# Quantum oscillations in the linear chain of coupled orbits: the organic metal with two cation layers -(Et)CoBr(CHCl)

###### Abstract

Analytical formulae for de Haas-van Alphen (dHvA) oscillations in linear chain of coupled two-dimensional (2D) orbits (Pippard’s model) are derived systematically taking into account the chemical potential oscillations in magnetic field. Although corrective terms are observed, basic () and magnetic breakdown-induced ( and 2 - ) orbits can be accounted for by the Lifshits-Kosevich (LK) and Falicov-Stachowiak semiclassical models in the explored field and temperature ranges. In contrast, the ’forbidden orbit’ - amplitude is described by a non-LK equation involving a product of two classical orbit amplitudes. Furthermore, strongly non-monotonic field and temperature dependence may be observed for the second harmonics of basic frequencies such as 2 and the magnetic breakdown orbit + , depending on the value of the spin damping factors. These features are in agreement with the dHvA oscillation spectra of the strongly 2D organic metal -(ET)CoBr(CHCl).

###### pacs:

71.18.+y###### pacs:

71.20.Rv###### pacs:

72.15.GdFermi surface: calculations and measurements; effective mass, g factor. Polymers and organic compounds. Galvanomagnetic and other magnetotransport effects.

## 1 Introduction

Quasi-two-dimensional (q-2D) multiband organic compounds (ET)X (where ET stands for the bis-ethylenedithio-tetrathiafulvalene molecule and X is a monovalent anion) exhibit very rich phase diagrams with ground states ranging from antiferromagnetic insulator to Fermi liquid. In particular, superconducting and magnetic ordered phases are observed close to each other which is reminiscent of cuprates and iron-based superconductors (for a recent review, see [1] and references therein). These compounds share the same Fermi surface (FS) topology which is an illustration of the linear chain of coupled orbits introduced by Pippard in the early sixties [2]. Such FS is liable to give rise to a network of orbits coupled by magnetic breakdown (MB) in large magnetic fields. Quantum oscillations in such systems have been extensively studied during the two last decades (for a review, see [3] and references therein). The main feature of the oscillation spectra of these compounds is the presence of frequency combinations that are forbidden in the framework of the semiclassical model of Falicov-Stachowiak [4]. Theoretical studies have demonstrated that these Fourier components result from both the formation of Landau bands due to coherent MB, instead of discrete levels, and the oscillation of the chemical potential enabled by the 2D character of the FS. However, easy to handle analytic tools necessary to quantitatively account for the data are still lacking. As a matter of fact, comprehensive theoretical calculations taking into account coherent MB and oscillation of the chemical potential have been reported in Ref. [5] and implemented in de Haas-van Alphen (dHvA) data of the above mentioned compound. Nevertheless, as pointed out in Ref. [5], equations are complex and data could be analyzed only numerically.

The aim of this paper is therefore to provide analytical formulae accounting for quantum oscillation spectra of these systems. In the first step, analytical calculation of the Fourier amplitude of the various dHvA oscillations predicted for the FS illustrating the Pippard’s model is reported. In the second step, the FS of the recently synthesized strongly 2D charge transfer salt -(ET)CoBr(CHCl) [6] is presented and, in the third step, analysis of the observed field- and temperature-dependent quantum oscillation spectra are reported. An excellent agreement with the calculations is obtained.

