Ian Jauslin
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-rw-r--r--Jauslin_Rutgers_2018.tex259
-rw-r--r--Makefile50
-rw-r--r--README43
-rw-r--r--figs/Millikan-Lauritsen_current.pngbin0 -> 29963 bytes
-rw-r--r--figs/animation/FN_base.jl170
-rw-r--r--figs/animation/Makefile12
-rw-r--r--figs/animation/animate.py128
-rw-r--r--figs/animation/animate_compute.jl66
-rw-r--r--figs/contour.fig/Makefile30
-rw-r--r--figs/contour.fig/contour.py82
-rw-r--r--figs/contour.fig/contour.tikz.tex34
-rw-r--r--figs/emitter.jpgbin0 -> 14324 bytes
-rw-r--r--figs/fowler-nordheim.fig/FN_base.jl170
-rw-r--r--figs/fowler-nordheim.fig/Makefile29
-rw-r--r--figs/fowler-nordheim.fig/asymptotic.gnuplot59
-rw-r--r--figs/fowler-nordheim.fig/asymptotic.jl46
-rw-r--r--figs/libs/Makefile28
l---------figs/potential.fig/Makefile1
-rw-r--r--figs/potential.fig/potential.tikz.tex15
-rw-r--r--figs/potential.fig/potential_square.tikz.tex15
-rw-r--r--libs/ian-presentation.cls187
21 files changed, 1424 insertions, 0 deletions
diff --git a/Jauslin_Rutgers_2018.tex b/Jauslin_Rutgers_2018.tex
new file mode 100644
index 0000000..50b9bea
--- /dev/null
+++ b/Jauslin_Rutgers_2018.tex
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+\documentclass{ian-presentation}
+
+\usepackage[hidelinks]{hyperref}
+\usepackage{graphicx}
+\usepackage{array}
+
+\begin{document}
+\pagestyle{empty}
+\hbox{}\vfil
+\bf\Large
+\hfil Solution of time-dependent Schr\"odinger equation\par
+\smallskip
+\hfil for field emission\par
+\vfil
+\large
+\hfil Ian Jauslin
+\normalsize
+\vfil
+\hfil\rm joint with {\bf Ovidiu Costin}, {\bf Rodica Costin}, and {\bf Joel L. Lebowitz}\par
+\vfil
+arXiv:{\tt \href{http://arxiv.org/abs/1808.00936}{1808.00936}}\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}}
+\eject
+
+\setcounter{page}1
+\pagestyle{plain}
+
+\title{Field emission}
+\vfill
+\hfil\includegraphics[height=5cm]{emitter.jpg}
+\vfill
+\eject
+
+\title{Emitter}
+$$
+ V(x)=U\Theta(x)
+$$
+\hfil\includegraphics[height=5cm]{potential_square.pdf}
+\vfill
+\eject
+
+\title{Semi-classical model}
+$$
+ V_t(x)=\Theta(x)(U-E_tx)
+ ,\quad
+ E_t=e_0+e_1\cos(\omega t)
+$$
+\hfil\includegraphics[height=5cm]{potential.pdf}
+\vfill
+\eject
+
+\title{Triangular barrier}
+$$
+ V(x)=\Theta(x)(U-Ex)
+$$
+\hfil\includegraphics[height=5cm]{potential.pdf}
+\vfill
+\eject
+
+\title{Field emission through a triangular barrier}
+\begin{itemize}
+ \item \href{https://doi.org/10.1073\%2Fpnas.14.1.45}{[Millikan, Lauritsen, 1928]}: experimental plot of the logarithm of the current against $1/E$
+\end{itemize}
+\hfil\includegraphics[height=4.5cm]{Millikan-Lauritsen_current.png}
+\vfill
+\eject
+
+\title{Field emission through a triangular barrier}
+\vfill
+\begin{itemize}
+ \item \href{https://doi.org/10.1098/rspa.1928.0091}{[Fowler, Nordheim, 1928]}: predicted that the current is, for small $E$,
+ $$
+ J\approx Ce^{-\frac aE}
+ $$
+ \item (\href{https://doi.org/10.1088/1751-8113/44/5/05530@}{[Rokhlenko, 2011]}: studied the range of applicability of the approximation, and found more accurate approximations for larger fields.)
