Topological superconductivity with orbital effects in magnetic skyrmion based heterostructures
TTopological superconductivity with orbital effects in magnetic skyrmion basedheterostructures
Maxime Garnier, ∗ Andrej Mesaros, and Pascal Simon Laboratoire de Physique des Solides, UMR 8502, CNRS,Universit´e Paris-Sud, Universit´e Paris-Saclay, 91405 Orsay, France (Dated: October 2, 2019)Proximitizing magnetic textures and s -wave superconductors is becoming a platform for engi-neering topological superconductivity and Majorana fermions by the means of exchange processes.However, the consequences of orbital effects have not yet been fully taken into account. In this work,we investigate the magnetic skyrmion texture-induced orbital effects using a Ginzburg-Landau ap-proach and clarify the conditions under which they can induce superconducting vortices. Theseorbital effects are then included in Bogoliubov-De-Gennes theory containing the exchange interac-tion, as well as superconducting vortices (when induced). We find that the topological phase islargely stable to all investigated effects, increasing the realistic promise of skyrmion-superconductorhybrid structures for realization of topological superconductivity. I. INTRODUCTION
The search for topological superconductors and Ma-jorana fermions has mostly relied on an approach com-bining conventional s -wave superconductivity, spin-orbitcoupling and magnetism to generate p -wave pairing.Even though great advances have been made using thisapproach using semiconducting wires (see e.g., Ref. 1 fora review) or magnetic atoms combined with supercon-ducting substrates, these experiments remain highlychallenging so that it may be worth considering remov-ing one of the ingredients. For example, one may considerremoving the spin-orbit coupling since the exchange in-teraction between conduction electrons and a magnetictexture induces an effective spin-orbit interaction.
This seems to be a viable platform for inducing topo-logical superconductivity.
In particular, particle-like topological spin configurations known as magneticskyrmions have recently been highlighted as primecandidates with interesting prospects for manipula-tion due to their high degree of controllability.
How-ever, in this context, the magnetic orbital effects havenot yet been fully taken into account.Generally, systems in presence of supercurrents andspin-orbit coupling may support topologically non-trivialphases without a spin coupling of the Zeeman form.
This suggests that orbital effects are sufficient to cre-ate topological phases in magnet-superconductor het-erostructures. In a more involved scenario, the Zeeman-form coupling (induced by the exchange interaction be-tween skyrmion and superconductor) combined with themagneto-electric effect due to the spin-orbit couplingleads to the appearance of vortices, which induce su-percurrents and topological superconductivity. If onenow considers alternatives based on removing the explicitspin-orbit coupling, which is a key coupling ingredient inprevious scenarios, one finds that topological supercon-ductivity is also predicted with only the Zeeman-formcoupling induced by the exchange interaction with theskyrmion.
However, even without explicit spin-orbit coupling, one should consider the effect of supercurrents,which may appear only via the orbital effects generatedby the skyrmion (since the magneto-electric couplingvanishes together with the spin-orbit coupling). The or-bital effects due to skyrmions cannot be generally ne-glected, as we argue below, and their effect on topologi-cal superconductivity has not been analyzed so far. As aconsequence, it is not clear if vortices are to be expectedand whether topological superconductivity persists whenboth exchange and orbital effects of the skyrmion are in-cluded in a superconductor without spin-orbit coupling.In conventional type-II superconducting thin films, theeffective penetration depth (or Pearl length) λ eff can beorders of magnitude larger than the film thickness d . This implies that the screening currents are weak andmay become negligible. Previous works have usedthis observation to set to zero the magnetic vector po-tential in the superconductor. However, we argue thatsince the screening is weak, the magnetic field generatedby the skyrmion penetrates the superconductor almostunaltered, and is thus not necessarily negligible. There-fore in contrast to previous works, we include the vectorpotential generated by the skyrmion as an orbital effecton the electrons in the superconductor. To fully under-stand the phase diagram, we consider the magnetic ex-change as an independent effect on the electrons, sincethis term could be experimentally tuned by an insulat-ing non-magnetic layer, which prevents the hopping ofelectrons between the skyrmion and the superconductorwhile not affecting the strength of the vector potential.The aim of the present work is twofold. First, we inves-tigate the orbital effects of the magnetic field created bythe skyrmion on a conventional type-II superconductorwithout spin-orbit coupling and clarify the conditions ofexistence of vortices in a Ginzburg-Landau framework.Second, using the Bogoliubov-de-Gennes formalism weaddress the robustness of the topological phase inducedby the exchange interaction to the orbital effects and thepossible presence of vortices. We find that the exchange-induced topological phase is robust to the inclusion of a r X i v : . [ c ond - m a t . s up r- c on ] O c t x
The system under investigation is composed of aconventional type-II superconducting film without spin-orbit coupling (such as Al) in proximity to an insulat-ing magnetic thin film harboring a skyrmion (such asCu OSeO ) as shown in Fig. 1. We denote by d the thickness of the superconducting layer and by h thethickness of the magnet. We focus on insulating magnetsso as to minimize feedback mechanisms on the magnetdue to the superconductor. Even though magneticskyrmions exist in two types, namely N´eel ( cf.
