129
Potential Density(cm^-3) : Electron and Ion + + + + + + – – – – – – – + + – + ~ Sheath Sheath ) cos( 0 t V • 1D Planar RF Voltage-Driven System Bul k Pla sma Substrate X direction
PotentialDensity(cm^-3): Electron and Ion+++++~SheathSheath 1D Planar RF Voltage-Driven SystemBulk PlasmaSubstrateX direction
Klystron Phase spaceDensityKinetic energyuz2 cm3 cm
PIC Overview Applications of PIC model Basic plasma physics: waves and instabilities Magnetic fusion Gaseous discharges Electron and ion optics Microwave-beam devices Plasma-filled microwave-beam devices
Congratulations! It's nice to get emails like this, isn't it. Cheers, Mike Lieberman On Sep 28, 2005, at 5:46 PM, wrote: > Subject: Your article has been downloaded 250 times! > Dear Professor Lee, > I am pleased to tell you that your article, "Particle and fluid > simulations of low-temperature plasma discharges: benchmarks and > kinetic effects", in Journal of Physics D: Applied Physics, Vol 38, > ppR283 (2005), has been downloaded 250 times so far. > > This was achieved in 12 days from the date of publication. To put > this into context, across all IOP journals 10% of articles were > accessed over 250 times this quarter. > You can link directly to your article at: > http://stacks.iop.org/0022-3727/38/R283 > I would like to thank you for supporting Journal of Physics D: > Applied Physics and I trust that you have found our publication > process to be friendly and efficient. I hope you will consider > submitting further papers to the journal and encourage your > colleagues to do likewise. > Our publication times are highly competitive and we use a fully > electronic editorial process from submission to peer review to > production. Electronic submission to Institute of Physics journals > can be done very simply at the following web page: > http://www.iop.org/journals/authorsubs > We look forward to working with you again. If you have any queries > about the journal, please don't hesitate to contact me. > If you would prefer not to receive further updates on the number of > times this article has been downloaded, please reply to this email > with the word remove in the subject line. > Kind regards > Sarah Quin , Publisher > Journal of Physics D: Applied Physics > Institute of Physics Publishing > Dirac House, Temple Back, Bristol BS1 6BE, England
PIC Overview PIC codes simulate plasma behavior of a large number of charges particles using a few representative super particles.
These type of codes solve the Newton-Lorentz equation of motion to move particles in conjunction with Maxwells equations (or a subset).
Boundary conditions are applied to the particles and the fields to solve the set of equations.
PIC codes are quite successful in simulating kinetic and nonlinear plasma phenomenon like ECR, stochastic heating, etc. PIC Codes Overview
PIC-MCC Flow ChartFig: Flow chart for an explicit PIC-MCC scheme Particles in continuum space Fields at discrete mesh locations in space Coupling between particles and fieldsIIIIIIIVIVV
I. Particle Equations of Motion Newton-Lorentz equations of motion In finite difference form, the leapfrog methodFig: Schematic leapfrog integration
I. Particle Equations of Motion Second order accurate Requires minimal storage Requires few operations Stable for
I. Particle Equations of Motion Boris algorithm
I. Particle Equations of MotionFinally,
II. Particle Boundary Conductor : absorb charge, add to the global Dielectric : deposit charge, weight q locally to mesh Absorption Reflection Physical reflection
Specular reflection 1st order error Thermionic Emission Fowler-Nordheim Field Emission Childs Law Field Emission Gausss Law Field Emission
II. Particle Boundary Secondary electron emission+, Ion impact secondary emission Electron impact secondary emission Important in processes related to high-power microwave sources Photoemission
III. Electrostatic Field Model Possions equation Finite difference form in 1D planar geometry Boundary condition : External circuitFig: Schematic one-dimensional bounded plasmawith external circuit
EECE695: Computer Simulation (2005)
Particle-in-Cell Techniques
HC Kim, SJ Kim, and JK LeePlasma Application Modeling Group, POSTECH References: Minicourse by Dr. J. P. Verboncoeur (PTS Group of UC Berkeley) in IEEE International Conference on Plasma Science (2002) Plasma Physics via Computer Simulation by C.K. Birdsall and A.B. Langdon (Adam Hilger, 1991)
XPDx1 Flow ChartFig: Flow chart for an explicit PIC-MCC schemeIIIIIIIVIVV
Collisions Electron-neutral collisions Elastic scattering (e + A e + A) Excitation (e + A e + A*) Ionization (e + A e + A+ + e) Ion-neutral collisions Elastic scattering (A+ + A A+ + A) Charge exchange (A+ + A A + A+)
V. Monte-Carlo Collision Model The MCC model statistically describes the collision processes, using cross sections for each reaction of interest. Probability of a collision event For a pure Monte Carlo method, the timestep is chosen as the time interval between collisions.where 0< R< 1is a uniformly distributed random number. However, this method can only be applied when space charge and self-field effects can be neglected.
