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Electronic Materials and
Processing IChapter 3
OVERVIEW OF ELECTRONIC
DEVICES
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3.1 DIFFUSION AND DRIFT OF CARRIERS
3.1.1 Chemical Potential
-Consider a thermodynamic system containing n constituent species. Its total internal energy U is to be a
function of the entropy S , the volume V , and the number
of particles of each species N 1, ..., N n
U : the internal energy ,
The chemical potential of the i -th species, µ i isdefined as the partial derivative
Where the subscripts emphasize that the entropy,volume, and the other particle numbers are to be
kept constant.
1 2( , , , ,....., )nU U S V N N N =
, , j i
i
i S V N
U
N µ
≠
⎛ ⎞∂= ⎜ ⎟
∂⎝ ⎠
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-In real systems, it is usually difficult to hold the entropy
fixed, since this involves good thermal insulation. It istherefore more convenient to define the Helmholtz energy
A, which is a function of the temperature T , volume, and
particle numbers:
In terms of the Helmholtz energy, the chemical
potential is
-Under conditions of constant temperature and pressure,
the chemical potential is the partial derivative of the Gibbsenergy with respect to number of particles
1 2( , , , ,....., )n
A A T V N N N =
, , j i
i
i T V N
A
N µ
≠
⎛ ⎞∂= ⎜ ⎟∂⎝ ⎠
, , j i
i
i T p N
G
N µ
≠
⎛ ⎞∂= ⎜ ⎟
∂⎝ ⎠
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-A chemical potentials associated with "external" force
fields (Electric potential energy differences, gravitationalpotential energy differences, etc.), and with "internal"
factors (density, temperature, etc.)
-Total chemical potential: internal chemical potential and
external chemical potentialµtot = µint + µext
where
i.e., the external potential is the sum of electric potential,
gravitational potential, etc. (q and m are the charge andmass of the species, respectively, V and h are the
voltage and height of the container, respectively, and g
is the acceleration due to gravity.)
...ext qV mghµ = + +
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-The chemical potential is the generalized potentialenergy of a particle at a given position that describes
the driving force for particle movement .
-Real particles always move to lower their chemical
potential.
-When any two systems are allowed to interact, they
will always exchange matter or energy in such a way as
to equalize the chemical potentials of the systems.-Equilibrium is defined as equal chemical potential
between the two systems.
-The Fermi energy in a solid gives a measure of thechemical potential of the lowest energy free electron or
the highest energy free hole.
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3.1.2 Carrier Motion in a Chemical Potential Gradient
-The force, F, on a particle is expressed through thecorresponding gradient in the chemical potential, µ:
where C is a constant.
-The force on an electron due to a chemical potential gradient
produces a current density J,
.
Here µn is the electron mobility , µ is the chemical potential
.
- Electron chemical potential gradients may result from
electric fields and concentration gradients.
-The presence of an electric field, E, corresponds to a voltage
(potential) gradient dV/dx and contributes to the chemical
potential gradient dµ /dx at constant concentration of electrons
as:
3.2
3.3
3.1
dV E
x dx
∂= − =∂
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-A change in electron concentration, n, in the absence of an
electric field causes a change in chemical potential:
-The general chemical potential gradient in the system :
-Substituting for chemical potential gradient in Equation 3.2
gives:
3.7
3.6
3.5
3.4
-Using the Einstein relation between mobility and diffusivity
D=µnkBT/q :
-The first term is a current due to a concentration gradient
:diffusion current
-The second term is due to the electric field: a drift current.
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3.2 SIMPLE DIODES
-To consider the effect of joining a semiconductor
doped n- with semiconductor p-type:
The n-type material has higher concentration of electrons.
The p-type material has higher concentration of holes
-Two materials are brought into contact:The large electron concentration gradient across
the junction causes electrons to diffuse from the n-type
to the p-type side, leaving positively-charged donor
dopant atoms behind.
The similar process for holes.
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Figure 3.1: Electrons diffuse and drift across a junction between unlike materials.Before contact the Fermi energies (chemical potentials) of
electrons in the two materials do not match.After contact, electrons flow until the Fermi energy is matched across the junction.This electron diffusion produces an electric field. At equilibrium the
drift current balances the diffusion current and no net electron transfer occurs.
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3.2.1 The Junction Contact Potential
-The diffusion of charged species causes an electrostatic potential tobuild up, which opposes further diffusion of electrons.
-The initial difference in chemical potential of the electrons on
the two sides of the junction is gradually reduced and finally
eliminated. At this point, diffusion stops.
-An accumulation of positive charge on the n-type side and negativecharge on the p-type side, causing an electric field at the junction.
-The magnitude of the field can be calculated by setting J=0 in
Equations 3.6 :
3.8
-Substituting E=-dV/dx, and integrating across all x in the electric field:
where Vbi is the “built in” voltage (contact potential) across the
junction ; nn and np are the electron concentrations on the n and p
sides of the junction,
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For a shallow dopant at high temperature:
nn~Nd
where Nd is the donor concentration on the n-type side of the junction.
np=ni2 /pp
or
Substituting nn~ND and using Equation 3.10 for np:
3.10
Or
Substituting 3.12 into 3.9
3.11
3.12
3.13
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-Vbi is zero volts if both sides of the junction are
undoped and increases with NA and ND.
-The built in voltage can be determined from a diagram of the band edges of a junction at equilibrium.
Figure 3.2: The difference in Fermi energies (dashed lines) before contact establishes a band edge bending and a contact potential V bi after contact.
-The built-in voltage can also be written in terms of the
intrinsic carrier concentration ni:3.14
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Figure 3.3: The depletion region and
the charge associated with it showing how the depletion is primarily on the lightly-doped side of the junction. The charges on the two sides of the
junction must balance. The electric field that results is due to the charge on
the two sides of the junction. The maximum electric field is directly related to the total charge on each side of the junction.