Fig. 1)and Bloch (as in Cu OSeO ), this distinction doesnot modify the physics described here as will be clari-fied.Since in-plane critical fields of superconducting filmsare usually larger than that of bulk superconductors ,in analyzing the magnetostatics of our system we focuson the magnetic field component perpendicular to thesuperconductor ( z axis). The orbital coupling to the su-perconductor is hence through the vector potential whichsatisfies ∇ × A ( r , z ) = B z ( r ) ˆz , where r labels the po-sition in the plane. We take B z to be the magnetic in-duction created by a lone skyrmion, thereby neglectingany feedback effect of the superconductor on the mag-netic material. This assumption is consistent with ne- glecting the screening currents and their fields. In thelimit where the skyrmion is confined to a plane, the mag-netic induction B z ( r , z ) it creates becomes equal to itsmagnetization component m z ( r ) at least near the plane(see Ref. 46 and Supplementary Material (SM) A). Moreprecisely, we find that B z decays away from the plane ona lengthscale given by the radius R sk of the skyrmion,and for z smaller than this lengthscale we can define A by ∇× A = µ m z ( r ) ˆz where µ is the vacuum magneticpermeability. The radius R sk is defined by the magne-tization profile m ( r ) of the skyrmion texture in polarcoordinates r = ( r, θ ) m ( r ) = M sin f ( r ) cos ( θ + γ )sin f ( r ) sin ( θ + γ )cos f ( r ) (1)The skyrmion is characterized by the radial winding num-ber p ∈ N that counts the number of spin flips as onemoves away from the core r = 0 along the radial direc-tion. f ( r ) is the radial profile of the skyrmion that wechoose to be f ( r ) = k s r for r ≤ R sk , where we have in-troduced k s = π/λ s with the spin-flip length λ s = R sk /p .The global angular offset γ called helicity allows todescribe both N´eel ( γ = 0, π ) and Bloch ( γ = ± π )skyrmions. The norm M of m ( r ) defines the saturationmagnetization of the magnet.Focusing on the N´eel ( γ = 0) case, the magnetizationEq. (1) can be written in cylindrical coordinates m ( r ) = M (sin ( k s r ) u r + cos ( k s r ) u z ) (2)A suitable vector potential is A ( r ) = µ Mk s (cid:18) cos ( k s r ) k s r + sin ( k s r ) (cid:19) u θ (3)which indeed gives B z = µ M cos ( k s r ). Note that al-though this expression was derived for a N´eel skyrmion,the magnetic induction generated by a Bloch skyrmionis qualitatively the same (see Ref. 46 and SM A).For conventional superconductors, the Ginzburg-Landau (GL) free energy functional F is F = B c µ (cid:90) d r (cid:20) − (cid:12)(cid:12)(cid:12) ˜∆ (cid:12)(cid:12)(cid:12) + 12 (cid:12)(cid:12)(cid:12) ˜∆ (cid:12)(cid:12)(cid:12) + ξ (cid:12)(cid:12)(cid:12) ˆ D ˜∆ (cid:12)(cid:12)(cid:12) + B B c (cid:21) (4)where ˜∆ is the superconducting order parameter nor-malized by its thermodynamic value in the absence offields or gradients, B c is the thermodynamic critical field,ˆ D = − i ∇ + e (cid:126) A is the covariant derivative with 2 e thecharge of Cooper pairs, and ξ is the superconducting co-herence length. Finally, B = ∇ × A is the magneticinduction inside the superconductor. The free energyEq. (4) is measured with respect to the normal-state freeenergy. As we are interested in the behavior of the su-perconductor in an external magnetic field, the correctthermodynamic potential to consider is the Gibbs free en-ergy G = F − (cid:82) d r H ( r ) · B ( r ) where H is the magneticfield. As mentioned above, the typical electromagneticresponse of a type-II superconducting film in a homoge-neous magnetic field occurs on a lengthscale λ eff muchlarger than any other lengthscale in the problem. Asa consequence, the inhomogeneous