V. Monte-Carlo Collision Model Computing the collision probability for each particle each timestep is computationally expensive. Null collision method 1. The fraction of particles undergoing a collision each time step is given by 3. The type of collisions for each particle is determined by choosing a random number, 2. The particles undergoing collisions are chosen at random from the particle list.Fig: Summed collision frequencies for the null collision method.Null collisionCollision type 3Collision type 1Collision type 2
+++++~SheathSheathj = 1, , N~ 1D Asymmetric Dual-Freq. Voltage-Driven SystemCapacitively Coupled Plasma 1D PIC-MCC MCC (Monte-Carlo Collision) Processes - Electron-Neutral Collisions (Ionization, Scattering, Excitation) - Ion-Neutral Collisions (Charge-exchange, Scattering)Bulk PlasmaSubstrate
Vx vs. x for electronsVx vs. x for ionsDensity vs. xPotential vs. xCapacitively Coupled Plasma 1D PIC-MCC
Ion Energy Distribution Function High frequency : 100MHz (100V) Low frequency :1MHz (102V) Symmetric Discharge L = 2.5 cm 10 mTorr20 mTorr50 mTorr30 mTorr
Electron Energy Distribution FunctionIn the discharge center
Single langmuir probe measurement result from KAISTElectron temperature is decreased as pressure is increasedElectron temperature is comparable to PIC simulationTable 1 Electron temperature from PIC simulation (pressure=110mT)Comparison Between PIC and Experiment (II)
27MHz/2MHz300/300300/900600/300600/900Te (eV)2.732.72.772.7
PIC-MCC versus Fluid Simulation (I) EEPF obtained from PIC/MCC is totally in disagreement with that of LFA because of nonlocal behavior of electrons.Due to Ramsauer minimum~ 4eV
IV. Coupling Fields to Particles Particle and force weighting : connection between grid and particle quantities Weighting of charge to grid Weighting of fields to particlesa point chargegrid point
IV. Coupling Fields to Particles Nearest grid point (NGP) weighting fast, simple bc, noisy Linear weighting : particle-in-cell (PIC) or cloud-in-cell (CIC) relatively fast, simple bc, less noisy Higher order weighting schemes slow, complicated bc, low noisyNGPLinear splineQuadratic spline1.00.50.0Cubic splineFig: Density distribution function of a particle atfor various weightings in 1DPosition (x)
IV. Coupling Fields to ParticlesFig: Charge assignment for linear weighting in 2DAreas are assigned to grid points; i.e., area a to grid point A, b to B, etc
PDP StructureAC PDPDischarge in PDP
Striation Profiles in PDP 2D PIC/MCC Pressure dependence of striations : Number of peaks depend on the pressure and electrode size.AnodeCathode
Input file(II) 1.001.00, -1.00The direction
The comparison of like-sign and opposite-sign results(I)Like-signOpposite-sign Like-sign : electron electron two stream case Opposite-sign : electron ion two stream case
The comparison of like-sign and opposite-sign results(II)Field energyKinetic energy Field & Kinetic energy distribution in time is different. The peak field energy value in like-sign case is higher than that of opposite- sign case and the time to reach to the peak value longer than that of opposite- sign case.
Simulation Domain of Klystron (SJ Kim)RF input portRF output portE-beamCylindrical Axis10.05 cm13.07 cm9.55 cm37.2 cm7.569 cm6.66 cm Simulation condition: Beam emitter: I= 12 kA, ud =2.48e8 m/s Input port : Rin=2300 , R=20 , f=7.69 GHz Output port : R=47.124
Example of Klystron Simulation Phase spaceDensityKinetic energyuz
Simulation Results at 10 ns
Simulation Results at 6 us
KE as a Function of Beam Current
KE as a Function of Beam Energy
Klystron Phase spaceDensityKinetic energyuz2 cm3 cm
Simulation Results at 0.5 ns and 2.5 ns
Simulation Results at 10 ns and 20 ns
Simulation Results at 6 us
Simulation Results at 0.5 ns and 2.5 ns
Simulation Results at 10 ns and 20 ns
Simulation Results at 6 us
Charge Conservation SchemeBoris-DADI correction (OOPIC)Langdon-Marder correction (MAGIC)
ECE586: Advanced E&M Simulation (2004)
On PDX1 Program
2004. 9. 16HyunChul Kim and J.K. LeePlasma Application Modeling, POSTECH References: Minicourse by Dr. J. P. Verboncoeur (PTS Group of UC Berkeley) in IEEE International Conference on Plasma Science (2002) Plasma Physics via Computer Simulation by C.K. Birdsall and A.B. Langdon (Adam Hilger, 1991)
A Series of XPDX1*r~LRCComputation Space* Developed by PTS group, UC BerkeleyAll are available at http://ptsg.eecs.berkeley.edu XPDx1: X window (using xgrafix library), Plasma Device, 1 Dimensional (1d3v), Bounded (with external circuit drive), Electrostatic Code XPDP1 (x=P) : Planar Configuration XPDC1 (x=C) : Cylindrical Configuration XPDS1 (x=S) : Spherical Configuration
PIC Overview Plasma behavior of a large number of charges particles are simulated by using a few representative super particles.
PIC codes solve fundamental equations, the Newton-Lorentz equation of motion to move particles in conjunction with Maxwells equations (or a subset) with few approximations.