-The number of free carriers decreases exponentially with
the distance of the Fermi energy from the band edge,
-A well-defined zone (the “depletion region”) around the
junction contains no free carriers.
-The electric field, present in this region, sweeps out the
mobile carriers (resulting in a drift current that balances
diffusion).
-The charge giving rise to the built-in voltage on each side of
the junction : the number of ionized impurities in the
depletion region on that side of the junction. (Figure 3.3).
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where xp is the depletion width on the p-type side,
NA is the ionized acceptor concentration
q is the acceptor charge,
ε is the dielectric constant of the semiconductor.
-The voltage is the integral of the field over a distance. Graphically,it is the area under the electric field vs. position .
-The electric field E0 of the junction is found by integration of the
charge Q across either side of the depletion region.
For example, on the p-type side:
As
W=xn+xp
The balance of the charges accumulated on the both sides of the
juntions
3.15
3.16
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Solving for xp
Substituting for xp in Equation 3.16 in turn, gives
3.18
3.17
Substituting for Vbi from Equation 3.14 and solving for
W leads eventually to:
3.19
The depletion lies on the lightly-doped side of the
junction
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-When a voltage is applied to a p-n junction, the
total voltage across the junction is
V= Vbi - Vapplied.
A voltage that is the opposite sign as the built in
voltage increases the depletion width.
A voltage of the same sign decreases thedepletion width
3.20
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- The resulting capacitance is voltage dependent and
depends primarily on the concentration of dopant on
the lightly-doped side of the junction
Where Nnet is the difference between the
number of ionized majority carrier dopant atoms andthe number of minority carrier dopants on the lightly-
doped side of the junction.
3.21
-Diode junctions have a significant capacitance. The
effective separation of the “capacitor plates” in the
junction is the depletion width, W.
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-The doping concentration on the lightly doped side
can be determined as a function of depth bymeasuring the capacitance of an asymmetric
junction as a function of voltage,
- For an applied reverse bias voltage VR:
3.22
- This relationship is used in the capacitance-voltage
technique for profiling carrier concentrations near diode
junctions
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Figure 3.4: Band edge diagrams for biased junctions
The band edge
diagrams for biased
junctions are shown inFigure 3.4.
3.2.2 Biased junctions
-To consider the biased junction.-The majority of an applied voltage adds to or subtracts from
the field in the depletion region
-The amount of band bending is increased or reduced.
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-The current density includes both carrier types to
completely describe the situation in a p-n junction:
where np and pn are the electron concentration on the p-type side
and the hole concentration on the n-type side: concentrations of the
minority carriers (np<<pp and pn <<nn ).
-The first term is the total diffusion current
-The second term is the total drift current.
-In a biased junction the chemical potential barrier
associated with the junction is changed.-Drift current flows in unchanged but diffusion current
changes dramatically with the applied bias.
3.23
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3.2.2.1 Drift current
-The drift current is relatively insensitive to the junction
electric field.
-This field sweeps away mobile electrons and holes.
-This sweeping effect does not depend upon how large the
field is as long as it is not too small.
-The drift current does not change under reverse bias
Figure 3.5: A schematic of the motion of minority carriers
by drift across a junction. Drift current does not depend significantly on the electric field magnitude.
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-The number of minority carriers reaching the edge of the
depletion region per unit area depends upon the equilibriumnumber of minority carriers on each side of the junction (pn or
np for the n- and p-type sides of the junction, respectively)
multiplied by the effective velocity at which they reach the
edge of the depletion region.-The effective velocity depends upon their diffusivity (or
mobility) and inversely on the average distance they can
diffuse before recombining with a majority carrier:
veff =Dp /Lp.The resulting drift minority carrier current density :
where Dp and Dn are the minority carrier diffusivities and Lp and Ln
are the carrier diffusion lengths for holes in the n-type and
electrons in the p-type materials
3.24
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-Reverse current density is important in determining the
power that a device consumes when in the “off ” state as
it determines the effective resistance of the device in
reverse bias.
-Reverse current is also referred to as leakage current.
-The reverse current is carried by drift of minoritycarriers from the lightly doped side of the junction
toward the heavily-doped side.
-Case: p+-n diode (the p-type side is very heavily doped)
:
-The hole current can be related to the doping level on
the n-type side and the intrinsic carrier concentration(nnpn=ni
2):
3.25
3.26
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-Reverse saturation current density increases
exponentially with decreasing energy gap.-Ge semiconductor: small energy gap of 0.6 eV,
produces a far higher reverse current density and
consequently much greater “off ” currents.
-Ge devices consume too much power.
Assuming n ~ND, and using expression of ni
3.27
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3.2.2.2 Diffusion current
-The diffusion current is a majority carrier current.
-Diffusion current is carried by electrons leaving the n-
type side of the junction and by holes leaving the p-type
side.
-It depends upon the potential barrier opposing motion
of majority carriers and on the velocity with which
these carriers can diffuse into the material of oppositetype, and their concentration gradient. As the Fermi
energy is at mid-gap, carriers switch from being in the
majority to being in the minority and are to recombine.
-Rapid recombination the region far end of thedepletion region where the majority carriers are not
depleted.
-No recombination occurs in the space-charge region in
a good diode.
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-The concept of the quasi-Fermi level:
The normal Fermi level or Fermi energy, EF, describesthe equilibrium situation in a junction, where np=ni
2.
When the junction is forward biased, electrons and
holes diffuse into the depletion region faster than they
can recombine locally. In this region np>ni2.
-The Fermi level for electrons cannot lie at the same
location as for holes in this case.
-To distinguish these non-equilibrium and bias-
dependent levels from the equilibrium Fermi level :
quasi-Fermi levels, Fn and Fp for electrons and holes.
Fn
− Ei
= kB
Tln(n /ni
)
Ei − Fp = kBTln(p /ni )
.
