Andreev bound states in Josephson junctions of semi-Dirac semimetals
Author
Ipsita Mandal
Title
Andreev bound states in Josephson junctions of semi-Dirac semimetals
Description
Andreev bound states in Josephson junctions of semi-Dirac semimetals
Category
Academic Articles & Supplements
Keywords
Andreev bound states, Josephson junctions, semi-Dirac semimetals
URL
http://www.notebookarchive.org/2024-06-46qd4rb/
DOI
https://notebookarchive.org/2024-06-46qd4rb
Date Added
2024-06-09
Date Last Modified
2024-06-09
File Size
1. megabyte
Supplements
Rights
Redistribution rights reserved
Download
Open in Wolfram Cloud
Andreev bound states in Josephson junctions of semi-Dirac semimetals
Andreev bound states in Josephson junctions of semi-Dirac semimetals
Ipsita Mandal
In[]:=
$PrePrint=#/.{Csc[z_]:>1/Defer@Sin[z],Sec[z_]:>1/Defer@Cos[z]}&;(*PRB86,075124(2012):
*)s1=PauliMatrix[1];s2=PauliMatrix[2];s3=PauliMatrix[3];id=IdentityMatrix[2];Clear[ham];ham[kx_,ky_]=s1+s3ky;ham[kx,ky]//MatrixForm{evals,evecs}=FullSimplify[Eigensystem[ham[kx,ky]],Assumptions{kx,ky,k0,k}∈Reals&&k0>0]
2
kx
k0
Out[]//MatrixForm=
ky | 2 kx k0 |
2 kx k0 | -ky |
Out[]=
-+,+,,1,,1
4
kx
2
k0
2
ky
k0
4
kx
2
k0
2
ky
k0
k0ky-+
4
kx
2
k0
2
ky
2
kx
k0ky++
4
kx
2
k0
2
ky
2
kx
In[]:=
{k0ky+ens,}/.{ky->,kx->,k0->1,en->"E"}+/.{ky->,kx->,k0->1,en->"E"}
2
kx
K
z
K
x
1
4
kx
2
(k0ky+ens)
K
z
K
x
Out[]=
{Es+,}
K
z
2
K
x
Out[]=
1
4
K
x
2
(Es+)
K
z
In[]:=
Clear[ham,Kx,kx,Kz,kz];(*
*)kx=2mvKx/hbar;kz=2mvKz/hbar;ham=PauliMatrix[1]+PauliMatrix[3]hbarvkz;MatrixFormEigensystem[ham](2mv)^2/(2m)(*
*)
2
hbar
2
kx
2m
ham
2m
2
v
Out[]//MatrixForm=
Kz | 2 Kx |
2 Kx | -Kz |
Out[]=
-2+m,2+m,-,1,-,1
4
Kx
2
Kz
2
v
4
Kx
2
Kz
2
v
-Kz++
4
Kx
2
Kz
2
Kx
-Kz-+
4
Kx
2
Kz
2
Kx
Out[]=
2m
2
v
In[]:=
-,1,-,1
2
Kx
-Kz++
4
Kx
2
Kz
2
Kx
2
Kx
-Kz-+
4
Kx
2
Kz
2
Kx
Out[]=
Kz-+,,Kz++,
4
Kx
2
Kz
2
Kx
4
Kx
2
Kz
2
Kx
In[]:=
Clear[v,euprt,huprt];v[s_]=,1;
2
kx
k0ky+sen
2
kx
In[]:=
FullSimplify[Normalize[v[s]],Assumptions{kx,ky,en,k0,s}∈Reals]
Out[]=
+,+
k0ky+ens
4
kx
2
(k0ky+ens)
2
kx
4
kx
2
(k0ky+ens)
(*
*)
In[]:=
Clear[u,v,ustar,vstar,μ,wv,mat,matr,φ,exp,ψ,hamil,hbdg];(*electronlikewvfn*)pair=id;Δ=d0id;Δdag=d0ConjugateTranspose[pair];μ=+k0-Ω;wv=k0ky++,;ψ=Joinwv,pair.wv/.ustar->v;mat=FullSimplify[{{ham[kx,ky]-μid,Δ},{Δdag,-ham[kx,ky]+μid}},Assumptions{kx,ky,d0,μ}∈Reals];mat//MatrixForm;hbdg=ArrayFlatten[mat];hamil=hbdg//FullSimplify;exp=hamil.ψ//FullSimplify;(exp-ϵψ)/.Ω->-//FullSimplifyeβψ/.{d0->(ϵ-Ω)eβ,α->θ/2}Clear[k,φ,Ω,Ef,μ];
