Ideal Diode Current-Voltage Characteristics
Qualitative View
Thermal Equilibrium
As we have said before, the electron and hole current across the PN junction in thermal
equilibrium is zero. Also as we have said before, this doesn’t mean that each component
of the electron or hole current are themselves zero. There is a balance between diffusion
and drift currents. These concepts are well explained in the diagram below:
In the top part of figure 3.1 which looks at the conduction band electron current, we see
that the drift component contributes to electrons progressing towards the right but a
diffusion component that causes electrons to travel towards the left. In thermal
equilibrium, these components are equal and opposite resulting in zero electron current.
A similar situation exists for valence band holes.
Forward Bias
Qualitatively speaking, it is easy to see what happens when a voltage is applied to the
device. We will first look at a forward bias in which a positive voltage is applied to the
p-type side of the PN junction. As we have said previously, the applied voltage is
expended at the pn junction with no voltage drop in the bulk region. The applied voltage
will tend to reduce the built in voltage across the junction thus reducing the barrier for
diffusion by an amount q(Vbi-VA), where VA is the applied voltage. This reducing in the
barrier for diffusion causes the diffusion component of the current to increase
tremendously because an exponentially increasing concentration of carriers have the
necessary energy to overcome this decreasing barrier to diffusion. The drift current on
the other hand, doesn’t change nearly as much as the diffusion current. Therefore, the
two components of current for both the electron and hole currents will not cancel each
other out and a nonzero current will result. These concepts are illustrated in Figure 3.3 of
Pierret:
It is easy to deduce a possible voltage dependence on this forward-biased current. As the
applied voltage is increased, the concentration of majority carriers on either side of the pn
junction that have the necessary energy to diffuse across the junction is increasing
exponentially. Hence the forward-biased current should be exponential dependence on
applied voltage.
Reverse Bias
A reverse bias is the situation when a negative voltage applied to the p-type side (relative
to the n-type side). As in the forward bias case, the components of both electron and hole
currents will change resulting in a net current. However, there are large differences in the
reverse bias case. As figure 3.4 of Pierret below shows, the diffusion currents decrease
because the applied voltage acts to increase the diffusion barrier. The drift currents may
grow somewhat because the electric field is indeed increasing across the junction but the
current will be greatly limited because the electrons from the n-type side drifting to the ptype side are minority carriers in the n-type side and are quickly depleted. Thermal
generation is required to create these electrons. This generation rate is small and
independent of applied voltage, hence the drift current quickly assumes some constant
value as the reverse bias is increased. Since the diffusion current is decreasing
exponentially, there immediately exists an imbalance in the drift and diffusion current
components resulting in a net current. As stated however, this current will be small and
largely independent of applied voltage.
Complete Circuit
The complete circuit for forward and reverse bias is shown in figure 3.5 of Pierret:
At the metal/p-type contact, under forward bias, electrons are given up to the metal
contact, thus producing holes in the p-type material which migrate to the junction and
diffuse to the n-type side. These injected minority carrier holes on the n-type side, then
quickly recombine with electrons that have been supplied by the metal/n-type contact and
migrating towards the pn junction. Additionally, electrons diffuse from the n-type
material into the p-type material, then quickly recombine with majority carrier holes that
were created at the metal/p-type contact when electrons were given up to the metal
contact.
Reverse bias is similar in many regards except for electrons are injected into the p-type
material from the metal contact. These electrons quickly recombine with the majority
carrier holes which have drifted towards the contact after they were thermally generated
near the pn junction. Their (holes) thermal generation was accompanied by the
generation of an electron which drifts across the pn junction into the n-type side where
they then drift to the metal/n-type contact and leave the device. Similarly, thermal
generation near the pn junction on the n-type side results in minority carrier holes that
drift across the junction into the p-type side and towards the metal/p-type contact. The
thermally generated electrons drift in the opposite direction towards the metal/n-type
contact.
Quantitative Analysis of the PN Junction
Pierret list seven assumptions that will be used:
1. There are no external sources of carrier generation; for example, no light.
2. The depletion approximation and the step junction are applicable.
3. The steady-state dc solution is desired; that is, all the d/dt terms in the continuity
equations are zeros.
4. No generation or recombination takes place in the depletion region (what goes in
must come out).
5. Low-level injection is maintained in the quasi-neutral (bulk) regions of the
device; this means that the number of minority carriers is always much less than
the number of majority carriers in the bulk regions.
6. The electric field for the minority carriers is zero in the bulk regions.
7. The bulk regions are uniformly doped; that is, NA and ND are constants.
These assumptions cause the minority carrier diffusion equations, the current density
equations and the carrier concentration equations to become:
n-type
0  DP
d 2 p n p n

p
dx 2
J P  qDP
dpn
dx
pn  pno  pn x 
p-type
0  DN
d 2 n p
dx 2
J N  qD N

n p
n
dn p
dx
n p  n po  n p x 
We will focus are analysis on the edges of the depletion region, i.e., x = -xp and x = xn,
We can do this because we assume that there is no carrier generation or recombination in
the depletion region. This means that JN and JP are independent of position within the
depletion region. Hence we can solve for JN and JP at these locations and obtain their
values throughout the entire depletion region. To start this procedure, we will now solve
for p n xn  and n p  x p  .
To do this, we will assume that for moderate voltages, the values for JN|drift and JN|diffusion
have not changed tremendously relative to their values. Hence we can still say:
J N  q n n  qD N
dn
0
dx
This allows us to write:
 
DN dn
KT d ln n

 n n dx
q dx
Hence, the junction voltage VJ is:
xn
VJ  Vbi  V A    dx 
xp
KT  nn xn  
ln
q  n p  x p  
This gives us:
n p  x p   nn xn e

q
Vbi VA 
KT
q
q
VA
VA
ni2
KT
KT
 nno
e
 n po e
nno p po
Hence we get the following result for n p  x p  :
 q VA

n p  x p   n po  e KT  1


Similarly for p n xn  , we have:
 q VA

p n x n   p no  e KT  1


Two other boundary conditions are the values of p n   and n p    which are zero
(i.e., in the bulk material, the excess charge carriers above their equilibrium value will go
to zero). Hence we have:
p n    0
n p     0
This is shown in the following figure 3.2 of Pierret:
Once we have these boundary conditions, we look at the minority carrier diffusion
equations in the bulk regions where the electric field is negligible. For the n-type region
we have:
d 2 p n p n
0  DP

p
dx 2
The solutions to this equation after imposing the boundary condition at x   and
x  xn is:
p n ~
x   p n x n e

~
x
Lp
~
x
A
 qV
 L
 p no  e KT  1e p


L p  D p p
Finding the current according to the equation:
~
x
A
 qV
  Lp
dp n qD p
~
KT

J p x   qD p

p no  e  1e
dx
Lp


and at x  xn , we have the hole current as:
J p xn  
qD p
Lp
A
 qV

p no  e KT  1


Similarly for the electron current in the bulk p-type material, we have at x   x p :
A
 qV

qDn
J n  x p  
n po  e KT  1
Ln


Hence the total current is I  Area  J n  x p   J p x p  is:
 qVA

I  I o  e KT  1


where Io is the saturation current and is given by:
D

Dp
I o  q  Area   n n po 
p no 
L

Lp
 n
