Dr. M.M.

Abdel Aziz

Chapter 1

The p-n Junction

The Contact Potential

Let us consider separate regions of p- and n-type semiconductor material, brought
together to form a junction (Fig. 1). Before they are joined, the n-material has a large
concentration of electrons and few holes, whereas the converse is true for the p- material.
Upon joining the two regions, (Fig. 1), holes diffuse from the p side into the n side, and
electrons diffuse from n to p.

Fig.1
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The resulting diffusion current cannot build up indefinitely, however, because an

opposing electric field is created at the junction Fig. b. If we consider that electrons
diffusing from n to p leave behind uncompensated donor ions (Nd) in the n material, and
holes leaving the p region leave behind uncompensated acceptors (Na), it is easy to visualize
the development of a region of positive space charge near the n side of the junction and
negative charge near the p side. The resulting electric field is directed from the positive
charge toward the negative charge. Thus ℰ is in the direction opposite to that of diffusion
current for each type of carrier (recall electron current is opposite to the direction of electron
flow). Therefore, the field creates a drift component of current from n to p, opposing the
diffusion current (Fig. c).

Since we know that no net current can flow across the junction at equilibrium, the
current due to the drift of carriers in the ℰ field must exactly cancel the diffusion current.
Furthermore, since there can be no net buildup of electrons or holes on either side as a
function of time, the drift and diffusion currents must cancel for each type of carrier:

Jp(drift) + J p(diff.) = 0

Jn(drift) + J n(diff.) = 0

Therefore, the electric field ℰ builds up to the point where the net current is zero at
equilibrium. The electric field appears in some region W about the junction, and there is an
equilibrium potential difference V 0 across W.

In the electrostatic potential diagram of Fig. b, there is a gradient in potential in the

direction opposite to ℰ, in accordance with the fundamental relation ℰ (x) = -d 𝒱(x) / dx. We
assume the electric field is zero in the neutral regions outside W. Thus there is a constant
potential 𝒱n in the neutral n material, a constant 𝒱p in the neutral p material, and a potential
differenceV0 = 𝒱n – 𝒱p between the two. The region W is called the transition region, and
the potential difference V 0 is called the contact potential. The contact potential appearing
across W is a built-in potential barrier, in that it is necessary to the maintenance of
equilibrium at the junction. Indeed, the contact potential cannot be measured by placing a
voltmeter across the devices, because new contact potentials are formed at each probe, just
canceling V0. By definition V0 is an equilibrium quantity, and no net current can result from
it.

The contact potential separates the bands as in Fig. b; the valence and conduction
energy bands are higher on the p side of the junction than on the n side by the amount qV0.
The separation of the bands at equilibrium is just that required to make the Fermi level
constant throughout the device.

To obtain a quantitative relationship between V0 and the doping concentrations on

each side of the junction, we must use the requirements for equilibrium in the drift and
diffusion current equations. For example, the drift and diffusion components of the hole
current just cancel at equilibrium:

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Use Einestein relation Dn = ( KT / q ) µn , and ξ(x) = - dV(x) / dx

pp and pn are the hole concentration at the edge of the transition region on either side.

Now, we have p p =Na , and n n = Nd,

But pn .nn = ni2 or pn . Nd = ni2

Another useful relation

Using the equilibrium condition p n .n n = n i2 = p p . n p

Example 5.1

Note: Use Fermi- Dirac distribution:

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Solution:

Equilibrium Fermi Levels

We have observed that the Fermi level must be constant throughout the device at
equilibrium. This observation can be easily related to the results of the previous section as
follows

The Fermi level and valence band energies are written with subscripts to indicate the
p side and the n side of the junction.
From Fig. 1b the energy bands on either side of the junction are separated by the contact
potential V0 times the electronic charge q.
Thus qV0 = Evp - Evn

This equation results from the fact that the Fermi levels on either side of the junction are
equal at equilibrium (EFn - EFp = 0) . When bias is applied to the junction, the potential
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barrier is raised or lowered from the value of the contact potential, and the Fermi levels on
either side of the junction are shifted with respect to each other by an energy in electron volts
numerically equal to the applied voltage in volts.
Equation 11.b gives : EFn – EFp = 0

Or EFn = EFp = EF ( at equilibrium)

Space Charge at a Junction

Within the transition region, electrons and holes are in transit from one side of the
junction to the other. Some electrons diffuse from n to p, and some are swept by the electric
field from p to n (and conversely for holes); there are, however, very few carriers within the
transition region at any given time, since the electric field serves to sweep out carriers.
To a good approximation, we can consider the space charge within the transition region as
due only to the uncompensated donor and acceptor ions. (depletion approximation.)
Since the dipole about the junction must have an equal number of charges on either side, (Q +
= ∙Q- ∙), the transition region may extend into the p and n regions unequally, depending on
the relative doping of the two sides. For a sample of cross-sectional area A, the total
uncompensated charge on either side of the junction is:

Where xp0 is the penetration of the space charge region into the p material, and xn0 is the
penetration into n. The total width of the transition region (W) is the sum of xp0 and xn0.

