Proximity effect in superconductor/conical magnet/ferromagnet heterostructures

At the interface between a superconductor and a ferromagnetic metal spin-singlet Cooper pairs can penetrate into the ferromagnetic part of the heterostructure with an oscillating and decaying spin-singlet Cooper pair density. However, if the interface allows for a spin-mixing effect, equal-spin spin-triplet Cooper pairs can be generated that can penetrate much further into the ferromagnetic part of the heterostructure, known as the long-range proximity effect. Here, we present results of spin-mixing based on self-consistent solutions of the microscopic Bogoliubov-de Gennes equations incorporating a tight-binding model. In particular, we include a conical magnet into our model heterostructure to generate the spin-triplet Cooper pairs and analyse the influence of conical and ferromagnetic layer thickness on the unequal-spin and equal-spin spin-triplet pairing correlations. It will be show that, in agreement with experimental observations, a minimum thickness of the conical magnet is necessary to generate a sufficient amount of equal-spin spin-triplet Cooper pairs allowing for the long-range proximity effect.


Introduction
At the interface between a normal metal and a superconductor (SC) an incoming electron with an energy above the Fermi energy (or chemical potential) µ can be reflected back into the metal as a hole with opposite spin orientation. This phenomenon, known as Andreev reflection [1], gives rise to the well-known proximity effect resulting in superconducting properties decaying into the normal metal part of the heterostructure. Replacing the normal metal by a ferromagnet (FM) drastically changes the behaviour at the interface. At first sight the phenomenona of ferromagnetism and superconductivity appear to be mutually exclusive due to the specific requirements concerning spin orientations. In ferromagnetic materials the Pauli principle requires the spins to orient parallel whereas superconducting spinsinglet Cooper pairs require an antiparallel orientation. Based on these intrinsic spin orientations there are interesting phenomena to be expected at the interface between a ferromagnet and a superconductor. The exchange interaction in the ferromagnet leads to different Fermi velocities of electrons in the spin-up and spin-down channel. Therefore, the centre of mass motion is modulated and superconducting correlations in the ferromagnet show an oscillating behaviour [2,3,4]. These are essentially the FFLO oscillations named after Fulde and Ferrell [5] and Larkin and Ochinikov [6], respectively. The penetration of these spin-singlet superconducting correlations are strongly supressed by the ferromagnetic exchange field and are only short-range. In addition, the exchange field also generates S z = 0 (unequal-spin) components of spin-triplet correlations, which also oscillate and decay. Similar to the spin-singlet correlations these are short-range as well.
However, it has been suggested theoretically by Bergeret et al. [7], that due to spin-flip processes at the interface equal-spin spin-triplet Cooper pairs can form which should be unaffected by the ferromagnetic exchange field thereby allowing much larger penetration depths. This phenomenon is called the long-range proximity effect and has triggered a lot of experimental and theoretical work. The proximity effect in superconductor-ferromagnet heterostructures is reviewed by Buzdin [8], whereas Bergeret et al. [9] review the physics behind this type of "odd-triplet" superconductivity. The symmetry relations between the different pairing correlations within the superconductor-ferromagnet heterostructures is discussed in detail by Eschrig et al. [10].
From the experimental side several multilayer setups have been suggested to observe spin-triplet proximity effect, but so far there exist only indirect proofs of the generation of spin-triplet supercurrents through the observation of supercurrents in Josephson junctions [11,12] or spin-valves [13].
Typical examples of multilayer systems to generate spin-triplet Cooper pairs experimentally involve noncollinear magnetisations within the different ferromagnetic layers [11,13,12], or helical (or conical) magnets in the multilayer setup [14,15,16]. This is accompanied by respective theoretical investigations of those sytems containing noncollinear magnetisations [17,18] and helical (or conical) magnetic material in the multilayer setup [19]. Additionally, also the effects of Bloch [20] or Néel [21] domain walls, spin-orbit coupling [22] or specific interface potentials [23,24] at the SC/FM interface on the generation of spin-triplet Cooper pairs have been investigated theoretically. Another route towards generating spin-triplet Cooper pairs leads to the inclusion of half-metallic ferromagnets such as CrO 2 into the heterostructures [25,26,27] which would pave the way for a marriage between supercurrents and spintronics applications [28].
The aim of the paper is as follows. A heterostructural setup similar to those used in the experiments of Robinson et al. [15] consisting of superconductor, conical magnet (CM) and ferromagnet will be investigated using self-consistent solutions to the spin-dependent microscopic Bogoliubov−de Gennes (BdG) equations [29]. One focus of the work lies on the influence of the conical magnet's opening and turning angles on the induced spin-triplet pairing correlations for which a detailed symmetry analysis will be provided. Secondly, we focus on the influence of conical magnetic layer thickness on the spin-triplet correlations. It will be shown that a minimum number of conical magnetic layers are necessary to efficiently generate equal-spin spin-triplet correlations, in agreement with experimental observations. The paper is organised as follows. Section 2 starts with a description of the theoretical method, namely the self-consistent solution of the spin-dependent BdG equations. This is followed by a detailed description of the multilayer structure used in the calculations, a setup for the conical magnetic structure, and finally the spindependent pairing correlations. Results are presented in section 3, where first a symmetry analysis of different conical magnet orientations is presented followed by the conical magnet's and ferromagnet's thickness dependence of the spin-dependent triplet pairing correlations. A concluding summary and an outlook will be given in section 4.

