Magnon-Cooparons in magnet-superconductor hybrids

Generation and detection of spinful Cooper pairs in conventional superconductors has been intensely pursued by designing increasingly complex magnet-superconductor hybrids. Here, we demonstrate theoretically that magnons with nonzero wavenumbers universally induce a cloud of spinful triplet Cooper pairs around them in an adjacent conventional superconductor. The resulting composite quasiparticle, termed magnon-cooparon, consists of a spin flip in the magnet screened by a cloud of the spinful superfluid condensate. Thus, it inherits a large effective mass, which can be measured experimentally. Furthermore, we demonstrate that two magnetic wires deposited on a superconductor serve as a controllable magnonic directional coupler mediated by the nonlocal and composite nature of magnon-cooparons. Our analysis predicts a quasiparticle that enables generation, control, and use of spinful triplet Cooper pairs in the simplest magnet-superconductor heterostructures.


INTRODUCTION
The widely available and used conventional superconductors consist of spin-singlet Cooper pairs which are devoid of a net spin. Unconventional superconductors, in contrast, host qualitatively distinct phenomena and Cooper pair properties [1]. Their limited experimental availability, however, has driven the scientific community to try and engineer heterostructures comprising conventional superconductors into effectively unconventional ones [2][3][4][5][6], e.g., in achieving Majorana bound states [7]. In particular, the highly desired spinful spintriplet Cooper pairs can be generated from a conventional superconductor if the latter interacts with two or more non-collinear magnetic moments [2][3][4][5][6]. With this design principle, a wide range of magnet-superconductor hybrids with multiple magnetic layers to generate and detect spinful Cooper pairs have been investigated [8][9][10][11]. The challenge of detecting a spin or its flow directly has resulted in the need for increasingly complex magnet-superconductor hybrids rendering their direct detection a highly demanding, debated, and pursued goal [12][13][14][15].
The ambition is to go beyond detection, and towards exploiting the fascinating physics of these unconventional Cooper pairs for phenomena that are otherwise out of reach [16].
Noncollinear ground states of magnets [17,18] and spin-orbit coupling [19,20] have been exploited in generating equilibrium spinful Cooper pairs. These have allowed a control over static properties, such as magnetic anisotropy [19,20] or superconducting critical temperature [21], of various superconductor-magnet hybrids. Nevertheless, on-demand steering and movement of spinful Cooper pairs is highly desired and has remained an outstanding challenge. For example, a directed flow of the spinful Cooper pairs could be used for delivering nondissipative spin transfer torques and magnetic switching [22][23][24][25][26][27]. Such goals face a similar challenge that even when a complex heterostructure generates spinful Cooper pairs, it becomes difficult to steer them. An injected charge current predominantly converts into conventional spinless supercurrent [22]. Due to such reasons, several advantages of magnetsuperconductor heterostructures realized in various concepts and devices are still dominated by the quasiparticle properties [6,[28][29][30][31][32][33][34]. The exploiting of spinful Cooper pairs for exciting physics and devices has been impeded by the complex hybrids needed to generate them and the difficulty of steering them.
In this work, we uncover a ubiquitous existence and control of spinful Cooper pairs in the simplest magnet-superconductor hybrid -a bilayer -that has escaped attention thus far. We find that a magnon, the quasiparticle of spin waves in a magnet, with nonzero wavevector induces a cloud of spinful Cooper pairs in the adjacent superconductor [ Fig. 1(a)]. This accompanying cloud screens the magnon spin giving rise to a composite heavy quasiparticle with an enhanced effective mass, which is termed 'magnon-cooparon' due to its similarity to the polaron quasiparticle as discussed below. This induction of spinful Cooper pairs in a conventional superconductor is caused by the noncollinear magnetization profile of a spin wave with finite wavevector [ Fig. 1(b)], an effect not seen when considering ferromagnetic resonance of the uniform magnon mode. Furthermore, we demonstrate theoretically that magnon-cooparons enable a magnonic directional coupler [35,36] composed of two separate ferromagnetic wires with coupling lengths shorter than previously feasible thereby allowing smaller devices. Thus, it enables a valuable application in magnon-based logic and circuits [37,38]. The magnon-cooparon is reminiscent of the fermionic polaron quasiparticle created by screening of an electron by a phonon cloud [39], although the magnon-cooparon is a bosonic excitation. Considering the gradual discovery of polaron and its variants in a wide range of phenomena [39,40], we expect magnon-cooparon to find a similar important role in a broad range of magnet-superconductor hybrids. This concept can also significantly expand the range of reported effects related to the mutual influence of superconductivity and magnons [10-14, 31-33, 41-49]. We consider a bilayer as depicted in Fig. 1(b), in which a ferromagnetic insulator FI (e.g., yttrium iron garnet) is interfaced with a conventional spin-singlet s-wave superconductor S (e.g., Nb). The two layers with thicknesses d FI and d S ( ξ S , the superconducting coherence length) are considered thin such that physical properties vary only in the in-plane direction.