+\end{itemize}
+\vfill
+\eject
+
+\title{Fowler-Nordheim equation}
+\begin{itemize}
+ \item Schr\"odinger equation
+ $$
+ i\partial_t\psi=-\Delta\psi+\Theta(x)(U-Ex)\psi
+ $$
+ \item Fowler-Nordheim: quasi-stationary solution: $\psi_{\mathrm{FN}}(x,t)=e^{-ik^2t}\varphi_{\mathrm{FN}}(x)$
+ $$
+ \varphi_{\mathrm{FN}}(x)=
+ \left\{ \begin{array}{l@{\ }l}
+ e^{ikx}+R_Ee^{-ikx} & x<0\\
+ T_E\mathrm{Ai}(e^{-\frac{i\pi}3}(E^{\frac13}x-E^{-\frac23}(U-k^2)) & x>0
+ \end{array}\right.
+ $$
+ $R_E$ and $T_E$ are chosen so that $\varphi_{\mathrm{FN}}$ and $\partial\varphi_{\mathrm{FN}}$ are continuous at $x=0$.
+\end{itemize}
+\vfill
+\eject
+
+\title{Fowler-Nordheim equation}
+\vfill
+\hfil\includegraphics[height=5.5cm]{asymptotic.pdf}
+\vfill
+\eject
+
+\title{Initial value problem}
+\begin{itemize}
+ \item Initial condition:
+ $$
+ \psi(x,0)=
+ \left\{ \begin{array}{l@{\ }l}
+ e^{ikx}+R_0e^{-ikx} & x<0\\
+ T_0 e^{-\sqrt{U-k^2}x} & x>0
+ \end{array}\right.
+ $$
+ $R_0$ and $T_0$ ensure that $\psi$ and $\partial\psi$ are continuous.
+ \item Behaves asymptotically like $\psi_{\mathrm{FN}}$:
+ $$
+ \psi(x,t)e^{ik^2t}\mathop{\longrightarrow}_{t\to\infty}\varphi_{\mathrm{FN}}(x)
+ $$
+\end{itemize}
+\vfill
+\eject
+
+\title{Initial value problem}
+\begin{itemize}
+ \item Laplace transform:
+ $$
+ \hat\psi_p(x):=\int_0^\infty dt\ e^{-pt}\psi(x,t)
+ $$
+ \item Schr\"odinger equation:
+ $$
+ (-\Delta+\Theta(x)V(x)-ip)\psi_p(x)=-i\psi(x,0)
+ ,\quad
+ V(x):=U-Ex
+ $$
+\end{itemize}
+\vfill
+\eject
+
+\title{Solution in Laplace space}
+\begin{itemize}
+ \item For simplicity, $R_0\equiv T_0\equiv0$.
+ \item Solution:
+ $$
+ \hat\psi_p(x)=
+ \left\{\begin{array}{>\displaystyle l@{\ }l}
+ C_1(p)e^{\sqrt{-ip}x}-\frac{ie^{ikx}}{-ip+k^2}
+ &\mathrm{if\ }x<0\\[0.5cm]
+ C_2(p)\varphi_p(x)
+ &\mathrm{if\ }x> 0
+ \end{array}\right.
+ $$
+ with
+ $$
+ (-\Delta+V(x)-ip)\varphi_p(x)=0
+ $$
+ $$
+ \varphi_p(x)=\mathrm{Ai}\left(e^{-\frac{i\pi}3}\left(E^{\frac13}x-E^{-\frac23}(U-ip)\right)\right)
+ $$
+\end{itemize}
+\vfill
+\eject
+
+\title{Solution in Laplace space}
+\begin{itemize}
+ \item $C_1$ and $C_2$ ensure that $\hat\psi_p(x)$ and $\partial\hat\psi_p(x)$ are continuous at $x=0$:
+ $$
+ C_1(p)=\frac{i(ik\varphi_p(0)-\partial\varphi_p(0))}{(-ip+k^2)(\sqrt{-ip}\varphi_p(0)-\partial\varphi_p(0))}
+ $$
+ $$
+ C_2(p)=-\frac{i}{(\sqrt{-ip}+ik)(\sqrt{-ip}\varphi_p(0)-\partial\varphi_p(0))}.
+ $$
+\end{itemize}
+\vfill
+\eject
+
+\title{Poles in Laplace plane}
+\vfill
+\hfil\includegraphics[height=5.5cm]{contour.pdf}
+\vfill
+\eject
+
+\title{Asymptotic behavior}
+\begin{itemize}
+ \item As $t\to\infty$:
+ $$
+ \psi(x,t)
+ =\psi_{\mathrm{FN}}(x,t)+\left(\frac{t}{\tau_E(x)}\right)^{-\frac32}+O(t^{-\frac52})
+ .
