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4. The p-n Junction Diode
4.1 p-n Junction in Equilibrium (For Homostructure: same Eg materials)
There are two important concepts:
(i) The distribution of electrons and holes.
(ii) Electron and hole currents due to diffusion and drift processes.
- Electrons are the majority carriers in n-type semiconductors
- Holes are the majority carriers in p-type semiconductors
- Electron affinity (e χ ) is defined as the energy difference between the conduction bandedge
and the vacuum level.
sp
and φ
sn
) is the energy required for an electron to reach the vacuum
level and leave the semiconductor crystal, and is equal to the difference between the Fermi
energy and vacuum level.
- In the p-type semiconductor: 𝑝
𝑝
𝑖𝑠 𝑡ℎ𝑒 majority holes 𝑎𝑛𝑑 𝑛
𝑝
𝑖𝑠 𝑡ℎ𝑒 minority electrons
- In the n-type semiconductor, 𝑛
𝑛
𝑖𝑠 the majority electrons 𝑎𝑛𝑑 𝑝
𝑛
𝑖𝑠 the minority holes
- When the junction is formed, the difference between the electron and hole densities at the
junction (𝑛
𝑛
𝑝
) for electrons and (𝑝
𝑝
𝑛
) for holes causes diffusion currents to flow
across the junction.
- Electrons diffuses from the n side to p side and holes are injected in the opposite direction.
- When an electron crosses the junction, it leaves behind positively charged ionised donor
atom so that the n-side of the junction is no longer electrically neutral but has a net positive
charge of + e. Upon arriving at the p-side, this free electron recombines with one free hole,
which results in one negatively charged ionised acceptor atom.
The p-n junction in equilibrium can be divided into three regions:
- Neutral p-type region : This region is away from the junction, where the electrostatic potential
is flat, Fermi energy lies close to the VB and the majority carriers are holes and their density is
equal to the acceptor doping density, 𝑝
𝑝
𝐴
. The minority electron density, 𝑛
𝑝
𝑛
𝑖
2
𝑝
𝑝
- Neutral n-type region : This region is away from the junction, where the electrostatic potential
is flat, Fermi energy lies close to the CB and the majority carriers are electrons and their density
is equal to the donor doping density, 𝑛
𝑛
𝐷
. The minority hole density, 𝑝
𝑛
𝑛
𝑖
2
𝑛
𝑛
or drift current is equal to diffusion current:
From Einstein’s relations
and
The constant potential values on the p and n-sides of the junction area are Vp and Vn , respectively.
The built-in potential (contact potential) is defined as
Similarly for electrons the built-in potential is:
Build-in potential can also be obtained from the energy bandprofile:
4.2 p-n Junction Under an External Electric Field
In the formation of the p-n junction, currents exist across the junction even at equilibrium, in the
form of diffusion and drift currents. However, these two currents balance each other out and the
net current becomes zero.
When we apply an external voltage, hence an external electric field, this balance is disturbed and
a finite current flows through the diode.
Under forward bias of V = Vf , the potential difference between the p and n-sides of the junction
is:
Under reverse bias of, V = − Vr,Vr > 0, the potential difference between the p and the n-sides is:
HETEROJUNCTIONS
- When the semiconductor material was homogeneous throughout the entire structure This
type of junction is called a homojunction
- When two different semiconductor materials are used to form a junction, the junction is
called a heterojunction
- The energy diagram energy levels of the step heterojunction exhibit discontinuities at the
junction interface
- In order to have a useful heterojunction, the lattice constants of the two materials must be
well matched
- Nn, Np, nP and pP heterojunctions are used in practice Here letters n and p denote a
semiconductor with a relatively narrow forbidden band and capital letters N and P are
related to a semiconductor with a wider forbidden band
- The heights of the potential barriers are different for electrons and holes
For example, germanium and gallium arsenide have lattice constants matched to within
approximately 0.13 percent.
