Marius Grundmann The Physics of Semiconductors

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1 Marius Grundmann The Physics of Semiconductors

2 Marius Grundmann The Physics of Semiconductors An Introduction Including Devices and Nanophysics With 587 Figures, 6 in Color, and 36 Tables 123

3 Marius Grundmann Institut für Experimentelle Physik II Universität Leipzig Linnéstraße Leipzig grundmann@physik.uni-leipzig.de Library of Congress Control Number: ISBN X Springer Berlin Heidelberg New York ISBN Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com Springer-Verlag Berlin Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Protago-TEX-Production GmbH, Berlin Production: LE-TEXJelonek,Schmidt&VöcklerGbR,Leipzig Cover design: estudio Calamar S.L., F. Steinen-Broo, Pau/Girona, Spain Printed on acid-free paper 57/3100/YL

4 To Michelle, Sophia Charlotte and Isabella Rose

5 Preface Semiconductor devices are nowadays commonplace in every household. In the late 1940s the invention of the transistor was the start of a rapid development towards ever faster and smaller electronic components. Complex systems are built with these components. The main driver of this development was the economical benefit from packing more and more wiring, transistors and functionality on a single chip. Now every human is left with about 100 million transistors (on average). Semiconductor devices have also enabled economically reasonable fiber-based optical communication, optical storage and highfrequency amplification and have only recently revolutionized photography, display technology and lighting. Along with these tremendous technological developments, semiconductors have changed the way we work, communicate, entertain and think. The technological sophistication of semiconductor materials and devices is progressing continuously with a large worldwide effort in human and monetary capital, partly evolutionary, partly revolutionary embracing the possibilities of nanotechnology. For students, semiconductors offer a rich, diverse and exciting field with a great tradition and a bright future. This book is based on the two semester semiconductor physics course taught at Universität Leipzig. The material gives the students an overview of the subject as a whole and brings them to the point where they can specialize and enter supervised laboratory research. For the interested reader some additional topics are included in the book that are taught in subsequent, more specialized courses. The first semester contains the fundamentals of semiconductor physics (Part I Chaps. 1 17). Besides important aspects of solid-state physics such as crystal structure, lattice vibrations and band structure, semiconductor specifics such as technologically relevant materials and their properties, electronic defects, recombination, hetero- and nanostructures are discussed. Semiconductors with electric polarization and magnetization are introduced. The emphasis is put on inorganic semiconductors, but a brief introduction to organic semiconductors is given in Chap. 16. In Chap. 17 dielectric structures are treated. Such structures can serve as mirrors, cavities and microcavities and are a vital part of many semiconductor devices. The second part (Part II Chaps ) is dedicated to semiconductor applications and devices that are taught in the second semester of the

6 VIII Preface course. After a general and detailed discussion of various diode types, their applications in electrical circuits, photodetectors, solar cells, light-emitting diodes and lasers are treated. Finally, bipolar and field-effect transistors are discussed. The course is designed to provide a balance between aspects of solid-state and semiconductor physics and the concepts of various semiconductor devices and their applications in electronic and photonic devices. The book can be followed with little or no pre-existing knowledge in solid-state physics. I would like to thank several colleagues for their various contributions to this book, in alphabetical order (if no affiliation is given, from Universität Leipzig): Klaus Bente, Rolf Böttcher, Volker Gottschalch, Axel Hoffmann (Technische Universität Berlin), Alois Krost (Otto-von-Guericke Universität Magdeburg), Michael Lorenz, Thomas Nobis, Rainer Pickenhain, Hans- Joachim Queisser (Max-Planck-Institut für Festkörperforschung, Stuttgart), Bernd Rauschenbach (Leibniz-Institut für Oberflächenmodifizierung, Leipzig), Bernd Rheinländer, Heidemarie Schmidt, Rüdiger Schmidt-Grund, Mathias Schubert, Gerald Wagner, Holger von Wenckstern, Michael Ziese, and Gregor Zimmermann. Their comments, proof reading and graphic material improved this work. Also, numerous helpful comments from my students on my lectures and on preliminary versions of the present text are gratefully acknowledged. I am also indebted to many other colleagues, in particular to (in alphabetical order) Gerhard Abstreiter, Zhores Alferov, Levon Asryan, Günther Bauer, Manfred Bayer, Immanuel Broser, Jürgen Christen, Laurence Eaves, Ulrich Gösele, Alfred Forchel, Manus Hayne, Frank Heinrichsdorff, Fritz Henneberger, Detlev Heitmann, Robert Heitz, Nils Kirstaedter, Fred Koch, Nikolai Ledentsov, Evgeni Kaidashev, Eli Kapon, Claus Klingshirn, Jörg Kotthaus, Axel Lorke, Anupam Madhukar, Bruno Meyer, David Mowbray, Hisao Nakashima, Mats-Erik Pistol, Fred Pollak, Volker Riede, Hiroyuki Sakaki, Lars Samuelson, Vitali Shchukin, Maurice Skolnick, Oliver Stier, Robert Suris, Volker Türck, Konrad Unger, Victor Ustinov, Leonid Vorob jev, Richard Warburton, Alexander Weber, Eicke Weber, Peter Werner, Ulrike Woggon, Roland Zimmermann and Alex Zunger, with whom I have worked closely, had enjoyable discussions with and who have posed questions that stimulated me. I reserve special thanks for Dieter Bimberg, who supported me throughout my career. I leave an extra niche as the Romans did, in order not to provoke the anger of a God missed in a row of statues for those who had an impact on my scientific life and that I have omitted to mention. Leipzig, January 2006 Marius Grundmann

