Optoelectronics and

FFFN25/FYST50

Dan Hessman, Solid State Cord Arnold, Atomic Physics

Optoelectronics The application of electronic devices that source, detect and control .

(Electronic devices = devices)

Optical Communication The use of light to transport information. Practical Info

Lectures Mondays 13-15 and Thursdays 8-10 K404 (or Rydberg)

H221, H421 Rydberg

H442,H443 Calculus exercises Fridays 13-15 (2 Feb: 15-17) H221 (H421) K404 Cord Arnold Dan Hessman Lab exercises

- The Diode (Q131) Q131 - Fiber (H443) - CCD & CMOS Cameras (H442)

Email with instructions for signing up Practical info

Literature • Fundamentals of , B. E. A. Saleh and M. C. Teich • ~880kr @ KFS • Can be found as e-book via the University library • Lecture notes • Hand out material. On website, password protected

Exam • 16/3 MA10, 14.00-19.00 • Book allowed, e-book not allowed!

Course Outline

Week 1: Optical Processes & Semiconductor Optics

Week 2: Semiconductor Photon Sources

Week 3: Fiber waveguides

Week 4: Semiconductor Photon Detectors

Week 5: Fiber communication

Week 6: Cameras and

Week 7: Coherent communication + repetition

Semiconductors Ch 16 12.345*2/*6/

6. Halvledare

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E EF 0 g

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-4 z Energy (eV) -6

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Direct- and Indirect-Bandgap Semiconductors Semiconductors for which the conduction-band minimum energy and the valence- band maximum energy correspond to the same value of the wavenumber k (same momentum) are called direct-bandgap materials. Semiconductors for which this is not the case are known as indirect-bandgap materials. As is evident in Fig. 16.1-5, GaAs is a direct-bandgap semiconductor whereas Si is an indirect-bandgap semiconductor. The distinction is important because a transition between the bottom of the conduction band and the top of the valence band in an indirect-bandgap semiconductor must accommodate a substantial change in the momentum of the electron. It will be shown subsequently that direct-bandgap semiconductors such as GaAs are efficient photon emitters, whereas indirect-bandgap semiconductors such as Si cannot serve as efficient light emitters under ordinary circumstances.

B. Semiconductor Materials Figure 16.1-6 reproduces the section of the periodic table that comprises most of the elements important in semiconductorSemiconductorelectronics and photonics. Both elemental and compound semiconductorsmaterialsplay crucial roles in these technologies.

• Group IV: Si, Ge Indirect; Detectors, CCD, photovoltaics • Group III-V: GaP, GaAs, InGaAsP… LEDs, , detectors

• GroupGas III-N: GaN, InGaN… Blue (&white) LEDs, UV lasers D Liquid • Group II-VI: HgCdTe… IR camerasSolid Figure 16.1-6 Section ofThethe periodic periodictable relating tableto semiconductors. Elements indicated in blue, yellow, and silver take the form of gases, liquids, and solids, respectively, at room temperature. The full periodic table is displayed(fromin Fig. a semiconductor13.1-3. physicist’s view)

We proceed to discuss elemental, binary, ternary, and quaternary semiconductors in turn, and then consider doped semiconductors.

Elemental Silicon (Si) and germanium (Ge) are important elemental Semiconductors semiconductors in column IV of the periodic table. Virtually all commercial electronic integrated circuits and devices are fabricated using Si. Both Si and Ge also find widespread use in photonics, principally as . These materials have traditionally not been used for the fabrication of light emitters because of their indirect bandgaps. However, some forms of Si are viable as light emitters and has come to the fore. The basic properties of Si and Ge are provided in Table 16.1-2. How do we produce semiconductor structures? Epitaxy Homostructure Heterostructure Add one atom layer at a Just one material Combinaons of materials me to substrate wafer Lace matching important!

