Optical detection of vortex spin-wave eigenmodes in microstructured

PHYSICAL REVIEW B 84, 174401 (2011)

Optical detection of vortex spin-wave eigenmodes in microstructured ferromagnetic disks K. Vogt,1,2 O. Sukhostavets,3 H. Schultheiss,1,4 B. Obry,1 P. Pirro,1 A. A. Serga,1 T. Sebastian,1,2 J. Gonzalez,3 K. Y. Guslienko,3,5 and B. Hillebrands1 1

Fachbereich Physik and Forschungszentrum OPTIMAS, Technische Universit¨at Kaiserslautern, D-67663 Kaiserslautern, Germany 2 Graduate School Materials Science in Mainz, Staudinger Weg 9, D-55128 Mainz, Germany 3 ´ Spain Departamento de F´ısica de Materiales, Universidad del Pa´ıs Vasco, E-20018 San Sebastian, 4 Material Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA 5 IKERBASQUE, The Basque Foundation for Science, E-48011 Bilbao, Spain (Received 14 June 2011; revised manuscript received 9 September 2011; published 1 November 2011) We examine the excitation of spin-wave eigenmodes in the vortex state of microsized ferromagnetic circular dots made of Permalloy (Ni81 Fe19 ) both theoretically and experimentally using Brillouin light scattering microscopy. We report on the detection of the radial spin-wave eigenmodes of single elements with high mode number (up to n = 13 for the largest disk radius 2.5 μm). Theoretically we obtain an equation for the eigenfrequencies valid for arbitrary dot aspect ratios (thickness/radius) within the magnetostatic approximation. We demonstrate the influence of the disk radius on the spatial mode profiles, in particular, changes in the pinning of the dynamical magnetization at the edges and in the center of the disks. The measured spin-wave eigenfrequencies are in good agreement with our calculations for the disks with different radii. DOI: 10.1103/PhysRevB.84.174401

PACS number(s): 75.40.Gb, 75.75.Jn

I. INTRODUCTION

The magnetization dynamics of patterned ferromagnetic elements on the micro- and nanometer scale has been investigated with extensive efforts during the last decade.1–7 The possibility of constructing future information-processing and high-density storage devices based on the magnetic properties of such elements is most intriguing for scientists working in the field of magnetism.8 Disk-shaped elements made of soft magnetic materials like Permalloy (Ni81 Fe19 ) are of particular interest due to the vortex magnetization structure they can exhibit for certain geometrical parameters (diameters and thicknesses) of the disk.9 In this distinct magnetization configuration, the magnetic moments curl along circular lines around the vortex core, a narrow region in the center of the disk, where the exchange coupling forces the magnetic moments to be in an orientation perpendicular to the disk’s plane. The reduction of stray fields by this flux closure magnetization configuration minimizes the dipolar coupling between single elements lined up in an array and, thus, allows for a dense packing in potential magnetic storage devices based on patterned media. In addition, information might be coded using both the polarity (direction of the out-of-plane vortex core magnetization) and the chirality (sense of the in-plane flux closure) of the vortex structure. Based on this potential for future applications, the vortex state dynamics in microstructured magnetic dots have been studied quite ambitiously. Recent investigations were focused on how the polarity of the vortex core can be reversed by spin-polarized currents10–12 or short magnetic field pulses,2,13 utilizing the sub-GHz gyrotropic motion of a vortex core displaced from its equilibrium position. In addition, the spin excitation spectra of the magnetic vortex state were intensively studied using variable magnetic field excitation.14 Due to the spatial confinement within the microsized magnetic elements, lateral quantization leads to a well defined set of the spin-wave eigenmodes whose discrete frequencies, intensities, and spatial 1098-0121/2011/84(17)/174401(6)

