g Diodes
`
`SECOND EDITION
`E. Fred Schubert
`
`Petitioner Apple Inc. - Ex. 1035, p. Cover 1
`
`

`

`•
`
`LIGHT-EMITTING DIODES
`SECOND EDITION
`
`Revised and fully updated, the Second Edition of this textbook offers a comprehensive
`explanation of the technology and physics of light-emitting diodes (LEDs) such as infrared,
`visible-spectrum, ultraviolet, and white LEDs made from III-V semiconductors. The
`elementary properties of LEDs such as electrical and optical characteristics are reviewed,
`followed by the analysis of advanced device structures.
`With nine additional chapters, the treatment of LEDs has been vastly expanded, including
`new material on device packaging, reflectors, UV LEDs, III-V nitride materials, solid-state
`sources for illumination applications, and junction temperature. Radiative and non-radiative
`recombination dynamics, methods for improving light extraction, high-efficiency and high(cid:173)
`power device designs, white-light emitters with wavelength-converting phosphor materials,
`optical reflectors, and spontaneous recombination in resonant-cavity structures, are dis(cid:173)
`cussed in detail. Fields related to solid-state lighting such as human vision, photometry,
`colorimetry, and color rendering are covered beyond the introductory level provided in the
`first edition. The applications of infrared and visible-spectrum LEDs in silica fiber, plas(cid:173)
`tic fiber, and free-space communication are also discussed. Semiconductor material data,
`device design data, and analytic formulae governing LED operation are provided.
`With exercises, solutions and illustrative examples, this textbook will be of interest to
`scientists and engineers working on LEDs, and to graduate students in electrical engineering,
`applied physics, and materials science.
`
`Additional resources for
`9780521865388.
`
`this
`
`title are available online at www.cambridge.org/
`
`E. FRED SCHUBERT received his Ph.D. degree with Honors in Electrical Engineering
`from University of Stuttgart in 1986 and is currently a Wellfleet Senior Constellation Pro(cid:173)
`fessor of the Future Chips Constellation at Rensselaer Polytechnic Institute. He has made
`several pioneering contributions to the field of LEDs, including the first demonstration of the
`resonant-cavity light-emitting diode (RCLED). He has authored or co-authored more than
`200 publications including Doping in III-V Semiconductors (Cambridge University Press,
`1993, 0-521-01784-X) for which he was awarded the VDE Literature Prize. He is inventor or
`co-inventor of 28 US Patents and a Fellow of the IEEE, APS, OSA, and SPIE. He received the
`Senior Research Award of the Humboldt Foundation, the Discover Award for Technological
`Innovation, the RD 100 Award, and Boston University's Provost Innovation Fund Award.
`
`Note: This book contains many figures in which color adds important information. For this reason, all
`figures are available in color on the Internet at the following websites: <http://www.cambridge.org/
`9780521865388> and < http://www.LightEmittingDiodes.org >.
`
`Petitioner Apple Inc. - Ex. 1035, p. Cover 2
`
`

`

`I
`
`LIGHT-EMITTING DIODES
`SECOND EDITION
`
`E. FRED SCHUBERT
`Rensselaer Polytechnic Institute,
`Troy, New York
`
`.,..:i,,,. CAMBRIDGE
`::: UNIVERSITY PRESS
`
`Petitioner Apple Inc. - Ex. 1035, p. Cover 3
`
`

`

`CAMBRIDGE
`UNIVERSITY PRESS
`
`University Printing House, Cambridge CB2 8BS, United Kingdom
`
`Cambridge University Press is part of the University of Cambridge.
`It furthers the University's mission by disseminating knowledge in the pursuit of
`education, learning and research at the highest international levels of excellence.
`
`www.cambridge.org
`Information on this title: www.cambridge.org/9780521865388
`First edition© E. Fred Schubert 2003
`Second edition© E. Fred Schubert 2006
`This publication is in copyright. Subject to statutory exception
`and to the provisions of relevant collective licensing agreements,
`no reproduction of any part may take place without the written
`permission of Cambridge University Press.
`First edition published 2003
`Second edition 2006
`Second edition reprinted 2007
`6th printing 2014
`A catalogue record for this publication is available from the British Library
`ISBN 978-0-521-86538-8 Hardback
`Cambridge University Press has no responsibility for the persistence or accuracy of
`URLs for external or third-party internet websites referred to in this publication,
`and does not guarantee that any content on such websites is, or will remain, accurate
`or appropriate.
`
`l
`
`Petitioner Apple Inc. - Ex. 1035, p. Cover 4
`
`

