View source for Physics:Superparamagnetism

{{Condensed matter physics}}
'''Superparamagnetism''' is a form of [[Physics:Magnetism|magnetism]] which appears in small [[Physics:Ferromagnetic|ferromagnetic]] or ferrimagnetic nanoparticles. In sufficiently small nanoparticles, magnetization can randomly flip direction under the influence of temperature. The typical time between two flips is called the [[Physics:Néel relaxation theory|Néel relaxation time]]. In the absence of an external magnetic field, when the time used to measure the magnetization of the nanoparticles is much longer than the Néel relaxation time, their magnetization appears to be in average zero; they are said to be in the superparamagnetic state. In this state, an external magnetic field is able to magnetize the nanoparticles, similarly to a paramagnet. However, their [[Physics:Magnetic susceptibility|magnetic susceptibility]] is much larger than that of paramagnets.

== The Néel relaxation in the absence of magnetic field ==
{{Main|Physics:Néel relaxation theory}}
Normally, any ferromagnetic or ferrimagnetic material undergoes a transition to a paramagnetic state above its [[Physics:Curie temperature|Curie temperature]]. Superparamagnetism is different from this standard transition since it occurs below the Curie temperature of the material.

Superparamagnetism occurs in nanoparticles which are [[Physics:Single domain (magnetic)|single-domain]], i.e. composed of a single [[Physics:Magnetic domain|magnetic domain]]. This is possible when their diameter is below 3–50&nbsp;nm, depending on the materials. In this condition, it is considered that the magnetization of the nanoparticles is a single giant magnetic moment, sum of all the individual magnetic moments carried by the atoms of the nanoparticle. Those in the field of superparamagnetism call this "macro-spin approximation".

Because of the nanoparticle’s [[Magnetic anisotropy|magnetic anisotropy]], the magnetic moment has usually only two stable orientations antiparallel to each other, separated by an energy barrier. The stable orientations define the nanoparticle’s so called “easy axis”. At finite temperature, there is a finite probability for the magnetization to flip and reverse its direction. The mean time between two flips is called the Néel relaxation time <math>\tau_\text{N}</math> and is given by the following Néel–Arrhenius equation:<ref name=Neel49>{{cite journal|first=L. |last=Néel|title=Théorie du traînage magnétique des ferromagnétiques en grains fins avec applications aux terres cuites|journal=Ann. Géophys.|volume=5|pages=99–136|year=1949}} (in French; an English translation is available in {{cite book|title=Selected Works of Louis Néel|editor-first=N.|editor-last=Kurti|publisher=Gordon and Breach|year=1988|isbn=978-2-88124-300-4|pages=407–427}}).</ref>

: <math>\tau_\text{N} = \tau_0 \exp \left(\frac{K V}{k_\text{B} T}\right)</math>,

where:
* <math>\tau_\text{N}</math> is thus the average length of time that it takes for the nanoparticle’s magnetization to randomly flip as a result of [[Physics:Thermal fluctuations|thermal fluctuations]].
* <math>\tau_0</math> is a length of time, characteristic of the material, called the ''attempt time'' or ''attempt period'' (its reciprocal is called the ''attempt frequency''); its typical value is between 10<sup>−9</sup> and 10<sup>−10</sup> second.
* ''K'' is the nanoparticle’s magnetic anisotropy energy density and ''V'' its volume. ''KV'' is therefore the energy barrier associated with the magnetization moving from its initial easy axis direction, through a “hard plane”, to the other easy axis direction.
* ''k''<sub>B</sub> is the [[Physics:Boltzmann constant|Boltzmann constant]].
* ''T'' is the temperature.

This length of time can be anywhere from a few nanoseconds to years or much longer. In particular, it can be seen that the Néel relaxation time is an exponential function of the grain volume, which explains why the flipping probability becomes rapidly negligible for bulk materials or large nanoparticles.

=== Blocking temperature ===

Let us imagine that the magnetization of a single superparamagnetic nanoparticle is measured and let us define <math>\tau_\text{m}</math> as the measurement time. If <math>\tau_\text{m} \gg \tau_\text{N}</math>, the nanoparticle magnetization will flip several times during the measurement, then the measured magnetization will average to zero. If <math>\tau_\text{m} \ll \tau_\text{N}</math>, the magnetization will not flip during the measurement, so the measured magnetization will be what the instantaneous magnetization was at the beginning of the measurement. In the former case, the nanoparticle will appear to be in the superparamagnetic state whereas in the latter case it will appear to be “blocked” in its initial state.