response on length-scales of the order of the skyrmion’s spin-flip length λ s cannot be inferred easily. As outlined above we neglectthe screening currents so that we can approximate B by µ H inside the superconductor. With this assumption, G simply reduces to F and is given by Eq. (4) where B is now the skyrmion-generated induction given belowEq. (3).We further suppose that all quantities are indepen-dent on the z coordinate, which is valid as long as the superconducting thin film thickness d is smaller than theskyrmion radius R sk , so that in this regime the thickness d can be factored out of the free energy.The final ingredient of our model is a superconduct-ing vortex. To establish its presence or absence we usea standard vortex ansatz ˜∆ ( r ) = ˜∆ α ( r ) e iαθ where˜∆ α ( r ) = tanh | α | ( r/ξ ) and α ∈ Z is the phase windingof the vortex. Note that even if this ansatz correspondsto an Abrikosov vortex ( i.e. in a bulk sample), we ex-pect that the details of the ansatz don’t matter muchas long as the order parameter amplitude decays on alengthscale ξ and vanishes at the vortex core. Includingthe vector potential due to the skyrmion Eq. (3) and thevortex ansatz, the total free energy G tot is: G tot = B c µ πdk s (cid:90) d˜ r ˜ r (cid:34) − (cid:12)(cid:12)(cid:12) ˜∆ α (cid:12)(cid:12)(cid:12) + 12 (cid:12)(cid:12)(cid:12) ˜∆ α (cid:12)(cid:12)(cid:12) + k s ξ (cid:16) ∂ ˜ r (cid:12)(cid:12)(cid:12) ˜∆ α (cid:12)(cid:12)(cid:12)(cid:17) + k s ξ ν ˜ r (cid:12)(cid:12)(cid:12) ˜∆ α (cid:12)(cid:12)(cid:12) (cid:16) αν + cos (˜ r ) + ˜ r sin (˜ r ) (cid:17) + p π (cid:18) µ MB c (cid:19) (cid:35) (5)where we have defined the dimensionless coordinate ˜ r = k s r and the parameter ν ≡ πµ Mk s φ where φ = h/ e isthe superconducting flux quantum. The ν parameter canbe interpreted as the ratio | φ S | / (2 φ ) of the flux of theskyrmion | φ S | through a disc of radius λ s (the single spinflip distance) to the normal-metal flux quantum 2 φ . III. PHASE DIAGRAM OF THESUPERCONDUCTOR WITHOUT EXCHANGEEFFECTS
The free energy, Eq. (5), depends on three parameters:( i ) the ratio of the characteristic skyrmion flux and theflux quantum ν = | φ S | φ ,( ii ) the ratio of the superconducting coherence lengthto the spin-flip length of the skyrmion k s ξ = pπξ/R sk ,( iii ) the ratio µ M/B c .Given some values of these parameters, our strategy is tonumerically compute the free energy for different wind-ings α (including α = 0 for absence of vortex) and findthe value that gives the lowest free energy. If the freeenergy is negative, the normal state is realized instead ofthe superconducting one. Furthermore, as the magneticenergy term (last term in Eq. (5)) doesn’t depend on α ,we start by neglecting it.In Fig. 2 we show the phase diagram obtained for a p =4 skyrmion where we have introduced a short-distancecutoff k s l for the dimensionless variable ˜ r to deal with thelogarithmic divergence in the α = 0 case. (This cutoff has ξ / λ s ν = | φ S | / (2 φ ) -1-201NormalGround state FIG. 2. Phase diagram obtained from Eq. (5) for a p = 4skyrmion with short-distance cutoff k s l = 10 − . If the groundstate is superconducting, the color represents the value of thevortex winding number α , with α = 0 meaning absence of vor-tex (red). The non-superconducting ground state is labeled“normal” (green). only minor consequences on our results, see SM B 1, andits value is chosen to be comparable to the lattice spacingin the Bogoliubov-de-Gennes Hamiltonian of Sec. IV).The phase diagram shows that there exists a super-conducting phase without vortex (red color) for a rela-tively large range of superconducting coherence lengthsand skyrmion fluxes ν (cid:46) .
