SEMICONDUCTOR HETEROJUNCTIONS
A p-n junction is formed when a p-type doped portion of the semiconductor is
d with an n-type doped portion. As a fundamental component for functions
such as rectification, the p-n junction forms the basic unit of a bipolar transistor.
If both the p-type and the n-type regions are of the same semiconductor
material, the junction is called a homojunction. If the junction layers are made of
different semiconductor materials, it is a heterojunction.
As a matter of convention, if the n-type doped semiconductor material has
larger energy gap than the p-type doped material, it is denoted a p-N
heterojunction.
The use of capital and lowercase Ietters connotes the relative size
the energy gap. Conversely, if the p-type doped material has a larger energy
gap than the n-type material, the junction is referred to as a P-n heterojunction.
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Semiconductor Device Theory
EEE 6397 –
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Heterojunctions -2
When semiconductors of different band gaps, work
functions, and electron affinities are brought
together to form a junction, we expect
discontinuities in the energy bands as the Fermi
levels line up at equilibrium . The discontinuities in
the conduction band EC and the valence band EV
accommodate the difference in band gap between
the two semiconductors Eg.
In an ideal case, EC would be the difference in
electron affinities q(2 - 1), and Ev would be found
from Eg - EC This is known as the Anderson
affinity rule.
Eg  Eg1  Eg 2
 EC  EV
In practice, the band discontinuities are found experimentally for particular semiconductor
pairs.
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Semiconductor Device Theory
EEE 6397 –
20
Heterojunctions -4
To draw the band diagram, we need:
 depend on semiconduc tor