`

`l
`
`Contents
`
`Preface
`
`pagex
`
`1
`
`2
`
`3
`
`4
`
`History of light-emitting diodes
`History of SiC LEDs
`1.1
`History of GaAs and AlGaAs infrared and red LEDs
`1.2
`History of GaAsP LEDs
`1.3
`History of GaP and GaAsP LEDs doped with optically active impurities
`1.4
`History of GaN metal-semiconductor emitters
`1.5
`History of blue, green, and white LEDs based on GalnN p-njunctions
`1.6
`History of AlGalnP visible-spectrum LEDs
`1.7
`LEDs entering new fields of applications
`1. 8
`References
`
`Radiative and non-radiative recombination
`Radiative electron-hole recombination
`2.1
`Radiative recombination for low-level excitation
`2.2
`Radiative recombination for high-level excitation
`2.3
`Bimolecular rate equations for quantum well structures
`2.4
`Luminescence decay
`2.5
`Non-radiative recombination in the bulk
`2.6
`Non-radiative recombination at surfaces
`2.7
`Competition between radiative and non-radiative recombination
`2.8
`References
`
`Theory of radiative recombination
`Quantum mechanical model ofrecombination
`3.1
`The van Roosbroeck-Shockley model
`3.2
`Temperature and doping dependence ofrecombination
`3.3
`The Einstein model
`3.4
`References
`
`LED basics: Electrical properties
`Diode current-voltage characteristic
`4.1
`Deviations from ideal I-V characteristic
`4.2
`Evaluation of diode parasitic resistances
`4.3
`Emission energy
`4.4
`Carrier distribution in p-n homojunctions
`4.5
`Carrier distribution in p-n heterojunctions
`4.6
`Effect ofheterojunctions on device resistance
`4.7
`Carrier loss in double heterostructures
`4.8
`Carrier overflow in double heterostructures
`4.9
`4.10 Electron-blocking layers
`4.11 Diode voltage
`References
`
`5
`
`LED basics: Optical properties
`Internal, extraction, external, and power efficiencies
`5 .1
`Emission spectrum
`5.2
`
`1
`1
`4
`8
`9
`15
`17
`19
`21
`23
`
`27
`27
`28
`32
`33
`33
`35
`41
`44
`46
`
`48
`48
`50
`54
`56
`57
`
`59
`59
`63
`67
`68
`69
`70
`71
`75
`78
`81
`83
`84
`
`86
`86
`87
`
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`6
`
`7
`
`8
`
`9
`
`5.3
`5.4
`5.5
`5.6
`5.7
`
`The light escape cone
`Radiation pattern
`The lambertian emission pattern
`Epoxy encapsulants
`Temperature dependence of emission intensity
`References
`
`Junction and carrier temperatures
`Carrier temperature and high-energy slope of spectrum
`6.1
`Junction temperature and peak emission wavelength
`6.2
`Theory of temperature dependence of diode forward voltage
`6.3
`6.4 Measurement of junction temperature using forward voltage
`Constant-current and constant-voltage DC drive circuits
`6.5
`References
`
`High internal efficiency designs
`Double heterostructures
`7 .1
`Doping of active region
`7 .2
`p-n junction displacement
`7 .3
`Doping of the confinement regions
`7.4
`Non-radiative recombination
`7.5
`Lattice matching
`7.6
`References
`
`Design of current flow
`Current-spreading layer
`8.1
`Theory of current spreading
`8.2
`Current crowding in LEDs on insulating substrates
`8.3
`Lateral injection schemes
`8.4
`Current-blocking layers
`8.5
`References
`
`High extraction efficiency structures
`Absorption ofbelow-bandgap light in semiconductors
`9.1
`Double heterostructures
`9.2
`Shaping of LED dies
`9.3
`Textured semiconductor surfaces
`9.4
`Cross-shaped contacts and other contact geometries
`9.5
`Transparent substrate technology
`9.6
`Anti-reflection optical coatings
`9.7
`Flip-chip packaging
`9.8
`References
`
`10 Reflectors
`10.1 Metallic reflectors, reflective contacts, and transparent contacts
`10.2 Total internal reflectors
`10.3 Distributed Bragg reflectors
`10.4 Omnidirectional reflectors
`Specular and diffuse reflectors
`10.5
`References
`
`VI
`
`91
`93
`94
`97
`98
`100
`
`101
`101
`103
`104
`108
`110
`112
`
`113
`113
`116
`118
`119
`122
`123
`126
`
`127
`127
`133
`136
`140
`142
`143
`
`145
`145
`149
`150
`154
`156
`157
`159
`160
`161
`
`163
`164
`168
`170
`181
`184
`189
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`11 Packaging
`11.1 Low-power and high-power packages
`11.2 Protection against electrostatic discharge (ESD)
`11.3 Thermal resistance of packages
`11.4 Chemistry of encapsulants
`11.5 Advanced encapsulant structures
`References
`
`12 Visible-spectrum LEDs
`12.1 The GaAsP, GaP, GaAsP:N, and GaP:N material systems
`12.2 The AlGaAs/GaAs material system
`12.3 The AlGainP/GaAs material system
`12.4 The GainN material system
`12.5 General characteristics of high-brightness LEDs
`12.6 Optical characteristics of high-brightness LEDs
`12.7 Electrical characteristics of high-brightness LEDs
`References
`
`13 The AlGalnN material system and ultraviolet emitters
`13.1 The UV spectral range
`13.2 The AlGainN bandgap
`13.3
`Polarization effects in III-V nitrides
`13.4 Doping activation in III-V nitrides
`13.5 Dislocations in III-V nitrides
`13.6 UV devices emitting at wavelengths longer than 360 nm
`13.7 UV devices emitting at wavelengths shorter than 360 nm
`References
`
`14
`
`Spontaneous emission from resonant cavities
`14.1 Modification of spontaneous emission
`14.2 Fabry-Perot resonators
`14.3 Optical mode density in a one-dimensional resonator
`14.4 Spectral emission enhancement
`14.5
`Integrated emission enhancement
`14.6 Experimental emission enhancement and angular dependence
`References
`
`15 Resonant-cavity light-emitting diodes
`15.1
`Introduction and history
`15.2 RCLED design rules
`15.3 GainAs/GaAs RCLEDs emitting at 930 nm
`15.4 AlGainP/GaAs RCLEDs emitting at 650 nm
`15.5 Large-area photon recycling LEDs
`15.6 Thresholdless lasers
`15.7 Other RCLED devices
`15.8 Other novel confined-photon emitters
`References
`
`16 Human eye sensitivity and photometric qualities
`16.1 Light receptors of the human eye
`
`191
`191
`193
`195
`196
`198
`199
`
`201
`201
`206
`209
`211
`213
`216
`218
`220
`
`222
`222
`223
`224
`226
`227
`231
`233
`236
`
`239
`239
`241
`244
`248
`249
`251
`253
`
`255
`255
`256
`260
`265
`268
`270
`271
`272
`273
`
`275
`275
`
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`Petitioner Apple Inc. - Ex. 1035, p. vii
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`