distributions were thoroughly analyzed using time-resolved magneto-optic Kerr effect microscopy.3,15 These spin-wave excitations determine switching fields and switching times of the magnetic elements. However, only low-order spin-wave modes were measured so far by the Kerr microscopy. Recently, it was shown how the sub-GHz gyrotropic motion of the vortex core and the spin-wave excitations in the gigahertz regime can be linked to achieve vortex core reversal utilizing spin-wave eigenmodes of certain symmetry.16 Therefore, it is of great importance to understand and predict the excitation of even high-order spin-wave eigenmodes in the vortex ground state. In previous works17,18 the vortex spin-wave eigenmodes were calculated analytically for the simple limiting case of the dot aspect ratio (thickness/radius) L/R  0.1. Then, in Ref. 19 the eigenfrequency calculation was extended for larger values of L/R, but the developed heuristic approach allows calculation of only the main mode eigenfrequency (the mode profile having no zeros along the radial direction). Therefore, a more general approach accounting for more complicated mode profiles (arbitrary mode numbers) is needed for large enough values of L/R. In this paper we present the theoretical calculations of the radial spin-wave eigenfrequencies and mode profiles for different mode numbers in the vortex-state ferromagnetic disks for intermediate (L/R < 0.1) and large disk aspect ratios (L/R > 0.1) within a unified approach of the magnetostatic Green’s functions. We report also on the optical detection of high-order radial spin-wave eigenmodes of the vortex ground state in Permalloy disks by Brillouin light scattering (BLS) microscopy. This allows us to measure directly the spinwave eigenfrequencies as well as the eigenmode distributions in single ferromagnetic elements with dimensions on the submicrometer scale and a spatial resolution of 250 nm.20–22 One of the goals of the present paper is the experimental observation of the numerous radial modes (with n up to 13) since before there were detected mostly low-order modes with the numbers below n = 5. By investigating the BLS excitation

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FIG. 1. (Color online) Schematic layout of the sample geometry. Individual, 40 nm thick, cylindrical Ni81 Fe19 disks of different radii ranging from 2.5 μm down to 250 nm are positioned in the center of a microcoil having an inner and outer radius of 8 μm and 11 μm, respectively. The driving microwave magnetic field is directed perpendicularly to the disk’s plane along the z axis as depicted by red arrows circulating around the microcoil. The magnetic moments within the disk are aligned along the ϕ direction following circular lines around the vortex core.

assumed (independent of the z coordinate along the thickness of the disk): m(ρ,t) = M(ρ,t)/Ms ; ρ = (ρ,ϕ). In our particular case of excitations of the spin waves by a practically uniform magnetic field oscillating perpendicularly to the disk’s plane (see Fig. 1), only the spin-wave modes which have nonzero dipolar moment in the z direction can be excited. Thus, we only consider spin waves having radial symmetry with respect to the center of the disk (azimuthal index m is equal to zero18 ). Accounting uniform distribution of the variable magnetization across the disk thickness and radial symmetry of the problem, we use averaging Gα,β (r,r ) over the thickness coordinates z,z as well as over the polar angle ˆ tensor are ϕ  . Only ρρ and zz components of the averaged G nonzero. The Landau-Lifshitz equation (1) is then linearized by the substitution m(ρ,t) = mν (ρ) + μ(ρ,t), where mν (ρ) is the centered vortex ground state. The vector μ = (μρ ,0,μz ) describes the small dynamic part of the magnetization oscillating mainly outside the vortex core. The components of the dynamic magnetization can be written as μρn (ρ,t) = ρn (ρ) cos(ωn t + φ0 ),

spectra of disks with different radii, we are able to determine spin-wave eigenmodes in a wide frequency and wave vector range and, thus, to compare theoretical calculations of the spin-wave dispersion in the vortex state with much higher accuracy. II. THEORY

For our analysis of the spin-wave eigenmodes of the vortex ground state, we consider ferromagnetic cylindrical Permalloy disks of thickness L and radius R and use a cylindrical coordinate system with r = (ρ,ϕ,z). As depicted in Fig. 1, the magnetization is aligned along the ϕ direction following circular lines around the vortex core, the narrow region in the center of the disk where the magnetization is pointing along the z axis. In the theoretical derivation below, the following  definitions are used: γ is the gyromagnetic ratio, Lex = 2A/Ms2 is the exchange length (γ /2π = 2.95 MHz/Oe and Lex = 18 nm for Ni81 Fe19 ), A is the exchange stiffness, Ms is the saturation magnetization, and ωM = γ 4π Ms . As the basis of our discussion, we use the Landau-Lifshitz equation describing the motion of the magnetization M dM/dt = γ M × δw/δM,

(1)

where w = A(∇mα )2 − 12 M · Hm is the energy density with Hm being the magnetostatic field. For micron disk radii R, we can neglect the exchange energy contribution to the spin-wave eigenfrequencies since it is proportional to the parameter (Lex /R)2 , which is small in this case. Therefore, we consider the spin waves to be magnetostatic ones. The time-dependent magnetization distribution m(r,t) = M(r,t)/Ms within the disk leads to the variable demagnetizing field h(r,t) = ˆ ˆ is a tensorial nonlocal integral operator G[m(r,t)], where G [the tensorial magnetostatic Green’s function Gα,β (r,r ); see ˆ in the cylindrical Ref. 17]. We use the components of G coordinates. A two-dimensional magnetization distribution is

