Quasiclassical Theory for Dirac Material -Superconductor Proximity StructuresHenning G. Hugdal1, Jacob Linder1, and Sol H. Jacobsen1,*

1Department of Physics, NTNU, Norwegian University of Science and Technology, N-7491 Trondheim, Norway*Corresponding author: sol.jacobsen@ntnu.no

ABSTRACT

We derive the quasiclassical non-equilibrium Eilenberger and Usadel equations valid for Dirac edge and surface electrons withspin-momentum locking p · σσσ, as relevant for topological insulators. We discuss in detail several of the key technical pointsand assumptions of the derivation, focusing on the role of the normalization condition and providing a Riccati-parametrizationof the equations. Solving first the equilibrium equations for S/N and S/F bilayers and Josephson junctions, we study thesuperconducting proximity effect in Dirac materials. Similarly to related works, we find that the effect of an exchange fielddepends strongly on the direction of the field. Only components normal to the transport direction lead to attenuation of theCooper pair wavefunction inside the F. Fields parallel to the transport direction lead to phase-shifts in the dependence on thesuperconducting phase difference for both the charge current and density of states in an S/F/S-junction. Moreover, we computethe differential conductance in S/N and S/F bilayers with an applied voltage bias, and determine the dependence on the lengthof the N and F regions and the exchange field. We discuss the importance of the normalization condition in terms of achievingfurther refinements of quasiclassical theory for Dirac electrons.

IntroductionThe study of materials featuring symmetry-protected topological states has in recent years attracted much attention. Topologicalinsulators represent a notable example of such systems, which are characterized by a topological invariant that is manifestedphysically e.g. via the presence or absence of robust edge-states for thin-films (2D) or surface-states for bulk materials (3D) (seereviews1–3). Much of the exotic physics predicted to occur in topological insulators requires proximity to a superconductinghost material, such as the appearance of Majorana zero modes.4 Therefore, it is of interest to establish a theoretical frameworkthat is accurate, yet practical to work with analytically, and capable of treating superconducting order in Dirac materials notonly in the idealized ballistic limit of transport, but also in the “dirty”, diffusive limit of frequent impurity scattering.

The quasiclassical theory of superconductivity is a suitable candidate for describing the diffusive limit of topologicalinsulators as it is known to account very well for phenomena such as the Josephson effect, bound-states, thermoelectric effects,and many more in conventional metallic hybrid structures.5–9 Recently, this theory has also been expanded to incorporate thepresence of strongly spin-polarized interfaces.10 The quasiclassical equations for topological insulator/superconductor structureswith strong impurity scattering, i.e. the Usadel equation, have recently been used in the study of the presence of vortices11

and helical magnetization.12 However, several basic features of the quasiclassical superconducting proximity effect in Diracmaterials have not yet been studied in detail, such as the fundamental superconductor-normal and superconductor-ferromagnetbilayer stuctures. Moreover, an analysis of technical aspects such as how to parametrize the quasiclassical distribution functionsthat provide the kinetic equations out-of-equilibrium, how to describe the full proximity effect regime with a numericallysuitable Ricatti-parametrization,13–15 and, importantly, the pivotal role played by the normalization condition for the Green’sfunction is also missing in the literature.

Here, we address these issues and more by providing a detailed derivation of the Eilenberger and Usadel equations valid forgeneric Dirac materials with spin-momentum locking p · σσσ in the normal-state Hamiltonian. By considering a superconduc-tor/normal (S/N) bilayer, a superconductor/ferromagnet (S/F) bilayer, and a superconductor/ferromagnet/superconductor (S/F/S)junction, we draw out features of the superconducting proximity effect that contrast with conventional metallic structures.We show that the effect of the exchange field on such systems depends greatly on the direction of the field,16, 17 and wedetail how this difference manifests in physical observables like the charge current and density of states. We also solve thenon-equilibrium equations, calculating the differential conductance and electron distribution function in S/N and S/F bilayers.Our emphasis is on providing a detailed working of the derivation and an explanation of the underlying physical assumptionsmade both here and implicitly in other works. We reproduce some existing experimental features of such systems, but areunable to reproduce certain previous theoretical predictions. More specifically, due to the assumption of the Fermi level µbeing the largest energy scale in the system, and the resulting fixed spin-structure of the Green’s function, we show that neither

arX

iv:1

606.

0124

9v1

[co

nd-m

at.m

es-h

all]

3 J

un 2

016

odd-frequency18 s-wave triplets nor a suppression of the p-wave component of the superconducting order (predicted to appear intopological insulator/superconductor structures in Ref.19 and Ref.,20 respectively) appear in our framework. We discuss how thenormalization condition for the Green’s function plays a crucial role in terms of capturing these phenomena. The quasiclassicalapproach developed here could provide a useful framework to explore phenomena in superconducting spintronics21 in thecontext of Dirac materials.

The remainder of the article is organised as follows. First we outline the quasiclassical theory and use it to provide thedetails of the derivation of the non-equilibrium Eilenberger equation for firstly edge, then surface Dirac electrons in the diffusivelimit. Particular attention is paid to the normalization condition, which impacts significantly on the physical predictions of thetheory. We then discuss the analytical solutions to the equations in the weak proximity limit, and present numerical resultsin the full proximity regime. We calculate physical observables and discuss the consequences of our findings. We concludewith a summary of the main points and broader impact of the results before providing a brief outlook for further developments.Further details of the key calculational steps are provided in the Methods section.

Theory

Dirac materials with spin-momentum locking can be described by a Hamiltonian resembling the relativistic Dirac Hamiltonian22

for massless fermions, written in second quantized form as

H =−ivF

∫dr∑

αβ

ψ†α(r)(∇− ieA) · σσσαβψ

β(r), (1)

where the Fermi velocity vF takes the role of the speed of light in vacuum, e =−|e| is the electron charge, A the vector potential,σσσ = (σ1, σ2, σ3) is the vector in coordinate space consisting of Pauli spin matrices in spin space, and ψ†(r) = (ψ†

↑ ψ†↓) and

ψ(r) = (ψ↑ ψ↓)T where ψ

†↑(↓) and ψ↑(↓) are the field operators creating or annihilating an electron with spin up (down) at

position r respectively. The subscripts α,β = 1,2 are used to specify the spin space elements of ψ†(r), ψ(r) and the matrix σσσ.As this is the only term differing from the non-Dirac case, we treat only this term explicitly here, with the full Hamiltonianprovided for reference in the Methods section. In order to describe both equilibrium and non-equilibrium properties of Diracmaterials with the above Hamiltonian, we will utilize the Keldysh Green’s function technique to find an equation of motion forthe Green’s functions. We will use the notation G for the full 8×8 Keldysh Green’s function matrices, G for 2×2 matrices inparticle-hole space, G for 2×2 matrices in spin space, and G for 4×4 matrices in particle-hole⊗spin space, i.e. G = G⊗ G,where ⊗ denotes a Kronecker product. We also define the notation G for 4× 4 matrices in Keldysh space where the spinstructure is excluded, i.e. G = G⊗ G.

