Editing Hall effect (section)

==Theory==
The Hall effect is due to the nature of the current in a conductor. Current consists of the movement of many small [[charge carrier]]s, typically [[electron]]s, [[Electron hole|holes]], [[ion]]s (see [[Electromigration]]) or all three. When a magnetic field is present, these charges experience a force, called the [[Lorentz force]].<ref>{{cite web|url=http://www.eeel.nist.gov/812/effe.htm|access-date=2008-02-28|title=The Hall Effect|publisher=[[NIST]]|archive-url=https://web.archive.org/web/20080307092429/http://www.eeel.nist.gov/812/effe.htm|archive-date=2008-03-07|url-status=dead}}</ref> When such a magnetic field is absent, the charges follow approximately straight paths between collisions with impurities, [[phonons]], etc. However, when a magnetic field with a perpendicular component is applied, their paths between collisions are curved; thus, moving charges accumulate on one face of the material. This leaves equal and opposite charges exposed on the other face, where there is a scarcity of mobile charges. The result is an asymmetric distribution of charge density across the Hall element, arising from a force that is perpendicular to both the straight path and the applied magnetic field. The separation of charge establishes an [[electric field]] that opposes the migration of further charge, so a steady [[electric potential]] is established for as long as the charge is flowing.<ref>{{Cite web|url=https://www.electronics-tutorials.ws/electromagnetism/hall-effect.html|title=Hall Effect Sensor|website=Electronic Tutorials|date=13 August 2013 }}</ref>

In [[classical electromagnetism]] electrons move in the opposite direction of the current {{math|''I''}} (by [[Electric current#Conventions|convention]] "current" describes a theoretical "hole flow"). In some metals and semiconductors it ''appears'' "holes" are actually flowing because the direction of the voltage is opposite to the derivation below.

[[File:Hall Effect Measurement Setup for Electrons.png|right|frame|Hall effect measurement setup for electrons. Initially, the electrons follow the curved arrow, due to the magnetic force. At some distance from the current-introducing contacts, electrons pile up on the left side and deplete from the right side, which creates an electric field {{math|''ξ<sub>y</sub>''}} in the direction of the assigned {{math|''V''<sub>H</sub>}}. {{math|''V''<sub>H</sub>}} is negative for some semiconductors where "holes" appear to flow. In steady-state, {{math|''ξ<sub>y</sub>''}} will be strong enough to exactly cancel out the magnetic force, thus the electrons follow the straight arrow (dashed).]]
[[File:Hall Sensor.webm|thumb|The animation shows the action of a magnetic field on a beam of electric charges in vacuum, or in other terms, exclusively the action of the [[Lorentz force]]. This animation is an illustration of a typical error performed in the framework of the interpretation of the Hall effect. Indeed, at stationary regime and inside a Hall-bar, the electric current is longitudinal whatever the magnetic field and there is no transverse current <math>{j_y = 0}</math>  (in contrast to the case of the corbino disc). Only the electric field is modified by a transverse component <math>{E_y}</math>.<ref>{{Cite journal|last1=Creff|first1=M.|last2=Faisant|first2=F.|last3=Rubì|first3=J. M.|last4=Wegrowe|first4=J.-E.|date=2020-08-07|title=Surface currents in Hall devices|url=https://aip.scitation.org/doi/10.1063/5.0013182|journal=Journal of Applied Physics|volume=128|issue=5|pages=054501|doi=10.1063/5.0013182|arxiv=1908.06282 |bibcode=2020JAP...128e4501C |hdl=2445/176859 |s2cid=201070551 |issn=0021-8979}}</ref>]]
For a simple metal where there is only one type of [[charge carrier]] (electrons), the Hall voltage {{math|''V''<sub>H</sub>}} can be derived by using the [[Lorentz force]] and seeing that, in the steady-state condition, charges are not moving in the {{math|''y''}}-axis direction. Thus, the magnetic force on each electron in the {{math|''y''}}-axis direction is cancelled by a {{math|''y''}}-axis electrical force due to the buildup of charges. The {{math|''v<sub>x</sub>''}} term is the [[drift velocity]] of the current which is assumed at this point to be holes by convention. The {{math|''v<sub>x</sub>B<sub>z</sub>''}} term is negative in the {{math|''y''}}-axis direction by the right hand rule.