The state of the nanoparticle (superparamagnetic or blocked) depends on the measurement time. A transition between superparamagnetism and blocked state occurs when <math>\tau_\text{m} = \tau_\text{N}</math>. In several experiments, the measurement time is kept constant but the temperature is varied, so the transition between superparamagnetism and blocked state is seen as a function of the temperature. The temperature for which <math>\tau_\text{m} = \tau_\text{N}</math> is called the ''blocking temperature'':

:<math>T_\text{B} = \frac{K V}{k_\text{B} \ln \left(\frac{\tau_\text{m}}{\tau_0}\right)}</math>

For typical laboratory measurements, the value of the logarithm in the previous equation is in the order of 20–25.

Equivalently, blocking temperature is the temperature below which a material shows slow relaxation of magnetization.<ref>{{Cite journal|last1=Cornia|first1=Andrea|last2=Barra|first2=Anne-Laure|last3=Bulicanu|first3=Vladimir|last4=Clérac|first4=Rodolphe|last5=Cortijo|first5=Miguel|last6=Hillard|first6=Elizabeth A.|last7=Galavotti|first7=Rita|last8=Lunghi|first8=Alessandro|last9=Nicolini|first9=Alessio|last10=Rouzières|first10=Mathieu|last11=Sorace|first11=Lorenzo|date=2020-02-03|title=The Origin of Magnetic Anisotropy and Single-Molecule Magnet Behavior in Chromium(II)-Based Extended Metal Atom Chains|url=https://doi.org/10.1021/acs.inorgchem.9b02994|journal=Inorganic Chemistry|volume=59|issue=3|pages=1763–1777|doi=10.1021/acs.inorgchem.9b02994|issn=0020-1669|pmc=7901656|pmid=31967457}}</ref> 

== Effect of a magnetic field ==

[[Image:Langevin function.png|300px|thumb|right|Langevin function (red line), compared with <math display="inline">\tanh\left(\frac{1}{3}x\right)</math> (blue line).]]

When an external magnetic field ''H'' is applied to an assembly of superparamagnetic nanoparticles, their magnetic moments tend to align along the applied field, leading to a net magnetization. The magnetization curve of the assembly, i.e. the magnetization as a function of the applied field, is a reversible S-shaped increasing function. This function is quite complicated but for some simple cases:
# If all the particles are identical (same energy barrier and same magnetic moment), their easy axes are all oriented parallel to the applied field and the temperature is low enough (''T''<sub>B</sub> &lt; ''T'' ≲ ''KV''/(10 ''k''<sub>B</sub>)), then the magnetization of the assembly is
#: <math>M(H) \approx n \mu \tanh\left(\frac{\mu_0 H \mu}{k_\text{B} T}\right)</math>.
# If all the particles are identical and the temperature is high enough (''T'' ≳ ''KV''/''k''<sub>B</sub>), then, irrespective of the orientations of the easy axes:
#: <math>M(H) \approx n \mu L\left(\frac{\mu_0 H \mu}{k_\text{B} T}\right)</math>

In the above equations:
* ''n'' is the density of nanoparticles in the sample
* <math display="inline">\mu_0</math> is the magnetic permeability of vacuum
* <math display="inline">\mu</math> is the magnetic moment of a nanoparticle
* <math display="inline">L(x) = \frac{1}{\tanh(x)} - \frac{1}{x}</math> is the Langevin function

The initial slope of the <math>M(H)</math> function is the magnetic susceptibility of the sample <math>\chi</math>:
: <math>\chi = \begin{cases}
  \displaystyle \frac{n \mu_0 \mu^2}{k_\text{B} T} & \text{for the 1st case} \\
  \displaystyle \frac{n \mu_0 \mu^2}{3k_\text{B} T} & \text{for the 2nd case}
\end{cases}</math>

The latter susceptibility is also valid for all temperatures <math>T > T_\text{B}</math> if the easy axes of the nanoparticles are randomly oriented.

It can be seen from these equations that large nanoparticles have a larger ''µ'' and so a larger susceptibility. This explains why superparamagnetic nanoparticles have a much larger susceptibility than standard paramagnets: they behave exactly as a paramagnet with a huge magnetic moment.

