Mercurial > forge
changeset 9872:e567b7ac3d1f octave-forge
new version of secs1d
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--- a/extra/secs1d/DESCRIPTION Sun Mar 25 22:35:18 2012 +0000 +++ b/extra/secs1d/DESCRIPTION Sun Mar 25 22:44:30 2012 +0000 @@ -1,12 +1,12 @@ Name: SECS1D -Version: 0.0.8 -Date: 2008-08-23 +Version: 0.0.9 +Date: 2012-03-25 Author: Carlo de Falco Maintainer: Carlo de Falco Title: SEmi Conductor Simulator in 1D Description: A Drift-Diffusion simulator for 1d semiconductor devices Categories: Electrical Engineering -Depends: octave (>= 2.9.17) +Depends: octave (>= 3.0), bim Autoload: no License: GPL version 2 or later
--- a/extra/secs1d/INDEX Sun Mar 25 22:35:18 2012 +0000 +++ b/extra/secs1d/INDEX Sun Mar 25 22:44:30 2012 +0000 @@ -1,23 +1,9 @@ secs1D >> SEmiConductor Simulator in 1D -DDG - DDGelectron_driftdiffusion - DDGgummelmap - DDGhole_driftdiffusion - DDGn2phin - DDGnlpoisson - DDGp2phip - DDGphin2n - DDGphip2p - DDGplotresults -DDN - DDNnewtonmap -Utilities - constants - Ubern - Ubernoulli - Ucompconst - Ucomplap - Ucompmass - Udriftdiffusion - Umediaarmonica - Uscharfettergummel +Drift-Diffusion solvers + secs1d_dd_gummel_map + secs1d_dd_newton +non-linear Poisson solver + secs1d_nlpoisson_newton +Physical constants and material data + secs1d_physical_constants.m + secs1d_silicon_material_properties.m \ No newline at end of file
--- a/extra/secs1d/README Sun Mar 25 22:35:18 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,1 +0,0 @@ - SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator ------------------------------------------------------------------- Copyright (C) 2004-2007 Carlo de Falco This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program (see the file COPYING); if not, see <http://www.gnu.org/licenses/>. For more information, you may also contact me by email at defalco@math.uni-wuppertal.de \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/bak/DDG/DDGelectron_driftdiffusion.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,89 @@ +## Copyright (C) 2004-2008 Carlo de Falco +## +## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator +## +## SECS1D is free software; you can redistribute it and/or modify +## it under the terms of the GNU General Public License as published by +## the Free Software Foundation; either version 2 of the License, or +## (at your option) any later version. +## +## SECS1D is distributed in the hope that it will be useful, +## but WITHOUT ANY WARRANTY; without even the implied warranty of +## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +## GNU General Public License for more details. +## +## You should have received a copy of the GNU General Public License +## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. +## +## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> + +## -*- texinfo -*- +## +## @deftypefn {Function File}@ +## {@var{n}} = DDGelectron_driftdiffusion(@var{psi},@var{x},@var{ng},@var{p},@var{ni},@var{tn},@var{tp},@var{un}) +## +## Solve the continuity equation for electrons +## +## Input: +## @itemize @minus +## @item psi: electric potential +## @item x: integration domain +## @item ng: initial guess and BCs for electron density +## @item p: hole density (for SRH recombination) +## @end itemize +## +## Output: +## @itemize @minus +## @item n: updated electron density +## @end itemize +## +## @end deftypefn + +function n = DDGelectron_driftdiffusion(psi,x,ng,p,ni,tn,tp,un) + +nodes = x; +Nnodes =length(nodes); + +elements = [[1:Nnodes-1]' [2:Nnodes]']; +Nelements=size(elements,1); + +Bcnodes = [1;Nnodes]; + +nl = ng(1); +nr = ng(Nnodes); +h=nodes(elements(:,2))-nodes(elements(:,1)); + +c=1./h; +Bneg=Ubernoulli(-(psi(2:Nnodes)-psi(1:Nnodes-1)),1); +Bpos=Ubernoulli( (psi(2:Nnodes)-psi(1:Nnodes-1)),1); + + +d0 = [c(1).*Bneg(1); c(1:end-1).*Bpos(1:end-1)+c(2:end).*Bneg(2:end); c(end)*Bpos(end)]; +d1 = [1000;-c.* Bpos]; +dm1 = [-c.* Bneg;1000]; + +A = spdiags([dm1 d0 d1],-1:1,Nnodes,Nnodes); +b = zeros(Nnodes,1);%- A * ng; + + ## SRH Recombination term +SRHD = tp * (ng + ni) + tn * (p + ni); +SRHL = p ./ SRHD; +SRHR = ni.^2 ./ SRHD; + +ASRH = Ucompmass (nodes,Nnodes,elements,Nelements,SRHL,ones(Nelements,1)); +bSRH = Ucompconst (nodes,Nnodes,elements,Nelements,SRHR,ones(Nelements,1)); + +A = A + ASRH; +b = b + bSRH; + + ## Boundary conditions +b(Bcnodes) = []; +b(1) = - A(2,1) * nl; +b(end) = - A(end-1,end) * nr; +A(Bcnodes,:) = []; +A(:,Bcnodes) = []; + +n = [nl; A\b ;nr]; + +endfunction +
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/bak/DDG/DDGgummelmap.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,153 @@ +## Copyright (C) 2004-2008 Carlo de Falco +## +## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator +## +## SECS1D is free software; you can redistribute it and/or modify +## it under the terms of the GNU General Public License as published by +## the Free Software Foundation; either version 2 of the License, or +## (at your option) any later version. +## +## SECS1D is distributed in the hope that it will be useful, +## but WITHOUT ANY WARRANTY; without even the implied warranty of +## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +## GNU General Public License for more details. +## +## You should have received a copy of the GNU General Public License +## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. +## +## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> + +## -*- texinfo -*- +## +## @deftypefn {Function File}@ +## {@var{odata},@var{it},@var{res}} = DDGgummelmap(@var{x},@var{idata},@var{toll},@var{maxit},@var{ptoll},@var{pmaxit},@var{verbose}) +## +## Solve the scaled stationary bipolar DD equation system using Gummel +## algorithm +## +## Input: +## @itemize @minus +## @item x: spatial grid +## @item idata.D: doping profile +## @item idata.p: initial guess for hole concentration +## @item idata.n: initial guess for electron concentration +## @item idata.V: initial guess for electrostatic potential +## @item idata.Fn: initial guess for electron Fermi potential +## @item idata.Fp: initial guess for hole Fermi potential +## @item idata.l2: scaled electric permittivity (diffusion coefficient in Poisson equation) +## @item idata.un: scaled electron mobility +## @item idata.up: scaled electron mobility +## @item idata.nis: scaled intrinsic carrier density +## @item idata.tn: scaled electron lifetime +## @item idata.tp: scaled hole lifetime +## @item toll: tolerance for Gummel iterarion convergence test +## @item maxit: maximum number of Gummel iterarions +## @item ptoll: tolerance for Newton iterarion convergence test for non linear Poisson +## @item pmaxit: maximum number of Newton iterarions +## @item verbose: verbosity level (0,1,2) +## @end itemize +## +## Output: +## @itemize @minus +## @item odata.n: electron concentration +## @item odata.p: hole concentration +## @item odata.V: electrostatic potential +## @item odata.Fn: electron Fermi potential +## @item odata.Fp: hole Fermi potential +## @item it: number of Gummel iterations performed +## @item res: total potential increment at each step +## @end itemize +## +## @end deftypefn + +function [odata,it,res] = DDGgummelmap (x,idata,toll,maxit,ptoll,pmaxit,verbose) + + odata = idata; +Nnodes=rows(x); + +D = idata.D; +vout(:,1) = idata.V; + +hole_density (:,1) = idata.p; +electron_density (:,1)= idata.n; +fermin (:,1)=idata.Fn; +fermip (:,1)=idata.Fp; + +for i=1:1:maxit + if (verbose>1) + fprintf(1,"*****************************************************************\n"); + fprintf(1,"**** start of gummel iteration number: %d\n",i); + fprintf(1,"*****************************************************************\n"); + endif + + if (verbose>1) + fprintf(1,"solving non linear poisson equation\n\n"); + endif + + [vout(:,2),electron_density(:,2),hole_density(:,2)] =\ + DDGnlpoisson (x,[1:Nnodes],vout(:,1),electron_density(:,1),hole_density(:,1),fermin(:,1),fermip(:,1),D,idata.l2,ptoll,pmaxit,verbose); + + if (verbose>1) + fprintf (1,"\n\nupdating electron qfl\n\n"); + endif + electron_density(:,3)=\ + DDGelectron_driftdiffusion(vout(:,2), x, electron_density(:,2),hole_density(:,2),idata.nis,idata.tn,idata.tp,idata.un); + + fermin(:,2) = DDGn2phin(vout(:,2),electron_density(:,3)); + fermin(1,2) = idata.Fn(1); + fermin(end:2) = idata.Fn(end); + + if (verbose>1) + fprintf(1,"updating hole qfl\n\n"); + endif + + hole_density(:,3) = \ + DDGhole_driftdiffusion(vout(:,2), x, hole_density(:,2),electron_density(:,2),idata.nis,idata.tn,idata.tp,idata.up); + + fermip(:,2) = DDGp2phip(vout(:,2),hole_density(:,3)); + fermip(1,2) = idata.Fp(1); + fermip(end,2) = idata.Fp(end); + + if (verbose>1) + fprintf(1,"checking for convergence\n\n"); + endif + + nrfn= norm(fermin(:,2)-fermin(:,1),inf); + nrfp= norm (fermip(:,2)-fermip(:,1),inf); + nrv = norm (vout(:,2)-vout(:,1),inf); + nrm(i) = max([nrfn;nrfp;nrv]); + + if (verbose>1) + fprintf (1," max(|phin_(k+1)-phinn_(k)| , |phip_(k+1)-phip_(k)| , |v_(k+1)- v_(k)| )= %d\n",nrm(i)); + endif + + if (nrm(i)<toll) + break + endif + + vout(:,1) = vout(:,end); + hole_density (:,1) = hole_density (:,end) ; + electron_density (:,1)= electron_density (:,end); + fermin (:,1)= fermin (:,end); + fermip (:,1)= fermip (:,end); + + + if(verbose) + DDGplotresults(x,electron_density,hole_density,vout,fermin,fermip); + endif + endfor + +it = i; +res = nrm; + +if (verbose>0) + fprintf(1,"\n\nInitial guess computed by DD: # of Gummel iterations = %d\n\n",it); + endif + +odata.n = electron_density(:,end); +odata.p = hole_density(:,end); +odata.V = vout(:,end); +odata.Fn = fermin(:,end); +odata.Fp = fermip(:,end); + +endfunction \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/bak/DDG/DDGhole_driftdiffusion.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,87 @@ +## Copyright (C) 2004-2008 Carlo de Falco +## +## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator +## +## SECS1D is free software; you can redistribute it and/or modify +## it under the terms of the GNU General Public License as published by +## the Free Software Foundation; either version 2 of the License, or +## (at your option) any later version. +## +## SECS1D is distributed in the hope that it will be useful, +## but WITHOUT ANY WARRANTY; without even the implied warranty of +## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +## GNU General Public License for more details. +## +## You should have received a copy of the GNU General Public License +## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. +## +## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> + +## -*- texinfo -*- +## +## @deftypefn {Function File}@ +## {@var{p}} = DDGhole_driftdiffusio(@var{psi},@var{x},@var{pg},@var{n},@var{ni},@var{tn},@var{tp},@var{up}) +## +## Solve the continuity equation for holes +## +## Input: +## @itemize @minus +## @item psi: electric potential +## @item x: spatial grid +## @item ng: initial guess and BCs for electron density +## @item n: electron density (for SRH recombination) +## @end itemize +## +## Output: +## @itemize @minus +## @item p: updated hole density +## @end itemize +## +## @end deftypefn + +function p = DDGhole_driftdiffusion(psi,x,pg,n,ni,tn,tp,up) + +nodes = x; +Nnodes =length(nodes); +elements = [[1:Nnodes-1]' [2:Nnodes]']; +Nelements=size(elements,1); +Bcnodes = [1;Nnodes]; + +pl = pg(1); +pr = pg(Nnodes); +h=nodes(elements(:,2))-nodes(elements(:,1)); +c=up./h; +Bneg=Ubernoulli(-(psi(2:Nnodes)-psi(1:Nnodes-1)),1); +Bpos=Ubernoulli( (psi(2:Nnodes)-psi(1:Nnodes-1)),1); + + +d0 = [c(1).*Bpos(1); c(1:end-1).*Bneg(1:end-1)+c(2:end).*Bpos(2:end); c(end)*Bneg(end)]; +d1 = [1000;-c.* Bneg]; +dm1 = [-c.* Bpos;1000]; + +A = spdiags([dm1 d0 d1],-1:1,Nnodes,Nnodes); + b = zeros(Nnodes,1);## - A * pg; + + ## SRH Recombination term +SRHD = tp * (n + ni) + tn * (pg + ni); +SRHL = n ./ SRHD; +SRHR = ni.^2 ./ SRHD; + +ASRH = Ucompmass (nodes,Nnodes,elements,Nelements,SRHL,ones(Nelements,1)); +bSRH = Ucompconst (nodes,Nnodes,elements,Nelements,SRHR,ones(Nelements,1)); + +A = A + ASRH; +b = b + bSRH; + + ## Boundary conditions +b(Bcnodes) = []; +b(1) = - A(2,1) * pl; +b(end) = - A(end-1,end) * pr; +A(Bcnodes,:) = []; +A(:,Bcnodes) = []; + +p = [pl; A\b ;pr]; + +endfunction + +
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/bak/DDG/DDGn2phin.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,36 @@ +## Copyright (C) 2004-2008 Carlo de Falco +## +## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator +## +## SECS1D is free software; you can redistribute it and/or modify +## it under the terms of the GNU General Public License as published by +## the Free Software Foundation; either version 2 of the License, or +## (at your option) any later version. +## +## SECS1D is distributed in the hope that it will be useful, +## but WITHOUT ANY WARRANTY; without even the implied warranty of +## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +## GNU General Public License for more details. +## +## You should have received a copy of the GNU General Public License +## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. +## +## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> + +## -*- texinfo -*- +## +## @deftypefn {Function File}@ +## {@var{phin}} = DDGn2phin(@var{V},@var{n}) +## +## Compute the qfl for electrons using Maxwell-Boltzmann statistics. +## +## @end deftypefn + +function phin = DDGn2phin (V,n); + + ## Load constants + nmin = 0; +n = n .* (n>nmin) + nmin * (n<=nmin); +phin = V - log(n) ; + +endfunction
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/bak/DDG/DDGnlpoisson.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,212 @@ +## Copyright (C) 2004-2008 Carlo de Falco +## +## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator +## +## SECS1D is free software; you can redistribute it and/or modify +## it under the terms of the GNU General Public License as published by +## the Free Software Foundation; either version 2 of the License, or +## (at your option) any later version. +## +## SECS1D is distributed in the hope that it will be useful, +## but WITHOUT ANY WARRANTY; without even the implied warranty of +## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +## GNU General Public License for more details. +## +## You should have received a copy of the GNU General Public License +## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. +## +## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> + +## -*- texinfo -*- +## +## @deftypefn {Function File}@ +## {@var{V},@var{n},@var{p},@var{res},@var{niter}} = @ +## DDGnlpoisson(@var{x},@var{sinodes},@var{Vin},@var{nin},@var{pin},@var{Fnin},@var{Fpin},@var{D},@var{l2},@var{toll},@var{maxit},@var{verbose}) +## +## Solve the non linear Poisson equation +## +## - lamda^2 *V'' + (n(V,Fn) - p(V,Fp) -D) = 0 +## +## Input: +## @itemize @minus +## @item x: spatial grid +## @item sinodes: index of the nodes of the grid which are in the +## semiconductor subdomain (remaining nodes are assumed to be in the oxide subdomain) +## @item Vin: initial guess for the electrostatic potential +## @item nin: initial guess for electron concentration +## @item pin: initial guess for hole concentration +## @item Fnin: initial guess for electron Fermi potential +## @item Fpin: initial guess for hole Fermi potential +## @item D: doping profile +## @item l2: scaled electric permittivity (diffusion coefficient) +## @item toll: tolerance for convergence test +## @item maxit: maximum number of Newton iterations +## @item verbose: verbosity level (0,1,2) +## @end itemize +## +## Output: +## @itemize @minus +## @item V: electrostatic potential +## @item n: electron concentration +## @item p: hole concentration +## @item res: residual norm at each step +## @item niter: number of Newton iterations +## @end itemize +## +## @end deftypefn + +function [V,n,p,res,niter] = DDGnlpoisson (x,sinodes,Vin,nin,pin,Fnin,Fpin,D,l2,toll,maxit,verbose) + + ## Set some useful constants +dampit = 10; +dampcoeff = 2; + + ## Convert grid info to FEM form +Ndiricheletnodes = 2; +nodes = x; +sielements = sinodes(1:end-1); +Nnodes = length(nodes); +totdofs = Nnodes - Ndiricheletnodes ; +elements = [[1:Nnodes-1]' , [2:Nnodes]']; +Nelements = size(elements,1); +BCnodes = [1;Nnodes]; + + ## Initialization +V = Vin; +Fn = Fnin; +Fp = Fpin; +n = DDGphin2n(V(sinodes),Fn); +p = DDGphip2p(V(sinodes),Fp); +if (sinodes(1)==1) + n(1)=nin(1); + p(1)=pin(1); + endif +if (sinodes(end)==Nnodes) + n(end)=nin(end); + p(end)=pin(end); + endif + + ## Compute LHS matrices +L = Ucomplap (nodes,Nnodes,elements,Nelements,l2.*ones(Nelements,1)); + + ## Compute Mv = ( n + p) +Mv = zeros(Nnodes,1); +Mv(sinodes) = (n + p); +Cv = zeros(Nelements,1); +Cv(sielements)= 1; +M = Ucompmass (nodes,Nnodes,elements,Nelements,Mv,Cv); + + ## Compute RHS vector +Tv0 = zeros(Nnodes,1); +Tv0(sinodes) = (n - p - D); +Cv = zeros(Nelements,1); +Cv(sielements)= 1; +T0 = Ucompconst (nodes,Nnodes,elements,Nelements,Tv0,Cv); + + ## Build LHS matrix and RHS of the linear system for 1st Newton step +A = L + M; +R = L * V + T0; + + ## Apply boundary conditions +A(BCnodes,:) = []; +A(:,BCnodes) = []; +R(BCnodes) = []; + + +normr(1) = norm(R,inf); +relresnorm = 1; +reldVnorm = 1; +normrnew = normr(1); + + + ## Start of the newton cycle +for newtit=1:maxit + if verbose + fprintf(1,"\n newton iteration: %d, reldVnorm = %e",newtit,reldVnorm); + endif + dV =[0;(A)\(-R);0]; + + ## Start of the damping procedure + tk = 1; + for dit = 1:dampit + if verbose + fprintf(1,"\n damping iteration: %d, residual norm = %e",dit,normrnew); + endif + Vnew = V + tk * dV; + n = DDGphin2n(Vnew(sinodes),Fn); + p = DDGphip2p(Vnew(sinodes),Fp); + if (sinodes(1)==1) + n(1)=nin(1); + p(1)=pin(1); + endif + if (sinodes(end)==Nnodes) + n(end)=nin(end); + p(end)=pin(end); + endif + + ## Compute LHS matrices + Mv = zeros(Nnodes,1); + Mv(sinodes) = (n + p); + Cv = zeros(Nelements,1); + Cv(sielements)= 1; + M = Ucompmass (nodes,Nnodes,elements,Nelements,Mv,Cv); + + ## Compute RHS vector (-residual) + Tv0 = zeros(Nnodes,1); + Tv0(sinodes) = (n - p - D); + Cv = zeros(Nelements,1); + Cv(sielements)= 1; + T0 = Ucompconst (nodes,Nnodes,elements,Nelements,Tv0,Cv); + + ## Build LHS matrix and RHS of the linear system for 1st Newton step + Anew = L + M; + Rnew = L * Vnew + T0; + + ## Apply boundary conditions + Anew(BCnodes,:) = []; + Anew(:,BCnodes) = []; + Rnew(BCnodes) = []; + + if ((dit>1)&(norm(Rnew,inf)>=norm(R,inf))) + if verbose + fprintf(1,"\nexiting damping cycle \n"); + endif + break + else + A = Anew; + R = Rnew; + endif + + ## Compute | R_{k+1} | for the convergence test + normrnew= norm(R,inf); + + ## Check if more damping is needed + if (normrnew > normr(newtit)) + tk = tk/dampcoeff; + else + if verbose + fprintf(1,"\nexiting damping cycle because residual norm = %e \n",normrnew); + endif + break + endif + endfor + + V = Vnew; + normr(newtit+1) = normrnew; + dVnorm = norm(tk*dV,inf); + + ## Check if convergence has been reached + reldVnorm = dVnorm / norm(V,inf); + if (reldVnorm <= toll) + if(verbose) + fprintf(1,"\nexiting newton cycle because reldVnorm= %e \n",reldVnorm); + endif + break + endif + + endfor + +res = normr; +niter = newtit; + +endfunction \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/bak/DDG/DDGp2phip.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,36 @@ +## Copyright (C) 2004-2008 Carlo de Falco +## +## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator +## +## SECS1D is free software; you can redistribute it and/or modify +## it under the terms of the GNU General Public License as published by +## the Free Software Foundation; either version 2 of the License, or +## (at your option) any later version. +## +## SECS1D is distributed in the hope that it will be useful, +## but WITHOUT ANY WARRANTY; without even the implied warranty of +## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +## GNU General Public License for more details. +## +## You should have received a copy of the GNU General Public License +## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. +## +## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> + +## -*- texinfo -*- +## +## @deftypefn {Function File}@ +## {@var{phip}} = DDGn2phin(@var{V},@var{p}) +## +## Compute the qfl for holes using Maxwell-Boltzmann statistics +## +## @end deftypefn + +function phip = DDGp2phip (V,p); + + ## Load constants + pmin = 0; + p = p .* (p>pmin) + pmin * (p<=pmin); + phip = V + log(p) ; + +endfunction
