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Ferromagnetic resonance modes in the exchange dominated limit in cylinders of finite length

We analyze the magnetic mode structure of axially-magnetized, finite-length, nanoscopic cylinders in a regime where the exchange interaction dominates, along with simulations of the mode frequencies of the ferrimagnet yttrium iron garnet. For the bulk modes we find that the frequencies can be represented by an expression given by Herring and Kittel by using wavevector components obtained by fitting the mode patterns emerging from these simulations. In addition to the axial, radial, and azimuthal modes that are present in an infinite cylinder, we find localized"cap modes"that are"trapped"at the top and bottom cylinder faces by the inhomogeneous dipole field emerging from the ends. Semi-quantitative explanations are given for some of the modes in terms of a one-dimensional Schrodinger equation which is valid in the exchange dominant case. The assignment of the azimuthal mode number is carefully discussed and the frequency splitting of a few pairs of nearly degenerate modes is determined through the beat pattern emerging from them.


Introduction
Measurements of the resonant microwave response of the spins of unpaired electrons in ferromagnetic materials have a long history, beginning with its initial observation by Griffiths 1 .
The vast majority of the subsequent experimental studies have involved "macroscopic" samples where the spins excited by an external microwave field precess in the combined field arising from an externally applied constant field together with the field produced by the sample's own magnetization (the so-called demagnetization field) under conditions where both are nearly uniform. Under such conditions the strength of the exchange interaction, though itself responsible for the materials magnetization, does not affect the resonance frequency.
In the decade after Griffiths' work, it was found that attempts to drive the resonance to larger amplitudes did not work, and that it saturated at values of the microwave power far below that expected from the observed resonance linewidth. The explanation is that small inhomogeneities in the magnetization of the sample are amplified by the inhomogeneities generated in turn in the dipolar or demagnetizing field, and leads to what are now called the Suhl instabilities 2, 3 . In addition to studies of the uniformly precessing mode, spatially-nonuniform 2 modes have also been intensively studied, both experimentally and theoretically, as we shall describe below.
In this paper, we report on an exhaustive numerical study, using the OOMMF micromagnetic simulation code 4 , of the resonance modes of Yttrium Iron Garnet (YIG) cylinders, primarily of diameter d = 75nm and height h = 300 nm, although some aspects have been studied for other values of h (7.5-1200 nm). Our methodology shares many features with the work of McMichael and Stiles 5 on two-dimensional elliptical disks and three dimensional thin cylindrical discs. Our three-dimensional geometry displays a much richer mode structure, however, requiring a more detailed theoretical framework. In addition, we have also developed techniques to resolve modes that are nearly degenerate in frequency.
The motivation for our work is as follows. Firstly, in recent years many authors have examined whether the magnetization of small particles (in the size range of order 10 2 nm) can be reversed by applying microwave fields (so-called microwave assisted switching) with a view to applications in magnetic storage 6,7,8,9 . The reasoning is that at small sizes, the exchange interaction favors parallel alignment of the spins and suppresses the dipolar instabilities. In previous work, we have used OOMMF to simulate YIG cylinders with (d, h) = (25,50)nm, (25,100)nm and (75,150)nm, where we have attempted to reverse the magnetic moment of the cylinder by applying pi-pulses 10,11 , as in NMR 12 . We found that by chirping the frequency of the pulse, we could achieve reversal in the 25 nm diameter samples, but not in the larger ones. If pipulses or other reversal protocols are to be successfully implemented in more realistic larger samples, then an understanding of the normal modes in restricted geometries is a necessary prior step. The problem is also of intrinsic interest, and we have found novel features not seen in ellipsoidal geometries (including degnerate forms thereof) that have been the subject of almost all other work to date.
Secondly, with recent advances in sub-micron patterning techniques, the study of arrays of objects (so as to have large signals) for which the largest dimension is few hundred nanometers or less is attracting increasing attention. With advanced techniques it is even possible to probe the magnetic properties of individual sub-micron particles 13,14,15 . Measurements on such samples can even be performed in the absence of an external field, i.e., solely in the presence of the internal demagnetization field (for shapes where such a field exists), provided the sample is small enough to be in a single domain state 16 . Modes with an odd number of maxima and minima 3 can be excited directly with a uniform microwave field; coupling to modes with higher wave numbers will be more challenging 17 .