(2a)

μzn (ρ,t) = zn (ρ) sin(ωn t + φ0 ), (2b) ρ,z where the functions n (ρ) are the eigenmode profiles and ωn

the eigenmode frequencies with n = 1,2, . . . being the mode number. ρ The equation for the radial profiles n (ρ) = μ(ρ) of the magnetization can be written as the Fredholm integral equation17    1 ωn 2 μ(ρ) = dρ  ρ  (ρ,ρ  )μ(ρ  ), (3) γ Ms 0 where ρ is given in units of R. The magnetostatic kernel is 1

(ρ,ρ  ) = 0 drrgzz (ρ,r)gρρ (r,ρ  ) and the Green’s function components averaged over z,z , and ϕ are  4π ∞ gzz (ρ,ρ  ) = − dk[1 − e−βk ]J0 (kρ)J0 (kρ  ), (4a) β 0  ∞  dkkf (βk)J1 (kρ)J1 (kρ  ), (4b) gρρ (ρ,ρ ) = −4π 0 −x

where f (x) = 1 − (1 − e )/x. Assuming that the dynamic m components are mainly concentrated outside the vortex core, Eq. (3) can be simplified to the following integral equation for small disk aspect ratios β  1 [in this case gzz (ρ,ρ  ) = −4π δ(ρ − ρ  )/ρ]:    1 ωn 2 1 μ(ρ) = − dρ  ρ  gρρ (ρ,ρ  )μ(ρ  ). (5) ωM 4π 0 The eigenfrequency ωn , which corresponds to the normalρ ized to unit n eigenfunction n (ρ), is given by    1  1 ωn 2 1 =− dρρ dρ  ρ  gρρ (ρ,ρ  )ρn (ρ)ρn (ρ  ). ωM 4π 0 0 (6) The kernel in Eq. (5) can be easily symmetrized and the standard Gilbert-Schmidt theory can be applied for its solution. However, the kernel (ρ,ρ  ), which is a convolution of the kernels gzz (ρ,ρ  ) and gρρ (ρ,ρ  ), is not symmetric and the

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solution of Eq. (3) is more complicated since no theory has been developed to solve integral equations with such kernels. For intermediate and large values of β (β > 0.1), Eq. (5) is not a satisfactory approximation and we need to solve the full integral equation (3). In order to do this, we expand the function  μ(ρ) in a Bessel-Fourier series μ(ρ) = N n=1 an Jν (ανn ρ), where in our case ν is equal to 0 or 1 according to the integral representations given by Eqs. (4a) and (4b). The values of ανn , roots of the equation Jν (α) = 0, correspond to strong magnetostatic pinning of the eigenfunctions at the element’s lateral edges ρ = R. Since J1 (α1n ρ) give the asymptotically exact solutions of Eq. (5) at β  1, we can set ν equal to one at intermediate values of β (β < 0.1). After choosing the basis functions J1 (α1n ρ), we apply the approximate substitution gρρ (ρ,ρ  ) → −4πf (βα1n )δ(ρ − ρ  )/ρ. In order to simplify the kernel in Eq. (3) for smaller disk radii (β > 0.1), we use the basis functions with ν = 0 and the approximate substitution gzz (ρ,ρ  ) → −4π [1 − f (βα0n )]δ(ρ − ρ  )/ρ. The decomposition for μ(ρ) can be substituted to Eqs. (3) or (5) which then can be reduced to the ˆ = λa, standard eigenvalue problem for a Hermitian matrix: a λ = (ω/ωM )2 with a = (a1 ,a2 , . . . ,an ). The eigenfrequencies can be found from the secular equation det( mn − λδmn ) = 0, ˆ n where δmn is the Kronnecker symbol and mn = m | | ˆ are the matrix elements of using the normalized basis functions 1 1 (7) nρ (ρ) = √ Jν (ανn ρ), Nn = [Jν+1 (ανn )]2 . 2 Nn For numerical diagonalization of the matrix mn , it is more convenient to use the explicit expression for the Green’s functions gzz (ρ,ρ  ) and gρρ (ρ,ρ  ) (see Appendix). As a result of the matrix diagonalization of mn , we obtain the discrete set of the frequency eigenvalues λn = ρ (ωn /ωM )2 and eigenfunctions n (ρ). In the main approximation for intermediate values of β, the diagonal components β)] and the eigenfrequencies are nn (β) = f (α1n β)[1 − f (α1n√ can be estimated as ωn (β) = ωM nn (β). Note that the nondiagonal components mn are not negligible with increasing β. However, the eigenvalue problem can be essentially simplified within the limit of small β ≈ 0.01 and we can only use the kernel gρρ (ρ,ρ  ) as was done in Ref. 17. In this limiting case of small disk aspect ratios β = L/R, the discrete frequencies of the radial magnetostatic spin waves can be calculated using the equation17,18   ωn 2 = f (βγn ), (8) ωM where γn is the nth root of the boundary equation J1 (γ ) + ηJ1 (γ ) = 0 at the disk lateral surface. Therefore, the frequencies of the radial spin √ waves satisfy √ the simple asymptotic relation ωn (β) = ωM γn β/2 ∼ β calculated in Refs. 17 and 23. Equation (8) assumes some pinning of dipolar origin of the dynamic magnetization at the disk lateral edges ρ = R. According to Ref. 18, the pinning parameter η is η(β,L) =

