Section A5: Current Flow in Semiconductors
Conductive behaviors in materials, defined by the parameter conductivity,
are a primary factor in the development of electronic and optoelectronic
devices. Electrical conductivity (denoted by the symbol σ and expressed
in units of (Ωcm)-1) is purely a material property that describes how easily
the material allows charges to move or through it, equivalently, how easily
electric current can flow. Alternatively, we can talk about the resistivity (ρ) of
a material, where ρ=1/σ and has units of Ωcm.
Since conductive behavior is intimately tied to charge movement, it makes
sense that we should take into consideration all free charges in the medium.
In the modified version of Figure 3.1, we saw that electrons may break the
covalent bonds that bind them to a parent atom and move throughout the
material. When each electron leaves its position in the valence band, a hole
is created – which can be considered as a unit of positive charge (not
negative is positive, right?). Looking at it from a slightly different view, and
using the band edges defined in Figure 3.3, we can look at the development
of charge movement (current flow) as follows:
Electron
Energy
EC
Eg
For a pure (intrinsic) semiconductor at
T=0oK, all electrons are associated with
their covalent bonds. The conduction band
(above energy EC) is empty of electrons,
while the valence band (below energy EV)
is full. NOTE: Eg is the forbidden gap and
is the difference between EC and EV.
EV
Electron
Energy
EC
Eg
EV
For a pure (intrinsic) semiconductor at
T>0oK, some electrons gain enough
energy to break their bonds and jump the
forbidden gap (only one shown). The
conduction band now contains free
electrons, while the valence band now has
free holes.
Electron
Energy
-
Electric Field
+
electron movement
EC
Eg
hole movement
EV
With a potential applied as shown,
electrons in the conduction band move
to the right. Electrons in the valence
band also move to the right, but move
by filling a hole. This process is
equivalent to holes moving to the left.
NOTE: An intrinsic semiconductor is
shown; i.e., the number of holes in the
valence band is equal to the number of
electrons in the conduction band.
electron movement
Electron
Energy
-
Electric Field
+
electron movement
EC
Eg
hole movement
The same process applies to an
n-type semiconductor; only now
electrons are the majority carrier
(i.e.,
there
are
more
free
electrons in the conduction band
than holes in the valence band).
EV
Electron
Energy
-
Electric Field
+
electron movement
EC
Eg
hole movement
EV
Likewise for a p-type semiconductor;
only now holes are the majority
carrier (i.e., there are more holes in
the valence band than electrons in
the conduction band).
Electron
Energy
Electric Field
-
electron flow
EC
Eg
EV
hole flow
+
Regardless of the semiconductor type,
with the potential applied as shown,
the net effect is as indicated in the
figure. Note that both electrons and
holes contribute to the total current
flow. Although they are moving in
opposite directions, they are of
opposite charge… two negatives is a
positive!
Conventional direction
for positive current
Remember that, at any temperature above absolute zero, a certain number
of electrons gain enough energy to break their covalent bonds and become
free to move in the conduction band (leaving holes free to move in the
valence band). This is not a static process and there is a continuous
exchange of energy between these electrons and the lattice in the form of
elastic and inelastic collisions. Without an externally applied potential, the
result is electron motion that is totally random, yielding a zero net
movement and therefore a zero net current.
Now… notice that carrier movement in the above figures is due to an applied
potential (and the resulting electric field), and that the charge of the carrier
determines the direction of movement. This illustrates the first of two ways
by which current can flow through a material and is called drift. Drift current
occurs with the application of an electric field to the medium. The electric
field causes a net, or directed, movement of carriers in the medium. The
drift velocity of the carriers in the medium is a direct function of the electric
field strength and the carrier mobility (µn for electrons and µp for holes), as
illustrated in Equation 3.17 of your text. The drift velocity is ultimately
limited by the saturation velocity in the material of interest.
So – how do we get current out of all this mess? We’ve got moving
charges, we’ve got mobilities, we’ve got an electric field due to an
applied potential, and somehow we’ve got to put it all together to
get something we can use! Believe it or not, if we take the number of
charges, the elemental charge and the mobility of the carrier type,
we can get the current density (J) – which is current per unit area
and by multiplying by area we can get the total current (I). Yeah,
right – well, let’s see how this goes…
Starting with Equation 3.24 (with slightly modified notation from your text to
be consistent with our discussions):
J = q ( pµ p + n µ n ) E
where:
J
q
n
p
µn
µp
E
is
is
is
is
is
is
is
the
the
the
the
the
the
the
(Equation 3.24: modified)
current density (in A/m2=C/m2s=V/Ωm2)
elemental charge (≈1.602x10-19 C)
number of free electrons in the medium
number of free holes in the medium
mobility of electrons in the medium (in m2/Vs)
mobility of holes in the medium (in m2/Vs)
electric field strength (in V/m)
Keep in mind that, although electrons and holes have opposite charge they
are also moving in opposite directions. This allows us to add the contribution
of each carrier type to the total current density.
Using the relationship for conductivity (σ=Nqµ) where N is the number of
free carriers and µ is the mobility of the carrier type, we can rewrite
Equation 3.24 in the form of Equation 3.21 by defining a composite
conductivity term (σ = σp + σn) which, believe it or not, is a form of our old
friend – Ohm’s law:
J = (σ p + σ n ) E = σE
(Equation 3.21: modified)
Let’s look at the units of what we’ve got,
V
A
⎛ 1 ⎞⎛ V ⎞
= 2
⎟⎜ ⎟ =
2
m
⎝ Ωm ⎠⎝ m ⎠ Ωm
σE in units of ⎜
Hmm.. looks like we just have to multiply by the area that the charges are
moving through to get the total drift current!
Another way of getting this result is illustrated by your author in Equation
3.22. All he’s done to Equation 3.21 here is multiply by area (to get an
expression in I instead of J) and multiply and divide by L. This allows the use
ρL L
=
R=
A σA , along with the units of electric
of the expression for resistance,
V
I=
R.
field, to be substituted and manipulated to come up with Ohm’s Law:
Pretty cool, huh? However you prefer to look at it –
•
•
•
If you’ve got an applied potential, you’ve got an electric field.
If you’ve got free charges in your medium, you’ve got directed carrier
movement.
If you’ve got directed carrier flow, you’ve got net (nonzero) current flow
that has possible contributions from both carrier types. At typical doping
levels and normal operating conditions, it is usually valid to make the
simplification:
o In an n-type material, electrons are the majority carriers and the
current contribution of holes is generally considered negligible
(n•ND, p•0).
o In a p-type material, holes are the majority carriers and the current
contribution of electrons is generally considered negligible (p•NA,
n•0).
Well, that’s drift for you. The second contribution to current is known as
diffusion. Unlike drift, diffusion has nothing to do with the charge of the
carrier and arises whenever there is a non-uniform concentration, or
concentration gradient, of charges in a medium. Using the classic
example, if you would spray perfume in the corner of a room, the smell (Can
you tell I’m not a perfume fan?) diffuses throughout the available space until
it is equally spread out. This is exactly what happens when a charge
concentration gradient exists – they want to move, or spread out, until an
equilibrium concentration is achieved.
Since we’ve been talking about intrinsic materials and either uniformly
doped n- or p-type extrinsic materials, this is a nice segue into what
happens when we put these two types of extrinsic materials together.