Here we will primarily be concerned with size-quantization effects arising from finite sample dimensions. In particular we will examine the mode spectrum in samples having a cylindrical shape with radius a (and corresponding diameter d = 2a) and height h, both analytically and numerically. Due to the ease of preparation of some materials as wires, such samples are widely studied experimentally, e.g., in permalloy Py 18 and in Ni 19,20 . Cylinders of finite length with h/d ratios of order unity and larger can be readily patterned using optical and ebeam lithography by creating hole arrays in a resist followed by deposition and liftoff 21 . here H is the total field at a given position within the sample arising from the external field as well as that produced by the magnetization itself and an effective field arising from quantum mechanical exchange; it can also include crystalline anisotropy, but this is suppressed in what follows. The second term on the right-hand side of Eq. (1) is incorporated to phenomenologically account for damping, which will largely be neglected in what follows. In addition to satisfying where N and N ⊥ are coefficients accounting for the effect of demagnetization perpendicular and parallel to the rotation axes (with 2N N 1 ⊥ += ), and 0 M is the internal magnetization, taken as a constant; note  may differ from the free-space value due to atomic and solid-state effects.
In addition to the uniformly precessing mode there exist non-uniform modes 24  )  k H z where ẑ is the plane normal; it has a positive group velocity but also has the unusual property that it propagates only on one side of the slab for a given field direction, switching to the opposite side on reversing the field. The second class is a family of bulk modes which are quantized in the film thickness direction; they are typically studied with 0 kH. They propagate in both directions and the lowest lying mode has the unusual property that its group velocity is negative for small k and therefore it is called a backward volume mode. The frequencies of all the volume modes asymptotically approach the orbital angular momentum mp l = , and there is a small admixture of s m1 =− and m p 2 l =+. Accordingly, we find it better to label the modes by the orbital angular momentum p of the major 7 component. This is especially so when examining the computer-generated mode patterns since the orbital behavior of any component of M is immediately apparent.
That the two components of a mode are so unequal goes hand in hand with the fact that modes with ml = p and −p are nearly degenerate, as we discuss in Sec. 2.2. A clear understanding of this issue is important, as this near degeneracy can lead to some confusion when looking at the mode patterns. In one case, we have resolved this degeneracy (see Sec. 3.4) by exciting and examining the beat pattern between the ±p modes.
In Sec. 2.3, we show that exchange dominated modes in long cylinders are approximately described via a Schrodinger-like equation for a particle in a cylindrical box with a modified boundary condition, such that axial and radial dependence of the mode function factorizes, and the resulting quantization gives rise to axial and radial mode numbers. In an infinite cylinder this separation is exact, which is exploited to good effect in the analyses of Joseph and Schlomann 34 and of Arias and Mills 35 . In a finite cylinder, the separation is approximate since the demagnetizing field is non-uniform, and flares away from the axis near the perimeter of caps at z = 0 and z = h. We give a semi-quantitative argument in Sec. 2.4 that the Schrodinger equation possesses bound state solutions near these caps, corresponding to "cap modes" which we see very clearly in our simulations. For any p, there are two such modes (one for each cap), whose frequencies lie below those of the bulk modes with the same p. This means that the uniform FMR or Kittel mode, which is the lowest bulk mode with p = 0, is not the lowest frequency mode of the body. For this case, we present numerical results for the solution of the Schrodinger equation in Sec. 2.5, and find good agreement with the simulations.
The cap modes are a novel and unexpected feature of our study, as they do not exist in an infinite cylinder or a finite sized ellipsoid of revolution. Similar "end modes" were found by McMichael and Stiles 5 , who did not however investigate their origin. We expect that such localized modes will exist near the surfaces of other sample shapes as well whenever the demagnetizing field departs significantly from uniformity. Our simulational approach is described in Sec. 3. It is based on the OOMMF code developed at the National Institute of Standards and Technology. After finding the static equilibrium magnetization, eq () Mr (Sec. 3.1), we can excite the system by applying pulses that are localized in either space, time or both 41 . The spatial center and the width and frequency bandwidth are varied depending on which mode(s) we wish to excite. The resulting time development of ( , t) Mr is Fourier transformed, and the point-wise power spectrum is added over all the cells. The resulting sum displays peaks at many mode frequencies, and by honing in on individual peaks, we can identify the magnetization patterns for each mode as explained in Sec.