Quasiclassical approximations – The Eilenberger equationThe derivation of the Eilenberger equation for the Dirac case follows the the same steps as in the conventional case, see Methodsfor further details. At sufficiently low temperatures, only electrons near the Fermi surface will take part in the dynamics ofthe system, giving p a pronounced peak at pF.6–8 To make the substitution p→ pF we introduce the quasiclassical Green’sfunction,

g(r, t,pF,ε)≡iπ

∫dξpG(r, t,p,ε), (2)

where ξp = vFp, and the structure of the 8×8 matrix in Keldysh space G is given in Methods. Since the Fermi wavelengthλF is much smaller than the superconducting correlation length ξS, we keep only terms to lowest order in ν = λF/ξS. Afterperforming the approximations known collectively as the quasiclassical approximations, we arrive at the Eilenberger equationin the Dirac case,

vF

2{

∇g, ρ3σσσ}= i[ερ

3 + vFeA · σσσ− vFpF · ρ3σσσ, g]◦. (3)

As in the conventional case, we can now add the contributions from a superconducting pair potential, impurity and spin-flipscattering potentials and an exchange field. This results in

vF

2{

∇g, ρ3σσσ}

= i[ερ

3 + ∆−Vimp1− vFpF · ρ3σσσ+(h−Vsfs+ vFeA) · σσσ, g

]◦, (4)

where ∆ = iτ1⊗ σ2∆ with the superconducting gap ∆, for simplicity chosen to be real, h is the exchange energy, and Vsfs andVimp are the spin-flip and impurity scattering potentials respectively.

2/15

In order to separate the particle-hole and spin space parts of the above anticommutator, we use the unitary transformation

U =

(σ0 00 σ2

). (5)

This gives an Eilenberger equation in terms of the transformed Green’s function G = U gU†,

vF

2{

∇G ,τ0⊗ σσσ}

= i[ετ

3⊗ σ0 + i∆τ

1⊗ σ0 +(h−Vsfs+ vFeA) · τ3⊗ σσσ−Vimpτ

0⊗ σ0− vFpF · τ0⊗ σσσ, G

]◦, (6)

where ⊗ here denotes the Kronecker product between the Pauli matrices τi in particle-hole space and the Pauli matrices σi inspin space. So far, our treatment is similar to Ref.12

Normalization conditionThe above Eilenberger equation must be supplemented by a normalization condition. In the quasiclassical limit the Fermienergy εF = vF|pF| is by far the largest energy scale in the system, and hence a bulk solution to Eq. (6) must commute withvFpF · τ0⊗ σσσ. In the non-superconducting state ∆ = 0, possible solutions for the different matrices in particle-hole⊗spin spaceobeying the necessary symmetries are

GR/A =±τ3⊗(σ

0 + pF · σσσ)

and GK = 2tanh(ε/2kBT)τ3⊗ (σ0 + pF · σσσ), (7)

where the last result is calculated directly from Eq. (46) and the definition of GK using only the Fermi-Dirac distribution.Collecting the above in G and calculating GG we get the possible normalization condition

G ◦ G = 2τ0⊗(σ

0 + pF · σσσ). (8)

In order to show that this is in fact a general normalization condition for Eq. (6), G ◦ G must obey an equation of the same formas Eq. (6), and the normalization in Eq. (8) must be a solution to this equation. Denoting everything in the commutator onthe right hand side of Eq. (6) by A, we proceed by multiplying Eq. (6) by G from the left and right, and adding the resultingequations:

vF

2[G ◦ (∇G) · τ0⊗ σσσ+ G ◦ τ

0⊗ σσσ · (∇G)+(∇G) · τ0⊗ σσσ◦ G + τ0⊗ σσσ · (∇G)◦ G

]= i[A, G ◦ G

]◦. (9)

We see that the right hand side is already of the desired form. However, in order to get the left hand side in the form{∇(G ◦ G),τ0⊗ σσσ

}we have to make further assumptions regarding the form of G .

In the general case where superconducting correlations may be present and the Green’s function depends on position, thequasiclassical approximation that the Fermi energy is the dominant energy scale dictates that the solution of Eq. (6) has tocommute with vFpF · τ0⊗ σσσ, and a natural parametrization of G is

G = G ′⊗ σ0 + G ′′⊗ pF · σσσ, (10)

where G ′ and G ′′ are functions in Keldysh space excluding the spin parts of the Green’s functions. Furthermore, we have toassume that the system is two-dimensional to get the wanted form of the left hand side (see Methods for further details). ForDirac electrons moving in two dimensions, the Fermi surface of the edge states is one-dimensional, i.e. pF =±x and ∇ = ∂xx,and hence all the spin matrices on the left hand side of Eq. (9) commute with each other. Thus we have obtained an equation forG ◦ G of exactly the same form as Eq. (6),

vF

2{

∇G ◦ G ,τ0⊗ σσσ}= i[ετ

3⊗ σ0 + i∆τ

1⊗ σ0 +(h−Vsfs+ vFeA) · τ3⊗ σσσ−Vimpτ

0⊗ σ0− vFpF · τ0⊗ σσσ, G ◦ G

]◦. (11)

We see that in the limit |hy/z|, |Ay/z|, |Vsfsy/z| � εF, the normalization condition is a solution of the above equation which joinsup smoothly to the bulk solution. Hence it is a valid normalization condition for the transformed Eilenberger equation in thecase of superconducting Dirac electrons moving along an edge (for instance manifested via a 2D topological insulator in contactwith an s-wave superconductor), i.e. the normalization is valid exactly for 1D Fermi surfaces.

Inserting the parametrization in Eq. (10) into the normalization condition yields

2τ0⊗(σ

0 + pF · σσσ)

= (G ′ ◦ G ′+ G ′′ ◦ G ′′)⊗ σ0 +{

G ′, G ′′}◦⊗ pF · σσσ.

3/15

Comparing the factors in front of the different spin matrices, one finds 2τ0 = G ′R/A ◦G ′R/A+G ′′R/A ◦G ′′R/A ={

G ′R/A,G ′′R/A}◦for the diagonal elements. This implies that G ′R/A = G ′′R/A, meaning that the spin structure of GR/A is locked and proportionalto the projector on to helical eigenstates.20 We also notice that the parametrization

GK = GR ◦ h− h◦ GA (12)

solves the off-diagonal part of the normalization, GR ◦ GK + GK ◦ GA = 0, when

h = h′⊗ σ0 +h′′⊗ (pF · σσσ). (13)

The reason h has to be parametrized this way is that the solution of GK has to commute with pF · σσσ, meaning that GK has tohave the same form as Eq. (10). In order for Eq. (12) to have this form, h must be parametrized as stated above. When insertingthe parametrization for h into Eq. (12), we find that

GK =(

GR−GA)(h′+h′′)⊗ (σ0 + pF · σσσ), (14)

and hence the spin structure of GK is locked in the same way as for GR/A, allowing us to simplify Eq. (10) in the followingway:

G = G ⊗ (σ0 + pF · σσσ), (15)

with G taking the same structure as the usual 8×8 Keldysh Green’s function, Eq. (47). Equation (13) is a new result whichestablishes how the non-equilibrium distribution function matrix can be parametrized for quasiclassical Dirac materials.

