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Hall effect
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Hall effect
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In diagram A, the flat conductor possesses a negative charge on the top (symbolized by the blue color) and a positive charge on the bottom (red color). In B and C, the direction of the electrical and the magnetic fields are changed respectively which switches the polarity of the charges around. In D, both fields change direction simultaneously which results in the same polarity as in diagram A.
  1. electrons
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  3. magnet
  4. magnetic field
  5. power source
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The Hall effect is the production of a potential difference, across an electrical conductor, that is transverse to an electric current in the conductor and to an applied magnetic field perpendicular to the current. Such potential difference is known as the Hall voltage. It was discovered by Edwin Hall in 1879.[1][2]

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

[edit]
See also: History of electromagnetic theory

Wires carrying current in a magnetic field experience a mechanical force perpendicular to both the current and magnetic field.

In the 1820s, André-Marie Ampère observed this underlying mechanism that led to the discovery of the Hall effect.[3] However it was not 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.[3] In 1879, he discovered this Hall effect while he was working on his doctoral degree at Johns Hopkins University in Baltimore, Maryland.[4] 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".[5][6][7]

Theory

[edit]

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.[8] 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.[9]

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 ξy in the direction of the assigned VH. VH is negative for some semiconductors where "holes" appear to flow. In steady-state, ξy will be strong enough to exactly cancel out the magnetic force, thus the electrons follow the straight arrow (dashed).
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 (in contrast to the case of the corbino disc). Only the electric field is modified by a transverse component .[10]

For a simple metal where there is only one type of charge carrier (electrons), the Hall voltage VH 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 vx term is the drift velocity of the current which is assumed at this point to be holes by convention. The vxBz term is negative in the y-axis direction by the right hand rule.

In steady state, F = 0, so 0 = Ey − vxBz, where Ey is assigned in the direction of the y-axis, (and not with the arrow of the induced electric field ξy 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 vx → −vx and q → −q. Also Ey = −⁠VH/w⁠. Substituting these changes gives

The conventional "hole" current is in the negative direction of the electron current and the negative of the electrical charge which gives Ix = ntw(−vx)(−e) where n is charge carrier density, tw is the cross-sectional area, and −e is the charge of each electron. Solving for and plugging into the above gives the Hall voltage:

If the charge build up had been positive (as it appears in some metals and semiconductors), then the VH assigned in the image would have been negative (positive charge would have built up on the left side).

The Hall coefficient is defined as or where j is the current density of the carrier electrons, and Ey is the induced electric field. In SI units, this becomes

(The units of RH 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).[11]

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.[12] 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

[edit]

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[13][14]

or equivalently with Here n is the electron concentration, p the hole concentration, μe the electron mobility, μh 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

[edit]

Although it is well known that magnetic fields play an important role in star formation, research models[15][16][17] indicate that Hall diffusion critically influences the dynamics of gravitational collapse that forms protostars.

Quantum Hall effect

[edit]
Main article: 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 quantized values.

Spin Hall effect

[edit]
Main article: 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:

Where is the spin current generated by the applied current density .[18]

Quantum spin Hall effect

[edit]
Main article: 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.[19]

Anomalous Hall effect

[edit]

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.[20] 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).[21]

Hall effect in ionized gases

[edit]

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, Ωe, and the electron-heavy particle collision frequency, ν: where

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:

Other Hall effects

[edit]

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[22][23] and exciton-polaritons[24] in 2D materials and quantum wells.

Applications

[edit]
Main article: Hall effect sensor

Hall sensors amplify and use the Hall effect for a variety of sensing applications. Hall-effect thrusters use the Hall effect to limit electrons' axial motion and use them to accelerate a propellant.

Corbino effect

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Corbino disc – dashed curves represent logarithmic spiral paths of deflected electrons.

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. The absence of the free transverse boundaries renders the interpretation of the Corbino effect simpler than that of the Hall effect.[25]

