# Spin-dependent boundary conditions for isotropic superconducting Green’s functions

###### Abstract

The quasiclassical theory of superconductivity provides the most successful description of diffusive heterostructures comprising superconducting elements, namely, the Usadel equations for isotropic Green’s functions. Since the quasiclassical and isotropic approximations break down close to interfaces, the Usadel equations have to be supplemented with boundary conditions for isotropic Green’s functions (BCIGF), which are not derivable within the quasiclassical description. For a long time, the BCIGF were available only for spin-degenerate tunnel contacts, which posed a serious limitation on the applicability of the Usadel description to modern structures containing ferromagnetic elements. In this article, we close this gap and derive spin-dependent BCIGF for a contact encompassing superconducting and ferromagnetic correlations. This finally justifies several simplified versions of the spin-dependent BCIGF, which have been used in the literature so far. In the general case, our BCIGF are valid as soon as the quasiclassical isotropic approximation can be performed. However, their use require the knowledge of the full scattering matrix of the contact, an information usually not available for realistic interfaces. In the case of a weakly polarized tunnel interface, the BCIGF can be expressed in terms of a few parameters, i.e. the tunnel conductance of the interface and five conductance-like parameters accounting for the spin-dependence of the interface scattering amplitudes. In the case of a contact with a ferromagnetic insulator, it is possible to find explicit BCIGF also for stronger polarizations. The BCIGF derived in this article are sufficienly general to describe a variety of physical situations and may serve as a basis for modelling realistic nanostructures.

###### pacs:

73.23.-b, 74.45.+c, 85.75.-d## I Introduction

The quantum mechanical spin degree of freedom is widely exploited to control current transport in electronic circuits nowadays. For instance, the readout of magnetic hard disks is based on the giant magnetoresistance effect, which provides the possibility to tune the conductance of e.g. a ferromagnet/normal metal/ferromagnet () trilayer by changing the magnetizations of the two layers from a parallel to an antiparallel configurationGMR (). However, many functionalities of hybrid circuits enclosing ferromagnetic elements remain to be explored. Presently, non-collinear spin transport is triggering an intense activity, due to spin-current induced magnetization torquesSlonczewski (), which offer new possibilities to build non-volatile memoriesRalphStiles (). Another interesting possibility is to include superconducting elements in hybrid circuits. When a layer is connected to a BCS superconductor (), the singlet electronic correlations characteristic of can propagate into because electrons and holes with opposite spins are coupled coherently by Andreev reflections occurring at the interfaceRefAndreev (). This so-called ”superconducting proximity effect” is among other responsible for strong modifications of the density of states of RefMcMillan (). In a ferromagnet (), the ferromagnetic exchange field , which breaks the symmetry between the two spin bands, is antagonistic to the Bardeen-Cooper-Schrieffer(BCS)-type singlet superconducting order. However, this does not exclude the superconducting proximity effect. First, when the magnetization direction is uniform in a whole circuit, superconducting correlations can occur between electrons and holes from opposite spin bands, like in the limit. These correlations propagate on a characteristic distance limited by the ferromagnetic coherence length , where is the diffusion coefficient. Furthermore, produces an energy shift between the correlated electrons and holes in the opposite spin bands, which leads to spatial oscillations of the superconducting order parameter in Buzdin1982 (), as recently observedTakisN (); Ryazanov (); TakisI (). These oscillations allow to build new types of electronic devices, such as Josephson junctions with negative critical currentsGuichard (), which promise applications in the field of superconducting circuitsIoffe (); Taro (). Secondly, when the circuit encloses several ferromagnetic elements with noncollinear magnetizations, spin precession effects allow the existence of superconducting correlations between equal spinsBergeret (). These correlations are expected to propagate in a on a distance much longer than opposite-spin correlations. This property could be used e.g. to engineer a magnetically switchable Josephson junction. These and many more effects have been reviewed recently Golubov (); Buzdin ().

