From b30f418f5be8b3d6994cd61e058572e5fcbe4ea1 Mon Sep 17 00:00:00 2001 From: LaurentNevou <35040499+LaurentNevou@users.noreply.github.com> Date: Wed, 26 Sep 2018 18:54:55 +0200 Subject: [PATCH] Add files via upload --- MultiLayersResults1.PNG | Bin 0 -> 66074 bytes MultiLayersResults2.PNG | Bin 0 -> 63548 bytes MultiLayersResults3.PNG | Bin 0 -> 65667 bytes Pot_MultiLayers.m | 53 +++++++++ Pot_Parabolic.m | 20 ++++ Pot_Sinus.m | 21 ++++ Schroed1D_Euler_f.m | 87 ++++++++++++++ Schroed1D_FEM_f.m | 50 +++++++++ Schroed1D_PWE_f.m | 91 +++++++++++++++ Schroed1D_TMM_f.m | 118 +++++++++++++++++++ Schroedinger1D_Main.m | 243 ++++++++++++++++++++++++++++++++++++++++ 11 files changed, 683 insertions(+) create mode 100644 MultiLayersResults1.PNG create mode 100644 MultiLayersResults2.PNG create mode 100644 MultiLayersResults3.PNG create mode 100644 Pot_MultiLayers.m create mode 100644 Pot_Parabolic.m create mode 100644 Pot_Sinus.m create mode 100644 Schroed1D_Euler_f.m create mode 100644 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+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +Vb=0.3; % potential barrier [eV] + +%zt = [ 10 10 10 9 10 8 10 7 10 6 10 ]*1E-9; % thickness of each layer [nm] +%Vt = [ 1 0 1 0 1 0 1 0 1 0 1 ]*Vb; % barrier height of each layer [eV] + +zt = [ 10 10 5 3 10 ]*1E-9; % thickness of each layer [nm] +Vt = [ 1 0 1 0 1 ]*Vb; % barrier height of each layer [eV] + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%% Discretisation %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% here, I descretize the grid z and the potential V0 + +for i=1:length(zt) + if i==1 + zv{1} = 0:dz:zt(1); + z=zv{1}; + V0=zv{1}*0+Vt(1); + else + zv{i} = (z(end)+dz):dz:(z(end) + zt(i)); + z = [ z zv{i} ]; + V0 = [ V0 (zv{i}*0+1) * Vt(i) ]; + end +end + +% here, I kind of patch the grid for the TMM + +if TM_Method==1 + clear zv + for i=1:length(zt) + zzt((i-1)*Ns+1:i*Ns)=zt(i)/Ns; + VVt((i-1)*Ns+1:i*Ns)=Vt(i); + end + zzt=[1e-12 zzt 1e-12]; + VVt=[10 VVt 10]; + + for i=1:length(zzt) + if i==1 + ZZ(1) = zzt(1); + zv{1} = 0:dz:zzt(1); + zz = zv{1}; + else + ZZ(i) = ZZ(end)+zzt(i); + zv{i} = (zz(end)+dz):dz:ZZ(end); + zz = [zz zv{i}]; + end + end + +end diff --git a/Pot_Parabolic.m b/Pot_Parabolic.m new file mode 100644 index 0000000..cbfd2d5 --- /dev/null +++ b/Pot_Parabolic.m @@ -0,0 +1,20 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +Vb=0.3; % potential barrier [eV] +a=1e15; +Ltot=50e-9; + +z=0:dz:Ltot; +V0=a*(z-Ltot/2).^2; + +V0(V0>Vb)=Vb; + +if TM_Method==1 + display('ERROR: TM_Method not valid for that potential') +end + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% diff --git a/Pot_Sinus.m b/Pot_Sinus.m new file mode 100644 index 0000000..963cbd7 --- /dev/null +++ b/Pot_Sinus.m @@ -0,0 +1,21 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +Vb=2; % potential barrier [eV] +a=1e15; +Ltot=50e-9; + +z=0:dz:Ltot; +V0=Vb/2*(sin(7*pi/Ltot*z) +1); +V0(1)=Vb; +V0(end)=Vb; +%V0(V0>Vb)=Vb; + +if TM_Method==1 + display('ERROR: TM_Method not valid for that potential') +end + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% diff --git a/Schroed1D_Euler_f.m b/Schroed1D_Euler_f.m new file mode 100644 index 0000000..e2b81a2 --- /dev/null +++ b/Schroed1D_Euler_f.m @@ -0,0 +1,87 @@ +function[E,psi]=Schroed1D_Euler_f(z,V0,Mass,n,dE,precision) + +method=2; % method "2" is much more acurate than "1" but take also more time... +e=min(V0); +Emax=max(V0)+0.1; + +C=0; N=0; E=[]; +psi=z*0+1; psi_old=psi; + + +while e1 % here, I catch a quantum state because