## 2 Experimental

The studied crystals were synthesized by electrocrystallization technique as reported in Ref. [6]. They are elongated hexagonal platelets with approximate dimensions 1.5 0.5 0.04 mm. X-ray diffraction data were collected both at room temperature in Chernogolovka with a KM-4 single-crystal diffractometer (Kuma Diffraction) and at 100 K with an Xcalibur diffractometer (Oxford Diffraction) at the Laboratoire de Chimie de Coordination in Toulouse. Tight-binding band structure calculations were based upon the effective one-electron Hamiltonian of the extended Hückel method [7], as reported in Ref. [8]. Magnetic torque and magnetoresistance were measured in pulsed magnetic fields of up to 55 T with a pulse decay duration of 0.32 s. Magnetic torque measurements were performed with a commercial piezoresistive microcantilever [9] in the temperature range from 1.4 K to 4.2 K. The size of the sample studied was approximately 0.12 0.1 0.04 mm. Variations of the cantilever piezoresistance was measured with a Wheatstone bridge with an excitation at a frequency of 63 kHz. The angle between the normal to the conducting plane and the magnetic field direction was = 7. Magnetoresistance was measured thanks to a contactless radio frequency measurement technique based on tunnel diode oscillator as reported in Ref. [10]. A pair of counter-wound coils made with copper wire of 50 m in diameter was used. As pointed out in [11], in the case where the interlayer resistivity is much larger than the in-plane resistivity, the contribution of the former parameter to the measured signal is negligible. The time evolution of the oscillator frequency during the field pulse is deduced from successive short-term fast Fourier transforms of the raw signal. In contrast, analysis of the oscillatory magnetoresistance and magnetic torque is based on discrete Fourier transforms of the field-dependent data, calculated with a Blackman window after removal of a non-oscillating background. Zero-field interlayer and in-plane 4-point resistance was measured with 1 A current at 77 Hz. Electrical contacts were made using annealed platinum wires of 20 m in diameter glued with graphite paste.

## 3 Calculation of de Haas-van Alphen amplitudes

We consider a two-band system with band extrema (see Fig. 1(c)) and effective masses . In the low temperature limit, the grand potential of a two-dimensional slab with area is given by a series of harmonics of classical frequencies :

(1) |

where = / is the quantum flux and = 2 = 14.694 T/K. Effective masses are expressed in units and chemical potential in Tesla. The index stands for all the possible classical orbits , , + , 2 - , , excluding their harmonics, with frequencies = ( - ) and effective masses . Frequencies depend on the chemical potential since they represent a filling factor corresponding to classical orbits. Coefficient is the symmetry factor of the orbit . For example, and . is the phase of the orbit , times the number of extrema of the orbit in the or direction, for example, and . These phases are important for determining the sign of amplitudes. In particular, we define two fundamental orbits with frequencies, = ( - ) ( = ) and = ( - ) + ( - ) which can be written as = ( - ) (see Fig. 1(d)). Therefore, + is identified to the effective mass of the MB orbit . The damping factors can be expressed as = where the thermal, Dingle, MB and spin damping factors are given by:

(2) | |||

(3) | |||

(4) | |||

(5) |

respectively. is the Dingle temperature ( = /2 where is the
relaxation time).
Integers and are the (even) number of MB tunnelings and reflections, respectively,
encountered by the quasiparticle along its closed trajectory. The MB tunneling and reflection probabilities are
given by = and = 1 - , respectively, where is the MB field.
is the effective Landé factor^{1}^{1}1FS warping can be accounted for by the additional damping factor J(4/), where is the interlayer transfer integral [12]. In most q-2D organic metals, this factor is close to 1. .
The electron density per surface area is given by d/d = -.
In zero-field, Eq. 3 yields = ( + ) - - .
In presence of a magnetic field, satisfies the following implicit equation:

(6) |

The oscillating part of the magnetization can be computed systematically by inserting Eq. 6 in Eq. 3 and deriving the resulting expression with respect to the magnetic field. Direct numerical resolution of this implicit equation could be done. Besides, a more user-friendly controlled expansion in term of damping factors can be derived systematically at any possible order. Keeping successive terms in powers of up to the second order, yields analytically the following expression for the oscillating part of the magnetization:

(7) | |||||

where frequencies are evaluated at = , namely, . Eq. 7 can be written as a sum of periodic functions with frequencies and Fourier amplitudes where the index stands for either classical orbits or ’forbidden’ orbits such as - . Relevant examples for experimental analysis are given below:

