Current and Electricity

Learning Objectives

• Understand the concept of mobility
• Explore the factors affecting mobility
• Examine how mobility affects resistivity
• Analyze the temperature dependence of resistivity

Introduction to Mobility

• Mobility is a measure of the ease with which charge carriers move through a material
• It is represented by the symbol μ (mu)
• Mobility is defined as the ratio of the drift velocity of charge carriers to the electric field applied to the material

Factors Affecting Mobility

• Charge carriers’ nature and mass
• Charge carriers’ concentration
• Presence of impurities and defects in the material
• Temperature of the material

Relationship between Mobility and Resistivity

• Resistivity (ρ) is inversely related to mobility (μ)
• Resistivity is defined as the resistance (R) of a material per unit length and unit area ρ = R * (A / L) R = ρ * (L / A)
• Conductivity (σ) is the reciprocal of resistivity σ = 1 / ρ

Example: Calculating Resistivity from Mobility

• Given:

  • Charge carrier mass (m) = 9.1 x 10^-31 kg
  • Charge carrier concentration (n) = 1 x 10^22 m^-3
  • Charge carrier mobility (μ) = 0.02 m^2 V^-1 s^-1
  • Length of material (L) = 0.1 m
  • Cross-sectional area (A) = 0.01 m^2

• Solution:

  • Calculate the drift velocity (v): v = μ * E (electric field)
  • Calculate the resistance (R): R = ρ * (L / A)
  • Calculate the resistivity (ρ): ρ = R * (A / L)

Temperature Dependence of Resistivity

• Conductivity decreases with increasing temperature for metals
• Resistivity increases with increasing temperature for metals
• Temperature affects the scattering of charge carriers by lattice vibrations, impurities, and defects
• Resistance increases due to increased scattering of charge carriers

Temperature Dependence of Resistivity (Contd.)

• Resistivity of metals (ρ) can be expressed as: ρ(T) = ρ(0) * [1 + α(T - T_0)]

  • ρ(0) is the constant resistivity at some reference temperature (T_0)
  • α is the temperature coefficient of resistivity

Example: Calculating Resistivity at a Given Temperature

• Given:

  • Temperature coefficient of resistivity (α) = 0.0039 K^-1
  • Resistivity at reference temperature (ρ_0) = 1.2 x 10^-8 Ωm
  • Temperature (T) = 300 K

• Solution:

  • Calculate the change in temperature (T - T_0)
  • Substitute the values in the resistivity equation to calculate ρ(T)

Summary

• Mobility is a measure of charge carriers’ ease of movement in a material
• Factors affecting mobility include charge carriers’ nature, mass, concentration, and presence of impurities
• Resistivity and mobility are inversely related
• Temperature affects resistivity due to increased scattering of charge carriers
• Resistivity can be calculated using mobility and resistivity equations

Factors Affecting Mobility (Contd.)

• Charge carriers’ nature can be either positive (holes) or negative (electrons)
• Charge carriers’ mass affects their mobility, with lighter particles having higher mobility
• Charge carriers’ concentration is the number of charge carriers per unit volume
• Presence of impurities and defects can increase the scattering of charge carriers, lowering mobility
• Temperature affects mobility due to increased lattice vibrations and collisions with other particles

Relationship between Mobility and Resistivity (Contd.)

• The relationship between resistivity and mobility can be expressed as: σ = nqμ ρ = 1 / σ ρ = 1 / (nqμ)
• n is the charge carrier concentration
• q is the charge of a single carrier (e.g., e = 1.6 x 10^-19 C for electrons)

Example: Calculating Resistivity with Charge Carrier Concentration

• Given:

  • Charge carrier concentration (n) = 5 x 10^28 m^-3
  • Charge carrier mobility (μ) = 0.03 m^2 V^-1 s^-1
  • Charge of a single carrier (q) = 1.6 x 10^-19 C

• Solution:

  • Calculate the conductivity (σ) using σ = nqμ
  • Calculate the resistivity (ρ) using ρ = 1 / σ

Temperature Dependence of Resistivity (Contd.)

• Some materials, such as semiconductors, have opposite temperature dependence compared to metals
• The resistivity of semiconductors decreases with increasing temperature
• This is due to increased charge carrier concentration and mobility with temperature

Temperature Dependence of Resistivity (Contd.)

• The temperature dependence of resistivity in semiconductors can be modeled using the following equation: ρ(T) = ρ(0) * e^(E_g / (2kT))

  • ρ(0) is the constant resistivity at some reference temperature
  • E_g is the energy bandgap of the semiconductor
  • k is the Boltzmann constant (1.38 x 10^-23 J/K)
  • T is the absolute temperature

Example: Calculating Resistivity in a Semiconductor

• Given:

  • Energy bandgap of the semiconductor (E_g) = 1.2 eV
  • Constant resistivity at reference temperature (ρ_0) = 5 x 10^-4 Ωm
  • Temperature (T) = 300 K

• Solution:

  • Calculate the expression e^(E_g / (2kT))
  • Multiply the result by ρ_0 to obtain ρ(T)

Temperature Dependence of Resistivity - Superconductors

• Superconductors are materials that exhibit zero electrical resistance below a certain critical temperature (T_c)
• Below T_c, resistivity drops abruptly to zero
• Superconductivity is a phenomenon that can only occur under specific conditions in certain materials

Temperature Dependence of Resistivity - Superconductors (Contd.)

