As presented at VirginiaTech on 2019-05-24HEADv1.0master

1 files changed, 170 insertions, 0 deletions

diff --git a/figs/fowler-nordheim.fig/FN_base.jl b/figs/fowler-nordheim.fig/FN_base.jl
new file mode 100644
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+++ b/figs/fowler-nordheim.fig/FN_base.jl
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+# 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