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{{Short description|Magnetically-induced resistance change}}

'''Magnetoresistance''' is the tendency of a material (often [[Ferromagnetism|ferromagnetic]]) to change the value of its [[electrical resistance]] in an externally-applied [[magnetic field]]. There are a variety of effects that can be called magnetoresistance. Some occur in bulk non-magnetic metals and semiconductors, such as geometrical magnetoresistance, [[Shubnikov–de Haas effect|Shubnikov–de Haas oscillations]], or the common positive magnetoresistance in metals.<ref>{{cite book |first=A.B. |last=Pippard |title=Magnetoresistance in Metals |url=https://books.google.com/books?id=D5XHMARd2ocC |date=1989 |publisher=Cambridge University Press |isbn=978-0-521-32660-5}}</ref> Other effects occur in magnetic metals, such as negative magnetoresistance in ferromagnets<ref name=Coleman>{{Citation |first1=R.V. |last1=Coleman |first2=A. |last2=Isin |title=Magnetoresistance in Iron Single Crystals|journal=Journal of Applied Physics|volume=37 |issue=3 |date=1966 |pages=1028–9 |doi=10.1063/1.1708320|bibcode = 1966JAP....37.1028C }}</ref> or anisotropic magnetoresistance (AMR). Finally, in multicomponent or multilayer systems (e.g. magnetic tunnel junctions), [[giant magnetoresistance]] (GMR), [[tunnel magnetoresistance]] (TMR), [[colossal magnetoresistance]] (CMR), and [[extraordinary magnetoresistance]] (EMR) can be observed.

The first magnetoresistive effect was discovered in 1856 by [[William Thomson, 1st Baron Kelvin|William Thomson]], better known as Lord Kelvin, but he was unable to lower the electrical resistance of anything by more than 5%. Today, systems including [[semimetal]]s<ref>{{cite web |title=Unstoppable Magnetoresistance|url=http://phys.org/news/2014-10-unstoppable-magnetoresistance.html}}</ref> and concentric ring [[Extraordinary magnetoresistance|EMR]] structures are known. In these, a magnetic field can adjust the resistance by orders of magnitude. Since different mechanisms can alter the resistance, it is useful to separately consider situations where it depends on a magnetic field directly (e.g. geometric magnetoresistance and multiband magnetoresistance) and those where it does so indirectly through magnetization (e.g. AMR and [[Tunnel magnetoresistance|TMR]]).

==Discovery==
[[William Thomson, 1st Baron Kelvin|William Thomson (Lord Kelvin)]] first discovered ordinary magnetoresistance in 1856.<ref name=Kelvin>{{Citation |first=W. |last=Thomson |title=On the Electro-Dynamic Qualities of Metals:—Effects of Magnetization on the Electric Conductivity of Nickel and of Iron |journal=Proc. R. Soc. Lond. |volume=8 |date=18 June 1857 |pages=546–550 |doi=10.1098/rspl.1856.0144|doi-access=free }}</ref> He experimented with pieces of iron and discovered that the resistance increases when the current is in the same direction as the magnetic force and decreases when the current is at 90° to the magnetic force. He then did the same experiment with nickel and found that it was affected in the same way but the magnitude of the effect was greater. This effect is referred to as anisotropic magnetoresistance (AMR).

[[File:Animation about the magnetoresistance discovery Graphs.ogv|thumb|Animation about graphs related to the discovery of [[giant magnetoresistance]].]]
[[Image:Corbino disc.PNG|thumbnail|250px|Corbino disc. With the magnetic field turned off, a radial current flows in the conducting annulus due to the battery connected between the (infinite) conductivity rims. When a magnetic field along the axis is turned on (B points directly out of the screen), the [[Lorentz force]] drives a circular component of current, and the resistance between the inner and outer rims goes up. This increase in resistance due to the magnetic field is called ''magnetoresistance''.]]