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/bak/DDG/DDGphin2n.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,35 @@ +## Copyright (C) 2004-2008 Carlo de Falco +## +## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator +## +## SECS1D is free software; you can redistribute it and/or modify +## it under the terms of the GNU General Public License as published by +## the Free Software Foundation; either version 2 of the License, or +## (at your option) any later version. +## +## SECS1D is distributed in the hope that it will be useful, +## but WITHOUT ANY WARRANTY; without even the implied warranty of +## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +## GNU General Public License for more details. +## +## You should have received a copy of the GNU General Public License +## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. +## +## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> + +## -*- texinfo -*- +## +## @deftypefn {Function File}@ +## {@var{n}} = DDGphin2n(@var{V},@var{phin}) +## +## Compute the electron density using Maxwell-Boltzmann statistics +## +## @end deftypefn + +function n = DDGphin2n (V,phin); + + nmin = 0; + n = exp ((V-phin)); + n = n .* (n>nmin) + nmin * (n<=nmin); + +endfunction
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/bak/DDG/DDGphip2p.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,36 @@ +## Copyright (C) 2004-2008 Carlo de Falco +## +## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator +## +## SECS1D is free software; you can redistribute it and/or modify +## it under the terms of the GNU General Public License as published by +## the Free Software Foundation; either version 2 of the License, or +## (at your option) any later version. +## +## SECS1D is distributed in the hope that it will be useful, +## but WITHOUT ANY WARRANTY; without even the implied warranty of +## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +## GNU General Public License for more details. +## +## You should have received a copy of the GNU General Public License +## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. +## +## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> + +## -*- texinfo -*- +## +## @deftypefn {Function File}@ +## {@var{p}} = DDGphip2p(@var{V},@var{phip}) +## +## Compute the hole density using Maxwell-Boltzmann statistic +## +## @end deftypefn + +function p = DDGphip2p (V,phip); + +## Load constants +pmin = 0; +p = exp ((phip-V)); +p = p .* (p>pmin) + pmin * (p<=pmin); + +endfunction
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/bak/DDG/DDGplotresults.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,52 @@ +## Copyright (C) 2004-2008 Carlo de Falco +## +## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator +## +## SECS1D is free software; you can redistribute it and/or modify +## it under the terms of the GNU General Public License as published by +## the Free Software Foundation; either version 2 of the License, or +## (at your option) any later version. +## +## SECS1D is distributed in the hope that it will be useful, +## but WITHOUT ANY WARRANTY; without even the implied warranty of +## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +## GNU General Public License for more details. +## +## You should have received a copy of the GNU General Public License +## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. +## +## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> + +## -*- texinfo -*- +## +## @deftypefn {Function File}@ +## DDGplotresults(@var{x},@var{n},@var{p},@var{V},@var{Fn},@var{Fp}) +## +## Plot densities and potentials +## +## @end deftypefn + +function DDGplotresults(x,n,p,V,Fn,Fp); + +subplot(2,3,1) +title('Electron Density') +semilogy(x,n) + +subplot(2,3,2) +title('Hole Density') +semilogy(x,p) + +subplot(2,3,4) +title('Electron QFL') +plot(x,Fn) + +subplot(2,3,5) +title('Hole QFL') +plot(x,Fp) + +subplot(2,3,6) +title('Electric Potential') +plot(x,V) +pause(.1) + +endfunction
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/bak/DDN/DDNnewtonmap.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,230 @@ +## Copyright (C) 2004-2008 Carlo de Falco +## +## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator +## +## SECS1D is free software; you can redistribute it and/or modify +## it under the terms of the GNU General Public License as published by +## the Free Software Foundation; either version 2 of the License, or +## (at your option) any later version. +## +## SECS1D is distributed in the hope that it will be useful, +## but WITHOUT ANY WARRANTY; without even the implied warranty of +## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +## GNU General Public License for more details. +## +## You should have received a copy of the GNU General Public License +## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. +## +## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> + +## -*- texinfo -*- +## +## @deftypefn {Function File}@ +## {@var{odata},@var{it},@var{res}} = DDNnewtonmap(@var{x},@var{idata},@var{toll},@var{maxit},@var{verbose}) +## +## Solve the scaled stationary bipolar DD equation system using a +## coupled Newton algorithm +## +## Input: +## @itemize @minus +## @item x: spatial grid +## @item idata.D: doping profile +## @item idata.p: initial guess for hole concentration +## @item idata.n: initial guess for electron concentration +## @item idata.V: initial guess for electrostatic potential +## @item idata.Fn: initial guess for electron Fermi potential +## @item idata.Fp: initial guess for hole Fermi potential +## @item idata.l2: scaled electric permittivity (diffusion coefficient in Poisson equation) +## @item idata.un: scaled electron mobility +## @item idata.up: scaled electron mobility +## @item idata.nis: scaled intrinsic carrier density +## @item idata.tn: scaled electron lifetime +## @item idata.tp: scaled hole lifetime +## @item toll: tolerance for Newton iterarion convergence test +## @item maxit: maximum number of Newton iterarions +## @item verbose: verbosity level: 0,1,2 +## @end itemize +## +## Output: +## @itemize @minus +## @item odata.n: electron concentration +## @item odata.p: hole concentration +## @item odata.V: electrostatic potential +## @item odata.Fn: electron Fermi potential +## @item odata.Fp: hole Fermi potential +## @item it: number of Newton iterations performed +## @item res: residual at each step +## @end itemize +## +## @end deftypefn + +function [odata,it,res] = DDNnewtonmap (x,idata,toll,maxit,verbose) + + odata = idata; + Nnodes = rows(x); +Nelements=Nnodes-1; +elements=[1:Nnodes-1;2:Nnodes]'; +BCnodesp = [1,Nnodes]; +BCnodes = [BCnodesp,BCnodesp+Nnodes,BCnodesp+2*Nnodes]; +totaldofs= Nnodes-2; +dampcoef = 2; +maxdamp = 2; + +V = idata.V; +n = idata.n; +p = idata.p; +D = idata.D; + + ## Create the complete unknown vector +u = [V; n; p]; + + ## Build fem matrices +L = Ucomplap (x,Nnodes,elements,Nelements,idata.l2*ones(Nelements,1)); +M = Ucompmass (x,Nnodes,elements,Nelements,ones(Nnodes,1),ones(Nelements,1)); +DDn = Uscharfettergummel(x,Nnodes,elements,Nelements,idata.un,1,V); +DDp = Uscharfettergummel(x,Nnodes,elements,Nelements,idata.up,1,-V); + + ## Initialise RHS +r1 = L * V + M * (n - p - D); +r2 = DDn * n; +r3 = DDp * p; + RHS = - [ r1; r2; r3 ]; + + ## Apply BCs +RHS(BCnodes,:)= []; +nrm = norm(RHS,inf); +res(1) = nrm; + + ## Begin Newton Cycle +for count = 1: maxit + if verbose + fprintf (1,"\n\n\nNewton Iteration Number:%d\n",count); + endif + Ln = Ucomplap (x,Nnodes,elements,Nelements,Umediaarmonica(idata.un*n)); + Lp = Ucomplap (x,Nnodes,elements,Nelements,Umediaarmonica(idata.up*p)); + Z = sparse(zeros(Nnodes)); + DDn = Uscharfettergummel(x,Nnodes,elements,Nelements,idata.un,1,V); + DDp = Uscharfettergummel(x,Nnodes,elements,Nelements,idata.up,1,-V); + + A = L; + B = M; + C = -M; + DDD = -Ln; + E = DDn; + F = Z; + G = Lp; + H = Z; + I = DDp; + + ## Build LHS + LHS =sparse([ + [A B C]; + [DDD E F]; + [G H I]; + ]); + + ## Apply BCs + LHS(BCnodes,:)=[]; + LHS(:,BCnodes)=[]; + + ## Solve the linearised system + dutmp = (LHS) \ (RHS); + dv = dutmp(1:totaldofs); + dn = dutmp(totaldofs+1:2*totaldofs); + dp = dutmp(2*totaldofs+1:3*totaldofs); + du = [0;dv;0;0;dn;0;0;dp;0]; + + ## Check Convergence + nrm_u = norm(u,inf); + nrm_du = norm(du,inf); + + ratio = nrm_du/nrm_u; + if verbose + fprintf (1,"ratio = %e\n", ratio); + endif + + if (ratio <= toll) + V = u(1:Nnodes); + n = u(Nnodes+1:2*Nnodes); + p = u(2*Nnodes+1:end); + res(count) = nrm; + break; + endif + + ## Begin damping cycle + tj = 1; + + for cc = 1:maxdamp + if verbose + fprintf (1,"\ndamping iteration number:%d\n",cc); + fprintf (1,"reference residual norm:%e\n",nrm); + endif + ## Update the unknown vector + utmp = u + tj*du; + Vnew = utmp(1:Nnodes); + nnew = utmp(Nnodes+1:2*Nnodes); + pnew = utmp(2*Nnodes+1:end); + ## Try a new RHS + DDn = Uscharfettergummel(x,Nnodes,elements,Nelements,idata.un,1,Vnew); + DDp = Uscharfettergummel(x,Nnodes,elements,Nelements,idata.up,1,-Vnew); + + r1 = L * V + M * (nnew - pnew - D); + r2 = DDn * nnew; + r3 = DDp * pnew; + + RHS =- [r1;r2;r3]; + + ## Apply BCs + RHS(BCnodes,:) = []; + nrmtmp=norm(RHS,inf); + + ## Update the damping coefficient + if verbose + fprintf(1,"residual norm:%e\n\n", nrmtmp); + endif + + if (nrmtmp>nrm) + tj = tj/(dampcoef*cc); + if verbose + fprintf (1,"\ndamping coefficients = %e",tj); + endif + else + break; + endif + endfor + + nrm_du = norm(tj*du,inf); + u = utmp; + + if (count>1) + ratio = nrm_du/nrm_du_old; + if (ratio<.005) + V = u(1:Nnodes); + n = u(Nnodes+1:2*Nnodes); + p = u(2*Nnodes+1:end); + res(count) = nrm; + break; + endif + endif + nrm = nrmtmp; + res(count) = nrm; + + ## Convert result vector into distinct output vectors + V = u(1:Nnodes); + n = u(Nnodes+1:2*Nnodes); + p = u(2*Nnodes+1:end); + nrm_du_old = nrm_du; + endfor + +odata.V = V; +odata.n = n; +odata.p = p; +Fn = V - log(n); +Fp = V + log(p); +it = count; + +endfunction + + + +
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/bak/PKG_ADD Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,19 @@ +dirlist = {"Utilities","DDG","DDN"}; + +dir = fileparts (mfilename ("fullpath")); + +if (! exist (fullfile (dir, "inst"), "dir")) + ## Run this if the package is installed + for ii=1:length(dirlist) + addpath ( [ canonicalize_file_name([dir "/.."]) "/" dirlist{ii}]) + endfor +else + ## Run this if we are testing the package without installation + for ii=1:length(dirlist) + addpath ([ canonicalize_file_name(dir) "/inst/" dirlist{ii}]) + endfor +endif + +warning ("off", "Octave:fopen-file-in-path"); +warning ("off", "Octave:load-file-in-path"); +clear dirlist dir
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/bak/Utilities/Ubern.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,87 @@ +## Copyright (C) 2004-2008 Carlo de Falco + ## + ## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator + ## + ## SECS1D is free software; you can redistribute it and/or modify + ## it under the terms of the GNU General Public License as published by + ## the Free Software Foundation; either version 2 of the License, or + ## (at your option) any later version. + ## + ## SECS1D is distributed in the hope that it will be useful, + ## but WITHOUT ANY WARRANTY; without even the implied warranty of + ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + ## GNU General Public License for more details. + ## + ## You should have received a copy of the GNU General Public License + ## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. +## +## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> + +## -*- texinfo -*- +## +## @deftypefn {Function File} {@var{bp},@var{bn}} = Ubern(@var{x}) +## +## Compute Bernoulli function for scalar x: +## +## @itemize @minus +## @item @var{bp} = @var{x}/(exp(@var{x})-1) +## @item @var{bn} = @var{x} + B( @var{x} ) +## @end itemize +## +## @end deftypefn + +function [bp,bn] = Ubern(x) + +xlim=1e-2; +ax=abs(x); + + ## Compute Bernoulli function for x = 0 + +if (ax == 0) + bp=1.; + bn=1.; + return + endif + + ## Compute Bernoulli function for asymptotic values + + if (ax > 80) + if (x > 0) + bp=0.; + bn=x; + return + else + bp=-x; + bn=0.; + return + endif + endif + + ## Compute Bernoulli function for intermediate values + + if (ax > xlim) + bp=x/(exp(x)-1); + bn=x+bp; + return +else + ## Compute Bernoulli function for small x + ## via Taylor expansion + + ii=1; + fp=1.; + fn=1.; + df=1.; + segno=1.; + while (abs(df) > eps), + ii=ii+1; + segno=-segno; + df=df*x/ii; + fp=fp+df; + fn=fn+segno*df; + bp=1./fp; + bn=1./fn; + endwhile + return + endif + +endfunction \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/bak/Utilities/Ubernoulli.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,47 @@ +## Copyright (C) 2004-2008 Carlo de Falco + ## + ## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator + ## + ## SECS1D is free software; you can redistribute it and/or modify + ## it under the terms of the GNU General Public License as published by + ## the Free Software Foundation; either version 2 of the License, or + ## (at your option) any later version. + ## + ## SECS1D is distributed in the hope that it will be useful, + ## but WITHOUT ANY WARRANTY; without even the implied warranty of + ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + ## GNU General Public License for more details. + ## + ## You should have received a copy of the GNU General Public License + ## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. +## +## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> + +## -*- texinfo -*- +## +## @deftypefn {Function File} {@var{b}} = Ubernoulli(@var{x},@var{sg}) +## +## Compute Bernoulli function for vector x: +## +## @itemize @minus +## @item @var{b} = @var{x}/(exp(@var{x})-1) if @var{sg} == 1 +## @item @var{b} = @var{x} + B( @var{x} ) if @var{sg} == 0 +## @end itemize +## +## @end deftypefn + +function b=Ubernoulli(x,sg) + + for count=1:length(x) + [bp,bn] = Ubern(x(count)); + bernp(count,1)=bp; + bernn(count,1)=bn; + endfor + + if (sg ==1) + b=bernp; + elseif (sg ==0) + b=bernn; + endif + +endfunction \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/bak/Utilities/Ucompconst.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,43 @@ +## Copyright (C) 2004-2008 Carlo de Falco + ## + ## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator + ## + ## SECS1D is free software; you can redistribute it and/or modify + ## it under the terms of the GNU General Public License as published by + ## the Free Software Foundation; either version 2 of the License, or + ## (at your option) any later version. + ## + ## SECS1D is distributed in the hope that it will be useful, + ## but WITHOUT ANY WARRANTY; without even the implied warranty of + ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + ## GNU General Public License for more details. + ## + ## You should have received a copy of the GNU General Public License + ## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. +## +## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> + +## -*- texinfo -*- +## +## @deftypefn {Function File}@ +## {@var{R}} = Ucompconst(@var{nodes},@var{Nnodes},@var{elements},@var{Nelements},@var{D},@var{C}) +## +## Compute P1 finite element rhs: +## +## @itemize @minus +## @item @var{nodes}: list of mesh nodes +## @item @var{Nnodes}: number of mesh nodes +## @item @var{elements}: list of mesh elements +## @item @var{Nelements}: number of mesh elements +## @item @var{D}: piecewise linear reaction coefficient +## @item @var{C}: piecewise constant reaction coefficient +## @end itemize +## +## @end deftypefn + +function R = Ucompconst (nodes,Nnodes,elements,Nelements,D,C) + + h = (nodes(2:end)-nodes(1:end-1)).*C; + R = D.*[h(1)/2; (h(1:end-1)+h(2:end))/2; h(end)/2]; + +endfunction \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/bak/Utilities/Ucomplap.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,49 @@ +## Copyright (C) 2004-2008 Carlo de Falco + ## + ## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator + ## + ## SECS1D is free software; you can redistribute it and/or modify + ## it under the terms of the GNU General Public License as published by + ## the Free Software Foundation; either version 2 of the License, or + ## (at your option) any later version. + ## + ## SECS1D is distributed in the hope that it will be useful, + ## but WITHOUT ANY WARRANTY; without even the implied warranty of + ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + ## GNU General Public License for more details. + ## + ## You should have received a copy of the GNU General Public License + ## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. +## +## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> + +## -*- texinfo -*- +## +## @deftypefn {Function File}@ +## {@var{R}} = Ucomplap(@var{nodes},@var{Nnodes},@var{elements},@var{Nelements},@var{coeff}) +## +## Compute P1 finite element approximation of the differential operator: +## +## - d ( coeff d (.)\dx)\dx +## +## @itemize @minus +## @item @var{nodes}: list of mesh nodes +## @item @var{Nnodes}: number of mesh nodes +## @item @var{elements}: list of mesh elements +## @item @var{Nelements}: number of mesh elements +## @item @var{coeff}: piecewise linear reaction coefficient +## @end itemize +## +## @end deftypefn + +function L = Ucomplap (nodes,Nnodes,elements,Nelements,coeff) + + h = nodes(2:end)-nodes(1:end-1); + d0 = [ coeff(1)./h(1); + (coeff(1:end-1)./h(1:end-1))+(coeff(2:end)./h(2:end)); + coeff(end)./h(end)]; + d1 = [1000; -coeff./h]; + dm1 = [ -coeff./h;1000]; + L = spdiags([dm1, d0, d1],-1:1,Nnodes,Nnodes); + +endfunction \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/bak/Utilities/Ucompmass.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,45 @@ +## Copyright (C) 2004-2008 Carlo de Falco + ## + ## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator + ## + ## SECS1D is free software; you can redistribute it and/or modify + ## it under the terms of the GNU General Public License as published by + ## the Free Software Foundation; either version 2 of the License, or + ## (at your option) any later version. + ## + ## SECS1D is distributed in the hope that it will be useful, + ## but WITHOUT ANY WARRANTY; without even the implied warranty of + ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + ## GNU General Public License for more details. + ## + ## You should have received a copy of the GNU General Public License + ## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. +## +## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> + +## -*- texinfo -*- +## +## @deftypefn {Function File}@ +## {@var{R}} = Ucompmass(@var{nodes},@var{Nnodes},@var{elements},@var{Nelements},@var{Bvect},@var{Cvect}) +## +## Compute P1 finite element mass-matrix: +## +## @itemize @minus +## @item @var{nodes}: list of mesh nodes +## @item @var{Nnodes}: number of mesh nodes +## @item @var{elements}: list of mesh elements +## @item @var{Nelements}: number of mesh elements +## @item @var{Bvect}: piecewise linear reaction coefficient +## @item @var{Cvect}: piecewise constant reaction coefficient +## @end itemize +## +## @end deftypefn + +function Bmat = Ucompmass (nodes,Nnodes,elements,Nelements,Bvect,Cvect); + + h = (nodes(2:end)-nodes(1:end-1)).*Cvect; + d0 = Bvect.*[h(1)/2; (h(1:end-1)+h(2:end))/2; h(end)/2]; + Bmat = spdiags(d0, 0, Nnodes,Nnodes); + +endfunction +
