Slide 1 - Mobility and Temperature Dependence of Resistivity

  • Introduction to mobility and resistivity
  • Definition of resistivity ($\rho$)
  • Role of temperature in determining resistivity
  • Concept of carrier mobility ($\mu$)
  • Relationship between resistivity and carrier mobility

Slide 2 - Resistivity

  • Definition: Resistivity is a measure of a material’s ability to oppose electric current.
  • Units: $\Omega \cdot m$ (ohm-meter)
  • Symbol: $\rho$
  • Formula: $\rho = \frac{R \cdot A}{L}$
    • $R$ is the resistance of the material
    • $A$ is the cross-sectional area
    • $L$ is the length of the material

Slide 3 - Temperature Dependence of Resistivity

  • Most materials exhibit a change in resistivity with temperature.
  • Two types of materials based on temperature dependence:
    1. Conductors: resistivity increases with temperature
    2. Semiconductors: resistivity decreases with temperature
  • This behavior is due to the change in carrier concentration and mobility with temperature.

Slide 4 - Conductivity and Carrier Mobility

  • Conductivity ($\sigma$) is the reciprocal of resistivity.
  • Conductivity formula: $\sigma = \frac{1}{\rho}$
  • Carrier mobility ($\mu$) is a measure of how freely carriers move in a material.
  • Mobility formula: $\mu = \frac{\text{Drift velocity}}{\text{Electric field}}$
    • Higher mobility means better conductivity.

Slide 5 - Carrier Mobility and Temperature

  • In conductors, carrier mobility decreases with increasing temperature.
  • Interactions between carriers and lattice ions increase at higher temperatures, impeding their motion.
  • This leads to an increase in resistivity.
  • Example: Copper has high mobility at low temperatures but decreases as the temperature rises.

Slide 6 - Semiconductor Behavior

  • In semiconductors, carrier mobility and resistivity show different behavior.
  • As temperature increases, carrier mobility also increases.
  • This is because higher temperatures provide more energy to the carriers, allowing them to overcome lattice imperfections and move more freely.
  • Example: Silicon is a typical semiconductor with a negative temperature coefficient of resistivity.

Slide 7 - Relationship Between Resistivity and Carrier Mobility in Conductors

  • In conductors, resistivity ($\rho$) is inversely proportional to carrier mobility ($\mu$).
  • Mathematically: $\rho \propto \frac{1}{\mu}$
  • This means that materials with high carrier mobility have low resistivity, and vice versa.

Slide 8 - Relationship Between Resistivity and Carrier Mobility in Semiconductors

  • In semiconductors, resistivity ($\rho$) is also inversely proportional to carrier mobility ($\mu$).
  • Mathematically: $\rho \propto \frac{1}{\mu}$
  • However, the relationship is more complex due to the dependence on carrier concentration as well.
  • Higher carrier mobility and lower carrier concentration result in lower resistivity.

Slide 9 - Example: Temperature Dependence of Resistivity in Copper

  • Copper is a good conductor of electricity and exhibits a positive temperature coefficient of resistivity.
  • As temperature increases, resistivity also increases.
  • This behavior can be explained by the increase in carrier-lattice interactions at higher temperatures, impeding the motion of carriers.
  • This phenomenon is utilized in devices like temperature sensors and thermistors.

Slide 10 - Example: Temperature Dependence of Resistivity in Silicon

  • Silicon is a semiconductor material commonly used in electronic devices.
  • It exhibits a negative temperature coefficient of resistivity.
  • As temperature increases, resistivity decreases, allowing for better conductivity.
  • This behavior is due to the increase in carrier mobility with temperature, allowing carriers to move more freely.
  • This characteristic is exploited in the operation of transistors and diodes.

