Abstract
Creating micro/nanoscale systems with one dimension substantially larger than the others is relatively straightforward, and convenient for studying thermal energy transport. This paper reports the theory for a novel noncontact micro/nanoscale temperature measurement technique named Suspended ThermoReflectance (STR). Here, the heat diffusion equation is applied to a cantilever beam of micro and nanoscale dimensions with Neumann and Dirichlet Boundary Conditions. The Neumann condition takes on constant, decaying, and periodic forms leading to steady-state, transient and harmonic solutions, respectively. Though general solutions are presented for multiple length to width ratios (L/w) in terms of geometry as well as thermal properties, silicon is studied due to its continued technological importance. The analytical solutions are compared to a 3D Finite Element Model to determine at what L/w ratio the analytical solutions accurately represent a 3D microcantilever beam. Thermal conductivity is determined using the steady-state solution. The transient solution yields the overall thermal diffusivity of the system irrespective of any frequency change, while the harmonic solution provides the phase difference along the length of the cantilever beam leading to a thermal diffusivity and the frequency dependent heat capacity. The analytical model can be used to analyze the aforementioned thermal properties for STR measurement at 4% accuracy for the system with length to width ratio of 20 and at 1% accuracy for length to width ratio of 100.
Highlights
There is a constant thrust in the semiconductor industry to reduce device sizes to attain increased operational speed, reduced power consumption and to package devices into smaller envelopes
It is assumed that the heat conduction in the lateral direction is several orders of magnitude larger than the penetration depth into the substrate in order to attribute a 1D model to Transient Thermoreflectance (TTR) and Time Domain Thermoreflectance (TDTR).7–10
The cantilever is initially at the same temperature as the fixed end and is instantaneously heated at its tip. It exists in a transient state until enough time has passed, after which it is identical to the steady-state regime. Analysis of this transient regime leads to the thermal diffusivity of the system, which, in conjunction with the thermal conductivity from the steady-state results, leads to the heat capacity of the material
Summary
There is a constant thrust in the semiconductor industry to reduce device sizes to attain increased operational speed, reduced power consumption and to package devices into smaller envelopes. Analysis of this transient regime leads to the thermal diffusivity of the system, which, in conjunction with the thermal conductivity from the steady-state results, leads to the heat capacity of the material.
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