More recently, gallium arsenide–aluminum gallium arsenide (GaAs–AlGaAs) junctions have
been investigated quite thoroughly, since the lattice constants of GaAs and the AlGaAs system
vary by no more than 0.14 percent.
- Heterojunctions with the dopant type changes at the junction are called anisotype We can
form nP or Np junctions
- Heterojunctions with the same dopant type on either side of the junction are called isotype
We can form nN and pP isotype heterojunctions
- The ratio of electronic and hole currents is determined by the heights of the barriers for
electrons and holes (In a homojunction this ratio depends on doping of n and p regions).
1. pN Heterojunction
Consider first a p-type narrow-gap semiconductor, such as GaAs, in contact with an N-type wide-
band-gap semiconductor, such as Al x
Ga
1 - x
As.
Let χ be the electron affinity, which is the energy required to take an electron from the conduction
band edge to the vacuum level, and
𝑞𝑁
𝑎
(𝑥+𝑥
𝑝
)
𝜀
𝑝
𝑝
𝑞𝑁
𝐷
( 𝑥−𝑥
𝑁
)
𝜀
𝑁
𝑁
The boundary condition states that the normal
displacement vector D = εE is continuous at x =
𝑝
−
𝑁
𝑎
𝑝
𝐷
𝑁
The electrostatic potential distribution ϕ( x ) across the junction is related to the electric field by
which means that the slope of the potential profile is given by the negative of the electric field
profile. If we choose the reference potential to be zero for x < - x p
we have
𝑝
𝑎
𝑝
𝑝
2
𝑝
𝑎
𝑝
2
𝑝
𝐷
𝑁
𝑁
2
𝑁
𝑜
𝑁
Where
𝑜
𝑜𝑝
𝑜𝑁
𝑜𝑝
𝑎
𝑝
2
𝑝
𝑜𝑁
𝑜
𝑜𝑝
𝐷
𝑁
2
𝑁
Vo is the total potential drop across the junction,
whereas Vop, is the portion of the voltage drop on the p side and
V
oN
is the portion of the voltage drop on the N side.
The contact potential (Vo) is evaluated using the bulk values of the Fermi levels F p
and F N
measured from the valence or conduction band edges E vp
and E
cN
respectively, before contact:
𝑐
𝑝
𝑁
𝑣
𝑣𝑝
𝑉𝑁
𝑐
𝐶𝑁
𝑐𝑝
𝑣
𝑐
𝐺𝑁
𝑔𝑝
𝑔
𝑣
𝑔
𝑐
𝑔
𝐺𝑁
𝑔𝑝
𝑜
𝑁
𝑝
𝑜
𝑁
𝐶𝑁
𝐶𝑁
𝑣𝑝
𝑣𝑝
𝑝
𝑜
𝑁
𝐶𝑁
𝐶𝑁
𝑣𝑝
𝑣𝑝
𝑝
𝐶𝑁
𝑣𝑝
𝑔𝑝
𝑐
𝑜
𝑔𝑝
𝑐
𝑁
𝐶𝑁
𝑣𝑝
𝑝
𝑐
𝐺𝑁
𝑔𝑝
𝑔
𝑣
𝑔
𝑔
𝑣
𝑣
𝑥
1 −𝑥
𝑤
𝑝
𝑁
𝑎
𝑝
𝐷
𝑁
𝑝
𝐷
𝑎
𝐷
𝑤
𝑁
𝑎
𝑎
𝐷
𝑤
𝑤
= [
𝑝
𝑜
𝑎
𝐷
𝐷
𝑝
𝑁
𝑎
]
1 / 2
𝑎
𝐷
The band edge E v
(x) from the p side to the N side is given by (choosing 𝐸
𝑣
= 0 as the
reference potential energy)
𝑣
𝑣
or
𝑣
𝑝
2
𝑎
𝑝
𝑝
2
𝑝
𝑣
2
𝑎
𝑝
2
𝑝
2
𝐷
𝑁
𝑁
2
𝑁
𝑣
𝑜
𝑁
The conduction band edge E c
(x) is above E
v
(x) by
an amount E gp
on the p side and by an amount E GN
on the N side. E c
(x) is always parallel to E
v
(x)
𝑐
𝑣
𝑔𝑝
𝑣
𝐺𝑁
Example:
A p-GaAs/N- Al x
Ga 1 - x
As (x = 0. 3) heterojunction is formed at thermal equilibrium without an
external bias at room temperature. The doping concentration is Na = 1 × 10
18
cm
in the p side
and N D
= 2 × 10
17
cm
in the N side. Assume that the density-of-states hole effective mass for
Al x
Ga
1 - x
As
ℎ
∗
𝑜
which accounts for both the heavy-hole and light-hole density of states. Other parameters are
𝑒
∗
𝑜
𝑔
𝑜
where x is the mole fraction of aluminum.
