Mobility And Temperature Dependence Of Resistivity- Current And Electricity - Mobility And Temperature Dependence Of Resistivity- Current And Electricity – [Lecture 3]

Mobility and Temperature Dependence of Resistivity

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