• The transition to superconductivity is characterized by the following temperature dependence: ρ(T) = ρ(0) * (1 - (T / T_c)^n)

  • ρ(0) is the resistivity of the normal state at T = 0 K
  • T_c is the critical temperature of the superconductor
  • n is an exponent that varies depending on the superconducting material

Example: Calculating Resistivity in a Superconductor

• Given:

  • Resistivity of the normal state at T = 0 K (ρ_0) = 5 x 10^-10 Ωm
  • Critical temperature of the superconductor (T_c) = 9 K
  • Exponent (n) = 2

• Solution:

  • Calculate (T / T_c)^n
  • Multiply the result by ρ_0 to obtain ρ(T)

Summary

• Mobility is influenced by charge carriers’ nature, mass, concentration, impurities, and temperature
• Resistivity and mobility have an inverse relationship
• The temperature dependence of resistivity varies for different types of materials (metals, semiconductors, and superconductors)
• Resistivity equations can be used to calculate resistivity at different temperatures for given material properties

Slide 21

• Resistivity of insulating materials is generally very high
• Insulators have very few or no charge carriers that can move freely
• Examples of insulating materials include rubber, glass, and ceramics
• In insulators, the energy required to move charge carriers is significantly higher compared to conductors
• The resistivity of insulators is in the order of 10^12 Ωm or higher

Slide 22

• Semiconductors have resistivity values between those of conductors and insulators
• Examples of semiconductors include silicon and germanium
• The resistivity of semiconductors can be controlled by doping (adding impurities)
• Doping introduces charge carriers that can increase or decrease the conductivity of the material
• Semiconductors are widely used in electronic devices such as transistors and integrated circuits

Slide 23

• Superconductors are materials that exhibit zero electrical resistance below a certain critical temperature (T_c)
• Superconductivity was first discovered in mercury by Heike Kamerlingh Onnes in 1911
• Below the critical temperature, superconductors can carry electric currents indefinitely without any energy loss due to resistance
• Superconducting materials are used in various applications, including magnets for magnetic resonance imaging (MRI) machines and particle accelerators

Slide 24

• The temperature dependence of resistivity is an important property in the study of materials

• The temperature coefficient of resistivity (α) quantifies this dependence

• α is defined as the change in resistivity per degree change in temperature

• It is given by the equation: α = (ρ₂ - ρ₁) / (ρ₁ * (T₂ - T₁))

• α has units of Ωm/K

Slide 25

• The temperature coefficient of resistivity (α) can be positive or negative depending on the material
• Materials with positive α have increasing resistivity with increasing temperature (e.g., most metals)
• Materials with negative α have decreasing resistivity with increasing temperature (e.g., semiconductors)
• Superconductors have α = 0 below their critical temperature since they have zero resistivity

Slide 26

• The temperature coefficient of resistivity (α) can vary among different materials
• Some materials have very small values of α, indicating weak temperature dependence
• Other materials, such as thermistors, have large values of α, making them highly sensitive to temperature changes
• The temperature coefficient is an important parameter for characterizing and selecting materials for specific applications

Slide 27

• The resistivity-temperature relationship can also be visualized using temperature-resistivity graphs
• For metals, the graph shows an increasing linear trend as temperature increases, indicating positive α
• For semiconductors, the graph shows a decreasing trend as temperature increases, indicating negative α
• Superconductors exhibit a sharp drop to zero resistivity below the critical temperature, forming a horizontal line on the graph

Slide 28

• The mobility of charge carriers is crucial for understanding electrical conduction in materials
• Higher mobility means charge carriers can move more freely, resulting in lower resistivity
• Mobility is directly proportional to the average drift velocity of charge carriers under an applied electric field
• Conductors typically have high mobility, whereas insulators generally have low mobility
• Doping in semiconductors can increase or decrease mobility, depending on the type of impurity introduced

Slide 29

• The concept of mobility and the temperature dependence of resistivity have practical implications
• The study of conductivity and resistivity helps in the design and improvement of electrical and electronic systems
• Understanding these properties is crucial for optimizing the performance of various devices and materials
• It also aids in the development of advanced technologies and applications, such as renewable energy and nanoelectronics

Slide 30

• In conclusion, the mobility and temperature dependence of resistivity play significant roles in electrical conduction
• Mobility determines the ease with which charge carriers move in a material
• Resistivity measures the resistance offered by a material to the flow of electric current
• Temperature affects resistivity through changes in scattering, energy levels, and lattice vibrations
• Knowledge of these concepts allows us to better understand and manipulate electrical properties to suit specific needs