Mattis-Bardeen calculation of (Subscript[σ, 1])(T)/(Subscript[σ, n]) and  (Subscript[σ, 2])(T)/(Subscript[σ, n])

Steven Anlage, University of Maryland
BCS weak-coupling theory with Δ(0)=1.76 (Subscript[k, B]) (Subscript[T, c])
Pippard limit: lambda << xi, mean free path (clean type-I superconductor)
Note that the coherence peak only appears for hbar*omega < Delta/5λℏ
Following O. Klein, et al., PRB 50, 6307 (1994).

hbar=1.05457148×10^(-34)

kB=1.3806503×10^(-23)

Δ[Δ0_,t_] := Δ0*y/.FindRoot[ y == Tanh[y/t],{y,0.5} ][[1]]  (* Simple BCS gap temperature dependence *)

Δ[1,.5]

Plot[Δ[1,t], {t,.001,0.999}, PlotRange ->{ {0,1}, {0,1}}, PlotPoints -> 100]

sig1bysignF[f_,Δ0_,t_,Tc_]:=(2/(hbar 2 π f)) NIntegrate[((1/(1+Exp[u/(kB Tc*t)]))-(1/(1+Exp[(u+hbar 2 π f)/(kB Tc*t)]))) (u^2+Δ[Δ0,t]^2+hbar 2 π f u)/(If[Abs[u]>Δ[Δ0,t], Sign[u] (Sqrt[u^2-Δ[Δ0,t]^2]),-I (Sqrt[Δ[Δ0,t]^2-u^2])] (Sqrt[(u+hbar 2 π f)^2-Δ[Δ0,t]^2]) ) ,{u,Δ[Δ0,t],100 Δ[Δ0,t]}]+(1/(hbar 2 π f)) NIntegrate[(1-(2/(1+Exp[(u+hbar 2 π f)/(kB Tc*t)]))) (u^2+Δ[Δ0,t]^2+hbar 2 π f u)/(If[Abs[u]>Δ[Δ0,t], Sign[u] (Sqrt[u^2-Δ[Δ0,t]^2]),-I (Sqrt[Δ[Δ0,t]^2-u^2])] (Sqrt[(u+hbar 2 π f)^2-Δ[Δ0,t]^2]) ) ,{u,Δ[Δ0,t]-hbar 2 π f,Δ[Δ0,t]}]

sig1bysignF[10^9,3*10^(-3)*1.602*10^(-19),0.99,9.25]

sig1bysignF[10*10^9,3*10^(-3)*1.602*10^(-19),0.85,9.25]

127/(1.19)

Plot[Re[sig1bysignF[10^9,3*10^(-3)*1.602*10^(-19),tr,9.25]], {tr,0.01,0.999}]

Plot[Re[sig1bysignF[60*10^9,1.76*kB*9.25,tr,9.25]], {tr,0.01,0.9999}]

Plot[Re[sig1bysignF[75*10^6,1.76*kB*9.25,tr,9.25]], {tr,0.01,0.9999}]

Table[i^2,{i,10}]

sigvst=Table[{tr,Re[sig1bysignF[75*10^6,1.76*kB*9.25,tr,9.25]]},{tr,.01,0.99, 0.01}]

ListPlot[sigvst]

Plot[Re[sig1bysignF[14*10^6,1.76*kB*9.25,tr,9.25]], {tr,0.01,0.9999}]

Plot3D[Re[sig1bysignF[f,1.76*kB*9.25,tr,9.25]], {tr,0.01,0.99},{f,0.01*2*1.76*kB*9.25/(hbar 2π),2*2*1.76*kB*9.25/(hbar 2 π)}]

Plot3D[Re[sig1bysignF[f,1.76*kB*9.25,tr,9.25]], {f,0.01*2*1.76*kB*9.25/(hbar 2π),2*2*1.76*kB*9.25/(hbar 2 π)},{tr,0.01,0.99}, PlotStyle->Directive[Orange, Specularity[White,50]],PlotPoints->50]

Now check the frequency dependence of (Subscript[σ, 1])/(Subscript[σ, n]) near the peak.  It is predicted to diverge logarithmically.

Plot[{Re[sig1bysignF[freq,1.76*kB*9.25,0.85,9.25]],Log[2 Δ[1.76*kB*9.25,0.85]/(hbar 2 π freq)]}, {freq,10^(-3),10^9}]

LogLinearPlot[{Re[sig1bysignF[freq,1.76*kB*9.25,0.85,9.25]],Log[2 Δ[1.76*kB*9.25,0.85]/(hbar 2 π freq)]}, {freq,10^(-6),10^9}]

Plot3D[Re[sig1bysignF[f,1.76*kB*9.25,tr,9.25]], {f,0.01*2*1.76*kB*9.25/(hbar 2π),2*2*1.76*kB*9.25/(hbar 2 π)},{tr,0.01,0.99}, PlotStyle->Directive[Red, Specularity[White,50]],PlotPoints->100]

Plot[Im[sig1bysignF[10^9,3*10^(-3)*1.602*10^(-19),tr,9.25]], {tr,0.01,0.999}]