ABSTRACT (original article): We extend the Mattis-Bardeen theory for the dynamical response of superconductors to include different types of Hall responses. This is possible thanks to a recent modification of the quasiclassical Usadel equation, which allows for analyzing Hall effects in disordered superconductors and including the precise frequency dependence of such effects. Our results form a basis for analyzing dynamical experiments especially on novel thin-film superconductors, where ordinary Hall and spin Hall effects can both show up. CITATION (original article): Alberto Hijano, Sakineh Vosoughi-nia, F. Sebastián Bergeret, Pauli Virtanen, Tero T. Heikkilä (2023), Dynamical Hall responses of disordered superconductors, Phys. Rev. B 108, 104506. https://doi.org/10.1103/PhysRevB.108.104506

Definitions


Conductivity

The advanced GF is related to the retarded one through
A
g
0
=-
R
g
0
,
A
f
0
=
R
f
0
In[]:=
gR=g0τ3+f0τ1;​​gA=-g0τ3+f0τ1;​​gRp=g0pτ3+f0pτ1;​​gAp=-g0pτ3+f0pτ1;​​gRm=g0pτ3+f0pτ1;​​gAm=-g0pτ3+f0pτ1;​​hm=-hp;​​gR2=g0τ3+f0τ1;​​gA2=-g0τ3+f0τ1;​​h2=-h;​​δgR=δgR2τ3.σ2+δfR2τ1.σ2;​​δgA=δgA2τ3.σ2+δfA2τ1.σ2;​​δgK=δgK2τ3.σ2+δfK2τ1.σ2;

Derivation of the integrand of the longitudinal conductivity (Eq. (17) in the paper)

Combining Eqs. (12) and (14), and Fourier transforming into the energy domain, the time convolutions become energy integrations. The integrand for the longitudinal conductivity is given by:
In[]:=
integrandx0=FullSimplify
σD
16ω0
Tr[(gR+gA).((gR-gA)h)-τ3.gR2.τ3.((gRm-gAm)hm)-τ3.gAm.τ3.((gR2-gA2)h2)]/.{g0->g0[ϵ],f0->f0[ϵ],g0p->g0[ϵ+ω],f0p->f0[ϵ+ω],h->h[ϵ],hp->h[ϵ+ω]}
Out[]=
1
4ω0
σD-
2
Conjugate[f0[ϵ]]
h[ϵ]+2Conjugate[f0[ϵ]]h[ϵ+ω]Im[f0[ϵ+ω]]+h[ϵ]-
2
Conjugate[g0[ϵ]]
+
2
f0[ϵ]
+
2
g0[ϵ]
+2f0[ϵ+ω]Im[f0[ϵ]]-2g0[ϵ+ω]Re[g0[ϵ]]+2Conjugate[g0[ϵ]]h[ϵ+ω]Re[g0[ϵ+ω]]
We rewrite the integrand in a compact form
In[]:=
dσx0dϵ=
σD
2ω0
((Re[g0[ϵ+ω]]Re[g0[ϵ]]+Im[f0[ϵ+ω]]Im[f0[ϵ]])(h[ϵ+ω]-h[ϵ])+((-Re[g0[ϵ+ω]]Im[g0[ϵ]]+Im[f0[ϵ+ω]]Re[f0[ϵ]])h[ϵ+ω]+(-Im[g0[ϵ+ω]]Re[g0[ϵ]]+Re[f0[ϵ+ω]]Im[f0[ϵ]])h[ϵ]+2(Re[g0[ϵ]]Im[g0[ϵ]]+Re[f0[ϵ]]Im[f0[ϵ]])h[ϵ]));FullSimplify[dσx0dϵ==integrandx0]
Out[]=
True

Derivation of the integrand of the Hall conductivity (Eq. (18) in the paper)

