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Lecture Notes – 09ECE 531 Semiconductor Devices

Dr. Andre Zeumault

Outline

1 Overview 1

2 Non-Ideal Behavior 12.1 Ideal Diode Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Comparing the Ideal Diode Equation to Experiment . . . . . . . . . . . . . . . . . . 22.3 Breakdown Under Reverse Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3.1 Impact Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3.2 Band-To-Band Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Current Due to Recombination and Generation . . . . . . . . . . . . . . . . . . . . . 72.4.1 Trap-Induced Recombination and Generation – Qualitative Visualization . . 72.4.2 Trap-Induced Recombination and Generation – Quantitative Description . . 82.4.3 Shockley-Read-Hall Recombination . . . . . . . . . . . . . . . . . . . . . . . . 112.4.4 R-G Current in the Depletion Region . . . . . . . . . . . . . . . . . . . . . . 14

1 Overview

In the previous lecture, we derived the ideal diode equation. In this lecture, we discuss sourcesof non-ideality in a pn-junction diode as a method of refining our simple treatment. The iterativeapproach, which involves (1) formulation of a simple theory, (2) comparing predictions of the simpletheory to experiment, (3) formulate a revision to the simple theory, (4) comparing the revised theorywith experiment, etc... is an indefinite process and is common to both engineering and science.Topics to cover include:

• Ideal Diode Equation

• Comparing the Ideal Diode Equation to Experiment

• Breakdown Under Reverse Bias: impact ionization and tunneling breakdown

• Current Due to Recombination and Generation: Trap-induced Recombination andGeneration

2 Non-Ideal Behavior

2.1 Ideal Diode Equation

Recall, from previous lecture, the ideal diode equation is given as follows:

I = I0

(e

eVAkbT − 1

)I0 = eA

(Dp

Lp

n2iN+D

+Dn

Ln

n2iN−A

)The reverse saturation current, I0, is a constant with respect to voltage, and the forward currentis exponential with voltage. Also, there is no mechanism for breakdown occurring at largenegative voltages.

In the depletion region of a diode, a non-negligble current component due to recombination-generation exists (R-G current) giving rise to a |I| ∝

√VR dependence. Additionally, breakdown –

a large spike in current beyond a particular value of VR exists. We will account for both of thesein this lecture.

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Dr. Andre Zeumault

Figure 1: Linear-scale plot of a measured IV characteristic of a Silicon diode.

2.2 Comparing the Ideal Diode Equation to Experiment

The measured current of a Silicon diode is shown in Figure 1 . At first glance, the measureddata corresponds well to the ideal theory. The current under forward bias rises exponentially withincreased voltage. Furthermore, the reverse saturation current appears to be roughly constant. Asa word of caution, it is not appropriate to evaluate diode data using a linear scale. This is becausethe magnitude of the reverse saturation current (≈ pA) is many orders of magnitude smaller thanthat of the forward current (µA - mA). The same data is plotted on a logarithmic scale, underforward, Figure 2 , and reverse bias, Figure 3 . This plot is referred to as a semi-log plot, sinceone of the axes is plotted in log scale and the other in linear scale. The main point here is thatthe current is evaluated logarithmically, since it is the current that varies by several orders ofmagnitude.

Always evaluate experimental diode data on log(|I|) vs VA plots

Under forward bias, there is a region below 0.35 V in which the slope of the current vs voltageis closer to e/2kbT than the ideal value of e/kbT , which is evident for voltages in the range of0.35 V to 0.7 V. At higher voltages above 0.7 V, the slope gradually decreases, as if the current issaturating. This is due to high-level injection, and we will discuss this later.

These observations of the forward current alone indicate several issues with the ideal theory,since there is only one predicted slope for the ideal diode equation – e/kbT . Investigation of thereverse current makes the need for a revised theory perhaps more evident. In Figure 3 , it is clearthat the reverse bias current is not constant with respect to voltage, but increases in magnitudeas the reverse bias voltage is increased. Note the estimated value of I0 for this device is 10 fA,which is clearly lower than the observed value by 3 to 4 orders of magnitude. This dependence on

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Dr. Andre Zeumault

Figure 2: Logarithmic-scale plot of a measured IV characteristic of a Silicon diode under forwardbias

Figure 3: Logarithmic-scale plot of a measured IV characteristic of a Silicon diode under reversebias

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Dr. Andre Zeumault

voltage is due to current stemming from generation-recombination in the depletion region – whatwe neglected in our ideal diode equation derivation.