To model the behavior of superconducting hybrid circuits, a proper description of the interfaces between the different materials is crucial. This article focuses on the so-called diffusive limit, which is appropriate for most nanostructures available nowadays. In this limit, a nanostructure can be separated into interfaces (or contacts) and regions characterized by isotropic Green functions , which do not depend on the direction of the momentum but conserve a possible dependence on spatial coordinates. The spatial evolution of the isotropic Green functions is described by Usadel equations Usadel (). One needs boundary conditions to relate the values of at both sides of an interface. For a long time, the only boundary conditions for isotropic Green’s functions (BCIGF) available were spin-independent BCIGF derived for a tunnel contactKuprianov (). The only interfacial parameter involved in these BCIGF was the tunnel conductance of the contact. Such a description is incomplete for a general diffusive spin-dependent interface. Spin-dependent boundary conditions have been first introduced in the ballistic regimeMillis (); Tokuyasu (); Kopu (); Zhao (). Recently, many references have used spin-dependent BCIGFDani1 (); Dani2 (); Cottet1 (); Cottet2 (); Cottet3 (); Linder1 (); Morten1 (); Morten2 (); Braude (); Linder2 (); DiLorenzo () to study the behavior of hybrid circuits enclosing BCS superconductors, ferromagnetic insulators, ferromagnets, and normal metals. These BCIGF, which have been first introduced in Ref. Dani1, , include the term of Ref. Kuprianov, . They furthermore take into account the spin-polarization of the interface tunnel probabilities through a term, and the spin-dependence of interfacial scattering phase shifts through terms. It has been shown that the and terms lead to a rich variety of effects. First, the terms can produce effective Zeeman fields inside thin superconducting or normal metal layersDani1 (); Dani2 (); Cottet2 (), an effect which could be used e.g. to implement an absolute spin-valveDani1 (). In thick superconducting layers, this effect is replaced by spin-dependent resonances occurring at the edges of the layersCottet3 (). Secondly, the terms can shift the spatial oscillations of the superconducting order parameter in ferromagnetsCottet2 (); Cottet1 (); Cottet3 (). Thirdly, the term can produce superconducting correlations between equal spins, e.g. in a circuit enclosing a BCS superconductor and several ferromagnetic insulators magnetized in noncollinear directionsBraude (). The terms have been taken into account for a chaotic cavity connected to a superconductor and several ferromagnetsMorten1 (); Morten2 (). In this system, crossed Andreev reflections and direct electron transfers are responsible for nonlocal transport properties. The ratio between these two kinds of processes, which determines e.g. the sign of the nonlocal conductanceFalci01 (); Sanchez03 (), can be controlled through the relative orientation of the ferromagnets magnetizations.

In this article, we present a detailed derivation of the spin-dependent BCIGF based on a scattering description of interfaces. Our results thus provide a microscopic basis for all future investigations of ferromagnet-superconductor diffusive heterostructures taking into account the spin-dependent interface scattering. To make the BCIGF comprehensive and of practical value, we make a series of sequential assumptions, starting from very general to more and more restrictive hypotheses. In a first part, we assume that the contact is fully metallic, i.e. it connects two conductors which can be superconductors, ferromagnets or normal metals. We consider ferromagnets with exchange fields much smaller than their Fermi energies, as required for the applicability of the quasiclassical isotropic description. We assume that the contact nevertheless produces a spin-dependent scattering due to a spin-dependent interfacial barrier . In this case, we establish general BCIGF which require the knowledge of the full contact scattering matrix. Then, we assume that the contact locally conserves the transverse channel index (specular hypothesis) and spins collinear to the contact magnetization. In the tunnel limit, assuming is weakly spin-dependent, we find that the BCIGF involve the , , and terms used in Refs. Cottet1, ; Cottet2, ; Cottet3, ; Linder1, ; Morten1, ; Morten2, ; Braude, ; Linder2, ; DiLorenzo, , plus additional terms which are usually disregarded. In a second part, we study a specular contact connecting a metal to a ferromagnetic insulator (). If we assume a weakly spin-dependent interface scattering, we obtain the BCIGF used in Refs. Dani1, ; Dani2, . We also present BCIGF valid beyond this approximation. Note that the various BCIGF presented in this article can be applied to noncollinear geometries.