See also

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References

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Sources

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

Further reading

[edit]
[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Hall effect

View on Grokipedia
from Grokipedia
The Hall effect is the production of a voltage difference, known as the Hall voltage, across an that is transverse to both an applied flowing through the conductor and an external applied perpendicular to the current. This phenomenon arises from the exerted on the moving charge carriers by the , which deflects them toward one side of the conductor, resulting in a buildup of charge and an that opposes further deflection. Discovered in 1879 by American physicist Edwin Hall while studying metallic conduction, the effect provides a direct method to determine the type (electrons or holes), density, and mobility of charge carriers in materials, particularly semiconductors. Theoretically, within the of electrical conduction, the Hall voltage VHV_H is described by the formula VH=IBnedV_H = \frac{IB}{ned}, where II is the current, BB is the strength, nn is the , ee is the , and dd is the thickness of the conductor. The Hall coefficient RH=1nqR_H = \frac{1}{nq}, where qq is the charge of the carriers, is negative for electrons (n-type materials) and positive for holes (p-type materials), enabling distinction between conduction mechanisms. In experiments, a thin sample such as a strip is subjected to a perpendicular (typically up to several tesla), with current passed longitudinally and voltage measured transversely, yielding Hall voltages on the order of for typical setups. Beyond its classical form, the Hall effect has notable extensions, including the , observed in 1980 by Klaus von Klitzing in two-dimensional electron gases at low temperatures and high , where the Hall conductivity exhibits quantized plateaus at integer multiples of e2/he^2/h (with ee the charge and hh Planck's constant). This quantization, explained through and topological principles, has profound implications for fundamental physics and , serving as a basis for precise resistance standards with accuracy to parts per billion. The Hall effect underpins diverse technological applications, from magnetic field sensors in automotive systems (e.g., speedometers, anti-lock brakes) and (e.g., keyboards, joysticks) to Hall effect thrusters for , where crossed electric and s accelerate ions efficiently. In semiconductors, it facilitates characterization essential for device fabrication, while emerging variants like the spin Hall and orbital Hall effects explore spin-orbit interactions for low-power .

History

Discovery

The Hall effect was first observed by American physicist Edwin Herbert Hall on October 28, 1879, while conducting experiments as part of his doctoral research at under the supervision of Henry A. Rowland. Hall's setup involved a thin strip of mounted flat on a plate and secured with brass clamps to ensure stability. A steady was passed longitudinally along the strip using a battery and adjustable resistance, while a perpendicular was applied across the plane of the foil via an powered by a separate battery. To detect any potential difference, Hall connected a sensitive Thomson reflecting across the transverse edges of the strip, perpendicular to both the current and the . During the experiment, Hall noted a clear deflection in the , indicating the development of a voltage across the width of the gold strip, with the polarity reversing when the direction of the was inverted. This transverse was persistent and independent of the magnet's motion, ruling out inductive effects, and its magnitude increased with stronger currents or magnetic fields. The observation marked the initial of what would later be termed the Hall effect, demonstrating a novel interaction between electric currents and magnetic fields in conductors. Hall conducted multiple trials over several days in late to confirm the result, achieving consistent measurements after refining the apparatus to minimize mechanical vibrations and thermal influences. Hall promptly reported his findings in a paper titled "On a New Action of the Magnet on Electric Currents," published in the September 1879 issue of the American Journal of Mathematics. A more detailed account followed in 1880 in the , where he described the experimental procedure and results using , as well as preliminary tests with other metals like iron. Despite the precision of the setup, the discovery faced initial from some contemporaries, who questioned whether the observed voltage arose from artifacts such as uneven heating or contact potentials rather than a fundamental magnetic action, owing to the absence of a contemporary theoretical framework. This doubt persisted until later explanations, including the , provided clarity. The effect's recognition grew over time, establishing its role in probing properties in materials.

Early theoretical developments

In 1895, provided the first comprehensive theoretical explanation of the Hall effect within the framework of electron theory, attributing the transverse voltage to the deflection of negatively charged carriers by the , balanced against the electric arising from charge accumulation. This model introduced the concept of carrier deflection under the combined influence of electric and s, laying the groundwork for understanding the effect as a manifestation of motion in conductors. Lorentz's approach emphasized the role of free electrons as the primary charge carriers, aligning with emerging ideas about atomic structure. The discovery of the electron by J.J. Thomson in 1897 offered crucial confirmation of Lorentz's ideas, as subsequent measurements around 1900 revealed a negative Hall coefficient in metals such as and , directly indicating that conduction occurs via negatively charged rather than positive ions. These early experiments, building on Hall's original setup, quantified the 's sign and magnitude, providing empirical support for electron-based transport and resolving ambiguities from pre-electron theories that assumed positive carriers. A key historical challenge was the initial assumption of positive charge carriers, which predicted a positive Hall coefficient, but measurements in metals consistently showed negative values, while some materials like cuprous oxide exhibited positive coefficients suggestive of hole-like conduction. This discrepancy was resolved through identification of carrier sign via the Hall effect, with pivotal experiments by Wilson in on semiconductors such as demonstrating both signs and linking them to material composition and defect structures. By the early 20th century, around 1900–1920, the Hall effect was fully integrated into the Drude model of electrical conduction, which treated electrons as a classical gas undergoing collisions and drifts under applied fields, yielding the Hall coefficient as RH=−1neR_H = -\frac{1}{ne} for electron-dominated transport and enabling quantitative predictions of carrier density nn. This framework bridged experimental observations with kinetic theory, facilitating broader applications in material characterization despite its classical limitations.