PN Junction Diodes exhibit material specific non-idealities (e.g. Si, GaAs, Ge)

This data, even though it comes from a specific Silicon device, can be taken as being represen-tative of the various non-idealities one might encounter in practice, although there are differencesthat depend on the material being used to fabricate the diode. For example, GaAs diodes tend toexhibit similar IV characteristics as Silicon diodes, whereas Ge diodes exhibit a near ideal diodebehavior at room temperature. Silicon, at elevated temperatures, resembles an ideal diode. Wewill understand why this is later.

2.3 Breakdown Under Reverse Bias

Theoretically speaking, when we speak of the term breakdown, we are referring to a failure of theassumptions made in the derivation of the ideal diode equation. Specifically, if in our derivationthere are no other processes assumed to exist, the fact that we are observing other processes is aclear “breakdown” in our logical reasoning. Practically speaking, breakdown refers to a point atwhich the current spikes, typically defined using some accepted value of the current beyond whichthe device is assumed to have “failed”. It is important to note that, in a diode, breakdownis completely reversible and does not cause any damage to the device.

Breakdown in a pn-junction diode is completely reversible!

There are two types of breakdown mechanisms that are relevant to the operation of pn-junctiondiodes – impact ionization (aka avalanche breakdown) and band to band Tunneling (akaZener breakdown).

2.3.1 Impact Ionization

Impact ionization is a process in which an electron (or hole) moving across the depletion re-gion transfers energy to the crystal due to collisions (i.e. scattering) with lattice vibrations (i.e.phonons). To understand impact ionization, we first consider the process of electron transportacross the depletion region in a reverse-biased pn-junction diode. As shown in Figure 4 , anelectron moving across the depletion region is periodically accelerated by the electric field(gaining energy) and scattered inelastically due to collisions with the lattice (losing en-ergy). As charge carriers move in a semiconductor, they scatter elastically and inelastically withother charge carriers, defects and phonons. The applied electric field accelerates charge carriersonly in-between scattering events. During acceleration, the charge carriers move elastically withoutchanging energy. Under normal circumstances, the energy released due to each collision with thelattice is relatively small and only contributes heat to the lattice locally (Joule heating).

Normal low-energy collisions with the lattice lead to Joule heating

When the magnitude of the reverse bias is increased, therefore increasing the electric field withinthe depletion region and the acceleration of carriers, electrons (and holes) move a larger distancein between scattering events. The height above the conduction band defines the energy releasedduring the scattering event. If this energy is larger than the bandgap energy, the released energy

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Dr. Andre Zeumault

Figure 4: An electron traversing the depletion region in a pn-junction, experiencing inelastic scat-tering.

can break Silicon bonds, therefore creating an electron and a hole. This process is known as impactionization and is illustrated in Figure 5 .

The increase in current associated with impact ionization is modeled by a multiplication factor,M which ratios the current due to impact ionization to the current without carrier multiplication.

M =|I|I0

The multiplication factor, M , accounts for the additional charge carriers generated due to impactionization, and is empirically fit using the following equation:

M =1

1−(|VA|VBR

)mTypically, the value of m takes on a value in the range of 3 to 6, depending on the semiconductor.The multiplication factor is a correction factor between the ideal current and the measured current,accounting for the impact ionization.

