# Hall Effect

### Definition:

In general , the Hall effect is the production across an electrical conductor of a voltage difference (also called Hall Voltages). Hall Effect was discovered in 1879 by Edwin Hall.

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.

With a semiconductor, it is necessary to know whether it is a n-type or p-type, its carrier concentration and the mobility of its charge carriers.  These parameters can be determined with using the Hall Effect.

If a metal or semiconductor carrying a current is placed in a magnetic field perpendicular to current, an electric field is induced in the direction perpendicular to bothe current and magnetic field.  This phenomenon is know as Hall Effect.

Hall Effect Diagram

Let assume that the current is the positive x-direction and the magnetic field is in the positive z-direction.  This exerts a force in the neqative y-direction.  If the material is an n-type semiconductro, then the charge carriers are electrons.  Because of the negative electrons are forced to the bottom.  This indicates that the bottom will be more negatively charged than the top.  This produces a voltage difference between the two surfaces.  Hence, a potential know as Hall Voltage (VH) where H indicated Hall.  Hall voltage appears between the two surfaces when in equilibrium , the electric field intensity (E) is equal to the magnetic froce .

Therefore, eE= Bev

where V is the drift velociy of the electrons; B is the magnetic field intensity.

E= Bv

The value of E= VH/d

where VH is the Hall voltage and d is the thickness of the semiconductor .

Let current density J= I/wd

where I is the current, w is the width and d is the thickness.

We have

VH= Ed and E=Bv
So VH= Bvd
But v= J/p     where p is the charge density.
= BJd/p
But J=I/wd
VH= BJd/pwd= Bj/pw
VH= BJ/pw

If the semiconductor is a p-type then the bottom of the field is more positive that the top.

We have p=n × e or p = p × e  depending upon the type of material.
ρ = BI/VHw
RH= VHw/ BI
Where RH is the Hall coeffiecient, It is equal to Vo For n-type semiconductor.
σ = neμ, but ne = ρ
σ = ρ × μ
= 1/RH × μ
So:
μ= σ × RH = σ × BI / VHw

The above experiment is obtained on the assumption that the drift velocity is the same for all carriers.  But this is not true for all practical considerations.  Experiments have shown that the results obtained are more accurate if I/RH is taken as 3Π/8ρ

μ = 8s/3Π × RH

The product Bev is known as the Lorentz force.  This shown as that, the majority of carriers is semiconductors will move in a direction perpendicular to the magnetic field.

### 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 becomes more complex in semiconductors where the carriers are generally both electrons and holes which may be present in different concentrations and have different liabilities. For moderate magnetic fields the Hall coefficient is

where  is the electron concentration,  the hole concentration,  the electron mobility ,  the hole mobility and  the absolute value of the electronic charge.

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

with

### 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, which is the quantization of the Hall voltage.

### 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 M.I. 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.

### Quantum spin Hall effect

For HgTe two dimensional quantum wells with strong spin-orbit coupling, in zero magnetic field, at low temperature, the Quantum Spin Hall effect has been recently observed.

### Anomalous Hall effect

In ferromagnetic materials (and para magnetic 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.) 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 very inferior to unity). In a plasma, the Hall parameter can take any value. The Hall parameter, β, in a plasma is the ratio between the electroencephalography, Ωe, and the electron-heavy particle collision frequency, ν:

where

e is the elementary charge (approx. 1.6 × 10-19 C)
B is the magnetic field (in teslas)
me is the electron mass (approx. 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 more colinear with the electric field vector, E. The two vectors J and E make the Hall angleθ, which also gives the Hall parameter:

β = tan(θ)