the effect of geometrical frustration have long been studied in single domain ferromagnets known as Artificial Spin Ice typically around 200-800nm in size. but I can't find any studies that goes below this limit and study the effect of geometrical features in smaller particles known as superparamagnetic particles and how their phase transition may or may not differ from their ferromagnetic counterparts. would this reduction in size help with reaching lower energy states? does lowering the blocking temperature by using such particles help accessing the dynamic energy landscape as these particles wouldn't be frozen at room temperature?
It is not clear to me what you mean by 'geometrical frustration', however for the particle size and magnetic behaviour in nanomagnets can refer you to [1] (see in particular Sec. 3.2 herein).
[1] PA Guimarães, Principles of Nanomagnetism, 2nd edition (Springer, Berlin, 2017).
Dear Parniz,
Although spin glass is quite different than the superparamagnetism, it is true that there were some common to understand their behavour under frustration and disorder. Let me to add an interesting reference on this issue:
https://arxiv.org/pdf/cond-mat/0310684.pdf
Behnam Farid thank you for taking the time to answer my question.I've read the book back and forth a couple times.
the effect of geometrical frustration have long been studied in single domain ferromagnets and has been able to mimic the spin configurations of spin ice materials typically known as "artificial spin ice". but I can't find any studies that goes below this limit and study the effect of geometrical features in smaller particles known as superparamagnetic and how their phase transition may or may not differ from their ferromagnetic counterparts. so my question is basically is there any reason beside limitation of fabrication methods that has caused this or are there other reasons why no one has ever been interested in such studies?
Daniel Baldomir thank you for your answer.
spin glass is caused by both randomness and frustration when the system can't satisfy all of the competing interactions. my question is more like spin ice model in which the frustration is caused by certain geometrical features and I want to know why these studies are limited to single domain ferromagnets which is around 200-800nm in size (above the superparamagnetism regime) and how the phase transition of such smaller systems (below 10nm) could be different than their ferromagnetic counterparts.
Paniz Rahmani : Superparamagnets cease to be superparamagnets by increasing the size of 'nanomagnets' towards macroscopic size. Theoretically, magnets are never in thermal equilibrium [1], so that magnetic order parameter is only approximately constant in time, an approximation that is valid over astronomically long durations of time for macroscopic magnets. Therefore, the phenomenon of superparamagnetism is only of practical / observational relevance when the dynamics of the magnetisation is sufficiently fast; the inverse of the magnetisation fluctuation frequency should be smaller than the duration of typical observations. This limits the size of superparamagnets to nano scale, for which each superparamagnet essentially consists of a single magnetic domain.
[1] Thermal equilibrium requires the system to have sampled the entire phase space, the average over which leads to zero magnetisation.
Dear Paniz,
Let me to tell you that your question is not well enough put if you were interested in superparamagnetism in fustration within the spin ice model. Your question is
Why the geometrical frustration has never been studied in superparamagnetic regime?