5. For sufficiently strongskyrmion flux and small ξ/R sk , there exist phases witha vortex. We have also checked that including the mag-netic energy term and varying the size of the skyrmiondon’t affect qualitatively our results, see SM B 2 and B 3.We thus conclude that vortices are not always expectedwhen proximitizing a conventional superconductor with-out spin-orbit coupling by a magnetic skyrmion. IV. IMPLICATIONS FOR TOPOLOGICALSUPERCONDUCTIVITY
We now turn to the consequences of the orbital ef-fects on the topological superconducting phase inducedby the exchange interaction.
As shown in the pre-vious section, the inclusion of the orbital effects due tothe skyrmion has two consequences: the first is that elec-trons see a magnetic vector potential and the second isthat vortices may be present.
A. Bogoliubov-de-Gennes setup
In the Bogoliubov-de-Gennes (BdG) formalism, the to-tal Hamiltonian H describing the electrons can be writ-ten H = (cid:82) d r Ψ † ( r ) H ( r ) Ψ ( r ) where H is the BdGHamiltonian. Throughout the article, we work in theNambu basis Ψ † ( r ) = (cid:16) ψ †↑ ( r ) , ψ †↓ ( r ) , ψ ↓ ( r ) , − ψ ↑ ( r ) (cid:17) where ψ † σ ( r ) is the field operator creating an electronwith spin projection σ = ↑ , ↓ at position r = ( r, θ ) in twodimensions (2D). Following the minimal coupling pro-cedure ˆ p → ˆ p + e A , the 2D BdG Hamiltonian in thepresence of both orbital and exchange effects reads H ( r ) = (cid:18) m (ˆ p + e A τ z ) − µ (cid:19) τ z + J σ · m ( r ) + ∆ τ x (6)where τ i and σ i , i = x, y, z , are Pauli matrices actingin particle-hole and spin space, respectively, ˆ p = − i (cid:126) ∇ is the momentum operator, m is the effective mass, µ the chemical potential, ∆ the bare s -wave pairing, J isthe exchange interaction, and orbital effects are due tothe vector potential A . The vector m ( r ) is the skyrmiontexture as parametrized in Eq. (1) and we set the he-licity γ = 0 since it can be unitarily removed from theHamiltonian. Hereafter, the exchange interaction J in-cludes the saturation magnetization M of the magnet.We emphasize that the strengths of orbital effects andexchange interactions can be tuned independently. Ex-perimentally, the coupling J can be reduced by an insu-lating layer between the magnet and the superconductor.The BdG Hamiltonian Eq. (6) has a generalized ro-tation symmetry and total angular momentum operatorwhich, in absence of any vortices, reads J z = L z + (cid:126) σ z where the orbital angular momentum reads L z = − i (cid:126) ∂ θ .The eigenvalues of J z provide a quantum number m J (in units of (cid:126) ) labeling independent blocks of the BdGHamiltonian. Angular momenta are half-odd-integer asrequired by the singlevaluedness of the wavefunction. Wediscretize the Hamiltonian according to r → r j = ja withlattice constant a chosen as the length unit ( a ≡
1) andthe hopping parameter t = (cid:126) / (2 ma ) is chosen as theenergy unit ( t ≡
1, see SM C and the Methods section of Ref. 27 for additional technical details). For complete-ness, the lengthy expression of the discretized Hamilto-nian H (1) m J corresponding to Eq. (6) is given in SM C 1. Inall computations we use hard wall boundary conditionsat the skyrmion’s edge. B. Orbital effects without exchange and vortices
Neglecting the exchange interaction ( J = 0 in Eq. (6)),the relevant angular momentum operator is J z = L z .The discretized m L -dependent Hamiltonian H (2) m L is givenin SM C 2, with m L ∈ Z the eigenvalue of L z (in unitsof (cid:126) ). The vector potential contributes two terms: (1) aspace-dependent renormalization of the chemical poten-tial via the A term, and (2) a term ∝ that dependson both space and angular momentum.Fig. 3 below shows the numerically obtained BdG spec-tum contrasting the cases of weak and strong orbitaleffects, i.e. ν = 1 (Fig. 3a) and ν = 5 (Fig. 3b). Inthe latter case, we neglect the predicted appearance ofvortices (see Fig. 2) to better isolate the orbital effects.The electronic local density of states (LDoS) along the r = 0 → r = R sk radial line in the ν = 5 case is shownin Fig. 4a. −
20 0 20 m J − . − . . . . E / ∆ −
20 0 20 m J − . − . . . . E / ∆ −
20 0 20 m L − . − . . . . E / ∆ (a) (c) −
20 0 20 m J − . − . . . . E / ∆ (d)(b) FIG. 3. Bogoliubov-de-Gennes spectra in absence of vortices.In absence of exchange coupling ( J = 0), the purely orbitaleffects are chosen weaker, ν = 1, in (a) and stronger, ν =5, in (b). In (c,d) topological superconductivity is causedby exchange coupling, J/t = 0 .