The dependence of the multiplication factor on the breakdown voltage stems from electrostatics.For a step junction doping profile the peak electric field is given by:

Epk = E (0) = −eN+

D

εxn = −

√2eN (φbi + VR)

ε1

N=

1

N−A+

1

N+D

Assuming that breakdown voltage VBR is much larger than the built-in potential, and occurs at a

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Dr. Andre Zeumault

Figure 5: Impact ionization process in a pn-junction diode under large reverse bias.

critical electric field Ecr that is a material constant independent of the doping concentration – ameasure of the bond strength:

Ecr = −√

2eNVBRε

Squaring both sides yields the following relationship:

E 2cr =

2eNVBRε

Therefore,

VBR =ε

2eNE 2cr ∝ ε ∝

1

N

From this result we can infer that the bandgap energy plays a critical role in defining the breakdownvoltage since the bandgap is the energy required to break a covalent bond and create an electron-hole pair – precisely the energy limiting process describing impact ionization. Therefore, we expectthat larger bandgap materials will be have larger critical electric fields. Also, for a given material,the breakdown voltage can be increased by reducing the doping concentration. This should beobvious, since, for an abrupt junction, the peak electric field within the depletion region increasesas the square-root of the doping concentration, leading to greater charge carrier accelerations.

Epk ∝√N

2.3.2 Band-To-Band Tunneling

Band-to-band Tunneling is a process by which an electron in the valence band “tunnels” into theconduction band elastically without changing its energy. Tunneling is a purely quantum-mechanicaleffect that has no classical analog. When we view an electron in a band diagram, the reference

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Dr. Andre Zeumault

Figure 6: A potential energy barrier and an electron.

energy levels refer to the electronic potential energy. Any electron having energy above the referencelevel has additional energy in the form of kinetic energy. Classically, if the kinetic energy of anelectron is less than the potential energy at some position along the band diagram x, the electronmust be reflected at that boundary. To visualize this, consider Figure 6 . Provided that thereare occupied and unoccupied states at the same energy level on both sides of the barrier withbarrier height (HB), and the barrier width (WB) is narrow, tunneling can occur. The electron willeffectively “spill” into the regions at either side of the junction rather than being limited to eitherside. Thus, in order for tunneling to occur:

• Filled and empty states must be at the same energy

• The tunneling barrier must be small, ≈WB < 100 A

All commercially-available diodes that make use of breakdown are referred to as Zener diodes, eventhough some of the diodes called Zener diodes, may in fact breakdown via impact ionization.

2.4 Current Due to Recombination and Generation

2.4.1 Trap-Induced Recombination and Generation – Qualitative Visualization

Before discussing the current component due to generation and recombination in the diode, wemust introduce an important factor in device analysis and fabrication – traps. Traps or impuritiesare trace chemicals which are incorporated unintentionally during device fabrication but can havea profound effect on device behavior.

A trap is an unintentional impurity, usually deep in the bandgap.

Traps are to be treated identically to dopants, from a mathematical point of view, using thesame framework, although what we call a trap and what we call a dopant depends mostly on therelative position of the trap level in the bandgap.

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Dr. Andre Zeumault

Figure 7: Illustration of recombination and generation processes due to a trap center.

Dopants are shallow impurities, traps are deep impurities.

Deep impurities tend to be referred to as traps whereas shallow states tend to be referred toas dopants. The presence of traps can modify:

• Carrier concentrations at equilibrium

• Non-equilibrium dynamics –(∂p∂t

)R−G

,(∂n∂t

)R−G

• Measured current

• Electrostatics – if traps are ionized (acceptor-like traps or donor-like traps)

In Figure 7 , the processes of generation and recombination due to a trap center located atET is shown. During generation, an electron (black dot) can gain thermal energy to move from thevalence band to the trap level or from the trap level to the conduction band. During recombination,an electron (black dot) can release thermal energy to move from the conduction band to the traplevel or from the trap level to the valence band.

2.4.2 Trap-Induced Recombination and Generation – Quantitative Description

The procedure for deriving an expression for the change in carrier concentration due to recombina-tion and generation with a trap center is very similar to what we covered when deriving minoritycarrier lifetimes. In the n-type semiconductor at equilibrium shown in Figure 8 , a deep trap levelat ET tend to be nearly fully occupied with electrons, as they are well below the Fermi level. Theconcentration of electrons occupying this state is denoted as nT and given by:

nT =

∫ EC

EV

gT (E)f(E)dE

≈∫ EC

EV

gT (E)dE

=

∫ EC

EV

NT δ(E − ET )dE

= NT

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Dr. Andre Zeumault

Figure 8: Low level injection assumption, applied to an n-type semiconductor with a trap centerlocated at the middle of the bandgap.