(8) | |||||

(9) | |||||

(10) | |||||

(11) | |||||

(12) | |||||

(13) | |||||

(14) |

In the framework of the LK model, the amplitude of a Fourier component linked to a basic orbit involves only one
damping factor ( ). This is also the case for a MB orbit, the effective mass
of which is given by the Falicov-Stachowiak model ( = 2 - )
[4]. According to this statement, the leading term of Eqs. 8 and 10 which accounts for the and components, respectively, reproduces the LK equation. Furthermore, deviations from the LK behaviour due to the high order terms involving products of damping factors are only significant in the low temperature and high field ranges. This statement also stands for 2- since is significantly higher than the product (see Eq. 14). As a consequence, the
Falicov-Stachowiak model should apply for this component as well. In contrast, algebraic sum of damping terms
in Eqs. 9 and 13, where minus signs accounts for dephasings, may
cancel at field and temperature values dependent on the effective masses, Dingle temperature, MB field,
as displayed in Fig. 2 relevant to 2. This point has already been reported for the
second harmonic of the basic orbits both for compensated [13] and un-compensated [14] metals. In
that respect, since algebraic sums are involved, care must be taken of the spin damping factor sign^{2}^{2}2Since the LK equation involves a single damping term, the absolute value of the spin damping factor
is considered in the case where data are analyzed on this basis. However, dephasing is observed on either
side of spin-zero angles as reported in [15].. Indeed, the value of the spin damping factor, which is changed
through the value of in Fig. 2, can have a drastic effect on the field and temperature
dependence of the amplitude in this case. Finally, the Fourier amplitude of the ’forbidden orbit’
- is accounted for by Eq. 12. Although a non-LK behaviour is evidenced, a monotonic field and temperature dependence is expected owing to the negligibly small value of the high order terms. The consistency of Eqs. 8 to 14 with experimental data is examined
in the next section.

## 4 Results and discussion

Even though the organic metal -(ET)CoBr(CHCl) undergoes a phase transition at 343 K [6], no additional phase transition is revealed by X-ray diffraction below room temperature, at which crystals are synthesized, down to 100 K. As displayed in Fig. 1, the central feature of the crystalline structure at room temperature is that the unit cell has two different donors planes labeled A and B. According to band structure calculations, layer A is insulating while layer B is metallic. FS of layer B is composed of one closed tube centered at Y with an area of 18 of the first Brillouin zone (FBZ) area (yielding orbit in magnetic field) and two q-1D sheets with rather small gap in between. Therefore, it is expected that the MB orbit , which corresponds to the hole tube centered at with an area equal to that of the FBZ is observed in moderate fields, giving rise to the textbook linear chain of coupled orbits. In addition, since layers A are insulating, conducting planes are strongly separated from each other which suggests a strongly 2D behaviour. In line with this statement, the room temperature resistivity anisotropy (interlayer)/ (in plane) is as large as 10 [6]. Furthermore, according to the data in Fig. 3, this anisotropy further increases up to (interlayer)/(in plane) = 10 as the temperature decreases down to liquid helium temperatures.

Magnetic torque (dHvA oscillations) and tunnel diode oscillator [Shubnikov-de Haas (SdH) oscillations] data are displayed in Fig. 4. Besides Fourier components linked to the closed orbit and to the MB orbit , few frequency combinations typical of the FS displayed in Fig. 1 are observed in the high field range of the torque spectra. In contrast, numerous combinations are observed in SdH spectra. As an example, the Fourier component 2 - 3 is clearly detected in Fig. 4(b). This frequency, which to our knowledge is observed for the first time, could correspond to a quantum interferometer path. However, its amplitude is unexpectedly large. The frequencies linked to the and orbits are ( = 0) = 4600 10 T, in agreement with the FBZ area at 100 K, and ( = 0) = 944 4 T, respectively. The latter value yields a cross section area of 20.6 % of the FBZ area, in satisfactory agreement with the band structure calculations of Fig. 1. Field- and temperature-dependent Fourier amplitudes can be studied at the light of the results given in the preceding section, keeping in mind that dHvA amplitudes A are related to torque oscillation amplitudes as /tan(). Effective masses, derived from Eq. 8 at various magnetic field values are field-independent: = 1.81 0.05, = 3.52 0.19 (see Fig. 5(a)). In agreement with results of Ref. [16], is twice within the error bars. At odds with torque results, effective masses deduced from SdH data, in particular those relevant to the component, definitely increase as the magnetic field increases. This behaviour indicates that the above calculations, relevant to dHvA oscillations, cannot account for the SdH data at high field.

