Do Fourier analysis yield reliable amplitude of quantum oscillations?

Quantum oscillations amplitude of multiband metals, such as high T c superconductors in the normal state, heavy fermions or organic conductors are generally determined through Fourier analysis of the data even though the oscillatory part of the signal is field-dependent. It is demonstrated that the amplitude of a given Fourier component can strongly depend on both the nature of the windowing (either flat, Hahn or Blackman window) and, since oscillations are obtained within finite field range, the window width. Consequences on the determination of the Fourier amplitude, hence on the effective mass are examined in order to determine the conditions for reliable data analysis.


Introduction
Quantum oscillations, the extrema of which are periodic in inverse magnetic field, are known to provide valuable information for the study of Fermi surface of metals. In particular, in addition to their frequencies which are directly related to the closed orbit areas on the Fermi surface, field and temperature dependence of their amplitudes allows for the determination of the effective masses and scattering rates [1]. Multiband metals such as heavy fermions [2] or high-T c superconducting iron chalcogenides [3][4][5][6] have a complex Fermi surface due to numerous sheets crossing the Fermi level, giving rise to many orbits under a magnetic field, hence to a complex quantum oscillation spectrum. Besides, in the case where magnetic breakdown (MB) or quantum tunneling of quasiparticles between orbits occurs, as in many organic metals [7,8], additional orbits are further generated. In such cases, data can be readily derived through a Fourier analysis, allowing discrimination between the various frequencies.
The point is that the amplitude of quantum oscillations is field dependent. Therefore, strictly speaking, they are not periodic in inverse field. The question that arises is then to determine to what extent reliable temperature-and fielddependent amplitudes can be derived from the Fourier analysis of such field-dependent data.
In the following, we consider the organic metal u-(ET) 4 ZnBr 4 (C 6 H 4 Cl 2 ), for which the de Haas-van Alphen (dHvA) and Shubnikov-de Haas (SdH) oscillations were extensively studied [9] in pulsed magnetic fields of up to 55 T. As it is the case of many compounds based on the bis(ethylenedithio)tetrathiafulvalene molecule (abbreviated as ET), this compound illustrates the model Fermi surface proposed by Pippard to compute magnetic breakdown amplitudes of multiband metals [10]. As reported in reference [9] and depicted in Figure 1, its Fermi surface is composed of one strongly two-dimensional closed orbit (a) and a pair of quasi-one-dimensional sheets giving rise in a magnetic field to the orbit b originated from MB. As a result, the oscillation spectrum is composed of many frequencies which are linear combinations of the frequencies linked to the a and b orbits. Amplitudes relevant to these combinations are strongly influenced by oscillations of the chemical potential in the magnetic field [9,11]. Nevertheless, it appears that this phenomenon has negligible influence on the amplitude of the dominant components a and b, allowing relevant data analysis on the basis of the Lifshitz-Kosevich formalism [12].
Rather than bringing additional information on this compound, the aim of this paper is to determine to what extent the Fourier analysis is able to yield reliable values of physical parameters of interest, in particular effective masses and scattering rates (through the Dingle temperature). To this purpose, we will consider the b orbit [9], with frequency f 0 = f b = 4534 T, effective mass m* = m b = 3.4 m e and T D = 0.8 K (this latter parameter being dependent on the considered crystal), and which involves no reflections and four tunnelings with MB field B 0 = 26 T. This component will serve as a basis to determine the influence of the windowing (nature and width) on the Fourier amplitude evaluation.

