Magnetic Normal Mode Calculations in Big Systems: A Highly Scalable Dynamical Matrix Approach Applied to a Fibonacci-Distorted Artificial Spin Ice

Round 1

Reviewer 1 Report

This is a very interesting paper that develops a technique to study the magnetic dyamics of large systems.  It performs approximations to the original equations-of-motion matrix to render it more numerically tractable.

 

The authors demonstrate this technique on a distorted honeycomb lattice.  The degree of disortion can be controlled with a parameter r.

 

I would be pleased if the authors can answer a few questions to clarify the ideas in the paper.

 

1. Why does the matrix A above Eq.(4) only contain phi and not theta? Does phi correspond to angles in the plane of the honeycomb and phi to angles from the perpendicular to the plane?
2. The honeycomb lattice in Fig.3 shows many spins in each side of the hexagon. How many spins are in each side?  What physically determines this number?  Does the separation between spins need to be smaller than any relevant wavelength?
3. I assume that the exchange A between spins is ferromagnetic. Is that right?
4. There is no anisotropy but a magnetic field that aligns the spins horizontally. What aligns the spins in the other sides of the hexagon, where they point at 60 degree angles from the x axis?
5. I assume that in Fig.4, green means no displacement out of the plane, red means positive displacement and blue means negative displacement. A color bar would be helpful.
6. What are the physical effects of the approximations used to simplify the dynamical matrices? What is left out?

 

I look forward to reading the authors’ answers to these questions.

Author Response

We thank the reviewer for his/her very valuable criticism and very plausible comments that allowed us to improve the manuscript. We also thank the Referee for finding the paper very interesting. We address below all the reviewer comments.  

1. We took advantage of the matrix A, introduced in order to cast the motion equations in the form of a generalized eigenvalue problem, to also include the phi terms present in the equations of motion; essentially, these terms appear in the motion equations due to the projection of the magnetization fluctuation of each cell in the direction of the magnetic field, which depends on phi only. We added in the manuscript the expression of the magnetization and its fluctuation, so that the origin of this term can be traced more easily (lines 124). We believe that this clarification, together with the reference to appendix A of Ref. 17 (where, however, the definition of theta and phi is interchanged), can satisfy the curiosity of the most interested readers. Theta refer to angles in the plane of the honeycomb and phi to angles from the perpendicular to the plane of the magnetization. We have revised the Appendix A in order to make this definition explicit (lines 372).  
2. As discussed in the Appendix, we make use of a micromagnetic approach, i.e., in order to calculate the magnetization and its fluctuations, the continuous magnetic sample is ideally parted into cells, small enough to consider the magnetization within them almost uniform and represented by a single variable, but not too small to generate an unmanageable number of independent variables. The spins thus obtained do not correspond to single atoms, but to micromagnetic cells which, in our case, contain about 10^5 atoms. This approach works well if the cell size does not exceed the typical variation length of the magnetization in the specific problem, as discussed at lines 250-256 of the revised manuscript. We have now explicitly stated the number of cells in the hexagon sides (4x30x1, lines 257); note also that the spins represented in Fig. 3 are subsampled with respect to the calculated mesh to increase visibility. 
3. Yes, the exchange constant of permalloy is ferromagnetic. We have revised the manuscript in several points to specify this fact.  
4. We have revised the manuscript in order to explicitly mention the shape anisotropy which is responsible, together with the external field, for the orientation of the ground state magnetization in the inclined segments (lines 305-315).
5. The Referee's assumptions are correct. We added to Fig. 4 the missing color scale and specified that the magnetization is expressed in arbitrary units. 
6. As discussed at lines 83-87 of the manuscript and beyond, the present approach differs from the previous one (DMM method, Ref. 17) only by the mathematical form in which the motion equation is written (taking advantage of some symmetry properties of the interactions) and the corresponding numerical technique used for its solution; it does not involve any additional approximation. The technique now introduced allow to deal with bigger systems, at the price of finding only some eigenmodes instead of the full spectrum. The abstract has been improved in order to immediately highlight the main nature of this approach (lines 2-4).

Reviewer 2 Report

This manuscricpt by Giovannini et al. introduces a method in calculating magnetic normal modes in magnetic nanostructures and nicely demonstrate it in a gradually distorted kagome lattice. The manuscript is very well written and results will be of interest for researchers in the field of artificial spin ice. I could not detect any flaws and the paper can already be published as is. 

Author Response

We thank the reviewer for finding our results interesting and the manuscript ready for publication.

