Pre Print 1 Micromagnetic and plane wave analysis of an antidot magnonic crystal with a ring defect G. Venkat, N. Kumar and A. Prabhakar. Abstract—We simulate spin wave (SW) dynamics in a thin film antidot magnonic crystal (MC) with a ring defect. An external magnetic field is applied perpendicular to the film plane, in a forward volume configuration. We initially use the plane wave method to obtain the SW band diagram for the antidot MC structure (without the defect) which gives us an idea of modes which are trapped in the crystal. We then use finite element micromagnetic simulations to study the SW propagation in the MC with the ring defect. The power spectral density of the magnetization at the top of the ring shows peaks that fall within the band gap of the MC. The SW energy in other frequency components are dissipated in the MC and do not reach the other end of the ring. Index Terms—Magnonic crystals, micromagnetic simulations, plane wave method I. I NTRODUCTION The field of magnonics is an upcoming and exciting area be- cause it is possible to construct novel devices while understanding various physical phenomena. In this context, spin waves (SWs) have been studied in various kinds of structures and materials [1], [2]. Magnonic crystals (MCs) have attracted a lot of interest because of their abilities to tune the SW band gap [3]–[9]. Defects in such structures have also been studied and the SWs propagating in these defects have showed an increase in group velocity [10], [11]. Different methods ranging from analytic (such as the plane wave method), computational (using micromagnetic packages) to experimental means have been used to study these structures. We consider a ring shaped defect in an antidot MC and study the SW propagation therein. Ferromagnetic rings have been studied for a fairly long time [12], [13] and have been shown to exhibit SW interference [14], SW coherence and decoherence [15] amongst other effects. An alternate to a ring is a MC with a ring defect, which is the subject of our current investigation. We use a finite element micromagnetic package Nmag [16] as well as the plane wave method (PWM) for our analyses. We expect that the ring defect will preferentially support spin wave frequencies that lie within the band gap of the MC. We use the PWM to identify these frequencies and corroborate our simulation results against the predictions of the PWM. II. THE PLANE WAVE METHOD A. The geometry Consider a periodic antidot lattice on a film of permalloy (Ni 80 Fe 20 ). A static field H 0 , along the z axis is applied (through out) the device so that M z is saturated. Three material parameters: saturation magnetization, exchange length and damping constant are assumed to be homogeneous along z and are varied in the (x, y) plane. G. Venkat, N. Kumar and A. Prabhakar are with the Dept. of Electrical Engineering, Indian Institute of Technology – Madras, Chennai 600036, India (email: ee11d037@ee.iitm.ac.in) Figure 1. Magnonic crystal of a square lattice formed by antidots of diameter 2R = 100 nm included in a permalloy sheet. The thickness of the magnonic crystal is d = 20 nm. The lattice constant is a = 150 nm. B. Plane wave method We primarily use the PWM detailed in [17] for our analysis. The Landau-Lifshitz-Gilbert equation of motion is ∂ M(r, t) ∂t = −γμ 0  M(r, t) × H eff (r,t)  + α(r) M s M (r, t) ×  ∂ M(r, t) ∂t  (1) where γ is the gyromagnetic ratio, α is the damping constant, and M is the magnetization vector. We consider the effective field H eff to be composed of a constant external magnetic field H 0 ˆ z, the exchange field H ex and the magnetostatic field H ms . The exchange field and magnetostatic field are space and time dependent terms. The exchange field has the form H ex (r,t)= ( ∇.λ 2 ex ∇ ) M (r,t) (2) where λ ex =  2A (r) μ 0 M 2 s (r) is the exchange length, and r =[x, y] is the position vector. Thus the total field is expressed as H eff (r,t)= H 0 ˆ z + h (r,t)+ ( ∇.λ 2 ex ∇ ) M (r,t) , (3) This is the forward volume SW geometry, and we assume that the total magnetization vector is given by M (r,t)= M S ˆ z + m (r,t) , (4) where M S is the saturation magnetization, and m (r,t) is the dynamic component which lies in (x, y) plane. We also assume time harmonic variations m (r,t)= m (r) e iωt , h (r,t)= h (r) e iωt , (5) In the antidot magnonic crystal presented here, the material parameters, M s , α and λ 2 ex are periodic functions of the in plane position vector r =[x, y] with their periods equal to the lattice constant a. We use Bloch’s theorem, which states that the