Thermal Modeling of Optical Power Absorption in Moving Multilayer Thin Films

A technique for computing the thermal profile in a multilayer moving medium is described. This technique is particularly suitable for studying the near field optical/thermal interplay in hybrid optical/magnetic recording because the boundaries of the computation space are effectively removed from the optical source. It is shown that a three layer medium can be designed with a thermal time constant which is suitable for high recording data rates and that minimizes the thermal bloom from motion of the medium with respect to the optical spot. However, the thermal spot is much larger than the optical spot which leads to a reduced storage density. INTRODUCTION Numerical techniques for efficiently computing the temperature rise [1-3] in a multilayer thin film stack [4-6] due to a focused laser beam have been developed for a variety of applications such as recordable optical data storage. The focused optical spot size for current optical storage products ranges from ~0.88 μm for CD’s at a wavelength of 780 nm to 0.24 μm for Blu-Ray disks at a wavelength of 405 nm. When the optical spot is much larger than or comparable to the thickness of the thin film stack, the heat sink/reflector layer in the disk ensures that the dominant direction for heat flow is perpendicular to the thin films. As a result, the thermal boundary conditions in the lateral direction are not difficult to handle with simple approximations, such as a quadratic temperature dependence at the boundary [4]. Future optical data storage products may require near field optics to achieve larger storage densities. Magnetic hard discs may also incorporate a hybrid optical/magnetic technology called “heat assisted magnetic recording” (HAMR) using near field optics to transfer the optical energy into the recording medium in a highly localized spot smaller than the total thickness of the film stack [7,8]. In such a case, the lateral heat flow can become as important as the perpendicular heat flow and thermal boundary conditions must be carefully applied. The advantages of the alternate direction implicit (ADI) technique for thermal modeling of optical data storage media have been described by Mansuripur et al. [4,5] They developed the ADI equations for a cylindrical coordinate system and a circularly symmetric optical spot in both the stationary and moving frame of reference. The ADI equations were subsequently applied by Peng et al. [6] to a stationary 3D Cartesian coordinate system to investigate the amorphization and crystallization dynamics of optical phase change media. *Address correspondence to this author at the Seagate Technology, 1251 Waterfront Place, Pittsburgh, PA 15222, USA; Tel. 412-918-7197; Fax: 412-918-7010; E-mail: [email protected] Itagi [9] extended the ADI equations to include variable layer thicknesses, thereby enabling a substantial decrease in computation time for film stacks with thick layers. He also derived the ADI equations for the moving frame in a cylindrical coordinate system [10]. In this paper we present the ADI equations for a Cartesian coordinate system in the moving frame of reference and a method for accurately handling the lateral thermal boundary conditions in a multilayer film stack, and then apply these results to a HAMR recording medium. BASIC THEORY The Fourier heat conduction equation is C r, t ( ) t T r, t ( ) = k r, t ( ) T r, t ( ) + g r, t ( ) (1) where C is the heat capacity, k is the thermal conductivity tensor, g is the input power, t is the time, and T is the temperature. Eq. (1) can be implemented numerically by subdividing the region of interest into many smaller cells to approximate the spatial gradients, and stepping the time in discrete increments. In the explicit method, the temperature at a specific point in the cell space at time t = n+1 is completely determined by the temperatures within the cell space for the previous time step, t = n. C T n+1 T n ( ) t = k T n ( ) + g . (2) There is only one unknown variable, T(i,j,k), at each point in the cell space and time step. However, there is a constraint on the size of the time step to ensure numerical stability of the calculation. It is generally necessary to choose very small time steps, and as a result, the simple explicit technique is not suitable for many problems of practical interest. The simplest implicit method for solving the heat conduction equation rewrites Eq. (1) as C T n+1 T n ( ) t = k T n+1 ( ) + g . (3) 68 The Open Optics Journal, 2008, Volume 2 Challener and Itagi In this case the change in temperature between time step n+1 and time step n is expressed in terms of the temperatures at time step n+1. There are seven unknown variables in this equation, the temperatures at t = n+1 at point (i,j,k) and its six neighbors, making the solution of the set of equations in cell space much more complex and time consuming, although this technique has the advantage of unconditional stability. The Crank-Nicolson (CN) technique is an average of the implicit and explicit techniques. The CN equation is C T n+1 T n ( ) t = k T n+1 + T n ( ) 2 + g . (4) This technique is also unconditionally stable and the accuracy is second order in t, which is greater than either that of the simple implicit or simple explicit techniques. However, there are still seven unknown quantities in this equation at each point and time step making it computationally intensive to solve directly. Fig. (1). Cell space in the horizontal plane with extended boundaries. The central red region is the standard computational space with uniform cell sizes. This region is ringed by multiple additional regions with varying cell sizes so that the final cell space is orders of magnitude larger than the central region. Douglas and Gunn [7] developed an ADI algorithm for the CN equation that maintains its unconditional stability and numerical accuracy but greatly reduces the computation time. In their technique Eq. (5) at each time step is subdivided into two or more equations with fewer unknowns that can be solved sequentially to determine the change in temperature. The general approach for the stationary coordinate system is discussed in Ref. [3]. In this paper we develop the ADI equations for the moving frame of reference in the 3D Cartesian coordinate system. A problem with the numerical solution to the heat flow equation is handling the boundaries of the finite computation space appropriately. Ideally the boundaries of the computation space are placed far enough from the region of interest that the boundaries have no effect on the thermal calculation within the time interval of interest. In some cases, however, this approach requires unreasonably large computation spaces. In particular, we have found that for modeling heat flow in HAMR media due to optical energy delivered to the medium in a tightly confined spot in the near field, the effects of boundaries in a finite computation space must be carefully handled. In the next section we develop a method of extending the boundaries by variable cell sizes to completely remove boundary effects. EXTENDED BOUNDARIES For convenience, the vertical direction will be defined to be perpendicular to the plane of the films. To implement extended boundaries the ADI equations are first modified to handle a variable cell size in the lateral direction. In the region of interest in the center of the computation space where the optical power is delivered to the films, the cells are discretized laterally with a fine grid. Outside of this center region are rings of cells with successively larger lateral dimensions as shown in Fig. (1). Four sets of rings that are each five cells thick have been chosen for the results described in section 4, and in each ring the lateral dimension(s) are chosen to be ten times greater than that of the ring immediately inside it. For example, if the central cell space region is 500 by 500 cells with (20 nm) cell areas corresponding to a 10 10 μm cell space, the extended cell space has 540 540 cells covering a 2 2 mm area. Thus the cell space boundaries are too far from the central heat source to effect the calculation for any time interval that is practical to compute. The equations for the numerical calculation are obtained using the approach of Itagi [10]. Following the notation in that reference, a specific point in the cell space is considered with coordinates (i,j,k) as shown in Fig. (2). The points are located at the corners of the cells and the temperatures are defined at each point. The thermal conductivities are defined on the cell edges between points while the heat capacities are defined for each cell volume. Some cells in the computation space are surrounded by other cells of the same material and cell dimensions. The thermal conductivity, heat capacity, and cell dimensions associated with the points on the corners of these cells are then clearly defined. On the other hand, some points occur at the boundary between layers of different materials, and some points occur at the boundary between cells of different dimensions as shown in Fig. (2). To compute the temperature at these points with the ADI equations some additional definitions are helpful. Fig. (2). Illustration of parameter definitions at boundaries of nonuniform cells within a thin film layer. Thermal Modeling of Thin Films The Open Optics Journal, 2008, Volume 2 69