<math display="block">\mathbf{F} = q\bigl(\mathbf{E} + \mathbf{v} \times \mathbf{B}\bigl)</math>

In steady state, {{math|1='''F''' = '''0'''}}, so {{math|1=0 = ''E<sub>y</sub>'' − ''v<sub>x</sub>B<sub>z</sub>''}}, where {{math|''E<sub>y</sub>''}} is assigned in the direction of the {{math|''y''}}-axis, (and not with the arrow of the induced electric field {{math|''ξ<sub>y</sub>''}} as in the image (pointing in the {{math|−''y''}} direction), which tells you where the field caused by the electrons is pointing).

In wires, electrons instead of holes are flowing, so {{math|''v<sub>x</sub>'' → −''v<sub>x</sub>''}} and {{math|''q'' → −''q''}}.  Also {{math|1=''E<sub>y</sub>'' = −{{sfrac|''V''<sub>H</sub>|''w''}}}}. Substituting these changes gives

<math display="block">V_\mathrm{H}= v_x B_z w</math>

The conventional "hole" current is in the negative direction of the electron current and the negative of the electrical charge which gives {{math|1=''I<sub>x</sub>'' = ''ntw''(−''v<sub>x</sub>'')(−''e'')}} where {{math|''n''}} is [[charge carrier density]], {{math|''tw''}} is the cross-sectional area, and {{math|−''e''}} is the charge of each electron.  Solving for <math>w</math> and plugging into the above gives the Hall voltage:

<math display="block">V_\mathrm{H} = \frac{I_x B_z}{n t e}</math>

If the charge build up had been positive (as it appears in some metals and semiconductors), then the {{math|''V''<sub>H</sub>}} assigned in the image would have been negative (positive charge would have built up on the left side).

The Hall coefficient is defined as
<math display="block">R_\mathrm{H} = \frac{E_y}{j_x B_z}</math>
or
<math display="block">\mathbf{E} = -R_\mathrm{H}(\mathbf{J}_c \times \mathbf{B})</math>
where {{mvar|j}} is the [[current density]] of the carrier electrons, and {{math|''E<sub>y</sub>''}} is the induced electric field. In SI units, this becomes
<math display="block">R_\mathrm{H} =\frac{E_y}{j_x B}= \frac{V_\mathrm{H} t}{IB}=\frac{1}{ne}.</math>

(The units of {{math|''R''<sub>H</sub>}} are usually expressed as m<sup>3</sup>/C, or Ω·cm/[[Gauss (unit)|G]], or other variants.) As a result, the Hall effect is very useful as a means to measure either the carrier density or the magnetic field.

One very important feature of the Hall effect is that it differentiates between positive charges moving in one direction and negative charges moving in the opposite. In the diagram above, the Hall effect with a negative [[charge carrier]] (the electron) is presented. But consider the same magnetic field and current are applied but the current is carried inside the Hall effect device by a positive particle. The particle would of course have to be moving in the opposite direction of the electron in order for the current to be the same—down in the diagram, not up like the electron is. And thus, mnemonically speaking, your thumb in the [[:File:Regla mano derecha Laplace.svg|Lorentz force law]], representing (conventional) current, would be pointing the ''same'' direction as before, because current is the same—an electron moving up is the same current as a positive charge moving down. And with the fingers (magnetic field) also being the same, interestingly ''the charge carrier gets deflected to the left in the diagram regardless of whether it is positive or negative.'' But if positive carriers are deflected to the left, they would build a relatively ''positive voltage'' on the left whereas if negative carriers (namely electrons) are, they build up a negative voltage on the left as shown in the diagram. Thus for the same current and magnetic field, the [[electric polarity]] of the Hall voltage is dependent on the internal nature of the conductor and is useful to elucidate its inner workings.

This property of the Hall effect offered the first real proof that electric currents in most metals are carried by moving electrons, not by protons. It also showed that in some substances (especially [[p-type semiconductor]]s), it is contrarily more appropriate to think of the current as positive "[[Electron hole|holes]]" moving rather than negative electrons. A common source of confusion with the Hall effect in such materials is that holes moving one way are really electrons moving the opposite way, so one expects the Hall voltage polarity to be the same as if electrons were the [[charge carriers]] as in most metals and [[n-type semiconductor]]s. Yet we observe the opposite polarity of Hall voltage, indicating positive charge carriers. However, of course there are no actual [[positrons]] or other positive [[elementary particle]]s carrying the charge in [[p-type semiconductor]]s, hence the name "holes". In the same way as the oversimplistic picture of light in glass as photons being absorbed and re-emitted to explain [[refraction]] breaks down upon closer scrutiny, this apparent contradiction too can only be resolved by the modern quantum mechanical theory of [[quasiparticles]] wherein the collective quantized motion of multiple particles can, in a real physical sense, be considered to be a particle in its own right (albeit not an elementary one).<ref>N.W. Ashcroft and N.D. Mermin "Solid State Physics" {{ISBN|978-0-03-083993-1}}</ref>

