A note on the planar surface sum of Heegaard splittings

Unstabilized dual Heegaard splittings of 3-manifolds

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216516500322 ◽

2016 ◽

Vol 25 (06) ◽

pp. 1650032 ◽

Cited By ~ 3

Author(s):

Kun Du

Keyword(s):

Heegaard Splitting ◽

Heegaard Splittings ◽

Planar Surface

Let [Formula: see text] be a [Formula: see text]-manifold, [Formula: see text] be an essential planar surface which cuts [Formula: see text] into two 3-manifolds [Formula: see text] and [Formula: see text]. Suppose [Formula: see text] [Formula: see text] is a Heegaard splitting of [Formula: see text], [Formula: see text] is the dual Heegaard splitting of [Formula: see text] and [Formula: see text] along [Formula: see text], where [Formula: see text] and [Formula: see text]. In this paper, we give a condition of unstabilized dual Heegaard splittings of [Formula: see text]-manifolds by using Hempel’s distance and the method of proof of Gordon’s Conjecture. Also, we give a counterexample for stabilized dual Heegaard splittings of [Formula: see text]-manifolds for any Hempel’s distance.

Download Full-text

QUANTIFYING THE EFFECTS OF FUEL COMPOSITIONS AND PROCESS VARIABLES ON PLANAR SURFACE AREA AND SPRAY NONUNIFORMITY VIA COMBINED MIXTURE-PROCESS DESIGN OF EXPERIMENT

Atomization and Sprays ◽

10.1615/atomizspr.2017019656 ◽

2017 ◽

Vol 27 (8) ◽

pp. 707-722 ◽

Cited By ~ 1

Author(s):

Longfei Chen ◽

Liuyang Feng ◽

Zhixin Liu ◽

Guangze Li ◽

Yanfei Li ◽

...

Keyword(s):

Surface Area ◽

Design Of Experiment ◽

Process Design ◽

Process Variables ◽

Planar Surface

Download Full-text

Automated Diagonal Slice and View Solution for 3D Device Structure Analysis

10.31399/asm.cp.istfa2018p0224 ◽

2018 ◽

Author(s):

Sang Hoon Lee ◽

Jeff Blackwood ◽

Stacey Stone ◽

Michael Schmidt ◽

Mark Williamson ◽

...

Keyword(s):

Cross Section ◽

Focused Ion Beam ◽

Ion Beam ◽

Image Data ◽

Planar Surface ◽

Device Structure ◽

Cross Sectional ◽

Device Structures ◽

The Cross ◽

Planar Analysis

Abstract The cross-sectional and planar analysis of current generation 3D device structures can be analyzed using a single Focused Ion Beam (FIB) mill. This is achieved using a diagonal milling technique that exposes a multilayer planar surface as well as the cross-section. this provides image data allowing for an efficient method to monitor the fabrication process and find device design errors. This process saves tremendous sample-to-data time, decreasing it from days to hours while still providing precise defect and structure data.

Download Full-text

Precision Xe Plasma FIB Delayering for Physical Failure Analysis of Sub-20 nm Microprocessor Devices

ISTFA 2017: Conference Proceedings from the 43rd International Symposium for Testing and Failure Analysis ◽

10.31399/asm.cp.istfa2017p0574 ◽

2017 ◽

Author(s):

D. Zudhistira ◽

V. Viswanathan ◽

V. Narang ◽

J.M. Chin ◽

S. Sharang ◽

...

Keyword(s):

Failure Analysis ◽

Dielectric Films ◽

Planar Surface ◽

Chemical Methods ◽

Root Cause ◽

Essential Step ◽

Dielectric Layers ◽

Wet Chemical ◽

20 Nm ◽

Low K

Abstract Deprocessing is an essential step in the physical failure analysis of ICs. Typically, this is accomplished by techniques such as wet chemical methods, RIE, and mechanical manual polishing. Manual polishing suffers from highly non-uniform delayering particularly for sub 20nm technologies due to aggressive back-end-of-line scaling and porous ultra low-k dielectric films. Recently gas assisted Xe plasma FIB has demonstrated uniform delayering of the metal and dielectric layers, achieving a planar surface of heterogeneous materials. In this paper, the successful application of this technique to delayer sub-20 nm microprocessor chips with real defects to root cause the failure is presented.

Download Full-text