LASER RATE EQUATIONS by P N V Anil Kumar,IIT Bombay

Solves the LASER RATE EQUATIONS
111 Downloads
Updated 7 Apr 2022

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%Transient Response of Coupled Rate Equations
% Copyright .This code is developed by P N V Anil Kumar,IIT Bombay & %SAC/ISRO,Ahmedabad
clc;
clear all;
close all;
tic
Time_initial = 0;
Time_final = 10^-9;
Time_steps = 10^-16;
Time = Time_initial:Time_steps:Time_final;
Frequency = 1.2*10^10;
N = zeros(1,length(Time));
I = 2.0*10^-2;
S = zeros(1,length(Time));
q = 1.6*10^-19;
V = 2*10^-11;
tau_n = 10^-9;
g0 = 1.5*10^-5;
Nth = 10^18;
epsilon = 1.5*10^-17;
tau_p = 10^-12;
beta = 10^-4;
gamma = 0.2;
alpha = 3.1;
phi = zeros(1,length(Time));
Optical_Output_Power_of_LASER = zeros(1,length(Time));
Quantum_Efficiency = [0.1 0.5 0.7 1];
h = 6.6261*10^-27;%in CGS unit
I_bias = linspace(10,50,10^4).*10^-3;%Bias Current in mA
Gain_Saturation_Parameter = 3.4*10^-17;
Relaxation_Oscillation_Frequency = zeros(1,length(I_bias));
Laser_Transfer_Function_Magnitude = zeros(1,length(Relaxation_Oscillation_Frequency));
Laser_Transfer_Function_Phase = zeros(1,length(Relaxation_Oscillation_Frequency));
for i =1:length(Time)
x1(1,i) = I/(q*V);
x2(1,i) = N(1,i)/tau_n;
x3(1,i) = (g0*(N(1,i) - Nth));
x4(1,i) = 1+(epsilon*S(1,i));
x5(1,i) = (x3(1,i)/x4(1,i))*S(1,i);
x6(1,i) = x1(1,i)-x2(1,i)- x5(1,i);
N(1,i+1) = N(1,i) + (Time_steps*x6(1,i));
y1(1,i) = (gamma*x3(1,i))-(1/tau_p);
y2(1,i) = (1/2)*alpha* y1(1,i);
phi(1,i+1) = phi(1,i) + (Time_steps*y2(1,i));
x7(1,i) = (((gamma*g0)*(N(1,i)-Nth))/x4(1,i))*S(1,i);
x8(1,i) = S(1,i)/tau_p;
x9(1,i) = (gamma*beta*N(1,i))/tau_n;
x10(1,i) = x7(1,i)- x8(1,i)+x9(1,i);
S(1,i+1) = S(1,i) + (Time_steps*x10(1,i));
z1(1,i) = S(1,i)*V*h*Frequency;
z2(1,i) = 2*gamma*tau_p;
Optical_Output_Power_of_LASER(1,i) = z1(1,i)/z2(1,i);
end
Pump_Rate = linspace(0,2,10^4);
tau_L = 1*10^-9;
for j = 1:length(Pump_Rate)
if Pump_Rate(j) < 1
Carrier_Concentration(j) = Pump_Rate(j)*Nth;
Photon_Concentration(j) = Pump_Rate(j)/(1-Pump_Rate(j));
else
Carrier_Concentration(j)=Nth;
Photon_Concentration(j) = (tau_p/tau_L)*Nth*(Pump_Rate(j)-1);
end
end
for k =1:length(I_bias)
z3 = (gamma*g0)/(q*V);
z4(1,k) = z3*(I_bias(1,k)-I);
z5 = (gamma*Gain_Saturation_Parameter*tau_p)/(q*V);
z6(1,k) = z5*(I_bias(1,k)-I);
z7(1,k) = 1-z6(1,k);
Capital_K(1,k) = z4(1,k)*z7(1,k);
z8 = z3*(tau_p+(Gain_Saturation_Parameter/g0));
z9(1,k) = z8*(I_bias(1,k)-I);
gamma_d(1,k) = (1/tau_n)+z9(1,k)*z7(1,k);
z10(1,k) = (1/2)*(gamma_d(1,k)^2);
z11(1,k) = Capital_K(1,k)-z10(1,k);
z12(1,k) = real(sqrt(z11(1,k)));