GaAs p AlGaAs GaAs GaAs n AlGaAs

Wafer Epitaxy machine Reactor cell 16.1 SEMICONDUCTORS 637

0.5 AIN 0.2

0.6 2.0 E E 0.7 :::i. OJ 0.3 0.8 ...-'"( 4 ...-'"( OJ ..c lLJ ..c >. >. bO 1.0 • Si 3 0.4 1.2 g- 1.0 g- 0.5 bO g- bO 0.6 g- -g bO -g 2 bO 2.0 ] -g 0.5 P:::l 1.0 3.0 2.0 10 0 0 5.4 5.6 5.8 6.0 6.2 6.4 6.6 3.0 3.1 3.2 3.3 3.4 3.5 Lattice constant (A) Lattice constant (A) (a) (b) 16.1 SEMICONDUCTORS 637 Figure 16.1-7 Bandgap energies, bandgap wavelengths, and lattice constants for Si, Ge, SiC, and 12 III-V binary compounds. Solid and dashed curves represent direct-bandgap and indirect- bandgap compositions, respectively. A material0.5 may have AINa direct bandgap for one mixing ratio0.2 and an indirect bandgap for a different mixing ratio. Ternary materials are represented along the line that joins two binary compounds. A quaternary0.6 compound is represented by the area formed by its 2.0 binary components. (a) Inl-xGaxAsl-yPy is represented by the stippled area with vertices at InP, 0.7 E E InAs, GaAs, and GaP, while (AlxGal-x)ylnl-yP is represented by the shaded area with vertices:::i. at OJ 0.3 AlP, InP, and GaP. Both are important quaternary0.8 ...-'"( compounds,4 the former in the near ...-'"( and OJ ..c lLJ ..c >.the latter in the visible. AlxGal-xAs is represented>. by points along the line connecting GaAsbOand 1.0 AlAs.• Si As x varies from 0 to 1, the point moves along3 the line from GaAs and AlAs. Since 0.4this line 1.2 g- is1.0nearly vertical, AlxGal-xAs is lattice matchedg-to GaAs. (b) Although the III-nitride compound0.5 bO InxGal-xN can, in principle, be compositionallyg- tunedbO to accommodate the entire visible spectrum,0.6 g- -g bO -g 2 bO this material becomes increasingly difficult2.0 to] grow as the composition of In becomes appreciable.-g 0.5 P:::l 1.0 InxGal-xN is principally used in the green,3.0 blue, and violet spectral regions, while AlxGal-xN and AlxlnyGal-x_yN serve the region. All compositions of these III-Nitride compounds2.0 are direct-bandgap semiconductors. 10 0 0 5.4 5.6 5.8 6.0 6.2 6.4 6.6 3.0 3.1 3.2 3.3 3.4 3.5 Lattice constant (A) Lattice constant (A) (a) (b) ZnS 3.5 Figure 16.1-7 Bandgap energies, bandgap wavelengths, and lattice constants for Si, Ge, SiC, 0.4 • Group IV: Si, Ge and 12 III-3.0 V binary compounds. Solid and dashed curves represent direct-bandgap and indirect- bandgap compositions,ZnSe respectively. A material mayE haveIndirect;a direct bandgapDetectors,for one CCD,mixing photovoltaicsratio and >' 2.5 CdS ZnTe 0.5 3 Ol an indirect bandgap for a different mixing ratio....-'"( Ternary materials are represented along the line lLJOl 2.0 0.6 ..c Figure 16.1-8 Bandgap energies, bandgap that joins>. two binary compounds. A quaternary bOcompound• Groupis represented III-V: GaP,by theGaAs,area InGaAsP…formed by its 0.7 c:: wavelengths, and lattice constants for various binary components.1.5 (a) Inl-xGaxAsl-yPy is representedLEDs,by the lasers,stippled detectorsarea with vertices at InP, > II-VI semiconductors (HgSe and HgTe are InAs, GaAs,g. and GaP, while (AlxGal-x)ylnl-yP1.0 is represented by the shaded area with vertices at bO 1.0 1.2 g. semimetals with small negative bandgaps). AlP, InP,] and GaP. Both are important quaternarybJ) compounds, the former in the near infrared and P:::l 2.0 "0 • GroupHgTe and III-N:CdTe GaN,are nearly InGaN…lattice matched, the latter0.5in the visible. AlxGal-xAs is represented§ by points along the line connecting GaAs and P:::l Blueas evidenced (&white)by LEDs,the vertical UV lasersline connect- 10 AlAs. As0.0x varies from 0 to 1, the point moves along theinglinethem,fromsoGaAsthat theandternaryAlAs. semiconductorSince this line is nearly vertical, AlxGal-xAs is lattice matched to GaAs. (b) Although the III-nitride compound • GroupHgxCd 1 -II-VI:xTe canHgCdTe…be grown without strain InxGal-xN5.4 can,5.6in principle,5.8 6.0be 6.2compositionally 6.4 6.6 tuned toonaccommodatea CdTe template.the entireIt is anvisibleimportantspectrum,mid- this material becomesLatticeincreasinglyconstant (A) difficult to grow as theIRinfrared compositioncamerasphotodetector of In material.becomes appreciable. InxGal-xN is principally used in the green, blue, and violet spectral regions, while AlxGal-xN and AlxlnyGal-x_yN serve the ultraviolet region. All compositions of these III-Nitride compounds are direct-bandgapUndopedsemiconductors.semiconductors (i.e., semiconductors devoid of intentional doping) are referred to as intrinsic materials, whereas doped semiconductors are called extrinsic materials. The concentrations of mobile electrons and holes are equal in an intrinsic n p ni, ni semiconductor,ZnS == == where the intrinsic concentration grows with in- 3.5 creasing temperature at an exponential rate. On the other hand, the concentration of 0.4 3.0 mobile electrons in an n-type semiconductor (majority carriers) is far greater than the concentration of holes (minority carriers), i.e., n » p. The opposite is true in ZnSe E >' 2.5 CdS ZnTe 0.5 3 Ol ...-'"( lLJOl 2.0 0.6 ..c Figure 16.1-8 Bandgap energies, bandgap >. bO 0.7 c:: wavelengths, and lattice constants for various 1.5 > g. 1.0 II-VI semiconductors (HgSe and HgTe are bO 1.0 1.2 g. semimetals with small negative bandgaps). ] bJ) P:::l 2.0 "0 HgTe and CdTe are nearly lattice matched, 0.5 § P:::l as evidenced by the vertical line connect- 10 0.0 ing them, so that the ternary semiconductor HgxCd 1- xTe can be grown without strain 5.4 5.6 5.8 6.0 6.2 6.4 6.6 on a CdTe template. It is an important mid- Lattice constant (A) infrared material.

