Hall Effect

Hall Effect, deflection of conduction carriers by an external magnetic field, was discovered in 1879 by Edwin Hall. It has been known that moving carriers in a magnetic field are accelerated by the Lorentz Force, and the magnitude and the direction of the applied force on the carriers are given as in Equation (1):

$\mathbf{F} = q\boldsymbol{v}\times \mathbf{B} \label{1}$

Where $$q$$ is electron charge, $$\mathbf{v}$$ is drift velocity and $$\mathbf{B}$$ is magnetic flux density. If $$\mathbf{B}$$ is perpendicular to current flow direction as seen in Figure $$\PageIndex{1}$$, then the direction of the force is found by right hand rule and magnitude is expressed as scalar multiplication of $$q$$, $$v$$ and $$B$$.

Figure $$\PageIndex{1}$$ 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 $$\xi_y$$ in the direction of the assigned VH. VH is negative for some semi-conductors where "holes" appear to flow. In steady-state, $$\xi_y$$ will be strong enough to exactly cancel out the magnetic force, so that the electrons follow the straight arrow (dashed). Image used with permission (public domain; Gnefgnoix).$$^{[2]}$$

Figure $$\PageIndex{1}$$ shows a conductive slab with a current $$I_x$$ in x-direction, and with width $$w$$ and thickness $$t$$. An external magnetic flux density ($$B_z$$) is applied in z direction, resulting a Lorentz force in –y direction for holes and electrons (due to negative charge of electrons). Material shown in Figure $$\PageIndex{1}$$ is a p-type semiconductor (doped with acceptor atoms). Without application of any magnetic field, holes flow through by experiencing no force along the y-direction. However, once $$B$$ is applied, holes are deflected in –y direction resulting a (+) charge accumulation on the side A. Since the material has to be neutral, same amount of charge with opposite sign (-) appears on the other side of the slab (side $$B$$). Source of this negative charge on side B is negatively charged dopant ions. An internal electrical field ($$\varepsilon_y$$) starts building. As more charge accumulates on the sides, (\varepsilon_y\) gets stronger and opposes the Lorentz force. Eventually, electrical force ($$F_y=q\varepsilon_y$$) cancels out the Lorentz force ($$-qvB_z$$) and results:

$\varepsilon_{y} = v B_{z} \label{2}$

Also, current density ($$J_x$$) and the drift velocity ($$v$$) are also related with:

$J_{x} = pqv \label{3}$

Where p is the hole density of the material. When $$v$$ in (3) is substituted in (2):

$\varepsilon _{y} = \dfrac{J_{x}B_{z}}{pq} \label{4}$

is achieved. Also, voltage potential difference between $$A$$ and $$B$$ ($$V_{AB}$$) can be written as:

$V_{AB} = \dfrac{R_{H}I_{x}B_{z}}{t} \label{5}$

Where $$R_H$$(=1/qp)is Hall coefficient. In (5), all parameters except $$R_H$$ are known or can be measured, which gives solution to $$R_H$$ , so p. If a similar derivation is performed for an n-type material (majority carriers are electrons), $$R_H= -1/ qn$$ will be achieved. Since $$R_H$$ is found to be positive for p-type material and negative for the n-type, Hall coefficient can be measured to analyze the type of majority carrier of an unknown material.

Furthermore, if the resistivity ($$\rho$$) of the material is known (or measured) mobility of the carriers can also be extracted as given:

$\mu_{h}=\dfrac{\sigma }{qp}=\dfrac{R_{h}}{\rho } \label{6}$

Where $$\mu_p$$ is hole mobility. Relation in (6) enables mobility measurements of majority carriers for single carrier materials.

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Figure $$\PageIndex{2}$$
Hall effect in presence of both holes (h) and electrons (e) $$^{[3]}$$.

If both holes and electrons are conduction carriers, then a different derivation has to be done to solve for Hall coefficient. Figure $$\PageIndex{2}$$ shows a semiconductor with holes and electrons flowing through. Magnetic flux density B deflects both carrier types to bottom surface of the material. In this case, Lorentz force on electrons and holes can be written as:

$F_{e}=qv_{e}B \label{7}$

$F_{h}=qv_{h}B \label{8}$

Which result in built in electric fields of:

$\varepsilon_{e}=\dfrac{J_{e}B}{qn} \label{9}$

$\varepsilon_{h}=\dfrac{J_{h}B}{qp} \label{10}$

In transient analysis, current density along y-axis (perpendicular to electrical current flow) can be written using (9) and (10):

$J_{y}=nq\mu_{e}\varepsilon_{e} + pq\mu_{h}\varepsilon_{h}+ (nq\mu_{e} + pq\mu_{h})\varepsilon_{H} \label{11}$

In steady state, current density (net charge flow) along the y-axis has to be zero, which results in Hall coefficient of:

$R_{H}=\dfrac{p\mu_{h}^{2}-n\mu_{e}^{2} }{q(p\mu_{h}+n\mu_{e})^{2}} \label{12}$

Equation (12) indicates that Hall coefficient of a material can either be positive or negative depending on mobility and density of the carriers (electrons and holes). Table $$\PageIndex{1}$$ shows Hall coefficient of a number of metals. Notice that $$R_H$$ is positive for some metals, showing holes are the dominant type.

Table $$\PageIndex{1}$$ Hall coefficient (Volt m3/Amp Weber) x 10-10 at room temperature.
Li Na Cu Ag Au Zn Cd Al
-1.7 -2.5 -0.55 -0.84 -0.72 +0.3 +0.6 -0.3

References

1. F. T. Ulaby, Electromagnetics for Engineers. ,4th ed.NJ: Pearson/Prentice Hall, 2005, pp. 397.
2. B. G. Streetman, Solid State Electronic Devices. ,4th ed.Prentice Hall, 1995.
3. M. A. Omar, Elementary Solid State Physics: Principles and Applications. MA: Addison Wesley Publishing Company, 1993, pp. 669.

Contributors

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