Most of the literature on superconducting hybrid circuits uses a spatially continuous description, i.e., in each conductor, the spatial dependence of the Green’s function is explicitly taken into account. The BCIGF presented in this article can also be used in the alternative approach of the so-called circuit theory. This approach is a systematic method to describe multiterminal hybrid structures, in order to calculate average transport propertiesRefCircuit (); Braatas (); RefYuli () but also current statisticsyuli:99 (); wn01 (). It relies on the mapping of a real geometry onto a topologically equivalent circuit represented by finite elements. The circuit is split up into reservoirs (voltage sources), connectors (contacts, interfaces) and nodes (small electrodes) in analogy to classical electric circuits. Each reservoir or node is characterized by an isotropic Green’s function without spatial dependence, which plays the role of a generalized potential. One can define matrix currents, which contain information on the flows of charge, spin, and electron/hole coherence in the circuit. Circuit theory requires that the sum of all matrix currents flowing from the connectors into a node is balanced by a “leakage” current which accounts for the non-conservation of electron/hole coherence and spin currents in the node. This can be seen as a generalized Kirchhoff’s rule and completely determines all the properties of the circuit. So far, circuit theory has been developed separately for RefCircuit () and circuitsRefYuli (). Throughout this article, we express the BCIGF in terms of matrix currents. Our work thus allows a straightforward generalization of circuit theory to the case of multiterminal circuits which enclose superconductors, normal metals, ferromagnets and ferromagnetic insulators, in a possibly noncollinear geometry.

This article is organized as follows. We first consider the case of a metallic contact, i.e. a contact between two conductors. Section II defines the general and isotropic Green’s functions and used in the standard description of hybrid circuits encompassing BCS superconductors. Section III introduces the ballistic Green’s function , which we use in our derivation. Section IV discusses the scattering description of the contact with a transfer matrix . Although we consider the diffusive limit, the scattering description is relevant for distances to the contact shorter than the elastic mean free path. On this scale, one can use to relate the left and right ballistic Green’s functions and . Section V presents an isotropization scheme which accounts for impurity scattering and leads to the isotropic Green’s functions away from the contact. Section VI establishes the general metallic BCIGF which relate , and . Section VII gives more transparent expressions of these BCIGF in various limits. Section VIII addresses the case of a contact with a side, in analogy with the treatment realized in the metallic case. Section IX concludes. Appendix A discusses the structure of the transfer matrix and Appendix B gives details on the calculation of the matrix current. Appendix C relates our BCIGF to the equations previously obtained in the normal-state limitRefCircuit (); Braatas (). Appendix D discusses the BCIGF obeyed by the retarded parts of in the collinear case. For completeness, Appendix E presents the Usadel equations in our conventions.

## Ii General and isotropic Green’s functions

From section II to VII, we consider a planar metallic contact between two diffusive conductors noted (left conductor) and (right conductor) (see Fig. II). The conductor [] can exhibit spin and/or superconducting correlations, due to its superconducting order parameter or exchange field , or due to the proximity effect with other conductors.

For the primary description of electronic correlations in and , one can use a general Green’s function defined in the KeldyshNambuSpinCoordinate space. In the stationary case, can be defined as

(1) |

withdef ()

(2) |

(3) |

and

(4) |

Here, and denote commutators and anticommutators respectively, , space coordinates, , time coordinates, and the energy. We use a spinor representation of the fermion operators, i.e.

(5) |

in the NambuSpin space. We denote by the third Nambu Pauli matrix, i.e. in the NambuSpin space. For later use, we also define the third spin Pauli matrix i.e. . With the above conventions, the Green’s function follows the Gorkov equations:

(6) |

and

(7) |

Here, corresponds to the gap matrix associated to a BCS superconductor (see definition in Appendix E). The Hamiltonian can be decomposed as

(8) |

with and the longitudinal and transverse components of . The part includes a ferromagnetic exchange field in the direction , and the Fermi energy , whereas the part includes a lateral confinement potential . The potential barrier describes a possibly spin-dependent and non-specular interface. It is finite in the area only. In the Born approximation, the impurity self-energy at side of the interface can be expressed as . Here, the impurity elastic scattering time in material can be considered as spin-independent due to . The Green’s function , which has already been mentioned in section I, corresponds to the quasiclassical and isotropic average of inside conductor . It can be calculated asRefWolfgang1 ()

(9) |

with the longitudinal component of and the density of states per spin direction and unit volume for free electrons. Note that we consider geometries where , and are independent of .

In this article, we consider the diffusive (i.e. quasiclassical and isotropic) limit, i.e.

(10) |

where is the temperature and the Boltzmann constant. In this regime, the spatial evolution of inside and is described by the Usadel equations which follow from Eqs. (6) and (7) [see Appendix E]. The characteristic distances occurring in the Usadel equations are , and for a ferromagnet , a normal metal and a superconductor , respectively, with and the diffusion constant and Fermi velocities of material . According to Eq. (10), the scale is much larger than the elastic mean free path . Importantly, the Usadel equations alone are not sufficient to describe the behavior of diffusive hybrid circuits. One also needs to relate the values of at both sides of an interface with BCIGF, which we derive in the next sections.

