Computer Solution of the Scattering Problem for a Groove in a Metallic Plane Using the Moda
BEF原理
LCD BEF
Solutions: reduced reflection, reduced coherence, rotation, matte. Common solution is top diffuser.
Polarization - Two Types
1. Unpolarized 2. Linear • Describe polarization by orientation of electric field vector. (Magnetic field vector is always perpendicular) • Human eye doesn’t “see” polarization, but we can see polarization effects. • Human eye responds to square of the electric field magnitude
• Provides controlled spacer for crossed sheets • Isolates coupling below eye resolution • BEF III “chaos pattern” varies height along prisms
BEF Moiré - Two Cases
46.3%
Recycled Flux
Summary of prism angle
and RI response
n = 1.6
1.5 1.4 b = 90° 110° 130° b
n
Increasing (n) refractive index shrinks the cut-off angle
F019-00-0304
SEMI F19-0304 © SEMI 1995, 20041SEMI F19-0304SPECIFICATION FOR THE SURFACE CONDITION OF THE WETTED SURFACES OF STAINLESS STEEL COMPONENTSThis specification was technically approved by the Global Gases Committee and is the direct responsibility of the North American Gases Committee. Current edition approved by the North American Regional Standards Committee on December 4, 2003. Initially available at February 2004; to be published March 2004. Originally published in 1995.NOTICE : This document was completely rewritten in 2004. 1 Purpose1.1 The purpose of this specification is to provide a standard for the quality of the wetted surfaces of stainless steel components used in the chemical (gas and liquid) distribution systems of semiconductor manufacturing facilities. 2 Scope2.1 This specification defines the wetted surface characterization requirements and the finish acceptance criteria for tubing and components fabricated in stainless steel per SEMI F20 and intended to control and/or contain gases and liquids used in semiconductor manufacturing.2.2 The surface characterization tests to be performedare specified herein, and the existing standards for performing these tests are referenced.2.3 Terms specific to this technology are either listed herein as they relate to the acceptance criteria of this specification or are defined in the referenced documents as they relate to a specific test method.NOTICE: This standard does not purport to address safety issues, if any, associated with its use. It is the responsibility of the users of this standard to establish appropriate safety and health practices and determine the applicability of regulatory or other limitations prior to use.3 Referenced Standards 3.1 SEMI StandardsSEMI F20 — Specification for 316L Stainless Steel Bar, Extruded Shapes, Plate, and Investment Castings for Components Used in High Purity Semiconductor Manufacturing ApplicationsSEMI F37 — Method for Determination of Surface Roughness Parameters for Gas Distribution System ComponentsSEMI F60 — Test Method for ESCA Evaluation of Surface Composition of Wetted Surfaces of Passivated 316L Stainless Steel ComponentsSEMI F70 — Test Method for Determination of Particle Contribution of Gas Delivery SystemSEMI F72 — Test Method for Auger Electron Spectroscopy (AES) Evaluation of Oxide Layer of Wetted Surfaces of Passivated 316L Stainless Steel ComponentsSEMI F73 — Test Method For Scanning ElectronMicroscopy (SEM) Evaluation of Wetted Surface Condition of Stainless Steel ComponentsSEMI F77 — Test Method for Electrochemical Critical Pitting Temperature Testing of Alloy Surfaces Used in Corrosive Gas Systems 3.2 ASTM 1 StandardASTM A 967 — Standard Specification for ChemicalPassivation Treatments for Stainless Steel Parts NOTICE: Unless otherwise indicated, all documents cited shall be the latest published versions. 4 Terminology4.1 Abbreviations and Acronyms 4.1.1 Å — Angstrom (= 0.1 nm)4.1.2 AES — Auger Electron Spectroscopy 4.1.3 Avg — Average4.1.4 CPT — Critical Pitting Temperature 4.1.5 Cr — Chemical symbol for Chromium4.1.6 Cr/Fe — Ratio of total Chromium to total Iron in the passive oxide layer, as defined in SEMI F60 4.1.7 CrOX — Chromium Oxide1 American Society for Testing and Materials, 100 Barr Harbor Drive, West Conshohocken, Pennsylvania 19428-2959, USA. Telephone: 610.832.9585, Fax: 610.832.9555 Website: SEMI F19-0304 © SEMI 1995, 2004 24.1.8 CrOX/FeOX — Ratio of Chromium Oxide to Iron Oxide in the passive oxide layer, as defined in SEMI F604.1.9 EDS — Energy Dispersive X-ray Spectroscopy 4.1.10 ESCA — Electron Spectroscopy for Chemical Analysis (sometimes referred to as XPS) 4.1.11 Fe — Chemical symbol for Iron 4.1.12 FeOX — Iron Oxide 4.1.13 Max — Maximum4.1.14 µin — Micro inch (= 10-6 inch) 4.1.15 µm — Micrometer (= 10-6 meter) 4.1.16 nm — Nanometer (= 10-9 meter)4.1.17 Ra — Roughness average of surface profile4.1.18 Ry — Maximum peak-to-valley roughness height of surface profile4.1.19 SEM — Scanning Electron Microscopy 4.1.20 XPS — X-Ray Photoelectron Spectroscopy (sometimes referred to as ESCA) 4.2 Definitions4.2.1 blistering — a localized delamination within the metal that has an appearance of chipped or flaked-off areas.4.2.2 defect — unintentional and undesirable irregularity in the part surface that could affect system performance. Examples of such defects include cracks, inclusions, blistering, dents, pits, stringers and scratches.4.2.3 dent — an isolated irregularity on the wetted surface made by an impact.4.2.4 electropolishing — a method of polishing metals and alloys in which material is removed from the surface by making the metal the anode in an electrolytic bath.4.2.5 etched — a surface that has alloy structuralfeatures revealed by preferential chemical or electrochemical attack. 4.2.6 flaw — synonymous with defect. 4.2.7 frostiness — a continuous surface pattern whose appearance is like that of a sparkly, very fine, sandy-textured surface.4.2.8 grain boundary — an interface separating two grains, where the orientation of the lattice changes from that of one grain to that of the other.4.2.8.1 stepped grain boundary — a form of preferential etching in which the grains are attacked at different rates, resulting in one grain to appear raised with respect to an adjacent grain, forming a “step” at the grain boundary.4.2.9 haze — a diminished surface brightness or specularity attributable to diffuse light scattering by concentrations of microscopic surface irregularities, or to chemical inhomogeneity.4.2.10 inclusion — indigenous or foreign material within the metal, usually referring to non-metallic compound particles such as oxides, alumina, sulfides or silicates.4.2.11 interrupted electropolishing — a break in the continuity of the electropolished surface appearance due to a change of electropolishing conditions at the interruption boundary; may be visible as a change inreflectivity across the boundary or a step in the surface. 4.2.12 lay — the direction of the predominant surfacepattern, ordinarily determined by the production method used. 4.2.13 machining Lines — a type of process line that results from machining processes.4.2.14 nominal surface — the intended surface contour, the shape and extent of which is shown and dimensioned on a drawing or descriptive definition. 4.2.15 orange peel — large-featured, roughened type of surface visible to the unaided eye whose surface appearance pattern is like that of an orange peel. 4.2.16 passivation — the chemical treatment of a stainless steel surface with an oxidizing solution for the purpose of enhancing the corrosion resistant surface film.4.2.17 pit — a surface cavity or crater with a defined edge not caused by impact.4.2.18 process Lines — surface features that are a result of the material removal or forming process used in manufacturing of a component.4.2.19 Ra Avg. — average Ra of a set of surface roughness measurements. 4.2.20 Ra Max. — maximum Ra of a set of surface roughness measurements.4.2.21 Ry Max. — maximum Ry of a set of surface roughness measurements.4.2.22 roughness — the finer irregularities of the surface texture, usually including those irregularities that result from the manufacturing process. These are considered to include traverse feed marks and otherSEMI F19-0304 © SEMI 1995, 20043irregularities within the limits of the roughness sampling length.4.2.23 scratch — an elongated mark or groove cut inthe surface by mechanical means, not associated withthe predominant surface texture pattern.4.2.24 stringer — a microstructural configuration ofalloy constituents or foreign nonmetallic material, ortrace thereof, oriented in the direction of mechanicalworking.4.2.25 surface texture — the repetitive or randomdeviations from the nominal surface which form thethree dimensional topography of the surface. Surfacetexture includes roughness, waviness, lay, and flaws.4.2.26 waviness — the more widely spaced componentof surface texture. Unless otherwise noted, waviness isto include all irregularities whose spacing is greaterthan the roughness sampling length. Waviness may result from such factors as machine or work deflections,vibrations, chatter, heat treatment, or warping strains.Roughness may be considered as superimposed on a“wavy” surface.4.2.27 wetted surface — surfaces of the components that are in contact with the contained gases and/orliquids used in semiconductor manufacturing processes.5 Ordering Information5.1 The purchase order for components to be supplied in compliance with this specification shall include thefollowing information:5.1.1 Purchase Order Number. 