Small Surface Height Research Articles

A comparison of perturbation theory and the small-slope approximation for acoustic scattering from a rough interface for a Biot medium

In this paper, the lowest order small-slope approximation (SSA) scattering cross section for Biot theory is derived. Numerical results are obtained for both backscattering and bistatic scattering using a modified power law spectrum, and these results are compared with those of lowest order perturbation theory (PT). Frequencies ranging from 100 Hz to 3 kHz are used for surfaces with RMS heights h of 0.1 and 1 m and a correlation length l of 10 m. The angle of incidence for the bistatic results is limited to 45/spl deg/. It is found that for the smaller surface height roughness (h = 0.1 m), the SSA and PT give the same results for frequencies up to almost 1 kHz for both backscattering and bistatic scattering. For h = 1 m, the SSA and PT backscatter results are in good agreement at all frequencies for incident grazing angles up to approximately 45/spl deg/. For the bistatic results, the SSA and PT results agree only at low grazing angles of scatter. In the specular region, the results differ significantly.

Modeling and Mapping of Nanotopography Interactions with CMP

AbstractAs the demand for planarity increases with advanced IC technologies, nanotopography has arisen as an important concern in shallow trench isolation (STI) chemical mechanical polishing (CMP) processes. Previous work has shown that nanotopography, or small surface height variations on raw wafers 20 to 50 nm in amplitude extending across millimeter scale lateral distances, can result in substantial CMP-induced localized thinning of surface films such as oxides or nitrides used in STI [1]. This interaction with CMP depends both on characteristics of the wafer such as heights and spatial wavelengths of the nanotopography, and characteristics of the CMP process including the planarization length or pad stiffness.In this paper we review and extend the previous work on modeling of nanotopography. Three approaches to predicting the post-CMP oxide thinning due to nanotopography are compared. The first approach is the simplest, where a statistical aggregate effect is computed. Following the work of Schmolke [2], a transfer coefficient α is found which captures the portion of the nanotopography that is correlated with the final oxide thinning. The second approach is the most detailed, depending on explicit numerical simulation of pad elastic properties. In this case, a contact wear simulation is used to produce a detailed map of oxide thickness corresponding to any given pre-measured nanotopography wafer surface. The third approach is a signal processing method, sitting somewhere between the previous two extremes in terms of approximation and complexity. In this last case, a two-dimensional transfer function is extracted which captures the spatial smoothing accomplished by CMP. This filter can then be applied efficiently to premeasured nanotopography maps for other wafers to predict the final oxide thicknesses.We also propose a predictive mapping of post-CMP oxide or nitride thicknesses to provide insight into the relative goodness of a wafer measured for nanotopography which is to be subjected to a CMP process. Specifically, we suggest that for post-CMP impact, maps and computation of areas having insufficient oxide clearing, or having final nitride thickness outside of required ranges, are useful and practical. Such device failure potential maps complement the fundamental nanotopography height map data and metrics based directly on that data, and enable evaluation, comparison, and development of improved wafers and STI CMP processes.

An investigation into the applicability of perturbation techniques to solve the boundary integral equations for a parallel-plate capacitor with a rough electrode

We derive a system of integral equations for the surface charge distribution on the electrodes of a parallel-plate capacitor for which the profile of one electrode is rough. We show that the solution to this system of equations is tractable using a perturbation technique assuming small surface heights compared with the mean plate separation. The accuracies of the first- and second-order perturbative approximations and of the local height approximation are evaluated for a few examples with a one-dimensional rough surface by comparing them with exact numerical results obtained by solving the boundary integral equations directly using a numerical procedure. Some general guidelines for when the first- and second-order approximations will be accurate are given. It is shown that the perturbative formulation provides approximations with a regime of validity that may extend over a larger region than the regime of validity of the local height approximation. These results could be useful for capacitive-sensor design purposes or in modelling solid-state electronic devices.

Comparison of surface polishing techniques used for InP wafers

InP wafers prepared using different polishing and mounting methods were compared by Makyoh topography, which is capable of detecting extremely small surface height variations. In some wafers a cellular surface ridge structure due to uneven mounting wax distribution was observed. It is shown that the ridge structure can be completely eliminated by the optimisation of the wafer mounting procedure or by the use of a “free floating” double sided polishing technique.

An examination of the “full‐wave” method for rough surface scattering in the case of small roughness

We examine the accuracy of Bahar's “full‐wave” method for rough surface scattering in the case of small roughness. The specific problem examined is scattering from one‐dimensional rough surfaces assuming the Dirichlet boundary condition. This corresponds to scattering of horizontally polarized electromagnetic waves from perfectly conducting one‐dimensional surfaces. By one dimensional surfaces we mean surfaces with constant height in the direction perpendicular to the incident wave direction. Our main result is that full‐wave predictions for the normalized bistatic radar cross section are found to disagree substantially with first‐order perturbation theory when the latter is known to be accurate through comparison with exact integral equation studies. This disagrees with Bahar's conclusions. The full‐wave predictions for the normalized radar cross section were computed using a Monte Carlo method and also by averaging formally; the two methods show excellent agreement. For the examples discussed in the paper, the full‐wave prediction agrees closely with the Kirchhoff approximation result, though the Kirchhoff approximation is not accurate away from the specular direction for these cases. In fact, for the special case of backscattering with shadowing neglected, the full‐wave predictions are shown to agree exactly with the Kirchhoff approximation results, independent of surface roughness properties. We examine Bahar's analytic argument for the reduction of full‐wave theory to perturbation theory and show that it is based on an inconsistent expansion in small surface height. When the small height expansion is done consistently, the full‐wave prediction does not reduce to first‐order perturbation theory. Finally, we consider the computational procedure given by Bahar to reduce the numerical effort in evaluating full‐wave theory. We find that numerical results obtained via this procedure approximately coincide with numerical results from first‐order perturbation theory for sufficiently small surface heights and slopes. This may at first seem confusing in light of our other findings. However, we show that certain simplifying assumptions in Bahar's computational procedure change the numerical results significantly from those computed from full‐wave theory without such assumptions. Thus these assumptions are not acceptable approximations for full‐wave theory, and results based on them should not be considered valid results of full‐wave theory.