3.3.
Once a particular mode pattern is obtained in a simulation it can be used as is or altered in some way, say by combining it with some other mode, to study the subsequent development in time. This is a potentially promising way to study mode-mode coupling or large amplitude responses which we hope to pursue in the future. As an application of this idea, and as noted In Sec. 5 we show that the mode frequencies that we obtain agree surprisingly well with the Herring-Kittel expression (1.3) provided we identify k ⊥ and z k in this formula correctly.
We give reasons why this agreement might be so good, explain how the wavevector components are found, and how this allows us to organize the normal mode spectrum into families of modes labeled by p.
Spatial patterns for a variety of modes are given in Sec's. 6  Finally, Sec. 8 summarizes our conclusions. Here we take the opportunity to emphasize the importance that simulations of the small amplitude mode structure in nano-structures have for present and possible future applications, some of which are currently speculative in character.

Modes in the exchange dominated limit
In the presence of an isotropic exchange interaction, and neglecting the effects of damping, Here H is the applied magnetic field, 0Ĥ z , plus the dipolar or demagnetizing field generated by M . In a cylinder of finite height, the dipolar field is not uniform, especially near the caps, and so the static equilibrium field, eq () Mr is not everywhere parallel to ẑ . A linearized normal mode analysis should therefore consider deviations eq ( , t) ⊥ M r M , which do not lie in the x-y plane. If exchange is strong, however, the non-uniformity in eq M is very small (this is true for all the simulations we have performed), and we may then take z 0 = M . This assumption makes it much easier to discuss the physics, and relaxing it only obscures the key ideas without adding substance. We stress that it is not essential to our argument, especially with respect to the symmetries and the azimuthal quantum number. With this assumption, we may write ( ) where m has only x and y components and is dimensionless since we have scaled out 0 M .
For small deviations, | | 1  m , the linearized LL equation can be cast as Here z H ( ,z) r consists of the applied field, 0Ĥ z , together with the position dependent longitudinal demagnetization field arising from the static magnetization, and We need only carry this analysis out to leading order in  . The components of the rotated vector  m are then (see Fig. 2 The coordinates  (x , y ,z) themselves are related to (x, y,z) as If we expand the left side of (2.4a,b) to first order in  , note that to zeroth order , and recall the definition of the (dimensionless) orbital angular momentum operator z l in quantum mechanics as can be rewritten in the form , but the presence of the last term in (2.8) mixes the two components of the vector field m . This can be interpreted as arising from an "internal" or "spin" angular momentum of = s m1 that is added to or subtracted from the orbital angular momentum m l associated with our vector field m (a tensor of rank 1). A similar separation exists in the description of light fields 44 . This field has orbital angular momentum  mp l , but has no definite spin angular momentum.
For it to have a definite spin a and b must be proportional according to It follows that an eigen mode with total angular momentum p1 + must be of the form where we have adopted an − it e time dependence following standard practice. The physical solution is obtained by taking the real part of this complex-valued solution. We shall see below that for positive frequency solutions, −+  FF in the strong exchange limit, and it is often convenient to neglect + F entirely. It is then more useful to label the modes by p, the orbital angular momentum of the dominant component, − F . This is especially so when looking at mode patterns generated by OOMMF, since we can read off p by seeing how many times m turns as we go around a circle in the x-y plane. For example, in a p = 0 mode (see Figs. 6.1a and 6.1b) m appears uniform, while in p = −1 ( Fig. 6.3) and p = 1 ( Fig. 6.4) modes m winds by 2π and −2π, respectively as we go anti-clockwise around a circle.
In the magnetostatic limit by contrast, +−  F / F O (1) , and the m labeling is better. Thus, for the sphere, while we would describe the uniform or Kittel mode as having p = 0, Walker 29 assigns m = 1 to it (see his Fig. 3 where the mode is labeled (110)).

