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Diffstat (limited to 'figs')
-rw-r--r-- | figs/Millikan-Lauritsen_current.png | bin | 0 -> 29963 bytes | |||
-rw-r--r-- | figs/animation/FN_base.jl | 170 | ||||
-rw-r--r-- | figs/animation/Makefile | 13 | ||||
-rw-r--r-- | figs/animation/animate-positive.py | 130 | ||||
-rw-r--r-- | figs/animation/animate.py | 128 | ||||
-rw-r--r-- | figs/animation/animate_compute.jl | 66 | ||||
-rw-r--r-- | figs/contour.fig/Makefile | 30 | ||||
-rw-r--r-- | figs/contour.fig/contour.py | 82 | ||||
-rw-r--r-- | figs/contour.fig/contour.tikz.tex | 34 | ||||
-rw-r--r-- | figs/emitter.jpg | bin | 0 -> 14324 bytes | |||
-rw-r--r-- | figs/fowler-nordheim.fig/FN_base.jl | 170 | ||||
-rw-r--r-- | figs/fowler-nordheim.fig/Makefile | 29 | ||||
-rw-r--r-- | figs/fowler-nordheim.fig/asymptotic.gnuplot | 59 | ||||
-rw-r--r-- | figs/fowler-nordheim.fig/asymptotic.jl | 46 | ||||
-rw-r--r-- | figs/libs/Makefile | 28 | ||||
l--------- | figs/potential.fig/Makefile | 1 | ||||
-rw-r--r-- | figs/potential.fig/potential.tikz.tex | 17 | ||||
-rw-r--r-- | figs/potential.fig/potential_square.tikz.tex | 17 | ||||
-rw-r--r-- | figs/potential.fig/potential_square_photonic.tikz.tex | 22 | ||||
-rw-r--r-- | figs/potential.fig/potential_square_thermal.tikz.tex | 17 |
20 files changed, 1059 insertions, 0 deletions
diff --git a/figs/Millikan-Lauritsen_current.png b/figs/Millikan-Lauritsen_current.png Binary files differnew file mode 100644 index 0000000..5645be8 --- /dev/null +++ b/figs/Millikan-Lauritsen_current.png diff --git a/figs/animation/FN_base.jl b/figs/animation/FN_base.jl new file mode 100644 index 0000000..af2a1ee --- /dev/null +++ b/figs/animation/FN_base.jl @@ -0,0 +1,170 @@ +# fractional power with an arbitrary branch cut +function pow(x,a,cut) + if(angle(x)/cut<=1) + return(abs(x)^a*exp(1im*angle(x)*a)) + else + return(abs(x)^a*exp(1im*(angle(x)-sign(cut)*2*pi)*a)) + end +end + +# asymptotic airy functions +# specify a branch cut for the fractional power +function airyai_asym(x,cut) + if(abs(real(pow(x,3/2,cut)))<airy_threshold) + return(exp(2/3*pow(x,3/2,cut))*airyai(x)) + else + ret=0 + for n in 0:airy_order + ret+=gamma(n+5/6)*gamma(n+1/6)*(-3/4)^n/(4*pi^(3/2)*factorial(n)*pow(x,3*n/2+1/4,cut)) + end + return ret + end +end +function airyaiprime_asym(x,cut) + if(abs(real(pow(x,3/2,cut)))<airy_threshold) + return(exp(2/3*pow(x,3/2,cut))*airyaiprime(x)) + else + ret=0 + for n in 0:airy_order + ret+=gamma(n+5/6)*gamma(n+1/6)*(-3/4)^n/(4*pi^(3/2)*factorial(n))*(-1/pow(x,3*n/2-1/4,cut)-(3/2*n+1/4)/pow(x,3*n/2+5/4,cut)) + end + return ret + end +end + +# solutions of (-\Delta+V-ip)phi=0 +# assume that p has an infinitesimal real part (and adjust the branch cuts appropriately) +function phi(p,x,E,V) + return(airyai_asym(exp(-1im*pi/3)*(E^(1/3)*x-E^(-2/3)*(V-1im*p)),pi)) +end +function dphi(p,x,E,V) + return(exp(-1im*pi/3)*E^(1/3)*airyaiprime_asym(exp(-1im*pi/3)*(E^(1/3)*x-E^(-2/3)*(V-1im*p)),pi)) +end +function eta(p,x,E,V) + return(exp(-1im*pi/3)*airyai_asym(-(E^(1/3)*x-E^(-2/3)*(V-1im*p)),pi/2)) +end +function deta(p,x,E,V) + return(-exp(-1im*pi/3)*E^(1/3)*airyaiprime_asym(-(E^(1/3)*x-E^(-2/3)*(V-1im*p)),pi/2)) +end + +# Laplace transform of psi +# assume that p has an infinitesimal real part (and adjust the branch cuts appropriately) +# for example, (1im*p-V)^(3/2) becomes pow(1im*p-V,3/2,-pi/2) because when 1im*p is real negative, its square root should be imaginary positive +function f(p,x,k0,E,V) + T=2im*k0/(1im*k0-sqrt(V-k0*k0)) + R=T-1 + + if x>=0 + C2=-1im*T/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))*((sqrt(V-k0*k0)+pow(-1im*p,1/2,pi/2))/(-1im*p+k0*k0)-2im*E^(-1/3)*pi*quadgk(y -> (pow(-1im*p,1/2,pi/2)*eta(p,0,E,V)-deta(p,0,E,V))*phi(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2))),0,Inf)[1]) + FT=2*E^(-1/3)*pi*(quadgk(y -> phi(p,x,E,V)*eta(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2)-pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2))),0,x)[1]+quadgk(y -> eta(p,x,E,V)*phi(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2))),x,Inf)[1]) + main=C2*phi(p,x,E,V)*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2)))+T*FT + + # subtract the contribution of the pole, which will be added back in after the integration + pole=psi_pole(x,k0,E,V)/(p+1im*k0*k0) + return(main-pole) + else + C1=-1im*T*((sqrt(V-k0*k0)*phi(p,0,E,V)+dphi(p,0,E,V))/(-1im*p+k0*k0)/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))+E^(-1/3)*quadgk(y -> phi(p,y,E,V)/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2))),0,Inf)[1]) + FI=-1im*exp(1im*k0*x)/(-1im*p+k0*k0) + FR=-1im*exp(-1im*k0*x)/(-1im*p+k0*k0) + main=C1*exp(pow(-1im*p,1/2,pi/2)*x)+FI+R*FR + + # subtract the contribution of the pole, which will be added back in after the integration + pole=psi_pole(x,k0,E,V)/(p+1im*k0*k0) + return(main-pole) + end +end +# its derivative +function df(p,x,k0,E,V) + T=2im*k0/(1im*k0-sqrt(V-k0*k0)) + R=T-1 + + if x>=0 + C2=-1im*T/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))*((sqrt(V-k0*k0)+pow(-1im*p,1/2,pi/2))/(-1im*p+k0*k0)-2im*E^(-1/3)*pi*quadgk(y -> (pow(-1im*p,1/2,pi/2)*eta(p,0,E,V)-deta(p,0,E,V))*phi(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2))),0,Inf)[1]) + dFT=2*E^(-1/3)*pi*(quadgk(y -> dphi(p,x,E,V)*eta(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2)-pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2))),0,x)[1]+quadgk(y -> deta(p,x,E,V)*phi(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2))),x,Inf)[1]) + main=C2*dphi(p,x,E,V)*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2)))+T*dFT + + # subtract the contribution of the pole, which will be added back in after the integration + pole=dpsi_pole(x,k0,E,V)/(p+1im*k0*k0) + return(main-pole) + else + C1=-1im*T*((sqrt(V-k0*k0)*phi(p,0,E,V)+dphi(p,0,E,V))/(-1im*p+k0*k0)/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))+E^(-1/3)*quadgk(y -> phi(p,y,E,V)/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2))),0,Inf)[1]) + dFI=k0*exp(1im*k0*x)/(-1im*p+k0*k0) + dFR=-k0*exp(-1im*k0*x)/(-1im*p+k0*k0) + main=C1*pow(-1im*p,1/2,pi/2)*exp(pow(-1im*p,1/2,pi/2)*x)+dFI+R*dFR + + # subtract the contribution of the pole, which will be added back in after the integration + pole=dpsi_pole(x,k0,E,V)/(p+1im*k0*k0) + return(main-pole) + end +end + +# psi (returns t,psi(x,t)) +function psi(x,k0,E,V,p_npoints,p_cutoff) + fft=fourier_fft(f,x,k0,E,V,p_npoints,p_cutoff) + # add the contribution of the pole + for i in 1:p_npoints + fft[2][i]=fft[2][i]+psi_pole(x,k0,E,V)*exp(-1im*k0*k0*fft[1][i]) + end + return(fft) +end +# its derivative +function dpsi(x,k0,E,V,p_npoints,p_cutoff) + fft=fourier_fft(df,x,k0,E,V,p_npoints,p_cutoff) + # add the contribution of the pole + for i in 1:p_npoints + fft[2][i]=fft[2][i]+dpsi_pole(x,k0,E,V)*exp(-1im*k0*k0*fft[1][i]) + end + return(fft) +end + +# compute Fourier transform by sampling and fft +function fourier_fft(A,x,k0,E,V,p_npoints,p_cutoff) + fun=zeros(Complex{Float64},p_npoints) + times=zeros(p_npoints) + + # prepare fft + for i in 1:p_npoints + fun[i]=p_cutoff/pi*A(1im*(-p_cutoff+2*p_cutoff*(i-1)/p_npoints),x,k0,E,V) + times[i]=(i-1)*pi/p_cutoff + end + + ifft!(fun) + + # correct the phase + for i in 2:2:p_npoints + fun[i]=-fun[i] + end + return([times,fun]) +end + +# asymptotic value of psi +function psi_pole(x,k0,E,V) + if x>=0 + return(1im*phi(-1im*k0*k0,x,E,V)*2*k0/(1im*k0*phi(-1im*k0*k0,0,E,V)+dphi(-1im*k0*k0,0,E,V))*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(k0*k0-V),3/2,-pi/2)-E^(-1)*pow(k0*k0-V,3/2,-pi/2)))) + else + return((1im*k0*phi(-1im*k0*k0,0,E,V)-dphi(-1im*k0*k0,0,E,V))/(1im*k0*phi(-1im*k0*k0,0,E,V)+dphi(-1im*k0*k0,0,E,V))*exp(-1im*k0*x)+exp(1im*k0*x)) + end +end +function dpsi_pole(x,k0,E,V) + if x>=0 + return(1im*dphi(-1im*k0*k0,x,E,V)*2*k0/(1im*k0*phi(-1im*k0*k0,0,E,V)+dphi(-1im*k0*k0,0,E,V))*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(k0*k0-V),3/2,-pi/2)-E^(-1)*pow(k0*k0-V,3/2,-pi/2)))) + else + return(-1im*k0*(1im*k0*phi(-1im*k0*k0,0,E,V)-dphi(-1im*k0*k0,0,E,V))/(1im*k0*phi(-1im*k0*k0,0,E,V)+dphi(-1im*k0*k0,0,E,V))*exp(-1im*k0*x)+1im*k0*exp(1im*k0*x)) + end +end + +# current +function J(ps,dps) + return(2*imag(conj(ps)*dps)) +end + +# complete computation of the current +function current(x,k0,E,V,p_npoints,p_cutoff) + ps=psi(x,k0,E,V,p_npoints,p_cutoff) + dps=dpsi(x,k0,E,V,p_npoints,p_cutoff) + Js=zeros(Complex{Float64},p_npoints) + for i in 1:p_npoints + Js[i]=J(ps[2][i],dps[2][i]) + end + return(Js) +end diff --git a/figs/animation/Makefile b/figs/animation/Makefile new file mode 100644 index 0000000..19696f8 --- /dev/null +++ b/figs/animation/Makefile @@ -0,0 +1,13 @@ +PROJECTNAME=animate + +all: animate.dat + +run: animate.dat + python3 animate.py animate.dat & + python3 animate-positive.py animate.dat & + +animate.dat: + julia animate_compute.jl > animate.dat + +clean: + rm -f animate.dat diff --git a/figs/animation/animate-positive.py b/figs/animation/animate-positive.py new file mode 100644 index 0000000..3a78f66 --- /dev/null +++ b/figs/animation/animate-positive.py @@ -0,0 +1,130 @@ +from matplotlib import pyplot as pl +from matplotlib import animation +import sys + +# read data +# time dependent data +frames=[] +# asymptotic data (located in the first block) +asym=[] +infile=open(sys.argv[1],'r') +row=[] +for line in infile: + # read first block + if len(asym)==0: + if line=='\n': + asym=row + row=[] + else: + dat=[] + for n in line.split(): + dat.append(float(n)) + if(dat[0]>0): + row.append(dat) + # read other blocks + else: + if line=='\n': + frames.append(row) + row=[] + else: + dat=[] + for n in line.split(): + dat.append(float(n)) + if(dat[1]>0): + row.append(dat) +infile.close() + + +# set up plot +fig = pl.figure() +pl.subplot(211) +axr=fig.gca() +asym_rho, = axr.plot([],[],linewidth=3.5,color='#00FF00') +rho, = axr.plot([],[],color='red') + +pl.subplot(212) +axJ=fig.gca() +asym_J, = axJ.plot([],[],linewidth=3.5,color='#00FF00') +J, = axJ.plot([],[],color='red') + +# plot ranges +xmax=0 +maxyr=0 +maxyJ=0 +for frame in frames: + for i in range(len(frame)): + if frame[i][1]>xmax: + xmax=frame[i][1] + if frame[i][2]>maxyr: + maxyr=frame[i][2] + if frame[i][3]>maxyJ: + maxyJ=frame[i][3] +for i in range(len(asym)): + if asym[i][0]>xmax: + xmax=asym[i][0] + if asym[i][1]>maxyr: + maxyr=asym[i][1] + if asym[i][2]>maxyJ: + maxyJ=asym[i][2] +xmin=0 +minyr=0 +minyJ=0 +for frame in frames: + for i in range(len(frame)): + if frame[i][1]<xmin: + xmin=frame[i][1] + if frame[i][2]<minyr: + minyr=frame[i][2] + if frame[i][3]<minyJ: + minyJ=frame[i][3] +for i in range(len(asym)): + if asym[i][0]<xmin: + xmin=asym[i][0] + if asym[i][1]<minyr: + minyr=asym[i][1] + if asym[i][2]<minyJ: + minyJ=asym[i][2] + + +# plot asymptotes +asym_rho_datax=[] +asym_rho_datay=[] +for i in range(len(asym)): + asym_rho_datax.append(asym[i][0]) + asym_rho_datay.append(asym[i][1]) +asym_rho.set_data(asym_rho_datax,asym_rho_datay) +asym_J_datax=[] +asym_J_datay=[] +for i in range(len(asym)): + asym_J_datax.append(asym[i][0]) + asym_J_datay.append(asym[i][2]) +asym_J.set_data(asym_J_datax,asym_J_datay) + +# animate +def init_plot(): + axr.set_ylim(minyr,maxyr) + axr.set_xlim(xmin,xmax) + axJ.set_ylim(minyJ,maxyJ) + axJ.set_xlim(xmin,xmax) + + axr.vlines(0,minyr,maxyr,linestyles="dotted") + axJ.vlines(0,minyJ,maxyJ,linestyles="dotted") + return rho,J +def update(frame): + axr.set_title("t=% .3f fs" % (frame[0][0])) + xdata=[] + ydata=[] + for i in range(len(frame)): + xdata.append(frame[i][1]) + ydata.append(frame[i][2]) + rho.set_data(xdata,ydata) + + xdata=[] + ydata=[] + for i in range(len(frame)): + xdata.append(frame[i][1]) + ydata.append(frame[i][3]) + J.set_data(xdata,ydata) + return rho,J +anim = animation.FuncAnimation(fig, update, frames=frames, blit=False, interval=100, repeat=True, init_func=init_plot) +pl.show() diff --git a/figs/animation/animate.py b/figs/animation/animate.py new file mode 100644 index 0000000..6203fb7 --- /dev/null +++ b/figs/animation/animate.py @@ -0,0 +1,128 @@ +from matplotlib import pyplot as pl +from matplotlib import animation +import sys + +# read data +# time dependent data +frames=[] +# asymptotic data (located in the first block) +asym=[] +infile=open(sys.argv[1],'r') +row=[] +for line in infile: + # read first block + if len(asym)==0: + if line=='\n': + asym=row + row=[] + else: + dat=[] + for n in line.split(): + dat.append(float(n)) + row.append(dat) + # read other blocks + else: + if line=='\n': + frames.append(row) + row=[] + else: + dat=[] + for n in line.split(): + dat.append(float(n)) + row.append(dat) +infile.close() + + +# set up plot +fig = pl.figure() +pl.subplot(211) +axr=fig.gca() +asym_rho, = axr.plot([],[],linewidth=3.5,color='#00FF00') +rho, = axr.plot([],[],color='red') + +pl.subplot(212) +axJ=fig.gca() +asym_J, = axJ.plot([],[],linewidth=3.5,color='#00FF00') +J, = axJ.plot([],[],color='red') + +# plot ranges +xmax=0 +maxyr=0 +maxyJ=0 +for frame in frames: + for i in range(len(frame)): + if frame[i][1]>xmax: + xmax=frame[i][1] + if frame[i][2]>maxyr: + maxyr=frame[i][2] + if frame[i][3]>maxyJ: + maxyJ=frame[i][3] +for i in range(len(asym)): + if asym[i][0]>xmax: + xmax=asym[i][0] + if asym[i][1]>maxyr: + maxyr=asym[i][1] + if asym[i][2]>maxyJ: + maxyJ=asym[i][2] +xmin=0 +minyr=0 +minyJ=0 +for frame in frames: + for i in range(len(frame)): + if frame[i][1]<xmin: + xmin=frame[i][1] + if frame[i][2]<minyr: + minyr=frame[i][2] + if frame[i][3]<minyJ: + minyJ=frame[i][3] +for i in range(len(asym)): + if asym[i][0]<xmin: + xmin=asym[i][0] + if asym[i][1]<minyr: + minyr=asym[i][1] + if asym[i][2]<minyJ: + minyJ=asym[i][2] + + +# plot asymptotes +asym_rho_datax=[] +asym_rho_datay=[] +for i in range(len(asym)): + asym_rho_datax.append(asym[i][0]) + asym_rho_datay.append(asym[i][1]) +asym_rho.set_data(asym_rho_datax,asym_rho_datay) +asym_J_datax=[] +asym_J_datay=[] +for i in range(len(asym)): + asym_J_datax.append(asym[i][0]) + asym_J_datay.append(asym[i][2]) +asym_J.set_data(asym_J_datax,asym_J_datay) + +# animate +def init_plot(): + axr.set_ylim(minyr,maxyr) + axr.set_xlim(xmin,xmax) + axJ.set_ylim(minyJ,maxyJ) + axJ.set_xlim(xmin,xmax) + + axr.vlines(0,minyr,maxyr,linestyles="dotted") + axJ.vlines(0,minyJ,maxyJ,linestyles="dotted") + return rho,J +def update(frame): + axr.set_title("t=% .3f fs" % (frame[0][0])) + xdata=[] + ydata=[] + for i in range(len(frame)): + xdata.append(frame[i][1]) + ydata.append(frame[i][2]) + rho.set_data(xdata,ydata) + + xdata=[] + ydata=[] + for i in range(len(frame)): + xdata.append(frame[i][1]) + ydata.append(frame[i][3]) + J.set_data(xdata,ydata) + return rho,J +anim = animation.FuncAnimation(fig, update, frames=frames, blit=False, interval=100, repeat=True, init_func=init_plot) +pl.show() diff --git a/figs/animation/animate_compute.jl b/figs/animation/animate_compute.jl new file mode 100644 index 0000000..452320e --- /dev/null +++ b/figs/animation/animate_compute.jl @@ -0,0 +1,66 @@ +using QuadGK +using SpecialFunctions +using FFTW + +# numerical values +hbar=6.58e-16 # eV.s +m=9.11e-31 # kg +Vn=9 # eV +#En=14e9 # V/m +En=10e9 # V/m +Kn=4.5 # eV + +V=1 +E=En*hbar/(2*Vn^1.5*m^0.5)*sqrt(1.60e-19) +k0=sqrt(Kn/Vn) + +# rescale x to nm +nm_scale=hbar*sqrt(1.6e-19)/sqrt(2*m*Vn)*1e9 + +# cutoffs +p_cutoff=20*k0 +p_npoints=256 + +# airy approximations +airy_threshold=30 +airy_order=5 + +# xbounds +xmax=10 +xmin=-10 +x_npoints=200 + +include("FN_base.jl") + +# compute wave function +ps=Array{Array{Array{Complex{Float64},1},1}}(undef,x_npoints) +dps=Array{Array{Array{Complex{Float64},1},1}}(undef,x_npoints) +for i in 1:x_npoints + print(stderr,i,'/',x_npoints,'\n') + x=xmin+(xmax-xmin)*i/x_npoints + ps[i]=psi(x,k0,E,V,p_npoints,p_cutoff) + dps[i]=dpsi(x,k0,E,V,p_npoints,p_cutoff) +end + +# compute asymptotic values +ps_asym=Array{Complex{Float64}}(undef,x_npoints) +dps_asym=Array{Complex{Float64}}(undef,x_npoints) +for i in 1:x_npoints + x=xmin+(xmax-xmin)*i/x_npoints + ps_asym[i]=psi_pole(x,k0,E,V) + dps_asym[i]=dpsi_pole(x,k0,E,V) +end + +# print asymptotic values +for i in 1:x_npoints + print((xmin+(xmax-xmin)*i/x_npoints)*nm_scale,' ',abs(ps_asym[i])^2,' ',J(ps_asym[i],dps_asym[i])/(2*k0),'\n') +end +print('\n') + +# print values at each time +for j in 1:p_npoints + for i in 1:x_npoints + print(real(ps[i][1][j])*hbar/Vn*1e15,' ',(xmin+(xmax-xmin)*i/x_npoints)*nm_scale,' ',abs(ps[i][2][j])^2,' ',J(ps[i][2][j],dps[i][2][j])/(2*k0),'\n') + end + print('\n') +end diff --git a/figs/contour.fig/Makefile b/figs/contour.fig/Makefile new file mode 100644 index 0000000..bade7d0 --- /dev/null +++ b/figs/contour.fig/Makefile @@ -0,0 +1,30 @@ +PROJECTNAME=contour + +PDFS=$(addsuffix .pdf, $(PROJECTNAME)) + +all: $(PDFS) + +$(PDFS): poles.tikz.tex + pdflatex -jobname $(basename $@) -file-line-error $(patsubst %.pdf, %.tikz.tex, $@) + +poles.tikz.tex: + python3 contour.py > poles.tikz.tex + +install: $(PDFS) + cp $^ $(INSTALLDIR)/ + +$(LIBS): + ln -fs libs/$@ ./ + +clean-libs: + rm -f $(LIBS) + +clean-aux: + rm -f $(addsuffix .aux, $(PROJECTNAME)) + rm -f $(addsuffix .log, $(PROJECTNAME)) + rm -f poles.tikz.tex + +clean-tex: + rm -f $(PDFS) + +clean: clean-libs clean-aux clean-tex diff --git a/figs/contour.fig/contour.py b/figs/contour.fig/contour.py new file mode 100644 index 0000000..102df86 --- /dev/null +++ b/figs/contour.fig/contour.py @@ -0,0 +1,82 @@ +#!