Superconducting magnetoelectric effects in mesoscopic hybrid structures
Mostafa Tanhayi Ahari Yaroslav Tserkovnyak






Materials Research Laboratory, The Grainger College of Engineering, University of Illinois Urbana-Champaign, Illinois 61801, USA
Department of Physics and Astronomy & Bhaumik Institute for Theoretical Physics, University of California, Los Angeles, California 90095, USA


Department of Physics and Astronomy & Bhaumik Institute for Theoretical Physics, University of California, Los Angeles, California 90095, USA


In superconductors that lack inversion symmetry, a supercurrent flow can lead to nondissipative magnetoelectric effects. We offer a straightforward formalism to obtain a supercurrent-induced magnetization in superconductors with broken inversion symmetry, 
which may have orbital, layer, sublattice, or valley degrees of freedom—multiband noncentrosymmetric superconductors. The nondissipative 
magnetoelectric effect may find applications in fabricating quantum computation platforms or efficient superconducting spintronic devices. 
We explore how the 
current-induced magnetization 
can be employed to create and manipulate Majorana zero modes in a simple hybrid structure.  
 

Superconducting magnetoelectric effects in mesoscopic hybrid structures
    Yaroslav Tserkovnyak
    April 29, 2024
=======================================================================




Introduction.—Hybrid structures involving magnetic materials and normal metals are one of the building blocks of spintronic devices. 
Motivated by mitigating dissipation and enhancing device performance, combining superconductivity with spintronics has led to 
interesting equilibrium and nonequilibrium phenomena that fuel interest in superconducting spintronics <cit.>. The injection of 
a spin-polarized (super)current into the superconductor, whose transport may be facilitated by an unconventional superconducting order, such as a triplet pairing, has been
of central interest. In this Letter, we primarily focus on the effect of the supercurrent-induced magnetization on an adjacent material in a hybrid structure 
with an emphasis on potential utility in superconducting spintronics and quantum computation devices. 

In multiband superconductors, where electrons have multiple degrees of freedom, such as orbital,
layer, sublattice, or valley, a supercurrent flow may lead to a sizeable induced magnetization.
The source of the induced magnetization is spin polarization <cit.> 
and/or orbital magnetic
moments <cit.>, where the former is due to spin-orbit coupling (SOC) and the latter is related to a Berry curvature of the Bloch electronic bands in these materials. Importantly, the supercurrent-induced
orbital magnetization can be orders of magnitude greater than that due to the
spin <cit.>. Due to the increasing accessibility 
of multiband superconductors with strong SOC, such as TMDs (transition metal dichalcogenides) <cit.>, 
or large orbital moments, such as twisted bilayer 
graphene <cit.>, the supercurrent-induced magnetoelectric effects may play a more 
functional role in mesoscopic hybrid structures, particularly in enhancing the efficiency of magnetic memory manipulations <cit.>.





In this Letter, we offer a simple formulation of superconducting magnetoelectric effects for multiband superconductors 
in the static regime, encompassing both orbital and spin magnetization contributions. 
The superconducting magnetoelectric effect has been studied in Ref. <cit.>, where
the evolution of the magnetoelectric effect across the superconductor-normal metal phase transition is discussed. As we show, our results do not match 
Ref. <cit.> (in the DC limit) and reveal a greater sensitivity to the characteristics of the pairing gap function. 
Therefore, we suggest that our approach more accurately accounts for the influence of anisotropic pairing gaps. 
Importantly, the magnetization induced by supercurrents in our study may provide additional means to probe the unconventional superconducting pair potentials. Furthermore, we provide an expression for supercurrent flow in multiband superconductors that agrees with earlier results, such as those in Refs. <cit.>, establishing connections between supercurrent density, Fermi surface parameters, and the superconducting energy gap. Additionally, we highlight the reciprocal nature of this phenomenon: while coupling to magnetization, such as that of an adjacent magnet or an applied magnetic field, can induce
supercurrent. Conversely, supercurrent-induced magnetization can exert torque on the adjacent magnet [see Fig. <ref>].



