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Semiconductor Physics pp 1–32Cite as

Carriers in Magnetic Fields and Temperature Gradients

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Abstract

The application of a magnetic field in addition to an electric field yields significant information on carrier polarity and mobility, on the effective mass, and on the origin of energy levels in paramagnetic centers. If a temperature gradient exists in addition to an electric field, thermoelectric effects occur with useful applications, such as the Seebeck effect rendering thermoelectricity used in thermocouples and the Peltier effect applied for cooling. If a magnetic field is added to the temperature gradient and to the electric field, several galvanomagnetic and thermomagnetic effects are observed.

In strong magnetic fields, the electronic density of states is changed: energy levels condense on quantized Landau levels with cylindrical equi-energy surfaces in k space. Quantities controlled by their vicinity to the Fermi energy then show an oscillatory dependence on the magnetic field, such as the DeHaas-van Alphen oscillations of the magnetic susceptibility and Shubnikov-DeHaas oscillations of the resistivity.

If scattering is suppressed in highly pure samples at very low temperature, a strong magnetic field forces carriers to propagate on edge states at the sample surface, creating a topological insulator with no conductance in the bulk. In a two-dimensional electron gas, this leads to the quantum Hall effect, which established an international metrological standard for the electrical resistance. The related fractional quantum Hall effect lead to the discovery of composite fermions, quasi-particles composed of an electron and flux quanta, which conjointly carry a fractional charge. The quantum spin Hall phase represents a third type of topological insulators, which require no external magnetic field.

K. W. Böer: deceased

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Notes

  1. 1.

    In the following sections the magnetic induction B is used, which is connected to the magnetic field H by B = μμ0H, with μ0 the permeability of free space and μ the relative permeability. Occasionally, the magnetization M is used, defined by B = μ0H + M with M = χmagμ0H, μ = 1 + χmag, and with χmag the magnetic susceptibility, see chapter “Magnetic Semiconductors”.

  2. 2.

    Although the electric and magnetic fields act as external forces, and we have e (F + v × B) as total force, the scalar product of (v × B) · j is zero since the vectors v × B and j are perpendicular to each other; in first approximation, there is no energy input into the carrier gas from a magnetic field.

  3. 3.

    Galvanomagnetic effects signify electrical and thermal phenomena occurring when a current passes through a solid placed in a magnetic field, see Table 2. It should be noted that often different sign conventions in defining the tensor coefficients are used in literature.

  4. 4.

    Experimental Seebeck coefficients S (in μeV/K) are −8.3 (Na), −15.6 (K), −4.4 (Pt), +1.7 (Au), +11.5 (Li), and +0.2 (W).

  5. 5.

    In the kz direction there are subbands; in the kx and ky directions there are discrete levels in E(k).

  6. 6.

    At finite temperature U must be replaced by U – TS, where S is the entropy of the system.

  7. 7.

    The area in k space has a unit of length−2.

  8. 8.

    This concept applies also for a two-dimensional hole gas (2DHG). However, the mobility of holes is usually much lower due to a larger effective mass.

  9. 9.

    In contrast to the case of vanishing magnetic induction where the motion proceeds in the x direction and, without scattering, is accelerated (ballistic transport, see Sect. 3.1 of chapter “Carrier Transport in Low-Dimensional Semiconductors”).

  10. 10.

    Electron injection relates to electrode properties not discussed in this book. It provides an experimental means of increasing the carrier density by simply increasing the bias, thereby injecting more carriers from an appropriate electrode. For a review, see Rose (1978).

  11. 11.

    The velocity of light contained in α is the best known of the three constants.

  12. 12.

    Later also even denominators were observed, see Willett et al. (1987).