Relaxation_Oscillation_Frequency(1,k) = (1/(2*pi))*z12(1,k);
end
I_bias1 = [30 40 50]*10^-3;
clear j;
for ii = 1:length(I_bias1)
for jj = 1:length(Relaxation_Oscillation_Frequency)
pp1 = (gamma*g0)/(q*V);
pp2(ii,jj) = pp1*(I_bias1(ii)-I);
pp3 = (gamma*Gain_Saturation_Parameter*tau_p)/(q*V);
pp4(ii,jj) = pp3*(I_bias1(ii)-I);
pp5(ii,jj) = 1-pp4(ii,jj);
Capital_K1(ii,jj) = pp2(ii,jj)* pp5(ii,jj);
pp6 = pp1*(tau_p+(Gain_Saturation_Parameter/g0));
pp7(ii,jj) = pp6*(I_bias1(ii)-I);
gamma_d1(ii,jj) = (1/tau_n)+pp7(ii,jj)*pp5(ii,jj);
pp8(ii,jj) = j*(2*pi*Relaxation_Oscillation_Frequency(1,jj));
pp9(ii,jj) = pp8(ii,jj)*(pp8(ii,jj)+gamma_d1(ii,jj));
pp10(ii,jj) = pp9(ii,jj)+Capital_K1(ii,jj);
Laser_Transfer_Function_Magnitude(ii,jj) =abs(Capital_K1(ii,jj)/pp10(ii,jj));
Laser_Transfer_Function_Phase(ii,jj) = angle(Capital_K1(ii,jj)/pp10(ii,jj));
end
end
figure(1)
ax = axes;
plot(Time,N(1:length(N)-1),'m','Linewidth',2)
grid minor
xlabel('Time in Seconds','Interpreter','latex','FontSize',13);
ylabel('Carrier Concentration','Interpreter','latex','FontSize',13);
title('LASER Coupled Rate Equation','Interpreter','latex','FontSize',13);
legend('Transient Carrier Concentration','Location','southeast');
ax.YAxis.Color = [0 0 0];
ax.XAxis.Color = [0 0 0];
figure(2)
ax = axes;
plot(Time,S(1:length(N)-1),'r','Linewidth',2)
grid minor
xlabel('Time in Seconds','Interpreter','latex','FontSize',13);
ylabel('Photon Concentration','Interpreter','latex','FontSize',13);
title('LASER Coupled Rate Equation','Interpreter','latex','FontSize',13);
legend('Transient Photon Concentration');
ax.YAxis.Color = [0 0 0];
ax.XAxis.Color = [0 0 0];
figure(3)
ax = axes;
plot(N,S,'b','Linewidth',2)
grid minor
xlabel('Carrier Concentration','Interpreter','latex','FontSize',13);
ylabel('Photon Concentration','Interpreter','latex','FontSize',13);
title('Photon Concentration Vs Carrier Concentration','Interpreter','latex','FontSize',13);
ax.YAxis.Color = [0 0 0];
ax.XAxis.Color = [0 0 0];
figure(4)
ax = axes;
yyaxis left
plot(Time,S(1:length(N)-1),'r','Linewidth',2)
xlabel('Time in Seconds','Interpreter','latex','FontSize',13);
ylabel('Photon Concentration','Interpreter','latex','FontSize',13);
yyaxis right
plot(Time,N(1:length(N)-1),'m','Linewidth',2)
xlabel('Time in Seconds','Interpreter','latex','FontSize',13);
ylabel('Carrier Concentration','Interpreter','latex','FontSize',13);
title('LASER Rate Equations - Transient Response','Interpreter','latex','FontSize',12);
grid minor
ax.YAxis(1).Color = [1 0 0];
ax.YAxis(2).Color = [1 0 1];
figure(5)
ax = axes;
yyaxis left
