Optoelectronics is a branch of physics and technology

OPTOELECTRONICS -MODULE 2 PART 1
Junctions between differently doped semiconductors,
between metals and semiconductors and between
different semiconductors provide the building blocks of
optoelectronic devices
The most important junction is the p-n junction in which
the nature of dopants is altered across a boundary to
create a region that is p-type next to a region which is
n-type
Most optoelectronic devices are based on p-n diode
structure

It is a junction between a p-type and n-type Semi-
conductor
If a p-n junction consists of p-type and n-type section of
the same semi-conductor material  it is called a
homojunction
If the junction consists of two different type of semi-
conductor material  it is called a heterojunction
p-n junction can be formed in different ways.
The simplest formation is bringing p and n type SC
together using the crystal growth technique
The p-n junction is fabricated either by growing p-type
material on a n-type material / vice-versa

The general technique used in crystal growth are
epitaxial growth, diffusion and ion-implantation
1. Epitaxial growth  where the dopant species
are simply switched at a particular instant of
time
2. Diffusion here dopants are selectively diffuses
into oppositely doped semiconductor
3. Ion-implantation in which the dopant ions are
implanted at high energies into the
semiconductor

A p-n junction consists of p-type and n-type
section of semi-conductor materials in
metallurgical contact.
The p-type has an abundance of holes (majority
carriers) and few mobile electrons (minority
carriers), and n-type region has an abundance of
mobile e-s (majority carriers) and few holes
(minority carriers)
Both charge carriers are in continuous and
random thermal motion in all directions

When two regions are brought into contact, the following
consequences of events take place:
1. Electrons and holes diffuse from areas of high concentrations
towards the areas of low concentrations. Thus e-s diffuse from n-
region into p-region leaving behind +vely charged ionized donor
atoms

In the p- region the e-s combine with the abundant e+s
Similarly the e+s diffuse from p-region into n-region
leaving behind –vely charged ionized acceptor atoms.
In the n-region these e+s recombines with abundant mobile
e-s
This diffusion process does not continue indefinitely,
however because it causes a disruption of the charge
carriers balance
2. As a result a narrow region on both sides of the junction
becomes nearly depleted of mobile charge carriers 
leads to a depletion region(layer)
The thickness of this layer is inversely proportional to the
concentration of dopants in the region

3. The fixed charges create an electric field in the
depletion layer that points from the n-side towards the
p-side of the junction. The built-in field obstructs the
diffusion of further mobile carriers through the junction
region
4. An equilibrium condition is established that results in
a net built-in potential difference Vo between the two
sides of the depletion region with n-side exhibiting high
potential than p-side
5. The built–in potential provides a lower potential
energy for an e- on n-side relative to p-side.
 As a result the energy bands bend as in fig. below

6. No current flows across the junction. The currents associated
with diffusion and built-in field cancel for both the e-s and e+s
The Biased p-n junction
An externally applied potential will alter the potential
difference between p and n regions
This in-turn will modify the flow of majority carriers, so that
the junction can be used as a gate. If the junction is forward
biased by applying a positive voltage to the p-region , its
potential is increased with respect to n-region so that the
electric field is produced in a direction opposite to that of
built-in field
The presence of the external bias voltage causes a departure
from equilibrium and a mis-alignment of Fermi-levels in the p
and n-regions , as well as in the depletion layer

The presence of two Fermi-levels in the depletion layer
Efc and Efv represents the state of quasi equilibrium

The net effect of the forward bias is to reduce the
height of the potential-energy hill by an amount eV
The majority carrier current turns out to increase by
an exponential factor  exp (eV/kT) so that the net
current becomes
ί = ίs exp (eV/kT) – ίs
where ίs is constant
The excess majority carriers , holes and electrons
that enter the n and p regions, become minority
carriers and recombine with the local majority
carriers this process is minority carrier- injection