Diffusive limitIn the experimentally common “dirty limit”, where the non-magnetic impurity scattering rate is high, we can expand the matrixG in spherical harmonics,

G ≈ Gs + pF · Gp. (16)

In this limit, we can also use the self-consistent Born approximation to simplify the regular impurity and spin-flip potentials inthe following way:9

Vimpτ0⊗ σ

0 ≈ − i2τ〈G〉F , (17)

Vsfs · τ3⊗ σσσ ≈ − i2τsf

τ3⊗ σ

0〈G〉F τ3⊗ σ

0, (18)

where 〈·〉F denotes averaging over the Fermi surface. Inserting these approximations into the Eilenberger equation, Eq. (6),together with the parametrization in Eq. (15), and performing a trace over the spin space matrices yields

vF ·∇G = i[ετ

3 + i∆τ1 +

i2τ

Gs +i

2τsfτ

3Gsτ3, G

]◦+ i[(hhh+ vFeAAA)τ3 +

iGp

2τ+

i2τsf

τ3Gpτ

3, pFG]◦. (19)

In order to seperate the even and odd terms, we average over the Fermi surface after multiplying with the identity and pFrespectively, the first case giving

vF∇ · Gp = i[ετ

3 + i∆τ1 +

i2τsf

τ3Gsτ

3, Gs

]◦+ i[(h+ vFeA)τ3 +

i2τsf

τ3Gpτ

3, Gp]◦, (20)

while first multiplying with pF and then averaging yields

vF∇Gs = i[ετ

3 + i∆τ1 +

i2τsf

τ3Gsτ

3, Gp

]◦+ i[(h+ vFeA)τ3 +

i2τsf

τ3Gpτ

3, Gs]◦. (21)

We see that the impurity scattering term has completely dropped out from the above equations. This means that the usualmethod of using the fact that τ→ 0 can not be used to express Gp in terms of Gs. However, exploiting the fact that we areconsidering a one-dimensional Fermi surface, we are able to combine the two above equations to a simplified equation for G ,

vF · ∇G =[iετ

3−∆τ1− 1

2τsfτ

3Gτ3, G

]◦, (22)

4/15

where we have defined the operator

∇G = ∇G − ivF

[(hhh+ vFeAAA)τ3, G

]◦. (23)

We see that this is almost identical to the regular Eilenberger equation23 without any spin structure. The most prominentdifference is that the exchange field hhh now enters the equation in the same way as the vector field AAA. Note that the restrictionregarding the strength of the fields for the validity of the normalization condition does not matter for the final equation, sincethe y- and z-components do not enter at all. The fact that the impurity scattering term does not enter the Eilenberger equation atall simply expresses that non-magnetic impurities cannot cause backscattering for Dirac electrons moving along an edge.

Dirac electrons moving on a 2D surfaceThe normalization condition Eq. (8) was only shown to be exactly valid in the case of a one-dimensional Fermi surface, withzero fields in the y- and z-directions. However, we proceeded with the normalization condition under the assumption thatthe fields were weak, meaning that we neglected terms of order ∼ |hhh|, |AAA|, |Vs f sss|. It can be shown that solutions to Eq. (22)vary across lengths L∼ vF|hhh|−1, vF|AAA|−1, vF|Vs f sss|−1, vF|ε|−1. If we assume that this is the case also for Dirac electrons with atwo-dimensional Fermi surface, we see that the gradient term in Eq. (6) is of the same order as the neglected terms. Hence theerror made when neglecting the gradient terms that are not consistent with the normalization condition, is of the same orderof magnitude as the error made when using Eq. (8) as a solution to Eq. (11) (see Methods for details). Hence we will in thefollowing proceed with the above normalization condition to obtain a working Usadel equation for Dirac electrons moving on asurface.

Using the normalization condition in Eq. (8), we arrive at the same conclusions regarding the spin-locking and parametriza-tion of G for Dirac electrons moving on a surface as we did for the edge case. In the high impurity limit we will again expandthe matrix G in spherical harmonics (Eq. (16)), and assume that |Gp| � |Gs|. Inserting this together with the approximationsfor the impurity potentials, Eqs. (17) and (18), into the transformed Eilenberger equation Eq. (6), we get an equation with termsboth even and odd in pF. Again we seperate the even and odd terms by averaging over the Fermi surface after multiplying withthe identity and pF respectively, which yields the equations

vF∇Gs⊗ σσσ+vF

2∇(Gp · σσσ)⊗ σσσ = i

[C, Gs⊗ σ

0 +12Gp⊗ σσσ

]◦(24)

and

vF∇(Gp⊗ σσσ) · σσσ+vF

2{

∇(Gs⊗ σσσ‖),τ0⊗ σσσ

}= i[C, Gs⊗ σσσ‖+ Gp⊗ σ

0]◦, (25)

where

C ≡ ετ3⊗ σ

0 + i∆τ1⊗ σ

0 +(h+ vFeA) · τ3⊗ σσσ+i

2τsfτ

3(Gs⊗ σ0 +

12Gp⊗ σσσ

)τ

3 +i

2τ

(Gs⊗ σ

0 +12Gp⊗ σσσ

),

and the symbol ‖ denotes that only the in-plane (x- and y- ) components of the vector enters the equation. Performing a traceover the spin-space matrices in Eqs. (24) and (25) we get

vF

2∇ · Gp = i

[ετ

3 + i∆τ1 +

i2τsf

τ3Gsτ

3, Gs

]◦+

i2[(hhh‖+ vFeAAA‖)τ

3, Gp]◦, (26)

and

vF∇Gs = i[ετ

3 + i∆τ1 +

i2τsf

τ3Gsτ

3 +i

4τGs, Gp

]◦+ i[(hhh‖+ vFeAAA‖)τ

3 +i

4τsfτ

3Gpτ3, Gs

]◦. (27)

We have also investigated the equations that arise when taking the trace of Eqs. (24) and (25) when multiplying with σ j

( j = 1,2,3) instead of σ0. In the case of 1D motion and with h = 0, the resulting equations are consistent with each other. Inthe case of 2D motion, one obtains a set of equations when taking the trace after multiplying with σ1 and σ2 which do notcontain the impurity scattering self-energy, but which are otherwise identical to the equations obtained when taking the tracedirectly on Eqs. (24) and (25) for h = 0. The discrepancy that arises for the impurity scattering term and the exchange fieldcould be related to the fact that the normalization condition for the Green’s function is, as we have mentioned previously, notexact in the 2D case. We therefore stick with Eqs. (26) and (27), obtained by applying the spin trace directly on Eqs. (24) and(25), since these equations capture the contribution from the impurity self-energy. Further work where an exact normalization

5/15

condition valid for 2D motion would be desirable. Nevertheless, we later show that the present framework is able to accuratelydescribe recent experimental results on supercurrents in Dirac systems, which suggests that our approach is reasonable.