φ
-φ
4
kx
2
k0
2
ky
4
kx
2
k0
2
ky
2
kx
ustar
v
φ
ϵ-Ω
d0
2
d0
2
ϵ
Out[]=
{0,0,0,0}
Out[]=
eβk0ky++,eβ,k0ky++,
4
kx
2
k0
2
ky
2
kx
-φ
4
kx
2
k0
2
ky
-φ
2
kx
In[]:=
Clear[u,v,ustar,vstar,wv,mat,matr,φ,exp,ψ,hamil,hbdg];(*holelikewvfn*)pair=id;Δ=d0id;Δdag=d0ConjugateTranspose[pair];μ=+k0+Ω;wv=k0ky++,;ψ=Joinwv,pair.wv/.ustar->v;mat=FullSimplify[{{ham[kx,ky]-μid,Δ},{Δdag,-ham[kx,ky]+μid}},Assumptions{kx,ky,d0,μ}∈Reals];mat//MatrixForm;hbdg=ArrayFlatten[mat];hamil=hbdg//FullSimplify;exp=hamil.ψ//FullSimplify;(exp-ϵψ)/.Ω->-//FullSimplifyClear[k,φ,Ω,Ef,μ];ψ/.{d0->(ϵ+Ω)/eβ,α->θ/2}
φ
-φ
4
kx
2
k0
2
ky
4
kx
2
k0
2
ky
2
kx
ustar
v
φ
ϵ+Ω
d0
2
d0
2
ϵ
Out[]=
{0,0,0,0}
Out[]=
k0ky++,,eβk0ky++,eβ
4
kx
2
k0
2
ky
2
kx
-φ
4
kx
2
k0
2
ky
-φ
2
kx
In[]:=
Solved0==(ϵ+Ω),Ω
β
Out[]=
{{Ωd0-ϵ}}
β
In[]:=
(ky-ef),,eβ(ky-ef),eβ/.eβ->,ky->,kx->,ef->
2
kx
-φ
-φ
2
kx
β
K
z
K
x
e
F
Out[]=
-+,,(-+),
e
F
K
z
2
K
x
β-φ
e
F
K
z
β-φ
2
K
x
In[]:=
Clear[e1,h1,es1,hs1,ef,v0];k0=1;e1[ef_,v0_,kx_,ky_]={k0ky+(ef-v0)k0,,0,0};h1[ef_,v0_,kx_,ky_]={0,0,k0ky+(-ef+v0)k0,};(*rtsc*)es1[ef_,kx_,ky_,eβ_,φ_]=eβk0(ky+ef),eβ,k0(ky+ef),;hs1[ef_,kx_,ky_,eβ_,φ_]=k0(ky-ef),,eβk0(ky-ef),eβ;Clear[k0];
2
kx
2
kx
2
kx
-φ
-φ
2
kx
2
kx
-φ
-φ
2
kx
In[]:=
4
kx
2
k
2
ky
K
z
K
x
Out[]=
4
K
x
2
k
2
K
z
In[]:=
Solve+==k0Ef,kx
4
kx
2
k0
2
ky
Out[]=
kx-,kx-,kx,kx
k0
1/4
(-)
2
Ef
2
ky
k0
1/4
(-)
2
Ef
2
ky
k0
1/4
(-)
2
Ef
2
ky
k0
1/4
(-)
2
Ef
2
ky
In[]:=
Clear[L,Ef,V0,bdy1,bdy2,wv2,wv1,eβ,arr,array,eβ,cmat,kxpe];kxpe=kxe;kxh=-kxe;kxph=-kxpe;wv1[x_]=ales1[ef,-kxe,ky,eβ,0]+blhs1[ef,-kxh,ky,eβ,0]+cles1[ef,-kxpe,ky,eβ,0]+dlhs1[ef,kxph,ky,eβ,0];wv2[x_]=ares1[ef,kxe,ky,eβ,ϕ]+brhs1[ef,kxh,ky,eβ,ϕ]++cres1[ef,kxpe,ky,eβ,ϕ]+drhs1[ef,-kxph,ky,eβ,ϕ];der1[x_]=D[wv1[x],x];der2[x_]=D[wv2[x],x];bdy1=wv1[0]-wv2[0];bdy2=der1[0]-der2[0]+V0wv1[0];bdy2//MatrixForm
-kxex
-kxhx
-kxpex
kxphx
kxe(x)
kxh(x)
kxpe(x)
-kxph(x)
Out[]//MatrixForm=
blkxe(-ef+ky)+brkxe(-ef+ky)+dlkxe(-ef+ky)+drkxe(-ef+ky)-aleβkxe(ef+ky)-areβkxe(ef+ky)+cleβkxe(ef+ky)+creβkxe(ef+ky)+(bl(-ef+ky)+dl(-ef+ky)+aleβ(ef+ky)+cleβ(ef+ky))V0 |
bl 3 kxe 3 kxe 3 kxe 3 kxe 3 kxe 3 kxe 3 kxe 3 kxe 2 kxe 2 kxe 2 kxe 2 kxe |
