DIODE APPLICATIONS: AN OVERVIEW

I. The injection of electron-hole pairs to generate light via recombination (eg. LEDs and LASERs) II. The separation of electron-hole pairs at the junction to constitute a current source (eg. solar cell) III. The temperature dependence of the I-V characteristic (eg. a temperature sensor) IV. The non-linear nature of the I-V characteristic (eg. frequency multipliers and mixers) V. The device as a switch (eg. rectifiers, inverters, power supplies etc) © Nezih Pala [email protected]

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Diode applications: An overview The diode has many uses when employed as a current source. The diode when operated under reverse bias has the properties of a current source (infinite output resistance or equivalently a constant current with voltage). Consider the figure. If a current source is available which can be controlled then it can form the basis of several critical and valuable applications. If large changes in the current source can be effected by a small change in input voltage, (ΔVin) then the resultant change in output voltage, ΔVout, could be large if the current is delivered to a large load resistance. The resultant voltage gain , ΔVout/ΔVin, forms the basis of transistor operation and explains why the output of a transistor is always represented by a current source. If the current source can be controlled by incident photons, then the resultant current is basis of the operation of a photodetector or a solar cell.

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Solar Cells

Solar cell inventors at Bell Labs (left to right) Gerald Pearson, Daryl Chapin and Calvin Fuller are checking a Si solar cell sample for the amount of voltage produced (1954). © Nezih Pala [email protected]

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Principle of operation of solar cells -1

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Principle of operation of solar cells -2

Photogenerated carriers within the volume Lh + W + Le give rise to a photocurrent Iph. The variation in the photegenerated EHP concentration with distance is also shown where α is the absorption coefficient at the wavelength of interest.

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Principle of operation of solar cells -3 The greater the light intensity the higher is the photogeneration rate and larger is the Iph. If Ilight is the light intensity then

I SC   I Ph   KI light where K is a constant that depends on the particular device. The photocurrent does not depend on the voltage across the pn junction because there is always some internal field to drift the photogenerated EHPs. We exclude the secondary effect of the voltage modulating the width of the depletion region. The photocurrent Iph therefore flows even when there is not a voltage across the device. © Nezih Pala [email protected]

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Principle of operation of solar cells -4  qV nkT  Id  I0  e  1  

 qV nkT  I   I ph  I 0  e  1  

Typical I-V characteristics of a Si solar cell. The short circuit current is Iph and the open circuit voltage is VOC. The I-V curves for positive current requires an external bias voltage. Photvoltaic operation is always in the negative current region. © Nezih Pala [email protected]

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Principle of operation of solar cells -5

Consider an ideal pn junction PV device connected to a resistive load R as shown in the figure. Note that I and V in the figure define the convention for the direction of positive current and positive voltage. If the load is a short circuit the only current in the circuit is that generated by the incident light. This is the photocurrent Iph which depends on the number of EHPs.

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Principle of operation of solar cells -6 I mVm I V  FF   I scVoc I scVoc

(a) When a solar cell drives a load R. R has the same voltage as the solar cell but the current through it is in the opposite direction to the convention that current flows from high to low potential. (b) The current I’ and voltage V’ in the circuit of (a) can be found from a load line construction. Point P is the operating point (I’, V’). The load line is for R = 30 W. © Nezih Pala [email protected]

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Solar Cells Generation Currents: p-n Junctions Illuminated With Light

Any minority carrier electrons generated within a diffusion length of the depletion edge can diffuse to the edge of the junction and be swept away. Minority electrons generated well beyond a length Ln will recombine with holes resulting in the equilibrium concentration, np0. Similarly holes generated within, Lp, a diffusion length, of the depletion region edge will be swept into the depletion region. In the event that there is light shining on the p-n junction, as shown in the figure, then the charge profile is perturbed in the following manner. Far in the bulk region, an excess minority carrier concentration is generated, where Δnp = GLτn and Δpn = GLτp. The new equation to be solved for reverse saturation current differs from the one previously used in that a light generation term is added.

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Solar Cells Remember the continuity equation from the last chapter:

p  2p p  Dp  2 t x p

d2p D p 2  Gth  R  GL  0 dx d 2 p pn 0  pn Dp 2   GL  0 dx p with boundary conditions similar to before.

pn ()  pn 0   p GL pn (Wn )  0 Solving these equations, we get

  x  Wn   pn ( x)   pn 0   p GL 1  exp    L p    © Nezih Pala [email protected]

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Solar Cells A similar set of equations for electrons gives us the following expression for np(x) in the neutral p-region:

  x  W p   n p ( x)  n p 0   nGL 1  exp   Ln  

The slope of the charge profile at the edge of the depletion region is

Therefore

dpn (Wn ) pn 0   p GL  dx Lp

 pn 0   p GL   similarly J ( x  W )  eD  n p 0   nGL  J p ( x  Wn )  eD p    n p n  Lp Ln     The reverse saturation current JR is then given by

 n p ,bulk pn,bulk   J R  e Dn  Dp   L L n p   where np,bulk (pn,bulk) is the minority carrier concentration in the bulk in non-equilibrium (steady state). Here np,bulk = np0 + τnGL. © Nezih Pala [email protected]

