PYRAMID: A Hierarchical Approach to E-beam Proximity Effect Correction

Transcription

1 PYRAMID: A Hierarchical Approach to E-beam Proximity Effect Correction Soo-Young Lee Auburn University leesooy@eng.auburn.edu

2 Presentation Proximity Effect PYRAMID Approach Exposure Estimation Correction Shape Modification Dose Modification Heterogeneous Substrates Recent Developments Grayscale Lithography Non-rectangular Features

3 Lithography Optical lithography Electron-beam lithography Direct-write Throughput Single or multiple beams

4 Terms Dose Energy given to a point (pixel) Exposure Energy deposited at a point (pixel) Critical point Point at which exposure is to be estimated

5 Proximity Effect - PSF Electron scattering Forward Backward PSF (Point Spread Function) Beam energy Substrate Blurring in the written pattern Proximity effect 50 KeV

6 Proximity Effect - Model f ( i, j) g( i, j) h( i, j) Space-invariant linear system g( i, j) = h( i, j) f ( i, j)

7 Proximity Effect Etec Pattern Minimum feature: 0.1 µm Circuit size: 50 50µ m 2 High density

8 Proximity Effect - Example A center region of the Etec pattern 500 nm PMMA on Si, 50 KeV Lift-off (Au)

9 Proximity Effect Correction f ( i, j) g( i, j) h( i, j) g( i, j) = h( i, j) f ( i, j) Modify f(i,j) modify shape or dose s.t. f (i,j) * h(i,j) = f(i,j)

10 History of PYRAMID PYRAMID approach: hierarchical and look-up table Shape modification Heterogeneous substrates Neural network Hierarchical pattern representation format Dose modification Grayscale lithography Distributed processing Non-rectangular features

11 Issues in Proximity Effect Correction Accuracy Speed

12 PYRAMID Approach Hierarchical approach to exposure estimation and correction Exposure estimation Local exposure and global exposure Correction Local correction and global correction Look-up tables Fast and accurate

13 Exposure Estimation Local exposure: exposure contributions from the features (regions) close to the critical point calculated by the exact and fast CDF (cumulative distribution function) table method time-consuming due to the large number of regions Global exposure: exposure contributions from the features located far from the critical point calculated by a coarse grain convolution where a pixel is a global exposure block

14 Exposure Estimation Local Exposure CDF (Cumulative Distribution Function) C ( k, l ) = k l i = 0 j = 0 P ( i 2 + j 2 ) l (k,l) where C(k,l) is the exposure contribution from a rectangle of size k x l to the critical point which is at the lower-left vertex of the rectangle and P(r) is the energy deposition profile. Critical point k ( 2 k 2, l ) k, l 2 ) ( 2 ( k1 1, l2 ) ( k 1, l1) - ( k, l ) ( k 1, l1 1 1) Exposure = C( k, l2 ) C( k1 1, l2 ) C( k2, l1 1) + C( k1 1, l1 2 1)

15 Exposure Estimation Local Exposure Complexity of computing exposure contribution from a rectangular feature of size K K The CDF table approach O(1) A conventional approach O( K 2 )

16 Exposure Estimation Global Exposure Coarse circuit image: block-wise circuit area distribution 2-D PSF: energy deposition profile sampled at the interval of block size

17 Correction Shape Modification Local correction Global correction IMR replacement Junction adjustment IMR adjustment Correction Hierarchy Each rectangle is replaced by its IMR (Inner Maximum Rectangle). IMR s and junctions are adjusted through iterations. acceptable No Yes

18 Correction Shape Modification IMR Replacement Each rectangle is replaced by its IMR (Inner Maximum Rectangle). IMR s and junctions are adjusted through iterations. IMR replacement

19 Correction Shape Modification IMR Adjustment Each side of IMR is adjusted based on the exposure estimate at the corresponding critical point.

20 Correction Shape Modification Junction Adjustment The size of the removed square is adjusted based on the exposure estimate at the corresponding critical point.

21 Correction Shape Modification Experimental Results (CNF) 500 nm PMMA on Si, 50 KeV, Gold lift-off Etec pattern with MFS of 0.1 um Uncorrected edge region Corrected edge region

22 Correction Shape Modification Experimental Results (CNF) 500 nm PMMA on Si, 50 KeV, Gold lift-off Etec pattern with MFS of 0.1 um Uncorrected center region Corrected center region

23 Correction Dose Modification Feature Partitioning Rectangle Partitioning Correction Hierarchy Circuit features are partitioned for spatial control of dose within each feature. Dose factor for each region is determined through iterations.