For the sake of concreteness, we give typical order of magnitudes for the different lenghtscales involved in the problem. These lenghtscales strongly depend on the detailed composition and structure of the materials and interfaces considered, so that the applicability of the quasiclassical isotropic description has to be checked in each case. The value of can strongly vary from a few atomic layers to a few nanometers if the two materials constituting the interface interdiffuseRobinson (). The mean free path, which strongly depends on the impurity concentration, can be of the order of a few nanometersRyazanov (). The superconducting lenghtscale is usually of the order of nm for NiobiumKim (); Moraru (). The Cooper pair penetration length can reach nm for a diluted magnetic allow like CuNiRyazanov (), or nm for a normal metal like Cu at mKgueron ().

It is worth to note, at this point, that the derivation presented below is not restricted to stationary problems on superconducting heterostructures. Actually most of the derivations made below do not rely on the specific Keldysh structure introduced in Eqs. (1)-(4) and our results can be directly used to describe full counting statistics in the extended Keldysh technique wn01 () or multiple Andreev reflections CuevasBelzig03 (). In fact, boundary conditions for arbitrary time-dependent scattering problems have been recently formulated in a similar spirit SnymanNazarov08 (). However, having in mind the many concrete applications of the boundary conditions in superconducting heterostructures and keeping the notation as simple as possible, we derive the BCIGF below in the framework of the stationary Keldysh-Nambu Greens functions.

## Iii Ballistic Green’s function

Considering the structure of Eqs. (6)-(8), for or , one can expand in transverse modes asRefZaitsev ()

(11) |

In this section, we use spin indices which correspond to spin directions parallel or antiparallel to the direction , and Nambu indices for electron and hole states. The indices account for the longitudinal direction of propagation (we use , , and in mathematical expressions). We introduce the wavefunction for the transverse channel , i.e. , and the corresponding longitudinal momentum and velocity, i.e. and . Importantly, we have disregarded the dependences of and on and due to Eq. 10. The decoration denotes that the Green’s function can have a general structure in the KeldyshNambuSpinChannelDirection space, noted in the following. In contrast, denotes the fact that has no structure in the ChannelDirection sub-space, noted in the following (see the summary of notations in Table 1).

Due to Eqs. (6) and (7), is not continuous at RefZaitsev (); RefYuli (). One can useShelankov (); DefSign ()

(12) |

with the third Pauli matrix in the direction of propagation space, i.e. . Equation (12) involves a ballistic Green’s function which is continuous at . We will see below that this quantity plays a major role in the derivation of the BCIGF.

For later use, we now derive the equations of evolution followed by . Inserting Eq. (11) into Eqs. (6-7), one can check that, for and () or (), follows the equations

(13) |

and

(14) |

We have introduced above a velocity matrix with a structure in the channels subspace only, i.e. , with the identity matrix in the Keldysh space. We have furthermore assumed that the so-called envelope function varies smoothly on the scale of the Fermi wave length, in order to neglect terms proportional to and in Eqs. (13-14) RefZaitsev ().

## Iv Scattering description of a metallic contact

We now define, at both sides of the barrier , two ballistic zones (with no impurity scattering) located at and , with (grey areas in Fig. II). In the region , we can disregard the superconducting gap matrix since . Therefore, the electron and hole dynamics can be described with the Schrödinger equation

(15) |

or, equivalently,

(16) |

whose solution has the formBlanterButtiker ()

(17) |

in the ballistic zones. Here, is a vector in the SpinNambuKeldysh space, and is a vector in the space. The index corresponds again to the longitudinal direction of propagation. We have introduced indices in the exponential factors of Eq. (17) because, for the same sign of wavevector, electrons and holes go in opposite directions. Therefore, in Eq. (17), systematically denotes the right/left going states. One can introduce a transfer matrix such that . The matrix and the Landauer-Büttiker scattering matrix can be considered as equivalent descriptions of a contact, provided one introduces small but finite transmission coefficients to regularize in case of perfectly reflecting channels. This regularization procedure does not affect practical calculations as illustrated in Section VII.D. Since does not couple electron and holes, has the structure