`

`16.2 Basic radiometric and photometric units
`16.3 Eye sensitivity function
`16.4 Colors of near-monochromatic emitters
`16.5 Luminous efficacy and luminous efficiency
`16.6 Brightness and linearity of human vision
`16.7 Circadian rhythm and circadian sensitivity
`References
`Appendix 16.1 Photopic eye sensitivity function
`Appendix 16.2 Scotopic eye sensitivity function
`
`17 Colorimetry
`17 .1 Color-matching functions and chromaticity diagram
`17.2 Color purity
`17 .3 LEDs in the chromaticity diagram
`17.4 Relationship between chromaticity and color
`References
`Appendix 17 .1 Color-matching functions (CIE 1931)
`Appendix 17 .2 Color-matching functions (CIE 1978)
`
`18 Planckian sources and color temperature
`18.1 The solar spectrum
`18.2 The planckian spectrum
`18.3 Color temperature and correlated color temperature
`References
`Appendix 18.1 Planckian emitter
`
`19 Color mixing and color rendering
`19 .1 Additive color mixing
`19.2 Colorrendering
`19.3 Color-rendering index for planckian-locus illumination sources
`19.4 Color-rendering index for non-planckian-locus illumination sources
`References
`Appendix 19.1 Reflectivity of test-color samples
`Appendix 19.2 Reflectivity oftest-color samples
`
`20 White-light sources based on LEDs
`20.1 Generation of white light with LEDs
`20.2 Generation of white light by dichromatic sources
`20.3 Generation of white light by trichromatic sources
`20.4 Temperature dependence of trichromatic LED-based white-light source
`20.5 Generation of white light by tetrachromatic and pentachromatic sources
`References
`
`21 White-light sources based on wavelength converters
`21.1 Efficiency of wavelength-converter materials
`21.2 Wavelength-converter materials
`21.3 Phosphors
`21.4 White LEDs based on phosphor converters
`21.5 Spatial phosphor distributions
`21.6 UV-pumped phosphor-based white LEDs
`
`viii
`
`277
`280
`283
`284
`286
`287
`289
`290
`291
`
`292
`292
`300
`301
`302
`302
`304
`305
`
`306
`306
`307
`309
`311
`312
`
`313
`313
`315
`323
`324
`327
`328
`330
`
`332
`332
`333
`338
`340
`344
`344
`
`346
`347
`349
`3 51
`353
`355
`357
`
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`