5.1.2 Supplier’s (preferred) or purchaser’s partnumber(s), including size(s).5.1.3 Quantity of components by part number and size.5.1.4 Grade of each part number and size per the gradeclassifications specified in this specification.5.1.5 Reference to this specification number and anyadditional applicable specifications.5.1.6 Documentation and certification requirements.5.1.7 Any special and/or supplementary requirements.6 Requirements6.1 Requirements of this specification are divided intothree grades of products:6.1.1 General Purpose Components (GP) — These are components intended for use in fluid distribution systems of semiconductor manufacturing facilities that do not have stringent cleanliness requirements. Examples are clean dry air and vacuum lines. 6.1.2 High Purity Components (HP) — These arecomponents intended for use in high performance capability chemical distribution systems ofsemiconductor manufacturing facilities.6.1.3 Ultra High Purity Components (UHP) — These are components intended for use in advanced chemicaldistribution systems of semiconductor manufacturingfacilities in which optimum performance capabilitiesare required.6.2 Requirements of this specification are shown inTable 1, along with the appropriate test methods forevaluating each.6.3 Surface Roughness Requirements — Surface roughness requirements for each grade of componentare shown in Table 1. Measurements shall be made perthe mechanical profilometry test method SEMI F37.6.3.1Welds, screw slots, intersections, and non-metallic parts are not measured. 6.4 Surface Defect Requirements — Surface defectrequirements for the HP and UHP grades ofcomponents are shown in Table 1. Measurements shallbe made per the SEM test method of SEMI F73. 6.5 Surface Contamination Requirements — Grosssurface contamination on the HP and UHP grades of components observed in the SEM determination ofsurface defects per SEMI F73 shall be evaluated byEDS. 6.5.1 Elements expected to be found on the surfaces of components by EDS include the primary alloyingelements Iron (Fe), Chromium (Cr), Nickel (Ni), andMolybdenum (Mo). The residual elements Manganese(Mn) and Silicon (Si), the Oxygen (O) of the passivelayer, and adsorbed carbon are also expected to be found. Any other elements detected by EDS are to be considered contaminants and are cause for rejection.6.6 Surface Chemistry Requirements — The preferred test method for evaluating the chemistry andcontamination of the oxide passive layer is the ESCAtest method of SEMI F60 for the HP and UHP grades of components. Alternatively, the evaluation may be performed by the AES test method of SEMI F72, peragreement between the supplier and user. 6.6.1 Elements expected to be found on the surfaces of components by ESCA or AES include the primaryalloying elements Iron (Fe), Chromium (Cr), Nickel(Ni), and Molybdenum (Mo), the residual elementsManganese (Mn), Silicon (Si), Sulfur (S), Phosphorous (P), Nitrogen (N) and Silicon (Si), and the Oxygen (O) of the passive layer. Adsorbed Carbon contamination shall be < 30 atomic percent, declining to base levelswithin 15 Angstrom of the initial surface.SEMI F19-0304 © SEMI 1995, 2004 46.6.2 Sulfur (S), Phosphorous (P), Nitrogen (N) and Silicon (Si) shall be < 2 atomic percent on the initial surface by ESCA or AES.6.6.3 All other contaminants shall be < 1 atomic percent by ESCA or AES.Table 1 Table of Specifications and Test Methods by Product GradeNOTE: Oxide thickness > 5 nm (50 Å) approximately, formed by electropolishing may have an outer layer enriched in Iron, and therefore exhibitpoor corrosion resistance.6.7 Corrosion Resistance Requirements — Corrosion resistance of the surface shall be evaluated by the CPT test method of ASTM G 150 or SEMI F77.6.8 Visual Acceptance Criteria — Visual inspection of the wetted surfaces for defects shall be performed under normal room lighting. Additional lighting may be used when appropriate to illuminate blind or darkened areas and to clarify questionable areas. Visual inspection shall be done with up to 4X magnification; if a possible problem is detected, an 8 to 10X magnifier may be used for clarification of the area in question.6.8.1 Process and machining lines are acceptable on GP grade surfaces if the surface roughness requirements are met. Process and machining lines are not acceptable on HP and UHP surfaces. Surface finishing processes for HP and UHP surfaces are expected to “round off” the lines to the degree that they are not apparent by the SEM test method SEMI F73 and are not well defined visually.6.8.2 Scratches and haze that meet the surface roughness requirements are acceptable on GP surfaces. These defects are not acceptable on HP and UHP surfaces.General Purpose GradeGPHigh Purity GradeHPUltra-High Purity GradeUHPCharacteristic Test Method ValueValueValueSurface RoughnessSEMI F37≤ 0.50 µm (20 µin) Ra Avg.≤ 0.62 µm (25 µin) Ra Max.≤ 3.75 µm (150 µin) Ry Max.≤ 0.25 µm (10 µin) Ra Avg. ≤ 0.38 µm (15 µin) Ra Max. ≤ 3.75 µm (150 µin) Ry Max.≤ 0.13 µm (5 µin) Ra Avg.≤ 0.25 µm (10 µin) Ra Max. ≤ 2.50 µm (100 µin) Ry Max. Surface Defects SEMI F73 (Not Applicable)≤ 30 Avg, ≤ 50 Max Counts/Photo ≤ 10 Avg, ≤ 20 MaxCounts/PhotoSurface Contamination SEMI F73 (Not Applicable) No elements detectable by EDS other than primary alloying elements Fe, Cr, Ni, Mo, residual elements Mn and Silicon (Si), the O of the passive layer, and adsorbed C. No elements detectable byEDS other than primary alloying elements Fe, Cr,Ni, Mo, residual elementsMn and Silicon (Si), the O of the passive layer, and adsorbed C.Cr/Fe and CrOX/FeOX ratios ≥ 1.0; oxide thickness t ≥ 1.5 nm ( t ≥ 15 Å). See Note. Cr/Fe ratio ≥ 1.5 andCrOX/FeOX ≥ 2.0; oxide thickness t ≥ 1.5 nm ( t ≥15 Å). See Note. Surface Chemistry SEMI F60 (or SEMI F72 by agreement) Product must be passivatedper ASTM A967Adsorbed Carbon contamination shall be <30 atomic percent, declining to base levels within 1.5 nm (15 Å) of the initial surface. Sulfur (S), Phosphorous (P), Nitrogen (N) and Silicon (Si) shall be < 2 atomic percent on the initial surface. All other contaminants shall be < 1 atomic percent. Adsorbed Carbon contamination shall be <30 atomic percent, declining to base levelswithin 1.5 nm (15 Å) ofthe initial surface. Sulfur (S), Phosphorous (P), Nitrogen (N) and Silicon (Si) shall be < 2 atomic percent on the initial surface. All other contaminants shall be < 1atomic percent.Corrosion Resistance ASTM G 150; SEMI F77 As agreed upon between user and supplier. As agreed upon between user and supplier. As agreed upon betweenuser and supplier.SEMI F19-0304 © SEMI 1995, 200456.8.3 Waviness is acceptable if the surface roughness requirements are met.6.8.4 Orange peel is acceptable if the surface roughness requirements are met. 6.8.5 Dents, frostiness, blistering and interrupted electropolishing are not acceptable.6.8.6 Contamination (water spots, rust, process residues, chemical staining, etc.) are not acceptable.6.8.7 Pitting, stringers, and inclusions apparent to theunaided eye are not acceptable in UHP grade surfaces. Acceptance of these defects in HP and UHP grade surfaces shall be determined by the SEM test method ofSEMI F73 and shall be in conformance with the specifications of Table 1. 7 Test Methods 7.1 Test methods are listed in Table 1 and described in Section 6, Requirements. 8 Sampling8.1 Sampling shall be per agreement between component supplier and purchaser. 9 Certification9.1 The component supplier shall provide the follow-ing reports and certifications with all shipments of processed components:9.1.1 Name/part number/description of component(s), 9.1.2 Quantity of components by size or description, 9.1.3 Product grade of each size or category of components per the classifications in this specification, and9.1.4 Certification that the manufacturing processes were performed per an appropriate quality control program and that the processes are qualified. 10 Product Labeling10.1 Product labeling shall be per agreement between component supplier and purchaser. 11 Related Documents11.1 SEMI StandardSEMI F55 — Test Method for Determining the Corrosion Resistance of Mass Flow Controllers 11.2 ASME 2Standards2 American Society of Mechanical Engineers: United Engineering Center, Three Park Avenue, New York, NY 10016-5990, USA.ASME B46.1 — Surface Texture (Surface Roughness, Waviness and Lay)11.3 ASTM 1Standards ASTM F 1372 — Test Method for Scanning Electron Microscope (SEM) Analysis of Metallic SurfaceCondition for Gas Distribution System Components ASTM F 1374 — Test Method for Ionic/OrganicExtractables of Internal Surfaces – IC/GC/FTIR for Gas Distribution System ComponentsASTM F 1375 — Test Method for Energy Dispersive X-Ray Spectrometer (EDX) Analysis of MetallicSurface Condition for Gas Distribution System Components ASTM F 1394 — Test Method for Determination ofParticle Contribution from Gas Distribution SystemValvesASTM F 1396 — Test Method for Determination of Oxygen Contribution by Gas Distribution SystemComponentsASTM F 1397 — Test Method for Determination ofMoisture Contribution by Gas Distribution System Components ASTM F 1398 — Test Method for Determination of Total Hydrocarbon Contribution by Gas Distribution System ComponentsASTM G 150 — Standard Test Method for Electrochemical Critical Pitting Temperature Testing of Stainless SteelsTelephone: 800.843.2763; Fax: 973.882.1717; Website: SEMI F19-0304 © SEMI 1995, 2004 6NOTICE: SEMI makes no warranties or representations as to the suitability of the standards set forth herein for any particular application. The determination of the suitability of the standard is solely the responsibility of the user. Users are cautioned to refer to manufacturer's instructions, product labels, product data sheets, and other relevant literature, respecting any materials or equipment mentioned herein. These standards are subject to change without notice.By publication of this standard, Semiconductor Equipment and Materials International (SEMI) takes no position respecting the validity of any patent rights or copyrights asserted in connection with any items mentioned in this standard. Users of this standard are expressly advised that determination of any such patent rights or copyrights, and the risk of infringement of such rights are entirely their own responsibility.Copyright by SEMI® (Semiconductor Equipment and Materials International), 3081 Zanker Road, San Jose, CA 95134. Reproduction of the contents in whole or in part is forbidden without express written consent of SEMI.。
