A Model for Estimating the Stress Induced During Oxidation of Sharp Silicon Structures
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INTRODUCTION In recent years there has been increased interest in nanoscale tips for use in VacuumMicroelectronic (VME) devices, Scanning Probe Microscopy (SPM) and Field Emission Guns (FEGs). One approach which has been used for the fabrication of these tips is to use silicon technology to produce Si tips through a series of etch steps. An extension of this technique which has been adopted to sharpen the tips further is based on oxidation of the silicon cone at relatively low temperatures (-900"C)[1] to produce a narrow silicon core inside a thick oxide coating. The oxide can then be removed using a selective etch to leave a sharpened silicon tip. It is thought that the stress concentration in the vicinity of the sharpest part of the tip plays a significant role in this process by inhibiting oxidation in these regions. High Resolution Electron Microscopy (HREM) investigations of tips treated in this way have revealed a significant occurrence of gross defects within the silicon [2], and mechanisms for their generation have been suggested which involve deformation under high stresses. The purpose of this investigation is to attempt to model this oxidation process, and obtain estimates of the stresses involved. The complex geometry and the interrelationship of many of the properties forbid an analytical solution to the problem, so a program has been developed which simulates the progress of the oxidation over a series of discrete steps.
MODELS FOR THE OXIDATION OF SILICON The Deal and Grove theory for the kinetics of the oxidation of silicon [3] produces a master equation for the growth rate dxox/dt which can be written as: dt
dXox
1 + xox
kL
(1)
kp
where Xox is the oxide thickness and kL and kp are the linear and parabolic rate constants respectively. This theory assumes steady state conditions and that oxidation occurs by the diffusion of oxidant through the oxide to react at the silicon/silicon oxide interface. Thus the linear rate constant relates to the reaction rate at the interface and the parabolic one relates to the diffusion of oxidant through the oxide. These constants can be expressed in terms of the volume change associated with oxidation by one mole of oxidant (v), the concentration of oxidant at the oxide surface (C), the diffusivity of the oxidant through the oxide (D) and the interfacial kinetic constant (kint) (which is a function of t). kp = 2vCD kL = vCkint(o)exp(-Eask')kT
(2) (3)
This model works well except for thin oxide layers and at low temperatures (
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