`

`11
`
`21.7
`21.8
`21.9
`21.10
`21.11
`
`White LEDs based on semiconductor converters (PRS-LED)
`Calculation of the power ratio of PRS-LED
`Calculation of the luminous efficiency of PRS-LED
`Spectrum of PRS-LED
`White LEDs based on dye converters
`References
`
`22 Optical communication
`22.1 Types ofoptical fibers
`22.2 Attenuation in silica and plastic optical fibers
`22.3 Modal dispersion in fibers
`22.4 Material dispersion in fibers
`22.5 Numerical aperture of fibers
`22.6 Coupling with lenses
`22.7 Free-space optical communication
`References
`
`23 Communication LEDs
`23.1 LEDs for free-space communication
`23.2 LEDs for fiber-optic communication
`23.3
`Surface-emitting Burrus-type communication LEDs emitting at 870 nm
`23.4 Surface-emitting communication LEDs emitting at 1300 nm
`23.5 Communication LEDs emitting at 650 nm
`23.6 Edge-emitting superluminescent diodes (SLDs)
`References
`
`24 LED modulation characteristics
`24.1 Rise and fall times, 3 dB frequency, and bandwidth in linear circuit theory
`24.2 Rise and fall time in the limit oflarge diode capacitance
`24.3 Rise and fall time in the limit of small diode capacitance
`24.4 Voltage dependence of the rise and fall times
`24.5 Carrier sweep-out of the active region
`24.6 Current shaping
`24.7
`3 dB frequency
`24.8 Eye diagram
`24.9 Carrier lifetime and 3 dB frequency
`References
`
`Appendix I
`Appendix2
`Appendix 3
`Appendix4
`Appendix 5
`Appendix 6
`Appendix 7
`Appendix 8
`
`Index
`
`Frequently used symbols
`Physical constants
`Room temperature properties oflII-V arsenides
`Room temperature properties of III-V nitrides
`Room temperature properties ofIII-V phosphides
`Room temperature properties of Si and Ge
`Periodic system of elements (basic version)
`Periodic system of elements ( detailed version)
`
`358
`359
`361
`363
`364
`364
`
`367
`367
`369
`371
`372
`374
`376
`379
`381
`
`382
`382
`382
`383
`384
`386
`388
`391
`
`393
`393
`395
`396
`397
`399
`400
`401
`401
`402
`403
`
`404
`408
`409
`410
`411
`412
`413
`414
`
`415
`
`IX
`
`Petitioner Apple Inc. - Ex. 1035, p. ix
`
`