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Solar Cells In a solar cell under optical excitation the forward current is unchanged and continues to be provided by the thermal injection of carriers across the junction (as has been described before) whereas the reverse current changes dramatically and is carried dominantly by photogenerated carriers. This is the reason why the net current is not zero at zero applied bias in an illuminated solar cell. This current is called the short circuit current, Isc. The forward voltage increases the forward thermionic/diffusion currents exponentially as given by the diode law whereas the reverse current remains a constant with the net current being given by

  eV  J  A  I  I f  I r  I s exp   1  I sc   k BT 

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Solar Cells In a solar cell configuration the forward bias is not explicitly applied across the cell. It is generated by the flow of the current across the load (which may be the resistance of a light bulb for instance). This is shown schematically in the figure along with the equivalent circuit of the solar cell. The total cell current goes to zero at a voltage, Voc, termed the open circuit voltage, when the forward diode current is equal and opposite to the generated current. From equation

k BT  I sc  Voc  ln   1 e  Is 

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Solar Cells To obtain the maximum power from a cell it is desirable to have the largest product of voltage and current possible in the fourth quadrant of the I −V plane. The maximum power point is that bias at which the maximum power is available from the cell, or, is the bias at which the largest rectangle can be accommodated within the I-V curve. The power at any bias point is given by the IV product

  eV   P  I V  I sc  I D  V   I sc  I s exp   1 V  k BT  

and the maximum power point is obtained by maximizing the product. This is left as an exercise. The maximum power is also alternately represented by

P  I sc Voc  F where F is the defined as the Fill Factor of the cell. Hence to get the maximum power from a cell it is desirable to obtain the largest Voc and Isc possible which is best achieved by using a tandem cell which comprise of a series connection of cells with different bandgaps that maximize solar absorption (while maintaining a large open circuit voltage) coupled with concentrator lenses that maximize input photon intensity. © Nezih Pala [email protected]

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Series and shunt resistance

Series and shunt resistances and various fates of photogenerated EHPs.

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Equivalent circuit

The equivalent circuit of a solar cell (a) Ideal pn junction solar cell (b) Parallel and series resistances Rs and Rp. © Nezih Pala [email protected]

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Effect of series resistance

The series resistance broadens the I–V curve and reduces the maximum available power and hence the overall efficiency of the solar cell. The example is a Si solar cell with η ≈ 1.5 and Io ≈ 3 × 10−6 mA. Illumination is such that the photocurrent Iph = 10 mA. © Nezih Pala [email protected]

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Reflection losses

Inverted pyramid textured surface substantially reduces reflection losses and increases absorption probability in the device. © Nezih Pala [email protected]

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Efficiencies

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Heterojunction solar cells -1

AlGaAs window layer on GaAs passivates the surface states and thereby increases the photgeneration efficiency.

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Heterojunction solar cells -2

A heterojunction solar cell between two different bandgap semiconductors (GaAs and AlGaAs) © Nezih Pala [email protected]

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Tandem solar cells

A tandem cell. Cell 1 has a wider bandgap and absorbs energetic photons with hf > Eg1. Cell 2 absorbs photons that pass cell 1 and have hf > Eg2.

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Example •

Consider a solar cell that is driving a load of 3Ω and has an area of 3cm x 3cm. Using the given plot, find the current and voltage in the circuit when it is illuminated with light of 700 W/m2 intensity. Find the power delivered to the load, the efficiency of the solar cell in this circuit and the fill factor of the solar cell.

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Example -1 The I-v characteristics of the load in the figure, is the load line given by I= - V/R that is I=-V/3. The line drawn in the figure with the slope 1/ 3. It cuts the I-V characteristics of the solar cell at I’=157mA and V’=0.475 V as apparent in the figure which are the current and voltage respectively, in the photovoltaic circuit. The power delivered to the load is

Pout  I  V   157 103  0.457  0.0746 W or 74.6 mW The input sunlight power is

Pin  (light intansity) (surface area)  700 Wm 2  (0.03 m) 2  0.63 W The efficiency is

 PV  (100%)

Pout 0.0746  (100%)  11.8% Pin 0.63

This will increase if the load is adjusted to extract the maximum power from the solar cell, but increase will be small as the rectangular area I’V’ is already quite close to the maximum. The fill factor can also be calculated since point P in the figure is close to the optimum operation , maximum output power, in which the rectangular area I’V’ is maximum:

I mVm I V  157mA  0.475V FF     0.722 I scVoc I scVoc 178mA  0.58V © Nezih Pala [email protected]

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Example -2 A solar cell under an illumination of 500W m-2 has short circuit current ISC of 150 mA and an open circuit output voltage VOC of 0.530 V. What are the short circuit current and open circuit voltage when the light intensity is doubled? Assume η=1.5. qV  The general I-V characteristics under illumination is given by I   I ph  I 0  e nkT  1   qV

Setting I=0 for open circuit

 I   I ph  I 0  e 

oc

nkT

  1  0 

Assuming that Voc>>nkT/q, (neglecting the -1 term) rearranging the above equation, we can find VOC

VOC 

nkT  I ph   ln  q  I0 

The photocurrent Iph depends on the light intensity I via Iph=KInt where K is a constant. This at a given temperature, the change in VOC is

nkT  I ph2  nkT  I ph1  nkT  I ph2  nkT  Int2       VOC 2  VOC1  ln  ln  ln  ln    q q q  Int1   I0  q  I0   I ph1  © Nezih Pala [email protected]

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Example -2 Assuming n=1.5, the new open circuit voltage is

Voc2

nkT  Int2     0.530V  (1.5).(0.026). ln( 2)  0.557V  Voc1  q  Int1 

This is a 5% increase compared with the 100 % increase in illumination and the short circuit current.

