^{1}, M. P. Stoykovich

^{1}, H. B. Cao

^{1,a)}, J. J. de Pablo

^{1}, P. F. Nealey

^{1,b)}and W. J. Drugan

^{2}

### Abstract

A model was developed for predicting the collapse behavior of photoresiststructures due to the drying of rinse liquids during wet chemical processing. The magnitude of the capillary forces was estimated using the classical thermodynamics of surface tension, and the deformation of the structure was modeled using beam bending mechanics that accounts for both elastic and plastic modes of deformation. The two-dimensional model can predict the critical beam height of collapse as a function of the wetting behavior of the rinse liquid on the beam, the elastic and plasticmechanical properties of the polymericphotoresist, and the beam dimensions. Collapse behavior was predicted for polymer nanostructures with elastoplastic mechanical properties similar to those of bulk poly(methyl methacrylate). We have compared the collapse predictions from our model with the results of models that account only for elastic or plastic deformation behavior. Regimes in the elastic-plasticmechanical property space for which it is necessary to use the developed beam bending model have been highlighted. It is shown that in some cases the inclusion of both elastic and plasticmechanical properties is necessary for modeling the collapse behavior of polymer beams fabricated using the lithographic process.

This work was supported by the Semiconductor Research Corporation (Grant No. 2002-MJ-985), the NSF Nanoscale Interdisciplinary Research Team program (Grant No. CTS-0210588), Sematech, and Intel Corporation. M.P.S. would like to thank the Graduate Fellowship Program of the Semiconductor Research Corporation.

I. INTRODUCTION

II. TWO-DIMENSIONAL MODELS OF ELASTIC AND PLASTIC BEAM BENDING

III. TWO-DIMENSIONAL ELASTOPLASTIC BEAM BENDING MODEL

IV. RESULTS AND DISCUSSION

V. CONCLUSIONS

## Figures

Cross-sectional scanning electron micrograph of collapsed Apex E photoresist structures. The pattern consists of dense lines wide with a period of .

Cross-sectional scanning electron micrograph of collapsed Apex E photoresist structures. The pattern consists of dense lines wide with a period of .

Geometry examined by the two-dimensional beam bending models. In each model the rinse liquid is trapped between two parallel polymer beams, while the outside pools are assumed to be very large and do not contain any rinse liquid. (a) A three-dimensional view of the polymer beam geometry prior to deformation indicates how the rinse liquid meniscus can be modeled as a portion of a cylindrical surface. The beams are initially separated by a distance and each beam has a width , a height , and a length . The initial capillary line force acting on the polymer beams is a function of the meniscus’ radius of curvature , the receding contact angle , and the rinse liquid surface tension . (b) The two-dimensional cross-section view of the beams during deformation demonstrates that the meniscus’ radius of curvature is dependent upon the maximum deflection and the angle of deflection of the beams.

Geometry examined by the two-dimensional beam bending models. In each model the rinse liquid is trapped between two parallel polymer beams, while the outside pools are assumed to be very large and do not contain any rinse liquid. (a) A three-dimensional view of the polymer beam geometry prior to deformation indicates how the rinse liquid meniscus can be modeled as a portion of a cylindrical surface. The beams are initially separated by a distance and each beam has a width , a height , and a length . The initial capillary line force acting on the polymer beams is a function of the meniscus’ radius of curvature , the receding contact angle , and the rinse liquid surface tension . (b) The two-dimensional cross-section view of the beams during deformation demonstrates that the meniscus’ radius of curvature is dependent upon the maximum deflection and the angle of deflection of the beams.

The simplified stress-strain behavior of the elastoplastic model. The elastoplastic model reduces the complex stress-strain behavior observed for polymers to an elastic regime at low strains (region 1—solid line) characterized by the Young’s modulus and a perfectly plastic regime at larger strains (region 2—dashed line) characterized by a constant yield stress, .

The simplified stress-strain behavior of the elastoplastic model. The elastoplastic model reduces the complex stress-strain behavior observed for polymers to an elastic regime at low strains (region 1—solid line) characterized by the Young’s modulus and a perfectly plastic regime at larger strains (region 2—dashed line) characterized by a constant yield stress, .