Near degeneracy of p and − p modes
When exchange dominates over dipole-dipole interactions, we may as a first approximation We seek solutions of the form it m ( , t) m ( )e −  = rr (2.16) and demand that 0  , with the understanding that the physical solution will be given by the real part. These solutions then obey This equation is like a one-particle Schrodinger equation, and since γ > 0 in our convention, the operator on the left is a positive operator which cannot have negative eigenvalues. Since we also demand that ω > 0, we must choose m0 + = . Finally, since our finite cylinder retains full azimuthal symmetry, the solution for m − takes the form can be chosen to be real, and ω has the same (positive) value for either sign of p.
In terms of the general form (2.12), this solution corresponds to putting F0 + = , and a physical solution If we now include the dipolar field d h as a perturbation, we can expect that F + will become nonzero, with also produces a solution. This operation is equivalent to φ → − φ, or alternatively to p → − p.
Inclusion of the dipole-dipole interaction destroys this invariance: the field d h produced by the operation is not the same field as before.
Strictly speaking therefore, modes differing only in the sign of p are not degenerate, although the non-degeneracy may be small. Indeed, as explained below, we have spent significant effort to numerically resolve the splitting and have not always succeeded. The physical origin of this non-degeneracy is just that the applied external field breaks time reversal and parity symmetries. In the magnetostatic limit, this point emerges directly from the solution in terms of the scalar potential. Joseph and Schlomann 45 ( The allowed values of k ⊥ are given by If the boundary condition is taken as ()0 = nm  , then the allowed z k values are Even parity is associated with even z  and odd parity with odd z  .
If instead the boundary condition is taken as 0 = m , the allowed values of z k are Now even parity is associated with odd z  and odd parity with even z  .
The boundary condition obeyed by OOMMF mode functions is closer to ()0 = nm  than to 0 = m . In addition, they do have definite parity. Except for the two lowest frequency modes, which we call "cap modes" and which require a separate discussion, they are well fit by We can refer to z  as an "end defect" analogous to the concept of a quantum defect in atomic spectroscopy 47 . With this correction Eq. (2.26) continues to be a good approximation to the mode frequencies.
We comment further on the boundary condition (2.24) that the normal derivative vanishes at the surface. The isotropic continuum exchange field 2 ex DM − arises from a microscopic ij  SS Heisenberg interaction, which has the property that for any pair of spins, the torque on i S due to j S cancels that on j S due to i S . Thus, the total exchange torque on the body vanishes and Eq. (2.24) is the continuum expression of this fact. This argument dates back to Ament and Rado 48 and has been used by many authors since. Aharoni 49 offers a different derivation. Thus, it would appear to be very general, and valid for any ex D , however small. For the magnetostatic limit, ex D0 = , there is however no such condition on M . Turning on ex D perturbatively would then appear to lead to a contradiction. This is not so for the following reason.
In the boundary value problem for the spatial form of the eigenmodes, ex D multiplies the highest derivative, and is thus a singular perturbation from the mathematical point of view. Such perturbations are known to lead to thin boundary layers where the solution changes character rapidly 50 . Thus, while the normal derivative at the surface may formally be zero, there could be large curvature in the boundary layer, and the derivative of m as we approach this layer need not be small. This is especially relevant for our OOMMF simulations, where the discretization into cells may: (a) be too coarse to reveal any boundary layer behavior, and (b) fundamentally preclude measurements of this derivative by fitting to the mode functions. In this case adoption of an end defect z  is an effective practical procedure.

A variational solution for a cylinder of finite length
In the second stage of our argument, we attempt to include the inhomogeneity in the static demagnetizing field by adopting the trial form 17 If we substitute this form in Eq. (2.17), together with some radially averaged z-dependent magnetic field H(z) , we obtain the following one-dimensional eigenvalue problem with the boundary conditions In the spirit of this variational approach we could obtain H(z) ) by averaging with respect to 2 p J (k r) ⊥ , which would lead to slight differences between modes with differing k ⊥ .
Alternatively, we can use the analytic expression for the dipole field along the cylinder axis