bleβkxe(-ef+ky)+dleβkxe(-ef+ky)+br -ϕ -ϕ -ϕ -ϕ |
-al 3 kxe 3 kxe -ϕ 3 kxe -ϕ 3 kxe 3 kxe 3 kxe -ϕ 3 kxe -ϕ 3 kxe 2 kxe 2 kxe 2 kxe 2 kxe |
In[]:=
sol=Solve[{bdy1[[1]]==0,bdy1[[2]]==0,bdy1[[3]]==0,bdy1[[4]]==0},{al,bl,cl,dl}]//Flatten//FullSimplifybdy20=(-1+)(-+)bdy2/.sol//FullSimplify;bdy20//MatrixForm
2
eβ
2
ef
2
ky
Out[]=
alar-1+(ef+ky)-(-1+)eβ(dref-brky),bl-br-(ef-ky)+(-1+)eβ(cref+arky),clcr-1+(ef+ky)-(-1+)eβ(bref-drky),dl-dr-(ef-ky)+(-1+)eβ(aref+crky)
-ϕ
ϕ
2
eβ
ϕ
(-1+)(ef+ky)
2
eβ
-ϕ
ϕ
2
eβ
ϕ
(-1+)(ef-ky)
2
eβ
-ϕ
ϕ
2
eβ
ϕ
(-1+)(ef+ky)
2
eβ
-ϕ
ϕ
2
eβ
ϕ
(-1+)(ef-ky)
2
eβ
Out[]//MatrixForm=
-ϕ ϕ 2 eβ 2 eβ 2 eβ 2 eβ 3 eβ 2 eβ |
- -ϕ 2 kxe ϕ 2 eβ 2 eβ 2 eβ 2 eβ ϕ 2 eβ 2 eβ 2 eβ 2 eβ ϕ 2 eβ 2 eβ |
-ϕ 2 eβ ϕ 2 eβ 2 eβ 2 eβ ϕ 2 eβ 2 eβ ϕ 2 eβ 2 eβ ϕ 2 eβ 2 eβ ϕ 2 eβ 2 eβ 2 eβ |
-ϕ 2 kxe ϕ 2 eβ 2 eβ ϕ 2 eβ 2 eβ 2 eβ ϕ 2 eβ 2 eβ ϕ 2 eβ ϕ 2 eβ 2 eβ 2 eβ 2 eβ 2 eβ |
In[]:=
arr2=CoefficientArrays[{bdy20[[1]]==0,bdy20[[2]]==0,bdy20[[3]]==0,bdy20[[4]]==0},{ar,br,cr,dr}];array2=Normal[arr2]//FullSimplify;cmat2=array2[[2]];Dimensions[cmat2]cmat2//MatrixForm
Out[]=
{4,4}
In[]:=
Out[]//MatrixForm=
(*
*)
In[]:=
Sin[x]-Cos[x]//TrigToExpSin[x]-Cos[x]//TrigToExp
Out[]=
-
x
Out[]=
-
-x
In[]:=
Clear[det];det=/.eβ->
Det[cmat2]//FullSimplify
-4
-2ϕ
2
(-1+e2b)
4
kxe
3
(ef-ky)
3
(ef+ky)
e2b
//FullSimplifyOut[]=
-(4(1+)+)+2(2kxe+V0)(8(1+)+4(1+(-1+e2b)e2b)V0+2kxe+)+(-8(2+e2b(-4+e2b(7+2(-2+e2b)e2b)))-16(1+(-1+e2b)e2b)V0-8-4kxe-)+8e2b(-(2kxe+V0)(4kxe+V0)+2kxe(2(1+(-1+e2b)e2b)kxe+V0))Cos[ϕ]-8Cos[2ϕ]
2
(-1+e2b)
4
ky
2
(2kxe+V0)
2
e2b
2
kxe
2
(-1+e2b)
2
V0
2
(-1+e2b)
2
ef
2
ky
2
e2b
3
kxe
2
kxe
2
(-1+e2b)
2
V0
2
(-1+e2b)
3
V0
4
ef
4
kxe
2
(-1+e2b)
3
kxe
4
(-1+e2b)
2
kxe
2
V0
4
(-1+e2b)
3
V0
4
(-1+e2b)
4
V0
2
kxe
2
(-1+e2b)
4
ky
2
(2kxe+V0)
2
(-1+e2b)
2
ef
2
ky
4
ef
2
(-1+e2b)
2
e2b
4
ef
4
kxe
In[]:=
Clear[det];det=-(4(1+)+)+2(2kxe+V0)(8(1+)+4(1+(-1+e2b)e2b)V0+2kxe+)+(-8(2+e2b(-4+e2b(7+2(-2+e2b)e2b)))-16(1+(-1+e2b)e2b)V0-8-4kxe-)+8e2b(-(2kxe+V0)(4kxe+V0)+2kxe(2(1+(-1+e2b)e2b)kxe+V0))Cos[ϕ]-8Cos[2ϕ];
2
(-1+e2b)
4
ky
2
(2kxe+V0)
2
e2b
2
kxe
2
(-1+e2b)
2
V0
2
(-1+e2b)
2
ef
2
ky
2
e2b
3
kxe
2
kxe
2
(-1+e2b)
2
V0
2
(-1+e2b)
3
V0
4
ef
4
kxe
2
(-1+e2b)
3
kxe
4
(-1+e2b)
2
kxe
2
V0
4
(-1+e2b)
3
V0
4
(-1+e2b)
4
V0
2
kxe
2
(-1+e2b)
4
ky
2
(2kxe+V0)
2
(-1+e2b)
2
ef
2
ky
4
