Heat Conduction to Photoresist on Top of Wafer during Post Exposure Bake Process: I. Numerical Approach

Kim, Do Wan; Lee, Jieun; Oh, Hye−Keun

doi:10.1143/jjap.47.8338

Cited by 3 publications

(2 citation statements)

References 31 publications

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“…The mathematical model (1) arises in various physical and engineering settings, in particular in hydrology [2], material sciences [22], heat transfer [16,24] and transport problems [25], etc. The inverse problem in determination of the source term has been studied intensively for decades (cf., e.g., [5,6,15,18,21]).…”

Section: Introductionmentioning

confidence: 99%

Numerical Methods for Reconstruction of the Source Term of Heat Equations From the Final Overdetermination

Deng¹,

Fang²,

Li³

2015

Bulletin of the Korean Mathematical Society

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Abstract. This paper deals with the numerical methods for the reconstruction of the source term in a linear parabolic equation from final overdetermination. We assume that the source term has the form f (x)h(t) and h(t) is given, which guarantees the uniqueness of the inverse problem of determining the source term f (x) from final overdetermination. We present the regularization methods for reconstruction of the source term in the whole real line and with Neumann boundary conditions. Moreover, we show the connection of the solutions between the problem with Neumann boundary conditions and the problem with no boundary conditions (on the whole real line) by using the extension method. Numerical experiments are done for the inverse problem with the boundary conditions.

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Section: Introductionmentioning

confidence: 99%

Numerical Methods for Reconstruction of the Source Term of Heat Equations From the Final Overdetermination

Deng¹,

Fang²,

Li³

2015

Bulletin of the Korean Mathematical Society

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“…3) We showed the numerical temperature change of PR surface on a hot plate heated wafer in our previous paper. 4) PR surface temperature changes were calculated. Temperature initially starts at a room temperature.…”

Section: Introductionmentioning

confidence: 99%

Heat Conduction to Photoresist on Top of Wafer during Post Exposure Bake Process: II. Application

Oh¹,

Kim

Lee³

2008

jjap

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Chemically amplified resists are used for 248 nm, 193 nm, immersion and extreme ultraviolet (UV) lithography. Among many process steps, post exposure bake (PEB) is the key process to make the desired small line width and critical dimension control. During PEB, the de-protection reaction and acid diffusion are determined by bake temperature and time. One of the key factors that determines the de-protection and acid diffusion is the initial temperature rising of the hot plate. The unpredictable temperature rising to the pre-set temperature is the main cause of line width variation. In order to predict the accurate PEB temperature and time dependency to the line width, the heat transfer from the hot plate to the resist on top of the silicon wafer is studied. Numerical approach is used to solve the heat conduction problem. Only the boundary temperature values are needed to solve this conduction, the information inside each layer is not required. We calculated the temperature rising characteristics of the photoresist on top of the several layers of the mask. The air conductivity, air gap, number of layers underneath the resist, thickness of the wafer, thickness of the layer including the resist, and different kind of layers are varied to see the characteristics of the bake temperature rising. We showed that there was small temperature difference at photoresist among the layer stack and thickness variation, even though it was very small. There is a strong possibility that this small PEB temperature difference would cause serious critical dimension (CD) control problem.

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Influence of deep reactive ion etching process parameters on etch selectivity and anisotropy in stacked silicon substrates for fabrication of comb-type MEMS capacitive accelerometer

Dutta,

Yadav,

Kumar

et al. 2023

J Mater Sci: Mater Electron

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Heat Conduction to Photoresist on Top of Wafer during Post Exposure Bake Process: I. Numerical Approach

Cited by 3 publications

References 31 publications

Numerical Methods for Reconstruction of the Source Term of Heat Equations From the Final Overdetermination

Numerical Methods for Reconstruction of the Source Term of Heat Equations From the Final Overdetermination

Heat Conduction to Photoresist on Top of Wafer during Post Exposure Bake Process: II. Application

Influence of deep reactive ion etching process parameters on etch selectivity and anisotropy in stacked silicon substrates for fabrication of comb-type MEMS capacitive accelerometer

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