2π

2  .  8 β 4 ln β − 12 + LLex

(9)

Decreasing the pinning parameter η leads to a change of the mode profiles and to a decrease of the eigenfrequencies. In our case, for the disk radii between R = 250 nm and 2.5 μm and the thickness of L = 40 nm, the parameter η varies from 2.84 for the smallest disk up to 17.0 for the disk with the largest radius. Accounting the finite values of η, the frequencies decrease by approximately 0.1 GHz in comparison to the case of strong pinning η 1. That means that the pinning is relatively strong allowing us to use the roots αn of the equation J1 (α) = 0 instead of the parameter γn defined above. As already mentioned above, the homogeneity of the oscillating magnetic field only allows for the excitation of radial spin-wave modes having an odd number n of antinodes ρ,z of the functions n (ρ) along the radial direction within the interval ρ = (0,R). ρ The eigenfunctions n (ρ) are proportional to zn (ρ) in the limit of small β. If the vortex core is neglected, they satisfy the strong pinning boundary conditions (3) at ρ = R and ρ = 0, ρ,z ρ,z i.e., n (R) ≈ 0 and n (0) ≈ 0. From analysis of the kernel 

(ρ,ρ ), it can be concluded that the eigenfunction profiles ρ n (ρ) and zn (ρ) become different with increasing β. In parρ ticular, n (0) = 0 whereas zn (0) = 0 is still true. The eigenρ functions n (ρ) satisfy the strong pinning conditions with the pinning parameter η given by Eq. (9), whereas the eigenfunctions zn (ρ) are unpinned at the disk lateral edge ρ = R. This is physically reasonable because the vortex core radius is very small (∼10 nm) in comparison to R, and mzν (0) = ±1, μz = 0 in the center of the core (ρ ≈ 0). There is no reason to demand μρ (0) = 0 even though this boundary condition formally follows from Eq. (5) within the limit of small disk aspect ratios. This implies qualitative changes of the radial eigenmode profiles with increasing disk aspect ratios β. We do not expect hybridization of the radial spin waves with the vortex gyrotropic mode due to different symmetries of these spin excitations. ρ,z In a physical sense, the eigenfunctions n (ρ) supply the eigenmode radial profiles which can be measured by spatially resolved BLS microscopy as is shown in the following section. III. EXPERIMENTAL RESULTS

Knowing the frequencies of the excited spin waves with high quantization numbers allows for a verification of the theoretical model described in the preceding section. This is essential for a deeper understanding of the spin-wave mode profiles and pinning mechanisms at the structure boundaries and at the position of the vortex core. The schematic layout of the experimental setup was already shown in Fig. 1. Single cylindrical dots with radii ranging from R = 2.5 μm down to 250 nm are fabricated from an L = 40 nm thick Ni81 Fe19 film using electron beam lithography and conventional lift-off procedures. Each disk is positioned in the center of an omega-shaped microcoil made of 500 nm thick copper with an inner and outer diameter of 8 μm and 11 μm, respectively. A microwave current flowing through the microcoil generates an oscillating magnetic field oriented parallel to the z axis, i.e., perpendicular to the disk’s plane as depicted by red arrows circulating around the microcoil in Fig. 1. Thus, the magnetization components of the flux closure vortex configuration, which are oriented along the ϕ direction in the plane of

