Meissner Currents Induced by Topological Magnetic Textures in Hybrid Superconductor/Ferromagnet Structures
Samme M. Dahir Anatoly F. Volkov Ilya M. Eremin
cond-mat.supr-con cond-mat.mes-hall cond-mat.str-el

	
	
	Institut für Theoretische Physik III,
		Ruhr-Universität Bochum, D-44780 Bochum, Germany
	
	
	
	
		Topological spin configurations in proximity to a superconductor have recently attracted great interest due to the potential application of the former in spintronics and also as another platform for realizing non-trivial topological superconductors. Their application in these areas requires precise knowledge of the existing exchange fields and/or the stray-fields which are therefore essential for the study of these systems.
		Here, we determine the effective stray-field H_str and the Meissner currents j_S in a Superconductor/Ferromagnet/Superconductor (S/F/S) junction produced by various nonhomogenous magnetic textures M(*r) in the F. The inhomogeneity arises either due to a periodic structure with flat domain walls (DW) or is caused by an isolated chiral . 
		We consider both Bloch– and  Sk and also analyze in detail the periodic structures of different types of DW's— that is  DW (BDW) and  DW  (NDW) of finite width with in- and out-of-plane magnetization vector M(x). The spatial dependence of the fields H_str(*r) and Meissner currents j_S(*r)  are shown to be qualitatively different for the case of Bloch– and  magnetic textures. While the spatial distributions in the upper and lower S are identical for  Sk and DW's they are asymmetric for the case of  magnetic textures.
		The depairing factor, which determines the critical temperature T_c and which is related to vector potential of the stray-field, can have its maximum at the center of a magnetic domain but also, as we show, above the DW. For Sk's the maximum is located at a finite distance within the Sk radius r_Sk. Based on this, we study the nucleation of superconductivity in the presence of DW's. Because of the asymmetry for  structures, the critical temperature T_c in the upper and lower S is expected to be different. The obtained results can also be applied to S/F bilayers.
 	
	
	Meissner Currents Induced by Topological Magnetic Textures in Hybrid Superconductor/Ferromagnet Structures
    Samme M. Dahir, Anatoly F. Volkov, and Ilya M. Eremin
    December 30, 2023
==========================================================================================================


Over the past decades, continuous efforts have been made to study superconductor-ferromagnet heterostructures due to a variety  of interesting features caused by the proximity effect, i.e., the penetration of Cooper pairs from the superconductor (S) into the ferromagnet (F). The most interesting and well established  effects are the sign reversal of the Josephson current in S/F/S junctions and the appearance of a long-ranged triplet component (see review articles <cit.> and references therein).
	
Other interesting features involve the interplay of various types of topological defects that, under certain conditions, can be present in the superconductor and/or ferromagnet. One of these topological defects are the Abrikosov vortices which occur in  superconductors<cit.> in the magnetic field interval, . 
There are also several different topological structures that can be found in ferromagnets. The most prominent ones are magnetic domain walls (DW), where the magnetization vector M rotates by an angle π across the DW. Another example  of a topological defect that has received much attention recently due to its potential application in spintronics are the so-called magnetic Skyrmions (Sk)<cit.>. These local whirl-like structures are topologically equivalent to two DW's as one can map the inner part of the Sk on the stripes between two domains via conformal transformation. Similar to flat DW's, where the magnetization vector M changes its direction by rotating either in the (x,z)-plane () or in the (y,z)-plane (), the winding of chiral Sk can either have a Bloch– or a Néel–like structure. Which type of chiral Sk is realized depends on the underlying chiral interaction. Note that there is already some work on the mutual interaction between topological defects occurring in ferromagnets and superconductors, see review<cit.> and references therein. In the absence of the direct proximity effect (no direct contact between S and F), this interaction is realized through the magnetic stray-field H_str  generated by the non-uniform magnetic textures in the F and the magnetic field associated with the superconducting vortices. The creation of Pearl and Abrikosov vortices in S/F structures with and without DW's has been analyzed in Refs.<cit.>. More recently, the spontaneous creation of vortices in S/F structures with Sk's with and without direct proximity effect was also studied theoretically<cit.>
	
As it is well known, there is no stray-field H_str outside of uniformly magnetized infinite film <cit.>. However, within the ferromagnet the magnetic induction B_F or the magnetic field H_F can still acquire finite values, i.e.,  and  for the in-plane magnetization and  and  for the out-of-plane magnetization. Therefore for a uniform magnetization M_0 in the F of a S/F/S structure, both the B_S, the H_S and the Meissner current j_S are equal zero in the superconductors  where . Thus, non-zero stray-fields and Meissner currents can only occur if the magnetization of an infinite F is non-homogeneous. This was studied in S/F structures with DW's of zero width in Refs.<cit.>, for DW's of finite width in Ref.<cit.> and for a magnetic vortex configuration in Ref.<cit.>. In the presence of a proximity effect and spin-orbit coupling, the Meissner current was recently calculated in a bilayer S/F structure with a particular  Sk in the F and a vortex in the S <cit.>.
	
Despite of existing literature, there are still no systematic studies of Meissner currents in S/F and S/F/S structures with different topological magnetic textures (Sk's or flat DW's) with different orientations of the magnetization vector M. This is particular interesting due to potential realization of Majorana fermions in such heterostructures<cit.>. In the present paper we address this topic, by analyzing S/F/S systems with an isolated magnetic Sk (Bloch– and  Sk) or with a periodic flat DW structure (out-of-plane and in-plane magnetization, Bloch and Néel DW's) in the ferromagnetic material.

