Semiconductor Physics – Band Theory, Doping, Devices, and Applications

Introduction

Semiconductors are the backbone of modern electronics. Unlike conductors, whose electrons are free, or insulators, whose electrons are tightly bound, semiconductors possess a band gap that can be manipulated to control charge transport. This unit explains the physics behind that control, the creation of p‑type and n‑type materials, and how these materials combine to form the devices that power smartphones, computers, and power grids.

Band Theory of Solids

In a crystalline solid, atomic orbitals overlap to form energy bands. The valence band is filled with electrons, while the conduction band is typically empty. The energy difference between the top of the valence band and the bottom of the conduction band is the band gap $E_g$.

• Metals: $E_g=0$ – valence and conduction bands overlap.
• Insulators: $E_g>4$ eV – very few electrons can be thermally excited.
• Semiconductors: $1\text{ eV}<E_g<4$ eV – moderate excitation possible.

The probability that an electron occupies an energy level $E$ at temperature $T$ follows the Fermi‑Dirac distribution:

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

where $E_F$ is the Fermi level and $k$ is Boltzmann’s constant.

Merkle Diagram: Band Structure Flow

flowchart TD
  "Atomic Orbitals" --> "Overlap → Energy Bands"
  "Energy Bands" --> "Valence Band"
  "Energy Bands" --> "Conduction Band"
  "Band Gap" --> "Control of Conductivity"

Intrinsic Semiconductors

An intrinsic semiconductor is pure, with no intentional impurities. At absolute zero, all electrons occupy the valence band. As temperature rises, electrons acquire enough thermal energy to cross the band gap, leaving behind holes in the valence band.

The intrinsic carrier concentration $n_i$ is given by:

$$n_i=\sqrt{N_c{N}_v}{e}^{-E_g/(2kT)}$$

where $N_c$ and $N_v$ are the effective density of states in the conduction and valence bands, respectively.

Worked Example – Silicon at 300 K

• $E_g=1.12$ eV
• $N_c=2.8\times 10^{19}{\text{cm}}^{-3}$
• $N_v=1.04\times 10^{19}{\text{cm}}^{-3}$

$$n_i=\sqrt{(2.8\times 10^{19})(1.04\times 10^{19})}{e}^{-1.12/(2\times 8.617\times 10^{-5}\times 300)}\approx 1.5\times 10^{10}{\text{cm}}^{-3}$$

Thus, intrinsic silicon at room temperature has about $1.5\times 10^{10}$ electrons and an equal number of holes per cubic centimeter.

Extrinsic Semiconductors and Doping

To achieve useful conductivity, semiconductors are doped with impurities that introduce energy levels close to the band edges.

n‑Type Doping

Adding a group‑V element (e.g., phosphorus) contributes an extra valence electron. This electron occupies a donor level just below the conduction band, making it easy to excite into the conduction band.

• Electron concentration $n\approx N_D$ (donor concentration).
• Hole concentration $p=n_i^2/n$.

p‑Type Doping

Adding a group‑III element (e.g., boron) creates an acceptor level just above the valence band, effectively generating holes.

• Hole concentration $p\approx N_A$ (acceptor concentration).
• Electron concentration $n=n_i^2/p$.

Mermaid Diagram: Doping Process

flowchart LR
  "Silicon Crystal" --> "Introduce Dopant"
  "Introduce Dopant" --> "Create Donor Level" 
  "Introduce Dopant" --> "Create Acceptor Level"
  "Donor Level" --> "n‑type Conductivity"
  "Acceptor Level" --> "p‑type Conductivity"

Carrier Concentration and Conductivity

Electrical conductivity $\sigma$ in a semiconductor is:

$$\sigma =q(n{\mu }_n+p{\mu }_p)$$

where $q$ is the elementary charge, ${\mu }_n$ and ${\mu }_p$ are electron and hole mobilities.

• Mobility depends on scattering mechanisms (phonon, impurity).
• In n‑type material, $\sigma \approx q{N}_D{\mu }_n$.
• In p‑type material, $\sigma \approx q{N}_A{\mu }_p$.

Temperature Dependence

• Intrinsic: $\sigma \propto e^{-E_g/(2kT)}$ – increases rapidly with $T$.
• Extrinsic: At low $T$, carriers freeze out; at moderate $T$, conductivity rises linearly with $T$ due to increased carrier concentration; at high $T$, mobility decreases due to phonon scattering.

Graph: Conductivity vs Temperature for intrinsic and extrinsic silicon

p‑n Junctions

A p‑n junction is formed by joining p‑type and n‑type regions. At the interface, electrons diffuse from n to p, and holes diffuse from p to n, creating a depletion region with an internal electric field.