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FIG. 3. (Color online) Measured two-dimensional intensity profiles of the radial spin-wave eigenmodes in a Permalloy disk with 2.5 μm radius. As can be clearly seen from the intensity distributions, the measured eigenmodes excited at the frequencies of 4.3, 6.9, and 8.5 GHz can be identified as mere radially symmetric modes having odd numbers of circular antinodes n = 1, 3, and 5, respectively. IV. DISCUSSION

FIG. 2. (Color online) Spin-wave spectra measured in Ni81 Fe19 disks with different radii R of (a) 2.5 μm, (b) 2 μm, (c) 1 μm, (d) 750 nm, and (e) 250 nm. In the largest disk with R = 2.5 μm, the radial spin-wave eigenmodes with mode numbers up to n = 13 can be clearly seen. The disk thickness is L = 40 nm.

the Permalloy disk (see Fig. 1), can be excited efficiently using the given antenna geometry. However, due to the homogeneous distribution of the exciting magnetic field across the entire disk, no net torque is applied to all azimuthal modes having indices m = 0 as well as to radial spin-wave modes with even mode numbers n. Therefore, we expect to excite only eigenmodes of radial symmetry having an odd number n of antinodes across the disk radius with the given antenna geometry. The spin-wave eigenmodes are determined by exciting the disks of different radii with microwave currents having frequencies between 3 and 16 GHz. For each applied microwave frequency, the BLS intensity is recorded at the center of each individual disk, and then plotted as a function of the applied microwave frequency. The resulting excitation spectra are summarized in Fig. 2. Especially in the two larger disks with R = 2.5 μm (a) and R = 2 μm (b), several well-defined peaks in the BLS intensity can be recognized. However, it can be directly seen that no spin-wave signal is measured for microwave frequencies above 12 GHz. In this frequency range, the excited spin-wave eigenmodes have wave vectors larger than 17 μm−1 which are not accessible with the given Brillouin light scattering microscope.20 In order to demonstrate that the investigated spin-wave eigenmodes exhibit radial character with odd mode numbers n only, we mapped the two-dimensional intensity distributions of the first three eigenmodes measured in the largest disk with 2.5 μm radius. As can be seen from the mode intensity profiles in Fig. 3, the spin-wave eigenmodes excited at the frequencies of 4.3, 6.9, and 8.5 GHz are mere radial spin-wave eigenmodes with mode numbers n = 1, 3, and 5, respectively.

By analyzing the spin-wave spectra in Fig. 2, we can notice that the frequency of eigenmodes with the same mode number increases with decreasing the disk radius. In addition, it is clear that for each disk radius higher mode numbers correspond to higher frequency values. Figure 4 summarizes these results. The frequencies of the spin-wave eigenmodes extracted from the spin-wave spectra in Fig. 2 (filled symbols) as well as the frequencies obtained from solving Eq. (3) (empty symbols) are plotted as a function of the disk aspect ratio β = L/R. Good agreement of the frequencies measured with BLS microscopy and the calculated ones can be observed, especially at small β  0.053 (disk radii R  750 nm) where up to 7 BLS intensity peaks were detected for a single aspect ratio. The spatial distributions of the BLS intensity very nicely corroborate the assumptions made for the theoretical calculations. We would like to note that we have managed to detect the radial spin-wave eigenmode frequencies with high mode numbers up to n = 13 and n = 9 in two larger

FIG. 4. (Color online) The radial spin-wave mode frequencies measured by Brillouin light scattering microscopy (filled symbols) and calculated using Eq. (3) (empty symbols) as a function of the disk aspect ratio β = L/R. The lines connecting the calculated frequency values are guides to the eye.

