Skip to main content
SpringerLink
Log in

Bands and Bandgaps in Solids

Semiconductor Physics

  • 818 Accesses

Abstract

Valence and conduction bands and the bandgap in between these bands determine the electronic properties of solids. For semiconductors, the band structure of the conduction band and the valence bands near the edge to the fundamental bandgap is of particular interest. Both the band structure and bandgap are influenced by external parameters such as temperature and pressure and can also be changed by alloying and heavy doping.

In low-dimensional semiconductors like superlattices and quantum wells, quantum wires and quantum dots anisotropic carrier confinement occurs. The effective gap and energies in conduction and valence bands can be varied by changing spatial dimensions and barrier height of the low-dimensional structure. The bands in amorphous semiconductors near the band edge are ill-defined since long-range periodicity is missing. Still the density-of-state distribution shows significant similarities to that of the same material in the crystallite state.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Notes

  1. 1.

    In a real crystal this number is reduced since electron scattering limits the coherence length of the electron wave (i.e., the length in which quantum-mechanical interaction can take place). With a mean free path λ the number of atoms responsible for the band-level splitting is of the order of (λ/a)3 for a primitive cubic lattice with lattice constant a. Each of these levels is broadened by collision broadening; therefore, even for λ approaching a, the result is still bands rather than discrete levels.

  2. 2.

    AB compounds containing these elements are referred to as pnictides or chalcogenides.

  3. 3.

    In a quantum-mechanical picture, any E(k) state represents a certain mass and velocity. In a filled band, all of them add up to zero.

  4. 4.

    A positive charge was arbitrarily related to the charge of a glass rod, rubbed with silk (by Benjamin Franklin); this charge was not caused by an added electron as it became known later, but by a missing electron on the glass rod. This electron was removed by the silk.

  5. 5.

    In this and all chapters dealing with electrical conductivity, the electric field is identified as F and the energy is identified as E.

  6. 6.

    The difference between metals, where the overlap range is allowed, and semiconductors, where the overlap range is forbidden (bandgap), depends on Wigner’s rules (Wigner 1959), which state that eigenstates belonging to different symmetry groups of the Hamiltonian cannot mix (metals). In semiconductors they do mix, yielding sp 3 hybridization for Si.

  7. 7.

    Each band contains a large number of energy levels (see Section 4.2 of chapter “Quantum Mechanics of Electrons in Crystals”). Increasing number of electrons first fills the levels at the lowest energy (for T = 0) and then successively higher and higher energies. This process is referred to as band filling.

  8. 8.

    There are also Dresselhaus parameters L, M, and N (Dresselhaus et al. 1955) to describe the valence band. They are related to the Luttinger parameters by

    $$ {\gamma}_1=-2{m}_0\left(L+2M\right)/3, \kern0.5em {\gamma}_2=-2{m}_0\left(L-M\right)/6, \kern0.5em {\gamma}_3=-2{m}_0N/6. $$
    (14)
  9. 9.

    Note that F represents the electric field and E an energy.

  10. 10.

    Here cyclotron resonance is discussed within the same band, and quantum effects are neglected. This can be justified when, neglecting scattering, each electron describes full circles which have to be integers of its de Broglie wavelength Eq. 6 of chapter “The Origin of Band Structure”. This integer represents the quantum number n q of the circle, and for the magnetic induction discussed here, it is a large number. Resonance means absorption (or emission) of one quantum ℏω c, hence changing n q by Δn q = ±1, which is the selection rule for cyclotron transitions. Since Δn q << n q , a change in circle diameter is negligible; hence, the classical approach is justified. At higher fields the circles become smaller, and when approaching atomic size, the quantum levels (Landau levels – see Section 2 of chapter “Carriers in Magnetic Fields and Temperature Gradients”) become wider spaced and a quantum-mechanical approach is required. For reviews, see Lax (1963), Mavroides (1972), and McCombe and Wagner (1975).

  11. 11.

