# De Haas–van Alphen effect

The de Haas–van Alphen effect, often abbreviated to dHvA, is a quantum mechanical effect in which the magnetic susceptibility of a pure metal crystal oscillates as the intensity of the magnetic field B is increased. Other quantities also oscillate, such as the electrical resistivity (Shubnikov–de Haas effect), specific heat, and sound attenuation and speed.[1][2][3] It is named after Wander Johannes de Haas and his student Pieter M. van Alphen.[4] The dHvA effect comes from the orbital motion of itinerant electrons in the material. An equivalent phenomenon at low magnetic fields is known as Landau diamagnetism.

## Description

The differential magnetic susceptibility of a material is defined as

${\displaystyle \chi ={\frac {\partial M}{\partial H}}}$

where ${\displaystyle H}$  is the applied external magnetic field and ${\displaystyle M}$  the magnetization of the material. Such that ${\displaystyle B=\mu _{0}(H+M)}$ , where ${\displaystyle \mu _{0}}$  is the vacuum permeability. For practical purposes, the applied and the measured field are approximately the same ${\displaystyle B\approx \mu _{0}H}$  (if the material is not ferromagnetic).

The oscillations of the differential susceptibility when plotted against ${\displaystyle 1/B}$ , have a period ${\displaystyle P}$  (in teslas−1) that is inversely proportional to the area ${\displaystyle S}$  of the external orbit of the Fermi surface (m−2), in the direction of the applied field, that is

${\displaystyle P\left(B^{-1}\right)={\frac {2\pi e}{\hbar S}}}$ ,

where ${\displaystyle \hbar }$  is Planck constant and ${\displaystyle e}$  is the elementary charge.[5]

The modern formulation allows the experimental determination of the Fermi surface of a metal from measurements performed with different orientations of the magnetic field around the sample.

## History

Experimentally it was discovered in 1930 by W.J. de Haas and P.M. van Alphen under careful study of the magnetization of a single crystal of bismuth. The magnetization oscillated as a function of the field.[4] The inspiration for the experiment was the recently discovered Shubnikov–de Haas effect by Lev Shubnikov and de Haas, which showed oscillations of the electrical resistivity as function of a strong magnetic field. De Haas thought that the magnetoresistance should behave in an analogous way.[6]

The theoretical prediction of the phenomenon was formulated before the experiment, in the same year, by Lev Landau,[7] but he discarded it as he thought that the magnetic fields necessary for its demonstration could not yet be created in a laboratory.[8][9][6] The effect was described mathematically using Landau quantization of the electron energies in an applied magnetic field. A strong homogeneous magnetic field — typically several teslas — and a low temperature are required to cause a material to exhibit the dHvA effect.[10] Later in life, in private discussion, David Shoenberg asked Landau why he thought that an experimental demonstration was not possible. He answered by saying that Pyotr Kapitsa, Shoenberg's advisor, had convinced him that such homogeneity in the field was impractical.[6]

After the 1950's, the dHvA effect gained wider relevance after Lars Onsager (1952),[11] and independently, Ilya Lifshitz and Arnold Kosevich (1956),[12] pointed out that the phenomenon could be used to image the Fermi surface of a metal.[6]

## References

1. ^ Zhang Mingzhe. "Measuring FS using the de Haas-van Alphen effect" (PDF). Introduction to Solid State Physics. National Taiwan Normal University. Retrieved 2010-02-11.
2. ^ Holstein, Theodore D.; Norton, Richard E.; Pincus, Philip (1973). "de Haas-van Alphen Effect and the Specific Heat of an Electron Gas". Physical Review B. 8 (6): 2649. Bibcode:1973PhRvB...8.2649H. doi:10.1103/PhysRevB.8.2649.
3. ^ Suslov, Alexey; Svitelskiy, Oleksiy; Palm, Eric C.; Murphy, Timothy P.; Shulyatev, Dmitry A. (2006). "Pulse-echo technique for angular dependent magnetoacoustic studies". AIP Conference Proceedings. 850: 1661.
4. ^ a b de Haas, W.J.; van Alphen, P.M. (1930). "The dependence of the susceptibility of diamagnetic metals upon the field" (PDF). Proc.Acad.Sci.Amst. 33: 1106–1118.
5. ^ Kittel, Charles (2005). Introduction to Solid-State Physics (8th ed.). Wiley. ISBN 978-0-471-41526-8.
6. ^ a b c d Shoenberg, David (1987). "Electrons at the Fermi Surface". In Weaire, D.L.; Windsor, C.G. (eds.). Solid state science : past, present, and predicted. Bristol, England: A. Hilger. p. 115. ISBN 978-0852745847. OCLC 17620910.
7. ^ Landau, L. D. "Diamagnetismus der Metalle." Zeitschrift für Physik 64.9 (1930): 629-637.
8. ^ Shoenberg, David (1965). "The de Haas-Van Alphen Effect". In Daunt, J.G.; Edwards, D.O.; Milford, F.J.; Yaqub, M. (eds.). Low Temperature Physics LT9. Boston: Springer. pp. 665–676. doi:10.1007/978-1-4899-6443-4_6. ISBN 978-1-4899-6217-1.
9. ^ Marder, Michael P. (2000). Condensed Matter Physics. Wiley.
10. ^ Harrison, Neil. "de Haas-van Alphen Effect". National High Magnetic Field Laboratory at the Los Alamos National Laboratory. Retrieved 2010-02-11.
11. ^ Onsager, Lars (1952). "Interpretation of the de Haas-van Alphen effect". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 43: 1006–1008 – via Taylor & Francis.
12. ^ Lifshitz, Ilya Mikhailovich; Kosevich, Arnold M. (1956). "Theory of magnetic susceptibility in metals at low temperatures" (PDF). Soviet Physics JETP. 2: 636–645 – via Journal of Experimental and Theoretical Physics.