Magnetotransport studies of the organic superconductor and ...Ludwi… · The BETS donors were...

Walther-Meißner-Institut Technische Universität Physik Departmentfür München Lehrstuhl E23

Tieftemperaturforschung

Magnetotransport studiesof the organic superconductor

and antiferromagnetκ-(BETS)2FeBr4

Master Thesis

Ludwig Schaidhammer

Themensteller: Prof. Dr. Rudolf GrossGarching, 07. November 2014

http://www.wmi.badw.de/

http://www.tum.de/

http://www.ph.tum.de/

Contents

1 Introduction 1

2 The organic metal κ-(BETS)2FeBr4 32.1 Synthesis of κ-(BETS)2FeBr4 single crystals . . . . . . . . . . . . . . . . 32.2 Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Band structure and electronic properties . . . . . . . . . . . . . . . . . 52.4 Phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4.1 Pressure-temperature phase diagram . . . . . . . . . . . . . . . 72.4.2 Magnetic phase diagrams . . . . . . . . . . . . . . . . . . . . . . 8

3 Theoretical background 133.1 Semiclassical Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . 13

3.1.1 Magnetoresistance in Conventional Metals . . . . . . . . . . . . 133.1.2 Magnetoresistance in Layered Organic Metals . . . . . . . . . . 15

3.2 Magnetic Quantum Oscillations . . . . . . . . . . . . . . . . . . . . . . 203.2.1 Shubnikov-de Haas Oscillations . . . . . . . . . . . . . . . . . . 223.2.2 Damping Factors . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.3 Magnetic Breakdown . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4 Magnetic States in κ-(BETS)2FeBr4 . . . . . . . . . . . . . . . . . . . . 26

3.4.1 Paramagnetism of localised magnetic moments . . . . . . . . . . 263.4.2 Antiferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Experimental setup 314.1 Experimental equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1.1 Superconducting magnet . . . . . . . . . . . . . . . . . . . . . . . 314.1.2 Two axes rotator . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.1.3 3He-insert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.1.4 Pressure cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2 Measurement techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2.1 Resistance measurements . . . . . . . . . . . . . . . . . . . . . . 354.2.2 Torque measurements . . . . . . . . . . . . . . . . . . . . . . . . 36

5 Results and discussion 395.1 The normal metallic state . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.1.1 Shubnikov-de Haas oscillations . . . . . . . . . . . . . . . . . . . 405.1.2 Angle-Dependent Magnetoresistance Oscillations and Lebed Magic-

Angle Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . 43

iii

Contents

5.1.3 Coherence Peak . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2 The antiferromagnetic state . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.2.1 Shubnikov-de Haas oscillations . . . . . . . . . . . . . . . . . . . 515.2.2 Magnetic phase diagrams . . . . . . . . . . . . . . . . . . . . . . 57

5.3 Superconducting state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6 Summary 67

Bibliography 70

Acknowledgements 79

iv

1 Introduction

In the year 1964, W. H. Little [1] stated the hypothesis based on the BCS theoryof superconductivity that in organic polymers with highly polarisable side chains thepairing of electrons to Cooper pairs should be highly favourable. Little assumed thatin such structures Cooper pairs are formed by the interaction of electrons with specialelectronic excitations within the molecules, so-called excitons. Within this model theexcitons would take over the role of phonons in the standard BCS theory. However,Little’s hypothesis could not be confirmed so far but led to the discovery of variousother organic superconductors of different kind.Fifteen years after Little’s hypothesis, the first organic superconductor (TMTSF)2PF6

was discovered [2]. TMTSF stands for tetramethyltetraselenafulvalene. However, atambient pressure, this material shows a metal-insulator transition to a spin-densitywave (SDW) state at low temperatures. By the application of a considerably largepressure of 12 kbar, superconductivity with a critical temperature Tc ≈ 0.9 K can beobserved. By replacing the anion PF6 by AsF6, SbF6, TaF6, ReO4, FSO3, and ClO4,a series of isostructural superconductors was discovered shortly afterwards [3]. Of allthese materials, only for (TMTSF)2ClO4 superconductivity was observed at ambientpressure.Within the next few years, a large variety of organic superconductors with different

building blocks showing larger critical temperatures in comparison to the TMTSF basedsalts was found. Among these the BEDT-TTF [bis(ethylenedithio)-tetrathialfulvaleneor ET for short] molecule plays an important role. The critical temperatures of the su-perconductors based on ET reach up to ∼11.6 K [4]. The ET salts are characterised bytheir two-dimensional layered structure, resulting in a high anisotropy of the electricalconductivity. The electronic bands are formed by the overlapping molecular π orbitalsof the electrons in the conducting layers. For these compounds many different kinds ofstacking of the ET molecules are possible. These polymorphic phases are denoted byα, β, κ, θ, and so on.In general, due to the low dimensionality of their electronic bands, and despite of

their rather complicated crystal structure, organic superconductors exhibit very simpleFermi surfaces. Together with their high electronic anisotropy and high crystal qual-ity they are perfectly suited for studying general properties of quasi-two-dimensionalmetallic systems. For example in high magnetic fields many novel effects concerningthe interlayer resistance were found [5]. Another aspect is the rich variety of phasediagrams, which can be observed for low temperatures, with for example supercon-ducting, Mott insulating, SDW, charge-density wave (CDW) and antiferromagneticstates. Often several of these states are observed together in the same compound.In the field of organic conductors showing quasi-two-dimensional conduction and

magnetic ordering, also compounds based on BETS were extensively studied. The

1

1 Introduction

BETS molecule was obtained as a modification of the BEDT-TTF compound by replac-ing Se atoms for S atoms in the TTF-fragment [6]. Typical examples for organic super-conductors with magnetic ions based on BETS are (BETS)2FeBr4 and (BETS)2FeCl4.The conducting and magnetic systems of these compounds are spatially separated andinteract via exchange coupling. (BETS)2FeCl4 can crystallise in the κ or λ phase,whereas for (BETS)2FeBr4 only the κ phase was observed so far. All three compoundsexhibit an antiferromagnetic ground state, but while the κ compounds remain metallicat low temperatures, λ-(BETS)2FeCl4 shows a Mott-insulating state below the antifer-romagnetic transition [7].The focus of this master thesis is on κ-(BETS)2FeBr4. The physical properties

were mainly investigated by electric transport but also by magnetic torque measure-ments. This very interesting antiferromagnetic superconductor was well studied byother groups (see refs. [8–20]), however, still several open questions remain. For in-stance the observation of a low frequency in the antiferromagnetic state was reportedby Konoike et al. [16]. In this thesis the angular dependence of this frequency is stud-ied, and we also plan to search for further frequencies. An open question is also thevalue of the energy gap between different parts of the Fermi surface. In addition thenormal state properties of the electrical conductivity will be investigated by angle-dependent magnetoresistance oscillations in order to clarify open questions in formerreports. For instance, characteristic oscillations originating from the one-dimensionalpart of the Fermi-surface do not follow the standard theory. In the superconductingstate magnetic-field-induced superconductivity will be investigated by torque measure-ments. Finally, there are open questions concerning the magnetic phase diagram ofκ-(BETS)2FeBr4. For instance, recent studies of the phase diagram of κ-(BETS)2FeCl4revealed interesting new results such as the observation of a spin-flop phase [21]. Thus,one focus will also be on the search for this phase in κ-(BETS)2FeBr4. Additionallythese physical properties will be investigated by interlayer resistance measurements un-der hydrostatic pressure, because tracing the physical properties under pressure oftenyields a better understanding of the ambient pressure results. In this thesis pressuresof 1.9 kbar and 4.5 kbar were applied.A quick overview of the structure of this thesis is given below:In the second chapter the organic superconductor κ-(BETS)2FeBr4 is introduced by

presenting the crystal structure, the electronic band structure and the phase diagram.The third chapter provides the theoretical background describing magnetoresistive

effects as far as they are discussed in this work, some aspects of superconductivity oflayered compounds and antiferromagnetism.The experimental setup will be described in Chapter 4.Finally, in Chapter 5 the results of the measurements are presented and analysed. At

the end of each section the conclusions are discussed and compared to previous reports.A summary of the most important results of this thesis is given in the last chapter.

2

2 The organic metal κ-(BETS)2FeBr4

2.1 Synthesis of κ-(BETS)2FeBr4 single crystalsThe κ-(BETS)2FeBr4 samples used in this experiment were provided by N. D. Kushch1.The single crystals of κ-(BETS)2FeBr4 (BETS = bis(ethylenedithio)-tetraselenafulvalene) were grown by electrochemical oxidation as described in refs.[6, 10]. The BETS donors were prepared as reported in ref. [22]. For electrochem-ical oxidation the BETS molecules are brought into a 10% ethanol-chlorobenzene so-lution together with the tetraethylammonium iron(III) tetrabromide, which serves assupporting electrolyte. Then an electrical current of 0.7 µA is applied via platinumelectrodes and kept constant for a period of 2-4 weeks under nitrogen atmosphere atroom temperature or at 40C. At the end black plate-like samples with typical sizes of0.5×0.4×0.03 mm2 appear on the platinum anode.

2.2 Crystal StructureCrystals of κ-(BETS)2FeBr4 are composed of conducting layers and insulating layers,which are arranged alternately along the b axis [10]. Figure 2.1 shows the crystalstructure of κ-(BETS)2FeBr4 viewed along the three crystallographic axes a, b and cat room temperature. The crystal structure is orthorhombic with the lattice constantsa = 11.787 Å, b = 36.607 Å and c = 8.504 Å [11].The conducting layers are composed of dimers of organic BETS molecules [see Fig.

2.2(a))]. BETS is analog to BEDT-TTF a basic donor molecule, which can be foundin most of the known layered organic superconductors [23]. Adjacent dimers of BETSare arranged in a roughly orthogonal manner [see Fig. 2.1(b))] which corresponds tothe κ-type packing structure.The BETS molecule and the FeBr4 anion with labeling of the elements are shown

in Figs. 2.2(a) and 2.2(b), respectively. In the insulating anion layer, the shortestFe-Fe distances along the a and c axis are 5.878 Å and 8.504 Å, respectively, and theshortest Br-Br distances between neighbouring FeBr−4 anions along the a and the caxis are 4.137 Å and 4.637 Å [24]. Both the Br-Br distances are longer than the sumof the van der Waals radii of bromines (3.90 Å). However, from the fact that the Br-Br distance along the a axis is much shorter than along the c axis one can considerthe direct intermolecular interaction to be stronger along the a axis than along the caxis. Several Br-S contacts between the BETS molecules and the anions almost equalto the sum of the van der Waals radii of bromine and sulfur atoms (3.80 Å) exist,with the shortest Br-S distance being 3.708 Å. In contrast to that in λ-(BETS)2FeCl4

1Institute of Problems of Chemical Physics, 142432 Chernogolovka, Russian Federation

3


(a)

(b) (c)

Figure 2.1: Crystal structure of κ-(BETS)2FeBr4 and unit cell viewed along the (a) caxis, (b) b axis with labeling of the overlap integrals and (c) a axis. In each case onlyone layer is presented.

several short contacts between the BETS molecules and the FeCl4 anions, which aresmaller than the corresponding van der Waals radii, can be found [24]. In particularin λ-(BETS)2FeCl4 one short Cl-Se contact exists, where the FeCl−4 anion reaches intothe donor layer. However, no such Br-Se contact was observed in κ-(BETS)2FeBr4

4

2.3 Band structure and electronic properties

(a) (b)

Figure 2.2: (a) Molecular structure of the BETS radical cation and (b) the FeBr−4

anion with labeling of the elements.

[10]. Therefore, the interaction between the π-electrons of the conduction layers andthe localised d-electrons of the anion layers is considered to be relatively weaker in κ-(BETS)2FeBr4 than in λ-(BETS)2FeCl4. In the conduction layers, short S-S contactstogether with short Se-S contacts provide the metallic bands due to the overlap of theirmolecular orbitals [10].The overlap integrals indicated in Fig. 2.1(b) are given by p = 77.33 eV, q =

8.11 · 10−3 eV, a = −22.41 eV and c = 35.14 eV [10], where p and q are the intradimerand interdimer overlap integrals, respectively.The BETS radical cations donate electrons to the FeBr−4 anion layers, therefore,

κ-(BETS)2FeBr4 is a member of the so-called charge transfer salts.

2.3 Band structure and electronic properties

The band dispersion and Fermi surface of κ-(BETS)2FeBr4 based on the crystal struc-ture at room temperature were calculated by Fujiwara et al. [10] using the extendedHückel approximation as shown in Fig. 2.3(a). Since one unit of the donor sheetcontains four BETS molecules, the energy dispersion has four bands with a midgap of0.68 eV between the upper two and lower two bands. This midgap energy is much largerthan that of κ-(BETS)2FeCl4 (0.48 eV), which suggests that the electron correlation ismuch stronger in κ-(BETS)2FeBr4 [6]. Since two BETS molecules donate one electronto one anion the upper two bands are effectively half filled.The anistotropy ratio of the resistivity is ρb

ρa≈ 200 at room temperature [18]. De-

spite of this the temperature dependences of the resistivities along the a and b axesbehave similar. In contrast to the normal metallic behaviour of κ-(BETS)2FeCl4, κ-(BETS)2FeBr4 shows an anomalous resistivity peak at low temperature, which is sup-pressed by increasing pressure to a few kbar [18].From the band calculations [10] it follows that the Fermi surface, analogous to that

of similar compounds like κ-(BETS)2FeCl4 and κ-(BETS)2GaCl4, consists of a cylinderwith a cross-sectional area of 100% of the first Brillouin zone (FBZ), which is foldedat the FBZ boundary. This folding of the Fermi surface produces a pair of open sheetsextending along the ka-direction and a weakly warped tube with a cross section of 19.1 %of the FBZ area. Therefore the Fermi surface of κ-(BETS)2FeBr4 is composed of quasi

5


(a) (b)

Figure 2.3: (a) Band structure (from [10]) and (b) 2D-Fermi surface of κ-(BETS)2FeBr4 at room temperature (from [12]).

one-dimensional and quasi two-dimensional parts. However, the interlayer transferintegral between the conducting layers is much larger than that for κ-(BETS)2FeCl4[12], suggesting that the warping of the Fermi surface is also larger.

Figure 2.4: (a) Fourier transform spectrum of SdH oscillations recorded in the fieldrange from 10 to 19.8 T (from [12]). (b) Experimentally derived Fermi surface ofκ-(BETS)2FeBr4 at low temperatures (from [16]).

From the measurement of the Shubnikov-de Haas oscillations at low temperatures andhigh fields three frequencies Fα = 850 T, Fβ = 4280 T and Fγ = 96.4 T in the Fouriertransform spectrum were revealed [9, 12, 16]. For the corresponding orbits Balicas et al.

6

2.4 Phase diagrams

[9] obtained cross sectional areas of 19.8, 99.9 and 2.4 % of the FBZ. As indicated inFig. 2.3(b), the α-orbit corresponds to the cross-sectional area of the cylindrical Fermisurface. Looking at the band calculation it follows that two bands are degenerate at thezone boundary. But since both α- and β-frequencies are observed at low temperaturesa gap between these two parts must exist, which in turn implies that the β-orbit is amagnetic breakdown orbit. The resulting Fermi surface at low temperatures is shownin Fig. 2.4(b). Possible explanations for the discrepancy between the calculated andthe experimental determined Fermi surface are given by Balicas et al. [9].An additional interesting fact is that the β-frequency actually splits up into two fre-

quencies Fβ1 and Fβ2 with the difference of frequencies Fβ2 − Fβ1 = 0.023Fβ. Thissplitting of the β frequency is caused by a large effective internal field produced by thespins of the localised d-electrons [25], which will be introduced later. The two frequen-cies Fβ1 and Fβ2 correspond to the Fermi surfaces of up- and down-spin electrons.The effective cyclotron masses for the α- and β-orbits were determined as mα =

5.2m0 and mβ = 7.9m0 [12] with m0 being the mass of a free electron.

2.4 Phase diagrams

2.4.1 Pressure-temperature phase diagram

In this section, the pressure-temperature phase diagram of κ-(BETS)2FeBr4 deter-mined by Otsuka et al. [18] is presented (see Fig. 2.5). The magnetic properties ofthis material are entirely dominated by the localised Fe3+ spins. At high tempera-

Figure 2.5: Pressure-temperature phase diagram of κ-(BETS)2FeBr4 (from [18]).

7


tures, κ-(BETS)2FeBr4 is paramagnetic, but below the Néel temperature TN = 2.5 Kthe spins of the localised electrons order antiferromagnetically. Finally the crystal ex-hibits a superconducting transition with Tc ≈ 1.4 K. With increasing the pressure, theantiferromagnetic transition temperature TN is reported to increase slightly, whereasthe critical temperature Tc decreases until the superconductivity is suppressed above apressure of 4 kbar.

2.4.2 Magnetic phase diagrams

κ-(BETS)2FeBr4 is one of the first organic metals showing an antiferromagnetic phase atambient pressure. The magnetic phase diagrams for B parallel to the a-axis and c-axis,respectively, are shown in Fig. 2.6. For B parallel to the c axis the transition lines bothfor the superconducting and the antiferromagnetic phase are clearly separated, whereasfor B parallel to the a axis, both lines coincide at low tempereatures. This particulartransition from the antiferromagnetic state to the paramagnetic state induced by themagnetic field has been referred to as a metamagnetic transition [15]. Also anotherinteresting phenomenon can be observed for magnetic fields along the in-plane axes,namely field induced superconductivity (FISC). The FISC phase appears for magneticfields 10 T . B . 15 T and temperatures below 0.8 K depending on the direction ofthe external magnetic field.

(a) (b)

Figure 2.6: B-T phase diagram for (a) B||a axis and (b) B||c axis (from [15]).

