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.
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- 1.
In the following sections the magnetic induction B is used, which is connected to the magnetic field H by B = μμ 0 H, with μ 0 the permeability of free space and μ the relative permeability. Occasionally, the magnetization M is used, defined by B = μ 0 H + M with M = χ mag μ 0 H, μ = 1 + χ mag, and with χ mag the magnetic susceptibility, see chapter “Magnetic Semiconductors”.
- 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.
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.
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.
In the k z direction there are subbands; in the k x and k y directions there are discrete levels in E(k).
- 6.
At finite temperature U must be replaced by U – TS, where S is the entropy of the system.
- 7.
The area in k space has a unit of length −2.
- 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.
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.
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.
The velocity of light contained in α is the best known of the three constants.
- 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)
Allen JW (1995) Spectroscopy of lattice defects in tetrahedral II-VI compounds. Semicond Sci Technol 10:1049
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
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
Beer AC (1963) Galvanomagnetic effects in semiconductors, vol Supplement 4, Solid state physics. Academic Press, New York
Bernevig BA, Hughes TL (2013) Topological insulators and topological superconductors. Princeton University Press, Princeton
Bernevig BA, Zhang S-C (2006) Quantum spin Hall effect. Phys Rev Lett 96:106802
Brooks H (1955) Theory of the electrical properties of germanium and silicon. Adv Electron Electron Phys 7:85
Cavenett BC (1981) Optically detected magnetic resonance (O.D.M.R.) investigations of recombination processes in semiconductors. Adv Phys 30:475
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
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
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
Firsov YA, Gurevich VL, Parfeniev RV, Shalyt SS (1964) Investigation of a new type of oscillations in the magnetoresistance. Phys Rev Lett 12:660
Friedman L (1971) Hall conductivity of amorphous semiconductors in the random phase model. J Non-Cryst Solids 6:329
Glicksman M (1958) The magnetoresistivity of germanium and silicon. In: Gibson AF (ed) Progress in semiconductors, vol 3. Wiley, New York, pp 1–26
Gu Z-C, Wen X-G (2009) Tensor-entanglement-filtering renormalization approach and symmetry protected topological order. Phys Rev B 80:155131
Gurevich VL, Firsov YA (1964) A new oscillation mode of the longitudinal magnetoresistance of semiconductors. Sov Phys JETP 20:489
Hasan MZ, Kane CL (2010) Topological insulators. Rev Mod Phys 82:3045
Haug A (1972) Theoretical solid state physics. Pergamon Press, Oxford
Herring C (1955) Transport properties of a many-valley semiconductor. Bell Syst Tech J 34:237
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
Jain JK (1989) Composite-fermion approach for the fractional quantum Hall effect. Phys Rev Lett 63:199
Jain JK (1990) Theory of the fractional quantum Hall effect. Phys Rev B 41:7653
Joseph AS, Thorsen AC (1965) Low-field de Haas-van Alphen effect in Ag. Phys Rev 138:A1159
Kane CL, Mele EJ (2005) Z 2 topological order and the quantum spin Hall effect. Phys Rev Lett 95:146802
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
Koyano M, Kurita R (1998) Magnetization of quasi-two-dimensional conductor η-Mo4O11. Solid State Commun 105:743
Landau LD (1933) Motion of electrons in a crystal lattice. Phys Z Sowjetunion 3:664
Laughlin RB (1981) Quantized Hall conductivity in two dimensions. Phys Rev B 23:5632
Laughlin RB (1983) Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations. Phys Rev Lett 50:1395
Lax B, Roth LM, Zwerdling S (1959) Quantum magneto-absorption phenomena in semiconductors. J Phys Chem Solid 8:311
Madelung O (1981) Introduction to solid state theory, vol 2. Springer, Berlin
McKelvey JP (1966) Solid state and semiconductor physics. Harper & Row, New York
Onsager L (1952) Interpretation of the de Haas-van Alphen effect. Philos Mag 43:1006
Ortmann F, Roche S, Valenzuela SO (eds) (2015) Topological insulators. Wiley-VCH, Weinheim
Pfeiffer LN, West KW, Störmer HL, Baldwin KW (1989) Electron mobilities exceeding 107 cm2/Vs in modulation-doped GaAs. Appl Phys Lett 55:1888
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
Pollock DD (1985) Thermoelectricity: theory, thermometry, tool. ASTM, Philadelphia
Rose A (1978) Concepts in photoconductivity and allied problems. RE Krieger Publication Co, New York
Roth LM, Lax B (1959) g factor of electrons in germanium. Phys Rev Lett 3:217
Seeger K (2004) Semiconductor physics, 9th edn. Springer, Berlin
Shen S-Q (2012) Topological insulators: Dirac equation in condensed matters. Springer, Heidelberg
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
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)
Smith RA (1952) The physical properties of thermodynamics. Chapman & Hall, London
Stormer HL, Tsui DC (1983) The quantized Hall effect. Science 220:1241
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
Tamura H, Ueda M (1996) Effects of disorder and electron–electron interactions on orbital magnetism in quantum dots. Physica B 227:21
Tauc J (1954) Theory of thermoelectric power in semiconductors. Phys Rev 95:1394
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
Tsui DC, Stormer HL (1986) The fractional quantum Hall effect. IEEE J Quantum Electron QE 22:1711
Tsui DC, Stormer HL, Gossard AC (1982) Two-dimensional magnetotransport in the extreme quantum limit. Phys Rev Lett 48:1559
Tsui DC, Janssen M, Viehweger O, Fastenrath U, Hajdu J (1994) Introduction to the theory of the integer quantum hall effect. VCH, Weinheim
von Klitzing K (1981) Two-dimensional systems: a method for the determination of the fine structure constant. Surf Sci 113:1
von Klitzing K (1986) The quantized Hall effect. Rev Mod Phys 58:519
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
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
Willardson RK, Harman TC, Beer AC (1954) Transverse Hall and magnetoresistance effects in p-type germanium. Phys Rev 96:1512
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
Wilson AH (1954) The theory of metals. Cambridge University Press, Cambridge, UK
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
Ziman JM (1972) Principles of the theory of solids. Cambridge University Press, Cambridge, UK
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Böer, K.W., Pohl, U.W. (2018). Carriers in Magnetic Fields and Temperature Gradients. In: Semiconductor Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-69150-3_25
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