In[]:=
integrandy0=FullSimplify-
σDωcτ
64ω
Tr[τ3.gRp.τ3.gRp.τ3.((gR-gA)h)+(τ3.gRp.τ3.((gRp-gAp)hp)+τ3.((gRp-gAp)hp).τ3.gAp).τ3.gA-τ3.gRp.(τ3.gR.τ3.((gR-gA)h)+τ3.((gR-gA)h).τ3.gA)-τ3.((gRp-gAp)hp).τ3.gA.τ3.gA+3τ3.((gR-gA)h-(gRp-gAp)hp)]/.{g0->g0[ϵ],f0->f0[ϵ],g0p->g0[ϵ+ω],f0p->f0[ϵ+ω],h->h[ϵ],hp->h[ϵ+ω]}
Out[]=
1
16ω
σDτωc-2f0[ϵ]f0[ϵ+ω](g0[ϵ]-2g0[ϵ+ω])-
2
f0[ϵ]
g0[ϵ+ω]+
2
g0[ϵ]
g0[ϵ+ω]h[ϵ]+Conjugate[g0[ϵ]]
2
Conjugate[f0[ϵ+ω]]
-
2
Conjugate[g0[ϵ+ω]]
-
2
f0[ϵ+ω]
+
2
g0[ϵ+ω]
h[ϵ+ω]+2Conjugate[f0[ϵ]](-Conjugate[g0[ϵ]]f0[ϵ+ω]h[ϵ]+Conjugate[f0[ϵ+ω]]Conjugate[g0[ϵ+ω]]h[ϵ+ω]+f0[ϵ+ω]g0[ϵ+ω]h[ϵ+ω]+2Conjugate[g0[ϵ]]h[ϵ+ω]Im[f0[ϵ+ω]])-4f0[ϵ+ω]g0[ϵ+ω]h[ϵ]Re[f0[ϵ]]+2-3+
2
f0[ϵ+ω]
-
2
g0[ϵ+ω]
h[ϵ]Re[g0[ϵ]]+6h[ϵ+ω]Re[g0[ϵ+ω]]+
2
Conjugate[f0[ϵ]]
(g0[ϵ+ω]h[ϵ]-2h[ϵ+ω]Re[g0[ϵ+ω]])+
2
Conjugate[g0[ϵ]]
(-g0[ϵ+ω]h[ϵ]+2h[ϵ+ω]Re[g0[ϵ+ω]])
We rewrite the integrand in a compact form
In[]:=
dσy0dϵ=
σDωcτ
16ω
6(Re[g0[ϵ+ω]]h[ϵ+ω]-Re[g0[ϵ]]h[ϵ])+2g0[ϵ]f0[ϵ]f0[ϵ+ω]+
2
g0[ϵ]
g0[ϵ+ω]-
2
f0[ϵ]
g0[ϵ+ω]+g0[ϵ]
2
g0[ϵ+ω]
-
2
f0[ϵ+ω]
+2f0[ϵ]g0[ϵ+ω]f0[ϵ+ω](h[ϵ+ω]-h[ϵ])+
2
g0[ϵ]
-
2
f0[ϵ]
g0[ϵ+ω]+g0[ϵ]
2
f0[ϵ+ω]
-
2
g0[ϵ+ω]
+2f0[ϵ]f0[ϵ+ω](g0[ϵ+ω]-g0[ϵ])h[ϵ]+
2
g0[ϵ]
-
2
f0[ϵ]
g0[ϵ+ω]+g0[ϵ]
2
f0[ϵ+ω]
-
2
g0[ϵ+ω]
+2f0[ϵ]f0[ϵ+ω](g0[ϵ+ω]-g0[ϵ])h[ϵ+ω];FullSimplify[dσy0dϵ==integrandy0]
Out[]=
True

Derivation of the integrand of the spin Hall conductivity (Eq. (25) in the paper)

In[]:=
integrandyz=FullSimplify
ℏσDmα
32
2
e
E0
Tr[σ3.(gRp.Commutator[σ1,δgK]+((gRp-gAp)hp).Commutator[σ1,δgA])]-​​
ℏσDτm
2
α
16eω
Tr[σ3.σ3.(τ3.((gR-gA)h)-((gRp-gAp)hp).τ3)]/.{g0->g0[ϵ],f0->f0[ϵ],g0p->g0[ϵ+ω],f0p->f0[ϵ+ω],h->h[ϵ],hp->h[ϵ+ω],δgR2->δgR2[ϵ,ω],δfR2->δfR2[ϵ,ω],δgA2->δgA2[ϵ,ω],δfA2->δfA2[ϵ,ω],δgK2->δgK2[ϵ,ω],δfK2->δfK2[ϵ,ω]}
Out[]=
1
4
2
e
E0ω
mασDℏ(-2eE0ατh[ϵ]Re[g0[ϵ]]-2ωh[ϵ+ω]Im[f0[ϵ+ω]]δfA2[ϵ,ω]+2h[ϵ+ω]Re[g0[ϵ+ω]](eE0ατ-ωδgA2[ϵ,ω])-ω(f0[ϵ+ω]δfK2[ϵ,ω]+g0[ϵ+ω]δgK2[ϵ,ω]))
We rewrite the integrand in a compact form
In[]:=
dσyzdϵ=-
ℏσDmα
4
2
e
E0
(2(Re[g0[ϵ+ω]]δgA2[ϵ,ω]+Im[f0[ϵ+ω]]δfA2[ϵ,ω])h[ϵ+ω]+g0[ϵ+ω]δgK2[ϵ,ω]+f0[ϵ+ω]δfK2[ϵ,ω])+
ℏσDτm
2
α
2eω
(Re[g0[ϵ+ω]]h[ϵ+ω]-Re[g0[ϵ]]h[ϵ]);FullSimplify[dσyzdϵ==integrandyz]
Out[]=
True

