5 currenst at pn junc

8/11/2019 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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