7 Contents Abbreviations...XXI Symbols... XXVII Physical Constants... XXXI 1 Introduction Timetable NobelPrizeWinners GeneralInformation... 9 Part I Fundamentals 2 Bonds Introduction CovalentBonds Electron-PairBond sp 3 Bond sp 2 Bond IonicBonds MixedBond Metallic Bond van-der-waalsbond HamiltonOperatoroftheSolid Crystals Introduction CrystalStructure Lattice UnitCell PointGroup SpaceGroup DBravaisLattices DBravaisLattices Polycrystalline Semiconductors... 39

8 X Contents Amorphous Semiconductors ImportantCrystalStructures RocksaltStructure CsClStructure DiamondStructure ZincblendeStructure WurtziteStructure ChalcopyriteStructure DelafossiteStructure PerovskiteStructure NiAsStructure Polytypism ReciprocalLattice ReciprocalLatticeVectors Miller Indices Brillouin Zone Alloys RandomAlloys PhaseDiagram VirtualCrystalApproximation LatticeParameter Ordering Defects Introduction PointDefects ThermodynamicsofDefects Dislocations StackingFaults Grain Boundaries AntiphaseandInversionDomains Disorder Mechanical Properties Introduction LatticeVibrations MonoatomicLinearChain DiatomicLinearChain Lattice Vibrations of a Three-Dimensional Crystal Phonons LocalizedVibrationalModes PhononsinAlloys ElectricFieldCreatedbyOpticalPhonons Elasticity Stress Strain Relation... 94

9 Contents XI BiaxialStrain Three-DimensionalStrain Substrate Bending Scrolling CriticalThickness Cleaving Band Structure Introduction Bloch stheorem Free-ElectronDispersion Kronig PenneyModel ElectronsinaPeriodicPotential Approximate Solution at the Zone Boundary Solution in the Vicinity of the Zone Boundary Kramer sdegeneracy Band Structure of Selected Semiconductors Silicon Germanium GaAs GaP GaN LeadSalts Chalcopyrites Delafossites Perovskites Alloy Semiconductors Amorphous Semiconductors Systematics of Semiconductor Bandgaps TemperatureDependenceoftheBandgap EquationofElectronMotion ElectronMass EffectiveMass PolaronMass NonparabolicityofElectronMass Holes HoleConcept HoleDispersionRelation Valence-BandFineStructure StrainEffectontheBandStructure StraineffectonBandEdges Strain Effect on Effective Masses DensityofStates GeneralBandStructure Free-ElectronGas

10 XII Contents 7 Electronic Defect States Introduction FermiDistribution CarrierConcentration Intrinsic Conduction ShallowImpurities,Doping Donors Acceptors Compensation AmphotericImpurities HighDoping Quasi-FermiLevels DeepLevels ChargeStates Jahn TellerEffect Negative-U Center DXCenter EL2Defect Semi-insulating Semiconductors SurfaceStates Hydrogen in Semiconductors Transport Introduction Conductivity Low-FieldTransport Mobility Microscopic Scattering Processes IonizedImpurityScattering DeformationPotentialScattering PiezoelectricPotentialScattering PolarOpticalScattering TemperatureDependence HallEffect High-FieldTransport Drift-SaturationVelocity VelocityOvershoot ImpactIonization High-FrequencyTransport Diffusion ContinuityEquation Heat Conduction CoupledHeatandChargeTransport SeebeckEffect PeltierEffect