Undoped semiconductors (i.e., semiconductors devoid of intentional doping) are referred to as intrinsic materials, whereas doped semiconductors are called extrinsic materials. The concentrations of mobile electrons and holes are equal in an intrinsic semiconductor, n == p == ni, where the intrinsic concentration ni grows with in- creasing temperature at an exponential rate. On the other hand, the concentration of mobile electrons in an n-type semiconductor (majority carriers) is far greater than the concentration of holes (minority carriers), i.e., n » p. The opposite is true in 640 CHAPTER 16 SEMICONDUCTOR OPTICS

polarization (i.e., two photon spin values), whereas in the semiconductor case there are two spin values associated with the electron state. In resonator optics the allowed electromagnetic solutions for k were converted into allowed frequencies via the linear frequency-wavenumber relation v == ck/21r. In semiconductor physics, on the other hand, the allowed solutions for k are converted into allowed energies via the quadratic energy-wavenumber relations given in (16.1-3) and (16.1-4). If (}e( E) represents the number of conduction-band energy levels (per unit volume) lying between E and E+ then, because ofthe one-to-one correspondence between E and k governed by (16.1-3), the densities (}e (E) and {}( k) must be related by (}e (E) dE == {}( k) dk. Thus, the density of allowed energies in the conduction band is (}e (E) == {}( k) / (dE/ dk ). Similarly, the density of allowed energies in the valence band is (}v(E) == (}(k)/(dE/dk), where E is given by (16.1-4). The approximate quadratic E-k relations (16.1-3) and (16.1-4), which are valid near the edges of the conduction band and valence band, respectively, are used to evaluate the derivative dE/ dk for each band. The result that obtains is

(16.1-7)

(16.1-8) 16.1 SEMICONDUCTORS 643Density of States Near Band Edges Our attention has beenThe directedsquare-rootto therelationmobileis acarriersresult ofinthedopedquadraticsemiconductors.energy-wavenumberThese formulas for materials are, of course,electronselectricallyand holesneutral,near theasbandassurededges.by theThefixeddependencedonor andof theacceptordensity of states on energy is illustrated in Fig. 16.1-10. It is zero at the band edge, and increases away ions, so thatEnergyn + NfromA ==BandsitP at+aNrateD , wherethat dependsNA andon NtheD effectiveare, respectively,masses ofthethe numberelectronsofand holes. The ionized acceptors andvaluesdonorsof meperandunitm vvolume.provided in Table 16.1-1 are actually averaged values suitable for calculating the density of states. E E E E