Parameters and functions

Self-consistent gap


Parameters

Parameters to be changed. All magnitudes are measured in units of
Δ
0
, e and m (with ℏ=1 and
k
B
=1). The configurations considered are a superconductor with a constant magnetic field and a superconductor with Rashba SOC. The longitudinal conductivity is the same for both configurations, while the Hall and spin Hall conductivities correspond to each configuration, respectively. The normalized conductivities depend on the following parameters: Γ, T and
ϵ
α
.
In[]:=
(*Units:*)​​ℏ=1;​​e=1;​​m=1;​​Δ0=1;​​Tc0=Δ0/N[Pi/Exp[EulerGamma]];​​(*Normalizedconductivitiesdonotdependonthefollowingparameters:*)​​σD=1;​​E0=1;​​D0=1;​​τ=5*^-2ℏ/Δ0;​​ωc=0.2Δ0/ℏ;​​ωp=3*^4Δ0/ℏ;​​(*Parametersofthemodel:*)​​Γ=1*^-4Δ0;​​T=1*^-4Tc0;​​Δ=Δfun[T,Tc0];​​α=Sqrt[0.25Δ0/(D0
2
m
)];

BCS Green’s function

In[]:=
g0[ϵ_?NumericQ]:=
-(ϵ+Γ)
2
Δ
-
2
(ϵ+Γ)
;​​f0[ϵ_?NumericQ]:=
Δ
2
Δ
-
2
(ϵ+Γ)
;​​gRBCS[ϵ_?NumericQ]:=g0[ϵ]τ3+f0[ϵ]τ1;​​gABCS[ϵ_?NumericQ]:=-g0[ϵ]τ3+f0[ϵ]τ1;​​h[ϵ_?NumericQ]:=Tanh
ϵ
2T
;

Correction to the GF

Retarded GF
In[]:=
δgRfun[ϵ_?NumericQ,ω0_?NumericQ]:=Chop4D0τ
2
m
3
α
e
E0
ω0
Inverse-4
2
m
2
α
D0-
2
Δ
-
2
(ϵ+Γ)
gRBCS[ϵ+ω0]+ω0τ3+(ϵ+Γ)τ3-Δτ1.(τ3.gRBCS[ϵ]-gRBCS[ϵ+ω0].τ3).σ2;​​δgR2[ϵ_?NumericQ,ω0_?NumericQ]:=
1
4
Tr[τ3.σ2.δgRfun[ϵ,ω0]];​​δfR2[ϵ_?NumericQ,ω0_?NumericQ]:=
1
4
Tr[τ1.σ2.δgRfun[ϵ,ω0]];
Advanced GF
In[]:=
δgAfun[ϵ_?NumericQ,ω0_?NumericQ]:=Chop4D0τ
2
m
3
α
e
E0
ω0
Inverse-4
2
m
2
α
D0-
2
Δ
-
2
(ϵ-Γ)
gABCS[ϵ+ω0]+ω0τ3+(ϵ-Γ)τ3-Δτ1.(τ3.gABCS[ϵ]-gABCS[ϵ+ω0].τ3).σ2;​​δgA2[ϵ_?NumericQ,ω0_?NumericQ]:=
1
4
Tr[τ3.σ2.δgAfun[ϵ,ω0]];​​δfA2[ϵ_?NumericQ,ω0_?NumericQ]:=
1
4
Tr[τ1.σ2.δgAfun[ϵ,ω0]];
Keldysh GF

Faraday and Kerr angles

Faraday angle
Kerr angle

Numerical computation

Conductivity computation

Faraday and Kerr angles computation

Plots

Longitudinal conductivity (used for Fig. 1 in the paper)

Hall conductivity (used for Fig. 2(a) in the paper)

Kerr angle (used for Fig. 2(b) in the paper)

Faraday angle (used for Fig. 2(c) in the paper)

Spin Hall conductivity (used for Fig. 4 in the paper)

CITE THIS NOTEBOOK

Dynamical Hall responses of disordered superconductors​
by Alberto Hijano, Sakineh Vosoughi-nia, F. Sebastián Bergeret, Pauli Virtanen and Tero T. Heikkilä
Wolfram Community, STAFF PICKS, May 20, 2024
​https://community.wolfram.com/groups/-/m/t/3179263