Reviewer 3 Report

The authors demonstrated the modification of the dynamical matrix method by presenting the outcomes of exemplary calculations for Fibonacci-distorted artificial spin ice. This modification is based on usage of the iterative solver to find the eigenmodes of the lowest frequencies. The presented methods allow solving effectively the large, non-periodic magnonic systems with dipolar interactions taken into account in a non-collinear magnetization state.

The paper discusses briefly the impact of geometrical distortion of the Kagome lattice (of interconnected ferromagnetic bars) on frequencies and localizations of spin wave eigenmodes.

The paper should be accepted for publication in MDPI Magnetochemistry after minor revision considered by the authors:

1. The linearized equation of motion gives the eigenmodes in pairs, differing in the signs of the omega. Can this property be used to speed up the computations or save the memory demands?
   The authors need to select the values form the middle of the spectrum of eigenvalues, i.e. the eigenvalues closest to zero. How is done?
   It is recommended to briefly outline to Krylov-Schur approach.
   Can be the authors more specific about eigensolver and the preconditioning of matrices?
2. Is it possible to compare quantitatively the efficiency of the presented approach (e.g.time of calculations) with the standard one?
3. How do the authors test if the ground states for the distorted structure is properly selected (possible degeneration, or a lot of shallow minima)?
4. Please relate the legend in Fig.6 to the type of modes presented in Fig.4.
5. Why the modes in Fig.4 do not occupy all equivalent branches or vertices of the structure (taking in to account the symmetry related to the application of external field)?

Author Response

We thank the reviewer for his/her very valuable criticism and very plausible comments that allowed us to improve the manuscript. We also thank the Referee for finding the paper acceptable for publication after minor revision. We address below all the reviewer comments.  

1. The Referee is right, since the C matrix is Hermitian and H is Hermitian and positive definite (being the Hessian of the system calculated at equilibrium), the eigenvalues (mode frequencies) are real and occur in pairs of opposite sign. Some eigensolvers are able to exploit symmetry, that is, they compute a solution for Hermitian problems with less storage and/or computational cost than other methods that ignore this property. Also, symmetric solvers may be more accurate. In particular, the Krylov-Schur method is among them: in other words, this symmetry property is already exploited. We have revised the manuscript to highlight this point (lines 124). We have also included in the manuscript some basic considerations on iterative numerical methods and the Krylov-Schur approach in particular (lines 139-149 and 169-177), without going into too much depth to avoid falling into excessive technicality (further details can be found in the cited papers). When dealing with extremal eigenvalues we have selected the largest in value (eigenvalues are the inverse of the frequency, Eq. (2)).
2. The mentioned comparison can hardly be made, because the typical application of an iterative solver consists in finding a small fraction of the eigenvalues/eigenvectors of a problem so big that cannot be afforded by a direct solver. For example, a direct solver could not deal with the Fibonacci structure investigated in our manuscript (at the computing facility we have access to). It is possible to apply both methods to find all eigenvalues of a smaller test system of course, and iterative solvers prove to be much less efficient (by a factor of some tens, depending on the dimension of the problem and numerical characteristics of the specific matrices, such as singularities, degeneracy, etc.); the results obtained in such inappropriate comparisons would  be much dispersed and not significant. The only lesson to learn, here, is that an iterative solver should be tried when much less than 10% of the eigenvalues are required or the system is too big to be investigated by direct methods. In order to give a very rough idea of the computer power required by a direct method we mentioned in the revised version of the manuscript the largest eigenvalue analysis carried out with a direct method we are aware of (lines 260-266).
3. In order to investigate and exclude the presence of other local minimums of energy, the ground states has been calculated starting from different initial states (uniform magnetization, several random magnetization). In all cases we found a single, well defined fundamental state. The non-zero applied field contributes in a decisive way to this uniqueness. We have included a brief discussion of this point in the revised manuscript (lines 244-247).
4. The mode labels used in Fig. 6 (a,b,c,d) already coincide with those used in Fig. 4, but appeared in reversed order. In the revised version we reversed the order of the labels in the legend in Fig. 6 in order to improve the presentation and amended the caption.
5. Each mode of Fig. 4 actually appears repeatedly in the spectrum, i.e. as a family of modes localized on all equivalent places (branches or vertices according to its localization) of the structure. When r=1, apart from border effects, these equivalent modes have about the same frequency and are practically degenerate. Therefore, solutions corresponding to a mix of modes localized in different equivalent places are also possible. It is just a matter of points of view: when there are several eigenvectors corresponding to the same eigenvalue, any of their linear combination is a valid solution. We have chosen to evidence the localization in a single branch or vertex of each family member, but other authors may just as well represent them as modes distributed in the whole array (each family member has different phase relations between the magnetization at the various branches or vertices). A brief mention of this property was included in the manuscript (lines 290-295).