Hall effect

The Hall effect is the production of a potential difference (the Hall voltage) across an electrical conductor that is transverse to an electric current in the conductor and to an applied magnetic field perpendicular to the current. It was discovered by Edwin Hall in 1879.<ref>Edwin Hall (1879). "On a New Action of the Magnet on Electric Currents". American Journal of Mathematics. 2 (3): 287–92. doi:10.2307/2369245. JSTOR 2369245. S2CID 107500183. Archived from the original on 2011-07-27. Retrieved 2008-02-28.</ref><ref>"Hall effect | Definition & Facts". Encyclopedia Britannica. Retrieved 2020-02-13.</ref>

The Hall coefficient is defined as the ratio of the induced electric field to the product of the current density and the applied magnetic field. It is a characteristic of the material from which the conductor is made, since its value depends on the type, number, and properties of the charge carriers that constitute the current.

Discovery

Wires carrying current in a magnetic field experience a mechanical force perpendicular to both the current and magnetic field. André-Marie Ampère in the 1820s observed this underlying mechanism that led to the discovery of the Hall effect.<ref name=":1">Ramsden, Edward (2011-04-01). Hall-Effect Sensors: Theory and Application. Elsevier. p. 2. ISBN 978-0-08-052374-3.</ref>

But it wasn't until a solid mathematical basis for electromagnetism was systematized by James Clerk Maxwell's "On Physical Lines of Force" (published in 1861–1862) that details of the interaction between magnets and electric current could be understood.

Edwin Hall then explored the question of whether magnetic fields interacted with the conductors or the electric current, and reasoned that if the force was specifically acting on the current, it should crowd current to one side of the wire, producing a small measurable voltage.<ref name=":1" /> In 1879, he discovered this Hall effect while he was working on his doctoral degree at Johns Hopkins University in Baltimore, Maryland.<ref name="bridgeman-momoir">Bridgeman, P. W. (1939). Biographical Memoir of Edwin Herbert Hall. National Academy of Sciences.</ref> Eighteen years before the electron was discovered, his measurements of the tiny effect produced in the apparatus he used were an experimental tour de force, published under the name "On a New Action of the Magnet on Electric Currents".<ref>Hall, E. H. (1879). "On a New Action of the Magnet on Electric Currents". American Journal of Mathematics. JSTOR. 2 (3): 287–292. doi:10.2307/2369245. ISSN 0002-9327. JSTOR 2369245.</ref><ref>"Hall Effect History". Archived from the original on 29 May 2015. Retrieved 2015-07-26.</ref><ref>Ramsden, Edward (2006). Hall-Effect Sensors. Elsevier Inc. pp. xi. ISBN 978-0-7506-7934-3.</ref>

Hall effect within voids

The term ordinary Hall effect can be used to distinguish the effect described in the introduction from a related effect which occurs across a void or hole in a semiconductor or metal plate when current is injected via contacts that lie on the boundary or edge of the void. The charge then flows outside the void, within the metal or semiconductor material. The effect becomes observable, in a perpendicular applied magnetic field, as a Hall voltage appearing on either side of a line connecting the current-contacts. It exhibits apparent sign reversal in comparison to the "ordinary" effect occurring in the simply connected specimen. It depends only on the current injected from within the void.<ref name=":0">Mani, R. G.; von Klitzing, K. (1994-03-07). "Hall effect under null current conditions". Applied Physics Letters. 64 (10): 1262–1264. Bibcode:1994ApPhL..64.1262M. doi:10.1063/1.110859. ISSN 0003-6951.</ref>

Hall effect superposition

Superposition of these two forms of the effect, the ordinary and void effects, can also be realized. First imagine the "ordinary" configuration, a simply connected (void-less) thin rectangular homogeneous element with current-contacts on the (external) boundary. This develops a Hall voltage, in a perpendicular magnetic field. Next, imagine placing a rectangular void within this ordinary configuration, with current-contacts, as mentioned above, on the interior boundary of the void. (For simplicity, imagine the contacts on the boundary of the void lined up with the ordinary-configuration contacts on the exterior boundary.) In such a combined configuration, the two Hall effects may be realized and observed simultaneously in the same doubly connected device: A Hall effect on the external boundary that is proportional to the current injected only via the outer boundary, and an apparently sign-reversed Hall effect on the interior boundary that is proportional to the current injected only via the interior boundary. The superposition of multiple Hall effects may be realized by placing multiple voids within the Hall element, with current and voltage contacts on the boundary of each void.<ref name=":0" /><ref>DE Patent 4308375 </ref>

Further "Hall effects" may have additional physical mechanisms but are built on these basics.

Theory

The Hall effect is due to the nature of the current in a conductor. Current consists of the movement of many small charge carriers, typically electrons, holes, ions (see Electromigration) or all three. When a magnetic field is present, these charges experience a force, called the Lorentz force.<ref>"The Hall Effect". NIST. Archived from the original on 2008-03-07. Retrieved 2008-02-28.</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>"Hall Effect Sensor". Electronic Tutorials.</ref>

In classical electromagnetism electrons move in the opposite direction of the current $I$ (by 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.