ATK基础教程_第1部分介绍如何构建模型和初步的计算
Atomistix ToolKit Tutorial Part 1: Two-probe geometry & convergence parametersOutlinePart 1:» Fundamental concepts of a two-probe system » Solving convergence problems » Based on a specific example which is hard to converge (FeMgO MTJ)Part 2: AnalysisFundamental but complex topics in ATKUnderstanding the fundamental concepts of a two-probe system is crucial to establish confidence in the simulations and their results Most calculations in ATK converge with default parameters» But some do not... (for a variety of reasons)Geometrical aspects» Related to certain “hidden” approximations and/or simplifications in ATKNumerical parameters» Accuracy » Solving convergence problemsBoth aspects are related to convergence, but may also compromise quality of results if not chosen appropriately (even if the calculation converges!)Three ways to improve performance Total calculation time = time/iteration × number of iterations Reduce time/iteration» Parallelization (scaling depends on system & parameters) » Code and algorithm improvements (ATK 2008.10)Reduce number of iterations» Algorithm improvements (new convergence criterion in ATK 2008.02) » Parameter tuningThird way?» Avoid running end-less calculations which do not converge! » Avoid re-running calculations because of poor quality resultsOur guinea pig Fe/MgO/Fe magnetic tunnel junction (MTJ) Anti-parallel electrode spin polarization» Among the most difficult systems to converge » Strong peak in the minority DOS at the Fermi levelFe MgO FeMany other interface systems exhibit similar convergence issuesAu/Si/Au two-probeElectrodes, the simple stuffTransport direction must be perpendicular to the interface plane» Still possible to consider transport at an angleElectrodes must not feel each other» Atoms in the left electrode may not have any basis set overlap with atoms in the right electrodeCrossed nanotubesElectrode cell must be periodic in the transport direction» Example: fcc [111] → 3, 6, 9, ... layersElectrodes, more simple stuff Periodic boundary conditions in the interface plane (x/y)» Allows study of true interfaces » Vacuum padding needed to allow electrostatic interactions to decay for 1D/2D systems » Sufficiently large metal surface cell needed for “broad” moleculesNickel – Graphene – Nickel spin filterNanotube two-probeAu [111] 3x3AlignmentInput geometries are relative (L/C/R)» Alignment is done by specially appointed alignment pairs (atom indices)Pros:» Homogeneous electrodes only need to be defined once » Easy to change the internal geometries of electrodes or central region » No need to think about absolute alignmentCons:» Difficult to assign alignment atoms by hand » Atom indices may change if the geometry is modified → must remember to update alignment » Easy to make non-obvious mistakesAlways inspect two-probe geometries in VNL (Nanoscope) before the calculation to ensure proper alignment!Alignment, the detailsAppoint 2 atom pairs to align the 3 components» (left electrode, central region) = (L,CL) » (right electrode, central region) = (R,CR) » NanoLanguage input is [(L,CL), (R,CR)]Convenient to use relative index (−1) for last atom in listtwoprobe_configuration = TwoProbeConfiguration( (left_electrode_configuration,right_electrode_configuration), scattering_elements, scattering_coordinates, equivalent_atoms = ([0,0],[3,5]) )1. Central region is aligned to left electrode such that rC(0) = rL(3) + uL3 (uL3 = 3rd unit cell vector for left electrode) 2. Right electrode is aligned such that rR(0) = rC(5) + uR3 (uR3 = 3rd unit cell vector for right electrode)Concrete alignment exampleAlignment atoms: [(3,0),(0,5)]uL3See the ATK manual on TwoProbeConfiguration for more details!−uR31. Left + central2. RightConcrete alignment exampleAlignment atoms: [(3,0),(0,5)]See the ATK manual onTwoProbeConfigurationfor more details!u L3−u R3Equivalent bulk unit cellElectrodes, the subtleties Electrode calculation:electrode is treated as abulk system»All interactions includedTwo-probe calculation: interactions extendingbeyond the nearest-neighboring cell (red) are truncated from the Hamiltonian»May shift the Fermi level »Excluded interactionsconstitute anapproximation electrode cellinteraction range (3a)aAlways ensure thatthe electrodes aredeep enough!Heterogeneous two-probesTwo-probe systems can be»Homogeneous(identical electrodes)»Heterogeneous(different electrodes)Electrostatics (Poisson equation)»Homogeneous: FFT»Heterogeneous: Multigrid(in z, FFT in x/y)Geometrically identical electrodes with different numerical parametersis a heterogeneous system»Example: MTJ anti-parallel caseMg OCoSiMnCo2MnSi/Mgo/Co2MnsiParallel or Anti-Parallel Nanotube heterojunction Heusler/MgO MTJHomogeneous vs. heterogeneousDeciding factor: two-probe system and method constructions in NanoLanguage»L/R electrode = same variable →homogeneous»L/R electrode = different variables →heterogeneoushomogeneous_twoprobe= TwoProbeConfiguration((electrode_configuration, electrode_configuration),...)heterogeneous_twoprobe = TwoProbeConfiguration((left_electrode_configuration, right_electrode_configuration),...)homogeneous_twoprobe_method= TwoProbeMethod(electrode_parameters= (electrode_parameters, electrode_parameters),...)heterogeneous_twoprobe_method= TwoProbeMethod(electrode_parameters= (left_electrode_parameters, right_electrode_parameters), ...)Initial densityTwo-probe systems can be hard to converge for many reasons»One reason is that the bulk density matrix from the convergedelectrodes is far from the “neutral atom”density matrixAn initial EquivalentBulk calculation can help in many cases»ATK will automatically create an equivalent bulk system fromL+C+R and perform a bulk calculation for this system»The converged density matrix is then used as a starting guessfor the two-probe calculation»InitialDensityType.EquivalentBulk is defaultAu-DTB-Au Requirement:the entire system L+C+R should be periodic,or at least without “bad”stacking faults»Examples, fcc[111]:•[ABC]AB–...–BC[ABC](no mirror, no stacking fault)•[ABC]AB–...–BA[CBA](mirror, bad stacking fault BAAB)•[ABC]AB–...–BAC[BAC](mirror, minor stacking fault ACAB)Homogeneous systems should always fulfill the requirement»Otherwise the electrode itself is not periodic!Heterogeneous systems may or may not»If left/right electrodes are really different (physically or withdifferent stacking), change to InitialDensityType.NeutralAtomsTransverse unit cellFor heterogeneous systems, X/Y unit cell commensurability must be carefully ensured»Trivial for 1D systems like CNTs and graphene (vacuum padding)»Challenge for interfaces which are not lattice matched (large supercell) Too small X/Y unit cell leads to electrostatic ”cross-talk”»May occur both for true 1D system and quasi-1D wires (molecular or metallic) between extended interfaces»Typical signal of residual electrostatic interactions is broken degeneracies, e.g. in high symmetry point of the band structure Z-shaped graphene transistorRu RuHfO 2Screening approximationWhy is it necessary to have a lot ofL−1L C R R+1“electrode atoms”in the centralregion?»These atoms form the screeninglayersTwo-probe boundary condition:match the bulk electrode effectivepotential at the outer electrode cellboundaries (arrows)»In the region in between (L/C/R),the effective potential and electrondensity are calculated self-consistentlySurface layers provide screening sothat the electrode region is close tobulk-like»The surface layers are part of thecentral region config!»Too few screening layers constitutesan approximationVoltage biasBias is applied across entire two-probe region (L+C+R)»FFT (homogeneous): linear ramp added to effective potential»Multigrid (heterogeneous): included in the boundary conditions Bias is relative»Can be applied symmetrically (±V/2) or asymmetrically (V,0)»Bias direction determines direction of current (diode characteristics) Convergence under bias is more difficult»Converge with zero bias first, then use zero-bias calculation to initialize