Magnetic Quantum Oscillations in the Organic Antiferromagnetic Superconductor -(BETS… · 2016. 5....

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Walther-Meißner-Institut Technische Universität Physik Department für München Lehrstuhl E23 Tieftemperaturforschung Bachelorthesis Magnetic Quantum Oscillations in the Organic Antiferromagnetic Superconductor κ-(BETS) 2 FeBr 4 Florian Michael Kollmannsberger Aufgabensteller: Prof. Dr. Rudolf Gross Abgabetermin: 25.4.2016

Transcript of Magnetic Quantum Oscillations in the Organic Antiferromagnetic Superconductor -(BETS… · 2016. 5....

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Walther-Meißner-Institut Technische Universität Physik Department

für München Lehrstuhl E23Tieftemperaturforschung

Bachelorthesis

Magnetic Quantum Oscillations in theOrganic Antiferromagnetic

Superconductor κ-(BETS)2FeBr4Florian Michael Kollmannsberger

Aufgabensteller: Prof. Dr. Rudolf Gross

Abgabetermin: 25.4.2016

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Contents

1 Introduction 1

2 The organic metal κ-(BETS)2FeBr4 32.1 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Band structure and electronic properties . . . . . . . . . . . . . . . . . . 5

3 Theoretical background 93.1 Electrodynamics in a magnetic field . . . . . . . . . . . . . . . . . . . . . 93.2 Shubnikov-de Haas oscillations . . . . . . . . . . . . . . . . . . . . . . . 10

3.2.1 Temperature damping RT . . . . . . . . . . . . . . . . . . . . . . 113.2.2 Impurity damping RD . . . . . . . . . . . . . . . . . . . . . . . . 113.2.3 Spin damping RS . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Experimental setup 154.1 Sample preparation and sample holder . . . . . . . . . . . . . . . . . . . 154.2 3He-system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.3 Superconducting magnet . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5 Results and discussion 175.1 Temperature and field dependent resistance . . . . . . . . . . . . . . . . 17

5.1.1 Superconducting transition . . . . . . . . . . . . . . . . . . . . . 175.1.2 Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.2 The antiferromagnetic state: Shubnikov-de Haas oscillations . . . . . . . 185.2.1 δ frequency orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2.2 η frequency orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.3 The normal metallic state . . . . . . . . . . . . . . . . . . . . . . . . . . 235.3.1 Field induced superconductivity . . . . . . . . . . . . . . . . . . . 235.3.2 β Shubnikov-de Haas oscillations . . . . . . . . . . . . . . . . . . 265.3.3 α Shubnikov-de Haas oscillations . . . . . . . . . . . . . . . . . . 28

6 Summary 31

Bibliography 31

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1 Introduction

In 1911 Heike Kamerlingh Onnes [1] discovered superconductivity by measuring theresistance of mercury down to 4.2 K. With time more and more materials showing su-perconductivity were discovered, and in 1958 Bardeen, Cooper and Shrieffer developedthe microscopic theory for superconductors (SC). They explained superconductivity bypairs of electrons, which are called Cooper pairs, forming a bosonic state and are heldtogether by an attractive exchange interaction. They showed that phonons mediatethis positive exchange interaction in conventional superconductors. Today many un-conventional superconductors like cuprat, heavy fermion and iron pnictides are known,where the pairing mechanism is not yet understood. In 1964 W. A. Little [2] proposedpossible superconductivity in organic superconductors, where a long organic aromaticmain chain with polarisable side chains forms a superconductor. The oscillation in theside chains could generate an attractive interaction between the electrons in the mainmolecule, and provide the attractive interaction giving rise to SC pairing at high criti-cal temperatures. However Little’s theory could so far not be realized, but lead to thediscovery of organic superconductors. These superconductors known today are quasi1D or quasi 2D organic charge transfer salts consisting of an inorganic anion and anorganic aromatic molecule, for examples see [3].In this thesis we will focus on κ-(BETS)2FeBr4. κ-(BETS)2FeBr4 is part of the familyof (BETS)2FeX4 where X stands for Cl or Br and the greek letter labels the packingmodification of the BETS molecules in the salt. The salts of this family are alternatinglayers of BETS and FeX4, where the π orbitals of the BETS molecules overlap and formconducting bands. The layered structure results in strongly quasi 2D conduction bands.The interest to this family started when superconductivity induced by magnetic fieldwas discovered in λ-(BETS)2FeCl4 [4, 5]. A strong interaction between the π electronsand the 3d spins of the Fe ions in λ-(BETS)2FeCl4 leads to an antiferromagnetic (AFM)type of ordering of the 3d spins and the π electrons at low temperatures [6] and evento an insulating ground state. As this insulating ground state is broken by magneticfield λ-(BETS)2FeCl4 shows an insulator-metal-superconductor transition at low tem-peratures with rising magnetic field. The main focus in this thesis is on the role of π-dinteractions on the conducting layers more precisely on the effect of internal magneticfields on the Shubnikov-de Haas quantum oscillations as well as the impact of the AFMordering on the Fermi surfaces. To investigate these questions we measured Shubnikov-de Haas oscillations in the antiferromagnetic state as well as in the paramagnetic stateof the salt, as the salt transfers to a paramagnetic state when the antiferromagneticordering is broken by magnetic field. In the AFM state, a slow oscillation has beenreported and which has been interpreted as a result of a superstructure due to theAFM ordering. Since only one low frequency Shubnikov-de Haas oscillation has beendetected, more experimental evidence is desirable to support this theory and to further

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1 Introduction

clarify the structure of the Fermi surface in the AFM state. Another issue addressed inthis thesis is related to the effect of AFM ordering on the different spin subbands of theπ-electron conduction layers. By theory it was proposed that there is no Zeeman split-ting for the carriers in antiferromagnetic conductors with small carrier pockets [7, 8].In the paramagnetic state all 3d iron spins are ordered and form an internal field, sowe expect Zeeman splitting for the energy levels of the carrier bands in the conductinglayers. We investigated the influence of Zeeman splitting on the Shubnikov-de Haasoscillations, and compared it to the behavior predicted by theory.