2, both in absence of orbitaleffects, ν = 0 in (c), and also in presence of orbital effects, ν = 1 in (d). The system parameters for all panels are p = 10, L/a = 1000, ∆ /t = 0 . µ/t = 0 (see SM C 2). We observe that the bare superconducting gap ∆ ispreserved when orbital effects are added. This is some-what expected since spin degeneracy is not lifted in theHamiltonian. In the ν = 5 case, a few localized statescome down in energy but stick to the top (resp. bot-tom) of the gap for m L < m L > (b) L D o S ( a . u . ) (a) FIG. 4. Local density of states in absence of vortices, con-trasting (a) purely orbital effects, J = 0 and ν = 5 with (b)both exchange, J/t = 0 .
2, and orbital effects, ν = 1. Themodels in (a), (b) are the same as in Fig. 3(b), (d), respec-tively. In both panels the system parameters are p = 10, L/a = 1000, ∆ /t = 0 . µ/t = 0 (see SM C 2). near the core of the skyrmion) resembling an analogue ofCaroli-de-Gennes-Matricon states. In fact, similar in-gap spectrum due to purely orbital effects was found ina self-consistent calculation of the magnetic field dueto a preformed normal metal dot in a superconductor(without vortex). C. Orbital effects and exchange interaction
When restoring the exchange interaction, the 2D BdGHamiltonian in Eq. (6) has total angular momentum J z = L z + (cid:126) σ z and the full radial tight-binding Hamiltonian H (1) m J is given in SM C 1.To assess the consequences of orbital effects, we first re-turn to the model with only exchange interaction ( ν = 0),which leads to topological superconductivity, seeFig. 3c. For completess, let us recall briefly the prop-erties of this topological superconductor. First, the ef-fective gap is of p -wave origin with an amplitude givenby ∆ eff = πλ s ∆ J √ J + µ where all energies are in unitsof t and λ s is expressed in units of a . This evaluates to∆ eff ≈ which is consistent with the numerical data.Within the gap, there are two types of states: namely anearly flat band located at the edge of the skyrmion to-gether with dispersing states located near the core. Thesestates are attributed to impurity-like states induced bythe discretized magnetic texture. In the absence of or-bital effects, the nearly-flat band is in fact slightly chiraland can be assigned a topological character thereby form-ing a chiral Majorana edge mode. We now turn to the case of combined exchange andorbital effects for ν = 1. Fig. 4b shows that the topolog-ical superconductor described above essentially survivesthe inclusion of orbital effects. Specifically, the strongesteffect on the spectrum is for the dispersing in-gap “im-purity states”. In contrast, the momentum m ∗ J at whichthe bulk gap closes still matches the value predicted forpurely exchange coupling (see Ref. 27 for details). Re-garding the chiral edge mode, interestingly its velocityincreases and changes sign. This can be phenomenolog- ically explained by the chiral symmetry interpretationof Ref. 27. In absence of orbital effects, the smallnessof the velocity was attributed to an only weakly brokenchiral symmetry given by S = σ y τ y . When including or-bital effects ( ν (cid:54) = 0), different chiral-symmetry-breaking(CSB) terms appear, e.g., the one proportional to . Thisterm spatially decays as r − , so it should affect chiralitystronger than the CSB term in the pure exchange model,which decays as r − . D. Adding superconducting vortices
The interplay between exchange effects and vorticesin a skyrmion proximitized by an s -wave superconductorwas studied in Ref. 28 in absence of orbital effects. Herewe analyse the full problem by including orbital effectstogether with the exchange interaction and a supercon-ducting vortex.Using the notation of the previous section, a super-conducting vortex of winding number α is representedby modifying the pairing Hamiltonian H SC = ∆ τ x inEq. (6) to H v = ∆ α ( r ) e iαθ τ + + ∆ α ( r ) e − iαθ τ − (7)where τ ± = ( τ x ± iτ y ) / We neglect the spatial vari-ation of ∆ α ( r ) and assume that the amplitude of theorder parameter is constant, independent of α and equalto ∆ . In this situation, the total angular momentumreads J z = L z + (cid:126) σ z − α (cid:126) τ z and we still denote itseigenvalue by m J . The momentum is quantized accord-ing to m J ∈ Z (resp. m J ∈ Z + ) if α odd (resp. α even).The total radial Hamiltonian H (3) m J is given in SM C 3.Fig. 5a, b show the excitation spectrum and the elec-tronic LDoS for a vortex with α = − ν = 2 ( cf. the phase diagram in Fig. 