`

`Preface
`During the last four decades, technical progress in the field of light-emitting diodes (LEDs) has
`been breathtaking. State-of-the art LEDs are small, rugged, reliable, bright, and efficient. At this
`time, the success story of LEDs still is in full progress. Great technological advances are
`continuously being made and, as a result, LEDs play an increasingly important role in a myriad
`of applications. In contrast to many other light sources, LEDs have the potential of converting
`electricity to light with near-unit efficiency.
`LEDs were discovered by accident in 1907 and the first paper on LEDs was published in the
`same year. LEDs became forgotten only to be re-discovered in the 1920s and again in the 1950s.
`In the 1960s, three research groups, one working at General Electric Corporation, one at MIT
`Lincoln Laboratories, and one at IBM Corporation, pursued the demonstration of the
`semiconductor laser. The first viable LEDs were by-products in this pursuit. LEDs have become
`devices in their own right and today possibly are the most versatile light sources available to
`humankind.
`The first edition of this book was published in 2003. The second edition of the book is
`expanded by the discussion of additional technical areas related to LEDs including optical
`reflectors, the assessment of LED junction temperature, packaging, UV emitters, and LEDs used
`for general lighting applications. No different than the first edition, the second edition is
`dedicated to the technology and physics of LEDs. It reviews the electrical and optical
`fundamentals of LEDs, materials issues, as well as advanced device structures. Recent
`developments, particularly in the field of III-V nitrides, are also discussed. The book mostly
`discusses LEDs made from III-V semiconductors. However, much of the science and technology
`discussed is relevant to other solid-state light emitters such as group-IV, II-VI, and organic
`emitters. Several application areas of LEDs are discussed in detail, including illumination and
`communication applications.
`Many colleagues and collaborators have provided information not readily available and have
`given valuable suggestions on the first and second editions of this book. In particular, I am
`deeply grateful to Enrico Bellotti (Boston University), Jaehee Cho (Samsung Advanced Institute
`of Technology), George Craford (LumiLeds Corp.), Thomas Gessmann (RPI), Nick Holonyak Jr.
`(University of Illinois), Jong Kyu Kim (RPI), Mike Krames (LumiLeds Corp.), Shawn Lin (RPI),
`Ralph Logan (retired, formerly with AT&T Bell Laboratories), Fred Long (Rutgers University),
`Paul Maruska (Crystal Photonics Corp.), Gerd Mueller (LumiLeds Corp.), Shuji Nakamura
`(University of California, Santa Barbara), N. Narendran (RPI), Yoshihiro Ohno (National
`Institute of Standards and Technology), Jacques Pankove (Astralux Corp.), Yongjo Park
`(Samsung Advanced Institute of Technology), Manfred Pilkuhn (retired, University of Stuttgart,
`Germany), Hans Rupprecht (retired, formerly with IBM Corp.), Michael Shur (RPI), Cheolsoo
`(Osram Opto
`Institute of Technology), Klaus Streubel
`(Samsung Advanced
`Sone
`Semiconductors Corp., Germany), Li-Wei Tu (National Sun Yat-Sen University, Taiwan),
`Christian Wetzel (RPI), Jerry Woodall (Yale University), and Walter Yao (Advanced Micro
`Devices Corp.). I would also like to thank my current and former post-doctoral fellows and
`students for their many significant contributions to this book.
`
`X
`
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`Petitioner Apple Inc. - Ex. 1035, p. x
`
`

`

`5
`
`LED basics: Optical properties
`
`5.1 Internal, extraction, external, and power efficiencies
`The active region of an ideal LED emits one photon for every electron injected. Each charge
`quantum-particle (electron) produces one light quantum-particle (photon). Thus the ideal active
`region of an LED has a quantum efficiency of unity. The internal quantum efficiency is defined
`as
`
`llint
`
`number of photons emitted from active region per second
`number of electrons injected into LED per second
`
`Pint /(hv)
`Ile
`
`(5.1)
`
`where Pint is the optical power emitted from the active region and I is the injection current.
`Photons emitted by the active region should escape from the LED die. In an ideal LED, all
`photons emitted by the active region are also emitted into free space. Such an LED has unity
`extraction efficiency. However, in a real LED, not all the power emitted from the active region is
`emitted into free space. Some photons may never leave the semiconductor die. This is due to
`several possible loss mechanisms. For example, light emitted by the active region can be
`reabsorbed in the substrate of the LED, assuming that the substrate is absorbing at the emission
`wavelength. Light may be incident on a metallic contact surface and be absorbed by the metal. In
`addition, the phenomenon of total internal reflection, also referred to as the trapped light
`phenomenon, reduces the ability of the light to escape from the semiconductor. The light
`extraction efficiency is defined as
`
`number of photons emitted into free space per second
`llextraction = -----~--------~-~----- =
`number of photons emitted from active region per second
`
`P!(hv)
`Pint I (hv)
`
`(5.2)
`
`where P is the optical power emitted into free space.
`The extraction efficiency can be a severe limitation for high-performance LEDs. It is quite
`difficult to increase the extraction efficiency beyond 50% without resorting to highly
`
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`