In the high-impurity limit, the mean time between scattering events τ becomes very small. Together with the assumptionthat |Gp| � |Gs|, this allows us to neglect all terms linear in G except the impurity scattering term in Eq. (27). Following theregular procedure24 to express Gp in terms of Gs, we arrive at

Gp =−2τvFGs ◦ ∇Gs, (28)

where ∇ is the operator defined in Eq. (23) with only the in-plane components of the fields. Since ∇Gs ∼ εv−1F , where ε is small

compared to the Fermi energy, we see that the assumption |Gp| � |Gs| holds in the high-impurity limit where τ→ 0. Insertingthe above into Eq. (26), and defining the diffusion constant D≡ τv2

F/2, we arrive at the Usadel equation for the isotropic matrix,

2Di∇ · (G ◦ ∇G) =[ετ

3 + i∆τ1 +

i2τsf

τ3Gτ

3, G]◦, (29)

where we have dropped the subscript s. The form is very similar to the regular Usadel equation, with the significant differencethat the exchange field enters in a way similar to the vector field.12

Riccati parametrizationThe retarded component of Eq. (29) can be solved numerically in the full proximity effect regime using the Riccati parametriza-tion.13–15 The particle-hole part of the transformed retarded Green’s function matrix GR has the following symmetries,

GR =

(G FF −G

), (30)

where the ˜(·) is the combined operation of complex conjugation and letting ε→−ε. We have used that G = G , which followsfrom the normalization condition. This differs slightly from the symmetries of the regular retarded matrix gR (due to the unitarytransformation conducted initially), but using the regular parametrization in terms of the arbitrary unknown functions γ and γ asan ansatz, G = N(1+ γγ) and F = 2Nγ with N = (1− γγ)−1, we find from the normalization for GR that F =−2Nγ. Since Fand F are related by the tilde operation, we must have γ =−γ. We therefore parametrize GR in the following way,

GR = N((1− γγ) 2γ

2γ −(1− γγ)

), (31)

where N = (1+ γγ)−1. GR can thus be found by determining γ and γ. Since G and F are functions and not matrices, theRiccati parametrization is greatly simplified since N, γ and γ all commute. Inserting this parametrization into (29), we get twodifferential equations for γ and γ:

D(∇2γ−2Nγ(∇γ)2) = −iεγ+

∆

2(γγ−1)+

1τsf

γ(2N−1)+2iDvF

(∇ ·hhh)γ+ 4iDvF

hhh ·∇γ+4Dh2

v2F

(2N−1)γ, (32)

D(∇2γ−2Nγ(∇γ)2) = −iεγ+

∆

2(γγ−1)+

1τsf

γ(2N−1)− 2iDvF

(∇ ·hhh)γ− 4iDvF

hhh ·∇γ+4Dh2

v2F

(2N−1)γ. (33)

Notice that the second equation is the tilde-conjugate of the first. The above equations are a new result which renders anumerical treatment of the Usadel equation for Dirac materials particularly efficient.

Results and discussionWe derived the Eilenberger equation for a Dirac material Eq. (4). Transforming this according to Eq. (5) and supplementing itwith the normalization condition Eq. (8) valid for Dirac electrons moving along an edge, we arrive at a simplified Eilenbergerequation for the particle-hole part of the Keldysh matrix, Eq. (22) supplemented by the normalization condition G ◦ G = τ0.Using the same normalization condition as an approximate normalization condition for Dirac electrons moving on a surface, wehave arrived at a Usadel equation for the isotropic particle-hole part of the Keldysh matrix, Eq. (29). In both the above equationswe have defined the operator ∇G = ∇G − ihhh‖/vF ·

[τ3,G

]◦, where the ◦-product reduces to regular matrix multiplication inequilibrium. In the following, we first solve the retarded components of Eqs. (22) and (29) for various systems in equilibrium.Afterwards, we proceed to solve the Keldysh-Usadel equation for the distribution function matrix h (see Methods for details),allowing us to study S/N and S/F structures brought out of equilibrium by an applied bias potential.

6/15

Bulk solution in N, F and SUsing Eqs. (22) and (29), we find the bulk solution in a normal Dirac material (N) and proximity induced ferromagnet (F) to beGR = τ3. Adding the spin structure by Kronecker multiplying with the matrix σ0 + pF · σσσ and using the unitary transformationU in Eq. (5), we find identical solutions for both N and F:

gRN/F =

(σ0 + pF · σσσ 0

0 −σ0 + pF · σσσ∗). (34)

We see that the above solution satisfies the necessary symmetries between the two diagonals. Furthermore, this solution isconsistent with the fact that backscattering is supressed even in high-impurity Dirac materials due to the spin-momentumlocking.1, 3 This can be seen by e.g. calculating the spin-dependent density of states for Dirac edge electrons with spin in the±x-direction for pF = ±x. Using the relations ψ

†↑x = (ψ†

↑z +ψ†↓z)/√

2 and ψ†↓x = (ψ†

↑z−ψ†↓z)/√

2 between the creation andannihilation field operators with spins in the x- and z-directions, we can express the density of states for spin up (down) in thex-direction as

N↑(↓)x =N0

2Re{

Tr [gR +(−)gRσ

1]}, (35)

where N0 is the density of states per spin level in the normal state. Using this result, we get N↑x = 2N0 and N↓x = 0 whenpF =+x, and N↑x = 0 and N↓x = 2N0 when pF =−x, consistent with the fact that there is no backscattering from non-magneticimpurities.

The bulk solution in a proximity induced superconducting Dirac material is similar to that of a normal superconductor, thedifference being the spin-structure of the resulting Green’s function matrix:

gRS =

(c(σ0 + pF · σσσ) seiφ(σ0 + pF · σσσ)iσ2

se−iφiσ2(σ0 + pF · σσσ) −c(σ0− pF · σσσ∗)

)(36)

where we have used the θ-parametrization,6, 7 with c = coshθ, s = sinhθ and θ = arctanh(|∆|/ε). From the spin-structure, wesee that both s- and p-wave pairing is present. We also see that a superconducting gap is present in the density of states, butwith the possibility for a finite density of states only for electrons with spins in the direction of motion.

Application: superconducting proximity effect for Dirac edge electrons in normal and ferromagnetic re-gionsIn the two-dimensional case, the simplified Eilenberger equation Eq. (22) can be solved exactly for both S/N and S/F structures.Using transparent boundaries, i.e. continuity of the Green’s functions, we find that the solution for the transformed retardedGreen’s function in an S/F structure is

GR(x) =

(c ise

2ivF

(εpx+hx)x

ise−2ivF

(εpx+hx)x −c

), (37)

where px =±1 depending on the direction of the Fermi momentum, and hx is the x-component of the exchange field. From theabove we see that there is no attenuation of the anomalous components of GR due to the exchange field.