24 Correction Dose Modification Feature Partitioning (A) A rectangle (feature) with multiple junctions (B) A large rectangle

25 Correction Dose Modification Rectangle Partitioning Each rectangle is partitioned into regions (center, edge, and corner regions) for spatial dose control within the rectangle. Each region is assigned one or two critical points.

26 Correction Dose Modification Simulation Results (Sequential vs. Simultaneous) Etec pattern of 50umx50um, 1000 nm PMMA on Si, 50 KeV

27 Heterogeneous Substrates - Region-wise Correction (A) A feature lies over two different regions. (B) Each part is corrected with the corresponding PSF. (C) An intermediate segment is set in the transition zone. (A) (B) (C)

28 Heterogeneous Substrates - Simulation Results Ring width of 0.1 um 50 KeV Ring pattern Uncorrected PMMA thickness (nm) Homogeneous correction Heterogeneous correction

29 Recent Developments Grayscale Proximity Effect Correction Non-Rectangular Features

30 Grayscale Lithography PBG crystals, DOE, etc. A circuit pattern (structure) contains multiple levels. Exposure within each feature is to be uniform. Ideal exposure (corresponding to etch depth) for each feature

31 Grayscale Lithography - Region Partitioning Original circuit Fixed edge partitioning width array [ ] (a) Circuit after edge partitioning (a) Fixed partitioning: may lead to unnecessary partitions (b) Simple adaptive partitioning: simple and effective Original circuit Simple adaptive partitioning (b) Circuit after edge partitioning (c) Deconvolution-based adaptive partitioning: higher adaptability but larger errors sometimes Deconvolution-based adaptive partitioning Original circuit (c) Circuit after edge partitioning

32 Grayscale Lithography - Simulation Ideal No-correction DC correction Simple adpt Deconv

33 Grayscale Lithography - Experiment Cross-section of remaining resist profile J. Kim and D. Joy, UT Knoxville

34 Non-Rectangular Features Most proximity effect correction schemes assume rectangular features only. Circuits (e.g., PBG crystals, bus lines, etc.) contain non-rectangular features such as circles, rotated rectangles, polygons, etc Efficient methods to handle such features are necessary.

35 Non-Rectangular Features - Hierarchical Approach Non-rectangular feature Non-rectangle feature Partitioned into correction shapes yes Basic Shapes? Correction shape no Partitioned into basic exposure shapes A set of basic exposure shapes Exposure shapes Basic shapes : Right triangle, Rectangle, and Circle

36 Non-Rectangular Features - Exposure Estimation: Direct Method Circle and Ring Exposure contribution from a circle or ring is derived from CDFC table. CDFC table CDFC(d,r): exposure contribution from a circle of which radius is r and whose center is at the distance d from the critical point. d d r CDFC CDFC( d, r) Table Exposure circle = CDFC( d, r) r 2 r 1 CDFC Table CDFC ( d, r 1) CDFC d, r ) ( 2 Exposure ring = CDFC ( d, r1 ) CDFC ( d, r2 )

37 Non-Rectangular Features - Exposure Estimation Direct Method Right triangle Exposure contribution from a right triangle is derived from the CDFT table. + h Each of w and h can be positive or + w negative to represent different orientations of a right triangle. -w -h CDFT table CDFT(x,y,w,h): exposure contribution from a right triangle whose right-angle vertex is at (x,y) and width and height are w and h, respectively. + h ( x, y) CDFT + w Table CDFT( x, y, w, h) Exposure triangle = CDFT ( x, y, w, h)

38 Slicing Method Non-Rectangular Features - Exposure Estimation Any (non-rectangular) correction shape may be decomposed into thin slices (exposure shapes) such that each slice is a rectangle. Then, the CDF table can be used to derive exposure contribution from each slice. Correction shapes Exposure shapes

39 Hybrid Method Non-Rectangular Features - Exposure Estimation A correction shape which is not a basic shape may be decomposed into a set of exposure shapes of rectangles and right triangles. In order to minimize the number of exposure shapes, a portion of correction shape may be represented by a rectangle and negative right triangles. Negative triangles Correction shapes Exposure shapes

40 Non-Rectangular Features - Exposure Estimation Coordinate Transformation A correction shape of rotated rectangle may be generated from a slanted bus line, a polygon, etc. θ x x The local coordinates (x,y) centered at the critical point may be rotated such that the rectangle is not slanted (i.e., horizontally or vertically oriented) in the new coordinates (x,y ). Then, the CDF table may be used for exposure estimation. y y

41 Summary Hierarchical & look-up table: fast, accurate, flexible Shape & dose modifications, heterogeneous substrates, grayscale lithography, Non-rectangular primitives, etc. Further developments

42 On-line PYRAMID Electrical & Computer Engineering Auburn University