If the junction is reverse biased by applying a negative
voltage V to the p-region, the height of the potential-
energy hill is augmented by eV. This impedes the flow
of majority carriers. The corresponding current is
multiplied by the exponential factor(eV/kT), when V
is negative
The net result for the current is
ί = ίs exp (-eV/kT) – ίs ⋍ ίs
So that a small current of magnitude ⋍ ίs flows in the
reverse direction
Hence the ideal I-V characteristics is given by
ί = ίs exp (eV/kT) – 1  Shockely equation

 In the ideal diode model, an abrupt junction is assumed
between the uniformly doped n- and p- regions
 In addition certain assumptions are also made:
1. The current flow across the diode is one-dimensional
2. All the applied voltage appears across the space
charge region and the neutral regions are field free.
Thus the minority carrier currents in the neutral n- and
p-regions can be described as diffusion currents only
3. There is no net generation or recombination of carriers
in the junction depletion region  which means that
the electron and hole currents remain constant
throughout this region

4. Low-level injection prevails on both sides of the
junction. This condition places a restriction on the
current that can be drawn through the ideal diode
and is not valid at high currents
5. The Boltzmann relation for electrons and holes is
valid throughout the depletion region. The
Boltzmann relation is a direct consequence of the
balance between the drift and diffusion processes in
the depletion region. Since the current drawn
through the junction is quite small compared to the
equilibrium diffusion or drift current, this balance is
approximately maintained

The response of a p-n junction to a dynamic applied
voltage is determined by solving the set of differential
equations governing the processes of electron and hole
diffusion, drift ( under the influence of the built-in and
external fields) and recombination.
Heterojunctions:
Junctions between different semi-conductor materials
are known as heterojunctions
Optical sources and detectors make extensive use of
heterojunctions and their designs , they are used not
only as active regions , but also as contact layers and
waveguiding regions
The importance of hetero junctions are the following:

1. Junction between materials of different band-gap create
localized jumps in the energy band diagram. A potential-
energy discontinuity provides a barrier that can be useful in
preventing selected charge carriers from entering the regions,
where they are undesired  This property may be used in p-n
junctions (ie, to reduce the proportion of the current carried
by minority carriers and thus to increase the injection
efficiency)

2. Discontinuities in the energy –band diagram created by
two heterojunctions can be useful for confining charge
carriers to a desired region of space. For example , a
layer of narrow band-gap material can be sandwiched
between two layers of a wider band-gap material, as
shown in the p-p-n structure above which consists of a
p-p heterojunction and a p-n heterojunction. This double
heterostructure(DH) configuration is used effectively in
the fabrication of LEDs
3. Heterojunctions are useful for creating energy-band
discontinuities that accelerate carriers at specific
locations. The additional kinetic energy suddenly
imparted to a carrier can be useful for selectively
enhancing the probability of impact ionization in a
multilayer heterojunctions as in Avalanche photodiode

4. semi-conductors of different band-gap type(direct and
indirect) can be used in the same device to select regions
of structure where light is emitted. Practically only direct
band-gap semiconductor type effectively emit light
5. semi-conductors of different bandgap can be used in
the same device to select regions of the structure where
light is absorbed. Semi-conductor materials whose
bandgap energy is larger than the photon energy incident
on them will be transparent , acting as a window layer
6. Heterojunctions of materials with different refractive
indexes can be used to create photonic structures and
optical waveguides that confine and direct photons in
certain optical devices

Junction Theory  p-n junction:
 The PN junction diode consists of a p-region and n-region
separated by a depletion region where charge is stored.
However, if we were to make electrical connections at the
ends of both the N-type and the P-type materials and then
connect them to a battery source, an additional energy
source now exists to overcome the potential barrier.
 The effect of adding this additional energy source results
in the free electrons being able to cross the depletion
region from one side to the other. The behaviour of the PN
junction with regards to the potential barrier’s width
produces an asymmetrical conducting two terminal
device, better known as the PN Junction Diode.