Ei is the intrinsic energy, near midgap,
3.28
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Figure 3.6: Shows the quasi-Fermi levels for a forward biased junction. The deviation of the quasi-Fermi levels from the Fermi level indicates the excess number of
injected carriers that changes exponentially with the difference in these levels.
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-The current density for the junction is given by theinjected carrier densities,
-The number of electrons increases exponentially as
the potential barrier decreases.
- As the barrier decreases the number of injected
carriers increases exponentially.
Diffusion current :
3.29
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3.2.2.3 Total current in a junction
The total current in a junction is just the sum of the driftcurrent and the diffusion current
This current/voltage relationship known as the diodeequation.
Figure 3.7: The current voltage curve for an ideal diode obeying the diode equation.
The current density in reverse bias is nearly constant,
while it rises exponentially in forward bias
3.30
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Figure 3.8: Comparison of forward currents asa function of reverse current J0.
Minority carrier devices: the magnitude of both the forward
and reverse currents in a homojunction diode depend uponthe minority carrier properties.
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3.2.3 Non-ideal diode behaviors
Series resistance:
Resistance of the semiconductor on the lightly-doped side of the junction outside of the depletion
region.
Contact resistances
Figure 3.9: Compares an ideal diode behavior with a non-ideal diode
exhibiting series resistance.
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- In reverse bias an ideal diode has an extremely high
resistance. Series resistance in the remainder of thedevice is irrelevant as the diode junction is the most
significant resistance in the circuit.
-In forward bias : the resistance of the diode will drop
below that of some other circuit element.- A series resistance leads to a linear current-voltage
curve in forward bias
-The final current : voltage divided by the net
resistance:
R=dV/dJ is the resistance of an ideal diode
3.31
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3.2.3.2 Shunt resistance
A shunt resistance in the diode :
A top contact punches through the device andmake a direct connection to the back contact,
Figure 3.10: A non-ideal behavior due to shunt resistance
compared with an ideal diode.
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-The origine:
physical defects such as holes in the device layer,high conductivity paths such as grain boundaries
or dislocations
-The shunt resistance allows current to flow in spite of
the high resistance of the good portion of the diode in
reverse bias.
-In forward bias the shunt only has an effect at very low
voltages where the diode resistance is still high.- Current density flowing through Shunt resistances :
Jshunt=Vapplied /Rshunt
- Total Current density3.32
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-No obvious effect on reverse (drift) current : the injected
carrier begins as a minority carrier and becomes a
majority carrier upon transiting the depletion region.
It effectively reduces J0.
-A a large effect on forward current: Recombination in
the space charge region dramatically increases thevoltage needed to achieve a given level of current
injection
The added term “a” is called the diode ideality factor
-The value of the ideality factor ranges from 1 for an
ideal diode to at most 2.-The latter occurs when the minority carrier diffusion
lengths (before recombination) are small relative to the
depletion width, W.
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3.2.3.4 Reverse breakdown
There are three forms of breakdown failure in reverse
bias:
tunneling breakdown,
avalanche breakdown,
the soft breakdown associated with current flowthrough defects.
-Avalanche breakdown is generally catastrophic because
the resistance of the diode drops as the process begins.The resulting increased current flow heats the diode
Tunneling breakdown is used in Zener diodes to provide
voltage control..
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Figure 3.12: A schematic of the avalanche breakdown process.
Avalanche breakdown occurs when the depletion width
is relatively large (light doping on both sides of the junction). It is most obvious when the semiconductor is
of high quality.
The basic process is shown in Figure 3.12
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Avalanche breakdown occurs when the depletion width
is relatively large (light doping on both sides of the
junction); the semiconductor is of high quality.The basic process :
A single minority carrier (1) enters the depletion region
and is accelerated.
The carrier gains more kinetic energy than the energy gapand experiences a collision with a valence band electron,
losing its kinetic energy and transferring the valence band
electron to the conduction band, resulting in an additional
free electron (2) and a free hole (3).Both the free electron and the free hole can go on to
experience additional ionizing collisions.
The original electron (1) and the newly free electron (2)
are accelerating and undergoing such additionalcollisions, creating two more electrons (4 and 5) and two
more holes (6 and 7).
Thus, beginning from a single electron, a continuously
increasing current is created.
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As the reverse bias rises, at some point the cascade begins.
The additional current and non-ionizing collisions heat the
system and increase minority carriers which increase the
the reverse current. This leads to the diode acting as a fuse
and burning out.
Th bilit f l t
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The mobility of an electron
µe=qτe /me*
where τe is the mean time between scattering events.A mean length between scattering events of electron
l = vτe
l = me* vµe /q.
For silicon, l = 1.1x10-6 cm (11 nm).The breakdown field EBR is that field at which an electron
accumulates at least the energy of energy gap over a
distance of l.
qEBR l = Egap.For silicon, the energy gap is ~1.1 eV at 300 K. EBR~ 1x106
V/cm.
EBR=VBR /W
where VBR is the breakdown voltage and is a negative value.As the doping rises and the depletion region shrinks, the
voltage for breakdown decreases.
Thus, the breakdown field is inversely related to the
square root of the doping concentration.
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The relationship of breakdown voltage to doping isshown schematically in Figure 3.13.
Assuming VBR>>Vbi and noting that VBR<0,
Figure 3.13: Typical
breakdown voltages as afunction of doping level
calculated from Equation
3.34 and 3.20. Note: the
curvature at low
breakdown voltage
shows the correction for
non-zero Vbi.
3.34
3.35
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Tunneling breakdown occurs in heavily-doped junctions
where the depletion region is relatively narrow.Before the avalanche breakdown process can begin,
electrons can tunnel through the depletion region from
filled states into empty adjacent states
Figure 3.14: A schematic of the tunneling
breakdown process.
The tunneling probability PT through the barrier is given
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The tunneling probability, PTu, through the barrier is given
by:
where C is a constant, and E is the electric field of the
junction [E=(Vbi -Vappl)/W].