Application: superconducting proximity effect for Dirac surface electrons in normal and ferromagneticregionsIn the case of Dirac electrons moving on a surface, we solve the Usadel equation Eq. (29) in the weak proximity regime. Thisregime is valid either when the interface transparency is low, or the system is close to the critical temperature, in both casesleading to weak superconducting correlations in the F region. Hence the solution in this case is expanded around the bulksolution τ3, G ≈ τ3 +δG , where δG has the matrix structure shown in Eq. (30). Using the normalization condition one canshow that the corrections δG , δG to the normal components of G is second order in the corrections δF , δF to the anomalouscomponents. Hence we will focus on finding the solution only for δF and δF rather than all elements in δG . Assuming thatthe system varies only along the x-direction, we insert the expansion into Eq. (29) and keep only terms to first order in δF , δF .Solving the resulting differential equations we find the general solutions

δF = e2ihxx

vF (A1 coshkx+A2 sinhkx), (38a)

δF = e−2ihxx

vF (A3 coshkx+A4 sinhkx), (38b)

7/15

Figure 1. Current-phase relation of a Dirac S/N/S structure at temperature T/Tc = 0.02 for different lengths of the weak-link.The length is given in units of ξS, the diffusive coherence length of a bulk superconductor. The current decreases for increasingL/ξS, and is skewed compared to the regular sinφ-dependence, as shown in the inset where the current is normalized to thecritical current.

where k =√

4h2y/v2

F− iε/D, and A j, j = 1,2,3,4 are x-independent functions which must be determined by the boundaryconditions.

In an S/F structure, where the solution must be equal to the superconducting bulk solution at the left boundary (x = 0), andequal to zero at the right boundary (x = L), we get the following solution for the ferromagnetic region

GF= τ

3−

(0 e

2ihxxvF

e−2ihxx

vF 0

)i sinhθ

sinhk(x−L)sinhkL

, (39)

From this we see that the Cooper pair correlation function oscillates and is damped in the F-region. Notice that the valueof hx does not affect the penetration length of Cooper pairs into the F-region. However, increasing the exchange field in they-direction increases the damping of the above functions, meaning that the Cooper pairs’ penetration length into the F-regiondepends on only hy, ξF ∼ |vF/hy|, compared to ∼

√D/|h| in the normal case.25 From this we see that the effect on the system

differs greatly depending on whether the field points parallel or perpendicular to the transport direction.In an S/F/S structure, where we assume that the absolute value of the superconducting gap is the same in the left and right

superconductors, we arrive at the solution

GF= τ

3−

0 e2ihxx

vF+iφL

[sinhk(x−L)− e−

2ihxLvF

+iφ sinhkx]

e−2ihxx

vF−iφL

[sinhk(x−L)− e

2ihxLvF−iφ sinhkx

]0

i sinhθ

sinhkL, (40)

where φ = φR−φL is the phase difference between the right and left superconductors, and L is the length of the weak-link.Starting from the expression for the probability current density in a Dirac material, j(r) = vF ∑αβ ψ

†α(r)σσσαβψ

β(r), we derive

the following expression for the charge current density given in terms of the transformed Green’s function matrices,

jq = N0eD∫

dεTr {τ3(G ◦ ∇G)K}, (41)

where the superscript K means that we take the Keldysh component of the matrix G ◦ ∇G . In equilibrium, this expression can

8/15

Figure 2. (a) Current-phase relation for different values of hxL/vF. We see that the x-component of the exchange field leads toa phase-shift δ =−2hxL/vF. This is also the case for the density of states (DOS), as seen when comparing the dependence on φ

(b) with hx = 0, and (c) hxL/vF = π/2. In the latter case, the dependence on φ is inverted compared to the regular dependence.All results are obtained at temperature T/Tc = 0.02.

be simplified to

jq = N0eD∫

dεTr {τ3GR∇GR +(GR

∇GR)†τ

3} tanhβε

2. (42)

Inserting the above results for the S/F/S-junction, we arrive at the following expression for the current density in the x-direction,

jxq = − 4N0eDsin

(φ− 2hxL

vF

)×

∫dεℑm

{k

sinhkL

}sinh2

θ tanhβε

2. (43)

We see that in the absence of exchange fields, the current follows the regular sinφ-dependence on the phase difference betweenthe superconductors. In an S/F/S-junction, however, the x-component of the exchange field leads to a shift in the current-phaserelation, consistent with previous findings.16 Since this shift depends on both hx and the length of the junction, the current atφ = 0 can in principle be tuned by the length of the weak-link.

Motivated by the experiment of Sochnikov et al.,26 we have solved the Usadel equation in the N-region of an S/N/S structurenumerically using the Riccati parametrization and analyzed the full proximity effect. Calculating the current at temperaturesclose to the critical temperature, we get results in good correspondence with the analytical weak proximity results. At lowtemperatures, we get the current-phase relation shown in Fig. 1 for different junction lengths in the absence of an exchangefield, where we have defined the constant I0 = N0eD∆A/L. The figure shows that the current-phase relation is skewed comparedto the regular sinφ-dependence, reproducing the experimental results reported in Ref.26 Using εF = 0.05 eV to estimate thedensity of states at the Fermi level, N0, and parameter values from Sochnikov et al., we find I0 = 0.2 µA. For e.g. junctionlength L = 400 nm, corresponding to L/ξS = 1, we get a numerical value for the critical current, IC = 2.8 µA, which is inreasonable agreement with Ref.26 Another interesting experimental finding was reported in Ref.,27 who found signatures ofinduced triplet superconductivity in a superconductor/3D topological insulator bilayer.

The weak-proximity results showed that the x- and y-components of the exchange field affect the system in very differentways. This is also found to be the case when considering the full proximity effect: increasing hy lowers the critical current,while increasing hx leads only to a phase-shift δ =−2hxL/vF in the current-phase relation without changing the critical current,as shown in Fig. 2(a). This is also the case for the density of states, where increasing hx affects the density of states only as aphase-shift in the φ-dependence. At the value hxL/vF = π/2 the phase-dependence of the density of states is inverted comparedto the normal case, as seen in Fig. 2(b) and (c). Thus, with a finite hx the energy ground state of the system might also be shiftedto a phase φ0 other than 0 or π,28–30 where φ0 can be tuned by the exchange field and the length of the junction. With a finitevalue of hy, however, the density of states approaches that of a normal metal.