References

  • Abstreiter G (1998) Die Entdeckung des fraktionalen Quanten-Hall-Effekts. Phys Bl 54:1098 (The discovery of the fractional quantum Hall effect, in German)

    Article  Google Scholar 

  • Allen JW (1995) Spectroscopy of lattice defects in tetrahedral II-VI compounds. Semicond Sci Technol 10:1049

    Article  ADS  Google Scholar 

  • Bagraev NT, Mashkov VA (1986) Optical nuclear polarization and spin-dependent reactions in semiconductors. In: von Bardeleben HJ (ed) Defects in semiconductors, Mater Sci Forum, vol 10–12, pp 435–443. Trans Tech Publ, Switzerland

    Google Scholar 

  • Becker WM, Fan HY (1964) Magnetoresistance oscillations in n-GaSb. In: Hulin M (ed) Proc 7th international conference physics of semiconductors, Paris. Dumond, Paris, pp 663–667

    Google Scholar 

  • Beer AC (1963) Galvanomagnetic effects in semiconductors, vol Supplement 4, Solid state physics. Academic Press, New York

    MATH  Google Scholar 

  • Bernevig BA, Hughes TL (2013) Topological insulators and topological superconductors. Princeton University Press, Princeton

    Book  MATH  Google Scholar 

  • Bernevig BA, Zhang S-C (2006) Quantum spin Hall effect. Phys Rev Lett 96:106802

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  • Cavenett BC (1981) Optically detected magnetic resonance (O.D.M.R.) investigations of recombination processes in semiconductors. Adv Phys 30:475

    Article  ADS  Google Scholar 

  • Chang AM, Berglund P, Tsui DC, Stormer HL, Hwang JCM (1984) Higher-order states in the multiple-series, fractional, quantum Hall effect. Phys Rev Lett 53:997

    Article  ADS  Google Scholar 

  • Conwell E (1982) Transport: the Boltzmann equation. In: Paul W, Moss TS (eds) Handbook on semiconductors, vol. 1. Band theory and transport properties. North Holland, Amsterdam, pp 513–561

    Google Scholar 

  • DeHaas WJ, van Alphen PM (1930) Note on the dependence of the susceptibility of diamagnetic metal on the field. Leiden Commun 208d and Leiden Commun 212a

    Google Scholar 

  • Firsov YA, Gurevich VL, Parfeniev RV, Shalyt SS (1964) Investigation of a new type of oscillations in the magnetoresistance. Phys Rev Lett 12:660

    Article  ADS  Google Scholar 

  • Friedman L (1971) Hall conductivity of amorphous semiconductors in the random phase model. J Non-Cryst Solids 6:329

    Article  ADS  Google Scholar 

  • Glicksman M (1958) The magnetoresistivity of germanium and silicon. In: Gibson AF (ed) Progress in semiconductors, vol 3. Wiley, New York, pp 1–26

    Google Scholar 

  • Gu Z-C, Wen X-G (2009) Tensor-entanglement-filtering renormalization approach and symmetry protected topological order. Phys Rev B 80:155131

    Article  ADS  Google Scholar 

  • Gurevich VL, Firsov YA (1964) A new oscillation mode of the longitudinal magnetoresistance of semiconductors. Sov Phys JETP 20:489

    Google Scholar 

  • Hasan MZ, Kane CL (2010) Topological insulators. Rev Mod Phys 82:3045

    Article  ADS  Google Scholar 

  • Haug A (1972) Theoretical solid state physics. Pergamon Press, Oxford

    Google Scholar 

  • Herring C (1955) Transport properties of a many-valley semiconductor. Bell Syst Tech J 34:237

    Article  Google Scholar 

  • Hübner J, Döhrmann S, Hägele D, Oestreich M (2009) Temperature-dependent electron Landé g factor and the interband matrix element of GaAs. Phys Rev B 79:193307

    Article  ADS  Google Scholar 

  • Jain JK (1989) Composite-fermion approach for the fractional quantum Hall effect. Phys Rev Lett 63:199

    Article  ADS  Google Scholar 

  • Jain JK (1990) Theory of the fractional quantum Hall effect. Phys Rev B 41:7653

    Article  ADS  Google Scholar 

  • Joseph AS, Thorsen AC (1965) Low-field de Haas-van Alphen effect in Ag. Phys Rev 138:A1159