The tunneling current is proportional to the tunnelingprobability and the number of empty states adjacent to
filled states.
It increases exponentially with applied voltage.
Higher energy gap materials have lower tunneling currents,as do junctions with higher depletion widths.
Semiconductors with higher densities of states at the band
edges have higher tunneling currents.
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The important points :-as current begins to flow, other resistances in the
circuit drop more of the applied voltage. This reduces
the voltage across the diode and consequently
limits the tunneling current.
-the tunneling current does not increase
significantly with increasing temperature. This is
less to cause the diode to fail in coparision toavalanche breakdown.
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Soft breakdown occurs by generation of carriers through
defect states in the energy gap.Carriers may tunnel through defect states rather than moving
directly band-to-band.
Collisions with energies too low to excite carriers across the
entire energy gap can move electrons into states in the middleof the gap or from there on into the higher energy band edge,
resulting in avalanche-like breakdown
As the reverse voltage is increased, more and more defects
come into the depletion region.
The Fermi energy, shifts with bias. This leads to changes
in the charge state of defects in the energy gap.
These effects combine to increase the amount of reverse
current flowing in a diode with increasing reverse bias.
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Figure 3.15: Combined effects of soft breakdown
and avalanche breakdown on the current/voltagecharacteristic of a diode. The soft breakdown is
shown as might be observed in a very poor diode.
Tunneling has a very similar behavior to avalanche
breakdown but is not self-amplifying.
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The avalanche and tunneling mechanisms have very abrupt
onsets.
where n normally ranges from 3 to 6.Higher quality materials produce more abrupt onsets and
larger values of n.
3 3 SCHOTTKY BARRIERS AND OHMIC CONTACTS
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3.3 SCHOTTKY BARRIERS AND OHMIC CONTACTS
Metal/semiconductor junctions have either linear (ohmic)
or diode-like current voltage characteristics : ohmic
contacts or Schottky diodes,
3.3.1 Ideal Metal/Semiconductor Junctions
The electron chemical potential in metals is the workfunction, Φ.
It measures the position of the Fermi energy with respect to
the vacuum level .
The work function of a semiconductor also measures theposition of the Fermi level.
However, this value changes with doping .
The energies of the band edges with respect to the vacuum
level do not depend significantly upon doping so they aremuch better choices.
The conduction band energy relative to the vacuum level
(the electron affinity, χs) is therefore used to quantify the
general properties of semiconductors .
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Figure 3.16: Shows the definitions of the electron affinity χs of the
semiconductor and the work functions of the semiconductor, Φs
and metal, Φm. Also indicated are the depletion width, W, the
Fermi levels, band edges, and the Schottky barrier heights, ΦB.
Note that Schottky barriers to both n and p-type semiconductorsexist but that to calculate the barrier for the p-type semiconductor
you need to compare the valence band edge with the metal work
function while for the n-type material it is the conduction band
that is compared to the work function.
Wh t t i d b t t l d
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When contact is made between a metal and a
semiconductor, electrons flow from one material to the other as determined by their relative Fermi levels. This flow can
lead to depletion or enhancement of the majority carriers.
There are four possible situations to consider:
a low work function metal in contact with (1) a p-type or (2)
an n-type semiconductor, and the same cases for a high work
function metal.
“High” and “low” work function here simply indicate whether
the Fermi level for the metal lies above or below the Fermi
level of the semiconductor before contact is made.
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Figure 3.17: The four possible metal-semiconductor cases
before and after contact. Two of the four produce ohmic
contacts (resistors) while the others produce Schottky
diodes.
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These four possibilities are shown schematically in Figure
3.17.When the direction of electron flow is into n-type material
or out of p-type material,
it increases the majority carrier concentration of the
semiconductor. This increasesthe conductivity of the semiconductor near the junction
and is described as "ohmic"
because the junction acts only as a resistor. Such
junctions are desired for contacts to microelectronicdevices.
Unfortunately, it is often impossible to find a metal of
sufficiently high or low work function to arrange the
electron flow in the correct direction to obtain an ohmiccontact, especially to wide-gap semiconductors and
organic electronic materials
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When charge flowing into the semiconductor is of the
opposite sign as the majority carrier type, the majoritycarriers are depleted near the junction forming a resistive
depletion region (Figure 3.17).
An applied bias voltage appears across this relatively
insulating depletion region, modifies the chemical potential of
electrons across the junction and induces current flow as
shown in Figure 3.18.
For positive applied voltages the barrier for majority electrons
is decreased, causing them to diffuse into the metal over the
barrier.
The diffusion current changes exponentially with bias
voltage, as it does in a p-n junction.
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Figure 3.18: The effects
of bias voltage onSchottky diode band
edges and band
bending.
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In reverse bias the barrier height for electron motion from an
n-type semiconductor or hole motion from a p-typesemiconductor to a metal is increased.
The flow of electrons/holes in this direction decreases
exponentially to near zero.
Charge flow from the metal to the semiconductor faces aconstant barrier height, ΦB (Fig 3.18).
This height is characteristic of any metal-semiconductor
junction independent of the semiconductor doping, and is
known as the Schottky barrier height,ΦB= ΦM - χS.
Compare the metal work function with the valence band
edge
ΦB= χS+Egap-ΦM.
The current voltage curve for the resulting metal-
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The reverse saturation current depends upon a
Boltzmann factor describing the probability that anelectron will have sufficient energy to overcome the
barrier height rather than the semiconductor energy gap.
Thermionic emission: electrons are emitted from a hot
filament.The final form of the reverse saturation current is:
g g
semiconductor junction has the same form as a
homojunction diode:
where A and B are constants.Significant advantage : diode operation does not depend
upon the minority carrier properties of the
semiconductor. The only carriers injected across the
unction are ma orit carriers.