plot(Pump_Rate,Carrier_Concentration,'r','Linewidth',2)
xlabel('Pump Rate','Interpreter','latex','FontSize',13);
ylabel('Carrier Concentration','Interpreter','latex','FontSize',13);
ylim([0,11*10^17])
xline(1,'--','Color','[0.4 0.6 0.2]','Linewidth',2);
y1 = yline(Nth,'--','Nth','Color','[1 0 0]','Linewidth',2);
y1.LabelHorizontalAlignment = 'left';
yl.Color = [1 0 1];
yyaxis right
plot(Pump_Rate,Photon_Concentration,'m','Linewidth',2)
xlabel('Pump Rate','Interpreter','latex','FontSize',13);
ylabel('Photon Concentration','Interpreter','latex','FontSize',13);
xline(1,'--','Color','[0.4 0.6 0.2]','Linewidth',2);
title('Steady State Solution Vs Pump Rate','Interpreter','latex','FontSize',12);
ax.YAxis(1).Color = [1 0 0];
ax.YAxis(2).Color = [1 0 1];
grid minor
figure(6)
ax = axes;
plot(Time,phi(1:length(phi)-1),'r','Linewidth',2)
xlabel('Time in Seconds','Interpreter','latex','FontSize',13);
ylabel('Optical Phase','Interpreter','latex','FontSize',13);
title('LASER Coupled Rate Equation','Interpreter','latex','FontSize',13);
ax.YAxis.Color = [0 0 0];
ax.XAxis.Color = [0 0 0];
grid minor
figure(7)
ax = axes;
yyaxis left
plot(Time,N(1:length(N)-1),'r','Linewidth',2)
xlabel('Time in Seconds','Interpreter','latex','FontSize',13);
ylabel('Carrier Concentration','Interpreter','latex','FontSize',13);
yyaxis right
plot(Time,phi(1:length(phi)-1),'m','Linewidth',2)
xlabel('Time in Seconds','Interpreter','latex','FontSize',13);
ylabel('Optical Phase','Interpreter','latex','FontSize',13);
title('LASER Coupled Rate Equation','Interpreter','latex','FontSize',13);
ax.YAxis(1).Color = [1 0 0];
ax.YAxis(2).Color = [1 0 1];
grid minor
figure(8)
ax = axes;
yyaxis left
plot(Time,S(1:length(N)-1),'r','Linewidth',2)
xlabel('Time in Seconds','Interpreter','latex','FontSize',13);
ylabel('Photon Concentration','Interpreter','latex','FontSize',13);
yyaxis right
plot(Time,phi(1:length(phi)-1),'m','Linewidth',2)
xlabel('Time in Seconds','Interpreter','latex','FontSize',13);
ylabel('Optical Phase','Interpreter','latex','FontSize',13);
title('LASER Coupled Rate Equation','Interpreter','latex','FontSize',13);
ax.YAxis(1).Color = [1 0 0];
ax.YAxis(2).Color = [1 0 1];
grid minor
figure(9)
ax = axes;
plot(Time,Optical_Output_Power_of_LASER.*Quantum_Efficiency(1),'r','Linewidth',2)
hold on
plot(Time,Optical_Output_Power_of_LASER.*Quantum_Efficiency(2),'g','Linewidth',2)
hold on
plot(Time,Optical_Output_Power_of_LASER.*Quantum_Efficiency(3),'b','Linewidth',2)
hold on
plot(Time,Optical_Output_Power_of_LASER.*Quantum_Efficiency(4),'m','Linewidth',2)
hold off