+ $$
+
+ \item If $k<0$ (reflected wave), then there is no pole on the imaginary axis, so there is no contribution as $t\to\infty$.
+ \item Similarly, the transmitted wave in the initial condition does not contribute.
+\end{itemize}
+\vfill
+\eject
+
+\title{Laser field}
+\begin{itemize}
+ \item Time dependent potential:
+ $$
+ V_t(x)=\Theta(x)(U-\epsilon\omega\cos(\omega t)x)
+ $$
+ \item Magnetic gauge:
+ $$
+ \Psi(x,t):=
+ :=\psi(x,t)e^{-ix\Theta(x)A(t)}
+ ,\quad
+ A(t):=\int_0^t ds\ \epsilon\omega\cos(\omega s)
+ =
+ \epsilon\sin(\omega t)
+ $$
+ satisfies
+ $$
+ i\partial_t\Psi(x,t)=\left((-i\nabla+\Theta(x)A(t))^2+\Theta(x)U\right)\Psi(x,t)
+ $$
+\end{itemize}
+\vfill
+\eject
+
+\title{Periodic solution}
+\begin{itemize}
+ \item A solution:
+ $$
+ \Psi(x,t)=\left\{\begin{array}{ll}
+ \Psi_I(x,t)+\Psi_R(x,t)&\mathrm{\ if\ }x<0\\
+ \Psi_T(x,t)&\mathrm{\ if\ }x>0
+ \end{array}\right.
+ $$
+ $$
+ \Psi_I(x,t)=e^{ikx}\exp\left(-ik^2t\right)
+ $$
+ $$
+ \Psi_R(x,t)=\sum_{M\in\mathbb Z}R_Me^{iq_Mx}\exp\left(-iq_M^2t\right)
+ $$
+ $$
+ \Psi_T(x,t)=\sum_{M\in\mathbb Z}T_Me^{ip_Mx}\exp\left(-iUt-i\int_0^td\tau\ (p_M+A(\tau))^2\right)
+ $$
+\end{itemize}
+\vfill
+\eject
+
+\title{Periodic solution}
+\begin{itemize}
+ \item Choose $q_M$ and $p_M$ to make the solution periodic (up to the phase $e^{ik^2t}$):
+ $$
+ q_M=\pm\sqrt{k^2+M\omega}
+ ,\quad
+ p_M=\pm\sqrt{k^2-U+M\omega-U_V}
+ $$
+ and
+ $$
+ U_V:=\frac\omega{2\pi}\int_0^{\frac{2\pi}\omega} d\tau\ A^2(\tau)
+ =\frac{\epsilon^2}2
+ .
+ $$
+\end{itemize}
+
+
+\end{document}
diff --git a/Makefile b/Makefile
new file mode 100644
index 0000000..4ed7b60
--- /dev/null
+++ b/Makefile
@@ -0,0 +1,50 @@
+PROJECTNAME=$(basename $(wildcard *.tex))
+LIBS=$(notdir $(wildcard libs/*))
+FIGS=$(notdir $(wildcard figs/*.fig))
+
+PDFS=$(addsuffix .pdf, $(PROJECTNAME))
+SYNCTEXS=$(addsuffix .synctex.gz, $(PROJECTNAME))
+
+all: $(PROJECTNAME)
+
+$(PROJECTNAME): $(LIBS) $(FIGS) images
+ pdflatex -file-line-error $@.tex
+ pdflatex -synctex=1 $@.tex
+
+$(SYNCTEXS): $(LIBS) $(FIGS) images
+ pdflatex -synctex=1 $(patsubst %.synctex.gz, %.tex, $@)
+
+
+$(LIBS):
+ ln -fs libs/$@ ./
+
+
+$(FIGS):
+ make -C figs/$@
+ for pdf in $$(find figs/$@/ -name '*.pdf'); do ln -fs "$$pdf" ./ ; done
+
+images:
+ for jpg in figs/*.jpg; do ln -fs "$$jpg" ./ ; done
+ for png in figs/*.png; do ln -fs "$$png" ./ ; done
+
+
+clean-aux: clean-figs-aux
+ rm -f $(addsuffix .aux, $(PROJECTNAME))
+ rm -f $(addsuffix .log, $(PROJECTNAME))
+ rm -f $(addsuffix .out, $(PROJECTNAME))
+
+clean-libs:
+ rm -f $(LIBS)
+
+clean-figs:
+ $(foreach fig,$(addprefix figs/, $(FIGS)), make -C $(fig) clean; )
+ rm -f $(notdir $(wildcard figs/*.fig/*.pdf))
+
+clean-figs-aux:
+ $(foreach fig,$(addprefix figs/, $(FIGS)), make -C $(fig) clean-aux; )
+
+
+clean-tex:
+ rm -f $(PDFS) $(SYNCTEXS)
+
+clean: clean-aux clean-tex clean-libs clean-figs
diff --git a/README b/README
new file mode 100644
index 0000000..6461b12
--- /dev/null
+++ b/README
@@ -0,0 +1,43 @@
+This directory contains the source files to typeset the presentation, and
+generate the figures. This can be accomplished by running
+ make
+
+An animation was prepared for this talk. It represents the time evolution of
+the density and the current. The source files for the animation are located in
+the 'figs/animation' directory. To run the animation, run
+ cd figs/animation
+ make run
+This can take a while the first time it is run.