Classical Hall Effect

Fundamental principles

The Hall effect arises in a conductor through which an flows when subjected to a perpendicular to the direction of the current. The moving charge carriers within the conductor experience a , which deflects them toward one side of the material, leading to a separation of charges across the transverse direction. This deflection was first observed by Edwin Hall in during experiments on thin metal foils. As charges accumulate on one side of the conductor, an , known as the Hall field, develops perpendicular to both the current and the . This field opposes further deflection of the carriers, and in the steady-state condition, it exactly balances the , resulting in no net transverse current flow. The potential difference established by this Hall field across the width of the conductor is termed the Hall voltage. A key feature of the Hall effect is the reversal of the deflection direction—and thus the polarity of the Hall voltage—depending on the sign of the charge carriers: negative for electrons and positive for holes. The magnitude of the deflection, and consequently the Hall voltage, increases with the strength of the and the density of the current passing through the conductor. The standard experimental geometry for observing the classical Hall effect involves a thin rectangular sample, such as a flat strip or bar, where the current flows along the length, the is applied perpendicular to the plane of the strip, and the Hall voltage is measured across the width. This setup allows for clear separation of the longitudinal current path from the transverse voltage .

Mathematical

The mathematical formulation of the classical Hall effect begins with the Lorentz force experienced by charged carriers in a conductor subjected to both an electric field and a magnetic field. For a charge carrier with charge qq, drifting with velocity v\mathbf{v}, the total force is given by F=q(E+v×B)\mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B}), where E\mathbf{E} is the electric field and B\mathbf{B} is the magnetic field. In the standard experimental setup, a rectangular sample carries a j\mathbf{j} along the xx-direction (jxj_x), with a B\mathbf{B} applied along the zz-direction (BzB_z). The carriers, assumed to have vxv_x along xx, experience a magnetic force qvxBzq v_x B_z in the yy-direction, leading to charge accumulation on the sample faces perpendicular to yy. This accumulation generates a Hall EyE_y (or EH\mathbf{E}_H) that opposes further deflection. In , the electric force qEyq E_y balances the magnetic force qvxBzq v_x B_z, yielding Ey=vxBzE_y = v_x B_z. To relate this to measurable quantities, express the drift velocity in terms of : jx=nqvxj_x = n q v_x, where nn is the carrier density. Substituting gives vx=jx/(nq)v_x = j_x / (n q), so the Hall field becomes EH=Ey=jxBznq.E_H = E_y = \frac{j_x B_z}{n q}. The Hall voltage VHV_H is then the potential difference across the sample width ww in the yy-direction: VH=EHwV_H = E_H w. In terms of total current I=jxwtI = j_x w t (with tt the thickness in the zz-direction), this simplifies to VH=IBznqt.V_H = \frac{I B_z}{n q t}. The sign of VHV_H depends on the carrier charge: negative for electrons (q=−eq = -e, where e>0e > 0 is the magnitude) and positive for holes (q=+eq = +e). The Hall coefficient RHR_H quantifies the material's response and is defined for this geometry as RH=VHtIBzR_H = \frac{V_H t}{I B_z}, yielding RH=1n[q](/page/Q)R_H = \frac{1}{n [q](/page/Q)} for a single carrier type. Thus, RHR_H is negative for conduction and positive for conduction, allowing determination of the dominant carrier type from measurements. In vector form, applicable to arbitrary geometries where fields are not necessarily aligned with sample axes, the steady-state condition requires E+v×B=0\mathbf{E} + \mathbf{v} \times \mathbf{B} = 0, so EH=−v×B\mathbf{E}_H = - \mathbf{v} \times \mathbf{B}. Substituting v=j/(nq)\mathbf{v} = \mathbf{j} / (n q) gives the general Hall field EH=j×Bnq=RH(j×B),\mathbf{E}_H = \frac{\mathbf{j} \times \mathbf{B}}{n q} = R_H (\mathbf{j} \times \mathbf{B}), with the Hall voltage obtained by integrating EH\mathbf{E}_H along the appropriate path transverse to j\mathbf{j} and B\mathbf{B}. This form assumes low magnetic fields where is negligible and holds for isotropic single-carrier systems.

Material dependence

In metals, the Hall effect arises predominantly from electron conduction, with the Hall coefficient approximated by RH≈−1neR_H \approx -\frac{1}{n e}, where nn is the and ee is the . The high carrier density in metals, typically 102810^{28} to 102910^{29} m−3^{-3}, results in a small RHR_H, producing a correspondingly small Hall voltage for typical currents and magnetic fields. For instance, in , RH≈−5.5×10−11R_H \approx -5.5 \times 10^{-11} m³/C at . In semiconductors, the presence of both electrons and holes leads to two-carrier conduction, complicating the Hall effect and requiring a two-carrier model for accurate description. The Hall coefficient in this regime is given by RH=pμh2−nμe2(pμh+nμe)2e,R_H = \frac{p \mu_h^2 - n \mu_e^2}{(p \mu_h + n \mu_e)^2 e}, where pp and nn are the hole and electron densities, respectively, and μh\mu_h and μe\mu_e are the corresponding mobilities. The value and sign of RHR_H depend strongly on the relative densities and mobilities of the carriers; high mobility amplifies the contribution of the dominant carrier type. In extrinsic semiconductors, where doping favors one carrier (electrons in n-type or holes in p-type), RHR_H approximates the single-carrier form, but in intrinsic semiconductors, balanced electron-hole pairs yield a smaller, often positive or negative RHR_H based on mobility differences. The Hall effect in semiconductors exhibits pronounced temperature dependence due to variations in carrier density and mobility. At low temperatures, carrier freeze-out occurs as thermally excited carriers bind to impurities, reducing nn or pp and thus increasing the magnitude of RHR_H. For example, in , significant freeze-out begins below approximately 100 , while in , it is observable below 50 . For more complex classical materials involving multiple conduction bands, such as certain alloys or semimetals, a two-band model extends the analysis to account for independent contributions from distinct or bands, modifying RHR_H through weighted sums of partial conductivities and . In superconductors, the Hall coefficient vanishes in the Meissner state owing to the complete expulsion of magnetic fields from the interior, eliminating the on carriers.