    The circle diameter (πv n /ω c) is typically of the order of 10−3 cm for a magnetic induction of 1 T; here v n is the thermal velocity of an electron. In metals, however, one also has to consider the skin penetration of the probing electromagnetic field. The skin depth of a metal is usually a very small fraction of the circle diameter, so that the probing ac field can interact only at the very top part of each electron cycle close to the surface. This enhances information about near-surface behavior in metals, while in semiconductors, probing extends throughout the bulk.

  12. 12.

    The conventional term alloy of metals also encompasses crystallite mixtures of nonintersoluble metals, such as lead and tin (solder). Here, however, only materials within their solubility ranges are discussed. The Hume-Rothery rule identifies these metals as having similar binding character, similar valency, and similar atomic radii (Hume-Rothery 1936). Corresponding guidelines apply to the intersolubility of cations or anions in compounds.

  13. 13.

    This Debye–Waller factor (W) is related to the probability of phonon emission during electron or x-ray diffraction and is given in the Debye approximation by

    $$ p= \exp \left(-2W\right)= \exp \left(-\frac{6{E}_R}{k\varTheta}\left[\frac{1}{4}+\frac{T}{\varTheta }{\displaystyle {\int}_0^{\varTheta /T}\frac{x\kern0.5em dx}{ \exp (x)-1}}\right]\right), $$
    (41)

    where E R is the recoil energy Mv 2/2.

  14. 14.

    At room temperature this is λ = h/p = h/ \( \sqrt{2{m}^{*}E} \) ≅ 17 nm for 300 K thermal energies and a typical effective mass of 0.2 × m 0, i.e., in the 10 nm range. For excitons the relevant Bohr radius is also in this range, e.g., 11.5 nm for GaAs and 3.2 nm for GaN.

  15. 15.

    Quantum wells are fabricated from a layer sequence BAB grown using epitaxy (Section 3.3 of chapter “Properties and Growth of Semiconductors”); the growth direction is usually referred to as z direction, which we designate here as the direction where the confinement occurs.

  16. 16.

    The alternating potential shown in Fig. 30a is of type I, i.e., a minimum of E c(z) coincides with a maximum of E v(z). Both minima and maxima coincide in a type-II superlattice (Fig. 30b). An example for type II is the GaAs x Sbl-x superlattice. For x below 0.25, the valence band of GaAs extends above the conduction band of GaSb, resulting in quasimetallic behavior. For a review of type-II superlattices, see Voos and Esaki (1981).

  17. 17.

    This is a typical band offset at heterointerfaces; for the important GaAs/Ga x A1-x As interface, 62 % of the x-dependent bandgap discontinuity ΔE g form the discontinuity at the conduction-band edge ΔE c and 38 % at the valence-band edge ΔE v (Watanabe et al. 1985).

  18. 18.

    For the concept of polaron quasiparticles, see Section 1.2 of chapter “Carrier-Transport Equations”.

  19. 19.

    The intermolecular interaction is expressed in terms of a transfer integral (see the chapter on transport later in the book). In the tight-binding approximation of a one-dimensional molecule chain, the total bandwidth equals four times the transfer integral between neighboring molecules. The bandwidth for any molecular packing can be expressed from the amplitude of the transfer integrals between the various interacting units (Brédas et al. 2002).

References

  • Abram RA, Res GJ, Wilson BLH (1978) Heavily doped semiconductors and devices. Adv Phys 27:799

    ADS  Google Scholar 

  • Aggarwal RL, Zuteck MD, Lax B (1969) Magnetopiezotransmission studies of the indirect transition in germanium. Phys Rev 180:800

    ADS  Google Scholar 

  • Akiyama H (1998) One-dimensional excitons in GaAs quantum wires. J Phys Condens Matter 10:3095

    ADS  Google Scholar 

  • Ando T, Fowler AB, Stern F (1982) Electronic properties of two-dimensional systems. Rev Mod Phys 54:437

