BCS fit to conductance spectrum (from STS) to get the superconducting energy gap
Mathematica Asked on December 28, 2020
Can someone help me fitting STS spectra using the Dynes formula (Phys. Rev. Lett. 41, 1509 (1978).) to find a superconducting gap?
I try to fit using FindFit in Mathematica but it gives several errors.
I followed a paper PRL 97, 077003 (2006). Below is data and code.
datasc = {{-0.014077264599286`,
0.621374900897037`}, {-0.013973006869221`,
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ListPlot[%]
kB = 8.617 10^-5;
q = 1.60217 10^-19;
T = 3.2;
DOS = NIntegrate[
No Re[(En - I [CapitalGamma])/
Sqrt[(En - I [CapitalGamma])^2 + [CapitalDelta]^2]] (-D[1/(
Exp[(En - V)/(kB T)] + 1), V]), {En, 0, 10},
MaxRecursion -> 100];
FindFit[datasc, DOS, {[CapitalGamma], [CapitalDelta], No}, V]
One Answer
Data can be scaled to mV as
data = Table[{datasc[[i, 1]] 10^3, datasc[[i, 2]]}, {i,
Length[datasc]}];
Then we use parameter $10^{-3}/(kT)=3.6265521643263314$ and function (we put x=Ea-V)
dos[No_?NumericQ, [CapitalGamma]_?NumericQ, [CapitalDelta]_?
NumericQ, V_?NumericQ] :=
No Sign[V] NIntegrate[
Re[(x + V - I [CapitalGamma])/
Sqrt[(x + V - I [CapitalGamma])^2 - [CapitalDelta]^2]] ( (
E^(3.6265521643263314 x))/(1 + E^(
3.6265521643263314 x))^2), {x, -60, 60},
PrecisionGoal -> 5] // Quiet;
The best fit with this function is
ff =
FindFit[data,
dos[No, [CapitalGamma], [CapitalDelta],
V], {No, [CapitalGamma], [CapitalDelta]}, V]
Out[]= {No -> 2.22042, [CapitalGamma] ->
0.249188, [CapitalDelta] -> 1.54661}
It looks with data as follows
Show[ListPlot[data, PlotRange -> All, Frame -> True,
FrameLabel -> {"Bias voltag (mV)", "Normalized Conductance"}],
Plot[dos[No, [CapitalGamma], [CapitalDelta], V] /. ff, {V, -15,
15}, PlotRange -> All]]
It is not the best fit since we used Sign[V]. Probably it can be corrected around V=0. For instance, we can use Abs instead of Re and Infinity as a limit of integration, but it is not much better than version shown in Figure 1:
dos1[
No_?NumericQ, [CapitalGamma]_?NumericQ, [CapitalDelta]_?NumericQ,
V_?NumericQ] := -No NIntegrate[
Abs[(x + V - I [CapitalGamma])/
Sqrt[(x + V - I [CapitalGamma])^2 - [CapitalDelta]^2]] ( (
E^(3.6265521643263314` x))/(1 + E^(
3.6265521643263314` x))^2), {x, -Infinity, Infinity},
PrecisionGoal -> 5] // Quiet;
f1 =
FindFit[data,
dos1[No, [CapitalGamma], [CapitalDelta],
V], {No, [CapitalGamma], [CapitalDelta]}, V]
(*Out[]= {No -> -2.08873, [CapitalGamma] ->
0.360637, [CapitalDelta] -> -2.04026}*)
Show[
ListPlot[data, PlotRange -> All, Frame -> True,
FrameLabel -> {"Bias voltag (mV)", "Normalized Conductance"}],
Plot[dos1[No, [CapitalGamma], [CapitalDelta], V] /. f1, {V, -15,
15}, PlotRange -> All]]
Now we can fixed $Gamma=0.2$ as in the paper and define new function
[CapitalGamma] = .2;
dos2[No_?NumericQ, N1_?NumericQ, [CapitalDelta]_?NumericQ,
V_?NumericQ] :=
NIntegrate[ (No Sign[
V] Re[(x + V - I [CapitalGamma])/
Sqrt[(x + V - I [CapitalGamma])^2 - [CapitalDelta]^2]] -
N1 Abs[(x + V - I [CapitalGamma])/
Sqrt[(x + V - I [CapitalGamma])^2 - [CapitalDelta]^2]]) ( (
E^(3.6265521643263314` x))/(1 + E^(
3.6265521643263314` x))^2), {x, -60, 60}, PrecisionGoal -> 5] //
Quiet;
With this function we have
f2 =
FindFit[data,
dos2[No, N1, [CapitalDelta], V], {No, N1, [CapitalDelta]}, V]
(*Out[]= {No -> 1.79784, N1 -> -0.386709, [CapitalDelta] -> 1.62088}*)
It is looks much better than dos, dos1, and we can improve it by varying $Gamma$
Show[ListPlot[data, PlotRange -> All, Frame -> True,
FrameLabel -> {"Bias voltag (mV)", "Normalized Conductance"}],
Plot[dos2[No, N1, [CapitalDelta], V] /. f2, {V, -15, 15},
PlotRange -> All]]
Answered by Alex Trounev on December 28, 2020
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