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/bak/Utilities/Udriftdiffusion.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,61 @@ +## Copyright (C) 2004-2008 Carlo de Falco + ## + ## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator + ## + ## SECS1D is free software; you can redistribute it and/or modify + ## it under the terms of the GNU General Public License as published by + ## the Free Software Foundation; either version 2 of the License, or + ## (at your option) any later version. + ## + ## SECS1D is distributed in the hope that it will be useful, + ## but WITHOUT ANY WARRANTY; without even the implied warranty of + ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + ## GNU General Public License for more details. + ## + ## You should have received a copy of the GNU General Public License + ## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. +## +## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> + +## -*- texinfo -*- +## +## @deftypefn {Function File}@ +## {@var{A}} = Udriftdiffusion(@var{x},@var{psi},@var{coeff}) +## +## Builds the Scharfetter-Gummel approximation of the differential +## operator +## +## - (coeff (n' - n psi'))' +## +## @itemize @minus +## @item @var{x}: list of mesh nodes +## @item @var{psi}: piecewise linear potential values +## @item @var{coeff}: piecewise linear diffusion coefficient +## @end itemize +## +## @end deftypefn + +function A = Udriftdiffusion(x,psi,coeff) + + nodes = x; + Nnodes =length(nodes); + + elements = [[1:Nnodes-1]' [2:Nnodes]']; + Nelements=size(elements,1); + + Bcnodes = [1;Nnodes]; + + h=nodes(elements(:,2))-nodes(elements(:,1)); + + c=coeff./h; + Bneg=Ubernoulli(-(psi(2:Nnodes)-psi(1:Nnodes-1)),1); + Bpos=Ubernoulli( (psi(2:Nnodes)-psi(1:Nnodes-1)),1); + + + d0 = [c(1).*Bneg(1); c(1:end-1).*Bpos(1:end-1)+c(2:end).*Bneg(2:end); c(end)*Bpos(end)]; + d1 = [1000;-c.* Bpos]; + dm1 = [-c.* Bneg;1000]; + + A = spdiags([dm1 d0 d1],-1:1,Nnodes,Nnodes); + +endfunction \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/bak/Utilities/Umediaarmonica.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,34 @@ +## Copyright (C) 2004-2008 Carlo de Falco + ## + ## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator + ## + ## SECS1D is free software; you can redistribute it and/or modify + ## it under the terms of the GNU General Public License as published by + ## the Free Software Foundation; either version 2 of the License, or + ## (at your option) any later version. + ## + ## SECS1D is distributed in the hope that it will be useful, + ## but WITHOUT ANY WARRANTY; without even the implied warranty of + ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + ## GNU General Public License for more details. + ## + ## You should have received a copy of the GNU General Public License + ## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. +## +## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> + +## -*- texinfo -*- +## +## @deftypefn {Function File}@ +## {@var{m}} = Umediarmonica(@var{w}) +## +## Return the harmonic mean value of @var{w} +## +## @end deftypefn + +function m = Umediaarmonica(w); + + dw = (1./w(1:end-1))+(1./w(2:end)); + m = 2 ./ dw; + +endfunction
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/bak/Utilities/Uscharfettergummel.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,57 @@ +## Copyright (C) 2004-2008 Carlo de Falco + ## + ## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator + ## +## SECS1D is free software; you can redistribute it and/or modify + ## it under the terms of the GNU General Public License as published by + ## the Free Software Foundation; either version 2 of the License, or + ## (at your option) any later version. + ## +## SECS1D is distributed in the hope that it will be useful, + ## but WITHOUT ANY WARRANTY; without even the implied warranty of + ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + ## GNU General Public License for more details. + ## + ## You should have received a copy of the GNU General Public License +## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. +## +## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> + +## -*- texinfo -*- +## +## @deftypefn {Function File}@ +## {@var{R}} = Uscharfettergummel(@var{nodes},@var{Nnodes},@var{elements},@var{Nelements},@var{acoeff},@var{bcoeff},@var{v}) +## +## Build the Scharfetter-Gummel matrix for the the discretization of +## the LHS of the Drift-Diffusion equation: +## +## -(a(x) (u' - b v'(x) u))'= f +## +## @itemize @minus +## @item @var{nodes}: list of mesh nodes +## @item @var{Nnodes}: number of mesh nodes +## @item @var{elements}: list of mesh elements +## @item @var{Nelements}: number of mesh elements +## @item @var{acoeff}: piecewise linear diffusion coefficient +## @item @var{bcoeff}: piecewise constant drift constant coefficient +## @item @var{v}: piecewise linear drift potential +## @end itemize +## +## @end deftypefn + +function A = Uscharfettergummel(nodes,Nnodes,elements,Nelements,acoeff,bcoeff,v) + + h=nodes(elements(:,2))-nodes(elements(:,1)); + + c=acoeff./h; + Bneg=Ubernoulli(-(v(2:Nnodes)-v(1:Nnodes-1))*bcoeff,1); + Bpos=Ubernoulli( (v(2:Nnodes)-v(1:Nnodes-1))*bcoeff,1); + + + d0 = [c(1).*Bneg(1); c(1:end-1).*Bpos(1:end-1)+c(2:end).*Bneg(2:end); c(end)*Bpos(end)]; + d1 = [1000;-c.* Bpos]; + dm1 = [-c.* Bneg;1000]; + + A = spdiags([dm1 d0 d1],-1:1,Nnodes,Nnodes); + +endfunction
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/bak/Utilities/constants.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,74 @@ +## Copyright (C) 2004-2008 Carlo de Falco +## +## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator +## +## SECS1D is free software; you can redistribute it and/or modify +## it under the terms of the GNU General Public License as published by +## the Free Software Foundation; either version 2 of the License, or +## (at your option) any later version. +## +## SECS1D is distributed in the hope that it will be useful, +## but WITHOUT ANY WARRANTY; without even the implied warranty of +## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +## GNU General Public License for more details. +## +## You should have received a copy of the GNU General Public License +## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. +## +## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> + +## -*- texinfo -*- +## +## @deftypefn {Script File} constants +## +## Compute global constants needed for Drift-Diffusion simulation +## +## @end deftypefn + + + +Kb = 1.3806503e-23; +q = 1.602176462e-19; +e0 = 8.854187817e-12; +esir = 11.7; +esio2r = 3.9; +esi = e0 * esir; +esio2 = e0 * esio2r; +hplanck = 6.626e-34; +hbar = ( hplanck/ (2*pi)); +mn0 = 9.11e-31; +mn = 0.26*mn0; +mh = 0.18*mn0; + + +qsue = q / esi; +T0 = 300 ; +Vth = Kb * T0 / q; +un = 1417e-4; +up = 480e-4; +tp = 1e-7; +tn = 1e-7; + +mnl = 0.98*mn0; +mnt = 0.19*mn0; +mndos = (mnl*mnt*mnt)^(1/3); + +mhh = 0.49*mn0; +mlh = 0.16*mn0; +mhdos = (mhh^(3/2)+mlh^(3/2))^(2/3); + +rn = .1; +aleph = hbar^2/(4*rn*q*mn); +alephn = aleph; +rp = .1; +alephp = hbar^2/(4*rp*q*mh); + +Nc = (6/4)*(2*mndos*Kb*T0/(hbar^2*pi))^(3/2); +Nv = (1/4)*(2*mhdos*Kb*T0/(hbar^2*pi))^(3/2); +Eg0 = 1.16964*q; +alfaEg = 4.73e-4*q; +betaEg = 6.36e2; +Egap = Eg0-alfaEg*((T0^2)/(T0+betaEg)); + +ni = sqrt(Nc*Nv)*exp(-Egap/(2*(Kb * T0))); +Phims = - Egap /(2*q);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/bak/secs1d_demo_pndiode.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,138 @@ +## Copyright (C) 2009 Carlo de Falco +## +## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator +## +## SECS1D is free software; you can redistribute it and/or modify +## it under the terms of the GNU General Public License as published by +## the Free Software Foundation; either version 2 of the License, or +## (at your option) any later version. +## +## SECS1D is distributed in the hope that it will be useful, +## but WITHOUT ANY WARRANTY; without even the implied warranty of +## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +## GNU General Public License for more details. +## +## You should have received a copy of the GNU General Public License +## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. +## +## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> + +function secs1d_demo_pndiode () + + constants + + len = 1e-6; + Nnodes = 1000; + + vmin = -2; + vmax = 2; + + vstep=.4; + + istep = 1; + va = vmin + + x = linspace(0,len,Nnodes)'; + xm = mean(x); + + Nd=1e25; + Na=1e25; + + D = Nd * (x>xm) - Na * (x<=xm); + + nn = (Nd + sqrt(Nd^2+4*ni^2))/2; + pp = (Na + sqrt(Na^2+4*ni^2))/2; + + xn = xm+1e-7; + xp = xm-1e-7; + + %% Scaling coefficients + xs = len; + ns = norm(D,inf); + idata.D = D/ns; + Vs = Vth; + us = un; + Js = xs / (us * Vs * q * ns); + + + while va <= vmax + + vvect(istep) = va; + + n(:,istep) = nn * (x>=xn) + (ni^2)/pp * (x<xn); + p(:,istep) = (ni^2)/nn * (x>xp) + pp * (x<=xp); + + Fn = va*(x<=xm); + Fp = Fn; + + V(:,istep) = (Fn - Vth * log(p(:,istep)/ni)); + + %% Scaling + xin = x/xs; + idata.n = n(:,istep)/ns; + idata.p = p(:,istep)/ns; + idata.V = V(:,istep)/Vs; + idata.Fn = (Fn - Vs * log(ni/ns))/Vs; + idata.Fp = (Fp + Vs * log(ni/ns))/Vs; + + lambda2(istep) = idata.l2 = (Vs*esi)/(q*ns*xs^2); + idata.nis = ni/ns; + idata.un = un/us; + idata.up = up/us; + + %% Solution of DD system + + %% Algorithm parameters + toll = 1e-3; + maxit = 20; + ptoll = 1e-10; + pmaxit = 100; + verbose = 0; + sinodes = [1:length(x)]; + idata.tn = inf; + idata.tp = inf; + + [odata,it,res] = DDGgummelmap (xin,idata,toll,maxit,ptoll,pmaxit,verbose); + [odata,it,res] = DDNnewtonmap (xin,odata,toll, maxit,verbose); + + n(:,istep) = odata.n; + p(:,istep) = odata.p; + V(:,istep) = odata.V; + + DV(istep) = odata.V(end) - odata.V(1); + Emax(istep) = max(abs(diff(odata.V)./diff(xin))) + + Bp = Ubernoulli(diff (V(:, istep)),1); + Bm = Ubernoulli(diff (V(:, istep)),0); + Jn(:,istep) = -odata.un * (n(2:end, istep).*Bp-n(1:end-1, istep).*Bm)./diff (xin); + Jp(:,istep) = odata.up * (p(2:end, istep).*Bm-p(1:end-1, istep).*Bp)./diff (xin); + + va = va+vstep + istep = istep+1; + + endwhile + + %% Descaling + n = n*ns; + p = p*ns; + V = V*Vs; + J = abs (Jp+Jn)*Js; + + close all + + figure(); + plot(x, n.') + xlabel("x") + ylabel("n") + + figure(); + plot(vvect, J) + xlabel("V") + ylabel("J") + + figure(); + plot(vvect, Emax) + xlabel("V") + ylabel("max(abs(\\phi^\'))") + +endfunction \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/bak/src/Makefile Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,10 @@ +PROGS = $(patsubst %.cc,%.oct,$(wildcard *.cc)) + +all: $(PROGS) + +$(PROGS): Makefile + +%.oct: %.cc + mkoctfile $< + +clean: ; -$(RM) *.o core octave-core *.oct *~ \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/bak/src/Ubern.cc Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,131 @@ +/* +% This file is part of +% +% SECS2D - A 2-D Drift--Diffusion Semiconductor Device Simulator +% ------------------------------------------------------------------- +% Copyright (C) 2004-2006 Carlo de Falco +% +% +% +% SECS2D is free software; you can redistribute it and/or modify +% it under the terms of the GNU General Public License as published by +% the Free Software Foundation; either version 2 of the License, or +% (at your option) any later version. +% +% SECS2D is distributed in the hope that it will be useful, +% but WITHOUT ANY WARRANTY; without even the implied warranty of +% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +% GNU General Public License for more details. +% +% You should have received a copy of the GNU General Public License +% along with SECS2D; If not, see <http://www.gnu.org/licenses/>. +*/ + +#include <iostream> +#include <octave/oct.h> + +//////////////////////////////////////////// +// Ubern function +// this function, though it does make use +// of liboctave, is not an octve command +// the wrapper to make the command is defined +// below +//////////////////////////////////////////// + + +int Ubern(const double x, double &bp, double &bn ) +{ + +double xlim=1e-2; +int ii; +double fp,fn,df,segno; +double ax; + +ax=fabs(x); + + +if (ax == 0.0) { +bp=1.; +bn=1.; +goto theend ; +} + +if (ax > 80.0){ +if (x >0.0){ + bp=0.0; + bn=x; + goto theend ; +}else{ + bp=-x; + bn=0.0; + goto theend ; +} +} + +if (ax > xlim){ +bp=x/(exp(x)-1.0); +bn=x+bp; +goto theend ; +} + +ii=1; +fp=1.0;fn=1.0;df=1.0;segno=1.0; +while (fabs(df) > 2.2204e-16){ +ii++; +segno= -segno; +df=df*x/ii; +fp=fp+df; +fn=fn+segno*df; +bp=1/fp; +bn=1/fn; +} + + +theend: +return 0; + +} + + +//////////////////////////////////// +// WRAPPER +//////////////////////////////////// +// DEFUN_DLD and the macros that it +// depends on are defined in the +// files defun-dld.h, defun.h, +// and defun-int.h. +// +// Note that the third parameter +// (nargout) is not used, so it is +// omitted from the list of arguments +// to DEFUN_DLD in order to avoid +// the warning from gcc about an +// unused function parameter. +//////////////////////////////////// + +DEFUN_DLD (Ubern, args, , +" [bp,bn]=Ubern(x)\n \ +computes Bernoulli function\n \ +B(x)=x/(exp(x)-1) corresponding to \n \ +to input values Z and -Z, recalling that\n \ +B(-Z)=Z+B(Z)\n") +{ + // The list of values to return. See the declaration in oct-obj.h + octave_value_list retval; + + + NDArray X ( args(0).array_value() ); + octave_idx_type lx = X.length(); + + NDArray BP(X),BN(X); + + for (octave_idx_type jj=0; jj<lx; jj++) + Ubern(X(jj),BP(jj),BN(jj)); + + retval (0) = BP; + retval (1) = BN; + + return retval ; + +} +
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/doc/COPYING.tex Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,345 @@ + +\begin{verbatim} + GNU GENERAL PUBLIC LICENSE + Version 2, June 1991 + + Copyright (C) 1989, 1991 Free Software Foundation, Inc. <http://fsf.org/> + Everyone is permitted to copy and distribute verbatim copies + of this license document, but changing it is not allowed. + + Preamble + + The licenses for most software are designed to take away your +freedom to share and change it. By contrast, the GNU General Public +License is intended to guarantee your freedom to share and change free +software--to make sure the software is free for all its users. This +General Public License applies to most of the Free Software +Foundation's software and to any other program whose authors commit to +using it. (Some other Free Software Foundation software is covered by +the GNU Library General Public License instead.) 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It is safest +to attach them to the start of each source file to most effectively +convey the exclusion of warranty; and each file should have at least +the "copyright" line and a pointer to where the full notice is found. + + <one line to give the program's name and a brief idea of what it does.> + Copyright (C) <year> <name of author> + + This program is free software; you can redistribute it and/or modify + it under the terms of the GNU General Public License as published by + the Free Software Foundation; either version 2 of the License, or + (at your option) any later version. + + This program is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + GNU General Public License for more details. + + You should have received a copy of the GNU General Public License + along with this program; if not, see <http://www.gnu.org/licenses/>. + +Also add information on how to contact you by electronic and paper mail. + +If the program is interactive, make it output a short notice like this +when it starts in an interactive mode: + + Gnomovision version 69, Copyright (C) year name of author + Gnomovision comes with ABSOLUTELY NO WARRANTY; for details type `show w'. + This is free software, and you are welcome to redistribute it + under certain conditions; type `show c' for details. + +The hypothetical commands `show w' and `show c' should show the appropriate +parts of the General Public License. Of course, the commands you use may +be called something other than `show w' and `show c'; they could even be +mouse-clicks or menu items--whatever suits your program. + +You should also get your employer (if you work as a programmer) or your +school, if any, to sign a "copyright disclaimer" for the program, if +necessary. Here is a sample; alter the names: + + Yoyodyne, Inc., hereby disclaims all copyright interest in the program + `Gnomovision' (which makes passes at compilers) written by James Hacker. + + <signature of Ty Coon>, 1 April 1989 + Ty Coon, President of Vice + +This General Public License does not permit incorporating your program into +proprietary programs. If your program is a subroutine library, you may +consider it more useful to permit linking proprietary applications with the +library. If this is what you want to do, use the GNU Library General +Public License instead of this License. + + + +\end{verbatim} +\null +