Zeros of mode functions and mode labels
The demagnetizing field plays the role of an external potential in the Schrodinger equation (2.32), and the strong decrease in this field near the end caps leads to surface bound states or cap states whose wave functions die off exponentially away from the caps. In principle there could be many bound states, but for our parameters we find only one state at each cap. All higher energy states are extended along the z direction, and since the demagnetization field is essentially uniform in the bulk of the cylinder, their wavefunctions behave approximately as sinusoidal standing waves.
Let us now recall that for a one-dimensional Schrodinger equation with a reflection symmetric potential, the states with successively higher energy alternate in parity and have successively increasing number of zeros, with the lowest energy state having no nodes and even parity. Further their wave functions must be mutually orthogonal. For our problem, these theorems are satisfied as follows. The two cap states are nearly degenerate but they are admixed by tunneling to form even and odd parity states with zero and one node respectively. ( for the end defect at the same time. For each value of p and r n , we could label the differently quantized modes along z by the number of their zeros. The lowest extended or bulk mode in any family with given p and r n As is evident, the semi-analytic results for Z(z) are surprisingly good for h 300nm  . The frequencies however are not. These could be improved by including the transverse dipolar field using perturbation theory, which will raise the frequency. We have not attempted this exercise since our approximate treatment is quite rough in the first place and it would not add to our qualitative understanding. GHz which is to be compared with the OOMMF value of 6.64 GHz. Here we encounter progressively higher mode frequencies scaling approximately as 2 z n .

Computational Approaches
The material studied here is YIG which was chosen for its long mode lifetimes. Whether  erg/cm. The applied field was 2 kOe along z direction. Damping was turned on to relax the system to its initial state, and turned off after the system was excited for most simulations. In the few that it was not, it was too small to have any significant effect.

Static equilibrium
Prior to exciting the system, the spins were initially aligned along the cylinder axis (parallel to the external field) after which the system was evolved in time (with damping) until it stabilizes in an equilibrium configuration. Various tests can be applied to determine that it is a global equilibrium state. This part of our simulations yields the static magnetic field distribution which could also be used for the calculations in Section 2.

Exciting the system
Several different excitation schemes were utilized. In the simplest of these, all spins were tipped by a small fixed angle relative to their equilibrium orientations in a plane containing the z axis as an initial condition. This favors the excitation of uniformly precessing modes. To drive a particular non-uniform mode the spins were tipped from their equilibrium positions in a manner that mimics the mode (such as that obtained as the mode pattern in a prior simulation) 54 . To drive a broader spectrum of modes that is localized around a time 0 t and some position 0 r we tip the spins in some direction according to the function 41 y0 0 x 0 x 0 , k , k and k     control the extent to which the excitation is localized in time and space. Here x, y, z denote cell coordinates. Such pulses can also be introduced at multiple times and positions to favor the excitation of modes with differing spatial and temporal properties. In particular inclusion of only the last factor induces modes propagating along z. Forms can be constructed that favor the excitation of radial or azimuthal modes. Finally, some simulations were performed in which individual spins were tipped in random directions within some specified average angular range. This excites a very broad range of modes and if the tipping angles are large (e.g., approaching 180 o ) generates a "hot" system, from which it is difficult to extract clear modes. Altogether, we tried more than ten different excitation pulses in an effort to identify modes with different symmetry and numbers of nodes. Despite this, our mode table (see Sec. 4) has gaps. In some cases, modes are nearly degenerate [for example modes (003), (-10g), and (-10u)] and cannot be easily resolved. In others, they were too high in frequency to be seen with the particular excitation pulse employed. We are confident that these modes exist and that our mode classification is complete. we can, respectively, obtain the real-space vector magnetization at some time and a quarter cycle later for the spatial pattern associated with some nominal mode at the specific frequency. By plotting these vector fields, we can get a highly visual depiction of the mode, permitting easy mode assignment and further analysis. In constructing the power spectrum, modes with higher frequencies than the inverse of the chosen integration time step, which violate the Nyquist sampling criterion and are then aliased to lower frequencies, must be identified and rejected. In the exchange dominated samples considered here some of them can be identified as spurious peaks with frequencies lower than the known uniform modes, but in dipole dominated larger samples genuine modes below the uniform modes are expected. More generally they must be identified by altering the time interval over which the transform is performed to determine if some mode moves its position. Most simulations were done with a time step of 10 ps and for a duration of 10.24 ns.

