Direct Energy, 2018a

92 5.2 Physics of the Hall Eect
The Hall eect occurs in both conductors and semiconductors. In conductors,
electrons are the charge carriers responsible for the eect while in
semiconductors, both electrons and holes are the charge carriers responsible
for the eect [9]. A hole is the absence of an electron. Consider a
piece of semiconductor oriented as shown in Fig. 5.1a. Assume the length
is specied by l, the width is specied w, and the thickness is specied
by d thick . For a typical Hall eect device, these dimensions may be in the
millimeter range. Furthermore, assume the semiconductor is p-type with
hole concentration p in units m −3 . The charge concentration represents the
net, or excess, charge density above a neutral material. Materials with a
net negative charge, excess valence electrons, will have a positive value for
the electron concentration n and are called n-type. Materials with a net
positive charge, an excess of holes, will have a positive value for the hole
concentration p which represents the density of holes in the material and
are called p-type. Overall charge density is related to n and p by
ρ ch = −qn + qp (5.2)
where q is the magnitude of the charge of an electron.
Assume the semiconductor is placed in an external magnetic eld oriented
in the â z direction, with magnetic ux density
−→ B = Bz â z .
Also assume a current is supplied through the semiconductor in the â x
direction. The positive charge carriers in the semiconductor, holes, move
with velocity −→ v = v x â x because current is the owof charge per unit
time. These measures are illustrated in Fig. 5.1b. Hall eect devices are
typically used as sensors as opposed to energy harvesting devices because
power must be supplied from this external current and because the amount
of electricity produced is typically quite small.
The force on the charges can be found from the Lorentz force equation.
The force due to the external magnetic eld on a charge of magnitude q is
given by
q −→ v × −→ B = qv x â x × B z â z = −qB z â y (5.3)
and is oriented in the −â y direction. Positive charges accumulate on one
side of the semiconductor as shown in Fig. 5.1c. This charge build up
causes an electric eld oriented in the â y direction which opposes further
charge build up. Charges accumulate until an equilibrium is reached when
the forces on the charges in the â y direction are zero.
−→ F =0=Q
( −→E +
−→ v ×
−→ B
)
(5.4)