/usr/bin/env python3 + +import cmath +import math +import scipy.special as sp +from scipy import optimize as op +import random +import sys + +# number of roots +nr_roots=4 + +# size of plot +plotsize_x=3 +plotsize_y=3 +# rescale plot (so roots are not too close together) +plotsize_scale_x=1 +plotsize_scale_y=6 + +# numerical values +hbar=6.58e-16 # eV.s +m=9.11e-31 # kg +Vn=9 # eV +En=20e9 # V/m + +V=1 +E=En*hbar/(Vn**1.5*m**0.5)*math.sqrt(1.60e-19) + +# sqrt with branch cut along iR_+ +def sqrt_p(x): + r,p=cmath.polar(x) + if(p<cmath.pi/2): + return(cmath.rect(math.sqrt(r),p/2)) + else: + return(cmath.rect(math.sqrt(r),(p-2*math.pi)/2)) + +# solution of (-\Delta+V-ip)phi=0 +def phi(p,x,E,V): + return(sp.airy(cmath.exp(-1j*math.pi/3)*(E**(1/3)*x-E**(-2/3)*(V-1j*p)))[0]) +# its derivative +def dphi(p,x,E,V): + return(cmath.exp(-1j*math.pi/3)*E**(1/3)*sp.airy(cmath.exp(-1j*math.pi/3)*(E**(1/3)*x-E**(-2/3)*(V-1j*p)))[1]) + + +# the function whose roots are to be computed and its derivative +def f(p): + return(sqrt_p(-1j*p)*phi(p,0,E,V)-dphi(p,0,E,V)) +def df(p): + return(-1j/(2*sqrt_p(-1j*p))*phi(p,0,E,V)+sqrt_p(-1j*p)*dp_phi(p,0,E,V)-dp_dphi(p,0,E,V)) + +# derivatives of phi with respect to p +def dp_phi(p,x,E,V): + return(1j*cmath.exp(-1j*math.pi/3)*E**(-2/3)*sp.airy(cmath.exp(-1j*math.pi/3)*(E**(1/3)*x-E**(-2/3)*(V-1j*p)))[1]) +def dp_dphi(p,x,E,V): + return(-1j*(x-(V-1j*p)/E)*phi(p,x,E,V)) + +# check whether the root was already found +def check(root,roots): + # reject if the root doesn't fit on the plot + if(plotsize_scale_x*root.real<-plotsize_x or abs(plotsize_scale_y*root.imag)>plotsize_y): + return(False) + for x in roots: + if(abs(root-x)<1e-12): + return(False) + return(True) + +# find roots +roots=[] +while len(roots)<nr_roots: + try: + root=op.newton(f, -random.random()*plotsize_x/plotsize_scale_x+(2*random.random()-1)*plotsize_y/plotsize_scale_y*1j, fprime=df) + if(check(root,roots)): + roots.append(root) + print("found "+str(len(roots))+" roots of "+str(nr_roots), file=sys.stderr) + except RuntimeError: + root=0 + except: + break + +# print result +for root in roots: + print("\\pole{(% .3f,% .3f)}" % (plotsize_scale_x*root.real,plotsize_scale_y*root.imag)) diff --git a/figs/contour.fig/contour.tikz.tex b/figs/contour.fig/contour.tikz.tex new file mode 100644 index 0000000..953d8f0 --- /dev/null +++ b/figs/contour.fig/contour.tikz.tex @@ -0,0 +1,34 @@ +\documentclass{standalone} + +\usepackage{tikz} + +\def\pole#1{ + \draw#1++(0.1,0.1)--++(-0.2,-0.2); + \draw#1++(-0.1,0.1)--++(0.2,-0.2); + \draw[color=red, line width=1pt]#1circle(0.3); +} + +\begin{document} +\begin{tikzpicture} + +% branch cut +\fill[color=gray](-3.3,-0.1)--++(3.3,0)--++(0,0.2)--++(-3.3,0)--cycle; + +% axes +\draw(-3.3,0)--++(6.6,0); +\draw(0,-3.3)--++(0,6.6); + +% -ik0^2 +\pole{(0,-2)} + +% other poles +\input{poles.tikz} + +% contour +\draw[color=red, line width=1pt](-3.3,-0.3)--++(3.3,0); +\draw[color=red, line width=1pt](0,-0.3)..controls(0.3,-0.3)and(0.3,0.3)..(0,0.3); +\draw[color=red, line width=1pt](-3.3,0.3)--++(3.3,0); + + +\end{tikzpicture} +\end{document} diff --git a/figs/emitter.jpg b/figs/emitter.jpg Binary files differnew file mode 100644 index 0000000..d84ad1f --- /dev/null +++ b/figs/emitter.jpg diff --git a/figs/fowler-nordheim.fig/FN_base.jl b/figs/fowler-nordheim.fig/FN_base.jl new file mode 100644 index 0000000..af2a1ee --- /dev/null +++ b/figs/fowler-nordheim.fig/FN_base.jl @@ -0,0 +1,170 @@ +# fractional power with an arbitrary branch cut +function pow(x,a,cut) + if(angle(x)/cut<=1) + return(abs(x)^a*exp(1im*angle(x)*a)) + else + return(abs(x)^a*exp(1im*(angle(x)-sign(cut)*2*pi)*a)) + end +end + +# asymptotic airy functions +# specify a branch cut for the fractional power +function airyai_asym(x,cut) + if(abs(real(pow(x,3/2,cut)))<airy_threshold) + return(exp(2/3*pow(x,3/2,cut))*airyai(x)) + else + ret=0 + for n in 0:airy_order + ret+=gamma(n+5/6)*gamma(n+1/6)*(-3/4)^n/(4*pi^(3/2)*factorial(n)*pow(x,3*n/2+1/4,cut)) + end + return ret + end +end +function airyaiprime_asym(x,cut) + if(abs(real(pow(x,3/2,cut)))<airy_threshold) + return(exp(2/3*pow(x,3/2,cut))*airyaiprime(x)) + else + ret=0 + for n in 0:airy_order + ret+=gamma(n+5/6)*gamma(n+1/6)*(-3/4)^n/(4*pi^(3/2)*factorial(n))*(-1/pow(x,3*n/2-1/4,cut)-(3/2*n+1/4)/pow(x,3*n/2+5/4,cut)) + end + return ret + end +end + +# solutions of (-\Delta+V-ip)phi=0 +# assume that p has an infinitesimal real part (and adjust the branch cuts appropriately) +function phi(p,x,E,V) + return(airyai_asym(exp(-1im*pi/3)*(E^(1/3)*x-E^(-2/3)*(V-1im*p)),pi)) +end +function dphi(p,x,E,V) + return(exp(-1im*pi/3)*E^(1/3)*airyaiprime_asym(exp(-1im*pi/3)*(E^(1/3)*x-E^(-2/3)*(V-1im*p)),pi)) +end +function eta(p,x,E,V) + return(exp(-1im*pi/3)*airyai_asym(-(E^(1/3)*x-E^(-2/3)*(V-1im*p)),pi/2)) +end +function deta(p,x,E,V) + return(-exp(-1im*pi/3)*E^(1/3)*airyaiprime_asym(-(E^(1/3)*x-E^(-2/3)*(V-1im*p)),pi/2)) +end + +# Laplace transform of psi +# assume that p has an infinitesimal real part (and adjust the branch cuts appropriately) +# for example, (1im*p-V)^(3/2) becomes pow(1im*p-V,3/2,-pi/2) because when 1im*p is real negative, its square root should be imaginary positive +function f(p,x,k0,E,V) + T=2im*k0/(1im*k0-sqrt(V-k0*k0)) + R=T-1 + + if x>=0 + C2=-1im*T/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))*((sqrt(V-k0*k0)+pow(-1im*p,1/2,pi/2))/(-1im*p+k0*k0)-2im*E^(-1/3)*pi*quadgk(y -> (pow(-1im*p,1/2,pi/2)*eta(p,0,E,V)-deta(p,0,E,V))*phi(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2))),0,Inf)[1]) + FT=2*E^(-1/3)*pi*(quadgk(y -> phi(p,x,E,V)*eta(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2)-pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2))),0,x)[1]+quadgk(y -> eta(p,x,E,V)*phi(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2))),x,Inf)[1]) + main=C2*phi(p,x,E,V)*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2)))+T*FT + + # subtract the contribution of the pole, which will be added back in after the integration + pole=psi_pole(x,k0,E,V)/(p+1im*k0*k0) + return(main-pole) + else + C1=-1im*T*((sqrt(V-k0*k0)*phi(p,0,E,V)+dphi(p,0,E,V))/(-1im*p+k0*k0)/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))+E^(-1/3)*quadgk(y -> phi(p,y,E,V)/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2))),0,Inf)[1]) + FI=-1im*exp(1im*k0*x)/(-1im*p+k0*k0) + FR=-1im*exp(-1im*k0*x)/(-1im*p+k0*k0) + main=C1*exp(pow(-1im*p,1/2,pi/2)*x)+FI+R*FR + + # subtract the contribution of the pole, which will be added back in after the integration + pole=psi_pole(x,k0,E,V)/(p+1im*k0*k0) + return(main-pole) + end +end +# its derivative +function df(p,x,k0,E,V) + T=2im*k0/(1im*k0-sqrt(V-k0*k0)) + R=T-1 + + if x>=0 + C2=-1im*T/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))*((sqrt(V-k0*k0)+pow(-1im*p,1/2,pi/2))/(-1im*p+k0*k0)-2im*E^(-1/3)*pi*quadgk(y -> (pow(-1im*p,1/2,pi/2)*eta(p,0,E,V)-deta(p,0,E,V))*phi(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2))),0,Inf)[1]) + dFT=2*E^(-1/3)*pi*(quadgk(y -> dphi(p,x,E,V)*eta(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2)-pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2))),0,x)[1]+quadgk(y -> deta(p,x,E,V)*phi(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2))),x,Inf)[1]) + main=C2*dphi(p,x,E,V)*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2)))+T*dFT + + # subtract the contribution of the pole, which will be added back in after the integration + pole=dpsi_pole(x,k0,E,V)/(p+1im*k0*k0) + return(main-pole) + else + C1=-1im*T*((sqrt(V-k0*k0)*phi(p,0,E,V)+dphi(p,0,E,V))/(-1im*p+k0*k0)/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))+E^(-1/3)*quadgk(y -> phi(p,y,E,V)/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2))),0,Inf)[1]) + dFI=k0*exp(1im*k0*x)/(-1im*p+k0*k0) + dFR=-k0*exp(-1im*k0*x)/(-1im*p+k0*k0) + main=C1*pow(-1im*p,1/2,pi/2)*exp(pow(-1im*p,1/2,pi/2)*x)+dFI+R*dFR + + # subtract the contribution of the pole, which will be added back in after the integration + pole=dpsi_pole(x,k0,E,V)/(p+1im*k0*k0) + return(main-pole) + end +end + +# psi (returns t,psi(x,t)) +function psi(x,k0,E,V,p_npoints,p_cutoff) + fft=fourier_fft(f,x,k0,E,V,p_npoints,p_cutoff) + # add the contribution of the pole + for i in 1:p_npoints + fft[2][i]=fft[2][i]+psi_pole(x,k0,E,V)*exp(-1im*k0*k0*fft[1][i]) + end + return(fft) +end +# its derivative +function dpsi(x,k0,E,V,p_npoints,p_cutoff) + fft=fourier_fft(df,x,k0,E,V,p_npoints,p_cutoff) + # add the contribution of the pole + for i in 1:p_npoints + fft[2][i]=fft[2][i]+dpsi_pole(x,k0,E,V)*exp(-1im*k0*k0*fft[1][i]) + end + return(fft) +end + +# compute Fourier transform by sampling and fft +function fourier_fft(A,x,k0,E,V,p_npoints,p_cutoff) + fun=zeros(Complex{Float64},p_npoints) + times=zeros(p_npoints) + + # prepare fft + for i in 1:p_npoints + fun[i]=p_cutoff/pi*A(1im*(-p_cutoff+2*p_cutoff*(i-1)/p_npoints),x,k0,E,V) + times[i]=(i-1)*pi/p_cutoff + end + + ifft!