The short circuit current is the photocurrent, so at double the intensity this is

I sc2

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 Int2    (150mA)  (2)  300mA  I sc1  Int  1

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Light emitting diode (LED) One of the most important applications is in the area of optoelectronic devices. Essentially all the semiconductor devices catering to optoelectronics use the diode concept. These include detectors, avalanche photodetectors, optical modulators, as well as light-emitting diodes and semiconductor lasers. In this section we discuss the operation of the light emitting diode. The simplicity of the light-emitting diode (LED) makes it a very attractive device for display and communication applications. The basic LED is a p-n junction that is forward biased to inject electrons and holes into the p- and n-sides respectively. The injected minority charge recombines with the majority charge in the depletion region or the neutral region. In direct band semiconductors, this recombination leads to light emission since radiative recombination dominates in high-quality materials. In indirect gap materials, the light emission efficiency is quite poor and most of the recombination paths are nonradiative, which generates heat rather than light. In the following section we will examine the important issues that govern the LED operation.

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History of LEDs -1

The first publication on electroluminescence from a semiconductor (Silicon carbide) light emitting diode. The article indicates that the first LED was a Schottky diode rather than a p-n junction. ( after by H. J. Round, Electrical World, Vol. 49, p.309, 1907)

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History of LEDs -2 Independently, Oleg Vladimirovich Losev published "Luminous carborundum [silicon carbide] detector and detection with crystals" in the Russian journal Telegrafiya i Telefoniya bez Provodov (Wireless Telegraphy and Telephony). Losev's work languished for decades.

In the period of 1924 and 1941, he published a number of articles detailing the function of a device that he developed, which would generate light via electroluminescence when electrons fall to a lower energy level. In the April 2007 issue of Nature Photonics, Nikolay Zheludev gives credit to Losev for inventing the LED. Specifically, Losev patented the "Light Relay" and foresaw its use in telecommunications.

Unfortunately, before this device could be developed, the Second World War intervened, and Losev died in 1942 during The Siege of Leningrad (now St. Petersburg), at the age of 39.

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History of LEDs -3 The First Patent For The Infrared LED In 1961, experimenters Bob Biard and Gary Pittman working at Texas Instruments, found that GaAs emitted infrared radiation when electric current was applied and received the patent for the infrared LED.

Visible LEDs The first practical visible-spectrum (red) LED was developed in 1962 by Nick Holonyak Jr., while working at General Electric Company. Holonyak is seen as the "father of the light-emitting diode". In addition to introducing the III-V alloy LED, Holonyak holds 41 patents. His other inventions include the red-light semiconductor laser, usually called the laser diode (used in CD and DVD players and cell phones) and the shorted emitter p-n-p-n switch (used in light dimmers and power tools).

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History of LEDs -4 The First Yellow LED M. George Craford, a former graduate student of Holonyak, invented the first yellow LED and improved the brightness of red and red-orange LEDs by a factor of ten in 1972.

The First Blue LED The first high-brightness blue LED was demonstrated by Shuji Nakamura of Nichia Corporation and was based on InGaN borrowing on critical developments in GaN nucleation on sapphire substrates and the demonstration of p-type doping of GaN (1989-1993).

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LEDs Practical Uses The first commercial LEDs were commonly used as replacements for incandescent indicators, and in seven-segment displays, first in expensive equipment such as laboratory and electronics test equipment, then later in such appliances as TVs, radios, telephones, calculators, and even watches. These red LEDs were bright enough only for use as indicators, as the light output was not enough to illuminate an area.

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LEDs A Light Emitting Diode (LED) is essentially a p-n junction diode typically made from a direct band gap semiconductor in which the electron-hole pair recombination results in the emission of a photon. The emitted photon energy Eph=hν is approximately equal to the bandgap energy Eg. When a forward bias V is applied, the built in potential V 0 is reduced to V0-V which allows the electrons from n+ side to diffuse (become injected) into the p side. The recombination of injected electrons in the depletion region and within the diffusion length Ln in the p-side leads a photon emission.

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Energy Band Diagram of LEDs

The energy band diagram of a p-n+ (heavily n-type doped) junction without any bias. Builtin potential V0 prevents electrons from diffusing from n+ to p side. The applied bias reduces V0 and thereby allows electrons to diffuse, be injected, into the pside. Recombination around the junction and within the diffusion length of the electrons in the p-side leads to photon emission.

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Injection Electroluminescence The phenomenon of light emission from EHP recombination as result of minority carrier injection is called injection electroluminescence.

Due to the statistical nature of the recombination process between electrons and holes, the emitted photons are in random directions. They result form spontaneous emission process. The LED structure should be designed such that the emitted photons can escape the device without being reabsorbed by the semiconductor material. This means the p-side has to be sufficiently narrow or we have to use heterostructure devices.

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Epilayer Structure of of LEDs -1 A schematic illustration of one possible LED device structure. First n+ is epitaxially grown on a substrate. A thin p layer is then epitaxially grown on the first layer.

First a doped layer is epitaxially grown on a suitable substrate (GaAs, GaP, SiC) . The p-n+ junction is formed by growing another layer but doped p type. The photons that are emitted toward the n-side become either absorbed or reflected back at the substrate interface depending on the substrate thickness and exact structure of the LED.

Any lattice mismatch causes defects hic can act as radiationless recombination centers. Therefore lattice matching between the p-n+ junction and the substrate is important.