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A Schottky diode can be made to operate as an ohmic
contact by increasing the doping level of thesemiconductor near the junction.
The depletion width in the semiconductor becomes very
small which lead to very low tunneling breakdown
voltages.
Once breakdown occurs the diode operates as an ohmic
contact.
3 3 2 Real schottky diodes
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3.3.2 Real schottky diodes
- Clean surfaces of semiconductors typically have a
large number of defect states.
Because of the limited doping and carrier
concentration of the semiconductor, the Fermi level atthe surface is determined by the energy of these
defects, rather than by the dopant atoms.
A surface Fermi level measurement usually gives
an unreliable estimate of the barrier height.The typical barrier potential in a real Schottky
diode is shown schematically in Figure 3.19.
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Figure 3.19: A defective Schottky contact, typical of real
barriers. The small lines indicate defect states in thesemiconductor.
-Thin layers of compounds can form at the junction or
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Thin layers of compounds can form at the junction or
short-range diffusion can broaden the effective interface.This spreads out or changes the potential at the
boundary and causes a change in the effective barrier.
Advantage of such reactions in forming metal-Si
Schottky barriers by reacting the metal with thesemiconductor to form a silicide.
This gives a very reliable contact because the
junction is formed by solid phase reaction within the Si
rather than at the original semiconductor surface resultingin few interfacial defects.
-Finally, interfacial defects can permit reverse current to
flow by tunneling through the resulting states (see Figure3.19), which can lead to an unexpectedly high current
(an anomalously low apparent barrier).
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3.4 SEMICONDUCTOR HETEROJUNCTIONS
Heterojunctions are any electrical junctions between
dissimilar materials.
Semiconductor heterojunctions are very useful : one
semiconductor can be used to emit light while a
second can be used as a top contact
3.4.1 Heterojunctions at equilibrium
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-The critical points in determining the electronicproperties of a semiconductor heterojunction :
The energy gaps,
Electron affinities,
Doping types and levels.
-To consider the case: the Fermi levels of two
semiconductors are arranged to be at the same energy
before contact.-This “flat band” condition means that upon contact there
is no net flow of electrons across the junction and the
band edges remain flat.
-The energy gaps and electron affinities of flat bandsemiconductor heterojunctions lead to three possible
band-edge configurations, shown in Figure 3.20.
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Figure 3.20: the band edges in
the flat band condition for thetypes of semiconductor
heterojunctions.
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-“Straddling” : the conduction and valence band edges
of the narrow-gap semiconductor lie within the energy
gap of the wide-gap semiconductor.
-To obtain a straddling gap :One gap to be significantly larger than the other:
EgA > EgB
The electron affinity of the smaller gap material isslightly larger than that of the wider gap material:
EgA-EgB > χB
- χA > 0.
A typical example of a straddling gap configuration is
the AlAs-GaAs heterojunction
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.- Offset-gap heterojunction :the energy gaps are roughly equal:
EgA~EgB
the electron affinities are differentχA > χB.
The difference is less than the energy gap of either
constituent: both band edges of one semiconductor lieabove both band edges of the other but where a
portion of the energy gap overlaps across the interface.
An example of an offset-gap heterojunction is
between InSb and InP.
- “Broken gap” semiconductor heterojunctions are
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composed of semiconductors with such extremedifferences in electron affinity that the band gaps do
not overlap at all.
-In this case the valence band edge of one
semiconductor lies above the conduction band edge of the other.
-An example of this type of junction is between InAs
and GaSb.
- Such semiconductor heterojunctions usually involveat least one relatively narrow-gap material in order to
achieve sufficient difference in electron affinities.
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-Transitivity principle:
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y
If the discontinuities between any twosemiconductors and a third semiconductor are
known, the discontinuity between the first two
can be inferred.
3.41
The Common Anion Rule:
When the anion (the electron accepting atomsuch as As in GaAs and InAs) is in common across a
semiconductor heterojunction, the change in the
conduction band edge is greater than the change in the
valence band edge across the semiconductor
heterojunction.
∆EV<∆EC.
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The Common Cation Rule:When the cation (e.g. Ga in GaAs or GaSb) is in
common across the junction, the valence band edge
energies scale with the anion electronegativities.
For example, the valence band edge of phosphide
semiconductors will lie below those for arsenides which
will lie below those of antimonides.
EV(CA1) < EV(CA2) < EV(CA3),
where C designates a cation, and A1, A2, and A3
designate three anions with decreasing
electronegativities.
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Figure 3.21: Recommended
steps in drawing asemiconductor heterojunction.
Step 5 shows the complete band
edges for this junction.
A series of steps for making sketches :
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1. Mark a vertical line on a page to indicate the junctionand a horizontal line to indicate the flat-band vacuum
level.
2. Based on a constant vacuum level, mark the band
edge positions on the vertical line using the electronaffinity and energy gap values for the two materials.
These positions are doping-independent and fix the band
edges in relation to one another at the junction.
3.Draw a horizontal dashed line indicating the Fermi levelat equilibrium.
4. Far from the junction, mark the band edges relative to
the Fermi level as appropriate to the doping in each
semiconductor.
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5. Connect the band edges far from the junction to the
band edges you marked in Step 2 as appropriate.
-Requirements to be met in this step are:
I. the slopes of the connecting segments must matchwhere they intercept the points marked in Step (2) to
satisfy the electrostatic continuity equation.
II. the energy gaps must be constant on each side of the
junction because the gap for a given semiconductor doesnot change.
III. the depletion widths are fixed by the doping levels.
These are calculated as in Equation 3.19.
-The major question is where to put the Fermi level with
respect to the flat band band edges (step 3)
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respect to the flat-band band edges (step 3).
Marking this position will determine on which side
of the heterojunction most of the band bending occurs.
When one semiconductor is very heavily doped :
Mark the Fermi level with respect to the flat band edge of
the heavily-doped material. There will be very little band
bending in that material (see Figure 3.22).