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 $ξ y$ in the direction of the assigned $V H$ . $V H$ is negative for some semiconductors where "holes" appear to flow. In steady-state, $ξ y$ will be strong enough to exactly cancel out the magnetic force, thus the electrons follow the straight arrow (dashed).

File:Hall Sensor.webm For a simple metal where there is only one type of charge carrier (electrons), the Hall voltage $V H$ can be derived by using the Lorentz force and seeing that, in the steady-state condition, charges are not moving in the $y$ -axis direction. Thus, the magnetic force on each electron in the $y$ -axis direction is cancelled by a $y$ -axis electrical force due to the buildup of charges. The $v x$ term is the drift velocity of the current which is assumed at this point to be holes by convention. The $v x B z$ term is negative in the $y$ -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, $F = 0$ , so $0 = E y - v x B z$ , where $E y$ is assigned in the direction of the $y$ -axis, (and not with the arrow of the induced electric field $ξ y$ as in the image (pointing in the $- y$ direction), which tells you where the field caused by the electrons is pointing).

In wires, electrons instead of holes are flowing, so $v x \to - v x$ and $q \to - q$ . Also $Ey = −.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}VH/w$ . 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 $I x = ntw (- v x)(- e)$ where $n$ is charge carrier density, $tw$ is the cross-sectional area, and $- e$ 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 $V H$ 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 $j$ is the current density of the carrier electrons, and $E y$ 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{B}{ne}.</math>

(The units of $R H$ are usually expressed as m³/C, or Ω·cm/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 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 semiconductors), it is contrarily more appropriate to think of the current as positive "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 semiconductors. 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 particles carrying the charge in p-type semiconductors, 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 configuration of electrodes. For example, a Hall effect consistent with positive carriers was observed in evidently n-type semiconductors.<ref>Ohgaki, Takeshi; Ohashi, Naoki; Sugimura, Shigeaki; Ryoken, Haruki; Sakaguchi, Isao; Adachi, Yutaka; Haneda, Hajime (2008). "Positive Hall coefficients obtained from contact misplacement on evident n-type ZnO films and crystals". Journal of Materials Research. 23 (9): 2293. Bibcode:2008JMatR..23.2293O. doi:10.1557/JMR.2008.0300. 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 holes, which may be present in different concentrations and have different mobilities. For moderate magnetic fields the Hall coefficient is<ref name='Kasap2001'>Kasap, Safa. "Hall Effect in Semiconductors" (PDF). Archived from the original (PDF) on 2008-08-21.</ref><ref>"Hall Effect". hyperphysics.phy-astr.gsu.edu. Retrieved 2020-02-13.</ref>

<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 $n$ is the electron concentration, $p$ the hole concentration, $μ e$ the electron mobility, $μ h$ the hole mobility and $e$ 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>Mark Wardle (2004). "Star Formation and the Hall Effect". Astrophysics and Space Science. 292 (1): 317–323. arXiv:astro-ph/0307086. Bibcode:2004Ap&SS.292..317W. CiteSeerX 10.1.1.746.8082. doi:10.1023/B:ASTR.0000045033.80068.1f. S2CID 119027877.</ref><ref>Braiding, C. R.; Wardle, M. (2012). "The Hall effect in star formation". Monthly Notices of the Royal Astronomical Society. 422 (1): 261. arXiv:1109.1370. Bibcode:2012MNRAS.422..261B. doi:10.1111/j.1365-2966.2012.20601.x. S2CID 119280669.</ref><ref>Braiding, C. R.; Wardle, M. (2012). "The Hall effect in accretion flows". Monthly Notices of the Royal Astronomical Society. 427 (4): 3188. arXiv:1208.5887. Bibcode:2012MNRAS.427.3188B. doi:10.1111/j.1365-2966.2012.22001.x. S2CID 118410321.</ref> indicate that Hall diffusion critically influences the dynamics of gravitational collapse that forms protostars.

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 conductance $σ$ undergoes quantum Hall transitions to take on the quantized values.