electron affinity (  ) material, NOT doping
band gap (E g )
and
depends on semiconduc tor
Work function 
material AND doping
The electron affinity and work function are
referenced to the vacuum level. The electron affinity
and work function are referenced to the vacuum
level.
The true vacuum level (or global vacuum level), Evac, is the potential energy reference when
an electron is taken out of the semiconductor to infinity, where it sees no forces. Hence, the
true vacuum level is a constant
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Semiconductor Device Theory
EEE 6397 –
21
Heterojunctions -5
However, since the electron affinity is a
material parameter and therefore constant
we need to introduce the new concept of
the local vacuum level, Evac(loc), which
varies along with and parallel to the
conduction band edge, thereby keeping the
electron affinity constant. The local vacuum
level tracks the potential energy of an
electron if it is moved just outside of the
semiconductor, but not far away.
P-side
n-side
The difference between the local and global vacuum levels is due to the electrical work done
against the fringing electric fields of the depletion region, and is equal to the potential energy
qV0 due to the built-in contact potential V0 in equilibrium. This potential energy can, of
course, be modified by an applied bias.
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Semiconductor Device Theory
EEE 6397 –
22
Heterojunctions -6
To draw the band diagram for a heterojunction accurately, we must not only use the proper
values for the band discontinuities but also account for the band bending in the junction.
To do this, we must solve Poisson's equation across the heterojunction, taking into account
the details of doping and space charge, which generally requires a computer solution. We
can, however, sketch an approximate diagram without a detailed calculation. Given the
experimental band offsets EV and EC, we can proceed as follows:
1. Align the Fermi level with the two
semiconductor bands separated.
Leave space for the transition region.
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Semiconductor Device Theory
EEE 6397 –
23
Heterojunctions -7
2. The metallurgical junction (x = 0) is located
near the more heavily
doped side. At x = 0 put EV and EC, separated
by the appropriate
band gaps.
3. Connect the conduction band and valence
band regions, keeping t he
band gap constant in each material.
Steps 2 and 3 of this procedure are where the exact band bending is important and must be
obtained by solving Poisson's equation. In step 2 we must use the band offset values EV and
EC for the specific pair of semiconductors in the heterojunction.
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Semiconductor Device Theory
EEE 6397 –
24
Heterojunctions -8
Consider heavily n-type AlGaAs is grown
on lightly doped GaAs where the
discontinuity in the conduction band
allows electrons to spill over from the N+AlGaAs into the GaAs, where they become
trapped in the potential well.
As a result, electrons collect on the GaAs side of the heterojunction and move the Fermi
level above the conduction band in the GaAs near the interface. These electrons are
confined in a narrow potential well in the- GaAs conduction band. If we construct a device
in which conduction occurs parallel to the interface, the electrons in such a potential well
form a two-dimensional electron gas with very negligible impurity scattering in the GaAs
well, and very high mobility controlled almost entirely by lattice scattering (phonons).
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Semiconductor Device Theory
EEE 6397 –
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p+-N Heterojunction under equilibrium -1
Consider, for example, a p-N heterojunction formed with a p-type GaAs layer doped at
3x1019 cm-3 and an N-type AIGaAs layer doped at 1x1016 cm-3
The band diagrams of individual layers are shown in the figure side by side with their
vacuum levels aligned to illustrate the difference in the Fermi levels in the chargeneutral condition .
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EEE 6397 – Semiconductor Device Theory
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p+-N Heterojunction under equilibrium -2
When the two regions are brought into contact, the Fermi level on the N side is
initially higher than that on the p-side. Therefore, the electrons in a p-N junction lend to
flow from regions of higher Fermi level (N regions) to regions or lower Fermi levels (p
regions).
On the other hand, the holes flow from the region with a lower Fermi level toward the
region with a higher Fermi level. Therefore, the holes flow from the p- type layer toward
the N-type layer consistent with the fact that there are more holes in the p-type
region and they tend to diffuse to the region with less holes.
As these mobile carriers move toward the other sides, they leave behind the
uncompensated dopant atoms near the junction. In the p-type region, the
uncompensated acceptors are negative ions. In the N-type region, the uncompensated
donors are positive ions. Therefore, the carrier diffusion results in a electric field pointing
from the N-type region to the p-type region. This electric field retards further electron
diffusion from the N-type toward the p-type as well as hole diffusion in the opposite
direction. Hence, there is a natural negative feedback mechanism such that the more
carrier movement takes place, the larger the electric field becomes and the tendency of
the carrier movement is reduced.
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EEE 6397 – Semiconductor Device Theory
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p+-N Heterojunction under equilibrium -3
Eventually, an equilibrium condition sets in and the tendency of the carrier diffusion is
exactly counterbalanced by the electric field that impedes the carrier movement.
At this precise movement, the Fermi levels at the two regions are aligned. The above
exchange of carriers occurs on a very short time scale. So the whole process can be
thought of as instantaneous.
The two side immediately adjacent to the junction where the dopants become
uncompensated are called the depletion region, meaning that they are depleted of
mobile carriers. (Another name for the depletion region is the space-charge region).
At the two extreme ends away from the actual junction, carrier movement has never
occurred. Their dopant atoms are still compensated by their respective electrons
or holes. These are called the neutral regions because the net charge concentrations
there are zero.
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EEE 6397 – Semiconductor Device Theory
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p+-N Heterojunction under equilibrium -3
Fermi level during thermal
equilibrium lines up throughout
the entire semiconductor.
At regions far away from the
junction where the semiconductor
remains neutral, the relative
positions of Ef with respect to
Ec and to Ev are unmodified from
those prior to the joining of the
two sides.
The conduction and valence band
edges at the sides are not at the
same level, since it is the Fermi
level that must be aligned.
The conduction and valence band edges across
the depletion region must be connected
somehow to form continuous curves.
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EEE 6397 – Semiconductor Device Theory
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p+-N Heterojunction under equilibrium -4
The Poisson equation must be solved to ascertain the manner in which the conduction and
valence band edges are connected across the depletion region:
d
q

( p  n  Nd  Na )
dx S
Let us define that x = 0 correspond to the junction boundary, with the p side in the
-x direction and the N side in the + x direction. Moreover, x = - Xpo is the boundary separating
the neutral and the depletion regions on the p side, and XN0 the boundary on the N side. The
subscript 0 emphasizes that we are considering the thermal equilibrium condition.
Considering that, in the p-side depletion region, the net charge concentration is the negative
acceptor density and in the N-side depletion region the net charge concentration
is the donor density (depletion approximation). Therefore, the last equation simplifies to
d
q