Assuming that there is no proximity effect present and that there are no Abrikosov vortices in the S/F/S structure , i.e., magnetic stray-fields are supposed to be less than the critical field H_c1, we find the effective magnetic stray-field from which we deduce the induced screening currents j_S. Note that the obtained spatial distribution of the current density j_S in the S in a Josephson system S/F/S is qualitatively similar to that in S/F bilayer. 
The knowledge of the effective stray-field H_str and the current density j_S allows one to estimate the region where superconductivity nucleates upon decreasing the temperature T below T_c: Either at the DW's or in the center of the domains. Available experimental data point out that the nucleation of superconductivity preferably occurs at the DW's<cit.>. However it will be shown that the exact location depends on the considered type of DW's.
In addition, we show that new interesting and non-trivial features arise in the system under consideration. For example, we find an pronounced asymmetry in the z-dependence of j_S(z), which occurs for both Néel DW's and  magnetic Sk's. This asymmetry is characterized by the in-plane dependence *r_⊥ of the Meissner current j_S(*r_⊥) and stray-field H_str(*r_⊥), which differs greatly above and below the ferromagnet and can even result in a local sign change of the Meissner current. In the absence of a superconductor, the asymmetry of the stray-field for DW's is already known<cit.>. For instance, it was recently demonstrated in experiments on artificial magnetic structures<cit.>. The asymmetry arises due to a non-zero ÷M term inside the ferromagnet. In the case of Bloch DW's such an asymmetry does not appear, since ÷M term vanishes. 

This difference between Bloch– or  DW's and Sk's follows from the different orientation of the vector ê_rot which describes the rotation axis of the magnetization M along the domain wall. For instance, in the case of Bloch– and  DW one can define a vector , where ê_x is a unit vector normal to the plane of the DW. For  DW's this vector is non-zero, while for  DW's the vector product is zero  because the rotation vector ê_rot is collinear to the vector ê_x. In the language of magnetic monopoles, which can be used for magnetic stray-fields, the presence of ê_N translates into the existence of magnetic bulk charges. In combination with the magnetic surface charges, the stray-field components of the bulk charges results in the aforementioned asymmetry. In the case of a S/F bilayer it generally makes no sense to speak about an asymmetry, but the spatial distribution of the Meissner j_S in the S still depends on the direction of the vector ê_N with respect to the F film (upward or downwards).


We will begin this paper by calculating a general expression for the effective magnetic stray-field H(*r) in an S/F/S structure generated by a nonhomogeneous two-component magnetization M, see . From the stray-field, we extract an expression for the Meissner current in the two superconducting region, which is then applied to describe induced currents in the presence of isolated Néel– and  Sk, see , as well as for various magnetic DW configurations, see . In , we use a Ginzburg-Landau model in the absence of external currents to estimate the nucleation of  superconductivity in the presence of the DW structures, we considered earlier. Note that the obtained results in this sections are independent of the type of S, as we are working with unscreened magnetic stray-fields. The universal expressions for these unscreened fields can be easily extracted from our results in the previous section. We end this work with a conclusion in .




§ STRAY FIELD AND MEISSNER CURRENT 

	
We consider an S/F/S structure, that is, a ferromagnetic film of thickness 2d_F interfaced by two superconductors at . The magnetization M(*r) inside the ferromagnet can be written in the form


    M(*r)=M_0n(*r)


where the unit vector n(*r) is a function of the position vector  with *r_⊥ lying in the (x,y)-plane. In the following the magnetization is assumed to be independent of the z-coordinate .
	
We will now begin with determining the spatial distribution of the screened stray-field in the superconducting regions. The superconducting order parameters (OP) are assumed to be homogeneous. Any magnetic field   inside the S must then satisfy the London equation which we write for the Fourier component 


    ∂_zz^2H_S(*k,z)-κ^2H_S(*k,z)=0,   S regions


where  and λ_L is the London penetration depth. In the general case, the two superconductors may have different London penetration depth λ_L^+ and λ_L^-.
The solution of Eq.(<ref>) is given by 


    H_S(*k,z)=C_±(*k)e^-κ_±z,   S regions


where the index ± of the constant  and κ^± indicates their values in the upper/lower superconducting regions, respectively.

	
The stray-field generated by the magnetization M inside the F has to fulfill the magnetostatic condition  which allows us to define a magnetic scalar potential U with


    H_F(*k,z)=-(i*kU(*k,z),∂_zU(*k,z))


In the absence of the proximity effect (PE) the potential U is related to the magnetization M via  so that we can write


    ∂_zz^2U(*k,z)-k^2U(*k,z)=4π M_0 ikn_⊥(*k),   F film


Solving Eq.(<ref>) for U(*k,z) we obtain

	
    U(*k,z)=   4π M_0{A(*k)sinh(kz)+B(*k)cosh(kz)
       -i*kn_⊥(*k)/k^2+C_0δ(*k)+i*kC_*r_⊥/k^2δ(*k)}


with the Dirac δ-function δ(*k). The last two terms are contributions to the homogeneous solutions of Eq.(<ref>). In the coordinate representation it has the form:  Note, the constant C_0 does not affect any physical quantity, so that we can set . The constant C_⊥ on the other hand, is related to a non-compensated magnetic moment M_un in the F which turns to zero for . Using Eq.(<ref>) we can determine the stray-field H_F in the F film 


    H_F^⊥(*k,z)=   -4π M_0 i*k{A(*k)sinh(kz)+B(*k)cosh(kz)
       -i*kn̅_⊥(*k)/k^2}
    H_F^z(*k,z)=   -4π M_0kA(*k)cosh(kz)+B(*k)sinh(kz)


where we defined


    n̅_⊥(*k):=(n_⊥(*k)-C_*r_⊥δ(k))