The trap state is modeled as a single energy level – delta function in energy having NT statesper unit volume. This expression tells us that, since the state is mostly occupied f(ET ) ≈ 1, thenumber of electrons in that state is only depends on the number of states available. This simplifiesour mathematical treatment considerably, although the full expression can be carried through, withits corresponding dependence on the Fermi energy and temperature. At equilibrium, the rates ofrecombination and generation balance such that:(

∂n0∂t

)R−G

=

(∂p0∂t

)R−G

=

(∂nT∂t

)R−G

= 0

We are interested in the total change in minority carrier concentration due to recombination andgeneration. This is just a simple matter of book-keeping to keep track of what carriers are beinggenerated and those which are recombining.(

∂p

∂t

)R−G

=

(∂p

∂t

)R

+

(∂p

∂t

)G

=

(∂p0∂t

)R

+

(∂∆p

∂t

)R

+

(∂p0∂t

)G

+

(∂∆p

∂t

)G

At equilibrium, the rates of recombination and generation of the equilibrium concentration exactlycancel each other. Under conditions of low-level injection, the new majority carrier electrostaticsare not modified by the presence of the minority carriers. Therefore, quasi-equilibrium is main-tained with respect to the hole concentration, which is unchanged due to the presence of additionalelectrons. Under these assumptions, we can drop these terms and we are left with the followingexpression: (

∂p

∂t

)R−G

=

(∂∆p

∂t

)R

+

(∂∆p

∂t

)G

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Dr. Andre Zeumault

Figure 9: Low level injection assumption, applied to a p-type semiconductor with a trap centerlocated at the middle of the bandgap.

Next, we note that the trap level is at midgap, and therefore doesn’t change its occupancy withrespect to slight changes in the Fermi level. If all states are occupied with electrons at the traplevel, there can be no generation processes whereby an electron moves from the valence band tothe trap level, thus creating additional excess holes. This allows us to drop the generation termand we are left with the recombination term only:(

∂p

∂t

)R−G

=

(∂∆p

∂t

)R

At this point, we need to evaluate the recombination rate of excess holes. We assume that therecombination rate will be proportional to the product of the occupied states and the unoccupiedstates involved in the transition. In this case, the occupied states are electrons located at the traplevel (nT ≈ NT ) and the unoccupied states are the holes in the valence band (p). Due to the Pauliexclusion principle, an electron transition is always from an occupied state to an unoccupied state.(

∂∆p

∂t

)R

∝ nT∆p = −cpNT∆p

The situation is similar for a p-type semiconductor as shown in Figure 9 . In this case, the excesselectrons created will recombine with holes located at ET . Since the Fermi level is close to thevalence band, we can say that ET will be mostly empty of electrons and nearly fully occupied withholes. The concentration of these holes is taken to be constant, since they are located at midgap.As a consequence, we do not write rate equations for them. However, they recombine with theexcess electrons located in the conduction band. Following a similar procedure:(

∂n

∂t

)R−G

=

(∂n

∂t

)R

+

(∂n

∂t

)G

=

(∂n0∂t

)R

+

(∂∆n

∂t

)R

+

(∂n0∂t

)G

+

(∂∆n

∂t

)G

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Dr. Andre Zeumault

Low level injection allows us to drop the equilibrium terms:(∂n

∂t

)R−G

=

(∂∆n

∂t

)R

+

(∂∆n

∂t

)G

Since the hole concentration is taken as constant, there can be no generation of holes, only recom-bination. (

∂n

∂t

)R−G

=

(∂∆n

∂t

)R

At this point, we need to evaluate the recombination rate of excess holes. As in the case of ann-type semiconductor, we assume that the recombination rate will be proportional to the product ofthe occupied states and the unoccupied states involved in the transition. In this case, the occupiedstates are excess electrons located at the conduction band (∆n) and the unoccupied states are theholes located at the trap level (pT ≈ NT ).(