Superconducting and antiferromagnetic phase

κ-(BETS)2FeBr4 is the first organic superconductor for which the superconductingand antiferromagnetic phase coexist [8]. In the antiferromagnetic phase below thetransition field BAFM, the Fe3+ 3d spins order antiferromagnetically with the easy spinaxis parallel to the crystallographic a-axis [10]. An estimation of the π-d interactionbased on the extended-Hückel molecular orbital method was done by Mori et al. [26]and the stable magnetic structure of the 3d-spins was reported to be of a horizontal

8

2.4 Phase diagrams

type as shown in Fig. 2.7 with the magnetic unit cell being (a, b, 2c) containing fourd-spins as well as four π-spins each.In general, the π-d interaction tends to destabilise the superconducting state because

of an effective internal magnetic field produced by the localised spins of the magneticions [25]:

Heff = H − JS

gµB, (2.1)

where g is the Landé factor and µB is the Bohr magneton, and J and S are thequantum numbers of the total angular momentum and the spin of the electrons. HJ =JSgµB

describes the exchange interaction between the π-electrons and the localised d-electrons. Due to the layered structure one can assume a two-dimensional system. ForB parallel to the layers, the orbital effect is strongly supressed and superconductivitycan only be broken due to the Zeeman effect (see Section 3.3). In the antiferromagneticstate, the effective internal field remains small because HJ is effectively zero. Nowone has to distinguish two cases: For B||a either a spin flop or a spin flip transitioncan occur (see Section 3.3). However, as soon as the paramagnetic state is reached,the spins of the localised d-electrons are lined up along the direction of the externalfield. As a consequence, the conduction electrons experience the full exchange field andconsequently superconductivity is broken. In the other case, B||c, the spins of localisedelectrons at some point are progressively tilted with increasing field until they finallyline up with the field at a considerably larger value BAFM than for B||a. In this casethe possibility exists that superconductivity is broken before B reaches the value BAFMeither due to the external field as long as the effective exchange field is small comparedto the external field or due to the exchange interaction produced by the canted spinsof the localised electrons.For the low field regime B < BAFM a very low Shubnikov-de Haas frequency Fδ = 60

T was reported by Konoike et. al [16] as shown in the Fourier spectrum in Fig. 2.8(a),which corresponds to 1.5% of the FBZ. Konoike et al. propose that this δ-frequencyoriginates from a reconstruction of the Fermi surface in the antiferromagnetic state asshown in Fig. 2.8(b). The reconstruction of the Fermi surface comes from the factthat the magnetic cell in the antiferromagnetic phase is (a, b, 2c) and therefore the FBZis divided by two in the kc direction. The δ-frequency is supposed to correspond tothe small circular orbits at the corners of the reconstructed Fermi surface indicated bythe diagonal circles in Fig. 2.8(b). However, two more closed orbits indicated by thedark and light gray areas as well as magnetic breakdown orbits are not yet observed.This means that either the reconstructed Fermi surface is not explained by this simplereconstruction model or the effective cyclotron masses of the orbits are too large inorder to be detected at low fields.

Field induced superconductivity

In general, s-wave superconductivity as well as d-wave superconductivity can be de-stroyed under magnetic fields by two effects: Orbital efffect and Zeeman effect (seeSection 3.3). However, a stable superconducting state under high magnetic fields hasbeen reported e.g. in the organic compound λ-(BETS)2FeCl4 [27–29]. The FISC phase

9


Figure 2.7: Stable magnetic structure of κ-(BETS)2FeBr4, constructed starting froma horizontal spin alignment of the π-electron system (from [26]).

Figure 2.8: (a) Fourier transform spectrum in low magnetic fields at 0.3 K. (b) Recon-structed Fermi surface in the AFM phase (from [16]). The FBZ of the antiferromag-netic phase is represented by the solid rectangle, the FBZ of the paramagnetic phaseis represented by the dashed rectangle. The δ orbit is indicated by red circles.

can be explained by the compensation mechanism proposed by Jaccarino and Peter [30].First indication for FISC in κ-(BETS)2FeBr4 was reported by Fujiwara et al. [13] and afull experimental report with certain evidence was done by Konoike et al. [15]. As men-

10

2.4 Phase diagrams

tioned above, the spins of the Fe3+-ions produce an effective internal magnetic field [seeEq. (2.1)]. As long as the spins are antiferromagnetically ordered, this effective internalfield is weak and therefore has no influence on the superconducting state. However, inthe paramagnetic state, when the Fe3+ spins line up with the external magnetic field,the superconductivity is supressed due to the Zeeman effect. But when the value of theexternal magnetic field matches the value of the exchange field, H = HJ , the effectivefield will be zero which allows the π-electron system to become superconducting again.For κ-(BETS)2FeBr4, Konoike et al. determined µ0HJ = 12.6T.It should be mentioned that this value is also proved by the SdH oscillations reported

above as pointed out by Cepas et al. [25]. In the presence of a large internal magneticfield the SdH frequencies should split up into two frequencies corresponding to theFermi surfaces of up and down spin electrons. The exchange field HJ can then directlybe determined from the difference of frequencies δF = 1

4 · g ·mcm0·HJ [25]. From the

splitting of the β-frequency Konoike et al. [15] obtained with g = 2 a value of 12.7T for the internal field which is in quite good agreement with their experimentallydetermined value.The value HJ = 12.6 T is about one-third of that of λ-(BETS)2FeCl4 (33 T) [24].

Although the crystal structures of both compounds are similar, in the ground stateλ-(BETS)2FeCl4 is an antiferromagnetic insulator, whereas for κ-(BETS)2FeBr4 theelectronic state stays metallic below the Néel temperature. Responsible for this differ-ence is the fact that in λ-(BETS)2FeCl4 the π-d interaction is much stronger than inκ-(BETS)2FeBr4, because of the absence of the Se-Br contact as described in Section2.2 [10]. Since the strong π-d coupling results in a large exchange field, the FISC isobserved at a much higher field in λ-(BETS)2FeCl4.Konoike et al. [15] investigated also the dependence of the FISC on the magnetic

field direction. Already when the field was tilted out of the ac-plane by an angle of1.2, the component of the magnetic field perpendicular to the layers became alreadyso large that the FISC was completely surpressed due to the orbital effect.

11

3 Theoretical background

3.1 Semiclassical Magnetoresistance

3.1.1 Magnetoresistance in Conventional Metals

In this section a brief introduction to the semiclassical model of magnetoresistance inconventional metals is given. For a more detailed review see ref. [5, 31–33]. In thismodel, only electrons in the vicinity of the Fermi energy εF responsible for conductingproperties are considered. Scattering processes are taken into account by a constantrelaxation time τ , which is independent of the electron’s momentum and magnetic field.When a magnetic field is applied to a metal, the conduction electrons are subject to

Lorentz force and the equation of motion takes the form

dpdt

= FL = −ev×B, (3.1)

where p = ~k, v and e are the electron’s momentum, velocity and charge. The velocityv is related to the energy E of the electron by

v(k) = 1~∇kE(k). (3.2)

From the above equations follows that dpBdt = 0, where pB is the projection of the

momentum on the field direction, and dEdt = FL ·v = 0, which means that pB = const.

and E(k) = const. Therefore, in the momentum space, the electron moves on an orbitwhich lies on a surface of constant energy, and the cross-sectional area of the orbit liesperpendicular to the field direction.We assume that the electrons are scattered elastically after the scattering time τ . The

conductivity is proportional to the mean free path ` = vF τ . Between two scatteringevents, the electrons move on slightly curved trajectories with the Lamor radius rL =pFeB , where pF is the Fermi momentum. In the case of low magnetic fields the mean freepath ` is much smaller than the radius rL, and it can be shown that the relative changein resistivity is

∆ρ(B)ρ(0) ∝

(B

ρ(0)

)2, (3.3)

which obeys Kohler’s rule.In the more important case of higher fields (` ≥ rL), the electron’s momentum is

considerably changed within the scattering time τ , which results in a varying electron’svelocity, which depends on the momentum and is always perpendicular to the Fermi

13


surface. From the linear Boltzmann equation in the semiclassical τ -approximation inthe presence of both electrical and magnetic fields,

eE ·v(k)(−df0dE

)= g(k)

τ+ e

~(v(k)×B) · ∇kg(k), (3.4)

where f0(k) is the equilibrium Fermi distribution and g(k) = f(k)−f0(k), the conduc-tivity tensor σij can be derived in the form

σij = 2e2τ

(2π)3~2

∫ (df0dE

)vi(k)vj(k)dk, (3.5)

with the velocity vj(k) averaged over the scattering time τ :

vj(k) = 1τ

∫ 0

∞vje− tτ dt (3.6)

From Eq. (3.5) one can see that the conductivity is determined by the average veloctiy,which depends on the magnetic field. In k-space, the electron motion along a closedorbit is characterised by the cyclotron frequency

ωc = 2πeB∂S/∂εpB

= eB

mc, (3.7)

where S is the area enclosed by the electron trajectory and

mc = 12π

(∂S

∂ε

)pB

(3.8)

is the cyclotron mass. In the high-field limit, ωcτ 1, the electron completes manyturns around the Fermi surface before being scattered. Therefore its velocity compo-nents perpendicular to the field oscillate rapidly around zero, which means that theirtime-average values tend to zero with increasing the field. According to Eq. (3.5) thisleads to a decrease of the corresponding conductivity components. In a coordinate sys-tem with B parallel to the z axis, the conductivity tensor can be expressed as follows[33]:

σij =

axxB2

axyB

axzB

−axyB

ayyB2

ayzB

−axzB −ayz

B azz

, (3.9)

where aij are coefficients determined by characteristics of the material and independenton the magnetic field. The resistivitiy tensor is obtained in the lowest order in 1/B bytaking the inverse tensor of Eq. (3.9). The diagonal components then are independentof B at high fields, as can be seen for example for ρxx:

ρxx =ayyazz + a2

yz

azza2xy + axxayyazz+axxa2

yz+ayya2xz

B2

(3.10)

14


In the high field limit, the expression (3.10) takes the form ρxx = ayyazz+a2yz

azza2xy

= const.Thus, the magnetoresistance comes to saturation with increasing fields, if all orbits areclosed.In the case of a Fermi surface geometry which allows open orbits, and under the as-sumption, that the open orbit is in kx direction, the conductivity tensor can be derivedin a similar way as [33]

σij =

axxB2

axyB

axzB

−axyB byy byz

−axzB −byz azz

. (3.11)

In this case, the y component of the velocity doesn’t oscillate around zero and theconductivity components σyy and σyz tend to finite values. The diagonal resisitivitycomponents then are ρxx ∝ B2 and ρyy,zz −→ const., which means, that the magne-toresistance measured in the direction of an open orbit increases quadratically with thefield.

3.1.2 Magnetoresistance in Layered Organic Metals

Connected with the special low-dimensional Fermi surface topology of layered organicmetals, new magnetoresistive effects were observed. Some of them are introduced inthe following sections as far as they are discussed in Chapter 5. For a detailed reviewof magnetoresistive effects in layered organic metals see ref. [5].For organic metals, the anisotropy ratio of the resistivities is typically ρ⊥

ρ||∼ 103 to

105 [5] and due to this very high anisotropy one can assume ρ⊥ ∝ 1σ⊥

. Therefore, onlythe interlayer resistance will be considered throughout this work.

Angle-Dependent Magnetoresistance Oscillations (AMROs)

The first observation of Angle-Dependent Magnetoresistance Oscillations was reportedin an experimental study of the magnetoresistance anisotropy in the layered supercon-ductor β-(ET)2IBr2 [34]. By turning the magnetic field in a plane perpendicular tothe layers, strong oscillations of the magnetoresistance were observed (see Fig. 3.1).The nature of these oscillations is strongly connected with the topology of the cylindri-cal Fermi surfaces appearing in layered organic metals. A schematic view of a weaklywarped Fermi surface cylinder is shown in Fig. 3.2(a). The energy dispersion is thendescribed by

ε(p) =p2x + p

2y

2m − 2t⊥ cos(pzd~

), (3.12)

where t⊥ is the interlayer overlap integral and d the interlayer period. As was firstpointed out by Yamaji [36], the difference ∆S between the largest and the smallestareas of the electron orbits oscillates with varying the field direction:

∆S(θ) ≈ 8πmt⊥cos θ J0

(pFd

~tan θ

)(3.13)

15


Figure 3.1: Angular magnetoresistance dependence of β-(ET)2IBr2 at B = 14 T (from[35]).

As can be seen, ∆S vanishes at angles corresponding to zeros of the zeroth order Besselfunction J0. For tan θ > 1, this condition can be approximated by

| tan θn| =π~pFd

(n− 1

4

); n = 1, 2, ..., (3.14)

which means that at field directions corresponding to this condition, all the electronorbits have the same area.

(a) (b)

Figure 3.2: (a) Schematic view of a warped cylindrical Fermi surface (from [4])(b)Schematic view of a transverse cross section of a cylindrical FS (from [5])

The origin of the AMROs can be qualitatively understood as follows [35]. Theinterlayer conductivity σzz is determined by the interlayer velocity vz averaged overthe period of the electron motion on the closed orbit, and vz is determined by the

16


dependence of the orbit area on its position in p-space:

vz = ∂ε

∂pz= −

∂S(Pz)∂Pz(∂S∂ε

)pz

= −∂S(Pz)∂Pz

2πmc, (3.15)

where Pz is the point at which the plane of the orbit intersects the pz axis. Thecyclotron mass mc defined by Eq. (3.8), monotically changes with θ. In general,the derivative ∂S(Pz)

∂Pzis finite, and the interlayer conductivity σzz saturates at a finite

value with increasing the field. However, at angles satisfying Yamaji’s condition (3.14),∂S(Pz)∂Pz

≈ 0, which leads to vz ≈ 0 and, hence, to σzz ≈ 0. It can be shown [37, 38] thatat Yamaji’s angles, the interlayer conductivity σzz decreases with the magnetic fieldas B−2, as long as 1 ωcτ εF /t⊥. Therefore, the resistivity ρzz ≈ 1/σzz increasesproportional to B2 at the angles θn.Yamaji’s condition (3.14) can be generalised for a more realistic dispersion relation:

ε(p) = ε(px, py)− 2t⊥ cos[(pzd+ pxux + pyuy)/~], (3.16)

where ε(px, py) is an even function of px,y, which corresponds to a convex cross sectionof the Fermi surface, and ux,y are the in-plane components of the direction vectorh = (ux, uy, d) of the interlayer hopping. Equation (3.14) then is modified to [35]

| tan θn| =π~(n− 1

4

)±(pmax|| ·u

)pmax|| d

. (3.17)

pmax|| is the in-plane Fermi momentum and pmaxB is its projection on the field rotationplane determined by ϕ as illustrated in Fig. 3.2(b). In the case of an elliptical crosssection of the Fermi surface, pmaxB can be expressed as [39, 40]:

pmaxB (ϕ) =[(p1 cosϕ)2 + (p2 sinϕ)2

] 12 (3.18)

Coherence peak

At high tilt angles θ of the magnetic field direction, with coming closer to π/2, a sharppeak was observed in the magnetoresistance as can be for example seen in Fig. 3.1. Adetailded investigation of the peak structure [41] revealed that its width is independentof the field strength. This suggests that the origin of the peak is geometric in nature.This effect can be explained as follows: When the angle θ reaches the critical valueθc ≈ arctan

(εF2t⊥

), the orbits on the warped cylindrical Fermi surface split up into open

orbits and small closed loops at the very side of the warped Fermi surface. Relevanthere is now the existence of self crossing orbits [42], which lay on the border separatingthe open and closed orbits as for example shown in Fig. 3.3. As one can see there,the Fermi velocity vF at the crossing point A is exactly parallel to the field direction.If the Fermi surface is only weakly warped, then also the velocities of the electrons inthe vicinity of A are nearly parallel to B. Thus, due to the vanishing Lorentz force in

17


this area, the electrons do not move on the Fermi surface and therefore, their velocitiesare conserved, whereas the interlayer velocity components of the electrons situated farfrom the self crossing orbits oscillate rapidly. These conserved velocities dominate inthe interlayer conductivity at high fields. In this way the interlayer conductivity ismaximum at θ = θc, which gives rise to a local minimum in the interlayer resistivity.As θ approaches π/2, the interlayer velocitiy component vz rapidly decreases becomingexactly zero at θ = π/2, which results in a sharp peak of the magnetoresistance atθc < θ < π − θc.

Figure 3.3: Schematic view of cyclotron orbits on a warped cylindrical FS. The mag-netic field is aligned almost parallel to the layers (from [5]).

From the width of the peak the anisotropy ratio 2t⊥EF

can be estimated [42, 43]:

2t⊥EF∼=π − 2θkFd

(3.19)

Typical values for the half width of the peak in organic metals are in the order of 1.The peak feature described above is called coherence peak, because it can be used asan evidence for coherent interlayer transport [44].

Lebed Magic-Angle Resonances

The above described effects are strongly connected with a cylindric Fermi surface.However, the Fermi surface of κ-(BETS)2FeBr4 includes also quasi-one-dimensionalsheets. These open Fermi sheets give rise to a series of other magnetoresistive effects,depending on the magnetic field orientation. The sheets are considered to extendperpendicular to the kx axis and to be warped along the ky axis much more stronglythan along the kz axis.In the following part a strong magnetic field is assumed to be applied in the kykzplane.

In such a field orientation, the electrons move due to the Lorentz force along the sheets,crossing many Brillouin zones. The frequencies of crossing one Brillouin zone in the ky

18


and kz directions are [5], respectively,

ωy = ay

∣∣∣∣dkydt∣∣∣∣ = evFayB cos θ (3.20)

andωz = az

∣∣∣∣dkzdt∣∣∣∣ = evFazB sin θ. (3.21)

(a) (b)

Figure 3.4: Schematic view of an electron trajectory on a one-dimensional Fermi sur-face sheet with the magnetic field applied (a) in an arbitrary direction and (b) alonga lattice vector (from [4]).