Focusing on dHvA oscillations, Dingle plots of and can be obtained plotting
/ where is given by Eq. 2, assuming the LK formalism accounts
for the data, i.e. the high orders terms of Eq. 8 can be neglected. As reported in
Fig. 5, all the data relevant to the and components lies, respectively,
on the same curve within the experimental scattering which suggests that the above assumption holds.
Dingle temperature and MB field can be derived for components which do not involve tunneling ( = 0
in Eq. 5). Although this is the case of , a very broad maximum is observed in
Fig. 5, leading to a very large uncertainty on the MB field. More specifically,
values ranging from 0 to 70 T can account for the data, in which case T ranges from 1.4 to 0.5 K,
respectively. Assuming the Dingle temperature linked to the orbit is the same as that of
considerably reduces the uncertainty since the well defined slope = -(+2) of the Dingle plot for yield an additional relationship between T and B. This assumption yields = 35 5 T and = 0.79
0.10 K. With these values of effective masses, Dingle temperature and MB field, it is possible to check
whether or not the high order terms of Eq. 8 can be neglected. Inset
of Fig. 5(b) displays the field dependence of / calculated
for the highest field range covered by the experiments, assuming further that = 2^{3}^{3}3Effective Landé factor as low as = 1.5 1.6 are reported for the strongly
correlated compound -(ET)Cu(NCS) [5], nevertheless = 2 is consistent with
data of other ET salts [17, 18].. Although the contribution of these factors rises as the field
increases, it remains smaller than the experimental scattering for both and .

The field and temperature dependence of the Fourier amplitude relevant to frequency combinations is displayed in Fig. 6. Since their amplitude is much smaller than that of the orbits and , only mass plots at high field and Dingle analysis at the lowest explored temperature are considered. As already discussed for the second harmonics 2 (see Fig. 2), amplitudes are strongly dependent on the spin damping factors value. Solid lines in Fig. 6 are the best fits of Eqs. 12 to 14, obtained with the same values of , , , and as above, with only a prefactor as a free parameter for each of the data: a very good agreement is obtained, in particular for the ’forbidden orbit’ - . It can be remarked that the behaviour of both 2 - and + are close to the predictions of the Falicov-Stachowiak model. Whereas this feature is mainly due to the small value of compared to in the former case, it can be attributed to the peculiar set of values of the spin damping factors (/ = -0.22) in the latter case.

## 5 Summary and conclusion

Analytical formulae for dHvA oscillations relevant to a class of FS illustrating the Pippard’s model and observed in many q-2D organic conductors have been derived systematically taking into account the chemical potential oscillations in magnetic field (Eqs. 8 to 14). Although high orders terms are observed, basic () and MB-induced ( and 2 - ) orbits can be accounted for by the LK and Falicov-Stachowiak semiclassical models, at least at moderate fields and temperatures. In contrast, the ’forbidden orbit’ - is accounted for by a non-LK equation, involving products of classical amplitudes relevant to and . Furthermore, strongly non-monotonic field and temperature dependence may be observed for the second harmonics of basic frequencies such as 2 and the MB orbit + , depending on the value of the spin damping factor .

These formulae are in agreement with the dHvA oscillation spectra of the strongly 2D organic metal -(ET)CoBr(CHCl), assuming = 2. However, SdH oscillations spectra exhibit a tremendous number of Fourier components suggesting, besides quantum interference contributions, more severe deviations from the LK model that remains to be explained within a SdH theory including quantum interference contribution.

###### Acknowledgements.

This work has been supported by EuroMagNET II under the EU contract number 228043, DGES-Spain (Projects FIS2009-12721-C04-03 and CSD2007-00041) and by the CNRS-RFBR cooperation under the PICS contract number 5708. We acknowledge the technical help of Laure Vendier at the X-ray facility of the LCC-Toulouse.## References

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