11 Contents XIII 9 Optical Properties SpectralRegionsandOverview ReflectionandDiffraction Electron PhotonInteraction Band BandTransitions JointDensityofStates DirectTransitions IndirectTransitions UrbachTail Intravalence-BandAbsorption Amorphous Semiconductors Excitons ExcitonPolariton Bound-Exciton Absorption Biexcitons Trions Burstein MossShift BandgapRenormalization Electron Hole Droplets Two-PhotonAbsorption ImpurityAbsorption Free-CarrierAbsorption LatticeAbsorption DielectricConstant Reststrahlenbande Polaritons Phonon PlasmonCoupling Recombination Introduction Band BandRecombination SpontaneousEmission Absorption StimulatedEmission NetRecombinationRate RecombinationDynamics Lasing Free-ExcitonRecombination Bound-Exciton Recombination AlloyBroadening PhononReplica Donor Acceptor Pair Transitions Inner-ImpurityRecombination AugerRecombination Band ImpurityRecombination

12 XIV Contents FieldEffect ThermallyActivatedEmission Direct Tunneling Assisted Tunneling MultilevelTraps SurfaceRecombination Excess-Carrier Profiles Heterostructures Introduction GrowthMethods MaterialCombinations PseudomorphicStructures Heterosubstrates BandLineupinHeterostructures EnergyLevelsinHeterostructures QuantumWell Superlattices Single Heterointerface Between Doped Materials RecombinationinQuantumWells IsotopeSuperlattices WaferBonding External Fields ElectricFields BulkMaterial QuantumWells MagneticFields Free-CarrierAbsorption EnergyLevelsinBulkCrystals EnergyLevelsina2DEG Shubnikov de Haas Oscillations QuantumHallEffect IntegralQHE FractionalQHE Weiss Oscillations Nanostructures Introduction QuantumWires PreparationMethods Quantization in Two-Dimensional Potential Wells QuantumDots Quantization in Three-Dimensional PotentialWells

13 Contents XV ElectricalandTransportProperties Self-Assembled Preparation OpticalProperties Polarized Semiconductors Introduction SpontaneousPolarization Ferroelectricity Materials SoftPhononMode PhaseTransition Domains OpticalProperties Piezoelectricity Magnetic Semiconductors Introduction Magnetic Semiconductors Diluted Magnetic Semiconductors Spintronics SpinTransistor SpinLED Organic Semiconductors Materials Properties Dielectric Structures Photonic-BandgapMaterials Introduction General1DScatteringTheory Transmission of an N-PeriodPotential TheQuarter-WaveStack Formationofa3DBandStructure DefectModes CouplingtoanElectronicResonance MicroscopicResonators Microdiscs PurcellEffect DeformedResonators Hexagonal Cavities

14 XVI Contents Part II Applications 18 Diodes Introduction Metal Semiconductor Contacts Band Diagram in Equilibrium Space-ChargeRegion SchottkyEffect Capacitance Current VoltageCharacteristic OhmicContacts Metal Contacts to Organic Semiconductors Metal Insulator Semiconductor Diodes BandDiagramforIdealMISDiode Space-ChargeRegion Capacity NonidealMISDiode BipolarDiodes BandDiagram Space-ChargeRegion Capacitance Current VoltageCharacteristics Breakdown ApplicationsandSpecialDiodeDevices Rectification Frequency Mixing VoltageRegulator ZenerDiodes Varactors Fast-RecoveryDiodes Step-RecoveryDiodes pin-diodes Tunneling Diodes BackwardDiodes HeterostructureDiodes Light-to-Electricity Conversion Photocatalysis Photoconductors Introduction Photoconductivity Detectors Electrophotography QWIPs BlockedImpurity-BandDetectors