In general, these frequencies are different for an arbitrary value of θ. However, as wasfirst pointed out by Lebed [45], the frequencies become commensurate, if the magneticfield directs along a translation vector of the crystal lattice, that is, when

tan θ = p

q

ayaz, (3.22)

where p and q are integers. In this case, the motion in k-space becomes periodic.An anomalous decrease of the magnetoresistance in the metallic state of the organic

metal (TMTSF)2ClO4 at the Lebed magic angles was first reported by Naughton et al.[46, 47] and Osada et al. [48]. At present, plenty of theoretical models explaining theLMA effect in quasi-one-dimensional conductors exist. Here, only the model of Osada[49] is shortly introduced. In this model, the following x direction linearised energydispersion near the Fermi level is considered:

ε(k) = ~vF (|kx| − kF )−∑pq

tpq cos(payky + qazkz), (3.23)

where tpq describe the effective electron hopping along the corresponding lattice vectorsRpq = (0, pay, qaz). It was found that the velocity in the y and z direction exhibitspeaks at the LMAs.

19


If a magnetic field is applied in a general direction in the kykz-plane, the trajectoryof an electron in the reduced first Brillouin zone fills the whole warped Fermi surface[shown in Fig. 3.4(a)]. Thus, the velocity components vy and vz can take all possiblevalues and because of that average to zero. However, if the field is aligned along oneof the lattice vectors, the electrons in k-space move along the reciprocal lattice vectorK = pky + qkz. The trajectory in the first Brillouin zone then consists only of a finiteset of lines [see Fig. 3.4(b). Hence, the velocity component vz takes only a limitednumber of values. Then, with the energy dispersion in Eq. (3.23), the average velocitybecomes non zero. As a consequence of that, dips of the transverse magnetoresistanceappear.

3.2 Magnetic Quantum OscillationsIn Section 3.1 it was derived that in a magnetic field the electrons, depending on theFermi surface geometry, move on closed orbits in k-space. From a quantum mechanicalpoint of view one would expect that the electron waves obey the Bohr-Sommerfeldquantisation, which means that the phase of an electron is only allowed to change byan integer value times 2π during a complete cycle. Thus, the electron orbits in k-spaceshould be quantised, which was first suggested by Landau [50].Starting with the Schrödinger equation for free electrons in a magnetic field,

12m

(~i∇− eA

)2Ψ = EΨ, (3.24)

where A = (0, Bx, 0) is the vector potential, one obtains the energy spectrum

E =(n+ 1

2

)~ωc + ~2

2mek2z , (3.25)

with ωc = eBme

being the cyclotron frequency of a free electron and n = 0, 1, 2, ....Thus, the electron states in the k-space are confined to tubes with radius k⊥,n =√

2m~2

(n+ 1

2

)ωc, the so-called Landau tubes (see Fig. 3.5). The cross-sectional areas

of these tubes perpendicular to B are:

Sn = πk2⊥,n =

(n+ 1

2

) 2πeB~

(3.26)

In the case of crystal electrons, Onsager proposed a quasiclassical calculation based onBohr’s correspondence principle, which lead to the Onsager relation [51]

Sn = πk2⊥,n = (n+ γ) 2πeB

~, (3.27)

where γ is a correction factor. In most cases, γ can be assumed to be close to 12 . Eq.

(3.27) is also valid, if the areas of cross sections of the Landau tubes are not circular.Thus, the effect of the magnetic field on the electrons can be described in this way

20

3.2 Magnetic Quantum Oscillations

that the equidistant distributed states of the electrons are forced in concentric Landauorbits, which lie in a plane perpendicular to the field axis, and all electrons with thesame quantum number n cycle around the field axis with the same cyclotron frequencyωc. The area S of these orbits is the same for all electrons with the quantum numbern, but the component of the wave vector parallel to the magnetic field can, of course,be different.

Figure 3.5: Schematic representation of the Landau tubes (thin lines) for a 3D isotropicFermi surface (left) and a quasi-2D warped cylindrical Fermi surface (right), (from[5]).

.

At T = 0 K, all states on the Landau tubes up to the Fermi level are occupied.If the magentic field increases, the number of occupied states on the largest Landautube inside the Fermi surface decreases and eventually vanishes infinitely rapidly at themoment the tube touches the extreme cross section Sextr of the Fermi surface. If themagnetic field is continuously increased, the Landau tubes one after another cross theextreme cross section of the Fermi surface. The period in the inverse field scale of thissubsequent crossing is

∆( 1B

)= 2πe

~Sextr. (3.28)

As a consequence, physical parameters, such as magnetisation, conduction, heatcapacity and so forth, which depend on the density of states at the Fermi energy, showa magnetooscillatory behaviour with a characteristic frequency in a 1

B -scale.The criteria for observing this oscillatory behaviour are on the one hand that the

thermal energy is smaller than the distance between neighbouring Landau tubes (~ωc >kBT ) to prevent thermal smearing, and on the other hand that the scattering time τis sufficiently large (ωcτ > 1). Therefore, low temperatures, sufficiently high fields andclean samples are required.From the period of oscillations [see Eq. (3.28)] the cross sectional area of the Fermi

surface can immediately be determined and also the oscillation amplitude gives infor-mation on some other important properties of the electronic system, such as for examplethe effective mass or Dingle temperature.

21


3.2.1 Shubnikov-de Haas Oscillations

The first experimental methods of probing Fermi surfaces were investigations of theShubnikov-de Haas (SdH) [52] and de Haas-van Alphen (dHvA) [53] effects, that isquantum oscillations of resisitivity and magnetisation, which are most extensively usedin Fermiology until today [54]. In contrast to the dHvA effect, the theory of theSdH effect is more complex. A first theoretical explanation of the phenomenon wasgiven by Adams and Holstein [55]. A qualitative description for the oscillations of theconductivity can be obtained using the argument of Pippard [56, 57] that the scatteringpropability and, thus, the electrical resisitivity are direct proportional to the densityof states at the Fermi level εF , which in turn is proportional to the field-derivative ofmagnetisation [31]:

D(EF ) ∝(mcB

Sextr

)2 ∂M

∂B(3.29)

The oscillatory part of the conductivity can then be expressed in the form

σ

σ0=∞∑r=1

1r

12ar cos

[2π(F

B− 1

2

)± π

4

], (3.30)

where

ar ∝mcB

12

(S′′)12extr

RT (r)RD(r)RS(r) (3.31)

and σ0 is the background conductivity. The damping factors RT , RD and RS areintroduced in the next section.

3.2.2 Damping Factors

The first damping factor, RT , originates from the fact that a metal with the Fermienergy ε0F for T > 0 can be considered as a superposition of metals with T = 0, whichhave different Fermi energies distributed around ε0F . Because of that, the oscillationfrequencies F ∝ Sextr corresponding to the different metals would slightly differ fromeach other. Therefore, this should result in a smearing of the phase of the oscillationsand, hence, in a damping of the amplitude. RT can be exactly expressed as [58]

RT (r) = KrµT

B sinh(KrµTB

) , (3.32)

where µ = mcme

is the normalized cyclotron mass and K = 2π2kBmc~e ≈ 14.7 T/K. For

high enough temperatures Eq. (3.32) is reduced to

RT (r) ∝ T

Be−

KrµTB . (3.33)

With Eq. (3.33), µ can directly be determined from the slope of a linear plot ln(ar/T )versus T .

22

3.2 Magnetic Quantum Oscillations

In a perfect crystal, the scattering time should be infinitely large (1/τ = 0), and theLandau levels should be infinitely sharp. However, in real crystals τ takes finite values,and the Landau levels are broadened due to the uncertainty principle. The resulting,so-called Dingle damping factor, is expressed as [59]

RD(r) = e−KrµTD

B , (3.34)

where TD = ~(2πkBτ)−1 is the Dingle temperature. TD directly provides informationabout the cleanness of a sample. By fitting Eq. (3.31), the Dingle temperature andrelaxation time τ , respectively, can be determined. For organic metals, typical Dingletemperatures are of the order of 1 K, which corresponds to τ ∼ 10−12 s.The third damping factor, RS , originates from the Zeeman splitting effect. In pres-

ence of a magnetic field, every Landau level splits into two levels with the energydifference

∆E = gβ0B, (3.35)

where g is the Landé factor and β0 = e~2me is the Bohr magneton. In real metals,

this leads to a phase shift between contributions from levels with opposite spins. Thecorresponding reduction factor takes the form [5]

RS(r) = cos(π

2 rgµ). (3.36)

As mentioned above, the cyclotron mass of an orbit on a cylindrical Fermi surface isdependent on the angle θ: µ(θ) = µ(0)/ cos θ. This leads to a periodic vanishing ofRS every time rgµ/ cos θ becomes an odd integer. Thus, the product gµ(0) can bedetermined by finding the angles where the oscillation amplitude becomes zero.

3.2.3 Magnetic BreakdownIn the description of the Landau quantisation it was assumed that the electrons inpresence of a magnetic field move along well defined trajectories both in real spaceand k-space. However, when two bands come close to each other at the Fermi levelεF , electrons can tunnel through a small energy gap εL between the bands, if themagnetic field is high enough. This phenomena is called magnetic breakdown (MB).The magnetic breakdown orbits in k-space then result from a combination of orbits onthe Fermi surface, which come close to each other at specific points.From the expression analogous to the Zener tunneling process [32], a condition for

the MB effect, the so-called Blount criterium [60] can be derived [33]:

~ωcεFε2L

> 1 (3.37)

The tunneling probability exponentially increases with the field and can be expressedas [60]:

Y = e−BMB/B, (3.38)

where BMB is the characteristic breakdown field. In the cases B BMB and B BMB,the quasiclassical approach can be used, because in the former case, the electrons just

23


stay on their semiclassical orbits, and in the latter case they can freely tunnel throughthe MB junctions producing the MB orbits. However, when B ∼ BMB, the trajectoriesare no longer well defined. In this case in the approach of Falicov and Stachowiak [61]it can be shown that an inital electron wave entering a MB junction is separated intoa transmitted wave with amplitude ν and a reflected wave with amplitude ξ [62, 63]:

ν =√Y , ξ =

√1− Y (3.39)

Taking into account magnetic breakdown, each contribution in Eq. (3.30) has to bemultiplied by the MB reduction factor [61]

RMBj = (iν)l1jξl2j , (3.40)

where l1j and l2j are the numbers of points at which the electron encircling the j-thorbit tunnels through and is reflected from a MB junction, respectively. The tunnelingamplitude was rewritten as iν under the assumption that the reflected phase preservesthe initial phase whereas the transmitted phase changes by π/2.

3.3 Superconductivity

As described in the previous chapter, the crystal structure of κ-(BETS)2FeBr4 con-sists of an alternating sequence of highly conducting and insulating layers. It can beimagined, that this layered structure influences the superconducting properties of κ-(BETS)2FeBr4 in presence of a magnetic field. Since superconductivity is a basic fieldof condensed matter physics today, in this section only the case of a type-II super-conductor with a layered structure in a magnetic field will be discussed. A detailedintroduction to superconductivity can be found e.g. in refs. [33, 64, 65].The main difference between a type-I and a type-II superconductor is that the former

stays in the Meißner phase until the thermodynamic critical value Bc,th is reached,whereas in the latter single flux lines can enter the bulk superconductor above thelower critical field Bc1 (see Fig. 3.6). This so-called Shubnikov phase survives untilthe upper critical field Bc2. From the solution of the Ginzburg-Landau equations oneobtains for the upper critical field Bc2 the following relation [64]:

Bc2 = Φ02πξ2

GL, (3.41)

where Φ0 = h2e is the flux quantum and ξGL is the Ginzburg-Landau coherence length.

ξGL is the characteristic length, over which the order parameter Ψ, which is introducedin the Ginzburg-Landau theory, can change.When now a magnetic field is applied exactly perpendicular to the conduction layers

of a layered type-II superconductor, the supercurrents, which shield the bulk of thesuperconductor from magnetic flux, can flow completely within the layers. Therefore,the expression for the upper critical field perpendicular to the layers Bc2,⊥ should bethe same as Eq. (3.41), with the exception that ξGL is replaced by ξ||, which is the

24

3.3 Superconductivity

(a) (b)

Figure 3.6: Schematic representation of (a) the effective internal magnetic field and (b)of the magnetisation as a function of the external field for a type-II superconductorand a type-I superconductor (blue graph) (from [33]).

coherence length parallel to the layers [64]:

Bc2,⊥ = Φ02πξ2||

(3.42)

However, if the field is applied exactly parallel to the layers, then, under the assumptionthat there is no coupling between neighbouring layers, the shielding currents are con-fined solely to cross sectional areas of the layers with thickness d. If the layer thicknessd is smaller than ξ⊥, Bc2 becomes dependent on d instead of ξ⊥ [66]:

Bc2,|| =√

6 Φ0πξ||d

, (3.43)

where ξ|| is the coherence length parallel to the layers. For d ξ||, the value for Bc2,||can get much higher than that for Bc2,⊥.For a general field orientation, the upper critical field can be obtained from the

quadratic equation [66]

Bc2(θ)∣∣∣∣∣ sin θBc2,⊥

∣∣∣∣∣+[Bc2(θ) cos θ

Bc2,||

]2

= 1, (3.44)

where θ is the angle between the magnetic field direction and the layers.In the case of a finite coupling between the layers, the superconducting properties also

depend on the field direction. In the so-called "anisotropic Ginzburg-Landau theory"the angle dependence of the upper critical field is derived as [64]

Bc2(θ) =

√√√√( sin θ

Bc2,⊥

)2

+(

cos θBc2,||

)2−1

. (3.45)

25


Bc2,⊥ is the same as in Eq. (3.42), and for Bc2,|| one gets

Bc2,|| =Φ0

2πξ||ξ⊥. (3.46)

According to Eq. (3.43), the upper critical field B2c,|| can become very large, if thethe layer thickness d becomes sufficiently small. However, under high magentic fieldsanother effect, which limits the upper critical field, arises. For the spin singlet couplingof the Cooper pairs, in presence of a magnetic field the energy levels of electrons withopposite spins are shifted due to the Zeeman effect. If the field gets high enough, thesplitting energy becomes so large that it is energetically favourable for the electrons toalign their spins parallel to the field direction and abandon supercondutivity. Also forthis behaviour an expression for the critical field can be derived [67–69]:

Bp = ∆0√2µB

≈ 1.84 [T/K] ·Tc, (3.47)

where ∆0 is the energy gap of superconductivity and µB is the Bohr magneton. Relation(3.43) is also called the "Chandrasekhar-Clogston limit" or "paramagnetic limit".

3.4 Magnetic States in κ-(BETS)2FeBr43.4.1 Paramagnetism of localised magnetic moments

Paramagnetism can be observed when the atoms of a solid possess a non-zero magneticmoment due to unpaired electrons and the single magnetic moments only interactweakly with each other. The application of a magnetic field will line them up. Thedegree of lining up depends on the field strength and on the temperature. For acloser look on the subject see for example ref. [70]. Here only the expression of themagnetisation M for a general quantum number J of the total angular momentum isgiven [70]:

M = MSBJ(y), (3.48)

where MS = ngµBJ is the saturation magnetisation and BJ is the Brillouin functiongiven by

BJ(y) = 2J + 12J coth

(2J + 12J y

)− 1

2J coth y

2J (3.49)

with the substitution y = gµBJBkBT

.

3.4.2 Antiferromagnetism

Antiferromagnetism is refered to as a magnetic ordering mechanism, which describes anantiparallel alignment of nearest neighbour magnetic moments in solids. This alignmentis energetically favourable, if the exchange interaction J is negative. Very often thesystem can be considered as two interpenetrating sublattices, on one of which the

26

3.4 Magnetic States in κ-(BETS)2FeBr4

magnetic moments point "up" (+) and on the other of which they point "down" (-).The molecular field on each sublattice is then given by

B± = − |λ|M±, (3.50)

where λ is the molecular field constant and |M+| = |M−| = M . Therefore, the molec-ular field M is given by

M = MSBJ

(gµBJ |λ|M

kBT

). (3.51)

The net magnetisation M+ +M− is zero. The temperature, above which the antiferro-magnetism disappears is known as the Néel temperature TN , which is defined by

TN = gµB(J + 1)|λ|MS

3kB. (3.52)

The magnetic susceptibility of an antiferromagnet can be derived as

χ = limB→0

µ0M

B∝ 1T + TN

, (3.53)

which satisfies the Curie-Weiss law.

Figure 3.7: The antiferromagnetic susceptibility as a function of T (from [70]).

One of the differences in comparison to ferromagnetism is that below TN the directionin which an external magnetic field is applied is now crucial, because an energeticadvantage for the moments to line up along the field direction is not given anymore.ForM+ = −M− the energy saving through lining up on one sublattice will be cancelledby the energy cost for the other sublattice.For T = 0 K |M+| = |M−| = MS . For the case of a small magnetic field aligned

parallel to the magnetisation direction of one of the sublattices, the magnetisation in-duced in both sublattices is zero χ|| = 0 since both sublattices are already in saturation.But if, instead, a small field is applied perpendicular to the magnetisation direction,

27


this causes the magnetic moments of both sublattices to tilt slightly. Therefore a com-ponent of the magnetisation is produced along the external field and χ⊥ 6= 0. If thetemperature is increased χ⊥ is independent of the temperature and χ⊥ rises from 0 toχ|| as T → TN . This behaviour is illustrated in Fig. 3.7.