• Built‑in potential $V_{bi}=\frac{kT}{q}\ln \left(\frac{N_A{N}_D}{n_i^2}\right)$.
• Depletion width $W=\sqrt{\frac{2{\epsilon }_s}{q}\left(\frac{N_A+N_D}{N_A{N}_D}\right)V_{bi}}$.

Diode I‑V Characteristic

The Shockley equation describes the current:

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

where $I_S$ is the saturation current.

Worked Example – Silicon Diode

• $N_A=10^{16}{\text{cm}}^{-3}$, $N_D=10^{15}{\text{cm}}^{-3}$, $T=300\text{K}$.
• $V_{bi}=\frac{(8.617\times 10^{-5}\text{eV/K})\times 300}{1.602\times 10^{-19}\text{C}}\ln \left(\frac{10^{16}\times 10^{15}}{(1.5\times 10^{10}{)}^2}\right)\approx 0.65\text{V}$.

Bipolar Junction Transistors (BJTs)

A BJT consists of two p‑n junctions: emitter–base (EB) and base–collector (BC). There are two types: NPN and PNP.

• Current gain $\beta =I_C/I_B$.
• Collector current $I_C\approx \beta I_B$ (for active region).

NPN Operation

1. Forward‑active: EB junction forward‑biased, BC junction reverse‑biased.
2. Saturation: Both junctions forward‑biased.
3. Cut‑off: Both junctions reverse‑biased.

Mermaid Diagram: BJT Operation Modes

stateDiagram-v2
  [*] --> ForwardActive : EB forward, BC reverse
  ForwardActive --> Saturation : EB & BC forward
  ForwardActive --> Cutoff : EB & BC reverse

Metal‑Oxide‑Semiconductor Field‑Effect Transistors (MOSFETs)

MOSFETs control current via an electric field applied to a gate electrode separated by an oxide layer.

• Types: Enhancement‑mode (normally off) and depletion‑mode (normally on).
• Key parameters: Threshold voltage $V_{th}$, transconductance $g_m$, drain current $I_D$.

Enhancement‑Mode nMOSFET Current Equation

$$I_D=\frac{W}{L}\frac{{\mu }_n{C}_{ox}}{2}(V_{GS}-V_{th}{)}^2\text{for }{V}_{GS}>V_{th}$$

where $W$ and $L$ are channel width and length, $C_{ox}$ is oxide capacitance per unit area.

Worked Example – nMOSFET

• $W=10\mu \text{m}$, $L=1\mu \text{m}$, ${\mu }_n=400{\text{cm}}^2/\text{V⋅s}$, $C_{ox}=10{\text{nF/cm}}^2$, $V_{th}=0.7\text{V}$.
• For $V_{GS}=2\text{V}$:

$$I_D=\frac{10}{1}\frac{400\times 10^{-4}}{2}(2-0.7{)}^2\times 10^{-9}\approx 1.2\text{mA}$$

Cross‑section of an nMOSFET showing gate oxide and channel (Image: Vegar Ottesen, CC BY-SA 3.0, via Wikimedia Commons)

Applications

In the real world

• eSewa uses MOSFET‑based power management ICs to regulate the voltage supplied to its servers, ensuring efficient data handling.
• Daraz relies on silicon‑based BJT switches in its warehouse automation robots for precise motion control.
• Ncell’s base stations incorporate high‑power MOSFETs to amplify RF signals, demonstrating the importance of carrier mobility and thermal management.

In the real world

1. Google Search Engine – Uses billions of MOSFETs in its data centers to switch and amplify signals, directly applying the field‑effect principle.
2. Pathao Delivery App – Relies on embedded microcontrollers with silicon BJTs for sensor interfacing and power regulation.
3. NEPSE Trading Platform – Employs high‑speed silicon photodiodes (p‑n junctions) for optical data links, showcasing the role of semiconductor physics in financial infrastructure.

Exam tip

• Understand the derivation of key equations (e.g., $V_{bi}$, $I_D$ for MOSFETs).
• Practice numerical problems: carrier concentration, depletion width, and I‑V curves.
• Draw band diagrams for intrinsic, n‑type, and p‑type semiconductors; label donor/acceptor levels.
• Know the differences between BJT and MOSFET operation modes and their characteristic curves.

Detailed cross‑section of a BJT with doping profiles (Image: Inductiveload, Public domain, via Wikimedia Commons)

Detailed cross‑section of a MOSFET showing gate oxide and channel (Image: Vegar Ottesen, CC BY-SA 3.0, via Wikimedia Commons)