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A Full Wafer-Scale PAML Modeling Approach for Predicting CMP

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Abstract

Chemical mechanical polishing (CMP) is a complex, multi-scale problem with several order of physics. As it is a subtractive manufacturing process, it involves cutting tool–workpiece interaction. The cutting tool in CMP are nanoscale abrasives trapped in the contact between the workpiece (wafer) and a soft, rotating pad. These abrasives are suspended in a liquid medium and transported across the interface to provide even material removal. The current model presents an expansive wafer-scale framework that not only accounts for the solid–solid contact mechanics and wear, but also utilizes the mechanics of the slurry through fluid and particle dynamics. Results from this work include the temporal evolution of hydrodynamic fluid pressure, contact stress and material removal at the die and wafer scales. Comparisons with published CMP experiments have been made, and the results are favorable. Parametric studies have been conducted to predict the influence of different polishing parameters on the material removal rate. With this new framework, the entire wafer–pad interface can be studied under the influence of the four major physical interactions (contact mechanics, fluid mechanics, particle mechanics, wear). The result is a significantly faster multi-physical model that can simulate realistic CMP conditions without sacrificing accuracy.

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Abbreviations

\(\alpha\) :

Orientation of the wafer with X-axis

\(\beta\) :

Orientation of the wafer with Y-axis

\(\delta _0\) :

Z-separation between the wafer center and mean plane of the pad

\(\eta\) :

Slurry viscosity

\(\nu _{\rm pad}\) :

Poisson’s ratio of the foundation (pad)

\(\varOmega _{\rm p}\) :

Angular velocity of the pad

\(\varOmega _{\rm w}\) :

Angular velocity of the wafer

\(\sigma (x,y)\) :

Contact stress at position (x, y)

\(\sigma _d\) :

Standard deviation in the diameter of abrasives

\(\tau\) :

Initial height of the foundation

\(\theta\) :

Tangential coordinate measured from X-axis of the wafer

\(\Updelta\) :

Combined indentation of the particle into the pad and wafer

\(\Updelta _{\rm p}\) :

Indentation of the abrasive into the pad surface

\(\Updelta _{\rm w}\) :

Indentation of the abrasive into the wafer surface

\(a_{\rm w}\) :

Width of contact between an abrasive and the wafer surface

d :

Diameter of an abrasive particle

\(d_{\mathrm{avg-a}}\) :

Average diameter of active particle

\(d_{\rm avg}\) :

Mean diameter of abrasives

\(E_{\rm pad}\) :

Elastic modulus of the foundation (pad)

F :

Force on the abrasive while indenting on the wafer

\(F_z\) :

Net force on the wafer, along the Z-axis

h :

Fluid film thickness

\(H_{\rm w}\) :

Hardness of the wafer

\(M_x\) :

Net moment on the wafer, along the X-axis

\(M_y\) :

Net moment on the wafer, along the Y-axis

N :

Number of abrasive particles

p :

Hydrodynamic fluid pressure

r :

Radial coordinate with wafer center as the origin

\(r_{\rm wp}\) :

Separation between the axes of rotation of wafer and pad

u(x, y):

Z-deflection at position (x, y)

\(v_{\theta ({\mathrm{p}})}\) :

Velocity of the pad in the tangential direction

\(v_{\theta ({\mathrm{w}})}\) :

Velocity of the wafer in the tangential direction

\(v_{r({\mathrm{p}})}\) :

Velocity of the pad in the radial direction

\(v_{r({\mathrm{w}})}\) :

Velocity of the wafer in the radial direction

\(v_{\rm rel}\) :

Relative velocity between an abrasive and wafer surface

\(\hbox {Vol}_{\rm avg}\) :

Average material removed by an active particle

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Acknowledgments

This work was partially supported by the NSF CISE Award # \(\tilde{0}\)811770 and the ICES Philip and Marsha Dowd Fellowship.

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Authors and Affiliations

  1. Carnegie Mellon University, 5000 Forbes Ave, Scaife Hall, Pittsburgh, PA, 15213, United States

    Gagan Srivastava & C. Fred Higgs III

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  1. Gagan Srivastava
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  2. C. Fred Higgs III
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Correspondence to Gagan Srivastava.

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Srivastava, G., Higgs, C.F. A Full Wafer-Scale PAML Modeling Approach for Predicting CMP. Tribol Lett 59, 32 (2015). https://doi.org/10.1007/s11249-015-0553-y

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