Alain Audouard

Jean-Yves Fortin

Vladimir Laukhin

David Vignolles

Tatyana G Prokhorova

Eduard B Yagubskii

Enric Canadell

oscillations in the compensated organic metal α-pseudo-κ'-(ET)4H3O[Fe(C2O4)3]×(C6H4Br2)

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Provided that no phase transition occurs as the temperature is lowered, Fermi surface (FS) of two-dimensional (2D) organic metals is generally rather simple and achieve model systems for quantum oscillations physics. Indeed, in the numerous cases where the compound possesses two carriers (generally holes) per unit cell, the FS originates from a single orbit with an area equal to that of the first Brillouin zone [1]. In an extended zone scheme, these orbits overlap either along (i) one or (ii) two directions yielding in magnetic field either (i) the model linear chain of orbits coupled by magnetic breakdown(MB)proposedb y Pippard in the early sixties [2,[START_REF] Shoenberg | Magnetic Oscillations in Metals[END_REF] for which all the orbits are of the same (hole) type or (ii) a set of compensated electron-and hole-type orbits (i.e. the sum of the hole-type orbits cross section is equal to that of the electron-type orbits).

The former case (i) has been widely studied: it is known that quantum oscillations spectra are strongly affected by field-induced chemical potential oscillations yielding many frequency combinations, the field and temperature dependence of which cannot be accounted for by the Lifshitz-Kosevich (LK) formula [START_REF] Wosnitza | Fermi Surfaces of Low-Dimensional Organic Metals and Superconductors[END_REF][5][6]. In contrast, to our best knowledge, the second case (ii) has only received little attention up to now even though MB between hole-and electron-type orbits is relevant, for a e-mail: alain.audouard@lncmi.cnrs.fr instance, for recently studied oxide superconductors [7][8][START_REF] Helm | s n i t z a ,R .G r o s s[END_REF]. Numerical resolution of the grand potential equation for the case (ii) reveals that chemical potential oscillations are strongly damped for compensated orbits, even in the case of 2D metals [START_REF] Fortin | [END_REF]11]. As a consequence, contrary to case (i), the LK formula is predicted to account for the field and temperature dependence of the oscillations amplitude in case (ii). However, depending on the MB probability and taking into account the opposite sign of electron and hole orbits, a given Fourier component can involve combinations of an infinite set of orbits, the contribution of which must be included in the Fourier amplitudes calculation. From the experimental side, only few quasi-2D compensated metals have been synthesized yet. Such FS are observed in e.g. (BEDO-TTF) 2 ReO 4 •H 2 O [ 12] (where BEDO-TTF stands for the bis-ethylenedioxy-tetrathiafulvalene molecule) and, more recently synthesized, (ET

) 4 H 3 O[Fe(C 2 O 4 ) 3 ]
• Solv (where ET stands for the bis-ethylenedithio-tetrathiafulvalene molecule and Solv is an organic solvent) [13,14].

As reported in Figure 1a, the unit cell of α-'pseudo- [13]containstwodifferent donor planes with different packing. One of them, with a 'pseudo-κ' structure is insulating while the other with an α-type structure is metallic. As a consequence, spacing between conducting layers is as large as 3.64 nm, ensuring negligibly small interlayer transfer integral, hence avoiding effects due to FS corrugation [15]. According to band structure calculations, the FS of the metallic plane, displayed in Figure 1b, is composed of two electron-type and one hole-type compensated orbits (i.e. the hole orbit area is twice the electron orbits area). In the extended zone scheme, this set of three orbits is isolated from the other sets. In other words, we are not dealing with a network of coupled orbits. In contrast, each of the orbits within a set is liable to be connected to the other by MB. This feature is shared by the FS studied in reference [START_REF] Fortin | [END_REF], which is composed of one hole and one electron compensated orbit.