Identifying modes
A curious spatial aliasing was also observed (as evidenced by a rapid spatial variation of the mode intensity on the scale of the cell period) in some patterns; it is thought to be associated with a spatial FFT that is performed to calculate the dipole field in the underlying program. Such modes must also be rejected.
Here again the spin direction rotates (in both senses) but now the amplitude is modulated in  .
A similar discussion applies to the solutions that are odd parity.
In Appendix B we discuss how a pure p mode pattern can be extracted from a superposition of +p and -p OOMMF patterns.

Separating nearly degenerate modes
As noted above, in the absence of the dipole interaction modes with +p and -p are degenerate. When this interaction is present such modes are split; i.e., according to the discussion of section 3.4 our eigen modes will be running waves in  . But the splitting rapidly decreases for larger mode numbers, and for running times less than 1 t −   , where  is the splitting, the power spectrum displays a single (slightly broadened) peak at the mode frequency.
In order to resolve the splitting in a power spectrum the OOMMF run times must be increased.
When the splitting is not resolved in the power spectrum the resulting mode patterns display a standing wave character. For a few of these we used the standing wave mode pattern as an initial configuration and ran the program long enough to display a beat pattern in time from which the splitting could be accurately determined. An example of this technique is shown in  A very small splitting also occurs between the symmetric (or gerade, denoted g) and antisymmetric (ungerade, denoted u) combinations of the cap modes. In the Schrodinger equation language of Sec. 2, this is a tunnel splitting between the surface bound states. This splitting is intrinsically small and hard to resolve in long cylinders, although we have resolved it for the r p 0, n 0 == , g and u cap modes for the cylinder with h = 300 nm. Now f 6.54 GHz = and f 14.4 MHz = . The corresponding beat pattern is shown in Fig. 3.2b.

Frequencies of low-lying modes
Most computations were carried out on a YIG sample with h = 4d =8a = 300 nm in a static field of 0 H 2000 Oe = .
To test the behavior at small and large z k some calculations were carried out for samples with h = 7.5, 37.5, 75, 150, 600 and 1200 nm. The material parameters used are typical for YIG, as given earlier in section 3. Table II lists  additional radial nodes can be present and their mode number is denoted as r n0  . Modes listed as p  are nearly degenerate in the sense discussed above, and the mode patterns display standing waves in  as described by Eq. (3.9a,b).

34
The symbols in Fig. 5.3 show the OOMMF simulations of p = 0 mode frequencies as a function of the dimensionless z k for r n 0, 1 and 2 = while the continuous curves show the predictions for the corresponding mode numbers of the H -K formula. Again, the agreement is excellent. Readers may notice that some modes are missing in these plots. This is because they were not excited with the protocols used, but we are confident they exist and that their mode patterns conform with the general framework presented here.

Why the H -K formula works so well
The H -K relation is sometimes referred to as a spin-wave dispersion relation, and indeed it is tantamount to saying that the normal modes of the body can be described by a continuous (or quasicontinuous) variable k. As explained in Appendix A, this assertion is grossly incorrect when the spatial variation of m is on a scale comparable to the dimensions of the body 56 ; the mode functions must then take account of the shape of the body, and be described by discrete sets of appropriate mode numbers, which may or may not be wavevector-like.
Nevertheless, if it should turn out that some discrete modes with slow spatial variation are well described by wavevector-like variables, then for those modes the H -K expression can be expected to give the frequency to good approximation, especially if one is in the exchange dominated limit. This is the situation in the present investigation. We have seen that for every bulk mode, we can identify a reasonably well-defined z k and an orbital angular momentum quantum number p. We also see a radial dependence in (r) m that pretty well matches a Bessel function p J (k r) ⊥ from which can obtain a k ⊥ .
That the mode functions should look this way is not an accident. We have some support for this functional behavior from the theory of the infinite cylinder in the exchange dominated limit.
We intend to publish the details of this theory separately, and here we only summarize the key results. In first order perturbation in the small parameter we find that the mode function is given by The corrections f  are both O( )  . We add here that we discovered the properties of the mode functions by experimenting with OOMMF first, and developed the theoretical framework later. There is a wealth of information in these figures as we now explain. They all depict various aspects of the frequency space Fourier amplitude m(r, ), which is a complex vector,