    ADS  Google Scholar 

  • Antončik E (1955) On the theory of temperature shift of the absorption curve in non-polar crystals. Czech J Phys 5:449

    ADS  Google Scholar 

  • Asada M, Miyamoto Y, Suematsu Y (1986) Gain and threshold of three-dimensional quantum-box lasers. IEEE J Quantum Electron QE 22:1915

    ADS  Google Scholar 

  • Bastard G, Brum JA (1986) Electronic states in semiconductor heterostructures. IEEE J Quantum Electron QE 22:1625

    ADS  Google Scholar 

  • Batra IP, Ciraci S, Nelson JS (1987) Confined states and stability of GaAs–AlAs superlattices. J Vac Sci Technol B 5:1300

    Google Scholar 

  • Berggren K-F, Sernelius BE (1984) Intervalley mixing versus disorder in heavily doped n-type silicon. Phys Rev B 29:5575

    ADS  Google Scholar 

  • Bernard JE, Zunger A (1987) Electronic structure of ZnS, ZnSe, ZnTe, and their pseudobinary alloys. Phys Rev B 36:3199

    ADS  Google Scholar 

  • Bimberg D, Grundmann M, Ledentsov NN (1999) Quantum dot heterostructures. J. Wiley, Chichester

    Google Scholar 

  • Bir GL, Pikus GE (1974) Symmetry and strain-induced effects in semiconductors. Wiley, New York

    Google Scholar 

  • Blakemore JS (1982) Semiconducting and other major properties of gallium arsenide. J Appl Phys 53:R123

    ADS  Google Scholar 

  • Bludau W, Onton A, Heinke W (1974) Temperature dependence of the band gap of silicon. J Appl Phys 45:1846

    Google Scholar 

  • Brédas JL, Calbert JP, da Silva Filho DA, Cornil J (2002) Organic semiconductors: a theoretical characterization of the basic parameters governing charge transport. Proc Natl Acad Sci U S A 99:5804

    ADS  Google Scholar 

  • Brooks H (1955) Theory of the electrical properties of germanium and silicon. Adv Electron Electron Phys 7:85

    Google Scholar 

  • Bullett DW, Kelly MJ (1975) The valence bands of model amorphous semiconductor structures. Solid State Commun 16:1379

    ADS  Google Scholar 

  • Chang LL, Esaki L, Howard WE, Ludeke R (1973a) The growth of a GaAs–GaAlAs Superlattice. J Vac Sci Technol 10:11

    ADS  Google Scholar 

  • Chang LL, Esaki L, Howard WE, Ludeke R, Schul G (1973b) Structures grown by molecular beam epitaxy. J Vac Sci Technol 10:655

    ADS  Google Scholar 

  • Cheng YC, Silbey RJ, da Silva Filho DA, Calbert JP, Cornil J, Brédas JL (2003) Three-dimensional band structure and bandlike mobility in oligoacene single crystals: a theoretical investigation. J Chem Phys 118:3764

    ADS  Google Scholar 

  • Cho AY (1971) Growth of periodic structures by the molecular-beam method. Appl Phys Lett 19:467

    ADS  Google Scholar 

  • Chuang SL, Chang CS (1996) Effective-mass Hamiltonian for strained wurtzite GaN and analytical solutions. Appl Phys Lett 68:1657

    ADS  Google Scholar 

  • Cohen ML (1962) Electron–phonon self-energies in many-valley semiconductors. Phys Rev 128:131

    ADS  Google Scholar 

  • Collins RT, Vina L, Wang WI, Mailhiot C, Smith DL (1987) Electronic properties of quantum wells in perturbing fields. Proc SPIE 792:2

    ADS  Google Scholar 

  • Connell GAN, Street RN (1980) Handbook on semiconductors. In: Moss TS, Keller SP (eds) Materials properties and preparation, vol 3. North Holland, Amsterdam

    Google Scholar 

  • Cracknell AP, Wong KC (1973) The fermi surface: its concepts, determination, and use in the physics of metals. Oxford University Press, Oxford

