Editing Magnetoresistance (section)

==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.