PIC codes are quite successful in simulating kinetic and nonlinear plasma phenomenon like ECR, stochastic heating, etc. PIC Codes Overview
Computer Simulation of PlasmaKinetic DescriptionFluid DescriptionVlasov, Fokker-PlanckCodesParticleCodesHybrid CodesFluidCodes The particle-particle model The particle-mesh model The particle-particleparticle-mesh model Particle codes
XPDx1 Flow ChartFig: Flow chart for an explicit PIC-MCC schemeIIIIIIIVIVV Particles in continuum space Fields at discrete mesh locations in space Coupling between particles and fields
II. Particle Boundary Secondary electron emission Ion impact secondary emission Electron impact secondary emission+ Conductor : absorb charge, add to the global Absorption
III. Electrostatic Field Model In electrostatics, Maxwells equation in vacuum(Poissons equation)Or Gauss law
III. Electrostatic Field Model Possions equation Finite difference form in 1D planar geometry Boundary condition : External circuitFrom Gausss law, Short circuit Open circuit
III. Electrostatic Field Model Voltage driven series RLC circuit From Kirchhoffs voltage law, One second order difference equation where
Weighting in 1D For linear weighting in cylindrical coordinates,( 0 < j < N )IV. Coupling Fields to Particles
XPDx1 Flow ChartFig: Flow chart for an explicit PIC-MCC schemeIIIIIIIVIVV
V. Monte-Carlo Collision Model There is a finite probability that the i-th particle will undergo more than one collision in the timestep.Thus, the total number of missed collisions (error in single-event codes)Hence, traditional PIC-MCC codes are constrained by for accuracy.
Numerical Parameters Choose x and t to resolve the smallest important physical feature Require x < Debye length, sheath length, wave length, Larmor radius, boundary feature, etc. Require for all species (particle Courant) for accurate sampling of fields. Require for accuracy of explicit leap frog mover or for accuracy when space charge forces are important. Require when collisions are important. Require # of superparticles per cell > 10. It should be larger in simulations where particles remain trapped for long times.
Example of XPDP1 Input FileRF DISCHARGE(IN MKS UNITS) Voltage-driven with electron-neutral collisions (Argon atom) -nsp---nc---nc2p---dt[s]---length[m]--area[m^2]--epsilonr---B[Tesla]---PSI[D]-- 2 400 1.8e6 8e-12 0.03 0.01 1.0 0 .0 0.0 -rhoback[C/m^3]---backj[Amp/m^2]---dde--extR[Ohm]--extL[H]---extC[F]---q0[C]- 0.0 0.0 0.0 0.0 0.0 1.0 0.0 -dcramped--source--dc[V|Amp]--ramp[(V|Amp)/s]---ac[V|Amp]---f0[Hz]--theta0[D]- 0 v 0.0 0.0 100 13.56e6 0 --secondary--e_collisional---i_collisional---reflux---nfft--n_ave--nsmoothing--ntimestep-- 1 1 2 0 0 276549 6 0 --seec(electrons)---seec(ions)---ion_species----Gpressure[Torr]---GTemp[eV]---imp-- 0.0 0.2 2 100e-3 0.026 0 ---GAS----psource--nstrt-- 1 0 0 SPECIES 1 ----q[C]-------m[Kg]---j0L[Amp/m^2]---j0R[Amp/m^2]----initn[m^-3]----k-- -1.602e-19 9.11e-31 0.0 0.0 5e14 1 --vx0L[m/s]---vxtL[m/s]--vxcL[m/s]---vxLloader(0=RNDM,1=QS)-- 0.0 4.19e5 0.0 1 --vx0R[m/s]---vxtR[m/s]--vxcR[m/s]---vxRloader 0.0 4.19e5 0.0 1 --v0y[m/s]---vty[m/s]---vyloader---v0z[m/s]---vtz[m/s]--vzloader-- 0.0 4.19e5 1 0.0 4.19e5 1 --nbin----Emin[eV]----Emax[ev]---maxnp 200 0.0 20.0 300000 -For-Mid-Diagnostic---nbin----Emin[eV]---Emax[eV]----XStart--XFinish 300 0.0 20.0 0.0 0.03 SPECIES 2 ----q[C] ------m[Kg]---j0L[Amp/m^2]---j0R[Amp/m^2]----initn[m^-3]----k- 1.602e-19 6.68e-26 0.0 0.0 5e14 1 -vx0L[m/s]---vxtL[m/s]--vxcL[m/s]---vxLloader(0=RNDM,1=QS)-- 0.0 97.8 0.0 1 --vx0R[m/s]---vxtR[m/s]--vxcR[m/s]---vxRloader 0.0 97.8 0.0 1 --v0y[m/s]---vty[m/s]---vyloader---v0z[m/s]---vtz[m/s]--vzloader-- 0.0 97.8 0 0.0 97.8 1 --nbin----Emin[eV]----Emax[ev]---maxnp-- 100 0.0 100.0 300000 -For-Mid-Diagnostic---nbin----Emin[eV]---Emax[eV]----XStart--XFinish-- 200 0.0 0.3 0.0 0.03
Some Input Parameters nsp : Number of species. nc: The number of spatial cells. x=length/nc nc2p: Superparticle to actual particle weight. The initial number of superparticles is N=initnarealength/nc2p. dt: Timestep for simulation in seconds. length: The length of the system (distance between electrodes) in meters. B: Applied homogeneous magnetic field in Tesla PSI: Angle of the B-field in degrees extC: The external circuit capacitance in Farads. Used in conjuction with extL, extR and the driving source. source: Either a voltage (v) or current (i) source f0: AC frequency of the source. GAS: The type of gas, important when collisions are turned on. Helium = 1, Argon = 2, Neon = 3, Oxygen = 4. Gpressure : Background gas pressure in Torr. q: Charge of the particle in Coulombs. m: Mass of the particle in Kgs. initn: Initial particle number density For details, refer the source code itself or the manual inside the package of source file.