2). Fig. 5a,bshow that the features of the topological superconduc-tor of the exchange model are almost unaffected by thepresence of the vortex (without orbital effects), in accordwith Ref. 28.Fig. 5c,d show that the inclusion of orbital couplingleads to similar effects as in the case of the absence ofvortex (previous subsection), namely, (1) it mixes the in-gap impurity states (similar impurity states were founddue to purely orbital effects in presence of vortex on ametallic dot in a superconductor ), and (2) it changesthe slope of the topological chiral mode. For these spe-cific parameters, note that the chiral symmetry breakingdue to the vortex (proportional to α/r , see Eq. (C3) inSM C 3) adds constructively to the symmetry breakingby the orbital term (proportional to ν/r , see Eq. (C1) inSM C 1), because the change in the slope of the chiralmode is significantly higher when the vortex is includedtogether with the orbital terms. −
20 0 20 m J − . − . . . . E / ∆ (c) −
20 0 20 m J − . − . . . . E / ∆ (b) (d)(a) L D o S ( a . u . ) FIG. 5. BdG spectrum and LDoS in presence of supercon-ducting vortex with winding number α = −
1. (a), (b) Spec-trum and LDoS for ν = 0 (no orbital effects). (c), (d) Spec-trum and LDoS for ν = 2. Fixed system parameters are p = 10, L/a = 1000,
J/t = 0 .
2, ∆ /t = 0 . µ/t = 0 (seeSM C 3). V. DISCUSSION
Our analysis shows that the exchange-induced topo-logical phase is robust to the inclusion of skyrmion-generated orbital effects as well as superconducting vor-tices. These effects have to be taken into account sincethe magnetic field generated by an isolated skyrmion canreach the mT range . We have shown using a Ginzburg-Landau approach that proximitizing a skyrmion with a superconductor does not necessarily lead to the forma-tion of vortices. However, even if the superconductordoes not develop vortices, the electrons still experience amagnetic vector potential whose effects on the topologicalsuperconducting phase were not fully understood in ma-terials without spin-orbit coupling. Our results demon-strate that the inclusion of orbital effects does not in-validate the previously established understanding of thetopological superconductor and contributes to makingthe skyrmion-superconductor hybrid structure a promis-ing platform for the realization of topological supercon-ductivity. Note however that we have neglected the Zee-man effect that would effectively render the exchange in-teraction anisotropic (see Eq. (A2)). Nevertheless thisrenormalization is far below the bare exchange interac-tion strength, so would not matter even if the exchangestrength was reduced by a non-magnetic insulating layerbetween the skyrmion and the superconductor.Because isolated skyrmions with arbitrary windingnumbers and helicity are theoretically more likely to in-duce topological superconductivity, a possible inter-esting direction would be to extend the present magne-tostatic calculations for such skyrmions. More generally,our work calls for a fully self-consistent calculation ofboth the magnetic and the superconducting order to con-firm all the features of the system. VI. ACKNOWLEDGMENTS
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Maxime Garnier, Andrej Mesaros and Pascal Simon
Laboratoire de Physique des Solides, UMR 8502, CNRS,Universit´e Paris-Sud, Universit´e Paris-Saclay, 91405 Orsay, France
Supplementary Material A: Magnetostatics of the skyrmion lattice
In this appendix we support our hypothesis that the perpendicular induction B z created by a single skyrmion isproportional to the z component of the magnetization. We do so by focusing on a skyrmion lattice following Ref. 46.A triangular skyrmion lattice can be approximated using a superposition of three helical spin orders with wavevectors Q i =1 , , yielding the so-called triple- Q parametrization . In three-dimensional space equiped with the unit vectors u x,y,z , consider an infinitely thin magnetic film that lies in the z = 0 plane. The three wavevectors Q i have the samenorm | Q i | = Q for all i and they make a 2 π/ Q = Q u x , Q = Q (cid:0) − u x + √ u y (cid:1) / Q = Q (cid:0) − u x − √ u y (cid:1) /