Ect----- I neE)Ec------Ec

:: : : ±:e: ±: : : ±: ±:e:-:±±±e: :±: : ---- peE) Ev------Ev .++++++++++++++++++++++++I .++++++++++++++++++++++++ .++++++++++++++++++++++++fJ-+ .++++++++++++++++++++++++ .++++++++++++++++++++++++11111111111111111111) k .++++++++++++++++++++++++ .++++++++++++++++++++++++ Density of states (a) (b) (c) o 1 feE) Canier concentration Figure 16.1-10 (a) Cross section of the E-k diagram (e.g., in the direction of the k1 component, with k2 and k3 fixed). (b) Allowed energy levelsFermi-function(at all k). (c) Density * DOSof states near the edges of the Figure 16.1-13• DensityEnergy-bandconduction of statesand (DOS)diagram,valence Fermibands. Thefunctionquantityf(E),(}c (E)anddE isconcentrationsthe number of quantumof mobilestates with energy electrons and holes, n(betweenE) and p(E andE),E+dE,respectively,per unitinvolume,an n-typein thesemiconductor.conduction band. The quantity (}v (E) has an analogous interpretation for the valence band. • Fermi-function & -level E BoltzmannProbability farof fromOccupancy E In the absence of thermalf excitation (at T == 0° K), all electrons occupy the lowest Steppossible at lowenergy T levels, subjectI to the Pauli exclusion principle. The valence band is then

E 1.. Acceptor level EAT

o 1 fCE) Carrier concentration Figure 16.1-14 Energy-band diagram, Fermi function f(E), and concentrations of mobile electrons and holes, n(E) and p(E), respectively, in a p-type semiconductor.

EXERCISE 16. 1-2

Exponential Approximation of the Fermi Function. When E - Ef » kT, the Fermi function f(E) may be approximated by an exponential function. Similarly, when Ef - E » kT, 1- f (E) may be approximated by an exponential function. These conditions apply when the Fermi level lies within the bandgap, but away from its edges by an energy of at least several times kT (at room temperature kT 0.026 eV whereas E g = 1.12 eV in Si and 1.42 eV in GaAs). Using these approximations, which apply for both intrinsic and doped semiconductors, show that (16.1-11) gives Ef n = Ncexp (- Ec:r ) (16.1-12)

f p = Nv exp( E :rEv) (16.1-13)

np = NcNvexp ( - :;), (16.1-14)

2 2 where Ne = 2(21fm e kT/h )3/2 and Nv = 2(21fm v kT/h )3/2. Verify that if Ef is closer to the conduction band and m v = me, then n > p, whereas if it is closer to the valence band, then p > n. 16.1 SEMICONDUCTORS 645

Quasi-Equilibrium Carrier Concentrations The occupancy probabilities and carrier concentrations considered above are applica- ble only for a semiconductor in thermal equilibrium. They are not valid when thermal equilibrium is disturbed. There are, nevertheless, situations in which the conduction- band electrons are in thermal equilibrium among themselves, as are the valence-band holes, but the electrons and holes are not in mutual thermal equilibrium. This can occur, for example, when an external electric current or photon flux induces band-to-band transitions at too high a rate for interband equilibrium to be achieved. This situation, which is known as quasi-equilibrium, arises when the relaxation (decay) times for transitions within each of the bands are much shorter than the relaxation time between the two bands. Typically, the intraband relaxation time < 10-12 s, whereas the radiative electron-hole recombination time 10-9 S. Under these circumstances, it is appropriate to use a separate Fermi function for each band; the two associated Fermi levels, denoted Efe and Efv, are known as quasi- Fermi levels (Fig. 16.1-15). When E fe and Efv lie well inside the conduction and valence bands,Quasi-Equilibriumrespectively, the concentration of both electrons and holes can be quite large.

E E

--, neE) ------E c I DOS*Fermi-function I ------E __ Efu - Efu v peE)

o 1 icCE) o Carrier concentration • Quasi-Fermi levels E & E Figure 16.1-15 A semiconductorfc fv in quasi-equilibrium. The probability that a particular conduction-band energy level E is occupied by an electron is le( E), a Fermi function with Fermi level E fe.• TheSeparateprobability Fermi-functionsthat a valence-band for energy level E is occupied by a hole is 1- Iv (E), where Iv (E)electronsis a Fermi andfunction holes with Fermi level Efv. The concentrations of electrons and holes are n(E) and p(E), respectively. Both can be large.

EXERCISE 16.1-3 Determination ofthe Quasi-Fermi Levels Given the Electron and Hole Concentrations.

