PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 1 of 35
Chapter 6.
Semiconductor Junctions
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 2 of 35
In chapter 5, we learned (in horrifying detail) about the physics of bulk
semiconductors, and how their conductivity can be altered by doping (permanently)
and by charge carrier generation (temporarily). So far, there’s not a whole lot of
interesting things that we can actually do with a semiconductor...I suppose that it can
be used as a temperature sensor (since n and p - and therefore σ - vary with
temperature). You can probably guess that it can be used as a light detector as well,
since incident light can create charge carriers, contributing to a measurable current.
The considerable benefit of semiconductors is only apparent when they are formed
into junctions; that is, when two dissimilar semiconductors share a physical
boundary. In this chapter, we will see what happens when such junctions exist. This
chapter will consider only electrical properties of junctions; we will look at optical
properties starting in chapter 7.
In a more practical sense, this chapter exists to partly compensate for the fact that
our department no longer offers PC322 (Solid State Device Physics).
While this chapter is contained for the most part in chapter 5 of Numai, some of the
material is taken from the following volume - it’s an excellent reference and can be
found in the WLU library (in a previous edition):
B.G. Streetman, Solid State Electronic Devices, 4th ed. Prentice Hall, 1995.
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 3 of 35
Outline
Table of Contents
1
The pn Junction
Drift, Diffusion, Equilibrium, and Space-Charge Layers
Abrupt Junctions - Fields and Voltages
Forward and Reverse Bias
Junction Capacitance
Graded Junctions
2
Diode Equations
Forward and Reverse Bias
Reverse Breakdown
3
Heterojunctions
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 4 of 35
The pn Junction
Depletion & Space-Charge Layers TN§5.3
What happens when a p-type semiconductor and an n-type semiconductor are
brought into physical contact? We will start by assuming that we have a
homojunction - that is, the semiconductor alloys are identical; they differ only in the
type of doping. From chapter 5, we recall that both materials will have the same
bandgap Eg , but the position of the Fermi level EF will be different. The physical
boundary between the materials is called the metallurgical junction, M .
Before the semiconductors are brought into contact, they are both individually in
thermodynamic equilibrium. Both are electrically neutral, and there is a uniform
density of electrons in the conduction band of the n-doped SC and a uniform density
of holes in the valence band of the p-doped SC (these 2 densities aren’t necessarily
the same - we’re not claiming that Nd in the n-type SC is equal to Na in the p-type
SC; in fact, that’s usually not the case).
When the semiconductors are brought into contact, they are no longer in equilibrium,
since there is a concentration gradient of both electrons and holes at the junction.
As a result, electrons will diffuse from the n-side to the p-side, while holes will
diffuse from the p-side to the n-side, in what is termed the “diffusion current”.
When equilibrium is restored, the Fermi level EF will be constant across M .
Figure 1(a) illustrates this diffusion.
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 5 of 35
The pn Junction
Depletion & Space-Charge Layers cont’
TN§5.3
As a result of this diffusion, the p-side is depleted of holes while the n-side is
depleted of electrons; in both cases, ionized acceptors (donors) of concentration
Na (Nd ) are left behind. This produces a narrow depletion layer or space-charge
layer around M , as shown in Fig. 1(b). The properties of this depletion layer are very
important to the use of this junction in a functional device - we will analyze them in
disturbing detail in the upcoming slides. The small number of electrons now on the
p-side, and holes now on the n-side, are termed “minority carriers”, while the large
number of electrons on the n-side and holes on the p-side are “majority carriers”.
Conduction Electron
Hole
n
p
Depletion Layer
(a)
Fig. 1:
Ionized Donor