The constants A(*k), B(*k) and C_±(*k) can be found using the matching conditions for the magnetic field and the magnetic induction at the S/F interfaces. They are reduced to the continuity of the tangential components of the in-plane field H^⊥(*k,z) and the normal component of the magnetic induction , i.e.,

	
    H_S^⊥(± d_F)=   H_F^⊥(± d_F)
    H_S^z(± d_F)=   H_F^z(± d_F)+4π M_0n_z(*k)


In addition, the in-plane component of H_S is coupled to the normal component via the equation  so that 


    C_±^⊥(*k)=∓i*k/kκ_±/kC_±^z(*k)



	
Using Eqs.(<ref>-<ref>) we can determine the coefficients A(*k) and B(*k), which are given by 

	
    A(*k)=   n_z(*k)κ_+D_2^-(k)+κ_-D_2^+(k)/kD(k)+n̅_⊥(*k)i*k/kD_2^-(k)-D_2^+(k)/D(k)
    
    		B(*k)=   n_z(*k)κ_+D_1^-(k)-κ_-D_1^+(k)/kD(k)+n̅_⊥(*k)i*k/kD_1^-(k)+D_1^+(k)/D(k)


with D(k)=D_1^-(k)D_2^+(k)+D_1^+(k)D_2^-(k), where


    D_1^±(k)=   ksinh(kd_F)+κ_±cosh(kd_F)
    
    		D_2^±(k)=   kcosh(kd_F)+κ_±sinh(kd_F)


The coefficient C_±(*k) is given by 
	
    C_±^z(k)=-4π M_0ksinh(kd_F)e^κ_± d_F   [±i *k/kn̅_⊥(*k)/D(k)(D_1^-(k)+D_1^+(k)±κ_--κ_+cosh(kd_F)).
       .-n_z(*k)/D(k)(D_2^-(k)+D_2^+(k)±κ_--κ_+sinh(kd_F))]


For the sake of simplicity, we will from now on consider two identical superconducting materials, i.e., λ_L^+=λ_L^-. In this case the expression for the coefficients can be reduced to 


    A(*k)=κ/kn_z(*k)/D_1(k),
    		      B(*k)=i*k/kn̅_⊥(*k)/D_2(k)


and 


    C_±^z(*k)=-4π M_0ksinh(kd_F)e^κ d_F[±i*k/kn̅_⊥(*k)/D_2(k)-n_z(*k)/D_1(k)]


With this we obtain the k-space representation of the screened stray-field in an S/F/S junction for two identical superconductors.

In the S region z>d_F:

	
    H̃_⊥^(S)(*k,z)=   κsinh(kd_F)i*k/k[i*k/kn̅_⊥(*k)/D_2(k)∓n_z(*k)/D_1(k)]e^-κz∓ d_F
    H̃_z^(S)(*k,z)=   -ksinh(kd_F)[±i*k/kn̅_⊥(*k)/D_2(k)-n_z(*k)/D_1(k)]e^-κz∓ d_F


	In the F film z<d_F:

	
    H̃_⊥^(F)(*k,z)=   -i*k/k[i*k/kn̅_⊥(*k)(kcosh(kz)/D_2(k)-1)+n_z(*k)κsinh(kz)/D_1(k)]
    H̃_z^(F)(*k,z)=   -
    		[i*k/kn̅_⊥(*k)ksinh(kz)/D_2(k)+n_z(*k)κcosh(kz)/D_1(k)]


where we expressed the results in terms of a dimensionless field H̃=H/4π M_0.


The obtained expressions describe the screened stray-field in an S/F/S structure. By taking , i.e., , we can also extract the unscreened stray-field which would be present in the absence of superconductors. In this limit, the result describes the general distribution of the stray-field created by a nonhomogeneous magnetization in a F in vacuum. The associated vector potential is later used to estimate the nucleation of superconductivity. The origin of the screening field that leads to the effective stray-field in Eq.(<ref>,<ref>), are the induced supercurrents inside the S. The supercurrent (Meissner current) j_S can be determined using Ampère's law .


    j_S(*r_⊥,z)=c/4π[(∇_*r_⊥+ê_z∂_z)×(H_⊥(*r_⊥,z)+ê_zH_z(*r_⊥,z))]


In the Fourier representation we further obtain


    j_S(*k,z)=c/4π[(i*k+ê_z∂_z)×(H_⊥(*k,z)+ê_zH_z(*k,z))]


It can easily be shown, that the supercurrent disappears within the F, which is the expected result when the PE is absent.  Outside the ferromagnet , we obtain the following expression
	
    j_S(*k,z)=cM_0λ_L^-2/k(ê_z×i*k/k)H̃_z^(S)(*k,z)


from which we can directly derive the vector potential A in the superconductor using 


    A^(S)(*k,z)=-4π M_0/k(ê_z×i*k/k)H̃_z^(S)(*k,z)


	
	


§ ISOLATED SKYRMION 

In this section, we will set the magnetization profile M to describe an isolated magnetic skyrmion (Sk) in a ferromagnetic background. It is assumed that the Sk's are stabilized by an underlying chiral interaction resulting in either Bloch– or  Sk's. The magnetization profile has a cylindrical symmetry and varies along the radial direction *ρ so that .
The unit vector n of a chiral Bloch or Néel Sk can then be written as


    n_⊥(*ρ)=   *ρ/ρsin(θ(ρ))Θ(r_Sk-ρ),         Néel Sk
    n_⊥(*ρ)=   ê_z*ρ/ρsin(θ(ρ))Θ(r_Sk-ρ),         Bloch Sk


for the in-plane component and

	
    n_z(*ρ)=[1+cos(θ(ρ))]Θ(r_Sk-ρ)-1 ,   Bloch & Nééĺ Sk


for the out-of-plane component. Here, θ(ρ) describes the angular variation of the magnetization w.r.t. the z-axis and Θ(r_Sk-ρ) is a Heaviside step function with r_Sk being the skyrmion radius. The Fourier components of n(*ρ) are equal to 