κ'-(ET) 4 H 3 O[Fe(C 2 O 4 ) 3 ]•(C 6 H 4 Br 2 )
The aim of this paper is, in the first step, to provide calculation of the Fourier amplitudes of de Haas-van Alphen (dHvA) oscillations spectra for the FS of Figure 1b. In the second step, magnetic torque oscillations of α-'pseudo-κ'-(ET

) 4 H 3 O[Fe(C 2 O 4 ) 3 ]•(C 6 H 4 Br 2 )
, yielding dHvA spectra, are studied in magnetic fields up to 55 T. Whereas magnetoresistance data of this organic metal, measured in magnetic fields below 16 T reveal only one frequency attributed to the hole orbit [13], the two frequencies corresponding to electron and hole orbits area are observed. It is shown that the field and temperature dependence of the Fourier amplitudes are in agreement with the reported calculations.

Experimental

Crystals were synthesized by electrocrystallization technique as reported by Zorina et al. [13]. Two crystals denoted as crystal #1 and #2 hereafter were studied. Their approximate dimensions were 0.12 × 0.1 × 0.04 mm 3 . Magnetic torque was measured with a commercial piezoresistive microcantilever, in pulsed magnetic fields of up to 55 T with a pulse decay duration of 0.32 s. Variations of the cantilever piezoresistance were measured in the temperature range from 1.4 K to 4.2 K with a Wheatstone bridge with an ac excitation at a frequency of 63 kHz. The angle between the normal to the conducting plane and the magnetic field direction was θ =1 5 • and 32 • for crystal #1 while θ was varied from 15 • to 71 • thanks to a rotating sample holder for crystal #2.

Calculations of the Fourier amplitudes

In the framework of the Lifshitz-Kosevich and Falicov-Stachowiak models [START_REF] Shoenberg | Magnetic Oscillations in Metals[END_REF]16], the oscillatory part of the magnetization M osc for a set of 2D orbits η can be written as:

M osc = - η p F η π A pη sin 2πp F η B cos θ , (1)
where θ is the angle between the normal to the conducting plane and the field direction. The index η stands for all the closed orbits allowed by the FS topology, including MB orbits, with fundamental frequencies F η .T h e y do not include orbits which can be associated with an harmonic of a simpler trajectory, the latter being accounted for by the index pF η (with p>1) where p is the harmonic order. The amplitude A pη of the Fourier component with frequency pF η depend on parameters such as temperature, magnetic field, effective masses (m η ), Dingle temperatures (T Dη ), MB field (B 0 ), and effective Landéf a c t o r s( g * η ). These amplitudes can be expressed as A pη =( -1) psη R MB pη R pη /(pm η ). MB damping factor is given by R MB pη = C pη (ip 0 ) ptη q pbη 0 ,w h e r eC pη is the symmetry factor of the orbit pη, t η and b η are the number of tunnelings and reflections, respectively, encountered by a quasi-particle during its path and 2s η is the number of turning points around the orbit η. The tunneling and reflection probabilities at a MB junction are given, in agreement with the Chambers approximation, by p 2 0 =e x p ( -B 0 /B cos θ)a n dq 2 0 =1-p 2 0 , respectively [START_REF] Chambers | Proc. Phys. Soc[END_REF]. The damping factor R pη c a nb ew r i t t e na s the product of thermal, Dingle and spin damping factors

(R pη = R T pη R D pη R s pη )w h i c ha r eg i v e nb yR T pη = X pη / sinh(X pη ), R D pη =e x p ( -u 0 T Dη pm η /B cos θ), and R s pη =c o s ( pπg * η m η /2cosθ), respectively, where X pη = u 0 Tpm η /B cos θ and u 0 =2π 2 k 2
B m e /e . In the case of the FS of Figure 1b, which is modeled by Figure 2, and owing to the orbits compensation predicted by band structure calculations, each electron (e) and hole (h) orbit contributes to the dHvA oscillations spectrum with the frequency F e and F h =2 F e , respectively. Besides, MB orbits composed of several individual orbits (η = n e e + n h h) are also liable to contribute to the spectrum. Indeed, all contributing trajectories for a given frequency are accounted for by considering the amplitude and phase variation exp(iS η )=e x p [ 2 iπhA η /eB)] of the wave-function, where A η is the area of the trajectory in the Brillouin zone, which is directly identified to the frequency F η = |A η |h/e. Taking into account the opposite sign of electron and hole surface area, S nee+n h h = n e S e + n h S h with S h = -2S e , all Fourier components with frequencies [START_REF] Shoenberg | Magnetic Oscillations in Metals[END_REF]16], where n e and n h are the (positive) numbers of electron and hole orbits involved, contribute. Therefore, the dominant frequencies F e ,2 F e = F h and 3F e we will consider in the following, arise from infinite orbits combinations with n e =2n h ± 1, n e =2n h ± 2andn e =2n h ± 3, respectively. To obtain the amplitude of these Fourier components, the Fermi surface is modeled in Figure 2 by a linear finite chain of e and h orbits.