density matrix for finite bias»Up to about 2 V bias is currently possible in ATK (above this, convergence becomes very difficult)Especially important to have enough screening surface layers under biasL −1L C R R+1μLμRBack to our guinea pig: GeometryUse MTJ Builder in VirtualNanoLab (new in 2008.02)Default geometry works forour demonstration» 4 electrode layers and 4surface layers» 6 would be better, costsmore time & memoryDrop system onNanoLanguage ScripterNanoLanguage Scripter Powerful tool for generating NanoLanguage code»Even QuantumWise expertsuse it!»Can also be used for post-SCF analysisGenerated code is»MPI safe»Explicit (all parametersvisible)»Verbose (prints details toconsole)Non-standard things canalways be added to the scriptby hand afterwardsBasis setATK uses localized atomic orbitals basis sets with finiterange (SIESTA)Most parameters are“advanced”(keep default),except:»Type (basis set size)»Energy shift (lower toincrease basis functionrange)SingleZetaPolarized orDoubleZetaPolarized areusually optimal»Balance time & memoryvs. accuracyPossible to use differentparameters for each elementBasis set, detailsNorm-conserving pseudopotentials describe the core electronsValence orbitals are expanded in localized basis functions with a finite range » A list of valence electrons for each element can be found in the manualThe range of the cut-off is determined implicitly by the ”Energy shift”»Energy shift = 0 means infinite range, basis orbital = atomic orbital (cannot be used)»Decrease with caution from default to increase range (to improve accuracy)»Range increases exponentially and thus the number of interacting pairs goes up fastBasis sets (ordered least to most accurate):»SingleZeta; 1 basis orbital for each valence orbital»SingleZetaPolarized; SingleZeta + 1 basis orbital for the first unoccupied shell»DoubleZeta; 2 basis orbitals for each valence orbital»DoubleZetaPolarized; DoubleZeta + 1 basis orbital for the first unoccupied shell»DoubleZetaDoublePolarized; DoubleZeta + 2 basis orbitals for the first unoccupied shell For details, see the manual for ”AtomicOrbitals”Brillouin zone integration K-points are crucial for good accuracy and convergence»Especially important forFeMgO due to sharp DOSpeak at Fermi levelAs always, larger cell →fewer k-points neededC-direction k-points only used in electrode calculationMemory/time increases with k-points, but slower (N) thanwith number of atoms (N2)Both electrodes must use the same number of k-points ATK is parallelized over the k-pointsFor details see next 2 pagesDetails on k-points A/BA/B k-points used both in electrodes, equivalent bulk, and two-probe calculationNumber required depends on system size (wider electrode in A/B, fewer points)In directions without dispersion/periodicity (vacuum padding), only 1 point needed»Examples: nanotubes, wires, graphene (perpendicularto the sheet)Some systems, like FeMgO, require more points to get accurate Fermi levelIncreased temperature can be used to reduce need for many k-points, at the expense ofsomewhat reduced accuracy in the resultsDetails on k-points CWhy k-points in the C (Z) direction?»Only used in the electrodes bulk calculation»NOT used in the two-probe calculation –no periodicity!!!»N A xN B x1 used in the equivalent bulk calculation•If NA =NB=1 the equivalent bulk calculation is performed with realmatrices (faster, less memory)•Also applies to regular bulk calculations, if NA =NB=NC=1Why so many k-points in the C direction?»Crucial to get an accurate determination of the Fermi level»Two-probe calculation assumes semi-infinite electrode, whileelectrode calculation is finite (different boundary conditions)»Thus, two-probe calculation corresponds, effectively, toINFINITELY many k-points»25, 50, 100 points are usual;longer electrode, fewer points neededEigenstate occupation Electron temperature»Used in the electrode Fermi distribution»No phonons...!Increase to cure convergence problems»Smoother Fermi function»1000−1300 K usually doesthe trickNOTE:Changing thetemperature may changephysical results (like thecurrent) slightly!»Converge at hightemperature, then anneal,to be safeElectron densityMesh cut-off»Controls electrostatic mesh for Poisson equation»Default usually fine for electronstructure calculations»d-elements like Fe need highercut-off for good geometries (inoptimizations)Spin»Initial = absolute, in h»Scaled = fraction of max spinpolarization for isolated atom−1 ≤s≤1»To set spin for individual atoms,edit the scriptEach can be given forleft/central/right independently»Forces the system to be treated as heterogeneousEmxe2hπ=Δ−0.56 for anti-parallel0.56 for parallelContour integrationIntegral lower bound»Increase (5–7 Ry) avoid polesbelow the contour, which leadto charge run-away(calculation converges to q=0) Circle points»Increase (50-70) not to loseaccuracy when increasing thelength of the contourReal axis infinitesimal»Can be increased in reallydifficult cases»Changes the results –anneal! Real axis point density»Lower to cure convergenceproblems»Only relevant for finite biasATK is parallelized over thecontour integration pointsFor details see M. Brandbyge et al., PRB 65, 165401 (2002)MixingTotal energy criterion can reduce iterations by a factor 2 withoutcompromising the resultsAlways use Hamiltonian mixing for two-probe systemsHistory Steps»Experiment with increasing to improve convergence (doesn’t always work,sometimes fewer history steps arebetter…)»Costs memory (but not a lot)Diagonal mixing parameter (β)»Reduce βto stabilize convergence»Increase βto speed up convergence(works well in simple systems likecarbon nanotubes)Advanced strategy for difficult cases:»Run with small βfor some iterations(say, 5 or 20), then break»Restart with larger βand run toconvergenceTwo-probe algorithmElectrode constraint» Complex issue » See separate slides for details » Default = ”Off”, best workhorse for transmission and current » Undocumented option ”DensityMatrix” best, but can be hard to converge; only option for good voltage dropInitial density type» EquivalentBulk• Bulk calculation used to initialize the two-probe density matrix • Often the best option!» NeutralAtom• Heterogeneous systemsCheckpoint file NetCDF file stores all details of the calculation» Restore for analysis » Initialize density matrix for other calculationsWARNING: By default, no NetCDF file is created!Verbosity level» Default in ATK = 0 (quiet) » Default in VNL = 10 (all info) » Level 1 is often bestSetting the initial spin Open NanoLanguage script» Internal Script Editor » External editor• We recommend SciTE or Notepad++ • Completion, highlightingInitial spin for each atom in the central region» ±0.56 for surface layers (4 atoms left and right) » Zero initial spin for MgOelectron_density_parameters = electronDensityParameters( mesh_cutoff = 150.0*Rydberg, initial_scaled_spin = [ 0.56, 0.56, 0.56, 0.56, 0., 0., 0., 0., 0., 0., 0., -0.56, -0.56, -0.56, -0.56 ] )0.,Run the calculations! Two calculations (special for MTJs)» Parallel: 0.56 initial spin (right electrode/surface atoms) » Anti-parallel: −0.56 initial spin (right electrode/surface atoms)Different filenames for the checkpoint files! Possible to run in serial» Calculation uses < 1 Gb of RAM » About 12 hours in serial (dual-core)Run in parallel!» Use as many nodes as possible » Excellent scaling because of many k-points and contour integration points » Can cut the calculation time by order of magnitudeSummaryTo converge the FeMgO system, modify the following parameters:» » » » » Basis Set• Type (for Fe) = SingleZetaPolarizedBrillouin Zone Integration• Number of k-points (A/B/C) = 10/10/100Eigenstate Occupation• Electron Temperature = 1200 KelvinElectron Density• Heterogeneous (check) • Initial Scaled Spin = 0.56 (left) and −0.56 (right)Energy Contour• Circle Points = 50 • Integral Lower Bound = 7 Rydberg • Real Axis Points Density = 0.005 eV (for finite bias)» » »Iteration Mixing• Diagonal mixing parameter = 0.05TwoProbe Algorithm• Initial Density = NeutralAtomRemaining parameters are left at defaultGeometry = default from MTJ Builder Set initial spin per atom in editor (parallel / anti-parallel) NetCDF file!OutlookPart 2: Analysis» K-point resolved transmission coefficients » Transmission spectrum » Current & conductance » Tunneling magneto-resistance (TMR) » Scattering states。
吴振森-波传播与散射数值分析-1
分离变量法（有时也称傅立叶方法）是解 决偏微分方程（PDE）最适合的方法，基本上 它导致了一个不用对函数积分的方法。
1.3.1 拉普拉斯方程
先考虑二维情况下的拉普拉斯方程，然后扩展到三维波动方程。
考虑一个无限长规则导 体的狄里克利问题，其横 截面如图Fig.2.1。这是一 个边界值问题，其PDE为：
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A better approach to wear-rate representation in non-conformal contacts
1. Introduction
Wear is a complex phenomenon, which depends on many variables w1x. The ability to devise better ways to combat wear depends much on the availability of a better understanding of the wearing process. Such an understanding invariably requires the combined effort of many who examined wear through different experimental conditions and testing configurations. Very often, such varied approaches in wear investigation led to conflicting wear data, defeating the very effort to gain this understanding. It is therefore desirable to first obtain a certain degree of harmony in the large amount of wear data generated before such an understanding on a global scale could be achieved.