    Article  ADS  Google Scholar 

  • Kane CL, Mele EJ (2005) Z2 topological order and the quantum spin Hall effect. Phys Rev Lett 95:146802

    Article  ADS  Google Scholar 

  • König M, Wiedmann S, Brüne C, Roth A, Buhmann H, Molenkamp LW, Qi X-L, Zhang S-C (2007) Quantum spin Hall insulator state in HgTe quantum wells. Science 318:766

    Article  ADS  Google Scholar 

  • Koyano M, Kurita R (1998) Magnetization of quasi-two-dimensional conductor η-Mo4O11. Solid State Commun 105:743

    Article  ADS  Google Scholar 

  • Landau LD (1933) Motion of electrons in a crystal lattice. Phys Z Sowjetunion 3:664

    Google Scholar 

  • Laughlin RB (1981) Quantized Hall conductivity in two dimensions. Phys Rev B 23:5632

    Article  ADS  Google Scholar 

  • Laughlin RB (1983) Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations. Phys Rev Lett 50:1395

    Article  ADS  Google Scholar 

  • Lax B, Roth LM, Zwerdling S (1959) Quantum magneto-absorption phenomena in semiconductors. J Phys Chem Solid 8:311

    Article  ADS  Google Scholar 

  • Madelung O (1981) Introduction to solid state theory, vol 2. Springer, Berlin

    Google Scholar 

  • McKelvey JP (1966) Solid state and semiconductor physics. Harper & Row, New York

    Google Scholar 

  • Onsager L (1952) Interpretation of the de Haas-van Alphen effect. Philos Mag 43:1006

    Article  Google Scholar 

  • Ortmann F, Roche S, Valenzuela SO (eds) (2015) Topological insulators. Wiley-VCH, Weinheim

    Google Scholar 

  • Pfeiffer LN, West KW, Störmer HL, Baldwin KW (1989) Electron mobilities exceeding 107 cm2/Vs in modulation-doped GaAs. Appl Phys Lett 55:1888

    Article  ADS  Google Scholar 

  • Pollmann F, Berg E, Turner AM, Oshikawa M (2012) Symmetry protection of topological phases in one-dimensional quantum spin systems. Phys Rev B 85:075125

    Article  ADS  Google Scholar 

  • Pollock DD (1985) Thermoelectricity: theory, thermometry, tool. ASTM, Philadelphia

    Book  Google Scholar 

  • Rose A (1978) Concepts in photoconductivity and allied problems. RE Krieger Publication Co, New York

    Google Scholar 

  • Roth LM, Lax B (1959) g factor of electrons in germanium. Phys Rev Lett 3:217

    Google Scholar 

  • Seeger K (2004) Semiconductor physics, 9th edn. Springer, Berlin

    Book  MATH  Google Scholar 

  • Shen S-Q (2012) Topological insulators: Dirac equation in condensed matters. Springer, Heidelberg

    Book  MATH  Google Scholar 

  • Shoenberg D (1969) Electronic structure: the experimental results. In: Ziman JM (ed) Physics of metals, vol 1. Cambridge University Press, Cambridge, UK, pp 62–112

    Google Scholar 

  • Shubnikov L, DeHaas WJ (1930) Magnetische Widerstandsvergrößerung in Einkristallen von Wismut bei tiefen Temperaturen. Leiden Commun 207a,c,d; and: Leiden Commun 210a (Magnetic increase of the resistance of bismuth at low temperatures, in German)

    Google Scholar 

  • Smith RA (1952) The physical properties of thermodynamics. Chapman & Hall, London

    Google Scholar 

  • Stormer HL, Tsui DC (1983) The quantized Hall effect. Science 220:1241

    Article  ADS  Google Scholar 

  • Stradling RA (1984) Studies of the free and bound magneto-polaron and associated transport experiments in n-InSb and other semiconductors. In: Devreese JT, Peeters FM (eds) Polarons and excitons in polar semiconductors and ionic crystals. Plenum Press, New York

