# Presentation: 484527

## predicted

- surface
- height

- lateral position

- Ion sputtering from Focused Ion Beams is used for micromachining tall, steep features.  A new regime of ion sputtering has been discovered for sufficiently steep slopes.  **High slopes propagate over large distances without dissipating the steepest features.**
**	**A feature, e.g., a metal line can be fabricated at a large length scale and, through uniform ion irradiation, **reduced to a smaller length scale while preserving, or even sharpening, the sharpest features.**
- H.H. Chen, O.A. Urquidez, S. Ichim, L.H. Rodriguez, M.P. Brenner, and M.J. Aziz, *Science*, in press (2005).

**Notes:**
Methods of micro and nanofabrication of three-dimensional (3D) structures are of increasing interest due to their importance for the creation of compact 3D devices and their assembly into functional 3D systems.  Examples include micro-electromechanical systems, 3D integration of electronic microprocessors, 3D information storage, and photonic band gap waveguides. Through theory and experiment, we have discovered a new regime of ion beam sputtering that occurs for sufficiently steep slopes. High slopes propagate over large distances without dissipating the steepest features. Both the propagation velocity and the dynamically selected slope are universal, independent of the details of the initial shape of the surface.  The SEM image at the top left shows a pit with steep side walls that has been micromachined into silicon using a Focused Ion Beam, and to the right the same pit, with sides that have not smeared out, after uniform ion irradiation of the entire wafer.  The middle figure shows predictions of a new theory of sputter morphology evolution that is valid for arbitrarily large slopes when the curvature is small.  Under uniform ion irradiation the pit wall, initially at t=0, propagates laterally and, for this particular set of conditions, evolves to maintain a uniform slope that is steeper than the original slope.  (The theory also predicts that sufficiently shallow slopes dissipate ,which is the conventionally observed behavior.) The mathematical structure of the solutions is the same as that previously observed for shock fronts in thin-film fluid flows.  The experimental sequence at the bottom, obtained with a confocal optical profilometer, shows striking confirmation of the predictions of the theory.  The arrows are for illustrative purposes and do not signify a quantitative one-to-one correspondence between theoretical and experimental profiles.  An important implication of the transition from dissipative behavior to propagative behavior at high slopes is that a pattern can be fabricated at a large length scale and, with uniform ion irradiation, reduced to a smaller length scale while preserving, or even sharpening, the steepest features. 

## Nonlinear Theory of Sputtering

**Our treatment:**small curvature, treat other terms exactly

- Rewrite in terms of slope b=dh/dx

- Erosion Rate
- (Sigmund
- sputter integral)

- Previous treatments  (Bradley-Harper, Makeev *et al.*)
- Small-slope expansions of Sigmund sputter integral:
- valid for small deviations from average slope and curvature.

**“****The Advection Diffusion Eqn”**

**Notes:**
	Our theoretical results utilize the classical theory of sputtering (Sigmund, 1973).  This theory posits that the erosion velocity, v_n, at any point on a surface is proportional to the total energy transmitted to that point by ion bombardment induced collision cascades everywhere on the surface.  Sigmund modeled the collision cascades as Gaussian ellipsoids centered at the mean penetration depth, a, below the surface everywhere.  The resulting equation for the erosion velocity, called the Sigmund sputter integral, is an integral equation that is, in general, impossible to solve analytically.
	Previous analytic treatments (Bradley and Harper, 1988; Makeev, Cuerno, and Barabasi, 2002) have performed the Sigmund sputter integral by expanding the surface topography in powers of slope and curvature about the average values.  These perturbative approaches are valid for small variations in slope but cannot predict the general behavior of steep features containing large variations in slope.  
	To treat tall, steep features, we developed from the Sigmund sputter integral a small-curvature approximation that is valid for all slopes. Combining this with simultaneous surface smoothening mediated by surface diffusion results in a nonlinear partial differential equation called the advection-diffusion equation.  The solutions of this equation predict behavior that can be understood as the propagation of a shock front that self-selects a stable slope; the mathematical structure of the solutions is the same as that previously observed in thin-film fluid flows.  The shock front behavior turns out to be a generic feature of the non-monotonic behavior of the sputter yield (atoms removed per incident ion) vs. slope.  As the slope increases, the center of the energy deposition gets closer to the surface; hence the sputter yield increases with slope.  But at high enough slope, the sputter yield turns down again due to effects such as deflection at grazing incidence, and possibly to deviations of the true collision cascade shape from Gaussian ellipsoids.  This qualitative behavior of the sputter yield vs. slope, combined with the kinetics of surface diffusion, turns out to be sufficient for propagative rather than dissipative solutions to the morphology evolution equation.
	For definitions of variables in equations in lower right, see notes on next page.

## the Advection-Diffusion Equation

- advection

- curvature effect,
- small for our cases

- where

- traveling wave speed

- surface
- diffusion

**Notes:**
dot means partial derivative with respect to time; prime means partial derivative with respect to x.
S and S-tilde are the curvature coefficient of the sputter yield expressed as a function of angle (on the previous slide) and slope, respectively.
F_full and F-tilde_full are the sputter yield of a flat surface expressed as a function of angle (on the previous slide) and slope, respectively.
B_full is a materials parameter describing the smoothening by surface diffusion; it includes the surface tension and the surface diffusivity.

## Slope Selection

- predicted evolution of h(x)

- conditions for formation

- undercompressive shock:

*b**=3.89

**Notes:**
predictions of the theory:
left:  high initial slopes (A) and lower initial slopes (B) evolve to the same slope (C); red solid curve is steady-state solution.
right:  for slopes below 2.5, dissipative behavior - no shock front.
for slopes between 2.5 and 3.89, slope spontaneously steepens, approaching 3.89.
for slopes above 3.89, slope spontaneously decreases, approaching 3.89.

## Strange Matter Illustratedin Boston Museum of Science ExhibitionMichael J. Aziz, Harvard University, DMR-0306997 

- Strange Matter Illustratedin Boston Museum of Science ExhibitionMichael J. Aziz, Harvard University, DMR-0306997

- In a collaboration with the Boston Museum of Science on the traveling exhibit “Strange Matter” developed by the Materials Research Society, the P.I. and graduate students made interactive presentations to visitors of all ages.  Visitors explored the unique properties of everyday materials such as honey, cornstarch and silly putty and were introduced to unheard-of materials such as metallic glass.  The exhibit exposed tens of thousands of visitors to the wonderment of materials science and gave them an enhanced appreciation of the value of materials in society.