`

`5.2 Emission spectrum
`
`sophisticated and costly device processes.
`
`The external quantum efficiency is defined as
`
`_ number of photons emitted into free space per second _ PI ( h v) _
`.
`.
`llext -
`. •
`•
`-
`- 11mt 1lextractJon ·
`I I e
`number of electrons mJected mto LED per second
`
`(
`
`)
`
`5 3
`·
`
`The external quantum efficiency gives the ratio of the number of useful light particles to the
`
`number of injected charge particles.
`
`The power efficiency is defined as
`
`1lpower =
`
`p
`IV
`
`(5.4)
`
`where IV is the electrical power provided to the LED. Informally, the power efficiency is also
`
`called the wallplug efficiency.
`
`Exercise: LED efficiency. Consider an LED with a threshold voltage of Vih = Egle = 2.0 V with a
`differential resistance of R, = 20 n, so that the 1-V characteristic in the forward direction is given by
`V = Vih + IR,. When the device is operated at 20 mA it emits a light power of 4 mW of energy h v = Eg,
`Determine the internal quantum efficiency, the external quantum efficiency, and the power efficiency,
`assuming that the extraction efficiency is 50%.
`
`5.2 Emission spectrum
`
`The physical mechanism by which semiconductor LEDs emit light is spontaneous recombination
`
`of electron-hole pairs and simultaneous emission of photons. The spontaneous emission process
`
`is fundamentally different from the stimulated emission process occurring in semiconductor
`
`lasers and superluminescent LEDs. Spontaneous recombination has certain characteristics that
`
`determine the optical properties of LEDs. The properties of spontaneous emission in LEDs will
`
`be discussed in this section.
`
`An electron-hole recombination process is illustrated schematically in Fig. 5.1. Electrons in
`
`the conduction band and holes in the valence band are assumed to have the parabolic dispersion
`relations
`
`and
`
`(for electrons)
`
`(5.5)
`
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`

`

`5 LED basics: optical properties
`
`E
`
`(for holes)
`
`(5.6)
`
`where me* and mh* are the electron and hole effective masses, ti is Planck's constant divided by
`2n, k is the carrier wave number, and Ev and Ee are the valence and conduction band edges,
`respectively.
`
`2 2
`E = Ee +ti k /(2111/)
`
`e
`
`Fig. 5.1. Parabolic electron and
`hole dispersion relations showing
`"vertical" electron-hole recom(cid:173)
`bination and photon emission.
`
`Wave vector k
`
`The requirement of energy and momentum conservation leads to further insight into the
`radiative recombination mechanism. It follows from the Boltzmann distribution that electrons
`and holes have an average kinetic energy of kT. Energy conservation requires that the photon
`energy is given by the difference between the electron energy, Ee, and the hole energy, Eh, i.e.
`
`(5.7)
`
`The photon energy is approximately equal to the bandgap energy, Eg, if the thermal energy is
`small compared with the bandgap energy kT << Eg, Thus the desired emission wavelength of an
`LED can be attained by choosing a semiconductor material with an appropriate bandgap energy.
`For example, GaAs has a bandgap energy of 1.42 eV at room temperature and thus GaAs LEDs
`emit at the infrared wavelength of 870 nm.
`It is helpful to compare the average carrier momentum with the photon momentum. A carrier
`with kinetic energy kT and effective mass m* has the momentum
`
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`

`

`5.2 Emission spectrum
`
`p
`
`m*v
`
`✓ 2m*½m*v
`
`2
`
`.J 2m*kT
`
`The momentum of a photon with energy Eg can be derived from the de Broglie relation
`
`p
`
`nk
`
`hv
`C
`
`C
`
`(5.8)
`
`(5.9)
`
`Calculation of the carrier momentum (using Eq. 5.8) and the photon momentum (using Eq. 5.9)
`yields that the carrier momentum is orders of magnitude larger than the photon momentum.
`Therefore the electron momentum cannot change significantly during the transition from the
`conduction to the valence band. The transitions are therefore "vertical" as shown in Fig. 5.1, i.e.
`electrons only recombine with holes that have the same momentum or k value.
`Using the requirement that electron and hole momenta are the same, the photon energy can
`be written as the joint dispersion relation
`
`li2 k2
`li2 k2
`hv = Ee + --*- - Ev + --*-
`2me
`2mh
`
`where m, * is the reduced mass given by
`
`1
`1
`= -*-+-*-.
`me
`~
`
`(5.10)
`
`(5.11)
`
`Using the joint dispersion relation, the joint density of states can be calculated and one obtains
`* )3/2
`
`p(E) = ~ 2
`~r
`Ii
`21t
`
`[
`
`✓E-Eg .
`
`(5.12)
`
`The distribution of carriers in the allowed bands is given by the Boltzmann distribution, i.e.
`
`fB(E) = e -E/(kT) .
`
`(5.13)
`
`The emission
`
`intensity as a function of energy
`
`is proportional
`
`to
`
`the product of
`
`Eqs. (5.12) and (5.13),
`
`J(E) oc ✓E - Eg e -El(kT)
`
`1 ·
`
`(5.14)
`
`89
`
`Petitioner Apple Inc. - Ex. 1035, p. 89
`
`