 




When a superconductor is in contact with a non-superconducting material, a weak superconductivity is induced
in the normal material over mesoscopic distances (superconducting proximity effect) <cit.>. We will show that the topological character of the 
induced superconductivity in the normal material can be controlled by the magnetoelectric effect in the parent
superconductor. 




The physics here is analogous to the induced topological phase in a quantum wire
discussed in Ref. <cit.>, where zero-energy Majorana bound states are formed in the topological phase. However, here we propose a setup in which the Majorana fermions are electrically created and manipulated via the magnetoelectric effect.

In particular, we propose a simple supercurrent protocol that realizes a braiding operation for the Majorana zero 
modes in a trijunction formed by the wires. Our setup can be relevant for fabricating a platform for quantum 
computation in a web of nanowires <cit.>. 


Magnetoelectric effect in multiband superconductors.—From the viewpoint of symmetry, a linear 
current-induced magnetization can occur if inversion symmetry is broken <cit.> 
(a noncentrosymmetric material). As a result, we consider a superconductor with broken inversion symmetry 
described by a time-reversal symmetric normal state Hamiltonian H_0(p). 
We assume that close to the Fermi surface, the Bloch band Kramers' splitting due to the broken symmetry  
is much larger than the superconducting gap, 
as is often a realistic limit for the multiband superconductors <cit.>. This assumption justifies
a semiclassical treatment of small perturbations (relative to the band splitting), under which the electron dynamics would be confined within the individual Bloch bands. 
Moreover, in the superconducting state, 
Cooper pairs are formed by time-reversal electron partners belonging to the same Bloch band with opposite momentum |u_ν,p⟩ and |u_ν,-p⟩ <cit.>, 
where ν is the band index. It has been shown that the resultant intraband Cooper pairing is the energetically
favorable (stable) superconducting state <cit.>. As a consequence of projecting physical quantities onto a single band, we focus on the intraband effects, with interband contributions related to quantum metric effects being overlooked <cit.> (see the Discussion for further details). The Bogoliubov-de Gennes Hamiltonian written in the
Bloch band basis reads

    Ĥ(p)=
    [   ξ(p)   Δ(p); Δ^*(p)  -ξ(p) ],
 
where ξ(p)=ξ(-p) is a diagonal matrix whose entries are the Bloch 
band energies ξ_ν(p) (measured from chemical potential), which are obtained by a unitary transformation
U_p  H_0(p) U^†_p=ξ(p). 
The pairing gap function 
Δ(p)=U_p Δ_0(p)U^T_-p is a diagonal matrix 
with elements Δ_ν(p), where Δ_0(p) is the usual pairing function written in the space of the electronic degrees of freedom such as 
spin and valley <cit.>. Finally, the quasiparticle energy for band ν is given by E_ν=√(ξ_ν^2+|Δ_ν|^2).   

Under a uniform supercurrent flow, quasiparticle energies get shifted (Doppler shift <cit.>)  as 
E_ν+q·v_ν, where v_ν =∇_pξ_ν(p), and 2q is the Cooper pair momentum (q/m_e is usually called the superfluid velocity, where m_e is the electron mass). 
It can be checked <cit.> that the total supercurrent and supercurrent-induced magnetization can be written as j_s= 𝕋·q and M= ℚ·q, respectively, where 


    𝕋_ij   = ∫ dτ∑_ν( ∂ f_ν/∂ E_νv_ν,j-∂ n_ν/∂ p_j) ev_ν,i, 
    ℚ_ij   =   ∫ dτ∑_ν( ∂ f_ν/∂ E_νv_ν,j-∂ n_ν/∂ p_j)  M_ν,i.