(a) Given the concentrations of electrons n and holes p in a semiconductor at T = 0° K, use (16.1- 10) and (16.1-11) to show that the quasi-Fermi levels are

2 3 Efc = Ec + n / (16.1-18a) 2me

2 Efv = Ev (31f )2/3 p2/3. (16.1-18b) 2mv (b) Show that these equations are approximately applicable for an arbitrary temperature T if nand p are sufficiently large so that E fe - E e » kT and Ev - E fv » kT, i.e., if the quasi-Fermi levels lie deep within the conduction and valence bands. 646 CHAPTER 16 SEMICONDUCTOR OPTICS

D. Generation, Recombination, and Injection Generation and Recombination in Thermal Equilibrium The thermal excitation of electrons from the valence band into the conduction band results in the electron-hole generation (Fig. 16.1-16). Thermal equilibrium requires that this generation process be accompanied by a simultaneous reverse process ofdeex- citation. This process, called electron-hole recombination, occurs when an electron Generation and decays from the conduction band to fill a hole in the valence band (Fig. 16.1-16). The energy released by the electron may take the form of an emitted photon, in which case Recombination the process is called radiative recombination.

• Thermal generation G0+injection R

• Recombination = r n p radiative and non-radiative

• Excess carrier concentration ∆n

• ∆n=injection rate*lifetime=R τ Figure 16.1-16 Electron-hole generation and recombination. • High and low level injection Nonradiative recombination can occur via a number of independent competing • Electrical injection R=current/(e*volume) processes, including the transfer of energy to lattice vibrations (creating one or more phonons) or to another free electron (Auger process). Recombination may also take place at surfaces and indirectly via traps or defect centers, which are energy levels associated with impurities or defects associated with grain boundaries, dislocations, or other lattice imperfections that lie within the forbidden band. An impurity or defect state can act as a recombination center if it is capable of trapping both an electron and a hole, thereby increasing their probability of recombining (Fig. 16.1-17). Impurity- assisted recombination may be radiative or nonradiative.

Figure 16.1-17 Electron-hole recombina- tion via a trap.

Because it takes both an electron and a hole for a recombination to occur, the rate of recombination is proportional to the product of the concentration of electrons and holes, i.e.,

rate of recombination == rnp, (16.1-19)

where the recombination coefficient r (cm3/s) depends on the characteristics of the material, including its composition and defect density, and on temperature; it also depends relatively weakly on the doping level. pn-Junction

16.1 SEMICONDUCTORS 651

• Built-in electric field (used in solar cells and detectors)

• Injection of electrons and holes when forward biased (used for light emitting devices)

Figure 16.1-19 A p-n junction in thermal equilibrium at T > 0° K. The depletion-layer, energy-band diagram, and concentrations (on a logarithmic scale) of mobile electrons n(x) and p(x) n(x) holes p (x) are shown as functions of ·SE the position x. The built-in potential 1 x------difference Va corresponds to an energy u8 ------" eVa, where e is the magnitude of the g x electron charge.

16.1-19. In thermal equilibrium there is only a single Fermi function for the entire structure so that the Fermi levels in the p- and n-regions must align. • No net current flows across the junction. The currents associated with diffusion and built-in field (drift current) cancel for both the electrons and holes.

The Biased p-n Junction An externally applied potential will alter the potential difference between the p- and n- regions. This in turn will modify the flow of majority carriers, so that the junction can be used as a "gate." If the junction is forward biased by applying a positive voltage V to the p-region (Fig. 16.1-20), its potential is increased with respect to the n-region, so that an electric field is produced in a direction opposite to that of the built-in field. The presence of the external bias voltage causes a departure from equilibrium and a misalignment of the Fermi levels in the p- and n-regions, as well as in the depletion layer. The presence of two Fermi levels in the depletion layer, Efe and E fv, represents a state of quasi-equilibrium.

.S p(x) n(x) 'C b Excess I/ electrons holes Figure 16.1-20 Energy-band diagram U § __-_-_-_-_-_-_..-t!_-_J and carrier concentrations for a forward- biased p-n junction. 654HeterostructuresCHAPTER 16 SEMICONDUCTOR OPTICS