    n_⊥(*k)=   -2πi*k/k m_⊥(k),         Néel Sk
    n_⊥(*k)=   -2πê_z*k/k m_⊥(k),         Bloch Sk
	

and 


    n_z(*k)=2π[m_z(k)-2πδ(*k)],   Bloch & Nééĺ Sk


The functions m_⊥(k) and m_z(k) are defined as

	
    m_⊥(k)=   ∫_0^r_Skρρ J_1(kρ)sin(θ(ρ))
    
    		m_z(k)=   ∫_0^r_Skρρ J_0(kρ)[1+cos(θ(ρ))]


where J_n(x) is the Bessel-function of the first kind of order n. The angular dependence of the Sk profile θ(ρ), is described using a circular  360^∘–domain wall Ansatz<cit.>.

	
    θ(ρ)=∑_±arcsin(tanh(-ρ∓ c/w/2))


with c being the size of the domain core and w is the domain wall width. For the remainder of this work, we set c=0 nm. Using Eq.(<ref>), one can estimate the radius r_Sk of the Sk. It should be noted that the expressions in this section can be used for any radially symmetric magnetization profile. 
		
Using the obtained result from the previous section, we will begin analyzing the effective stray-field generated by a Sk in our S/F/S structure in the case of two identical superconductors. Afterwards we will determine the corresponding induced Meissner currents. Taking into account that for a Bloch Sk  (see Eq.(<ref>)), we see that the first term in Eqs.(<ref>-<ref>) vanishes. This means that the individual components of the stray-field are either symmetric   or anti-symmetric functions  of z. On the other hand, the in-plane magnetization of a Néel Sk . Hence, in this case H(z)≠H(-z) which describes an asymmetry of the magnetic stray-field. This asymmetry is a typical feature of stray-fields  generated by magnetic textures with Néel–like magnetization<cit.>.

In order to fully determine the magnetic stray-field and the Meissner current, we first need to specify the value of the constant C_*r_⊥. Using the condition that the spatial average of the in-plane component of the stray-field vanishes, i.e., , we get an additional equation for C_*r_⊥. For the case of an isolated Sk this constant is equal to zero . The real-space representation of the screened stray-field in Eqs.(<ref>-<ref>) can now be easily expressed as
	



	
In the S region z>d_F:

	
    H̃_⊥^(S)(*ρ,z)=   -*ρ/ρ∫_0^∞kkJ_1(kρ)κsinh(kd_F)[m̅_⊥(k)/D_2(k)∓m_z(k)/D_1(k)]e^-κz∓ d_F
    H̃_z^(S)(*ρ,z)=   -∫_0^∞kkJ_0(kρ)ksinh(kd_F)[±m̅_⊥(k)/D_2(k)-m_z(k)/D_1(k)]e^-κz∓ d_F


In the F film :


    H̃_⊥^(F)(*ρ,z)=   *ρ/ρ∫_0^∞kkJ_1(kρ)[m̅_⊥(k)(kcosh(kz)/D_2(k)-1)+m_z(k)κsinh(kz)/D_1(k)]
    H̃_z^(F)(*ρ,z)=   -∫_0^∞kkJ_0(kρ)[m̅_⊥(k)ksinh(kz)/D_2(k)+m_z(k)κcosh(kz)/D_1(k)]+1/2




where we inserted the magnetization profile Eq.(<ref>-<ref>) and defined 


    m̅_⊥(k)=
    		m_⊥(k) ,      Néel Sk
    
    		0,       Bloch Sk
	

Analogously, the Meissner current can be found using Eq.(<ref>), which has the following form in real-space

	
    j_S(ρ,z)=   cM_0λ_L^-2∫_0^∞kkJ_1(kρ)sinh(kd_F)
       ×[±m̅_⊥(k)/D_2(k)-m_z(k)/D_1(k)]e^-κz∓ d_Fê_φ


The stray-field induces circulating supercurrents pointing in ê_φ-direction. Since the supercurrent is linked to the stray-field, the Meissner current also features the asymmetry which is related to the magnetization profile of the Néel Sk. Using Eq.(<ref>), this asymmetry can be identified by the changing sign in the term associated with the in-plane contribution of the magnetization. In  we show the spatial dependence of the Meissner current j_S(ρ,± d_F) in the upper (a) and c)) and the lower (b) and d)) superconductors in the presence of a  Sk in the ferromagnetic material. The curves are displayed for the parameters  and . As expected, we observe a strong asymmetry in the dependence j_S(ρ, d_F) and j_S(ρ,- d_F) in the upper and lower superconductors. The current j_S(ρ,d_F) in the upper S changes sign at some finite distance within the Sk region whereas the current j_S(ρ,- d_F) remains negative for all ρ. Note that the sign reversal of the Meissner current in S/F systems has been found earlier <cit.>, but its underlying mechanism was different as it was related to the proximity effect. In the case of Bloch Sk, all the mentioned features are missing and the Meissner current in both superconducting regions is the same, see .





	



§ FLAT DOMAIN WALLS


	In this section we consider the magnetization profiles of several different periodic flat DW's,  as illustrated in Fig. <ref>. The alignment of magnetization changes across the DWs as a function of the x-coordinate, i.e., *r_⊥=xê_x with ê_x being the corresponding unit-vector.  The period of the structures is 2L_0. This enables us to expand all function as a Fourier series: For example, the vector n(x,z) is represented as

	
    n(x,z)=∑_n=-∞^∞n(k_n,z)exp(ik_nx)


	with

	
    n(k_n,z)=1/2L_0∫_-L_0^L_0xn(x,z)exp(-ik_nx)


	where . Below we drop the subindex n for brevity.
	
