4.

PN Junctions Summary
Review of semiconductor transport theory
• Continuity equation
• Current density equations
• Mobilities
• Diffusion
• Recombination mechanisms
• Diffusion length
• Debye length
• Dielectric relaxation time
• p-n junction diodes
• Depletion approximation

EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 1

Recombination
There are three main classes of conduction band to valence band
recombination processes

• Radiative recombination
• Recombination through traps
• Auger recombination.

EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 2

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 Auger recombination
Auger recombination is a 3 carrier process which is the
“reciprocal” of Avalanche Multiplication in that there are 3
carriers in the initial state and only a single free carrier in the
final state. The initial state can be either two electrons and
one hole, or two holes and one electron, leaving the last free
carrier with ~Eg kinetic energy after the recombination.

EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 3

Auger recombination

In an Auger recombination event, e.g., with two electrons and

one hole, one electron recombines with the hole and the energy
of recombination is transferred to the remaining electron rather
than a photon as in radiative recombination.
Recombination rates in Auger processes therefore have form

rAuger = C2h p 2 n + C2e pn 2 (4.28)

where C2h and C2e are coefficients corresponding to the 2-

holes--1-electron process and 2-electrons--1-hole process
respectively.

EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 4

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 Auger recombination (2)
Because Auger recombination depends upon the product of the
square of the majority carrier density x the minority carrier
density (n2p or np2, but in most cases, n3 or p3 since the
electron and hole densities are equal in undoped QW regions),
it is not important at low carrier densities. It is, however, quite
important at carrier densities required for the operation of
semiconductor laser diodes.

This is especially true for longer wavelength laser diodes used

for long distance telecommunications (e.g., 1.3 µm and 1.5 µm)
In some cases, the bandgap energy is comparable to
separation between valence bands (SO band), leading to a
resonance enhancement of the 2-hole--1-electron
recombination process.
EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 5

Laser Recombination Characterization

I ∝ An or Ap Trap Recombination

I ∝ Bnp Spontaneous Radiative

Recombination

€ I ∝ Cn 2 p or Cnp 2 Auger Recombination

I = eVa ( An1 + Bn 2 + Cn 3 ) I ~ nZ TSE ∝ Bn 2

€
END VIEW d (ln( I )) I Mono I Rad I Aug
€ WINDOW EDGE Z≡ = 1 + 2 + 3
d (ln(TSE 1/ 2 )) I Tot I Tot I Tot
p-metal
p-AlGaAs mesa
GaAs waveguide Measure the total spontaneously
Active region
GaAs waveguide
emitted light (top or edges, not end
n-AlGaAs
facets) as a function of current and
GaAs substrate
n-metal
plot log (SPE) vs log I
EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 6

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 Local Z-Parameter Theory
n/Nc
0.0 0.5 1.0 1.5 2.0 2.5 3.0 I Process
4.5 Z= ∑Z Effective
(f) I Tot
4.0 AllProcesses
Effective Z-Parameter

(g)
3.5 a) Mid-bandgap QW traps
3.0 (d) b) Mid-bandgap barrier traps
2.5
c) (N-N)As barrier traps
(e) d) (N-As)As barrier traps
2.0
(b,c) e) Radiative
1.5
(c) (a) f) Auger CHCC
1.0 g) Auger CHHS
0.0 0.8 1.6 2.4 3.2 4.0 4.8
Carrier Concentration (1018 cm-3)

Z < 1.5 indicates at least 50% of current is monomolecular

i.e Recombination through traps in QW or barriers
EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 7

Measured Laser Z Parameters

InGaAs lasers GaInNAsSb lasers

3 4.0
Previous
3.5 New Devices Devices
(55｡ C) (55｡ C)
Z-Parameter

1550nm
Z-Parameter

InGaAs/InP 3.0

2 2.5

980nm 2.0
InGaAs/GaAs
1.5

1 1.0
10 20 30 40 50 60 70 0.0 0.5 1.0 1.5 2.0
Current (mA) Current Density (kA/cm2)

EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 8

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 GaInNAsSb Threshold Density Distribution
1800
Monomolecular
1600
Radiative
1400 Leakage/Auger
1200

Jth (A/cm )
2
1000
800
600
400
200
0
Older Newer Older Newer
20°C 55°C
• > 50% of 1st
Gen threshold current is monomolecular recombination
• New sample has 4x less monomolecular recombination
• Significant room for improvement by reducing Leakage Current

EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 9

Diffusion length
Important application of continuity equations - diffusion length
Consider semiconductor, with no electric field, with excess
carrier density (e.g., electrons) being injected, say at one end of
the sample and assume:
in steady state
no generation in the volume
recombination is characterized by a simple lifetime, τe
From the electron continuity equation, for the excess electron
density Δn in the steady state,
∂Δn 1 Δn (4.29)
= 0 = ∇ • Je −
∂t e τe

€
EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 10

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 Diffusion length (2)
Since there is no electric field, only term in the electron current
density equation is the diffusion term. Substituting, we have,
considering just the one-dimensional problem,
d 2 Δn Δn
De = (4.30)
dz 2 τe
with solution Δn( z) = Δn( 0) exp( −z Le ) (4.31)
Where Δn(0) is excess electron density at the end where
electrons are injected
Parameters Le and Lh are electron and hole diffusion lengths
Le = Deτ e (4.32)
Lh = Dhτ h (4.33)
characteristic length the particle diffuses before recombining.

EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 11

Debye length
Suppose we have a doped semiconductor, e.g., uniform n-type
material, with the electron charge density identical magnitude
and opposite in sign to ionized impurity charge density
Suppose now the doping density changes abruptly, we might
imagine electron charge density would also change equally
suddenly, but if it did, we would find very large concentration
gradient in electron density, leading to net diffusion down the
density gradient, leading to build up of electric field because now
locally, there is a net charge density because electrons have
diffused away from their associated dopants

EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 12

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 Debye length
Hence, near region of abrupt change in doping density, a net
dipole is created, giving rise to an electric field that opposes
diffusion. Net result - instead of the mobile electron density
exactly following the change in dopant density, the effect of
diffusion "smoothes out" resulting change in electron density.
Characteristic length scale of the "smoothing" is called Debye
length, LD, (when in the classical limit of low occupation numbers
for carriers (i.e., fe and fh << 1)
εrε o kBT
LDe = (4.34)
ne2
is electron Debye length
For a p-doped material, the Debye length would similarly be
εε k T
LDh = r o 2B (4.35)
pe
EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 13

Debye length (2)

Because of Poissonʼs Equation
changes in charge density,
electric field, and voltage all
have exponential forms with
the same characteristic length
(the Debye length).

d 2V ρ
2
=−
dx ε

EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 14

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 Debye length (3)
In semiconductor electronic and optoelectronic devices, we
usually find Debye lengths in two limiting situations;
1. Inside, or near the edge of, heavily doped material,
relatively short Debye lengths (e.g. <100 Å)
2. Semiconductor with its chemical potential (Fermi energy)
in the middle of the gap, i.e. in materials deliberately
made semi-insulating through introduction of defects (As
antisite, EL2) or doping with materials that produce "deep
levels" (e. g., iron in InP, or chromium or oxygen in GaAs)
or in the depletion regions inside diodes, which have very
long Debye lengths, >1 mm

EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 15

Debye length examples

Short Debye lengths
For a semiconductor with electron or hole density of 1018 cm-3
Debye length 4.2 nm (42 Å)
For 1016 cm-3, Debye length is 42 nm (420 Å)

Long Debye lengths

For a 1.5 eV bandgap semiconductor, with a Fermi level at the
middle of the bandgap, the carrier concentrations n = p= ni ~
106 cm-3, leading to Debye lengths in the range of millimeters

In such cases, there is essentially no screening of any

fluctuations in fixed charge density.
Ionized deep levels, for example, are essentially completely
"bare" in electrostatic terms (other than experiencing the
background dielectric constant of the material.)
EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 16

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 Semiconductor carrier transport--
Dielectric relaxation time
What is the characteristic time in which the carrier density can
respond to perturbations (other than recombination times)
i.e., how do intrinsic conductivity and dielectric constant lead to
a characteristic speed of response for recovery of perturbations
in the carrier density?

Answer – dielectric relaxation time

• for times >> dielectric relaxation time

material behaves like a conductor
• for times << dielectric relaxation time
material behaves like an insulator

EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 17

Dielectric relaxation time (2)

We start with continuity equation, for example Eq. (4.7), for the
case of electrons in an n-doped material, neglecting generation
and recombination terms
More generally, we can work with net mobile charge density,
ρmobile = e( p − n) (4.36)
For small perturbation δρmobile from the condition where there is
no net charge density, if no recombination or generation, we
have (from conservation of charge)
∂δρmobile 
= −∇ ⋅ J mobile (4.37)
∂t
where Jmobile is total electrical current density for mobile charge.
If the material is Ohmic, then
  (4.38)
J = σE
assuming€ a constant conductivity σ, we have
 
∇ ⋅ J mobile = σ∇ ⋅ E (4.39)
EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 18
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 Dielectric relaxation time (3)
Assuming the fixed charge densities from the dopant atoms are
uniform, we can substitute using Maxwellʼs equation for ∇.E , in the
form ∇.E=ρfree/εrεo, and hence obtain
∂δρ σ
=− δρ (4.40)
∂t ε rε o
solution δρ( t ) = ρ o exp( −t / τ d ) (4.41)

ε rε o (4.42)
where τd =
σ
τd is the dielectric relaxation time
Dielectric relaxation time is the "internal" material resistive-
capacitive (RC) time constant
Product of resistivity ([1/σ] ohm-meters), and permittivity (εrεo
Farads/meter).

EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 19

Dielectric relaxation time (4)

In a doped semiconductor, τd can be very short e.g., as fast as
a few femtoseconds for n-type GaAs doped at 1018 cm-3

In an ideal intrinsic semiconductor, τd can be quite long

e.g., with perhaps 106 cm-3 carriers, as slow as milliseconds

In insulators, τd can be very long--1010 YEARS

Important, for example, for photorefractive materials for optical
storage since the information is stored in the electric field
distributions resulting from fixed, charged traps.
Semiconductors are unlikely to be suitable for any long term
storage because the dielectric relaxation time is usually far too
short, requiring the use of larger band gap insulators.

EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 20

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 Diodes in optoelectronics
In optoelectronics, diodes are used for three purposes,
(1) Photodetectors
Diode structure provides a good way to collect electrons and
holes generated by optical absorption and turn the result of
absorption into an electrical current (photocurrent)
Reverse biased photodiode
Relatively little current in the absence of light and very
efficiently generates photocurrent from the absorbed photons
therefore, a detector with low background noise
Can operate with large electric fields to produce electrical
gain in avalanche photodiodes.
Can be operated in forward bias and used to generate
electrical power from light, as in a solar cell

EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 21

Diodes in optoelectronics (2)

(2) Light emitters (LEDs and lasers)
Convenient structure for controllably injecting electrons and
holes into a particular volume so that they can recombine
radiatively and efficiently to emit light
Under forward bias, holes are injected from the p-region and
electrons from the n-region into the volume between or near
(within a diffusion length) of the edges of the p and n regions.

(3) Modulators
Can apply necessary large electric fields by reverse-biasing a
p-i-n diode in several types of modulators. The p-i-n structure
provides a UNIFORM electric field in the depletion region such
that all QWs will see the same field, independent of location in
the depletion region.
EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 22

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 pn junction - review
Semiconductor pn junction diode
Interface between two semiconducting materials
One p-doped--chemical potential (Fermi energy) near or in
the valence band
One n-doped
Chemical potential near or in the conduction band

We start out by analyzing the case of thermal equilibrium.

This is valuable because non-equilibrium is just a perturbation
of equilibrium

EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 23

Equality of chemical potentials in equilibrium

From basic thermodynamics, chemical potentials must be
equal throughout a system if it is in thermal and diffusive
equilibrium, otherwise, if there is a gradient, the particles will
move so as to equalize the chemical potentials
Equality of chemical potentials ensures
Occupation probability of states of the same energy in
different parts of the system is the same
Hence, there is no net transition rate between states of the
same energy
In common semiconductor terminology, Fermi levels (chemical
potentials) must be equal
This holds even if materials are different (heterojunctions)
No matter how many materials of different kinds we join
together, hence it holds for any kind of heterostructure, no
matter, for example, what the bandgap energies or dopants are.

EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 24

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 Fermi levels in diode at thermal equilibrium
Well away from the junction between any two materials (+∞ & -∞,
the materials should look just like they did before we joined them,
hence

We have not yet discussed what happens in the middle region

We need to make some approximations and assumptions, e.g.
Depletion Approximation and band tilting with an electric field
but the built-in voltage is fixed by thermodynamics
EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 25

Formation of the depletion region

Because of imbalance of number of electrons between the n and
p materials, electrons diffuse from n material into p material,
leaving behind a net positive charge in the n material

In the p material, electrons will tend to recombine with holes

reducing the number of holes near the edge of the p material
creating a net negative charge just inside the p material.
Similarly for holes diffusing into the n material

EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 26

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 Formation of the depletion region (2)

We have neither added nor removed charge overall so net +ve

charge in n material is balanced by net -ve charge in p material
Hence a dipole forms near the junction which leads to an
electric field in this region.
This electric field stops transfer of electrons and holes between
the materials--the chemical diffusive force is balanced by this
electrostatic force. Important result - states with same total
energies have equal occupation probabilities

EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 27

Depletion approximation
Depletion Approximation
There is a region of the material lying between the p and n
regions, which is largely depleted of mobile carriers (we
ASSUME TOTALLY) In that region, we find ionized doping (and
possibly other impurity) atoms
At the edges of this region, we ASSUME immediate return to the
uniform n or p carrier concentrations in the n or p materials
The requirement of uniform chemical potential throughout the
system tells us what voltage has to be dropped across the
depletion region (the so-called "built-in" voltage, Vbi)
The width of the depleted regions on either side can then be
deduced from simple electrostatics, given the (presumably
known) dopant concentrations, with the requirement that we
achieve a diffusive equilibrium defined voltage difference, Vbi,
from one side of the depletion region to the other
EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 28

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 Depletion approximation fields and voltage

We already know charge

neutrality overall, even
without making the
Depletion Approximation.
In the depletion
approximation,
charge neutrality
requires equal total
numbers of ionized
donors and acceptors in
the depletion region
This ensures the electric field comes back to zero in the
n conducting material
EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 29

Band tilting with electric field

Approximation--view band structure of semiconductor (e.g.,
densities of states, effective masses) as unaffected by presence
of electric field, except to add electrostatic potential to the whole
band structure at each point. This is not strictly correct (we will
discuss the Franz-Keldysh effect later), but it is a reasonable
approximation because there is relatively little change in potential
energy across a unit cell for fields which are interesting
Hence, we draw conduction
and valence bands in
presence of an electric field,
E, with magnitude of the
slope of the conduction and
valence bands as |E|
electron-volts per unit
distance.
EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 30

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 Validity of depletion approximation -
absence of mobile charge
Is the mobile carrier density negligible when calculating fields
and voltages in depletion region?
Yes, at least in thermal equilibrium over most of the distance.
Voltage dropped (~ bandgap energy in eV, e.g., ~ 1.4 eV in
GaAs) >> thermal energy (kBT/e in eV, e.g., ~ 26 meV at room
temperature) hence throughout bulk of depletion region, carrier
concentration is very low because chemical potential is many
kBT inside bandgap region e.g., when band edge has risen 100
meV in depletion region near conducting n region, the electron
concentration will have dropped by ~ e-4 (~ 1/ 50) making
electron charge density << ionized donor density (i.e., about 1/
50 of the ionized donor density)
Hence the mobile charge is negligible for calculation of
electrostatics (voltages and fields), even though it is sufficient to
give rise to substantial currents when the diode is forward
biased.
EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 31

Validity of depletion approximation -

absence of mobile charge (2)
Hence, the vast majority of voltage drop between conducting n
and p regions occurs in material with essentially negligible free
carrier density, justifying the depletion approximation except
possibly very close to the back edges of the conducting n and p
regions
Under bias - the depletion approximation is likely valid as long as
not injecting substantial charge densities by strong forward current
or strong optical absorption.

Note injected carrier densities used for running semiconductor

lasers (e.g., 1018 cm-3) are comparable to doping densities and the
depletion approximation is far from valid in such heavily
forward-biased devices.

EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 32

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 Validity of depletion approximation (3)
At edges of depletion region, depletion approximation assumes
abrupt transition from zero mobile carrier density to density given
by the doping concentration
BUT mobile charge density does not change abruptly, because
carriers diffuse into regions of low density (e.g., depletion region)
actual solution for carrier density near the edge of the depletion
region is not as simple as idealized situation considered above
In the derivation of Debye length, the change in mobile carrier
concentration comparable to the carrier or dopant concentration
First approximation - at edge of depletion region, the change in
mobile charge density is smeared out over ~ Debye length
associated with the doping concentration
The Debye length is usually small (~10%) compared to typical
size of depletion region e.g, 50 nm, compared to typical thickness
of depletion regions for practical carrier concentrations (e.g., 100ʼs
of nm to microns)
EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 33

Debye and depletion lengths

Log-Log Plot of Debye Length and Depletion Length vs.
Electron Concentration (GaAs)
Depletion length depends upon Vbi, Debye length is independent of bandgap
1.00E+05

Debye Length (Å)

Depl Length (Å)
1.00E+04

1.00E+03
Length (A)

1.00E+02

1.00E+01

1.00E+00
1.00E+14 1.00E+15 1.00E+16 1.00E+17 1.00E+18 1.00E+19 1.00E+20
Electron Concentration (cm^-3)

EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 34

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 Gentle Reminder

MIDTERM
Thursday, February 19, 2009

Material thru Recombination

Mechanisms, Ch 4 p 94 of class notes

EE243. Semiconductor Optoelectronic Devices (Winter 2009) Prof. J. S. Harris 35

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