Advanced Hall Effects

Quantum Hall effect

The quantum Hall effect (QHE) emerges in two-dimensional electron systems subjected to strong perpendicular magnetic fields at cryogenic temperatures, where the Hall conductivity becomes precisely quantized, differing fundamentally from the classical Hall effect observed at higher temperatures. This quantization arises due to the formation of Landau levels, discrete energy states for electrons in the magnetic field, leading to plateaus in the Hall resistance and zero longitudinal resistance. The integer quantum Hall effect (IQHE) was discovered in 1980 by Klaus von Klitzing while studying metal-oxide-semiconductor field-effect transistors, revealing Hall resistance plateaus at values RH=hie2R_H = \frac{h}{i e^2}, where hh is Planck's constant, ee is the , and ii is an corresponding to the number of filled . This discovery earned von Klitzing the 1985 . In the IQHE, the Hall conductivity is σxy=ie2h\sigma_{xy} = i \frac{e^2}{h}, with the longitudinal resistivity ρxx\rho_{xx} vanishing at these plateaus due to the insulating nature of the bulk and dissipationless edge transport. The fractional quantum Hall effect (FQHE), observed in 1982 by Daniel Tsui, Horst Störmer, and Arthur Gossard in high-mobility GaAs heterostructures, extends this quantization to fractional values of the filling factor ν\nu, such as ν=1/3\nu = 1/3, where σxy=νe2h\sigma_{xy} = \nu \frac{e^2}{h}. Robert Laughlin provided the theoretical framework in 1983, proposing a variational wavefunction that describes the as an incompressible with excitations carrying fractional charge, notably e/3e/3 for the ν=1/3\nu = 1/3 state. Subsequent hierarchy models, building on Laughlin's work, explain higher-order fractional states through successive condensation of quasiparticles into new effective . Tsui, Störmer, and Laughlin shared the 1998 for these contributions. Experiments require high-mobility two-dimensional gases, typically in modulation-doped GaAs/AlGaAs heterostructures, with electron densities around 101110^{11} cm−2^{-2} and mobilities exceeding 10610^6 cm²/Vs to minimize disorder. Measurements are conducted at temperatures below 1 , often in dilution refrigerators reaching millikelvin, and greater than 5 T to resolve the Landau level splitting and observe the plateaus. Post-2020 developments have highlighted enhanced FQHE states in , where fractional plateaus appear at higher temperatures (up to 10 ) and lower fields due to the material's Dirac fermion band structure and reduced effective mass. These graphene-based systems show promise for standards, achieving quantized Hall resistance with uncertainties below 2 at moderate fields around 10 T, enabling more practical realizations of the in the SI system. The spin Hall effect (SHE) is a phenomenon in which a longitudinal charge current in a material generates a transverse pure spin current due to spin-orbit coupling, resulting in spin accumulation at the sample edges without requiring an external . This effect arises from two primary classes of mechanisms: intrinsic contributions from the band structure's curvature dipole, which are independent of impurities, and extrinsic mechanisms involving disorder, such as skew scattering—where spin-orbit interaction asymmetrically deflects scattering paths—and side-jump scattering, where electrons experience a transverse displacement during collisions. The SHE was theoretically predicted in 1971 by Mikhail Dyakonov and Vladimir Perel, who described spin accumulation near boundaries in semiconductors driven by current-induced spin relaxation. Experimental observation came in 2004, when optically detected spin polarization at the edges of GaAs/AlGaAs heterostructures confirmed the effect in p-doped semiconductors. The quantum spin Hall effect (QSHE) extends the SHE into a topological regime, featuring dissipationless edge states in two-dimensional topological insulators where spin and momentum are helically locked, meaning opposite spins propagate in opposite directions along the edges, protected by time-reversal symmetry. This state was theoretically proposed in by Charles Kane and Eugene Mele using a model for that incorporates spin-orbit coupling, predicting a bulk with robust, spin-polarized chiral edge modes. The first experimental realization occurred in in HgTe/CdTe quantum wells, where transport measurements revealed a quantized longitudinal conductance of 2e2/h2e^2/h (one channel per spin) and suppressed backscattering, confirming the topological phase for well thicknesses above a critical value of about 6.3 nm. Recent advances have explored QSHE in two-dimensional materials like stanene, a buckled tin , with post-2020 studies demonstrating large band gaps exceeding 0.3 eV under strain or substrate , positioning it as a candidate for room-temperature topological applications. The strength of the SHE is quantified by the spin Hall conductivity σsH\sigma_{sH}, which relates the spin current density JsJ_s to the electric field EE via Jsy=σsHExJ_s^y = \sigma_{sH} E^x, typically on the order of 10−210^{-2} to 10−1(e/ℏ)10^{-1} (e/\hbar) in heavy metals like Pt. The inverse spin Hall effect (ISHE) reciprocally converts a spin current into a transverse charge current, enabling electrical detection of spin accumulation and serving as a key tool for spintronic devices. Like the anomalous Hall effect, the SHE originates from spin-orbit interactions but emphasizes spin currents over charge deflection in non-magnetic materials.

