IntRoduction to solid state physics

Week 5: Magnetism in Solids

Lecture Topics:

1. Introduction to Magnetism in Solids

   • Magnetic materials: Overview of different magnetic properties found in solids.

   • Origins of magnetism: Role of electron spin and orbital motion in generating magnetic moments.

2. Diamagnetism

   • Definition: A form of magnetism that occurs in all materials, but is only observable in substances with no net magnetic moment.

   • Behavior: When an external magnetic field is applied, diamagnetic materials develop a weak, opposing magnetic field.

   • Example materials: Bismuth, copper, gold.

   • Key characteristics: Magnetic susceptibility is negative and small. Diamagnetism is temperature-independent.

3. Paramagnetism

   • Definition: Occurs in materials with unpaired electrons, which align with an external magnetic field, resulting in a weak attraction.

   • Behavior: Magnetic susceptibility is positive and small, and the magnetization is proportional to the applied field.

   • Curie's Law: Describes the temperature dependence of paramagnetic susceptibility:

where χ is the magnetic susceptibility, C is the Curie constant, and T is the temperature.

1. Example materials: Aluminum, platinum, and some transition metal ions.

2. Temperature dependence: The magnetic susceptibility decreases with increasing temperature.

Ferromagnetism

• Definition: A strong form of magnetism in which materials have spontaneous magnetization without an external field.

• Exchange interactions: The quantum mechanical interaction aligning neighboring magnetic moments.

• Domains: Ferromagnetic materials comprise small regions called magnetic domains, where the magnetic moments are aligned. Domains can reorient when an external field is applied.

• Hysteresis: Ferromagnetic materials exhibit hysteresis, meaning they retain a magnetic moment after removing the external field.

• Curie Temperature: The temperature above which a ferromagnetic material loses its magnetic properties and becomes paramagnetic.

• Example materials: Iron, cobalt, nickel.

Curie’s Law and Curie-Weiss Law

• Curie’s Law: Explains the behavior of paramagnetic materials and shows that their magnetic susceptibility is inversely proportional to temperature.

• Curie-Weiss Law: Describes the magnetic susceptibility of ferromagnetic materials in the paramagnetic region (above the Curie temperature):

where θ is the Weiss constant and is related to the strength of the internal field.

1. Exchange Interactions and Ferromagnetism

   1. Quantum mechanical exchange: The mechanism behind the strong coupling of neighboring magnetic moments in ferromagnetic materials.

   2. Types of exchange:

      1. Direct exchange: Involves the overlap of electron orbitals, contributing to the alignment of spins.

      2. Superexchange: Involves an intermediate non-magnetic atom (e.g., oxygen) between two magnetic ions.

      3. Double exchange: Involves electron transfer between ions of different valences, typically in mixed-valence compounds like manganites.

2. Magnetic Domains and Hysteresis

   1. Magnetic domains: Regions within a ferromagnetic material where all the magnetic moments are aligned in the same direction.

   2. Domain wall movement: When an external magnetic field is applied, domains grow or shrink, leading to changes in the material's magnetization.

   3. Hysteresis loop:

      1. Coercivity: The field required to reduce the magnetization to zero after saturation.

      2. Remanence: The magnetization that remains after the external field is removed.

      3. Applications of hysteresis: Magnetic storage devices (hard drives) and transformers.

Examples:

• Calculation of the magnetic susceptibility of a paramagnetic material using Curie’s Law.

• Plotting a hysteresis loop for a ferromagnetic material and identifying key parameters like coercivity and remanence.

• Explanation of the effect of temperature on the magnetic behavior of materials using Curie-Weiss Law.

Homework/Exercises:

1. Explain the role of exchange interactions in ferromagnetism and describe the difference between direct and superexchange interactions.

2. Using Curie’s Law, calculate the magnetic susceptibility of a paramagnetic material at a given temperature.

3. Draw and label a hysteresis loop for a ferromagnetic material, explaining the significance of the coercivity and remanence.

4. Compare and contrast the magnetic behavior of diamagnetic, paramagnetic, and ferromagnetic materials.

Suggested Reading:

• Charles Kittel, Introduction to Solid State Physics, Chapter 14: Magnetic Properties of Solids.

Key Takeaways:

• Magnetism in solids arises from the magnetic moments of electrons and their interactions within the material.

• Different types of magnetism (diamagnetism, paramagnetism, ferromagnetism) arise from different electron configurations and interactions.

• Ferromagnetic materials exhibit unique properties like spontaneous magnetization, magnetic domains, and hysteresis, which are critical for practical applications in data storage and electronics.

• Temperature plays a significant role in determining the magnetic properties of materials, as described by Curie’s Law and Curie-Weiss Law.

This week introduces students to the fundamental concepts of magnetism in solids, focusing on the quantum mechanical origins of magnetic behavior and the real-world applications of ferromagnetic materials.