/usr/share/octave/packages/secs1d-0.0.9/secs1d_nlpoisson_newton.m is in octave-secs1d 0.0.9-2.
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%%
%% 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/>.
%% Solve the non-linear Poisson problem using Newton's algorithm.
%%
%% [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')
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