 The depletion layer widens with an increase in the
application of a reverse voltage and narrows with an
increase in the application of a forward voltage.
 This is due to the differences in the electrical
properties on the two sides of the PN junction
resulting in physical changes taking place.
 One of the results produces rectification as seen in
the PN junction diodes static I-V (current-voltage)
characteristics.
 Rectification is shown by an asymmetrical current
flow when the polarity of bias voltage is altered as
shown below.

 There are two operating regions and three possible
“biasing” conditions for the standard Junction
Diode and these are:
• 1. Zero Bias – No external voltage potential is applied
to the PN junction diode.
• 2. Reverse Bias – The voltage potential is connected
negative, (-ve) to the P-type material and positive,
(+ve) to the N-type material across the diode which
has the effect of Increasing the PN junction diode’s
width.
• 3. Forward Bias – The voltage potential is connected
positive, (+ve) to the P-type material and negative, (-
ve) to the N-type material across the diode which has
the effect of Decreasing the PN junction diodes width.

 1. The p-n junction in Equilibrium (Zero Biased Junction
Diode)
 A p-n junction may be fabricated as a single crystal of
semiconductor by a no. of different techniques. Indeed
the exact behavior of a junction depends to a large
extent on the fabrication process used, which in turn
determines the distances over which the change from p-
to n- type nature occurs
For mathematical convenience we shall assume that the
junction is abrupt , that is there is a step change in
impurity type as shown in fig. below, which also shows
the corresponding carrier concentrations also

Let Pp and np be the carrier concentrations of holes and
electrons on p-side and nn and np that of n-side of the
junction respectively. These apply only at relatively large
distances from the junction ; close to the junction they
are modified
Let us assume that the junction is formed by bringing
initially isolated pieces of n-type and p-type materials
into intimate contact
Then since there are many more holes in p-type than in n-
type material , holes will diffuse from the p- to n- region
The holes diffusing out of p-type side leave behind ionized
acceptors, thereby building up a negative space charge
layer in p-type side close to the junction

Similarly electrons diffusing into the p-side, leave behind a
positive space charge layer of ionized donors as in fig.(c)
below
This double space charge layer causes an electric field to be
set up across a narrow region on either side of the junction,
directed from the n- to p-type region as in fig. below
The direction of the junction electric field is such as to
inhibit further diffusion of the majority carriers , though such
diffusion is not prevented altogether
This must so, since the electric field will sweep minority
carriers across the junction so that there is a drift current of
electrons from p- to the n- type side and holes from n- to p-
type side, which is in opposite direction to the diffusion
current

The junction field thus builds up until these two
current flows are equal, at which stage the Fermi-level
is constant across the junction as shown in fig.(b)
above, indicating that equilibrium has been reached
within the crystal as a whole. Thus as there is no net
current flow in equilibrium
Jh (drift) + Jh (diff) = 0
and Je (drift) + Je (diff) = 0
The induced electric field establishes a contact or
diffusion potential Vo between the two regions and the
energy bands of p-type side are displaced relative to
those of n-type side as shown in fig.(b)

The magnitude of the contact potential depends on the
temperature and the doping levels
The contact potential is established across the space charge
region, which is also referred to as the transition or depletion
region so-called because this region has been almost
depleted of its majority carriers.
As a consequence it is very resistive relative to the other
(so-called bulk) regions of the space)
An expression relating the contact potential to the doping
levels can be obtained , adopting the notations used in the figs
above, we can write the electron concentration in the
conduction band of p-type as
np = Nc exp [-(Ecp-Efp/kT)]
where Nc is the effective density of states in the conduction
band of p-type

Similarly, the electron concentration in the n-type side is,
nn = Nc exp [-(Ecn-Efn/kT)]
where Nc is the effective density of states in the conduction band
of n-type
Now as mentioned , the Fermi-level is a constant everywhere
in equilibrium so that Efp = Efn =EF
Hence eliminating Nc we get,
Ecp – Ecn = kT ln (nn/np) =eVo
Therefore Vo = (kT/e) ln (nn/np)
At temperatures in the range 100K ≾ T ≾ 400K, the majority
carrier concentrations are equal to the doping levels, that is
nn = Nd and Pp = Na and remembering np = ni
2