Ionized Acceptor
n
p
Depletion Layer (Space-Charge Layer)
(b)
(a) Carrier diffusion at the interface of a pn junction. (b) The resulting depletion layer. The thickness of the depletion
layer, ℓD , is not necessarily divided evenly on each side of M, if Na , Nd . See TN Fig. 5.7.
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 6 of 35
The pn Junction
Diffusion and Built-in Potential TN§5.3
From chapter 5, we can define the electrical potential of the two semiconductors
based on how far their Fermi level shifts from its intrinsic value due to the doping. If
we define the intrinsic Fermi levels of the n- and p-doped regions as Ein and Eip ,
respectively† , then the electrical potential of the n-type SC is
φn ≡
kB T Nd
1
(EF − Ein ) =
ln
,
e
e
ni
(6.1)
and, equivalently for the p-type SC,
φp ≡
1
kB T Na
(EF − Eip ) = −
ln
.
e
e
ni
(6.2)
Here, EF is the Fermi level of the pn junction, which is constant across the
junction since it’s now in thermodynamic equilibrium.
At the interface, a potential difference φD is generated; we can express it in two
forms:
Eg
kB T Na Nd
kB T Na Nd
φD = φn − φp =
ln
≡
+
ln
.
(6.3)
e
e
e
Nc Nv
ni2
† in the case of a homojunction,
Ein = Eip by definition, since they’re both the same intrinsic material. Numai is simply
anticipating the case where the n- and p-type semiconductors might be different alloys.
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 7 of 35
The pn Junction
Diffusion and Built-in Potential cont’
TN§5.3
φD is called the diffusion potential or built-in potential (other texts give it the
subscript “b” or “bi” or “0”). The presence of this potential causes an electric field to
be present within the depletion region. This in turn produces a drift current of the
carriers, which exactly balances the diffusion current. As a result, zero net electric
current flows across the pn junction (unless, of course, we apply an external
bias...but that comes later).
Abrupt Junctions - Fields and Voltages TN§5.3
Here, we consider abrupt junctions, in which there is a sharp transition between
the n- and p-doped regions. These would occur if an n-doped material is deposited
onto a p-doped substrate (or vice versa), for example. Abrupt junctions are fairly
common in optoelectronic devices, less so in electronic devices such as diodes and
transistors. Fig. 2 illustrates the dopant distribution in an abrupt pn junction. By
convention, we place the origin (x = 0) at M .
I should point out here that “pn” has become the common term...“np” would work just
as well. In Numai’s figures, he consistently shows the n-doped region on the left and
the p-doped region on the right. Be aware that other texts may not follow this
convention, in which case their figures will appear to be flipped with respect to M .
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 8 of 35
The pn Junction
n
p
n
p
0
Depletion Layer
Fig. 2:
Impurity concentration distribution in an abrupt pn
junction. See TN Fig. 5.8.
Fig. 3: Carrier concentration in an abrupt pn junction. See
TN Fig. 5.9.
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 9 of 35
The pn Junction
Diffusion and Built-in Potential cont’
TN§5.3
In Fig. 3, we show the spatial distribution of the carrier concentrations n and p . The
boundaries of the depletion layer in the n- and p-regions are labelled x = −ℓn and
x = ℓp , respectively. We assume that all donors and acceptors are ionized, and thus
the region −ℓn ≤ x ≤ ℓp is depleted completely; the values of ℓn and ℓp will be derived
shortly.
Fig. 4 illustrates the spatial distribution of charge density ρ. ρ will be positive in the
n-doped region due to the depletion of electrons, and negative in the p-doped region
due to the depletion of holes. The magnitude of ρ depends on the dopant
concentrations Nd and Na . Since in this case we have Nd > Na (Fig. 2), this
magnitude is greater on the n-side. However, we are still operating under the
condition of electrical neutrality. Therefore, the depletion region on the n-side is
proportionally narrower than that on the p-side (ℓn /ℓp = Na /Nd ).
The charge densities are, of course, simply the donor and acceptor densities
multiplied by the charge of the respective carriers (we assume that all dopant atoms
are ionized):