ef
2
(-1+e2b)
2
e2b
4
ef
4
kxe
In[]:=
Exponent[det,e2b]
Out[]=
4
In[]:=
det1=det/.{e2b->c2b+s2b};{real,imag}=ComplexExpand[ReIm[det1]]//Simplify;(*reim*)Exponent[real,{c2b,s2b}]Exponent[imag,{c2b,s2b}]
Out[]=
{4,4}
Out[]=
{3,3}
In[]:=
Clear[yre,yim,phi,ell,kp];yre[V0_,ef_,ky_,{c2b_,s2b_},ϕ_,kxe_]=real;yim[V0_,ef_,ky_,{c2b_,s2b_},ϕ_,kxe_]=imag;yre[V0,ef,ky,{c2b,s2b},ϕ,kxe];yim[V0,ef,ky,{c2b,s2b},ϕ,kxe];
In[]:=
(*wetakeV0>fen*)Clear[Ef,V0,L,eβ,φ,kxe,kxh,kxpe,kxph,eβ,pow,re,im,twob,s2b,c2b,soln1,i,ebyd];trs[Ef_,V0_,ϕ_,kperp_]:=Block{kxe,kxh,kxpe,kxph,eβ,pow,re,im,twob,s2b,c2b,soln1,i,len,ebyd={},k0=1},(*β=ArcCos[ϵ/d0]=>ϵ=d0Cosβ*)kxe=;re=yre[V0,Ef,kperp,{c2b,s2b},ϕ,kxe];im=yim[V0,Ef,kperp,{c2b,s2b},ϕ,kxe];(*Print[re];Print[im];*)soln1=Quiet[Solve[re==0&&im==0,{c2b,s2b},Reals]];len=Length[soln1];Fori=1,i<=len,i++,If(Abs[+-1]/.soln1[[i]])<=&&-1<=(c2b/.soln1[[i]])<=1&&-1<=(s2b/.soln1[[i]])<=1,ebyd=Appendebyd,/.soln1[[i]],0;;ebyd;fen=1;v0=80;phi=1.0*π/5;kp=fen*0.1;trs[fen,v0,phi,kp]
k0
1/4
(-)
2
(Ef)
2
kperp
2
s2b
2
c2b
-1
10
1+c2b
2
Out[]=
{0.9981,0.9981}
Plots
Plots
Clear[kp,list,len,dat];fen=1;v0=80;phi=1.0*π/5;list=Table[{kp,trs[fen,v0,phi,kp]},{kp,fen*1.0*,fen,fen*0.01}]//Quiet;list=DeleteCases[list,{_,{}}];len=Length[list];dat11=Table[{list[[i,1]],list[[i,2]][[1]]},{i,1,len}];dat12=Table[{list[[i,1]],If[Length[list[[i,2]]]>1,list[[i,2]][[2]],{}]},{i,1,len}];ListLinePlot[dat11]
-4
10
In[]:=
dat11dat12
In[]:=
Clear[kp,list,len,dat];fen=1;v0=90;phi=1.0*π/5;list=Table[{kp,trs[fen,v0,phi,kp]},{kp,fen*1.0*,fen,fen*0.01}]//Quiet;list=DeleteCases[list,{_,{}}];len=Length[list];dat21=Table[{list[[i,1]],list[[i,2]][[1]]},{i,1,len}];dat22=Table[{list[[i,1]],If[Length[list[[i,2]]]>1,list[[i,2]][[2]],{}]},{i,1,len}];
-4
10
In[]:=
Clear[kp,list,len,dat];fen=1;v0=100;phi=1.0*π/5;list=Table[{kp,trs[fen,v0,phi,kp]},{kp,fen*1.0*,fen,fen*0.01}]//Quiet;list=DeleteCases[list,{_,{}}];len=Length[list];dat31=Table[{list[[i,1]],list[[i,2]][[1]]},{i,1,len}];dat32=Table[{list[[i,1]],If[Length[list[[i,2]]]>1,list[[i,2]][[2]],{}]},{i,1,len}];
-4
10
In[]:=
Clear[kp,list,len,dat];fen=1;v0=110;phi=1.0*π/5;list=Table[{kp,trs[fen,v0,phi,kp]},{kp,fen*,fen,fen*0.01}]//Quiet;list=DeleteCases[list,{_,{}}];len=Length[list];dat41=Table[{list[[i,1]],list[[i,2]][[1]]},{i,1,len}];dat42=Table[{list[[i,1]],If[Length[list[[i,2]]]>1,list[[i,2]][[2]],{}]},{i,1,len}];