Anomalous and topological variants

The anomalous Hall effect (AHE) is characterized by the emergence of a transverse voltage in ferromagnetic materials without an applied external , driven instead by the material's intrinsic . This effect produces a Hall resistivity ρ_xy = R_AH M_z, where M_z denotes the component of magnetization perpendicular to both the and the measurement direction, and R_AH is the anomalous Hall coefficient proportional to the magnetization. First observed in ferromagnetic samples during the 1880s shortly after the discovery of the classical Hall effect, the AHE puzzled researchers until theoretical explanations emerged in the 1950s. A comprehensive understanding was later solidified through semiclassical treatments incorporating quantum geometric effects. The AHE arises from a combination of intrinsic and extrinsic mechanisms, distinguished by their dependence on disorder and processes. Intrinsic mechanisms, independent of impurities in the clean limit, originate from the Berry curvature in the , which imparts an anomalous transverse velocity to charge carriers akin to a topological in momentum space. This Berry phase contribution, first proposed in the and reformulated topologically in the 1980s, scales with the square of the resistivity ρ_xx^2. Extrinsic mechanisms include skew , where spin-orbit coupling causes asymmetric deflection of electrons (linear in ρ_xx), and side-jump , involving transverse displacements during collisions (also ∝ ρ_xx^2). These contributions can be separated experimentally via scaling laws plotting the anomalous Hall conductivity σ_AH_xy against the longitudinal conductivity σ_xx: skew dominates in highly conductive regimes (σ_xx > 10^6 Ω^{-1} cm^{-1}) with σ_AH_xy ∝ σ_xx, while intrinsic and side-jump effects yield σ_AH_xy nearly independent of σ_xx in intermediate regimes (10^4–10^6 Ω^{-1} cm^{-1}). Topological variants of the AHE extend these concepts to materials with nontrivial band , where the integrates to a quantized value. In Chern insulators, the manifests as a fully quantized transverse conductivity σ_xy = C e^2 / h, with C as the integer Chern number representing the topological invariant of the occupied bands. This dissipationless state, theoretically predicted in the and first realized experimentally in in thin films of Cr-doped (Bi,Sb)_2Te_3 magnetic topological insulators, requires broken time-reversal via intrinsic . Beyond ferromagnets, large AHE has been observed in non-collinear antiferromagnets like Mn_3Sn, where noncoplanar spin textures generate substantial hotspots, yielding room-temperature Hall conductivities up to several hundred (Ω cm)^{-1} without net . Recent post-2020 studies in Weyl semimetals, such as Co_3Sn_2S_2, have linked AHE enhancements to the , where parallel electric and magnetic fields pump charge between Weyl nodes of opposite , enabling detection of these topological features.