the last point of psi change sign + + N=N+1; de=dE; + + while abs(de)>precision + if sign( psi(end) ) ~= sign( psi_old(end) ) + de = -de/2; + end + e=e+de; + psi_old=psi; + psi=Schroed1D_Euler_Eval(z,V0,Mass,e,method); + end + + E(N,:)=e; PSI(:,N)= psi; + C=0; + end + + e=e+dE; +end + +psi=PSI; + +end + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +function[psi]=Schroed1D_Euler_Eval(z,V0,Mass,ee,method) + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +h=6.62606896E-34; %% Planck constant [J.s] +hbar=h/(2*pi); +e=1.602176487E-19; %% electron charge [C] +me=9.10938188E-31; %% electron mass [kg] + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +dz=z(2)-z(1); +y = [0 1]; + +if method==1 + + for i = 1:length(z) %% number of z steps to take + dy(2) = -2*e/(hbar^2) * ( ee - V0(i) ) * y(1) ; %% Equation for dv/dz + dy(1) = Mass*me*y(2); %% Equation for dx/dz + y = y + dz*dy ; %% integrate both equations with Euler + psi(i)= y(1); + end + +elseif method==2 + + for i = 1:length(z) %% number of z steps to take + + dy(2) = -2*e/(hbar^2) * ( ee - V0(i) ) * y(1) ; %% Equation for dv/dz + dy(1) = Mass*me*y(2); %% Equation for dx/dz + + K = y + 0.5*dz*dy; + dK(2) = -2*e/(hbar^2) * ( ee - V0(i) ) * K(1) ; %% Equation for dv/dz + dK(1) = Mass*me*K(2); %% Equation for dx/dz + + y = y + dz*dK; + psi(i)= y(1); + end +end + +end \ No newline at end of file diff --git a/Schroed1D_FEM_f.m b/Schroed1D_FEM_f.m new file mode 100644 index 0000000..5679452 --- /dev/null +++ b/Schroed1D_FEM_f.m @@ -0,0 +1,50 @@ +function[E,psi]=Schroed1D_FEM_f(z,V0,Mass,n) + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%% Constants %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +h=6.62606896E-34; %% Planck constant [J.s] +hbar=h/(2*pi); +e=1.602176487E-19; %% electron charge [C] +me=9.10938188E-31; %% electron mass [kg] + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +Nz=length(z); +dz=z(2)-z(1); + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%% Building of the operators %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +DZ2 =(-2)*diag(ones(1,Nz)) + (1)*diag(ones(1,Nz-1),-1) + (1)*diag(ones(1,Nz-1),1); +DZ2=DZ2/dz^2; + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%% Building of the Hamiltonien %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +H = -hbar^2/(2*me*Mass) * DZ2 + diag(V0*e) ; + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%% Diagonalisation of the Hamiltonien %%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +H = sparse(H); +[psi,Energy] = eigs(H,n,'SM'); +E = diag(Energy)/e ; +E=real(E); + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%% Normalization of the Wavefunction %%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +for i=1:n + psi(:,i)=psi(:,i)/sqrt(trapz(z',abs(psi(:,i)).^2)); % normalisation at 1 +end + +psi=psi(:,end:-1:1); +E=E(end:-1:1); + +end \ No newline at end of file diff --git a/Schroed1D_PWE_f.m b/Schroed1D_PWE_f.m new file mode 100644 index 0000000..2045e50 --- /dev/null +++ b/Schroed1D_PWE_f.m @@ -0,0 +1,91 @@ +function[E,psi]=Schroed1D_PWE_f(z,V0,Mass,n,Nz,NG) + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Constants %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +h=6.62606896E-34; %% Planck constant [J.s] +hbar=h/(2*pi); +e=1.602176487E-19; %% electron charge [C] +me=9.10938188E-31; %% electron mass [kg] + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%% Interpolation on a grid that have 2^N points %%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +NG = 2*floor(NG/2); % round