eNd (n-side, −ℓn ≤ x < 0),



−eNa (p-side, 0 ≤ x ≤ ℓp ),
ρ=
(6.4)



0 otherwise.
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 10 of 35
The pn Junction
n
p
n
Depletion Layer
Fig. 4: Electric charge density in an abrupt pn junction.
See TN Fig. 5.10.
p
Depletion Layer
Fig. 5:
5.11.
Electric field in an abrupt pn junction. See TN Fig.
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 11 of 35
The pn Junction
Diffusion and Built-in Potential cont’
TN§5.3
To determine the voltage and field profiles within the depletion region, we turn to
Poisson’s equation, ∇2 φ = −ρ/ε. In our case, assuming that there is no y − or
z −dependence in ρ,
 eN

 − εn εd (n-side, −ℓn ≤ x < 0),

d2φ 
 eNa0
(6.5)
=
(p-side, 0 ≤ x ≤ ℓp ),

εp ε0

dx 2


0 otherwise.
Here, εn and εp are the relative permittivities of the two sides; Numai again allows for
the possibility of dissimilar alloys, εn , εp , but in the homojunction case they will be
the same.
The electric field Ex in the depletion region is found by integrating the previous
equations with respect to x , then applying the boundary conditions that Ex = 0 in the
electrically neutral sections outside of this region. This results in
 eN

 εn εd (x + ℓn ) (n-side, −ℓn ≤ x < 0),

dφ 
 eNa0
Ex = −
=
(6.6)
− εp ε (x − ℓp ) (p-side, 0 ≤ x ≤ ℓp ),

0

dx


0 otherwise;
This is illustrated in Fig. 5. If εn = εp , Ex must be continuous at the boundary, which
reaffirms our result that ℓn /ℓp = Na /Nd .
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 12 of 35
The pn Junction
Diffusion and Built-in Potential cont’
TN§5.3
Next, we can integrate the previous equations to find the electrical potential profile
within the depletion region. Since absolute values of potential are meaningless, we
can use the arbitrary condition that φ = 0 at M , with the result that (see Fig. 6)


φn (neutral n-type SC, x < −ℓn ),



eNd
2


 − 2εn ε0 (x + 2ℓn x ) (n-side, −ℓn ≤ x < 0),
φ=
(6.7)