N a for -X p 0  x  0
dx
p
d
q

Nd
dx N
© Nezih Pala npala@fiu.edu
for 0  x  X N 0
EEE 6397 – Semiconductor Device Theory
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p+-N Heterojunction under equilibrium -5
After integrating and applying the boundary conditions that the electric fields at – Xp0 and at
XN0 are zero, the electric field within the depletion region can be found as
 ( x)  
q
N a (x  X p 0 ) for -X p 0  x  0
p
 ( x)  
q
N d ( X N 0 - x)
N
for 0  x  X N 0
In the p-side, the potential profile is obtained by integrating (x)
q
x2
V ( x)     ( x)dx 
N a (  X p 0 x)  C for -X p 0  x  0
p
2
Since it is the relative value or potentials rather than their absolute values that is of
importance, we arbitrarily define the zero potential at a convenient location, namely,
at x =-Xpo .With the boundary condition that V(- Xpo) = 0, V(x) is written as
2
X p0
q
x2
V ( x) 
Na (  X p0 x 
)
p
2
2
© Nezih Pala npala@fiu.edu
for -X p 0  x  0
EEE 6397 – Semiconductor Device Theory
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p+-N Heterojunction under equilibrium -6
The built-in potential on the p side (p0 ) is the difference in the potentials at x=0 and x=-Xp0. It
is readily verified that
 p0 
q
N a X 2p0
2 p
where p0 is a positive number. The band diagram, being an energy diagram for the
negatively charged electron, shows that the electron energy decreases with the associated
increase in V(x).
Similarly the potential profile on the N side can be obtained by integrating the appropriate
electric profile. Taking the boundary condition that V(0) = p0,

q
x2 
V ( x) 
N d  X N 0 x -   p0
N
2 

for 0  x  X N 0
The built-in potential on the N side ( N0 ) is the difference in the potentials at
x = 0 and x = XN0. V(XN0) - V(0) is then equal to
N 0
© Nezih Pala npala@fiu.edu
q

N d X 2N0
2 N
EEE 6397 – Semiconductor Device Theory
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p+-N Heterojunction under equilibrium -7
The overall built-in junction potential (bi ) is the potential difference from the neutral region
on one side the neutral region on the other. It is equal to
bi  N 0   p 0 
q
q
N d X 2N0 
N a X 2p0
2 N
2 p
Xp0 and XN0 can be determined by solving two linearly independent equations relating these
two variables. One of the required equations is obtained by enforcing the continuity of the
electric flux density D = in the absence of an interfacial charge density at the junction:
N d X N0  N a X p0
The equality ensures that charge neutrality exists in the overall p-N junction. The second
equation relating Xp0 and XN0 is found by taking the ratio of the built ·in potentials on the N
side to those on the p side:
N 0 p N d X N0

 p 0 N N a X 2p0
2
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EEE 6397 – Semiconductor Device Theory
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p+-N Heterojunction under equilibrium -8
Substituting the charge conservation relationship into the above equation, we rewrite the
ratio as
 N 0 p N a

 p 0 N N d
From this equation, the built-in potential of the junction is
 N 0 
 p N a 


 p 0
bi  1 
 p 0  1 

 N N d 
  p0 
Alternatively, from an examination
of the band diagram bi is equal to
bi 
Egp  Ec   p  N
q
where Egp is the energy gap of the p-side
material
© Nezih Pala npala@fiu.edu
EEE 6397 – Semiconductor Device Theory
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p+-N Heterojunction under equilibrium -9
p and N represent the differences of the Fermi levels with respect to the valence band edge
and the conduction band edge, respectively, in the neutral regions. That is,
 p  E f  Ev | x 
N  Ec  E f | x 
Both p and N are calculated from the equilibrium statistics equations given in
n  Nce
 E f  Ec

 kT



p  Nve
 Ev  E f

 kT



Once the doping levels are specified,
bi for the heterojunction is readily
calculated.
Other quantities such as N0 and
V(x) are then obtained from the
Equations derived above.
© Nezih Pala npala@fiu.edu
EEE 6397 – Semiconductor Device Theory
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p+-N Heterojunction under equilibrium -10
EXAMPLE
Let us draw a band diagram for a (p) GaAs/(N) AI0.35Ga0.65As. As heterojunction at thermal
equilibrium. The doping level and relevant material parameters are
Egap (GaAs) =1.42 eV Egap (AlAs) =2.16 eV
Nd=1x1016 cm-3 Na=3x1019 cm-3
n  NC e
p 
 E f  Ec