    Google Scholar 

  • Dexter RN, Zeiger HJ, Lax B (1956) Cyclotron resonance experiments in silicon and germanium. Phys Rev 104:637

    ADS  Google Scholar 

  • Dimmock JO, Melngailis I, Strauss AJ (1966) Band structure and laser action in Pb x Sn1-x Te. Phys Rev Lett 16:1193

    ADS  Google Scholar 

  • Dingle R (1975) Confined carrier quantum states in ultrathin semiconductor heterostructures. In: Queisser HJ (ed) Festkörperprobleme, advances in solid state physics, vol 15. Pergamon/Vieweg, Braunschweig, p 21

    Google Scholar 

  • Dingle R, Wiegmann W, Henry CH (1974) Quantum states of confined carriers in very thin AlxGa1-xAs–GaAs–AlxGa1-xAs heterostructures. Phys Rev Lett 33:827

    ADS  Google Scholar 

  • Djurovich PI, Mayo EI, Forrest SR, Thompson ME (2009) Measurement of the lowest unoccupied molecular orbital energies of molecular organic semiconductors. Org Electron 10:515

    Google Scholar 

  • Dresselhaus G, Kip AF, Kittel C (1955) Cyclotron resonance of electrons and holes in silicon and germanium crystals. Phys Rev 98:368

    ADS  Google Scholar 

  • D’yakonov MI, Perel’ VI (1971) On spin orientation of electrons in interband absorption of light in semiconductors. Sov Phys JETP 33:1053

    ADS  Google Scholar 

  • Economou N, Cohen MH, Freed KF, Kirkpatrick ES (1974) In: Tauc J (ed) Amorphous and liquid semiconductors. Plenum Press, New York, p 101

    Google Scholar 

  • Edwards AL, Slykhouse TE, Drickamer HG (1959) The effect of pressure on zinc blende and wurtzite structures. J Phys Chem Solids 11:140

    ADS  Google Scholar 

  • Esaki L (1985) Compositional superlattices. In: Parker EMC (ed) The technology and physics of molecular beam epitaxy. Plenum Press, New York, p 143

    Google Scholar 

  • Esaki L, Tsu R (1970) Superlattice and negative differential conductivity in semiconductors. IBM J Res Dev 14:61

    Google Scholar 

  • Fan HY (1951) Temperature dependence of the energy gap in semiconductors. Phys Rev 82:900

    ADS  MATH  Google Scholar 

  • Ferhat M, Bechstedt F (2002) First-principles calculations of gap bowing in In x Ga1-x N and In x Al1-x N alloys: relation to structural and thermodynamic properties. Phys Rev B 65:075213

    ADS  Google Scholar 

  • Froyen S, Wood DM, Zunger A (1987) Structural and electronic properties of epitaxial thin-layer SinGen superlattices. Phys Rev B 37:6893

    ADS  Google Scholar 

  • Hannewald K, Stojanovi VM, Schellekens JMT, Bobbert PA, Kresse G, Hafner J (2004) Theory of polaron bandwidth narrowing in organic molecular crystals. Phys Rev B 69:075211

    ADS  Google Scholar 

  • Hansen GL, Schmit JL, Casselman TN (1982) Energy gap versus alloy composition and temperature in Hg1−x Cd x Te. J Appl Phys 53:7099

    ADS  Google Scholar 

  • Hayes W, Stoneham AM (1984) Defects and defect processes in nonmetallic solids. Wiley, New York

    Google Scholar 

  • Heine V, van Vechten JA (1976) Effect of electron–hole pairs on phonon frequencies in Si related to temperature dependence of band gaps. Phys Rev B 13:1622

    ADS  Google Scholar 

  • Hill IG, Kahn A, Soos ZG, Pascal RA Jr (2000) Charge-separation energy in films of π-conjugated organic molecules. Chem Phys Lett 327:181

    ADS  Google Scholar 

  • Hurt NE (2000) Mathematical physics of quantum wires and devices – from spectral resonances to Anderson localization. Kluwer, Dordrecht

