The origin of combination frequencies in quantum magnetization oscillations of two-dimensional multiband metals

The influence of chemical potential oscillations on the magnetization oscillations in two-dimensional multiband metals is investigated. In the regime of small chemical potential oscillations, the Fourier series of magnetization oscillations is derived. As a result, combination frequencies with significant amplitudes are found if the individual band frequencies differ significantly. The analytical derivation of Alexandrov and Bratkovsky [Phys. Rev. B 63, 033105 (2001)] of magnetic quantum oscillations in two-dimensional multiband metals in the presence of chemical potential oscillations is shown to be irrelevant. Their mechanism for explaining the presence of combination frequencies in the Fourier spectrum of magnetization oscillations makes no sense. 71.18.+y, 71.10.Ay, 71.70.Di, 71.10.Pm Typeset using REVTEX 1 Quantum magnetization oscillations (the de Haas-van Alphen effect) is one of many manifestations of the quantization of the electronic energy spectrum in a magnetic field H . This quantum effect is observable both in two-dimensional (2D) and three-dimensional (3D) metals. It provides an interesting thermodynamical insight into the specificity of the 2D system, where the whole electronic energy spectrum is discrete (Landau levels structure), with regard to the 3D one. On account of the importance of chemical potential oscillations, the de Haas-van Alphen (dHvA) effect in 2D metals is quite different from its 3D counterpart. It becomes dependent on experimental conditions : the magnetization oscillations show different characteristics according to either a constant electron density for which the chemical potential μ oscillates with the magnetic field or to a fixed chemical potential1–3. For example, in 2D multiband metals, unusual combination frequencies of the ordinary frequencies of the individual bands appear in the Fourier spectrum of magnetization oscillations in presence of chemical potential oscillations. The observation of these additional contributions to the dHvA effect with significative amplitudes enables a quantitative analysis. For this purpose, an analytical description of the dHvA effect in 2D multiband metals including chemical potential oscillations on a quantitative basis is needed. Recently, Alexandrov and Bratkovsky have proposed an analytical derivation for the amplitudes of the combination frequencies in magnetic quantum oscillations. They argued that, for a fixed number of electrons N , the combination frequencies originate from an additional field oscillating term, due to chemical potential oscillations, only present in the expression of the free energy F in the canonical ensemble. After calculating the grand canonical thermodynamic potential Ω, they obtained the explicit form of the free energy [Eq. (13) of Ref. 5] using the relation F = Ω + μN . From their expression (13), they got straightforwardly the Fourier harmonics of the combination frequencies. According to Ref. 5, the field oscillating part F̃ of the free energy F is related to the field oscillating part Ω̃ of the grand canonical thermodynamic potential Ω through F̃ = Ω̃− 1 2ρ (