The quantization of the electron energy spectrum in magnetic field leads to the quantum oscillations not only of the magnetization but almost of all electronic properties [[1], chap. 4]. The conductivity also reveals quantum oscillations as magnetic field is swept. This effect was discovered by Shub-nikov and de Haas [1] even several months earlier than the dHvA effect. The standard 3D theory of the Shubnikov - de Haas effect (SdH) [[10], for a re-view see [8], chap. 11] is based on the same quasi-classical consideration of the electron motion along the Fermi surface as the dHvA theory is. From the Onsager quantization rule one determines the electron spectrum and the density of statesρ(E). The oscillations of the density of states at the Fermi level make the main contribution to the conductivity because they lead to the oscillations of the electron relaxation time τ to which the conductivity is proportional. The scattering rate 1/τ on the point-like impurities in Born approximation is proportional to the density of electron states at the Fermi level ρ(µ). The DoS can be found as whereδ(x) is the Dirac delta function. The summations in (1.25) are similar to those in (1.3). Applying the same mathematical tricks as in Sec. 1.1 one finds that the oscillating part ˜σ(B) of conductivity is proportional to the derivative of magnetization [[8], chap. 11]:
˜ Even more than in the case of magnetization, the 3D theory of the Shub-nikov - de Haas effect fails to describe the 2D or quasi-2D magnetotransport in strong fields. The problem of the 2D magnetotransport is known to differ completely from the 3D case. The quantum Hall effect (Q.H.E.) certainly can not be described by the 3D magnetotransport theory. But even when the electron dispersion in the third dimension is larger than the cyclotron energy, the new qualitative effects appear that can not be explained in the framework of the 3D theory. Namely, these are the slow oscillations of con-ductivity or the phase shift of the beats of concon-ductivity oscillations. These effects were observed, for example, in strongly anisotropic organic metals.
We shall discuss these problems and their solutions in detail in chapter 4 of this thesis.
1.3. THE SHUBNIKOV - DE HAAS EFFECT IN 3D METALS. 21 In chapter 2 we shall emphasize the difference between the 3D and 2D cases and illustrate the physics by many pictures. There we shall study the 2D chemical potential oscillations and their effect on magnetization os-cillations in the simplest case of sharp Landau levels and low temperature.
Practically, chapter 2 is also introductory (although it contains some new results) and the main results are given in chapters 3 and 4. In the third chapter we shall consider the dHvA effect under more general conditions in-cluding also the LL broadening, finite kz dispersion and chemical potential oscillations at arbitrary electron reservoir. This chapter requires more math-ematics, but we shall invariably indicate the physics of the phenomena. The fourth chapter is devoted to magnetoresistance oscillations where the theory of the quasi-two-dimensional Shubnikov-de Haas effect is developed. Actu-ally, this theory still uses many approximations. These limitations and the further perspectives of the development of the theories are then discussed.
Chapter 2
2D dHvA effect (the model study)
2.0.1 Comparison between 2D and 3D dHvA cases
From the general point of view, oscillations in the two-dimensional (2D) case are much sharper and even stronger than in the 3D case. This is because the kz dispersion smears out the effect of the quantization of Landau levels. As a result (see chapter 1) only the extreme cross sections of the Fermi surface effectively contribute to the magnetic quantum oscillations. In the 2D case the Fermi surface is a cylinder that goes along the LLs (see Fig. 2.1). Hence, in the 2D case the entire FS is extreme.
Figure 2.1: The Fermi surface and Landau levels in the three- and two-dimensional cases.
The area of the extreme parts of the FS in the 3D case is aboutp
~ωc/EF
times smaller than the total Fermi surface area. Hence, at the same elec-tron concentration the magnetic quantum oscillations in the 2D case are
23
∼p
EF/~ωc times larger than in the 3D case. The smearing from thekz dis-persion leads also to a much stronger harmonic damping of the oscillations.
Comparison of the 2D Shoenberg formula (1.24) with the Lifshitz-Kosevich formula (1.7) gives that the coefficient before the k-th harmonics in the 3D case has an additional k−1/2 power of the harmonic number. This leads to a different shape of oscillations in the weak harmonic damping limit (at low temperature and in pure compounds; see Fig. 2.3). Magnetization oscilla-tions in the 2D case without any smearing have the saw-tooth shape (Fig.
2.3 b or Fig. 2.4 b). This can be easily derived. At zero temperature the magnetization M(B) is given by the derivative of the total electron energy E(B) as a function of magnetic field:
M(B) =−dE(B)/dB.
The electron energy is calculated as a sum over all Landau levels at a constant electron density N. Assume the electrons to be spinless and the LLs to be sharp. The last occupied LL with a numbernF has N−nFg electrons where g = B/Φ0 is the degeneracy of LLs and Φ0 = 2π~c/e. The total electron energy becomes
E(B) = g
nF−1
X
n=0
~ωc(n+ 1/2) + (N −nFg(B))~ωc(nF +1 2) =
= n2F
2 g~ωc+ (N −nFg(B))~ωc(nF +1 2) =
= −n2F
2 g~ωc+N~ωc(nF + 1
2)−nFg(B)~ωc/2. (2.1) It has a quadratic dependence on the magnetic field on each dHvA period (see Fig. 2.2).
The magnetization is M(B) = −dE(B)
dB = n2F
B g~ωc −N~ωc
B (nF +1
2) +nFg(B)~ωc/B =
= ~ωcnF
B (nFg−N) + (nFg−N/2)~ωc/B (2.2) When many LLs are occupied (nF 1) the first term in (2.2) gives the main contribution to magnetization. The magnetization is always linear in B except at points where an integer number of LLs is occupied and the number of the highest occupied LL nF jumps by unity. The magnetization at these points jumps by ≈ EF/Φ0. This dependence is plotted on the graphs 2.3b and 2.4b. Such magnetization behavior has a relation to the