eNa

(x 2 − 2ℓp x ) (p-side, 0 ≤ x ≤ ℓp ),


 2εp ε0

φp (neutral p-type SC, x > ℓp ).
(It should be pointed out here that most other texts choose to let φ = 0 in the neutral
p-side, far from the junction).
Finally, recall that the energy of the electron / hole is represented by ∓e φ. Therefore,
the conduction band and the valence band must bend throughout the
depletion region. This is illustrated in Fig. 7.
Incidentally, we noted earlier that Numai’s convention is to draw the n material on the
left and the p material on the right. If it were drawn the other way around, the electric
field in Fig. 5 would be negative, since it would be oriented in the −x direction.
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 13 of 35
The pn Junction
n
p
Depletion Layer
Fig. 6:
Fig. 5.12.
Electrical potential in an abrupt pn junction. See TN
n
p
Depletion Layer
Fig. 7: Bending of energy bands in an abrupt pn junction.
See TN Fig. 5.13.
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 14 of 35
The pn Junction
Depletion Layer Thickness TN§5.3
Up until now, we have been describing the depletion layer using ℓD = ℓn + ℓp . We
are now in a position to calculate these values.
Assume that the relative dielectric constants on the n- and p-sides are equal:
εn = εp ≡ εs (I can’t see why Numai doesn’t just use εr here, but I’m following him
nonetheless). The resulting continuity of Ex at M tells us that
EM =
eNd
eNa
ℓn =
ℓp .
εs ε0
εs ε0
Next, since Ex = −d φ/dx , we have
Z
(6.8)
ℓp
−φD =
Ex dx ;
(6.9)
−ℓn
that is, the negative of the built-in potential is the area under the Ex vs. x triangle;
EM ℓD
eNd ℓn ℓD
φD =
=
.
(6.10)
2
2εs ε0
However, since ℓn Nd = ℓp Na , and ℓD = ℓn + ℓp , we can write ℓn = ℓD Na /(Na + Nd ),
and thus
q
Na Nd
φD =
(6.11)
ℓ2 .
2εs ε0 Na + Nd D
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 15 of 35
The pn Junction
Depletion Layer Thickness cont’
TN§5.3
Finally, we can solve for the depletion layer thickness as a function of the built-in
potential and the doping concentrations:
s
s
2εs ε0 φD Na + Nd 2εs ε0 φD 1
1
=
+
.
(6.12)
ℓD =
e
Na Nd
e
Na
Nd
While this form of ℓD will prove useful in a moment, it is also possible to express ℓD
purely as a function of the dopant concentrations, with the aid of Eq. (6.3):
s
!
1
1
2εs ε0 kB T
Na Nd
ln
ℓD =
+
.
(6.13)
2
e
Na
Nd
ni
We can also calculate the individual contributions to ℓD :
s
"
#
2εs ε0 φD
ℓD Nd
Nd
ℓp =
=
,
Na + Nd
e
Na (Na + Nd )
s
"
#
2εs ε0 φD
ℓD Na
Na
ℓn =
=
.
Na + Nd
e
Nd (Na + Nd )
(6.14)
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 16 of 35
The pn Junction
Depletion Layer Thickness cont’
TN§5.3
The preceding equations reaffirm our previous result that the depletion region
reaches farther into the n-side when Na > Nd , and vice versa.
Orders of Magnitude
For most structures of interest, Nd and Na have magnitudes of 1016 ∼ 1019 cm−3 (by
way of comparison, the atomic density of most semiconductors is 1022 ∼ 1023 cm−3 ,
and the intrinsic carrier concentrations are ni = 106 ∼ 1010 cm−3 ). The built-in
voltage will be on the order of 1 V - and hence Ec and Ev will bend by about 1 eV while the depletion region width will be on the order of a few hundred nm. When one
side is much more heavily doped than the other, the width of that portion of the
depletion region can be as small as a few Angstroms. Of future interest to us is the
fact that depletion region widths are, in general, smaller than the wavelength of
visible / IR light.
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 17 of 35
The pn Junction
Forward and Reverse Bias
So far, we’ve only examined the band diagrams for pn junctions in equilibrium; that
is, with no voltage applied across the junction. Here, we will take a mostly qualitative
look at what happens to Ec , Ev , and EF when a bias voltage exists. Later in the
chapter, we will see how this bias results in a current flow.
First, we need to recall that energy bands tilt in the presence of an electric field (the
direction of the field points to the higher-energy end of the tilt). The slope of the tilt is
proportional to the field magnitude; we can already see this in the equilibrium case
by comparing Figs. 5 and 7. This tilting supports our knowledge that electrons will
drift to regions of lowest energy (i.e. in the direction opposite to the field), and vice
versa for holes.
When a voltage V is applied across the junction, it is dropped almost entirely across