Here, f_ν≡ f(E_ν) 
is the Fermi-Dirac distribution function which determines the quasiparticle occupancy, n_ν=1/2(1-ξ_ν/E_ν(1-2f_ν) ) is the 
occupancy of the electronic single-particle state at band ν with momentum p in the superconducting state <cit.>, 
and dτ=d^dp/(2πħ)^d with d being the dimension of the system. M_ν(p) is the 
total magnetization pertaining to Bloch band ν that can be decomposed into spin and orbital <cit.> components, 
M_ν(p)= s_ν(p)+m_ν(p), where 

    s_ν(p)=⟨ u_ν,p|gμ_Bσ/2|u_ν,p⟩

is the spin magnetic moment, and m_ν(p) is the orbital
magnetic moment <cit.> for a 3D crystal 

    m_ν(p)   =  eħ/2Im⟨∇_pu_ν,p|×[H_0(p)-ξ_ν(p)]|∇_pu_ν,p⟩.

A nonzero orbital magnetic moment is an intrinsic property of the band that can roughly be interpreted as
a self-rotation of the electron wave function 
around its center of mass <cit.>, which could indicate a nonzero 
Berry curvature Ω_ν,p=iħ^2⟨∇_pu_ν,p|× |∇_pu_ν,p⟩ in the band structure.


The terms proportional to ∂ n_ν/∂ p_j in Eqs. (<ref>) differ from those obtained
in Ref. <cit.> (in the DC limit), 
wherein the normal state occupancy of the 
single-particle state at band ν, denoted as f(ξ_ν), is employed instead of n_ν <cit.>. 
However, Eq. (<ref>) concurs with the conventional (intraband) superfluid density as established in prior studies <cit.>, where the terms proportional to ∂ f_ν/∂ E_ν and ∂ n_ν/∂ p_j are identified, respectively, as the paramagnetic and diamagnetic contributions to the supercurrent <cit.>. This alignment offers reassurance regarding the validity of Eq. (<ref>) in capturing the intraband current-induced magnetization. 


Application to two and four-band models.—Let us first consider a two-band system described by a normal state Hamiltonian 
H_0(p)=ξ_pσ_0+ g_p·σ, 
where σ is the vector of Pauli matrices acting on the spin basis, and σ_0 is a 2× 2 identity matrix. As a result of broken inversion symmetry, g_p=- g_-p, the spin degeneracy of electrons is lifted suggesting
a two-band description of the Fermi surface ξ_ν (p)=ξ_p+ν | g_p|, 
with ν=±1 labeling the bands. Finally, it can be checked that the spin magnetic moment of band ν is given by s_ν(p)=⟨ u_ν,p|gμ_Bσ/2|u_ν,p⟩=ν/2gμ_Bĝ_p.

H_0(p) can also describe a minimal model for novel quantum systems such as Weyl
semimetals, in which σ acts on orbital space (valence and conduction orbitals mixed with spin), e.g., see <cit.>. In this case, the orbital magnetic moment is given by m_ν,i(p)=-ħ e/2ε_ijl 1/| g_p|^2 g_p· (∂_p_j g_p×∂_p_l g_p), 
where ε_ijl is the rank-3 Levi-Civita tensor <cit.>. 

Next, we consider a bilayer system with a spin-independent coupling between layers, 

    H_0=[ H_1   V;   V H_2 ],

where H_i is the Hamiltonian of layer i=1,2, and V=λσ_0 with λ being the interlayer interaction <cit.>. As illustrative examples, 
we consider each layer to be described by a 2D Rashba or Dresselhous Hamiltonian H_i=(p^2/2m_e-μ)σ_0+ g^i_p·σ, where g^i_p=α_i(p_y,-p_x) 
and g^i_p=α_i(p_x,-p_y) for the Rashba and Dresselhous systems, respectively. 
The band dispersion relation is given by

    ξ_ν(p)=p^2/2m_e-μ +a_1|g^+_p|+a_2√(|g^-_p|^2+λ^2),

where g^±_p=(g^1_p±g^2_p)/2 and a_1,a_2=± 1. Here we identify indices (a_1,a_2)=(1,1), (-1,1), (1,-1), and (-1,-1), with ν=1,2,3 and 4, respectively. For the Rashba system, Fig. <ref>(a) shows the
normal state energy spectrum. 