(a) We obtain for x = 0.
𝑒
∗
𝑜
ℎ
∗
𝑜
𝑝
𝑜
𝑔𝑝
- 3
- 7
𝑒
∗
𝑜
ℎ
∗
𝑜
𝑁
𝑜
𝑔𝑁
𝑔
𝑔𝑁
𝑔𝑝
𝑐
𝑔
= 0. 67 × 374 𝑚𝑒𝑉 = 250. 58 𝑚𝑒𝑉
𝑣
𝑔
= 0. 33 × 374 𝑚𝑒𝑉 = 123. 42 𝑚𝑒𝑉
(b) We calculate the quasi-Fermi levels F p
and F
N
for the bulk semiconductors for the given N a
and N D
separately.
p-GaAs region
𝑐
𝑒
∗
2
3 / 2
= 2. 51 × 10
19
𝑒
∗
𝑜
3 / 2
− 3
𝑜
3 / 2
2
𝑐
= 4. 31 × 10
17
− 3
𝑣
2 𝜋𝑚
ℎ
∗
𝑘𝑇
ℎ
2
3 / 2
= 2. 51 × 10
19
𝑚
ℎ
∗
𝑇
𝑚
𝑜
300
3 / 2
− 3
𝑁
𝑥
𝑝
𝑁
𝐷
𝑎
𝑤
𝑝
𝑁
Band diagram of a p-GaAs/N-Alo 0.
Ga 0.
As heterojunction with 𝑁
𝑎
= 1 × 10
18
− 3
in the p
region and 𝑁
𝐷
= 2 × 10
17
− 3
In the N region
Light Emitting Diodes and Semiconductor Lasers
Electroluminescence is the emission of light when a current passes through a semiconductor
device and the injected non-equilibrium electrons and holes recombine across the bandgap or via
localized levels within the bandgap.
LEDs and lasers are widely used in current technologies and have applications in lighting, optical
data processing, optical communications, medicine and spectroscopy.
They emit light at wavelengths corresponding to the bandgap energies of the semiconductor
materials from which they were fabricated. Therefore, the range of wavelengths they emit varies
between UV for the wide bandgap semiconductors, for example InGaN (λ ~ 400 nm), and IR for
narrow bandgap semiconductors such as InGaAsP (λ ~ 3300 nm).
1. Absorption and Emission Rates: Einstein Relations
In semiconductors there are three types of optical processes involving the interaction between
photons and electrons:
- Absorption: a photon with energy greater than the bandgap of the semiconductor transfers its
energy to an electron in the valance band and excites it into the conduction band
- Spontaneous emission: An electron in the conduction band recombines with a hole in the
valence band and gives up its excess energy as an emitted photon.
- Stimulated emission: An incident photon causes an electron in the conduction band to
recombine with a hole in the valence band, emitting a photon whose properties are identical
(coherent or in-phase) to those of the incident photon.