the depletion region, which is much less conductive than the neutral regions
(remember, they are doped, so their conductivity is relatively high). By definition,
forward bias represents the case when the external potential is positive on the
p-side relative to the n-side. Reverse bias represents the opposite case. As we
will (eventually) see, a pn junction easily passes current under forward bias, but not
under reverse bias.
Figure 8 illustrates the concepts of the upcoming discussion...
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 18 of 35
The pn Junction
Fig. 8: Effects of an applied bias at a pn junction. (a)
equilibrium, (b) forward bias, (c) reverse bias. Top to bottom:
depletion region and electric field, electrostatic potential,
energy band diagram, particle flow and current directions.
This source uses different symbols than Numai: φD → V0 ,
V → Vf , Vr , e → q. Note as well that this figure, and all
figures taken from Streetman, have the p-side on the left
and the n-side on the right (Streetman).
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 19 of 35
The pn Junction
Forward and Reverse Bias cont’
Electrical Potential
The potential barrier at the junction is lowered by a forward bias, from φD to φD − V ,
since the forward bias raises the potential on the p-side relative to the n-side. For a
reverse bias, the potential barrier is increased (also to φD − V , but in this case V is
negative).
Electric Field Ex
Since a forward bias reduces the potential barrier, it must reduce the electric field
strength as well. A reverse bias increases the field strength.
Depletion Region Width ℓD
A change in the field strength at the junction M requires a change in ℓD , since a
proper number of ionized donors and acceptors must still be present even with the
changed value of Ex . Without proof, we’ll just agree that Eqns. (6.13) and (6.14) are
still valid, if φD is replaced by φD − V .
Energy Bands Ec , Ev
Since the position of the energy bands is proportional to the potential, we can say
that the height of the potential barrier, which is e φD in the equilibrium case, is
changed to e (φD − V ); that is, it is lowered under forward bias and raised under
reverse bias.
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 20 of 35
The pn Junction
Forward and Reverse Bias cont’
Fermi Energy EF
As for the Fermi energy under bias, far away from the depletion region EF it is still at
its equilibrium value (since there’s negligible field here). Therefore, when Ec and Ev
shift, there must be a separation of EF on either side of the junction. Under forward
bias, the Fermi level is higher on the n-side by a magnitude eV , while under reverse
bias, it is lower on the n-side.
Diffusion Current
The diffusion current consists of majority carriers (electrons in the n-side and holes
in the p-side) that surmount the potential energy barrier to diffuse to the other side.
Recall from the last chapter that the density of states and the Fermi-Dirac
distribution determine the energy distribution of the electrons and holes (n(E ) and
p (E )). Under equilibrium, a small number of these majority carriers reside far from
the bandedges, and they can contribute to an electric field in spite of the barrier.
Under forward bias, the barrier is lowered to φD − V , and many more carriers can
contribute to the diffusion current. Under reverse bias, the barrier is raised, and the
diffusion current is strongly suppressed. Given that n(E ) and p (E ) decay
exponentially away from the bandedges, one can imagine that the change in
diffusion current with applied voltage is highly nonlinear.
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 21 of 35
The pn Junction
Forward and Reverse Bias cont’
Drift Current
The drift current is actually insensitive to the height of the potential barrier. For one
thing, it’s composed of minority carriers, for whom the “barrier” is more like a
waterslide (electrons in the p-side, for example, easily drift down the barrier to the
n-side, and vice versa for holes). While a steeper barrier does increase the velocity
of these carriers, this is irrelevant - drift current depends on how many carriers are
contributing per unit time. Therefore, as shown in the figure, the drift currents do not
depend on the bias conditions.
Total Current
The total current is the sum of the 4 individual current components (drift and