For various different values of the exchange field, we have not been able to produce a zero-energy peak in the DOS, asignature of odd-frequency spin-triplet pairing (although exceptions exist31), which has been theoretically predicted to bepresent in such structures.19 Using the unitary transformation in Eq. (5) to transform solutions to Eq. (29) back to the regularspin basis, we find the spin-structure of the anomalous matrix f to be

f =−iF(−px + ipy 1−1 px + ipy

), (44)

9/15

Figure 3. (a) Normalized differential conductance in an S/N structure with potential bias V for different lengths of theN-region. (b) Distribution function f (ε) for electrons in the N-region of an SN bilayer with potential bias eV/∆ = 0.5 appliedto the boundary at x/L = 1. The length of the N-region is L/ξS = 1, and T/Tc = 0.02. (c) Normalized differential conductancein an S/F structure with hx = 0, hy/∆ = 10.

where F is the particle-hole part of the solution, and px and py are the components of the unit vector pF. First of all, wenotice that the solution has both spin-singlet and -triplet components. However, due to the factors of px, py, the spin-tripletcomponents have p-wave pairing, not odd-frequency s-wave pairing. Getting spin-triplet solutions with odd-frequency s-wavepairing would require the introduction of a factor of both ε and px or py. However, the latter is impossible from the pF-averagedUsadel equation, and it therefore seems that it is not possible to get solutions including s-wave odd-frequency triplet Cooperpairs from the Usadel equation describing Dirac electrons moving on a surface.

The presence of p-wave pairing in proximity induced superconducting TIs has been found theoretically also in e.g.Refs.,19, 32, 33 while it has been theoretically predicted that the p-wave component is suppressed compared to the s-wavecomponent in a disordered TI.20 Since both components are described by the same particle-hole function in our solution, therecan be no such suppression of the p-wave component using this model.

The common reason for the lack of odd-frequency s-wave components and the lack of a suppression of the p-wavecomponent is the imposition of spin-locking by the approximate normalization condition Eq. (8). In neglecting terms onthe grounds that the Fermi energy is by far the largest energy scale in the system, we lose the possibility of changing thespin-structure of the Green’s functions. The implications of this approximation with regard to e.g. the absence of odd-frequencycorrelations has not been discussed in previous works.12 We note that our results are consistent with Ref.19 in the quasiclassicallimit µ� h, since the odd-frequency amplitude is smaller than the even-frequency one by a factor (h/µ)2. Nevertheless, furtherwork towards a more accurate normalization condition would be necessary in attempting to resolve the different predictions.

Application: proximity effect in non-equilibrium normal and ferromagnetic regionsIn order to study non-equilibrium systems, we solve the Keldysh component of the Usadel equation [Eq. (29)] using theparametrization in terms of the matrix h [Eq. (14)] for S/N and S/F bilayers with a potential bias V applied to the boundaryat x/L = 1. Using the parametrization h = hLτ0 +hT τ3 this amounts to solving two uncoupled equations for hL and hT usingthe solutions for the retarded and advanced Green’s functions (see Methods for details). For an S/N structure, the differentialconductance σ = dI/dV (normalized against its normal-state value obtained at eV � ∆), shown for different lengths of thenormal region in Fig. 3a, displays behaviour similar to the non-Dirac case.34 The distribution function for electrons, defined byn = (1−hL−hT )/2, at potential bias eV/∆ = 0.5 is shown in Fig. 3b. This differs from the non-equilibrium N/N case in thatthe step in Fig. 3b has twice the width but only half the height compared to the N/N case.6

Including an exchange field in the x-direction does not alter the above result, since the field neither changes the solutionof the retarded and advanced Green’s functions, nor directly enters the transport equations for the elements of h. However,increasing hy does affect the solution, as shown in Fig. 3c for hy/∆ = 10. We see that increasing hy leads to a small reduction ofthe peaks of σ around eV/∆ = 1, and a suppression of the low-bias conductance feature for longer sample lengths. The effectof the exchange field on the distribution function is shown in Fig. 4 in the case eV/∆ = 0.3.

Concluding remarksIn summary, we have derived the quasiclassical non-equilibrium Eilenberger and Usadel equations for Dirac edge and surfaceelectrons with spin-momentum locking. By studying S/N, S/N/S, S/F and S/F/S structures, we have shown that both singlets-wave and triplet p-wave superconductivity is induced in the normal and ferromagnetic regions. Moreover, we have shown that

10/15

Figure 4. Electron distribution function along the F-region of an S/F structure at different values of hy with eV/∆ = 0.3.

the different directions of the exchange field affect the systems in significantly different ways, the penetration length of Cooperpairs into the F-region depending only on the fields perpendicular to the transport direction, ξF ∼ |vF/hhh⊥|. This difference isalso clearly seen in the results for the density of states and charge current in an S/F/S-junction, where the exchange field inthe transport direction leads to a phase shift.16 We have also shown that the charge current for an S/N/S-junction is skewedcompared to the regular sinφ-dependence, in agreement with experimental results.26 Moreover, we have found results for thedifferential conductivity which resemble the non-Dirac case for S/N stuctures with a potential bias,34 and showed how theseresults are changed by increasing the y-component of the exchange field.

An important purpose of our work has been to provide an in-depth analysis of technical aspects such as how to parametrizethe quasiclassical distribution functions that provide the kinetic equations out-of-equilibrium, how to describe the full proximityeffect regime with a numerically suitable Ricatti-parametrization,13, 14 and, importantly, the pivotal role played by the normal-ization condition for the Green’s function. Due to the approximations made when constructing the normalization condition,we have not been able to find signatures of odd-frequency s-wave pairing19 or suppression of the p-wave component of thesuperconducting order parameter.20 This problem might be solved by finding a more accurate normalization condition if thismakes it possible to avoid the spin-locking of the Green’s functions. Further work is also needed in deriving more generalboundary conditions valid in Dirac materials, since the spin-momentum locking has consequences when introducing boundariesbetween different materials.

11/15

References1. Hasan, M. & Kane, C. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045 (2010)

2. Qi, X. & Zhang, S. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057 (2011)

3. Wehling, T., Black-Schaffer, A. & Balatsky, A. Dirac materials. Adv. Phys. 63, 1 (2014)

4. Fu, L. & Kane, C. Superconducting proximity effect and Majorana fermions at the surface of a topological insulator. Phys.Rev. Lett. 100, 096407 (2008).

5. Eschrig, M. Spin-polarized supercurrents for spintronics: a review of current progress. Rep. Prog. Phys. 78, 104501(2015).

6. Chandrasekhar, V. Proximity-Coupled Systems: Quasiclassical Theory of Superconductivity in Superconductivity:Conventional and Unconventional Superconductors (eds Bennemann, K. H. & Ketterson, J. B.) Ch. 8, 279-313 (SpringerBerlin Heidelberg, 2008)

7. Belzig, W., Wilhelm, F., Bruder, C., Schon, G. & Zaikin, A. D. Quasicalssical Green’s function approach to mesoscopicsuperconductivity. Superlattices and Microstructures 25, 1251 (1999)

8. Morten, J. P. Spin and charge transport in dirty superconductors, M. Sc. thesis, Norwegian University of Science andTechnology (2003).

9. Rammer, J. & Smith, H. Quantum field-theoretical methods in transport theory of metals. Rev. Mod. Phys. 58, 323-359(1986).

10. Eschrig, M., Cottet, A., Belzig, W. & Linder, J. General boundary conditions for quasiclassical theory of superconductivityin the diffusive limit: application to strongly spin-polarized systems. New J. Phys. 17, 083037 (2015).