F nee+n h h = |n e F e -n h F h | = |n e -2n h |F e ,

and effective masses m nee+n

h h = n h m h + n e m e
To count all the allowed paths from one of the arbitrary starting and ending points α, β, β, γ in Figure 2, we consider paths of length n,w h e r en is an integer corresponding to the number of elementary steps from one point to another that form and define the complete closed trajectory. Turning points are located at the arrows in Figure 2 and going through one of these is tantamount to adding a factor i in the wave-function amplitude. We introduce the vector amplitude |α, β, β, γ corresponding to the state of the quasi-particle. Any closed path is therefore defined by its length n and its starting point and state s 0 among the set (α, β, β, γ). Initially the state |s 0 is filled with zeroes except for the component corresponding to s 0 , for example |α = |1, 0, 0, 0 . We then define each step factor of the trajectory by moving from one point among the set (α, β, β, γ) to the next one, by following the direction imposed by the field, with the adequate changes in the phase and amplitude. For example, as shown in Figure 2, β = -ip 0 α exp(iS e /2+iS h /2) + iq 0 β exp(iS h /2). Introducing parameters x =e x p ( iS e /2) and y =e x p ( iS h /2), with y =1 /x 2 , we can write a transfer matrix for all the elementary steps on the Fermi surface:

T (x, y)= ⎛ ⎜ ⎝ -q 0 x 2 -ip 0 xy 00 00 iq 0 y -p 0 x -p 0 xi q 0 y 00 00 -ip 0 xy -q 0 x 2 ⎞ ⎟ ⎠ .
Since we consider closed paths and conserved current, the output vector is the same as the input vector |s 0 .T h e n the number of all possible paths of length n and starting from a point s 0 is given by the number s 0 |T (x, y) n |s 0 . The generalized damping factor A pη in equation (1)i s computed for all possible orbit combinations contributing to the same frequency pF η . The combinatorial factor (-1) psη R MB pη in A pη , combining the different ways of drawing the orbits pη on the FS, is precisely defined as the coefficient of x 2ne y 2n h ,a sw e l la sx -2ne y -2n h ,i nt h e following polynomial function

A(x, y)= 1 0 dz z s0 n≥0 W s0 s 0 |T n ( √ zx, zy)|s 0 (2)
weighted by damping factors for each orbit, and corresponding to integers (n e ,n h ). W s0 is the weight of the point s 0 in the quasi-particle path. We will take W α = W γ =1a n dW β = W β =1 /2. Indeed points β and β are equivalent in the sense that they define the same set of paths and belong to the same orbit. Finally, the integration over z i spe r f o r m e ds ot h a tt or e m o v et h en u m be r of cyclic permutations n e +2n h of the same fundamental orbit by adding a compensatory factor 1/(n e +2n h ), and to take into account the harmonics coefficients with a weighting factor 1/p. Each path can be simply decomposed using e (or x) for a portion of trajectory around the electron orbits, and √ h (or y) around the hole orbit, and z is added for each e or √ h encountered on the trajectory. For example the orbit ee √ he √ h (corresponding to n e =3,n h = 1) with effective mass 3m e + m h and phase S e (corresponding to the frequency F e ) has a weight proportional to 10p 4 0 q 0 /5=2 p 4 0 q 0 (the factor 5 = n e +2n h is the number of cyclic permutations of the same orbit ee