diffusion, electrons and holes). As shown in the figure, both diffusion currents are
directed from p to n (the electrons and holes move in opposite directions, of course,
but their currents line up). Likewise, both drift currents are from n to p. In equilibrium,
the net current crossing the junction is zero, since the drift and diffusion components
cancel. Under reverse bias, both diffusion components are negligible due to the
large barrier, and the only current is the relatively small (essentially
voltage-independent) drift current directed from n to p. Under forward bias, the small
drift current is overwhelmed by a large diffusion current in the direction of p to n.
We’ll take a closer quantitative look at this a bit later in the chapter.
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 22 of 35
The pn Junction
Junction Capacitance TN§5.3
The pn junction consists of a spatial separation of positive and negative charges,
which fits the description of a capacitor. To calculate the value of the junction
capacitance, Cj , we return to Eq. (6.8), and note that the charge stored in the
depletion layer is
Q = eNd ℓn A = eNa ℓp A ,
(6.15)
where A is the area of the resistor (in the plane of the junction). Note that Numai
omits A and discusses capacitance “per unit area”; other than that, these derivations
are identical. By definition, capacitance is
s
e εs ε0 A 2
Na Nd
dQ Cj = ,
(6.16)
=
dV 2(φD − V ) Na + Nd
where we used Eqs. (6.12)-(6.14), including the case of non-zero bias V . It can be
easily shown that
εs ε0 A
Cj =
,
(6.17)
ℓD
as expected. From Eq. (6.16), we see that a pn junction is a voltage-controlled
capacitor, or varactor. More relevantly to us, since the junction has finite Cj and σ,
there will be an RC time constant that limits its high-frequency response.
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 23 of 35
The pn Junction
Graded Junctions TN§5.3
Many pn junctions aren’t abrupt; rather, there is a smooth transition between the
p-doped and n-doped regions. Numai derives and plots the profiles of dopant
concentrations, carrier concentrations (n and p ), charge density ρ, electric field Ex ,
electrical potential φ, and energy E for a linearly graded junction in pp. 109-113. We
won’t look into this concept any further in PC421.
The concept of such graded junctions is important, however. In many
microelectronic devices, junctions are not formed by growing one material on top of
another. Rather, they are formed by diffusing a dopant into a region with the
opposite doping. For example, group III atoms can be diffused into a small region of
an n-doped substrate. The spatial profile of the group III dopants will decrease
gradually from a peak value to zero, resulting in a smooth transition between p-type
and n-type material. Finding ρ(x ), φ(x ), Ex (x ), etc. requires a numerical solution of
Poisson’s equations after first modeling the fabrication process to find Nd (x ) and
Na (x ).
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 24 of 35
Diode Equations
Diode Equations TN§5.3
We’re finally ready to see why a pn junction acts as a diode (that is, why it passes
current in one direction but not in the other). First, let’s have a look at the minority
carrier diffusion equations in the neutral regions outside of the space charge layer.
These equations represent the time evolution of the hole concentration on the n-side
(denoted pn ) and the electron concentration on the p-side (np ):
∂p n
∂t
=
−
pn − pn 0
∂2 pn
+ Dpn
,
τpn
∂x 2
(6.18)
∂n p
∂t
=
−
np − np 0
∂2 n p
+ Dnp
,
τnp
∂x 2
(6.19)
where pn0 and np 0 are the equilibrium values of pn and np , τpn and τnp are the
lifetimes of holes in the n-region and of electrons in the p-region, and Dpn and Dnp
are the appropriate diffusion coefficients.
In the steady state, we set ∂/∂t = 0, and define the excess minority carrier
concentrations as
(6.20)
pn′ = pn − pn0 , np′ = np − np 0 .
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 25 of 35
Diode Equations
Diode Equations cont’
TN§5.3
Furthermore, assuming that all donors and acceptors are ionized, the majority
carrier concentrations are simply the initial dopant concentrations,
pp 0 = Na , nn0 = Nd .
(6.21)
Combining this with Eq. (6.3), we can write
φD =
kB T pp 0 nn0
ln
e
ni2
(6.22)
where, by definition (assuming that we have a homojunction), ni2 = pp 0 np 0 = nn0 pp 0 .
Inverting this equation results in the following equations for the equilibrium minority
carrier concentrations.
