%Fig2_6 %Essential Electron Transport for Device Physics %close all clear all; clf; %plotting parameters + fontsizes FS = 12; %label fontsize 18 FSN = 12; %number fontsize 16 LW = 1; %linewidth % Change default axes fonts. set(0,'DefaultAxesFontName', 'Times'); set(0,'DefaultAxesFontSize', FSN); % Change default text fonts. set(0,'DefaultTextFontname', 'Times'); set(0,'DefaultTextFontSize', FSN); hbar=1.054571596e-34; echarge=1.602176462e-19; baremass=9.10938188e-31; meff = 0.07; me = meff*baremass; FS = 14; LW = 1.5; MS = 8; numsol = 4; length = 10e-9; %length of potential well [m] 10, use 20 with step change in potential N = 0.3e3+1; %number of spatial samples dx = length/(N-1); x = linspace(0,length,N); v = zeros(1,N); %rectangular potential with ininfite barrier energy %step change in potential %v(1,1:N/2)=echarge*0.3;%step change in potential 0.3 [E,psi] = solve_sch(length,N,v,me,numsol);%solve for rectangular potential EeV = E/echarge; psi = [zeros(numsol,1) psi zeros(numsol,1)]; psi0 = psi(1,:); %ground-state wave function dpsi0 = gradient(psi0)/dx; %numberically evaluate gradient E0 = E(1,1); %ground-state energy eigenvalue vn = v-E0; %subtract E0 to create vn potential %create supersymmetric pair potential vp numerically vp = -vn+(hbar^2/me)*(dpsi0./psi0).^2; %create supersymmetric pair potential vp analytically for rectangular well %vp = -vn+(pi^2*hbar^2/(length^2*me))*((cot(x*pi/length)).^2); [En,psin] = solve_sch(length,N,vn,me,numsol);%solve for vn potential [Ep,psip] = solve_sch(length,N,vp,me,numsol);%solve for vp potential %break symmetry by applying uniform electric field Nstark = 50;%100+1 total potential drop is Nstark * E_0 Astark = linspace(0,Nstark,Nstark); E1n = zeros(1,Nstark); E1p = zeros(1,Nstark); Enstark = zeros(numsol,Nstark); Epstark = zeros(numsol,Nstark); psinstark=zeros(numsol,N); psipstark=zeros(numsol,N); for ii = 1:Nstark vstark = (ii-1)*E0*(x-length/2)/length; vnstark = vn+vstark; vpstark = vp+vstark; [Enstarktemp,psinstark] = solve_sch(length,N,vnstark,me,numsol); [Epstarktemp,psipstark] = solve_sch(length,N,vpstark,me,numsol); diagEn = diag(En).'; diagEp = diag(Ep).'; diagEnstark = diag(Enstarktemp).'; diagEpstark = diag(Epstarktemp).'; E1n(ii) = diagEn(2); E1p(ii) = diagEp(1); Enstark(:,ii) = diagEnstark; Epstark(:,ii) = diagEpstark; end %************************************************************************** %**************** plot results ******************************************** %************************************************************************** disp(diag(En/echarge)) disp(diag(Ep/echarge)) X = ['E0 = ',num2str(E0/echarge),' eV'];disp(X) ttl=['\rmm_{eff} = ',num2str(meff),'\timesm_0, L = ',... num2str(length*1e9),' nm, E = 0 V/m']; ttl2=['\rm Fig2.6 m_{eff} = ',num2str(meff),'\timesm_0, L = ',... num2str(length*1e9),' nm, E_0 = ',num2str(E0/echarge),... ' eV, \Delta_{max} = ',num2str(Nstark*E0/echarge),' eV']; % % figure(1) % plot(1e9*x(2:end-1),psin,'linewidth',LW) % grid on % title(ttl); % xlabel('Position, $x$ (nm)','FontSize',14,'Interpreter','latex') % ylabel('$\psi^{-}$','FontSize',14,'Interpreter','latex') % % figure(2) % plot(1e9*x(2:end-1),psip,'linewidth',LW) % grid on % title(ttl); % xlabel('Position, $x$ (nm)','FontSize',14,'Interpreter','latex') % ylabel('$\psi^{+}$','FontSize',14,'Interpreter','latex') % % figure(3) % plot(1e9*x,vn/echarge,'linewidth',LW) % grid on % hold on % plot(1e9*x,(vn+vstark)/echarge,'r','linewidth',LW) % title(ttl2); % xlabel('Position, $x$ (nm)','FontSize',14,'Interpreter','latex') % ylabel('Potential energy, $V^{-} (eV)$','FontSize',14,'Interpreter','latex') % ylim([-Nstark*E0/echarge 2*Nstark*E0/echarge]) % hold off % % figure(4) % plot(1e9*x,vp/echarge,'linewidth',LW) % grid on % hold on % plot(1e9*x,(vp+vstark)/echarge,'r','linewidth',LW) % title(ttl2); % xlabel('Position, $x$ (nm)','FontSize',14,'Interpreter','latex') % ylabel('Potential energy, $V^{(+)} (eV)$','FontSize',14,'Interpreter','latex') % ylim([-Nstark*E0/echarge 2*Nstark*E0/echarge]) % hold off % % figure(5) % plot(1e9*x(2:end-1),psinstark,':','linewidth',LW) % grid on % title(ttl2); % xlabel('Position, $x$ (nm)','FontSize',14,'Interpreter','latex') % ylabel('$\psi^{-}$','FontSize',14,'Interpreter','latex') % % figure(6) % plot(1e9*x(2:end-1),psipstark,':','linewidth',LW) % grid on % title(ttl2); % xlabel('Position, $x$ (nm)','FontSize',14,'Interpreter','latex') % ylabel('$\psi^{+}$','FontSize',14,'Interpreter','latex') xplot=Astark*E0/echarge;%setup x-axis as potential energy perturbation, W figure(1) plot(xplot,Enstark/echarge,xplot,Epstark(1:3,:)/echarge,'--','linewidth',LW); grid on title(ttl2); xlabel('$W$ ($e|\textbf{E}|L$) (eV)','FontSize',14,'Interpreter','latex') ylabel('Eigenenergy, $E$ (eV)','FontSize',14,'Interpreter','latex') xlim([0 max(xplot)]) % legend('- stark','+ stark'); % figure(8) % plot(xplot,Enstark(2,:)/echarge,xplot,Epstark(1,:)/echarge,'linewidth',LW); % grid on % title(ttl2); % xlabel('$W$ ($e|\textbf{E}|L$) (eV)','FontSize',14,'Interpreter','latex') % ylabel('Eigenenergy, $E$ (eV)','FontSize',14,'Interpreter','latex') % legend('- stark','+ stark'); % xlim([0 max(xplot)]) % % figure(9) % plot(xplot,Enstark(3,:)/echarge,xplot,Epstark(2,:)/echarge,'linewidth',LW); % grid on % title(ttl2); % xlabel('$W$ ($e|\textbf{E}|L$) (eV)','FontSize',14,'Interpreter','latex') % ylabel('Eigenenergy, $E$ (eV)','FontSize',14,'Interpreter','latex') % legend('- stark','+ stark'); % xlim([0 max(xplot)]) % % figure(7) % plot(xplot,4*del2(Enstark/echarge),xplot,4*del2(Epstark/echarge),'--','linewidth',LW); % grid on % title(ttl2); % xlabel('$W$ ($e|\textbf{E}|L$) (eV)','FontSize',14,'Interpreter','latex') % ylabel('4*del2, $E^{(-)}$ solid line, $E^{(+)}$ dashed line','FontSize',14,'Interpreter','latex') % xlim([0 max(xplot)]) % %% function [E,psi] = solve_sch(length,N,v,me,numsol) hbar=1.054571596e-34; echarge=1.602176462e-19; dx = length/(N-1); H = zeros(N-2,N-2); const = hbar^2/(me*dx^2); for ii = 1:N-2 H(ii,ii) = const+v(ii+1); if(ii