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(a) Photon emission in a direct bandgap semiconductor. (b) GaP is an indirect bandgap semiconductor. When doped with nitrogen there is an electron recombination center at EN. Direct recombination between a captures electron at EN and a hole emits a photon. © Nezih Pala [email protected]

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External efficiency The external efficiency of an LED quantifies the efficiency of conversion of electric energy into an emitted external optical energy. It incorporates the internal efficiency of the radiative recombination process and the subsequent efficiency of photon extraction form the device.

The input electric power into en LED is simply the diode current and diode voltage product (IV).

Pout (optical) external  100% IV For indirect bandgap materials external efficiency is generally less than 1% whereas for direct bandgap semiconductors with right device design it could be substantial.

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Heterojunction High-Intensity LEDs -1 A p-n junction between two differently doped semiconductors that are of the same material, that is, the same bandgap is called homojunction. A junction between two different bandgap semiconductors is called heterojunction. A semiconductor device structure that has junction between different bandgap materials is called a heterostructure device. LED constructions for increasing the intensity of the output light make use of the double heterostructure. In the following example the semiconductors are AlGaAs with Eg=2eV and GaAs with Eg=1.4 eV. The double heterostructure has n+-p heterojunction between n+ AlGaAs and pGaAs. There is another heterojunction between p-GaAs and p-AlGaAs. The p-GaAs region is thin layer and is lightly doped.

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Heterojunction High-Intensity LEDs -2 When a forward bias is applied, most of the voltage drops between the n+ AlGaAs and p GaAs and reduces the potential barrier eV0 just as in a normal p-n junction. This allows electrons in the conduction band of n+ AlGaAs to be injected into p-GaAs. These electrons however are confined to the condcution band of p-GaAs since there is also a barrier between p-GaAs and p-AlGaAs.

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Heterojunction High-Intensity LEDs -3 The wide bandgap AlGaAs layers therefore act as confining layers that restrict injected electrons to the p-GaAs layer. The recombination of the injected electrons and holes already present in the p-GaAs layer results in spontaneous photon emission. Since the bangap Eg of AlGaAs is greater than GaAs, the emitted photons do not get reabsorbed as they escape the active region and can reach the surface of the device. Since the light is also not absorbed in pAlGaAs it can be reflected back to increase the light output.

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Epilayer Structure of GaN-based Heterostructure LEDs

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LED Characteristics -1 The energy of an emitted photon from an LED is not simply equal to the band gap energy Eg because electrons in the conduction band are distributed in energy and so are the holes in the valence band. The electron concentration in conduction band as a function of energy is asymmetrical and has a peak at kT/2 above Ec. The energy spread of these electrons is typically ~2kT from Ec. The hole concentration is similarly spread from Ev. Probability of radiative recombination of electrons and holes is directly proportional with the carrier concentration. Therefore transition #2 in the figure has the maximum probability.

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LED Characteristics -2

The relative intensity of emitted light corresponding to the energy of transition #2 (hv2) is the maximum. Distribution of intensity of the emitted radiation with the emitted photon energy is given in the Fig. (c) and with the wavelength of the emitted radiation in Fig. (d) above.

The linewidth of the output spectrum Δλ or Δν is defined as FWHM of these curves.

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LED Characteristics -3 Typical characteristics of a 655nm red LED is shown in the Fig. It exhibits less asymmetry than the idealized spectrum.

The linewidth is 24nm which corresponds to 2.7kT in the energy distribution of the emitted photons.

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LED Characteristics -4 As the LED current increases so does the injected minority carrier concentration and thus the rate of recombination and hence the output light intensity. The increase in the output light power however is not linear with the LED current. At high current levels a strong injection leads to the recombination time depending on the injected carrier concentration and hence on the current itself. This leads to a nonlinear recombination rate with current.

Turn-on or cut-in voltage of an LED depends on the semiconductor and increases with the bandgap. It is ~2V for yellow LEDs,~1V for GaAs infrared LEDs and ~3.5-4.5 V for blue LEDs.

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Example -1 If one makes an LED in a semiconductor with bandgap of 2.5 eV what wavelength of light will it emit?

hc 4.14 1015 eVs  3 108 m E h     4.968 107 m  496.8nm  E 2.5eV c

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Example -2 We know that a spread in the output wavelength is related to a spread in the emitted photon energies as shown before. The emitted photon energy Eph=hc/λ and the spread in the photon energies ΔEph =Δ(h) ≈ 3 kT between the half intensity points . Show the corresponding linewidth Δ λ between the half intensity points in the output spectrum is

3kT    hc 2

What is the spectral bandwidth of an optical communications LED operating at 1550 nm at 300K?

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Example -2 First consider the relationship between the photon frequency  and 

Where h  is the photon energy. We can differentiate this

c

hc    h

d hc 2   2 d (h ) hc h  The negative sign implies that increasing the photon energy decreases the wavelength. We are only interested in changes or spreads thus

 d  h  d (h )

 

2 hc

(h ) 

2 hc

3kT

Where we used (h)=3kT and obtained the equation in the question. We can substitute =1550nm and T=300K to calculate the line width of the 1550 nm LED 23 3 kT 3 ( 1 . 38  10 )300   2  (1550 109 )  1.50 107 m  150nm 34 8 6.626 10  3 10  hc

The spectral linewidth of an LED output is due to the spread in the photon energies, which is fundamentally about 3kT. The only option for decreasing  at a given wavelnegth is to reduce the temperature. The output spectrum of a laser, on the other hand has much narrower linewidth ( In the laser, this is encouraged by providing an optical resonant cavity in which the photon density can build up to large value through multiple internal reflections at certain frequencies.

