View source for Landau–Lifshitz–Gilbert equation

In physics, the '''Landau–Lifshitz–Gilbert equation''', named for [[Lev Landau]], [[Evgeny Lifshitz]], and [[T. L. Gilbert]], is a name used for a [[differential equation]] describing the [[precession|precessional motion]] of  [[magnetization]] {{math|'''M'''}} in a [[Solid-state physics|solid]]. It is a modification by Gilbert of the original equation of Landau and Lifshitz.

The various forms of the equation are commonly used in [[micromagnetics]] to model the effects of a [[magnetic field]] on [[ferromagnetic material]]s. In particular it can be used to model the time domain behavior of magnetic elements due to a magnetic field.<ref>{{cite web|last=Yang|first=Bo|title=Numerical Studies of Dynamical Micromagnetics|url=http://physics.ucsd.edu/~drf/pub/bo-thesis.ps.gz|accessdate=8 August 2011}}</ref> An additional term was added to the equation to describe the effect of spin polarized current on magnets.<ref>{{Cite web | url=http://wpage.unina.it/mdaquino/PhD_thesis/main/node47.html | title=2.6.1 Landau-Lifshitz-Gilbert equation with Slonczewski spin-transfer torque term}}</ref>

==Landau–Lifshitz equation==
[[File:Damped Magnetization Precession.jpg|thumb|upright|The terms of the Landau–Lifshitz–Gilbert equation: precession (red) and damping (blue). The trajectory of the magnetization (dotted spiral) is drawn under the simplifying assumption that the effective field '''H'''<sub>eff</sub> is constant.]]

In a [[ferromagnet]], the [[magnetization]] {{math|'''M'''}} can vary internally but at each point its magnitude is equal to the [[saturation magnetization]] {{math|''M''<sub>s</sub>}}. The Landau–Lifshitz–Gilbert equation predicts the rotation of the magnetization in response to torques. An earlier, but equivalent, equation (the Landau–Lifshitz equation) was introduced by {{harvtxt|Landau|Lifshitz|1935}}:<ref name=Aharoni96>{{harvnb|Aharoni|1996}}</ref><ref>{{harvnb|Brown, Jr.|1978}}</ref><ref>{{harvnb|Chikazumi|1997}}</ref>
{{NumBlk|:|<math>\frac{d\mathbf{M}}{d t}= -\gamma \mathbf{M} \times \mathbf{H_\mathrm{eff}} - \lambda \mathbf{M} \times \left(\mathbf{M} \times \mathbf{H_{\mathrm{eff}}}\right)</math>|{{EquationRef|1}}}}
where {{math|''γ''}} is the electron [[gyromagnetic ratio]]. and {{math|''λ''}} is a phenomenological damping parameter, often replaced by
:<math>\lambda = \alpha \frac{\gamma}{M_\mathrm{s}},</math>
where {{math|''α''}} is a dimensionless constant called the damping factor. The [[Micromagnetics#The effective field|effective field]] {{math|'''H'''<sub>eff</sub>}} is a combination of the external magnetic field, the ''[[demagnetizing field]]'' (magnetic field due to the magnetization), and some quantum mechanical effects. To solve this equation, additional equations for the demagnetizing field must be included.

Using the methods of [[Irreversible process|irreversible]] [[statistical mechanics]], numerous authors have independently obtained the Landau–Lifshitz equation.<ref>T. Iwata, J. Magn. Magn. Mater. 31–34, 1013 (1983); T. Iwata, J. Magn. Magn. Mater. 59, 215 (1986);  V.G. Baryakhtar, Zh. Eksp. Teor. Fiz. 87, 1501 (1984);  S. Barta (unpublished, 1999);  W. M. Saslow, J. Appl. Phys. 105, 07D315 (2009).</ref>