xlabel('Time in Seconds','Interpreter','latex','FontSize',13);
ylabel('Optical Output Power of LASER','Interpreter','latex','FontSize',13);
title('Optical Output Power of LASER Vs Time','Interpreter','latex','FontSize',13);
legend('Quantum Efficiency = 0.1','Quantum Efficiency = 0.5','Quantum Efficiency = 0.7','Quantum Efficiency = 1','Interpreter','latex','FontSize',11,'Location','northeast');
ax.YAxis.Color = [0 0 0];
ax.XAxis.Color = [0 0 0];
ylim([0,22.5])
grid minor
figure(10)
ax = axes;
plot(I_bias.*10^2,Relaxation_Oscillation_Frequency,'m','Linewidth',2)
xlabel('$I_{bias}$ (in mA)','Interpreter','latex','FontSize',13);
ylabel('Relaxation Oscillation Frequency (Hz)','Interpreter','latex','FontSize',13);
title('Relaxation Oscillation Frequency Vs $I_{bias}$','Interpreter','latex','FontSize',13);
grid minor
ax.YAxis.Color = [0 0 0];
ax.XAxis.Color = [0 0 0];
x1 = xline(I*10^2,'--','Threshold Current','Color','[0 0 1]','Linewidth',2);
x1.Interpreter = 'latex';
x1.Color = [0 0 1];
x1.FontSize =11;
M_0_e = 9.1*10^-31;
%Mn = 1.062*M_0_e;%4K Effective Mass
Mn = 1.182*M_0_e;%300K Effective Mass
T =300;
Kb = 1.38*10^-23;
x1 = (Kb*T);
h = 6.634*10^-34;
hbar = h/(2*pi);
deltaE_e = (linspace(0,6*x1,10^5));
gc = (Mn.*sqrt(2.*Mn.*deltaE_e))/((pi^2)*(hbar^3));
x = 1+exp((deltaE_e+(1.6*10^-20))./(Kb*T));
f = 1./x;
y = (gc.*f)*(10^-6)*(1.6*10^-19);
figure(11)
plot(y,(deltaE_e/(1.6*10^-19)),'b','linewidth',2)
xlabel('g_{C}(E) * f(E)');
ylabel('E-E_{C} in (eV)');
title('g_{C}(E) * f(E) Vs E-E_{C}');
yline((0.0259/2),'--r');
set(gca,'YTick',[(0.0259/2) 0.0259,0.0776,0.1294,0.1811,0.2329]);
set(gca,'YTickLabel',{'{KT}/_{2}','KT','3KT','5KT','7KT','9KT'});
grid minor
M_0_p = 9.1*10^-31;
%Mp =0.590*M_0_p;%4K Effective Mass
Mp =0.81*M_0_p;%300K Effective Mass
deltaE_p = (linspace(0,6*x1,10^5));
gv = (Mp.*sqrt(2.*Mp.*deltaE_p))/((pi^2)*(hbar^3));
x2 = 1+exp((deltaE_p+(1.6*10^-20))./(Kb*T));
f_p = 1./x2;
y2 = (gv.*f_p)*(10^-6)*(1.6*10^-19);
figure(12)
plot(y2,(deltaE_p/(1.6*10^-19)),'b','linewidth',2)
xlabel('g_{V}(E) * [1-f(E)]');
ylabel('E_{V} - E in (eV)');
title('g_{V}(E) * [1-f(E)] Vs E_{V} - E');
yline((0.0259/2),'--r');
set(gca,'YTick',[(0.0259/2) 0.0259,0.0776,0.1294,0.1811,0.2329]);
set(gca,'YTickLabel',{'{KT}/_{2}','KT','3KT','5KT','7KT','9KT'});
grid minor
figure(13)
ax = axes;
plot(y,(deltaE_e/(1.6*10^-19)),'b','linewidth',2)
hold on
plot(y2,(deltaE_p/(1.6*10^-19)),'m','linewidth',2)
hold off
set(gca,'YTick',[(0.0259/2) 0.0259,0.0776,0.1294,0.1811,0.2329]);
set(gca,'YTickLabel',{'{KT}/_{2}','KT','3KT','5KT','7KT','9KT'});
yline((0.0259/2),'--r');