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/doc/function/secs1d_dd_gummel_map.tex Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,183 @@ +\begin{verbatim} + + + [n, p, V, Fn, Fp, Jn, Jp, it, res] = secs1d_dd_gummel_map (x, D, Na, Nd, + pin, nin, Vin, Fnin, + Fpin, l2, er, u0n, + uminn, vsatn, betan, + Nrefn, u0p, uminp, vsatp, + betap, Nrefp, theta, tn, tp, + Cn, Cp, an, ap, Ecritnin, Ecritpin, + toll, maxit, ptoll, pmaxit) + + This function solves the scaled stationary bipolar DD + equation system using Gummel algorithm + + input: + x spatial grid + D, Na, Nd doping profile + pin initial guess for hole concentration + nin initial guess for electron concentration + Vin initial guess for electrostatic potential + Fnin initial guess for electron Fermi potential + Fpin initial guess for hole Fermi potential + l2 scaled Debye length squared + er relative electric permittivity + u0n, uminn, vsatn, Nrefn electron mobility model coefficients + u0p, uminp, vsatp, Nrefp hole mobility model coefficients + theta intrinsic carrier density + tn, tp, Cn, Cp, + an, ap, + Ecritnin, Ecritpin generation recombination model parameters + toll tolerance for Gummel iterarion convergence test + maxit maximum number of Gummel iterarions + ptoll convergence test tolerance for the non linear + Poisson solver + pmaxit maximum number of Newton iterarions + + output: + n electron concentration + p hole concentration + V electrostatic potential + Fn electron Fermi potential + Fp hole Fermi potential + Jn electron current density + Jp hole current density + it number of Gummel iterations performed + res total potential increment at each step + + +\end{verbatim} + + + + +\subsection{Demo 1 for unction secs1d\_dd\_gummel\_map} +\begin{verbatim} + + % physical constants and parameters + secs1d_physical_constants; + secs1d_silicon_material_properties; + + % geometry + L = 10e-6; % [m] + xm = L/2; + + Nelements = 1000; + x = linspace (0, L, Nelements+1)'; + sinodes = [1:length(x)]; + + % dielectric constant (silicon) + er = esir * ones (Nelements, 1); + + % doping profile [m^{-3}] + Na = 1e23 * (x <= xm); + Nd = 1e23 * (x > xm); + + % avoid zero doping + D = Nd - Na; + + % initial guess for n, p, V, phin, phip + V_p = -1; + V_n = 0; + + Fp = V_p * (x <= xm); + Fn = Fp; + + p = abs (D) / 2 .* (1 + sqrt (1 + 4 * (ni./abs(D)) .^2)) .* (x <= xm) + ... + ni^2 ./ (abs (D) / 2 .* (1 + sqrt (1 + 4 * (ni ./ abs (D)) .^2))) .* (x > xm); + + n = abs (D) / 2 .* (1 + sqrt (1 + 4 * (ni ./ abs (D)) .^ 2)) .* (x > xm) + ... + ni ^ 2 ./ (abs (D) / 2 .* (1 + sqrt (1 + 4 * (ni ./ abs (D)) .^2))) .* (x <= xm); + + V = Fn + Vth * log (n / ni); + + % scaling factors + xbar = L; % [m] + nbar = norm(D, 'inf'); % [m^{-3}] + Vbar = Vth; % [V] + mubar = max (u0n, u0p); % [m^2 V^{-1} s^{-1}] + tbar = xbar^2 / (mubar * Vbar); % [s] + Rbar = nbar / tbar; % [m^{-3} s^{-1}] + Ebar = Vbar / xbar; % [V m^{-1}] + Jbar = q * mubar * nbar * Ebar; % [A m^{-2}] + CAubar = Rbar / nbar^3; % [m^6 s^{-1}] + abar = 1/xbar; % [m^{-1}] + + % scaling procedure + l2 = e0 * Vbar / (q * nbar * xbar^2); + theta = ni / nbar; + + xin = x / xbar; + Din = D / nbar; + Nain = Na / nbar; + Ndin = Nd / nbar; + pin = p / nbar; + nin = n / nbar; + Vin = V / Vbar; + Fnin = Vin - log (nin); + Fpin = Vin + log (pin); + + tnin = tn / tbar; + tpin = tp / tbar; + + u0nin = u0n / mubar; + uminnin = uminn / mubar; + vsatnin = vsatn / (mubar * Ebar); + + u0pin = u0p / mubar; + uminpin = uminp / mubar; + vsatpin = vsatp / (mubar * Ebar); + + Nrefnin = Nrefn / nbar; + Nrefpin = Nrefp / nbar; + + Cnin = Cn / CAubar; + Cpin = Cp / CAubar; + + anin = an / abar; + apin = ap / abar; + Ecritnin = Ecritn / Ebar; + Ecritpin = Ecritp / Ebar; + + % tolerances for convergence checks + toll = 1e-3; + maxit = 1000; + ptoll = 1e-12; + pmaxit = 1000; + + % solve the problem using the full DD model + [nout, pout, Vout, Fnout, Fpout, Jnout, Jpout, it, res] = ... + secs1d_dd_gummel_map (xin, Din, Nain, Ndin, pin, nin, Vin, Fnin, Fpin, ... + l2, er, u0nin, uminnin, vsatnin, betan, Nrefnin, ... + u0pin, uminpin, vsatpin, betap, Nrefpin, theta, ... + tnin, tpin, Cnin, Cpin, anin, apin, ... + Ecritnin, Ecritpin, toll, maxit, ptoll, pmaxit); + + % Descaling procedure + n = nout*nbar; + p = pout*nbar; + V = Vout*Vbar; + Fn = V - Vth*log(n/ni); + Fp = V + Vth*log(p/ni); + dV = diff(V); + dx = diff(x); + E = -dV./dx; + + % band structure + Efn = -Fn; + Efp = -Fp; + Ec = Vth*log(Nc./n)+Efn; + Ev = -Vth*log(Nv./p)+Efp; + + plot (x, Efn, x, Efp, x, Ec, x, Ev) + legend ('Efn', 'Efp', 'Ec', 'Ev') + axis tight +\end{verbatim} + +\begin{figure}\centering +\includegraphics[width=.7\linewidth]{function/images/secs1d_dd_gummel_map_205.png} +\caption{Figure produced by demo number 1 for function secs1d\_dd\_gummel\_map} +\label{fig:secs1d_dd_gummel_map_figure_1} +\end{figure} +\clearpage
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/doc/function/secs1d_dd_newton.tex Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,169 @@ +\begin{verbatim} + + + [n, p, V, Fn, Fp, Jn, Jp, it, res] = secs1d_dd_newton (x, D, Vin, nin, + pin, l2, er, un, + up, theta, tn, tp, + Cn, Cp, toll, maxit) + + Solve the scaled stationary bipolar DD equation system using Newton's method + + input: + x spatial grid + D doping profile + pin initial guess for hole concentration + nin initial guess for electron concentration + Vin initial guess for electrostatic potential + l2 scaled Debye length squared + er relative electric permittivity + un electron mobility model coefficients + up electron mobility model coefficients + theta intrinsic carrier density + tn, tp, Cn, Cp generation recombination model parameters + toll tolerance for Gummel iterarion convergence test + maxit maximum number of Gummel iterarions + + output: + n electron concentration + p hole concentration + V electrostatic potential + Fn electron Fermi potential + Fp hole Fermi potential + Jn electron current density + Jp hole current density + it number of Gummel iterations performed + res total potential increment at each step + + +\end{verbatim} + + + + +\subsection{Demo 1 for function secs1d\_dd\_newton} +\begin{verbatim} + + % physical constants and parameters + secs1d_physical_constants; + secs1d_silicon_material_properties; + + % geometry + L = 1e-6; % [m] + x = linspace (0, L, 10)'; + sinodes = [1:length(x)]; + + % dielectric constant (silicon) + er = esir * ones (numel (x) - 1, 1); + + % doping profile [m^{-3}] + Na = 1e20 * ones(size(x)); + Nd = 1e24 * ones(size(x)); + D = Nd-Na; + + % externally applied voltages + V_p = 10; + V_n = 0; + + % initial guess for phin, phip, n, p, V + Fp = V_p * (x <= L/2); + Fn = Fp; + + p = abs(D)/2.*(1+sqrt(1+4*(ni./abs(D)).^2)).*(D<0)+... + ni^2./(abs(D)/2.*(1+sqrt(1+4*(ni./abs(D)).^2))).*(D>0); + + n = abs(D)/2.*(1+sqrt(1+4*(ni./abs(D)).^2)).*(D>0)+... + ni^2./(abs(D)/2.*(1+sqrt(1+4*(ni./abs(D)).^2))).*(D<0); + + V = Fn + Vth*log(n/ni); + + % scaling factors + xbar = L; % [m] + nbar = norm(D, 'inf'); % [m^{-3}] + Vbar = Vth; % [V] + mubar = max(u0n, u0p); % [m^2 V^{-1} s^{-1}] + tbar = xbar^2/(mubar*Vbar); % [s] + Rbar = nbar/tbar; % [m^{-3} s^{-1}] + Ebar = Vbar/xbar; % [V m^{-1}] + Jbar = q*mubar*nbar*Ebar; % [A m^{-2}] + CAubar = Rbar/nbar^3; % [m^6 s^{-1}] + abar = xbar^(-1); % [m^{-1}] + + % scaling procedure + l2 = e0*Vbar/(q*nbar*xbar^2); + theta = ni/nbar; + + xin = x/xbar; + Din = D/nbar; + Nain = Na/nbar; + Ndin = Nd/nbar; + pin = p/nbar; + nin = n/nbar; + Vin = V/Vbar; + Fnin = Vin - log(nin); + Fpin = Vin + log(pin); + + tnin = tn/tbar; + tpin = tp/tbar; + + % mobility model accounting scattering from ionized impurities + u0nin = u0n/mubar; + uminnin = uminn/mubar; + vsatnin = vsatn/(mubar*Ebar); + + u0pin = u0p/mubar; + uminpin = uminp/mubar; + vsatpin = vsatp/(mubar*Ebar); + + Nrefnin = Nrefn/nbar; + Nrefpin = Nrefp/nbar; + + Cnin = Cn/CAubar; + Cpin = Cp/CAubar; + + anin = an/abar; + apin = ap/abar; + Ecritnin = Ecritn/Ebar; + Ecritpin = Ecritp/Ebar; + + % tolerances for convergence checks + ptoll = 1e-12; + pmaxit = 1000; + + % solve the problem using the Newton fully coupled iterative algorithm + [nout, pout, Vout, Fnout, Fpout, Jnout, Jpout, it, res] = secs1d_dd_newton (xin, Din, + Vin, nin, pin, l2, er, + u0nin, u0pin, theta, tnin, + tpin, Cnin, Cpin, ptoll, pmaxit); + % Descaling procedure + n = nout*nbar; + p = pout*nbar; + V = Vout*Vbar; + Fn = V - Vth*log(n/ni); + Fp = V + Vth*log(p/ni); + dV = diff(V); + dx = diff(x); + E = -dV./dx; + + % compute current densities + [Bp, Bm] = bimu_bernoulli (dV/Vth); + Jn = q*u0n*Vth .* (n(2:end) .* Bp - n(1:end-1) .* Bm) ./ dx; + Jp = -q*u0p*Vth .* (p(2:end) .* Bm - p(1:end-1) .* Bp) ./ dx; + Jtot = Jn+Jp; + + % band structure + Efn = -Fn; + Efp = -Fp; + Ec = Vth*log(Nc./n)+Efn; + Ev = -Vth*log(Nv./p)+Efp; + + plot (x, Efn, x, Efp, x, Ec, x, Ev) + legend ('Efn', 'Efp', 'Ec', 'Ev') + axis tight +\end{verbatim} + +\begin{figure}\centering +\includegraphics[width=.7\linewidth]{function/images/secs1d_dd_newton_819.png} +\caption{Figure produced by demo number 1 for function secs1d\_dd\_newton} +\label{fig:secs1d_dd_newton_figure_1} +\end{figure} +\clearpage
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/doc/function/secs1d_nlpoisson_newton.tex Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,121 @@ +\begin{verbatim} + + + [V, n, p, res, niter] = secs1d_nlpoisson_newton (x, sinodes, Vin, nin, pin, + Fnin, Fpin, D, l2, er, toll, maxit) + + input: + x spatial grid + sinodes index of the nodes of the grid which are in the semiconductor subdomain + (remaining nodes are assumed to be in the oxide subdomain) + Vin initial guess for the electrostatic potential + nin initial guess for electron concentration + pin initial guess for hole concentration + Fnin initial guess for electron Fermi potential + Fpin initial guess for hole Fermi potential + D doping profile + l2 scaled Debye length squared + er relative electric permittivity + toll tolerance for convergence test + maxit maximum number of Newton iterations + + output: + V electrostatic potential + n electron concentration + p hole concentration + res residual norm at each step + niter number of Newton iterations + + +\end{verbatim} + + + + +\subsection{Demo 1 for function secs1d\_nlpoisson\_newton} +\begin{verbatim} + + secs1d_physical_constants + secs1d_silicon_material_properties + + tbulk= 1.5e-6; + tox = 90e-9; + L = tbulk + tox; + cox = esio2/tox; + + Nx = 50; + Nel = Nx - 1; + + x = linspace (0, L, Nx)'; + sinodes = find (x <= tbulk); + xsi = x(sinodes); + + Nsi = length (sinodes); + Nox = Nx - Nsi; + + NelSi = Nsi - 1; + NelSiO2 = Nox - 1; + + Na = 1e22; + D = - Na * ones (size (xsi)); + p = Na * ones (size (xsi)); + n = (ni^2) ./ p; + Fn = Fp = zeros (size (xsi)); + Vg = -10; + Nv = 80; + for ii = 1:Nv + Vg = Vg + 0.2; + vvect(ii) = Vg; + + V = - Phims + Vg * ones (size (x)); + V(sinodes) = Fn + Vth * log (n/ni); + + % Scaling + xs = L; + ns = norm (D, inf); + Din = D / ns; + Vs = Vth; + xin = x / xs; + nin = n / ns; + pin = p / ns; + Vin = V / Vs; + Fnin = (Fn - Vs * log (ni / ns)) / Vs; + Fpin = (Fp + Vs * log (ni / ns)) / Vs; + + er = esio2r * ones(Nel, 1); + l2(1:NelSi) = esi; + l2 = (Vs*e0)/(q*ns*xs^2); + + % Solution of Nonlinear Poisson equation + + % Algorithm parameters + toll = 1e-10; + maxit = 1000; + + [V, nout, pout, res, niter] = secs1d_nlpoisson_newton (xin, sinodes, + Vin, nin, pin, + Fnin, Fpin, Din, l2, + er, toll, maxit); + + % Descaling + n = nout*ns; + p = pout*ns; + V = V*Vs; + + qtot(ii) = q * trapz (xsi, p + D - n); + end + + vvectm = (vvect(2:end)+vvect(1:end-1))/2; + C = - diff (qtot) ./ diff (vvect); + plot(vvectm, C) + xlabel('Vg [V]') + ylabel('C [Farad]') + title('C-V curve') +\end{verbatim} + +\begin{figure}\centering +\includegraphics[width=.7\linewidth]{function/images/secs1d_nlpoisson_newton_85.png} +\caption{Figure produced by demo number 1 for function secs1d\_nlpoisson\_newton} +\label{fig:secs1d_nlpoisson_newton_figure_1} +\end{figure} +\clearpage
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/doc/function/secs1d_physical_constants.m.tex Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,18 @@ +\begin{verbatim} + + + some useful physical constants + + Kb = Boltzman constant + q = quantum of charge + e0 = permittivity of free space + hplanck = Plank constant + hbar = Plank constant by 2 pi + mn0 = free electron mass + T0 = temperature + Vth = thermal voltage + + +\end{verbatim} + +\clearpage
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/doc/function/secs1d_silicon_material_properties.m.tex Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,42 @@ +\begin{verbatim} + + + material properties for silicon and silicon dioxide + + esir = relative electric permittivity of silicon + esio2r = relative electric permittivity of silicon dioxide + esi = electric permittivity of silicon + esio2 = electric permittivity of silicon dioxide + mn = effective mass of electrons in silicon + mh = effective mass of holes in silicon + + u0n = low field electron mobility + u0p = low field hole mobility + uminn = parameter for doping-dependent electron mobility + betan = idem + Nrefn = idem + uminp = parameter for doping-dependent hole mobility + betap = idem + Nrefp = idem + vsatn = electron saturation velocity + vsatp = hole saturation velocity + tp = electron lifetime + tn = hole lifetime + Cn = electron Auger coefficient + Cp = hole Auger coefficient + an = impact ionization rate for electrons + ap = impact ionization rate for holes + Ecritn = critical field for impact ionization of electrons + Ecritp = critical field for impact ionization of holes + Nc = effective density of states in the conduction band + Nv = effective density of states in the valence band + Egap = bandgap in silicon + EgapSio2 = bandgap in silicon dioxide + + ni = intrinsic carrier density + Phims = metal to semiconductor potential barrier + + +\end{verbatim} + +\clearpage
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/doc/manual.tex Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,266 @@ +\documentclass[10pt]{article} +\usepackage{geometry} +\geometry{a4paper} +\hoffset=-1cm +\usepackage{graphicx} +\usepackage{amssymb} +\usepackage{epstopdf} +\usepackage{cprotect} +\usepackage{float} +\floatstyle{plain} +\newfloat{demo}{thp}{dem} +\floatname{demo}{Demo} +\newfloat{demoout}{thp}{deo} +\floatname{demoout}{Demo Output} +\newcommand{\unit}[1]{\mathrm{#1}} +\newcommand{\electronvolt}{\unit{eV}} +\newcommand{\kelvin}{\unit{K}} +\newcommand{\nano}{\unit{n}} +\newcommand{\meter}{\unit{m}} +\newcommand{\second}{\unit{s}} +\newcommand{\volt}{\unit{V}} +\newcommand{\Ampere}{\unit{A}} + + +\title{secs1d} +\author{Carlo de Falco \and Riccardo Sacco} +\begin{document} +\maketitle +\tableofcontents + +\begin{table} +\caption{secs1d Package Description} +\centering +\begin{tabular}{|l|l|} +\hline +{\bf Name: } & secs1d\\ \hline +{\bf Description: } & +A Drift-Diffusion simulator for 1d semiconductor devices\\ \hline +{\bf Version: } & 0.0.9\\ \hline +{\bf Release Date: } & 2012-03-25\\ \hline +{\bf Author: } & Carlo de Falco\\ \hline +{\bf Maintainer: } & Carlo de Falco\\ \hline +{\bf License: } & GPL version 2 or later\\ \hline +{\bf Depends on: } & +octave ($>=$ 3.0.0), bim ($>=$ 0.0.0), \\ \hline +{\bf Autoload: } &No\\ \hline +\end{tabular} +\end{table} + +\part{Mathematical models} + +\section{Full model} +\subsection{Conservation laws} + +\begin{equation}\label{eq:conservation} +\left\{ +\begin{array}{ll} +-\lambda^{2}\mathrm{div}\ \left(\varepsilon_{r} \mathrm{grad}\ +\varphi \right) = p - n + N_{D} - N_{A} \\[5mm] +-\mathrm{div}\ \left(J_{n} \right) + R_{n} \, n = G_{n} \\[5mm] +\phantom{-}\mathrm{div}\ \left(J_{p} \right) + R_{p} \, p = G_{p} +\end{array} +\right. +\end{equation} + +\section{Constitutive relations} + +\subsection{Currents} + +\begin{equation}\label{eq:currents} +\left\{ +\begin{array}{ll} +J_{n} = \phantom{-}\mu_{n} \left( \mathrm{grad}\ n - n\ \mathrm{grad}\ \varphi\right) +\\[5mm] +J_{p} = -\mu_{p} \left( \mathrm{grad}\ p + p\ \mathrm{grad}\ \varphi\right) +\end{array} +\right. +\end{equation} + +\subsection{Mobilities} + +\begin{equation}\label{eq:mobilities} +\left\{ +\begin{array}{ll} +\mu_{n} = \displaystyle \frac{2\bar{\mu}_{n}} +{1 + \sqrt{1 + 4 \left( \displaystyle \frac{\bar{\mu}_{n}|E|}{v_{sat,n}}\right)^{2}}} +; \qquad +\bar{\mu}_{n} = \mu_{min, n} + +\displaystyle \frac{\mu_{0,n} - \mu_{min,n}} +{1 +\displaystyle \left(\frac{N_{D}+N_{A}}{N_{ref,n}}\right)^{\beta_{n}}} +\\[10mm] +\mu_{p} = \displaystyle \frac{2\bar{\mu}_{p}} +{1 + \sqrt{1 + 4 \left( \displaystyle \frac{\bar{\mu}_{p}|E|}{v_{sat,p}}\right)^{2}}} +; \qquad +\bar{\mu}_{p} = \mu_{min, p} + +\displaystyle \frac{\mu_{0,p} - \mu_{min,p}} +{1 +\displaystyle \left(\frac{N_{D}+N_{A}}{N_{ref,p}}\right)^{\beta_{p}}} +\end{array} +\right. +\end{equation} + +\subsection{Production terms} + +\begin{equation}\label{eq:recombination} +\left\{ +\begin{array}{ll} +R_{n} = \displaystyle +\frac{p}{\tau_{n} (p + \theta) + \tau_{p} (n + \theta)} ++ p \left(C_{n} n + C_{p} p \right) +\\[5mm] +R_{p} = \displaystyle +\frac{n}{\tau_{n} (p + \theta) + \tau_{p} (n + \theta)} ++ n \left (C_{n} n + C_{p} p \right) +\end{array} +\right. +\end{equation} + +\begin{equation}\label{eq:generation} +G_{n} = G_{p} = +\displaystyle +\frac{\theta^{2}}{\tau_{n} (p + \theta) + \tau_{p} (n + \theta)} ++ \theta^{2} \left(C_{n} n + C_{p} p \right) ++ \left(\alpha_{n} |J_{n}|+ \alpha_{p} |J_{p}| \right) +\end{equation} + +%\subsection{Ionization coefficients} + +\begin{equation}\label{eq:ioniz_coeff} +\left\{ +\begin{array}{ll} +\alpha_{n} = \displaystyle +\alpha_{n}^{\infty} \exp \left( -\frac{E_{crit,n}}{|E|} \right) +\\[5mm] +\alpha_{p} = \displaystyle +\alpha_{p}^{\infty} \exp \left( -\frac{E_{crit,p}}{|E|} \right) +\end{array} +\right. +\end{equation} + +\newpage + +\section{Simplified model used for Newton's method} +\subsection{Conservation laws} + +\begin{equation}\label{eq:conservationN} +\left\{ +\begin{array}{ll} +-\lambda^{2}\mathrm{div}\ \left(\varepsilon_{r} +\mathrm{grad}\ \varphi \right) = p - n + N_{D} - N_{A} \\[5mm] +-\mathrm{div}\ \left(J_{n} \right) + R_{n} \, n = G_{n} \\[5mm] +\phantom{-}\mathrm{div}\ \left(J_{p} \right) + R_{p} \, p = G_{p} +\end{array} +\right. +\end{equation} + +\section{Constitutive relations} + +\subsection{Currents} + +\begin{equation}\label{eq:currentsN} +\left\{ +\begin{array}{ll} +J_{n} = \phantom{-}\mu_{n} \left( \mathrm{grad}\ n - n\ \mathrm{grad}\ \varphi\right) +\\[5mm] +J_{p} = -\mu_{p} \left( \mathrm{grad}\ p + p\ \mathrm{grad}\ \varphi\right) +\end{array} +\right. +\end{equation} + +\subsection{Production terms} + +\begin{equation}\label{eq:recombinationN} +\left\{ +\begin{array}{ll} +R_{n} = \displaystyle \frac{p}{\tau_{n} (p + \theta) + \tau_{p} (n + \theta)} ++ p \left(C_{n} n + C_{p} p \right) +\\[5mm] +R_{p} = \displaystyle \frac{n}{\tau_{n} (p + \theta) + \tau_{p} (n + \theta)} ++ n \left (C_{n} n + C_{p} p \right) +\end{array} +\right. +\end{equation} + +\begin{equation}\label{eq:generationN} +G_{n} = G_{p} = +\displaystyle \frac{\theta^{2}}{\tau_{n} (p + \theta) + \tau_{p} (n + \theta)} ++ \theta^{2} \left(C_{n} n + C_{p} p \right) +\end{equation} + +\newpage + +\section{Scaling factors/adimensional parameters} + +Given any generic quantity $u$ having units $U$, we +define the {\em scaled} quantity $\widehat{u}$ as +$$ +\widehat{u} : = \displaystyle \frac{u}{\overline{u}} +$$ +where $\overline{u}$ is the scaling factor associated with $u$ +and having the same units as $u$. + +\begin{table}[h!] +\begin{center} +\begin{tabular}{lll}\hline +\textbf{Scaling factor} & \textbf{Value} & \textbf{Units}\\ \hline +$\overline{x}$ & $L$ & $\meter$ \\[1mm] +$\overline{n}$ & $\| N_D^+ - N_A^-\|_{L^{\infty}(0,L)}$ +& $\meter^{-3}$ \\[1mm] +$\overline{\varphi}$ & $K_B T / q \simeq 26 \cdot 10^{-3}$ +& $\volt$ \\[1mm] +$\overline{\mu}$ & $\max\left\{ \mu_{0,n}, \, \mu_{0,p}\right\}$ +& $\meter^2\,\volt^{-1}\,\second^{-1}$ \\[1mm] +$\overline{t}$ & $\overline{x}^2/(\overline{\mu} \, \overline{\varphi})$ +& $\second$ \\[1mm] +$\overline{R}$ & $\overline{n}/\overline{t}$ +& $\meter^{-3} \second^{-1}$ \\[1mm] +$\overline{E}$ & $\overline{\varphi}/\overline{x}$ +& $\volt \meter^{-1}$ \\[1mm] +$\overline{J}$ & $q \, \overline{\mu} \, \overline{n} \, +\overline{E}$ & $\Ampere \meter^{-2}$ \\[1mm] +$\overline{\alpha}$ & $\overline{x}^{-1}$ & $\meter^{-1}$ \\[1mm] +$\overline{C}_{Au}$ & $\overline{R}/\overline{n}^3$ & +$\meter^{6} \second^{-1}$ \\[1mm] +\hline +\end{tabular} +\caption{Scaling factors for the Drift-Diffusion model equations.} +\label{tab:model_param_1d} +\end{center} +\end{table} + +We also introduce the following adimensional numbers +$$ +\lambda^2:= \displaystyle \frac{\varepsilon_0 \overline{\varphi}} +{q \, \overline{n} \, \overline{x}^2}, \qquad +\theta:= \displaystyle \frac{n_i}{\overline{n}} +$$ +having the meaning of squared normalized Debye length and +normalized intrinsic concentration, respectively. + +\part{Function reference} + +\section{Drift-Diffusion solvers} + +\subsection{secs1d\_dd\_gummel\_map} +\input{function/secs1d_dd_gummel_map.tex} + +\subsection{secs1d\_dd\_newton} +\input{function/secs1d_dd_newton.tex} + +\section{Non-linear Poisson solver} +\subsection{secs1d\_nlpoisson\_newton} +\input{function/secs1d_nlpoisson_newton.tex} + +\section{Physical constants and material properties} +\subsection{secs1d\_physical\_constants.m} +\input{function/secs1d_physical_constants.m.tex} + +\subsection{secs1d\_silicon\_material\_properties.m} +\input{function/secs1d_silicon_material_properties.m.tex} + +\appendix +\section{Licence} +\input{COPYING.tex} + + +\end{document} \ No newline at end of file
