# 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.
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.
In the 1820s, André-Marie Ampère observed this underlying mechanism that led to the discovery of the Hall effect. 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. In 1879, he discovered this Hall effect while he was working on his doctoral degree at Johns Hopkins University in Baltimore, Maryland. 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".

## 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. 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.
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.
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.
$\mathbf {F} =q{\bigl\mathbf {E} +\mathbf {v} \times \mathbf {B} {\bigl )}$
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$_x$ → −v$_x$ and q → −q. Also E$_y$ = −⁠V$_H$/w⁠. Substituting these changes gives
$V_{\mathrm {H} }=v_{x}B_{z}w$
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 $w$ and plugging into the above gives the Hall voltage:
$V_{\mathrm {H} }={\frac {I_{x}B_{z}}{nte}}$
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 $R_{\mathrm {H} }={\frac {E_{y}}{j_{x}B_{z}}}$ or $\mathbf {E} =-R_{\mathrm {H} }(\mathbf {J} _{c}\times \mathbf {B} )$ where j is the current density of the carrier electrons, and E$_y$ is the induced electric field. In SI units, this becomes $R_{\mathrm {H} }={\frac {E_{y}}{j_{x}B}}={\frac {V_{\mathrm {H} }t}{IB}}={\frac {1}{ne}}.$
(The units of R$_H$ are usually expressed as m$^3$/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).
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. 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
$R_{\mathrm {H} }={\frac {p\mu _{\mathrm {h} }^{2}-n\mu _{\mathrm {e} }^{2}}{e(p\mu _{\mathrm {h} }+n\mu _{\mathrm {e} })^{2}}}$ or equivalently $R_{\mathrm {H} }={\frac {p-nb^{2}}{e(p+nb)^{2}}}$ with $b={\frac {\mu _{\mathrm {e} }}{\mu _{\mathrm {h} }}}.$ 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 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 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:
$\theta _{SH}={\frac {2e}{\hbar }}{\frac {|j_{s}|}{|j_{e}|}}$
Where $j_{s}$ is the spin current generated by the applied current density $j_{e}$.

### 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.

### 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. 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).

### 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: $\beta ={\frac {\Omega _{\mathrm {e} }}{\nu }}={\frac {eB}{m_{\mathrm {e} }\nu }}$ 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: $\beta =\tan(\theta ).$

### 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 and exciton-polaritons in 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. The absence of the free transverse boundaries renders the interpretation of the Corbino effect simpler than that of the Hall effect.