11. Ioselevich, P. A., Ostrovsky, P. M. & Feigel’man, M. V. Majorana state on the surface of a disordered 3D topologicalinsulator. Phys. Rev. B 86, 035441 (2012)

12. Zyuzin, A., Alidoust, M. & Loss, D. Josephson junction through a 3D topological insulator with helical magnetization.arXiv:1511.01486

13. Schopohl, N. & Maki, K. Quasiparticle spectrum around a vortex line in a d-wave superconductor. Phys. Rev. B 52, 490(1995).

14. Schopohl, N. Transformation of the Eilenberger equations of superconductivity to a scalar Riccati equation. arXiv:cond-mat/9804064

15. Jacobsen, S. H., Ouassou, J. A. & Linder, J. Critical temperature and tunneling spectroscopy of superconductor-ferromagnet hybrids with intrinsic Rashba-Dresselhaus spin-orbit coupling. Phys. Rev. B 92, 024510 (2015).

16. Tanaka, Y., Yokoyama, T. & Nagaosa, N. Manipulation of the Majorana fermion, Andreev reflection, and Josephsoncurrent on topological insulators. Phys. Rev. Lett. 103, 107002 (2009)

17. Linder, J., Tanaka, Y., Yokoyama, T., Sudbø, A. & Nagaosa, N. Interplay between superconductivity and ferromagnetismon a topological insulator. Phys. Rev. B 81, 184525 (2010)

18. Berezinskii, V. L. New model of the anisotropic phase of superfluid He3. JETP Lett. 20, 287 (1975).

19. Yokoyama, T. Josephson and proximity effects on the surface of a topological insulator. Phys. Rev. B. 86, 075410 (2012).

20. Tkachov, G. Suppression of surface p-wave superconductivity in disordered topological insulators. Phys. Rev. B 87,245422 (2013)

21. Linder, J. & Robinson, J. W. A. Superconducting spintronics. Nat. Phys. 11, 307 (2015)

22. Dirac, P. A. M. The Quantum Theory of the Electron. Proc. R. Soc. A. 117, 610 (1928).

23. Eilenberger, G. Transformation of Gor’kov’s equation for type II superconductors into transport-like equations. Z. Phys.214, 195 (1968).

24. Usadel, K. D. Generalized diffusion equation for superconducting alloys. Phys. Rev. Lett. 25, 507 (1970).

25. Buzdin, A. I. Proximity effects in superconductor-ferromagnet heterostructures. Rev. Mod. Phys. 77, 935 (2005)

26. Sochnikov, I. et al. Nonsinusoidal current-phase relationship in Josephson junctions from the 3D topological insulatorHgTe. Phys. Rev. Lett. 114, 066801 (2015)

27. Koren, G., Kirzhner, T., Kalcheim, Y. & Millo, O. Signature of proximity-induced px + ipy triplet pairing in the dopedtopological insulator Bi2Se3 by the s-wave superconductor NbN. EPL, 103, 67010 (2013).

12/15

28. Buzdin, A. Direct coupling between magnetism and superconducting current in the Josephson φ0 junction. Phys. Rev. Lett.101, 107005 (2008)

29. Grein, R., Eschrig, M., Metalidis, G. & Schon, G. Spin-dependent Cooper pair phase and pure spin supercurrents instrongly polarized ferromagnets. Phys. Rev. Lett. 102, 227005 (2009).

30. Kulagina, I. & Linder, J. Spin supercurrent, magnetization dynamics, and φ-state in spin-textured Josephson junctions.Phys. Rev. B 90, 054504 (2014)

31. Linder, J. & Robinson, J. W. A. Strong odd-frequency correlations in fully gapped Zeeman-split superconductors. Sci.Rep. 5, 15483 (2015).

32. Stanescu, T. D., Sau, J. D., Lutchyn, R. M. & Das Sarma, S. Proximity effect at the superconductor–topological insulatorinterface. Phys. Rev. B 81, 241310(R) (2010).

33. Black-Schaffer, A. M. Self-consistent superconducting proximity effect at the quantum spin Hall edge. Phys. Rev. B 83,060504(R) (2011).

34. Tanaka, Y., Golubov, A. A. & Kashiwaya, S. Theory of charge transport in diffusive normal metalOconventionalsuperconductor point contacts. Phys. Rev. B 68, 054513 (2003).

MethodsThe full system Hamiltonian reads

H = − ivF

∫dr∑

αβ

ψ†α(r)(∇− ieA) · σσσαβψ

β(r)+

∫dr(

∆(r)ψ†↑(r)ψ

†↓(r)+∆

†(r)ψ↓(r)ψ↑(r))

+∫

dr∑α

Vimp(r)ψ†α(r)ψα(r)+

∫dr∑

αβ

ψ†α(r)Vsfs(r) · σσσαβψ

β(r)−

∫dr∑

αβ

ψ†α(r)h(r) · σσσαβψ

β(r), (45)

where ∆ is the superconducting pair potential, Vimp the impurity potential, Vsfs the spin-flip potential, and h the exchange field.The normal and anomalous retarded (R), advanced (A) and Keldysh (K) Green’s functions are defined by

GRαβ(r, t;r′, t ′) = −i〈

{ψα(r, t),ψ

†β(r′, t ′)

}〉θ(t− t ′),

FRαβ(r, t;r′, t ′) = −i〈

{ψα(r, t),ψβ

(r′, t ′)}〉θ(t− t ′),

GAαβ(r, t;r′, t ′) = +i〈

{ψα(r, t),ψ

†β(r′, t ′)

}〉θ(t ′− t),

FAαβ(r, t;r′, t ′) = +i〈

{ψα(r, t),ψβ

(r′, t ′)}〉θ(t ′− t),

GKαβ(r, t;r′, t ′) = −i〈

[ψα(r, t),ψ

†β(r′, t ′)

]〉,

FKαβ(r, t;r′, t ′) = −i〈

[ψα(r, t),ψβ

(r′, t ′)]〉.

Using the Heisenberg equation of motion for an operator O, i∂tO =[O,H

], we find the time-derivatives of the field operators.