We can write the above equation as
Vo = kT/e ln (Na Nd/ ni
2)
From the above equations we get a very
useful relationship between the carrier
concentrations on the two sides of the
junction, that is given by
np = nn exp (-eVo /kT)
and similarly Pn = Pp exp (-eVo /kT)

 An “Equilibrium” or balance will be established
when the majority carriers are equal and both
moving in opposite directions, so that the net
result is zero current flowing in the circuit. When
this occurs the junction is said to be in a state of
“Dynamic Equilibrium“.
 The minority carriers are constantly generated due
to thermal energy so this state of equilibrium can
be broken by raising the temperature of the PN
junction causing an increase in the generation of
minority carriers, thereby resulting in an increase
in leakage current but an electric current cannot
flow since no circuit has been connected to the PN
junction.

 2. Current flow in Forward Biased PN Junction Diode:
 if the equilibrium situation is disturbed by connecting
a voltage source externally across the junction there will
be a net current flow.
The junction is said to be forward biased, if the p-region
is connected to the positive terminal of the voltage
source as shown in fig.(a)
As we mentioned in the last section, the depletion region
is very resistive in comparison to the bulk regions so that
the external voltage V is dropped almost entirely across
the depletion region
This has the effect of lowering the height of the potential
barrier to (Vo-V) as in fig.(b)

Consequently, majority carriers are able to surmount the
potential barrier much more easily than in equilibrium
case so that the diffusion current becomes much larger
than the drift current.
There is now a net current from p- to the n- region in the
conventional forward sense and carriers flow in from the
external circuit to restore equilibrium in the bulk regions
We note that with the application of an external potential
the Fermi-levels are no longer aligned across the junction
The reduction of the height of the potential barrier leads
to majority carriers being injected across the junction

On being so injected, these carriers immediately
become minority carriers, and the minority carrier
concentrations near to the junction rise to new values
np` and Pn` This establishes excess minority carrier
concentration gradients as shown in fig., so that the
injected carriers diffuse away from the junction.
This is the situation of minority carrier injection and
diffusion So considering the n- region, the injected
holes diffuse away from the junction recombing as they
do so.
The electrons lost in this recombination are replaced by
the external voltage source so that current flows in the
external circuit

A similar argument applies to the p-region , with roles of
electrons and holes reversed, it should be noted that the
majority carrier concentrations are not noticeably changed as
a consequence of the injection, unless the bias voltage is
almost equal to Vo, resulting in a very large current flow
The drift current is relatively insensitive to the height of the
potential barrier since all the minority carriers generated
within about a diffusion length of the edge of the depletion
region may diffuse to the depletion region and be swift across
it, whatever the size of the electric field there
With these arguments we can write the relationship between
the minority carrier concentrations in the bulk regions
adjacent to the depletion layer as

np` = nn exp[-e(Vo-V)/kT]
and pn` = pp exp[-e(Vo-V)/kT]
but for steady state equilibrium we have
Pn = Pp exp (-eVo /kT)
So, pn` = Pn exp(eV/kT)
This shows the excess minority carrier concentration will
decrease owing to the recombination in accordance with
steady-state equilibrium
so that we may write the excess hole concentration in the n-
region at a distance x from the edge of depletion layer as
∆p(x)= ∆p(0)exp(-x/Lh), where Lhis the hole diffusion length
So, in this case we can write ∆p(x) = pn`(x) – pn
and ∆p(0) is the value of ∆p(x) at x=0, that is (pn`- pn )

We can therefore write
∆p(0) = pn [exp(eV/kT) -1]
We have assumed a one dimensional carrier flow in the
X-direction , which is an acceptable approximation even
though in practice carrier flow occurs in three
dimensions
Now we have argued that the electric field in the bulk
regions are very small and therefore , adjacent to the
depletion layer in the n-region , and particularly at x=0,
the total current density will be due to diffusion only
Thus the current density due to hole motion in this case
becomes Jh = (eDh/Lh )∆p(0) exp(- x/Lh)