15 Contents XVII 19.3 Photodiodes Introduction pnphotodiodes pinphotodiodes Position-SensingDetector MSMPhotodiodes AvalanchePhotodiodes Traveling-WavePhotodetectors ChargeCoupledDevices PhotodiodeArrays SolarCells SolarRadiation IdealSolarCells RealSolarCells DesignRefinements Solar-CellTypes CommercialIssues Electricity-to-Light Conversion RadiometricandPhotometricQuantities RadiometricQuantities PhotometricQuantities Scintillators CIEChromaticityDiagram DisplayApplications RadiationDetection Luminescence Mechanisms Light-EmittingDiodes Introduction SpectralRanges QuantumEfficiency DeviceDesign Lasers Introduction Applications Gain OpticalMode LossMechanisms Threshold SpontaneousEmissionFactor OutputPower TemperatureDependence ModeSpectrum LongitudinalSingle-ModeLasers Tunability

16 XVIII Contents Modulation Surface-emittingLasers Optically Pumped Semiconductor Lasers QuantumCascadeLasers Hot-HoleLasers Semiconductor Optical Amplifiers Transistors Introduction BipolarTransistors CarrierDensityandCurrents CurrentAmplification Ebers MollModel Current VoltageCharacteristics BasicCircuits High-FrequencyProperties HeterobipolarTransistors Light-emittingTransistors Field-EffectTransistors JFETandMESFET GeneralPrinciple StaticCharacteristics NormallyOnandNormallyOffFETs Field-Dependent Mobility High-FrequencyProperties MOSFETs OperationPrinciple Current VoltageCharacteristics MOSFETTypes ComplementaryMOS Large-ScaleIntegration NonvolatileMemories Heterojunction FETs Thin-Film Transistors Part III Appendices A Tensors B Kramers Kronig Relations C Oscillator Strength D Quantum Statistics

17 Contents XIX E Thek p Perturbation Theory F Effective-Mass Theory References Index

18 Abbreviations 2DEG AAAS AB ac AFM AIP AM APD APD APS AR ASE AVS BC bcc BEC BGR CAS CCD CD CEO CIE CIS CL CMOS CMY COD CPU CRT CVD two-dimensional electron gas American Association for the Advancement of Science antibonding (position) alternating current atomic force microscopy American Institute of Physics air mass antiphase domain avalanche photodiode American Physical Society antireflection amplified spontaneous emission American Vacuum Society (The Science & Technology Society) bond center (position) body-centered cubic Bose Einstein condensation bandgap renormalization calorimetric absorption spectroscopy charge coupled device compact disc cleaved-edge overgrowth Commission Internationale de l Éclairage CuInSe 2 material cathodoluminescence complementary metal oxide semiconductor cyan-magenta-yellow (color system) catastrophical optical damage central processing unit cathode ray tube chemical vapor deposition

19 XXII cw CZ Abbreviations continuous wave Czochralski (growth) DAP DBR dc DFB DH(S) DLTS DMS DOS DPSS DRAM DVD EEPROM EHL EL ELO EMA EPROM ESF EXAFS fcc FeRAM FET FIR FKO FPA FQHE FWHM FZ GLAD GRINSCH GSMBE HBT hcp HCSEL HEMT HIGFET HJFET hh donor acceptor pair distributed Bragg reflector direct current distributed feedback double heterostructure deep level transient spectroscopy diluted magnetic semiconductor density of states diode-pumped solid-state (laser) dynamic random access memory digital versatile disc electrically erasable programmable read-only memory electron hole liquid electroluminescence epitaxial lateral overgrowth effective mass approximation erasable programmable read-only memory extrinsic stacking fault extended X-ray absorption fine structure face-centered cubic ferroelectric random access memory field-effect transistor far infrared Franz Keldysh oscillation focal plane array fractional quantum Hall effect full width at half-maximum float-zone (growth) glancing-angle deposition graded-index separate confinement heterostructure gas-source molecular beam epitaxy heterobipolar transistor hexagonally close packed horizontal cavity surface-emitting laser high electron mobility transistor heterojunction insulating gate FET heterojunction FET heavy hole