-4
10.0
In[]:=
SetDirectory[NotebookDirectory[]];th=0.007;pl=ListLinePlot[{dat11,dat21,dat31,dat41},PlotRange->{All,{Automatic,1}}(**),PlotStyle(**){Directive[Darker[Red],Thickness[th]],Directive[Orange,Thickness[th]],Directive[Darker[Cyan],Thickness[th]],Directive[Darker[Blue],Thickness[th]]}(**),FrameTrue,Axes->None,BaseStyle{FontFamily"LM Roman 10",Bold},Frame->True,FrameStyle->{Directive[Black,24],Directive[Black,22]},FrameLabel{"/","|ε|/"},FrameTicksStyleDirective[Darker[Gray],16],ImageSize500(******),PlotLabelStyle["{, ϕ} = {1, π/5}",22,FontFamily->"LM Roman 10",Lighter[Brown],Bold],(******)PlotLegendsPlaced[LineLegend[{" = 80"," = 90"," = 100"," = 110"},LabelStyle->{FontFamily->"LM Roman 10",Bold,18},LegendLayout{"Row",1}],{Right,Below}]]Export["plot1.pdf",Rasterize[pl,ImageResolution500],ImageSize500];
K
z
E
F
Δ
0
E
F
V
0
V
0
V
0
V
0
Out[]=
 | |||||||||
|
In[]:=
Clear[kp,list,len,dat];fen=1;v0=80;kp=fen*0.5;mag=Abs[fen];list=Quiet[Table[{φ,trs[fen,v0,φ,kp]},{φ,0,2.0π+2π/50,2.0π/100}]];Clear[len,dat11];len=Length[list];dat11=Table[{list[[i,1]],list[[i,2]][[1]]},{i,1,len}];dat12=Table[{list[[i,1]],If[Length[list[[i,2]]]>1,list[[i,2]][[2]],{}]},{i,1,len}];ListLinePlot[{dat11}]
Out[]=
In[]:=
Clear[kp,list,len,dat];fen=1;v0=90;kp=fen*0.5;mag=Abs[fen];list=Quiet[Table[{φ,trs[fen,v0,φ,kp]},{φ,0,2.0π+2π/50,2.0π/100}]];list=DeleteCases[list,{_,{}}];len=Length[list];dat21=Table[{list[[i,1]],list[[i,2]][[1]]},{i,1,len}];dat22=Table[{list[[i,1]],If[Length[list[[i,2]]]>1,list[[i,2]][[2]],{}]},{i,1,len}];
In[]:=
Clear[kp,list,len,dat];fen=1;v0=100;kp=fen*0.5;mag=Abs[fen];list=Quiet[Table[{φ,trs[fen,v0,φ,kp]},{φ,0,2.0π+2π/50,2.0π/100}]];list=DeleteCases[list,{_,{}}];len=Length[list];dat31=Table[{list[[i,1]],list[[i,2]][[1]]},{i,1,len}];dat32=Table[{list[[i,1]],If[Length[list[[i,2]]]>1,list[[i,2]][[2]],{}]},{i,1,len}];
In[]:=
Clear[kp,list,len,dat41];fen=1;v0=110;kp=fen*0.5;mag=Abs[fen];list=Quiet[Table[{φ,trs[fen,v0,φ,kp]},{φ,0,2.0π+2π/50,2.0π/100}]];list=DeleteCases[list,{_,{}}];len=Length[list];dat41=Table[{list[[i,1]],list[[i,2]][[1]]},{i,1,len}];dat42=Table[{list[[i,1]],If[Length[list[[i,2]]]>1,list[[i,2]][[2]],{}]},{i,1,len}];
In[]:=
ListPlot[dat41,PlotRangeAll]
In[]:=