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Stimulated Emission -3 Similarly to obtain more stimulated emission than absorption we must have n2>n1 :

Stimulated emission rate B21n2  (12 ) B21 n 2   Absorption rate B12n1 (12 ) B12 n1 Thus if stimulated emission is to dominate over absorption of photons from the radiation field, we must have way of maintaining more electrons in the upper level than in the lower level.

This condition of n2>n1 is quite unnatural and is called population inversion.

As a summary to obtain effective stimulated emission for lasers, we must provide: 1. Optical resonant cavity for photon field build up 2. Population inversion

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Heterostructure Laser Diodes -1 All practical semiconductor laser diodes are double heterostructures.

In the case of AlGaAs and GaAs laser, the p-GaAs region is layer (~0.1-0.2 µm) an constitutes the active layer in which the stimulated emission take place. Both p-GaAs and p-AlGaAs are heavily doped and are degenerate. When a sufficiently large forward bias is applied, Ec of nAlGaAs moves very close to the Ec of p-GaAs which leads to a large injection of electrons. In fact, with a sufficiently large forward bias, Ec of AlGaAs can be moved above of the CB of pGaAs. These injected electrons however, are confined.

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Heterostructure Laser Diodes -2

The p-GaAs layer is degenerately doped. Thus, the top of its valance band (VB) is full of holes, or all the electronic states are above the Fermi level E Fp in this layer. The large forward bias injects very large concentration of electrons from n-AlGaAs into the conduction band of p-GaAs. Consequently there is a large concentration of electrons in the CB and totally empty states at the top of the VB, which means that there is population inversion.

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Heterostructure Laser Diodes -3

An incoming photon with energy hv0 just above Eg can stimulate an electron in the CB of pGaAs to fall down to VB and emit a photon by stimulated emission. Such a transition is a photon-stimulated e-h recombination or lasing recombination. Thus an avalanche of stimulated emissions in the active layer provides an optical amplification of photons with hv0 in this layer.

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Heterostructure Laser Diodes -4

To construct a semiconductor laser with self sustained lasing emission we have to incorporate the active layer into an optical cavity to build up high energy optical field. Distributed Bragg reflector (DBR) is the most commonly used method to achieve highly reflective mirrors for high quality optical cavities.

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Heterostructure Laser Diodes -5 DBR reflects only one desirable wavelength that falls within the optical gain of the active layer. This wavelength selective reflection leads to only one possible electromagnetic radiation mode existing in the cavity which leads to a very narrow output spectrum.

Semiconductor lasers that operate with only one mode in the radiation output are called single-mode or single frequency lasers. The spectral linewidth of a single-mode laser output is typically ~0.1 nm which should be compared with an LED spectral linewidth of 150nm operating at 1550nm emission wavelength.

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Heterostructure Laser Diodes -6 To achieve the necessary stimulated emission from laser diode and build up the necessary optical oscillations in the cavity (to overcome all the optical losses) the current must exceed a certain threshold current Ith.

The optical power output at a current I is then very roughly proportional to I- Ith

There is still some weak optical power output below Ith but this is simply due to spontaneous recombination of injected electrons and holes. The laser diode behaves like a “poor” LED below Ith. It should be reemphasized that the output form a laser diode is coherent radiation, whereas that from an LED is a stream of incoherent photons. © Nezih Pala [email protected]

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Emission Energy The light emitted from the device is very close to the semiconductor bandgap, since the injected electrons and holes are described by quasi-Fermi distribution functions. The desire for a particular emission energy may arise from a number of motivations. The figure shows the bandgaps of some semiconductors along with the color they correspond to. If a color display is to be produced that is to be seen by people, one has to choose an appropriate semiconductor. Very often one has to choose an alloy, since there is a greater flexibility in the bandgap range available.

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Emission Energy The figure shows the loss characteristics of an optical fiber. As can be seen, the loss is least at 1.55 μm and 1.3 μm. If optical communication sources are desired, one must choose materials that can emit at these wavelengths. This is especially true if the communication is long haul, i.e., over hundreds or even thousands of kilometers. InP-based materials are used for these applications. Materials like GaAs that emit at 0.8 μm can still be used for local area networks (LANs), which involve communicating within a building or local areas. The area of displays and lighting is filled dominantly by GaN-based materials using InGaN as the emission region for blue and green and GaAs-based AlGaInP for the red region. © Nezih Pala [email protected]

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Substrate Availability Almost all optoelectronic light sources depend upon epitaxial crystal growth techniques where a thin active layer (a few microns) is grown on a substrate (which is ∼ 200 μm). The availability of a high-quality substrate is extremely important in epitaxial technology. If a substrate that lattice-matches to the active device layer is not available, the device layer may have dislocations and other defects in it. These can seriously hurt device performance. One of the most important opto-electronic materials for LEDs that has emerged lately is GaN. In spite of the lack of a native substrate, GaN-based LEDs grown on either sapphire or SiC have become multi-billion dollar industry.