20 Abbreviations XXIII HOMO HR HRTEM HWHM IC IDB IF IPAP IQHE IR ISF ITO JDOS JFET KKR KTP LA LCD LDA LEC LED lh LO LPE LPCVD LPP LST LT LUMO LVM MBE MEMS MESFET MIGS MIOS MIR MIS MHEMT ML MMIC highest occupied molecular orbital high reflection high-resolution transmission electron microscopy half-width at half-maximum integrated circuit inversion domain boundary intermediate frequency Institute of Pure and Applied Physics, Tokyo integral quantum Hall effect infrared intrinsic stacking fault indium tin oxide joint density of states junction field-effect transistor Kramers Kronig relation KTiOPO 4 material longitudinal acoustic (phonon) liquid crystal display local density approximation liquid encapsulated Czochralski (growth) light-emitting diode light hole longitudinal optical (phonon), local oscillator liquid phase epitaxy low-pressure chemical vapor deposition longitudinal phonon plasmon (mode) Lyddane Sachs Teller (relation) low temperature lowest unoccupied molecular orbital local vibrational mode molecular beam epitaxy micro-electro-mechanical system metal semiconductor field-effect transistor midgap (surface) states metal insulator oxide semiconductor mid-infrared metal insulator semiconductor metamorphic HEMT monolayer millimeter-wave integrated circuit

21 XXIV Abbreviations MO MODFET MOMBE MOPA MOS MOSFET MOVPE MRS MS MSM MWQ NDR NEP NIR NMOS NTSC OPSL PA PBG pc PFM PHEMT PL PLD PLE PMMA PMOS PPC PPLN PV PWM PZT QCL QCSE QD QHE QW QWIP QWR master oscillator modulation-doped FET metal-organic molecular beam epitaxy master oscillator power amplifier metal oxide semiconductor metal oxide semiconductor field-effect transistor metal-organic chemical vapor deposition Material Research Society metal semiconductor (diode) metal semiconductor metal (diode) multiple quantum well negative differential resistance noise equivalent power near infrared n-channel metal oxide semiconductor (transistor) national television standard colors optically pumped semiconductor laser power amplifier photonic bandgap primitive cubic piezoresponse force microscopy pseudomorphic HEMT photoluminescence pulsed laser deposition photoluminescence excitation (spectroscopy) poly-methyl methacrylate p-channel metal oxide semiconductor (transistor) persistent photoconductivity perodically poled lithium niobate photovoltaic pulsewidth modulation PbTi x Zr 1 x O 3 material quantum cascade laser quantum confined Stark effect quantum dot quantum Hall effect quantum well quantum-well intersubband photodetector quantum wire

22 Abbreviations XXV RAM RAS REI RF RGB RHEED RKKY rms ROM SAGB SAM sc SCH SEL SEM SET SGDBR SHG si SIA SIMS SL s-o SOA SPD SPIE SPS srgb SRH SSR STM TA TCO TE TEGFET TEM TES TF TFT TM TO TOD TPA random access memory reflection anisotropy spectroscopy random element isodisplacement radio frequency red-green-blue (color system) reflection high-energy electron diffraction Ruderman Kittel Kasuya Yoshida (interaction) root mean square read-only memory small-angle grain boundary separate absorption and amplification (structure) simple cubic separate confinement heterostructure surface-emitting laser scanning electron microscopy single-electron transistor sampled grating distributed Bragg reflector second-harmonic generation semi-insulating Semiconductor Industry Association secondary ion mass spectroscopy superlattice spin-orbit (or split-off) semiconductor optical amplifier spectral power distribution International Society for Optical Engineering short-period superlattice standard RGB Shockley Read Hall (kinetics) side-mode suppression ratio scanning tunneling microscopy transverse acoustic (phonon) transparent conductive oxide transverse electric (polarization) two-dimensional electron gas FET transmission electron microscopy two-electron satellite thermionic field emission thin-film transistor transverse magnetic (polarization) transverse optical (phonon) turn-on delay (time) two-photon absorption

23 XXVI Abbreviations UHV UV ultrahigh vacuum ultraviolet VCA VCO VCSEL VFF VGF VIS VLSI WGM WKB WS XSTM virtual crystal approximation voltage-controlled oscillator vertical-cavity surface-emitting laser valence force field vertical gradient freeze (growth) visible very large scale integration whispering gallery mode Wentzel Kramer Brillouin (approximation or method) Wigner Seitz (cell) cross-sectional STM