SetDirectory[NotebookDirectory[]];th=0.007;pl=ListLinePlot[{dat11,dat21,dat31,dat41},PlotRange->{{0,2π},{Automatic,1}}(**),PlotStyle(***){Directive[Darker[Red],Thickness[th]],Directive[Orange,Thickness[th]],Directive[Darker[Cyan],Thickness[th]],Directive[Darker[Blue],Thickness[th]]}(***),FrameTrue,Axes->None,BaseStyle{FontFamily"LM Roman 10",Bold},FrameStyle->{Directive[Black,24],Directive[Black,24]},FrameLabel{"ϕ","|ε|/"},(**)FrameTicks{{0,π/2,π,3π/2,2π},Automatic},(**)FrameTicksStyleDirective[Darker[Gray],18],ImageSize500(******),PlotLabelStyle["{, } = {1, /2}",20,FontFamily->"LM Roman 10",Lighter[Brown],Bold],(******)PlotLegendsPlaced[LineLegend[{" = 80"," = 90"," = 100"," = 110"},LabelStyle->{FontFamily->"LM Roman 10",Bold,18},LegendLayout{"Row",1}],{Right,Below}]]Export["plot2.pdf",Rasterize[pl,ImageResolution500],ImageSize500];
Δ
0
E
F
K
z
E
F
V
0
V
0
V
0
V
0
Out[]=
 | |||||||||
|
3d Plots
3d Plots
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fen=1;v0=100;list=Quiet[Table[{φ,kp,trs[fen,v0,φ,kp]},{φ,0,2π,2.0π/100},{kp,fen*1.0*,fen*1.0,0.1}]];list0=DeleteCases[list,{_,{}}];list=Flatten[list0,1];len=Length[list];dat100=DeleteCases[Table[{list[[i,1]],list[[i,2]],If[Length[list[[i,3]]]>0,list[[i,3]][[1]],{}]},{i,1,len}],{_,{}}]//Quiet;
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SetDirectory[NotebookDirectory[]];pl=ListPlot3D[{dat100},InterpolationOrder2,PlotRange{{0,2π},Full,Full},BaseStyle{FontFamily"LM Roman 10",Bold},AxesLabel{"ϕ","","|ε|"},AxesStyle{{Bold,Black,30},{Bold,Black,26},{Bold,Black,30}},TicksStyleDirective[Darker[Gray],18],Ticks{{0,π/2,π,3π/2,2π},Automatic,{0.99,1}},ImageSize500,PlotLabelStyle["{, } = {1, 100}",22,FontFamily->"LM Roman 10",Lighter[Brown],Bold],ColorFunction->(ColorData[{"TemperatureMap","Reverse"}][#3]&),MeshStyleDarker[Gray],Mesh15,BoxRatios{0.7,0.7,1},ColorFunctionScalingTrue,ViewPoint{1.549475762943073`,-2.669330389483107`,1.3870112226777205`},ImagePadding->{{Automatic,10},{Automatic,10}}]Export["3dplot.pdf",Rasterize[pl,ImageResolution500],ImageSize500];
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dat1