=== Time dependence of the magnetization ===

There is no time-dependence of the magnetization when the nanoparticles are either completely blocked (<math>T \ll T_\text{B}</math>) or completely superparamagnetic (<math>T \gg T_\text{B}</math>). There is, however, a narrow window around <math>T_\text{B}</math> where the measurement time and the relaxation time have comparable magnitude. In this case, a frequency-dependence of the susceptibility can be observed. For a randomly oriented sample, the complex susceptibility<ref>{{cite journal |title=Superparamagnetism and relaxation effects in granular Ni-SiO<sub>2</sub> and Ni-Al<sub>2</sub>O<sub>3</sub> films  |first1=J. I. |last1=Gittleman |first2=B. |last2=Abeles |first3=S. |last3=Bozowski |journal=[[Physics:Physical Review B|Physical Review B]] |volume=9 |issue = 9|pages=3891&ndash;3897 |year=1974 |doi=10.1103/PhysRevB.9.3891|bibcode = 1974PhRvB...9.3891G }}</ref> is:

: <math>\chi(\omega) = \frac{\chi_\text{sp} + i \omega \tau \chi_\text{b}}{1 + i\omega\tau}</math>

where
* <math display="inline">\frac{\omega}{2\pi}</math> is the frequency of the applied field
* <math display="inline">\chi_\text{sp} = \frac{n \mu_0 \mu^2}{3k_\text{B} T}</math> is the susceptibility in the superparamagnetic state
* <math display="inline">\chi_\text{b} = \frac{n \mu_0 \mu^2}{3KV}</math> is the susceptibility in the blocked state
* <math display="inline">\tau = \frac{\tau_\text{N}}{2}</math> is the relaxation time of the assembly

From this frequency-dependent susceptibility, the time-dependence of the magnetization for low-fields can be derived:

:<math>\tau \frac{\mathrm{d}M}{\mathrm{d}t} + M = \tau \chi_\text{b} \frac{\mathrm{d}H}{\mathrm{d}t} + \chi_\text{sp} H</math>

==Measurements==
A superparamagnetic system can be measured with [[Physics:Magnetic susceptibility|AC susceptibility]] measurements, where an applied magnetic field varies in time, and the magnetic response of the system is measured. A superparamagnetic system will show a characteristic frequency dependence: When the frequency is much higher than 1/τ<sub>N</sub>, there will be a different magnetic response than when the frequency is much lower than 1/τ<sub>N</sub>, since in the latter case, but not the former, the ferromagnetic clusters will have time to respond to the field by flipping their magnetization.<ref>{{cite web |url=http://www.qdusa.com/resources/pdf/1078-201.pdf |last1=Martien |first1=Dinesh |title=Introduction to: AC susceptibility |publisher=Quantum Design | access-date= 15 Apr 2017}}</ref> The precise dependence can be calculated from the Néel–Arrhenius equation, assuming that the neighboring clusters behave independently of one another (if clusters interact, their behavior becomes more complicated). It is also possible to perform magneto-optical AC susceptibility measurements with magneto-optically active superparamagnetic materials such as iron oxide nanoparticles in the visible wavelength range.<ref>{{cite journal |doi=10.1063/1.4801837|last1=Vandendriessche |first1=Stefaan|title=Magneto-optical harmonic susceptometry of superparamagnetic materials|journal=Applied Physics Letters|volume=102|number=16|pages=161903–5|bibcode = 2013ApPhL.102p1903V |display-authors=0|year=2013 |url=https://lirias.kuleuven.be/handle/123456789/402755 }}</ref>

==Effect on hard drives==
Superparamagnetism sets a limit on the storage density of [[Engineering:Hard disk drive|hard disk drive]]s due to the minimum size of particles that can be used. This limit on [[Areal density (computer storage)|areal-density]] is known as the '''superparamagnetic limit'''.