Unrelatedly, inhomogeneity in the conductive sample can result in a spurious sign of the Hall effect, even in ideal [[Van der Pauw method|van der Pauw]] configuration of electrodes. For example, a Hall effect consistent with positive carriers was observed in evidently n-type semiconductors.<ref>{{Cite journal | doi=10.1557/JMR.2008.0300|bibcode = 2008JMatR..23.2293O|title = Positive Hall coefficients obtained from contact misplacement on evident ''n''-type ZnO films and crystals| journal=Journal of Materials Research| volume=23| issue=9| pages=2293|last1 = Ohgaki|first1 = Takeshi| last2=Ohashi| first2=Naoki| last3=Sugimura| first3=Shigeaki| last4=Ryoken| first4=Haruki| last5=Sakaguchi| first5=Isao| last6=Adachi| first6=Yutaka| last7=Haneda| first7=Hajime| year=2008| s2cid=137944281 }}</ref> Another source of artefact, in uniform materials, occurs when the sample's aspect ratio is not long enough: the full Hall voltage only develops far away from the current-introducing contacts, since at the contacts the transverse voltage is shorted out to zero.

===Hall effect in semiconductors===
When a current-carrying [[semiconductor]] is kept in a magnetic field, the charge carriers of the semiconductor experience a force in a direction perpendicular to both the magnetic field and the current. At equilibrium, a voltage appears at the semiconductor edges.

The simple formula for the Hall coefficient given above is usually a good explanation when conduction is dominated by a single [[charge carrier]]. However, in semiconductors and many metals the theory is more complex, because in these materials conduction can involve significant, simultaneous contributions from both [[electrons]] and [[Electron hole|holes]], which may be present in different concentrations and have different [[Electron mobility|mobilities]]. For moderate magnetic fields the Hall coefficient is<ref name='Kasap2001'>{{cite web|url=http://mems.caltech.edu/courses/EE40%20Web%20Files/Supplements/02_Hall_Effect_Derivation.pdf |title=Hall Effect in Semiconductors |last=Kasap |first=Safa |archive-url=https://web.archive.org/web/20080821202757/http://mems.caltech.edu/courses/EE40%20Web%20Files/Supplements/02_Hall_Effect_Derivation.pdf |url-status=dead |archive-date=2008-08-21 }}</ref><ref>{{Cite web|url=http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/Hall.html|title=Hall Effect|website=hyperphysics.phy-astr.gsu.edu|access-date=2020-02-13}}</ref><!-- O.V. Emelyanenko, T.S. Lagunova, D.N. Nasledov and G.N. Talakin, Sov. Phys. Sol. Stat. '''7''' 1063 (1965).-->

<math display="block">R_\mathrm{H}=\frac{p\mu_\mathrm{h}^2 - n\mu_\mathrm{e}^2}{e(p\mu_\mathrm{h} + n\mu_\mathrm{e})^2}</math>
or equivalently
<math display="block">R_\mathrm{H}=\frac{p-nb^2}{e(p+nb)^2}</math>
with
<math display="block">b=\frac{\mu_\mathrm{e}}{\mu_\mathrm{h}}.</math>
Here {{math|''n''}} is the electron concentration, {{math|''p''}} the hole concentration, {{math|''μ''<sub>e</sub>}} the electron mobility, {{math|''μ''<sub>h</sub>}} the hole mobility and {{math|''e''}} the elementary charge.

For large applied fields the simpler expression analogous to that for a single carrier type holds.