    MATH  Google Scholar 

  • Hume-Rothery W (1936) The structure of metals and alloys. The Institute of Metals, London

    Google Scholar 

  • Ioffe AF, Regel AR (1960) Non-crystalline, amorphous, and liquid electronic semiconductors. Progr Semicond 4:237

    Google Scholar 

  • Jaffe JE, Zunger A (1984) Electronic structure of the ternary pnictide semiconductors ZnSiP2, ZnGeP2, ZnSnP2, ZnSiAs2, and MgSiP2. Phys Rev B 30:741

    ADS  Google Scholar 

  • Jain SC, Roulston DJ (1991) A simple expression for band gap narrowing (BGN) in heavily doped Si, Ge, GaAs and Ge x Si1-x strained layers. Sol State Electron 34:453

    ADS  Google Scholar 

  • Jaros M (1985) Electronic properties of semiconductor alloy systems. Rep Prog Phys 48:1091

    ADS  Google Scholar 

  • Jen HR, Cherng MJ, Stringfellow GB (1986) Ordered structures in GaAs0.5Sb0.5 alloys grown by organometallic vapor phase epitaxy. Appl Phys Lett 48:1603

    ADS  Google Scholar 

  • Jeon J-B, Sirenko YM, Kim KW, Littlejohn MA, Stroscio MA (1996) Valence band parameters of wurtzite materials. Sol State Commun 99:423

    ADS  Google Scholar 

  • Kalt H (1996) Optical properties of III–V semiconductors – the influence of multi-valley band structures, vol 120, Springer series in solid-state sciences. Springer, Berlin

    Google Scholar 

  • Kamimura H, Nakayama T (1987) Comments. Cond Mat Phys 13:143

    Google Scholar 

  • Kane EO (1957) Band structure of indium antimonide. J Phys Chem Sol 1:249

    ADS  Google Scholar 

  • Kane EO (1979) Proceedings of the international summer school physics narrow gap semiconductor nimes. Springer, Berlin, p 13

    Google Scholar 

  • Klingshirn CF (2012) Semiconductor optics, 4th edn. Springer, Heidelberg/New York

    Google Scholar 

  • Kroemer H (1983) Heterostructure devices: a device physicist looks at interfaces. Surf Sci 132:543

    ADS  Google Scholar 

  • Lambrecht WRL, Rodina AV, Limpijumnong S, Segall B, Meyer BK (2002) Valence-band ordering and magneto-optic exciton fine structure in ZnO. Phys Rev B 65:075207

    ADS  Google Scholar 

  • Landsberg PT (1987) The band-to-band Auger effect in semiconductors. Solid State Electron 30:1107

    ADS  Google Scholar 

  • Landsberg PT, Neugroschel A, Lindholm FA, Sah CT (1985) A model for band-gap shrinkage in semiconductors with application to silicon. Phys Stat Sol (b) 130:255

    ADS  Google Scholar 

  • Larach S, Shrader RE, Stocker CF (1957) Anomalous variation of band gap with composition in zinc sulfo- and seleno-tellurides. Phys Rev 108:587

    ADS  Google Scholar 

  • Lautenschlager P, Allen PB, Cardona M (1985) Temperature dependence of band gaps in Si and Ge. Phys Rev B 31:2163

    ADS  Google Scholar 

  • Lax B (1963) In: Smith (ed) Enrico Fermi school of physics semiconductors. Academic, New York, p 240

    Google Scholar 

  • Luttinger JM (1956) Quantum theory of cyclotron resonance in semiconductors: general theory. Phys Rev 102:1030

    ADS  MATH  Google Scholar 

  • Mahan GD (1997) Good thermoelectrics. Solid State Phys 51:81

    Google Scholar 

  • Mahan GD (1980) Energy gap in Si and Ge: impurity dependence. J Appl Phys 51:2634

