Fundamentals of Quantum Physics Study Notes

Course Overview

• Course Name: Fundamentals of Quantum Physics

• Course Code: TEE7244

• Course Credit: 3+1

• Instructors: Prof. (Dr.) Meena Laad & Dr. Kuldeep Mishra

Unit 3: Semiconductors

• Duration: 09 Lectures

• Key Topics:

  • Intrinsic and extrinsic semiconductors

  • Dependence of Fermi level on carrier concentration and temperature (equilibrium carrier statistics)

  • Carrier generation and recombination

  • Carrier transport

  • Hall effect

  • Hall effect sensor

  • Magnetic field detection

  • Magnetic switching

Properties of Semiconductors

• Semiconductors:

  • Definition: A solid material with electrical conductivity between a conductor and an insulator.

  • Applications: used as optical detectors, light-emitting diodes (LEDs), lasers, amplifiers, waveguides, modulators, sensors, and nonlinear optical elements.

  • Behavior at Temperature: Materials that are semiconductors at room temperature behave as perfect insulators at absolute zero (0 K).

  • Energy Bands at Room Temperature: 25 eV - represents the energy absorbed by an electron when subjected to a 1 V potential difference at room temperature.

Charge Carriers in Semiconductors

• Types of Charge Carriers: Electrons and holes.

• At Absolute Zero (T = 0 K):

  • Valence Band (VB) is completely filled.

  • Conduction Band (CB) is completely empty.

  • Result: The material does not conduct electricity.

• As Temperature Increases:

  • Some electrons are thermally excited into the empty Conduction Band, forming a free electron in CB and a free hole in VB.

  • Definition: An empty state in the VB is referred to as a hole; the excitation of a VB electron to the CB creates an electron-hole pair (EHP).

Intrinsic Semiconductor

• Definition: A perfect semiconductor crystal with no impurities or lattice defects.

• At T = 0 K:

  • No charge carriers are present.

  • VB filled with electrons, CB empty.

• At T > 0 K:

  • Electron-hole pairs (EHPs) are generated.

  • EHPs serve as the only charge carriers in intrinsic materials.

  • Relationship: The electron concentration in the conduction band (n) equals the concentration of holes in the valence band (p), both denoted as intrinsic carrier concentration (ni).

  • Equation:
    n = p = n_i

Characteristics of Intrinsic Semiconductors

• Concentration of EHPs (ni): At any given temperature, there is a certain concentration of EHPs ni.

• Recombination Rate (ri):

  • EHPs recombine at the same rate they are generated.

  • Recombination occurs when an electron makes a transition to an empty state in the valence band, thereby annihilating the pair.

  • Equilibrium condition:
    ri = gi
    where g_i is the generation rate of EHPs.

  • Rates are temperature dependent.

• Temperature Influence: As temperature increases, the generation rate gi increases, establishing a new carrier concentration ni where the higher recombination rate ri balances it.

• Mathematical Relation:

  • Rate of recombination of electrons and holes:
    ri = \alphar n0 p0 = \alphar ni^2 = gi where \alphar is a constant of proportionality depending on the recombination mechanism.

Conductivity and Temperature

• Effect of Temperature:

  • As temperature increases, the number of free electrons and holes generated grows exponentially, affecting semiconductor conductivity.

• Carrier Concentration vs Temperature Example (Silicon) at Different Temperatures:

  • ( ext{Temperature (K)}): 150, 200, 250, 300, 350, 400, 450, 500

  • ( ext{Intrinsic Concentration (cm}^{-3} ext{)}): 1E3, 1E17, 1E10, … 1E17

Carrier Density in Intrinsic Semiconductor

• Mathematical Expression for Carrier Density:

  • ni^2 = \frac{Ne e^{(Ec - EF)/kT}}{N1 e^{(EF - E_v)/kT}}

  • Where:

  • N_c: Effective density of states in CB

  • N_v: Effective density of states in VB

  • E_c: Conduction Band edge

  • E_v: Valence Band edge

  • E_F: Fermi energy level

  • k: Boltzmann constant

  • T: Absolute temperature

• Effective Density of States for Silicon:

  • At Room Temperature:

  • N_c = 2.8 imes 10^{25} ext{ m}^{-3}

  • N_v = 1.0 imes 10^{25} ext{ m}^{-3}

Example of Charge Carrier Concentration

• Example Problem:

  • Given Eg = 1.1 eV. Calculate the carrier concentrations at 300 K, taking effective masses as 0.27 m0 for holes and 0.13 m0 for electrons.

Doping in Semiconductors

• Purpose of Doping:

  • To increase the number of charge carriers (electrons or holes) in a semiconductor material and modify its electrical properties.

• Process:

  • Mixing pure semiconductors with impurities (dopants).

  • Can introduce additional charge carriers (donors or acceptors).

• Types of Doped Semiconductors:

  • n-type: Predominantly contains electrons (donor impurities such as Phosphorus, Arsenic).

  • p-type: Predominantly contains holes (acceptor impurities such as Boron, Aluminum).

Doping Elements Characteristics

• Group V Elements (n-type dopants):

  • Elements like Phosphorus, Antimony introduce additional electrons.

• Group III Elements (p-type dopants):

  • Elements like Boron and Aluminum create holes in the valence band.

Doping and Conductivity Changes

• Effects of Temperature on Doping:

  • Conductivity can vary widely with temperature in doped semiconductors.

• Summary of Doping Impacts:

  • Doping enhances the concentration of mobile charge carriers significantly, thus increasing conductivity.

Doping Ionization Energy

• Ionization Energy for Donor and Acceptor Levels:

  • Ionization energies for common dopants at room temperature are remarkably low (order of 0.01 - 0.06 eV).

Fermi Level in Semiconductors

• Fermi Level Position:

  • The position of the Fermi level is crucial for determining the conductivity type.

  • In n-type: Fermi level is closer to the conduction band.

  • In p-type: Fermi level is closer to the valence band.

Carrier Transport Phenomena

• Drift Current Definition: The current due to charge carriers moving in response to the applied electric field.

  • Current defined as:
    J = \sigma E

  • Where:

  • J: Current density (A/m²)

  • \sigma: Conductivity (S/m)

  • E: Electric field strength (V/m)

• Diffusion Current Definition: The current resulting from the movement of charge carriers from regions of high concentration to low concentration.

  • Governed by Fick's laws:

  • J_D = -D \nabla n

  • Where D is the diffusion coefficient.

Hall Effect

• Definition: The voltage created across a conductor when an electric current flows through it in a magnetic field.

  • Governed by the Lorentz force, given by:
    F = q[E + (v \times B)]

• Applications of Hall Effect:

  • Used to determine semiconductor type, carrier concentration, measurement of magnetic fields.

Hall Sensor Technology

• Types of Hall Sensors:

  • Automotive, Industrial, and Consumer applications, including digital switches and automotive functions.

• Example Problem: Calculate the current density in an n-type semiconductor under an electric field using given parameters.

• Example Calculations: Show calculations for determining J, based on the current density formula and provided doping parameters.

Conclusion

• Key Takeaways:

  • Understanding semiconductors, doping mechanisms, and their applications is foundational for electronic device design.

  • The interplay between temperature, carrier concentration, and external fields shapes the behavior of semiconductor materials.