pn 0
np 0
=
=
pp 0 e −e φD /kB T ,
nn 0 e
−e φD /kB T
.
(6.23)
(6.24)
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 26 of 35
Diode Equations
Diode Equations cont’
TN§5.3
Next, let’s apply a bias voltage V across the junction. As was the case earlier in the
chapter, we simply replace φD with φD − V . Thus, the (now non-equilibrium) minority
carrier concentrations are
pn
=
pp 0 e −e (φD −V )/kB T = pn0 e −eV /kB T ,
(6.25)
np
=
nn0 e −e (φD −V )/kB T = np 0 e −eV /kB T .
(6.26)
Substituting these equations into Eq. (6.20), we find that the steady-state excess
minority carrier concentrations are
pn′ = pn0 e eV /kB T − 1 ,
np′ = np 0 e eV /kB T − 1 .
(6.27)
Now, we re-introduce the diffusion equations for these steady-state excess minority
carrier concentrations:
pn′
p′
∂2 pn′
=
≡ 2n ,
2
Dpn τpn
∂x
Lpn
∂2 np′
∂x 2
=
np′
Dnp τnp
≡
np′
Ln2p
,
(6.28)
p
where Lpn = Dpn τpn is the diffusion length for a hole in the n-region and similar
for Lnp ; these parameters will prove to be important later in the course when we
discuss photodetectors.
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 27 of 35
Diode Equations
Diode Equations cont’
TN§5.3
The solution to the diffusion equations gives us a spatial profile for the excess
minority carrier concentrations outside of the depletion region:
pn′ (x ) = pn0 e eV /kB T − 1 e (x +ℓn )/Lpn , np′ (x ) = np 0 e eV /kB T − 1 e −(x −ℓp )/Lnp . (6.29)
The diffusion current densities are then (see Eq. 5.74)
Jpn (x )
=
Jnp (x )
=
Dpn
dpn′
=e
pn0 e eV /kB T − 1 e (x +ℓn )/Lpn ,
dx
Lpn
dnp′
Dnp
−eDnp
=e
np 0 e eV /kB T − 1 e −(x −ℓp )/Lnp .
dx
Lnp
eDpn
(6.30)
(6.31)
The total electric current density is then given by the sum of Jnp (ℓp ) and Jpn (−ℓn ):
J = Js e eV /kB T − 1 ,
(6.32)
where we have defined the reverse saturation current density
!
Dnp
Dpn
Js = e
pn 0 +
np 0 .
Lpn
Lnp
(6.33)
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 28 of 35
Diode Equations
Diode Equations cont’
TN§5.3
Thus, we have shown that in a pn junction, current flows very easily in the p-to-n
direction when the junction is forward biased (the p-side has a positive bias
relative to the n-side), whereas virtually no current flows under conditions of
reverse bias. We say that the device is non-Ohmic, in that the current-voltage
relationship does not obey Ohm’s law (which states that the I/V relationship is linear,
and independent of direction). This allows us to classify the pn junction as a diode.
It is useful as a current rectifier. Multiple diodes are used to form transistors,
although this is beyond the scope of this course. And, as we will see in a later
chapter, diodes are essential elements of semiconductor light-emitting devices,
photodetectors, and photovoltaic cells.
For actual diodes, there is a non-ideality factor n in the denominator of the
exponent, where 1 < n < 2 (and where, in reality, n = n(I ) to a certain extent). n is
governed primarily by HSR recombination:
J = Js e eV /nkB T − 1 .
(6.34)
It should also be pointed out that the reverse saturation current density depends
critically on temperature, through pn0 and np 0 (and - to a lesser extent - through the
diffusion coefficients). Furthermore, pn0 and np 0 can be altered by the absorption of
photons, as we saw in chapter 5.
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 29 of 35
Diode Equations
Reverse Breakdown
The preceding model suggests that under reverse bias, a pn junction exhibits a
small, mostly voltage-independent saturation current. This neglects a couple of
physical effects which conspire to cause a reverse breakdown that occurs at a
critical reverse bias voltage Vbr . Beyond Vbr , the reverse current through the junction
increases dramatically, as shown in Fig. 9
Fig. 9:
Reverse breakdown in a pn junction (Streetman).
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 30 of 35
Diode Equations
Reverse Breakdown
Reverse breakdown occurs by two distinct physical mechanisms. The first, occurring
at a reverse bias of just a few volts is the Zener effect, shown in Fig. 10. Here, the
reverse bias has bent the energy bands to a sufficient degree that Ec in the n-side
lines up with - and then falls below - Ev in the p-side. If ℓD is sufficiently narrow,
quantum-mechanical tunneling of carriers across the depletion region can occur.
Tunneling of electrons from the p-side to the n-side (and of holes from the n-side to