A new approach to the analysis of rough surface scattering

A novel approach to the problem of scattering by a randomly rough surface is developed based on combining normalization and the method of smoothing. Previous uses of smoothing have been forced to assume small surface heights in order to guarantee the dominance of the scattered field by its average or specular part. The term that causes the average scattered field to be so sensitive to the height is identified, and is normalized out of the basic integral equation. Smoothing is then applied to the normalized integral equation with a subsequent reintroduction of the normalizing factor after smoothing. The results are shown to extend beyond existing approximations, and they contain none of the divergent integral behavior that has recently been observed with a pure second order iterative approximation. An examination of the conditions under which the technique provides very accurate results shows that it is essentially an extension of the so-called composite surface scattering model. >

A scattering result for rough surfaces having small height but arbitrary slope

An integral equation for the fluctuating part of the field scattered by a perfectly conducting randomly rough surface is derived. The Born term in this equation contains the Kirchhoff approximation and a new factor. This Born term is valid as long as the fluctuating field is small compared to the average scattered field. Translation of this field constraint to conditions on the rough surface shows that the new Born term will be accurate as long as the surface has a small surface height irrespective of the surface slopes, curvatures, etc. In the limit of small surface slopes, this result is shown to reduce to the classical Rice surface perturbation result. Applications of this new result are discussed.

Numerical evaluation of the statistics of acoustic scattering from a rough surface

In this article, a formulation is given for two-dimensional scattering of a plane wave incident on an infinitely long surface with a Gaussian distribution of surface height. The surface height may have any physically reasonable correlation function. This method generalizes previous work by the present authors, in which a finite length surface was treated numerically by Fourier methods. For each numerically generated random surface, an integral equation must be solved to find the pressure gradient on the surface. The reflection coefficients, describing the angular distribution of scattered energy, are then formed. It is shown how the various moments of the scattered pressure may be obtained efficiently from averages of these reflection coefficients. Using these techniques, numerical results are presented for the angular distribution of energy for a surface with a Gaussian correlation function of surface height and for a correlation function modeling a more realistic sea surface. Results are presented showing that it is not necessary to taper the incoming wave field in order to obtain accurate answers. For large values of the ratio of the standard deviation of height to the surface correlation length, it is found that the mode of scattering is very different to that found at small surface heights. For some cases, the maximum in the angular distribution is found in the backscatter direction, with the specular peak not at all apparent. It appears that this phenomenon is due to multiple scattering of the acoustic wave at the surface. Finally, it is shown that for the class of problems treated here, the ensemble average intensity is constant at all normal distances from the surface, while the normalized covariance of intensity varies with normal distance from the surface in a way that is, in some limiting cases, very similar to the scattering of a wave passing through a phase modulating screen or an extended random medium.

The numerical approach to understanding rough surface scattering

Numerical methods for solving simplified rough surface scattering problems “exactly” provide new insights into the rough surface scattering problem. Exact solutions for 1-D surfaces with the pressure release (Dirichlet) boundary condition are found with integral equation and Monte Carlo techniques. These studies provide a better understanding of the validity of the two basic approximations used in rough surface scattering: the Kirchhoff approximation and first-order perturbation theory. From this work, a resolution can be given for the difference in the Kirchhoff and first-order perturbation theory predictions at small surface heights and at very small surface slopes, where both approximations have generally been considered valid. Studies with the Kirchhoff approximation show that shadowing corrections based on geometric optics lead to inaccurate predictions. This work indicates the need for tractable multiple scattering theory for low grazing angle applications. Comparisons with exact results show that “phase perturbation” theory predictions account for a significant part of the multiple scattering associated with shadowing. In addition to better understanding theory with numerical techniques, the accuracy of approximate numerical methods can also be examined. Attempts are now being made to include rough surface scattering in marching solutions of the parabolic wave equation (PE). Methods of verifying the accuracy of marching solutions using exact and PE integral equations will be briefly discussed.

The Kirchhoff approximation and first-order perturbation theory for rough surface scattering

The conventional Kirchhoff approximation for rough surface scattering does not reduce to perturbation results in the limit of small surface heights. It is shown here how to modify the conventional Kirchhoff approximation by using appropriate half-space Green’s functions. The resulting expressions for the differential cross section for scattering from rough surfaces reduce to the conventional Kirchhoff results and the perturbation results in the appropriate limits.

Scattering from statistically rough surfaces: A 3-D numerical study of the Kirchhoff approximation

The angular spread of acoustic radiation scattered from a rough surface is computed using the Kirchhoff approximation. Both the dependence on azimuth and elevation of the scattered intensity are plotted for model surface statistics. Results are compared to those of perturbation theory and to those of an extended Kirchhoff approximation which formally reduces to the perturbation results in the small surface height limit. In all cases studied, the perturbation result for the scattered intensity near the specular direction dominates the Kirchhoff result. Both hard and soft surfaces are considered and an example of non-Gaussian statistics is presented. The Kirchhoff integral is also compared to an approximation of that integral proposed by M. V. Berry [J. Phys. A 8, 566–584 (1975)].