5 Currenst at Pn Junc
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Transcript of 5 Currenst at Pn Junc
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Currents At PN Junction
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Continuity equation
The continuity equation describes a basicconcept, namely that a change in carrier density
over time is due to the difference between the
incoming and outgoing flux of carriers plus the
generation and minus the recombination.
The flow of carriers and recombination and
generation rates are illustrated with Figure .
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The rate of change of the carriersbetween x and x + dx equals the
difference between the incoming flux and
the outgoing flux plus the generation andminus the recombination:
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where Jn(x,t) is the carrier density,Ais thearea, Gn(x,t) is the generation rate
and Rn(x,t) is the recombination rate.
Using a Taylor series expansion,
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this equation can be formulated as afunction of the derivative of the current:
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and similarly for holes one finds:
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A solution to these equations can beobtained by substituting the expression for
the electron and hole current.
This then yields two partial differentialequations as a function of the electron
density, the hole density and the electric
field. The electric field itself is obtained
from Gaussslaw.
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A generalization in three dimensions yieldsthe following continuity equations for
electrons and holes:
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Carrier generation and recombination
Recombination of electrons and holes is aprocess by which both carriers annihilate each
other:
the electrons fall in one or multiple steps into the
empty state which is associated with the hole.
Both carriers eventually disappear in theprocess.
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The energy difference between the initial andfinal state of the electron is given off.
This leads to one possible classification of the
recombination processes: In the case of
radiative recombination this energy is emitted in
the form of a photon.
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In the case of non-radiative recombination it ispassed on to one or more photons and in Auger
recombination it is given off in the form of kinetic
energy to another electron.
Another classification scheme considers the
individual energy levels and particles involved.
These different processes are further illustratedwith the figure below.
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Band to Band
The recombination occurs when an electronfalls from its state in the conduction band into
the empty state in the valence band which is
associated with the hole.
This band-to-band transition is typically also a
radiative transition in direct bandgap
semiconductors.
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Trap Assisted
The recombination occurs when an electron fallsinto a "trap", an energy level within the bandgap
caused by the presence of a foreign atom or a
structural defect.
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Once the trap is filled it can not accept anotherelectron. The electron occupying the trap energy
can in a second step fall into an empty state in
the valence band, thereby completing the
recombination process
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One can envision this process either as a two-step transition of an electron from the conduction
band to the valence band or also as the
annihilation of the electron and hole which meet
each other in the trap.
We will refer to this process as Shockley-Read-
Hall (SRH) recombination.
http://ecee.colorado.edu/~bart/book/recomb.htmhttp://ecee.colorado.edu/~bart/book/recomb.htmhttp://ecee.colorado.edu/~bart/book/recomb.htmhttp://ecee.colorado.edu/~bart/book/recomb.htmhttp://ecee.colorado.edu/~bart/book/recomb.htmhttp://ecee.colorado.edu/~bart/book/recomb.htmhttp://ecee.colorado.edu/~bart/book/recomb.htmhttp://ecee.colorado.edu/~bart/book/recomb.htmhttp://ecee.colorado.edu/~bart/book/recomb.htmhttp://ecee.colorado.edu/~bart/book/recomb.htmhttp://ecee.colorado.edu/~bart/book/recomb.htm -
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Auger
The recombination is a process in which anelectron and a hole recombine in a band-to-band
transition, but now the resulting energy is given
off to another electron or hole.
The involvement of a third particle affects the
recombination rate so that we need to treat
Auger recombination differently from band-to-band recombination.
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Each of these recombination mechanisms canbe reversed leading to carrier generation rather
than recombination.
A single expression will be used to describe
recombination as well as generation for each of
the above mechanisms.
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Carrier generation due to lightabsorptionoccurs if the photon energy is large
enough to lift an electron from the valence band
into an empty state in the conduction band.
