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 (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. . 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 {j_y = 0} (in contrast to the case of the corbino disc). Only the electric field is modified by a transverse component {E_y}. For a simple metal where there is only one type of
charge carrier (electrons), the Hall voltage can be derived by using the
Lorentz force and seeing that, in the steady-state condition, charges are not moving in the -axis direction. Thus, the magnetic force on each electron in the -axis direction is cancelled by a -axis electrical force due to the buildup of charges. The term is the
drift velocity of the current which is assumed at this point to be holes by convention. The term is negative in the -axis direction by the right hand rule. \mathbf{F} = q\bigl(\mathbf{E} + \mathbf{v} \times \mathbf{B}\bigl) In steady state, , so , where is assigned in the direction of the -axis, (and not with the arrow of the induced electric field as in the image (pointing in the direction), which tells you where the field caused by the electrons is pointing). In wires, electrons instead of holes are flowing, so and . Also . 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 where is
charge carrier density, is the cross-sectional area, and 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}{n t e} If the charge build up had been positive (as it appears in some metals and semiconductors), then the 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 is the
current density of the carrier electrons, and 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 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). 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 is the electron concentration, the hole concentration, the electron mobility, the hole mobility and 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 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 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 (-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, , in a plasma is the ratio between the electron
gyrofrequency, , and the electron-heavy particle collision frequency, : \beta=\frac {\Omega_\mathrm{e}}{\nu}=\frac {eB}{m_\mathrm{e}\nu} where • is the
elementary charge (approximately ) • is the magnetic field (in
teslas) • is the
electron mass (approximately ). 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, , is no longer collinear with the
electric field vector, . The two vectors and make the
Hall angle, , 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 ==