In its ground state, the FI is assumed to be magnetized along the z direction. For S/FI structures the effective induced exchange field in the superconductor is well-documented experimentally by measurements of the spin-split DOS [6]. At the same time for S/FM heterostructures, where FM means a ferromagnetic metal, the well-pronounced homogeneous spin-split DOS was not reported. The physical reason for this can be related to the leakage of Cooper pairs into the ferromagnet and, consequently, much stronger suppression of superconductivity at S/FM interfaces. Therefore, we expect that the renormalization of the magnon spin and stiffness by the cloud of triplet pairs, generated in the superconductor should be smaller in S/FM structures. From the other hand, in S/FM heterostructures there is a proximity effect, that is a penetration of Cooper pairs into the ferromagnet. In principle, in this case the cloud of triplet pairs, screening the magnon, could be generated directly in the ferromagnetic metal. We expect that qualitative physics of the renormalization should be similar.
We wish to examine wavevector-resolved excitations of the hybrid, and thus obtain the complete information needed for examining arbitrary wavepackets generated by a given experimental method. To this end, we assume existence of a spin wave with wavevector ke z in the FI [ Fig. 1(b)] such that the magnetization unit vector m(r, t) = m 0 + δm(r, t) consists of the equilibrium part m 0 = e z and the excitation part δm(r, t) = δm [cos(kz + ωt)e x + sin(kz + ωt)e y ] exp(−κt). While we consider an excitation with wavevector along e z , our analysis is general and valid for any in-plane wavevector. The magnetization dynamics is described within the Landau-Lifshitz-Gilbert framework aṡ where −γ with γ > 0 is the FI gyromagnetic ratio, α is the Gilbert damping parameter, H eff is the effective magnetic field in the FI, andJ ≡ J/d FI with J parameterizing interfacial exchange interaction between FI and S. The last term on the right hand side of Eq. (1) accounts for the spin torque exerted on the magnetization by the spin density s it induces in S [50]. Expressing s = s 0 m 0 + δs δm + δs ⊥ (δm × m 0 ) and substituting the expressions for s and m in Eq. (1) above, we obtain where D m is the FI spin wave stiffness and K parameterizes a uniaxial anisotropy. Thus, the spin density s induced in the S may renormalize both the excitation frequency and its lifetime.
We now evaluate the induced spin density s treating S using the quasiclassical Green's functions framework [4,6,51]. Working in the dirty limit, we need to solve the Usadel equation for the 8 × 8 matrix Green's functionǧ in spin, particle-hole, and Keldysh spaces: whereÎ is the 2 × 2 identity matrix, outer-product between the 2 × 2 matrices (decorated by overheadˆ) in obtaining an 8 × 8 matrix (identified via an overheadˇ) is implied, and we set = 1 throughout this Letter. Further, working in the mixed ( , t) representation, we employ the notation where h 0 and δh respectively capture the static and dynamic components of the induced where Tr 4 denotes trace over a 4 × 4 matrix (decorated by an overhead˘),g K is the 4 × 4 Keldysh component of the full 8 × 8 Green's functionǧ, and N F is the normal state density of states at the Fermi level in S.  that we discuss first. Further, quasiparticles are found to not play an important role in this limit leaving the focus on the superfluid condensate. We find δs ⊥ → 0 in this limit such that the excitation decay rate [Eq. (3)] is not influenced by S. δs − s 0 is found to scale as ∼ k 2 in this limit, such that the excitation frequency becomes ω =D m k 2 + γK with D m ≡ D m + δD m and where x = h 0 /2T c , and T c is the superconducting critical temperature taking into account the static exchange field h 0 . In obtaining Eq. (7), we further worked in the limit |T − T c | T c .