Eqs. (<ref>) and (<ref>) can now be used to evaluate the spin and orbital magnetization in the band basis. For the Rashba system <cit.>, 
we get


    s_ν   =a_1/2gμ_Bĝ_R , 
    m_ν(p)   =sign[α_+]a_1/4et λ^2α_-/|g^-_p|^2+λ^2ĝ_R,


where ĝ_R=(p_y,-p_x)/p, α_±=α_1±α_2,
and t is the distance between the layers <cit.>. We note that the magnitude of the spin magnetic moment is solely determined by the g-factor, while the orbital moment is influenced by multiple parameters, offering greater tunability. Fig. <ref>(b) shows the
the orbital magnetization in the y-direction vs momentum p_x for bands ν=3 and 4. Note that m_ν(p) 
for bands ν=1 and 2 is identical to bands ν=3 and 4, respectively. For the Dresselhous system, we get


    s_ν   =a_1/2gμ_Bĝ_D, 
    m_ν(p)   =sign[α_+]a_1/4et λ^2α_-/|g^-_p|^2+λ^2ĝ_R,


where ĝ_D=(p_x,-p_y)/p. It is interesting to note that, unlike the Rashba SOC, in the Dresselhous SOC, the magnetic moment perpendicular to the current direction is 
only due to the orbital one. Moreover, the orbital moment has the same direction for both systems, as it can 
be seen from Eqs. (<ref>) and (<ref>). In both systems, a non-zero orbital magnetic moment arises 
from the fact that the electron state in a band can be a superposition of two layers with different Fermi 
velocities. Consequently, a similar Fermi pocket shift due to Rashba or Dresselhous SOC can lead to an equivalent orbital magnetic moment. 
To obtain the current-induced magnetization, 
we consider the induced magnetization in the y-direction by a current in the x-direction. 
Fig. <ref>(c) and (d) show the orbital and spin component of ℚ̅_xy≡ℚ_xy/ℚ_xy|_α_2=0
vs relative strength of SOC. We close this section by pointing out that any uniform (momentum-independent) spinor rotation that maps Rashba to another SOC
would not change the orbital magnetic moment given in Eq. (<ref>). However, the spin magnetic moment transforms as a_1/2gμ_B Rĝ_pR^T, where R is the 3D orthogonal rotation corresponding to the spinor rotation. 




Topological superconductivity manipulated by the magnetoelectric effect.—Consider a semiconductor wire with Rashba spin-orbit coupling (such as electron-doped InAs <cit.>), where proximity to a superconducting substrate induces superconducting pairing. In the presence of an external Zeeman field, the spin degeneracy of electronic bands in the wire is broken, potentially leading to a transition to a topological phase, as discussed in Ref. <cit.>. This transition corresponds to symmetry class D in the Altland-Zirnbauer classification, characterized by a nontrivial Z_2 topological number and zero-energy Majorana edge modes.

Additionally, as outlined in Ref. <cit.>, a supercurrent flowing in the substrate superconductor can induce such a transition in the wire. Interestingly, the supercurrent can lower the critical Zeeman field required for the topological phase transition. This suggests the possibility of a topological phase transition solely driven by the supercurrent, without the need for an external Zeeman field, which breaks time-reversal symmetry. Various model superconductors with diverse spin-orbit couplings have been examined in Ref. <cit.>, showing that a finite supercurrent alone can trigger a topological phase transition. However, in simple s-wave superconductors, a relatively high supercurrent density is necessary for this transition, resulting in the quasiparticle spectrum intersecting the zero-energy level and burying the Majorana states within the bulk continuum.