    Google Scholar 

  • Tamura H, Ueda M (1996) Effects of disorder and electron–electron interactions on orbital magnetism in quantum dots. Physica B 227:21

    Article  ADS  Google Scholar 

  • Tauc J (1954) Theory of thermoelectric power in semiconductors. Phys Rev 95:1394

    Article  ADS  Google Scholar 

  • Thomson W (1857) On the electro-dynamic qualities of metals: effects of magnetization on the electric conductivity of nickel and of iron. Proc Roy Soc London 8:546

    Article  Google Scholar 

  • Tsui DC, Stormer HL (1986) The fractional quantum Hall effect. IEEE J Quantum Electron QE 22:1711

    Article  ADS  Google Scholar 

  • Tsui DC, Stormer HL, Gossard AC (1982) Two-dimensional magnetotransport in the extreme quantum limit. Phys Rev Lett 48:1559

    Article  ADS  Google Scholar 

  • Tsui DC, Janssen M, Viehweger O, Fastenrath U, Hajdu J (1994) Introduction to the theory of the integer quantum hall effect. VCH, Weinheim

    Google Scholar 

  • von Klitzing K (1981) Two-dimensional systems: a method for the determination of the fine structure constant. Surf Sci 113:1

    Article  Google Scholar 

  • von Klitzing K (1986) The quantized Hall effect. Rev Mod Phys 58:519

    Article  ADS  Google Scholar 

  • von Klitzing K, Dorda G, Pepper M (1980) New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Phys Rev Lett 45:494

    Article  ADS  Google Scholar 

  • White CRH, Davies M, Henini M, Davidson BR, Main PC, Owers-Bradley JR, Eaves L, Hughes OH, Heath M, Skolnick MS (1990) The observation of the fractional quantum Hall effect in a single (AlGa)As/GaAs/(AlGa)As quantum well. Semicond Sci Technol 5:792

    Article  ADS  Google Scholar 

  • Willardson RK, Harman TC, Beer AC (1954) Transverse Hall and magnetoresistance effects in p-type germanium. Phys Rev 96:1512

    Article  ADS  Google Scholar 

  • Willett R, Eisenstein JP, Störmer HL, Tsui DC, Gossard AC, English JH (1987) Observation of an even-denominator quantum number in the fractional quantum Hall effect. Phys Rev Lett 59:1776

    Article  ADS  Google Scholar 

  • Wilson AH (1954) The theory of metals. Cambridge University Press, Cambridge, UK

    Book  Google Scholar 

  • Xu HF, Shen ZZ (1994) Mechanism of refractive-index change in a Ce:SBN single crystal illuminated by nominal homogeneous light. Opt Lett 19:2092

    Article  ADS  Google Scholar 

  • Ziman JM (1972) Principles of the theory of solids. Cambridge University Press, Cambridge, UK

    Book  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Naples, FL, USA

    Karl W. Böer

  2. Institut für Festkörperphysik, EW5-1, Technische Universität Berlin, Berlin, Germany

    Udo W. Pohl

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  1. Karl W. Böer
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Correspondence to Udo W. Pohl .

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Böer, K.W., Pohl, U.W. (2020). Carriers in Magnetic Fields and Temperature Gradients. In: Semiconductor Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-06540-3_25-3

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  • DOI: https://doi.org/10.1007/978-3-319-06540-3_25-3

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  1. Latest

    Carriers in Magnetic Fields and Temperature Gradients
    Published:
    16 June 2022

    DOI: https://doi.org/10.1007/978-3-319-06540-3_25-4

  2. Carriers in Magnetic Fields and Temperature Gradients
    Published:
    27 March 2020

    DOI: https://doi.org/10.1007/978-3-319-06540-3_25-3

  3. Carriers in Magnetic Fields and Temperature Gradients
    Published:
    28 September 2017

    DOI: https://doi.org/10.1007/978-3-319-06540-3_25-2

  4. Original

    Carrier in Magnetic Fields and Temperature Gradients
    Published:
    13 February 2017

    DOI: https://doi.org/10.1007/978-3-319-06540-3_25-1

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