Hall effects in plasmas

In low-density plasmas, such as those in columns, the Hall effect arises from the differing responses of ions and electrons to an applied . Electrons, with their lower mass, magnetize more readily than ions, leading to differential drift velocities perpendicular to both the and the . This separation of charges generates a Hall that drives a Hall current, observable as a transverse voltage. In the low-density limit, where collisions are infrequent, the effect is pronounced, and the Hall voltage can be derived from the two-fluid equations, yielding ΔV=−e−1/mR[1−1+mR−11+mRηR]IB0n0q0h\Delta V = -e^{-1/m_R} \left[1 - \frac{1 + m_R^{-1}}{1 + m_R \eta_R}\right] \frac{I B_0}{n_0 q_0 h}, where mRm_R is the ion-to-electron mass ratio and ηR\eta_R the viscosity ratio. The strength of the Hall effect in such plasmas is characterized by the Hall parameter β=ωcτ\beta = \omega_c \tau, the ratio of the cyclotron frequency ωc=∣q∣B/m\omega_c = |q| B / m to the inverse collision time τ=1/ν\tau = 1/\nu. High β\beta regimes (β≫1\beta \gg 1) indicate that particles complete many gyro-orbits between collisions, resulting in strong Hall fields that dominate transport perpendicular to the magnetic field. In the high-density limit, this approaches the classical Hall coefficient RH=1/(ne)R_H = 1/(n e). In fully ionized plasmas, the Hall effect plays a key role in (MHD) through the generalized , which includes the Hall term J×Bne\frac{\mathbf{J} \times \mathbf{B}}{n e} to account for the decoupling of and motions. This term, arising from the , becomes significant when the electron gyro-frequency exceeds the (ωge/νc≫1\omega_{ge} / \nu_c \gg 1), enabling phenomena like on scales comparable to the ion inertial length. The full law is E+u×B=ηJ+1neJ×B+\mathbf{E} + \mathbf{u} \times \mathbf{B} = \eta \mathbf{J} + \frac{1}{n e} \mathbf{J} \times \mathbf{B} + higher-order terms, where the Hall contribution modifies ideal MHD by introducing non-ideal effects. Astrophysically, Hall-dominated regimes influence in protoplanetary disks, where the Hall effect alters the (MRI). In weakly ionized disks, the Hall term can stabilize or destabilize MRI modes depending on field orientation, enhancing transport outward while allowing accretion inward, thus facilitating disk evolution and formation. Simulations show that Hall-MRI saturates into large-scale zonal and flows, reducing turbulent transport by factors of 100 or more compared to resistive cases when the Hall lengthscale exceeds 0.2 times the disk . Recent non-ideal MHD simulations post-2020 highlight these effects in protoplanetary contexts, such as transition disks. Global 2D axisymmetric models incorporating Hall diffusion alongside Ohmic resistivity and reveal ring-like structures and varying accretion rates (e.g., 7.91×10−8M⊙7.91 \times 10^{-8} M_\odot yr−1^{-1} for aligned fields versus 2.63×10−8M⊙2.63 \times 10^{-8} M_\odot yr−1^{-1} for anti-aligned), driven by Hall-induced flows in cavities. Such simulations underscore the Hall effect's role in reviving magnetically active layers in otherwise dead zones. In practical examples like Hall thrusters, the effect confines electrons azimuthally to sustain , though detailed applications are covered elsewhere.

Applications

Magnetic sensing devices

Hall effect sensors utilize the classical Hall voltage, which is proportional to the applied magnetic field strength BB, to detect and measure magnetic fields in various applications. These devices typically employ geometries such as the van der Pauw configuration or cross-shaped (Greek cross) structures to ensure uniform current distribution and accurate field measurement across the sensor plane. The , in particular, allows for precise determination of the Hall coefficient without requiring knowledge of the sample's exact dimensions, provided contacts are small relative to the sample size. Hall effect sensors are classified into two primary types: linear (analog) sensors, which produce a continuous output voltage directly proportional to the intensity, and switch (digital) sensors, which provide a binary on/off signal when the field exceeds a predefined threshold. Linear sensors are often integrated with on-chip amplifiers to enhance signal strength and reduce , enabling precise analog readout for applications requiring field magnitude and direction. Digital switches, incorporating via Schmitt triggers, ensure reliable operation in noisy environments by preventing false triggering near the threshold. Semiconductor materials like (InSb) are favored for their high , yielding superior sensitivity compared to silicon-based devices. Typical sensitivity for Hall sensors ranges around 100 mV/T, while InSb variants can achieve up to 5 mV/mT (equivalent to 5000 mV/T), making them ideal for detecting weaker fields. Key performance metrics include offset voltage, which represents the output at zero field due to asymmetries or thermal effects (typically minimized to microvolts through spinning current techniques), and temperature drift, often on the order of ±0.02%/°C for offset and sensitivity. of these sensors relies on measuring the Hall coefficient RHR_H, which relates carrier type, , and mobility to the observed voltage, allowing correction for material variations and environmental factors. The first commercial Hall effect sensors emerged in the early , with developing a germanium-based Hall generator patented in 1960 for industrial measurement. By the late , introduced integrated Hall switches using , initially for solid-state keyboards and proximity detection. Modern iterations feature integration, enabling compact, low-power designs suitable for automotive applications like crankshaft position sensing and such as compasses and current monitors. Despite their versatility, Hall effect sensors exhibit limitations in low-field resolution, typically effective down to millitesla ranges, whereas superconducting quantum interference devices (SQUIDs) achieve femtotesla sensitivity for ultra-weak fields in specialized . However, Hall sensors remain highly cost-effective, with production costs under a dollar per unit in high volumes, making them preferable for widespread commercial use over more expensive cryogenic SQUIDs.