When the doping levels are nearly equal (see
Figure 3.21), the resulting electric field will be distributed
roughly equally across the heterojunction and the bandbending is nearly symmetric. In this case the Fermi level
must be drawn close to the middle of the distribution of
flat band points (Figure 3.21).
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Figure 3.22: Shows the formulas for calculation of
various energies in the heterojunction. This case issimilar to that shown in Figure 3.21, but with very
heavy doping in the AlAs layer to produce the electron
gas.
-The potentials (as shown in Figure 3.21) are distributed
approximately, assuming fully ionized dopants, as:
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where ΦD indicates the potential energy difference
associated with band bending on the designated side of
the junction.
- The contact potential (difference in Fermi energies of thesemiconductors before contact).
approximately, assuming fully ionized dopants, as:
As in the flat band picture there are discontinuities in
the band edges at the hetero-junction.
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The discontinuities result from the fact that not onlythe doping and the electron density are changing
across the interface, but that the atoms themselves are
changing.
The effective charges on the cores of the atoms makingup the lattice on the two sides of the junction change as
one goes across the interface. This leads to a step
function in electron potential. The magnitudes of the
band edge discontinuities can be calculated from the
electron affinities and energy gaps of the two materials,
as shown in Figure 3.21
3.4.2 Heterojunctions as diodes
To consider a perfect abrupt semiconductor
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heterojunction (p-GaAs/n-AlAs),-A “forward” applied bias will reduce the potential
between the layers, narrow the depletion region, and lower
the barrier for carrier injection (by diffusion) across the
junction.
-As the opposite signs of the band gap discontinuities
(one is a step upward, the other a step downward),
electrons emitted into the GaAs, will have a much lower barrier to overcome than will holes injected from the GaAs
into the AlAs.
Thus, most of the forward diffusion current will be carried
by electrons.
-In reverse bias the currents are simply the thermally-
generated carriers in the GaAs which reach the depletion
region and are swept into the AlAs.
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g
The much larger energy gap of the AlAs :few minority carriers in this material
no significant contribution of minority carrier drift
from the AlAs to the GaAs.
The barriers at the heterojunction will usually have little
effect on the reverse current drift process in high-quality
materials, as the carriers will generally gain enough
energy to pass over them.In a lightly-doped junction where the depletion region is
broad, and where the scattering of carriers is rapid for
any reason, the spike in the conduction band can
influence the behavior of the device.The minority carrier properties in the GaAs will generally
affect this type of device as current injection and
recombination processes are very similar to the
homo unction case.
- The behavior of the junction shown in Figure 3.22
responds in general in the same way as described above
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except that now the AlAs is very heavily doped.This further enhances the dominance of injection of
electrons into the GaAs.
The injected electrons come from the electron gas
induced in the GaAs, and the spike at the interface is
truly irrelevant (except that it is responsible for the
formation of the electron gas in the first place).
The behavior now becomes that of an n
++
-p GaAshomojunction.
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Figure 3.23: An n-n heterojunction with offset gap. Thisheterojunction may show Schottky diode-like behavior
based on injection of electrons across the spike in the
conduction band.
The larger potential barrier in the valence band reduceshole injection from the narrow gap material into the wider
gap material. An applied voltage reduces band bending in
both semiconductors.
a heterojunction such as shown in Figure 3.23, the
current is carried primarily by injection of electrons
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across the spike.both semiconductors are n-type but even so, a depletion
region occurs on one side of the junction
Electrons moving from the electron gas into the wide-gap
material face a barrier that does not change significantlyfor modest bias voltages.
Forward bias reduces the scale of the field causing the
electron gas to accumulate and reduces the barrier for
electron injection from the wide-gap material across thespike.
In reverse bias the behavior is determined by injection of
electrons from the electron gas across the spike.
Thus, there will be a Schottky-like behavior for at least
modest bias voltages.
3.5 TRANSISTORS
3.5.1 Bipolar junction transistors
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Figure 3.24: An example of the
connection of an n-p-n typetransistor. The circuit schematic
shows the transistor in forward
bias. Electrons are injected
from emitter to base due to theforward bias. Some electrons
flow across the base and reach
the collector. A minority are lost
through recombination in thebase. These are few and far
more reach the collector.
The bipolar transistor consists of two diodes joined back
to back by a thin common semiconductor layer (the
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base).bipolar transistors are referred to as p-n-p or n-p-n
depending upon whether the common base layer is n or p
type.
Consider a typical n-p-n bipolar junction transistor,shown schematically in Figure 3.24
When the emitter junction is forward biased, electrons are
injected into the base and holes into the emitter.
The holes injected into the emitter recombine there andare of no value to operation of the device.
The electrons emitted into the base may either
recombine, contributing to base current, or transit the
base.
If the base is thin, most of the electrons pass through it
without recombining, reach the reverse-biased collector
junction, and are accelerated into the collector
As the time required for the injected electrons to diffuse
through the base to the collector is less than the
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minority carrier lifetime, the transistor will permit morecurrent to flow emitter to collector, icollector than from
emitter to base, iba
The current gain of the transistor is approximately:
where τn and τn are the electron lifetime in the baseregion and the transit time required for an electron to
pass through the base.
If the base is wider than the mean-free-path between
collisions, then the carrier must diffuse across the base.By removal of the forward biased emitter junction, there
is no injection and therefore no current emitter to
collector.
3.43
the current emitter to collector is controlled by current
emitter to base as much as by the voltage emitter to base.
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The collector current increases rapidly with base currentbut the base current increases exponentially with base
voltage.
Therefore the emitter-to-base resistance is moderate and
decreases with increasing voltage.A bipolar junction transistor has a low input impedance
(resistance) and is a current amplifier .
-BJT’s are known as minority carrier devices:
performances are controlled by the minority carrier
ti
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properties.-Two require:
The minority carrier lifetime in the base is critical
to achieving high gains.