to lower even number + +zz=linspace(z(1),z(end),Nz); +V=interp1(z,V0,zz); + +dz=z(2)-z(1); +dzz=zz(2)-zz(1); +Ltot=zz(end)-zz(1); + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%% Building Potential in Fourier space %%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +Vk = fftshift(fft(V))*dzz/Ltot; +Vk =Vk(Nz/2-NG+1:Nz/2+NG+1); + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%% Reciprocal lattice vectors %%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +G = (-NG/2:NG/2)*2*pi/Ltot; +NG=length(G); + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%% Building Hamiltonien %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +idx = ones(NG,1)*(1:NG); % matrix with elements equal to the column index idx(i,j) = j +idx = idx' - idx; % matrix with elements equal to row - column idx(i,j) = i - j; +idx = idx + NG ; % idx(i,j) = i - j + N + +D2=diag(G(1:NG)).^2; % Operator second derivative + +H = hbar^2/(2*me*Mass)*D2 + Vk(idx)*e; + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%% Solving Hamiltonien %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +H = sparse(H); +[psik, Ek] = eigs(H,n,'SM'); +E = diag(Ek) / e; +E=abs(E); +%E=real(E); + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%% Transforming & Scaling the waves functions %%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +for i=1:n + psizz = invFFT1D(psik(:,i).',Nz)/dzz ; % take care that the transpose does not take the complex conjugate + psi_temp = interp1(zz,psizz,z); + psi(:,i) = psi_temp/sqrt(trapz(z,abs(psi_temp).^2)); % normalisation at 1 +end + +% in Octave, the order of the eigen values are reversed... +%psi=psi(:,end:-1:1); +%E=E(end:-1:1); + +end + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +function [Vz] = invFFT1D(Vk,Nz) + +Nk=length(Vk); +N00=Nz/2-ceil(Nk/2); + +Vk00= [zeros(1,N00+1) Vk zeros(1,N00)] ; % adding zeros on the missing Fourier components +Vz=ifft(ifftshift(Vk00)); + +end \ No newline at end of file diff --git a/Schroed1D_TMM_f.m b/Schroed1D_TMM_f.m new file mode 100644 index 0000000..c86f78e --- /dev/null +++ b/Schroed1D_TMM_f.m @@ -0,0 +1,118 @@ +function[E,psi]=Schroed1D_TMM_f(zz,zv,Vt,Mass,n,dE,precision) + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +h=6.62606896E-34; %% Planck constant [J.s] +hbar=h/(2*pi); +q=1.602176487E-19; %% electron charge [C] +me=9.10938188E-31; %% electron mass [kg] + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +DEBUG=0; + +e=min(Vt+dE); +Emax=max(Vt); + +C=0; N=0; E=[]; +b0=0; b0_old=b0; +aN=0; aN_old=aN; +PSI=[]; + +while e1) || ((sign( aN ) ~= sign(aN_old ) ) && C>1) + % here, I catch a quantum state because the last point of psi change sign + + N=N+1; de=dE; + + while abs(de)>precision + if (sign( b0 ) ~= sign( b0_old )) || (sign( aN ) ~= sign( aN_old )) + de = -de/2; + end + e=e+de; + if DEBUG==1 + ee(end+1)=e; + end + b0_old=b0; + aN_old=aN; + + kt=sqrt( 2*Mass*q*me*(e-Vt) ) / hbar; + [A,B]=Schroed1D_TMM_Eval(zz,kt,Mass); + b0=real(B(1)); + aN = real(A(end)); + + end + + psi=[]; + for j=1:length(Vt) + psi= [ psi A(j+1)*exp(1i*kt(j)*zv{j}) + B(j+1)*exp(-1i*kt(j)*zv{j}) ]; + end + E(N,:)=e; PSI(:,N)= psi; + C=0; + end + e=e+dE; + if DEBUG==1 + ee(end+1)=e; + end +end + +psi=PSI; + +if DEBUG==1 + figure + hold on;grid on; + plot(ee,'b.