--- a/extra/secs1d/inst/DDG/DDGelectron_driftdiffusion.m Sun Mar 25 22:35:18 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,89 +0,0 @@ -## Copyright (C) 2004-2008 Carlo de Falco -## -## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator -## -## SECS1D is free software; you can redistribute it and/or modify -## it under the terms of the GNU General Public License as published by -## the Free Software Foundation; either version 2 of the License, or -## (at your option) any later version. -## -## SECS1D is distributed in the hope that it will be useful, -## but WITHOUT ANY WARRANTY; without even the implied warranty of -## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the -## GNU General Public License for more details. -## -## You should have received a copy of the GNU General Public License -## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. -## -## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> - -## -*- texinfo -*- -## -## @deftypefn {Function File}@ -## {@var{n}} = DDGelectron_driftdiffusion(@var{psi},@var{x},@var{ng},@var{p},@var{ni},@var{tn},@var{tp},@var{un}) -## -## Solve the continuity equation for electrons -## -## Input: -## @itemize @minus -## @item psi: electric potential -## @item x: integration domain -## @item ng: initial guess and BCs for electron density -## @item p: hole density (for SRH recombination) -## @end itemize -## -## Output: -## @itemize @minus -## @item n: updated electron density -## @end itemize -## -## @end deftypefn - -function n = DDGelectron_driftdiffusion(psi,x,ng,p,ni,tn,tp,un) - -nodes = x; -Nnodes =length(nodes); - -elements = [[1:Nnodes-1]' [2:Nnodes]']; -Nelements=size(elements,1); - -Bcnodes = [1;Nnodes]; - -nl = ng(1); -nr = ng(Nnodes); -h=nodes(elements(:,2))-nodes(elements(:,1)); - -c=1./h; -Bneg=Ubernoulli(-(psi(2:Nnodes)-psi(1:Nnodes-1)),1); -Bpos=Ubernoulli( (psi(2:Nnodes)-psi(1:Nnodes-1)),1); - - -d0 = [c(1).*Bneg(1); c(1:end-1).*Bpos(1:end-1)+c(2:end).*Bneg(2:end); c(end)*Bpos(end)]; -d1 = [1000;-c.* Bpos]; -dm1 = [-c.* Bneg;1000]; - -A = spdiags([dm1 d0 d1],-1:1,Nnodes,Nnodes); -b = zeros(Nnodes,1);%- A * ng; - - ## SRH Recombination term -SRHD = tp * (ng + ni) + tn * (p + ni); -SRHL = p ./ SRHD; -SRHR = ni.^2 ./ SRHD; - -ASRH = Ucompmass (nodes,Nnodes,elements,Nelements,SRHL,ones(Nelements,1)); -bSRH = Ucompconst (nodes,Nnodes,elements,Nelements,SRHR,ones(Nelements,1)); - -A = A + ASRH; -b = b + bSRH; - - ## Boundary conditions -b(Bcnodes) = []; -b(1) = - A(2,1) * nl; -b(end) = - A(end-1,end) * nr; -A(Bcnodes,:) = []; -A(:,Bcnodes) = []; - -n = [nl; A\b ;nr]; - -endfunction -
--- a/extra/secs1d/inst/DDG/DDGgummelmap.m Sun Mar 25 22:35:18 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,153 +0,0 @@ -## Copyright (C) 2004-2008 Carlo de Falco -## -## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator -## -## SECS1D is free software; you can redistribute it and/or modify -## it under the terms of the GNU General Public License as published by -## the Free Software Foundation; either version 2 of the License, or -## (at your option) any later version. -## -## SECS1D is distributed in the hope that it will be useful, -## but WITHOUT ANY WARRANTY; without even the implied warranty of -## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the -## GNU General Public License for more details. -## -## You should have received a copy of the GNU General Public License -## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. -## -## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> - -## -*- texinfo -*- -## -## @deftypefn {Function File}@ -## {@var{odata},@var{it},@var{res}} = DDGgummelmap(@var{x},@var{idata},@var{toll},@var{maxit},@var{ptoll},@var{pmaxit},@var{verbose}) -## -## Solve the scaled stationary bipolar DD equation system using Gummel -## algorithm -## -## Input: -## @itemize @minus -## @item x: spatial grid -## @item idata.D: doping profile -## @item idata.p: initial guess for hole concentration -## @item idata.n: initial guess for electron concentration -## @item idata.V: initial guess for electrostatic potential -## @item idata.Fn: initial guess for electron Fermi potential -## @item idata.Fp: initial guess for hole Fermi potential -## @item idata.l2: scaled electric permittivity (diffusion coefficient in Poisson equation) -## @item idata.un: scaled electron mobility -## @item idata.up: scaled electron mobility -## @item idata.nis: scaled intrinsic carrier density -## @item idata.tn: scaled electron lifetime -## @item idata.tp: scaled hole lifetime -## @item toll: tolerance for Gummel iterarion convergence test -## @item maxit: maximum number of Gummel iterarions -## @item ptoll: tolerance for Newton iterarion convergence test for non linear Poisson -## @item pmaxit: maximum number of Newton iterarions -## @item verbose: verbosity level (0,1,2) -## @end itemize -## -## Output: -## @itemize @minus -## @item odata.n: electron concentration -## @item odata.p: hole concentration -## @item odata.V: electrostatic potential -## @item odata.Fn: electron Fermi potential -## @item odata.Fp: hole Fermi potential -## @item it: number of Gummel iterations performed -## @item res: total potential increment at each step -## @end itemize -## -## @end deftypefn - -function [odata,it,res] = DDGgummelmap (x,idata,toll,maxit,ptoll,pmaxit,verbose) - - odata = idata; -Nnodes=rows(x); - -D = idata.D; -vout(:,1) = idata.V; - -hole_density (:,1) = idata.p; -electron_density (:,1)= idata.n; -fermin (:,1)=idata.Fn; -fermip (:,1)=idata.Fp; - -for i=1:1:maxit - if (verbose>1) - fprintf(1,"*****************************************************************\n"); - fprintf(1,"**** start of gummel iteration number: %d\n",i); - fprintf(1,"*****************************************************************\n"); - endif - - if (verbose>1) - fprintf(1,"solving non linear poisson equation\n\n"); - endif - - [vout(:,2),electron_density(:,2),hole_density(:,2)] =\ - DDGnlpoisson (x,[1:Nnodes],vout(:,1),electron_density(:,1),hole_density(:,1),fermin(:,1),fermip(:,1),D,idata.l2,ptoll,pmaxit,verbose); - - if (verbose>1) - fprintf (1,"\n\nupdating electron qfl\n\n"); - endif - electron_density(:,3)=\ - DDGelectron_driftdiffusion(vout(:,2), x, electron_density(:,2),hole_density(:,2),idata.nis,idata.tn,idata.tp,idata.un); - - fermin(:,2) = DDGn2phin(vout(:,2),electron_density(:,3)); - fermin(1,2) = idata.Fn(1); - fermin(end:2) = idata.Fn(end); - - if (verbose>1) - fprintf(1,"updating hole qfl\n\n"); - endif - - hole_density(:,3) = \ - DDGhole_driftdiffusion(vout(:,2), x, hole_density(:,2),electron_density(:,2),idata.nis,idata.tn,idata.tp,idata.up); - - fermip(:,2) = DDGp2phip(vout(:,2),hole_density(:,3)); - fermip(1,2) = idata.Fp(1); - fermip(end,2) = idata.Fp(end); - - if (verbose>1) - fprintf(1,"checking for convergence\n\n"); - endif - - nrfn= norm(fermin(:,2)-fermin(:,1),inf); - nrfp= norm (fermip(:,2)-fermip(:,1),inf); - nrv = norm (vout(:,2)-vout(:,1),inf); - nrm(i) = max([nrfn;nrfp;nrv]); - - if (verbose>1) - fprintf (1," max(|phin_(k+1)-phinn_(k)| , |phip_(k+1)-phip_(k)| , |v_(k+1)- v_(k)| )= %d\n",nrm(i)); - endif - - if (nrm(i)<toll) - break - endif - - vout(:,1) = vout(:,end); - hole_density (:,1) = hole_density (:,end) ; - electron_density (:,1)= electron_density (:,end); - fermin (:,1)= fermin (:,end); - fermip (:,1)= fermip (:,end); - - - if(verbose) - DDGplotresults(x,electron_density,hole_density,vout,fermin,fermip); - endif - endfor - -it = i; -res = nrm; - -if (verbose>0) - fprintf(1,"\n\nInitial guess computed by DD: # of Gummel iterations = %d\n\n",it); - endif - -odata.n = electron_density(:,end); -odata.p = hole_density(:,end); -odata.V = vout(:,end); -odata.Fn = fermin(:,end); -odata.Fp = fermip(:,end); - -endfunction \ No newline at end of file
--- a/extra/secs1d/inst/DDG/DDGhole_driftdiffusion.m Sun Mar 25 22:35:18 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,87 +0,0 @@ -## Copyright (C) 2004-2008 Carlo de Falco -## -## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator -## -## SECS1D is free software; you can redistribute it and/or modify -## it under the terms of the GNU General Public License as published by -## the Free Software Foundation; either version 2 of the License, or -## (at your option) any later version. -## -## SECS1D is distributed in the hope that it will be useful, -## but WITHOUT ANY WARRANTY; without even the implied warranty of -## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the -## GNU General Public License for more details. -## -## You should have received a copy of the GNU General Public License -## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. -## -## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> - -## -*- texinfo -*- -## -## @deftypefn {Function File}@ -## {@var{p}} = DDGhole_driftdiffusio(@var{psi},@var{x},@var{pg},@var{n},@var{ni},@var{tn},@var{tp},@var{up}) -## -## Solve the continuity equation for holes -## -## Input: -## @itemize @minus -## @item psi: electric potential -## @item x: spatial grid -## @item ng: initial guess and BCs for electron density -## @item n: electron density (for SRH recombination) -## @end itemize -## -## Output: -## @itemize @minus -## @item p: updated hole density -## @end itemize -## -## @end deftypefn - -function p = DDGhole_driftdiffusion(psi,x,pg,n,ni,tn,tp,up) - -nodes = x; -Nnodes =length(nodes); -elements = [[1:Nnodes-1]' [2:Nnodes]']; -Nelements=size(elements,1); -Bcnodes = [1;Nnodes]; - -pl = pg(1); -pr = pg(Nnodes); -h=nodes(elements(:,2))-nodes(elements(:,1)); -c=up./h; -Bneg=Ubernoulli(-(psi(2:Nnodes)-psi(1:Nnodes-1)),1); -Bpos=Ubernoulli( (psi(2:Nnodes)-psi(1:Nnodes-1)),1); - - -d0 = [c(1).*Bpos(1); c(1:end-1).*Bneg(1:end-1)+c(2:end).*Bpos(2:end); c(end)*Bneg(end)]; -d1 = [1000;-c.* Bneg]; -dm1 = [-c.* Bpos;1000]; - -A = spdiags([dm1 d0 d1],-1:1,Nnodes,Nnodes); - b = zeros(Nnodes,1);## - A * pg; - - ## SRH Recombination term -SRHD = tp * (n + ni) + tn * (pg + ni); -SRHL = n ./ SRHD; -SRHR = ni.^2 ./ SRHD; - -ASRH = Ucompmass (nodes,Nnodes,elements,Nelements,SRHL,ones(Nelements,1)); -bSRH = Ucompconst (nodes,Nnodes,elements,Nelements,SRHR,ones(Nelements,1)); - -A = A + ASRH; -b = b + bSRH; - - ## Boundary conditions -b(Bcnodes) = []; -b(1) = - A(2,1) * pl; -b(end) = - A(end-1,end) * pr; -A(Bcnodes,:) = []; -A(:,Bcnodes) = []; - -p = [pl; A\b ;pr]; - -endfunction - -
--- a/extra/secs1d/inst/DDG/DDGn2phin.m Sun Mar 25 22:35:18 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,36 +0,0 @@ -## Copyright (C) 2004-2008 Carlo de Falco -## -## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator -## -## SECS1D is free software; you can redistribute it and/or modify -## it under the terms of the GNU General Public License as published by -## the Free Software Foundation; either version 2 of the License, or -## (at your option) any later version. -## -## SECS1D is distributed in the hope that it will be useful, -## but WITHOUT ANY WARRANTY; without even the implied warranty of -## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the -## GNU General Public License for more details. -## -## You should have received a copy of the GNU General Public License -## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. -## -## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> - -## -*- texinfo -*- -## -## @deftypefn {Function File}@ -## {@var{phin}} = DDGn2phin(@var{V},@var{n}) -## -## Compute the qfl for electrons using Maxwell-Boltzmann statistics. -## -## @end deftypefn - -function phin = DDGn2phin (V,n); - - ## Load constants - nmin = 0; -n = n .* (n>nmin) + nmin * (n<=nmin); -phin = V - log(n) ; - -endfunction
--- a/extra/secs1d/inst/DDG/DDGnlpoisson.m Sun Mar 25 22:35:18 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,212 +0,0 @@ -## Copyright (C) 2004-2008 Carlo de Falco -## -## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator -## -## SECS1D is free software; you can redistribute it and/or modify -## it under the terms of the GNU General Public License as published by -## the Free Software Foundation; either version 2 of the License, or -## (at your option) any later version. -## -## SECS1D is distributed in the hope that it will be useful, -## but WITHOUT ANY WARRANTY; without even the implied warranty of -## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the -## GNU General Public License for more details. -## -## You should have received a copy of the GNU General Public License -## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. -## -## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> - -## -*- texinfo -*- -## -## @deftypefn {Function File}@ -## {@var{V},@var{n},@var{p},@var{res},@var{niter}} = @ -## DDGnlpoisson(@var{x},@var{sinodes},@var{Vin},@var{nin},@var{pin},@var{Fnin},@var{Fpin},@var{D},@var{l2},@var{toll},@var{maxit},@var{verbose}) -## -## Solve the non linear Poisson equation -## -## - lamda^2 *V'' + (n(V,Fn) - p(V,Fp) -D) = 0 -## -## Input: -## @itemize @minus -## @item x: spatial grid -## @item sinodes: index of the nodes of the grid which are in the -## semiconductor subdomain (remaining nodes are assumed to be in the oxide subdomain) -## @item Vin: initial guess for the electrostatic potential -## @item nin: initial guess for electron concentration -## @item pin: initial guess for hole concentration -## @item Fnin: initial guess for electron Fermi potential -## @item Fpin: initial guess for hole Fermi potential -## @item D: doping profile -## @item l2: scaled electric permittivity (diffusion coefficient) -## @item toll: tolerance for convergence test -## @item maxit: maximum number of Newton iterations -## @item verbose: verbosity level (0,1,2) -## @end itemize -## -## Output: -## @itemize @minus -## @item V: electrostatic potential -## @item n: electron concentration -## @item p: hole concentration -## @item res: residual norm at each step -## @item niter: number of Newton iterations -## @end itemize -## -## @end deftypefn - -function [V,n,p,res,niter] = DDGnlpoisson (x,sinodes,Vin,nin,pin,Fnin,Fpin,D,l2,toll,maxit,verbose) - - ## Set some useful constants -dampit = 10; -dampcoeff = 2; - - ## Convert grid info to FEM form -Ndiricheletnodes = 2; -nodes = x; -sielements = sinodes(1:end-1); -Nnodes = length(nodes); -totdofs = Nnodes - Ndiricheletnodes ; -elements = [[1:Nnodes-1]' , [2:Nnodes]']; -Nelements = size(elements,1); -BCnodes = [1;Nnodes]; - - ## Initialization -V = Vin; -Fn = Fnin; -Fp = Fpin; -n = DDGphin2n(V(sinodes),Fn); -p = DDGphip2p(V(sinodes),Fp); -if (sinodes(1)==1) - n(1)=nin(1); - p(1)=pin(1); - endif -if (sinodes(end)==Nnodes) - n(end)=nin(end); - p(end)=pin(end); - endif - - ## Compute LHS matrices -L = Ucomplap (nodes,Nnodes,elements,Nelements,l2.*ones(Nelements,1)); - - ## Compute Mv = ( n + p) -Mv = zeros(Nnodes,1); -Mv(sinodes) = (n + p); -Cv = zeros(Nelements,1); -Cv(sielements)= 1; -M = Ucompmass (nodes,Nnodes,elements,Nelements,Mv,Cv); - - ## Compute RHS vector -Tv0 = zeros(Nnodes,1); -Tv0(sinodes) = (n - p - D); -Cv = zeros(Nelements,1); -Cv(sielements)= 1; -T0 = Ucompconst (nodes,Nnodes,elements,Nelements,Tv0,Cv); - - ## Build LHS matrix and RHS of the linear system for 1st Newton step -A = L + M; -R = L * V + T0; - - ## Apply boundary conditions -A(BCnodes,:) = []; -A(:,BCnodes) = []; -R(BCnodes) = []; - - -normr(1) = norm(R,inf); -relresnorm = 1; -reldVnorm = 1; -normrnew = normr(1); - - - ## Start of the newton cycle -for newtit=1:maxit - if verbose - fprintf(1,"\n newton iteration: %d, reldVnorm = %e",newtit,reldVnorm); - endif - dV =[0;(A)\(-R);0]; - - ## Start of the damping procedure - tk = 1; - for dit = 1:dampit - if verbose - fprintf(1,"\n damping iteration: %d, residual norm = %e",dit,normrnew); - endif - Vnew = V + tk * dV; - n = DDGphin2n(Vnew(sinodes),Fn); - p = DDGphip2p(Vnew(sinodes),Fp); - if (sinodes(1)==1) - n(1)=nin(1); - p(1)=pin(1); - endif - if (sinodes(end)==Nnodes) - n(end)=nin(end); - p(end)=pin(end); - endif - - ## Compute LHS matrices - Mv = zeros(Nnodes,1); - Mv(sinodes) = (n + p); - Cv = zeros(Nelements,1); - Cv(sielements)= 1; - M = Ucompmass (nodes,Nnodes,elements,Nelements,Mv,Cv); - - ## Compute RHS vector (-residual) - Tv0 = zeros(Nnodes,1); - Tv0(sinodes) = (n - p - D); - Cv = zeros(Nelements,1); - Cv(sielements)= 1; - T0 = Ucompconst (nodes,Nnodes,elements,Nelements,Tv0,Cv); - - ## Build LHS matrix and RHS of the linear system for 1st Newton step - Anew = L + M; - Rnew = L * Vnew + T0; - - ## Apply boundary conditions - Anew(BCnodes,:) = []; - Anew(:,BCnodes) = []; - Rnew(BCnodes) = []; - - if ((dit>1)&(norm(Rnew,inf)>=norm(R,inf))) - if verbose - fprintf(1,"\nexiting damping cycle \n"); - endif - break - else - A = Anew; - R = Rnew; - endif - - ## Compute | R_{k+1} | for the convergence test - normrnew= norm(R,inf); - - ## Check if more damping is needed - if (normrnew > normr(newtit)) - tk = tk/dampcoeff; - else - if verbose - fprintf(1,"\nexiting damping cycle because residual norm = %e \n",normrnew); - endif - break - endif - endfor - - V = Vnew; - normr(newtit+1) = normrnew; - dVnorm = norm(tk*dV,inf); - - ## Check if convergence has been reached - reldVnorm = dVnorm / norm(V,inf); - if (reldVnorm <= toll) - if(verbose) - fprintf(1,"\nexiting newton cycle because reldVnorm= %e \n",reldVnorm); - endif - break - endif - - endfor - -res = normr; -niter = newtit; - -endfunction \ No newline at end of file
--- a/extra/secs1d/inst/DDG/DDGp2phip.m Sun Mar 25 22:35:18 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,36 +0,0 @@ -## Copyright (C) 2004-2008 Carlo de Falco -## -## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator -## -## SECS1D is free software; you can redistribute it and/or modify -## it under the terms of the GNU General Public License as published by -## the Free Software Foundation; either version 2 of the License, or -## (at your option) any later version. -## -## SECS1D is distributed in the hope that it will be useful, -## but WITHOUT ANY WARRANTY; without even the implied warranty of -## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the -## GNU General Public License for more details. -## -## You should have received a copy of the GNU General Public License -## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. -## -## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> - -## -*- texinfo -*- -## -## @deftypefn {Function File}@ -## {@var{phip}} = DDGn2phin(@var{V},@var{p}) -## -## Compute the qfl for holes using Maxwell-Boltzmann statistics -## -## @end deftypefn - -function phip = DDGp2phip (V,p); - - ## Load constants - pmin = 0; - p = p .* (p>pmin) + pmin * (p<=pmin); - phip = V + log(p) ; - -endfunction