2. Presentation of metallic wear rates
量子计算机入门(日文)
Jyväskylä Summer School 2004, Finland
1.Superpositions: 2.Interference: 3.Entanglement:
Such as the many-body states in: superconductors, strongly correlated systems, quantum critical systems, quantum computers, ...
Jyväskylä Summer School 2004, Finland
Encounters with Non-Locality: Aharonov-Bohm Effects
Quantum Transport in Mesoscopic Systems: Sample-specific conductance replaces traditional material-specific conductivity.
Jyväskylä Summer School 2004, Finland
Weak Localization Theory
Jyväskylä Summer School 2004, Finland
Weak Localization: Magnetic Field Dependence
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c e o f t h e s p u t t e r i n g i o n s.A p l a s m a i s a r o u g h l y n e u t r a l c l o u d o f i o n s a n d e l e c t r o n s.T h e i o n s a r e c r e a t e d b y c o l l i s i o n s w i t he n e r g e t i c e l e c t r o n s;t h e e l e c t r o n s c o m ef r o m p r i o rg a s i o n i z a t i o n o r f r o m s e c o n d a r y e m i s s i o n o f th et a r g e t.T h e f i r s t i o n i s f o r m e d b y a c o s m i c r a y,o r b y c o l l i s i o n w i t h a n e l e c t r o n c r e a t e d b y a f i e l de m i s s i o nf r o m t h e t a rg e t.O n c e th e fi r s t f e w i o n s a r e f o r m e d,a c a s c a d e o c c u r s i f t h e e l e c t r o m a g n e t i cf i e l d(a n d p r e s s u r e)a r e w i t h i n w e l l-d e f i n e d r a ng e s.Thi s c a s c a d e f o r m s a s e l e c t r o n s a r e t r a p p e d b yt h e c r o s s e d f i e l d s a b o v e t h e t a r g e t;e a c h e l e c t r o n c a n i o n i z e m a n y n e u t r a l g a s a t o m s b e f o r e b e i n g l o s t t h r o u g h s o m e l o s s m e c h a n i s m.N o r m a l l y,t h e p l a s m a d e n s i t y i s d e t e r m i n e d b y t h e e l e c t r o n d e n s i t y,w h i c h i n t u r n i s a f u n c t i o n o f t h e m a g n e t i c f i e l d.I f f o r a n y r e a s o n a s m a l l r e g i o n o f t h e t a r g e t f e e d s e l e c t r o n s a t a h i g h r a t e i n t o t h ep l a s m a,i o n s a r e i m m e d i a t e l y a t t r a c t e d t o c r e a t e a s h e a t h a r o u n d t h e i n c o m i n g e l e c t r o n s,a n d a s m a l l t h r e a d o f i o n s a n d e l e c t r o n s b r i d g e s t h e c a t h o d e g a p.T h i s i s w h a t w e c a l l a n"a r c."T h e c o n d i t i o n c a n b e c a u s e d b y a f i e l d e m i s s i o n f r o m t h e t a r g e t,b y h e a t i n g o f a t a r g e t s p o t t o t h e r m i o n i c e m i s s i o n,b y a r e l e a s e o f g a s t r a p p e d i n a g a s p o c k e t,o r b y c a t a s t r o p h i c b r e a k d o w n o f t h e t a r g e t d u e t o e l e c t r i cf o r c e s.T h i s l a s t i s t h e f o c u s o f t h i s p a p e r;t h e c o n d i t i o n o c c u r s i f a n i n s u l a t i ng a r e a e x i s t s o n th et a r g e t s u r f a c e.C h a r g i n g t h e i n s u l a t o r b y b o m b a r d i n g i o n s c a n c r e a t e i n c r e d i b l e f o r c e s o n t h e a t o m s. '& 63877(5,1* 2) ,168/$7256C o n t r a r y t o c o n v e n t i o n a l w i s d o m,i n s u l a t o r s c a n b e s p u t t e r e d w i t h a d c p o t e n t i a l.T h e p r o b l e m i st h a t i t c a n n o t b e s u s t a i n e d f o r a n y l e n g t h o f t i m e.I n d c s p u t t e r i n g o f m e t a l s,t h e t a r g e t i s c o n d u c t i v e.A n e g a t i v e p o t e n t i a l i s a p p l i e d t o t h e t a r g e t a n d t h i s a t t r a c t s p o s i t i v e i o n s f r o m t h e p l a s m a(g e n e r a l l y a r g o n i o n s).A s t h e i o n s a r r i v e a t t h e t a r g e tt h e y k n o c k l o o s e a t a r g e t a t o m(t h i s i s,o f c o u r s e,t h e s p u t t e r i n g p r o c e s s)a n d a l s o c a p t u r e a n e l e c t r o n f r o m t h e t a r g e t b e c a u s e t h e r e a r e m a n y i n t h e m e t a l’s c o n d u c t i o n b a n d.T h e i o n t h e n b e c o m e s an e u t r a l a t o m a n d d r i f t s a w a y f r o m t h e t a r g e t s u r f a c e t o j o i n t h e o t h e r b a c k g r o u n d g a s a t o m s.T h ee l e c t r o n i s r e p l a c e d b y t h e p o w e r s u p p l y t h r o u g h t h e c o n d u c t i v e l e a d s t o t h e t a r g e t a n d t h i s b e c o m e s t h e p o w e r s u p p l y c u r r e n t.I f t h e i o n a r r i v e s i n a r e g i o n o f a m e t a l t a r g e t c o v e r e d b y a n i n s u l a t i n g f i l m,i t c a n n o t c a p t u r e a ne l e c t r o n,b e c a u s e t h e r e a r e n o e l e c t r o n s i n t h e c o n d u c t i o n b a n d of t h e f i l m.N e v e r t h e l e s s,s i n c es p u t t e r i n g i s p u r e l y a m o m e n t u m t r a n s f e r p r o c e s s,s p u t t e r i n g o f t h e i n s u l a t i n g l a y e r d o e s t a k e p l a c e, t h e s a m e a s w i t h t h e m e t a l s u r f a c e.A p o s i t i v e c h a r g e o n t h e s u r f a c e o f t h e i n s u l a t i n g r e g i o n,h o w e v e r,a t t r a c t s a"m i r r o r"e l e c t r o n o n t h e c o n d u c t i v e t a r g e t s u r f a c e o r b a c k i n g p l a t e,a n d t h i sm i r r o r c h a r g e i n t u r n a t t r a c t s t h e p o s i t i v e i o n,h o l d i n g i t f i r m l y i n p l a c e.O n e c a n t h i n k o f t h ei n s u l a t i n g l a y e r a s a c a p a c i t o r,w i t h o n e p l a t e b e i n g t h e m e t a l t a r g e t a n d t h e o t h e r t h e s u r f a c e o f t h e i n s u l a t i n g l a y e r.I o n s c o n t i n u e t o a r r i v e a t t h e s u r f a c e i n t h i s w a y,c o n t i n u a l l y s p u t t e r i n g a w a y t h e i n s u l a t o r,b u t a l s o c o n t i n u a l l y"c h a r g i n g u p"t h e c a p a c i t o r.T h i s c h a r g i n g p r o c e s s i n c r e a s e s t h e v o l t a g e a c r o s s t h ei n s u l a t i n g l a y e r.B e c a u s e t h e t o t a l v o l t a g e f r o m b a c k i n g p l a t e t o g r o u n d i s f i x e d b y t h e p o w e r s o u r c e, a n y v o l t a g e b u i l d i n g u p a c r o s s t h e i n s u l a t i n g l a y e r m u s t b e s u b t r a c t e d f r o m t h e v o l t a g e a c r o s s t h ec a t h ode d a r k s p a c e.P u t a n o t h e r w a y,a s t h e i o n s c h a r g e u p t h e s u rf a c e,t h e y f o r c e t h e s u r f a c ep o t e n t i a l c l o s e r a n d c l o s e r t o t h e p l a s m a p o t e n t i a l.E v e n t u a l l y t h e r e w i l l b e n o e l e c t r i c f i e l d l e f tb e t w e e n t h e p l a s m a a n d t h e t a r g e t s u r f ac e t o a t t r a c t m o r e i o n s;t h e c h a r g i n g o f t h e t a r g e t"c a p a c i t o r" w i l l h a v e u s ed u p a l l o f t he p o w e r s u p p l y p o t e n t i a l.A t t h i s p o i n t,s p u t t e r i n g of t h e i