FigureHeterostructures16.1-23 The p-p-n can:double heterojunc- n tion structure. The middle layer is of narrower bandgap• Createthan the barrierouter layers. forIn chargeequilibrium, carriersthe Fermi levels align so that the edge of the conduc- tion•band Confinedrops sharply chargeat the p-pcarriersjunction and the ••••------·----1 edge of the valence band drops sharply at the p-n t junction. The conduction- and valence-band dis- Eg2 Eg3 continuities• Accelerateare known ascarriersband offsets. When the . ! device is forward biased, these jumps act as barri- Ii ers that• Controlconfine the whereinjected minority emissioncarriers canto the take place region of lower bandgap. Electrons injected from the n-region,combiningfor example, direct-indirectare prevented materialfrom diffusing beyond the barrier at the p-p junction. Similarly,• Controlholes injected wherefrom absorptionthe p-region takesare place (and not) not permitted to diffuse beyond the energy barrier at the p-n junction. This double-heterostructure configuration• Combinetherefore diforcesfferentelectrons refractiveand holes indexes to occupy a narrow common region. This sub- stantially• Giveincreases quantumthe efficiency confinementof light-emitting diodes, semiconductor optical amplifiers, and laser diodes (see Chapter 17).

• Discontinuities in the energy-band diagram created by two heterojunctions can be useful for confining charge carriers to a desired region of space. For example, a layer of narrow-bandgap material can be sandwiched between two layers of a wider bandgap material, as shown in the p-p-n structure illustrated in Fig. 16.1-23 (which consists of a p-p heterojunction and a p-n heterojunction). This double-heterostructure (DH) configuration is used effectively in the fabrica- tion of LEDs, semiconductor optical amplifiers, and laser diodes, as explained in Chapter 17. • Heterojunctions are useful for creating energy-band discontinuities that accelerate carriers at specific locations. The additional kinetic energy suddenly imparted to a carrier can be useful for selectively enhancing the probability ofimpact ionization in a multilayer avalanche (see Sec. 18.4A). • Semiconductors of different bandgap type (direct and indirect) can be used in the same device to select regions of the structure where light is emitted. Only semi- conductors of the direct-bandgap type can efficiently emit light (see Sec. 16.2). • Semiconductors of different bandgap can be used in the same device to select regions of the structure where light is absorbed. Semiconductor materials whose bandgap energy is larger than the photon energy incident on them will be trans- parent, acting as a window layer. • Heterojunctions ofmaterials with different refractive indexes can be used to create photonic structures and optical waveguides that confine and direct photons, as discussed in Chapters 7 and 8.

G. Quantum-Confined Structures Heterostructures of thin layers of semiconductor materials can be grown epitaxially, i.e., as layers of one semiconductor material over another, by using techniques such as molecular-beam epitaxy (MBE); liquid-phase epitaxy (LPE); and vapor-phase epi- taxy (VPE), of which common variants are metal-organic chemical vapor deposition (MOCVD) and hydride vapor-phase epitaxy (HVPE). Homoepitaxy is the growth of 16.1 SEMICONDUCTORS 659

(a) (b) (c) Figure 16.1-28 Energy-band diagrams of MQW and superlattice structures fabricated from alternating layers of materials with different bandgaps, such as AlGaAs and GaAs. (a) Unbiased MQW structure. (b) Biased MQW structure. (c) Biased superlattice structure with minibands and minigap.

Quantum Wires A semiconductor material that takes the form of a thin wire surrounded by a material of wider bandgap is called a quantum-wire structure (Fig. 16.1-29). The wire acts as a potential well that narrowly confines electrons (and holes) in two directions, x and y. Assuming that the wire has a rectangular cross section of area d 1 d 2 , the energy- momentum relation in the conduction band is

(16.1-38) where

DOS in Quantum ql, q2 == 1,2,3, ... (16.1-39) Structuresand k is the vector component in the z direction (along the axis of the wire).

z Y }---/----} / x E E E E

/ / / / / "-C

Ec ---_.... Ec

Ev ------

Bulk Quantum well Quantum wire Figure 16.1-29 The density of states in different confinement configurations. The conduction and valence bands split into overlapping subbands that become successively narrower as the electron motion is restricted in a greater number of dimensions. Band-to-Band

664 TransitionsCHAPTER 16 SEMICONDUCTOR OPTICS

E

1111111111111111111 k 1111111111111111111 k 1111111111111111111 ) k (a) (b) (c)