The reason is that the InGaN quantum well which is used as the emission region has fluctuations which cause local energy minima for electron and holes. Thus radiative recombination is encouraged within this region and diffusion to and non-radiative recombination at a dislocation minimized. © Nezih Pala [email protected]

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GaN Blue LEDs

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Substrate availability Furthermore, the dislocation propagation and generation of dislocation sin GaN is very high because of the high bond energies in the material. This eliminates one of the failure mechanisms in conventional LEDs and lasers, that of generation and propagation of dislocations caused by absorption of emitted the photon energy. The important substrates that are available for conventional light-emitting technology (which do not benefit from the above mentioned advantages of GaN and InGaN) are GaAs and InP. A few semiconductors and their alloys can match these substrates. The lattice constant of an alloy is the weighted mean of the lattice constants of the individual components, i.e., the lattice constant of the alloy AxB1−x is

aall  xa A  (1  x)aB where aA and aB are the lattice constants of A and B. Semiconductors that cannot latticematch with GaAs or InP have an uphill battle for technological success. The crystal grower must learn the difficult task of growing the semiconductor on a mismatched substrate without allowing dislocations to propagate into the active region. Important semiconductor materials exploited in optoelectronics are the alloy GaxAl1−xAs, and AlGaInP which is a quaternary material which is lattice-matched very well to GaAs substrates; In0.53Ga0.47As and In0.52Al0.48As, which are lattice-matched to InP; InGaAsP, whose composition can be tailored to match with InP and can emit at 1.55 μm; and GaAsP, which has a wide range of bandgaps available. © Nezih Pala [email protected]

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Substrate availability In general, the electron-hole recombination process can occur by radiative and nonradiative channels. Under the condition of minority carrier recombination or high injection recombination, one can define a lifetime for carrier recombination. If τr and τnr are the radiative and nonradiative lifetimes, the total recombination time is (for, say, an electron)

1

n



1

r



1

 nr

The internal quantum efficiency for the radiative processes is then defined as

Qr 

1/ r 1  1 /  r  1 /  nr 1   r /  nr

In high-quality direct gap semiconductors, the internal efficiency is usually close to unity. In indirect materials the efficiency is of the order of 10−2 to 10−3. Before starting the discussion of light emission, let us remind ourselves of some important definitions and symbols used in this chapter: Iph : photon current = number of photons passing a cross-section/second. Jph : photon current density = number of photons passing a unit area/second. Pop : optical power intensity = energy carried by photons per second per area. © Nezih Pala [email protected]

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Carrier Injection and Spontaneous Emission The LED is essentially a forward-biased p-n diode, with a quantum well emission region as shown in the figure. The reason for using a quantum well is to (i) increase the electrons and hole density in the recombination region increasing the direct recombination rate and leading to higher light output, (ii) having an emission region that is lower in energy that the injection (cladding) regions which allows the generated photons to escape without being re-absorbed in the injection regions, (iii) minimizing the overflow of electrons into the cladding regions where the injected carriers either recombine non-radiatively or generate light of an undesired wavelength.

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Carrier injection and spontaneous emission The current flow in a p-n junction was discussed in detail earlier in this chapter. The basis of that derivation was that electrons and holes are injected across the junction and recombine either in the bulk(long base case) or at contacts (short base case). Neither of those conditions apply to an LED. Here the current flow occurs via recombination in the quantum well. The turn-on voltage of the LED is therefore given by the bandgap of the emission region and is not explicitly related to the built-in voltage of the p-n junction.

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Carrier injection and spontaneous emission An example of this is an InGaN LED grown within GaN p-and n regions. The built-in voltage of this device is close to the bandgap of GaN (3.4V) though the turn-on voltage is 2.8V close to the emission energy of the photons. The current flow mechanism is shown in last figure. The current is given by J = e · Rspon where Rspon is the spontaneous recombination rate in the well. The efficiency of the process is the ratio of the current generating photons of the desired wavelength to the total current. The current calculated for the p-n junction in the earlier sections are the wasted currents in the LED as they calculate currents in the bulk and due to non-radiative centers. What remains to be calculated is the spontaneous recombination rate Rspon . As discussed earlier, the radiative process is “vertical,” i.e., the k-value of the electron and that of the hole are the same in the conduction and valence bands, respectively. From figure in the next slide we see that the photon energy and the electron and hole energies are related by

 2k 2   Eg  2

1 1   2k 2  *  * *  me mh  2mr

where mr* is the reduced mass for the e-h system. © Nezih Pala [email protected]

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Carrier injection and spontaneous emission

A schematic of the E-k diagram for the conduction and valence bands. Optical transitions are vertical; i.e., the k-vector of the electron in the valence band and in the conduction band is the same. © Nezih Pala [email protected]

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Carrier injection and spontaneous emission If an electron is available in a state k and a hole is also available in the state k (i.e., if the Fermi functions for the electrons and holes satisfy fe(k) = fh(k) = 1), the radiative recombination rate is found to be



Wem  1.5 109  eVs 1



and the recombination time becomes (ħω is expressed in electron volts)

0.67 o  ns eV  The recombination time discussed above is the shortest possible spontaneous emission time since we have assumed that the electron has a unit probability of finding a hole with the same k-value.