-') + xlabel('steps') + ylabel('Energy (eV)') +end + +end + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +function[A,B]=Schroed1D_TMM_Eval(zz,kt,Mass) + +AmplitudeInput=1; +kL=kt(1); +kR=kt(end); + +M(1,1:3) = [ 1 -1 -1 ]; +M(2,1:3) = [ -kL -kt(1) kt(1) ]; + +for j=1:length(kt)-1 + M(j*2+1,2*j:2*j+3) = [ exp(1i*kt(j)*zz(j)) exp(-1i*kt(j)*zz(j)) -exp(1i*kt(j+1)*zz(j)) -exp(-1i*kt(j+1)*zz(j))]; + M(2*j+2,2*j:2*j+1) = kt(j) *[ +exp(1i*kt(j) *zz(j)) -exp(-1i*kt(j) *zz(j)) ]; + M(2*j+2,2*j+2:2*j+3) = -kt(j+1) *[ +exp(1i*kt(j+1)*zz(j)) -exp(-1i*kt(j+1)*zz(j)) ]; +end + +M(length(kt)*2+1,2*length(kt):2*length(kt)+2) = [exp(1i*kt(end)*zz(end)) exp(-1i*kt(end)*zz(end)) -exp(1i*kR*zz(end)) ]; +M(length(kt)*2+2,2*length(kt):2*length(kt)+2) = [ kt(end)*exp(1i*kt(end)*zz(end)) -kt(end)*exp(-1i*kt(end)*zz(end)) -kR*exp(1i*kR*zz(end)) ]; + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +M=sparse(M); + +D=zeros(length(M),1); +D(1)=-sqrt(AmplitudeInput); +D(2)=-sqrt(AmplitudeInput)*kL; + +AB=inv(M)*D; +A=[1 ; AB(2:2:end)]; +B=[AB(1:2:end-1) ; 0]; + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +end \ No newline at end of file diff --git a/Schroedinger1D_Main.m b/Schroedinger1D_Main.m new file mode 100644 index 0000000..5e778c2 --- /dev/null +++ b/Schroedinger1D_Main.m @@ -0,0 +1,243 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%% last update 26Sept2018, lne %%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +clear all +close all +clc +warning off +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%% Model activation %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% 0 for turn off +% 1 for turn on + +FE_Method=1; % Diagonalization of the Hamiltonian (FEM) +Euler_Method=0; % Scanning in Energy method (Euler) +PWE_Method=0; % Plane Wave Expansion (PWE) +TM_Method=1;Ns=3; % Transfer Matrix Method (TMM), Ns=number of segment per layer !!!WORK ONLY with Pot_MultiLayers!!! + +saveV=0; +savePSI=0; + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +dz=2E-10; % resolution of the grid [m] +n=4; % number of solution asked + +Mass = 0.067; % effective mass, constant over all the structure... +F0=0;%-1e7; % Electric field [V/m] +ScF=0.1; % scaling factor to plot the wave function [Without Dimension] + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%% Potential definition %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Two vectors of the same length must be defined +% z [meter] +% V0 [eV] +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Choose your between the next 3 potentials or build your own! + +Pot_MultiLayers +%Pot_Parabolic +%Pot_Sinus + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%% NOTHING TO CHANGE ANYMORE!!! %%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +V0=(F0*z)+V0; % adding the electric field to the potential +V0=V0-min(V0); + +if TM_Method==1 + VVt=(F0*ZZ)+VVt; % adding the electric field to the potential + VVt=VVt-min(VVt); +end + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%% Selection of the model %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +E1=[];E2=[];E3=[];E4=[]; +display('=======================================') + +if FE_Method==1 + tic + [E1,psi1] = Schroed1D_FEM_f(z,V0,Mass,n); % m=cste + display(strcat('-> Finite Elements method =',num2str(toc),'sec')) +end + +if Euler_Method==1 + dE=0.005; % minimum step [eV] between 2 eigen values (bigger => faster) + precision=1e-7; % precision [eV] of the eigen values (bigger => faster) + + tic + [E2,psi2] = Schroed1D_Euler_f(z,V0,Mass,n,dE,precision); + display(strcat('-> Euler method =',num2str(toc),'sec')) +end + +if PWE_Method==1 + Nz = 256 ; % number of points on the z grid % has to be a power of 2 (32,64,128,256,512,...) (smaller => faster) + NG = Nz/2-1 ; % number of harmonics % has