The photon energy needs to be at least equal to
the bandgap energy to satisfy this condition..
http://ecee.colorado.edu/~bart/book/recomb.htmhttp://ecee.colorado.edu/~bart/book/recomb.htmhttp://ecee.colorado.edu/~bart/book/recomb.htmhttp://ecee.colorado.edu/~bart/book/recomb.htmhttp://ecee.colorado.edu/~bart/book/recomb.htmhttp://ecee.colorado.edu/~bart/book/recomb.htmhttp://ecee.colorado.edu/~bart/book/recomb.htmhttp://ecee.colorado.edu/~bart/book/recomb.htmhttp://ecee.colorado.edu/~bart/book/recomb.htmhttp://ecee.colorado.edu/~bart/book/recomb.htmhttp://ecee.colorado.edu/~bart/book/recomb.htm -
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The photon is absorbed in this process and theexcess energy, Eph-Eg is added to the electron
and the hole in the form of kinetic energy
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Impact ionization
Finally there is a generation processcalled impact ionization, the generation
mechanism which is the counterpart of Auger
recombination.
Impact ionization is caused by an electron (hole)
with an energy which is much larger (smaller)
than the conduction (valence) band edge.
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A simple model for the recombination-generationmechanisms states that the recombination-
generation rate is proportional to the excess
carrier density.
It acknowledges the fact that no recombination
takes place if the carrier density equals the
thermal equilibrium value.
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The resulting expression for therecombination of electrons in a p-type
semiconductor is given by:
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similarly for holes in an n-typesemiconductor:
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Auger recombination
Auger recombination involves three particles: an
electron and a hole which recombine in a band-to-band transition and give off the resulting
energy another electron or hole.
is the coefficient representing interactions
in which the carrier is an electron or hole.
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Diode Analysis
The equations of diode are solved by makingseveral simplifying assumptions.
In addition to the assumption of a one-
dimensional device, the most important
simplifying assumption in determining a closed
form solution to the above equations is the
depletion approximation.
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According to this assumption, the device canthen be broken up into regions that have an
electric field and those that do not.
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Divide the device into regions with an electricfield and without an electric field.
Solve for electrostatic properties in the depletion
region (Region II on the diagram). This solution
depends on the doping profile assumed.
Solve for the carrier concentration and current inthe quasi-neutral regions (Regions I and III on
the diagram) under steady-state condition
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Assumptions are The electric field is confined to the junction
region and there is no electric field in the quasi-
neutral regions.
No free carriers (n(x), p(x) = 0 ) in depletion
region.
We can assume no free carriers since the
electric field sweeps them out of the depletionregion quickly
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The only equation left to solve is PoissonsEquation, with n(x)andp(x)=0, abrupt
doping profile and ionized dopant atoms.
Poissons equation then becomes:
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The total current in the device must be constant,independent of distance as long as there is not a contact
that can extract or inject carriers and as long as the
device is under steady state conditions. This can be
shown by:
dJT/dx=d(Jp+Jn)/dx
=dJp/dx+dJn/dx=q(Up+Gp)+q(Un+Gn)
=q(GnGp)q(UnUp)
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Current across the depletion region
The total current as the sum of thecurrents at the edges of Region I and III,
as shown below:
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A more accurate solution includes thechange in Jnand Jpacross the depletion
region, and we find the total current by:
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The continuity equations,
and in the depletion region this becomes
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Often, the recombination term is ignoredand Gis assumed to be a constant, such
that
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Wide Base P-N Junction
The change in the current across thedepletion region is:
Assuming that there is no generation and recombination, then Jn= 0 and
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Typically, we write the equation in theform:
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Narrow Base Diode
The change in the current across thedepletion region is given by the general
equation:
If there is a constant generation across the depletion region and no
recombination, then
wherexnis the depletion width in the p-type material.
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Jnat the edge of the depletion region inthe p-type material is:
Jnat the edge of the depletion region in the n-type material is:
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the total current is:
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Small signal equivalent of diode
The diode is modeled by a resistanceequal to the inverse of the slope of the
tangent to the i-v curve at the bias point.
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IC Diode
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