n s u l a t i ng r e g i o n m u s t s t o p(s e e F i g u r e1).F i g u r e1.C h a r g e b u i l d u p o n i n s u l a t i n g l a y e r s.d e s c r i b e d p r o v i d e d t h a t t h e i n s u l a t i n g f i l mi s a b l e t o w i t h s t a n d t h e e l e c t r i c f i e l d c r e a t e db y t h ec h a r g i n g o n i t s s u r f a c e.I n p r a c t i c et h i s i s f r e q u e n t l y n o t t h e c a s e,a n d t h e f i l mb r e a k s d o w n w h e n t h e e l ec t r i c f i e ld re a c h e st h e d i e l e c t r i c s t r e n g t h o f t h e m a t e r i a l.T h i se l e c t r i c a l b r e a k d o w n i s of t e n t h e i n i t i a t o rf o r a n a r c.I n a d d i t i o n,a s t h e f i l m i s t o r na s u n d e r,p a r t i c u l a t e s a r e f o r m e d.T h e s ep a r t i c u l a t e s a r e o f t e n c h a r g e d a n d c a n d e f yg r a v i t y a s t h e y a r e m o v e d l o n g d i s t a n c e s b ye l e c t r i cf i e l d s i n t h e c h a m b e r.T h e s ep r o j e c t i l e s c a n b e i n c l u d e d i n t h e f i l mg r o w i n g o n t h e s u b s t r a t e o r c a n r e m a i n o nt h e t a r g e t o r a d j a c e n t p a r t s t o b e c o m e t h es o u r c e o f f u r t h e r a r c i n g.O u r g o a l i s t op r e v e n t t h e e l e c t r i c a l b r e a k d o w n,t h u sp r e v e n t i n g a r c i n g a n d p a r t i c u l a t e s.B e f o r e Id e s c r i b e h o w t h i s c a n b e d o n e,h o w e v e r,I’dl i k e t o d i s c u s s t h e c a s e o f n o n u n i f o r m f i l m s b r i e f l y.I n r e a c t i v e s p u t t e r i n g,a t h i n f i l m i s c r e a t e d o n t h e s u r f a c e w h i c h i s n o t o f u n i f o r m t h i c k n e s s(s e eF i g u r e1).I n r e a c t i v e s p u t t e r i n g,t h e m e t a l l i c t a r g e t i s s p u t t e r e d a w a y a s u s u a l i n t h e e r o s i o n z o n e, b u t a n o n u n i f o r m i n s u l a t i n g f i l m b u i l d s u p i n t h e t r a n s i t i o n r e g i o n b e t w e e n t h e e r o s i o n z o n e a n d t h e u n s p u t t e r e d r e g i o n o f t h e t a r g e t(s e e F i g u r e2).T h i s f i l m’s t h i c k n e s s b u i l d s u p f r o m z e r o t os u b s t a n t i a l v a l u e s o v e r a f a i r l y s m a l l d i s t a n c e,a n d o n e m i g h t e x p e c t t h a t t h e f i r s t p l a c e b r e a k d o w n m i g h t o c c u r w o u l d b e i n t h e t h i n n e s t s p o t o f t h e f i l m,s i n c e t h i n n e r f i l m s b r e a k d o w n a t l o w e rv o l t a g e s.T h i s t u r n s o u t n o t t o b e t h e c a s e.F i g u r e 2. T h e e r o s i o n z o n e i n r e a c t i v e s p u t t e r i n g .F i g u r e 3 i s a n e x p a n d e d v i e w o f t h et r a n s i t i o n r e g i o n f r o m m e t a l t o i n s u l a t i n g f i l m . I m a g i n e a s m a l l a r e a A o f t h i s i n s u l a t i n g f i l m b e i n g b o m b a r d e d b y a c u r r e n t d e n s i t y J . T h e a r e a A f o r m s ac a p a c i t o r w i t h t h e u nde r l y i n g m e t a l w i t h a c a p a c i t a n c e o f(1) C = εεr d0AI n e q u a t i o n (1), C i s t h e c a p a c i t a n c e , εr i s t h e d i e l e c t r i c c o n s t a n t o f t h e f i l m , εr i s t h e p e r m i t t i v i t y o f f r e e s p a c e , a n d d i s t h e f i l m t h i c k n e s s .T h e v o l t a g e o n t h i s c a p a c i t o r i s r e l a t e d t o t h e c h a r g e o n t h e a r e a A a s f o l l o w s :(2) Q = J A t = C Vw h e r e t i s t h e t i m e s i n c e t h e c a p a c i t o r w a s l a s t d i s c h a r g e d .F i g u r e 3. T r a n s i t i o n z o n e f r o m m e t a l t o i n s u l a t o r .C o m b i n i n g e q u a t i o n s (1) a n d (2) y i e l d s(3) E = V /d = Jtr 0εεw h e r e E i s t h e e l e c t r i c f i e l d i n t h ef i l m .T h u s t h e e l e c t r i c f i e l d i n t h e i n s u l a t o r i s n o t d e p e n d e n t u p o n t h e t h i c k n e s s d . R e w r i t i n g e q u a t i o n (3), s u b s t i t u t i n g E b f o r t h e v a l u e o f t h e b r e a k d o w n f i e l d f o r t h e i n s u l a t i n g m a t e r i a l (i .e ., i t s d i e l e c t r i c s t r e n g t h ) a n d s o l v i n g f o r t h e t i m e t o r e a c h t h i s b r e a k d o w n y i e l d s(4) t b = εεr bE J0E q u a t i o n (4) g i v e s t h e t i m e t o b r e a k d o w n , a g a i n i n d e p e n d e n t o f d , t h e f i l m t h i c k n e s s . T h e r e a s o n f o r t h i s i n d e p e n d e n c e i s t h a t , w h i l e t h e b r e a k d o w n v o l t a g e i s l o w e r a t t h e t h i n n e r c r o s s -s e c t i o n s , t h e c a p a c i t a n c e t o t h e t a r g e t i s l a r g e r a s w e l l , a n d s o t h e v o l t a g e b u i l d s u p m o r e s l o w l y t h e r e . T h e t w o e f f e c t s c a n c e l a n d a l l r e g i o n s o f t h e f i l m b r e a k d o w n a t t h e s a m e m o m e n t , t h e t i m e o f w h i c h i s g i v e n b y e q u a t i o n (4).T h i s a s s u m e s t h a t t h e c u r r e n t d e n s i t y J i s c o n s t a n t , w h i c h o f c o u r s e i t i s n o t ; t h e c u r r e n t d e n s i t y i s h i g h e r n e a r t h e t h i n n e r r e g i o n s , w h i c h w o u l d c a u s e t h e s e r e g i o n s t o b r e a k d o w n f i r s t . T h e p o i n t o f t h e d e r i v a t i o n i s t o s h o w t h a t t h e r e i s a c o m p e n s a t i n g e f f e c t w h i c h h o l d s o u t h o p e t h a t i f t h e f i l m c a n b e d i s c h a r g e d p e r i o d i c a l l y (w i t h a p e r i o d l e s s t h a n t b ), b r e a k d o w n c a n b e a v o i d e d .7+( 62/87,21A c i r c u i t h a s b e e n d e s i g n e d w h i c h r e v e r s e s t h e t a r g e t v o l t a g e p e r i o d i c a l l y t o a v a l u e a b o u t o n e-t e n t h o f t h e o p e r a t i n g t a r g e t v o l t a g e.T h e v o l t a g e r e v e r s a l a t t r a c t s e l e c t r o n s t o t h e t a r g e t s u r f a c e,d i s c h a r g i n g a n y i o n s w h i c h m a y re s i d e o n a n y i n s u l a t i n g l a y e r s.A t t h e s a m e t i m e,t h e v o l t a g e i s n o t s o l a r g e n o r t h e t i m e s o l o n g a s t o d i s t u r b t h e i o n c l o u d.O n c e t h e s u rf a c e i s d i s c h a rg e d s p u t t e r i n gc a n b e g i n a g a i n.T h e v o l t a g e r e v e r s a l i s h e ld f o r5µs,a n d t he p e r i o d b e t w e e n r e v e r s a l s i s50µs.U s i n g e q u a t i o n(4),a n d a s s u m i n g m a t e r i a l c h a r a c t e r i s t i c s o f A l2O3(E b=108v/m,εr=10),t h ed i s c h a r ge p e r i o d t b o f50µs w i l l s uf f i c e t o d i s c h a rg e a s u r f a c e b o m b a r d e d w i th a c u r r e n t d e n si t y o fa p p r o x i m a t e l y103A/m2(100m A/c m2).T h e m e t h o d u s e d t o