To calculate the electric field distribution within the transition region, we begin with
Poisson’s equation, which relates the gradient of the electric field to the local space charge
at any point x:

.
if we neglect the contribution of the carriers (p-n) to the space charge. With this
approximation, we have two regions of constant space charge:

assuming complete ionization of the impurities (Nd+= Nd) and (Na- =Na ). We can see from
these two equations that a plot of ℰ(x) vs. x within the transition region has two slopes,
positive (ℰ increasing with x) on the n side and negative (ℰ becoming more negative as x
increases) on the p side. There is some maximum value of the field ℰ 0 at x = 0 (the

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Fig.2

metallurgical junction between the p and n materials), and ℰ(x) is everywhere negative
within the transition region (Fig. c).

To get the value of ℰ 0

Therefore, the maximum value of the electric field is

Thus the negative of the contact potential is simply the area under the ℰ(x) vs.x triangle.
This relates the contact potential to the width of the depletion region:
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To get the transition region width W, x n0 , and x p0

Since the balance of charge requirement is xn0Nd = xp0Na, and W is simply xp0 + xn0, we can
write xn0 = W. Na / (Na+ Nd) in Eq. (5–19):

We can also calculate the penetration of the transition region into the n and p materials:

Thus , we have W α √V0, for applied bias W α √(V0 ± VB )

So, applied bias can increase or decrease the potential across the junction and consequently,
increase or decrease the transition region width.

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Forward- and Reverse-Biased Junctions

(Steady State Conditions)

One useful feature of a p-n junction is that current flows quite freely in the p to n direction
when the p region has a positive external voltage bias relative to n (forward bias and forward
current), whereas virtually no current flows when p is made negative relative to n (reverse
bias and reverse current).
This asymmetry of the current flow makes the p-n junction diode very useful as a rectifier.
While rectification is an important application, it is only the beginning of a host of uses for
the biased junction. Biased p-n junctions can be used as voltage-variable capacitors,
photocells, light emitters, and many more devices which are basic to modern electronics.
Two or more junctions can be used to form transistors and controlled switches.

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Qualitative Description of Current Flow at a Junction

We assume that an applied voltage bias V appears across the transition region of the
junction rather than in the neutral n and p regions.
We shall take V to be positive when the external bias is positive on the p side relative to the
n side.
Since an applied voltage changes the electrostatic potential barrier and thus the
electric field within the transition region, we would expect changes in the various
components of current at the junction (Fig.3). In addition, the separation of the energy bands

Fig.3

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is affected by the applied bias, along with the width of the depletion region. Let us begin by
examining qualitatively the effects of bias on the important features of the junction.

Effects of bias on the important features of the junction.:

 The electrostatic potential barrier at the junction is lowered by a forward bias Vf

from the equilibrium contact potential V0 to the smaller value V0 - Vf. This lowering
of the potential barrier occurs because a forward bias (p positive with respect to n)
raises the electrostatic potential on the p side relative to the n side. For a reverse bias
(V = -Vr) the opposite occurs; the electrostatic potential of the p side is depressed
relative to the n side, and the potential barrier at the junction becomes larger (V0 +
Vr).

 The electric field: within the transition region decreases with forward bias, since the
applied electric field opposes the built-in field. With reverse bias the field at the
junction is increased by the applied field

 The transition region width W: we would expect the width W to decrease under
forward bias (smaller ℰ, fewer uncompensated charges) and to increase under reverse
bias.

 The separation of the energy bands: is a direct function of the electrostatic potential
barrier at the junction. The height of the electron energy barrier is simply the
electronic charge q times the height of the electrostatic potential barrier. Thus the
bands are separated less [q(V0 - Vf)] under forward bias than at equilibrium, and more
[q(V0 + Vr)] under reverse bias. We assume the Fermi level deep inside each neutral
region is essentially the equilibrium value (we shall return to this assumption later);
therefore, the shifting of the energy bands under bias implies a separation of the
Fermi levels on either side of the junction. Under forward bias, the Fermi level on
the n side EFn is above EFp by the energy qVf for reverse bias, EFp is qVr joules
higher than EFn.

 The diffusion current: is composed of majority carrier electrons on the n- side and
holes on the p-side. With forward bias, however, the barrier is lowered (to V0 - Vf),
and many more electrons in the n-side conduction band have sufficient energy to
diffuse from n to p over the smaller barrier. Therefore, the electron diffusion
current can be quite large with forward bias. Similarly, more holes can diffuse from
p to n under forward bias because of the lowered barrier. For reverse bias the barrier
becomes so large (V0 + Vr) that virtually no electrons in the n-side conduction band
or holes in the p-side valence band have enough energy to surmount it. Therefore,
the diffusion current is usually negligible for reverse bias.