xlabel('g_{C}(E) * f(E) & g_{V}(E) * [1-f(E)]');
ylabel('E-E_{C} in (eV) & E_{V} - E in (eV)');
title('Density of States (DOS) of CB & VB');
legend('g_{C}(E) * f(E) Vs E-E_{C} in (eV)','g_{V}(E) * [1-f(E)] Vs E_{V} - E in (eV)');
grid minor
ax.YAxis.Color = [0 0 0];
ax.XAxis.Color = [0 0 0];
Kb = 1.38*10^-23;
T = linspace(1,650,10^5);
M0 = 9.1*10^-31;
Mn = 1.182 *M0;
Mp = 0.81*M0;
h = 6.62607015 *10^-34;
ND = 10^15;
NA = 0;
EG_0K = 1.166;%eV For Silicon
alpha = 4.73*10^-4;%eV per Kelvin
beta = 636;%Kelvin
gD = 2;
EC_ED = 50*10^-3;%in eV
for i=1:length(T)
Kb_T_in_eV(i) = (Kb*T(i))/(1.6*10^-19);
x1(i) = (2*pi*Mn*Kb*T(i))/(h^2);
x2(i) = (x1(i))^(3/2);
Nc(i) = (2*x2(i))/(1*10^6);%in cm3
x3(i) = (2*pi*Mp*Kb*T(i))/(h^2);
x4(i) = (x3(i))^(3/2);
Nv(i) = (2*x4(i))/(1*10^6);%in cm3
x6(i) = alpha/(T(i)+beta);
x7(i) = (x6(i))* (T(i)^2);%in eV
Eg(i) = (EG_0K - x7(i));%in eV
x8(i) = sqrt(Nc(i)* Nv(i));%in cm3
x9(i) = -(Eg(i)/(2*Kb_T_in_eV(i)));%dimensionless
x10(i) = (x8(i))*exp(x9(i)); %in cm3
x11(i) = x10(i)/ND;%dimensionless
x12(i) = ((EC_ED)/Kb_T_in_eV(i));
N_zita(i) = ((Nc(i))/(gD))*(exp(-x12(i)));
x13(i) = 1+((4*ND)/(N_zita(i)));
x14(i) = (sqrt(x13(i)))-1;
n1(i) = ((N_zita(i))/2)*x14(i);
x15(i) = n1(i)/ND;
x16(i) = (((ND-NA)/2)^2) + ((x10(i))^2);
x17(i) = sqrt(x16(i));
n2(i) = ((ND-NA)/2)+ x17(i);
x18(i) = n2(i)/ND;
end
figure(14)
plot(T,Eg,'b','linewidth',2)
xlabel('Temperature in Kelvin');
ylabel('Band Gap in eV for Silicon');
title('Band Gap Vs Temperature For Silicon');
grid minor
Mn = 0.553 *M0;
Mp = 0.357*M0;
h = 6.62607015 *10^-34;
ND = 10^15;
NA = 0;
EG_0K = 0.7437;%eV
alpha = 4.77*10^-4;%eV per Kelvin
beta = 235;%Kelvin
gD = 2;
EC_ED = 12*10^-3;%in eV
for i=1:length(T)
Kb_T_in_eV(i) = (Kb*T(i))/(1.6*10^-19);
x1(i) = (2*pi*Mn*Kb*T(i))/(h^2);
x2(i) = (x1(i))^(3/2);
Nc(i) = (2*x2(i))/(1*10^6);%in cm3
x3(i) = (2*pi*Mp*Kb*T(i))/(h^2);
x4(i) = (x3(i))^(3/2);
Nv(i) = (2*x4(i))/(1*10^6);%in cm3
x6(i) = alpha/(T(i)+beta);
x7(i) = (x6(i))* (T(i)^2);%in eV
Eg(i) = (EG_0K - x7(i));%in eV
x8(i) = sqrt(Nc(i)* Nv(i));%in cm3
x9(i) = -(Eg(i)/(2*Kb_T_in_eV(i)));%dimensionless
x10(i) = (x8(i))*exp(x9(i)); %in cm3
x11(i) = x10(i)/ND;%dimensionless
x12(i) = ((EC_ED)/Kb_T_in_eV(i));
N_zita(i) = ((Nc(i))/(gD))*(exp(-x12(i)));
x13(i) = 1+((4*ND)/(N_zita(i)));
x14(i) = (sqrt(x13(i)))-1;
n1(i) = ((N_zita(i))/2)*x14(i);
x15(i) = n1(i)/ND;
x16(i) = (((ND-NA)/2)^2) + ((x10(i))^2);
x17(i) = sqrt(x16(i));
n2(i) = ((ND-NA)/2)+ x17(i);
x18(i) = n2(i)/ND;
end
figure(15)
plot(T,Eg,'b','linewidth',2)
xlabel('Temperature in Kelvin');
ylabel('Band Gap in eV for Germanium');