Bogoliubov-de Gennes equations and tight-binding Hamiltonian
All our calculations are based on self-consistent solutions of the microscopic BdG equations which for the spin-dependent case read [29,30,31] where ε n denote the eigenvalues of the matrix equation, and u nσ and v nσ are the quasiparticle and quasihole amplitudes for spin σ, respectively. H 0 is the tight-binding Hamiltonian, which for a system of two-dimensional layers can be written as with t being the next-nearest neighbour hopping parameter setting the energy scale with t = 1, and µ = 0 being the chemical potential (Fermi energy) set to half-filling. c † nk and c nk are electronic creation and destruction operators at layer index n with momentum k within the layers, respectively. Since the main focus of the present work lies on the presence of an interface within the multilayer or heterostructure, the only valid k values in (2) are to be found within the interface plane. For each of these k values the BdG equations (1) leads to a one-dimensional inhomogeneous problem in the layer index n [32]. For the sake of simplicity and since this would only lead to a parametrical dependence of the Hamiltonian on a discretised k mesh in the present work we neglect this k dependence. Equation (2) then simplifies to The implications of this simplification will be taken into account when discussing the obtained results in section 3. The pairing matrix can be rewritten according to the Balian−Werthamer transformation [33,34] utilising the Pauli matrices σ which effectively describes the superconducting order parameter comprising of a singlet (scalar) part ∆ and a triplet (vector) part d, respectively. In the present work ∆ is restricted to the s-wave singlet pairing potential in the superconductor sides of the heterostructure, to be determined self-consistently from the condition where the summation is performed over the positive eigenvalues ε n . f (ε n ) is the Fermi distribution function and g(r) the effective superconducting coupling set to 1 in our α β Figure 1. (a) Multilayer structural setup consisting of a spin-singlet s-wave superconductor (n SC layers), a conical magnet (n CM layers), a ferromagnetic metal (n FM layers), and a conical magnet and superconductor of the same thickness to the right. (b) Opening angle α and turning angle β of the conical magnet. From (6) it follows that α is measured from the positive y axis towards the positive z axis, whereas β is measured from the positive z axis towards the positive x axis.
calculations. It is assumed to be constant within the superconductor and to vanish elsewhere. Finally, h x , h y , and h z generally describe the vector components of a noncollinear exchange field to be added to the tight-binding Hamiltonian in the form hσ, with the vector components of σ being the Pauli matrices, respectively. In section 2.2 h will be defined to describe the conical magnetic structure within the multilayer setup.

Multilayer structural setup
The multilayer setup used in the present work is schematically shown in figure 1(a). It consists of a spin-singlet s-wave superconductor of n SC = 250 layers, a conical magnet of n CM = 1 · · · 25 layers, a ferromagnetic metal of up to n FM = 500 layers, followed by the same number of layers of conical magnet n CM and spin-singlet s-wave superconductor n SC to the right, respectively. The description of the conical magnet is chosen according to Wu et al. [19] h = h 0 cos αy + sin α sin βy a x + cos βy a z , with h 0 being the strength of the conical magnet's exchange field and a being the lattice constant (set to unity a = 1). As can be seen from (6) and figure 1(b), the opening angle α is measured from the positive y axis towards the positive z axis, whereas the turning angle β is measured from the positive z axis towards the positive x axis. Here these angles have been kept fixed to the values α = 80 • and β = 30 • to represent the conical magnet Holmium, a transition metal routinely used in similar experimental investigations [15]. Since the experimental geometry of how the conical structure is oriented with respect to the ferromagnetic region is unknown [35] our first set of calculations will examine the effects of different orientations and turning angle directions of the conical structure with respect to the two different ferromagnetic interfaces in section 3.1.