Methodology
At a fixed temperature T, a given Fourier component of the oscillatory part of magnetization (dHvA oscillations) and conductivity (SdH oscillations) can be generally written as A(x) = A 0 (x)sin(2pf 0 x + f), where x = 1/B, f 0 is the frequency and f is, for normal metals, the Onsager phase, generally equal to the product of p and the harmonic number. In the framework of the Lifshitz-Kosevich and Falicov-Stachowiak formalism [1,12,13], the amplitude is given by a product of amplitude damping factors A 0 (x) ∝ R T R D R MB for a given field direction (in which case the spin damping factor is a fieldand temperatureindependent prefactor). For a two-dimensional orbit, the thermal, Dingle and MB damping factors are given by the expressions R T = u 0 Tm*x/ sinh (u 0 Tm*x), R D ¼ e Àu 0 T D m Ã x and R MB ¼ e Àn t B 0 x=2 ð1 À e ÀB 0 x Þ n r =2 , respectively, where u 0 = 2p 2 k B m e (e h) À1 = 14.694 T/K, m * is the effective mass and T D is the Dingle temperature where t is the relaxation time). n t and n r are the number of tunneling and reflections the quasi-particles are facing during their travel along a MB orbit with a MB gap B 0 . In the following, we will consider dHvA oscillations relevant to the above-mentioned b orbit. Since measured magnetic torque t is related to magnetization M as t = M Â B, the Fourier amplitude can be written as At high-enough values of u 0 Tm*x, A 0 (x) can be approximated as where a 0 is a temperature-dependent prefactor (a 0 ∝ T) and l = u 0 (T + T D )m b + 2B 0 . 1 This approximation provides a single parameter characterizing the field dependence of the amplitude: the largest the l, the steepest the field dependence. For u-(ET) 4 ZnBr 4 (C 6 H 4 Cl 2 ), explored l values are within 194 T at 2 K and 305 T at 4.2 K. Due to large Dingle temperature, even larger values are obtained for the high-T c superconductor FeSe, for which l varies from 250 T at 1.6 K to 370 T at 4.2 K [6].
Since the signal amplitude is field-dependent, windowing [14][15][16][17][18] is mandatory in order to determine the Fourier amplitude at a given inverse field value x. The inverse field range Dx is within x m and x M (Dx = x M À x m ) and centered on x ¼ ðx m þ x M Þ=2. In order to explore the influence of windowing on the Fourier amplitude, flat, Hahn and Blackman windows are considered in the following: w(x) = 1, wðxÞ ¼ {1 þ cos ½2pðx À xÞ=Dx}=2, and wðxÞ ¼ 0:42 þ 0:5 cos ½2pðx À xÞ=Dx þ 0:08 cos ½4pðx À xÞ=Dx, respectively, within the range x m to x M and w(x) = 0 everywhere else. We can write more generally the window function as wðxÞ ¼ P p n ≥ 0 c n cos ½2pnðx À xÞ=Dx, where p = 0, 1, 2 for a flat, Hahn and Blackman window, respectively, but this can be generalized for higher values of p with appropriate coefficients. Discrete Fourier transforms are obtained as Analytical solution of equation (3) A F ðxÞ can also be obtained by numerical resolution of equation (3), where A 0 ðxÞ is either given by equation (2) or by experimental data of reference [9]. Available frequencies are bounded by the Raleigh frequency (f min = 1/Dx) and by the Nyquist frequency (f max = 1/2dx, for data sampled at evenly spaced dx values). Accordingly, Dx is kept above 1/f 0 , and dx is always small enough to ensure that f max is much higher than f 0 [19].