Here Dh is the hole diffusion coefficient
At x=0, we can write this, (substituting for ∆p(0)
Jh = (eDh/Lh) Pn[exp(eV/kT) -1]
There is a similar contribution due to electron flow, so
current density due to electron motion is given by
Je = (eDe/Le) np[exp(eV/kT) -1]
Hence the total current density
J = Jo [exp(eV/kT) -1]
where Jo = e[(Dh/Lh)Pn + (De/Le)np]
Jo is called the saturation current density

 In forward biased case, the negative voltage
pushes or repels electrons towards the junction
giving them the energy to cross over and combine
with the holes being pushed in the opposite
direction towards the junction by the positive
voltage.
 This results in a characteristics curve of zero
current flowing up to this voltage point, called the
“knee” on the static curves and then a high
current flow through the diode with little increase
in the external voltage as shown below.

The application of a forward biasing voltage on the junction diode results in the
depletion layer becoming very thin and narrow which represents a low impedance
path through the junction thereby allowing high currents to flow. The point at
which this sudden increase in current takes place is represented on the static I-V
characteristics curve above as the “knee” point.

3. Current flow in a reverse biased p-n junction:
In this case the external bias is applied so that the p-
region is connected to the negative terminal of the
voltage source as in fig.(a). This has the effect of
increasing the height of the potential barrier to Vo+V
(fig.(b), thereby reducing the diffusion current to
negligible proportions

The net current flow is therefore the drift current which is
directed in the conventional reverse sense , that is from
the n- to p- region
This results in carrier extraction rather than injection
because the minority carriers generated near the junction
diffuse to it and are swept across the depletion region.
The nearer the carriers are generated to the junction, the
greater is the probability of this occurring so that a
concentration gradient is formed towards the junction as
shown in fig.
The drift of the carriers across the junction is therefore fed
by the diffusion so that we may use precisely the same
argument to derive the current-voltage relationship as we
used in the previous case.

The same equation also apply in this case , only
difference being , of course that sign of V is changed

 The net result is that the depletion layer grows
wider due to a lack of electrons and holes and
presents a high impedance path, almost an insulator
and a high potential barrier is created across the
junction thus preventing current from flowing
through the semiconductor material.
 This condition represents a high resistance value to
the PN junction and practically zero current flows
through the junction diode with an increase in bias
voltage.
 However, a very small reverse leakage current does
flow through the junction which can normally be
measured in micro-amperes  ( μA ).

 One final point, if the reverse bias voltage VR applied to the
diode is increased to a sufficiently high enough value, it will
cause the diode’s PN junction to overheat and fail due to the
avalanche effect around the junction.
 This may cause the diode to become shorted and will result in the
flow of maximum circuit current, and this shown as a step
downward slope in the reverse static characteristics curve below.

Deviations from the simple theory
The theory discussed now is reasonable agreement with
what is observed in practice, there are several points of
difference
For our purposes one of the most important deviations is
that , at sufficiently large values of reverse bias ,
breakdown occurs. That is there is a sudden and rapid
increase in reverse current at a particular value of
reverse bias voltage as in fig.(b)
Reverse breakdown occurs by two mechanisms.
The first is called Zener effect  is due to quantum
mechanical tunnelling

This takes place most readily in heavily doped junctions,
which leads to narrow depletion layers and therefore high
junction fields
In effect we can see from fig.(a), the energy bands on two
sides of the junction become crossed so that the filled
states in the valence band of p-side are aligned with
empty states in the conduction band of n-side. Electrons
therefore tunnel from the p- to n- side vastly increasing
the reverse current.
The second mechanism is Avalanche breakdown  occurs
in lightly doped junctions with wide depletion layers
This mechanism involves impact ionization of the host
atoms by the energetic carriers

If the carriers crossing the depletion layer acquire
sufficient energy from the electric field between the
collisions, they may ionize lattice atoms on colliding with
them.
The electrons and holes so produced may, in turn, cause
further ionizing collisions and so on to generate an
avalanche of carriers
Neither breakdown mechanism is in itself destructive to
the junction. If, however, the reverse current is allowed
to become too large then Joule heating may cause
damage to the device
Other important deviations arise from such factors as
carrier generation and recombination within the depletion
layer