(fun) + + # correct the phase + for i in 2:2:p_npoints + fun[i]=-fun[i] + end + return([times,fun]) +end + +# asymptotic value of psi +function psi_pole(x,k0,E,V) + if x>=0 + return(1im*phi(-1im*k0*k0,x,E,V)*2*k0/(1im*k0*phi(-1im*k0*k0,0,E,V)+dphi(-1im*k0*k0,0,E,V))*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(k0*k0-V),3/2,-pi/2)-E^(-1)*pow(k0*k0-V,3/2,-pi/2)))) + else + return((1im*k0*phi(-1im*k0*k0,0,E,V)-dphi(-1im*k0*k0,0,E,V))/(1im*k0*phi(-1im*k0*k0,0,E,V)+dphi(-1im*k0*k0,0,E,V))*exp(-1im*k0*x)+exp(1im*k0*x)) + end +end +function dpsi_pole(x,k0,E,V) + if x>=0 + return(1im*dphi(-1im*k0*k0,x,E,V)*2*k0/(1im*k0*phi(-1im*k0*k0,0,E,V)+dphi(-1im*k0*k0,0,E,V))*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(k0*k0-V),3/2,-pi/2)-E^(-1)*pow(k0*k0-V,3/2,-pi/2)))) + else + return(-1im*k0*(1im*k0*phi(-1im*k0*k0,0,E,V)-dphi(-1im*k0*k0,0,E,V))/(1im*k0*phi(-1im*k0*k0,0,E,V)+dphi(-1im*k0*k0,0,E,V))*exp(-1im*k0*x)+1im*k0*exp(1im*k0*x)) + end +end + +# current +function J(ps,dps) + return(2*imag(conj(ps)*dps)) +end + +# complete computation of the current +function current(x,k0,E,V,p_npoints,p_cutoff) + ps=psi(x,k0,E,V,p_npoints,p_cutoff) + dps=dpsi(x,k0,E,V,p_npoints,p_cutoff) + Js=zeros(Complex{Float64},p_npoints) + for i in 1:p_npoints + Js[i]=J(ps[2][i],dps[2][i]) + end + return(Js) +end diff --git a/figs/fowler-nordheim.fig/Makefile b/figs/fowler-nordheim.fig/Makefile new file mode 100644 index 0000000..a6e5e53 --- /dev/null +++ b/figs/fowler-nordheim.fig/Makefile @@ -0,0 +1,29 @@ +PROJECTNAME=asymptotic + +PDFS=$(addsuffix .pdf, $(PROJECTNAME)) +TEXS=$(addsuffix .tikz.tex, $(PROJECTNAME)) + +all: $(PDFS) + +$(PDFS): $(addsuffix .dat, $(PROJECTNAME)) + gnuplot $(patsubst %.pdf, %.gnuplot, $@) > $(patsubst %.pdf, %.tikz.tex, $@) + pdflatex -jobname $(basename $@) -file-line-error $(patsubst %.pdf, %.tikz.tex, $@) + +asymptotic.dat: + julia asymptotic.jl > asymptotic.dat + +install: $(PDFS) + cp $^ $(INSTALLDIR)/ + +clean-aux: + rm -f $(addsuffix .aux, $(PROJECTNAME)) + rm -f $(addsuffix .log, $(PROJECTNAME)) + +clean-dat: + rm -f $(addsuffix .tikz.tex, $(PROJECTNAME)) + rm -f short-time.dat + +clean-tex: + rm -f $(PDFS) + +clean: clean-dat clean-aux clean-tex diff --git a/figs/fowler-nordheim.fig/asymptotic.gnuplot b/figs/fowler-nordheim.fig/asymptotic.gnuplot new file mode 100644 index 0000000..3296380 --- /dev/null +++ b/figs/fowler-nordheim.fig/asymptotic.gnuplot @@ -0,0 +1,59 @@ +datafile="asymptotic.dat" + +## can also set the following options +#set title "" +set ylabel "$|\\psi_{\\mathrm{FN}}|^2$" tc ls 1 #norotate +set y2label "$J_{\\mathrm{FN}}$" tc ls 2 #norotate +set xlabel "$x$" +# +#set xrange[:] +#set yrange [:] +set y2range [0:0.004] +# +## start ticks at 0, then every x +#set xtics 0,x +#set ytics 0,x +## puts 4 minor tics between tics (5 intervals, i.e. every 0.01) +set mxtics 5 +set mytics 5 +set my2tics 5 + +# default output canvas size: 12.5cm x 8.75cm +set term lua tikz size 12.5,8.75 standalone +# run +## gnuplot gnuplots && gnuplot_tikz out/latext/minimizer.tex + +set key off + +# 3=1+2 draw bottom and left sides of the box +#set border 3 +# don't show tics on opposite sides +set xtics nomirror +set ytics nomirror tc ls 1 +set y2tics nomirror tc ls 2 + +# Mathematica colors: +## 3f3d99 (dark blue) +## 9c4275 (dark pink) +## 9a8d3f (dark yellow) +## 3d9956 (dark green) +# My colors +## 4169E1 (pastel blue) +## DC143C (bright red) +## 32CD32 (bright green) +## 4B0082 (deep purple) +## DAA520 (ochre) + +# set linestyle +set style line 1 linetype rgbcolor "#4169E1" linewidth 3 +set style line 2 linetype rgbcolor "#DC143C" linewidth 3 +set style line 3 linetype rgbcolor "#32CD32" linewidth 3 +set style line 4 linetype rgbcolor "#4B0082" linewidth 3 +set style line 5 linetype rgbcolor "#DAA520" linewidth 3 + +set pointsize 0.6 + +set arrow to 0, graph 1 nohead lt 0 + +plot datafile using 1:2 with lines linestyle 1 ,\ + datafile using 1:3 with lines linestyle 2 axes x1y2 diff --git a/figs/fowler-nordheim.fig/asymptotic.jl b/figs/fowler-nordheim.fig/asymptotic.jl new file mode 100644 index 0000000..fd1d492 --- /dev/null +++ b/figs/fowler-nordheim.fig/asymptotic.jl @@ -0,0 +1,46 @@ +using QuadGK +using SpecialFunctions +using FFTW + +# numerical values +hbar=6.58e-16 # eV.s +m=9.11e-31 # kg +Vn=9 # eV +En=14e9 # V/m +Kn=4.5 # eV + +V=1 +E=En*hbar/(2*Vn^1.5*m^0.5)*sqrt(1.60e-19) +k0=sqrt(Kn/Vn) + +# rescale x to nm +nm_scale=sqrt(2*m*Vn)/hbar*1e9*sqrt(1.60e-19) + +# cutoffs +p_cutoff=20*k0 +p_npoints=256 + +# airy approximations +airy_threshold=30 +airy_order=5 + +# xbounds +xmax=10 +xmin=-10 +x_npoints=200 + +include("FN_base.jl") + +# compute asymptotic values +ps_asym=Array{Complex{Float64}}(undef,x_npoints) +dps_asym=Array{Complex{Float64}}(undef,x_npoints) +for i in 1:x_npoints + x=xmin+(xmax-xmin)*i/x_npoints + ps_asym[i]=psi_pole(x,k0,E,V) + dps_asym[i]=dpsi_pole(x,k0,E,V) +end + +# print asymptotic values +for i in 1:x_npoints + print((xmin+(xmax-xmin)*i/x_npoints)*nm_scale,' ',abs(ps_asym[i])^2,' ',J(ps_asym[i],dps_asym[i]),'\n') +end diff --git a/figs/libs/Makefile b/figs/libs/Makefile new file mode 100644 index 0000000..33b81e2 --- /dev/null +++ b/figs/libs/Makefile @@ -0,0 +1,28 @@ +PROJECTNAME=$(basename $(basename $(wildcard *.tikz.tex))) +LIBS=$(notdir $(wildcard libs/*)) + +PDFS=$(addsuffix .pdf, $(PROJECTNAME)) + +all: $(PDFS) + +$(PDFS): $(LIBS) + echo $(LIBS) + pdflatex -jobname $(basename $@) -file-line-error $(patsubst %.pdf, %.tikz.tex, $@) + +install: $(PDFS) + cp $^ $(INSTALLDIR)/ + +$(LIBS): + ln -fs libs/$@ ./ + +clean-libs: + rm -f $(LIBS) + +clean-aux: + rm -f $(addsuffix .aux, $(PROJECTNAME)) + rm -f $(addsuffix .log, $(PROJECTNAME)) + +clean-tex: + rm -f $(PDFS) + +clean: clean-libs clean-aux clean-tex diff --git a/figs/potential.fig/Makefile b/figs/potential.fig/Makefile new file mode 120000 index 0000000..704310e --- /dev/null +++ b/figs/potential.fig/Makefile @@ -0,0 +1 @@ +../libs/Makefile
\ No newline at end of file diff --git a/figs/potential.fig/potential.tikz.tex b/figs/potential.fig/potential.tikz.tex new file mode 100644 index 0000000..ac1ba36 --- /dev/null +++ b/figs/potential.fig/potential.tikz.tex @@ -0,0 +1,17 @@ +\documentclass{standalone} + +\usepackage{tikz} + +\begin{document} +\begin{tikzpicture} + +\draw(-3,0)--(3,0); +\draw(0,-1.5)--(0,3); + +\draw[color=gray,line width=1pt, densely dotted](-2.5,1.25)--++(2.5,0); + +\draw[line width=1.5pt](-2.5,0)--(0,0)--(0,2.5)--(1.975,-1.25); + +\end{tikzpicture} +\end{document} + diff --git a/figs/potential.fig/potential_square.tikz.tex b/figs/potential.fig/potential_square.tikz.tex new file mode 100644 index 0000000..bd13467 --- /dev/null +++ b/figs/potential.fig/potential_square.tikz.tex @@ -0,0 +1,17 @@ +\documentclass{standalone} + +\usepackage{tikz} + +\begin{document} +\begin{tikzpicture} + +\draw(-3,0)--(3,0); +\draw(0,-1.5)--(0,3); + +\draw[color=gray,line width=1pt, densely dotted](-2.5,1.25)--++(2.5,0); + +\draw[line width=1.5pt](-2.5,0)--(0,0)--(0,2.5)--(2.5,2.5); + +\end{tikzpicture} +\end{document} + diff --git a/figs/potential.fig/potential_square_photonic.tikz.tex b/figs/potential.fig/potential_square_photonic.tikz.tex new file mode 100644 index 0000000..8a35fb4 --- /dev/null +++ b/figs/potential.fig/potential_square_photonic.tikz.tex @@ -0,0 +1,22 @@ +\documentclass{standalone} + +\usepackage{tikz} + +\begin{document} +\begin{tikzpicture} + +\draw(-3,0)--(3,0); +\draw(0,-1.5)--(0,3); + +\draw[color=gray,line width=1pt, densely dotted](-2.5,1.25)--++(2.5,0); +\draw[color=gray,line width=1pt, densely dotted](-1,2.00)--++(1,0); +\draw[color=gray,line width=1pt, densely dotted](-1,2.75)--++(3.5,0); + +\draw[->,line width=0.75pt,color=red](-0.5,1.25)--++(0,0.75); +\draw[->,line width=0.75pt,color=red](-0.5,2.00)--++(0,0.75); + +\draw[line width=1.5pt](-2.5,0)--(0,0)--(0,2.5)--(2.5,2.5); + +\end{tikzpicture} +\end{document} + diff --git a/figs/potential.fig/potential_square_thermal.tikz.tex b/figs/potential.fig/potential_square_thermal.tikz.tex new file mode 100644 index 0000000..3e8fd22 --- /dev/null +++ b/figs/potential.fig/potential_square_thermal.tikz.tex @@ -0,0 +1,17 @@ +\documentclass{standalone} + +\usepackage{tikz} + +\begin{document} +\begin{tikzpicture} + +\draw(-3,0)--(3,0); +\draw(0,-1.5)--(0,3); + +\draw[color=gray,line width=1pt, densely dotted](-2.5,2.75)--++(5,0); + +\draw[line width=1.5pt](-2.5,0)--(0,0)--(0,2.5)--(2.5,2.5); + +\end{tikzpicture} +\end{document} + |