∂∆n

∂t

)R

∝ pT∆n = −cnNT∆n

We generally write the results as follows:(∂p

∂t

)R−G

= −cpNT∆p = −∆p

τp(∂n

∂t

)R−G

= −cnNT∆n = −∆n

τn

Where the time constants τn and τp are defined as follows:

τn =1

cnNT

τp =1

cpNT

The parameters cn and cp were introduced without justification or definition. Upon examination, itis evident that the units are [cm3 s−1]. These are known as the capture coefficients and dependon the nature of the trap, which we assess through an effective size denoted as the capture crosssection which has units of area [cm−2] defined in terms of the thermal velocity, vth.

cp = vthσp

cn = vthσn

2.4.3 Shockley-Read-Hall Recombination

We have so-far derived the recombination-generation rates due to a trap center located at ET underconditions of low-level injection on either an n-type or p-type semiconductor. It can be shown thatthe more general result applies at arbitrary injection levels and for both carrier types, referred to

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Dr. Andre Zeumault

as Shockley-Read-Hall (SRH) recombination:(∂p

∂t

)R−G

=

(∂n

∂t

)R−G

=n2i − np

τp(n+ nT ) + τn(p+ pT )

Where n1 and p1 are defined as:

nT = nieET−Ei

kbT

pT = nieEi−ET

kbT

To show this, we begin by lifting the assumption that traps can only participate in recombination byallowing their occupancy to change with the position of the Fermi level. We also no longer assumeone type of semiconductor – n or p – but treat the most general case simultaneously, arriving at anexpression that is valid for n and p type doping provided that the doping level is non-degenerate(we will still make the Boltzmann approximation).

The rate of change in electron concentration due to recombination and generation is given by:(∂n

∂t

)R−G

=

(∂n

∂t

)R

+

(∂n

∂t

)G

= −cnNT (1− f(ET ))n+ANT f(ET )

The constant, A, is a proportionality constant that makes the units work out. We can estimate itsvalue using the principle of detailed balance. We solve for A by forcing the generation rate to beequal to the recombination rate at equilibrium and use that definition of A under non-equilibriumsituations. At thermal equilibrium, (

∂n

∂t

)R−G

= 0

n = n0

Therefore, we can solve for A as follows:

cnNT (1− f(ET ))n0 = ANT f(ET )

A = cnn01− f(ET )

f(ET )

= cn

(nie

ET−EikbT

)= cnnT

In the last line, we’ve defined nT as follows:

nT = nieET−Ei

kbT

Substituting this value for A into the general non-equilibrium expression yields the following:(∂n

∂t

)R−G

= cnNT (nT f(ET )− n(1− f(ET )))

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Dr. Andre Zeumault

The rate of change in hole concentration due to recombination and generation is given by:(∂p

∂t

)R−G

=

(∂p

∂t

)R

+

(∂p

∂t

)G

= −cpNT f(ET )p+BNT (1− f(ET ))

The constant, B, is a proportionality constant that makes the units work out, as was the previouscase for A. We can estimate its value using the principle of detailed balance. We solve for Bby forcing the generation rate to be equal to the recombination rate at equilibrium and use thatdefinition of B under non-equilibrium situations. At thermal equilibrium,(

∂p

∂t

)R−G

= 0

p = p0

Therefore, we can solve for B as follows:

cpNT f(ET )p0 = BNT (1− f(ET ))

B = cpp0f(ET )

1− f(ET )

= cp

(nie

Ei−ETkbT

)= cppT

In the last line, we’ve defined pT as follows:

pT = nieEi−ET

kbT

Substituting this value for B into the general non-equilibrium expression yields the following:(∂p

∂t

)R−G

= cpNT (pT (1− f(ET ))− pf(ET ))