`

`5 LED basics: optical properties
`
`The lineshape ofan LED, as given by Eq. (5.14), is shown in Fig. 5.2. The maximum emission
`intensity occurs at
`
`(5.15)
`
`Boltzmann
`distribution
`
`\
`
`.
`
`/ \
`, ~<
`t oc exp (-ElkD
`.I <lJ u c:: .,
`
`...,
`
`u
`~ c::
`-~
`-l
`
`, ; , , ;~
`Density of states
`oc (E-E/ 12
`
`Fig. 5.2. Theoretical emis(cid:173)
`sion spectrum of an LED.
`The full-width at half(cid:173)
`maximum (FWHM) of the
`emission line is 1. 8kT.
`
`Theoretical
`emission spectrum
`
`Eg + kT/2
`
`Energy E
`
`The full-width at half-maximum of the emission is
`
`l.8kT
`
`or
`
`(5.16)
`
`For example, the theoretical room-temperature linewidth of a GaAs LED emitting at 870 nm is
`till = 46 me V or ti'A, = 28 nm.
`
`The spectral linewidth of LED em1ss1on is important in several respects. Firstly, the
`linewidth of an LED emitting in the visible range is relatively narrow compared with the range of
`the entire visible spectrum. The LED emission is even narrower than the spectral width of a
`single color as perceived by the human eye. For example, red colors range in wavelength from
`625 to 730 nm, which is much wider than the typical emission spectrum of an LED. Therefore,
`LED emission is perceived by the human eye as monochromatic.
`Secondly, optical fibers are dispersive, which leads to a range of propagation velocities for a
`light pulse consisting of a range of wavelengths. The material dispersion in optical fibers limits
`the "bit rate x distance product" achievable with LEDs.
`The spontaneous lifetime of carriers in LEDs in direct-gap semiconductors is of the order of
`1-100 ns depending on the active region doping concentration ( or carrier concentrations) and the
`
`90
`
`Petitioner Apple Inc. - Ex. 1035, p. 90
`
`

`

`5.3 The light escape cone
`
`material quality. Thus, modulation speeds up to 1 Gbit/s are attainable with LEDs.
`
`5.3 The light escape cone
`Light generated inside a semiconductor cannot escape from the semiconductor if it is totally
`internally reflected at the semiconductor-air interface. If the angle of incidence of a light ray is
`close to normal incidence, light can escape from the semiconductor. However, total internal
`reflection occurs for light rays with oblique and grazing-angle incidence. Total internal reflection
`reduces the external efficiency significantly, in particular for LEDs consisting ofhigh-refractive(cid:173)
`index materials.
`Assume that the angle of incidence in the semiconductor at the semiconductor-air interface is
`given by <p. Then the angle of incidence of the refracted ray, ct>, can be inferred from Snell's law
`
`n8 sin <I> = riair sin Cl>
`
`(5.17)
`
`where n s and n air are the refractive indices of the semiconductor and air, respectively. The
`critical angle for total internal reflection is obtained using ct>= 90°, as illustrated in Fig. 5.3 (a).
`Using Snell's law, one obtains
`
`and
`
`sin <l>c
`
`n·
`_jfil_ sin 90°
`ris
`
`<l>c
`
`n·
`arcsin _Jfil_ •
`ris
`
`(5.18a)
`
`(5.18b)
`
`The refractive indices of semiconductors are usually quite high. For example, GaAs has a
`refractive index of 3.4. Thus, according to Eq. (5.18), the critical angle for total internal
`reflection is quite small. In this case, we can use the approximation sin <l>c "" <l>c• The critical angle
`for total internal reflection is then given by
`
`(5.19)
`
`The angle of total internal reflection defines the light escape cone. Light emitted into the cone
`can escape from the semiconductor, whereas light emitted outside the cone is subject to total
`internal reflection.
`
`91
`
`Petitioner Apple Inc. - Ex. 1035, p. 91
`
`

