082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

7.6 The vector product 251
Engineeringapplication7.11
TheHalleffectinasemiconductor
A frequent requirement in the semiconductor industry is to be able to measure the
density of holes in a p-type semiconductor and the density of electrons in an n-type
semiconductor. This can be achieved by using the Hall effect. We will consider the
case of a p-type semiconductor but the derivation for an n-type semiconductor is
similar.
area, A
z
y
x
B
L
I
hole
E H
+
V
–
B
p-type
semiconductor
+ –
Meter
V H
Figure7.32
Hall effect in ap-type semiconductor.
Consider the piece of semiconductor shown in Figure 7.32. A d.c. voltage,V, is
applied to the ends of the semiconductor. This gives rise to a flow of current composed
mainly of holes as they are the majority carriers for a p-type semiconductor.
This current can be represented by a vector pointing in thexdirection and denoted
by I. A magnetic field, B, is applied to the semiconductor in the y direction. The
moving holes experience a force, F B
, per unit volume, caused by the magnetic field
given by
F B
= 1 A I ×B
where A is the cross-sectional area of the semiconductor. This causes the holes to
drift in thezdirection and so causes an excess of positive charge to appear on one
side of the semiconductor. This gives rise to a voltage known as the Hall voltage,
V H
. As this excess charge builds up it creates an electric field, E H
, in the negativez
direction,whichinturnexertsanopposingforceontheholes.Thisforceisgivenby
F E
= qp 0
E H
whereq = elementary charge = 1.60 ×10 −19 C, and p 0
= density of
holes (holes per cubic metre). Equilibrium is reached when the two forces are equal
inmagnitude, that is |F B
| = |F E
|. Now,
|F E
| =qp 0
|E H
| |F B
| = |I×B|
A
= IB A
➔