+
+
+This document uses a custom class file, located in the 'libs' directory, which
+defines a number of commands.
+
+
+* Dependencies:
+
+ pdflatex
+ TeXlive packages:
+ amsfonts
+ array
+ graphics
+ hyperref
+ latex
+ pgf
+ standalone
+ GNU make
+ julia
+ python3
+ gnuplot
+
+* Files:
+
+ Jauslin_Rutgers_2018.tex:
+ main LaTeX file
+
+ libs:
+ custom LaTeX class file
+
+ figs:
+ source code for the figures
+
diff --git a/figs/Millikan-Lauritsen_current.png b/figs/Millikan-Lauritsen_current.png
new file mode 100644
index 0000000..5645be8
--- /dev/null
+++ b/figs/Millikan-Lauritsen_current.png
Binary files differ
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..29098b3
--- /dev/null
+++ b/figs/animation/Makefile
@@ -0,0 +1,12 @@
+PROJECTNAME=animate
+
+all: animate.dat
+
+run: animate.dat
+ python3 animate.py animate.dat
+
+animate.dat:
+ julia animate_compute.jl > animate.dat
+
+clean:
+ rm -f animate.dat
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..b9c2d02
--- /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=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 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
new file mode 100644
index 0000000..d84ad1f
--- /dev/null
+++ b/figs/emitter.jpg
Binary files differ
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..8254e5d
--- /dev/null
+++ b/figs/potential.fig/potential.tikz.tex
@@ -0,0 +1,15 @@
+\documentclass{standalone}
+
+\usepackage{tikz}
+
+\begin{document}
+\begin{tikzpicture}
+
+\draw(-3,0)--(3,0);
+\draw(0,-1.5)--(0,3);
+
+\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..0fc8875
--- /dev/null
+++ b/figs/potential.fig/potential_square.tikz.tex
@@ -0,0 +1,15 @@
+\documentclass{standalone}
+
+\usepackage{tikz}
+
+\begin{document}
+\begin{tikzpicture}
+
+\draw(-3,0)--(3,0);
+\draw(0,-1.5)--(0,3);
+
+\draw[line width=1.5pt](-2.5,0)--(0,0)--(0,2.5)--(2.5,2.5);
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/libs/ian-presentation.cls b/libs/ian-presentation.cls
new file mode 100644
index 0000000..91bd487
--- /dev/null
+++ b/libs/ian-presentation.cls
@@ -0,0 +1,187 @@
+%%
+%% Ian's presentation class
+%%
+
+%% TeX format
+\NeedsTeXFormat{LaTeX2e}[1995/12/01]
+
+%% class name
+\ProvidesClass{ian-presentation}[2017/09/29]
+
+\def\ian@defaultoptions{
+ \pagestyle{plain}
+ \RequirePackage{color}
+ \RequirePackage{amssymb}
+}
+
+%% paper dimensions
+\setlength\paperheight{240pt}
+\setlength\paperwidth{427pt}
+
+%% fonts
+\input{size11.clo}
+\DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm}
+\DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf}
+\DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt}
+\DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf}
+\DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit}
+
+%% text dimensions
+\textheight=208pt
+\textwidth=370pt
+\hoffset=-1in
+\voffset=-1in
+\oddsidemargin=24pt
+\evensidemargin=24pt
+\topmargin=8pt
+\headheight=0pt
+\headsep=0pt
+\marginparsep=0pt
+\marginparwidth=0pt
+\footskip=16pt
+
+
+%% remove default skips
+\parindent=0pt
+\parskip=0pt