(a) (b)

Figure 3.8: Sublattice magnetisation for (a) B parallel to one of the sublattice mag-netisations in the antiferromagnetic phase and (b) B perpendicular to the sublatticemagnetisations for the spin-flop phase (from [70]).

Figure 3.9: Energy of the antiferromagnetic phase and the spin-flop phase as a functionof the magnetic field (after [70]).

If now a strong magnetic field is applied perpendicular to the magnetisation of oneof the sublattices at T = 0 K, the magnetic moments are tilted more and more withincreasing field until they line up along the field direction. However, if the applied field

28

3.4 Magnetic States in κ-(BETS)2FeBr4

is parallel to the sublattice magnetisation, at low fields the moments aren’t tilted butstay in line [see Fig. 3.8(a)]. When the external field now reaches a critical value, aso called spin-flop transition can occur, that means the system suddenly snaps into adifferent configuration shown in Fig. 3.8(b). After the spin-flop transition, the magneticmoments are progressively tilted into the field direction.The spin-flop effect can be calculated quantitatively [70]. One can find, that mag-

netisations prefer to lie parallel to a certain crystal axis. For the total energy in theantiferromagnetic case one gets

E = −AM2 −∆, (3.54)

where A and ∆ are constants due to the exchange coupling and magnetic anisotropy,respectively. In the spin flop case [φ = θ in [Fig. 3.8(b)]] one gets

E = −2MB cos θ +AM2 cos 2θ −∆ cos2 θ, (3.55)

where θ is the angle indicated in Fig. (3.8). These results are plotted in Fig. 3.9.Below the critical field Bspin−flop the antiferromagnetic phase has the lower energy, butabove Bspin−flop the spin-flop phase if preferred.If the anisotropy effect is very strong (large ∆), the magnetisation of one sublattice

suddenly is reversed when B reaches a critical value, and the antiferromagnetism isbroken in one single step. In this case the transition is called a spin-flip.

29

4 Experimental setup

4.1 Experimental equipment4.1.1 Superconducting magnetFor applying magnetic fields to the samples superconducting magnets from Cryogenicswere used. These magnets consist of two superconducting coils, namely a NbTi-coil anda Nb3Sn-coil, which are mounted coaxially and coupled in series. In order to becomeand stay superconducting, the coils have to be kept in a liquid 4He bath in a cryostat.For the two magnets used in the experiment, the maximum magnetic field strenghtsat the evaporation temperature 4.2 K of liquid 4He were 14 T and 15 T, respectively.The electrical current which is needed for inducing the magnetic field is provided byexternal power supplies. In our case for the 14 T-magnet a "Mercury IPS" device andfor the 15 T-magnet an "IPS 120-10" power supply, both from "Oxford Instruments",were used. The systems are also equipped with a switch heater which is simply aheated shunt coupled parallel to the magnets. When heated, the shunt has a finiteohmic resistance and the external current is applied to the coils. When the heater isswitched off, the shunt becomes superconducting and the electrical current producingthe magnetic field is imprinted to the coils. In this case after driving out the externalcurrent the magnet is in the persistent mode, which is favourable for measurementswhich require a constant and stable magnetic field.

4.1.2 Two axes rotatorSince the magnets available for this experiment were simple solenoids, the use of asample holder with a rotatable sample stage was necessary in order to apply magneticfields in any direction. The two axes rotator used for this experiment was constructedby D. Andres within the framework of his PhD thesis [71]. For this sample holder(see Fig. 4.1) the rotation is provided by two worm gear units. With these gears thesample stage can be turned in two rotational directions corresponding to the angles θand φ indicated in Fig. 4.1. The rotation in θ-direction is driven by a long rod whichis coupled to a piezo-electric motor on top of the insert, whereas φ has to be controlledmanually with a screwdriver which has to be pushed on a screw at the side of the sampleplatform. With the screwdriver decoupled from the platform a continouus rotation inθ-direction with a sweeping rate of 0.003 - 25/s is possible. The sweeping rate can bechanged via a mechanical gear and the variable speed of the motor. With this rotatorboth angles can be set with a resolution of < 0.05.The sample holder was additionally equipped with a 60 Ω heater, a calibrated Cernox

thermometer and a calibrated Ruthenium Oxide thermometer, which were measuredby a "LakeShore 340 Temperature Controller".

31


Figure 4.1: Bottom part of the two axes rotator with rotatable sample stage.

4.1.3 3He-insert

In order to cool the samples down to low temperatures, the two axes rotator describedabove was inserted into a homemade 3He evaporation insert. A schematic drawing ofthe bottom part of the insert is shown in Fig. 4.2. With this 3He-insert temperaturesdown to 0.4 K can be accessed.For cooling down the samples the 3He-insert is lowered into the cryostat. In the lowest

position of the 3He-insert the position of the samples in the sample holder coincideswith the field centre of the magnet.The 3He-insert is mainly composed of three parts: The isolation vacuum chamber,

the 4He-pot and the sample space. The isolation vacuum surrounds almost the complete3He-insert and shields the inner parts of the 3He-insert thermally against the outside.But in order to precool the 3He, liquid 4He can flow into the cold plate via a thincapillary with a large flow impedance. By pumping the 4He-pot the temperature isreduced from 4.2 K to 1.4 K according to the vapour pressure curve of 4He. In orderto get to the lowest temperature of 0.4 K, the sample space can then be filled withgaseous 3He from a tank on the outside. At 1.4 K the 3He condenses at the cold plateand drops down to the bottom of the sample space. As a consequence the sample holderis submerged in liquid 3He. The typical vapour pressure of 3He after condensing was50 mbar. By pumping the 3He back into the tank via a pumping stand the pressurecan be further reduced until the lowest temperature of 0.4 K is reached. By regulatingthe pumping power all temperatures between 0.4 K and 1.0 K could be stabilised.Temperatures above 1.4 K up to 3 K were stabilised by regulating the pressure in the4He-pot using a manostat.The 3He-insert was also equipped with two heaters, one for the 4He-pot and one

for the sample space. Heating out the liquid 4He from the 4He-pot became importantfor increasing the time over which the lowest temperature of 0.4 K could be kept inthe sample space. Due to the superfluidity of 4He the surface of liquid 4He in the4He-pot is quite large resulting in a higher vapour pressure. If the liquid 4He is heatedout, the surface is momentarily strongly reduced resulting in a decrease of pressure

32

4.1 Experimental equipment

Figure 4.2: Sketch of the bottom part of the 3He-insert (after [72], dimensions not inscale).

and temperature until liquid 4He has again flooded the 4He-pot through the capillaryafter some time. Via applying this procedure during condensing the 3He the amountof liquid 3He could be increased and a measurement time of ∼ 8 hours at 0.4 K couldbe achieved.

4.1.4 Pressure cellFor measuring the physical properties under pressure, the samples were mounted in apressure cell working with a liquid pressure medium. A schematic cross section of sucha pressure cell can be seen in Fig. 4.3. Since the pressure cell is used in high magneticfields, the cell consists only of nonmagnetic materials. All the metal parts of the cell aremade of a copper beryllium alloy except the piston which is made of tungsten carbide.In the first step, the samples are mounted for a four probe resistance measurement on

top of the sample holder, which is equipped with a manganin coil for determining thepressure. Twelve copper wires, which serve as current and voltage leads for the samplesand the manganin coil, are fed trough a central bore in the sample holder. The wires arefixed by encapsulating them in Stycast 2850FT. The samples are connected via shortplatinum wires to the copper wires. A teflon cup filled with the liquid pressure mediumis then put on the samples. The pressure medium, which was used in our case, was thesilicon oil "GKZh", which has a very low compressibility and solidifies, depending onthe pressure, at 150-220 K into an amorphous structure providing hydrostatic pressurealso in the solid phase. The sample holder then is inserted into the central bore of

33


the pressure cell and fixed by a nut. Two washers on the top and on the bottom ofthe teflon cup help sealing the interior of the cup and prevent the teflon from flowingbetween the piston and the inner wall of the cell.

Figure 4.3: Schematic drawing of a pressure cell (from [71]).

The pressure is applied by pushing the steel "mushroom" into the cell with an hy-draulic press until the desired pressure is reached. The mushroom hereby is pressingon the piston which is in turn pressing on the teflon cup. The low compressibility ofthe pressure medium has the advantage that the length the piston has to be presseddown is rather short. Before releasing the external pressure, the piston is fixed by anut. The pressure in the teflon cup can be determined from the resistance change ofthe calibrated manganin coil, whose resistance changes with 0.25% per kbar.With this technique pressures up to 15 kbar at room temperature can be reached. In

this experiment pressures up to 6 kbar were applied. The pressure at low temperatureswas determined by measuring the manganin resistance at 14 K and comparing it withthe corresponding resistance for ambient pressure. For this purpose the manganinresistance was recorded during cooling down and warming up of the experiment. Theresistance was measured using the conventional four-probe method. The current wasapplied with a "Keithley 2400 SourceMeter" and the voltage drop was measured witha "Keithley 2000 Multimeter". A current of 1 mA was applied in both directions and

34

4.2 Measurement techniques

then the average of the two absolute values was taken in order to avoid errors causedby thermoelectic voltages. This procedure was done by the computer.


4.2.1 Resistance measurements

In this part of the experiment the interlayer resistance of the κ-(BETS)2FeBr4 sampleswas investigated. For that purpose a current was applied perpendicular to the con-ducting planes of the samples and the resistance was measured using the conventionala.c. four probe method (see Fig. 4.4). The samples were contacted in each case by

Figure 4.4: Circuit diagram for resistance measurement and four probe method.

two pairs of 10 µm or 20 µm platinum wires, depending on the sample size. Wires ofthe same pair were attached to opposite sides of the sample and glued with graphitepaste. With this method it was possible to mesasure the sample resistance without thecontact and lead resistances. The samples used in this experiment had typical edgelengths from 0.3 to 0.8 mm and thicknesses from 5 to 200µm. After contacting thesamples they were oriented in a manner which was favourable for the later measure-ments. For measurements at ambient pressure the sample holder shown in Fig. 4.5(a)was used, whereas for measurements under pressure the sample holder of the pressure

35


cell described in Section 4.1.4 was used [see Fig. 4.5(b)].The resistance of the samples was measured with "Stanford Research Systems DSP

model SR 830" Lock-In amplifiers. The contact pair the current was applied with wasconnected with the internal oscillator of the Lock-In amplifier via a box containing ahigh ohmic resistor R2 ∼ 101-106 Ω, which was coupled in series with the sample. Withthis resistor the applied current could be calculated and set by adjusting the outputvoltage of the Lock-In amplifier. Due to its high resistance the other resistances, likefor example contact and lead resistances, could be neglected and a stable current wasguaranteed throughout the experiment. In order to adjust the current more preciselyadditionally a reference resistor R1 of 10 Ω and 100 Ω, respectively, was built in. Byswitching to the reference resistor also the out-of-phase component of the signal origi-nating from capacitive contributions could be checked. With the high ohmic resistanceR2 = 10 kΩ or 100 kΩ and a maximum output voltage of 5 V of the Lock-In, currentsbetween 10 µA and 100 µA were applied. Since a higher signal quality can be achievedwith a higher signal amplitude, higher currents were favourable in this respect. Butalso one had to take into consideration possible overheating of the samples at low tem-peratures. The values of the currents were chosen taking these issues into account.

(a) (b)

Figure 4.5: (a) Sample holder for the two axes rotator. (b) Sample holder for thepressure cell.

4.2.2 Torque measurementsIn addition to the interlayer resistance, also the magnetisation in a field range of theexternal magnetic field up to 14 T was studied using cantilever torque magnetometry.For this purpose the sample is mounted to a cantilever tip. If an external magneticfield is applied, the sample experiences a torque

τ = M×B (4.1)

and the resultant small deflection of the cantilever can be detected electrically.

36


Figure 4.6: Circuit diagram for torque measurement (from [73]).

For this experiment commercially available silicon piezoresistive microcantileversfrom Seiko Instruments Inc. were used. This model (PRC400) was originally designedfor atomic force microscopy. A schematic diagram of the cantilever is shown in Fig.4.6. The dimensions are t = 4-5 µm, l = 400 µm and w = 50 µm. Typical values forthe spring constant k and eigenfrequency f0 are k = 2-4 N/m and f0 = 35-40 kHz. Thedeflection of the cantilever is detected via a piezoresistive path, which is implanted onthe two legs of the cantilever. The cantilever is incorporated together with a referencelever to the same platform. For measuring the resistance change due to the cantileverdeflection, a Wheatstone bridge circuit is used. Through this arrangement precisemeasurements are possible since the background signal originating from the magneticfield dependence of the piezoresistive path can be effectively canceled out. The voltagesignal was measured with an "Stanford Research Systems DSP model SR 830" lock-inamplifier.

(a) (b)

Figure 4.7: (a) Microcantilever in rotator. (b) Cantilever with sample.

The samples used in this experiment had to be very small with typical sizes of

37


0.1 × 0.2 × 0.03 mm3. A larger sample size would result in a higher signal amplitude,but one has to consider that the cantilevers are very brittle and they can easily be brokenby a strong deflection. In first experiments the samples were glued with Apiezon to thecantilever tip. But since the samples tended to jump off the tip, later on silver pastewas used to fix the samples. The samples were oriented with the a or c crystal axisparallel to the long side of the cantilever. The cantilever together with a glued sampleis shown in Fig. 4.7.

38

5 Results and discussion

In this chapter the experimental results are presented and discussed. In the first sectionproperties of the normal metallic state of κ-(BETS)2FeBr4 are investigated, such asShubnikov-de Haas quantum oscillations and angle-dependent magnetoresistance. Fromthe contributions of the 1D Fermi surface to the angle-dependent magnetoresistanceconclusions are drawn about the charge transfer perpendicular to the layers. Also thequestion whether coherent charge transfer is observed or not is dealt with.

(a) (b)

Figure 5.1: (a) Crystal axes with respect to the sample shape [7].(b) Direction ofmagnetic field with respect to the crystal axes.

In Section 5.2 the physical properties of the antiferromagnetic phase are shown. Fromthe SdH oscillations measured at different pressures, the effective cyclotron masses willbe determined. Also the angle dependence of the SdH oscillations and the effect of themagnetic breakdown are investigated. In the second part of the section magnetic phasediagrams for different pressures are presented.The last section covers some aspects of the interaction of superconductivity and anti-

ferromagnetic ordering in κ-(BETS)2FeBr4, and the first indication for the observationof field-induced superconductivity in magnetic torque measurements will be reported.Before presenting the results, some remark should be made about the direction of the

magnetic field with respect to the crystal axes. The samples investigated in this workwere characterised by their rhombic shape with the side edges parallel to the diagonalsof the unit cell in the ac plane. By measuring the angles between the outer edges ofthe rhombic surface the orientation of the crystal could be determined [shown in Fig.5.1(a)]. In this chapter the direction of the magnetic field applied to the samples withrespect to the crystal axes will be expressed via the angles ϕ and θ as indicated in Fig.

39


5.1(b).

5.1 The normal metallic state

5.1.1 Shubnikov-de Haas oscillations

As reported in Section 2.3, in the normal metallic phase several SdH frequencies canbe observed in high magnetic fields. Also in this experiment, the three fundamentalfrequencies Fα, Fβ and Fγ were observed. In Fig. 5.2 SdH oscillations measured fortwo samples from different growth batches in the field range 9 T ≤ B ≤ 15 T forB||b are shown together with the corresponding Fourier spectra. The Fourier spectrawere obtained by subtracting the background signal and conducting a Fast FourierTransform (FFT).

(a) (b)

(c) (d)

Figure 5.2: High field SdH oscillations [(a) and (c)] and corresponding Fast FourierTransform spectra [(b) and (d)] of two different samples for B||b.

In the FFT spectrum of the first sample [see Fig. 5.2(b)] the dominant frequenciesFα = 853 T and Fγ = 103 T are observed. The values of these frequencies are in

40


agreement with previous reports [9, 12, 16]. Fβ is absent in the spectrum, but asone can see two more peaks are visible. The frequencies of these peaks are in goodagreement with the values Fβ − Fα and Fβ − 2Fα, and therefore they will be referedto as Fβ−α and Fβ−2α in the following part. The origin of these frequencies can beexplained with the Stark quantum interference effect or with the frequency mixingeffect, which will be both discussed below. One can also see that both Fβ−α and Fβ−2αare split up into two frequencies, which is due to the splitting of Fβ into Fβ1 and Fβ2

with the difference of frequencies δFβ ≡ Fβ2 − Fβ1 (see Section 2.3). This is supportedby the fact that the difference of frequencies δFβ−α ≡ Fβ2−α − Fβ1−α = 111 T andδFβ−2α ≡ Fβ2−2α−Fβ1−2α = 107 T are in good agreement with δFβ = 0.023 ·Fβ = 98Tas reported in ref. [12]. The frequencies Fβ1,2−α as well as the frequency F2α ≡ 2 ·Fαand Fβ1−2α were reported before in ref. [9, 16], in contrast to Fβ2−2α, which is firstseen in this experiment.In the FFT spectrum of the second sample [see Fig. 5.2(d)], beside the dominant α

frequency Fα = 845 T also the β frequency Fβ = 4258 T is clearly visible together withFβ1−α = 3348 T and Fβ2−α = 3448 T with a difference of frequencies δFβ−α = 100 T.However, no γ frequency is observered for this sample.With Eq. (3.27) also the areas of the cross sections of the α and β orbit in k space

can be calculated. For sample 1 one gets Sα = 0.081 Å−2, Sβ1 = 0.402 Å−2 andSβ2 = 0.413 Å−2. For sample 2 one calculates for the α orbit Sα = 0.081 Å−2 andSβ = 0.406 Å−2.