Propulsion and plasma technologies

Hall-effect thrusters (HETs) represent a key application of the Hall effect in electric space propulsion, where the effect enables efficient to produce . In HET operation, a crossed electric and radial configuration causes electrons to exhibit azimuthal drift, forming a Hall current that sustains a quasi-neutral plasma while allowing axial acceleration toward the exhaust. This process ionizes and accelerates propellant atoms, typically , to achieve specific impulses of 1500–3000 seconds, far exceeding chemical rockets. The design of an HET features a annular discharge channel with an at the upstream end for gas injection and , surrounded by electromagnets generating a radial of 0.01–0.1 tesla. Electrons emitted from a hollow cathode at the downstream end are trapped by the , creating the Hall current, while unmagnetized ions are accelerated by the resulting axial . Overall thruster efficiency typically reaches 50–60%, with levels from millinewtons to newtons depending on power input, which ranges from hundreds of watts to kilowatts. As of 2025, ongoing developments include higher-efficiency HETs for deep-space missions, with laboratory efficiencies approaching 65%. Development of HETs originated in the during the 1960s, with early prototypes like the stationary plasma thruster (SPT) tested for applications. The first in-space demonstration occurred in 1971 aboard the meteorological , marking the beginning of operational use for maintenance. In modern contexts, HETs power numerous missions, including SpaceX's constellation, where upgraded argon-fueled variants deployed post-2020 provide enhanced thrust of about 170 millinewtons and around 2500 seconds for station-keeping and deorbiting. Beyond propulsion, Hall currents influence magnetohydrodynamic (MHD) stability in fusion devices, where the Hall effect in two-fluid plasma models modifies resistive instabilities like tearing and kink modes, potentially accelerating reconnection processes and affecting confinement. In plasma processing technologies, such as , the Hall effect governs transport in magnetized discharges, enabling higher plasma densities and more uniform fluxes for anisotropic material removal. Key challenges in HETs include wall erosion from high-energy sputtering in the discharge channel and plume divergence that can contaminate surfaces. Recent advances, such as magnetic shielding techniques introduced in the , redirect trajectories to minimize wall bombardment, extending thruster lifetime beyond 10,000 hours while maintaining performance. The plasma Hall parameter, the product of gyrofrequency and collision time, often exceeds 10 in these systems, ensuring effective confinement essential for operation.

Emerging uses in spintronics

In , the inverse spin Hall effect (ISHE) enables efficient spin-torque switching for magnetic (MRAM) devices by converting spin currents into charge currents that drive reversal in adjacent ferromagnetic layers. In Pt/ferromagnet bilayers, such as Pt/CoFeB, the strong spin-orbit coupling in Pt generates pure spin currents via the , which, through ISHE, produce damping-like torques sufficient for deterministic switching at low current densities, enhancing energy efficiency over traditional spin-transfer torque methods. Recent prototypes demonstrate switching times below 1 ns with critical currents reduced by up to 50% compared to single-layer systems, paving the way for scalable, beyond 2025. The quantum spin Hall effect (QSHE) in topological insulators supports edge-state conduction for dissipationless spin transport, enabling prototypes of low-power spintronic logic and interconnects. In thin-film Bi₂Se₃ devices, helical edge states carry spin-polarized currents without backscattering, achieving conductances approaching 2e²/h per edge, with advances in epitaxial growth enabling observations at temperatures up to a few and improving scalability to micron-scale structures. Between 2021 and 2025, interface engineering with capping layers has reduced bulk leakage, facilitating integration into hybrid quantum-classical circuits for fault-tolerant computing. Antiferromagnetic materials exhibiting anomalous Hall effect (AHE) offer high-sensitivity magnetometry without stray fields, as their zero net minimizes external interference while large curvatures yield Hall resistivities up to 1 μΩ·cm. In non-collinear antiferromagnets like Mn₃Sn, AHE signals detect fields below 0.1 mT with noise floors under 1 nT/√Hz, surpassing ferromagnetic sensors in stability for applications like biomedical imaging. Post-2020 developments include bilayer structures with compensated anisotropies, achieving thermal stability over 300 K and sensitivities exceeding 1000 Ω/T. Emerging integrations address key gaps by combining spin Hall effects with AI hardware, such as neuromorphic synapses where spin-orbit torques mimic for efficient . Room-temperature spin Hall torque devices using Ta-based structures, like Ta/CoFeB/MgO, achieve field-free switching with efficiencies over 2, enabling low-power accelerators with 10x reduced energy per operation compared to equivalents, as demonstrated in 2023 prototypes. These advances support scalable spintronic processors for edge AI, with synaptic weights tuned via current-induced at densities exceeding 10^{12} cm^{-2}.

