VOLUME 80, NUMBER 14 PHYSICAL REVIEW LETTERS 6APRIL 1998 Frequency Mixing of Magnetic Oscillations: Beyond Falicov-Stachowiak Theory Jean-Yves Fortin* Laboratoire de Physique Théorique, ENSLAPP, URA 14-36 du CNRS, associée à l’ENS de Lyon, et à l’Université de Savoie, Ecole Normale Supérieure de Lyon, 46 allée d’Italie, 69007 Lyon, France Timothy Ziman † Institut Laue Langevin, B.P. 156 X, 38042 Grenoble, France and Laboratoire de Physique Quantique, CNRS-UMR5646, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex, France (Received 2 December 1997) The interpretation of de Haas – van Alphen oscillations in the presence of magnetic breakdown is usually based on the semiclassical theory of Falicov and Stachowiak (FS). There are now glaring discrepancies between its predictions and experiments, especially in quasi-two-dimensional organic conductors. We present an extension of the theory, using the appropriate constraints of conserved electron density, which explains the occurrence of frequencies not predicted by FS, and makes explicit the amplitudes as a function of Fermi surface parameters. The results involve a tunneling amplitude between different sheets as in FS, but other parameters as well, such as the areas of different orbits. [S0031-9007(98)05725-1] PACS numbers: 71.18. + y, 74.70.Kn Magnetic breakdown of oscillations in the magnetiza- tion or transport is an important tool in exploring Fermi surfaces. At weak magnetic fields oscillations are visible and correspond to extremal cross sections of semiclassi- cally closed orbits perpendicular to the field direction. If there are several bands there are frequencies correspond- ing to each closed orbit but not to orbits which traverse the first Brillouin zone and are open. As the magnetic field increases in a system with several bands, there may be magnetic field-induced tunneling from band to band, giving rise to larger orbits with frequencies that are com- binations of the original frequencies or which may include parts of open parts of the Fermi surface, giving new fre- quencies. The characteristic fields at which oscillations appear give information on local properties of the bands: separation of the disjoint parts of the Fermi surface and the local curvature. In the case of bidimensional organic con- ductors such as the family BEDT-TTF 2 X SCN 4 where BEDT-TTF is (bis)-ethyleneditho-tetrathiafulvalene [1,2], information on the open parts of the Fermi surface is a key to understanding low temperature instabilities. In ad- dition to extract physical parameters, renormalized masses and g factors in particular, we need a reliable and detailed theory of oscillations. In the past this has been pro- vided by semiclassical theory [3], which culminated in the theory of Falicov and Stachowiak (FS) [4]. In 1982, fre- quencies were observed [5] for pure magnesium that are, however, forbidden in such a theory. In the organic met- als there are more and more violations in the frequency spectrum from experiment. The existence of the “forbid- den” b-a frequency (see below) in the magnetoresistance of k-BEDT-TTF 2 CuSCN 2 [6] was attributed to a Stark interference [3,7] but has since been seen clearly in de Haas – van Alphen (dHvA) [8,9], where Stark interference does not apply. Numerical simulation [10 –12] showed that the discrepancy is a single particle effect. It has been argued [12 – 15] that the reason for observations of frequen- cies corresponding to classically disallowed orbits is that the different frequencies are coupled by the constraint that the total number of electrons across all bands is conserved as the field varies. This global constraint leads to coupling between different frequencies and frequencies forbidden in the FS theory. Such a constraint, without the effects of breakdown, has been shown numerically [14,15] to change the harmonic content of a single frequency. What we lack are explicit calculations of the amplitudes of breakdown when this effect is taken into account. These should be substituted for the FS expressions. In this paper we shall show how FS theory must be amended and we calculate, for a simple case, the correct forms. We also give explicit account of the spin splitting used to extract the renormal- ized Landé factor g, at least in the low-field limit. We now consider a simple case of breakdown in two dimensions between open and closed orbits. The Fermi surface, close to that of k-BEDT-TTF 2 CuSCN 2 is shown in Fig. 1. The theory describing the tunneling process at each junction is given in Ref. [16]. We adapt the theory of Ref. [16] to describe the tunneling process at each junction in terms of a gap k g and curvature k g 2l 2 , k x 6 q l 2 k 2 y 1 k 2 g 4  0. If p and q are the amplitudes of tunneling and reflection ( p 2 1 q 2  1), and v the phase the wave function takes during the reflection process, then the transfer relations between the wave amplitudes before and after the junction points are given [16], μ g d ∂  μ q expi v ip ip q exp2i v ∂μ a b ∂ 0031-9007 98 80(14) 3117(4)$15.00 © 1998 The American Physical Society 3117