    ADS  Google Scholar 

  • Maltezopoulos T, Bolz A, Meyer C, Heyn C, Hansen W, Morgenstern M, Wiesendanger R (2003) Wave-function mapping of InAs quantum dots by scanning tunneling spectroscopy. Phys Rev Lett 91:196804

    ADS  Google Scholar 

  • Martinez G (1980) Handbook on semiconductors. In: Moss TS, Balkanski M (eds) Optical properties of solids, vol 2. North Holland, Amsterdam

    Google Scholar 

  • Martins JL, Zunger A (1986) Ordering and decomposition in semiconductor alloys. J Mater Res 1:523

    ADS  Google Scholar 

  • Mavroides JG (1972) Magneto-optical properties. In: Abeles F (ed) Optical properties of solids. North Holland, Amsterdam

    Google Scholar 

  • McCombe BD, Wagner RJ (1975) Intraband magneto-optical studies of semiconductors in the far infrared. Adv Electron Electron Phys 37:1

    Google Scholar 

  • Mott NF, Davis EA (1979) Electronic processes in non-crystalline materials. Claredon Press, Oxford

    Google Scholar 

  • Norris DJ, Bawendi MG (1996) Measurement and assignment of the size-dependent optical spectrum in CdSe quantum dots. Phys Rev B 53:16338

    ADS  Google Scholar 

  • Ourmazd A, Bean JC (1985) Observation of order–disorder transitions in strained-semiconductor systems. Phys Rev Lett 55:765

    ADS  Google Scholar 

  • Ousset JC, Leotin J, Askenazy S, Skolnik MS, Stradling RA (1976) Cyclotron resonance measurements of the non-parabolicity of the conduction bands in silicon and germanium. J Phys C (London) 9:2803

    ADS  Google Scholar 

  • Pantelides ST, Selloni A, Car R (1985) Energy-gap reduction in heavily-doped silicon: causes and consequences. Solid State Electron 28:17

    ADS  Google Scholar 

  • Pantelides ST (1978) The electronic structure of impurities and other point defects in semiconductors. Rev. Mod. Phys 50:797

    ADS  Google Scholar 

  • Pearsall TP (ed) (1982) GaInAsP-alloy semiconductors. Wiley, New York

    Google Scholar 

  • Pearsall TP, Bevk J, Feldman LC, Bonar JM, Mannaerts JP, Ourmazd A (1987) Structurally induced optical transitions in Ge-Si superlattices. Phys Rev Lett 58:729; Phys Rev Lett 58:1053

    Google Scholar 

  • Phillips J (1970) Ionicity of the chemical bond in crystals. Rev Mod Phys 42:317

    ADS  Google Scholar 

  • Pidgeon CR (1969) In: Haidemanakis ED (ed) Electron structure in solids. Plenum Press, New York, p 47

    Google Scholar 

  • Pidgeon CR (1980) Free carrier optical properties of semiconductors. In: Balkanski M (ed) Handbook on semiconductors, vol 2, optical properties of solids. North Holland, Amsterdam, p 223

    Google Scholar 

  • Pippard AB (1965) Dynamics of conduction electrons. Gordon and Breach, London

    Google Scholar 

  • Richardson D (1973) PhD thesis, University of Newcastle. See also: Hill R (1974) Energy-gap variations in semiconductor alloys. J Phys C 7:521

    Google Scholar 

  • Robertson J (1983) Electronic structure of amorphous semiconductors. Adv Phys 32:361

    ADS  Google Scholar 

  • Rontani M, Molinari E, Maruccio G, Janson M, Schramm A, Meyer C, Matsui T, Heyn C, Hansen W, Wiesendanger R (2007) Imaging correlated wave functions of few-electron quantum dots: theory and scanning tunneling spectroscopy experiments. J Appl Phys 101:081714

    ADS  Google Scholar 

  • Rössler U (1984) Nonparabolicity and warping in the conduction band of GaAs. Solid State Commun 49:943