Carrier generation in the depletion layer gives
rise to a larger value of reverse bias current
than the simple theory predicts.
Optical generation of carriers within the
depletion layer may rise to an increase in
reverse current or initiate avalanche
breakdown if the reverse bias is sufficiently
great.
These phenomena form the basis of the
working of Photodiodes

Carrier Distribution in p-n Homojunctions:
The carrier distribution in p-n junctions depends on the
diffusion constant of the carriers. The diffusion constant of
the carriers is not easily measured ,more common is the
measurement of the carrier mobility and the diffusion
constant can be inferred from carrier mobility by Einstein
relation which for non-degenerate semiconductors , is
given by
Dn = (kT/e)μn & Dp = (kT/e)μn
Carriers injected into a neutral semiconductor , with no
external electric field applied , propagate by diffusion.
If the carriers are injected into a region with opposite
conductivity type, the minority carriers will eventually
recombine

The mean distance a minority carrier diffuses before
recombination is the diffusion length
Electrons injected into a p-type region will, on
average diffuse over the diffusion length Ln, before
recombining with holes.
This diffusion length is given by
Ln = √Dnτn & Lp = √Dpτp
Where τn and τp are electron and hole minority
carrier life-times respectively
In typical semiconductors , the diffusion lengths is of
the order of several micrometers

The distribution of carriers in a p-n junction under zero
and under forward bias is as shown in fig.(a) and (b)
below

Note that minority carriers are distributed over quite a
large distance
Furthermore the minority carrier concentration
decreases as these carriers diffuse further into
adjacent region . Thus recombination occurs over a
large region , with a strongly changing minority carrier
concentration
As shown, the large recombination region in
homojunctions is not beneficial for efficient
recombination
While in heterojunctions, carriers are confined by the
heterojunction barriers (fig.(c))

Carrier Distribution in p-n Heterojunctions:
All high-intensity light –emitting diodes do not use
homojunction design but rather employ heterojunctions,
which have clear advantages over homojunction devices
Heterojunction devices employ two types of
semiconductors , namely a small bandgap active region
and a large bandgap barrier region. If a structure consists
of two barriers , ie, two large bandgap semiconductors ,
then the structure is called a double heterostructure,
abbreviated as DH
The effect of heterojunctions on the carrier distribution
is shown in fig.(c) above

Carriers injected into the active region of the double
heterostructure are confined to the active region by
means of the barriers As a result the thickness of the
region in which carriers recombine is given by the
thickness of the active region rather the diffusion length
The consequences of this change are significant. We
assume that the thickness of the active region is much
smaller than the typical diffusion length
Diffusion lengths may range from 1μm to 20 μm. Thus
the carriers in the active region of a double
heterostructure have a much higher concentration than
carriers in homojunctions, which are distributed over
several diffusion lengths

It is also noted that a high concentration of carriers in the
active region increases the radiative recombination rate
and decreases the recombination life-time
For this reason, all high-efficiency LED designs employ
double heterostructure or quantum well designs.
The Effect of Heterojunctions on Device resistance:
The employment of heterostructures allows one to
improve the efficiency of LEDs by confining carriers to
the active region , thereby avoiding diffusion of minority
carriers over long distances
Heterostructures can also be used to confine the light to
waveguide regions ; in particular in Edge emitting LEDs

Generally modern semiconductor LEDs and Lasers have
many heterojunctions for contact layers, active regions
and waveguiding regions
Even though heterojunctions are useful for improved
LED designs, there are certain problems associated with
them as:
The resistance caused by the heterointerface
The origin of the resistance is illustrated in fig. (a),
which shows the band diagram of a hetero structure
The heterostructure consists of two semiconductors with
different band gap energy and it is assumed that both
sides of a heterostructure are of n-type conductivity