* Older hard disk technology uses longitudinal recording. It has an estimated limit of 100 to 200 Gbit/in<sup>2</sup>.<ref>{{cite conference |last1=Kryder |first1=M. H. |title=Magnetic recording beyond the superparamagnetic limit |work=Magnetics Conference, 2000. INTERMAG 2000 Digest of Technical Papers. 2000 IEEE International |year=2000 |page=575 |doi=10.1109/INTMAG.2000.872350|isbn=0-7803-5943-7 }}</ref>
* Current hard disk technology uses [[Perpendicular recording|perpendicular recording]]. {{As of|2020|7}} drives with densities of approximately 1 Tbit/in<sup>2</sup> are available commercially.<ref>{{cite web| url = https://www.computerhistory.org/storageengine/hdd-areal-density-reaches-1-terabit-sq-in/| title = Computer History Museum: HDD Areal Density reaches 1 terabitper square inch}}</ref> This is at the limit for conventional magnetic recording that was predicted in 1999.<ref>{{cite journal| url = https://ieeexplore.ieee.org/document/824422| title = R. Wood, "The feasibility of magnetic recording at 1 Terabit per square inch", IEEE Trans. Magn., Vol. 36, No. 1, pp. 36-42, Jan 2000| journal = IEEE Transactions on Magnetics| date = January 2000| volume = 36| issue = 1| pages = 36–42| doi = 10.1109/20.824422| last1 = Wood| first1 = R.}}</ref><ref>{{cite press release |title=Hitachi achieves nanotechnology milestone for quadrupling terabyte hard drive |publisher=[[Company:Hitachi|Hitachi]] |date=October 15, 2007 |url= http://www.hitachi.com/New/cnews/071015a.html |access-date=1 Sep 2011 }}</ref>
* Future hard disk technologies currently in development include: heat-assisted magnetic recording (HAMR) and microwave-assisted magnetic recording (MAMR), which use materials that are stable at much smaller sizes.<ref>{{cite journal| url = https://ieeexplore.ieee.org/document/5257351| title = Y. Shiroishi et al., "Future Options for HDD Storage", IEEE Trans. Magn., Vol. 45, No. 10, pp. 3816-22, Sep. 2009| journal = IEEE Transactions on Magnetics| date = October 2009| volume = 45| issue = 10| pages = 3816–3822| doi = 10.1109/TMAG.2009.2024879| s2cid = 24634675| last1 = Shiroishi| first1 = Y.| last2 = Fukuda| first2 = K.| last3 = Tagawa| first3 = I.| last4 = Iwasaki| first4 = H.| last5 = Takenoiri| first5 = S.| last6 = Tanaka| first6 = H.| last7 = Mutoh| first7 = H.| last8 = Yoshikawa| first8 = N.}}</ref> They require localized heating or microwave excitation before the magnetic orientation of a bit can be changed. Bit-patterned recording (BPR) avoids the use of fine-grained media and is another possibility.<ref>{{cite magazine |url=https://www.pcmag.com/article2/0,2817,2368023,00.asp |title=Will Toshiba's Bit-Patterned Drives Change the HDD Landscape? |last1=Murray |first1=Matthew |magazine=PC Magazine |date=2010-08-19 |access-date=21 Aug 2010}}</ref> In addition, magnetic recording technologies based on topological distortions of the magnetization, known as [[Physics:Skyrmion|skyrmion]]s, have been proposed.<ref>{{Cite journal|last1=Fert|first1=Albert|last2=Cros|first2=Vincent|last3=Sampaio|first3=João|date=2013-03-01|title=Skyrmions on the track|journal=Nature Nanotechnology|language=en|volume=8|issue=3|pages=152–156|doi=10.1038/nnano.2013.29|pmid=23459548|issn=1748-3387|bibcode=2013NatNa...8..152F}}</ref>

==Applications==

===General applications===
* [[Physics:Ferrofluid|Ferrofluid]]: tunable viscosity

===Biomedical applications===
* Imaging: contrast agents in [[Physics:Magnetic resonance imaging|magnetic resonance imaging]] (MRI)
* Magnetic separation: cell-, DNA-, protein- separation, RNA fishing
* Treatments: [[Chemistry:Targeted drug delivery|targeted drug delivery]], magnetic hyperthermia, [[Biology:Magnetofection|magnetofection]]

==See also==
* Iron oxide nanoparticles
* [[Physics:Single-molecule magnet|Single-molecule magnet]]

==References==
===Notes===
{{reflist}}

===Sources===
{{Refbegin}}
* {{cite journal |first1=L. |last1=Néel |url=http://hal.archives-ouvertes.fr/docs/00/23/42/17/PDF/ajp-jphysrad_1950_11_2_49_0.pdf |title=Théorie du traînage magnétique des ferromagnétiques en grains fins avec applications aux terres cuites |journal=Ann. Géophys. |volume=5 |pages=99&ndash;136 |year=1949 |language=fr}} An English translation is available in {{cite book|title=Selected Works of Louis Néel |editor1-first=N. |editor1-last=Kurti |publisher=Gordon and Breach |location=New York |year=1988 |isbn=978-2-88124-300-4 |pages=407&ndash;427}}
* {{cite journal |first1=D. |last1=Weller |first2=A. |last2=Moser |title=Thermal Effect Limits in Ultrahigh Density Magnetic Recording |journal= [[Physics:IEEE Transactions on Magnetics|IEEE Transactions on Magnetics]]  |volume=35 |issue=6 |pages=4423&ndash;4439 |year=1999 |doi=10.1109/20.809134|bibcode = 1999ITM....35.4423W }}
{{Refend}}

==External links==
* [http://www.mff.cuni.cz/veda/konference/wds/proc/pdf05/WDS05_090_f3_Vejpravova.pdf Superparamagnetism of Co-Ferrite Nanoparticles]
* [http://lmis1.epfl.ch/webdav/site/lmis1/shared/Files/Lectures/Nanotechnology%20for%20engineers/Archives/2004_05/Superparamagnetism.pdf Powerpoint presentation on Superparamagnetism in pdf]
{{magnetic states}}

[[Category:Magnetic ordering]]
[[Category:Statistical mechanics]]

{{Sourceattribution|Superparamagnetism}}