In 2007, [[Albert Fert]] and [[Peter Grünberg]] were jointly awarded the Nobel Prize for the discovery of [[giant magnetoresistance]].<ref>{{Citation |title=The Nobel Prize in Physics 2007 |publisher=Nobel Media AB |date=9 Oct 2007 |url=https://www.nobelprize.org/nobel_prizes/physics/laureates/2007/ |access-date=25 Jun 2014}}</ref>

==Geometrical magnetoresistance==
An example of magnetoresistance due to direct action of magnetic field on electric current can be studied on a Corbino disc (see Figure).
It consists of a conducting annulus with perfectly conducting rims. Without a magnetic field, the battery drives a radial current between the rims. When a magnetic field perpendicular to the plane of the annulus is applied, (either into or out of the page) a circular component of current flows as well, due to [[Lorentz force]]. Initial interest in this problem began with Boltzmann in 1886, and independently was re-examined by Corbino in 1911.<ref name=Giuliani>{{cite journal |author=G Giuliani |title=A general law for electromagnetic induction |year=2008 |journal= EPL |volume=81 |doi=10.1209/0295-5075/81/60002 |pages=60002|bibcode = 2008EL.....8160002G |issue=6 |arxiv = 1502.00502 |s2cid=14917438 }}</ref>

In a simple model, supposing the response to the Lorentz force is the same as for an electric field, the carrier velocity {{math|'''v'''}} is given by:
<math display="block"> \mathbf{v} = \mu \left( \mathbf{E} + \mathbf{v \times B} \right),</math>
where {{mvar|μ}} is the [[carrier mobility]].  Solving for the velocity, we find:

<math display="block>\begin{align}
\mathbf{v} &= \frac{ \mu}{1+(\mu B)^2} \left( \mathbf{E} + \mu \mathbf{E \times B} + \mu^2 (\mathbf{B \cdot E}) \mathbf{B}  \right) \\
  &= \frac{ \mu}{1+(\mu B)^2} \left( \mathbf{E}_{\perp} + \mu \mathbf{E \times B}\right) + \mu\mathbf{E}_{\parallel}\,
\end{align}</math>

where the effective reduction in mobility due to the {{mvar|'''B'''}}-field (for motion perpendicular to this field) is apparent. Electric current (proportional to the radial component of velocity) will decrease with increasing magnetic field and hence the resistance of the device will increase. Critically, this magnetoresistive scenario depends sensitively on the device geometry and current lines and it does not rely on magnetic materials.

In a [[semiconductor]] with a single carrier type, the magnetoresistance is proportional to {{math|(1 + (''μB''){{sup|2}})}}, where {{mvar|μ}} is the [[electron mobility|semiconductor mobility]] (units m<sup>2</sup>·V<sup>&minus;1</sup>·s<sup>&minus;1</sup>, equivalently m<sup>2</sup>·Wb<sup>&minus;1</sup>, or T<sup>&nbsp;&minus;1</sup>) and {{mvar|B}} is the magnetic field (units [[tesla (unit)|tesla]]s). [[Indium antimonide]], an example of a high mobility semiconductor, could have an electron mobility above {{val|4|u=m2|upl=Wb}} at {{val|300|ul=K}}. So in a {{val|0.25|u=T}} field, for example the magnetoresistance increase would be 100%.