√ he √ h = e √ he √ he = ...).
It is different from the orbit ee √ h √ he for example, which has a weight proportional to -2p 3 0 q 2 0 . We find that the first three amplitudes can be written as:

A e = - 2q 0 m e R e - 2p 2 0 q 0 m e + m h R e+h + 2 p 4 0 q 0 -p 2 0 q 3 0 3m e + m h R 3e+h + ..., (3)
A 2e = q 2 0 2m e R 2e - q 2 0 m h R h + 3p 4 0 q 2 0 -2p 2 0 q 4 0 2(m e + m h ) R 2e+2h + 2p 2 0 q 4 0 -3p 4 0 q 2 0 4m e + m h R 4e+h + ..., (4)
A 3e = - 2q 3 0 9m e R 3e + 2p 2 0 q 3 0 m e +2m h R e+2h + ... (5)
where

R pη = R T pη R D pη R s pη .
A sa ne x a m p l e ,t h efi r s tt e r m -(2q 3 0 /9m e )R 3e of A 3e in equation ( 5), comes from the third harmonics of the e orbit. Indeed, according to equation (1), this factor is given by F e (-2q 3 0 )R 3e /(3m e )whic h can be rewritten as 3F e (-2q 3 0 )R 3e /(9m e ). Magnetization can then be expanded as:

M osc = - F e π A e sin 2π F e B - 2F e π A 2e sin 2π 2F e B - 3F e π A 3e sin 2π 3F e B + ... (6)
Examples of Fourier spectra deduced from equation ( 6) are reported in Figure 3a for various temperatures and MB fields. Landéf a c t o r s( g * e =2 ,g * h = 2) and effective masses (m e =1 ,m h = 1), which otherwise are close to those deduced from the data reported in the next section, are chosen so that the absolute value of the spin Long dashed, short dashed, dotted and dash-dotted lines are the components linked to the h,2 e,2 e +2h and 4e + h orbits, respectively. The magnetic breakdown damping factors for the orbits 2e +2h and 4e + h are given by R 2e+2h =3p 4 0 q 2 0 -2p 2 0 q 4 0 and R 4e+h =2p 2 0 q 4 0 -3p 4 0 q 2 0 ,w h e r eq0 and p0 are the reflection and magnetic breakdown probability, respectively. damping factors is equal to 1 (|R s pη | =1 )i no r d e rt o avoid any spurious effect due to spin-zero phenomenon. First, A 3e is always small compared to A e and A 2e .C o rresponding field-dependent amplitudes A 2e are given in Figures 3b-3d. Despite the effective mass m η of a given η orbit increases as the number of individual orbits involved increases, a clear contribution of the orbits 2e,2e+2h and 4e + h to the amplitude is observed in Figure 3b. Their relative contributions decrease as the Dingle temperatures (not shown), the temperature (see Fig. 3c) and the MB field (see Fig. 3d) increase which may lead to errors in effective mass determination [11]. Contribution of the orbit 2e is substantial in any case, which indicates that this orbit needs to be considered, even in the case of large MB field, scattering rate and temperature, for correct data analysis. In contrast, complex orbits such as 2e +2h have significant contribution for clean compounds with moderate MB field at low temperature, only. Similar conclusions can be derived regarding the component with frequency F e for which the contribution of e + h is always significant in the range explored, except for large MB fields, while Field-dependent magnetic torque data and corresponding Fourier analysis are reported in Figure 4.A n g l e dependence of the two observed frequencies (not shown) follows the cosine law predicted for a 2D FS with F h (θ = 0) = 183 ± 3Ta n dF e (θ =0 )=9 1± 5 T. Consistently with the FS topology reported in Figure 1b, they correspond to the hole and electron orbits cross section area S h =8.9 ± 0.2%a n dS e =4.4 ± 0.3 %, respectively, of the FBZ area. These data are in agreement with magnetoresistance data (S h =8.9 % of the FBZ area) and band structure calculations at room temperature (S h =7 .6% of the FBZ area and S e = S h /2) [13]. Compared to the Fourier component with the frequency F h , that with the frequency F e , which was not observed in reference [13], has a relatively small amplitude at θ =1 5 • and 32 • ,a n d can only be reliably studied at θ =49 • .