We propose an alternative approach to topological superconductivity induced and manipulated by 
supercurrent-induced magnetization. We consider a semiconducting wire with Rashba SOC placed on top of a 
superconducting TMD, as shown in Fig. <ref>(a). Quasi-2D superconducting TMDs with C_1, C_2, C_1v, or C_2v point group symmetry possess an in-plane polar axis, which allows an in-plane supercurrent to induce an out-of-plane magnetization M(q) <cit.>. 
The substrate superconductor affects the wire band structure in two ways: an effective Zeeman field on the wire ∼M(q) and proximity
induced superconductivity with a pairing Δ(q)=Δ_0  e^i2q·r_W/ħ, 
where r_W is the position vector along the wire, see Fig. <ref>(a). 
The extra phase factor leads to a Doppler shift in the quasiparticle spectrum of the wire. Considering
a supercurrent protocol satisfying q·r_W=0, i.e.,
when the supercurrent flow is perpendicular to the wire direction, the Doppler shift effect can be minimized. 
As a result, the topological phase would occur at some 
transition momentum, denoted by |q|=q_T, at which the wire spectrum remains gapped. To enhance the effective Zeeman field on the wire and to maintain q_T below the critical depairing momentum in the substrate superconductor, a semiconductor wire with a large g-factor can be employed <cit.>. 

The supercurrent-induced topological phase may enable the implementation of quantum computation operations using Majorana modes. 
To see this, consider a trijunction fabricated from the wires on the substrate superconductor <cit.>. When a finite supercurrent 
|q|>q_T is crossing a branch of wire in the trijunction, the wire is in the nontrivial phase. Consequently, as sketched in Fig. <ref>(b), 
an adiabatically slow switching of the supercurrent flow between points A, B, and C leads to the exchange of the Majorana zero modes. The supercurrent switching steps here leads to a similar adiabatic exchange of Majorana fermions first proposed in Ref. <cit.>, where tunable local gate voltages were employed to perform an exchange operation. However, here we propose a supercurrent-induced exchange operation. We note that a double exchange (braiding) operation could realize a σ_z gate <cit.>.
If the two Majorana fermions comprising a qubit share no
quasiparticle (even electron parity) the braiding leaves the qubit unchanged. However, if the qubit has an odd electron parity, 
the braiding gives a minus sign. 



Discussion.—In a superconductor-magnet hybrid structure, exchange coupling the superconductor to an adjacent magnet, the 
supercurrent-induced magnetization can exert the so-called reactive torque <cit.> on the magnet, 
τ_r∼M(q)×n (even under time-reversal), where n is the 
magnetization. Reciprocally, the exchange interaction can modify the supercurrent near the 
interface <cit.>. 
Note that in addition to the reactive torque, a current carrying normal metal with SOC (broken inversion symmetry)
could in general also exert a dissipative torque on the adjacent magnet, 
τ_d∼n×M(q)×n (odd under time-reversal). 
However, the dissipative torque is absent in the superconductor-magnet hybrid structure <cit.>. 



In Ref. <cit.>, a spintronic Josephson phase qubit based on spin superfluidity and spin Hall phenomena is proposed 
in a metal-magnet hybrid structure. The basic idea is that the qubit state can be manipulated by injecting spin current 
(torque), engendered from an electric current flow in the normal metal with SOC <cit.>. Motivated by eliminating
the Joule heating, a natural generalization of the proposal 
in Ref. <cit.> would be to replace the metal with a superconductor. 
The supercurrent can be used for magnetic qubit manipulations and readout. 

In our treatment, we are projecting physical quantities onto a single band, and in doing so, we are neglecting interband coherent effects linked to the quantum metric of the Bloch wave function, discussed in previous 
works <cit.>. Consequently, without considering these interband effects, 
under a uniform supercurrent superconducting pair potential remains approximately unchanged and the quasiparticle energies experience a Doppler shift.
However, incorporating interband effects introduces an anomalous term for superfluid density, particularly prominent in flat or nearly flat band superconductors <cit.>. Exploring the impact of these geometric effects on supercurrent-induced 
magnetization using our formalism would be an interesting future step. In particular, investigating supercurrent-induced magnetization
in flat-band superconductors could offer an additional means to explore and probe unconventional superconductivity <cit.>. 