Now suppose that the vector n(x) depends only on the x-coordinate, i.e. it is completely described by its  x-component . In this case, the expression for the normalized magnetic stray-field  and the Meissner current j_S can be obtained in the same manner as in .  For instance, for two identical S, we obtain the magnetic stray-field H^(S)(k,z) by substituting  and  in Eq.(<ref>-<ref>). For periodic DW's, one further needs to replace , which follows from the finite range of integration in Eq.(<ref>). Finally, the normalized field components in the superconducting regions  are:


    H̃_x^(S)(k,z)=   -κsinh(kd_F)[n̅_x(k)/D_2(k)±ik/kn_z(k)/D_1(k)]e^-κz∓ d_F
    H̃_z^(S)(k,z)=   -ksinh(kd_F)[±ik/kn̅_x(k)/D_2(k)-n_z(k)/D_1(k)]e^-κz∓ d_F


with


    n̅_x(k):=n_x(k)-C_xsin(kL_0)/kL_0


and analogously within the ferromagnet z<d_F:


    H̃_x^(F)(k,z)=   [n̅_x(k)(kcosh(kz)/D_2(k)-1)-ik/kn_z(k)κsinh(kz)/D_1(k)]
    H̃_z^(F)(k,z)=   -[ik/kn̅_x(k)ksinh(kz)/D_2(k)+n_z(k)κcosh(kz)/D_1(k)]
	

The Meissner current can be extracted from Eq.(<ref>). The supercurrent flows in y-direction  and has the magnitude


    j_S(k,z)=cM_0λ_L^-2sinh(kd_F)[±n̅_x(k)/D_2(k)+ik/kn_z(k)/D_1(k)]e^-κz∓ d_F
    

where once again the ± indicates the solution in the upper or lower S region, respectively.
The current in coordinate representation j(x,z) can be calculated using

    
    j(x,z)=∑_kj(k,z)exp(ikx)


Having determined the expressions for H_str and j_S for an arbitrary type of DW, we need to specify the precise magnetic texture. Its components can be expressed in terms of the function n_even and n_odd which are characterized by an even or odd dependency on x or k. Since we are interested in a qualitative spatial dependence of all quantities (the fields and the Meisner currents), we approximate n_even,odd by


	
    n_odd(x)=   cos(π/2(x-L)/w)θ(w-x-L)-cos(π/2(x+L)/w)θ(w-x+L)
    
    		n_even(x)=   [1-sin(π/2(x-L)/w)]θ(w-x-L)+[1+sin(π/2(x+L)/w)]θ(w-x+L)+2θ(L-w-x)-1


with the domain wall width 2w and the size of the domain . That is, we assume that the rotation angle of the vector n outside the DW's  remains constant whereas it changes linearly inside the DW's . This approximation allows us to present results in a simple analytical form. Outside the interval , n(x) is a periodic function of x: . The Fourier components of n_odd(x) and n_even(x) are equal to
	
    n_odd(k)=   2π iw/2L_0cos(kw)sin(kL)/k^2w^2-(π/2)^2=2π iw/2L_0 f(k)
    
    		n_even(k)=   π^2/2L_0kcos(kw)sin(kL)/k^2w^2-(π/2)^2-2sin(kL_0)/2L_0k=2π w/2L_0 F(k)


	with 

	
    f(k)=   cos(kw)sin(kL)/k^2w^2-(π/2)^2
    
    		F(k)=   π/2kwf(k)-sin(kL_0)/π kw


Obviously, f(k) is also an odd function, whereas F(k) is an even function of k. It should be noted that the limiting case of the DW width w=0 was analyzed in Ref.<cit.>.

In our model, the vector n has two non-zero components that allow the construction of six different magnetic textures (see ). They are characterized by vectors n with the following components: (0,n_even,n_odd), (0,n_odd,n_even), (n_even,0,n_odd), (n_odd,0,n_even) and (n_even,n_odd,0), (n_odd,n_even,0). Note that we are working with the underlying assumption of fixed chirality, i.e, the vector n rotates in the same direction within the DW's, which is either clock-wise or counter-clockwise. Another chirality may be obtained if the rotation of the vector n in adjacent DW's occurs in different directions; then the function n_odd should be replaced by , where ñ_even is equal to

		
    ñ_even(x)=   cos(π/2(x-L)/w)θ(w-x-L)+cos(π/2(x+L)/w)θ(w-x+L)
    ñ_even(q)=   -2π w/2L_0cos(kw)cos(kL)/k^2w^2-(π/2)^2


	

	
In order to obtain our final result for the magnetic stray-field and the Meissner current from Eq.(<ref>-<ref>), we need to determine the constant C_x. As mentioned in , the average over the in-plane component H_x^(F) has to vanish, i.e, . From this follows
	
    H_x^(F)(x,z)=   1/2L_0∫_-L_0^L_0x1/2d_F∫_-d_F^d_FzH_x^(F)(x,z)
    
    		=   4π M_0λ_L^-1d_F/1+λ_L^-1d_F[n_x(k=0)-C_x]!=0


where we used . Hence, the constant C_x is given by


    C_x=n_x(k=0,z)


The quantity  can be either  or . In the latter case, n_x(k=0,z)=0 (see Eq.(<ref>,<ref>)) and therefore C_x=0. The other case is only realized for certain  DW's and results in


    C_x=2L/L_0-1


i.e., the constant C_x vanishes for L_0=2L. Otherwise,  if ,  the domains with positive and negative magnetization differ in size, which leads to an uncompensated total magnetization M_un and .
We will now examine the various possible magnetic textures that exhibit a chirality as defined in Eq.(<ref>). Note that, the type of DW in a ferromagnetic sample is determined by the existing magnetic interaction and material specific parameters (temperature, thickness of the F film etc). Accordingly, the actual magnetic texture in the F corresponds to the configuration associated with the minimum of the thermodynamic potential. Nevertheless, we will find the spatial distribution of the Meissner currents for all possible configurations, bearing in mind that some of these textures might not be  energetically favorable, but could be achieved in artificial magnetic structures <cit.>.