The same stiffness renormalization [Eq. (7)] is obtained from purely energy considerations within the Ginzburg-Landau framework [58]. The effective mass m eff of the composite quasiparticle is obtained as m eff = 1/2D m = 1/(2D m + 2δD m ). Since δD m < 0 [Eq. (7)], the effective mass of the composite quasiparticle is enhanced as compared to that of a magnon.
Numerically evaluated δD m , without making the adiabatic approximation, plotted in Fig. 2 versus temperature further shows the direct role of the superconducting condensate and suggests temperature as a handle to control the quasiparticle effective mass. The material parameters assumed in Fig. 2 are detailed further below, together with the discussion on experimental detection.
Thus, this composite quasiparticle shares some similarities with the polaron [39]. The latter, predicted almost a century ago [59,60] and having found numerous applications throughout condensed matter physics [39,40], is formed when an electron is screened by the Finally, going beyond the adiabatic approximation ω T , we find a nonzero renormalization of the k = 0 mode frequency and the decay rate ∼ k 2 , as detailed in the Supplementary Note 2. Similar effects are expected based on spin pumping into a normal metal or quasiparticles in a superconductor [31,61,62]. Specifically, when quasiparticle spin relaxation is disregarded, the increase in decay rate requires a spin sink, which may be provided by the noncollinear magnetic moment in a second magnet [63]. In our case, a spatially distinct part of the same magnet provides the noncollinear spin absorption channel, thereby shortcircuiting the spin wave. Hence, in this case too, the unique dynamic noncollinearity of the finite-k spin wave results in novel effects.

Spin of magnon-cooparons
The cloud of spinful Cooper pairs screening the magnon spin that increases its effective mass further implies that (i) the total spin of magnon-cooparon is reduced from 1, and (ii) a magnon spin current j m e z in FI is accompanied by a superfluid spin current j S e z in S. We now address these effects and ascertain the net spin of the magnon-cooparon.
Since the dc spin current j m accompanying a spin wave or magnon scales as δm 2 , we anticipate j S to scale as δh 2 , confirming this via a rigorous calculation detailed in the Supplementary Note 3. As a result, we now need to solve Eq. (4) for the matrix Green's function up to the second order in δh. Since this is a more demanding calculation than that carried out above, we restrict ourselves to the adiabatic approximation ω T and the limit |T − T c | T c in the rest of our analysis. The spin current flowing along the direction of magnon propagation (e z here) in S is then obtained as [6] j where the direction of j S pertains to the spin space. On explicit evaluation shown in the Supplementary Note 3, j S is found to bear only a z component, as can be expected from its screening the magnon spin, which itself bears only a z component. The total spin current may thus be expressed as where v k = 2D m k is the magnon-cooparon group velocity, n k is the number of excitations, and S becomes its net spin evaluated via where ω n are the fermionic Matsubara frequencies. Since j Sz /j m < 0, the net spin of the magnon-cooparon is reduced from 1 as per our expectation from the screening. Equation (10) shows that the dynamical induction of spinful Cooper pairs always causes screening, and thus, a reduction in the excitation net spin. Further, similar to the relative change in the spin stiffness (Fig. 2), the spin reduction |j Sz /j m | 1 for typical material parameters, as estimated further below.

Superfluid-mediated magnonic directional coupler
Since the Cooper pairs cloud comprising a magnon-cooparon extends over a length ∼ ξ S , it enables transfer of energy from a spin wave in one FI wire to another, nonlocally (Fig. 3).
Thus, two FI wires deposited on a conventional superconductor within ∼ ξ S from each other act as a magnonic directional coupler [35,36], proposed to be a key building block in wave-based logic and computing [64,65]. The magnon-cooparon based design that we demonstrate here offers stronger coupling strengths, smaller footprint, additional control (e.g., via temperature), and universality (e.g., for antiferromagnets [55,66]) as compared to the dipole-interaction based designs considered previously [35,36].