Figure• Absorption,16.2-5 (a) The Spontaneousabsorption of a photon and resultsStimulated in the generation emissionof an electron-hole pair. This process is used in the photodetection of light. (b) The recombination of an electron-hole pair results in the spontaneous emission of a photon. Light-emitting diodes (LEDs) operate on this basis. (c) Electron-hole• Determinedrecombination by can be induced by a photon. The result is the stimulated emission of an identical photon. -OpticalThis Jointis the underlyingDOS (combinesprocess responsible CB andfor VB)the operation of semiconductor laser diodes. -Occupation • Conservation -Transitionof Energy. probabilityThe absorption or emission of a photon of energy hv requires that the energies of the two states involved in the interaction (say E1 • andEnergyE2 in andthe valencemomentumband andconservationconduction band, respectively, as depicted in Fig. 16.2-5) be separated by hv. Thus, for photon emission to occur by electron- hole recombination, for example, an electron occupying an energy level E 2 must interact with a hole occupying an energy level E1, such that energy is conserved, i.e.,

(16.2-2)

• Conservation ofMomentum. Momentum must also be conserved in the process of photon emission/absorption, so that P2 - PI == hv/ c == h/A, or k2 - kI == 21f / A. The photon-momentum magnitude h/A is, however, very small in comparison with the range of momentum values that electrons and holes can assume. The semiconductor E-k diagram extends to values of k of the order 21f / a, where the lattice constant a is much smaller than the wavelength A, so that 21f / A « 21f / a. The momenta of the electron and the hole participating in the interaction must therefore be approximately equal. This condition, k2 kI , is called the k- selection rule. Transitions that obey this rule are represented in the E-k diagram (Fig. 16.2-5) by vertical lines, indicating that the change in k is negligible on the scale of the diagram. • Energies and Momenta ofthe Electron and Hole with Which a Photon Interacts. As is apparent from Fig. 16.2-5, conservation of energy and momentum require that a photon of frequency v interact with electrons and holes of specific energies and momenta determined by the semiconductor E-k relation. Using (16.1-3) and (16.1-4) to approximate this relation for a direct-bandgap semiconductor by two parabolas, and writing Ec - Ev == Eg , (16.2-2) may be written in the form

li2 k2 li2 k2 E2 - E I == -- + Eg + -- == hv, (16.2-3) 2mv 2mc from which

(16.2-4) 662 CHAPTER 16 SEMICONDUCTOR OPTICS

Wavelength Ao (p,m) 10 1.0 0.2

-- GaAs , --- Si , I I I I I I I L'Band- to- band Figure 16.2-2 Observed optical absorption coefficient a versus pho- ton energy and wavelength for Si and GaAs in thermal equilibrium at T = 3000 K. The bandgap energy Egis 1.12 eV for Si and 1.42 eV for GaAs. 10 Silicon is relatively transparent in the band Ao 1.1 to 12 /-Lm, whereas in- 1L...----l...--l...--'--'--'-'-...... -"--....I...I....----'--...... trinsic GaAs is relatively transparent 0.01 0.1 1.0 10.0 in the band Ao 0.87 to 12 /-Lm (see Photon energy hv (eV) Absorption Fig. 5.5-1).

Wavelength Ao (p,m) 10 54 321.5 1.00.9 0.8 0.7 0.6 0.5 0.4 0.3

InP

10L..--....I...... ---L..-----l...-----L..__.l...--..I...--..I...--1...... L..- ..L..----L_---I- ..I...--_-----'----L..- ...L--__----' o 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Photon energy hv (eV) •Figure Absorption16.2-3 alsoAbsorption with indirectcoefficient bandgapversus photonbut noenergy emissionand wavelength for Ge, Si, GaAs, GaN and selected other III-V binary semiconductors at T = 300 0 K, on an expanded scale.

B. Band-to-Band Transitions in Bulk Semiconductors We proceed to develop a simple theory of direct band-to-band photon absorption and emission in bulk semiconductors, ignoring the other types of transitions.

Bandgap Wavelength Direct band-to-band absorption and emission can take place only at frequencies for which the photon energy hv > Eg . The minimum frequency v necessary for this to occur is vg == Eg/h, so that the corresponding maximum wavelength is Ag == co/vg == hco / Eg' If the bandgap energy is given in eV (rather than in J), the bandgap wavelength 666 CHAPTER 16 SEMICONDUCTOR OPTICS

to reside) requires an exchange of momentum that cannot be accommodated by the emitted photon. Momentum may be conserved, however, by the participation of phonons in the interaction. Phonons can carry relatively large momenta but typically have small energies 0.01-0.1 eV; see Fig. 16.2-2), so their transitions appear horizontal on the E-k diagram (see Fig. 16.2-7). The net result is that momentum is conserved, but the k-selection rule is violated. Because phonon- assisted emission involves the participation of three bodies (electron, photon, and phonon), the probability of its occurrence is quite low. Thus, Si, which is an indirect-bandgap semiconductor, has a substantially lower radiative recombina- tion coefficient than does GaAs, which is a direct-bandgap semiconductor (see Table 16.1-4). Silicon is therefore not an efficient light emitter, whereas GaAs is.