(Triplet) Pairing correlations
The general expression for the on-site superconducting pairing correlation of spins α and β for times t = τ and t ′ = 0 reads Therein,Ψ σ (r, τ ) denotes the many-body field operator for spin σ at time τ , and the time-dependence is governed by the Heisenberg equation of motion. Notice that this pairing correlation is local in space and so the triplet contributions vanish automatically in the case τ = 0 according to the Pauli principle [17]. Therefore, such a pairing field is only non-zero at finite times τ , an example of odd-frequency triplet pairing [9]. Substituting the field operators valid for our setup and phase convention the spin-dependent triplet pairing correlations read depending on the time parameter τ and with ζ n (τ ) given by These triplet pairing correlations correspond to S z = 0 (f ↑↓ + f ↓↑ ), +1 (f ↑↑ ), and -1 (f ↓↓ ), respectively.

Influence of conical magnet
As soon as a conical magnetic structure is included in the multilayer setup there are several ways to orient the magnetic moments with respect to the direction perpendicular to the interface layer (being the y-axis in our multilayer setup). As mentioned earlier, the opening angle α = 80 • and turning angle β = 30 • of the conical magnet are chosen to represent the magnetic structure of Holmium, routinely used in experimentally available multilayer structures [35]. Experimental evidence shows that the magnetic coupling at the CM/FM interface is most likely antiferromagnetic. Looking for the moment at the right FM/CM interface ( figure 1(a)) and assuming the ferromagnetic moments to orient along the +z axis, the conical magnetic moment closest to the interface can have two different antiferromagnetic-like orientations, namely pointing slightly towards the FM side (α R = 260 • case, with the cone opening into the FM layer) or slightly towards the CM side of the interface (α R = 280 • case, with the cone opening away from the FM layer). These angles are reversed at the left CM/FM interface, respectively. Furthermore, the handedness of the respective cone is determined not only by the turning angle β R (30 • and −30 • ) but also influenced by the respective opening angle. Looking again at the right FM/CM interface and an opening angle α R = 280 • (cone opening away from the FM layer) a turning angle β R = 30 From the symmetry discussion of the two cones it's apparent that the conical magnetic structure left of the FM interface is opening away from the interface towards the −y direction with a clockwise rotation. The same holds for the right side of the interface; the cone is opening away from the interface towards the y direction with a clockwise rotation. For this setup both conical magnetic structures seem to be identical; they both open away from the FM interface into the CM layers with a clockwise rotation of the conical magnetisation. But the results for this setup show a sign change between the left and right side CM/FM interfaces (figure 2 and table 1) Table 1.
Superconductor-conical magnet interface symmetry properties depending on the two conical magnet's angles. Given is the relation of the leftside triplet pairing correlations f L ↑↑ and f L ↓↓ (first column) to the corresponding right-side triplet pairing correlation f R ↑↑ and f R ↓↓ depending on the opening angles α L and α R , and the turning angles β L and β R , respectively. As can also be seen from figure 2 the given dependencies apply equally for the real and imaginary part. case 1: case 2: case 3: At first this looks like a discrepancy, but in fact this stems from the underlying symmetry of the d(r) vector describing the triplet pairing correlations which will be discussed now. Although in case 1 and β L = β R = 30 • both conical magnetic structures seem to be identical, in fact they can be transformed into one another by a C 2 rotation about the z axis located in the middle of the FM layers. According to Tinkham [36] the transformation of an arbitrary vector r under a symmetry operation described by a transformation matrix R(u) reads Applying the transformation matrix for a C 2 rotation given by to the superconducting order parameter written in the Balian−Werthamer way as of (4) yields and expresses exactly what is displayed in figure 3 and given in table 1, namely f L ↑↑ = −f R ↑↑ and f L ↓↓ = −f R ↓↓ , respectively. Looking now at case 1 but for the two cones having different handednesses (lower left panels of figure 2 and table 1) one notices a mixture between ↑↑ and ↓↓ contributions, namely f L ↑↑ = −f R ↓↓ and f L ↓↓ = −f R ↑↑ , respectively. In this case the transformation between the left and right conical magnetic structure is realised by a σ xz mirror plane again located in the middle of the FM layers with the respective transformation matrix Applying R(σ xz ) to the superconducting order parameter yields in agreement with results displayed in figure 2  There is nearly no influence of different αs and βs on the f ↑↓ + f ↓↑ spin-triplet pairing correlation corresponding to the S z = 0 case (as depicted in the middle panel of figure 3).

