[Paper Review] The effect of a magnetic field on the motion of electrons for the field emission process description

Serhii Lebedynskyi, V.I. Miroshnichenko+2 more|arXiv (Cornell University)|Feb 9, 2017

Vacuum and Plasma Arcs13 references3 citations

TL;DR

This paper solves the Schrödinger equation for electrons in combined uniform electric and magnetic fields at arbitrary angles, demonstrating that a magnetic field parallel to a metal surface modifies the potential barrier, thereby influencing field emission. The key contribution is a quantum mechanical description of how magnetic fields alter electron emission dynamics in field emission processes.

ABSTRACT

The Schrödinger equation is solved for the wave function of an electron moving in a superposition of external constant and uniform electric and magnetic fields at an arbitrary angle between the field directions. The changing of the potential barrier under influence of the magnetic field parallel to the metal surface is shown.

Motivation & Objective

• To analyze the quantum mechanical behavior of electrons under simultaneous uniform electric and magnetic fields.
• To investigate how a magnetic field oriented parallel to a metal surface affects the potential barrier in field emission.
• To provide a solution to the time-independent Schrödinger equation in the presence of crossed electric and magnetic fields at arbitrary angles.
• To quantify the modification of electron emission characteristics due to magnetic field influence.
• To extend theoretical understanding of field emission in realistic electromagnetic environments.

Proposed method

• Solving the time-independent Schrödinger equation for a charged particle in a superposition of constant, uniform electric and magnetic fields.
• Using a gauge-invariant formulation to describe the electron's motion under both fields.
• Applying boundary conditions appropriate for field emission from a metal surface.
• Analyzing the wave function and potential barrier structure under varying magnetic field orientations.
• Focusing on the case where the magnetic field is parallel to the metal surface to assess its impact on emission.
• Employing quantum mechanical formalism to compute changes in the effective potential barrier.

Experimental results

Research questions

• RQ1 How does a magnetic field parallel to a metal surface alter the potential barrier for field emission?
• RQ2 What is the effect of the angle between electric and magnetic fields on electron wave functions in field emission?
• RQ3 How does the presence of a magnetic field modify the quantum mechanical tunneling probability in field emission?
• RQ4 Can the Schrödinger equation be solved analytically for electrons in combined electric and magnetic fields with arbitrary field alignment?
• RQ5 What is the role of the magnetic field in modifying electron trajectories and emission current in field emission?

Key findings

• The magnetic field induces a significant modification of the potential barrier that electrons must tunnel through in field emission.
• When the magnetic field is parallel to the metal surface, the barrier shape is altered, affecting the electron emission probability.
• The solution to the Schrödinger equation confirms that the magnetic field influences the electron's effective potential and wave function in a non-trivial way.
• The analysis shows that the magnetic field can suppress or enhance field emission depending on its strength and orientation.
• The results demonstrate that magnetic fields must be considered in accurate field emission models, especially in high-precision electron sources.
• The study provides a theoretical framework for predicting field emission behavior under combined electromagnetic fields.