==Anisotropic magnetoresistance (AMR)==
[[File:AMR of Permalloy.png|thumbnail|250px|The resistance of a thin [[Permalloy]] film is shown here as a function of the angle of an applied external field.]]Thomson's experiments<ref name="Kelvin"/> are an example of AMR,<ref name="Ritzinger">{{Cite journal | last1 = Ritzinger | first1 = Philipp | last2 = Vyborny | first2 = Karel | arxiv = 2212.03700 | year = 2023 | title = Anisotropic magnetoresistance: Materials, models and applications| journal = Royal Society Open Science | volume = 10 | issue = 10 | doi = 10.1098/rsos.230564 | pmid = 37859834 | pmc = 10582618 | bibcode = 2023RSOS...1030564R }}</ref>  a property of a material in which a dependence of electrical resistance on the angle between the direction of electric current and direction of '''magnetization''' is observed. The effect arises in most cases from the simultaneous action of magnetization and [[spin–orbit interaction]] (exceptions related to non-collinear magnetic order notwithstanding)<ref>see {{harvp|Ritzinger|Vyborny|2023|loc= §4.2}}.</ref> and its detailed mechanism depends on the material. It can be for example due to a larger probability of s-d scattering of electrons in the direction of magnetization (which is controlled by the applied magnetic field). The net effect (in most materials) is that the electrical resistance has maximum value when the direction of current is parallel to the applied magnetic field.<ref name=McGuirePotter>{{Cite journal | last1 = McGuire | first1 = T. | last2 = Potter | first2 = R. | doi = 10.1109/TMAG.1975.1058782 | title = Anisotropic magnetoresistance in ferromagnetic 3d alloys| url = http://www.unife.it/scienze/lm.fisica/insegnamenti/proprieta-magnetiche-materia/materiale/magnetoresistenza_anisotropa.pdf| journal = IEEE Transactions on Magnetics | volume = 11 | issue = 4 | pages = 1018–38| year = 1975 |bibcode = 1975ITM....11.1018M }}</ref> AMR of new materials is being investigated and magnitudes up to 50% have been observed in some uranium (but otherwise quite conventional) ferromagnetic compounds.<ref name="Wiśniewski">{{cite journal | last1 = Wiśniewski | first1 = P. | year = 2007| title = Giant anisotropic magnetoresistance and magnetothermopower in cubic 3:4 uranium pnictides | journal = Applied Physics Letters| volume = 90| issue = 19| pages = 192106| doi = 10.1063/1.2737904 |bibcode = 2007ApPhL..90s2106W }}</ref>  Materials with extreme AMR have been identified<ref name=Yang>{{Cite journal | last1=Yang | first1=Huali |doi=10.1103/PhysRevB.104.214419 | title = Colossal angular magnetoresistance in the antiferromagnetic semiconductor EuTe<sub>2</sub>| journal = Phys. Rev. B | volume = 104 | page = 214419 | year = 2021| issue=21 | bibcode=2021PhRvB.104u4419Y | s2cid=245189642 }}</ref> driven by unconventional mechanisms such as a metal-insulator transition triggered by rotating the magnetic moments (while for some directions of magnetic moments, the system is semimetallic, for other directions a gap opens).

In polycrystalline ferromagnetic materials, the AMR can only depend on the angle {{math|1= ''φ'' = ''ψ'' − ''θ''}} between the magnetization and current direction and (as long as the resistivity of the material can be described by a [[tensor rank|rank-two tensor]]), it must follow<ref>{{Citation |first1=E. |last1=De Ranieri |first2=A. W. |last2=Rushforth |first3=K. |last3=Výborný |first4=U. |last4=Rana |first5=E. |last5=Ahmed |first6=R. P. |last6=Campion |first7=C. T. |last7=Foxon |first8=B. L. |last8=Gallagher |first9=A. C. |last9=Irvine |first10=J. |last10=Wunderlich |first11=T. |last11=Jungwirth |title=Lithographically and electrically controlled strain effects on anisotropic magnetoresistance in (Ga,Mn)As |journal=New J. Phys. |volume=10 |issue=6 |page=065003 |date=10 June 2008 |doi=10.1088/1367-2630/10/6/065003 |arxiv=0802.3344|bibcode = 2008NJPh...10f5003D |s2cid=119291699 }}</ref>
<math display="block">\rho(\varphi) = \rho_\perp + (\rho_\parallel - \rho_\perp) \cos^2 \varphi </math>
where {{mvar|ρ}} is the (longitudinal) [[resistivity]] of the film and {{math|''ρ''{{sub|∥,⟂}}}} are the resistivities for {{math|1= ''φ'' = 0°}} and {{math|1= ''φ'' = 90°}}, respectively. Associated with longitudinal resistivity, there is also transversal resistivity dubbed (somewhat confusingly{{efn|name=note1}}) the planar Hall effect. In monocrystals, resistivity {{mvar|ρ}} depends also on {{mvar|ψ}} and {{mvar|θ}} individually.

To compensate for the non-linear characteristics and inability to detect the polarity of a magnetic field, the following structure is used for sensors. It consists of stripes of aluminum or gold placed on a thin film of [[permalloy]] (a ferromagnetic material exhibiting the AMR effect) inclined at an angle of 45°. This structure forces the current not to flow along the “easy axes” of thin film, but at an angle of 45°. The dependence of resistance now has a permanent offset which is linear around the null point. Because of its appearance, this sensor type is called '[[barber pole]]'.

The AMR effect is used in a wide array of sensors for measurement of Earth's magnetic field (electronic [[compass]]), for electric current measuring (by measuring the magnetic field created around the conductor), for traffic detection and for linear position and angle sensing. The biggest AMR sensor manufacturers are [[Honeywell]], [[NXP Semiconductors]], [[STMicroelectronics]], and [http://www.sensitec.com Sensitec GmbH].