To achieve low transit times by diffusion of carriers
through the base, a high carrier mobility is needed.
-Very good quality material with relatively low doping
concentrations and no unintended defects.- Some impurities such as Fe in Si cause such significant
loss of minority carrier properties that degradation in
bipolar transistor performances can be detected at Fe
concentrations of parts per trillion.This has been a major driving force for improved purity in
Si wafers.
-BJT gains and switching speed are controlled by base
t it ti G A d i
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transit times : GaAs devices.-GaAs has a higher electron mobility than Si, which
should lead to higher speed n-p-n devices.
-It is difficult to work with very brittle GaAs and to form a
good contact to and insulator on it- Si transistors outperform most GaAs devices. : the
devices can be made smaller , contact resistance and
surface and interface states are less of an issue.
-This compensates for the reduced carrier mobility and
results in similar net performances.
-Only with great care and effort can GaAs outperform Si,
which makes GaAs devices expensive and their applications limited.
3.5.2 Field-effect transistors
-A typical structure of the field-effect transistor (FET) is
h i Fi 3 25
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shown in Figure 3.25.
Figure 3.25: A schematic of a typical enhancement-mode n-p-n
“metal-insulator-semiconductor” field effect transistor. The device is
normally off and conduction is enhanced by application of a negative
gate voltage relative to the source, creating an n-type channel. Thesource-to-drain voltage adds to the gate bias when turned on, which
is why the channel is wider at one side. The hatched area under the
gate indicates an optional heavily-doped
polycrystalline semiconductor region as part of the gate .
T o hea il n t pe regions in a lightl p t pe s bstrate
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-Two heavily n-type regions in a lightly p-type substrate,fabricated in such a way as to leave a short p-type barrier
separating the n-type areas.
-The heavily doped materials are called the “source”
(from which electrons are flowing when on) and the“drain” (into which the electrons drain).
-A gate electrode provides the current control by causing
majority carrier depletion and ultimately inversion of the
channel in the substrate, drawing charge into the hannelas in a capacitor.
This allows current to flow easily from source to drain.
-More bias voltage produces a wider and more conductive
channel, leading to a lower resistance from source to
drain.
-The FET is considered a majority carrier device because,
the source, drain, and channel all conduct majority
carriers
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carriers-FETs are sensitive to majority carrier properties and are
less influenced by defects in the materials than are BJTs.
“junction” field effect transistors (JFET’s):
The type conversion, providing the conducting channel,
can be achieved by creating a heterojunction or
Schottky contact to the surface of the transistor
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Schottky contact to the surface of the transistor.The depletion region of such a junction provides type
conversion in the channel.
This depletion region can be increased or decreased
depending upon the gate bias voltage used.
These are used for GaAs-based devices : high quality
heterojunction diodes are relatively easy to fabricate
The major problem with the JFET design is that somegate current can occur even in a reverse-biased junction.
Therefore, the gate resistance is limited by the
conductivity of the reverse biased gate junction.
The most common form of the field-effect transistor uses
a metal-insulator-semiconductor (MISFET) structure .
Metal oxide semiconductor FET’s or MOSFETS when
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Metal-oxide-semiconductor FET s or MOSFETS whenSiO2 is used for the gate insulator
Silicon oxynitride gate dielectric can be a remarkable
insulator, withstanding fields of more than 107 V cm-1 with
very low leakage currents.Gate dielectrics less than 5 nm thick , as thin as 1 nm.
The effects of bias voltage
on the electric fields in the
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Figure 3.26: Shows the change in carrier concentration in a
semiconductor near the gate of a MISFET for three applied fields.
Under accumulation conditions the majority carrier concentration of the channel region is enhanced. Depletion conditions reduce the
majority carriers without causing the majority carrier type to change.
In the inversion regime the majority carrier type near the insulator is
reversed.
on the electric fields in the
gate insulator and the
underlying semiconductor
are shown in Figure 3.26.
FETs can operate in a number of modes:
Enhancement mode devices are normally off: Therei h l d th t d t d t fl
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Enhancement mode devices are normally off: Thereis no channel under the gate and current does not flow
unless a bias voltage is applied to turn conduction on.
Depletion-mode devices are normally on, having a
pre-existing channel. In these devices a voltage must beapplied to deplete the channel of charge and turn them
off.
Depletion-mode device must have an underlying
insulating layer against which the channel can be closed.
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Figure 3.27: A schematic diagram of a typical MISFET structure with a
conventional gate metal with spontaneous band bending due to
electron flow (right) and the situation with a heavily-doped
semiconductor gate using the same semiconductor as in the channel
(left). The strong depletion of carriers in the contact to the gate metal
now happens at the contact with the doped gate semiconductor. The
latter is doped sufficiently that the contact breaks down at essentially
zero volts, leading to an effectively ohmic contact.
The control of threshold voltage has largely been solved
by making the gate of a very-heavily-doped
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by making the gate of a very-heavily-doped
semiconductor of the same type as the barrier between
source and drain (Figure 3.27).
In silicon devices, polycrystalline silicon (poly-Si) can beused.
It is polycrystalline because it is deposited on a non-
single-crystal substrate (typically amorphous SiO2) and
so there is no template for formation of a single crystalmaterial.
Fortunately, the polycrystalline form works well when
heavily doped.
Using a poly-Si gate, the equilibrium Fermi energy isessentially constant between the gate and the channel
(Figure 3.27).
The most significant advantages of poly-Si gates in Si
devices :
(1) They can be doped the same type as the underlying
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(1) They can be doped the same type as the underlyingmaterial, resulting in very little and very predictable band
bending in the adjoining Si channel region for either
channel type.
(2) Any of a wide variety of metals can be used for contactto the semi-conductor gate material, regardless of the
carrier type, because the gate semiconductor is heavily
doped and so the resulting Schottky diode breaks down
at very low voltages.