Example of Result (driven by RF)Vx vs. x for electronsDensity vs. xVx vs. x for ionsPotential vs. xIon flux vs. Ion EnergyElectron Temperature vs. x
Semi-conductor Plasma Processing
H.S. Ko and J.K. Lee Department of Electronic and Electrical Engineering,POSTECH( Comparison of Plasma Kinetic Properties of Various Equipments )
CCP (Capacitively Coupled Plasma)
Outline of Charge-up SimulationPlasmawaferCharge-up & etching Simulation space Plasma Etching Reactor (CCP, ICP, etc) Plasma source simulation spaceIons are accelerated in sheath region
Single Frequency CCP: Effect of pressure (20, 45, 100mTorr)
Plasma potential profiles, plasma density and corresponding ion energy distributions at the smaller (LHS) electrode. The IEDF spread corresponds to the mean potential drop at the left electrode.
27 MHz
800 V
2 cm
20 mTorr,
45 mTorr
100 mTorr
Electron temperature profiles, electron energy probability function and power absorbed by electrons and ions for different values of pressure
J = 2.65 LangmuirProbe by GodyakThomson Scattering by Elsabbagh, MuraokaJ = 3.8 mA/cm2 Using PIC-MCCSimulationJ = 3.8EEDF Comparison for a Small-Gap CCP
Comparison with Experimental Result argon gas current density : gap distance : 2 cm
Depending on SEEC, the main mechanism for EEDF change differs. HC Kim & JK Lee, PRL (to appear)
Low-Freq. Current Effect on EEDF (Summary) W/O SEEC : from collisional (Ohmic) to collisionless (stochastic) With SEEC : - mode transition
SEEC: ION-ENERGY and ANGLE DEPENDENT SECONDARY EMISSIONFor argon:For oxygen:For each variant below SS is reachedNew SEEC for argon is incorporatedin XPDC1 code
IEDF (RHS) for Xe/Ar mixtures, 27 MHz, 800 V, 50 mTorr Comparison with experiment (Natalia B.)Experimental results for IEDFThe curves for Xe+ and Ar+ ionsare clipped from here and shown in the same scaleSimulations: IEDF for Xe:Ar=20:80 mixture, 27 MHz, 800 V, 50 mTorrQualitative comparison with experimentExperimentExperiment
1260m160m140manodecathodeElectron density and temperature: PIC simul.Striation density and temperature: out of phase
How to calculate potential and electric field in 3D charge-up simulationS.J. Kim, H.J. Lee, and J.K. LeePlasma Application Modeling Lab.Department of Electronic and Electrical EngineeringPohang University of Science and TechnologyEECE 586
What is charge-up effect?High charge-up potentialBecause of the electron shading effect in high aspect ratio etching, most of the ions reach the bottom of trench. High potential is generated at the bottom of trench. Trajectory of ions is changed.
Simulation routine and basic assumptionsIonElectronSolve LaplaceEquation
Update Potential0.2umMove particles(by E-field)All particles arrive at boundary Due to the small size of simulation domain( ~ 1um) Particle flight time is much shorter than the time interval of each entering particle. Number of space charge particle is small in the simulation domain. Ignore space charge effect in the E-field calculation. Collisions in the simulation domain are neglected. mean free path of ions or electrons(~mm) is much longer than the simulation domain size. The potential at the top and bottom boundary is the same.Surface current is neglected.
Flow chart
Simulation domain and boundary conditions
Solving electrostatic potential in 3D charge-up simulation(1)Space gradient of dielectric constant being considered, Poissons equation is as follows: Surface charges in the dielectric surfaces is only considered in right-hand.For solving PDE numerically, Poissons equation is represented as follows:
Solving electrostatic potential in 3D charge-up simulation(2)Poissons equation is described as Matrix equation
SOR algorithm for 2nd-order elliptic PDE From 2D finite difference model,Iterative procedure:Residual is calculated as follows:Finding optimal in SOR with Chebyshev acceleration
Solving electric field in 3D charge-up simulationElectric field is calculated by using Gausss law.i) Cases without changes in ii) Cases with changes in where,x-directional electric field in the plane of yz1 :
Main.c
/***********************************************************//* The main physics loop */
void XGMainLoop(){ int isp, phi_flag; it++; phi_flag=0; /* to determine if it is necessary to solve Laplace's Equaton */ for(isp=0; isp
Motions of electrons and ionsElectronIonInitialInitialSaturatedSaturated
Potential profilesAspect ratio = 7AR=3AR=3, long z-direction
3D Radiation Transport Simulation for Plasma Display Panels
HyunChul Kim and J. K. LeePlasma Application Modeling, POSTECH References: A.F. Molisch and B.P. Oehry, Radiation Trapping in Atomic Vapours (Oxford, New York, 1998). H.C. Kim, S.S. Yang, and J.K. Lee, J. Appl. Phys. 93, 9516 (2003).