2. For a N´eel skyrmion lattice, the magnetization m ( r , z )with r = ( x, y ) reads m lat ( r , z ) = m δ ( z ) u z + A (cid:88) i =1 [cos ( Q i · r ) u z + sin ( Q i · r ) u i ] δ ( z ) (A1)where we have defined the unit vectors u i = Q i /Q . We also define the skyrmion radius R sk as R sk = 2 π/ (cid:0) Q √ (cid:1) (seecaption of Fig. A 1). Eq. (A1) is an approximation in the sense that it is not a proper micromagnetic solution sinceit is not normalized.In terms of the thickness h and the saturation magnetization M lat of the magnetic film, the parameter A reads A = M lat h . In this approach, we define the skyrmion radius R sk as R sk = 2 π/ (cid:0) Q √ (cid:1) . An example of such aN´eel skyrmion lattice is presented in Fig. A 1. One way to compute the magnetic field B lat ( r , z ) created by the -2-1012-10 -5 0 5 10-10-50510 m z x y Q
755 and Q = 0 . magnetization distribution is to compute the magnetic vector potential A lat ( r , z ) since B lat = ∇ × A lat . Then, A lat is found by solving the Poisson equation ∇ A lat ( r , z ) = − µ J m ( r , z ) in the Coulomb gauge ∇ · A lat = 0, where J m ( r , z ) = ∇ × m lat ( r , z ) is the Amperean current density that contains both “bulk” and “surface” contributions.Taking the curl of the solution A lat ( r , z ) of this equation yields B lat ( r , z ) = µ A e − Q | z | (cid:88) i =1 [ Q (1 − sgn ( z )) cos ( Q i · r ) u z + ( Q sgn ( z ) − Q + 2 δ ( z )) sin ( Q i · r ) u i ] (A2)This field displays the “single-sided flux” phenomenon or Halbach effect meaning that the magnetic field isonly present on one side of the plane. The apparent discontinuity in the perpendicular component of the magneticinduction is an artifact of the model and can be regularized by taking into account the finite thickness of the magneticfilm so that this effect is indeed physical . The perpendicular decay length of the magnetic field is given by Q − and can be expressed in terms of the skyrmion radius R sk as Q − = √ R sk / (2 π ) ≈ . R sk .With the idea to proximitize the skyrmion lattice by a superconductor, we are only interested in the magnetic fieldnear the plane i.e. | Qz | →
0. In the case discussed here, the relevant limit is Qz → − . In this limit, the z componentof the magnetic field reads B z lat (cid:0) r , − (cid:1) = µ AQ (cid:88) i =1 cos ( Q i · r ) (A3) i.e. close to the plane B z lat is proportional to the z component of the magnetization. Focusing on z (cid:28) Q − , we arguethat this result applies to our case of the isolated skyrmion as long as the thickness d of the superconductor is muchsmaller than the skyrmion radius since R sk ∝ Q − . Additionally, we have A = M lat h , we find that the amplitude ofthe magnetic field near the surface of the magnet is given by µ M lat Qh . Note that this result also holds in the caseof a Bloch skyrmion lattice with the only difference that the magnetic field is evenly shared between the two sides ofthe plane . Supplementary Material B: Influence of parameters on the phase diagram
The phase diagram displayed in Fig. 2 in the main text was computed for a p = 4 skyrmion with a cutoff k s l = 10 − and we neglected the magnetic energy term proportional to µ M/B c where B c is the thermodynamic critical field ofthe superconductor. We now discuss their effects.
1. Effect of the short-distance cutoff
We expect that reducing the cutoff will make the superconducting state with winding α = 0 energetically defavorablesince is will give more weight to the divergence as r →
0. This is indeed what we obtain in Fig. B 1 by changing k s l = 10 − to k s l = 10 − . The normal-superconductor transition line is almost unaffected by this change but the ν = | φ S | / (2 φ ) ξ / λ s -10-21NormalGround state-3 Fig. B 1. Phase diagram for the same parameters as in Fig. 2 but with a smaller cutoff k s l = 10 − . α = 0 phase still exists for weak enough skyrmions.0
2. Effect of the magnetic energy term
We now reinstate the magnetic energy term we have neglected so far. As it is α -independent, it’s only effect will beto move the normal-superconducting transition line depending on the ratio µ M/B c . However, in our parametrizationthe parameter ν is not independent from M . In order to estimate the effects of this term, we set µ M/B c = 1 forall ν . This is a crude approximation that largely overestimates the strength of this term on a large area of the phasediagram. The phase diagram is shown in Fig. B 2 ν = | φ S | / (2 φ ) ξ / λ s -201NormalGround state Fig. B 2. Phase diagram for the same parameters as in Fig. 2 but the purely magnetic term is included with the approximation µ M/B c = 1 for all ν . This shows that the inclusion of the magnetic term does not qualitatively affect our results. It further shows thatthe inhomogeneity of the skyrmion yields a stable superconducting phase even if µ M > B c .