--- a/extra/secs1d/inst/DDG/DDGphin2n.m Sun Mar 25 22:35:18 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,35 +0,0 @@ -## Copyright (C) 2004-2008 Carlo de Falco -## -## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator -## -## SECS1D is free software; you can redistribute it and/or modify -## it under the terms of the GNU General Public License as published by -## the Free Software Foundation; either version 2 of the License, or -## (at your option) any later version. -## -## SECS1D is distributed in the hope that it will be useful, -## but WITHOUT ANY WARRANTY; without even the implied warranty of -## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the -## GNU General Public License for more details. -## -## You should have received a copy of the GNU General Public License -## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. -## -## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> - -## -*- texinfo -*- -## -## @deftypefn {Function File}@ -## {@var{n}} = DDGphin2n(@var{V},@var{phin}) -## -## Compute the electron density using Maxwell-Boltzmann statistics -## -## @end deftypefn - -function n = DDGphin2n (V,phin); - - nmin = 0; - n = exp ((V-phin)); - n = n .* (n>nmin) + nmin * (n<=nmin); - -endfunction
--- a/extra/secs1d/inst/DDG/DDGphip2p.m Sun Mar 25 22:35:18 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,36 +0,0 @@ -## Copyright (C) 2004-2008 Carlo de Falco -## -## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator -## -## SECS1D is free software; you can redistribute it and/or modify -## it under the terms of the GNU General Public License as published by -## the Free Software Foundation; either version 2 of the License, or -## (at your option) any later version. -## -## SECS1D is distributed in the hope that it will be useful, -## but WITHOUT ANY WARRANTY; without even the implied warranty of -## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the -## GNU General Public License for more details. -## -## You should have received a copy of the GNU General Public License -## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. -## -## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> - -## -*- texinfo -*- -## -## @deftypefn {Function File}@ -## {@var{p}} = DDGphip2p(@var{V},@var{phip}) -## -## Compute the hole density using Maxwell-Boltzmann statistic -## -## @end deftypefn - -function p = DDGphip2p (V,phip); - -## Load constants -pmin = 0; -p = exp ((phip-V)); -p = p .* (p>pmin) + pmin * (p<=pmin); - -endfunction
--- a/extra/secs1d/inst/DDG/DDGplotresults.m Sun Mar 25 22:35:18 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,52 +0,0 @@ -## Copyright (C) 2004-2008 Carlo de Falco -## -## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator -## -## SECS1D is free software; you can redistribute it and/or modify -## it under the terms of the GNU General Public License as published by -## the Free Software Foundation; either version 2 of the License, or -## (at your option) any later version. -## -## SECS1D is distributed in the hope that it will be useful, -## but WITHOUT ANY WARRANTY; without even the implied warranty of -## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the -## GNU General Public License for more details. -## -## You should have received a copy of the GNU General Public License -## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. -## -## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> - -## -*- texinfo -*- -## -## @deftypefn {Function File}@ -## DDGplotresults(@var{x},@var{n},@var{p},@var{V},@var{Fn},@var{Fp}) -## -## Plot densities and potentials -## -## @end deftypefn - -function DDGplotresults(x,n,p,V,Fn,Fp); - -subplot(2,3,1) -title('Electron Density') -semilogy(x,n) - -subplot(2,3,2) -title('Hole Density') -semilogy(x,p) - -subplot(2,3,4) -title('Electron QFL') -plot(x,Fn) - -subplot(2,3,5) -title('Hole QFL') -plot(x,Fp) - -subplot(2,3,6) -title('Electric Potential') -plot(x,V) -pause(.1) - -endfunction
--- a/extra/secs1d/inst/DDN/DDNnewtonmap.m Sun Mar 25 22:35:18 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,230 +0,0 @@ -## Copyright (C) 2004-2008 Carlo de Falco -## -## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator -## -## SECS1D is free software; you can redistribute it and/or modify -## it under the terms of the GNU General Public License as published by -## the Free Software Foundation; either version 2 of the License, or -## (at your option) any later version. -## -## SECS1D is distributed in the hope that it will be useful, -## but WITHOUT ANY WARRANTY; without even the implied warranty of -## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the -## GNU General Public License for more details. -## -## You should have received a copy of the GNU General Public License -## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. -## -## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> - -## -*- texinfo -*- -## -## @deftypefn {Function File}@ -## {@var{odata},@var{it},@var{res}} = DDNnewtonmap(@var{x},@var{idata},@var{toll},@var{maxit},@var{verbose}) -## -## Solve the scaled stationary bipolar DD equation system using a -## coupled Newton algorithm -## -## Input: -## @itemize @minus -## @item x: spatial grid -## @item idata.D: doping profile -## @item idata.p: initial guess for hole concentration -## @item idata.n: initial guess for electron concentration -## @item idata.V: initial guess for electrostatic potential -## @item idata.Fn: initial guess for electron Fermi potential -## @item idata.Fp: initial guess for hole Fermi potential -## @item idata.l2: scaled electric permittivity (diffusion coefficient in Poisson equation) -## @item idata.un: scaled electron mobility -## @item idata.up: scaled electron mobility -## @item idata.nis: scaled intrinsic carrier density -## @item idata.tn: scaled electron lifetime -## @item idata.tp: scaled hole lifetime -## @item toll: tolerance for Newton iterarion convergence test -## @item maxit: maximum number of Newton iterarions -## @item verbose: verbosity level: 0,1,2 -## @end itemize -## -## Output: -## @itemize @minus -## @item odata.n: electron concentration -## @item odata.p: hole concentration -## @item odata.V: electrostatic potential -## @item odata.Fn: electron Fermi potential -## @item odata.Fp: hole Fermi potential -## @item it: number of Newton iterations performed -## @item res: residual at each step -## @end itemize -## -## @end deftypefn - -function [odata,it,res] = DDNnewtonmap (x,idata,toll,maxit,verbose) - - odata = idata; - Nnodes = rows(x); -Nelements=Nnodes-1; -elements=[1:Nnodes-1;2:Nnodes]'; -BCnodesp = [1,Nnodes]; -BCnodes = [BCnodesp,BCnodesp+Nnodes,BCnodesp+2*Nnodes]; -totaldofs= Nnodes-2; -dampcoef = 2; -maxdamp = 2; - -V = idata.V; -n = idata.n; -p = idata.p; -D = idata.D; - - ## Create the complete unknown vector -u = [V; n; p]; - - ## Build fem matrices -L = Ucomplap (x,Nnodes,elements,Nelements,idata.l2*ones(Nelements,1)); -M = Ucompmass (x,Nnodes,elements,Nelements,ones(Nnodes,1),ones(Nelements,1)); -DDn = Uscharfettergummel(x,Nnodes,elements,Nelements,idata.un,1,V); -DDp = Uscharfettergummel(x,Nnodes,elements,Nelements,idata.up,1,-V); - - ## Initialise RHS -r1 = L * V + M * (n - p - D); -r2 = DDn * n; -r3 = DDp * p; - RHS = - [ r1; r2; r3 ]; - - ## Apply BCs -RHS(BCnodes,:)= []; -nrm = norm(RHS,inf); -res(1) = nrm; - - ## Begin Newton Cycle -for count = 1: maxit - if verbose - fprintf (1,"\n\n\nNewton Iteration Number:%d\n",count); - endif - Ln = Ucomplap (x,Nnodes,elements,Nelements,Umediaarmonica(idata.un*n)); - Lp = Ucomplap (x,Nnodes,elements,Nelements,Umediaarmonica(idata.up*p)); - Z = sparse(zeros(Nnodes)); - DDn = Uscharfettergummel(x,Nnodes,elements,Nelements,idata.un,1,V); - DDp = Uscharfettergummel(x,Nnodes,elements,Nelements,idata.up,1,-V); - - A = L; - B = M; - C = -M; - DDD = -Ln; - E = DDn; - F = Z; - G = Lp; - H = Z; - I = DDp; - - ## Build LHS - LHS =sparse([ - [A B C]; - [DDD E F]; - [G H I]; - ]); - - ## Apply BCs - LHS(BCnodes,:)=[]; - LHS(:,BCnodes)=[]; - - ## Solve the linearised system - dutmp = (LHS) \ (RHS); - dv = dutmp(1:totaldofs); - dn = dutmp(totaldofs+1:2*totaldofs); - dp = dutmp(2*totaldofs+1:3*totaldofs); - du = [0;dv;0;0;dn;0;0;dp;0]; - - ## Check Convergence - nrm_u = norm(u,inf); - nrm_du = norm(du,inf); - - ratio = nrm_du/nrm_u; - if verbose - fprintf (1,"ratio = %e\n", ratio); - endif - - if (ratio <= toll) - V = u(1:Nnodes); - n = u(Nnodes+1:2*Nnodes); - p = u(2*Nnodes+1:end); - res(count) = nrm; - break; - endif - - ## Begin damping cycle - tj = 1; - - for cc = 1:maxdamp - if verbose - fprintf (1,"\ndamping iteration number:%d\n",cc); - fprintf (1,"reference residual norm:%e\n",nrm); - endif - ## Update the unknown vector - utmp = u + tj*du; - Vnew = utmp(1:Nnodes); - nnew = utmp(Nnodes+1:2*Nnodes); - pnew = utmp(2*Nnodes+1:end); - ## Try a new RHS - DDn = Uscharfettergummel(x,Nnodes,elements,Nelements,idata.un,1,Vnew); - DDp = Uscharfettergummel(x,Nnodes,elements,Nelements,idata.up,1,-Vnew); - - r1 = L * V + M * (nnew - pnew - D); - r2 = DDn * nnew; - r3 = DDp * pnew; - - RHS =- [r1;r2;r3]; - - ## Apply BCs - RHS(BCnodes,:) = []; - nrmtmp=norm(RHS,inf); - - ## Update the damping coefficient - if verbose - fprintf(1,"residual norm:%e\n\n", nrmtmp); - endif - - if (nrmtmp>nrm) - tj = tj/(dampcoef*cc); - if verbose - fprintf (1,"\ndamping coefficients = %e",tj); - endif - else - break; - endif - endfor - - nrm_du = norm(tj*du,inf); - u = utmp; - - if (count>1) - ratio = nrm_du/nrm_du_old; - if (ratio<.005) - V = u(1:Nnodes); - n = u(Nnodes+1:2*Nnodes); - p = u(2*Nnodes+1:end); - res(count) = nrm; - break; - endif - endif - nrm = nrmtmp; - res(count) = nrm; - - ## Convert result vector into distinct output vectors - V = u(1:Nnodes); - n = u(Nnodes+1:2*Nnodes); - p = u(2*Nnodes+1:end); - nrm_du_old = nrm_du; - endfor - -odata.V = V; -odata.n = n; -odata.p = p; -Fn = V - log(n); -Fp = V + log(p); -it = count; - -endfunction - - - -
--- a/extra/secs1d/inst/Utilities/Ubern.m Sun Mar 25 22:35:18 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,87 +0,0 @@ -## Copyright (C) 2004-2008 Carlo de Falco - ## - ## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator - ## - ## SECS1D is free software; you can redistribute it and/or modify - ## it under the terms of the GNU General Public License as published by - ## the Free Software Foundation; either version 2 of the License, or - ## (at your option) any later version. - ## - ## SECS1D is distributed in the hope that it will be useful, - ## but WITHOUT ANY WARRANTY; without even the implied warranty of - ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the - ## GNU General Public License for more details. - ## - ## You should have received a copy of the GNU General Public License - ## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. -## -## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> - -## -*- texinfo -*- -## -## @deftypefn {Function File} {@var{bp},@var{bn}} = Ubern(@var{x}) -## -## Compute Bernoulli function for scalar x: -## -## @itemize @minus -## @item @var{bp} = @var{x}/(exp(@var{x})-1) -## @item @var{bn} = @var{x} + B( @var{x} ) -## @end itemize -## -## @end deftypefn - -function [bp,bn] = Ubern(x) - -xlim=1e-2; -ax=abs(x); - - ## Compute Bernoulli function for x = 0 - -if (ax == 0) - bp=1.; - bn=1.; - return - endif - - ## Compute Bernoulli function for asymptotic values - - if (ax > 80) - if (x > 0) - bp=0.; - bn=x; - return - else - bp=-x; - bn=0.; - return - endif - endif - - ## Compute Bernoulli function for intermediate values - - if (ax > xlim) - bp=x/(exp(x)-1); - bn=x+bp; - return -else - ## Compute Bernoulli function for small x - ## via Taylor expansion - - ii=1; - fp=1.; - fn=1.; - df=1.; - segno=1.; - while (abs(df) > eps), - ii=ii+1; - segno=-segno; - df=df*x/ii; - fp=fp+df; - fn=fn+segno*df; - bp=1./fp; - bn=1./fn; - endwhile - return - endif - -endfunction \ No newline at end of file
--- a/extra/secs1d/inst/Utilities/Ubernoulli.m Sun Mar 25 22:35:18 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,47 +0,0 @@ -## Copyright (C) 2004-2008 Carlo de Falco - ## - ## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator - ## - ## SECS1D is free software; you can redistribute it and/or modify - ## it under the terms of the GNU General Public License as published by - ## the Free Software Foundation; either version 2 of the License, or - ## (at your option) any later version. - ## - ## SECS1D is distributed in the hope that it will be useful, - ## but WITHOUT ANY WARRANTY; without even the implied warranty of - ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the - ## GNU General Public License for more details. - ## - ## You should have received a copy of the GNU General Public License - ## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. -## -## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> - -## -*- texinfo -*- -## -## @deftypefn {Function File} {@var{b}} = Ubernoulli(@var{x},@var{sg}) -## -## Compute Bernoulli function for vector x: -## -## @itemize @minus -## @item @var{b} = @var{x}/(exp(@var{x})-1) if @var{sg} == 1 -## @item @var{b} = @var{x} + B( @var{x} ) if @var{sg} == 0 -## @end itemize -## -## @end deftypefn - -function b=Ubernoulli(x,sg) - - for count=1:length(x) - [bp,bn] = Ubern(x(count)); - bernp(count,1)=bp; - bernn(count,1)=bn; - endfor - - if (sg ==1) - b=bernp; - elseif (sg ==0) - b=bernn; - endif - -endfunction \ No newline at end of file
--- a/extra/secs1d/inst/Utilities/Ucompconst.m Sun Mar 25 22:35:18 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,43 +0,0 @@ -## Copyright (C) 2004-2008 Carlo de Falco - ## - ## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator - ## - ## SECS1D is free software; you can redistribute it and/or modify - ## it under the terms of the GNU General Public License as published by - ## the Free Software Foundation; either version 2 of the License, or - ## (at your option) any later version. - ## - ## SECS1D is distributed in the hope that it will be useful, - ## but WITHOUT ANY WARRANTY; without even the implied warranty of - ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the - ## GNU General Public License for more details. - ## - ## You should have received a copy of the GNU General Public License - ## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. -## -## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> - -## -*- texinfo -*- -## -## @deftypefn {Function File}@ -## {@var{R}} = Ucompconst(@var{nodes},@var{Nnodes},@var{elements},@var{Nelements},@var{D},@var{C}) -## -## Compute P1 finite element rhs: -## -## @itemize @minus -## @item @var{nodes}: list of mesh nodes -## @item @var{Nnodes}: number of mesh nodes -## @item @var{elements}: list of mesh elements -## @item @var{Nelements}: number of mesh elements -## @item @var{D}: piecewise linear reaction coefficient -## @item @var{C}: piecewise constant reaction coefficient -## @end itemize -## -## @end deftypefn - -function R = Ucompconst (nodes,Nnodes,elements,Nelements,D,C) - - h = (nodes(2:end)-nodes(1:end-1)).*C; - R = D.*[h(1)/2; (h(1:end-1)+h(2:end))/2; h(end)/2]; - -endfunction \ No newline at end of file
--- a/extra/secs1d/inst/Utilities/Ucomplap.m Sun Mar 25 22:35:18 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,49 +0,0 @@ -## Copyright (C) 2004-2008 Carlo de Falco - ## - ## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator - ## - ## SECS1D is free software; you can redistribute it and/or modify - ## it under the terms of the GNU General Public License as published by - ## the Free Software Foundation; either version 2 of the License, or - ## (at your option) any later version. - ## - ## SECS1D is distributed in the hope that it will be useful, - ## but WITHOUT ANY WARRANTY; without even the implied warranty of - ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the - ## GNU General Public License for more details. - ## - ## You should have received a copy of the GNU General Public License - ## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. -## -## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> - -## -*- texinfo -*- -## -## @deftypefn {Function File}@ -## {@var{R}} = Ucomplap(@var{nodes},@var{Nnodes},@var{elements},@var{Nelements},@var{coeff}) -## -## Compute P1 finite element approximation of the differential operator: -## -## - d ( coeff d (.)\dx)\dx -## -## @itemize @minus -## @item @var{nodes}: list of mesh nodes -## @item @var{Nnodes}: number of mesh nodes -## @item @var{elements}: list of mesh elements -## @item @var{Nelements}: number of mesh elements -## @item @var{coeff}: piecewise linear reaction coefficient -## @end itemize -## -## @end deftypefn - -function L = Ucomplap (nodes,Nnodes,elements,Nelements,coeff) - - h = nodes(2:end)-nodes(1:end-1); - d0 = [ coeff(1)./h(1); - (coeff(1:end-1)./h(1:end-1))+(coeff(2:end)./h(2:end)); - coeff(end)./h(end)]; - d1 = [1000; -coeff./h]; - dm1 = [ -coeff./h;1000]; - L = spdiags([dm1, d0, d1],-1:1,Nnodes,Nnodes); - -endfunction \ No newline at end of file
--- a/extra/secs1d/inst/Utilities/Ucompmass.m Sun Mar 25 22:35:18 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,45 +0,0 @@ -## Copyright (C) 2004-2008 Carlo de Falco - ## - ## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator - ## - ## SECS1D is free software; you can redistribute it and/or modify - ## it under the terms of the GNU General Public License as published by - ## the Free Software Foundation; either version 2 of the License, or - ## (at your option) any later version. - ## - ## SECS1D is distributed in the hope that it will be useful, - ## but WITHOUT ANY WARRANTY; without even the implied warranty of - ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the - ## GNU General Public License for more details. - ## - ## You should have received a copy of the GNU General Public License - ## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. -## -## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> - -## -*- texinfo -*- -## -## @deftypefn {Function File}@ -## {@var{R}} = Ucompmass(@var{nodes},@var{Nnodes},@var{elements},@var{Nelements},@var{Bvect},@var{Cvect}) -## -## Compute P1 finite element mass-matrix: -## -## @itemize @minus -## @item @var{nodes}: list of mesh nodes -## @item @var{Nnodes}: number of mesh nodes -## @item @var{elements}: list of mesh elements -## @item @var{Nelements}: number of mesh elements -## @item @var{Bvect}: piecewise linear reaction coefficient -## @item @var{Cvect}: piecewise constant reaction coefficient -## @end itemize -## -## @end deftypefn - -function Bmat = Ucompmass (nodes,Nnodes,elements,Nelements,Bvect,Cvect); - - h = (nodes(2:end)-nodes(1:end-1)).*Cvect; - d0 = Bvect.*[h(1)/2; (h(1:end-1)+h(2:end))/2; h(end)/2]; - Bmat = spdiags(d0, 0, Nnodes,Nnodes); - -endfunction -
--- a/extra/secs1d/inst/Utilities/Udriftdiffusion.m Sun Mar 25 22:35:18 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,61 +0,0 @@ -## Copyright (C) 2004-2008 Carlo de Falco - ## - ## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator - ## - ## SECS1D is free software; you can redistribute it and/or modify - ## it under the terms of the GNU General Public License as published by - ## the Free Software Foundation; either version 2 of the License, or - ## (at your option) any later version. - ## - ## SECS1D is distributed in the hope that it will be useful, - ## but WITHOUT ANY WARRANTY; without even the implied warranty of - ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the - ## GNU General Public License for more details. - ## - ## You should have received a copy of the GNU General Public License - ## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. -## -## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> - -## -*- texinfo -*- -## -## @deftypefn {Function File}@ -## {@var{A}} = Udriftdiffusion(@var{x},@var{psi},@var{coeff}) -## -## Builds the Scharfetter-Gummel approximation of the differential -## operator -## -## - (coeff (n' - n psi'))' -## -## @itemize @minus -## @item @var{x}: list of mesh nodes -## @item @var{psi}: piecewise linear potential values -## @item @var{coeff}: piecewise linear diffusion coefficient -## @end itemize -## -## @end deftypefn - -function A = Udriftdiffusion(x,psi,coeff) - - nodes = x; - Nnodes =length(nodes); - - elements = [[1:Nnodes-1]' [2:Nnodes]']; - Nelements=size(elements,1); - - Bcnodes = [1;Nnodes]; - - h=nodes(elements(:,2))-nodes(elements(:,1)); - - c=coeff./h; - Bneg=Ubernoulli(-(psi(2:Nnodes)-psi(1:Nnodes-1)),1); - Bpos=Ubernoulli( (psi(2:Nnodes)-psi(1:Nnodes-1)),1); - - - d0 = [c(1).*Bneg(1); c(1:end-1).*Bpos(1:end-1)+c(2:end).*Bneg(2:end); c(end)*Bpos(end)]; - d1 = [1000;-c.* Bpos]; - dm1 = [-c.* Bneg;1000]; - - A = spdiags([dm1 d0 d1],-1:1,Nnodes,Nnodes); - -endfunction \ No newline at end of file