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>Deng, Yongcheng; Yang, Meiyin; Ji, Yang; Wang, Kaiyou (2020-02-15). "Estimating spin Hall angle in heavy metal/ferromagnet heterostructures". Journal of Magnetism and Magnetic Materials. 496: 165920. Bibcode:2020JMMM..49665920D. doi:10.1016/j.jmmm.2019.165920. ISSN 0304-8853. S2CID 209989182.</ref>

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>König, Markus; Wiedmann, Steffen; Brüne, Christoph; Roth, Andreas; Buhmann, Hartmut; Molenkamp, Laurens W.; Qi, Xiao-Liang; Zhang, Shou-Cheng (2007-11-02). "Quantum Spin Hall Insulator State in HgTe Quantum Wells". Science. 318 (5851): 766–770. arXiv:0710.0582. Bibcode:2007Sci...318..766K. doi:10.1126/science.1148047. ISSN 0036-8075. PMID 17885096. S2CID 8836690.</ref>

Anomalous Hall effect

In ferromagnetic materials (and 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>Robert Karplus and J. M. Luttinger (1954). "Hall Effect in Ferromagnetics". Phys. Rev. 95 (5): 1154–1160. Bibcode:1954PhRv...95.1154K. doi:10.1103/PhysRev.95.1154.</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-dependent scattering of the charge carriers, or an intrinsic effect which can be described in terms of the Berry phase effect in the crystal momentum space ( $k$ -space).<ref name="sinitsyn-08jpa">N. A. Sinitsyn (2008). "Semiclassical Theories of the Anomalous Hall Effect". Journal of Physics: Condensed Matter. 20 (2): 023201. arXiv:0712.0183. Bibcode:2008JPCM...20b3201S. doi:10.1088/0953-8984/20/02/023201. S2CID 1257769.</ref>

Hall effect in ionized gases

The Hall effect in an ionized gas (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, $β$ , in a plasma is the ratio between the electron gyrofrequency, $Ω e$ , and the electron-heavy particle collision frequency, $ν$ : <math display="block">\beta=\frac {\Omega_\mathrm{e}}{\nu}=\frac {eB}{m_\mathrm{e}\nu}</math> where

$e$ is the elementary charge (approximately 1.6×10⁻¹⁹ C)
$B$ is the magnetic field (in teslas)
$m e$ is the electron mass (approximately 9.1×10⁻³¹ 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 or ion) is almost linear. But if the Hall parameter is high, the electron movements are highly curved. The current density vector, $J$ , is no longer collinear with the electric field vector, $E$ . The two vectors $J$ and $E$ make the Hall angle, $θ$ , 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>Onga, Masaru; Zhang, Yijin; Ideue, Toshiya; Iwasa, Yoshihiro (December 2017). "Exciton Hall effect in monolayer MoS2". Nature Materials. 16 (12): 1193–1197. doi:10.1038/nmat4996. ISSN 1476-4660.</ref><ref>Kozin, V. K.; Shabashov, V. A.; Kavokin, A. V.; Shelykh, I. A. (21 January 2021). "Anomalous Exciton Hall Effect". Physical Review Letters. 126 (3): 036801. arXiv:2006.08717. doi:10.1103/PhysRevLett.126.036801.</ref> and exciton-polaritons<ref>Kavokin, Alexey; Malpuech, Guillaume; Glazov, Mikhail (19 September 2005). "Optical Spin Hall Effect". Physical Review Letters. 95 (13): 136601. doi:10.1103/PhysRevLett.95.136601.</ref> n 2D materials and quantum wells.

Applications

Hall sensors amplify and use the Hall effect for a variety of sensing applications.

Corbino effect

The Corbino effect, named after its discoverer Orso Mario Corbino, is a phenomenon involving the Hall effect, but a disc-shaped metal sample is used in place of a rectangular one. Because of its shape the Corbino disc allows the observation of Hall effect–based magnetoresistance without the associated Hall voltage.

A radial current through a circular disc, subjected to a magnetic field perpendicular to the plane of the disc, produces a "circular" current through the disc.<ref name="Adams1915">Adams, E. P. (1915). The Hall and Corbino effects. Vol. 54. pp. 47–51. Bibcode:1916PhDT.........2C. ISBN 978-1-4223-7256-2. Retrieved 2009-01-24. {{cite book}}: |journal= ignored (help)</ref>

The absence of the free transverse boundaries renders the interpretation of the Corbino effect simpler than that of the Hall effect.

References

Sources

Introduction to Plasma Physics and Controlled Fusion, Volume 1, Plasma Physics, Second Edition, 1984, Francis F. Chen

External links

U.S. patent 1,778,796, P. H. Craig, System and apparatus employing the Hall effect
U.S. patent 3,596,114, J. T. Maupin, E. A. Vorthmann, Hall effect contactless switch with prebiased Schmitt trigger
US Patent 5646527, R. G. Mani & K. von Klitzing, "Hall-effect device with current and Hall-voltage connections"
Understanding and Applying the Hall Effect
Hall Effect Thrusters Alta Space
Hall effect calculators
Interactive Java tutorial on the Hall effect Archived 2020-07-09 at the Wayback Machine National High Magnetic Field Laboratory
Science World (wolfram.com) article.
"The Hall Effect". nist.gov.
Table with Hall coefficients of different elements at room temperature Archived 2014-12-21 at the Wayback Machine.
Simulation of the Hall effect as a Youtube video
Hall effect in electrolytes
Bowley, Roger (2010). "Hall Effect". Sixty Symbols. Brady Haran for the University of Nottingham.

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