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Carrier injection and spontaneous emission When carriers are injected into the semiconductors, the occupation probabilities for the electron and hole states are given by the appropriate quasi-Fermi levels. The emitted photons leave the device volume so that the photon density never becomes high in the e-h recombination region. In a laser diode the situation is different. The photon emission rate is given by integrating the emission rate Wem over all the electron-hole pairs after introducing the appropriate Fermi functions. There are several important limits of the spontaneous rate: I. In the case where the electron and hole densities n and p are small (non degenerate case), the Fermi functions have a Boltzmann form (exp(−E/kBT)). The recombination rate is found to be 3/ 2

1  2 2 mr*    Rspon  * *  2 0  k BTme mh 

np

The rate of photon emission depends upon the product of the electron and hole densities. If we define the lifetime of a single electron injected into a lightly doped (p=Na ≤ 1017cm−3) p-type region with hole density p, it would be given from the last equation by

1 1  2 2 mr*      * *  n  r 2 0  k BTme mh 

Rspon

3/ 2

p

The time τr in this regime is very long (hundreds of nanoseconds), as shown in the next figure, and becomes smaller as p increases. © Nezih Pala [email protected]

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Carrier injection and spontaneous emission

Radiative lifetimes of electrons or holes in a direct gap semiconductor as a function of doping or excess charge. The figure gives the lifetimes of a minority charge (a hole) injected into an n-type material. The figure also gives the lifetime behavior of electron-hole recombination hen excess electrons and holes are injected into a material as a function of excess carrier concentration.

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Carrier injection and spontaneous emission II. In the case where electrons are injected into a heavily doped p-region (or holes are injected into a heavily doped n-region), the function fh(fe) can be assumed to be unity. The spontaneous emission rate is

1  mr*  Rspon ~  *   0  mh 

3/ 2

n

for electron concentration n injected into a heavily doped p-type region and

1  mr*  Rspon ~  *   0  mh 

3/ 2

p

for hole injection into a heavily doped n-type region.

The minority carrier lifetimes (i.e., n/Rspon) play a very important role not only in LEDs but also in diodes and bipolar devices. In this regime the lifetime of a single electron (hole) is independent of the holes (electrons) present since there is always a unity probability that the electron (hole) will find a hole (electron). The lifetime is now essentially τo, as shown in the last figure. © Nezih Pala [email protected]

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Carrier injection and spontaneous emission III. Another important regime is that of high injection, where n = p is so high that one can assume fe = fh = 1 in the integral for the spontaneous emission rate. The spontaneous emission rate is n p

Rspon ~

0

~

0

and the radiative lifetime (n/Rspon = p/Rspon) is τo. IV. A regime that is quite important for laser operation is one where sufficient electrons and holes are injected into the semiconductor to cause “inversion”. As will be discussed later, this occurs if fe + fh ≥ 1. If we make the approximation fe ∼ fh = 1/2 for all the electrons and holes at inversion, we get the relation

n Rspon ~ 4 0 or the radiative lifetime at inversion is

~

0 4

This value is a reasonable estimate for the spontaneous emission rate in lasers near threshold. © Nezih Pala [email protected]

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Carrier injection and spontaneous emission The radiative recombination depends upon the radiative lifetime τr and the non-radiative lifetime τnr. To improve the efficiency of photon emission one needs a value of τr as small as possible and τnr as large as possible. To increase τnr one must reduce the material defect density. This includes improving surface and interface qualities. The LED current is then given by

J  eRspontQW

 eVbi  Vturnon  J 0 exp    J SNS k BT  

The parasitic currents are the second and third terms in the expression. The second term represents current injected over the barrier and the third term the current recombining at the maximum recombination plane.

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The uses of diode non-linearity (Mixers, Multipliers, Power Detectors) A mixer is a frequency translation device that translates an input signal band of frequencies to a different band of output frequencies. There are two main uses of the mixer: down conversion and up conversion. Down conversion, used in receivers, takes a higher input RF frequency and shifts it down to a lower frequency where the channel selection can be performed and interfering signals can be filtered out. Up conversion takes a lower frequency band limited signal and shifts it to a higher frequency. This is typically the transmitter application. A mixer does not really “mix” or sum signals; it multiplies them. For example, the analog multiplier performs the frequency translation function:

A  A sin 1t

B  B sin 2t

 A sin 1t B sin 2t    AB / 2cos(1  2 )t  cos(1  2 )t  Note that both sum and difference frequencies are obtained by the multiplication of the two input sinusoidal signals as shown in the equation. One of these is the input signal (A) whose amplitude and phase generally vary with time. The other input (B) is a reference signal, locally generated, called the local oscillator, normally with fixed amplitude and phase.

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The uses of diode non-linearity (Mixers, Multipliers, Power Detectors) With the ideal analog multiplication process shown in the figure, no harmonics or spurious signals are produced. Also, there is no feed through of A or B to the output. But, in reality, mixers always produce many spurious outputs that consist of harmonics of A and B and additional mixing products mω1 ± nω2, where m and n are integers.

VRF VLO

RS

Vo(t) RL

A “good” mixer is designed such that it suppresses these spurious outputs and provides a highly linear amplitude and phase relationship between signal input (A) and the output.   eV   1 1   I D  I s exp  D   1  I s a1VD  a2VD2  a3VD3  ... 2 6     k BT   1   VRF  VLO  VD  I S  a1VD  a2VD2 RS  RL   0 2   1   Vo (t )  I S  a1VD  a2VD2  RL 2  



xn e  n  0 n! for small x (that is small voltages) x

x2 e 1  x  2 x

a2VD2 sin 2 t  a2VD2 1  cos 2t 

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The uses of diode non-linearity (Mixers, Multipliers, Power Detectors) Now, suppose that two inputs are summed as shown in the figure and the diode current produces an output Vo(t) across resistor RL. One input VRF is the signal; the other VLO is the reference local oscillator. The diode voltage, VD, can be found using the series approximation equation, and the output voltage, Vo(t), is calculated from the diode current ID. If only first and second order terms are used, a quadratic equation is easily solved.