c r e a t e t h e v o l t a g e r e v e r s a l i s s i m p l e:A n e l e c t r o n i c s w i t c h a n d t r a n s f o r m e rc i r c u i t a c t t o r e v e r s e t h e t a r g e t v o l t a g e t o a v a l u e a p p r o x i m a t e l y e q u a l t o o n e-t e n t h o f t h e o p e r a t i n g t a r g e t v o l t a g e.T h i s i s g e n e r a l l y s l i g h t l y m o r e p o s i t i v e t h a n t h e p l a s m a p o t e n t i a l.T h i s p o s i t i v ev o l t a g e a c t s t o a t t r a c t e l e c t r o n s f r o m t h e p l a s m a t o t h e t a r g e t.W h i l e t h e e l e c t r o n s a c t u a l l y a r r i v e i n a f e w n a n o s e c o n d s,t h e p o s i t i v e p o t e n t i a l i s m a i n t a i n e d f o r a b o u t5µs b e c a u s e s t r a y i n d u c t a n c e s a n d c a p a c i t a n c e s c a n c a u s e p o t e n t i a l d e l a y s i n t h e a p p e a r a n c e o f t h e v o l t a g e o n t h e t a r g e t s u r f a c e.W h i l e t h e p o s i t i v e p o t e n t i a l i s b e i n g a p p l i e d t o t h e t a r g e t s u r f a c e,a f a i r l y l a r g e i n d u c t a n c e,p r e l o a d e d t o b e c a r r y i n g a c u r r e n t j u s t e q u a l t o t h e p o w e r s u p p l y c u r r e n t,i s s h u n t e d a c r o s s t h ep o w e r s u p p l y l e a d s.T h i s c o m p l e t e l y i s o l a t e s t h e p o w e r s u p p l y f r o m t h e t a r g e t v o l t a g e r e v e r s a l.T h u s t h e s e r v o l o o p s o f t h e p o w e r s u p p l y d o n o t s e e t h e a b r u p t c h a n g e i n v o l t a g e a t t h e t a r g e t,a n d n e e dn o t s l e w t o a t t e m p t a c o r r e c t i o n.T h i s m e a n s t h a t t h e p o w e r s u p p l y i s r e a d y t o r e s u m e n o r m a lo p e r a t i o n w i t h i n a v e r y s h o r t t i m e a f t e r t h e p u l s e i s t e r m i n a t e d.B e c a u s e t h e f u l l o u t p u t v o l t a g e o f t h e p o w e r s u p p l y i s a p p l i e d t o t h e a b o v e-m e n t i o n e d i n d u c t a n c e,t h e c u r r e n t i n t h e i n d u c t o r,i n i t i a l l y e q u a l t o t h e s t e a d y-s t a t e p r o c e s s c u r r e n t,w i l l r a m p u p s l i g h t l y. T h i s r e p r e s e n t s s t o r a g e o f t h e e n e r g y d e l i v e r e d b y t h e p o w e r s u p p l y d u r i n g t h e p u l s e.A t10k W,t h i s a m o u n t s t o a b o u t50m J.A s p e c i a l"s n u b b e r"c i r c u i t s h u n t s t h i s e x c e s s e n e r g y t o a p o w e r r e s i s t o rd u r i n g t h e"o f f"t i m e.$5& +$1'/,1*T h e u n i t w i l l a l s o h a n d l e a r c s i f t h e y d o o c c u r.T h i s a r c h a n d l i n g a c t i o n i s a c t u a l l y i d e n t i c a l t o t h a t d e s c r i b e d a b o v e.W h e n a n a r c b e g i n s,t h e r e i s a n a b r u p t d r o p i n v o l t a g e a c r o s s t h e c a t h o d e d a r ks p a c e.A t r i g g e r c i r c u i t h a s b e e n d e s i g n e d t o m o n i t o r t h e t a r g e t v o l t a g e a n d p r o d u c e a t r i g g e r e v e n t w h e n t h e v o l t a g e d r o p s b e l o w a n o m i n a l a r c t h r e s h o l d l e v e l.T h i s t r i g g e r i s u s e d t o t u r n o n t h ee l e c t r o n i c s w i t c h i n b e t w e e n"s e l f-t r i g g e r e d"e v e n t s.S h u n t i n g t h e p o w e r s u p p l y c u r r e n t a n dr e v e r s i n g t h e v o l t a g e n o t o n l y e x t i n g u i s h e s t h e a r c b u t a l s o g r e a t l y r e d u c e s t h e e n e r g y d e l i v e r e d t o i t.T y p i c a l l y t h e e n e r g y d e l i v e r e d t o a n a r c i s u n d e r1m J.$33/,&$7,216T h e s y s t e m c a n b e a p p l i e d i n a w i d e v a r i e t y o f p r o c e s s e s.I n a p p l i c a t i o n s w h e r e r e d u c t i o n o fp a r t i c u l a t e s i s o f c o n s i d e r a b l e i n t e r e s t,t h e u n i t s c a n i n c r e a s e y i e l d s b e c a u s e o f t h e m u c h l o w e re n e r g y d u m p e d i n t o t h e a r c.I n r e a c t i v e s p u t t e r i n g a p p l i c a t i o n s,t h e u n i t s c a n p e r m i t m u c h h i g h e rr a t e s t h a n w e r e p r e v i o u s l y p o s s i b l e w i t h a s i n g l e c a t h o d e s y s t e m.I n a d d i t i o n,t h e l o w e r e n e r g yc o n t e n t o f t h e a r c h a nd l i n g i n s u c h s y s te m s m a k e s t h e i n c l u s i o n of p a r t i c u l a t e s i n t h e f i l m(w h i c ho f t e n d e s t r o y s i t s o p t i c a l p r o p e r t i e s)a m u c h s m a l l e r p r o b l e m.T h i s d e s i g n h a s b e e n u s e d i n p r o c e s s e s w h e r e i n t h e f i n a l f i l m w a s h i g h l y i n s u l a t i n g,i n c l u d i n g S i O2 a n d A l2O3.I n s u c h c a s e s t h e p r o b l e m o f t h e"d i s a p p e a r i n g a n o d e"m u s t b e d e a l t w i t h.E v e r y s y s t e m m u s t h a v e a n e l e c t r o d e w h i c h c o l l e c t s e l e c t r o n s l o s t f r o m t h e p l a s m a;t h e r e s u l t i n g c u r r e n t f l o w i sa c t u a l l y t h e r e t u r n c u r r e n t t o t h e p o w e r s u p p l y.T h e a n o d e m a yb e a s e p a r a t e e l ec t r ode o r t h ec h a m b e r w a l l s.I f,a s i s u s u a l l y t h e c a s e,t h e a n ode i s e x p o s e d t o d e p o s i t i o n,a n d if t h e d e p o s i t i sh i g h l y i n s u l a t i n g,e v e n t u a l l y t h e p r o c e s s m u s t s t o p f o r l a c k o f a n e f f e c t i v e a n o d e.F r o m t h e p o w e rs u p p l y’s p o i n t o f v i e w,t h e i m p e d a n c e o f t h e p l a s m a a p p e a r s t o i n c r e a s e o v e r t i m e u n t i l t h e s u p p l yr e a c h e s i t s v o l t a g e l i m i t.I t i s p o s s i b l e t o d e s i g n a n o d e s w h i c h a r e s t i l l e f f e c t i v e w h e n e x p o s e d t oc o a t i n g;t h e s e a r e u s u a l l y a r e e n t r a n t s t r u c t u r e o f s o m e s o r t.U s e r s i n t e nd i n g t o u se d e s i g n s s u c h a s t h i s t o d e p o s i t h i g h l y i n s u l a t i n gf i l m s f r o m a s i ng l e c a th o d e m u s t c a r e f u l l y c o n si d e r t h i s p r o b l e mf r o m t h e o u t s e t w h e n d e s ig n i n g s u ch s y s t e m s.27+(5 0(7+2'6S e v e r a l o t h e r m e t h o d s r e l a t i n g t o a r c c o n t r o l h a v e b e e n d e s c r i b e d i n t h e l i t e r a t u r e.T h e s e i n c l u d e u s e o f d u a l s o u r c e s[2,3,4],s o-c a l l e d"m i x e d m o d e"s p u t t e r i n g[5],l o w-f r e q u e n c y a c t e c h n i q u e s[6],a n du s e o f r a d i o f r e q u e n c i e s e i t h e r a l o n e o r i n c o m b i n a t i o n w i t h d c p o w e r.T h e p r i n c i p a l a d v a n t a g e o f t h e S p a r c a n d S p a r c-l e t e c h n i q u e i s t h a t i t c a n b e a d d e d t o e x i s t i n g s y s t e m s a t l o w i n s t a l l e d c o s t a n dw i t h a m i n i m u m o f s y s t e m c h a n g e s.