===Relationship with star formation===
Although it is well known that magnetic fields play an important role in star formation, research models<ref>{{cite journal|title = Star Formation and the Hall Effect|author = Mark Wardle|journal = Astrophysics and Space Science|volume = 292|year = 2004|pages = 317–323|doi = 10.1023/B:ASTR.0000045033.80068.1f|issue = 1|arxiv = astro-ph/0307086 |bibcode = 2004Ap&SS.292..317W |citeseerx = 10.1.1.746.8082|s2cid = 119027877}}</ref><ref>{{Cite journal |arxiv = 1109.1370|bibcode = 2012MNRAS.422..261B|title = The Hall effect in star formation|journal = Monthly Notices of the Royal Astronomical Society|volume = 422|issue = 1|pages = 261|last1 = Braiding|first1 = C. R.|last2 = Wardle|first2 = M.|year = 2012|doi = 10.1111/j.1365-2966.2012.20601.x| doi-access=free |s2cid = 119280669}}</ref><ref>{{Cite journal |arxiv = 1208.5887|bibcode = 2012MNRAS.427.3188B|title = The Hall effect in accretion flows|journal = Monthly Notices of the Royal Astronomical Society|volume = 427|issue = 4|pages = 3188|last1 = Braiding|first1 = C. R.|last2 = Wardle |first2 = M.|year = 2012|doi = 10.1111/j.1365-2966.2012.22001.x| doi-access=free |s2cid = 118410321}}</ref> indicate that Hall diffusion critically influences the dynamics of gravitational collapse that forms protostars.

===Quantum Hall effect===
{{Main|Quantum Hall effect}}
For a two-dimensional electron system which can be produced in a [[MOSFET]], in the presence of large [[magnetic field]] strength and low [[temperature]], one can observe the quantum Hall effect, in which the Hall [[Electrical conductance|conductance]] {{mvar|σ}} undergoes [[quantum Hall transitions]] to take on quantized values.

===Spin Hall effect===
{{Main|Spin Hall effect}}
The spin Hall effect consists in the spin accumulation on the lateral boundaries of a current-carrying sample. No magnetic field is needed. It was predicted by [[Mikhail Dyakonov]] and [[V. I. Perel]] in 1971 and observed experimentally more than 30 years later, both in semiconductors and in metals, at cryogenic as well as at room temperatures.

The quantity describing the strength of the Spin Hall effect is known as Spin Hall angle, and it is defined as:

<math>\theta_{SH}=\frac{2e}{\hbar}\frac{|j_s|}{|j_e|}</math>

Where <math>j_s</math> is the spin current generated by the applied current density <math>j_e</math>.<ref>{{Cite journal |last1=Deng |first1=Yongcheng |last2=Yang |first2=Meiyin |last3=Ji |first3=Yang |last4=Wang |first4=Kaiyou |date=2020-02-15 |title=Estimating spin Hall angle in heavy metal/ferromagnet heterostructures |url=https://www.sciencedirect.com/science/article/pii/S0304885318337077 |journal=Journal of Magnetism and Magnetic Materials |language=en |volume=496 |pages=165920 |doi=10.1016/j.jmmm.2019.165920 |bibcode=2020JMMM..49665920D |s2cid=209989182 |issn=0304-8853|url-access=subscription }}</ref>

===Quantum spin Hall effect===
{{Main|Quantum spin Hall effect}}
For [[mercury telluride]] two dimensional quantum wells with strong spin-orbit coupling, in zero magnetic field, at low temperature, the quantum spin Hall effect has been observed in 2007.<ref>{{Cite journal|last1=König|first1=Markus| last2=Wiedmann | first2=Steffen|last3=Brüne|first3=Christoph| last4=Roth|first4=Andreas|last5=Buhmann|first5=Hartmut| last6=Molenkamp | first6=Laurens W.|last7=Qi|first7=Xiao-Liang| last8=Zhang|first8=Shou-Cheng| date=2007-11-02| title=Quantum Spin Hall Insulator State in HgTe Quantum Wells|url=https://www.science.org/doi/10.1126/science.1148047|journal=Science|language=en | volume=318 | issue=5851|pages=766–770|doi=10.1126/science.1148047|issn=0036-8075|pmid=17885096|arxiv=0710.0582|bibcode=2007Sci...318..766K |s2cid=8836690}}</ref>