    ADS  Google Scholar 

  • Saxena AK (1981) Non-Γ deep levels and the conduction band structure of Ga1−xAlxAs alloys. Phys Stat Sol (b) 105:777

    ADS  Google Scholar 

  • Schliwa A, Winkelnkemper M, Bimberg D (2007) Impact of size, shape, and composition on piezoelectric effects and electronic properties of In(Ga)As/GaAs quantum dots. Phys Rev B 76:205324

    ADS  Google Scholar 

  • Schmit JL, Stelzer EL (1973) Semiconductor opto-electronics. Butterworths, Stoneham, p 398

    Google Scholar 

  • Scholes GD (2008) Controlling the optical properties of inorganic nanoparticles. Adv Funct Mater 18:1157

    Google Scholar 

  • Schulman JN, McGill TC (1981) Complex band structure and superlattice electronic states. Phys Rev B 23:4149

    ADS  Google Scholar 

  • Smith SD (1967) Magneto-Optics in crystals. In: S Flﺰgge (ed) Handbuch der Physik vol XXV, 2a, Springer, Berlin/Heidelberg/New York, p 234

    Google Scholar 

  • Srivastava GP, Martins JL, Zunger A (1986) Atomic structure and ordering in semiconductor alloys. Phys Rev B 31:2561

    ADS  Google Scholar 

  • Stier O, Grundmann M, Bimberg D (1999) Electronic and optical properties of strained quantum dots modeled by 8-band kp theory. Phys Rev B 59:5688

    ADS  Google Scholar 

  • Suzuki M, Uenoyama T, Yanase A (1995) First-principles calculations of effective-mass parameters of AlN and GaN. Phys Rev B 52:8132

    ADS  Google Scholar 

  • Suzuki K, Kanisawa K, Janer C, Perraud S, Takashina K, Fujisawa T, Hirayama Y (2007) Spatial imaging of two-dimensional electronic states in semiconductor quantum wells. Phys Rev Lett 98:136802

    ADS  Google Scholar 

  • Tsidilkovski IM, Harus GI, Shelushinina NG (1985) Impurity states and electron transport in gapless semiconductors. Adv Phys 34:43

    ADS  Google Scholar 

  • Urbieta A, Grandidier B, Nys JP, Deresmes D, Stiévenard D, Lemaître A, Patriarche G, Niquet YM (2008) Scanning tunneling spectroscopy of cleaved InAs/GaAs quantum dots at low temperatures. Phys Rev B 77:155313

    ADS  Google Scholar 

  • van Overstraeten RJ (1982) Proceedings in physics semiconductor devices. Wiley, New Delhi, p 69

    Google Scholar 

  • Varshni YP (1967) Temperature dependence of the energy gap in semiconductors. Physica 34:149

    ADS  Google Scholar 

  • Voos M, Esaki L (1981) InAs–GaSb superlattices in high magnetic fields. c Ser Sol State Sci 24:292

    Google Scholar 

  • Vouilloz F, Oberli DY, Dupertuis MA, Gustafsson A, Reinhardt F, Kapon E (1997) Polarization anisotropy and valence band mixing in quantum wires. Phys Rev Lett 78:1580

    ADS  Google Scholar 

  • Vouilloz F, Oberli DY, Dupertuis MA, Gustafsson A, Reinhardt F, Kapon E (1998) Effect of lateral confinement on valence-band mixing and polarization anisotropy in quantum wires. Phys Rev B 57:12378

    ADS  Google Scholar 

  • Vrehen QHF (1968) Interband magneto-optical absorption in gallium arsenide. J Phys Chem Sol 29:129

    ADS  Google Scholar 

  • Vurgaftman I, Meyer JR (2003) Band parameters for nitrogen-containing semiconductors. J Appl Phys 94:3675

    ADS  Google Scholar 

  • Vurgaftman I, Meyer JR, Ram-Mohan LR (2001) Band parameters for III-V compound semiconductors and their alloys. J Appl Phys 89:5815