the p-side) constitutes a reverse current. Since the tunneling probability depends
critically on ℓD , it is apparent that a small magnitude of Vbr requires an abrupt,
highly-doped junction. It also requires a relatively small bandgap, in order to facilitate
the band overlap.
Fig. 10:
The Zener effect. (left) Heavily-doped junction at equilibrium. (center) Reverse bias with electron tunneling from p
to n. (right) The resulting I-V curve (Streetman).
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 31 of 35
Diode Equations
Reverse Breakdown
The second breakdown mechanism is only observable in lightly-doped and/or
large-bandgap pn junctions, where the Zener effect is negligible, and requires a very
high reverse voltage. If the resulting electric field across the depletion region is
sufficiently large, an electron entering from the p-side may acquire enough kinetic
energy to cause an ionizing collision with the lattice, generating an additional
electron-hole pair. The carriers in this generated EHP are swept in opposite
directions by the field, where they contribute to the current. However, before leaving
the depletion region, they may cause subsequent ionizing collisions. This process is
known as avalanche breakdown.
Fig. 11:
Avalanche breakdown. (left) A single ionizing collision by an incoming electron. (right) Primary, secondary, and
tertiary electrons (Streetman).
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 32 of 35
Heterojunctions
Heterojunctions
Up to this point, we have only considered homojunctions, in which the p-doped and
n-doped semiconductors are the same alloy (the only difference is in the doping). In
a heterojunction, the two semiconductors have different alloys; for example, GaAs
and Alx Ga1−x As, where 0 ≤ x ≤ 0.45 (when x >0.45, Alx Ga1−x As transitions from
direct to indirect bandgap). When the semiconductors are dissimilar, they have
different energy gaps and electron affinities (χ), and this produces some interesting
features in the energy band diagram of the heterojunction - there will be
discontinuities in both Ec and Ev , both at equilibrium and under an applied voltage.
These are labelled ∆Ec and ∆Ev , where ∆Ec + ∆Ev = ∆Eg .
We will not attempt to derive formulas for the spatial dependence of φ, Ex , etc.
Instead, we will simply mention that these discontinuities can produce very narrow
regions in which electrons or holes can be “trapped” in the CB or VB, respectively, as
evident in Fig. 12.
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 33 of 35
Heterojunctions
Fig. 12: An ideal heterojunction between a p-type, wide bandgap semiconductor and an n-type, narrower bandgap
semiconductor. (a) Band diagrams before joining, (b) Band diagrams after joining, at equilibrium (Streetman).
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 34 of 35
Heterojunctions
Heterojunctions cont’
We’ve glossed over one rather critical aspect of heterojunctions - we can’t just grow
any alloy on top of any other alloy. They must be lattice matched; that is, their
lattice constant (atomic separation) must be nearly identical. Otherwise, there will be
strain at the interface; this would lead to cracking, dislocations, interstices, etc., all of
which produce charge centers (localized regions where charge carriers can be
trapped). For the same reason, it’s somewhat important that the materials have
similar thermal expansion coefficients.
As it turns out, it’s OK to grow ultra-thin layers of materials with dissimilar lattice
constants, since the strain will remain below the physical threshold for the
aforementioned effects. In fact, laser diode designers very often intentionally
incorporate strain into the design of quantum well gain regions. Time permitting, we
will learn why this is the case later on in the course.
Figure 13 is a bandgap diagram; it displays various common semiconductor alloys
used in optoelectronics in terms of their energy gap Eg and their lattice constant. For
lattice matching, alloys should be aligned vertically. In the diagram, binary
semiconductors are illustrated as points and ternary alloys follow the lines (solid lines
are direct gap, dotted lines are indirect). The diagram also indicates that In.53 Ga.47 As
is lattice-matched to InP and that the Alx Ga1−x As system is well matched for any x .
PC421 Winter 2013 — Chapter 6. Semiconductor Junctions — Slide 35 of 35
Heterojunctions
Fig. 13:
Bandgap diagram for III-V semiconductors.