3. Influence of the skyrmion size
The results for a p = 10 skyrmion are displayed in Fig. B 3. These results show that the conclusions of the maintext hold qualitatively except that winding number different from -1 have now disappeared from the phase diagram. ν = | φ S | / (2 φ ) ξ / λ s -10NormalGround state ν = | φ S | / (2 φ ) ξ / λ s -10NormalGround state (a) (b) Fig. B 3. Phase diagram of a p = 10 skyrmion with cutoff k s l = 10 − . (a) without the magnetic energy term and (b) with themagnetic energy term and µ M/B c = 1 for all ν . Supplementary Material C: Radial tight-binding Hamiltonians
In this section we give the expression of the second-quantized discretized radial Hamiltonians used to obtain theresults of the main text. In all Hamiltonians we have introduced t = (cid:126) / (cid:0) ma (cid:1) , k s = k s a where a ≡ C † j = (cid:16) c †↑ ( ja ) , c †↓ ( ja ) , c ↓ ( ja ) , − c ↑ ( ja ) (cid:17) and c σ ( ja )1to denote the discretized versions of the spinor Ψ ( r ) and the field operator ψ σ ( r ). We have chosen j = 0 . , . . . , L + 0 .
1. Orbital effects and exchange interaction
This case corresponds to Eq. (6) of the main text. The total angular momentum is J z = L z + (cid:126) σ z with eigenvalue m J (in units of (cid:126) ). The discretized Hamiltonian reads H (1) m J = (cid:88) j − t C † j +1 τ z C j + h . c . + C † j (cid:34) t − µ − t j (cid:0) − m J − q (cid:1) − tj qm J σ z + t s ν (cid:18) cos (k s j )k s j + sin (k s j ) (cid:19) (cid:35) τ z C j + C † j (cid:20) t k s νj (cid:18) cos (k s j )k s j + sin (k s j ) (cid:19) (cid:16) m J − q σ z (cid:17) + J cos (k s j ) σ z + J sin (k s j ) σ x + ∆ τ x (cid:21) C j (C1)
2. Orbital effects without exchange and vortices
This case corresponds to setting J = 0 in Eq. (6) of the main text. or equivalently J = q = 0 in Eq. (C1). Thetotal angular momentum is J z = L z with eigenvalue m L (in units of (cid:126) ). The discretized Hamiltonian is H (2) m L = (cid:88) j − t C † j +1 τ z C j + h . c . + C † j (cid:34) t − µ − t j (cid:0) − m L (cid:1) + t s ν (cid:18) cos (k s j )k s j + sin (k s j ) (cid:19) (cid:35) τ z C j + C † j (cid:20) t k s νj (cid:18) cos (k s j )k s j + sin (k s j ) (cid:19) m L + ∆ τ x (cid:21) C j (C2)
3. Orbital effects, exchange interaction and superconducting vortex
Replacing the superconducting pairing term in Eq. (6) by Eq. (7) (both in the main text), the total angularmomentum is J z = L z + (cid:126) σ z − α (cid:126) τ z with eigenvalue m J (in units of (cid:126) ) where α is the vortex winding. Thediscretized Hamiltonian reads H (3) m J = L (cid:88) j =1 − t C † j +1 τ z C j + h . c . + C † j (cid:34) t − µ − t j (cid:0) − m J − q − α (cid:1) + t s ν (cid:18) cos (k s j )k s j + sin (k s j ) (cid:19) (cid:35) τ z C j + C † j (cid:20) t k s νj (cid:18) cos (k s j )k s j + sin (k s j ) (cid:19) (cid:16) m J − q σ z + α τ z (cid:17) + tj (cid:16) − qm J σ z + αm J τ z − αq σ z τ z (cid:17) τ z + J cos (k s j ) σ z + J sin (k s j ) σ x + ∆ α ( r ) τ x ] C jj