Figure 5.3: Schematic representation of a Fermi surface similar to that ofκ-(BETS)2FeBr4 (from [5]). The α orbit and the MB orbit β are labeled. Thedirection of the cyclotron motion is indicated by arrows and the magnetic breakdownjunctions are indicated by crosses.

If one wants to understand the origin of the difference frequencies, one could for ex-ample describe the electron orbit corresponding to the frequency Fβ−α by the Falicov-Stachowiak theory [61]. In the schematic Fermi surface representation in Fig. 5.3 theclosed electron orbit a-1-2-c-3-4-a would result in such a difference frequency. How-ever, as indicated by the arrows, the motion along the path 2-c-3 is only allowed inthe opposite direction. Therefore, Fβ−α should not exist after the Falicov-Stachowiakmodel. In order to describe the difference frequencies Fβ−α and Fβ−2α correctly, onehas to take into account quantum interference between the electron wave packets trav-eling from one point on the Fermi surface to another via different paths. In gen-

41


eral terms, if one considers an electron moving from an arbitrary point A to anotherpoint B, then the probability W for the electron propagating from A to B is given byW = |

∑i Pi|

2 =∑i |Pi|

2 +∑i6=j PiP

∗j [33], where Pi are the probability amplitudes

of the possible paths. The first term∑i |Pi|

2 represents the sum of the probabilitiesof the different paths, whereas the second term

∑i6=j PiP

∗j describes the interference

between the different paths. Because of the different lenghts of the paths, the electronswill arrive at B with different phase differences ∆φ = 1

~∫ AB pds. Thus, the contribu-

tions of the different paths will be cancelled out at the end. The only exceptions areself-crossing orbits. In this case one gets a positive interference at point B.Following the argumentation of ref. [5], we consider only the case of two paths of

electrons crossing the Brillouin zone from point e′ to e in Fig. 5.3: Path 1 (e′-1-2-d-3-4-a-1-2-e) and path 2 (e′-1-2-d-3-c-2-e). The resulting wave functions γ1,2 correspondingto the paths 1 and 2 can be expressed as

γ1 = ξ2ν4e[i(φe′e+φβ)] (5.1)

andγ2 = ξ2ν2e[i(φe′e+φα)], (5.2)

with ξ and ν being the amplitudes of the transmitted and reflected wave at the MBjunction according to Eq. (3.39), φe′e being the phase change aquired between e′ ande and φα,β = Sα,β

~eB . The probability for an electron to travel from e′ to e is given by

We′e = γ21 + γ2

2 + γ1γ∗2 + γ2γ

∗1 . (5.3)

With this the sum of the last two terms is given by

γ1γ∗2 + γ2γ

∗1 ∝ cos(φβ − φα) = cos

(2πFβ − Fα

B

). (5.4)

Thus, the probabilityWe′e affects the open orbit contribution to the magnetoresistance.Because of that one should expect the oscillating term with freqency Fβ − Fα to con-tribute to the SdH oscillations. In the same way also other possible paths can be takeninto account leading for example to the frequency Fβ − 2Fα.The quantum interference effect as the origin of the difference frequencies in κ-

(BEDT-TTF)2Cu(NCS)2 was first suggested by Caulfield et al. [74] and in detail inves-tigated by Kartsovnik et al. [75] and Harrison et al. [76]. It should be noted that thequantum interference effect does not influence the density of states at the Fermi level,and therefore it can only explain the difference frequencies seen in SdH oscillations.However, difference frequencies were also observed in de Haas-van Alphen oscillations[77]. Therefore, one should consider an alternative explanation, namely the frequencymixing effect. This effect describes the difference frequencies in two-dimensional sys-tems, where more than one band is subject ot the Landau quantisation. Qualitativelyone can say that this effect originates from the communication between two Landau2D bands via the chemical potential. For a detailed explanation see ref. [5].The SdH oscillations were also investigated under pressure using the pressure cell

described in Section 4.1.4. At a pressure of 1.9 kbar the α frequency could still be

42


(a) (b)

Figure 5.4: (a) FFT spectra of the α frequency for various temperatures. (b) Tempera-ture dependence of the FFT amplitude of the α frequency divided by the temperature.

observed, and with Fα(1.9 kbar) = 835 T it was only changed by ∼ 1%. In contrast tothat the β frequency was not visible in fields B < 15 T any more.From measuring the SdH oscillations at various temperatures, the effective cyclotron

mass of the α frequency was evaluated. For this purpose, the amplitude of Fα in theFFT spectrum in Fig. 5.4(a) was determined for various temperatures. After Eq. (3.33)the temperature dependence of the SdH amplitude a(T ) follows by approximation theexpression

a(T ) ∝ Te−KµTB . (5.5)

Therefore, by plotting BK ln a(T )

T over T , where a(T ) is replaced by the FFT amplitudeof Fα, the normalised effective cyclotron mass µ = mc

m0can immediately be obtained

from a linear fit. As shown in Fig. 5.4(b), the effective cyclotron mass at 1.9 kbar wasdetermined to mc(1.9 kbar) = 3.3m0. This value is considerably lower than the valuemc = 5.2 for ambient pressure, which is a typical result for organic metals: The strongelectron correlations are weakened by applying hydrostatic pressure.At a pressure of 4.5 kbar, no SdH oscillations at all were observed for B ≤ 15 T.

5.1.2 Angle-Dependent Magnetoresistance Oscillations and LebedMagic-Angle Resonances

In this experiment, it was possible to observe Angle-Dependent Magnetoresistance Os-cillations (AMROs) on one of the samples. A report on AMROs in κ-(BETS)2FeBr4was given earlier by Konoike et al. [16]. For various orientations of the in-plane com-ponent of the magnetic field the dominant contribution to AMROs observed by theseauthors originated from the β orbit. The contributions from the α orbit were also ob-served, but only for azimuthal angles ϕ ≤ 60. For 70 < ϕ < 90, another feature wasfound, which was attributed to the Lebed resonance. However, these results were not

43


discussed in detail by Konoike et al [16]. Therefore, in this section mainly the AMROsfor ϕ ≥ 70 will be investigated.

Figure 5.5: AMRO curves for different angles ϕ at 15 T and 1.4 K. For clarity thecurves are vertically shifted.

The AMRO curves for various azimuthal angles ϕ at 15 T and 1.4 K measured inthis experiment are shown in Fig. 5.5. As discussed in Section 3.1.2, the oscillationsoriginating from the cylindrical Fermi surfaces are periodic in tan θ. After plottingthe resistance over tan θ the corresponding frequencies were determined analogous tothe SdH frequencies by subtracting the background and conducting a Fast FourierTransform. The frequencies determined in this way are shown in Fig. 5.6 in dependenceon the angle ϕ together with the calculated frequencies caused by the α and β orbitsand the calculated LMA frequencies. In the following part the frequencies for the α andβ orbits as well as for the LMAs will be referred to as fα, fβ and fLMA, respectively,in order to distinguish them from the SdH frequencies. It should also be mentionedthat the AMRO and LMA frequencies are not identical with the frequencies of theSdH oscillations. In contrast to the SdH oscillations, which are periodic in 1

B , theAMROs and LMAs are periodic in tan θ. The corresponding frequencies are thereforedimensionless.

44


The frequencies fα,β were calculated according to Eq. (3.14):

f =kmax|| · dπ

=kmax|| · b2π , (5.6)

where d = b2 is the interlayer distance. Under the assumption of an elliptical shape of

the Fermi surface cross-sectional aerea, kmax|| can be determined via Eq. (3.18) [39, 40]:

kmax|| =

[(k1 cosϕ)2 + (k2 sinϕ)2

] 12 , (5.7)

with k1, k2 being the semiaxes of the ellipse. For fα the semiaxes k1,α = 0.118 Å−1

and k2,α = 0.216 Å−1, and for fβ k1,β = 0.385 Å−1 and k2,β = 0.339 Å−1 were used.The values for the semiaxes were obtained from the Fermi surface shown in Fig. 2.4(b).The semiaxes of the elliptical orbits α and β are illustrated in Fig. 5.7. The frequencies

Figure 5.6: Calculated frequencies for the α and β orbit and for the LMAs (solidsquares) and experimentally obtained frequencies (empty circles) from the AMROcurves shown in Fig. 5.5.

of the LMAs were calculated using Eq. (3.22):

fLMA = cosϕda

= cosϕ b

2a (5.8)

By comparing the calculated and measured frequencies, it becomes evident that thedominant frequency in the AMROs is caused by the β orbit. For ϕ = 0, the α frequency

45


Figure 5.7: Fermi surface of κ-(BETS)2FeBr4 in the normal metallic state, with la-beling of the elliptic α and β orbits and the corresponding semiaxes. The magneticbreakdown junctions are indicated by green circles

is observed, but for 70 < ϕ < 90 the α frequency vanishes, whereas the frequencyfLMA caused by the Lebed resonance is found. This behaviour is in agreement withthe above mentioned report of Konoike et al [16]. The vanishing of fα may be causedby the fact that for ϕ approaching π/2, both frequencies fα and fβ are coming closertogether, and therefore in the FFT spectrum fα is covered by the large peak of thedominant frequency fβ. From fitting the measured β frequencies one obtains for thesemiaxes of the elliptical orbit k1,β = 0.394 Å−1 and k2,β = 0.327 Å−1, which results ina cross sectional area Sβ = 0.405 Å−2. This value is in good agreement with the valuedetermined from the SdH oscillations.The angles θn, for which one would expect a dip in the AMRO curve due to the LMA

resonance, can now be calculated according to Eq. (3.22):

| tan θn| =n

cosϕ ·a

d= 2n

cosϕ ·a

b, n = 0, 1, 2, ... (5.9)

However, by comparing the calculated positions of the LMA dips with the measuredones, we find that they are rather described by

| tan θn| =2n+ 1cosϕ ·

a

b= 2n

cosϕ ·a

b+ 1

cosϕ ·a

b(5.10)

instead of Eq. (5.9). In order to demonstrate this behaviour, the AMROs for ϕ = 70are shown in Fig. 5.8, where the resistance R is plotted against tan θ. Beside the

46


dominant frequency fβ one can clearly distinguish a lower frequency at higher anglesθ. The positions of the dips caused by this lower frequency correspond to the LebedMagic Angles. In the inset in Fig. 5.8 the inverse Fast Fourier Transform for fLMA isshown. The inverse FFT was obtained by subtracting the background from the originalcurve and conducting a Fast Fourier Transform. Then, after the LMA frequency wasidentified, a narrow frequency window containing fLMAwas chosen and for this windowan inverse FFT was conducted. The dip positions calculated with Eq. (5.9) andindicated by red arrows in the inset of Fig. 5.8 are in good agreement with the dipsobserved in the inverse FFT indicated by black arrows.

Figure 5.8: AMRO curve for ϕ = 70 with R plotted against tan θ and correspondinginverse FFT (inset) of fLMA. Calculated and experimentally obtained positions ofLMA dips are indicated by red and black arrows.

In Section 3.1.2 an explanation for the LMAs was given based on the electron move-mement on the quasi-one-dimensional Fermi surface sheets. However, the physicalorigin of LMAs can also be understood in a much simpler way [4]: For the followingwe consider a current applied perpendicular to the layers (I||b). If now a magneticfield is applied, the conduction electrons in real space are deflected due to the Lorentzforce. However, if the magnetic field is applied exactly parallel to the direction of thecharge transport across the layers, the conduction electrons will not be affected by theLorentz force and therefore keep their velocity component parallel to the field direction.Therefore, the conductivity should be maximum in this direction. This is the case if Bis directed along a translation vector of the crystal lattice.If one considers a field parallel to the ab plane, the Lebed Magic Angles θn are

given by Eq. (5.9). The resulting directions of the charge transfer in the ab planeare illustrated in Fig. 5.9 by the red lines. As discussed above, the positions of the

47


Figure 5.9: Directions of the interlayer charge transport as obtained from Eq. 5.9 (redlines) and Eq. 5.10 (blue lines). Orthorhombic unit cells of the crystal lattice areindicated by black rectangulars.

magnetoresistance dips in Fig. 5.8 satisfy Eq. (5.10), which is illustrated by the bluelines in Fig. 5.9. However, no clear conclusion can be drawn concerning the exact waythe electron is taking for crossing the layers. If one, for example, considers an electronfollowing the first blue line in Fig. 5.9 with tan θ0 = a

b , starting at point A, in orderto arrive at point B, the electron has to cross the insulating barrier of the anion layertwice. Looking along the b direction in an arbitrary ab plane, one can see that onlyevery second insulating barrier is filled with FeBr4 anions. This is due to the fact thatthe FeBr4 anions of neighbouring layers are shifted by half a period in the c directionwith respect to each other [see Fig. 2.1(c)]. Also one can see that the distance betweenneighbouring BETS layers in the ab plane is shorter for the case with no FeBr4 ionsbetween the layers. Therefore, for an electron following the path from A to B it is likelythat it first crosses the insulating barrier without any anions but in order to cross thesecond insulating layer it has to switch to the next plane below or above the initial abplane, where it can again cross the insulating layer without FeBr4 anions. Switchingbetween neighbouring ab planes should be easily possible because of the relatively largeinterdimer overlap integral p = 77.33 eV. Also by taking a look at the Fermi surface inFig. 2.4(b) one can see that for an electron situated on the large 1D Fermi sheet thevelocity component parallel to the c direction always dominates in comparison to theother components and therefore the wave function of the electron should be delocalisedalong the c direction, which means the position of the electron in the c direction is notdefined anyway.

48


5.1.3 Coherence Peak

In this section the dependence of the interlayer magnetoresistance on the polar angle θnear π

2 is investigated for azimuthal angles 0 ≤ ϕ ≤ 90. Fig. 5.10 shows a completeset of angle-dependent resistance curves at a constant field of 15 T and at a temperatureof 1.4 K for various angles ϕ. For B||a (ϕ = 0) the magnetoresistance exhibits a steepincrease for θ approaching π

2 . With turning the field towards the c axis, a clear peakstructure becomes visible at ϕ ≈ 40. With increasing ϕ further, the peak becomeseven narrower but is also reduced in height until for ϕ ≥ 55 a minimum with a localsmall peak at θ = 90 appears. For 62.5 ≤ ϕ ≤ 70 a small peak is still visible,although it is now situated on a broad hump structure. The resistance for ϕ ≥ 75 justincreases for θ → π

2 but shows no peak.For ϕ = 60 the peak structure is shown for various fields in Fig. 5.11(a) and the

same peak structure but normalised to the height of the peak is shown in Fig. 5.11(b).From this data one can see that the width of the peak does not change for differentfield strengths. The half width of the peak at 15 T differs only by 0.05 from that at7 T. This behaviour indicates that the peak indeed can be explained as a coherencepeak (see Section 3.1.2). The width of the peak W = 0.9 is a typical value one wouldexpect for organic metals. With Eq. (3.19) the anisotropy ratio 2t⊥

εFcan be estimated:

t⊥εE∼= (π − 2θ) 1

kF b(5.11)

Because of the elliptical shape of the Fermi surface cross sectional area the Fermik-vector for ϕ = 60 can be calculated using the in Section 5.1.2 experimentally deter-mined values of the semiaxes k1β = 0.394 Å−1 and k2β = 0.327 Å−1:

kF (ϕ = 60) = k1βk2β√k2

1β sin2 ϕ+ k22β cos2 ϕ

= 0.341 Å−1 (5.12)

With this one estimates for the anisotropy ratio a value of t⊥εF

= 9.1 · 10−4 and withεF = 0.06 eV one obtains for the transfer integral t⊥ = 5.5 · 10−5 eV.Still the question remains why no clear peak is seen for ϕ = 0. By taking a look

at the Fermi surface in Fig. 2.4(b), one can see, that on the small cylinder due to theelliptical shape a large area exists where the velocity vF is approximately parallel tothe field direction. For comparison this area is much smaller for for B||c. Therefore, alarge amount of electrons should contribute to the interlayer conductivity for B||a.In the previous report of Konoike et al. [16], also a coherence peak was reported,

which was most clearly seen for ϕ = 0. The peak became visible at B ≈ 9 T, butbecame really pronounced only in fields above 15 T. In contrast to that in this experi-ment a clear coherence peak is observed for intermediate ϕ angles. The anisotropy ratiodetermined above is a little bit smaller than the ratio 0.0015 determined by Konoikeet al. Eventually one should note that analysing the peak structure in κ-(BETS)2FeBr4is difficult because closed orbits on both of the 1D and 2D parts of the Fermi surfacecontribute to the resistance. But it is proved that the interlayer transport is coherent.

49


Figure 5.10: Angle dependent interlayer resistance near θ = 90 for various azimuthalangles ϕ at 15 T and 1.4 K and complete AMRO curve for ϕ = 0. The curves arevertically shifted for clarity.

50

5.2 The antiferromagnetic state

(a) (b)

Figure 5.11: (a) Angle-dependent interlayer resistance for ϕ = 60 and 65 < θ < 105.(b) Interlayer resistance from (a) normalised to the height of the coherence peak.