 §.§ Out-of-plane n (Néel and Bloch DW's)
	
For an out-of-plane magnetization, both  Néel and/or  Bloch DW's (see  can exist within the F. The Néel DW (NDW^(z)) is described by the following configuration n(x)
	
    n(x)=(n_odd,0,n_even),     NDW^(z)

	
	
The superscript (z) indicates the alignment of the vector n(x) across the domains, which is oriented along the z-axis. The Meissner current at the interfaces  is obtain by inserting the corresponding Fourier components in Eq.(<ref>).
	
    j_S(k,± d_F)=cM_0λ_L^-2sinh(kd_F)2π iw/2L_0[±f(k)/D_2(k)+sgn(k)F(k)/D_1(k)]

where f(k) and F(k) are given in Eq.(<ref>,<ref>). One can easily see that the current is an odd function of k. In the coordinate representation, we obtain the following result

    j_S(x,± d_F)=-4π M_0cwλ_L^-2/2L_0∑_k=0^∞sin(kx)sinh(k d_F)[±f(k)/D_2(k)+F(k)/D_1(k)]




		
In Fig.  we plot the dependence of the normalized current  for , ,  and . This plot shows a strong asymmetry between the upper and lower superconductors with currents flowing above/below the DW regions. The direction of the supercurrent depends on the direction of rotation of the DW. Varying the value for the London penetration depth  reveals a sign change for the supercurrent within DW's in the lower supercurrent (see . One can see  that the Meissner currents at different DW's flow in opposite directions. This means that the currents flow along closed loops. Unlike the case of Abrikosov vortices, there is no phase change along these loops. This sign change is similar to the behavior described for the Néel Sk's. As in the case of the Sk, the asymmetry follows from the non-vanishing ÷M term inside the FM, which means that both the bulk and the surface magnetic charges are present. Note that in the zero-width domain wall limit, the n_x component vanishes, i.e., ÷M=0. As a result the asymmetry would disappear.
	
In the case of a Bloch DW (BDW^(z)) the vector n(x) has the components
	
    n(x)=(0,n_odd,n_even),     BDW^(z)

where the magnetization in the domain is once again oriented along the z-direction. The Meissner current in Fourier representation is given by
	
    j_S(k,± d_F)=cM_0λ_L^-2sinh(kd_F)2π iw/2L_0sgn(k)F(k)/D_1(k)


and in the coordinate representation by
	
    j_S(x,± d_F)=-4π M_0cwλ_L^-2/2L_0∑_k=0^∞sin(kx)sinh(k d_F)F(k)/D_1(k)


One can directly deduce that the resulting Meissner currents are identical in the upper and lower superconductors, which is due to the missing x-component of the magnetization. This is once again, similar to the Sk case, as there was also no asymmetry present for Bloch Sk's. For both NDW^(z) and BDW^(z) follows that j_S(x,z) is an odd function of x so that the total current  vanishes. In , we plot the dependence of the Meissner current for the considered case of a BDW^(z). The parameter are the same as in . 



 §.§ In-plane n (Néel and Bloch DW's)


Let us first consider a  DW where the magnetization vector n(x) at the domains is oriented along the  (see ), then
	
    n(x)=(n_odd,n_even,0),     NDW^(y)

	The Fourier component of the Meissner current is equal to
	
    j_S(k,± d_F)=± cM_0λ_L^-2sinh(kd_F)2π iw/2L_0f(k)/D_2(k)

	and in the coordinate representation
	
    j_S(x,± d_F)=∓4π M_0cwλ_L^-2/2L_0∑_k=0^∞sin(kx)sinh(k d_F)f(k)/D_2(k)


The functions j_S(x,d_F) and j_S(x,-d_F) are shown in . Once again the currents differ in the two superconducting regions. The magnitude of the currents is the same, but the currents flow in opposite direction, resulting in an antisymmetric behavior. 
	
The Bloch type DW (see ) is described by
	
    n(x)=(0,n_even,n_odd),     BDW^(y)

	The occurring currents j_S(x,z) for BDW^(y) are even function of x given by
	
    j_S(x,± d_F)=-4π M_0cwλ_L^-2/2L_0∑_k=0^∞cos(kx)sinh(k d_F)f(k)/D_1(k)


    For both  the currents are equal. In , we plot the x-dependence of the functions j_S(x,d_F) and j_S(x,-d_F). The Meissner currents in the upper and lower S near the BDW flow in the same direction. The total current j_av is zero. The results for j_S(x,z) for the BDW^(y) are similiar to those obtained by Burmistrov and Chtchelkatchev<cit.>.

    
    

 §.§ Other types of NDW

    Other types of NDW's correspond to a magnetization profile n(x) in which the alignment in the domain is along the x-direction (see ). The rotation of the vector n(x) occurs either in the (x,z)-plane or in the (x,y)-plane. Thus, the vector n(x) has the components 
    
    n(x)=   (n_even,n_odd,0),     NDW_y^(x)
    n(x)=   (n_even,0,n_odd),     NDW_z^(x)

    Remember, that for the case  the constant C_x has a finite value given by . It follows that 
    
    n̅_x(k)=n_even(x)-C_xsin(kL_0)/kL_0=2π w/2L_0F̅(k)

    with 
    
    F̅(k)=π/2kwf(k)-2L/L_0sin(kL_0)/π kw

    
    
    With this expression, we obtain the Meissner current
    
    j_S^1(x,± d_F)=   ±4π M_0cwλ_L^-2/2L_0∑_k=0^∞cos(kx)sinh(k d_F)F̅(k)/D_2(k)
    
        	j_S^2(x,± d_F)=   4π M_0cwλ_L^-2/2L_0∑_k=0^∞cos(kx)sinh(k d_F)[±F̅(k)/D_2(k)-f(k)/D_1(k)]
  
    
    
    
	We continue to consider a compensated magnetization where  so that .