History of FL3P Simulator2000/12 : Initial cut of FL3P (FLuid simulator for 3-d Plasma Display Panels) as a generalization of FL2P
2001/09 : Implementation of self-consistent radiation model in FL2P2001/10 : Its generalization to FL3P
The first journal publication on 3D fluid PDP simulation2002/02 : H.C. Kim et al., IEEE Trans. Plasma Sci. 30, 188.2002/06 : H.C. Kim et al., J. Appl. Phys. 91, 9513.
The journal publication on RT simulation in PDP2002/06 : 2D self-consistent radiation transport model for Plasma Display Panels H.J. Lee et al., Phys. Plasmas 9, 2822. 2003/06 : 3D self-consistent radiation transport model H.C. Kim et al., J. Appl. Phys. 93, 9516. 1
Resonance Radiation Trapping (I)i) Emission of Independent Excited-State Atomsii) Interaction between Excited-State Atoms (Radiation trapping: absorption/reemission processes) Level 1 Level 2 Atom 1Atom 2Imprisonment of resonant radiation2
i) Natural broadeningii) Doppler broadeningiii) Pressure broadening Caused by the finite lifetime of the atomic levels and uncertainty principle Lorentz line shape with FWHM Caused by the thermal motion of the atoms and Doppler effect Doppler line shape with FWHM Caused by the collisions with other atoms Lorentz line shape with FWHM Particle diffusion vs. radiation trapping
Particle: Flight time >> Scattering timePhoton: Scattering time >> Flight time Spectral Line BroadeningResonance Radiation Trapping (II)3
4[Ref] H.J. Lee and J.P. Verboncoeur, Phys. Plasmas 8, 3077 (2001) Frequency redistribution at absorption/reemission processResonance Radiation Trapping (III)
Holstein equation (Continuity eqn. with radiation transport)
Kernel function (Prob. that photon emitted at r is reabsorbed at r.)
Transmission factor (Freq-avg. prob. of traversing a distance R)
where R=|r-r|.
5Holstein Equation (I)
i) When is spatially uniform, it is reduced toEscape factor (escape prob. of a photon) : : Average time that a photon spends in the vapor before it escapes.Holstein Equation (II) Holstein Equationii) When distribution is identical to the lowest-order eigenmode, : Effective lifetime : Effective lifetime: Trapping factor (mean number of absorption (or emission) processes that a photon suffers)6
Numerical Methods for Holstein Equation7 Propagator Function Method (PFM) Numerical method used for our research Monte Carlo Simulation (MC) Easy to program but slower than PFM Piecewise-Constant Approximation (PCA) Piecewise-constant approximation of the eigenfunctions Superior to PFM only when subcell # is low (i.e. < 300): Fredholm integral equation of the 2nd kind
Calculation of Resonant State Density (I) Escape factor method cannot treat the redistribution of excited state density profile. In this study, we use Propagator Function Method with full calculation of Holstein equation.Propagator Function MethodFrom cell ( ) to cell ( )8
PDP Structure and DischargeAC PDPDischarge in PDP Color AC PDPs operate on the same principle as fluorescent lamps converting UV to visible light by the use of phosphors.9
Xe*(3P1) Density Profiles (I)21288 um864 umCathodeAnodeBarrier RibsSolution of Holstein Equation10600.22.01.0(Unit: #/cm )Escape Factor Method30220140540100200.4us1.0usTop View
Energy Diagram26(10.9% of total input power)(5.47%)(0.433%)(35.1%)(4.0%)(59.8%)(50.1%)(5.1%)(45.9%)(31.7%)(4.7%)(63.6%)(1.43%)(31.8%)(4.7%)(63.5%)Standard