===Anomalous Hall effect===
In [[ferromagnetism|ferromagnetic]] materials (and [[paramagnetism|paramagnetic]] materials in a [[magnetic field]]), the Hall resistivity includes an additional contribution, known as the '''anomalous Hall effect''' (or the '''extraordinary Hall effect'''), which depends directly on the [[magnetization]] of the material, and is often much larger than the ordinary Hall effect. (Note that this effect is ''not'' due to the contribution of the [[magnetization]] to the total [[magnetic field]].) For example, in nickel, the anomalous Hall coefficient is about 100 times larger than the ordinary Hall coefficient near the Curie temperature, but the two are similar at very low temperatures.<ref>{{cite journal| journal=Phys. Rev. |volume=95 |pages=1154–1160 |year=1954 |author=Robert Karplus and J. M. Luttinger |title=Hall Effect in Ferromagnetics |doi=10.1103/PhysRev.95.1154| issue=5|bibcode = 1954PhRv...95.1154K }}</ref> Although a well-recognized phenomenon, there is still debate about its origins in the various materials. The anomalous Hall effect can be either an ''extrinsic'' (disorder-related) effect due to [[Spin (physics)|spin]]-dependent [[scattering]] of the [[charge carrier]]s, or an ''intrinsic'' effect which can be described in terms of the [[geometric phase|Berry phase]] effect in the crystal momentum space ({{math|''k''}}-space).<ref name="sinitsyn-08jpa">{{cite journal|title=Semiclassical Theories of the Anomalous Hall Effect|author=N. A. Sinitsyn|journal=Journal of Physics: Condensed Matter|volume=20|year=2008|pages=023201|arxiv=0712.0183|doi=10.1088/0953-8984/20/02/023201|bibcode = 2008JPCM...20b3201S | issue=2 |s2cid=1257769}}</ref><!-- N.A. Sinitsyn 2008 J. Phys.: Condens. Mater. '''20''' 023201 -->

=== Hall effect in ionized gases ===
The Hall effect in an ionized gas ([[Plasma (physics)|plasma]]) is significantly different from the Hall effect in solids (where the '''Hall parameter''' is always much less than unity). In a plasma, the Hall parameter can take any value. The Hall parameter, {{math|''β''}}, in a plasma is the ratio between the electron [[Gyroradius|gyrofrequency]], {{math|''Ω''<sub>e</sub>}}, and the electron-heavy particle collision frequency, {{mvar|ν}}:
<math display="block">\beta=\frac {\Omega_\mathrm{e}}{\nu}=\frac {eB}{m_\mathrm{e}\nu}</math>
where
* {{math|''e''}} is the [[elementary charge]] (approximately {{val|1.6e-19|ul=C}})
* {{math|''B''}} is the magnetic field (in [[tesla (unit)|teslas]])
* {{math|''m''<sub>e</sub>}} is the [[electron|electron mass]] (approximately {{val|9.1e-31|u=kg}}).

The Hall parameter value increases with the magnetic field strength.

Physically, the trajectories of electrons are curved by the [[Lorentz force]]. Nevertheless, when the Hall parameter is low, their motion between two encounters with heavy particles ([[Neutral particle|neutral]] or [[ion]]) is almost linear. But if the Hall parameter is high, the electron movements are highly curved. The [[current density]] vector, {{math|'''J'''}}, is no longer collinear with the [[electric field]] vector, {{math|'''E'''}}. The two vectors {{math|'''J'''}} and {{math|'''E'''}} make the '''Hall angle''', {{mvar|θ}}, which also gives the Hall parameter:
<math display="block">\beta = \tan(\theta).</math>

=== Other Hall effects ===
The Hall Effects family has expanded to encompass other quasi-particles in semiconductor nanostructures. Specifically, a set of Hall Effects has emerged based on excitons<ref>{{cite journal |last1=Onga |first1=Masaru |last2=Zhang |first2=Yijin |last3=Ideue |first3=Toshiya |last4=Iwasa |first4=Yoshihiro |title=Exciton Hall effect in monolayer MoS2 |journal=Nature Materials |date=December 2017 |volume=16 |issue=12 |pages=1193–1197 |doi=10.1038/nmat4996 |pmid=28967914 |url=https://www.nature.com/articles/nmat4996 |language=en |issn=1476-4660|url-access=subscription }}</ref><ref>{{cite journal |last1=Kozin |first1=V. K. |last2=Shabashov |first2=V. A. |last3=Kavokin |first3=A. V. |last4=Shelykh |first4=I. A. |title=Anomalous Exciton Hall Effect |journal=Physical Review Letters |date=21 January 2021 |volume=126 |issue=3 |pages=036801 |doi=10.1103/PhysRevLett.126.036801 |pmid=33543953 |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.126.036801|arxiv=2006.08717 |bibcode=2021PhRvL.126c6801K }}</ref> and exciton-polaritons<ref>{{cite journal |last1=Kavokin |first1=Alexey |last2=Malpuech |first2=Guillaume |last3=Glazov |first3=Mikhail |title=Optical Spin Hall Effect |journal=Physical Review Letters |date=19 September 2005 |volume=95 |issue=13 |pages=136601 |doi=10.1103/PhysRevLett.95.136601 |pmid=16197159 |bibcode=2005PhRvL..95m6601K |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.95.136601|url-access=subscription }}</ref> in 2D materials and quantum wells.