Finite Element Analysis of Nanoscale Thermal Measurements of Superlattices
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Finite Element Analysis of Nanoscale Thermal Measurements of Superlattices Jason R. Foley and C. Thomas Avedisian Thermal Sciences Laboratory Sibley School of Mechanical and Aerospace Engineering Cornell University Ithaca, New York, 14853 ABSTRACT A finite element analysis applicable to two- and three-dimensional heat flow in samples of arbitrary geometry and composition is presented for use in a thermal wave experiment. The finite element formulation is summarized, including the use of symmetry to simplify the problem, and the governing differential equations for the heat transport are found to be in the form of the Helmholtz equation for the specific case of a modulated heat source. Simulated data for a Nb/Si superlattice is calculated using the finite element code and is shown to agree with predictions from an analytical model, validating the approach taken. INTRODUCTION Finite element analysis (FEA) is a powerful tool for approximating solutions to problems for which an analytic solution is either intractable or nonexistent. These problems can include calculating fields from non-uniform sources, fluid flow over an irregularly shaped surface, or the propagation of waves in an inhomogeneous medium. One such field equation is the Helmholtz equation, which has the general form ∂ ∂φ ∂ ∂φ ∂ ∂φ ky + kx + kz + λφ + f ( x, y, z ) = 0 , ∂x ∂x ∂y ∂y ∂z ∂z
(1)
where λ is a constant, φ is the field that is to be calculated and f represents a source (or body force) term. The Helmholtz equation describes many physical systems, including waves in shallow water, electromagnetic waves in a waveguide, or acoustic vibrations [1]. One specific application where the Helmholtz equation arises naturally is in thermal diffusion wave thermometry [2], for which the theoretical basis can be traced to the pioneering work of Anders J. Ångström [3]. Thermal diffusion waves, which are generated through harmonic heating, have several interesting characteristics (e.g., an infinite propagation speed). However, the most useful property is the characteristic thermal penetration depth lT [4], which defines the physical distance over which the thermal oscillations are significant: lT =
2k , ωC
(2)
where k is the thermal conductivity, C is the volumetric heat capacity, and ω is the angular frequency of the modulated heat source. This feature allows the thermal field to be “tuned” to a desired size simply by changing the modulation frequency of the heat source and experimentally
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results in a variable probe that is sensitive to spatial variations in the material properties (which must be determined through inverse techniques). Implementing FEA allows much greater freedom in selection of test geometry, etc., and represents an enabling tool that allows non-standard geometries to be thermally characterized. For example, specific properties (e.g., components of anisotropic thermal conductivity) can be measured by nanofabricating structures so that the heat flow has the desired characteristics, such a
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