`

`5 LED basics: optical properties
`
`Next, we calculate the surface area of the spherical cone with radius r in order to determine
`the total fraction of light that is emitted into the light escape cone. The surface area of the
`calotte-shaped surface shown in Figs. 5.3 (b) and (c) is given by the integral
`A = f dA =
`
`(5.20)
`
`Let us assume that light is emitted from a point-like source in the semiconductor with a total
`power of Psource• Then the power that can escape from the semiconductor is given by
`
`Pescape
`
`Psource
`
`2n r 2 (1- cos ~c)
`7t r2
`4
`
`(5.21)
`
`where 4nr2 is the entire surface area of the sphere with radius r.
`
`(a)
`
`(b)
`
`(c)
`
`Fig. 5.3. (a) Definition of the escape cone by the critical angle <Pc, (b) Area element dA.
`( c) Area of calotte-shaped section of the sphere defined by radius r and angle <l>c.
`
`The calculation indicates that only a fraction of the light emitted inside a semiconductor can
`escape from the semiconductor. This fraction is given by
`
`Pescape
`
`Psource
`
`__!_(1
`2
`
`"')
`- COS'l'c
`
`(5.22)
`
`Because the critical angle of total internal reflection for high-index materials is relatively small,
`the cosine term can be expanded into a power series. Neglecting higher-than-second-order terms
`yields
`
`92
`
`Petitioner Apple Inc. - Ex. 1035, p. 92
`
`

`

`Pescape
`
`Psource
`
`_!_ <1>2
`4 C •
`
`Using the approximation ofEq. (5.19), one obtains
`
`P,
`escape
`Fsource
`
`1 -2
`,::; _ nair
`2
`4 n8
`
`5.4 Radiation pattern
`
`(5.23)
`
`(5.24)
`
`The escape problem is a significant problem for high-efficiency LEDs. In most semiconductors,
`the refractive index is quite high ( > 2.5 ) and thus only a small percentage of the light generated
`in the semiconductor can escape from a planar LED. The problem is less significant in
`semiconductors with a small refractive index and for polymers, which have refractive indices of
`the order of 1.5.
`
`Exercise: Light escape from planar GaAs, GaN, and polymer LED structures. The refractive indices of
`GaAs, GaN, and light-emitting polymers are 3.4, 2.5, and 1.5, respectively. Calculate the critical angle of
`total internal reflection for GaAs, GaN, and for polymers. Also calculate the fraction of light power that
`can escape from a planar GaAs and GaN semiconductor structures and a polymer LED structure.
`What improvement can be attained if a planar GaAs LED is encapsulated in a transparent polymer of
`refractive index 1.5, if the reflection at the polymer-air interface is neglected?
`Solution:
`Critical angle for total internal reflection:
`<l>c = 17.1 °
`<l>c = 23.6°
`GaAs
`GaN
`Fraction of light that can escape:
`GaAs 2.21%
`GaN 4.18%
`Polymer 12.7%.
`Improvement of the GaAs planar LED due to polymer encapsulation: 232%.
`
`Polymer <l>c = 41.8°.
`
`5.4 Radiation pattern
`All LEDs have a certain radiation pattern or far-field pattern. The intensity, measured in
`W/cm2
`, depends on the longitudinal and azimuth angle and the distance from the LED. The total
`optical power emitted by the LED is obtained by integration over the area of a sphere.
`p = L I,,, l(A) dA dA
`
`(5.25)
`
`where /(A) is the spectral light intensity (measured in W per nm per cm2
`) and A is the surface
`area of the sphere. The integration is carried out over the entire surface area.
`
`93
`
`Petitioner Apple Inc. - Ex. 1035, p. 93
`
`

`

`5 LED basics: optical properties
`
`5.5 The lambertian emission pattern
`The index contrast between the light-emitting material and the surrounding material leads to a
`non-isotropic emission pattern. For high-index light-emitting materials with a planar surface, a
`lambertian emission pattern is obtained. Figure 5.4 illustrates a point-like light source located a
`short distance below a semiconductor-air interface. Consider a light ray emitted from the source
`at an angle $ with respect to the surface normal. The light ray is refracted at the semiconductor(cid:173)
`air interface and the refracted light ray has an angle <l> with respect to the surface normal. The
`two angles are related by Snell's law, which, for small angles of$ (for which sin$~$), can be
`written as
`
`Light emitted into the angle d$ in the semiconductor is emitte