==Landau–Lifshitz–Gilbert equation==
In 1955  Gilbert replaced the damping term in the Landau–Lifshitz (LL) equation by one that depends on the time derivative of the magnetization:
{{NumBlk|:|<math>\frac{d \mathbf{M}}{d t}=-\gamma \left(\mathbf{M} \times \mathbf{H}_{\mathrm{eff}} - \eta \mathbf{M}\times\frac{d \mathbf{M}}{d t}\right)</math>|{{EquationRef|2b}}}}
This is the Landau–Lifshitz–Gilbert (LLG) equation, where {{math|''η''}} is the damping parameter, which is characteristic of the material. It can be transformed into the Landau–Lifshitz equation:<ref name=Aharoni96/>
{{NumBlk|:|<math>\frac{d \mathbf{M}}{d t} = -\gamma' \mathbf{M} \times \mathbf{H}_{\mathrm{eff}} - \lambda \mathbf{M} \times (\mathbf{M} \times \mathbf{H}_{\mathrm{eff}})</math>|{{EquationRef|2a}}}}
where
:<math>\gamma' = \frac{\gamma}{1 + \gamma^2\eta^2M_s^2} \qquad \text{and} \qquad\lambda = \frac{\gamma^2\eta}{1 + \gamma^2\eta^2M_s^2}. </math>

In this form of the LL equation, the precessional term {{math|''γ'''}} depends on the damping term. This better represents the behavior of real ferromagnets when the damping is large.<ref>For details of Kelly's non-resonant experiment, and of Gilbert's analysis (which led to Gilbert's modifying the damping term), see T. L. Gilbert and J. M. Kelly, "Anomalous rotational damping in ferromagnetic sheets", Conf. Magnetism and Magnetic Materials, Pittsburgh, PA, June 14–16, 1955 (New York: American Institute of Electrical Engineers, Oct. 1955, pp. 253–263).  http://people.physics.tamu.edu/saslow/MMMConf55_253GilbertKelly.pdf Text references to Figures 5 and 6 should have been to Tables 1 and 2. Gilbert could not fit Kelly's experiments with fixed usual gyromagnetic ratio {{math|''γ''}} and a frequency-dependent {{math|''λ''{{ = }}''αγ''}}, but could fit that data for a fixed Gilbert gyromagnetic ratio {{math|''γ''<sub>G</sub>{{ = }}''γ''/(1+''α''<sup>2</sup>)}} and a frequency-dependent {{math|''α''}}.  Values of {{math|''α''}} as large as 9 were required, indicating very broad absorption, and thus a relatively low-quality sample.  Modern samples, when analyzed from resonance absorption, give {{math|''α''}}'s on the order of 0.05 or less.</ref>

==Landau–Lifshitz–Gilbert–Slonczewski equation==
In 1996 [[John Slonczewski|Slonczewski]] expanded the model to account for the [[spin-transfer torque]], i.e. the torque induced upon the magnetization by [[Spin (physics)|spin]]-polarized current flowing through the ferromagnet.  This is commonly written in terms of the unit moment defined by {{math|'''m''' {{=}} '''M''' / ''M''<sub>S</sub>}}:
:<math>\dot{\mathbf{m}}=-\gamma \mathbf{m}\times \mathbf{H}_{\mathrm{eff}}+\alpha \mathbf{m}\times \dot{\mathbf{m}}+\tau _{\parallel}\frac{\mathbf{m}\times (\mathbf{x}\times \mathbf{m})}{\left|\mathbf{x}\times \mathbf{m}\right|}+\tau _{\perp}\frac{\mathbf{x}\times \mathbf{m}}{\left|\mathbf{x}\times \mathbf{m}\right|}</math>
where <math>\alpha</math> is the unitless damping parameter, <math>\tau_\perp</math> and <math>\tau_\parallel</math> are driving torques, and {{math|'''x'''}} is the unit vector along the polarization of the current.<ref>
{{Cite journal
| volume = 159
| issue = 1
| pages = –1–L7
| last = Slonczewski
| first = John C.
| title = Current-driven excitation of magnetic multilayers
| journal = Journal of Magnetism and Magnetic Materials
| date = 1996
| doi=10.1016/0304-8853(96)00062-5
|bibcode = 1996JMMM..159L...1S }}</ref><ref>{{cite journal|last1=Wolf|first1=S. A.|title=Spintronics: A Spin-Based Electronics Vision for the Future|journal=Science|date=16 November 2001|volume=294|issue=5546|pages=1488–1495|doi=10.1126/science.1065389|pmid=11711666|url=http://www.dtic.mil/get-tr-doc/pdf?AD=ADA516289|bibcode=2001Sci...294.1488W}}</ref>