Considering the setup depicted in Fig. 3, we now assume existence of spin waves with the wavevector ke z in both FIs, assumed identical for simplicity. As a result, there exists the same static exchange field h 0 in S below both the FI wires. However, distinct dynamic exchange fields δh l,r , similar to Eq. (5), exist in S below each of the FI wires. These are proportional to the respective spin wave amplitudes δm l,r in the two FIs. Solving the Usadel equation (4) under this exchange field profile as detailed in the SM, we obtain the spin density in S: Here, the contributions s 0 and s loc are due to the static and dynamic exchange fields induced by the FI directly above the S region. Thus, these are identical to our analysis of the magnoncooparon in a FI/S bilayer. The nonlocal contribution s nl (x) characterizes the spin density generated in S below the left FI by the right one, and vice versa. Relegating its detailed expression to the SM, we note that for d ξ S , s nl is comparable to the spin density s loc accompanying a magnon-cooparon.
The induced nonlocal spin density leads to a fieldlike spin torque with the contributioñ Js nl m l,r × m r,l to the magnetization dynamicsṁ l,r in the two FIs [46], wheres nl is s nl (x) averaged over the width t of the FI wire (Fig. 3), ad detailed further in the Supplementary Note 4. The resulting eigenmodes are magnon-cooparons distributed over the two FIs and the S layer with dispersion: ω ± = γK +D m k 2 ∓Js nl . Hence, a pure spin wave injected with frequency ω into the left FI transfers its energy via the spinful superfluid to the right FI after traveling the so-called [65] coupling length L: where k ± are the wavenumbers corresponding to the frequency ω of the injected spin wave. A smaller L allows transfer of energy and the concomitant implementation of logic operations in smaller devices and thus is desirable.

Numerical estimates and experimental detection
We now employ material parameters pertinent to yttrium iron garnet [67] as FI and Nb as S in finding the effects discussed above to be large. We consider D m = 5 · 10 −29 erg · cm 2 , This can be measured using, for example, the Brillouin Light Scattering technique [68] employed regularly in measuring magnon group velocities [69]. Furthermore, the enhanced effective mass, and thus an altered spin conductivity, will manifest itself in the typical nonlocal magnonic spin transport experiment [70,71]. As the magnon spin conductivity scales as ∼ 1/ √ D m [72], its fractional modification due to the magnon-cooparon formation is given by −δD m /2D m and is expected to be large (Fig. 2). Besides the in-situ control via, for example, temperature, the FI thickness can be used to engineerD m ex-situ. A negative value ofD m signifies that our assumed uniformly ordered magnetization is no longer the ground state [73].
With the material parameters above, h 0 = 0.61∆ 0 , and T = 0.9T c , the net spin of the magnon-cooparon [Eq. (9)] is evaluated as 0.4, reduced from spin 1 of the bare magnon. Further, assuming t = 10ξ S , d = ξ S , and (k + +k − )/2 = 10 7 m −1 , the coupling length L [Eq. (12)] of the magnon-cooparon based directional coupler is evaluated as ∼ 100 nm. This is an order of magnitude smaller than the coupling length afforded by dipolar-interaction based designs [35,36]. The experimental realization of the magnon-cooparon based directional coupler can follow the procedure similar to its dipole-interaction based counterpart [36,64] with the FI layers deposited on a superconductor instead of a substrate. Magnons with nonzero k are often generated by applying ac voltage to a narrow conductor deposited on the FI [36,64]. The resulting spatially varying Oersted magnetic field bears a broadband k spectrum and excites the finite-k magnon that matches the frequency of the exciting voltage.
Several other techniques can also generate finite-k magnons by exploiting the same lack of k-conservation in hybrid systems [74].
where [A, B] ⊗ = A ⊗ B − B ⊗ A and we work in the mixed (ε, t) representation with .τ x,y,z andσ x,y,z are Pauli matrices in particle-hole and spin spaces, respectively. ∆ is the superconducting order parameter. The explicit structure of the Green's function in the Keldysh space takes the form: whereg R(A) is the retarded (advanced) component of the Green's function andg K is the Keldysh component. Further we express the Keldysh part of the Green's function via the retarded, advanced Green's function and the distribution functionφ as follows: The exchange field is taken in the form of a time-independent component and a circularly polarized magnon: h = h 0 e z + δh cos(kr + ωt)e x + δh sin(kr + ωt)e y . Then The quasiclassical Green's function is to be found in the form:ǧ =ǧ 0 + δǧ, whereǧ 0 is the Green's function in the absence of the magnon andδg is the first order correction with respect to δh. Taking into account that ∇ǧ 0 = 0 (we assume that in the absence of the magnon the bilayer is spatially homogeneous along the interface) from Eq. (13) we obtain the following equation for δǧ: Introducing the unitary operatorÛ = e −i(kr+ωt)σz/2 we can transform the Green's function as follows: In case if the system is spatially homogeneous except for the magnon, δǧ m does not depend on coordinates. Then where when passing to the second equality it is used that δǧ = δg xσx + δg yσy and has no z-component in the spin space according to the spin structure of the magnon exchange field δh.