This in turn can be used to find the t and t ′-derivatives of the above Green’s functions, which are collected in the following way,

iρ3∂tG = δ(t− t ′)δ(r− r′)+ KG, (46a)

−i∂t ′Gρ3 = δ(t− t ′)δ(r− r′)+ GK′†. (46b)

Here we have defined the 8×8-matrix Keldysh space,

G =

(GR GK

0 GA

), (47)

consisting of the 4×4-matrices in particle-hole⊗spin space,

GR/A =

(GR/A FR/A

FR/A∗ GR/A∗

), GK =

(GK FK

−FK∗ −GK∗

), (48)

and the 2×2 Green’s function matrices in spin space,

GR/A/K =

(GR/A/K↑↑ GR/A/K

↑↓GR/A/K↓↑ GR/A/K

↓↓

), FR/A/K =

(FR/A/K↑↑ FR/A/K

↑↓FR/A/K↓↑ FR/A/K

↓↓

). (49)

13/15

In addition we have defined the matrices ρ3 ≡ τ3⊗ σ0 = diag(1,1,−1,−1), K =−ivF(∇− ieA) · σσσ, and K = diag(K, K∗) fornotational simplicity, where ∇ acts to the left or right according to the matrix with which it is multiplied. Note that Kroneckerproducts with identity matrices are implied to resolve products between matrices of different dimensions in Eq. (46). In addition,a prime (e.g. K′) denotes that the matrix function is a function of the primed coordinates r′ and t ′. Subtracting Eq. (46b) fromEq. (46a) yields

iρ3∂tG(r, t;r′, t ′)+ i∂t ′G(r, t;r′, t ′)ρ3 = KG(r, t;r′, t ′)− G(r, t;r′, t ′)K′†. (50)

Since we are interested in the two-particle wave functions describing superconductivity, we perform a coordinate transformationto the mixed representation, expressing the above equation in terms of the center-of-mass coordinates rCOM = (r+ r′)/2 andT = (t+ t ′)/2, and the relative coordinates rrel = r−r′ and τ = t− t ′. Fourier transforming with respect to the relative variables,the above equation can be expressed as[

ερ3, G(r, t,p,ε)

]⊗= − ivF

2

{∇G(r, t,p,ε), ρ3σσσ

}+ vF

[p · ρ3σσσ, G(r, t,p,ε)

]− vFe

[A · σσσ, G(r, t,p,ε)

]⊗, (51)

where we have defined the matrix σσσ = diag(σσσ, σσσ∗), and let rCOM → r, T → t. The symbol ⊗ in the superscript denotes aconvolution over the variables,7 which can be expressed as

A⊗B = ei2 (∇

Ar ∇B

p−∇Ap∇B

r )A◦B = ei2 (∇

Ar ∇B

p−∇Ap∇B

r )e−i2 (∂

At ∂B

ε−∂Aε ∂B

t )AB, (52)

which also defines the ◦-product. Note that a dot product between ∇ and σσσ is implied in the first term on the right hand side ofEq. (51). Performing the quasiclassical approximations, we arrive at the Eilenberger equation, Eq. (4).

Normalization condition for Dirac surface electronsWe here outline the steps followed to obtain an equation for G ◦ G for Dirac electrons moving on a surface, and state the termsneglected in order to prove the validity of the approximate normalization condition, Eq. (8). Starting from the transformedEilenberger equation, Eq. (6), we want to obtain an equation for G ◦ G of the same form as Eq. (6). Using the notation

[A, G

]◦for the commutator in Eq. (6), we calulate

vF

2[G ◦{

∇G ,τ0⊗ σσσ}+{

∇G ,τ0⊗ σσσ}◦ G]= G ◦

[A, G

]◦+[A, G

]◦ ◦ G =[A, G ◦ G

]◦. (53)

The right hand side is already of the wanted form. To proceed with the left hand side we have to make some assumptionsregarding G . Using the parametrization in Eq. (10) we rearrange two of the terms on the left hand side of Eq. (53):

(∇G) · τ0⊗ σσσ◦ G = (∇G)◦ G · τ0⊗ σσσ+2i(∂xG)◦ G ′′py⊗ σ3−2i(∂yG)◦ G ′′px⊗ σ

3,

G ◦τ0⊗ σσσ ·∇G = τ

0⊗ σσσ · G ◦∇G −2iτ0⊗ σ3 pyG ′′ ◦∂xG +2iτ0⊗ σ

3 pxG ′′ ◦∂yG ,

where px and py are the elements of pF. The two extra terms in each of the above equations are due to the fact that different Paulispin matrices do not commute, but obey the commutation relation

[σi, σ j

]= 2iεi jkσk, where εi jk is the totally antisymmetric Levi-

Cevita tensor and i, j,k ∈ {1,2,3}. Inserting this into the left hand side of Eq. (53), and using ∇(G ◦ G) = (∇G)◦ G + G ◦(∇G)we get

vF

2[G ◦{

∇G ,τ0⊗ σσσ}+{

∇G ,τ0⊗ σσσ}◦ G]

=vF

2{

∇G ◦ G ,τ0⊗ σσσ}+2i(∂xG)◦ G ′′py⊗ σ

3−2i(∂yG)◦ G ′′px⊗ σ3

− 2iτ0⊗ σ3 pyG ′′ ◦∂xG +2iτ0⊗ σ

3 pxG ′′ ◦∂yG . (54)

Due to the four extra terms, we do not get an equation for G ◦ G in the same form as the Eilenberger equation, Eq. (6). However,in order to proceed we neglect these terms under the assumption that the error made is of the same order of magnitude as whenusing the normalization condition Eq. (8) as a solution to the Eilenberger equation for G ◦ G in the case of Dirac edge electrons,where the four above terms do not enter since ∂yG = py = 0 for 1D motion.

Usadel equation for hInserting the parametrization for GK in Eq. (14) with h = h′+h′′ into the Keldysh component of the Usadel equation, Eq. (29),we get

2iD∇

(∇h−GR(∇h)GA +(GR

∇GR)h−h(GA∇GA)

)=[ετ

3,GRh−hGA], (55)

14/15

where we for simplicity have kept only the first term on the right hand side of the Usadel equation. Inserting the parametrizationh = hLτ0 +hT τ3,6, 7 we proceed by multiplying with the identity and τ3 and taking the trace.6 This gives the two equations

0 = ∇ ·{

∇hLTr {τ0−GRGA}−∇hT Tr {GRτ

3GA}+hT Tr {τ3GR∇GR− τ

3GA∇GA}

}, (56)

and

0 = ∇ ·{

∇hT Tr {τ0−GRτ

3GAτ

3}−∇hLTr {GRGAτ

3}+hLTr {τ3GR∇GR− τ

3GA∇GA}

}. (57)

In situations where the last two traces in both the above equations are zero (for instance, the third term corresponds to asupercurrent and is absent in S/N or S/F bilayers), we get two decoupled second order equations for hL and hT ,

Tr {τ0−GRGA}∇2hL = (∇hL)Tr {(∇GR)GA +GR(∇GA)}, (58)

Tr {τ0−GRτ

3GAτ

3}∇2hT = (∇hT )Tr {(∇GR)τ3GAτ

3 +GRτ

3(∇GA)τ3}. (59)

In this case the expression for the charge current simplifies to

jjjq = N0eD∫

dε ∇hT Tr {τ0−GRτ

3GAτ

3}, . (60)

AcknowledgementsJ.L was supported by the Research Council of Norway, Grants No. 205591, 216700, 240806 and the ”Outstanding AcademicFellows” programme at NTNU.

Author contributions statementH.H. did the analytical and numerical calculations with support from S.J. and J.L. All authors contributed to the discussion ofthe results and the writing of the manuscript.

Additional informationCompeting financial interests The authors declare no competing financial interests.

15/15