Examples of mode patterns
i.e., its x, y, and z components are all complex numbers. The real parts make a vector, and so do the imaginary parts, which we can call the real and imaginary parts of the complex vector, we have described the panels at z = 3 nm and z = 147 nm as lying at z = 0 and z = h/2, as this is more natural and intuitively easier to understand.
In the interest of clarity, we shall repeat these points as necessary, and add further information about the patterns as we discuss them one by one.

37
We start with the lowest lying modes: the nominal bulk uniform precession or cylindrical Kittel (0, 0, 0) mode with f = 7.813 GHz, which is concentrated within the body of the cylinder, away from the caps, together with the even (0, 0, g) and odd (0, 0, u) cap modes concentrated on the top and bottom cylinder faces with a mean frequency of 6.543 GHz and a splitting that is too small to be resolved.  . These are also the modes for which we resolved the splitting via the beat pattern in Fig. 3.2a.
As an example of a mode with a larger azimuthal mode number and mixed p character,

Evolution of the low-lying mode behavior with cylinder height
Although the majority of our simulations for mode patterns were carried out for a YIG cylinder with d = 2a = 75 nm and h = 300 nm, the behavior of the three lowest lying p = 0 modes, (000), (00g), and (00u), was studied over the much wider range of h extending from 7.5nm to 1200 nm 57 . Fig. 7.1 shows the frequencies emerging from the OOMMF simulations together with the predictions based on the Kittel expression, Eq. (1.2), according to the demagnetization coefficients found by Joseph and Schlomann 58 . Importantly we see that the (00g) "cap mode" evolves into the dominant mode in the thin-disc limit, while the (000) bulk mode becomes dominant (although lying slightly higher in frequency) in the long cylinder limit.
The Kittel formula, which is a single equation, actually describes two different modes in these limits and does not apply well to any mode for 150 nm h 300 nm. The level crossing of the (00g) and (00u) modes does not violate the Wigner-von Neumann anti-crossing theorem due to their even and odd character. Furthermore, the apparent violation of the ordering of energy levels of 1d Schrodinger equations is resolved by noting that the dipole-dipole interaction is a non-local perturbation which puts the eigenvalue problem outside the Sturm-Liouville class. The contours of constant amplitude run approximately parallel to the faces and we obtain a quite regular sine wave with one complete wavelength along z.

Conclusions and possible applications
We have explored the mode structure of nanoscopic cylinders of yttrium iron garnet, both analytically and through many-spin simulations, in a regime where the effects of exchange dominate the response. In addition to the extended (bulk) modes, which can be classified in terms of the azimuthal, radial, and axial mode numbers designated as rz p, n , n , we find symmetric and antisymmetric combinations of cap modes, that are localized at each of the cylinder faces. In all cases they lie lower in frequency than the accompanying family of modes z n for given azimuthal and radial mode numbers r p and n . When examining the height dependence, we find that the dominant FMR response cannot be precisely described by a simple Kittel-like formula. How this picture would change in passing to the dipole dominated limit is deserving of additional study.
By way of applications, there is a growing effort directed at using magnetic bits for computation as well as data storage. In particular it has been demonstrated that the required logic operations can be accomplished with lines and arrays of dipole-coupled, single-domain, bar magnets with dimensions of a few hundred nanometers 59 . While promising, this approach is still restricted to what can be done using binary macro-spin flips (and cascades thereof) of the individual magnets.
Looking farther ahead it is natural to ask if logic functions can be performed by exploiting the internal dynamics of nano particles. For the case of wave-guide based operations this is already a world-wide activity 60   We wish to recover the running wave patterns from a knowledge of the standing wave pattern.
To do this, we first transform the latter by rotating the amplitude by p, and the vector by direction by . The transformed field is, If these combinations are used as initial conditions in OOMMF, they should evolve into p + and p − running waves as indicated. In this way we can obtain positive confirmation that these running waves are indeed eigenmodes; this is how we separated the p + and p − modes shown in