Generalization of Faraday's Law to include nonconservative spin forces

Area of Science:

• Condensed matter physics
• Electromagnetism

Background:

• Faraday's Law describes electromotive force (E) based on electron charge and magnetic flux (Phi).
• Ferromagnetic materials possess nonconservative spin forces that also influence E.
• Existing models do not fully account for spin forces in electromotive force calculations.

Purpose of the Study:

• To incorporate nonconservative spin forces into Faraday's Law.
• To develop a more comprehensive understanding of energy conservation in ferromagnets.

Main Methods:

• Modified Faraday's Law by replacing magnetic flux (Phi) with a term involving Planck's constant, elementary charge (-e), and averaged Berry phase (gamma).
• Analyzed energy conservation requirements in itinerant ferromagnets with time-dependent order parameters.

Main Results:

• Introduced a modified Faraday's Law that includes contributions from electron spin forces.
• Demonstrated that the Berry phase (gamma) averaged over electron spin direction is key to this modification.
• Showcased how these spin contributions satisfy energy conservation principles.

Conclusions:

• The revised Faraday's Law provides a more complete description of electromotive force in ferromagnetic materials.
• Incorporating spin forces and Berry phase is crucial for understanding energy dynamics in these systems.
• This work advances the study of electromagnetism in magnetic materials.