5.2.1 Shubnikov-de Haas oscillations

In the antiferromagnetic state at T < 2.5 K and for fields B < 5 T (see Section 2.4.2)SdH oscillations were investigated. SdH oscillations of two samples at 0.4 K are shownin Fig. 5.12. The oscillations start at B ≈ 2 T, and the amplitude increases with themagnetic field until B ≈ 4.5 T, where the oscillations vanish due to the transition fromthe antiferromagnetic to the paramagnetic phase. From the fact that the oscillationsof sample 2 already appear at 2 T at 3He temperatures, one can conclude that thissample was very clean. The frequencies measured via a FFT for both the samples inFig. 5.12, 60 T for sample 1 and 63 T for sample 2, respectively, are in good agreementwith Konoike et al. [16].In Fig. 5.12 also the hysteretic behaviour of the magnetoresistance is presented. The

appereance of the hysteresis was qualitatively different depending on which sample wasmeasured. For each sample up- and downsweeps of the magnetic field are identicalin the paramagnetic phase but within the antiferromagnetic state a large hysteresisappears. For sample 1 in the antiferromagnetic phase the field-dependent resistance ofthe downsweep stays below the resistance of the upsweep. Also the amplitude of theoscillations observed in the downsweep at T = 0.4 K is smaller than in the upsweep.This behaviour is shown in Fig. 5.12(a).In contrast to that the hysteresis for sample 2 shown in Fig. 5.12(b)is inverted in

comparison to sample 1. As one can see, the field-dependent resistance of the down-sweep is larger than in the upsweep. Also no oscillations are visible in the downsweep atT = 0.41 K. It is also noteworthy that in a second upsweep without changing temper-ature, the resistance followed the previous downsweep, showing no oscillations. Thus,in order to observe the SdH oscillations in this sample, it was necessary to heat thesample above the Néel temperature TN = 2.5 K before each field sweep.

51


(a) (b)

Figure 5.12: (a) and (b): Hysteretis of the magnetoresistance of two different samplesin the AFM state. Up- and downsweeps are indicated by solid and dashed lines, andsweep directions are indicated by arrows.

Figure 5.13: Low field SdH oscillations at various temperatures. The construction ofthe antiferromagnetic transition field is exemplarily shown for T = 0.9 K.

Effective cyclotron mass of δ oscillations

From the temperature dependence of the SdH oscillations shown in Fig. 5.13, theeffective mass for the δ oscillation was determined as described in Section 3.2.2. In

52


the effective mass plot shown in Fig. 5.14(a) an effective mass of mc = 1.08m0 wasobtained, which is in agreement with the value of mc = 1.1m0 from [16].The effective mass was also determined for a pressure of 1.9 kbar. In the mass plot in

Fig. 5.14(a), an effective mass of mc(1.9 kbar) = 0.83m0 was obtained. The frequencyat 1.9 kbar was determined to 62 T, which is within the experimental error bar thesame value as with no pressure applied. For a pressure of 4.5 kbar neither the highfield α and β oscillations nor the low field δ oscillations could be detected.

(a) (b)

Figure 5.14: Temperature dependence of the FFT amplitude of the δ-oscillations (a)at ambient pressure and (b) under a pressure of 1.9 kbar. Inset in (b) shows the FFTfor low field SdH oscillations at 1.9 kbar.

θ-dependence of δ oscillations

In order to study the dependence of the δ oscillations on the field direction, the resis-tance was measured for various angles θ as shwon in Fig. 5.15 for ϕ = 0 and ϕ = 90,respectively, at 0.4 K. For tilting the field in the ab plane from the direction perpen-dicular to the layers towards the direction parallel to the layers, the antiferromagnetictransition shifts to lower fields more rapidly than it is the case for tilting the field in thebc plane. As one can see, for turning the field in the ab plane, already at θ = 30 thetransition field is 2.5 T and the oscillations can’t be observed in the antiferromagneticphase for much higher angles θ. Therefore, the field rotation in the bc plane is morefavourable for this experiment.The FFT spectra for the curves in Fig. 5.15(b) are shown in Fig. 5.16(a). As

one can see, the frequency of the δ oscillation shifts to higher values with increasingangle θ, whereas the amplitude decreases. Until θ = 40, the frequency Fδ follows therelation Fδ(θ) = Fδ(0)

cos θ which is typical for cylindrical Fermi surfaces [see Fig. 5.16(b)].However, for higher angles a deviation is observed. The frequency θ = 60 can onlybe determined with a large uncertainty because the peak in the FFT spectrum is verybroad and not larger than the background signal. But for θ = 50 and θ = 55 the peaks

53


in the FFT spectrum are clearly pronounced. The deviation may be caused from thefact that at these angles the warping of the small Fermi cylinder becomes important,leading to small extremal closed orbits at the side of the warped Fermi surface cylinder.

(a) (b)

Figure 5.15: SdH oscillations in the AFM state at various field directions for (a) ϕ = 0

and (b) ϕ = 90.

In Fig. 5.16(c), the θ-dependence of the FFT amplitude of the δ oscillations for twosamples is shown. The amplitude in dependence on the angle θ is simulated accordingto Eq. (3.31). For a Landé factor of 2 and a Dingle temperature of 1 K, one wouldexpect a zero of the amplitude at θ ≈ 43 due to the spin factor RS (see Section 3.2.2).A fit of the measured data following Eq. (3.31), suggests a Landé factor of 1.92 anda Dingle temperature of 0.59 K, which would be a reasonable value for the Dingletemperature of organic metals. The Fit also suggests a spin zero at θ ≈ 45. However,the combined data of the two samples leads to the conclusion that no such spin zerois observed. The Dingle temperature will be discussed in the following section, wherealso the magnetic breakdown will be considered.

Dingle temperature

Since the Dingle temperature TD is an indicator of the purity of a sample, in order todetermine the Dingle temperatures, Dingle plots described in Section 3.2.2 were per-formed. However, the Dingle plots revealed very low temperatures of TD ' 0.3 K forthe δ oscillations and TD ' 0.6 K for the α oscillations. Because of the discrepancybetween the Dingle temperatures obtained for α and δ oscillations, the Dingle tempera-ture considering magnetic breakdown (MB) was investigated. Therefore, the amplitudewas plotted over the field as for example shown in Fig. 5.17 and fitted with a curvedescribed by:

y (B) =√BRT (B)RD (B) ·

(1− e−

BMBB

)n2. (5.13)

54


(a) (b)

(c)

Figure 5.16: (a) FFT spectra of the magnetoresistance curves shown in Fig. 5.15(b).(b) Values of the δ frequencies in (a) as a function of the polar angle θ and (c) thecorresponding amplitudes.

In Eq. 5.13 the spin factor RS can be omitted because it is not dependent on the field.

The last factor(

1− e−BMBB

)n2describes the decrease of the oscillation amplitude due to

the magnetic breakdown orbit as described in Section 3.2.3. The integer n correspondsto the number of breakdown junctions for a particular orbit. For the α orbit twojunctions exist, therefore one has to chose n = 2, whereas for the δ orbit four junctionshave to be considered (n = 4) [see also Fig. 2.4(b) and 2.8(b)]. The values for the

55


MB field BMB and the Dingle temperature TD obtained from these plots ranged fromBMB = 0.002 T to BMB = 0.01 T and from TD = 0.42 K to TD = 0.57 K. By fittingthe amplitude with the function (5.13), with BMB fixed and only TD being the freeparameter, the diagram in Fig. 5.17(b) was obtained. One can notice that the Dingletemperatures for small and high breakdown fields are independent of the field. For bothsamples for small breakdown fields the Dingle temperatues are by ∼10-20% smaller forthe δ oscillations than for the α oscillations, whereas for large breakdown fields thevalues of TD are by ∼80-95% smaller for the δ oscillations than for the α oscillations.For different samples TD can of course be different, but for one particular sampleit is natural to expect that TD is equal for α and δ oscillations, since TD describesthe scattering time τ which should not depend on the magnetic phase. Therefore,because of the big discrepancy of TD between the low and high field oscillations andalso due to the unphysical low value of TD for the δ oscillations (TD < 0.1 K) for largemagnetic breakdown fields, one can assume that in the diagram in Fig. 5.17(b) thelow breakdown field regime (BMB < 1 T) is relevant, where the discrepancy betweenthe Dingle temperatures for the different oscillations is lower. There, one would expectDingle temperatures TD ≈ 0.45 K for sample 1 and TD ≈ 0.5 K for sample 2.

(a) (b)

Figure 5.17: (a) Amplitude of δ oscillations plotted against 1B . The fit curve cor-

responds to Eq. (5.13). (b) Dingle temperatures obtained from Dingle plots withdifferent values of BMB.

From the discussion above it is clear that it is not possible to determine the magneticbreakdown field and the Dingle temperatures with sufficient accuracy. However, BMBshould be in the order of 1 T. The higher Dingle temperature for sample 2 is alsoplausible, since the quality of sample 1 was found to be better than that of sample2. With the Dingle temperatures determined above one gets for the relaxation timeτ = 2.7 · 10−12 s for sample 1 and τ = 2.4 · 10−12 s for sample 2. The considerationsabove are of course only valid under the assumption that BMB is the same for α andβ oscillations, which is indeed suggested by the FS reconstruction model by Konoikeet al. [16] .

56


(a) (b)

Figure 5.18: (a) Temperature and (b) field dependent torque measurements for θ = 5.The transition temperature TAFM and the transition field BAFM indicated in (a) andin the inset in (b) are determined as described in the text.

5.2.2 Magnetic phase diagrams

Former measurements on κ-(BETS)2FeBr4 [10, 15] but also recent experiments at theWMI on κ-(BETS)2FeCl4 [21] have shown that the phase boundaries of the antiferro-magnetic state can be resolved by characteristic anomalies in the interlayer resistance.The phase diagrams for the magnetic field directions B||a and B||c, which were studiedby Konoike et al. [15], were already presented in Section 2.4.2. However, the phasediagram for fields perpendicular to the layers (B||b) is published only very near to TN[10]. In this master thesis the magnetic phase diagram at ambient pressure for B||band B||a was sudied to confirm and complete the data. However, the important partis the investigation of the phase diagram under hydrostatic pressures of 1.9 kbar and4.5 kbar.

Phase diagram at ambient pressure

The magnetic phase diagrams at ambient pressure for B||b and B||a are summarised inFig. 5.19. The data for B||b was extracted from torque measurements shown in Fig.5.18 and from resistance measurements shown in Fig. 5.13. For B||a the transitionfield was determined from resistance measurements shown in Fig. 5.26.In Fig. 5.18(a) the torque data are presented as τ/B as function of temperature at

various magnetic fields and in Fig. 5.18(b) τ/B is presented as function of magneticfield at various temperatures. The magnetic field is not aligned exactly parallel to theb axis, but tilted by an angle θ = 5 with respect to b, since the torque vanishes formagnetic fields applied along symmetry directions of the crystal. As one can see in Fig.5.18(a), the torque signal increases until a hump structure is forming and then decreasesagain with increasing temperature. In this case, the transition temperature TAFM wasdetermined as the crossing point of two lines corresponding to the inclinations of the

57


curve below and above the hump structure. An example for the construction of TAFMis illustrated in Fig. 5.18(a).Above 2.5 K in the field sweeps τ/B shows a linear increase and goes through a

maximum in the paramagnetic state. For the field sweeps below the Néel temperatureτ/B shows a characteristic upturn with a linear part at the steepest slope. The deter-mination of a transition point is not straightforward. Best coincidence with the dataobtained from the temperature sweeps was achieved by taking the upper end of thelinear part as shown by the arrow in Fig. 5.18(b) for T = 0.5 K. These torque data arepresented in Fig. 5.19.

Figure 5.19: Magnetic phase diagram at ambient pressure for B||a and B||b obtainedfrom torque and resistance measurements. Data of torque and resistance measure-ments is indicated by diamonds and triangles, respectively. For comparison data ofKonoike et al. [15] for B||a and B||b is shown (empty circles).

In the phase diagram 5.19 also the data for the antiferromagnetic transition obtainedby Konoike et al. [15] is shown for comparison. For B||a the critical fields BAFMobtained from the field sweeps in Fig. 5.26 are in good agreement with the data ofKonoike et al. Also the values BAFM are a little bit smaller in comparison to Konoike’sdata. The sample from which the transition fields BAFM for B||a were determinedshowed also a by ∼ 0.1 K smaller Néel temperature TN than that obtained by Konoikeet al. This may be also due to the fact that in the resistance measurements of Konoikeet al. two features were observed. The first large steplike feature was attributed tothe superconductiong transition, the second smaller feature to the antiferromagnetictransition (see phase diagram Fig. 2.6). However, in this measurement only one largesteplike feature was observed. Therefore, the feature seen in this experiment may

58


(a) (b)

Figure 5.20: (a) Field and (b) temperature sweeps for B||a axis at 1.9 kbar. Thevalues TAFM and BAFM were determined analogous to TAFM in Fig. 5.18(a). Forsome curves the antiferromagnetic transition is indicated by black arrows.

correspond to the feature of the superconducting transition seen by Konoike et al.However, the feature seen in this experiment is attributed to the antiferromagnetictransition, which will be discussed in more detail in Section 5.3.For B||b one can see that the phase line coincides with that obtained for B||c by

Konoike et al. This isotropic behaviour was also observed by Fujiwara et al. [10]. Fromthat one can conclude that the antiferromagnetic transition field is probably isotropicwith respect to the bc plane. Otherwise BAFM would exhibit a minimum or maximumin the bc plane which would violate the orthorhombic symmetry.

Phase diagrams under pressure

A previous report on the pressure effect on the antiferromagnetic and superconductingphase transitions was done by Otsuka et al. [18]. The resulting pressure-temperaturephase diagram was already presented in Section 2.4. In this report it was foundthat the Néel temperature slightly increases for increasing pressure, which is not acommon behaviour expected in comparison with other compounds like for exampleλ-(BETS)2FeBr4 [78]. Therefore, in order to study the influence of pressure on theantiferromagnetic transition pressures of 1.9 kbar and 4.5 kbar at low temperatureswere applied with the pressure cell described in Section 4.1.4. Figures 5.20 and 5.21show temperature and magnetic field sweeps for B||a at the two different pressures.At 1.9 kbar, the magnetic field dependence of the resistance in Fig. 5.20(a) follows aB2 dependence above the Néel temperature. With lowering the temperature a clearsteplike feature is observed and broadens with further decreasing the temperature. Asimilar feature is also observed in the temperature sweeps shown in Fig. 5.20(b). Thefeatures were analysed by the same construction as shown for TAFM in Fig. 5.18(a). At4.5 kbar (see Fig. 5.21) the features are much less pronounced, but still recognisable.The values for TAFM and BAFM were obtained by subtracting polynoms.

59


(a) (b)

Figure 5.21: (a) Field and (b) temperature sweeps for B||a axis at 4.5 kbar. Thefeatures corresponding to the antiferromagnetic transition were determined by sub-tracting polynoms. For some curves the features are indicated by black arrows.

The resulting phase diagrams for both pressures and for field orientations along thea and b axes are shown in Fig. 5.22. In the phase diagram for B||a the phase lineat 1.9 kbar starts with a slightly larger Néel temperature of 2.6 K than reported byKonoike et al. [16] for ambient pressure but then intersects their phase line. The phaseline then continues in a linear way and stays below Konoike’s data. For example, atT = 0.5 K, the value of the critical field is about 0.4 T smaller than the value at ambientpressure reported in ref. [15]. In contrast to that, the phase line at 4.5 kbar starts ata considerably higher Néel temperature TN = 3 K than reported for ambient pressure,showing no linear characteristic, but converging to the phase line at ambient pressurefor low temperatures.For B||c both phase lines are found to follow the convex characteristic of the phase

line at ambient pressure [see Fig. 5.22(b)]. The phase line for 1.9 kbar approximatelymatches the reported [15] ambient pressure phase line, whereas at 4.5 kbar it is con-siderably higher for all temperatures.The temperature and magnetic field sweeps for B||b (i. e. perpendicular to the

layers) at the pressure 1.9 kbar as well as the corresponding phase diagram for ambientpressure and 1.9 kbar and 4.5 kbar, respectively, are shown in Fig. 5.23 and Fig. 5.24,respectively. In the temperature sweeps at 1.9 kbar, the feature corresponding to theantiferromagnetic phase transition becomes very weak already at low fields. In contrastto that, in the field sweeps, starting from high temperatures, a clear dip-like featurecould be traced until 1.4 K. Below 1.4 K a clear feature could not be defined. Herethe feature in the low temperature regime was defined as the inflection point of theinclination above the field value corresponding to the maximum of the hump structure.For P = 4.5 kbar the transition features were much weaker. With the exception of a

few values of the transition temperatures obtained from temperature sweeps, a certaindetermination of the values for the transition fields was not possible for this pressure.In the resulting phase diagram, the data points for 1.9 kbar obtained from the tem-

60


(a) (b)

Figure 5.22: Magnetic phase diagrams for (a) B||a axis and (b) B||c axis under apressure of 1.9 kbar and 4.5 kbar. For comparison, the ambient pressure data fromref. [15] are also presented (empty circles).

(a) (b)

Figure 5.23: (a) Field and (b) temperature sweeps for B||b axis at 1.9 kbar. For somecurves the transition features are indicated by black arrows.

61


perature sweeps qualitatively follow the data for ambient pressure, whereas the phaseline obtained from the field sweeps for B||a intersects the ambient pressure phase lineand then shows a linear behaviour down to lowest temperatures, suggesting also ahigher critical field at the lowest measured temperature.

Figure 5.24: Magnetic phase diagram for B||b for pressures P = 0, 1.9 and 4.5 kbar.

From the results above it also shows that the Néel temperature at 1.9 kbar TN = 2.6Kis in good agreement with Otsuka et al. [18], whereas for 4.5 kbar a considerably largerNéel temperature TN ≈ 3 K was obtained in comparison to ref. [18].It should also be noted that resistance measurements dependent on the polar angle

θ and azimuthal angle ϕ gave no evidence for a change of the easy axis under pressure.Summarising one can say that no significant effect on the magnetic phase diagram

was seen for P = 1.9 kbar. For P = 4.5 kbar TN was considerably shifted by ≈ 0.5 K toa higher value and for B||c also a higher critical field BAFM was obtained at T = 0.7 K.Responsible for the stabilisation of the antiferromagnetic phase under pressure may bethe structural compression which leads to a stronger overlap of the localised electronsof neighbouring FeBr4 anions resulting in a stronger exchange interaction.

