In Chapter 1, we discussed the angular momenta and magnetic moments of individual atoms and ions. When these atoms or ions are constituents of a solid, it is important to take into consideration the ways in which the angular momenta on different sites interact with one another. For simplicity, we will restrict our attention to the case when the angular momentum on each site is entirely due to spin.The elementary excitations of coupled spin systems in solids are called spin waves. In this chapter, we will introduce the quantum theory of these excitations at low temperatures. The two primary interaction mechanisms for spins are magnetic dipole-dipole coupling and a mechanism of quantum mechanical origin referred to as the exchange interaction. The dipolar interactions are of importance when the spin wavelength is very long compared to the spacing between spins, and the exchange interaction dominates when the spacing between spins becomes significant on the scale of a wavelength. In this chapter, we focus on exchange-dominated spin waves, while dipolar spin waves are the primary topic of subsequent chapters.We begin this chapter with a quantum mechanical treatment of a single electron in a uniform field and follow it with the derivations of Zeeman energy and Larmor precession. We then consider one of the simplest exchangecoupled spin systems, molecular hydrogen. Exchange plays a crucial role in the existence of ordered spin systems. The ground state of H 2 is a two-electron exchange-coupled system in an embryonic antiferromagnetic state. It serves to illustrate the origins of the exchange interaction and also provides a vehicle for our discussion on the Heisenberg spin Hamiltonian in ferromagnetic solids. Charged Particle in an Electromagnetic FieldWe obtained the time-independent Schrödinger equation (1.46) by quantizing the total energy of a particle moving in an electrostatic potential. Since we also want to understand the effects of a magnetic field, we now concern
Abstract-We propose a standard micromagnetic problem, of a nanostripe of permalloy. We study the magnetization dynamics and describe methods of extracting features from simulations. Spin wave dispersion curves, relating frequency and wave vector, are obtained for wave propagation in different directions relative to the axis of the waveguide and the external applied field. Simulation results using both finite element (Nmag) and finite difference (OOMMF) methods are compared against analytic results, for different ranges of the wave vector.
Micromagnetic simulations are used to investigate the effects of different absorbing boundary layers (ABLs) on spin waves (SWs) reflected from the edges of a magnetic nano-structure. We define the conditions that a suitable ABL must fulfill and compare the performance of abrupt, linear, polynomial and tan hyperbolic damping profiles in the ABL. We first consider normal incidence in a permalloy stripe and propose a transmission line model to quantify reflections and calculate the loss introduced into the stripe due to the ABL. We find that a parabolic damping profile absorbs the SW energy efficiently and has a low reflection coefficient, thus performing much better than the commonly used abrupt damping profile. We then investigated SWs that are obliquely incident at 26.6, 45 and 63.4 degrees on the edge of a yttrium-iron-garnet film. The parabolic damping profile again performs efficiently by showing a high SW energy transfer to the ABL and a low reflected SW amplitude.Comment: Journal of Magnetism and Magnetic Materials, 201
Organoids are biological systems grown in vitro and are observed to self-organize into 3D cellular tissues of specific organs. Brain organoids have emerged as valuable models for the study of human brain development in health and disease. Researchers are now in need of improved culturing and imaging tools to capture the in vitro dynamics of development processes in the brain. Here, we describe the design of a microfluidic chip and bioreactor, to enable in situ tracking and imaging of brain organoids on-chip. The low-cost 3D printed microfluidic bioreactor supports organoid growth and provides an optimal imaging chamber for live-organoid imaging, with drug delivery support. This fully isolated design of a live-cell imaging and culturing platform enables long-term live-imaging of the intact live brain organoids as it grows. We can thus analyze their self-organization in a controlled environment with high temporal and spatial resolution.
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