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fen=1;v0=80;list=Quiet[Table[{φ,kp,trs[fen,v0,φ,kp]},{φ,0,2π,2.0π/100},{kp,fen*1.0*,fen*1.0,0.1}]];list0=DeleteCases[list,{_,{}}];list=Flatten[list0,1];len=Length[list];dat1=DeleteCases[Table[{{list[[i,1]],list[[i,2]]},If[Length[list[[i,3]]]>0,list[[i,3]][[1]],{}]},{i,1,len}],{_,{}}]//Quiet;Clear[φ,kp];ifn1=Interpolation[dat1];ifnde1[φ_,kp_]=D[ifn1[φ,kp],φ];j1[φ_]:=Block[{int,ans},int=2NIntegrate[ifnde1[φ,kp]Tanh[ifn1[φ,kp]/0.01]kp,{kp,fen*1.0*,fen*1.0}]//Quiet;If[NumericQ[int]==False,ans=0,ans=int];ans];jdat80=Quiet[Table[{φ,SetPrecision[Re[j1[φ]],5]},{φ,-0.1,2π+2.0π/100,2.0π/100}]];(*ListLinePlot[jdat80,PlotRange{{0,2π+0.1},All}]*)
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fen=1;v0=60;list=Quiet[Table[{φ,kp,trs[fen,v0,φ,kp]},{φ,0,2π,2.0π/100},{kp,fen*1.0*,fen*1.0,0.1}]];list0=DeleteCases[list,{_,{}}];list=Flatten[list0,1];len=Length[list];dat1=DeleteCases[Table[{{list[[i,1]],list[[i,2]]},If[Length[list[[i,3]]]>0,list[[i,3]][[1]],{}]},{i,1,len}],{_,{}}]//Quiet;Clear[φ,kp];ifn1=Interpolation[dat1];ifnde1[φ_,kp_]=D[ifn1[φ,kp],φ];j1[φ_]:=Block[{int,ans},int=2NIntegrate[ifnde1[φ,kp]Tanh[ifn1[φ,kp]/0.01]kp,{kp,fen*1.0*,fen*1.0}]//Quiet;If[NumericQ[int]==False,ans=0,ans=int];ans];jdat60=Quiet[Table[{φ,SetPrecision[Re[j1[φ]],5]},{φ,-0.1,2π+2.0π/100,2.0π/100}]];(*ListLinePlot[jdat60,PlotRange{{0,2π+0.1},All}]*)
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fen=1;v0=90;list=Quiet[Table[{φ,kp,trs[fen,v0,φ,kp]},{φ,0,2π,2.0π/100},{kp,fen*1.0*,fen*1.0,0.1}]];list0=DeleteCases[list,{_,{}}];list=Flatten[list0,1];len=Length[list];dat1=DeleteCases[Table[{{list[[i,1]],list[[i,2]]},If[Length[list[[i,3]]]>0,list[[i,3]][[1]],{}]},{i,1,len}],{_,{}}]//Quiet;Clear[φ,kp];ifn1=Interpolation[dat1];ifnde1[φ_,kp_]=D[ifn1[φ,kp],φ];j1[φ_]:=Block[{int,ans},int=2NIntegrate[ifnde1[φ,kp]Tanh[ifn1[φ,kp]/0.01]kp,{kp,fen*1.0*,fen*1.0}]//Quiet;If[NumericQ[int]==False,ans=0,ans=int];ans];jdat90=Quiet[Table[{φ,SetPrecision[Re[j1[φ]],5]},{φ,-0.1,2π+2.0π/100,2.0π/100}]];(*ListLinePlot[jdat90,PlotRange{{0,2π+0.1},All}]*)
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Clear[φ,kp,list];fen=1;v0=100;list=Quiet[Table[{φ,kp,trs[fen,v0,φ,kp]},{φ,0,2π,2.0π/100},{kp,fen*1.0*,fen*1.0,0.1}]];list0=DeleteCases[list,{_,{}}];list=Flatten[list0,1];len=Length[list];dat=DeleteCases[Table[{{list[[i,1]],list[[i,2]]},If[Length[list[[i,3]]]>0,list[[i,3]][[1]],{}]},{i,1,len}],{_,{}}]//Quiet;ifn1=Interpolation[dat];ifnde1[φ_,kp_]=D[ifn1[φ,kp],φ];j1[φ_]:=Block[{int,ans},int=2NIntegrate[ifnde1[φ,kp]Tanh[ifn1[φ,kp]/0.01]kp,{kp,fen*1.0*,fen*1.0}]//Quiet;If[NumericQ[int]==False,ans=0,ans=int];ans];jdat100=Quiet[Table[{φ,SetPrecision[Re[j1[φ]],5]},{φ,-0.1,2π+2.0π/100,2.0π/100}]];(*ListLinePlot[jdat100,PlotRange{{0,2π+0.1},All}]*)