+\baselineskip=0pt
+
+%% something is wrong with \thepage, redefine it
+\gdef\thepage{\the\c@page}
+
+%% correct vertical alignment at the end of a document
+\AtEndDocument{
+ % save total slide count
+ \immediate\write\@auxout{\noexpand\gdef\noexpand\slidecount{\thepage}}
+ \vfill
+ \eject
+}
+
+
+%% footer
+\def\ps@plain{
+ \def\@oddhead{}
+ \def\@evenhead{\@oddhead}
+ \def\@oddfoot{\tiny\hfill\thepage/\safe\slidecount\hfill}
+ \def\@evenfoot{\@oddfoot}
+}
+\def\ps@empty{
+ \def\@oddhead{}
+ \def\@evenhead{\@oddhead}
+ \def\@oddfoot{}
+ \def\@evenfoot{\@oddfoot}
+}
+
+
+%% title of slide
+\def\title#1{
+ \hfil{\bf\large #1}\par
+ \hfil\vrule width0.75\textwidth height0.3pt\par
+ \vskip5pt
+}
+
+
+%% hyperlinks
+% hyperlinkcounter
+\newcounter{lncount}
+% hyperref anchor
+\def\hrefanchor{%
+\stepcounter{lncount}%
+\hypertarget{ln.\thelncount}{}%
+}
+
+%% define a command and write it to aux file
+\def\outdef#1#2{%
+ % define command%
+ \expandafter\xdef\csname #1\endcsname{#2}%
+ % hyperlink number%
+ \expandafter\xdef\csname #1@hl\endcsname{\thelncount}%
+ % write command to aux%
+ \immediate\write\@auxout{\noexpand\expandafter\noexpand\gdef\noexpand\csname #1\endcsname{\csname #1\endcsname}}%
+ \immediate\write\@auxout{\noexpand\expandafter\noexpand\gdef\noexpand\csname #1@hl\endcsname{\thelncount}}%
+}
+
+%% can call commands even when they are not defined
+\def\safe#1{%
+ \ifdefined#1%
+ #1%
+ \else%
+ {\color{red}\bf?}%
+ \fi%
+}
+
+
+%% itemize
+\newlength\itemizeskip
+% left margin for items
+\setlength\itemizeskip{20pt}
+\newlength\itemizeseparator
+% space between the item symbol and the text
+\setlength\itemizeseparator{5pt}
+% penalty preceding an itemize
+\def\itemizepenalty{0}
+% counter counting the itemize level
+\newcounter{itemizecount}
+
+% item symbol
+\def\itemizept#1{
+ \ifnum#1=1
+ \textbullet
+ \else
+ $\scriptstyle\blacktriangleright$
+ \fi
+}
+
+\newlength\current@itemizeskip
+\setlength\current@itemizeskip{0pt}
+\def\itemize{
+ \par\penalty\itemizepenalty\medskip\penalty\itemizepenalty
+ \addtocounter{itemizecount}{1}
+ \addtolength\current@itemizeskip{\itemizeskip}
+ \leftskip\current@itemizeskip
+}
+\def\enditemize{
+ \addtocounter{itemizecount}{-1}
+ \addtolength\current@itemizeskip{-\itemizeskip}
+ \par\leftskip\current@itemizeskip
+ \medskip
+}
+\newlength\itempt@total
+\def\item{
+ \settowidth\itempt@total{\itemizept\theitemizecount}
+ \addtolength\itempt@total{\itemizeseparator}
+ \par
+ \medskip
+ \hskip-\itempt@total\itemizept\theitemizecount\hskip\itemizeseparator
+}
+
+%% enumerate
+\newcounter{enumerate@count}
+\def\enumerate{
+ \setcounter{enumerate@count}0
+ \let\olditem\item
+ \let\olditemizept\itemizept
+ \def\item{
+ % counter
+ \stepcounter{enumerate@count}
+ % set header
+ \def\itemizept{\theenumerate@count.}
+ % hyperref anchor
+ \hrefanchor
+ % define tag (for \label)
+ \xdef\tag{\theenumerate@count}
+ \olditem
+ }
+ \itemize
+}
+\def\endenumerate{
+ \enditemize
+ \let\item\olditem
+ \let\itemizept\olditemizept
+}
+
+
+%% end
+\ian@defaultoptions
+
+\endinput