--- a/extra/secs1d/inst/Utilities/Umediaarmonica.m Sun Mar 25 22:35:18 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,34 +0,0 @@ -## Copyright (C) 2004-2008 Carlo de Falco - ## - ## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator - ## - ## SECS1D is free software; you can redistribute it and/or modify - ## it under the terms of the GNU General Public License as published by - ## the Free Software Foundation; either version 2 of the License, or - ## (at your option) any later version. - ## - ## SECS1D is distributed in the hope that it will be useful, - ## but WITHOUT ANY WARRANTY; without even the implied warranty of - ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the - ## GNU General Public License for more details. - ## - ## You should have received a copy of the GNU General Public License - ## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. -## -## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> - -## -*- texinfo -*- -## -## @deftypefn {Function File}@ -## {@var{m}} = Umediarmonica(@var{w}) -## -## Return the harmonic mean value of @var{w} -## -## @end deftypefn - -function m = Umediaarmonica(w); - - dw = (1./w(1:end-1))+(1./w(2:end)); - m = 2 ./ dw; - -endfunction
--- a/extra/secs1d/inst/Utilities/Uscharfettergummel.m Sun Mar 25 22:35:18 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,57 +0,0 @@ -## Copyright (C) 2004-2008 Carlo de Falco - ## - ## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator - ## -## SECS1D is free software; you can redistribute it and/or modify - ## it under the terms of the GNU General Public License as published by - ## the Free Software Foundation; either version 2 of the License, or - ## (at your option) any later version. - ## -## SECS1D is distributed in the hope that it will be useful, - ## but WITHOUT ANY WARRANTY; without even the implied warranty of - ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the - ## GNU General Public License for more details. - ## - ## You should have received a copy of the GNU General Public License -## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. -## -## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> - -## -*- texinfo -*- -## -## @deftypefn {Function File}@ -## {@var{R}} = Uscharfettergummel(@var{nodes},@var{Nnodes},@var{elements},@var{Nelements},@var{acoeff},@var{bcoeff},@var{v}) -## -## Build the Scharfetter-Gummel matrix for the the discretization of -## the LHS of the Drift-Diffusion equation: -## -## -(a(x) (u' - b v'(x) u))'= f -## -## @itemize @minus -## @item @var{nodes}: list of mesh nodes -## @item @var{Nnodes}: number of mesh nodes -## @item @var{elements}: list of mesh elements -## @item @var{Nelements}: number of mesh elements -## @item @var{acoeff}: piecewise linear diffusion coefficient -## @item @var{bcoeff}: piecewise constant drift constant coefficient -## @item @var{v}: piecewise linear drift potential -## @end itemize -## -## @end deftypefn - -function A = Uscharfettergummel(nodes,Nnodes,elements,Nelements,acoeff,bcoeff,v) - - h=nodes(elements(:,2))-nodes(elements(:,1)); - - c=acoeff./h; - Bneg=Ubernoulli(-(v(2:Nnodes)-v(1:Nnodes-1))*bcoeff,1); - Bpos=Ubernoulli( (v(2:Nnodes)-v(1:Nnodes-1))*bcoeff,1); - - - d0 = [c(1).*Bneg(1); c(1:end-1).*Bpos(1:end-1)+c(2:end).*Bneg(2:end); c(end)*Bpos(end)]; - d1 = [1000;-c.* Bpos]; - dm1 = [-c.* Bneg;1000]; - - A = spdiags([dm1 d0 d1],-1:1,Nnodes,Nnodes); - -endfunction
--- a/extra/secs1d/inst/Utilities/constants.m Sun Mar 25 22:35:18 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,74 +0,0 @@ -## Copyright (C) 2004-2008 Carlo de Falco -## -## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator -## -## SECS1D is free software; you can redistribute it and/or modify -## it under the terms of the GNU General Public License as published by -## the Free Software Foundation; either version 2 of the License, or -## (at your option) any later version. -## -## SECS1D is distributed in the hope that it will be useful, -## but WITHOUT ANY WARRANTY; without even the implied warranty of -## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the -## GNU General Public License for more details. -## -## You should have received a copy of the GNU General Public License -## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. -## -## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> - -## -*- texinfo -*- -## -## @deftypefn {Script File} constants -## -## Compute global constants needed for Drift-Diffusion simulation -## -## @end deftypefn - - - -Kb = 1.3806503e-23; -q = 1.602176462e-19; -e0 = 8.854187817e-12; -esir = 11.7; -esio2r = 3.9; -esi = e0 * esir; -esio2 = e0 * esio2r; -hplanck = 6.626e-34; -hbar = ( hplanck/ (2*pi)); -mn0 = 9.11e-31; -mn = 0.26*mn0; -mh = 0.18*mn0; - - -qsue = q / esi; -T0 = 300 ; -Vth = Kb * T0 / q; -un = 1417e-4; -up = 480e-4; -tp = 1e-7; -tn = 1e-7; - -mnl = 0.98*mn0; -mnt = 0.19*mn0; -mndos = (mnl*mnt*mnt)^(1/3); - -mhh = 0.49*mn0; -mlh = 0.16*mn0; -mhdos = (mhh^(3/2)+mlh^(3/2))^(2/3); - -rn = .1; -aleph = hbar^2/(4*rn*q*mn); -alephn = aleph; -rp = .1; -alephp = hbar^2/(4*rp*q*mh); - -Nc = (6/4)*(2*mndos*Kb*T0/(hbar^2*pi))^(3/2); -Nv = (1/4)*(2*mhdos*Kb*T0/(hbar^2*pi))^(3/2); -Eg0 = 1.16964*q; -alfaEg = 4.73e-4*q; -betaEg = 6.36e2; -Egap = Eg0-alfaEg*((T0^2)/(T0+betaEg)); - -ni = sqrt(Nc*Nv)*exp(-Egap/(2*(Kb * T0))); -Phims = - Egap /(2*q);
--- a/extra/secs1d/inst/secs1d.m Sun Mar 25 22:35:18 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,15 +0,0 @@ -# Run this only if the package is installed -## PKG_ADD: if (! exist (fullfile (fileparts (mfilename ("fullpath")), "inst"), "dir")) -## PKG_ADD: dirlist= {"Utilities","DDG","DDN"}; -## PKG_ADD: for ii=1:length(dirlist) -## PKG_ADD: addpath ( [ fileparts( mfilename("fullpath")) "/" dirlist{ii}]) -## PKG_ADD: end -## PKG_ADD: end - -# Run this only if the package is installed -## PKG_DEL: if (! exist (fullfile (fileparts (mfilename ("fullpath")), "inst"), "dir")) -## PKG_DEL: dirlist= {"Utilities","DDG","DDN"}; -## PKG_DEL: for ii=1:length(dirlist) -## PKG_DEL: rmpath ( [ fileparts( mfilename("fullpath")) "/" dirlist{ii}]) -## PKG_DEL: end -## PKG_DEL: end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/inst/secs1d_dd_gummel_map.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,302 @@ +%% Copyright (C) 2004-2012 Carlo de Falco +%% +%% This file is part of +%% SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator +%% +%% SECS1D is free software; you can redistribute it and/or modify +%% it under the terms of the GNU General Public License as published by +%% the Free Software Foundation; either version 3 of the License, or +%% (at your option) any later version. +%% +%% SECS1D is distributed in the hope that it will be useful, +%% but WITHOUT ANY WARRANTY; without even the implied warranty of +%% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +%% GNU General Public License for more details. +%% +%% You should have received a copy of the GNU General Public License +%% along with SECS1D; If not, see <http://www.gnu.org/licenses/>. + +%% [n, p, V, Fn, Fp, Jn, Jp, it, res] = secs1d_dd_gummel_map (x, D, Na, Nd, +%% pin, nin, Vin, Fnin, +%% Fpin, l2, er, u0n, +%% uminn, vsatn, betan, +%% Nrefn, u0p, uminp, vsatp, +%% betap, Nrefp, theta, tn, tp, +%% Cn, Cp, an, ap, Ecritnin, Ecritpin, +%% toll, maxit, ptoll, pmaxit) +%% +%% This function solves the scaled stationary bipolar DD +%% equation system using Gummel algorithm +%% +%% input: +%% x spatial grid +%% D, Na, Nd doping profile +%% pin initial guess for hole concentration +%% nin initial guess for electron concentration +%% Vin initial guess for electrostatic potential +%% Fnin initial guess for electron Fermi potential +%% Fpin initial guess for hole Fermi potential +%% l2 scaled Debye length squared +%% er relative electric permittivity +%% u0n, uminn, vsatn, Nrefn electron mobility model coefficients +%% u0p, uminp, vsatp, Nrefp hole mobility model coefficients +%% theta intrinsic carrier density +%% tn, tp, Cn, Cp, +%% an, ap, +%% Ecritnin, Ecritpin generation recombination model parameters +%% toll tolerance for Gummel iterarion convergence test +%% maxit maximum number of Gummel iterarions +%% ptoll convergence test tolerance for the non linear +%% Poisson solver +%% pmaxit maximum number of Newton iterarions +%% +%% output: +%% n electron concentration +%% p hole concentration +%% V electrostatic potential +%% Fn electron Fermi potential +%% Fp hole Fermi potential +%% Jn electron current density +%% Jp hole current density +%% it number of Gummel iterations performed +%% res total potential increment at each step + +function [n, p, V, Fn, Fp, Jn, Jp, it, res] = secs1d_dd_gummel_map (x, D, Na, Nd, pin, nin, Vin, + Fnin, Fpin, l2, er, u0n, uminn, + vsatn,betan,Nrefn, u0p, + uminp,vsatp,betap,Nrefp, + theta, tn, tp, Cn, Cp, an, ap, + Ecritnin, Ecritpin, toll, maxit, + ptoll, pmaxit) + + p = pin; + n = nin; + V = Vin; + Fp = Fpin; + Fn = Fnin; + dx = diff (x); + dxm = (dx(1:end-1) + dx(2:end)); + + Nnodes = numel (x); + Nelements = Nnodes -1; + + Jn = zeros (Nelements, 1); + Jp = zeros (Nelements, 1); + + for it = 1:maxit + + [V(:,2), n(:,2), p(:,2)] = secs1d_nlpoisson_newton (x, [1:Nnodes], V(:,1), n(:, 1), + p(:,1), Fn(:,1), Fp(:,1), D, l2, + er, ptoll, pmaxit); + + dV = diff (V(:, 2)); + E = - dV ./ dx; + [Bp, Bm] = bimu_bernoulli (dV); + + [Rn, Rp, Gn, Gp, II] = generation_recombination_model (x, n(:, end), p(:, end), + E, Jn, Jp, tn, tp, theta, + Cn, Cp, an, ap, Ecritnin, Ecritpin); + + mobility = mobility_model (x, Na, Nd, Nrefn, E, u0n, uminn, vsatn, betan); + + A = bim1a_advection_diffusion (x, mobility, 1, 1, V(:, 2)); + M = bim1a_reaction (x, 1, Rn) + bim1a_reaction (x, II, 1); + + R = bim1a_rhs (x, 1, Gn); + + A = A + M; + + n(:,3) = nin; + n(2:end-1,3) = A(2:end-1, 2:end-1) \ (R(2:end-1) - A(2:end-1, [1 end]) * nin ([1 end])); + Fn(:,2) = V(:,2) - log (n(:, 3)); + Jn = mobility .* (n(2:end, 2) .* Bp - n(1:end-1, 2) .* Bm) ./ dx; + + [Rn, Rp, Gn, Gp, II] = generation_recombination_model (x, n(:, end), p(:, end), + E, Jn, Jp, tn, tp, theta, + Cn, Cp, an, ap, Ecritnin, Ecritpin); + + mobility = mobility_model (x, Na, Nd, Nrefp, E, u0p, uminp, vsatp, betap); + + A = bim1a_advection_diffusion (x, mobility, 1, 1, -V(:, 2)); + M = bim1a_reaction (x, 1, Rp) + bim1a_reaction (x, II, 1); + R = bim1a_rhs (x, 1, Gp); + A = A + M; + + p(:,3) = pin; + p(2:end-1,3) = A(2:end-1, 2:end-1) \ (R(2:end-1) - A(2:end-1, [1 end]) * pin ([1 end])); + Fp(:,2) = V(:,2) + log (p(:,3)); + Jp = -mobility .* (p(2:end, 2) .* Bm - p(1:end-1, 2) .* Bp) ./ dx; + + nrfn = norm (Fn(:,2) - Fn(:,1), inf); + nrfp = norm (Fp(:,2) - Fp(:,1), inf); + nrv = norm (V(:,2) - V(:,1), inf); + res(it) = max ([nrfn; nrfp; nrv]); + + if (res(it) < toll) + break + endif + + V(:,1) = V(:,end); + p(:,1) = p(:,end) ; + n(:,1) = n(:,end); + Fn(:,1) = Fn(:,end); + Fp(:,1) = Fp(:,end); + + endfor + + n = n(:,end); + p = p(:,end); + V = V(:,end); + Fn = Fn(:,end); + Fp = Fp(:,end); + +endfunction + +function u = mobility_model (x, Na, Nd, Nref, E, u0, umin, vsat, beta) + + Neff = Na + Nd; + Neff = (Neff(1:end-1) + Neff(2:end)) / 2; + + ubar = umin + (u0 - umin) ./ (1 + (Neff ./ Nref) .^ beta); + u = 2 * ubar ./ (1 + sqrt (1 + 4 * (ubar .* abs (E) ./ vsat) .^ 2)); + +endfunction + +function [Rn, Rp, Gn, Gp, II] = generation_recombination_model (x, n, p, E, Jn, Jp, tn, tp, + theta, Cn, Cp, an, ap, Ecritn, + Ecritp) + + denomsrh = tn .* (p + theta) + tp .* (n + theta); + factauger = Cn .* n + Cp .* p; + fact = (1 ./ denomsrh + factauger); + + Rn = p .* fact; + Rp = n .* fact; + + Gn = theta .^ 2 .* fact; + Gp = Gn; + + II = an * exp(-Ecritn./abs(E)) .* abs (Jn) + ap * exp(-Ecritp./abs(E)) .* abs (Jp); + +endfunction + + +%!demo +%! % physical constants and parameters +%! secs1d_physical_constants; +%! secs1d_silicon_material_properties; +%! +%! % geometry +%! L = 10e-6; % [m] +%! xm = L/2; +%! +%! Nelements = 1000; +%! x = linspace (0, L, Nelements+1)'; +%! sinodes = [1:length(x)]; +%! +%! % dielectric constant (silicon) +%! er = esir * ones (Nelements, 1); +%! +%! % doping profile [m^{-3}] +%! Na = 1e23 * (x <= xm); +%! Nd = 1e23 * (x > xm); +%! +%! % avoid zero doping +%! D = Nd - Na; +%! +%! % initial guess for n, p, V, phin, phip +%! V_p = -1; +%! V_n = 0; +%! +%! Fp = V_p * (x <= xm); +%! Fn = Fp; +%! +%! p = abs (D) / 2 .* (1 + sqrt (1 + 4 * (ni./abs(D)) .^2)) .* (x <= xm) + ... +%! ni^2 ./ (abs (D) / 2 .* (1 + sqrt (1 + 4 * (ni ./ abs (D)) .^2))) .* (x > xm); +%! +%! n = abs (D) / 2 .* (1 + sqrt (1 + 4 * (ni ./ abs (D)) .^ 2)) .* (x > xm) + ... +%! ni ^ 2 ./ (abs (D) / 2 .* (1 + sqrt (1 + 4 * (ni ./ abs (D)) .^2))) .* (x <= xm); +%! +%! V = Fn + Vth * log (n / ni); +%! +%! % scaling factors +%! xbar = L; % [m] +%! nbar = norm(D, 'inf'); % [m^{-3}] +%! Vbar = Vth; % [V] +%! mubar = max (u0n, u0p); % [m^2 V^{-1} s^{-1}] +%! tbar = xbar^2 / (mubar * Vbar); % [s] +%! Rbar = nbar / tbar; % [m^{-3} s^{-1}] +%! Ebar = Vbar / xbar; % [V m^{-1}] +%! Jbar = q * mubar * nbar * Ebar; % [A m^{-2}] +%! CAubar = Rbar / nbar^3; % [m^6 s^{-1}] +%! abar = 1/xbar; % [m^{-1}] +%! +%! % scaling procedure +%! l2 = e0 * Vbar / (q * nbar * xbar^2); +%! theta = ni / nbar; +%! +%! xin = x / xbar; +%! Din = D / nbar; +%! Nain = Na / nbar; +%! Ndin = Nd / nbar; +%! pin = p / nbar; +%! nin = n / nbar; +%! Vin = V / Vbar; +%! Fnin = Vin - log (nin); +%! Fpin = Vin + log (pin); +%! +%! tnin = tn / tbar; +%! tpin = tp / tbar; +%! +%! u0nin = u0n / mubar; +%! uminnin = uminn / mubar; +%! vsatnin = vsatn / (mubar * Ebar); +%! +%! u0pin = u0p / mubar; +%! uminpin = uminp / mubar; +%! vsatpin = vsatp / (mubar * Ebar); +%! +%! Nrefnin = Nrefn / nbar; +%! Nrefpin = Nrefp / nbar; +%! +%! Cnin = Cn / CAubar; +%! Cpin = Cp / CAubar; +%! +%! anin = an / abar; +%! apin = ap / abar; +%! Ecritnin = Ecritn / Ebar; +%! Ecritpin = Ecritp / Ebar; +%! +%! % tolerances for convergence checks +%! toll = 1e-3; +%! maxit = 1000; +%! ptoll = 1e-12; +%! pmaxit = 1000; +%! +%! % solve the problem using the full DD model +%! [nout, pout, Vout, Fnout, Fpout, Jnout, Jpout, it, res] = ... +%! secs1d_dd_gummel_map (xin, Din, Nain, Ndin, pin, nin, Vin, Fnin, Fpin, ... +%! l2, er, u0nin, uminnin, vsatnin, betan, Nrefnin, ... +%! u0pin, uminpin, vsatpin, betap, Nrefpin, theta, ... +%! tnin, tpin, Cnin, Cpin, anin, apin, ... +%! Ecritnin, Ecritpin, toll, maxit, ptoll, pmaxit); +%! +%! % Descaling procedure +%! n = nout*nbar; +%! p = pout*nbar; +%! V = Vout*Vbar; +%! Fn = V - Vth*log(n/ni); +%! Fp = V + Vth*log(p/ni); +%! dV = diff(V); +%! dx = diff(x); +%! E = -dV./dx; +%! +%! % band structure +%! Efn = -Fn; +%! Efp = -Fp; +%! Ec = Vth*log(Nc./n)+Efn; +%! Ev = -Vth*log(Nv./p)+Efp; +%! +%! plot (x, Efn, x, Efp, x, Ec, x, Ev) +%! legend ('Efn', 'Efp', 'Ec', 'Ev') +%! axis tight \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/inst/secs1d_dd_newton.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,263 @@ +%% Copyright (C) 2004-2012 Carlo de Falco +%% +%% This file is part of +%% SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator +%% +%% SECS1D is free software; you can redistribute it and/or modify +%% it under the terms of the GNU General Public License as published by +%% the Free Software Foundation; either version 3 of the License, or +%% (at your option) any later version. +%% +%% SECS1D is distributed in the hope that it will be useful, +%% but WITHOUT ANY WARRANTY; without even the implied warranty of +%% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +%% GNU General Public License for more details. +%% +%% You should have received a copy of the GNU General Public License +%% along with SECS1D; If not, see <http://www.gnu.org/licenses/>. + +%% [n, p, V, Fn, Fp, Jn, Jp, it, res] = secs1d_dd_newton (x, D, Vin, nin, +%% pin, l2, er, un, +%% up, theta, tn, tp, +%% Cn, Cp, toll, maxit) +%% +%% Solve the scaled stationary bipolar DD equation system using Newton's method +%% +%% input: +%% x spatial grid +%% D doping profile +%% pin initial guess for hole concentration +%% nin initial guess for electron concentration +%% Vin initial guess for electrostatic potential +%% l2 scaled Debye length squared +%% er relative electric permittivity +%% un electron mobility model coefficients +%% up electron mobility model coefficients +%% theta intrinsic carrier density +%% tn, tp, Cn, Cp generation recombination model parameters +%% toll tolerance for Gummel iterarion convergence test +%% maxit maximum number of Gummel iterarions +%% +%% output: +%% n electron concentration +%% p hole concentration +%% V electrostatic potential +%% Fn electron Fermi potential +%% Fp hole Fermi potential +%% Jn electron current density +%% Jp hole current density +%% it number of Gummel iterations performed +%% res total potential increment at each step + +function [n, p, V, Fn, Fp, Jn, Jp, it, normr] = secs1d_dd_newton (x, D, Vin, nin, pin, l2, er, un, up, theta, tn, tp, Cn, Cp, toll, maxit) + + dampit = 10; + dampcoeff = 2; + Nnodes = numel (x); + + V = Vin; + n = nin; + p = pin; + + [res, jac] = residual_jacobian (x, D, V, n, p, l2, er, un, up, theta, tn, tp, Cn, Cp); + + normr(1) = norm (res, inf); + normrnew = normr(1); + + for it = 1:maxit + + delta = - jac \ res; + + tk = 1; + for dit = 1:dampit + + Vnew = Vin; + Vnew(2:end-1) = V(2:end-1) + tk * delta(1:Nnodes-2); + + nnew = nin; + nnew(2:end-1) = n(2:end-1) + tk * delta((Nnodes-2)+(1:Nnodes-2)); + + pnew = pin; + pnew(2:end-1) = p(2:end-1) + tk * delta(2*(Nnodes-2)+(1:Nnodes-2)); + + [resnew, jacnew] = residual_jacobian (x, D, Vnew, nnew, pnew, l2, er, un, up, theta, tn, tp, Cn, Cp); + + normrnew = norm (resnew, inf); + if (normrnew > normr(it)) + tk = tk / dampcoeff; + else + jac = jacnew; + res = resnew; + break + endif + + endfor + + V = Vnew; n = nnew; p = pnew; + normr(it+1) = normrnew; + + if (normr(it+1) <= toll) + break + endif + endfor + + dV = diff (V); + dx = diff (x); + [Bp, Bm] = bimu_bernoulli (dV); + Jn = un .* (n(2:end) .* Bp - n(1:end-1) .* Bm) ./ dx; + Jp = -up .* (p(2:end) .* Bm - p(1:end-1) .* Bp) ./ dx; + Fp = V + log (p); + Fn = V - log (n); +endfunction + +function [res, jac] = residual_jacobian (x, D, V, n, p, l2, er, un, up, theta, tn, tp, Cn, Cp) + + denomsrh = tn .* (p + theta) + tp .* (n + theta); + factauger = Cn .* n + Cp .* p; + fact = (1 ./ denomsrh + factauger); + nm = (n(2:end) + n(1:end-1))/2; + pm = (p(2:end) + p(1:end-1))/2; + + Nnodes = numel (x); + + A11 = bim1a_laplacian (x, l2 .* er, 1); + A12 = bim1a_reaction (x, 1, 1); + A13 = - A12; + R1 = A11 * V + bim1a_rhs (x, 1, n-p-D); + + A21 = - bim1a_laplacian (x, un .