	Once again, we find the typical asymmetry associated with  magnetic textures. The plots for the two magnetization NDW_y^(x) and NDW_z^(x) can be found in  and , respectively.
	
	

§ NUCLEATION OF SUPERCONDUCTIVITY

	
In this section, we analyze qualitatively the nucleation of superconductivity in S films in an S/F/S structure when the temperature drops below the critical value in a bulk superconductor T_cB. The value of the critical temperature T_c differs from its bulk value in the presence of a local depairing factor V(r_⊥). This problem was analyzed in the case of ferromagnetic superconductors with DW's both theoretically <cit.> and experimentally<cit.>. Results of intensive studies of hybrid S/F structures are summarized in a review<cit.>. Theoretical studies were carried out under approximation of zero DW width. In particular the authors of Ref.<cit.> have studied the so-called diode effect, that is an asymmetrical dependence of the critical current j_cr in the S film with respect to the "magnetic current" j_M=M. Near the critical temperature at which superconductivity nucleates or disappears, the Ginzburg-Landau equation for the order parameter (OP) ψ  can be linearized. In the studies<cit.>, the OP ψ is presented in the form ψ=fexp(iχ), where in a one-dimensional case the phase of the OP is χ=kx. However, in a single-connected superconductor and in the absence of vortices one can choose a gauge with χ=0. On the other hand, if we consider our system in the form of a ring (annular geometry) with x=rφ, then the phase is χ=nφ, where φ is the azimuthal angle, n is an integer number of fluxons and R is the radius of the ring. Even in this case the gauge is not established in a unique way: one can add  in principle an arbitrary constant A_ext to the vector potential A without changing the observable quantity, the magnetic induction B=A. From the physical point of view, adding a constant A_ext means adding a uniform external or spontaneous (in the annular setup) current.
Observe that calculating the spatial dependence A(*r_⊥), we assumed, unlike previous theoretical studies, that A_ext=0. This means that the total current in the system vanishes j_S(*r_⊥)=-(c/4π)A(*r_⊥)=0.






	
Thus, assuming A_ext=0, we consider the Ginzburg-Landau equations for the order parameter f  (see, for example, Ref. <cit.>) near T_c.
	
    -d_S∇_*r_⊥^2f+V(*r_⊥)f-Ef=-Ef^3

where . Here A_0(*r_⊥) is not a gauge invariant quantity, so any observable quantity like the magnetic induction B=A will not change by adding a constant A_ext. However, in doing so it is necessary to analyze the excitation mechanism of the persistent current and to estimate the condensation energy E_S=Δ^2N(0)V in comparison to the magnetic energy F_I=I_ext^2ℒ/c^2 of the current I_ext, where N(0) is the density-of-states in the normal state, V=2π R(2_S)L_x is the volume of the superconductors and ℒ is the inductance of the superconducting ring.  The ratio γ of these two energies depends on various parameters of the system such as the London penetration depth λ_L and the DW width w and may be both smaller or larger than 1. Thus, the excitation of the current I_ext may not be necessarily energetically favorable. In addition, for a single-connected superconducting system, the problem should be then solved under the condition of the absence of the total current in a finite S/F/S structure disconnected from external circuits. Our solution with A_ext=0 would immediately satisfy these constrains. A more detailed analysis of the problem for a finite current I_ext should be left for future studies.   

The vector potential A_0 defines the stray-field in absence of superconductivity, which can be extracted from Eq.(<ref>) by taking the limit λ_L→∞. 


    A_0(*k,z)=4π M_0/ksinh(kd_F)(ê_z×i*k/k)[±i*k/kn_⊥(*k)-n_z(*k)]e^-kz




For simplicity, we assume that the thickness of the S films d_S is smaller than ξ_S, so that the order parameter (OP) f depends only on the in-plane  coordinates. At larger d_S, the factor V(*r_⊥,z) depends on the coordinate z and the effect of this depairing factor on the nucleation of superconductivity becomes weaker. Eq.(<ref>) is called the time-independent Gross-Pitaevskii <cit.> equation or nonlinear Schrödinger equation.  The "energy" E is related to the coherence length , . This equation is also used to analyze the nucleation of superconductivity near the critical magnetic field H_c2 (see Abrikosov's book <cit.> and <cit.>) and also near DW in a S/F system <cit.>. Note also Ref.<cit.>, where this equation is applied for studying the appearance of an OP in a system with two competing OPs.

In the following, we will focus on DW structures where in Eq.(<ref>) . In this case the real space expression for the vector potential at the interface  is given by
	
    A_0(x,± d_F)=-4π M_0∑_ksinh(kd_F)/ke^ikx[±n_x(k)+ik/kn_z(k)]ê_y

	
	

The associated depairing potential  is shown in  for the different magnetization configurations. The potential V(x) has minima located either at the DW's  or in the center of the domains . The critical temperature T_c is determined by the condition , where E_min is the minimal "energy" at which Eq.(<ref>) has a non-trivial solution. We assume that the domain size 2L is much larger than the width of the DW 2w. In the following we consider two possible cases.
	
