%Fig7_10a.m %Essential Electron Transport for Device Physics %Zero temperature Lindhard dielectric funtion in GaAs with optical phonons; %inelastic scattering rate as function of electron %injection energy (meV) clear; clf; 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=['\fontname{MT Extra}h\fontname{Arial}']; Emax=300; %maximum energy of injected electron (meV) 300 n1=1.e18; %electron carrier concentration (cm^-3) n=n1*1.e6; %convert to m^-3 m0=9.1095e-31; %bare electron mass mass=0.07; %effective electron mass in conduction band wLO=36.3; %longitudinal optic phonon energy (meV) wTO=33.3; %transverse optic phonon energy (meV) einf=11.1; %high frequency dielectric constant eye=complex(0,1); %square root of minus one hb=1.05459e-34; %Plank constant (J s) echarge=1.60219e-19; %electron chanrge (C) a0=0.529177e-10; %Bohr radius (m) kf=(3*(pi^2)*n)^(1/3); %Fermi wave vector (m^-1) kf1=kf*1e-2; %Fermi wave vector (cm^-1) Ef=((hb*kf)^2)/(2*m0*mass); %Fermi energy (eV) Ef=(Ef*1e3)/echarge; %Fermi energy (meV) wLO2=(wLO/Ef)^2; wTO2=(wTO/Ef)^2; zeta=mass/(pi*kf*a0); %Lindhard function prefactor gamma=0.11; %Energy broadening (meV) 0.52meV=0.01Ef when n1=1e18 % 0.11meV=0.0Ef when n1=1e17 gamma1=(eye*gamma)/Ef; %normalized broadening %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %setup y1=hw/Ef and x1=q/kf arrays dE=0.15; %energy increment (meV) 0.35 npoints=int16(Emax/dE); %number of integration points dx1=2*(sqrt(Emax/Ef))/(double(npoints)); %increment in x1 dy1=dE/Ef; %increment in y1 x1=[dx1:dx1:(double(npoints))*dx1]; y1=[dy1:dy1:(double(npoints))*dy1]+gamma1; f1=zeros(length(y1),npoints); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% for i1=1:npoints x=x1(i1); zeta1=zeta/x^3; a4=2*x; for j=1:length(y1); y=y1(j); a1=(1-1/4*(x-y/x).^2).*log((y-x*(x+2))./(y-x*(x-2))); a2=(1-1/4*(x+y/x).^2).*log((-y-x*(x+2))./(-y-x*(x-2))); a3=zeta1*(a4+a1+a2); %epsi=((y./(y-i*gamma)).*a3)+einf; %Electrons only and fix a3 %epsi=a3+einf; %Electrons only %epsi=a3+(einf*((y.^2-wLO2)./(y.^2-wTO2))); %Electrons+LO epsi=((y./(y-gamma1)).*a3)+(einf*((y.^2-wLO2)./(y.^2-wTO2))); %Electrons+LO and fix a3 f1(j,i1)=-imag(1./epsi); %loss function f1(j,i1)=f1(j,i1)/x1(i1); %loss function weighted by 1/q end end %truncated parabola of integration parameters Etruncate=Emax-Ef; %truncation energy (meV) Emax1=Emax/Ef; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % set up injection energy array %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Emaxtmp=Emax1; nmax=100; %plot 100 values of electron injection enegy Integral =zeros(1,nmax); xplot =zeros(1,npoints); yplot =zeros(1,npoints); energyplot=zeros(1,nmax); efactor=Emax/Ef; for jj=1:1:nmax; energyplot(jj)=(double(jj)*Ef*efactor)/(nmax); Emax1=Emaxtmp*(double(jj))/nmax; Etruncate1=Emax1-1; %truncation for i=1:npoints; xplot(i)=x1(i); yplot(i)=Emax1-((x1(i)-sqrt(Emax1))^2); %parabola of integration if yplot(i) > Etruncate1 %tuncate parabola yplot(i)=Etruncate1; %truncate parabola end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %set up meshgrid %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [X1,Y1]=meshgrid(x1,y1); sumindicator=Y1<=ones(length(y1),1)*yplot; %find valaues of sum indicator for each column Integral(jj)=sum(sum(f1(sumindicator)))/(sqrt(Emax1)); %this is the integral for current Emax end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% prefactor=(2.0*mass*dx1*dy1*Ef*0.001*echarge)/(pi*a0*kf*hb); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure(1); plot(energyplot,Integral*prefactor/1e12); xlabel('Energy, \itE_k\rm (meV)'); ylabel('Inelastic scattering rate, 1/ \tau_{in} (ps^{-1})'); ttl2=['\rmFig7.10a, GaAs, \itT\rm=0 K, \itm\rm^*_e=',num2str(mass),... '\times\itm\rm_0, \itn\rm_0=',... num2str(n1/1e18),'\times10^{18} cm^{-3}, \itE\rm_F=',num2str(Ef,'%5.1f'),... ' meV, \gamma=',num2str(gamma),' meV, \itE\rm_{max}=',... num2str(Emax1*Ef),' meV']; title(ttl2); return