 kT
Ev  E f
kT



 n
 n  E f  Ec  kT ln 
 Nd
 p
 ln 
 NV

 p
  A1 

 NV

 p
  A2 

 NV

 3.72 1017 
  0.0259 ln 
  0.093eV
16
 110


2

 p
  A3 

 NV
3

 p
  A4 

 NV
4

  0.103eV

With A1=3.53x10-1, A2=-4.95x10-3, A3=1.48x10-4, A4=-4.42x10-6 Nv=4.7x1018 cm-3
© Nezih Pala npala@fiu.edu
EEE 6397 – Semiconductor Device Theory
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p+-N Heterojunction under equilibrium -11
Energy gap of the AlxGa1-xAs material is determined by the  valley at x<0.45, but is
determined by the X valley at x>0.45. The energy gap (in eV) is given as:
1.424  1.247 x for 0  x  0.45 
Eg ( Al xGa1 x As )  
  1.86eV
2
1.90  0.125 x  0.143x for x  0.45
The energy gap difference (Eg) in an AlxGa1-xAs/GaAs heterojunction is shared between the
conduction band and the valence band. That is Eg= Ec+ Ev where Ec is the conduction
band discontinuity and Ev is the valence band discontinuity. The amount of Ev is linearly
dependent on the aluminum mole fraction (x) for all values of x. Ev and Ec in an AlxGa1xAs/GaAs heterojunction are therefore given by
Ev ( x)  0.55x
0  x  0.45
1.247  0.55 x
Ec ( x)  
2
0
.
476

(
0
.
125

0
.
55
)
x

0
.
143
x
x  0.45

E0. is 0.244 eV. Egp the energy gap of the narrower gap material (GaAs), is 1.424 eV.
Therefore, the built-in voltage is found
bi  1.424  0.244  (0.103)  0.093  1.678eV
© Nezih Pala npala@fiu.edu
EEE 6397 – Semiconductor Device Theory
37
p+-N Heterojunction under equilibrium -12
The relative dielectric constants are 13.18 for GaAs and 12.09 for Al0.35Ga0.65As. p0 can be
found as
 p N a 
 p N a 



bi  1 
 p 0   p 0  bi /1 

 N N d 
 N N d 
 13.18(3 1019 ) 
4

 1.678 / 1 

5
.
13

10
V
16 
 12.09(110 ) 
N0 which is equal to bi - p0 is 1.678 - 5. 13 x 10-4  1.6775 V. It is clear from this calculation
that if one side is significantly more heavily doped than the other practically all of the built in potential drops across the depletion region of the lightly doped side. Once N0 and p0 are
determined, XN0 and Xp0 are calculated from
N 0
2 N
q
2(12.09)(8.85 1014 )
2
-5

N d X N0  X N0 
N 0 
1
.
677

4.74

10
cm
19
16
2 N
qN d
(1.6 10 )(110 )
 p0
2 p
q
2(13.18)(8.85 1014 )
2

N a X p0  X p0 
 p0 
5.13 104  1.58 10-8 cm
19
19
2 p
qN a
(1.6 10 )(3 10 )
© Nezih Pala npala@fiu.edu
EEE 6397 – Semiconductor Device Theory
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p+-N Heterojunction under equilibrium -13
The depiction thickness of the heavily
doped side is merely 1.58 which is
practically zero. With the calculated
parameters, the band diagram is drawn as
shown in the figure.
The band profiles vary parabolically with
position as given by equations:
2
X p0
q
x2
V ( x) 
Na (  X p0 x 
)
p
2
2

q
x2 
V ( x) 
N d  X N 0 x -   p0
N
2 

© Nezih Pala npala@fiu.edu
EEE 6397 – Semiconductor Device Theory
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Semiconductor Heterojunctions
Abrupt p-n heterojunction
N-type
p-type
To construct a band diagram for an abrupt p-n heterojunction, we proceed in the same
manner as for the p-n homojunction. We begin with two separate materials. Since the
materials have different bandgaps, there must exist a discontinuity in the conduction band
(ΔEc) and/or the valence band (ΔEv) at the interface . The difference in the bandgap between
the two materials is equal to the sum of the conduction band and valence band discontinuities
Eg  Eg1  Eg 2  Ec  Ev
© Nezih Pala npala@fiu.edu
EEE 6397 – Semiconductor Device Theory
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Graded p-n heterojunction
Although the abrupt p-n heterostructure discussed in the previous section did result in an
increase in the barrier to hole injection, the notch in the conduction band at the interface
also caused an undesirable increase in the barrier to electron injection. While the net effect
was still an increase in the ratio In/Ip, eliminating this notch further increases In/Ip to a value
 In 
DN L
 E
   n d p exp  g
k T
I 
D
N
L
p a n
 B
 p  het
© Nezih Pala npala@fiu.edu
EEE 6397 – Semiconductor Device Theory