to be at least 2 times -1 smaller than Nz (smaller => faster) + + tic + [E3,psi3] = Schroed1D_PWE_f(z,V0,Mass,n,Nz,NG); + display(strcat('-> PWE method =',num2str(toc),'sec')) + if F0~=0 + display('Warning: The PWE method is not appropriate for non-periodic potential') + end +end + +if TM_Method==1 + + dE=0.001; % minimum step [eV] between 2 eigen values (bigger => faster) + precision=1e-16; % precision [eV] of the eigen values (bigger => faster) + + tic + [E4,psi4] = Schroed1D_TMM_f(ZZ,zv,VVt,Mass,n,dE,precision); + display(strcat('-> TMM method =',num2str(toc),'sec')) +end + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Display Results %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +E=nan(n,4); +E(1:length(E1),1)=E1; +E(1:length(E2),2)=E2; +E(1:length(E3),3)=E3; +E(1:length(E4),4)=E4; + +display('=======================================') +display('Results:') +display('=======================================') + +display(strcat('E(eV)=')) +display(strcat(num2str(E))) + +%break +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figures %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +if FE_Method==1 + for i=1:length(E1) + psi1(:,i)=abs(psi1(:,i)).^2/max(abs(psi1(:,i)).^2)*ScF + E1(i); % normalisation for the plotting + end +end + +if Euler_Method==1 + for i=1:length(E2) + psi2(:,i)=abs(psi2(:,i)).^2/max(abs(psi2(:,i)).^2)*ScF + E2(i); % normalisation for the plotting + end +end + +if PWE_Method==1 + for i=1:length(E3) + psi3(:,i)=abs(psi3(:,i)).^2/max(abs(psi3(:,i)).^2)*ScF + E3(i); % normalisation for the plotting + end +end + +if TM_Method==1 + for i=1:length(E4) + psi4(:,i)=abs(psi4(:,i)).^2/max(abs(psi4(:,i)).^2)*ScF + E4(i); % normalisation for the plotting + end +end + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +figure('position',[10 100 1000 700]); +LW=2; +FS=15; +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +subplot(1,1,1,'fontsize',FS) +hold on;%grid on; + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +plot(z*1e9,V0, 'b-','linewidth',LW) + +s{1}='\fontsize{20}\color{blue}Potential'; +s{2}='\fontsize{20}\color{blue}----------'; + +if FE_Method==1 + for i=1:length(E1) + plot(z*1e9,psi1(:,i),'r-','linewidth',LW) + end + s{end+1}='\fontsize{20}\color{red}FE-Method'; +end + +if Euler_Method==1 + for i=1:length(E2) + plot(z*1e9,psi2(:,i),'k--','linewidth',LW) + end + s{end+1}='\fontsize{20}\color{black}Euler-Method'; +end + +if PWE_Method==1 + for i=1:length(E3) + plot(z*1e9,psi3(:,i),'m--','linewidth',LW) + end + s{end+1}='\fontsize{20}\color{magenta}PWE-Method'; +end + +if TM_Method==1 + for i=1:length(E4) + plot(z*1e9,psi4(:,i),'g--','linewidth',LW) + end + s{end+1}='\fontsize{20}\color{green}TM-Method'; +end + +xlabel('z (nm)'); +ylabel('Energy (eV)'); + +ylim([min(V0)-0.05 max(V0)+0.1]) +title('Multi layers potential') + +xl=get(gca,'xlim'); +yl=get(gca,'ylim'); + +text(0.02*xl(2),yl(2),s) + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Data save %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +if saveV==1; + MV = [z' V0' ]; + save('data_V.txt','MV','-ascii') +end + +if savePSI==1; + + if FE_Method==1 + M1 = [z' psi1]; + save('data_PSI_FEM.txt','M1','-ascii') + end + if Euler_Method==1 + M2 = [z' psi2]; + save('data_PSI_Euler.txt','M2','-ascii') + end + if PWE_Method==1 + M3 = [z' psi3]; + save('data_PSI_PWE.txt','M3','-ascii') + end + if TM_Method==1 + M4 = [z' psi4]; + save('data_PSI_TMM.txt','M4','-ascii') + end + +end + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \ No newline at end of file