Let’s examine the two results from a different perspective.(∂p

∂t

)R−G

=

(∂p0∂t

)R

+

(∂∆p

∂t

)R

+

(∂p0∂t

)G

+

(∂∆p

∂t

)G(

∂n

∂t

)R−G

=

(∂n0∂t

)R

+

(∂∆n

∂t

)R

+

(∂n0∂t

)G

+

(∂∆n

∂t

)G

In equilibrium, electrons and holes are generated and recombine at the same rates. Thus, if we canassume that the carrier concentrations are always separable into two components – an equilibriumvalue and an excess value – then the net recombination and generation rates of electrons and holesare always equal. (

∂p0∂t

)R

=

(∂n0∂t

)R

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Dr. Andre Zeumault

(∂p0∂t

)G

=

(∂n0∂t

)G

(∂∆p

∂t

)R

=

(∂∆n

∂t

)R

(∂∆p

∂t

)G

=

(∂∆n

∂t

)G

These considerations lead to the important result that :(∂p

∂t

)R−G

=

(∂n

∂t

)R−G

Using this result, we can equate the two rates and solve for f(ET ) and then evaluate either of theabove expressions for the net recombination-generation rate.

cpNT (pT (1− f(ET ))− pf(ET )) = cnNT (nT f(ET )− n(1− f(ET )))

This expression, when solved for f(ET ) and 1− f(ET ) yields the following:

f(ET ) =cppT + cnn

cn(nT + n) + cp(p+ pT )

1− f(ET ) =cnnT + cpp

cn(nT + n) + cp(p+ pT )

We can now evaluate the net recombination-generation rate as follows, which after some simplealgebra yields the following:(

∂n

∂t

)R−G

=

(∂p

∂t

)R−G

=cncpNT (n2i − np)

cn(n+ nT ) + cp(p+ pT )

Using the definitions of the capture coefficients, we can re-write the above expression as:(∂n

∂t

)R−G

=

(∂p

∂t

)R−G

=n2i − np

τp(n+ nT ) + τn(p+ pT )

This is an important equation to remember and understand the derivation. It has importantpractical implications and is referred to often in discussion of devices.

2.4.4 R-G Current in the Depletion Region

Returning back to our discussion of diodes, we are interested in calculating the current due torecombination and generation is given. Since each net generation -recombination event creates apair of electrons and holes, which is swept rapidly away from the depletion region, we have the

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Dr. Andre Zeumault

following expression for the current:

IR−G = −2eA

∫ xn

−xp

∂n

∂t R−Gdx

= −2eA

∫ xn

−xp

n2i − npτp(n+ nT ) + τn(p+ pT )

dx

The factor 2 accounts for the fact that an electron and a hole are generated simultaneously, con-tributing equally towards the current. In the depletion region (n = p ≈ 0), which allows us toevaluate this expression simply leading to the following expression, valid under reverse bias:

IR−G = −2eA

∫ xn

−xp

n2i − npτp(n+ nT ) + τn(p+ pT )

dx

≈ −2eA

∫ xn

−xp

n2iτpnT + τnpT

dx

=−2eAn2i

τpnT + τnpTW

=−2eAn2i

τpnT + τnpT

√2εs (φbi + VR)

eN

∝√VR

To first order, this predicts a roughly square-root dependence of the recombination-generationcurrent on the reverse bias. Comparing this to the experimental data shown Figure 3 , we can seea quite reasonable agreement to the experimental data.

Under forward bias, the electron and hole concentrations cannot be neglected. Including thesein the calculation above yields the following expression for the current due to recombination-generation.

IR−G =−2eAn2i

τpnT + τnpT

√2εs (φbi + VR)

eN

1− eeVAkbT

1 + e(φbi−VA)kbT

√τnτp2τ0

eeVA2kbT

Where τ0 is defined as:

τ0 =1

4

(τpnTni

+ τnpTni

)Having introduced a new current component to the diode equation, we re-label our previous resultas the diffusion current, such that the total diode current is the sum of the diffusion current andthe current due to recombination-generation.

I = IDIFF + IR−G

Where the diffusion current IDIFF corresponds to the ideal diode equation.

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