Carriers in the large band gap material will diffuse over
to small band gap material where they occupy
conduction band states of lower energy
As a result of electron transfer, an electrostatic dipole
forms , consisting of a positively charged depletion layer
with ionized donors in the large band gap material, and
a negatively charged electron accumulation layer in the
small band gap material
The charge transfer leads to the band bending
illustrated as in fig.(a) above
Carriers transferring from one semiconductor to the
other must overcome this barrier by either tunneling or
by thermal emission over the barrier

The resistance caused by heterojunctions can have a
strong deleterious effect on device performance,
especially in high power devices
The thermal power produced by heterostructure
resistances leads to heating of the active region, thereby
decreasing the radiative efficiency
It has been shown that these type of heterostructure
band discontinuities can be completely eliminated by
grading of the chemical composition of the
semiconductor in the vicinity of the heterostructure
The band diagram of a graded heterostructure is shown in
fig.(b) above

Inspection of the above figure reveals that there is no
longer a spike in the conduction band which hinders the
electron flow
It has been shown that the resistance of parabolically
graded heterostructures is comparable to bulk material
resistance.
Thus, the additional resistance introduced by abrupt
heterostructures can be completely eliminated by parabolic
grading.
The shape of the graded region should be parabolic due to
the following reason:
The large band gap material will be depleted of free
carriers due to electron transfer to small band gap
material.

Thus the charge concentration in the large band-gap material
will be donor concentration.
Assuming that the donor concentration ND is a constant
throughout the heterostructure, the solution of Poisson
equation yields the electrostatic potential Φ as
Φ = (eND/2ε) x2
where ε  the dielectric permittivity of semiconductor
The equation reveals that the potential depends quadratically
on the spatial co-ordinate x, ie, the potential has a parabolic
shape
In order to compensate for the parabolic shape of the
depletion potential , the composition of the semiconductor is
varied parabolically as well , so that an overall flat potential
results

It is assumed here that the parabolic variation of the
chemical composition results in a parabolic change of
the bandgap energy , ie, that the band gap energy
depends linearly on the chemical composition and that
bandgap bowing can be neglected.
Next, assume that the conduction band discontinuity
of an abrupt heterojunction is given by ∆Ec and that
the structure is uniformly doped with doping
concentration ND
Let us assume that carriers have transferred to the
small bandgap semiconductor, thus causing a depletion
region of thickness WD in the large bandgap
semiconductor

If the potential created in the depletion region is equal
to ∆Ec/e then electrons will no longer transfer to the
small bandgap material .
The thickness of the depletion region can be inferred
from the above equation as
WD = √ (2ε ∆Ec/e2ND)
A heterostructure interface should be graded over
distance WD in order to minimize the resistance
introduced by an abrupt heterostructure
This approximation provides an excellent guidance for
device design, steps can be taken to refine the
calculation

Grading is useful for all heterostructures adjoining the
active region. The effect of grading in a double
heterostructure is shown in fig. below

The composition and band diagram of an ungraded structure is
shown in fig. (a).
At both heterointerfaces barriers develop as a result of free
charge transferring to the active region.
These barriers increase the device resistance under forward
bias conditions
The case of graded heterointerfaces is shown in fig.(b)
shows two linearly graded regions cladding the active region
The band diagram illustrates that barriers at the
heterointerfaces can be effectively reduced or completely
eliminated by grading.
Note that the linear grading shown in fig.(b), results in small
spikes at the interfaces between the linearly graded and non-
graded regions

These spikes are a result of linear grading assumed and
would not result for parabolic grading.
Generally, the transport of carriers in heterostructures is
such that , the carrier transport within semiconductor
device should not generate unnecessary heat.
This is particularly true for high-power devices where the
additional heat generated inside the device leads to a
performance loss due to increased operating temperature
Finally , note that lattice matching is a desirable in all
heterostructure devices.
It is also desirable in graded structures in order to
minimize the number of misfit dislocations that act as
non-radiative recombination centers

Optoelectronics is a branch of physics and technology

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