Influence of time parameter τ on spin-triplet pairing correlations
Chosing a specific fixed setup (case 1 as mentioned in section 3.1) this section deals with the influence of the time parameter τ entering the evaluations of the spindependent triplet pairing correlations (8) utilising (9). In addition to figure 2 the real and imaginary parts of f ↑↓ + f ↓↑ , f ↑↑ , and f ↓↓ are shown in figure 3 for different times τ ranging from 5 to 20. The upper panels display the unequal spin-triplet pairing amplitudes f ↑↓ + f ↓↑ . They clearly exhibit the oscillating behaviour associated with FFLO oscillations inside the FM region of the multilayer, whereas these oscillations are absent for the spin-equal triplet pairing amplitudes f ↑↑ (middle panels) and f ↓↓ (lower panels) of figure 3. Concentrating for the moment on the middle panels of figure 3 for τ = 10 one immediately recognises a change by a factor of 1/2 (left panels) and 2 (right panels) in the spin triplet pairing correlations when comparing with the left and right panels showing results obtained for times τ which are also changed by a factor of 1/2 and 2, respectively. Recognising this essentially linear dependence on the time factor τ in the present regime τ × ∆ << 1 entering the calculation of the spin triplet pairing correlations via (9), further calculations are restricted to a time parameter τ = 10. A more detailed investigation of the influence of τ on the spin triplet pairing correlations will be part of a later work. But here we simply note again that this triplet pairing correlation which is spatially local but retarded in time, vanishes at τ = 0, corresponding to the "odd triplet" pairing state derived by quasiclassical arguments by Bergeret et al. [9] and Eschrig et al. [10].

Influence of ferromagnetic layer thickness n FM on spin-triplet pairing correlations
Using the same geometries as above (case 1, case 2, and case 3 of section 3.1) and one full conical magnetic structure on either side of the ferromagnetic layer now the influence of the ferromagnet's layer thickness on the spin-dependent triplet pairing correlations shall be investigated. Figure   investigations the number of ferromagnetic layers will be fixed to n FM = 100 layers, respectively.

Influence of conical magnetic layer thickness n CM on spin-triplet pairing correlations
The results presented in this section allow for a deeper understanding of the influence of the conical magnetic layer thickness on the spin-triplet pairing correlations.
Since the influence of the overall conical magnetic layer orientation and number of ferromagnetic layers can be understood from the results already presented in section 3.2 and section 3.3 the multilayer setup will now be fixed to case 1 with n FM = 100 ferromagnetic layers, but with a conical magnetic layer thickness ranging from n CM = 0 to n CM = 25 layers (representing two full turns of the conical magnet along the growth direction). Figure 5 shows the influence of the conical magnetic layer thickness on the real (left panels) and imaginary parts (right panels) of the and f ↑↑ (lower panels) spin-triplet correlations in full view (left panels) and as a top view (right panels), respectively. From the magnitude of the f ↑↓ + f ↓↑ spin-triplet correlation one again notices the FFLO oscillations within the ferromagnetic region (upper panels), whereas the lower panels clearly show that the conical magnetic layer thickness strongly affects the strength of the f ↑↑ spin-triplet correlation. To get even more insight figure 7 finally shows sideviews of the real part (left panel) and the magnitude (right panel) of the f ↑↑ spin-triplet correlations. It is apparent that a minimum number of conical magnetic layers are necessary to generate equal-spin spintriplet correlations, in agreement with experimental observations. Keeping in mind that the multilayer setup is always starting with an antiferromagnetic-like coupling between the conical magnet and the ferromagnetic layer, an increasing number of conical magnetic layers as displayed in the results of figure 5 to figure 7 includes a different orientation of the conical magnetic structure at the superconductor / conical magnet interface. This detail requires more investigations as to whether the specific orientation of the conical magnet at the superconductor / conical magnet interface is partly responsible for the oscillating behaviour shown in the results with increasing conical magnetic layer thickness.

Summary and outlook
In summary, we presented a detailed analysis of spin-triplet pairing correlations within a superconductor/conical magnet/ferromagnet/conical magnet/superconductor heterostructure, similar to the ones investigated experimentally by Robinson et al. [15].
The results have been obtained by self-consistent solutions to the microscopic spin- dependent Bogoliubov−de Gennes equations which easily incorporate noncollinear exchange fields required to model the conical magnetic structure of Holmium also used in the experimental multilayers. While using a similar approach as Wu et al. [19] we extended their conical magnet/superconductor bilayer investigation to cover the whole heterostructure mentioned above. A detailed symmetry analysis of the equal-spin spin-triplet correlations from both, the left and the right hand side conical magnetic structure in the heterostructure, revealed at first sight surprising relations. These relations have been traced back to the specific underlying symmetry of our heterostructure setup. In addition, it has been shown that, in agreement with experimental observations, a certain minimum number of conical magnetic layers is necessary to sufficiently generate equal-spin spin-triplet Cooper pairs required for the long-range triplet proximity effect.