Direct Energy, 2018a

5 HALL EFFECT 95
An analogous expression can be found if electrons instead of holes are the
dominant charge carrier. The sign of this measured voltage is also used to
determine whether a piece of semiconductor is n-type or p-type [58].
The Hall resistance R H is a parameter inversely proportional to the
charge concentration, and it has the units of ohms [9] [59]. For the assumptions
above, the Hall resistance is dened as
R H = B z
qp ·
w
l · d thick
. (5.13)
By combining Eqs. 5.12 and 5.13, it can be written in terms of the measured
voltage and applied current.
R H = V AB
I x
· w
l
(5.14)
As an example, suppose that a piece of silicon with a hole concentration
of p =10 17 cm −3 is used as a Hall eect device. The device has dimensions
l =1cm, w =0.2 cm, and d thick =0.2 cm, and it is oriented as shown
in Fig. 5.1. The material has a resistivity of ρ =0.9 Ω·cm. A current
of I =1mA is applied in the â x direction. The device is in an external
magnetic eld of −→ B =10 −5 â Wb
z cm . If a voltmeter is connected as shown
2
in the gure, what voltage V AB is measured?
V AB =
I x B z
q · d thick · p =
1 −3 · 10 −5
1.6 · 10 −19 · 0.2 · 10 17 =3.1 · 10−6 V (5.15)
Signals in the millivolt range are easily detected with a standard voltmeter,
yet signals in the microvolt range often can be measured with some ampli-
cation. What output power is generated by this device? We can calculate
the resistance along the â y direction. The resistivity of the silicon was given
in the problem, and resistance R and resistivity ρ are related by
R = ρ · length . (5.16)
area
The resistance across the width of the device is
R width =
ρw
ld thick
=
0.09 · 0.2
1 · 0.2
=0.09 Ω (5.17)
We can use this calculated resistance and the measured voltage to nd the
power converted from the magnetic eld to electrical power of the device.
P =
V 2 AB
R width
=1.1 · 10 −11 W (5.18)