Perpendicular standing spin waves in films with interfacial Dzyaloshinskii-Moriya interaction

The interfacial Dzyaloshinskii-Moriya interaction (DMI) is known to induce nonreciprocity in spin wave propagation within the plane of magnetic thin films. This nonreciprocity can be utilized to design spin wave devices such as signal isolators and also provides an important means to quantitatively measure the magnitude of the DMI. Analytical expressions have been developed to estimate the effects of DMI on spin waves by assuming the magnetization dynamics are uniform throughout a film thickness. Here, we relax this assumption and use a so-called atomic layer method to calculate spin wave dispersion relations for not just the uniform thickness mode, but also for the perpendicular standing spin waves (PSSWs). Coupled magnetic equations of motion are written for each atomic plane within a thin film made of one or more materials. The method is tested against known analytic and numerical theories, and is then used to calculate the PSSW frequencies and mode profiles for single and bilayer magnetic films with interfacial DMI applied to the outer interfaces. These calculations show that the PSSW mode frequencies depend nonlinearly on the interfacial DMI strength and on the inverse thicknesses of the magnetic films. We develop an analytic estimate for the lowest two PSSW frequencies to explain this nonlinear dependence. Our method has the advantage that it can be used to calculate spin wave frequencies and depth profiles in any thin-film magnetic stack with atomic-scale variations in magnetic parameters, such as multiple DMI interfaces, or gradient properties.