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equation solving - Find Root doesn't work properly


I use a FindRoot that doesn't find the right solution in some range, the FindRoot is the following:


tabexp = ParallelTable[{Temp,V /.FindRoot[Ifix - Isis[V/2, 0.2, Temp, 1.5, 1.4, \[Gamma]0, \[Gamma]0,RSISIS], {V, 4.05 d0Al[1.5]/q, 1. d0Al[1.5]/q,4.1 d0Al[1.5]/q}, AccuracyGoal -> 25,  PrecisionGoal -> 25,MaxIterations -> 2000]}, {Temp, 0.2, 1.25, 0.9/40}];

This is the plot of the FindRoot output tabexp:


enter image description here


The point is that the graphical solution is easy:


Plot[Evaluate[Table[{Ifix -Isis[V/2 d0Al[1.5]/q, 0.2, Tx, 1.5, 1.4,[Gamma]0, \[Gamma]0,RSISIS]}, {Tx, 0.6, 0.8, 0.05}]], {V, 1 , 4.3},PlotRange -> All]


enter image description here


I don't understand why it's not working properly. If you need all the code it is the following:


q = 1.602176487` 10^-19;
kB = 1.3806504` 10^-23;
\[HBar] = 1.054571628251774` 10^-34;
TcAluminio = 1.55;
d0Al[TcAl_] := 1.764 kB TcAl;
d0max = 1.764 kB 1.65;
delta[T_] := Tanh[1.74 Sqrt[1/T - 1]];
\[CapitalDelta]Al[T_, TcAl_] := delta[T/TcAl] d0Al[TcAl];


ngammaAl[energy_, T_, gamma_, TcAl_] := 
  Abs[Re[(energy + I gamma d0Al[TcAl])/
    Sqrt[(energy + I gamma d0Al[TcAl])^2 - \[CapitalDelta]Al[T, 
      TcAl]^2]]];
Isis[V_, T1_, T2_, TcAlP_, TcAlI_, gamma1_, gamma2_, Rjunction_] := 
  Re[1/(    q Rjunction)
     NIntegrate[
     ngammaAl[energy - q V, T1, gamma1, TcAlP] ngammaAl[energy, T2, 
       gamma2, 

       TcAlI] (1/(Exp[(energy - q V)/(kB T1)] + 1) - 1/(
        Exp[energy/(kB T2)] + 
         1)), {energy, -10 d0max, -\[CapitalDelta]Al[T1, 
        TcAlP], \[CapitalDelta]Al[T2, TcAlI], \[CapitalDelta]Al[T2, 
       TcAlI], \[CapitalDelta]Al[T1, TcAlP], 10 d0max}, 
     MinRecursion -> 4, 
     Method -> {"GlobalAdaptive", "MaxErrorIncreases" -> 100000, 
       "SymbolicProcessing" -> 0, "SingularityHandler" -> None}, 
     PrecisionGoal -> 5]];
\[Gamma]0 = 10^-3;

RSISIS = 55000;
Ifix = Isis[0.0006/2, 1.1, 1.1, 1.5, 1.4, \[Gamma]0, \[Gamma]0, 
  RSISIS]

Thanks in advance and sorry for the bad English.



Answer



V_?NumericQ removes some errors. WorkingPrecision -> 32 removes others. NDSolve tracks the solution.


ClearAll[Isis];
Isis[V_?NumericQ, T1_, T2_, TcAlP_, TcAlI_, 
   gamma1_, gamma2_, Rjunction_] := Re[1/(q Rjunction) NIntegrate[

     SetPrecision[
      ngammaAl[energy - q V, T1, gamma1, TcAlP] ngammaAl[energy, T2, 
        gamma2, TcAlI] (1/(Exp[(energy - q V)/(kB T1)] + 1) - 
         1/(Exp[energy/(kB T2)] + 1)),
      50], {energy, -10 d0max, -ΔAl[T1, TcAlP], ΔAl[T2, TcAlI], ΔAl[T2, 
       TcAlI], ΔAl[T1, TcAlP], 10 d0max}, 
     MinRecursion -> 4, 
     Method -> {"GlobalAdaptive", "MaxErrorIncreases" -> 100000, 
       "SymbolicProcessing" -> 0, "SingularityHandler" -> None}, 
     PrecisionGoal -> 5, WorkingPrecision -> 32]];


sol = NDSolve[{Ifix == 
    Isis[V[Temp]/2, 0.2, Temp, 1.5, 1.4, γ0, γ0, 
     RSISIS], x'[Temp] == 1, x[0.2] == 0.2,
   V[0.2] == (VV /. 
      FindRoot[
       Ifix - Isis[VV/2, 0.2, 0.2, 1.5, 1.4, γ0, γ0, 
         RSISIS], {VV, 4.05 d0Al[1.5]/q, 3. d0Al[1.5]/q, 
        4.1 d0Al[1.5]/q}])},
  V,

  {Temp, 0.2, 1.25}]

ListLinePlot[V /. First@sol]

enter image description here


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