==References and footnotes==
{{reflist}}

==Further reading==
{{Refbegin}}
*{{cite book |last        = Aharoni
 |first       = Amikam
 |author-link = Amikam Aharoni
 |title       = Introduction to the Theory of Ferromagnetism
 |publisher   = [[Clarendon Press]]
 |year        = 1996
 |isbn        = 978-0-19-851791-7
 |df          = 
 |ref         = harv
 |url-access  = registration
 |url         = https://archive.org/details/introductiontoth00ahar
}}
*{{cite book
  |last = Brown, Jr.
  |first = William Fuller
  |title = Micromagnetics
  |author-link = William Fuller Brown, Jr.
  |publisher = Robert E. Krieger Publishing Co.
  |year = 1978
  |origyear = Originally published in 1963
  |isbn = 978-0-88275-665-3
  |ref = harv
}}
*{{cite book
  |last = Chikazumi
  |first = Sōshin
  |title = Physics of Ferromagnetism
  |publisher = [[Clarendon Press]]
  |year = 1997
  |isbn = 978-0-19-851776-4
  |ref = harv
}}
*{{cite journal|first=T.L.|last= Gilbert|title=A Lagrangian formulation of the gyromagnetic equation of the magnetic field|journal= Physical Review|volume= 100 |issue= 4|year=1955|page= 1243|doi=10.1103/PhysRev.100.1235|bibcode = 1955PhRv..100.1235. }}   This is only an abstract; the full report is "Armor Research Foundation Project No. A059, Supplementary Report, May 1, 1956", but was never published. A description of the work is given in {{cite journal|first=T. L.|last= Gilbert|journal = IEEE Trans. Mag.
|volume=40|issue=6
|pages= 3443–3449
|year=2004
|doi=10.1109/TMAG.2004.836740|title=A phenomenological theory of damping in ferromagnetic materials|bibcode = 2004ITM....40.3443G }}
*{{cite journal|authorlink1=Lev Landau|authorlink2=Evgeny Lifshitz|first=L.D. |last=Landau|first2= E.M.|last2= Lifshitz|title= Theory of the dispersion of magnetic permeability in ferromagnetic bodies|journal= Phys. Z. Sowjetunion|volume= 8, 153 |year=1935|ref=harv}}
*{{cite journal|title=The Landau–Lifshitz equation revisited
|first=G V|last= Skrotskiĭ |year=1984 |journal=Sov. Phys. Usp.|volume= 27 |pages=977–979   |doi=10.1070/PU1984v027n12ABEH004101|issue=12|bibcode = 1984SvPhU..27..977S }}
*{{cite book|isbn= 978-981-277-875-8	
|title=Landau–Lifshitz Equations |series=Frontiers of Research With the Chinese Academy of Sciences
|first=Boling |last=Guo |first2= Shijin |last2=Ding|year=2008
|publisher=World Scientific Publishing Company}}
*{{cite journal|title=A Survey on the Numerics and Computations for the Landau–Lifshitz Equation of Micromagnetism
|first=Ivan |last= Cimrak |year=2007 |journal=Archives of Computational Methods in Engineering |volume= 15 |pages=1–37   |doi=10.1007/BF03024947 |issue=3 |url=http://www.kst.fri.uniza.sk/~icimrak/publications/surveyLL.pdf }}
*{{cite journal|title=The fascinating world of the Landau–Lifshitz–Gilbert equation: an overview
|first=Lakshmanan |last= M |year=2010 |journal=Phil. Trans. R. Soc. A |volume= 369 |pages=1280–1300  |doi=10.1098/rsta.2010.0319 |pmid=21320917 |issue= 1939|arxiv = 1101.1005 |bibcode = 2011RSPTA.369.1280L }}
{{Refend}}

== External links ==
* [http://www.bama.ua.edu/~tmewes/Java/dynamics/MagnetizationDynamics.shtml Magnetization dynamics applet]

{{DEFAULTSORT:Landau-Lifshitz-Gilbert equation}}
[[Category:Magnetic ordering]]
[[Category:Partial differential equations]]
[[Category:Equations of physics]]