(18) |

in the Nambu subspace. Moreover, is proportional to the identity in the Keldysh space, like . For later use, we point out that flux conservation leads toRefStone ()

(19) |

We now connect the above scattering approach with the Green’s function descriptionMillis (). With the assumptions done in this section, Eqs. (6-7) give, for and

(20) |

and

(21) |

We recall that in the ballistic zones, takes the form (11). In the domain , a comparison between Eqs. (20-21) and (15-16) gives, in terms of the decompositions (11) and (17)

We have introduced above the transformation to compensate the fact that the indices do not occur in the exponential terms of Eq. (11). Using Eq. (12), we obtain

(22) |

with , , and

(23) |

in the Nambu subspace. Note that due to Eq. (19), one has

(24) |

We now discuss how spin-dependences arise in our problem. Due to the hypotheses required to reach the diffusive limit [see Eq. (10)], we have neglected the dependence of and on the exchange field and the energy . Accordingly, we have to disregard the dependence of on and . This does not forbid that depends on spin. Indeed, in the general case, when an interface involves a material which is ferromagnetic in the bulk, the transfer matrix can depend on spin for two reasons: first, the wavectors of the electrons scattered by the barrier can depend on spin due to , and second, the interface barrier potential can itself be spin-dependent. Importantly, one can check that and occur independently in Eqs. (6-7). The value of and the spin-dependence of are not directly related, because the second depend on properties like interfacial disorder or discontinuities in the electronic band structure, which do not influence far from the interface. Therefore, nothing forbids to have simultaneously (this can occur e.g. in a diluted ferromagnetic alloy like PdNi) and a spin-dependent , due to a spin-dependent interface potential . It is even possible to obtain this situation artificially, by fabricating e.g. a contact with a very thin barrier separating two normal metals or superconductors. Note that in spite of , the exchange field can play a major role in diffusive hybrid circuits by modifying drastically the spatial evolution of the isotropic Green’s function inside a ferromagnetic metal on the scale [see Appendix E].

## V Isotropization scheme

In this section, we show that the Green’s function becomes isotropic in momentum space (i.e. proportional to the identity in the subspace) due to impurity scattering, when moving further away from the contact. One can consider that this process occurs in ”isotropization zones” with a size of the order of a few for side of the contactscale () (dotted areas in Fig. II). Beyond the isotropization zones, quasiparticles reach diffusive zones (purple areas in Fig. II) characterized by isotropic Green’s functions with no structure in the subspace. We show below that tends to at the external borders of the isotropization zones. Note that the results presented in this section do not depend on the details of the isotropization mechanism.

We study the spatial evolution of in the isotropization zones located at and , using Eqs. (13) and (14). The superconducting gap matrix can be neglected from these Eqs. due to . We thus obtain, for the isotropization zone of side and

(25) |

and

(26) |

Due to , one can disregard the space-dependence of in the above equations. We will thus replace by its value at the beginning of the diffusive zone , i.e. . For later use, we recall that and fulfill the normalization condition

(27) |

with the identity in the space. In the isotropization zone of side , Eqs. (12) and (25-27) give

(28) | ||||

with and

(29) |

for . Note that the choice of the coordinate in Fig.1 is somewhat arbitrary, i.e. defined only up to an uncertainty of the order of , because there is a smooth transition between the isotropization and diffusive zones of the contact. As a result, must tend continuously to its limit value in the diffusive zones. The function must vanish for (see e.g. Ref. Abrikosov, ). This imposes to cancel the ”exponentially divergent” terms in Eq. (28) , which requiresRefYuli ()

(30) |

(31) |

(32) |

(33) |

For we obtain from Eqs. (28)-(33) that finally approaches

(34) |

so that tends to . As required, the expression (34) of does not depend on the exact choice of the coordinate and vanishes for . Equations (28-33) indicate that the decay length for the isotropization of is , as anticipated above. Moreover, inserting Eq. (34) into Eq. (11) leads to an expression of whose semiclassical and isotropic average corresponds to , as expectedQCaverage (). Importantly, from Eqs. (28-33), one sees explicitly that is smooth on a scale of the Fermi wave length, which justifies a posteriori the use of the approximated Eqs. (13) and (14) in this section.