Slide 11 - Relationship Between Resistivity and Carrier Mobility (Conductors)

  • Resistivity ($\rho$) is inversely proportional to carrier mobility ($\mu$) in conductors.
  • Higher carrier mobility leads to lower resistivity.
  • Conductors with high carrier mobility such as silver and copper have low resistivity.
  • Examples:
    • Copper has high carrier mobility (conduction electrons), therefore low resistivity.
    • Aluminum has lower carrier mobility than copper, leading to higher resistivity.

Slide 12 - Relationship Between Resistivity and Carrier Mobility (Semiconductors)

  • Resistivity ($\rho$) is also inversely proportional to carrier mobility ($\mu$) in semiconductors.
  • Higher carrier mobility and lower carrier concentration result in lower resistivity.
  • Example:
    • In a silicon-based n-type semiconductor, the addition of a pentavalent impurity increases electron concentration and carrier mobility, resulting in lower resistivity.

Slide 13 - Temperature Dependence of Resistivity in Copper

  • Copper exhibits a positive temperature coefficient of resistivity.
  • As temperature increases, the resistivity of copper also increases.
  • This behavior is due to increased carrier-lattice interactions impeding the motion of carriers.
  • Example:
    • Copper wires used for power transmission have a higher resistivity when operating at higher temperatures, resulting in more power loss in the form of heat.

Slide 14 - Temperature Dependence of Resistivity in Silicon

  • Silicon exhibits a negative temperature coefficient of resistivity.
  • As temperature increases, the resistivity of silicon decreases.
  • This behavior is due to the increase in carrier mobility with temperature.
  • Example:
    • In a silicon-based diode, as temperature increases, the resistivity decreases, resulting in improved conductivity and better diode performance.

Slide 15 - Applications of Temperature Dependence of Resistivity

  • Temperature sensors: Utilize resistivity changes in conductors to measure temperature variations.
  • Thermistors: Devices made from temperature-sensitive materials, such as semiconductors, used for temperature sensing.
  • Heating elements: As the resistance of a conductor increases with temperature, it can be used to produce heat in devices like electric heaters.
  • Thermoelectric devices: Utilize the Seebeck effect, which relates temperature difference to voltage, to generate electricity.

Slide 16 - Resistivity and Conductivity in Metals

  • Metals have low resistivity and high conductivity.
  • This is due to their high carrier concentration and high carrier mobility.
  • Good electrical conductors in daily life, such as copper and aluminum, are metals.
  • Metals are often used in electrical wiring, circuitry, and as conductors in various appliances and machinery.

Slide 17 - Resistivity and Conductivity in Insulators

  • Insulators have high resistivity and low conductivity.
  • This is due to their low carrier concentration and low carrier mobility.
  • Examples of insulators include rubber, glass, and plastic.
  • Insulators are used to prevent the flow of electricity and are commonly found in electrical insulation and insulating coatings.

Slide 18 - Resistivity and Conductivity in Semiconductors

  • Semiconductors have intermediate resistivity and conductivity compared to metals and insulators.
  • Their resistivity is highly dependent on temperature and impurities.
  • Examples of semiconductors include silicon and germanium.
  • Semiconductors are widely used in electronic devices, such as transistors, diodes, and integrated circuits.

Slide 19 - Superconductivity and Zero Resistivity

  • Some materials, at very low temperatures, exhibit superconductivity.
  • Superconductors have zero resistivity, allowing for the free flow of electrical current.
  • Zero resistivity is achieved when the material’s temperature is below its critical temperature.
  • Superconductors have numerous applications, such as in medical imaging devices (MRI machines) and particle accelerators.

Slide 20 - Summary

  • Resistivity is a measure of a material’s ability to oppose electric current.
  • Resistivity and carrier mobility are inversely proportional in both conductors and semiconductors.
  • The temperature coefficient of resistivity determines how resistivity changes with temperature.
  • Conductors usually exhibit a positive temperature coefficient, while semiconductors can have either a positive or negative temperature coefficient.
  • Understanding the temperature dependence of resistivity is crucial for various applications in electrical and electronic devices.