#define SIDE_ONN_C14#define SIDE_POP_C15#define SIDE_PON_C16#define SIDE_NOP_C17#define SIDE_NON_C18#define CORN_PPP_C19#define CORN_PNP_C20#define CORN_NPP_C21#define CORN_NNP_C22#define CORN_PPN_C 23#define CORN_PNN_C 24#define CORN_NPN_C 25#define CORN_NNN_C 26#define SIDE_PPO_V 27#define SIDE_PNO_V 28#define SIDE_NPO_V 29#define SIDE_NNO_V 30#define SIDE_OPP_V 31#define SIDE_OPN_V 32#define SIDE_ONP_V 33#define SIDE_ONN_V 34#define SIDE_POP_V 35#define SIDE_PON_V 36#define SIDE_NOP_V 37#define SIDE_NON_V 38#define CORN_PPP_V 39#define CORN_PNP_V 40#define CORN_NPP_V 41#define CORN_NNP_V 42#define CORN_PPN_V 43#define CORN_PNN_V 44#define CORN_NPN_V 45#define CORN_NNN_V 46// F means flat in xz surface#define CORN_PPP_F 47#define CORN_NPP_F 48#define CORN_PPN_F 49#define CORN_NPN_F 50// U means there is a side in up-direction#define CORN_PPP_U 51#define CORN_NPP_U 52#define CORN_PPN_U 53#define CORN_NPN_U 54/**************************************//* defining internal structures */#define ACCUMULATION 0#define INJECTION 1#define OPEN 3
Main.c/******************************************//* def.h */
#define EPS0 8.8542e-12 /* (F/m) */#define NperTORR 8.3221e20#define HISTMAX 2048#define HISTMAX2 4096
#define NSMAX 2
#define QUIET_START 0#define RANDOM 1
#define GROUNDED 0#define VOLTAGE_D 1#define CURRENT_D 2
#define DIRICHLET 0#define NEUMANN 1#define PERIODIC 2#define SYMMETRIC 1
#define LEFT 0#define RIGHT 1#define UP 2#define DOWN 3
#define SIDES 2
/* ---- PNO means positive, negative, and ordinary in each directions, and C and V means concave and convex. --*/
#define NO_FACE0#define FACE_X_UP1#define FACE_X_DOWN2#define FACE_Y_UP3#define FACE_Y_DOWN4#define FACE_Z_UP5#define FACE_Z_DOWN6#define SIDE_PPO_C7#define SIDE_PNO_C8#define SIDE_NPO_C9#define SIDE_NNO_C10#define SIDE_OPP_C11#define SIDE_OPN_C12#define SIDE_ONP_C13
field.c (3)/**********************************************/ /* Calculation of surface charge of conductor */
/* Divergence theorem (i,j,k) grid Cell Q = epsilon* \int E ds - \int rho dV charge . area surface charge . */ for (i=0; i
field.c (4)/*******************************************/ /*** calculate the E field... ***/
for(i=0; i
field.c (5) else if (face[i][j][k]==FACE_Y_UP) ey[i][j][k] = sigma[i][j][k] /epsy[i][j][k]; else if (face[i][j][k]==FACE_Y_DOWN)ey[i][j][k] = -sigma[i][j][k] /epsy[i][jm1][k];
else if (face[i][j][k]==FACE_X_UP)ex[i][j][k] = sigma[i][j][k] /epsx[i][j][k];
else if (face[i][j][k]==FACE_X_DOWN)ex[i][j][k] = -sigma[i][j][k] /epsx[im1][j][k];
else if (face[i][j][k]==CORN_PPP_F || face[i][j][k]==CORN_NPP_F || face[i][j][k]==CORN_PPN_F || face[i][j][k]==CORN_NPN_F) ey[i][j][k] = sigma[i][j][k] /epsy[i][j][k];
else if (face[i][j][k]==SIDE_PPO_V) { epsi2=(eps_array[im1][j][km1]+eps_array[im1][j][k]+ eps_array[i][jm1][km1]+eps_array[i][jm1][k]+ eps_array[i][j][km1]+eps_array[i][j][k])/6.0; ex_mag = (phi[i][j][k] -phi[ip1][j][k])*idx; ey_mag = (phi[i][j][k] -phi[i][jp1][k])*idy; e_mag = sqrt(ex_mag*ex_mag + ey_mag*ey_mag); ex[i][j][k] = fabs(sigma[i][j][k])*my_div(ex_mag, e_mag)/epsi2; ey[i][j][k] = fabs(sigma[i][j][k])*my_div(ey_mag, e_mag)/epsi2; } else if (face[i][j][k]==SIDE_NPO_V) { epsi2=(eps_array[im1][jm1][km1]+eps_array[im1][jm1][k]+ eps_array[im1][j][km1]+eps_array[im1][j][k]+ eps_array[i][j][km1]+eps_array[i][j][k])/6.0; ex_mag = (phi[im1][j][k] -phi[i][j][k])*idx; ey_mag = (phi[i][j][k] -phi[i][jp1][k])*idy; e_mag = sqrt(ex_mag*ex_mag + ey_mag*ey_mag); ex[i][j][k] = fabs(sigma[i][j][k])*my_div(ex_mag, e_mag)/epsi2; ey[i][j][k] = fabs(sigma[i][j][k])*my_div(ey_mag, e_mag)/epsi2; } else if (face[i][j][k]==SIDE_OPP_V) { epsi2=(eps_array[i][j][km1]+eps_array[im1][j][km1]+ eps_array[i][j][k]+eps_array[im1][j][k]+ eps_array[i][jm1][k]+eps_array[im1][jm1][k])/6.0; ez_mag = (phi[i][j][k] -phi[i][j][kp1])*idz; ey_mag = (phi[i][j][k] -phi[i][jp1][k])*idy; e_mag = sqrt(ez_mag*ez_mag + ey_mag*ey_mag); ez[i][j][k] = fabs(sigma[i][j][k])*my_div(ez_mag, e_mag)/epsi2; ey[i][j][k] = fabs(sigma[i][j][k])*my_div(ey_mag, e_mag)/epsi2; }