A 2e -q 0 2 M h q 0 2 M 2e 2e+2h M 2e+2h 4e+h M 4e+h c) T = 5 K, B 0 = 2 T A 2e -q 0 2 M h q 0 2 M 2e 4e+h M 4e+h d) T = 1.5 K, B 0 = 100 T 2e+2h M 2e+2h a) b d c b) T = 1.5 K, B 0 = 2 T A 2e -q 0 2 M h q 0 2 M 2e 2e+2h M 2e+2h 4e+h M 4e+h
It must be noticed that the studied compound contains magnetic Fe 3+ ions. As reported in the case of λ-(BETS) 2 FeCl 4 (where BETS stands for bisethylenedithio-tetraselenafulvalene) [19,20], these magnetic ions induce an exchange field leading to angledependent splitting of the oscillation frequency which may alter the oscillations amplitude. Nevertheless, such a splitting is not observed neither in the considered compound nor in the compound with the same composition and β" structure in fields of up to 55 T [21][22][23]. This could be due to the lower, by a factor of two, Fe 3+ concentration in the present case hence to a reduced exchange coupling constant.

A naive analysis of the temperature dependence of A τ h , relevant to F h , assuming that only one orbit contributes (i.e. through Eq. ( 1)) yields an effective mass m h (θ) × cos(θ)of0.96±0.05, 0.93±0.02 and 0.69±0.07 at θ =15 • , 32 • and 49 • , respectively. Within the same hypothesis, the magnetoresistance data at θ =0 • of reference [13] yield m h =1 .11 ± 0.04 1 . In other words, the product m h (θ) × cos(θ) would monotonously decreases as θ increases. Hence, at variance with the angle dependence of the frequency, the cosine law (m h (θ)=m h (0)/ cos θ)c o nsistent with a 2D FS, would not be followed for the effective mass within this assumption. This result strongly suggests that equation ( 1) is unable to account for the temperature dependence of the amplitude. Besides, it can be checked that the field dependence of the amplitude cannot be accounted for by equation (1) as well. As discussed in the preceding section and reported hereafter, other orbits (such as 2e,2e+2h, etc.) with effective masses different from each other enter the oscillatory spectra.

A c c o r d i n gt oe q u a t i o n( 6), oscillatory torque amplitudes A τ e and A τ h , of the Fourier components with frequencies F e and F h =2F e observed in Figure 4,a r eg i v e nb y :

A τ e = τ 0 tan(θ)B F e π A e ,A τ h = τ 0 tan(θ)B 2F e π A 2e , (7)
respectively, where τ 0 is a prefactor depending on the cantilever stiffness, crystal geometry, etc. and θ is the angle between the field direction and the normal to the conducting plane. Amplitudes A e and A 2e in equation (7) are given by equations (3)a n d( 4), respectively. We will limit ourselves to small n h and n e values. Namely, in addition to the basic electron and hole orbits, only the second harmonic of the electron orbit (2e)a n dt h eM Bo r b i t s 1 The, even larger, effective mass value m h =1 .26, reported in reference [13] is deduced from a high temperature asymptotic approximation of the temperature damping factor. composed of one hole and one electron orbit (e + h), 3 electron and one hole orbits (3e + h), 2 electron and 2 hole orbits (2e +2h) and 4 electron and one hole orbits (4e + h) are taken into account. In short, the contribution of the MB orbits and harmonics composed of more than 5 individual orbits are neglected. It must be kept in mind that spin damping factors may influence the sign of a given contribution or, in other words, induce a π dephasing, as observed on either side of a spin zero angle [5].