If you want to know the behaviour of size particles which "are limited to single domain ferromagnets which is around 200-800nm in size". These particles are never superparamagnetic as you ask in your question and in fact they have many magnetic domains, around 15-60, which is a huge number if you were to have a superspin for such particles. Respect to the spin ice model you have papers made recently on this topic
https://www.semanticscholar.org/paper/Superferromagnetism-and-Domain-Wall-Topologies-in-Li-Paterson/e03bc9e4cd657e5bd223ae7671aade958c3d9ca8
Behnam Farid thank you Dr.Farid, you're right, superparamagnetic particles are not typically stable due to thermal fluctuations. however they show collective behaviors with slow dynamics (e.g spin glass) when they become at close proximity. so what could we be expecting to see, if we could eliminate the randomness in spin glass systems, and self assemble superparamagnets (e.g 10nm iron oxides) in a frustrated superlattice structure? my guess is they spin configuration should follow something more like spin ice model
Daniel Baldomir I'm sorry I might have not explained my question well. I'm trying to study the effect of geometrical frustration (something like spin ice model) in nanoparticles that are around 10nm in size and they lay within the superparamagnetic size range. however when I look for papers on this matter, I don't find any studies that have tried this effect on superparamagnetic particles. my initial thoughts were that it might have been due to limitations of fabrication methods. as with typical methods it's not quite possible to arrange such small particles in certain geometries and with controlled distances. So I would like to know if this is the only reason or they might have been more insights in not trying to make artificial spin ices within superparamagnetic sizes? and how could the magnetic phases be different if we decrease the size down to this range
Paniz Rahmani : You are welcome. If the nanomagnets are so closely packed as to form a macroscopic system, then they can no longer be superparamagnets (this was implicit in my previous comment on this page). In such case, magnetic frustration is always a possibility. For details, consult [1] (Sec. 6.3.4. herein) for frustration of the magnetic coupling of magnetic ions on non-bipartite lattices (specifically a triangular lattice in two dimensions). Further, Chapter 8 of [2] is wholly devoted to spin liquid states. This chapter begins by considering how introduction of a next-nearest neighbour interaction between the spins of an antiferromagnet on a square lattice (which is bipartite) gives rise to magnetic frustration and ultimately degenerate classical ground states.
Ps: You may wish also to consult [3].
[1] P Fazekas, Lecture Notes on Electron Correlation and Magnetism (World Scientific, Singapore, 2003).
[2] E Fradkin, Field Theories of Condensed Matter Systems, 2nd edition (Cambridge University Press, 2013).
[3] C Lacroix, Ph Mendels, and F Mila (Editors), Introduction to Frustrated Magnetism: Materials, Experiments, Theory (Springer, Berlin, 2011).
Dear Paniz:
In a paramagnetic compound at high temperature, the magnetic susceptibility tends to zero as the result of the disorder, at low temperature this susceptibility tends to zero as a result of the quenching of the magnetic moments, between these two limits we have necessary a maximum.
If we introduce the energy Ed as the energy to cross, for the magnetic moment, to become disordered we need to determine the probability of finding the energy of a particle in the vicinity of a given energy Ed see “The most probable distribution of the thermal energy”.
This statistical distribution allows the calculation of the paramagnetic susceptibility see “Paramagnetism: An alternative view”.
With this approach you can introduce - the effect of geometrical frustration- with the energy Ed.
On the other hand this approach have allows to better understand the “High Tc superconductors” and also “The black body and the Dulong and petit law”
Paniz Rahmani : Thank you and good luck.
Ps: The following paper may interest you:
J Schnack, Frustration Effects in Magnetic Molecules, J. Low Temp. Phys. 142, 279 (2006). https://doi.org/10.1007/s10909-006-9184-x
Something close to the original question was investigated in article: Broken vertex symmetry and finite zero-point entropy in the artificial square ice ground state, Phys. Rev. B 92, 060413(R) (2015).
I have also seen similar investigations on antidot square lattices, but I can't find them immediately. Nice continuation of Behnam Farid's comments is in: Effects of magnetic dipolar interactions on the specific time constant in superparamagnetic nanoparticle systems, J. Phys. D: Appl. Phys. 49 (2016) 295001 (8pp)
Other publications often deal with ferrofluids, sometimes frozen:
Demagnetization experiments on frozen ferrofluids, phys. stat. sol. (c) 1, No. 12, 3596-3602 (2004)
Peculiar magnetic aftereffect of highly diluted frozen magnetic fluids, Journal of Magnetism and Magnetic Materials 307 (2006) 77-84,
Interaction and size effects in magnetic nanoparticles, Journal of Magnetism and Magnetic Materials 296 (2006) 124-133
Effect of the shape and lateral dimensions on the magnetization reversal in permalloy nanofilms, Physica B 486 (2016) 77--80
One of the first papers in this area was:
DYNAMIC AND STATIC PROPERTIES OF INTERACTING FINE FERROMAGNETIC PARTICLES, Journal of Magnetism and Magnetic Materials 40 (1983) 1-11
Hope this help a bit,
MWG
Marek Wojciech Gutowski thank you so much these articles are very useful
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