The normal stress distributions at the base of the beam at different stages of the deformation process. The relative magnitudes of the stresses are represented by the lengths of the arrows. (a) The corners of the beam begin to undergo plastic deformation when the stress reaches at an applied line force of . (b) Greater portions of the beam will yield and undergo plastic deformation as the applied line force is further increased. (c) The entire beam will eventually undergo plastic deformation when the line force reaches . The shaded areas at the bottom of (b) and (c) represent the regions of the beam that are yielding plastically.

The normal stress distributions at the base of the beam at different stages of the deformation process. The relative magnitudes of the stresses are represented by the lengths of the arrows. (a) The corners of the beam begin to undergo plastic deformation when the stress reaches at an applied line force of . (b) Greater portions of the beam will yield and undergo plastic deformation as the applied line force is further increased. (c) The entire beam will eventually undergo plastic deformation when the line force reaches . The shaded areas at the bottom of (b) and (c) represent the regions of the beam that are yielding plastically.

Comparison of the beam bending models that consider elastic, plastic, and elastoplastic mechanical properties. Plots of (a) and (b) as a function of beam height have been calculated for the case of and . Also included are (c) and (d) versus for the case of and . Both sets of plots have been constructed using the parameters , , , and .

Comparison of the beam bending models that consider elastic, plastic, and elastoplastic mechanical properties. Plots of (a) and (b) as a function of beam height have been calculated for the case of and . Also included are (c) and (d) versus for the case of and . Both sets of plots have been constructed using the parameters , , , and .

Comparison between the critical heights of collapse predicted by the elastic, plastic, and elastoplastic models as a function of the beam separation . (a) When and the estimated by the elastic and elastoplastic models are nearly equivalent for the entire range of . (b) In the case of low yield stresses such as and the predictions of the elastic and elastoplastic models become significantly different. Both plots utilize , , and .

Comparison between the critical heights of collapse predicted by the elastic, plastic, and elastoplastic models as a function of the beam separation . (a) When and the estimated by the elastic and elastoplastic models are nearly equivalent for the entire range of . (b) In the case of low yield stresses such as and the predictions of the elastic and elastoplastic models become significantly different. Both plots utilize , , and .

Comparison between the predicted by the elastic, plastic, and elastoplastic models as a function of beam width . (a) The elastic and elastoplastic models estimate nearly identical deformation behavior when and , but in the case of (b) and a lower yield stress, , the model predictions deviate significantly. Both plots utilize , , and .

Comparison between the predicted by the elastic, plastic, and elastoplastic models as a function of beam width . (a) The elastic and elastoplastic models estimate nearly identical deformation behavior when and , but in the case of (b) and a lower yield stress, , the model predictions deviate significantly. Both plots utilize , , and .

Regimes in which the elastic and elastoplastic models predict similar collapse behavior. The elastoplastic beam bending model can be applied to materials having any combination of and . However, the elastic model predicts exactly the same critical height of collapse as the elastoplastic model only for materials having a large (shaded region denoted by ). The shaded region encloses the sets of and for which the critical height of collapse calculated by the elastic model differs by less than 5% from the predictions of the elastoplastic model. The regions and enclose the space in which the elastic model predicts the critical height to within 25% and 50%, respectively, of the elastoplastic result. In addition, typical literature values for the and of PMMA (+), PS (●), novolak (∎), and Si (×) are included. These data were compiled for the case of , , , and .

Regimes in which the elastic and elastoplastic models predict similar collapse behavior. The elastoplastic beam bending model can be applied to materials having any combination of and . However, the elastic model predicts exactly the same critical height of collapse as the elastoplastic model only for materials having a large (shaded region denoted by ). The shaded region encloses the sets of and for which the critical height of collapse calculated by the elastic model differs by less than 5% from the predictions of the elastoplastic model. The regions and enclose the space in which the elastic model predicts the critical height to within 25% and 50%, respectively, of the elastoplastic result. In addition, typical literature values for the and of PMMA (+), PS (●), novolak (∎), and Si (×) are included. These data were compiled for the case of , , , and .

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