Supplementary Note 2: RENORMALIZATION OF THE EXCITATION DISPER-SION AND DAMPING
Following the methodology detailed in the previous section, the electron spin polarization s in the superconductor can be calculated as It can be written in the form: where s 0 is the equilibrium value of the electron spin polarization in the superconductor, corresponding to the absence of the magnon. δs and δs ⊥ describe the dynamic corrections to the spin polarization due to the magnon.
As per the main text, δs − s 0 accounts for the renormalization of the magnon dispersion.
It can be shown that in the adiabatic limit ω T if one neglects ω with respect to the superconducting energies δs − s 0 ∝ Dk 2 . Consequently, it only renormalizes the magnon stiffness. The explicit expression for the stiffness correction in the limit T → T c takes the form The same result for the stiffness renormalization has been obtained from the consideration of the total energy of the bilayer in the framework of the Ginzburg-Landau theory [58].
If one does not make the adiabatic approximation, then the zero-momentum value of the magnon energy is also slightly renormalized. The renormalization correction takes the form: where The renormalization of the zero-momentum magnon frequency in the limit of h 0 → 0 has already discussed [62], and our result above is consistent with this previous work in the It is seen that δα N is positive and vanishes for zero-momentum magnons. The result is natural because in the framework of our model we consider the only source of spin relaxation processes in superconductor -the finite momentum of the magnon, which results in the spin relaxation rate Dk 2 .
We do not consider any other spin relaxation processes such as spin-orbit relaxation and relaxation at magnetic impurities. The corresponding relaxation rates would additionally increase the correction to the Gilbert damping. Also the spin-flip scattering suppresses superconducting order parameter [75]. In its turn, the suppression of the order parameter leads to the suppression of the triplets, which are generated from the singlets. This fact results in weakening of the magnon screening by the triplet cloud. To support these qualitative arguments, in Fig. 5 we have plotted the dependence of the renormalized stiffness on the Dynes parameter Γ, which roughly models the effect of spin-flip scattering on the spectral functions and order parameter. We can also roughly estimate the realistic value of Γ ∼ τ −1 s , where τ s is the spin flip scattering time. Taking τ s ∼ 25 − 100ps, as it was reported for Al thin films [76] we obtain Γ ∼ 10 −1 K ∼ 10 −1 ∆ 0,Al or Γ ∼ 10 −2 ∆ 0,Nb , where ∆ 0,Al(Nb) is the superconducting order parameter at zero temperature for Al(Nb) superconductor. In our calculations we focus on Nb parameters.
where the distribution functionφ = tanh[ε/2T ] +φ (2) (1 + τ z )/2 +φ (2) (1 − τ z )/2 contains the seconds order correction with respect to the anomalous Green's function due to the non-equilibrium generation of the triplet pairs. At first let us calculateφ (2) . The equation for the distribution function is obtained from the Keldysh part of Eq. (13) and up to the second order with respect to the anomalous Green's function takes the form: In order to simplify the calculations further we work in the quasistatic approximation, when we neglect the correction of the order of ω/T c and higher orders of this parameter. In this limit all the ⊗-products in Eq. (46) can be changed by the usual multiplication because the neglected corrections are of the order of ω/T c , which is assumed to be small. To this approximation the correction ϕ (2) z to the distribution function (which could contribute to the spin current) is zero.