24 Symbols α Madelung constant, disorder parameter, linewidth enhancement factor α(ω) absorption coefficient α m mirror loss α n electron ionization coefficient α p hole ionization coefficient β used as abbreviation for e/(k B T ), spontaneous emission coefficient γ, Γ broadening parameter γ 1, γ 2, γ 3 Luttinger parameter Δ 0 spin-orbit splitting ɛ(ω) dielectric function ɛ 0 permittivity of vacuum ɛ r relative dielectric function ɛ xy strain components η quantum efficiency η d differential quantum efficiency η w wall-plug efficiency Θ D Debye temperature κ imaginary part of index of refraction, heat conductivity λ wavelength μ mobility μ 0 magnetic susceptibility of vacuum μ h hole mobility μ n electron mobility ν frequency Π Peltier coefficient ρ mass density, charge density, resistivity σ standard deviation, conductivity σ n electron capture cross section σ p hole capture cross section σ P polarization charge σ xy stress components τ lifetime, time constant

25 XXVIII Symbols φ φ Bn χ χ(r) Ψ(r) ω Ω phase Schottky barrier height electron affinity, electric susceptibility envelope wavefunction wavefunction angular frequency interaction parameter a hydrostatic deformation potential a accelaration A area A, A vector potential A Richardson constant A effective Richardson constant a 0 (cubic) lattice constant b bowing parameter, deformation potential b Burger s vector B bimolecular recombination coefficient, bandwidth B, B magnetic field c velocity of light in vacuum, lattice constant (along c-axis) C capacity, spring constant C n, C p Auger recombination coefficient C ij elastic constants d distance, shear deformation potential D density of states, diffusion coefficient D, D displacement field D e (E) electron density of states D h (E) hole density of states D n electron diffusion coefficient D p hole diffusion coefficient e elementary charge E energy E, E, E electric field E A energy of acceptor level EA b acceptor ionization energy E C energy of conduction-band edge E D energy of donor level ED b donor ionization energy E F Fermi energy E Fn electron quasi-fermi energy E Fp hole quasi-fermi energy E g bandgap E P energy parameter E V energy of valence-band edge exciton energy E X

26 Symbols XXIX EX b exciton binding energy f oscillator strength F free energy F, F force F (M) excess noise factor f e Fermi Dirac distribution function f i ionicity F n electron quasi-fermi energy F p hole quasi-fermi energy g degeneracy, g-factor, gain G free enthalpy, generation rate G vector of reciprocal lattice g m transconductance h Planck constant H enthalpy H, H magnetic field H Hamiltonian h/(2π) i imaginary number I current I s saturation current j current density, orbital momentum j s saturation current density k, k B Boltzmann constant k wavevector k F Fermi wavevector l angular orbital momentum L length of line element L line vector (of dislocation) L D diffusion length L z quantum-well thickness m mass m effective mass M mass, multiplication factor M, M magnetization m e effective electron mass m h effective hole mass m j magnetic quantum number n electron concentration (in conduction band), ideality factor n normal vector N(E) number of states n complex index of refraction (= n r +iκ) N A acceptor concentration critical doping concentration N c

27 XXX Symbols N C N D n i n r n s N t n tr n thr N V conduction-band edge density of states donor concentration intrinsic electron concentration index of refraction (real part) sheet electron density trap concentration transparency electron concentration threshold electron concentration valence-band edge density of states p pressure, free hole density p, p momentum P power P, P electric polarization p cv momentum matrix element p i intrinsic hole concentration q charge q, q heat flow Q charge, quality factor r radius r spatial coordinate R resistance, radius, recombination rate R vector of direct lattice r H Hall factor R H Hall coefficient s spin S entropy, Seebeck coefficient, total spin S ij stiffness coefficients t time T temperature u displacement, cell-internal parameter u nk Bloch function U energy v, v velocity v s drift-saturation velocity V volume, voltage, potential V (λ) (standardized) sensitivity of human eye V a unit-cell volume v g group velocity v s velocity of sound v th thermal velocity w depletion-layer width W m work function X electronegativity Y Young s module, CIE brightness parameter Z partition sum, atomic order number

28 Physical Constants constant symbol numerical value unit speed of light in vacuum c ms 1 permeability of vacuum μ 0 4π 10 7 NA 2 permittivity of vacuum ɛ 0 =(μ 0 c 2 0) Fm ev K 1 elementary charge e C electron mass m e kg Planck constant h Js = h/(2π) Js ev s Boltzmann constant k B JK 1 von-klitzing constant R H Ω Rydberg constant ev Bohr radius a B m