field.c (6) else if (face[i][j][k]==SIDE_OPN_V) { epsi2=(eps_array[i][j][km1]+eps_array[im1][j][km1]+ eps_array[i][j][k]+eps_array[im1][j][k]+ eps_array[i][jm1][km1]+eps_array[im1][jm1][km1])/6.0; ez_mag = (phi[i][j][km1] -phi[i][j][k])*idz; ey_mag = (phi[i][j][k] -phi[i][jp1][k])*idy; e_mag = sqrt(ez_mag*ez_mag + ey_mag*ey_mag); ez[i][j][k] = fabs(sigma[i][j][k])*my_div(ez_mag, e_mag)/epsi2; ey[i][j][k] = fabs(sigma[i][j][k])*my_div(ey_mag, e_mag)/epsi2; }
}/* end of conductor surface */
else if (grid_mask[i][j][k]==2) { ex[i][j][k] = hdx*(phi[im1][j][k] - phi[ip1][j][k]); ey[i][j][k] = hdy*(phi[i][jm1][k] - phi[i][jp1][k]); ez[i][j][k] = hdz*(phi[i][j][km1] - phi[i][j][kp1]);
if (face[i][j][k]==FACE_X_UP) { /* For the case that epsilon doesn't change in y,z directions */ epsi1=epsx[im1][j][k]; epsi2=epsx[i][j][k]; ex[i][j][k]=sigma[i][j][k] +epsi1*( (phi[im1][j][k]-phi[i][j][k])*idx - 0.5*dx* ( idy*(2*phi[i][j][k]-phi[i][jp1][k]-phi[i][jm1][k])*idy +idz*(2*phi[i][j][k]-phi[i][j][kp1]-phi[i][j][km1])*idz) ); ex[i][j][k]/=epsi2; } else if (face[i][j][k]==FACE_X_DOWN) { epsi2=epsx[im1][j][k]; epsi1=epsx[i][j][k]; ex[i][j][k]=sigma[i][j][k] + epsi1*( (phi[ip1][j][k]-phi[i][j][k])*idx - 0.5*dx* ( idy*(2*phi[i][j][k]-phi[i][jp1][k]-phi[i][jm1][k])*idy +idz*(2*phi[i][j][k]-phi[i][j][kp1]-phi[i][j][km1])*idz) ); ex[i][j][k]/=-epsi2; } else if (face[i][j][k]==FACE_Y_UP) { epsi1=epsy[i][jm1][k]; epsi2=epsy[i][j][k]; ey[i][j][k]=sigma[i][j][k] + epsi1*( (phi[i][jm1][k]-phi[i][j][k])*idy - 0.5*dy* ( idx*(2*phi[i][j][k]-phi[ip1][j][k]-phi[im1][j][k])*idx +idz*(2*phi[i][j][k]-phi[i][j][kp1]-phi[i][j][km1])*idz) ); ey[i][j][k]/=epsi2; } else if (face[i][j][k]==FACE_Y_DOWN) { epsi2=epsy[i][jm1][k]; Note
field.c (7) epsi1=epsy[i][j][k]; ey[i][j][k]=sigma[i][j][k] + epsi1*( (phi[i][jp1][k]-phi[i][j][k])*idy - 0.5*dy* ( idx*(2*phi[i][j][k]-phi[ip1][j][k]-phi[im1][j][k])*idx +idz*(2*phi[i][j][k]-phi[i][j][kp1]-phi[i][j][km1])*idz) ); ey[i][j][k]/=-epsi2; } else if (face[i][j][k]==FACE_Z_UP) { epsi1=epsz[i][j][km1]; epsi2=epsz[i][j][k]; ez[i][j][k]=sigma[i][j][k] + epsi1*( (phi[i][j][km1]-phi[i][j][k])*idz - 0.5*dz* ( idx*(2*phi[i][j][k]-phi[ip1][j][k]-phi[im1][j][k])*idx +idy*(2*phi[i][j][k]-phi[i][jp1][k]-phi[i][jm1][k])*idy) ); ez[i][j][k]/=epsi2; } else if (face[i][j][k]==FACE_Z_DOWN) { epsi2=epsz[i][j][km1]; epsi1=epsz[i][j][k]; ez[i][j][k]=sigma[i][j][k] + epsi1*( (phi[i][j][kp1]-phi[i][j][k])*idz - 0.5*dz* ( idx*(2*phi[i][j][k]-phi[ip1][j][k]-phi[im1][j][k])*idx +idy*(2*phi[i][j][k]-phi[i][jp1][k]-phi[i][jm1][k])*idy) ); ez[i][j][k]/=-epsi2; }}else { ex[i][j][k] = hdx*(phi[im1][j][k] - phi[ip1][j][k]); ey[i][j][k] = hdy*(phi[i][jm1][k] - phi[i][jp1][k]); ez[i][j][k] = hdz*(phi[i][j][km1] - phi[i][j][kp1]);} }}/***************************************************************/void set_voltage(){ register int i, k;
for (i=0;i
SOR3d.c(1)#include "et3d.h"
/* Successive Overrelaxation (SOR) with Chebyshev acceleration */
int sor_cheb(float ***a, float ***b, float ***c, float ***d, float ***e, float ***f, float ***g, float ***s, float ***u, double omega){ register int i,j,k; int im1,ip1,jm1,jp1,km1,kp1; int iterations; float residue, change, max_change; int i_order,j_order,k_order; int i_max2, j_max2, k_max2; float anorm, old_anorm=0.0;
/* rjac is input as the spectral radius of the Jacobi iteration, or an estimate of it */
if (bc_x == SYMMETRIC) i_max2 = ncx; else if (bc_x == PERIODIC) i_max2 = ncx-1; if (bc_y == SYMMETRIC) j_max2 = ncy; else if (bc_y == PERIODIC) j_max2 = ncy-1; if (bc_z == SYMMETRIC) k_max2 = ncz; else if (bc_z == PERIODIC) k_max2 = ncz-1;