5.3 Superconducting state

As reported in Section 2.4, the interlayer resistance of κ-(BETS)2FeBr4 shows a super-conducting transition below 1.4 K. In Fig. 5.25, the interlayer resistance of two samplesfor the low temperature range is shown. For sample 1 the superconducting transition

62


Figure 5.25: Temperature dependence of the interlayer resistance of two different sam-ples. Tc and TN are indicated by arrows.

occurs at Tc = 1.5 K, whereas for sample 2 the critical temperature is Tc = 0.6 K.Also the shapes of the transition are different. Sample 1 resembles the temperaturedependence as reported in [10]. The feature at T ≈ 2.5 K in both samples is causedby the antiferromagnetic transition as discussed in Section 5.2.2. The difference inTc between the two samples may be a hint for unconventional superconductivity inκ-(BETS)2FeBr4, because Tc obviously depends on the sample quality [79]. A differ-ence between the qualities of sample 1 and 2 was indeed confirmed by the determinationof the Dingle temperatures in Section 5.2.1, but the difference being quite small withTD = 0.45 K for sample 1 and TD = 0.5 K for sample 2.In Fig. 5.26 the interlayer resistance is shown as a function of the magnetic field

for various temperatures in the low temperature range for B||a. For T = 0.5 K aclear steplike increase of the resistance at 1.7 T can be seen for increasing field. In theprevious report by Konoike et. al. [15], two features were observed for B||a: A largesteplike increase of the resistance when superconductivity was broken and a smallerfeature which was believed to originate from the antiferromagnetic transition. However,in this experiment only one large steplike feature was observed.For B aligned parallel to the layers, the orbital pair breaking effect is strongly sup-

pressed. Therefore, superconductivity will be destroyed due to a magnetic field actingon the spins of the conduction electrons. As presented in Section 2.4.2, the effective fieldon the conduction electrons is given by Heff = Hext − HJ , where HJ is the exchangefield produced by the localised spins of the FeBr−4 anions. In the antiferromagneticstate the exchange field HJ is zero. Let us first consider the case B||c: In this casesuperconductivity is broken at ∼ 3 T for lowest temperatures, whereas the antiferro-

63


Figure 5.26: Interlayer resistance of sample 1 as a function of the magnetic field forvarious temperatures for B||a. Inset: Low field regime. Magnetic field values of AFMtransition for some temperatures are marked by arrows.

Figure 5.27: θ dependence of interlayer resistance for B = 12 T at T = 0.4 K.

64


Figure 5.28: Magnetic torque as function of the polar angle θ at B = 12 T for T = 1.5K and T = 0.45 K.

magnetic state is broken at ∼ 4.5 T (see phase diagram for B||c in Fig. 2.6). Therefore,one can conclude that superconductivity for B||c is broken either due to the canting ofthe spins of the localised electrons resulting in a finite exchange field HJ 6= 0 or due tothe external field reaching the paramagnetic limit [see Eq. (3.47)].In the case B||a superconductivity is broken already at ∼ 2 T. Two scenarios for B||a

can generally be considered: In one case, because the magnetic field is aligned along theeasy axis, a spin flop may occur at a certain value of the magnetic field. With a furtherincrease of the field, the spins are more and more canted, until the antiferromagneticordering is broken. Due to the canting of the localised spins it is possible that theexchange field becomes sufficiently large to break superconductivity even before theantiferromagnetic state is suppressed. In the other case only a spin flip occurs. At thespin flip transition suddenly all localised spins are saturated along the external field,producing an exchange field of 12.7 T and as a consequence of that, superconductivityis immediately broken. The shape of the transition for 0.4 K in Fig. 5.26 suggests, thatfor B||a at low fields superconductivity can only exist in the antiferromagnetic stateand is broken due to a spin flip transitionFor T = 0.5 K, also field induced superconductivity was observed. The minimum

of the resistance is located at 12.7 T, which is in good agreement with ref. [15]. Inaddition also the value BP of the paramagnetic limit can be estimated according toBP = B − BJ , where B is the lower critical field of the FISC (illustrated in Fig.5.26). With that one obtains BP = 3.10 T. This value is in good agreement withthe value of B ≈ 3 T at 0.5 K, for which superconductivity is broken for B||c in the

65


antiferromagnetic phase. From this one can conclude that superconductivity in thiscase is broken due to the external field. This suggests also that the exchange field isstill compensated in the AFM state, even in the the case of canted spins.The FISC in an angular sweep at 12 T and 0.4 K is shown in Fig. 5.27. From there

one can also see that the magnetic field has to be almost perfectly aligned parallel tothe layers in order to observe FISC. If the magnetic field is tilted only 1 out of theplane, the FISC is already destroyed due to the orbital pair-breaking effect.So far the FISC state in κ-(BETS)2FeBr4 has only been observed in resistive mea-

surements [13, 15]. In this thesis the FISC state has possibly been observed in magnetictorque measurements. Figure 5.28 shows the magnetic torque as a function of the polarangle θ at 1.5 K and 0.45 K for a constant field of 12 T. The linear background of thesignal is produced by the localised spins of the magnetic anions, which produce a largesignal changing sign at the symmetry direction (θ = 90). The measurement at 1.5 Kabove the superconducting transition just shows the background signal. But at 0.45 K,a pronounced loop structure near θ = 90 can be recognised, which can probably onlyoriginate from the FISC state. The width of the loop structure is 0.3, which is evensmaller than the width of the anomaly in the angle dependent resistive measurementshown in Fig. 5.27. However, it is not surprising that the superconducting signal inthe thermodynamic quantity is less pronounced.

66

6 SummaryIn this master thesis the normal metallic, antiferromagnetic and superconducting statesof the organic superconductor κ-(BETS)2FeBr4 were investigated. The results andconlucsion are summarised in the following chapter.

Shubnikov-de Haas oscillations at ambient pressure

At ambient pressure the SdH oscillations for the normal metallic state and the antifer-romagnetic state were investigated by measuring the interlayer resistance in magneticfield. The frequencies of the oscillations, Fα, Fβ and Fδ, were found to be consistentwith previous reports [9, 12, 16]. Also the effective cyclotron mass mc,δ = 1.08m0 de-termined from the temperature dependence of the δ oscillations is in good agreementwith [16].The fact that the δ oscillations were only observed in the antiferromagnetic phase

confirmes the assumption of a reconstruction of the Fermi surface in the antiferromag-netic state. However, the specific simple Fermi surface reconstruction model of Konoikeet al. [16] could neither be confirmed nor disproved. The absence of additional SdH fre-quencies, which one would expect due to the reconstruction model of Konoike et al. inthe antiferromagnetic state leads to the conclusion that either the Fermi surface is notexplained by this simple reconstruction model or the effective masses of the additionalorbits are too large in order to be detected at low fields.In this work the angular range, where δ oscillations could be observed, was extended

up to θ = 60 compared to θ = 40 in ref. [16]. However, no sign for a spin zero wasobserved. To confirm this result, further studies are recommended in this respect.The field dependence of the resistance for B||b revealed a large and sample dependent

hysteresis in the antiferromagnetic state, which was not reported before. The hysteresisis also observed in the amplitude of the SdH oscillations.From the amplitude of low and high field oscillations, for the samples Dingle temper-

atures of TD ≈ 0.5K were determined, and from the dependence of TD on the magneticbreakdown field, the value BMB was estimated to be of the order of 1 T.

Shubnikov-de Haas oscillations under pressure

The SdH oscillations were also observed under pressure. For a pressure of 1.9 kbar,α and β oscillations were still visible, but the β oscillations could not be observed formagnetic fields B ≤ 15 T. At a pressure of 4.5 kbar, no oscillations at all were observedfor B ≤ 15 T. The effective cyclotron masses of the α and δ oscillations, determined atP = 1.9 kbar, mc,α(1.9 kbar) = 3.3m0 and mc,δ(1.9 kbar) = 0.83m0, were found to besmaller than for ambient pressure, emphasising a decrease of electron correlations forhigh pressures.

67

6 Summary

Angle-Dependent Magnetoresistance Oscillations

From measuring the angle-dependent magnetoresistance conclusions on the Fermi sur-face and on the charge transport across the layers could be drawn. The main contri-bution to the angle-dependent magnetoresistance oscillations was found to originatefrom the magnetic breakdown orbit. From the angle dependence of the correspondingfrequency the semiaxes k1,β = 0.394 Å−1 and k2,β = 0.327 Å−1 of the breakdown orbitin k space were determined.In agreemente with ref. [16] for θ ≥ 70 no contributions of the α orbit were found.

Instead dips caused by the Lebe-Magic-Angles (LMA) resonance were observed. Thisbehaviour was also reported by Konoike et al [16]. However, the positions of the dipsare not described by the usual LMA condition but rather an offset of half a period hasto be considered. This result implies a tilted direction of the interlayer hopping vectorh = (ka/2, kb/2, 0) (where kb is the direction perpendicular to the layers), suggestingan important role of the FeBr−4 anions in bridging the conducting layers.For θ = 90 also a small coherence peak was observed. Therefore, one can conclude

that the charge transfer across the layers is coherent. From the width of the coherencepeak the anisotropy ratio t⊥

εF= 9.1 · 10−4 was determined in good agreement with ref.

[16]. This result also allowed to calculate the value of the interlayer transfer integralt⊥ = 5.5 · 10−5 eV.

Antiferromagnetic phase diagram

From the resistance and magnetic torque measurements the phase diagram for B||b atambient pressure could be determined. It showed that the value of the antiferromagnetictransition field is isotropic with respect to B aligned parallel to the bc plane. Forpressures of 1.9 kbar and 4.5 kbar also the magnetic phase diagrams for B parallel tothe three crystal axes were presented. A clear effect on the Néel temperature was seen,which increased by about 20 % at 4.5 kbar in comparison to ambient pressure. Thisvalue is considerably larger than that reported by Otsuka et al. [18].For κ-(BETS)2FeBr4 the easy axis lies along the crystallographic a axis at ambient

pressure. By applying a magnetic field in this direction, a clear metamagnetic transitioncan be observed, but it is not clear if this is a spin flop or a spin flip transition. Upto now no indication for a spin flop phase above this transition was observed so farat ambient pressure. The present measurements under pressure also give no evidencethat the direction of the easy axis is changed. However, the broadening of the featurecorresponding to the antiferromagnetic transition for B||a at lowest temperatures undera pressure of 1.9 kbar may be hint to a spin flop phase occurring under pressure.However, further measurements under pressure are required in this respect.

Superconductivity

Finally, also the interplay between the superconducting and antiferromagnetic stateswas investigated. In our experiment it was found that for B||a superconductivityonly exists in the antiferromagnetic state and is broken exactly at the metamagnetictransition. Furthermore field-induced superconductivity was observed in resistance and

68

magnetic torque measurements. The magnetic field corresponding to the paramagneticlimit was determined to BP = 3.1 T at T = 0.4 K, which is in good agreement withthe field value of the superconducting transition for B||c. It should also be mentionedthat the critical temperature Tc was found to be dependent on the crystal quality, thereason for which is possibly unconventional superconductivity.

69

Bibliography

[1] W. A. Little. Possibility of Synthesizing an Organic Superconductor. Phys. Rev.134, A1416–A1424 (1964). URL http://link.aps.org/doi/10.1103/PhysRev.134.A1416.

[2] D. Jérome, A. Mazaud, M. Ribault & K. Bechgaard. Superconductivity in asynthetic organic conductor (TMTSF)2PF6. J. Physique Lett. 41, 95–98 (1980).URL http://dx.doi.org/10.1051/jphyslet:0198000410409500.

[3] H. Mori. Overview of organic superconductors. International Journal of ModernPhysics B 8, 1–45 (1994).

[4] J. Wosnitza. Fermi Surfaces of Low-Dimensional Organic Metals and Superconduc-tors, Vol. 134 of Springer Tracts in Modern Physics (Springer Berlin / Heidelberg,1996). URL http://dx.doi.org/10.1007/BFb0048480. 10.1007/BFb0048480.

[5] M. V. Kartsovnik. High Magnetic Fields: A Tool for Studying Electronic Prop-erties of Layered Organic Metals. Chem. Rev. 104, 5737–5782 (2004). URLhttp://pubs.acs.org/doi/pdf/10.1021/cr0306891.

[6] H. Kobayashi, H. Tomita, T. Naito, A. Kobayashi, F. Sakai, T. Watan-abe & P. Cassoux. New BETS Conductors with Magnetic Anions (BETS =bis(ethylenedithio)tetraselenafulvalene). Journal of the American Chemical So-ciety 118, 368–377 (1996). URL http://dx.doi.org/10.1021/ja9523350.

[7] A. Kobayashi, T. Udagawa, H. Tomita, T. Naito & H. Kobayashi. NewOrganic Metals Based on BETS Compounds with MX Anions (BETS =bis(ethylenedithio)tetraselenafulvalene; M = Ga, Fe, In; X = Cl, Br). ChemistryLetters 22, 2179–2182 (1993).

[8] E. Ojima, H. Fujiwara, K. Kato, H. Kobayashi, H. Tanaka, A. Kobayashi, M. Toku-moto & P. Cassoux. Antiferromagnetic Organic Metal Exhibiting SuperconductingTransition, κ-(BETS)2FeBr4 [BETS = Bis(ethylenedithio)tetraselenafulvalene].Journal of the American Chemical Society 121, 5581–5582 (1999). URL http://dx.doi.org/10.1021/ja990894r.

[9] L. Balicas, J. Brooks, K. Storr, D. Graf, S. Uji, H. Shinagawa, E. Ojima, H. Fuji-wara, H. Kobayashi, A. Kobayashi & M. Tokumoto. Shubnikov-de Haas effect andYamaji oscillations in the antiferromagnetically ordered organic superconductorκ-(BETS)2FeBr4: a fermiology study. Solid State Communications 116, 557–562(2000).

71

http://link.aps.org/doi/10.1103/PhysRev.134.A1416

http://link.aps.org/doi/10.1103/PhysRev.134.A1416

http://dx.doi.org/10.1051/jphyslet:0198000410409500

http://dx.doi.org/10.1007/BFb0048480

http://pubs.acs.org/doi/pdf/10.1021/cr0306891

http://dx.doi.org/10.1021/ja9523350

http://dx.doi.org/10.1021/ja990894r

http://dx.doi.org/10.1021/ja990894r

Bibliography

[10] H. Fujiwara, E. Fujiwara, Y. Nakazawa, B. Z. Narymbetov, K. Kato, H.Kobayashi, A. Kobayashi, M. Tokumoto & P. Cassoux. A Novel An-tiferromagnetic Organic Superconductor κ-(BETS)2FeBr4 [Where BETS =Bis(ethylenedithio)tetraselenafulvalene]. Journal of the American Chemical So-ciety 123, 306–314 (2001). URL http://dx.doi.org/10.1021/ja002439x.

[11] H. Kobayashi, H. Tanaka, E. Ojima, H. Fujiwara, T. Otsuka, A. Kobayashi,M. Tokumoto & P. Cassoux. Coexistence of antiferromagnetic order and su-perconductivity in organic conductors. Polyhedron 20, 1587–1592 (2001). URLhttp://www.sciencedirect.com/science/article/pii/S0277538701006581.

[12] S. Uji, H. Shinagawa, Y. Terai, T. Yakabe, C. Terakura, T. Terashima, L.Balicas, J. Brooks, E. Ojima, H. Fujiwara, H. Kobayashi, A. Kobayashi &M. Tokumoto. Two-dimensional Fermi surface for the organic conductor κ-(BETS)2FeBr4. Physica B: Condensed Matter 298, 557–561 (2001). URLhttp://www.sciencedirect.com/science/article/pii/S0921452601003829.

[13] H. Fujiwara, H. Kobayashi, E. Fujiwara & A. Kobayashi. An Indication ofMagnetic-Field-Induced Superconductivity in a Bifunctional Layered Organic Con-ductor, κ-(BETS)2FeBr4. Journal of the American Chemical Society 124, 6816–6817 (2002). URL http://dx.doi.org/10.1021/ja026067z.

[14] M. Tanatar, M. Suzuki, T. Ishiguro, H. Tanaka, H. Fujiwara, H. Kobayashi, T.Toito & J. Yamada. Thermal conductivity of organic superconductors in ori-ented magnetic field. Synthetic Metals 137, 1291–1293 (2003). URL http://www.sciencedirect.com/science/article/pii/S0379677902011451.

[15] T. Konoike, S. Uji, T. Terashima, M. Nishimura, S. Yasuzuka, K. Enomoto, H.Fujiwara, B. Zhang & H. Kobayashi. Magnetic-field-induced superconductivity inthe antiferromagnetic organic superconductor κ-(BETS)2FeBr4. Physical ReviewB 70, 094514 (2004). URL http://link.aps.org/doi/10.1103/PhysRevB.70.094514.

[16] T. Konoike, S. Uji, T. Terashima, M. Nishimura, S. Yasuzuka, K. Enomoto, H.Fujiwara, E. Fujiwara, B. Zhang & H. Kobayashi. Fermi surface reconstructionin the magnetic-field-induced superconductor κ-(BETS)2FeBr4. Physical ReviewB 72, 094517 (2005). URL http://link.aps.org/doi/10.1103/PhysRevB.72.094517.

[17] T. Konoike, S. Uji, M. Nishimura, K. Enomoto, H. Fujiwara, B. Zhang & H.Kobayashi. Magnetic properties of field-induced superconductor κ-(BETS)2FeBr4.Physica B: Condensed Matter 359–361, 457–459 (2005).

[18] T. Otsuka, H. Cui, H. Fujiwara, H. Kobayashi, E. Fujiwara & A. Kobayashi.The pressure effect on the antiferromagnetic and superconducting transitions ofκ-(BETS)2FeBr4. Journal of Materials Chemistry 14, 1682–1685 (2004). URLhttp://pubs.rsc.org/en/content/articlelanding/2004/jm/b404004j.