title('Band Gap Vs Temperature For Germanium');
grid minor
Eph = linspace(1.55,1.8,10^4);
T1 = 293;
T2 = 323;
Eg1 = 1.55;
Eg2 = 0.8;
Kb = 1.38*10^-23;
rr1 = (Eph-Eg1);
rr2 = -rr1/(0.0253);
rr3 = -rr1/(0.0279);
rr4 = (Eph-Eg2);
rr5 = -rr4/(0.0253);
rr6 = -rr4/(0.0279);
Power1 = rr1.*exp(rr2);
Power2 = rr1.*exp(rr3);
power3 = rr4.*exp(rr5);
power4 = rr4.*exp(rr6);
figure(16)
plot(Eph,Power1,'r','Linewidth',2)
hold on
plot(Eph,Power2,'m','Linewidth',2)
hold off
grid minor
xlabel('Photon Energy ($E_{ph})$','Interpreter','latex','FontSize',13);
ylabel('Normalized Power','Interpreter','latex','FontSize',13);
title('Emission Spectrum for $\lambda$ = 800 nm','Interpreter','latex','FontSize',13);
legend('T = 20 C','T = 50 C','Interpreter','latex','FontSize',11,'Location','northeast');
figure(17)
ax = axes;
plot(Relaxation_Oscillation_Frequency,Laser_Transfer_Function_Phase(1,:),'r','Linewidth',2)
hold on
plot(Relaxation_Oscillation_Frequency,Laser_Transfer_Function_Phase(2,:),'g','Linewidth',2)
hold on
plot(Relaxation_Oscillation_Frequency,Laser_Transfer_Function_Phase(3,:),'b','Linewidth',2)
hold off
xlabel('Relaxation Oscillation Frequency (Hz)','Interpreter','latex','FontSize',12);
ylabel('Laser Transfer Function Phase','Interpreter','latex','FontSize',12);
legend('$I_{bias}$ = 30 mA','$I_{bias}$ = 40 mA','$I_{bias}$ = 50 mA','Interpreter','latex','FontSize',12)
grid minor
ax.YAxis.Color = [0 0 0];
ax.XAxis.Color = [0 0 0];
figure(18)
ax = axes;
plot(Relaxation_Oscillation_Frequency,Laser_Transfer_Function_Magnitude(1,:),'r','Linewidth',2)
hold on
plot(Relaxation_Oscillation_Frequency,Laser_Transfer_Function_Magnitude(2,:),'g','Linewidth',2)
hold on
plot(Relaxation_Oscillation_Frequency,Laser_Transfer_Function_Magnitude(3,:),'b','Linewidth',2)
hold off
xlabel('Relaxation Oscillation Frequency (Hz)','Interpreter','latex','FontSize',12);
ylabel('Laser Transfer Function Magnitude','Interpreter','latex','FontSize',12);
legend('$I_{bias}$ = 30 mA','$I_{bias}$ = 40 mA','$I_{bias}$ = 50 mA','Interpreter','latex','FontSize',12)
grid minor
ax.YAxis.Color = [0 0 0];
ax.XAxis.Color = [0 0 1];
clear all;
toc

Cite As

Nagavenkata Anil Peddiraju (2024). LASER RATE EQUATIONS by P N V Anil Kumar,IIT Bombay (https://www.mathworks.com/matlabcentral/fileexchange/109540-laser-rate-equations-by-p-n-v-anil-kumar-iit-bombay), MATLAB Central File Exchange. Retrieved .

MATLAB Release Compatibility
Created with R2022a
Compatible with any release
Platform Compatibility
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Version Published Release Notes
1.0.0