The term “stimulated” underlines the fact that this kind of radiation only occurs if there is photon
density present in active region of a light emitting device and the upper energy level must have
higher carrier density. The amplification arises due to the similarities between the incident and
emitted photons.
Spontaneous emission is required for LED operation while stimulated emission is the process
used in laser operation.
The possible optical transitions in a simple two-level atomic system with two energy states, E
and E2. The frequency of the incident photon is ν , and its energy satisfies the condition for
absorption: E = E 2
– E
1
= hν.
In 1917, Einstein worked out the relationship between the three processes, starting from the fact
that in thermal equilibrium the upward transition rates in a two-level system must be equal to the
downward transition rates.
ii. Absorption: a photon is absorbed by an atom, causing an electron to jump from a lower energy
level to a higher one.
1
12
𝑖
𝑣
12
1
The transition rate 𝑊
21
𝑖
is given by the Einstein coefficient B 12
which is the probability per unit
time, per unit spectral energy density that an electron in state 1 with energy E1 will absorb a
photon with an energy E 2 − E 1 = hν and jump to state 2 with energy E2.
ρν is the spectral energy density at a photon frequency of ν.
iii. Stimulated emission: is the process where an electron is induced to jump from a higher energy
level to a lower one by the presence of a photon at the same energy as the transition.
2
21
𝑖
𝑣
21
2
The stimulated emission rate, 𝑊
21
𝑖
is described by the Einstein coefficient B 21
, which is defined
as the probability per unit time per unit spectral energy density that an electron in state 2 with
energy E2 will decay to state 1 with energy E1 , emitting a photon with an energy E 2 − E 1 = hν.
- The main difference between the spontaneous emission and stimulated emission is that the
former is an arbitrary emission process whilst the latter is coherent. Therefore, the emitted
photons are in the same phase and have the same polarization and the same frequency.
In thermal equilibrium the rate of upward transitions, 𝑊
12
′
is equal to that of downward
transitions, 𝑊
21
′
as
12
′
12
𝑖
𝑣
1
12
21
′
21
21
𝑖
2
21
𝑣
2
21
𝑣
1
12
2
21
𝑣
2
21
𝑣
2
21
1
12
2
21
𝑣
21
21
12
1
2
21
The relative densities of atoms in the two states E1 and E2 can be found using Maxwell–
Boltzmann statistics as:
g 1 and g 2 are the degeneracy of levels E 1 and E 2, respectively.
If we assume that our two-level energy system is not degenerate g1 = g2 = 1 ,
𝑣
21
21
12
1
2
21
𝐵
The atomic system considered here is in thermal equilibrium therefore, the emitted energy density
is identical to that of blackbody radiation. Compare it with the spectral energy density of
blackbody radiation:
12
1
2
21
12
2
1
21
21
21
3
3
3
12
2
1
21
are known as 𝑬𝒊𝒏𝒔𝒕𝒆𝒊𝒏’𝒔 𝒓𝒆𝒍𝒂𝒕𝒊𝒐𝒏𝒔
If the degeneracy of two levels is equal (g 1 = g 2 ) , then the absorption and stimulated emission
rates will be the same. Furthermore, the ratio of the stimulated and spontaneous emission rates
will be:
21
𝑖
21
𝑣
21
21
𝐵
𝑖
12
𝑖
21
𝑖
21
3
3
3
3
3
3
21
This equation gives the transition rate for a uniform (white) spectrum with energy density per
unit frequency, ρ(ν). However, we are interested in the transition rate for a monochromatic light
at a single frequency 𝑊
𝑖
The transition with spontaneous emission from level 2 to 1 (2→1) is not in fact completely
monochromatic but broadened into a small frequency band as described by the line-shape
function , g(ν) which comprises of both homogenous broadening and inhomogeneous
broadening.
The Heisenberg uncertainty principle dictates that the broadening of the atomic level E is:
This equation implies that the broadening in the energy level is inversely proportional to the
lifetime of the atom at the excited level.