## Vi Matrix current and general boundary conditions

Our purpose is to establish a relation between and . To complete this task, it is convenient to introduce the matrix currentRefYuli ()

(35) |

This quantity characterizes the transport properties of the circuit for coordinate and energy . It contains information on the charge current (see section VII.5) but also on the flows of spins and electron-hole coherence. Note that in this article, denotes the absolute value of the electron charge. Using Eq. (11) and the orthonormalization of the transverse wave functions , the matrix current is written as

(36) |

for or . Here denotes the trace in the sub-space and is the conductance quantum. Inside the isotropization zones, using Eq. (28), one obtainsDefSign ()

(37) |

Considering that has a structure in the sub-space only, and that , one finds

(38) |

at any point in the left(right) isotropization zone. We conclude that, quite generally, the matrix current is conserved inside each isotropization zone. We will see in next paragraph that this property is crucial to derive the BCIGF.

In order to express in terms of and and , we multiply Eq. (30) by from the left and Eq. (32) by from the left and by from the right. Then, we add up the two resulting equations after simplifications based on Eqs. (22), (24), and (27). This leads to

(39) |

with . A similar calculation leads to

(40) |

with . Equations (39) and (40) represent the most general expression for in terms of the isotropic Green’s functions and the transfer matrix . The conservation of the matrix current up to the beginning of the diffusive zones allows to identify these expressions with

(41) |

Here, denotes the resistivity of conductor and the junction area. Formally speaking, Eqs. (39), (40) and (41) complete our task of finding the general BCIGF for spin-dependent and diffusive metallic interfaces. We recall that to derive these equations, we have assumed a weak exchange field in ferromagnets (), as required to reach the diffusive limit [see Eq. (10)]. However, we have made no restriction on the structure of the contact transfer matrix . In particular, can be arbitrarily spin-polarized, and it is not necessarily spin-conserving or channel-conserving. However, at this stage, a concrete calculation requires the knowledge of the full (or equivalently the full scattering matrix). Usually this information is not available for realistic interfaces and one has to reduce Eqs. (39)-(40) to simple expressions, using some simplifying assumptions. For a spin-independent tunnel interface, Eqs. (39)-(40) can be expressed in terms of the contact tunnel conductance only, which is a formidable simplificationKuprianov (). Another possibility is to disregard superconducting correlations. In this case, Eq. (39) and (40) lead to the normal-state BCIGF introduced in Refs. RefCircuit, ; Braatas, (see appendix C for details). The normal-state BCIGF involve the conductance but also a coefficient which accounts for the spin-dependence of the contact scattering probabilities, and the transmission and reflection mixing conductances and which account for spin-torque effects and interfacial effective fieldsrevuesFN (). We will show below that for a circuit enclosing superconducting elements, the BCIGF can also be simplified in various limits.

Note that since the transition between the ballistic, isotropization and diffusive zones is smooth, the choice of the coordinates and in Fig.1 is somewhat arbitrary, i.e. defined only up to an uncertainty of the order of or a fraction of respectively. However, one can check that this choice does not affect the BCIGF. First, a change of and by quantities and of the order of a fraction of requires to replace the matrix appearing in Eqs. (39-41) by , where the matrices and have a non-trivial (i.e. diagonal) structure in the subspace only, with diagonal elements and . Since commutes with , this leaves the BCIGF unchanged. Second, due to Eqs. (30-33), the BCIGF do not depend either on the exact values of and .

## Vii Case of a weakly spin-dependent contact

### vii.1 Perturbation scheme

In the next sections, we assume that the transverse channel index and the spin index corresponding to spin components along are conserved when electrons are scattered by the potential barrier between the two ballistic zones (we use for instance ). In this case, one can describe the scattering properties of the barrier with parameters , , , and defined from

(42) |

and

(43) |

with the transmission amplitude from side to side of the barrier and the reflection amplitude at side . The parameter corresponds to the spin-polarization of the transmission probability . The parameters and characterize the Spin Dependence of Interfacial Phase Shifts (SDIPS), also called in other references spin mixing angleTokuyasu (); Kopu (); Zhao (). In our model, and can be finite due to the spin-dependent interface potential . Due to flux conservation and spin conservation along , the parameters , , , and are sufficient to determine the value of the whole matrix (see Appendix A for details). Then, using Eq. (23), one can obtain an expression for . We will work below at first order in and . In this case, can be decomposed as

(44) |

The diagonal element of in the transverse channel subspace has the form, in the propagation direction subspace,

(45) |

with . Accordingly, the matrix is, in the propagation direction subspace,