Figure 5 displays field dependence at various temperatures of either A τ e or A τ h for θ =15 • and 32 • (crystal #1) and 49 • (crystal #2) while Figure 6 displays the angle dependence of A τ h at T =1.5Ka n dB =40Tforcrystal #2. Solid lines in these figures are the best fits of equation (7) to the Fourier amplitudes: the same set of parameters stands for all the data, namely m e =0 .93 ± 0.04, m h =0 .88 ± 0.04, g * e =2 .27 ± 0.12, g * h =2 .06 ± 0.09. Dingle temperatures and MB field, which jointly govern the field dependence of the amplitude are obtained with a large uncertainty. Dingle temperatures are rather large, of the order of few K, whereas B 0 is in the range between 0 and 4 T, which is rather small, in agreement with the FS of Figure 1. Owing to the small size of e and h orbits, compared to that of the FBZ, effective mass values are large which suggests significant renormalization due to manybody effects.

According to Dharma-wardana [24], electron correlations are predicted to yield large Landéf a c t o r .C o nsistently, large values are observed, although they are within the spread range usually reported for organic metals [START_REF] Wosnitza | Fermi Surfaces of Low-Dimensional Organic Metals and Superconductors[END_REF]. In that respect, puzzling data can be found in the literature since, for example, values as small as g * α = 1.6a n dg * β =1 .5 are observed for the strongly correlated κ-(ET) 2 Cu(SCN) 2 compound [25]. Oppositely values larger than 2 are reported for e.g. κ-(ET) 2 I 3 (g * β = 2.27 [26]).

Composites orbits, e + h and 2e have significant contributions to A τ e and A τ h , respectively, as observed in Figure 5c. Nevertheless, higher order terms (linked to 2e +2h and 3e + h) are small, due to rather large Dingle temperatures. Finally, as reported in Figure 5c for the data at 49 • , contributions of e and 3e + h to A e have opposite signs. This result which also holds at 15 • and 32 • , explains why this amplitude is small compared to A τ h .

Summary and conclusion

Fourier spectra of de Haas-van Alphen oscillations of compensated two-dimensional metals with Fermi surface composed of one hole and two electron components have been considered. The two main Fourier components observed have frequencies F e and F h =2 F e , corresponding to the electron and hole orbits area. Nevertheless, it is demonstrated that, taking into account the opposite sign of electron and hole orbits, a given Fourier component involves combination of several orbits, the contribution of which must be included in the calculations. Such FS, which is a textbook case, is observed in the strongly 2D organic metal α-'pseudo-κ'-

  Fig. 1. (a) Crystal structure of α-'pseudo-κ'-(ET)4H3O[Fe(C2O4)3]•(C6H4Br2). α-type and 'pseudo-κ'-type layers are conducting and insulating, respectively. (b) Fermi surface relevant to the α-t y p el a y e r s ,a c c o r d i n gt oZ o r i n a et al.[13]. Blue and green areas mark electron-and hole-type orbits, respectively. Ellipses in dashed lines mark the orbits from which the Fermi surface originates. Labels Γ ,Xa n dY refer to the (0, 0), (a * /2, 0) and (0, b * /2) points of the first Brillouin zone, displayed as a rectangle in solid line.

Fig. 2 .

 2 Fig. 2. Compensated Fermi surface with classical representation of the quasi-particle orbits. Phase variation of the wavefunction is given by exp(iSe) around the electronic orbit e and exp(iS h )a r o u n dt h eh o l eo r b i th,w h e r eS e(h) is the electron (hole) surface area. Either magnetic breakdown occurs with the probability amplitude ip0 between two orbits or reflection is allowed with the probability amplitude q0 =(1p 2 0 ) 1/2 .

Fig. 3 .

