Stochastics and the Phenomenon of Line-Edge Roughness Chris Mack February 27, 2017 Tutorial talk at the SPIE Advanced Lithography Symposium, San Jose, California
What s so Hard about Roughness? Roughness is Hard to Measure SEM images have systematic and random errors The statistics of roughness is tricky Roughness is Hard to Understand Think with a random, correlated mindset Some physics is not well understood Roughness is Hard to Reduce Is there a better resist? Does smoothing work? What are the ultimate limits? 1
Randomness in Lithography Photon count PAG positions Absorption/acid generation Polymer chain length Blocking position Reaction-diffusion Dissolution Photon Absorption Ionization e - e - e - PAG Acid 2
The Importance of Correlations White noise: uncorrelated, each random event is independent Photon shot noise, absorption, chemical concentration, acid generation Produces a flat power spectral density (PSD) Correlating mechanisms: random events that are not independent Secondary electron generation, reaction-diffusion, development front propagation Lowers (smooths) the PSD on length scales below the correlation length (i.e., high frequency roughness) 3
What Gives the PSD its Shape? 1000 100 Uncorrelated white noise PSD (nm 3 ) 10 1 Correlation Length Acid diffusion 0.1 0.0001 0.001 0.01 0.1 1 Frequency (1/nm) 4
The Power Spectral Density PSD(0) Correlation Length x Slope roughness exponent H Variance = area under the curve (Derived from other three parameters) 5
Frequency of Roughness Are these edges different? 6
Frequency of Roughness Knowing the roughness standard deviation is not enough x = 10 Dx H = 0.5 x = 10 Dx H = 1.0 x = 100 Dx H = 0.5 The 3s roughness is the same for all of these x = 0.1 Dx H = 0.5 L = 512 Dx, s = fixed 7
Finite-Length Features Within-feature roughness s LWR L s LWR s CDU L LCDU: Feature-to-feature variation of mean CD L 8
Conservation of Roughness For all features of the same CD and pitch, for any length L, s 2 CDU L s 2 L s 2 LWR LWR Different line lengths partition the total roughness into within-feature and feature-to-feature variation PSD(0) s CDU 1 L x L 2 L 2 PSD(0) / 2 H s 1 x LWR 9
Conservation of Roughness We need to measure s( ), PSD(0), and x to understand roughness for device features 10
Measuring Roughness is Hard We need to determine the PSD parameters to understand how roughness impacts device features Measuring noise tends to be noisy SEM images contain both random and systematic errors that bias our results Random noise in the image produces white noise Systematic field variations (intensity, distortion) increase the apparent low-frequency roughness 11
SEM Images are Noisy Can you pick out the edges from this linescan? 12
SEM Images are Noisy Average Linescan = average of column of pixels 13
To Achieve Robust Edge Detection In general, we must apply averaging (e.g., a Gaussian filter) in X and sometimes Y to make our edge detection robust No Filter 7X2 Gaussian Filter Threshold Edge Detection Problem: Averaging to reduce SEM noise also smoothes away the roughness we are trying to see (the feature roughness) 14
Filtering Changes the Measured PSD No Filter 7X2 Gaussian Filter 15
A Better Way The Analytical Linescan Model (ALM) is a physics-based prediction of a linescan given a wafer feature Run in reverse, the ALM can be fit to an experimental linescan to estimate the edge positions We can achieve robust noise rejection and edge detection without any filtering We still must remove SEM errors after edge detection (random and systematic) Thursday, 2:00pm: Chris Mack & Ben Bunday, Analytical linescan model for SEM metrology 16
Removing SEM Errors SEM Random Image Noise Caused by electron shot noise (white noise) Resist shrinkage limits allowable electron dose Noise interacts with linescan edge slope to produce edge uncertainty Right and left edges have different linescan slopes due to scan effects 17
Before and After Noise Subtraction 18
Before and After Noise Subtraction 19
SEM Field Distortion Even sub-nanometer amounts of SEM field distortion can cause significant changes in the PSD Define distortion based on max error in the corner Trapezoid Distortion Pincushion Distortion 20
Result: Increase in Low-Frequency LER and PPR Wednesday, 8:20am Barton Lane, et al., Global minimization line-edge roughness analysis of top down SEM images 21
Background Intensity Variation Result: increase in low-frequency LWR, LER, and PPR 22
Randomness in Lithography Photon count PAG positions Absorption/acid generation Polymer chain length Blocking position Reaction-diffusion Dissolution Photon Absorption Ionization e - e - e - PAG Acid 23
What is the EUV Image? Here is a typical aerial image from an EUV scanner or is it? 18nm HP 24
What is the EUV Image? 25
How to Reduce Roughness Increase Photon Efficiency Reduce Resist Information Loss Magic 26
Increase Photon Efficiency We maximize the number of photons absorbed at the bottom of the resist when (reasonable goal: ) Due to pattern collapse, D max ~ Pitch min New resist scaling law: 27
Reduce Resist Information Loss You can t add information to the wafer from a bottle of photoresist (DSA aside) Resist can throw information away and add noise Preserve information from the absorbed image: High resist contrast, low resist blur (correlation length) Add very little resist noise: High concentrations (non-random positions) Large integration volume (correlation length) The optimum blur/integration/correlation length scales with feature size (called RLS trade-off) 28
Using Magic Magic resists are those that don t obey the laws of statistics The Applicable Laws of Statistics Increasing the number of independent events N reduces the relative noise as 1/ Increasing the number of correlated events does nothing No subsequent process step can ever decrease PSD(0) that existed from the previous step Two examples: Post-process smoothing and EUV resist acid amplifiers or quantum yield 29
The Fundamental Smoothing Constraint The zero-frequency PSD cannot be lowered by post-processing (including etch) PSD(0) = constant Why? This frequency component represents uncertainty in the mean CD of the feature To lower PSD(0), the smoothing process must increase the mean CD of too-narrow lines, and decrease the mean CD of too-wide lines, in order to reduce the variation of linewidth 30
What Can Smoothing Do? Post-Processing (including etch) should be characterized as changes in PSD model parameters (PSD(0), x, H) Increasing the correlation length is very effective at reducing within-feature variation Increasing the roughness exponent also works Since LCDU (feature-to-feature variation) is only a function of PSD(0), smoothing can t help Lowering PSD(0) is magic 31
Improving EUV Resists with Magic One proposal to reduce roughness without increasing exposure dose is to increase the quantum yield (number of acids per absorbed photon) above 1 But this does not work: all the acids generated from one absorbed photon are correlated! Two correlated acids behave statistically like one You can t amplify your way out of a signal to noise problem The optimum quantum yield is 1 32
Conclusions We can t improve roughness without first understanding stochastics better Be mindful of correlated statistics Build first-principle models We can t understand roughness better without first measuring it better It s not just about 3s, it s about PSD(0) and correlation length Roughness is hard, so let s work together 33
Thanks to Eric Panning and Ken Goldberg for taking a chance on me John Biafore for amazingly fruitful discussions My many coauthors at this year s Symposium 34
Thank You Fractilia, LLC Austin, Texas 512 887-3646 info@fractilia.com www.fractilia.com