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The SECS1D Package
* Example, a PIN diode
<code>
% 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
</code>
[[Category:Octave-Forge]]
<code>
% 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
</code>
[[Category:Octave-Forge]]