Then the spin current is expressed by the first two lines of Eq. (45). In order to calculate the spin current we need to calculatê the second order with respect to δh contribution toÎ takes the form: The anomalous Green's function isf R =f R 0 + δf R , where δf R =Û δf R mÛ † . δf R m can be obtained from Eq. (21) and takes the form: In Eq. (49) f R 0,s = ∆ε/(ε 2 − h 2 0 ) and ε has an infinitesimal positive imaginary part δ. Substituting Eq. (48) into Eq. (45) and taking into account thatf A = −f R * , we obtain: The spin current in the FI, carried by the magnons takes the form: In order to catch the physical essence of the effect basing on the simplest equations, we again work in the linearized with respect to ∆ and adiabatic approximation. Making use of the unitary transformation Eq. (18) we come to the following equation for the dynamical triplet anomalous Green's function where is now also spatially inhomogeneous and should be found from the system of coupled equations: The singlet anomalous Green's function obeying these equations takes the form: where ∆ 0 and ∆ are superconducting order parameters in the middle superconducting region −d/2 < x < d/2 and in the left and right regions covered by the FIs, respectively. ∆ < ∆ 0 because of the order parameter suppression by the exchange field h 0 . λ = −2iε/D, λ ± = −2i(ε ± h 0 )/D. Constants C ±l,r take the form: The solution of Eq.
where λ k = −2i(ε + iDk 2 /2)/D and constants take the form: The spin polarization created by the triplet pairs is calculated from Eq. (45). In the superconducting region under the left(right) FI it can be written as follows: where the first term is the polarization induced by the equilibrium FI magnetization, s loc is the polarization induced in the left (right) covered superconducting region by the magnon travelling in the left (right) FI and s nl is the nonlocal part of the polarization induced by the magnon travelling in the right(left) covered superconducting region via the nonlocal triplet correlations penetrating from the other covered superconducting region. As it was mentioned above, we assume that the FI magnetization is homogeneous along the x-direction in each of the FIs. Consequently, only the averaged over the FI width value of the polarization enters the LLG equation. Therefore, we need to average Eq. (60) over the FI width in each of the superconducting regions under the FIs. Then s 0 and s loc contain terms of zero and first order with respect to the parameter ξ S /t and we can neglect the terms ∼ ξ S /t 1 for simplicity without loss of important qualitative physics. At the same time s nl is ∼ ξ S /t because it is entirely determined by the correlations coming from the second FI region and decaying at ξ S . Under the described simplifying assumptions we obtain that s 0 and s loc δh/h 0 are described by the linearized versions of Eqs. (37) and (38), respectively, and the nonlocal contribution to the polarization takes the form: Taking into account the torque resulting from the nonlocal polarization, we obtain the coupled system of the LLG equations for the both FIs: ∂m l,r ∂t = −γm l,r × H eff + αm l,r × ∂m l,r ∂t + J d FI s nl m l,r × m r,l .
The last term couples the both magnetizations. Then we express m l,r = m 0 + δm l,r as a sum of the equilibrium magnetization m 0 and the magnon contribution δm l,r . Linearizing Eqs. (62) with respect to δm l,r and solving them for the magnon dispersion, we obtain: where ω 0 +D m k 2 is the magnon dispersion in a separate FI/S bilayer with the renormalized that is, the eigenmodes of the system are represented by the symmetric and antisymmetric combination of the uncoupled magnons. The value of the frequency splitting ∆ω = (J/d FI )s nl can be estimated as ∆ω ∼ 10 9 exp(−d/ξ S ) Hz for the material and geometric parameters used above and t/ξ S ∼ 10.
The coupling between the uniform modes of two ferromagnetic insulators via a superconductor layer has recently been investigated [46]. However, the nonzero-wavevector excitations considered here allow for a realization of magnon directional coupler based on a fundamentally different physical principle than it has been proposed earlier [36,65,77].
Indeed magnons of a given frequency injected into the coupled region have different wave vectors k ± ≈ k 0 ± ∆k, where ∆k = k 0 ∆ω/D m k 2 0 . The coupling length admitted by this design is given in Eq. (12) of the main text. Among the other advantages of this coupling principle we can mention more compactness of the proposed setup, because the strength of the dipole-dipole coupling is strongly reduced with lowering the thickness d FI of the FI layers along the z-direction [36], while in the framework of the proposed mechanism the coupling strength is ∝ d −1 FI , that is ultra-thin ferromagnetic films are more favorable. Second, the superconducting coupling can be switched on/off by any means, which are known to control superconductivity: temperature, magnetic field, voltage. An interesting perspective is to investigate the possibility to control the coupling strength via the superconducting phase.
Our proposed design also enables an analogous coupler for magnons in antiferromagnetic lines, because the dynamical triplets should also be generated there, but the stray fields are weak and, therefore, the dipole-dipole coupling principle does not work well.