41
Graded p-n heterojunction -2
In order to reduce this notch, the bandgap of the
p material can be graded upwards from the
junction, as shown in the figure. For example, in
an n-AlGaAs/p-GaAs graded heterostructure, the
n material is GaAs at the junction and is graded to
the final AlGaAs composition over a short
distance. The final shape of the notch depends on
the length and profile of the grade; longer grading
typically gives a smaller notch. However, it is
important that the grade is contained well within
the depletion region. If the grade ends outside the
depletion region, then the barrier seen by holes
decreases, thus reducing the benefits of the
heterojunction. Note that the barrier to holes in
both abrupt and graded heterojunctions is the
same. It is just the barrier for electron flow that is
reduced in the graded structures, allowing for the
increased ratio of In/Ip.
© Nezih Pala npala@fiu.edu
EEE 6397 – Semiconductor Device Theory
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Example: Designing a p-n heterojunction grade
Consider four different n-p+ Al0.3Ga0.7As/GaAs heterojunctions with ND = 1017 and
NA = 5× 1018. The AlGaAs in these junctions is graded from x = 0 to x = 0.3 over
XGrade = 0(abrupt), XGrade = 100°A, XGrade = 300°A, and XGrade = 1μ. Calculate and plot the energy
band diagrams for the above four cases.
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EEE 6397 – Semiconductor Device Theory
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Example: Designing a p-n heterojunction grade
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EEE 6397 – Semiconductor Device Theory
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Quasi-electric fields
In a homogeneous semiconductor, the
separation between the conduction and
valence bands is everywhere equal to the
semiconductor bandgap. Any electric field
applied to the material therefore results in an
equal slope in the conduction and valence
bands, as indicated in the figure.
When a hole or electron is placed in this
structure, a force of magnitude eE will act on
the particle.
The magnitude of the force is equal to the
slope of the bands and is the same for both
electrons and holes. However, the direction of
the force is opposite for the two particles.
© Nezih Pala npala@fiu.edu
EEE 6397 – Semiconductor Device Theory
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Quasi-electric fields -2
An interesting phenomenon arises in
semiconductors with graded bandgaps, such as
the bipolar transistor emitter-base structure
shown at the top figure .
In the graded region, the bandgap is not constant,
so the slopes in the conduction and valence bands
are no longer equal. Hence the forces acting on
electrons and holes in this region are no longer
equal in magnitude.
It is in general possible for a force to act on only
one type of carrier, as shown in the top figure, or
for forces to act in the same direction for both
electrons and holes, as in the bottom figure. Such
behavior cannot be achieved by pure electric
fields in homogeneous materials. These fields,
which were first described by Herbert Kroemer in
1957, are therefore referred to as quasi-electric
fields.
© Nezih Pala npala@fiu.edu
EEE 6397 – Semiconductor Device Theory
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Quasi-electric fields -3
In a given material, the total field acting on a hole or an electron is always the sum of the
applied field and the quasi field, or
 e,tot   app   e,quasi
 h,tot   app   h,quasi
The applied field, which results from applying a voltage difference between the ends of the
material, will always be the same for both electrons and holes, but the quasi field could be
different for both. The band profiles in figures in the last slide can therefore be achieved in a
number of different ways. For example, the profile in top figure could be achieved in the
following two ways:
1. An undoped (intrinsic) material with a graded composition and zero applied electric field
typically results in the profile in the lower figure. If an electric field app = −e,quasi is then
applied to this material, the resulting profile will be the one shown in the upper figure.
2. A uniformly doped n-type material with a graded composition and zero applied electric
field will also result in the profile in the upper figure. In this case, the doping ensures that
the separation between the conduction band and the Fermi level remains approximately
constant. Notice that the resulting quasi-electric field in this structure acts only on minority
carriers.
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EEE 6397 – Semiconductor Device Theory
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Quasi-electric fields -4
Quasi-electric fields provide engineers with additional tools that can be
exploited in device design. They have proven to be very useful in decreasing
transit times in devices that rely on minority carrier transport.
For example, in bipolar technology, a highly doped graded base layer is often
used to speed up the transport of minority carriers from the emitter to the
collector. For a base with uniform bandgap, minority carriers injected from the
emitter must diffuse across the base, a process that is generally slow. By using a
highly doped graded base to generate a quasi-electric field, such as was
described in the second example above, minority carriers can be swept across
much more quickly, thus reducing the base transit time and improving the
device RF performance.
© Nezih Pala npala@fiu.edu
EEE 6397 – Semiconductor Device Theory
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