(3) Metal contacts to poly-Si are relatively stable and
predictable, and the poly-Si/SiO2 interface is very stable
and reproducible.
3.6 LIGHT-EMITTING DEVICES
3.6.1 Light-emitting diodesOne of the most basic solid-state optical devices is the
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3.6.1 Light emitting diodesOne of the most basic solid-state optical devices is the
light emitting diode: a p-n homojunction
In forward bias, minority carriers are injected across the
junction.Recombination increases within the depletion region
when the quasi-Fermi levels are more widely separated.
In a direct-gap semiconductor this recombination often
releases the excess energy in the form of light.The critical issues relate to the probability of light
emission and the possible recombination of carriers
through non-radiative processes.
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Figure 3.28: (a) A band edge diagram for light emission in a forward-
biased homojunction diode. Fn and Fp indicate the quasi Fermi
levels. The difference between these levels and the equilibrium Fermi
level Ef indicates the excess hole or electron minority carrier
populations.
(b) A schematic of a typical light emitting diode in a standardmounting. The reflector and lens focus almost all of the emitted light
out of the front of the lens providing high efficiency and brightness in
the forward direction.
3.6.1.1 Radiative recombination
Light emission requires that electrons and holes
recombine in a single step, satisfying requisiteconservation laws and with allowed quantum-mechanical
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g p, y g qconservation laws and with allowed quantum-mechanical
selection rules.
The rate at which radiative recombination, Rradiative, takes
place between free electrons and holes across the energygap of a semiconductor depends upon the product of the
number of electrons and the number of holes in the
material.
The rate at which these interact depends also upon their thermal velocity vth (the velocity of carriers with a kinetic
energy near kBT), the cross section for the recombination
process SR
, and a factor due to internal reflection in the
material.
where η is the index of refraction of the semiconductor. -
-The cross section is the apparent size of an electron to a
moving hole, or vice versa, and gives a measure of howeffectively one particle is captured by the other
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g , , geffectively one particle is captured by the other.
where T is the temperature in Kelvin, and α is the
absorption coefficient in cm-1
For Si the radiative recombination cross section at 300
K is 1.8x10-19
cm2
for GaP : 9.3x10-18 cm-2
-Some points to notice are the following:
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p gThe radiative recombination rate scales as
np. Therefore, the more carriers that can be
injected across the junction and the more heavily-
doped the junction, the higher the radiative
recombination rate.
Materials with higher energy gaps and
lower absorption coefficients have higher crosssections , more rapid radiative recombination.
Finally, radiative recombination goes up as
the temperature goes down.
-Radiative recombination normally occurs from band
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edge to band edge.
- Radiative recombination by excitons
- An exciton: the electron and hole becoming bound
together by electrostatic attraction.
A rough approximation of the exciton, as a
hydrogen atom in a dielectric medium
Typically the exciton binding energy is of theorder of 10±6 meV for common inorganic
semiconductors. In organic materials it can be 1 eV.
The emitted photon has a lower energy than
the gap of the semiconductor and is therefore notsignificantly reabsorbed as it leaves the material.
3.6.1.2 Non-radiative recombinationAuger processes : the electron recombining with a
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- Auger processes : the electron recombining with a
hole passes its energy to
another electron which then gives up the energy to
heat-Trap mediated recombination : Trap-related
recombination only occurs when the material contains
a defect of some sort.
3.6.2 Laser diodes
Stimulated emission : wave emitted with samel th h d i di ti th
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wavelength, phase and in same direction as the
driving wave.
the phenomenon of population inversion needed for
lasing
3
21
3
21
(stim)( )
(spon) 8
R ch
R h
ρ ν
π ν
=
-The ratio of stimulated to spontaneous emission
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107
-The ratio of stimulated emission to absorption is
-There are two important conclusions:
For stimulated photon emission to exceed photonabsorption we need to achieve population inversion, that is
N 2 > N 1 .
-For stimulated emission to far exceed spontaneousemission, we must have a large photon concentration which
is achieved by building an optical cavity to contain the
photons.
21 2
12 1
(stim)
(absorp)
N
N =
It is important to point that the population inversionrequirement N > N means that we depart from thermal
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108
requirement N 2 > N 1 means that we depart from thermal
equilibrium.
According to Boltzmann statistics in :
N 2 > N 1 implies a negative absolute temperature!
The laser principle is based on non-thermal equilibrium.
2 2 1
1
( )exp
B
N E E
N k T
⎡ ⎤−= −⎢ ⎥
⎣ ⎦
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In a Fabry-Perot laser, feedback is provided by
mirrors separated by length d. This results in the
modal spacing in frequency of c/2d.
-Laser oscillation condition (amplification).
1 1lnhg γ
⎛ ⎞= + ⎜ ⎟
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-Phase Condition and Laser Modes
The round-trip phase change ∆φround-trip must be a
multiple of 2π
∆φround-trip =m2πwhere m is an integer, 1, 2,…. This condition ensures
self-replication
1 2
ln2
thg L R R
γ + ⎜ ⎟⎝ ⎠
where γ is the attenuation coefficient of the medium threshold
gain g th, reflectances R 1 and R 2 of the facets
(a) A double heterostructure
diode has two junctions which
are between two differentbandgap semiconductors (GaAs
and AlGaAs)
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111
and AlGaAs).
(b) Simplified energy band
diagram under a large forward
bias. Lasing recombination
takes place in the p-GaAs layer,
the active layer .
(c) Larger bandgap materials
have a lower refractive index.
(d) AlGaAs layers provide
lateral optical confinement.
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Figure 15 Schematic illustration of the structure of a double heterojunction stripe contact laser
diode.
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•The active layer (GaAs) is effectivelyburied
within a wider bandgapmaterial (AlGaAs) and the structure is hence called buried double
heterostructure laser diode.
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