De Haas van Alphen effect

In solid state physics , the De Haas van Alphen effect describes certain changes in the magnetic properties of a metal when a static magnetic field is applied . It is very useful for detailed studies of electronic tape structure .

According to Landau's theoretical prediction in June 1930, the effect was first observed in December 1930 by Wander Johannes de Haas and Pieter Marinus van Alphen on bismuth ; its importance for band structure studies was not recognized until 1952 by Lars Onsager . At low temperatures and very pure samples, a fluctuation in the magnetic susceptibility is observed as a function of the applied magnetic field: As a function of the inverse field strength, the magnetic susceptibility shows a superposition of periodic oscillations.

Explanation

The applied magnetic field exerts a Lorentz force on the conduction electrons, which leads to a change in the electronic density of states: In a semiclassical description, it can be explained by the fact that, due to the Lorentz force, the kinetic energy of the movement component is quantized perpendicular to the field direction. This results in a split into so-called Landau levels . The density of states in the vicinity of the Fermi energy is decisive for most of the electronic properties of a metal . It can be shown that the density of states at the Fermi energy becomes singular (and therefore makes the dominant contribution) if an extremal electron orbit (perpendicular to the field direction) on the Fermi surface fulfills the quantization condition that is enforced by the magnetic field. An “extreme orbit” is to be understood here as a closed electron orbit with a minimum or maximum enclosed area. The quantization condition for an extremal electron trajectory is fulfilled for different field strengths; the difference between the inverses of two neighboring field strengths for which the quantization condition is fulfilled is a constant. It essentially depends on the area enclosed by the extreme electron orbit: ${\displaystyle S}$

${\displaystyle \Delta \left({\frac {1}{B}}\right)={\frac {2\pi e}{\hbar S}}}$

Thus, all physical quantities (especially the magnetic susceptibility) that depend on the density of states at the Fermi energy should have magnetic field-dependent oscillations that are periodic as a function of 1 / H. This includes oscillations in electrical conductivity ( quantum Hall effect and Schubnikow-de-Haas effect ), magnetostriction (change in sample dimensions) and other quantities. The number of superimposed oscillations corresponds to the number of extremal orbits oriented perpendicular to the field direction on the Fermi surface.

If one considers a quantity oscillating in 1 / B (for example the magnetic moment of a sample at absolute zero) for magnetic fields in the same direction but with different strengths, one can determine the period of the oscillation. Because of this, the area that is enclosed by the extreme orbit living on the Fermi surface can be inferred . The extremal path (and thus also the area S) is perpendicular to the magnetic field . The Fermi surface can thus be reconstructed by scanning different directions. ${\displaystyle {\vec {B}}}$ ${\displaystyle \Delta \left({\frac {1}{B}}\right)}$ ${\displaystyle \Delta \left({\frac {1}{B}}\right)={\frac {2\pi e}{\hbar S}}}$ ${\displaystyle S}$ ${\displaystyle {\vec {B}}}$

Experiments

One way of observing the De Haas van Alphen effect is to precisely measure changes in the sample's magnetic moment using a torsion balance .

In another method, which is particularly suitable for investigations with strong magnetic fields , the sample is located in a coil and the voltage induced in the event of a rapid change in the magnetic field is measured. With the help of the measured induction voltage and the time-resolved induction current , the inductance of the coil can be determined. ${\displaystyle B(t)}$ ${\displaystyle u(t)}$ ${\displaystyle i(t)}$ ${\displaystyle L}$

${\displaystyle u=L{\frac {\mathrm {d} i}{\mathrm {d} t}}\,}$

The functional relationship of the inductance can now be used to resolve what the magnetic susceptibility is obtained. ${\displaystyle \mu _{r}}$ ${\displaystyle \chi _{m}=\mu _{r}-1}$

literature

• Ch. Kittel: Introduction to Solid State Physics . Oldenbourg Verlag GmbH, Munich 1993.
• NW Ashcroft, ND Mermin: Solid State Physics . Saunder College Publishing, Fort Worth (et al.).
• WJ de Haas, PM van Alphen: The dependence of the susceptibility of diamagnetic metals upon the field . In: Proceedings of the Academy of Science of Amsterdam . tape 33 , 1930, pp. 1106-1118 .

Individual evidence

1. ↑ LD Landau: Diamagnetism of Metals. In: Journal of Physics . tape 64 , no. September 9 , 1930, p. 629-637 .
2. ^ WJ De Haas, PM van Alphen: The dependence of the susceptibility of diamagnetic metals upon the field. In: Proceedings of the Academy of Science of Amsterdam . tape 33 , 1930, pp. 1106–1118 ( knaw.nl [PDF]).
3. ^ L. Onsager: Interpretation of the de Haas-van Alphen effect . In: Philosophical Magazine . tape 7 , no. 43 , 1952 ( informaworld.com ).
4. ↑ That the crystal electrons move perpendicular to the B-field in k-space can be seen from their equation of motion:, where for the crystal electron the dispersion relation applies, where is the effective mass tensor . Thus applies to the speed . Overall, the following applies : the crystal electron moves in k-space perpendicular to the gradient of the Fermi surface and the B-field. ${\displaystyle {\vec {F}}={\dot {\vec {p}}}\Leftrightarrow -e{\vec {v}}\times {\vec {B}}=\hbar {\frac {d{\vec {k}}}{dt}}}$ ${\displaystyle E={\frac {1}{2}}m^{*-1}{\vec {k}}^{2}}$ ${\displaystyle m^{*}}$ ${\displaystyle {\vec {v}}=m^{*-1}{\vec {p}}={\frac {1}{\hbar }}\nabla _{\vec {k}}E({\vec {k}})}$ ${\displaystyle {\frac {d{\vec {k}}}{dt}}=-e{\frac {1}{\hbar ^{2}}}\nabla _{\vec {k}}E({\vec {k}})\times {\vec {B}}}$