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Clear[φ,kp,list];fen=1;v0=110;list=Quiet[Table[{φ,kp,trs[fen,v0,φ,kp]},{φ,0,2π,2.0π/100},{kp,fen*1.0*,fen*1.0,0.1}]];list0=DeleteCases[list,{_,{}}];list=Flatten[list0,1];len=Length[list];dat=DeleteCases[Table[{{list[[i,1]],list[[i,2]]},If[Length[list[[i,3]]]>0,list[[i,3]][[1]],{}]},{i,1,len}],{_,{}}]//Quiet;ifn1=Interpolation[dat];ifnde1[φ_,kp_]=D[ifn1[φ,kp],φ];j1[φ_]:=Block[{int,ans},int=2NIntegrate[ifnde1[φ,kp]Tanh[ifn1[φ,kp]/0.01]kp,{kp,fen*1.0*,fen*1.0}]//Quiet;If[NumericQ[int]==False,ans=0,ans=int];ans];jdat110=Quiet[Table[{φ,SetPrecision[Re[j1[φ]],5]},{φ,-0.1,2π+2.0π/100,2.0π/100}]];(*ListLinePlot[jdat110,PlotRange{{0,2π+0.1},All}]*)
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ListLinePlot[{jdat80,jdat90,jdat100,jdat110},PlotRangeAll,PlotLegendsAutomatic]
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th=0.007;pl=ListLinePlot[{jdat80,jdat90,jdat100,jdat110},PlotRange{{0,2π},{-0.0055,0.0055}},PlotStyle{(**){Darker[Red],Thickness[th]},(**){Orange,Thickness[th]}(**),{Darker[Cyan],Thickness[th]},(**){Darker[Blue],Thickness[th]}},FrameTrue,Axes->None,BaseStyle{FontFamily"LM Roman 10",Bold,24},Frame->True,FrameStyleThick,LabelStyle->Directive[Black],FrameLabel{"ϕ",""},FrameTicksStyleDirective[Darker[Gray],18],FrameTicks{{Automatic,None},{{0,π/2,π,3π/2,2π},None}},ImageSize500,GridLines{{0,π/2,π,3π/2,2π},Automatic},GridLinesStyleDirective[Darker[Gray]],(******)PlotLegendsPlaced[LineLegend[{" = 80"," = 90"," = 100"," = 110"},LabelStyle->{FontFamily->"LM Roman 10",Bold,18},LegendLayout{"Row",1}],{Right,Below}](**)]SetDirectory[NotebookDirectory[]];Export["jjc.pdf",Rasterize[pl,ImageResolution500],ImageSize500];
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Cite this as: Ipsita Mandal, "Andreev bound states in Josephson junctions of semi-Dirac semimetals" from the Notebook Archive (2024), https://notebookarchive.org/2024-06-46qd4rb
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