	
	

  §.§.§ V(x) has a minimum at the DW

	Consider first the case when the potential V(x) has a sharp minimum at the DW, for example, at  where . Since we are interested in a qualitative picture, we approximate the dependence V(x) near the DW with a rectangular potential well: . Then, neglecting the r.h.s. in Eq.(<ref>) and using the matching conditions at  (f(x) and ∂_xf(x) are continuous) we can write a solution in the form
	
    f(x)=C_incos(K_Sx̃),    x̃<w
 
	
    f(x)=C_outexp(-K_out(x̃-w)),        x̃> w
    exp(-K_out(x̃+w)),        x̃< -w

	where , and . The integration constants C_in and C_out are related to each other. In the limiting cases of small and large , we have for T_c and C_in, C_out
	
a) ; 

b) .


Thus, if the depairing potential V(x) has a dip at the DW's, superconductivity is nucleated at the DW's. This happens in the following DW configurations: BDW^(y), NDW_y^(x), NDW_z^(x) (the potentials are shown in ). The opposite case is realized for the magnetization profiles:  NDW^(z), BDW^(z), NDW^(y) (see also  for the respecting potentials) and is considered in the next section.

For simplicity, we neglect the width w in comparison with L. The solution outside the DW has the form
	
    f(x)=C_outcos(K_Sx),     x<L

The critical temperature T_c is found from the matching condition . The constant  is not zero provided that the condition
	
    θ_ctan(θ_c)=λ

is fulfilled, where . Eq.(<ref>) yields for the critical temperature T_c:

a) 

b) 

	The constant C is found analogously to the case considered in Ref.<cit.>
	
    C^2=T-T_c/T_cB-T_cr(θ_L)

	with
	
    r(θ_L)=cos^2(K_Sx)/cos^4(K_Sx)=2θ_L+sin(2θ_L)/(3/2)θ_L+sin(2θ_L)+(1/8)sin(4θ_L)

	where  and . For  the coefficient r is equal to: .
	

  §.§.§ V(x) has a mimimum at the center of the domain

	We assume that the potential V(x) in Eq.(<ref>) has the form
	
    V(x)=V_0∑_nδ(x-2nL)

	First, we linearize Eq.(<ref>) and find the minimum "energy" E of the Schrödinger like equation in the interval 
	
    -∂_x̃x̃^2f+V_0δ(x̃)f(0)=Ẽf

	where x̃=x/ξ_S0, Ẽ=E(ξ_S0/d_S)=1-T/T_cB. A periodic solution (f(x̃=f(x̃+2L/ξ_S0))) can be represented in the form
	
    f(x̃)=
    		acos(qx̃)+bsin(qx̃),       0<x̃<L/ξ_S0
    a̅cos(qx̃)+b̅sin(qx̃),       -L/ξ_S0<x̃<0

	where . The function f(x̃) has to fulfill the matching conditions
	
    [f]= 0       [∂_x̃f]=V(x̃)f(0)
    
    		f(L)=f(-L)       ∂_x̃f(x̃)|x=L= ∂_x̃f(x̃)|x=-L

	The solution (<ref>) exists if the condition
	
    θtan(θ)=v ≡ VL/2

	is satisfied where . The coefficients a and b are coupled by the relations: , . From Eq.(<ref>) we find the critical temperature
	
    T_c/T_cB=
    		1-(vξ_S0/L)^2,       v≪ 1
    
    		1-(πξ_S0/2L)^2,       v≫ 1

	If , superconductivity is suppressed completely.
	
	

§ CONCLUSION

	To conclude, in this manuscript we calculated the magnetic stray-field H_str and Meissner current j_S in a superconductors S created by various non-homogeneous magnetic texture in a F film incorporated in an S/F/S system. The total current in the system is assumed to be zero. Two types of topological structures were considered: isolated chiral magnetic skyrmions and periodic flat domain walls of Bloch (BDW) or  (NDW). Considering a two-dimensional two-component magnetization M(*r_⊥), we investigated six different magnetic DW textures as well as magnetic Sk of Bloch and . Each of these different magnetic textures possesses a particular spatial dependence of the stray-field H_str(*r_⊥,z) and the induced Meissner current j_S(*r_⊥,z).
	The most apparent difference appears between the Bloch– and the  magnetic structures. While the Neel-type structure yields a strong asymmetry j_S(*r_⊥,z)≠ j_S(*r_⊥,-z), the  remains always symmetric w.r.t z-component.  For certain parameter, this asymmetry can be strong enough to cause a sign change of the Meissner current for *r_⊥ within the DW region or within the Sk radius *r_⊥.
	Note that a similar sign change can be obtained in S/F or S/F/S systems that feature a proximity effect<cit.>. 
	
	The Meissner current j_S is connected to the vector potential A via  which enters the Ginzburg-Landau equation and acts as a depairing factor  where A_0 is the vector potential in absence of superconductivity. This factor determines the critical temperature of the superconducting transition in bulk superconductors<cit.> and in S/F heterostructures<cit.> and the superconductivity emerges first at places where A_0 has a minimum. As can be seen in  the locations of the minima or maxima of  depends on the type of DW's. For magnetic skyrmions the depairing potential V(*r_⊥) has its minimum in the center of the Sk. However, it can exhibit an additional local minimum for finite *r_⊥ within the radius of a Néel Sk which is not present for  skyrmions. 
	Thus, by measuring the location of the superconducting nucleation like it was done previously <cit.>, one can determine the type of the DW or distinguish between Bloch– and  skyrmions.
    	
   	

§ ACKNOWLEDGEMENT
 The authors acknowledge support from the Deutsche Forschungsgemeinschaft Priority Program SPP2137,
   	Skyrmionics, under Grant No. ER 463/10.


	 
	
	
	
	
	
	
	
	
	

	
	

