \rceil$) and add reset logic.

MCQ Assessment — Unit V

Introduction to Arduino & Sensors

Analog vs Digital Signals

ADC (Analog-to-Digital Converter): Converts continuous signals to discrete digital values. An $n$-bit ADC has $2^n$ levels. Arduino's 10-bit ADC → $2^{10} = 1024$ levels (0 to 1023).

DAC (Digital-to-Analog Converter): Converts digital values back to analog. Arduino uses PWM (Pulse Width Modulation) to simulate DAC.

Arduino Uno Board

Pin Configuration

Arduino Program Structure

// Every Arduino program has two mandatory functions

void setup() {
  // Runs ONCE when board powers on or resets
  pinMode(13, OUTPUT);    // Set pin 13 as output
  Serial.begin(9600);    // Start serial communication
}

void loop() {
  // Runs REPEATEDLY forever
  digitalWrite(13, HIGH);  // LED ON
  delay(1000);              // Wait 1 second
  digitalWrite(13, LOW);   // LED OFF
  delay(1000);              // Wait 1 second
}

Sensors

IR Sensor

Working Principle: An IR LED emits infrared light. A photodiode detects the reflected IR. When an obstacle is present, IR bounces back → photodiode receives signal → output goes LOW (active low).

Connections: VCC → 5V, GND → GND, OUT → Digital pin

// IR Sensor — Obstacle Detection
const int irPin = 2;

void setup() {
  pinMode(irPin, INPUT);
  Serial.begin(9600);
}

void loop() {
  int val = digitalRead(irPin);
  if (val == LOW) {
    Serial.println("Obstacle Detected!");
  } else {
    Serial.println("No Obstacle");
  }
  delay(200);
}

LDR (Light Dependent Resistor)

Working Principle: Made of Cadmium Sulfide (CdS). Resistance decreases with increasing light intensity. In dark: ~1 MΩ. In bright light: ~1 kΩ.

Circuit: Use a voltage divider with a fixed resistor (10kΩ). Connect the junction to an analog pin.

// LDR — Automatic Street Light
const int ldrPin = A0;
const int ledPin = 13;

void setup() {
  pinMode(ledPin, OUTPUT);
  Serial.begin(9600);
}

void loop() {
  int lightLevel = analogRead(ldrPin);  // 0-1023
  Serial.print("Light: ");
  Serial.println(lightLevel);
  
  if (lightLevel < 300) {  // Dark → turn ON street light
    digitalWrite(ledPin, HIGH);
  } else {
    digitalWrite(ledPin, LOW);
  }
  delay(500);
}

Ultrasonic Sensor (HC-SR04)

Working Principle: Sends a 40 kHz ultrasonic pulse via the Trigger pin. The pulse bounces off an object and returns to the Echo pin. The time taken for the round trip determines the distance.

Formula:

Pins: VCC (5V), Trig (digital out), Echo (digital in), GND

// HC-SR04 Ultrasonic Distance Measurement
const int trigPin = 9;
const int echoPin = 10;

void setup() {
  pinMode(trigPin, OUTPUT);
  pinMode(echoPin, INPUT);
  Serial.begin(9600);
}

void loop() {
  // Send 10μs pulse
  digitalWrite(trigPin, LOW);
  delayMicroseconds(2);
  digitalWrite(trigPin, HIGH);
  delayMicroseconds(10);
  digitalWrite(trigPin, LOW);
  
  // Measure echo duration
  long duration = pulseIn(echoPin, HIGH);
  float distance = duration * 0.034 / 2;
  
  Serial.print("Distance: ");
  Serial.print(distance);
  Serial.println(" cm");
  delay(500);
}

Temperature Sensor (DHT11 / DHT22)

Measures both temperature and humidity using a capacitive humidity sensor and a thermistor.

Data Format: 40-bit serial data = 8-bit humidity integer + 8-bit humidity decimal + 8-bit temperature integer + 8-bit temperature decimal + 8-bit checksum.

// DHT11 Temperature & Humidity (using DHT library)
#include "DHT.h"

#define DHTPIN 2
#define DHTTYPE DHT11

DHT dht(DHTPIN, DHTTYPE);

void setup() {
  Serial.begin(9600);
  dht.begin();
}

void loop() {
  float humidity = dht.readHumidity();
  float tempC = dht.readTemperature();
  
  if (isnan(humidity) || isnan(tempC)) {
    Serial.println("Sensor read failed!");
    return;
  }
  
  Serial.print("Humidity: ");
  Serial.print(humidity);
  Serial.print("% | Temp: ");
  Serial.print(tempC);
  Serial.println("°C");
  delay(2000);
}

Mini-Projects

MCQ Assessment — Unit VI