We close this Letter by noting that while we have studied the supercurrent-induced magnetization in a clean superconductor (where physically 
the superconducting gap is much larger than the disorder scattering rate), 
we expect that our results remain qualitatively valid at a low concentration of impurities. 
Indeed as shown in Ref. <cit.>, the magnetoelectric effect 
(spin magnetization), although weakened by impurity scattering, is not destroyed in dirty superconductors where the superconducting gap is
smaller than the disorder scattering rate. 


It is a pleasure to acknowledge discussions with Allan H. MacDonald. This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences under Award No. DE-SC0012190. 











 Supplementary Material for 

Superconducting magnetoelectric effects in mesoscopic hybrid structures





Mostafa Tanhayi Ahari^1,2, Yaroslav Tserkovnyak^2

^1Materials Research Laboratory, The Grainger College of Engineering, University of Illinois, Urbana-Champaign, IL 
61801, USA 

^2Department of Physics and Astronomy & Bhaumik Institute for Theoretical Physics, University of California, Los Angeles, California 90095, USA
















 §.§ Magnetoelectric effects in multiband superconductors: static limit


Here, employing the Bogoliubov-de Gennes formalism, we offer a straightforward derivation of supercurrent-induced magnetization in multiband superconductors. Additionally, we provide an expression for supercurrent flow in multiband superconductors, aligning with the findings of Ref.<cit.> and Ref.<cit.> (excluding the quantum metric effects). 

Consider the normal state Hamiltonian H_0(p). The Bloch band basis representation of the Hamiltonian is achieved by an unitary transformation as 

    U(p)H_0(p)U^† (p)≡ξ̂(p)=diag[ξ_1(p),  ξ_2(p), … ],

where ξ_ν(p) is the single-particle energy of Bloch band ν (measured from the chemical potential in the system). Similarly, we apply the unitary transformation to the Bogliubov-de Gennes (BdG) Hamiltonian of the corresponding superconductor  

    [    U(p)       0;       0 U^*(-p) ][     H_0(p)       Δ(p);     Δ^†(p) -H_0^*(-p) ][   U^†(p)        0;        0 U^*†(-p) ]   =
    [   ξ̂(p)   Δ̂(p); Δ̂^*(p) -ξ̂(-p) ],

where U(p) Δ(p)U^*†(-p)≡Δ̂(p). The energetically favorable pairing function in the superconducting phase Δ(p) is obtained <cit.> by the fitness criteria H_0(p)Δ(p)-Δ(p)H^*_0(-p)=0. Assuming that the electrons with opposite momentum are degenerate, ξ_ν(p)=ξ_ν(-p), the fitness criteria becomes ξ̂(p)Δ̂(p)-Δ̂(p)ξ̂(p)=0. This, in general, implies that the pair potential is diagonal in the band basis (intraband Cooper pairing), Δ̂(p)=diag[Δ_1(p),  Δ_2(p),  …]. As a result, interband pairings are ignored here. Within an isolated band, the BdG Hamiltonian for a supercurrent-carrying superconductor reads

    Ĥ=1/2V∑_p,ν(c_p+q,ν^†  c_-p+q,ν)
        [   ξ_ν(p+q)   Δ_ν,q(p); Δ_ν,q^*(p) -ξ_ν(-p+q) ][    c_p+q,ν; c_-p+q,ν^† ],

where 2q is the Cooper momentum. 
To diagonalize this Hamiltonian, we can now apply the Bogoliubov-Valatin transformation, 

    [    c_p+q,ν; c_-p+q,ν^† ]=
        [ 𝚞_ν^*  -𝚟_ν; 𝚟^*_ν   𝚞_ν ][   α_p,ν; β_p,ν^† ],

where α_p,ν and β_p,ν are quasiparticle annihilation operators, and |𝚞_ν|^2=1-|𝚟_ν|^2 with

    |𝚟_ν|^2=1/2(1-(ξ_ν(p+q)+ξ_ν(p-q))/2/√(1/4(ξ_ν(p+q)+ξ_ν(p-q))^2+|Δ_ν,q(p)|^2)).