for(iterations=1;iterations j_order = 1, odd j is swept i is odd => j_order = 0, even j is swept */
SOR3d.c(3)if (bc_x == PERIODIC && i == 0) u[ncx][j][k] = u[0][j][k]; if (bc_y == PERIODIC && j == 0) u[i][ncy][k] = u[i][0][k]; if (bc_z == PERIODIC && k == 0) u[i][j][ncz] = u[i][j][0];
max_change = max(max_change, fabs(change)); } } /* 0 , 1 swap */ j_order = 1 - j_order; } /* 0 , 1 swap */ k_order = 1 - k_order; }
if (iterations==1 && k_order==0) omega=1.0/(1.0-0.5*rjac_square); else omega=1.0/(1.0-0.25*rjac_square*omega);
} if (bc_x == PERIODIC && bc_z == PERIODIC) for (j=0;j 1) { opt_omega=obtain_opt_omega(anorm, old_anorm, omega); } old_anorm = anorm;
}#endif
} /* End of Iteration Loop */
fprintf(stderr, " Max. # of Iteration exceeded in SOR\n"); return -1;}
NoteFinding optimal in SOR with Chebyshev acceleration
SOR3d.c(4)/* rjac is input as the spectral radius of the Jacobi iteration, or an * estimate of it */void set_rjac_square(){ double rjac,factor; factor = dx*dx/dy/dy;
if (bc_x == SYMMETRIC && bc_y == SYMMETRIC) rjac = (cos(M_PI/ncx)+factor*cos(M_PI/ncy)); else if (bc_x == PERIODIC && bc_y == PERIODIC) rjac = (cos(2.*M_PI/ncx)+factor*cos(2.*M_PI/ncy)); else if (bc_x == PERIODIC && bc_y == SYMMETRIC) rjac = (cos(2.*M_PI/ncx)+factor*cos(M_PI/ncy)); else if (bc_x == SYMMETRIC && bc_y == PERIODIC) rjac = (cos(M_PI/ncx)+factor*cos(2.*M_PI/ncy));
rjac /= (1.+factor); rjac_square = rjac*rjac;}
double obtain_opt_omega(double anorm, double old_anorm, double omega){ double mu_W, mu_J, opt_omega; mu_W = sqrt(anorm/old_anorm); mu_J = (mu_W+omega-1.0)/(omega*sqrt(mu_W)); opt_omega = 2.0/(1.0+sqrt(1.0-mu_J*mu_J)); return opt_omega;}
void set_poisson_coefficient(){register int i, j, k;int ii, im1, jm1, jj, km1, kk, isp;
for (i=0;i
SOR3d.c(5)aa[i][j][k]= eps_array[im1][jj][kk] + eps_array[im1][jm1][kk] + eps_array[im1][jj][km1]+ eps_array[im1][jm1][km1];bb[i][j][k]= eps_array[ii][jj][kk] + eps_array[ii][jm1][kk] + eps_array[ii][jj][km1]+ eps_array[ii][jm1][km1];cc[i][j][k]= eps_array[ii][jm1][kk] + eps_array[im1][jm1][kk] + eps_array[ii][jm1][km1]+ eps_array[im1][jm1][km1];dd[i][j][k]= eps_array[ii][jj][kk] + eps_array[im1][jj][kk] + eps_array[ii][jj][km1]+ eps_array[im1][jj][km1];ee[i][j][k]= eps_array[ii][jj][km1] + eps_array[im1][jj][km1] + eps_array[ii][jm1][km1]+ eps_array[im1][jm1][km1];ff[i][j][k]= eps_array[ii][jj][kk] + eps_array[im1][jj][kk] + eps_array[ii][jm1][kk]+ eps_array[im1][jm1][kk]; } aa[i][j][k]*=idx2; bb[i][j][k]*=idx2; cc[i][j][k]*=idy2; dd[i][j][k]*=idy2; ee[i][j][k]*=idz2; ff[i][j][k]*=idz2; gg[i][j][k]= -(aa[i][j][k] + bb[i][j][k] + cc[i][j][k] + dd[i][j][k] + ee[i][j][k] + ff[i][j][k]); }}
int adjust_x(int i){ if (bc_x == SYMMETRIC) { /* not valid for all integer range */ if (i < 0) i = (-i-1) % ncx; else if (i >= ncx) i = ncx - 1 - i % ncx; } else if (bc_x == PERIODIC) { if (i < 0) i = ncx - 1 - (-i-1) % ncx; else if (i >= ncx) i = i % ncx;
}
return i;}
Noteidx2=0.25*idx*idx
SOR3d.c(6)int adjust_y(int j){ if (bc_y == SYMMETRIC) { if (j < 0) j = (-j-1) % ncy; else if (j >= ncy) j = ncy - 1 - j % ncy; } else if (bc_y == PERIODIC) { if (j < 0) j = ncy - 1 - (-j-1) % ncy; else if (j >= ncy) j = j % ncy;
} return j;}
int adjust_z(int k){ if (bc_z == SYMMETRIC) { if (k < 0) k = (-k-1) % ncz; else if (k >= ncz) k = ncz - 1 - k % ncz; } else if (bc_z == PERIODIC) { if (k < 0) k = ncz - 1 - (-k-1) % ncz; else if (k >= ncz) k = k % ncz;
} return k;}
RIE (Reactive Ion Etcher) RF(Radio frequency) CCP (Capacitively Coupled Plasma) . etch rate, target selectivity . rf CCP (Dual-Frequency Capacitively Coupled Plasma) CCP . , etching , , , . .
Hybrid Code H2EK 1 3 .