    ADS  Google Scholar 

  • Wagner J (1984) Photoluminescence and excitation spectroscopy in heavily doped n- and p-type silicon. Phys Rev B 29:2002

    ADS  Google Scholar 

  • Watanabe MO, Yoshida J, Mashita M, Nakanisi T, Hojo A (1985) Band discontinuity for GaAs/AlGaAs heterojunction determined by C-V profiling technique. J Appl Phys 57:5340

    ADS  Google Scholar 

  • Weaire D, Thorpe MF (1971) Electronic properties of an amorphous solid. I. A simple tight-binding theory. Phys Rev B 4:2508

    ADS  Google Scholar 

  • Thorpe MF, Weaire D (1971) Electronic properties of an amorphous solid. II. Further aspects of the theory. Phys Rev B 4:3518

    ADS  Google Scholar 

  • Wegscheider W, Pfeiffer LN, West KW (1996) GaAs/AlGaAs quantum wire lasers and other low-dimensional structures fabricated by cleaved edge overgrowth. Festkörperprobleme/Adv Solid State Phys 35:155

    Google Scholar 

  • Wellenhofer GA, Rössler U (1997) Global band structure and near-band-edge states. Phys Status Solidi (b) 202:107

    ADS  Google Scholar 

  • Wigner EP (1959) Group theory and its application to quantum mechanics of atomic spectra. Academic, New York

    MATH  Google Scholar 

  • Wilson AH (1931) The theory of electronic semiconductors. Proc R Soc London A133:458

    Google Scholar 

  • Woggon U (1997) Optical properties of semiconductor quantum dots. Springer, Berlin

    Google Scholar 

  • Woodall JM (1972) Solution grown Ga1-xAlxAs superlattice structures. J Crystal Growth 12:32

    ADS  Google Scholar 

  • Yu ET, McCaldin JO, McGill TC (1992) Band offsets in semiconductor heterojunctions, vol 46, Solid State Physics. Academic, New York, pp 1–146

    Google Scholar 

  • Zawadzki W (1974) Electron transport phenomena in small-gap semiconductors. Adv. Phys 23:435

    Google Scholar 

  • Zawadzki W (1982) Handbook on semiconductors. In: Moss TS, Paul W (eds) Band theory and transport properties. North Holland, Amsterdam

    Google Scholar 

  • Ziman JM (1969) The physics of metals, vol 1, Electrons. Cambridge University Press, London

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. 138 Moorings Park Rd. O 301, Naples, FL, 34105, USA

    Karl W. Böer

  2. Institut für Festkörperphysik, Technische Universität Berlin, Hardenbergstr. 36, 10623, Berlin, Germany

    Udo W. Pohl

Authors
  1. Karl W. Böer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Udo W. Pohl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Udo W. Pohl .

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this entry

Cite this entry

Böer, K.W., Pohl, U.W. (2014). Bands and Bandgaps in Solids. In: Semiconductor Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-06540-3_8-1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-06540-3_8-1

  • Received:

  • Accepted:

  • Published:

  • Publisher Name: Springer, Cham

  • Online ISBN: 978-3-319-06540-3

  • eBook Packages: Springer Reference Physics and AstronomyReference Module Physical and Materials ScienceReference Module Chemistry, Materials and Physics

Publish with us

Policies and ethics

Chapter history

  1. Latest

    Bands and Bandgaps in Solids
    Published:
    29 July 2022

    DOI: https://doi.org/10.1007/978-3-319-06540-3_8-4

  2. Bands and Bandgaps in Solids
    Published:
    28 March 2020

    DOI: https://doi.org/10.1007/978-3-319-06540-3_8-3

  3. Bands and Bandgaps in Solids
    Published:
    29 September 2017

    DOI: https://doi.org/10.1007/978-3-319-06540-3_8-2

  4. Original

    Bands and Bandgaps in Solids
    Published:
    08 September 2014

    DOI: https://doi.org/10.1007/978-3-319-06540-3_8-1

Search

Navigation