 3 Fig. 3. (a) Fourier spectra of the oscillatory magnetization calculated at a mean field of 40 T from equation (6)w i thme =1, m h =1 ,g * e =2 ,g * h =2 ,TDe =1K ,T Dh =1K .L a b e l sb, c, d correspond to T =1 .5Ka n dB0 =2T ,T =5Ka n d B0 =2T ,T =1 .5Ka n dB0 = 100 T, respectively. (b)-(d) Field dependence of the amplitude of the Fourier component with frequency F h =2 Fe corresponding to labels b, c, d,r espectively, of (a). Solid lines are calculated from equation (4).Long dashed, short dashed, dotted and dash-dotted lines are the components linked to the h,2 e,2 e +2h and 4e + h orbits, respectively. The magnetic breakdown damping factors for the orbits 2e +2h and 4e + h are given by R 2e+2h =3p 4 0 q 2 0 -2p 2 0 q 4

Fig. 4 .

 4 Fig. 4. Fourier analysis in the field range 25-55 T of the magnetic torque data reported in the insert for various directions of the magnetic field with respect to the normal to the conducting plane (angle θ).

  3e + h is negligible at high temperature and large scattering rate. Of course, all the considered amplitudes also depend on the field-and temperature-independent spin damping factor R s pη through the product g * η m η . 4Resultsanddiscusion This section is devoted to de Haas-van Alphen oscillations of α-'pseudo-κ'-(ET) 4 H 3 O[Fe(C 2 O 4 ) 3 ] •(C 6 H 4 Br 2 ), the FS of which (see Fig. 1b) is relevant to the above calculations. Note that this FS can be considered as resulting from the hybridization of a series of ellipses centered at the Γ point. The area of these ellipses, which is related to the total number of holes per unit cell in the HOMO bands, is twice the area of the cross section of the FBZ because the repeat unit of the layer contains eight ET molecules with an average charge of +1/2, i.e. a total of four electrons per repeat unit. It is interesting to point out how this FS differs from that of other α-type ET salts, such as the α-(ET) 2 [MHg(SCN) 4 ] family. As pointed out by Mori et al. [18], the closed and open portions of the latter can also be considered to arise from the hybridization of a series of ellipses with an area equal to the cross section of the first Brillouin zone (FBZ) and centered at Y (using the axes notation of Fig. 1b). Since the repeat unit of these salts contains only four ET molecules, the FS of Figure 1b is just a folded version along the b * direction of such FS. The different kind of overlap of the ellipses generated by the folding thus leads to the very different FS for the present salt (referred to as case (ii) above) and that of the α-(ET) 2 [MHg(SCN) 4 ] family (referred to as case (i) above).

Fig. 5 .

 5 Fig. 5. Ln(|A τ |)v s .1 / B at various temperatures for (a) A τ h (solid circles) and A τ e (solid squares) at θ =4 9 • ,( b )A τ h at θ =1 5 • (solid squares) and 32 • (solid circles) and (c) field dependence of A τ h (solid circles) and A τ e (solid squares) at θ = 49 • at 1.5 K. Solid lines are fits of equations (7), obtained with me =0 .93, m h =0 .88, g * e =2 .27, g * h =2 .06, TDe =5K , T Dh =4Ka n dB0 = 2 T. Long dashed, short dashed, dotted and dash-dotted lines in (c) are the contributions of the various components entering the fittings (see text and Fig. 3).

Fig. 6 .

 6 Fig. 6. Angle dependence of the amplitude |A τ h | at T =1.5K and B = 40 T. Solid line is the best fit of equation (7)t ot h e data, obtained with the same set of parameters as in Figure5. Long dashed, short dashed and dotted lines in the insert are the contributions of h,2 e and 2e +2h, respectively.

  ET) 4 H 3 O[Fe(C 2 O 4 ) 3 ]•(C 6 H 4 Br 2 ). Magnetic torque oscillations of this compound have been studied up to 55 T for various directions of the magnetic field with respect to the conducting plane. It is demonstrated that data analysis performed assuming that only single electron and single hole orbits contribute to Fourier components with frequency F e and 2F e , respectively, cannot account for the data. In other words, additional orbits generated by tunneling and reflection at magnetic breakdown junctions must be taken into account. Calculations are in good quantitative agreement with the data. This work has been supported by EuroMagNET II under the EU Contract No. 228043, and MINECO-Spain (Projects FIS2012-37549-C05-05 and CSD 2007-00041).