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disks with R = 2.5 μm and R = 2 μm, correspondingly. The modes with such high numbers have never been detected experimentally before. We can detect by the BLS technique the intensity distributions (the nodes and antinodes) for the radial modes with relatively small n = 1,3,5 as shown in Fig. 3. However, distinguishing between neighboring antinodes of the eigenmodes with mode numbers n > 5 was not possible due to the limited spatial resolution of the BLS microscope. The BLS spatial resolution is also not sufficient to measure the intensity profile of the radial spin modes distribution for the smallest radius R = 250 nm (L/R = 0.16) that could prove the qualitative changes in the radial mode profiles for intermediate dot aspect ratio predicted in the end of Sec. II. Nevertheless, we assume that all the measured spin-wave eigenmodes are mere radial modes having an odd number of antinodes in radial direction only. The comparison of our experimental results to the theoretical model described above supports this assumption. The main theoretical interest in this paper was to describe the frequencies in the disks with high aspect ratios including experimentally measured ones for β = 0.16 (R = 250 nm). Unfortunately, experimental resolution allows us to measure only the first mode for such dot aspect ratio. Within our theoretical approach we overestimate the frequency for β = 0.16 at about 10% as can be seen in Fig. 4. The estimation of the frequency for β = 0.16 using the theory developed in Ref. 19 give us the value of 12.6 GHz which is about 40% higher than the frequency 9.2 GHz measured experimentally. The equation for the eigenfrequencies obtained in this paper is much more complicated than in Refs. 17 and 19, but it allows one to calculate the frequencies for intermediate and large disk aspect ratios and for both high and low order mode numbers with quite good precision. V. SUMMARY

We have detected the radial spin-wave eigenmodes in the vortex state of circular micron-sized ferromagnetic Permalloy dots using highly sensitive Brillouin light scattering microscopy in a wide frequency range from 3 to 12 GHz. The spatial distributions of the dynamical magnetization as well as the frequencies of the excited radial eigenmodes up to the mode number of n = 13 were investigated. The measured quantized spin-wave eigenfrequencies in disks of different radii varying from 250 nm to 2.5 μm were understood by calculations of the radially symmetric magnetostatic spin waves in the vortex ground state. We have used the solutions of the magnetostatic equations for the radial magnetostatic spin waves in the vortex ground state. These equations are valid for arbitrary dot aspect ratios which would be important in the future due to the tendency to decrease the lateral parameters

1

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of the ferromagnetic elements. Good coincidence between the experimental and theoretical results proves that both the low and high number spin-wave modes can be described by the theoretical approach developed in this paper. We have shown by the calculations that with decreasing disk radius the pinning of the eigenmodes weakens for the ρ component of the dynamical magnetization at the disk’s center as well as the z component at the disk’s edges. However, the pinning remains strong for the ρ-component of the magnetization at the edge and for the z component at the center of the disk. ACKNOWLEDGMENTS

The authors would like to thank A. Beck and the Nano + Bio Center of the Technische Universit¨at Kaiserslautern for their assistance in sample preparation. K.V. gratefully acknowledges financial support by the Carl-Zeiss-Stiftung. B.O. would like to thank the Graduiertenkolleg 792 for financial support. K.Y.G. acknowledges support by IKERBASQUE (the Basque Foundation for Science). The Spanish team work was partially supported by SAIOTEK Grant No. S-PC09UN03 and MICINN Grants No. PIB2010US-00153 and No. FIS201020979-C02-01. APPENDIX

To solve the integral Eq. (3) we used explicit expressions for the magnetostatic Green’s functions gzz (ρ,ρ  ) [see Eq. (4a)] and gρρ (ρ,ρ  ) [see Eq. (4b)] in the coordinate representation:  √  2 ρρ 1 8  gzz (ρ,ρ ) = − K  β ρ+ρ ρ + ρ √   1 2 ρρ  −  K  , (ρ + ρ  )2 + β 2 (ρ + ρ  )2 + β 2 where K(k) is the full elliptical integral of the first kind.24 For finite β, the kernel gzz (ρ,ρ  ) has the integrable logarithmic singularity at ρ = ρ  , gzz (ρ,ρ  ) = −(4/βρ) ln(8ρ/|ρ − ρ  |) which is typical for two-dimensional magnetostatic problems. The explicit expression for the Green’s function gρρ (ρ,ρ  ) via K(k) and the full elliptical integral of the second kind E(k)24 is more complicated and shown here: 4π 4π δ(ρ − ρ  ) + [F (ρ,ρ  ,0) − F (ρ,ρ  ,β)], ρ β √ where F (ρ,ρ  ,x) = (1/ ρρ  )[(2 − k 2 )K(k) − 2E(k)] and 2   2 k = 4ρρ /[(ρ + ρ ) + x 2 ], x = 0,β. To represent the radial Green’s functions gzz (ρ,ρ  ) and gρρ (ρ,ρ  ) by the elliptical functions above, we used the integrals from Bessel functions taken from Ref. 25. gρρ (ρ,ρ  ) = −

3

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