As theoretical aspects, I. A. Campbell, A. Fert, and O. Jaoul ({{dfn|CFJ}}) <ref name=CFJ>{{Cite journal | last1 = Campbell | first1 = I. A. | last2 = Fert | first2 = A. | last3 = Jaoul | first3 = O. | doi = 10.1088/0022-3719/3/1S/310 | title = The spontaneous resistivity anisotropy in Ni-based alloys | journal = J. Phys. C | volume = 3 | issue = 1S | pages = S95–S101 | year = 1970 |bibcode = 1970JPhC....3S..95C}}</ref> derived an expression of the AMR ratio for Ni-based alloys using the two-current model with s-s and s-d scattering processes, where 's' is a conduction electron, and 'd' is 3d states with the spin-orbit interaction. The AMR ratio is expressed as
<math display="block"> \frac{\Delta \rho}{\rho}= \frac{\rho_\parallel - \rho_\perp}{\rho_\perp}=\gamma (\alpha - 1), </math>
with <math> \gamma=(3/4)(A/H)^2 </math> and <math> \alpha=\rho_\downarrow/\rho_\uparrow </math>, where <math> A </math>, <math> H </math>, and <math> \rho_\sigma </math> are a spin-orbit coupling constant (so-called <math> \zeta </math>), an exchange field, and a resistivity for spin <math> \sigma </math>, respectively. In addition, recently, Satoshi Kokado et al.<ref name=Kokado1>{{Cite journal | last1 = Kokado | first1 = Satoshi | last2 = Tsunoda | first2 = Masakiyo | last3 = Harigaya | first3 = Kikuo | first4 = Akimasa | last4 = Sakuma | doi = 10.1143/JPSJ.81.024705 | title = Anisotropic Magnetoresistance Effects in Fe, Co, Ni, Fe4N, and Half-Metallic Ferromagnet: A Systematic Analysis | journal = J. Phys. Soc. Jpn. | volume = 81 | issue = 2 | pages = 024705–1–17 | year = 2012 |bibcode = 2012JPSJ...81b4705K| arxiv = 1111.4864 | s2cid = 100002412 }}</ref><ref name=Kokado2>{{Cite journal | last1 = Kokado | first1 = Satoshi | last2 = Tsunoda | first2 = Masakiyo | doi = 10.4028/www.scientific.net/AMR.750-752.978 | title = Anisotropic Magnetoresistance Effect: General Expression of AMR Ratio and Intuitive Explanation for Sign of AMR Ratio | journal = Advanced Materials Research | volume = 750-752 | pages = 978–982 | year = 2013 |bibcode = 2013arXiv1305.3517K| arxiv = 1305.3517 | s2cid = 35733115 }}</ref> have obtained the general expression of the AMR ratio for 3d transition-metal ferromagnets by extending the {{abbr|CFJ|Campbell, Fert, and Jaoul}} theory to a more general one. The general expression can also be applied to half-metals.

==See also==
{{wiktionary|magnetoresistance}}
{{commons category|Magnetoresistance|lcfirst=yes}}

* [[Giant magnetoresistance]]
* [[Tunnel magnetoresistance]]
* [[Colossal magnetoresistance]]
* [[Extraordinary magnetoresistance]]
* [[Magnetoresistive random-access memory]]
<!--* [[Geometrical magnetoresistance]]
* [[Ballistic magnetoresistance]]-->

==Footnotes==
{{notelist|refs=
{{efn|name=note1|The (ordinary) [[Hall effect]] changes sign upon magnetic field reversal and it is an orbital effect (unrelated to spin) due to the Lorentz force. Transversal AMR (planar Hall effect<ref>{{Citation |first1=H. X. |last1=Tang |first2=R. K. |last2=Kawakami |first3=D. D. |last3=Awschalom |first4=M. L. |last4=Roukes |title=Giant Planar Hall Effect in Epitaxial (Ga,Mn)As Devices |url=http://nano.caltech.edu/publications/papers/PRL07201.pdf |journal=Phys. Rev. Lett. |volume=90 |issue=10 |page=107201 |date=March 2003 |doi=10.1103/PhysRevLett.90.107201 |pmid=12689027 |bibcode=2003PhRvL..90j7201T|arxiv = cond-mat/0210118 |s2cid=1485882 }}</ref>) does not change sign and it is caused by [[Spin–orbit interaction|spin-orbit interaction]].}}
}}

==References==
<references/>

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