While only the sum and difference frequencies are desired as output, the mixer output will also contain a DC term, RF and LO feed through, and terms at all harmonics of the RF and LO frequencies. Only the second-order product term produces the desired outputs. It can be seen in the last set of equations that the second-order nonlinearity also produces a second harmonic and a DC term. The second harmonic generation is the property used in frequency multipliers. Also, the DC term amplitude is proportional to the square of the input voltage, hence input power. This is the principle of operation of diode power detectors.

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Power devices A DC-to-DC converter is a module that accepts a DC input voltage and produces a DC output voltage typically at a different voltage level or of different polarity. These modules have become ubiquitous in modern electronic systems. For example, laptops use them to convert the mains power supply voltage to the battery voltage (18 V), which in turn is converted to the supply voltage for the computing electronics (1.5-3.5 V) and the voltage for the display (voltage variable depending on type of display). All are different! There are several topologies to achieve the desired conversion and we will briefly discuss a Buck or Step- Down Converter to appreciate the functional requirements of the transistor switch and diode that this employed. As in most power conversion circuits it is imperative to not have current flow with a large voltage across dissipative elements such as transistor switches. This will cause power dissipation and excessive heating in the circuit. To reduce the voltage across a switch while it is conducting, an inductor is typically employed in circuits. Furthermore a capacitor is used at the output to stabilize the output voltage through the switching cycle.

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Power devices – Buck converter In the Buck/Step-Down circuit, an input power is turned on causing the input voltage Vin (which has to be stepped-down) to appear at one end of the inductor while the other remains at the output. This voltage will cause the inductor current to rise, storing energy as magnetic flux. During this process the diode is reverse biased and turned off and the current flows through the transistor and the inductor to the output capacitor and load. When the input power is turned off, the current through the inductor will continue flowing but now be forced through the diode causing the diode to turn on. This process is called free-wheeling. The voltages Vx and Vo will follow standard L and C charging/discharging relationships as shown in the figure.

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Power devices – Buck converter

When the switch pictured above is closed, the voltage across the inductor is VL = Vi − Vo. The current through the inductor rises linearly. As the diode is reverse-biased by the voltage source V, no current flows through it; When the switch is opened, the diode is forward biased. The voltage across the inductor is VL = − Vo (neglecting diode drop). Current IL decreases. © Nezih Pala [email protected]

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Power devices – Buck converter The energy stored in inductor L is E 

1 L  I L2 2

Therefore, it can be seen that the energy stored in L increases during On-time (as IL increases) and then decreases during the Off-state. L is used to transfer energy from the input to the output of the converter. The rate of change of IL can be calculated from:

dI L VL  L dt With VL equal to Vi − Vo during the On-state and to − Vo during the Off-state. Therefore, the increase in current during the On-state is given by:

I Lon  

ton

0

(Vi  Vo ) VL dt  ton L L

Identically, the decrease in current during the Off-state is given by:

I Loff  

toff

0

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Vo VL dt   toff L L

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Power devices – Buck converter If we assume that the converter operates in steady state, the energy stored in each component at the end of a commutation cycle T is equal to that at the beginning of the cycle. That means that the current IL is the same at t=0 and at t=T (see the figure). So we can write from the above equations:

(Vi  Vo ) V ton  o toff  0 L L It is worth noting that the above integrations can be done graphically: In the figure, ILon is proportional to the area of the yellow surface, and Iloff to the area of the orange surface, as these surfaces are defined by the inductor voltage (red) curve. As these surfaces are simple rectangles, their areas can be found easily: (Vi  Vo )ton for the yellow rectangle and

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 Votoff for the orange one.

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Power devices – Buck converter As can be seen on the figure

ton  DT and toff  (1  D)T D is a scalar called the duty cycle with a value between 0 and 1. This yields:

(Vi  Vo ) DT  Vo (1  D)T  0  Vo  DVi  0 Vo D Vi From this equation, it can be seen that the output voltage of the converter varies linearly with the duty cycle for a given input voltage. As the duty cycle D is equal to the ratio between tOn and the period T, it cannot be more than 1. Therefore,

Vo  Vi This is why this converter is referred to as step-down converter. So, for example, stepping 12 V down to 3 V (output voltage equal to a fourth of the input voltage) would require a duty cycle of 25%, in our theoretically ideal circuit. © Nezih Pala [email protected]

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Power devices In the presence of a reverse leakage current Irev, additional loss of IrevVin occurs in the freewheeling diode when the transistor is ON; and if the forward diode drop is nonnegligible, IoVdiode will be lost in the diode when the transistor is OFF. Both these losses are important since in the first case Vin is large and in the second Io is large.

It is imperative that in these applications the device behave as a nearly- perfect diode with Vdiode and Irev both being as small as possible. Furthermore, the diode should switch off faster than the transistor, to reduce transient dissipation in it. Schottky diodes which are unipolar and have short switching times are emerging as preferred diodes in freewheeling applications.

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Example 6.5 Calculate the e-h recombination time when an excess electron and hole density of 1015cm−3 is injected into a GaAs sample at room temperature.

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Example 6.6 In two n+p GaAs LEDs, n+ p so that the electron injection efficiency is 100% for both diodes. If the nonradiative recombination time is 10−7s, calculate the 300 K internal radiative efficiency for the diodes when the doping in the p-region for the two diodes is 1016 cm−3 and 5 × 1017 cm−3.

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Example 6.7

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Example 6.8

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