 The drift current: is relatively insensitive to the height of the potential barrier. This
because the electron drift current does not depend on how fast an individual electron
is swept from p to n, but rather on how many electrons are swept down the barrier per
second. From the figure, the electrons and holes participate the drift current on both
n and p sides of the junction. To a good approximation, therefore, the electron and
hole drift currents at the junction are independent of the applied voltage.

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 The total current: crossing the junction is composed of the sum of the diffusion and
drift components. As Fig. 3 indicates, the electron and hole diffusion currents are
both directed from p to n (although the particle flow directions are opposite to each
other), and the drift currents are from n to p. The net current crossing the junction is
zero at equilibrium, since the drift and diffusion components cancel for each type of
carrier (the equilibrium electron and hole components need not be equal, as in Fig.3,
as long as the net hole current and the net electron current are each zero). Under
reverse bias, both diffusion components are negligible because of the large barrier at
the junction, and the only current is the relatively small (and essentially voltage-
independent) generation current from n to p. This generation current is shown in
Fig.4, in a sketch of a typical I–V plot for a p-n junction.

Fig.4

The only current flowing in this p-n junction diode for negative V is the small
current I(gen.) due to carriers generated thermally near the junction , diffuse to the junction,
drift across the junction to the other side ( drift current ). The current at V = 0 (equilibrium)
is zero.
For an applied forward bias V = Vf, the probability that a carrier can diffuse across the
junction, increases by the factor exp(qVf /kT). Thus the diffusion current under forward bias
is given by its equilibrium value multiplied by exp(qV/kT); similarly, for reverse bias the
diffusion current is the equilibrium value reduced by the same factor, with V = -Vr. Since the
equilibrium diffusion current is equal in magnitude to |I(gen.) | , the diffusion current with
applied bias is simply |I(gen.)|.exp(qV/kT). The total current I is then the diffusion current
minus the absolute value of the generation current, which we will now refer to as I0. The
total current I is given by :

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Diode Current Equation

Carrier Injection

The minority carrier concentration on each side of a p-n junction to vary with the
applied bias because of variations in the diffusion of carriers across the junction.
The equilibrium ratio of hole concentrations on each side

Neglecting changes in minority carrier concentration, the ratio of the two equations

Fig.5

The exponential increase of the hole concentration at xn0 with forward bias is an example of
minority carrier injection. As Fig. 5 suggests, a forward bias V results in a steady state
injection of excess holes into the n region and electrons into the p region. We can easily
calculate the excess hole concentration Δp n at the edge of the transition region xn0 by
subtracting the equilibrium hole concentration from Eq. (5–28),

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Similarly, on the p-side

As the holes diffuse deeper into the n region, they recombine with electrons in the n
material. If the n region is long compared with the hole diffusion length Lp ,then the
distribution of excess carriers are:

The hole diffusion current at any point xn in the n material can be calculated as:

A is the cross-sectional area of the junction.

The hole current injected into the n-material is obtained at xn0

If we neglect recombination in the transition region, i.e we consider that each injected
electron reaching -xp0 must pass through xn0. Thus the total diode current I at xn0 can be
calculated as the
sum of Ip(xn= 0) and -In(xp= 0):

This is the diode equation giving the total current through the diode for either forward or
reverse bias. We can calculate the current for reverse bias by letting V = -Vr:

If Vr is larger than a few kT/q, the total current is just the reverse saturation current

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One implication of Eq. (5–36) is that the total current at the junction is dominated by
injection of carriers from the more heavily doped side into the side with lesser doping.
For example, if the p material is very heavily doped and the n region is lightly doped, the
minority carrier concentration on the p side (n p) is negligible compared with the minority
carrier concentration on the n side (p n). Thus the diode equation can be approximated by
injection of holes only, as in Eq. (5–33). This means that the charge stored in the minority
carrier distributions is due mostly to holes on the n side. For example, to double the hole
current in this p+-n junction one should not double the p+ doping, but rather reduce the n-
type doping by a factor of two. This structure is called a p+- n junction, where the +
superscript simply means heavy doping.

Quasi-Fermi levels

Figure 5–15b shows the quasi-Fermi levels as a function of position for a p-n junction in
forward bias. The equilibrium EF is split into the quasi-Fermi levels Fn and Fp which are
separated within W by an energy qV caused by the applied bias, V. This energy represents
the deviation from equilibrium. In forward bias in the depletion region we thus get

On either side of the junction, it is the minority carrier quasi-Fermi level that varies
the most. The majority carrier concentration is not affected much, so the majority carrier
quasi-Fermi level is close to the original EF.
The quasi-Fermi levels are nearly flat within the transition region.

What happens ?

 The electron and hole currents are constants across the transition region ( no
recombination), and the product of the gradient of quasi-Fermi-level and carrier
concentration are independent of position.
 For a given current, the carrier concentration of minority carriers varies significantly
on both sides of the transition region, so quasi-Fermi level of minority carriers
changes most while the majority carriers are nearly constant , so its quasi-Fermi level
changes little (or no change).

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