E Figure 16.2-7 Photon emISSIon in an indirect-bandgap semiconductor. The recom- bination of an electron near the bottom of the conduction band with a hole near the top of the valence band requires the exchange of energy and momentum. The energy may be carried off by a photon, but one or more phonons are also required to conserve momentum. This type of multiparticle inter- 666 CHAPTER 16 SEMICONDUCTOR OPTICS action is therefore unlikely. to reside) requires an exchange of momentum that cannot be accommodated by the emitted photon. Momentum may• Photonbe conserved,Absorptionhowever,Is NotbyUnlikelythe participationin an Indirect-Bandgap Semiconductor. Al- of phonons in the interaction. Phononsthoughcanphotoncarry relativelyabsorptionlargealso requiresmomentaenergybut and momentum conservation in an typically have small energies 0.01-0.1indirect-bandgapeV; see Fig. semiconductor,16.2-2), so theirthistransitionsis readily achieved by means of a two-step appear horizontal on the E-k diagramprocess(see(Fig.Fig. 16.2-8).16.2-7).TheTheelectronnet resultis firstis thatexcited to a high energy level within the momentum is conserved, but the k-selectionconductionrulebandis violated.by a k-conservingBecause phonon-vertical transition. It then quickly relaxes to assisted emission involves the participationthe bottomof threeof thebodiesconduction(electron,bandphoton,by aprocessand called thermalization in which its phonon), the probability of its occurrencemomentumis quiteis transferredlow. Thus,to phonons.Si, whichTheis generatedan hole behaves similarly. Since indirect-bandgap semiconductor, hasthea substantiallyprocess occurslowersequentially,radiative recombina-it does not require the simultaneous presence tion coefficient than does GaAs, whichof threeis a bodiesdirect-bandgapand is thussemiconductornot unlikely. (seeSilicon is therefore an efficient photon TableIndirect16.1-4). bandgapSilicon is therefore notdetector,an efficientas islightGaAs.emitter, whereas GaAs is.

E Figure 16.2-8 Photon absorption in an Figure 16.2-7 Photon emISSIon in indirect-bandgapan semiconductor via a ver- indirect-bandgap semiconductor. The recom-tical (k-conserving) transition. The photon bination of an electron near the bottomgeneratesof an excited electron in the con- the conduction band with a hole near the ductiontop band, leaving behind a hole in the of the valence band requires the exchangevalence band. The electron and hole then of energy and momentum. The energy mayundergo fast transitions - to the lowest and be carried off by a photon, but one highestor possible levels in the conduction and more phonons are also required to conservevalence bands, respectively, releasing their momentum. This type of multiparticle inter-energy in the form of phonons. Since the action is therefore unlikely. process is sequential it is not unlikely. • Very weak emission… • …but good absorption • Photon Absorption Is Not Unlikely in an Indirect-Bandgap Semiconductor. Al- though photon absorption also requires energy and momentum conservation in an indirect-bandgap semiconductor,c.thisAbsorption,is readily achievedEmission,by meansandofGaina two-stepin Bulk Semiconductors process (Fig. 16.2-8). The electronWeisnowfirst proceedexcited totoa determinehigh energythelevelprobabilitywithin thedensities of a photon of energy hv conduction band by a k-conservingbeingverticalemittedtransition.or absorbedIt thenby aquicklybulk semiconductorrelaxes to material in a direct band-to-band the bottom of the conduction band by aprocess called thermalization in which its momentum is transferred to phonons. The generated hole behaves similarly. Since the process occurs sequentially, it does not require the simultaneous presence of three bodies and is thus not unlikely. Silicon is therefore an efficient photon detector, as is GaAs.

Figure 16.2-8 Photon absorption in an indirect-bandgap semiconductor via a ver- tical (k-conserving) transition. The photon generates an excited electron in the con- duction band, leaving behind a hole in the valence band. The electron and hole then undergo fast transitions - to the lowest and highest possible levels in the conduction and valence bands, respectively, releasing their energy in the form of phonons. Since the process is sequential it is not unlikely.

c. Absorption, Emission, and Gain in Bulk Semiconductors We now proceed to determine the probability densities of a photon of energy hv being emitted or absorbed by a bulk semiconductor material in a direct band-to-band