* nm, 1); + A22 = bim1a_advection_diffusion (x, un, 1, 1, V) + bim1a_reaction (x, 1, p .* fact); + A23 = bim1a_reaction (x, 1, n .* fact); + R2 = A22 * n + bim1a_rhs (x, 1, (p .* n - theta .^ 2) .* fact); + + A31 = bim1a_laplacian (x, up .* pm, 1); + A32 = bim1a_reaction (x, 1, p .* fact); + A33 = bim1a_advection_diffusion (x, up, 1, 1, -V) + bim1a_reaction (x, 1, n .* fact); + R3 = A33 * p + bim1a_rhs (x, 1, (p .* n - theta .^ 2) .* fact); + + res = [R1(2:end-1); R2(2:end-1); R3(2:end-1)]; + + jac = [A11(2:end-1, 2:end-1), A12(2:end-1, 2:end-1), A13(2:end-1, 2:end-1); + A21(2:end-1, 2:end-1), A22(2:end-1, 2:end-1), A23(2:end-1, 2:end-1); + A31(2:end-1, 2:end-1), A32(2:end-1, 2:end-1), A33(2:end-1, 2:end-1)]; + +endfunction + + +%!demo +%! % physical constants and parameters +%! secs1d_physical_constants; +%! secs1d_silicon_material_properties; +%! +%! % geometry +%! L = 1e-6; % [m] +%! x = linspace (0, L, 10)'; +%! sinodes = [1:length(x)]; +%! +%! % dielectric constant (silicon) +%! er = esir * ones (numel (x) - 1, 1); +%! +%! % doping profile [m^{-3}] +%! Na = 1e20 * ones(size(x)); +%! Nd = 1e24 * ones(size(x)); +%! D = Nd-Na; +%! +%! % externally applied voltages +%! V_p = 10; +%! V_n = 0; +%! +%! % initial guess for phin, phip, n, p, V +%! Fp = V_p * (x <= L/2); +%! Fn = Fp; +%! +%! p = abs(D)/2.*(1+sqrt(1+4*(ni./abs(D)).^2)).*(D<0)+... +%! ni^2./(abs(D)/2.*(1+sqrt(1+4*(ni./abs(D)).^2))).*(D>0); +%! +%! n = abs(D)/2.*(1+sqrt(1+4*(ni./abs(D)).^2)).*(D>0)+... +%! ni^2./(abs(D)/2.*(1+sqrt(1+4*(ni./abs(D)).^2))).*(D<0); +%! +%! V = Fn + Vth*log(n/ni); +%! +%! % scaling factors +%! xbar = L; % [m] +%! nbar = norm(D, 'inf'); % [m^{-3}] +%! Vbar = Vth; % [V] +%! mubar = max(u0n, u0p); % [m^2 V^{-1} s^{-1}] +%! tbar = xbar^2/(mubar*Vbar); % [s] +%! Rbar = nbar/tbar; % [m^{-3} s^{-1}] +%! Ebar = Vbar/xbar; % [V m^{-1}] +%! Jbar = q*mubar*nbar*Ebar; % [A m^{-2}] +%! CAubar = Rbar/nbar^3; % [m^6 s^{-1}] +%! abar = xbar^(-1); % [m^{-1}] +%! +%! % scaling procedure +%! l2 = e0*Vbar/(q*nbar*xbar^2); +%! theta = ni/nbar; +%! +%! xin = x/xbar; +%! Din = D/nbar; +%! Nain = Na/nbar; +%! Ndin = Nd/nbar; +%! pin = p/nbar; +%! nin = n/nbar; +%! Vin = V/Vbar; +%! Fnin = Vin - log(nin); +%! Fpin = Vin + log(pin); +%! +%! tnin = tn/tbar; +%! tpin = tp/tbar; +%! +%! % mobility model accounting scattering from ionized impurities +%! u0nin = u0n/mubar; +%! uminnin = uminn/mubar; +%! vsatnin = vsatn/(mubar*Ebar); +%! +%! u0pin = u0p/mubar; +%! uminpin = uminp/mubar; +%! vsatpin = vsatp/(mubar*Ebar); +%! +%! Nrefnin = Nrefn/nbar; +%! Nrefpin = Nrefp/nbar; +%! +%! Cnin = Cn/CAubar; +%! Cpin = Cp/CAubar; +%! +%! anin = an/abar; +%! apin = ap/abar; +%! Ecritnin = Ecritn/Ebar; +%! Ecritpin = Ecritp/Ebar; +%! +%! % tolerances for convergence checks +%! ptoll = 1e-12; +%! pmaxit = 1000; +%! +%! % solve the problem using the Newton fully coupled iterative algorithm +%! [nout, pout, Vout, Fnout, Fpout, Jnout, Jpout, it, res] = secs1d_dd_newton (xin, Din, +%! Vin, nin, pin, l2, er, +%! u0nin, u0pin, theta, tnin, +%! tpin, Cnin, Cpin, ptoll, pmaxit); +%! % Descaling procedure +%! n = nout*nbar; +%! p = pout*nbar; +%! V = Vout*Vbar; +%! Fn = V - Vth*log(n/ni); +%! Fp = V + Vth*log(p/ni); +%! dV = diff(V); +%! dx = diff(x); +%! E = -dV./dx; +%! +%! % compute current densities +%! [Bp, Bm] = bimu_bernoulli (dV/Vth); +%! Jn = q*u0n*Vth .* (n(2:end) .* Bp - n(1:end-1) .* Bm) ./ dx; +%! Jp = -q*u0p*Vth .* (p(2:end) .* Bm - p(1:end-1) .* Bp) ./ dx; +%! Jtot = Jn+Jp; +%! +%! % band structure +%! Efn = -Fn; +%! Efp = -Fp; +%! Ec = Vth*log(Nc./n)+Efn; +%! Ev = -Vth*log(Nv./p)+Efp; +%! +%! plot (x, Efn, x, Efp, x, Ec, x, Ev) +%! legend ('Efn', 'Efp', 'Ec', 'Ev') +%! axis tight \ No newline at end of file
--- a/extra/secs1d/inst/secs1d_demo_pndiode.m Sun Mar 25 22:35:18 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,138 +0,0 @@ -## Copyright (C) 2009 Carlo de Falco -## -## SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator -## -## SECS1D is free software; you can redistribute it and/or modify -## it under the terms of the GNU General Public License as published by -## the Free Software Foundation; either version 2 of the License, or -## (at your option) any later version. -## -## SECS1D is distributed in the hope that it will be useful, -## but WITHOUT ANY WARRANTY; without even the implied warranty of -## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the -## GNU General Public License for more details. -## -## You should have received a copy of the GNU General Public License -## along with SECS1D; If not, see <http://www.gnu.org/licenses/>. -## -## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> - -function secs1d_demo_pndiode () - - constants - - len = 1e-6; - Nnodes = 1000; - - vmin = -2; - vmax = 2; - - vstep=.4; - - istep = 1; - va = vmin - - x = linspace(0,len,Nnodes)'; - xm = mean(x); - - Nd=1e25; - Na=1e25; - - D = Nd * (x>xm) - Na * (x<=xm); - - nn = (Nd + sqrt(Nd^2+4*ni^2))/2; - pp = (Na + sqrt(Na^2+4*ni^2))/2; - - xn = xm+1e-7; - xp = xm-1e-7; - - %% Scaling coefficients - xs = len; - ns = norm(D,inf); - idata.D = D/ns; - Vs = Vth; - us = un; - Js = xs / (us * Vs * q * ns); - - - while va <= vmax - - vvect(istep) = va; - - n(:,istep) = nn * (x>=xn) + (ni^2)/pp * (x<xn); - p(:,istep) = (ni^2)/nn * (x>xp) + pp * (x<=xp); - - Fn = va*(x<=xm); - Fp = Fn; - - V(:,istep) = (Fn - Vth * log(p(:,istep)/ni)); - - %% Scaling - xin = x/xs; - idata.n = n(:,istep)/ns; - idata.p = p(:,istep)/ns; - idata.V = V(:,istep)/Vs; - idata.Fn = (Fn - Vs * log(ni/ns))/Vs; - idata.Fp = (Fp + Vs * log(ni/ns))/Vs; - - lambda2(istep) = idata.l2 = (Vs*esi)/(q*ns*xs^2); - idata.nis = ni/ns; - idata.un = un/us; - idata.up = up/us; - - %% Solution of DD system - - %% Algorithm parameters - toll = 1e-3; - maxit = 20; - ptoll = 1e-10; - pmaxit = 100; - verbose = 0; - sinodes = [1:length(x)]; - idata.tn = inf; - idata.tp = inf; - - [odata,it,res] = DDGgummelmap (xin,idata,toll,maxit,ptoll,pmaxit,verbose); - [odata,it,res] = DDNnewtonmap (xin,odata,toll, maxit,verbose); - - n(:,istep) = odata.n; - p(:,istep) = odata.p; - V(:,istep) = odata.V; - - DV(istep) = odata.V(end) - odata.V(1); - Emax(istep) = max(abs(diff(odata.V)./diff(xin))) - - Bp = Ubernoulli(diff (V(:, istep)),1); - Bm = Ubernoulli(diff (V(:, istep)),0); - Jn(:,istep) = -odata.un * (n(2:end, istep).*Bp-n(1:end-1, istep).*Bm)./diff (xin); - Jp(:,istep) = odata.up * (p(2:end, istep).*Bm-p(1:end-1, istep).*Bp)./diff (xin); - - va = va+vstep - istep = istep+1; - - endwhile - - %% Descaling - n = n*ns; - p = p*ns; - V = V*Vs; - J = abs (Jp+Jn)*Js; - - close all - - figure(); - plot(x, n.') - xlabel("x") - ylabel("n") - - figure(); - plot(vvect, J) - xlabel("V") - ylabel("J") - - figure(); - plot(vvect, Emax) - xlabel("V") - ylabel("max(abs(\\phi^\'))") - -endfunction \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/inst/secs1d_nlpoisson_newton.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,205 @@ +%% Copyright (C) 2004-2012 Carlo de Falco +%% +%% This file is part of +%% SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator +%% +%% SECS1D is free software; you can redistribute it and/or modify +%% it under the terms of the GNU General Public License as published by +%% the Free Software Foundation; either version 3 of the License, or +%% (at your option) any later version. +%% +%% SECS1D is distributed in the hope that it will be useful, +%% but WITHOUT ANY WARRANTY; without even the implied warranty of +%% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +%% GNU General Public License for more details. +%% +%% You should have received a copy of the GNU General Public License +%% along with SECS1D; If not, see <http://www.gnu.org/licenses/>. + +%% [V, n, p, res, niter] = secs1d_nlpoisson_newton (x, sinodes, Vin, nin, pin, +%% Fnin, Fpin, D, l2, er, toll, maxit) +%% +%% input: +%% x spatial grid +%% sinodes index of the nodes of the grid which are in the semiconductor subdomain +%% (remaining nodes are assumed to be in the oxide subdomain) +%% Vin initial guess for the electrostatic potential +%% nin initial guess for electron concentration +%% pin initial guess for hole concentration +%% Fnin initial guess for electron Fermi potential +%% Fpin initial guess for hole Fermi potential +%% D doping profile +%% l2 scaled Debye length squared +%% er relative electric permittivity +%% toll tolerance for convergence test +%% maxit maximum number of Newton iterations +%% +%% output: +%% V electrostatic potential +%% n electron concentration +%% p hole concentration +%% res residual norm at each step +%% niter number of Newton iterations + +function [V, n, p, res, niter] = secs1d_nlpoisson_newton (x, sinodes, Vin, nin, pin, + Fnin, Fpin, D, l2, er, toll, maxit) + + dampit = 10; + dampcoeff = 2; + + sielements = sinodes(1:end-1); + Nnodes = numel (x); + Nelements = Nnodes - 1; + + V = Vin; + Fn = Fnin; + Fp = Fpin; + n = exp ( V(sinodes) - Fn); + p = exp (-V(sinodes) + Fp); + + L = bim1a_laplacian (x, l2 .* er, 1); + + b = zeros (Nelements, 1); + b(sielements) = 1; + + a = zeros (Nnodes, 1); + a(sinodes) = (n + p); + + M = bim1a_reaction (x, b, a); + + a = zeros (Nnodes,1); + a(sinodes) = (n - p - D); + + N = bim1a_rhs (x, b, a); + + A = L + M; + R = L * V + N; + + normr(1) = norm (R(2:end-1), inf); + normrnew = normr(1); + + for newtit = 1:maxit + + dV = zeros (Nnodes, 1); + dV(2:end-1) = - A(2:end-1, 2:end-1) \ R(2:end-1) ; + + tk = 1; + for dit = 1:dampit + Vnew = V + tk * dV; + + n = exp ( Vnew(sinodes) - Fn); + p = exp (-Vnew(sinodes) + Fp); + + a = zeros (Nnodes, 1); + a(sinodes) = (n + p); + M = bim1a_reaction (x, b, a); + Anew = L + M; + + a = zeros (Nnodes,1); + a(sinodes) = (n - p - D); + N = bim1a_rhs (x, b, a); + Rnew = L * Vnew + N; + + normrnew = norm (Rnew(2:end-1), inf); + if (normrnew > normr(newtit)) + tk = tk / dampcoeff; + else + A = Anew; + R = Rnew; + break + endif + + endfor + + V = Vnew; + normr(newtit+1) = normrnew; + reldVnorm = norm (tk*dV, inf) / norm (V, inf); + + if (reldVnorm <= toll) + break + endif + + endfor + + res = normr; + niter = newtit; + +endfunction + +%!demo +%! secs1d_physical_constants +%! secs1d_silicon_material_properties +%! +%! tbulk= 1.5e-6; +%! tox = 90e-9; +%! L = tbulk + tox; +%! cox = esio2/tox; +%! +%! Nx = 50; +%! Nel = Nx - 1; +%! +%! x = linspace (0, L, Nx)'; +%! sinodes = find (x <= tbulk); +%! xsi = x(sinodes); +%! +%! Nsi = length (sinodes); +%! Nox = Nx - Nsi; +%! +%! NelSi = Nsi - 1; +%! NelSiO2 = Nox - 1; +%! +%! Na = 1e22; +%! D = - Na * ones (size (xsi)); +%! p = Na * ones (size (xsi)); +%! n = (ni^2) ./ p; +%! Fn = Fp = zeros (size (xsi)); +%! Vg = -10; +%! Nv = 80; +%! for ii = 1:Nv +%! Vg = Vg + 0.2; +%! vvect(ii) = Vg; +%! +%! V = - Phims + Vg * ones (size (x)); +%! V(sinodes) = Fn + Vth * log (n/ni); +%! +%! % Scaling +%! xs = L; +%! ns = norm (D, inf); +%! Din = D / ns; +%! Vs = Vth; +%! xin = x / xs; +%! nin = n / ns; +%! pin = p / ns; +%! Vin = V / Vs; +%! Fnin = (Fn - Vs * log (ni / ns)) / Vs; +%! Fpin = (Fp + Vs * log (ni / ns)) / Vs; +%! +%! er = esio2r * ones(Nel, 1); +%! l2(1:NelSi) = esi; +%! l2 = (Vs*e0)/(q*ns*xs^2); +%! +%! % Solution of Nonlinear Poisson equation +%! +%! % Algorithm parameters +%! toll = 1e-10; +%! maxit = 1000; +%! +%! [V, nout, pout, res, niter] = secs1d_nlpoisson_newton (xin, sinodes, +%! Vin, nin, pin, +%! Fnin, Fpin, Din, l2, +%! er, toll, maxit); +%! +%! % Descaling +%! n = nout*ns; +%! p = pout*ns; +%! V = V*Vs; +%! +%! qtot(ii) = q * trapz (xsi, p + D - n); +%! end +%! +%! vvectm = (vvect(2:end)+vvect(1:end-1))/2; +%! C = - diff (qtot) ./ diff (vvect); +%! plot(vvectm, C) +%! xlabel('Vg [V]') +%! ylabel('C [Farad]') +%! title('C-V curve')
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/inst/secs1d_physical_constants.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,37 @@ +%% Copyright (C) 2004-2012 Carlo de Falco +%% +%% This file is part of +%% SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator +%% +%% SECS1D is free software; you can redistribute it and/or modify +%% it under the terms of the GNU General Public License as published by +%% the Free Software Foundation; either version 3 of the License, or +%% (at your option) any later version. +%% +%% SECS1D is distributed in the hope that it will be useful, +%% but WITHOUT ANY WARRANTY; without even the implied warranty of +%% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +%% GNU General Public License for more details. +%% +%% You should have received a copy of the GNU General Public License +%% along with SECS1D; If not, see <http://www.gnu.org/licenses/>. + +%% some useful physical constants +%% +%% Kb = Boltzman constant +%% q = quantum of charge +%% e0 = permittivity of free space +%% hplanck = Plank constant +%% hbar = Plank constant by 2 pi +%% mn0 = free electron mass +%% T0 = temperature +%% Vth = thermal voltage + +Kb = 1.3806503e-23; +q = 1.602176462e-19; +e0 = 8.854187817e-12; +hplanck = 6.626e-34; +hbar = hplanck/ (2*pi); +mn0 = 9.11e-31; +T0 = 300; +Vth = Kb * T0 / q;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/extra/secs1d/inst/secs1d_silicon_material_properties.m Sun Mar 25 22:44:30 2012 +0000 @@ -0,0 +1,104 @@ +%% Copyright (C) 2004-2012 Carlo de Falco +%% +%% This file is part of +%% SECS1D - A 1-D Drift--Diffusion Semiconductor Device Simulator +%% +%% SECS1D is free software; you can redistribute it and/or modify +%% it under the terms of the GNU General Public License as published by +%% the Free Software Foundation; either version 3 of the License, or +%% (at your option) any later version. +%% +%% SECS1D is distributed in the hope that it will be useful, +%% but WITHOUT ANY WARRANTY; without even the implied warranty of +%% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +%% GNU General Public License for more details. +%% +%% You should have received a copy of the GNU General Public License +%% along with SECS1D; If not, see <http://www.gnu.org/licenses/>. + +%% material properties for silicon and silicon dioxide +%% +%% esir = relative electric permittivity of silicon +%% esio2r = relative electric permittivity of silicon dioxide +%% esi = electric permittivity of silicon +%% esio2 = electric permittivity of silicon dioxide +%% mn = effective mass of electrons in silicon +%% mh = effective mass of holes in silicon +%% +%% u0n = low field electron mobility +%% u0p = low field hole mobility +%% uminn = parameter for doping-dependent electron mobility +%% betan = idem +%% Nrefn = idem +%% uminp = parameter for doping-dependent hole mobility +%% betap = idem +%% Nrefp = idem +%% vsatn = electron saturation velocity +%% vsatp = hole saturation velocity +%% tp = electron lifetime +%% tn = hole lifetime +%% Cn = electron Auger coefficient +%% Cp = hole Auger coefficient +%% an = impact ionization rate for electrons +%% ap = impact ionization rate for holes +%% Ecritn = critical field for impact ionization of electrons +%% Ecritp = critical field for impact ionization of holes +%% Nc = effective density of states in the conduction band +%% Nv = effective density of states in the valence band +%% Egap = bandgap in silicon +%% EgapSio2 = bandgap in silicon dioxide +%% +%% ni = intrinsic carrier density +%% Phims = metal to semiconductor potential barrier + +esir = 11.7; +esio2r = 3.9; +esi = e0 * esir; +esio2 = e0 * esio2r; +mn = 0.26*mn0; +mh = 0.18*mn0; + +qsue = q / esi; + +u0n = 1417e-4; +u0p = 480e-4; +uminn = u0n; % ref. value: 65e-4; +uminp = u0p; % ref. value: 47.7e-4; +betan = 0.72; +betap = 0.76; +Nrefn = 8.5e22; +Nrefp = 6.3e22; +vsatn = inf; % ref. value: 1.1e5; +vsatp = inf; % ref. value: 9.5e4; + +tp = inf; % ref. value: 1e-6; +tn = inf; % ref. value: 1e-6; + +Cn = 0; % ref. value: 2.8e-31*1e-12; +Cp = 0; % ref. value: 9.9e-32*1e-12; +an = 0; % ref. value: 7.03e7; +ap = 0; % ref. value: 6.71e7; +Ecritn = 1.231e8; +Ecritp = 1.693e8; + +mnl = 0.98*mn0; +mnt = 0.19*mn0; +mndos = (mnl*mnt*mnt)^(1/3); + +mhh = 0.49*mn0; +mlh = 0.16*mn0; +mhdos = (mhh^(3/2)+mlh^(3/2))^(2/3); + +Nc = (6/4)*(2*mndos*Kb*T0/(hbar^2*pi))^(3/2); +Nv = (1/4)*(2*mhdos*Kb*T0/(hbar^2*pi))^(3/2); +Eg0 = 1.16964*q; +alfaEg = 4.73e-4*q; +betaEg = 6.36e2; +Egap = Eg0-alfaEg*((T0^2)/(T0+betaEg)); +Ei = Egap/2+Kb*T0/2*log(Nv/Nc); +EgapSio2 = 9*q; +deltaEcSio2 = 3.1*q; +deltaEvSio2 = EgapSio2-Egap-deltaEcSio2; + +ni = sqrt(Nc*Nv)*exp(-Egap/(2*(Kb * T0))); +Phims = - Egap /(2*q);
--- a/extra/secs1d/src/Makefile Sun Mar 25 22:35:18 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,10 +0,0 @@ -PROGS = $(patsubst %.cc,%.oct,$(wildcard *.cc)) - -all: $(PROGS) - -$(PROGS): Makefile - -%.oct: %.cc - mkoctfile $< - -clean: ; -$(RM) *.o core octave-core *.oct *~ \ No newline at end of file
--- a/extra/secs1d/src/Ubern.cc Sun Mar 25 22:35:18 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,131 +0,0 @@ -/* -% This file is part of -% -% SECS2D - A 2-D Drift--Diffusion Semiconductor Device Simulator -% ------------------------------------------------------------------- -% Copyright (C) 2004-2006 Carlo de Falco -% -% -% -% SECS2D is free software; you can redistribute it and/or modify -% it under the terms of the GNU General Public License as published by -% the Free Software Foundation; either version 2 of the License, or -% (at your option) any later version. -% -% SECS2D is distributed in the hope that it will be useful, -% but WITHOUT ANY WARRANTY; without even the implied warranty of -% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the -% GNU General Public License for more details. -% -% You should have received a copy of the GNU General Public License -% along with SECS2D; If not, see <http://www.gnu.org/licenses/>. -*/ - -#include <iostream> -#include <octave/oct.h> - -//////////////////////////////////////////// -// Ubern function -// this function, though it does make use -// of liboctave, is not an octve command -// the wrapper to make the command is defined -// below -//////////////////////////////////////////// - - -int Ubern(const double x, double &bp, double &bn ) -{ - -double xlim=1e-2; -int ii; -double fp,fn,df,segno; -double ax; - -ax=fabs(x); - - -if (ax == 0.0) { -bp=1.; -bn=1.; -goto theend ; -} - -if (ax > 80.0){ -if (x >0.0){ - bp=0.0; - bn=x; - goto theend ; -}else{ - bp=-x; - bn=0.0; - goto theend ; -} -} - -if (ax > xlim){ -bp=x/(exp(x)-1.0); -bn=x+bp; -goto theend ; -} - -ii=1; -fp=1.0;fn=1.0;df=1.0;segno=1.0; -while (fabs(df) > 2.2204e-16){ -ii++; -segno= -segno; -df=df*x/ii; -fp=fp+df; -fn=fn+segno*df; -bp=1/fp; -bn=1/fn; -} - - -theend: -return 0; - -} - - -//////////////////////////////////// -// WRAPPER -//////////////////////////////////// -// DEFUN_DLD and the macros that it -// depends on are defined in the -// files defun-dld.h, defun.h, -// and defun-int.h. -// -// Note that the third parameter -// (nargout) is not used, so it is -// omitted from the list of arguments -// to DEFUN_DLD in order to avoid -// the warning from gcc about an -// unused function parameter. -//////////////////////////////////// - -DEFUN_DLD (Ubern, args, , -" [bp,bn]=Ubern(x)\n \ -computes Bernoulli function\n \ -B(x)=x/(exp(x)-1) corresponding to \n \ -to input values Z and -Z, recalling that\n \ -B(-Z)=Z+B(Z)\n") -{ - // The list of values to return. See the declaration in oct-obj.h - octave_value_list retval; - - - NDArray X ( args(0).array_value() ); - octave_idx_type lx = X.length(); - - NDArray BP(X),BN(X); - - for (octave_idx_type jj=0; jj<lx; jj++) - Ubern(X(jj),BP(jj),BN(jj)); - - retval (0) = BP; - retval (1) = BN; - - return retval ; - -} -