Consequently, we obtain

    Ĥ=1/2∑_p,ν(E^+_ν(p,q) α_p,ν^†α_p,ν+E^-_ν(p,q)β_p,ν^†β_p,ν)+const.

where the positive energies E^±_ν(p,q)>0 (we assume sufficiently small q so that a stable
Cooper pair state exists <cit.>) are given by

    E^±_ν(p,q) =±1/2(ξ_ν(p+q)-ξ_ν(p-q))+ √(1/4(ξ_ν(p+q)+ξ_ν(p-q))^2+|Δ_ν,q(p)|^2).


In the presence of a uniform supercurrent, we express a single-particle operator Â in the Nambu space as 

    Â=1/2V∑_p,ν(c_p+q,ν^†  c_-p+q,ν)
        [     A_ν(p+q )             0;             0 -A^*_ν(-p+q ) ][    c_p+q,ν; c_-p+q,ν^† ],

where A_ν(p) is the intraband component of operator Â pertaining to band ν. The examples of A_ν(p ) include νĝ_p for spin, ∇_pξ_ν for velocity, and m_ν (p) for orbital magnetization. Applying the Bogoliubov transformation to Eq. (<ref>) and taking the thermal average, where ⟨α_p,ν^†α_p,ν⟩ =f(E_ν^+), ⟨β_p,ν^†β_p,ν⟩ =f(E_ν^-), and ⟨α_p,ν^†β_p,ν⟩ =⟨α_p,νβ_p,ν⟩  =0, we obtain (assuming A_ν(p )=A^*_ν(p )) 

    ⟨Â⟩ =1/2V∑_p,ν[(𝚞_νA_ν(p+q) 𝚞_ν^* -𝚟_νA_ν(-p+q) 𝚟_ν^*) f(E_ν^+)+ (𝚟_ν^* A_ν(p+q) 𝚟_ν -𝚞_ν^* A_ν(-p+q) 𝚞_ν) (1-f(E_ν^-))]

For small q, we obtain E^±_ν(p,q) ≈ E_ν±q·∇_pξ_ν+O(q)^2, where E_ν≡ E^±_ν(p,0), ξ_ν≡ξ_ν(p,0), and |Δ_ν,q(p)|^2≈ |Δ_ν(p)|^2+O(q)^2 <cit.>. Assuming that A_ν(-p )=-A_ν(p ) (odd under time-reversal symmetry), Taylor expanding and keeping terms linear in q, we present the (thermal averaged) quantity as

    ⟨Â⟩=1/V∑_p,ν( ∂ f_ν/∂ E_νq·v_ν-q·∇_pn_ν)A_ν(p),

where f_ν≡ f(E_ν) and n_ν=|𝚞_ν|^2 f_ν+|𝚟_ν|^2 (1-f_ν)=1/2(1-ξ_ν/E_ν(1-2f_ν) ) is the occupancy of the single-particle state at momentum p and band ν in the superconducting state, where |𝚞_ν|^2 is the probability that the pair state at momentum p is empty, and |𝚟_ν|^2 is the probability that it is occupied <cit.>. 

Now taking the infinite-volume approximation 1/V∑_p→1/(2πħ)^2∫ d^2p and substituting A_ν(p)=M_ν(p) and ev_ν(p), we obtain the expressions given in Eqs. (<ref>). 
We also observe that when dealing with a momentum-independent pairing potential, such as the examples discussed in the main text, the expression above can be rewritten as follows:

    ⟨Â⟩=1/V∑_p,ν( ∂ f_ν/∂ E_ν-∂ n_ν/∂ξ_ν)(q·v_ν)A_ν(p).