72

http://dx.doi.org/10.1021/ja002439x

http://www.sciencedirect.com/science/article/pii/S0277538701006581


http://dx.doi.org/10.1021/ja026067z



http://link.aps.org/doi/10.1103/PhysRevB.70.094514




http://pubs.rsc.org/en/content/articlelanding/2004/jm/b404004j

Bibliography

[19] S. Fujiyama, M. Takigawa, J. Kikuchi, H.-B. Cui, H. Fujiwara & H. Kobayashi.Compensation of Effective Field in the Field-Induced Superconductor κ-(BETS)2FeBr4. Observed by 77Se NMR. Physical Review Letters 96, 217001(2006). URL http://link.aps.org/doi/10.1103/PhysRevLett.96.217001.

[20] N. H. Khah, G. V. S. Rao, M. Reedyk, H. Fujiwara, H. Kobayashi, T. Nakamura,K. Yakushi & M. A. Tanatar. Low-temperature far-infrared absorption in theantiferromagnetic organic superconductor κ-(BETS)2FeBr4. Physical Review B 81,092508 (2010). URL http://link.aps.org/doi/10.1103/PhysRevB.81.092508.

[21] M. Kunz. Unpublished data.

[22] R. Kato, H. Kobayashi & A. Kobayashi. Synthesis and properties ofbis(ethylenedithio)tetraselenafulvalene (BEDT-TSeF) compounds. Synthetic Met-als 42, 2093–2096 (1991).

[23] T. Ishiguro, K. Yamaji & G. Saito. Organic Superconductors, Vol. 88 of SpringerSeries in Solid-State Sciences (Springer Verlag Berlin Heidelberg, 1998), 2nd edn.

[24] H. Kobayashi, H. Cui & A. Kobayashi. Organic Metals and Superconductors Basedon BETS (BETS = Bis(ethylenedithio)tetraselenafulvalene). Chemical Reviews104, 5265–5288 (2004). URL http://dx.doi.org/10.1021/cr030657d.

[25] O. Cépas, R. H. McKenzie & J. Merino. Magnetic-field-induced superconductivityin layered organic molecular crystals with localized magnetic moments. Phys-ical Review B 65, 100502 (2002). URL http://link.aps.org/doi/10.1103/PhysRevB.65.100502.

[26] T. Mori & M. Katsuhara. Estimation of πd-Interactions in Organic ConductorsIncluding Magnetic Anions. Journal of the Physical Society of Japan 71, 826–844(2002).

[27] S. Uji, H. Shinagawa, T. Terashima, T. Yakabe, Y. Terai, M. Tokumoto, A.Kobayashi, H. Tanaka & H. Kobayashi. Magnetic-field-induced superconductiv-ity in a two-dimensional organic conductor. Nature 410, 908–910 (2001). URLhttp://dx.doi.org/10.1038/35073531.

[28] L. Balicas, J. S. Brooks, K. Storr, S. Uji, M. Tokumoto, H. Tanaka, H. Kobayashi,A. Kobayashi, V. Barzykin & L. P. Gor’kov. Superconductivity in an OrganicInsulator at Very High Magnetic Fields. Phys. Rev. Lett. 87, 067002 (2001). URLhttp://link.aps.org/doi/10.1103/PhysRevLett.87.067002.

[29] S. Uji, H. Kobayashi, L. Balicas & J. Brooks. Superconductivity in an OrganicConductor Stabilized by a High Magnetic Field. Advanced Materials 14, 243–245(2002). URL http://dx.doi.org/10.1002/1521-4095(20020205)14:3<243::AID-ADMA243>3.0.CO;2-F.

[30] V. Jaccarino & M. Peter. Ultra-High-Field Superconductivity. Phys. Rev. Lett. 9,290–292 (1962). URL http://link.aps.org/doi/10.1103/PhysRevLett.9.290.

73

http://link.aps.org/doi/10.1103/PhysRevLett.96.217001


http://dx.doi.org/10.1021/cr030657d



http://dx.doi.org/10.1038/35073531


http://dx.doi.org/10.1002/1521-4095(20020205)14:3<243::AID-ADMA243>3.0.CO;2-F

http://dx.doi.org/10.1002/1521-4095(20020205)14:3<243::AID-ADMA243>3.0.CO;2-F


Bibliography

[31] A. Abrikosov. Fundamentals of the Theory of Metals (North-Holland, Amsterdam,1988).

[32] J. Ziman. Principles of the Theory of Solids (Cambridge University Press, Cam-bridge 1972).

[33] R. Gross & A. Marx. Festkörperphysik (Oldenbourg Verlag München, (2012)).

[34] M. V. Kartsovnik, P. A. Kononovich, V. N. Laukhin & I. V. Shchegolev. Anisotropyof magnetoresistance and the Shubnikov-de Haas oscillations in the organic metalβ-(ET)2IBr2. JETP Letters 48, 541 (1998).

[35] M. V. Kartsovnik, V. N. Laukhin, S. I. Pesotskii, I. F. Schegolev & V. M.Yakovenko. J. Phys. I 2, 89 (1992).

[36] K. Yamaji. On the Angle Dependence of the Magnetoresistance in Quasi-Two-Dimensional Organic Superconductors. Journal of the Physical Society of Japan58, 1520–1523 (1989). URL http://dx.doi.org/10.1143/JPSJ.58.1520.

[37] V. G. Peschansky, J. A. Lopes & T. G. Yao. J. Phys. I 1, 1469 (1991).

[38] P. D. Grigoriev. Angular dependence of the Fermi surface cross-section area andmagnetoresistance in quasi-two-dimensional metals. Phys. Rev. B 81, 205122(2010). URL http://link.aps.org/doi/10.1103/PhysRevB.81.205122.

[39] A. A. House, N. Harrison, S. J. Blundell, I. Deckers, J. Singleton, F. Herlach, W.Hayes, J. A. A. J. Perenboom, M. Kurmoo & P. Day. Oscillatory magnetoresistancein the charge-transfer salt β′′-(BEDT-TTF)2AuBr2 in magnetic fields up to 60 T:Evidence for field-induced Fermi-surface reconstruction. Phys. Rev. B 53, 9127–9136 (1996). URL http://link.aps.org/doi/10.1103/PhysRevB.53.9127.

[40] M. S. Nam, S. J. Blundell, A. Ardavan, J. A. Symington & J. Singleton. Fermisurface shape and angle-dependent magnetoresistance oscillations. Journal ofPhysics: Condensed Matter 13, 2271 (2001). URL http://iopscience.iop.org/0953-8984/13/10/319.

[41] V. N. Laukhin, E. E. Kostyuchenko, Y. V. Sushko, I. F. Shchegolev & Y. E.B. Effect of pressure on the superconductivity of β-(BEDT-TTF)2I3. Sov. Phys.JETP Lett. 41 (1985).

[42] V. G. Peschansky & M. V. Kartsovnik. Comment on "Contribution of small closedorbits to magnetoresistance in quasi-two-dimensional conductors". Phys. Rev. B60, 11207–11209 (1999). URL http://link.aps.org/doi/10.1103/PhysRevB.60.11207.

[43] N. Hanasaki, S. Kagoshima, T. Hasegawa, T. Osada & N. Miura. Contributionof small closed orbits to magnetoresistance in quasi-two-dimensional conductors.Phys. Rev. B 57, 1336–1339 (1998). URL http://link.aps.org/doi/10.1103/PhysRevB.57.1336.

74

http://dx.doi.org/10.1143/JPSJ.58.1520



http://iopscience.iop.org/0953-8984/13/10/319

http://iopscience.iop.org/0953-8984/13/10/319





Bibliography

[44] M. V. Kartsovnik, The Physics of Organic Superconductors and Conductors. A. G.Lebed (ed.), chap. 8, Layered Organic Conductors in Strong Magnetic Fields, 185–246 (Springer Verlag, 2008), 2nd edn.

[45] A. G. Lebed. Anisotropy of an instability for a spin density wave induced by amagnetic field in a q1d conductor. Sov. Phys.-JETP 43, 174–177 (1986).

[46] M. Naughton, O. Chung, L. Chiang, S. Hannahs & J. Brooks. Angular Dependenceof the Magnetoresistance in (TMTSF)2ClO4 173 (1989). URL http://journals.cambridge.org/article_S1946427400456885.

[47] M. J. Naughton, O. H. Chung, M. Chaparala, X. Bu & P. Coppens. Commensuratefine structure in angular-dependent studies of (TMTSF)2ClO4. Phys. Rev. Lett.67, 3712–3715 (1991). URL http://link.aps.org/doi/10.1103/PhysRevLett.67.3712.

[48] T. Osada, A. Kawasumi, S. Kagoshima, N. Miura & G. Saito. Commensurabil-ity effect of magnetoresistance anisotropy in the quasi-one-dimensional conductortetramethyltetraselenafulvalenium perchlorate, (TMTSF)2ClO4. Phys. Rev. Lett.66, 1525–1528 (1991). URL http://link.aps.org/doi/10.1103/PhysRevLett.66.1525.

[49] T. Osada, S. Kagoshima & N. Miura. Resonance effect in magnetotransportanisotropy of quasi-one-dimensional conductors. Phys. Rev. B 46, 1812–1815(1992). URL http://link.aps.org/doi/10.1103/PhysRevB.46.1812.

[50] L. D. Landau. Diamagnetismus der Metalle. Z. Phys. 64, 629–637 (1930).

[51] L. Onsager. Interpretation of the de Haas-van Alphen effect. Philosophical Mag-azine Series 7 43, 1006–1008 (1952). URL http://www.tandfonline.com/doi/abs/10.1080/14786440908521019.

[52] L. W. Shubnikov & W. J. de Haas. Proc. Netherlands Royal Academic Socitey 30,130 and 160 (1930).

[53] W. J. de Haas & P. M. van Alphen. Proc. Netherlands Royal Academic Socitey33, 1106 (1930).

[54] A. P. Cracknell & K. C. Wong. The Fermi Surface (Oxford University Press,London, 1973).

[55] E. N. Adams & T. D. Holstein. Quantum Theory of Transverse GalvanomagneticPhenomena. J. Phys. Chem. Solids 10, 254–276 (1959).

[56] A. B. Pippard. The Dynamics of Conduction Electrons (Gordon and Breach, NewYork, 1965).

[57] A. B. Pippard. Magnetoresistance in Metals (Cambridge University Press, 1989).

[58] I. M. Lifshitz & A. M. Kosevich. Sov. Phys.-JETP 2, 636 (1956).

75

http://journals.cambridge.org/article_S1946427400456885

http://journals.cambridge.org/article_S1946427400456885






http://www.tandfonline.com/doi/abs/10.1080/14786440908521019

http://www.tandfonline.com/doi/abs/10.1080/14786440908521019

Bibliography

[59] R. B. Dingle. Some Magnetic Properties of Metals. II. The Influence of Collisionson the Magnetic Behaviour of Large Systems. Proc. Roy. Soc. 211, 517–525 (1952).

[60] E. I. Blount. Bloch Electrons in a Magnetic Field. Phys. Rev. 126, 1636–1653(1962). URL http://link.aps.org/doi/10.1103/PhysRev.126.1636.

[61] L. M. Falicov & H. Stachowiak. Theory of the de Haas-van Alphen Effect in aSystem of Coupled Orbits. Application to Magnesium. Phys. Rev. 147, 505–515(1966). URL http://link.aps.org/doi/10.1103/PhysRev.147.505.

[62] A. B. Pippard. Proc. R. Soc. London A270 (1962).

[63] A. B. Pippard. Philos. Trans. R. Soc. London A256 (1964).

[64] W. Buckel & R. Kleiner. Supraleitung (Whiley-VCH, (2013)), 7th edn.

[65] M. Tinkham. Introduction to Superconductivity (Dover Puplications, Inc., (1996)),2nd edn.

[66] M. Tinkham. Effect of Fluxoid Quantization on Transitions of SuperconductingFilms. Phys. Rev. 129, 2413–2422 (1963). URL http://link.aps.org/doi/10.1103/PhysRev.129.2413.

[67] B. S. Chandrasekhar. A note on the maximum critical field of high-filed supercon-ductors. Appl. Phys. Lett. 1, 7–8 (1962). URL http://dx.doi.org/10.1063/1.1777362.

[68] A. M. Clogston. Upper Limit for the Critical Field in Hard Superconductors.Phys. Rev. Lett. 9, 266–267 (1962). URL http://link.aps.org/doi/10.1103/PhysRevLett.9.266.

[69] N. R. Werthamer, E. Helfand & P. C. Hohenberg. Temperature and Purity Depen-dence of the Superconducting Critical Field, Hc2. III. Electron Spin and Spin-OrbitEffects. Phys. Rev. 147, 295–302 (1966). URL http://link.aps.org/doi/10.1103/PhysRev.147.295.

[70] S. Blundell. Magnetism in Condensed Matter (Oxford University Press, 2012),10th edn.

[71] D. Andres. Effects of High Magnetic Fields and Hydrostatic Pressure onthe Low-Temperature Density-Wave State of the Organic Metal α-(BEDT-TTF)2KHg(SCN)4. Ph.D. thesis, Technische Universität München (2004).

[72] P. Böhm. Vergleichende Untersuchung der Schwingungsamplitude. Diplomarbeit,Technische Universit München (1996).

[73] E. Ohmichi & T. Osada. Torqumeter magnetometry in pulsed magnetic fields withuse of a commercial microcantilever. Rev. Sci. Instrum. 73, 3022–3026 (2002).

76

http://link.aps.org/doi/10.1103/PhysRev.126.1636




http://dx.doi.org/10.1063/1.1777362

http://dx.doi.org/10.1063/1.1777362





Bibliography

[74] J. Caulfield, J. Singleton, F. Pratt, M. Doporto, W. Lubczynski, W. Hayes, M.Kurmoo, P. Day, P. Hendriks & J. Perenboom. The effects of open sections of theFermi surface on the physical properties of 2D organic molecular metals. SyntheticMetals 61, 63 – 67 (1993). URL http://www.sciencedirect.com/science/article/pii/037967799391200L. Proceedings of Symposium H on MolecularElectronics: Doping and Recognition in Nanostructured Materials of the E-MRSSpring Conference.

[75] M. V. Kartsovnik, G. Y. Logvenov, T. Ishiguro, W. Biberacher, H. Anzai & N. D.Kushch. Direct Observation of the Magnetic-Breakdown Induced Quantum Inter-ference in the Quasi-Two-Dimensional Organic Metal κ-(BEDT-TTF)2Cu(NCS)2.Phys. Rev. Lett. 77, 2530–2533 (1996). URL http://link.aps.org/doi/10.1103/PhysRevLett.77.2530.

[76] N. Harrison, J. Caulfield, J. Singleton, P. H. P. Reinders, F. Herlach, W. Hayes,M. Kurmoo & P. Day. Magnetic breakdown and quantum interference in thequasi-two-dimensional superconductor κ-(BEDT-TTF)2Cu(NCS)2 in high mag-netic fields. Journal of Physics: Condensed Matter 8, 5415 (1996).

[77] F. A. Meyer, E. Steep, W. Biberacher, P. Christ, A. Lerf, A. G. M. Jansen, W.Joss, P. Wyder & K. Andres. High-Field de Haas-Van Alphen Studies of κ-(BEDT-TTF)2Cu(NCS)2. Journal of Physics: Condensed Matter 32, 681 (1995).

[78] L. Balicas, V. Barzykin, K. Storr, J. S. Brooks, M. Tokumoto, S. Uji, H. Tanaka, H.Kobayashi & A. Kobayashi. The effect of pressure on the phase diagram of the mag-netic field-induced superconducting state of λ-(BETS)2FeCl4. J. Phys. IV France114, 199–203 (2004). URL http://dx.doi.org/10.1051/jp4:2004114044.

[79] R. Balian & N. R. Werthamer. Superconductivity with Pairs in a Relative pWave. Phys. Rev. 131, 1553–1564 (1963). URL http://link.aps.org/doi/10.1103/PhysRev.131.1553.

77

http://www.sciencedirect.com/science/article/pii/037967799391200L

http://www.sciencedirect.com/science/article/pii/037967799391200L



http://dx.doi.org/10.1051/jp4:2004114044



Acknowledgements

At the end of this thesis I want to thank all the people who helped me accomplish thismaster thesis.I especially want to express my gratitude to

• Prof. Dr. Rudolf Gross for giving me the opportunity of doing this work at theWalther-Meißner-Institut.

• Dr. Mark Kartsovnik, for introducing me to the subject of organic superconduc-tors, supervising this thesis, always giving advice and sharing his knowledge.

• Dr. Werner Biberacher, who helped me throughout this thesis with many aspectsand whose door was always open for asking advice.

• Michael Kunz, who introduced me to the experimental part of this thesis, forcountless advice, for sharing his knowledge, for his help with everyday problems,and especially for contacting the smallest samples.

• Natasha D. Kushch for providing the samples.

• the other members of this group, Vasileios Tzanos and Luzia Höhlein, and all theother students and members of the WMI for the nice working atmosphere.

• the technical and administrative staff of the WMI.

• my family for supporting me all over the years.

79

Erklärung

des Masteranden

Name:

Vorname:

Mit der Abgabe der Masterbeit versichere ich, dass ich die Arbeit selbständig verfasstund keine anderen als die angegebenen Quellen und Hilfsmittel benutzt habe.

............................................ .............................................

(Ort, Datum) (Unterschrift)

81

Magnetotransport studies of the organic superconductor and ...Ludwi… · The BETS donors were...

Documents

Transcript of Magnetotransport studies of the organic superconductor and ...Ludwi… · The BETS donors were...