Corbino effect

The Corbino effect refers to a galvanomagnetic observed in a thin, disk-shaped conductive sample featuring concentric electrodes at the center and periphery, through which a radial current flows under an applied axial perpendicular to the disk plane. In this configuration, charge carriers experience Lorentz deflection, but the annular with closed current paths prevents their accumulation at lateral boundaries, suppressing the transverse Hall voltage that characterizes the standard linear Hall effect. Instead, the deflection induces circulating eddy currents orthogonal to the primary radial flow, resulting in an increase in the overall electrical resistance known as . This effect was discovered in by Italian physicist Orso Mario Corbino, who described it in experiments on metal disks such as , emphasizing the role of the circular geometry in producing secondary circular currents alongside the radial primary current. Corbino's observations challenged the prevailing monistic theory by supporting a dualistic model involving both positive and negative charge carriers, and he utilized the setup to probe carrier concentrations and mobilities through combined and Hall measurements. The original reports appeared in preliminary form in the Rendiconti dell'Accademia dei Lincei and were detailed in s in Il Nuovo Cimento (1, 397, ) and Physikalische Zeitschrift (12, 561, ). Mechanistically, the closed-loop current paths in the Corbino disk eliminate the buildup of a transverse that would otherwise oppose the and restore straight-line carrier motion, as occurs in open geometries. This leads to a geometric suppression of the Hall effect, where carriers follow spiraling trajectories, enhancing path lengths and thus resistivity. In the low-magnetic-field limit, assuming a single carrier type and isotropic scattering, the relative resistivity change is approximated by Δρρ=(μB)2,\frac{\Delta \rho}{\rho} = (\mu B)^2, where μ\mu is the carrier mobility and BB is the magnetic field strength; this quadratic dependence arises directly from the deflection angle scaling with μB\mu B. The Corbino effect has been applied primarily to extract carrier mobility in semiconductors and metals without the contact alignment challenges inherent in direct Hall voltage measurements, as the radial current simplifies electrode placement and avoids short-circuiting the Hall field. Unlike the van der Pauw method, which employs a cloverleaf or uniform sample geometry to simultaneously determine resistivity and Hall coefficient from multiple voltage probes, the Corbino approach isolates longitudinal magnetoconductivity by inherently bypassing Hall voltage contributions, making it advantageous for high-mobility materials or scenarios where edge effects complicate interpretations. This distinction allows precise mobility evaluation in Corbino disks even under conditions where traditional Hall setups suffer from misalignment or boundary scattering artifacts.

Magnetoresistance connections

Ordinary magnetoresistance (OMR) refers to the change in electrical resistivity of a material under an applied due to the orbital motion of charge carriers, which alters their paths. This effect manifests in both longitudinal configurations, where the is parallel to the current direction, and transverse configurations, where the field is perpendicular to the current. Longitudinal OMR is typically small in metals and arises from intrinsic band structure modifications, while transverse OMR can be more pronounced in semiconductors with high carrier mobility. OMR can be classified into physical and geometric components. Physical OMR stems from fundamental changes in the material's resistivity tensor caused by the on carriers, independent of sample shape. In contrast, geometric OMR arises from finite sample dimensions, where boundary conditions prevent full development of the Hall electric field, leading to current path distortions and an apparent increase in resistance. In an ideal Hall setup with infinite width, the physical transverse OMR for a single isotropic carrier band follows the , yielding a longitudinal resistivity ρxx=ρ0\rho_{xx} = \rho_0, unchanged by the field, as the Hall field balances the to maintain current direction. However, if the Hall voltage is suppressed (e.g., via shorted contacts), ρxx=ρ0[1+(μB)2]\rho_{xx} = \rho_0 [1 + (\mu B)^2], where μ\mu is the carrier mobility and BB is the strength; Hall measurements enable separation of these contributions by independently determining μ\mu from the Hall . Extraordinary magnetoresistance (EMR) represents a dramatic enhancement of OMR in inhomogeneous semiconductors, where spatial variations in conductivity—such as metallic inclusions or doping fluctuations—cause extreme current path redirection under a . This geometric effect can yield non-saturating resistance changes exceeding 105%10^5\% at , far surpassing typical OMR. Experimental demonstrations in (InSb) structures, like van der Pauw disks with embedded metal contacts, have achieved magnetoresistance ratios up to 50,000% at fields of 1 T, attributed to the high mobility of InSb (μ>104\mu > 10^4 cm²/V·s). Historically, early investigations into in metallic wires, starting with William Thomson (Lord Kelvin) in 1857, often encountered confusion with the Hall effect due to transverse voltages misinterpreted as resistance changes in asymmetric geometries. Edwin Hall's discovery of the transverse Hall voltage was initially pursued while seeking longitudinal , highlighting the intertwined nature of these phenomena. In modern contexts, (GMR) observed in ferromagnetic multilayers since 1988 is distinct from EMR, as GMR relies on spin-dependent scattering rather than classical orbital or geometric effects. EMR's robustness to field orientation and position makes it suitable for field-independent magnetic sensors, where uniform response is critical without precise alignment. These devices, often based on InSb hybrids, offer high sensitivity at low fields (<0.1 T) for applications like non-destructive evaluation and . The Corbino exemplifies a pure configuration without observable Hall voltage, amplifying geometric effects.

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