This discussion will follow a semiclassical approach, i.e. the motion of the particle is subject to quantisation, but the forces are not.
1.1.1 Relation of Real Space and Momentum Space
An electron of elementary chargeqe moving in a magnetic fieldB~ with a velocity~v expe-riences a Lorentz force that gives the rate of change of its momentum ¯h~k:˙
¯
h~k˙ =−qe
~ v×B~
. (1.1)
The integration of equation (1.1) with respect to time directly relates the trajectories in real and momentum space one to another:
¯ h
~k−k~0
=−qe(~r−r~0)×B.~ (1.2) 1.1.2 Quantisation of the Motion and Onsager Relation
Closed orbits in real and momentum space correspond to periodic motions to which the Bohr-Sommerfeld quantisation rule can be applied:
I
~
p·d~q= 2π¯h(n+γ) (1.3)
15
Figure 1.1: Illustration of Landau cylinders in momentum space. (a) Free electrons: the surfaces of constant energy are spherical. (b) Ellipsoidal surfaces of constant energy. From [90].
with~pand~q the canonically conjugate momentum and position respectively and nan integer, γ a phase factor. Using equation (1.2) and Stokes’ theorem, this can be written in terms of the magnetic flux Φ:
Φ≡BA= 2π¯h
qe (n+γ) (1.4)
withAthe surface of the orbit in real space perpendicular to the magnetic field. Using equation (1.2), this can be re-expressed as the Onsager relation:
an= 2πqeB
¯
h (n+γ) (1.5)
giving the areaan of an orbit in~k-space corresponding to the orbit indexed by n.
1.1.3 Degeneracy and the Concept of Landau Cylinders
As seen before, the motion is confined to orbits enclosing surfaces quantised in units of 2πqeB/¯h in the plane perpendicular to the magnetic field. The motion parallel to the magnetic field is not quantised since there is no symmetry breaking by the field in that direction.
This gives rise to a degeneracy of the states corresponding to the orbits enumbered byn as a result of the non-quantised degree of freedom in the direction parallel to the magnetic field which allows for several electrons having the same quantum number n without in the same single particle state. The set of states described by equation (1.5) for a specific value of ntherefore corresponds to a cylinder in~k-space as illustrated in figure1.1. Such a cylinder in~k-space due to a quantisation of orbits as result of a magnetic field is usually referred to as aLandau cylinder, the quantisation of a motion on Landau cylinders when a magnetic field is present, is calledLandau quantisation, the energy of the electron system is quantised inLandau levels, each of which corresponds to one Landau cylinder in~k-space.
1.1.4 Onsager Relation and Periodicity
The Onsager Relation (1.5) shows that for increasing magnetic field B the surface of all Landau cylinders in~k-space increases. However, the Fermi surface of the electrons remains constant to a first approximation. As the the thermodynamic properties as well as other important parameters – for example the electric resistivity – of a metallic system are largely controlled by the electrons at the Fermi surface only, it is of interest to know with what regularity the Landau cylinders will cross the Fermi surface since this will change the properties of the whole system with the same periodicity.
The degeneracy of each Landau level is such that each Landau tube accomodates as many electron states as there would be in the annual cross sectional area ∆abetween the Ladau tube in question and the neighbouring one in~k-space at zero magnetic field:
∆a= 2πqeB
¯
h (1.6)
according to the Onsager relation (1.5).
As shown by equations (1.5) and (1.6), both, an and ∆a increase with increasing B, thus, a Landau level can accomodate more electron states the more intense the magnetic field is.
When the magnetic field is increased, the Landau cylinders are increased in their diameter and will cross the Fermi surface. At some point, they will have crossed the outermost parts of the Fermi surface, this means they are now bigger than the extreme diameter of the Fermi surface Aextr. Being thus totally out of the Fermi surface, they cannot be occupied by electrons anymore and all electrons that were accomodated in states associated with these particular Landau cylinders must be accomodated in states lying within the Fermi surface. Ultimately, only the last Landau tube will remain within the Fermi surface and no more states within the Fermi surface will be availiable to accomodate the electrons when that Landau tube crosses the Fermi surface. This is the limit of validity of the theory sketched here. When the field is increased even more, other phenomena have to be taken into account as they are for example described by theories of composite fermions. This limit is usually named thequantum limit.
For the case of sufficiently high n, that means unaffected by effects rising from ap-proaching the quantum limit, the difference ∆B between the fields of two Landau tubes indexed n and n−1 crossing the extremal Fermi surface section area Aextr is given ac-cording to equation (1.5) as: The Landau cylinders’ crossing of the Fermi surface is thus periodic in 1/B and we can expect the density of states at the Fermi level to oscillate with that periodicity and hence all other parameters depending on that density of states.
A Fourier analysis yields the oscillation frequencyFosc on a 1/B-scale:
F = ¯h
2πqeAextr. (1.8)
1.1.5 Usefulness and Limits of Quantum Oscillations as Experimental Probes of the Fermi Surface
Since equations (1.7) and (1.8) are valid for any direction of the magnetic field, the obser-vation of quantum oscillations is a useful tool to investigate the shape of Fermi surfaces experimentally. Indeed this has been and still is a widely used tool for that purpose. It should however be kept in mind that this method yields information about the extreme diametres of the Fermi surface only, that means about the parts of the Fermi surface where its first derivative becomes zero and this averaged over the whole cross sectional area. In practise, it is virtually always necessary to fit the data obtained experimentally to a theoretical model in order to obtain a useful result.
Within the framework of this thesis, some experiments were done with the objective to explore an unknown Fermi surface, but an important part of this work is rather consecrated to the attempt to learn something about the deviations from the ideal behaviour as it is described by the theory and thus something about the more subtle, underlying interactions of the electronic system with itself or other parts of the entire system.