Results and discussion
The Fourier analysis displayed in Figure 1 shows that the largest (smallest) secondary lobes and the smallest (largest) peak width are obtained for the flat (Blackman) window, while the Hahn window provides intermediate behaviour, as widely reported [14][15][16][17][18].
Discrepancy between amplitude A F ðxÞ deduced from the Fourier analysis within a finite field range 1/x M to 1/x m and the actual Fourier amplitude A 0 ðxÞ given by equation (2) can be evaluated through the ratio A F ðxÞ=A 0 ðxÞ, which should be equal to 1. According to the data in Figure 2, a strong increase of this ratio is observed as Dx increases. Furthermore, for a given window width Dx, it increases as l increases, e.g. by increasing the temperature, while as the mean magnetic field (1=x) decreases, it grows staying on the same curve, as reported in Figure 2c. The most dramatic effect is observed for the flat window, indicating that smooth windowing is necessary to get amplitudes as reliable as possible since, more specifically, A F ðxÞ=A 0 ðxÞ grows as A F ðxÞ=A 0 ðxÞ ≃ sinhðlDx=2Þ=ðlDx=2Þ in this case.
In line with equation (4), the ratio AðxÞ=A 0 ðxÞ depends only on the product lDx for a given window type. Hence, strictly speaking, the Fourier analysis yields reliable amplitudes for finite Dx in the case of fieldindependent signal (l = 0) only. Unfavourably, moderate oscillations of the Fourier amplitude is however observed for small Dx, in particular for the flat window. It can be checked that these oscillations are periodic in Dx, their frequency being just f 0 , in agreement with equation (A.5). This feature brings us to consider the influence of the quantum oscillation frequency on the data. As reported in Figure 3, the Fourier amplitude A F ðxÞ is dominated by the monotonous term of equation (A.5), yielding equations (A.4) and (4), in the case of large-enough frequency and Dx. In contrast, large oscillations of both the Fourier amplitude and the frequency of the Fourier peaks (which is no more equal to f 0 in this case) are observed for low frequencies.
This effect is relevant in the case of slow magnetic oscillations, such as superconducting iron-based chalcogenides [3][4][5][6] and in semimetal BaNiS 2 [20], where frequencies as low as 37 T are reported. Also, slow frequency phenomena are present in magnetoresistance oscillations of organic conductors [21] or high-T c superconductors [22] due to the mixing of two close frequencies, the amplitude of which is still in debate [21]. In these cases, Fourier amplitudes can depend strongly on the windowing process.
In addition, whereas only the envelope of A F ðxÞ, i.e. A 0 ðxÞ, is relevant for the Fourier amplitude at high Dx, Onsager phase-dependent data are observed in Figure 3 for low frequencies. In short, Dx must be both small enough to avoid the amplitude overestimation predicted by equation (A.5) and large enough to avoid the undulations reported in Figure 3 in this case. As a consequence, reliable data can hardly been deduced from the Fourier analysis of low-frequency quantum oscillations.
Since l depends on temperature, the discrepancy between the actual and Fourier amplitudes for large Dx depends on temperature as well. This may lead to a significant error on the effective mass deduced from temperature dependence of the amplitude (the so-called mass plot), as evidenced in Figure 4a, and hence on the determination of the scattering rate through Dingle plots as well. As reported in Figure 4b and c, underestimation of m b by about 30% is obtained for a flat window at x = 1/32 T À1 for Dx = 0.026 T À1 (i.e. in the field range 23-56 T). About 50% would be reached at x = 1/56 T À1 for the same Dx value (field range within 32 and 193 T). Smaller although significant errors are obtained for Hahn (not shown) and Blackman windows, e.g. 15 and 13%, respectively, for x = 1/32 T À1 and Dx = 0.026 T À1 .

Conclusion
Amplitude of field-dependent quantum oscillations deduced from the Fourier analysis is overestimated even though it is widely used, as reported in the literature. Analytical formulas accounting for this phenomenon are discussed in this manuscript. Most dramatic effects are observed for steep field-dependent amplitudes No-p3 determined using flat windows with large width. Nevertheless, acceptable discrepancy with actual amplitude is obtained with Blackman window of moderate width for high-enough frequencies. In contrast, oscillations with low frequencies such as that observed in iron-based chalcogenides superconductors must be considered with care since Dx must be both small enough to avoid overestimated amplitude and large enough to avoid spurious effects observed coming close to the inverse of the Raleigh frequency.   Appendix: Analytical expression of the Fourier transforms In general, we can write the window function wðxÞ ¼ P p n ≥ 0 c n cos ½2pnðx À xÞ=Dx, where p = 0, 1, 2 stands for a flat, Hahn and Blackman window, respectively, and the condition P p n¼0 c n ¼ 1. These coefficients are given numerically by {c 0 ¼ 1} Flat , {c 0 ¼ 0:5; c 1 ¼ 0:5} Hahn and {c 0 ¼ 0:42; c 1 ¼ 0:5; c 2 ¼ 0:08} Blackman . Equations (2) and (3) lead to Writing F ðf 0 ;xÞ ¼ P n c n P e¼ ± 1 F ne in equation (A.1), one can compute individually F ne , which leads after integration to Â e Àl ne x sinhðl ne Dx=2Þ l ne À e ÀL ne x sinhðL ne Dx=2Þ L ne ; ðA:2Þ where we have defined l ne = l + 2ipne/Dx and L ne = l + 4ipf 0 + 2ipne/Dx. This expression does not depend on f up to a global sign, for the values f = 0, p. Assuming f 0 ≫ l, only the first term in bracket will contribute to F ne . Since sinh(l ne Dx/2) = (À 1) n sinh(lDx/2), one obtains Furthermore, as discussed in Section 2, A F ðxÞ=A 0 ðxÞ depends only on l at a given Dx.