&21&/86,21I t i s p o s s i b l e t o d e s i g n a c t i v e s w i t c h i n g c i r c u i t r y w h i c h c a n g r e a t l y r e d u c e t h e i n c i d e n c e o f a r c i n g i n r e a c t i v e s p u t t e r i n g p r o c e s s e s,p a r t i c u l a r l y w h e n t h e r e s u l t i n g f i l m i s h i g h l y i n s u l a t i n g.T h e s a m et e c h n i q u e s c a n b e u s e d t o l o w e r t h e e n e r g y a v a i l a b l e t o a n a r c,a n d t h e r e b y g r e a t l y d e c r e a s e t h ed e t r i m e n t a l e f f e c t s o f a r c i n g,i n c l u d i n g p a r t i c u l a t e g e n e r a t i o n a n d t a r g e t a n d s u b s t r a t e d a m a g e.T h e t e c h n i q u e c a n g e n e r a l l y b e a p p l i e d q u i c k l y a n d i n e x p e n s i v e l y t o e x i s t i n g s y s t e m s w i t h o u t s y s t e mr e d e s i g n.$&.12:/('*0(176T h e a u t h o r w o u l d p a r t i c u l a r l y l i k e t o c r e d i t G e o f f D r u m m o n d,t h e i n v e n t o r o f t h e d e v i c e,f o r h i sc o n t r i b u t i o n t o t h e f i e l d,a nd a l s o L a r r y T r u be l l,w h o d e s i g n e d t h e t r i g g e r c i r c u i t r y a n df i n a lp r o d u c t s.C r e d i t i s a l s o d u e t o e x p e r i m e n t e r s a t m a n y b e t a t e s t s i t e c o m p a n i e s;u n f o r t u n a t e l y,m o s t o f t h e m a r e w o r k i n g o n h i g h l y p r o p r i e t a r y p r o c e s s e s a n d h a v e a s k e d t h a t t h e i r w o r k b e h e l dc o n f ide n t i a l.5()(5(1&(6[1]L.A n d e r s o n,"A N e w T e c h n i q u e o f A r c C o n t r o l i n D C S p u t t e r i n g",35t h A n n u a l S V C T e c h n i c a lC o n f e r e n c e P r o c e e d i n g s(1992),p p.325-329.[2]G.E s t e,"A Q u a s i-D i r e c t C u r r e n t S p u t t e r i n g T e c h n i q u e f o r D e p o s i t i o n o f D i e l e c t r i c s a t E n h a n c e d R a t e s",J o u r n a l V a c u u m S c i.&T e c h.A,(J u n e1988),p.1845f f.[3]S.S h i l l e r,K.G o e d i c k e,a n d C h.M e t z n e r,"A d v a n c e s i n P u l s e d M a g n e t r o n S p u t t e r i n g(P M SP r o c e s s)",p a p e r p r e s e n t e d a t t h e I n t e r n a t i o n a l C o n f e r e n c e o n M e t a l l u r g i c a l C o a t i n g s a n d T h i n F i l m s (I C M C),25/29A p r i l1994,S a n D i e g o.。
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A b s t r a c t - - T h e roots of the complex transcendental equations that result from the application of the modal method to the scattering problem for a metallic groove are obtained iteratively as fixed points of entire functions of the form Fc(z), where c, z E E. Iterations are performed with Fc(z) or an appropriate branch of its multiple-valued inverse function, that is, zj+x = Fc(zj) or Zj+l = F~-l(zj), respectively. Since convergence fails near double roots, an insightful study of the problem is made and high-precision solutions near double roots are obtained by interpolation. Examples are given to illustrate the behaviour of the methods in different situations, with a connection to fractai theory. © 1998 Elsevier Science Ltd. All rights reserved. K e y w o r d s - - M o d a i method, Iterative solution of transcendental equations, Scattering problem, Metallic groove, Helmholtz equation, Fractals.
des@dr, uba. ar
R. VAILLANCOURT Department of Mathematics and Statistics, University of Ottawa Ottawa, Ontario, Canada K1N 6N5
remiv~mathstat, uottawa, ca
D. C. SKIGIN
Grupo de Electromagnetismo Aplicado, Departamento de Fisica Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires Ciudad Universitaria, PabellSn I, 1428 Buenos Aires, Argentina
Pergamon
PII: S 0 8 9 8 - 1 2 2 1 ( 9 8 ) 0 0 0 8 8 - 1
Computers Math. Applic. Vol. 35, No. 11, pp. 98-119, 1998 © 1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0898-1221/98 $19.00 + 0.00
Computer Solution of the Scattering Problem for a G r o o v e in a M e t a l l i c P l a n e Using the Modal Method
A. M. C. RUEDIN Departamento de ComputaciSn, Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires, Ciudad Universitaria PabellSn I, 1428 Buenos Aires, Argentina anita@de, uba. ax
Computer Solution of the tering Problem
99
(dielectrics, metals, etc.), to find the modal functions is a very complicated process, making the use of the modal method inconvenient in such cases. However, it is well known that an arbitrary profile can be approximated by layers of rectangular shape. This idea was first suggested by Peng et al. [1] and used by many authors [2,3]. In each layer, the fields can be expanded in modal functions corresponding to a rectangular groove [4-6], these functions being combinations of sines and cosines. Then, the problem can be solved by matching the fields at the interfaces. Recently, Li [7] developed a multilayer modal method for infinite periodic gratings of arbitrary profile, which can be applied to any kind of isotropic materials. Employing this idea, suggested in [7] and developed in [8] for a perfectly conducting surface, we propose to solve the scattering problem for a metallic surface with a groove of arbitrary shape. The first step consists in finding the modal eigenfunctions of a rectangular groove in a metallic surface. This calculation leads to transcendental equations that must be solved numerically. The purpose of this work is to find the roots of the complex-valued transcendental equations appearing in the calculation of the modal functions of a rectangular groove in a metallic surface. These equations can be reduced to the transcendental equations cos z = c z and sin z = c z , where z, c E C. We survey previously proposed [9,10] numerical methods and add a new interpolation algorithm to obtain high-precision solutions near double roots. The general procedure consists in dividing the complex plane into fundamental regions and iterating in each region with the corresponding function or a branch of its multiple-valued inverse. The root of smallest modulus of cos z --- c z is found by Newton's method. A connection with fractal theory is brought up. In Section 2, we describe the scattering problem and derive the two transcendental equations to be solved. The numerical methods used to solve both equations are detailed in Sections 3 and 4, where different algorithms are given to deal with all possible situations, together with several illustrative figures. Examples of the results obtained and a comparison of the efficiency of the different methods in some particular cases are presented in Section 5. Section 6 deals with the eigenvalues of the scattering problem. A conclusion is found in Section 7.