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2 The organic metal κ-(BETS)2FeBr4

2.1 Sample preparation

The single crystals of κ-(BETS)2FeBr4, where BETS stands for the organic moleculeBis(ethylenedithio)tetraselenafulvalene, were prepared electrochemically as describedin [9]. Therefore, the sections about the crystal structure and band structure, followsclosely [9]. BETS and tetraethylammonium iron(III) tetrabromide were dissolved in a10% ethanol/chlorobenzene solution and for 2-4 weeks a constant current of 0.7µA wasapplied between the electrodes at room temperature or at 40C. The grown crystals werethin black plates with typical dimensions of 0.7x0.7x0.05 mm3. The κ-(BETS)2FeBr4samples used in this experiment were provided by H.Fujiwara.

2.2 Crystal structure

The κ-(BETS)2FeBr4 crystals consist of conducting and insulating layers arranged al-ternating along the b-axis. The crystal has an orthorombic structure, with the unit cellparameters a = 11.5Å, b = 36.4Å and c = 8.5Å. The unit cell is large along b direction,as it contains two BETS and anion layers. The isolating layers consist of FeBr−4 an-ions that carry a magnetic moment and act as an acceptor, while the BETS molecules,which form the conducting layers, act as donor cations. Due to this transfer we call κ-(BETS)2FeBr4 a charge transfer salt. Fig. 2.1 shows the BETS molecule and the FeBr4

anion as well as atom labeling, and Fig. 2.2 shows the crystal structure. Fig 2.2(a) showsthe layered structure of the compound and in Fig 2.2 we see the structure of the BETSlayers which are aliened in the typical way of κ-structure, where the BETS molecules arestrongly dimerised. Compared to the λ-(BETS)2FeCl4 salt, the interaction between theπ-conducting electrons of the BETS layer and the Fe3+ 3d localized spins are weaker inthe κ-(BETS)2FeBr4 salt. In the conducting layer short S-S and Se-S contacts providethe inter molecular overlap that forms the metallic bands.

FeCl4.9b,11 The analyses of the BETS intermolecular distancesshow that there is only one Se-Se type short contact (<3.80Å), 3.796(3) Å [c1: Se(1)-Se(2)], and the shortest Se-S andS-S distances are 3.593(6) [c2: Se(3)-S(4)] and 3.362(8) Å[c4: S(1)-S(3)], respectively. Two-dimensional conductionlayers (theac-plane) composed of the BETS dimers and theinsulating layers of the tetrahedral anions with localizedmagnetic Fe3+ moments are arranged alternately along thebdirection (Figure 1a). In the anion layer, the shortest Fe-Fedistance is 5.921(3) Å along thea-axis and 8.504(5) Å alongthe c-axis. Between the FeBr4

- anion molecules, the shortestBr-Br distance along thea-axis is 4.137(4) Å [d3: Br(1)-Br-(3)], which is much shorter than that along thec-axis, 4.637(5)Å [d4: Br(2)-Br(3)]; however, these distances are still longerthan the sum of the van der Waals radii of bromines (3.7 Å).From these results the direct intermolecular interaction betweenthe FeBr4- anion molecules is considered to be weak butrelatively stronger along thea-axis than along thec-axis. Thereare several short Br-S contacts almost equal to the sum of thevan der Waals radii of bromine and sulfur atoms (3.7 Å) betweenBETS and the anion molecules as indicated in Figure 1a. Theshortest Br-S distance is 3.693 (6) Å [d1: Br(3)-S(2)]. In theλ-(BETS)2FeCl4 salt, the FeCl4

- anion gets into the donorlayer and there is a short Cl-Se contact (3.528 Å) between

BETS and the FeCl4- anion;9b however, no Br-Se contact is

observed in theκ-(BETS)2FeBr4 salt. Therefore, the interac-tion between theπ conduction electrons of the BETS layer andthe Fe3+ 3d localized spins of the anion molecules in theκ-(BETS)2FeBr4 salt is weaker than the case of theλ-(BETS)2FeCl4 salt. The temperature dependence of the latticeparameters ofκ-(BETS)2FeBr4 in the temperature range 270-7K is shown in Figure 3, where changes of the lattice constantsat cooling (filled squares) and warming (opened circles) cyclesare presented. Figure 3 displays the presence of some structuralanomalies in the temperature range of 60-140 K. Around 100K, the b-parameter shows a hysteresis-like behavior, i.e., thereis a decrease of theb-parameters for about 0.1 Å between 110and 90 K at cooling, and practically the same reverse changebetween 120 and 140 K at warming run. On the other hand,thea-parameter of the lattice has a remarkable change at around60 K both for cooling and for warming cycles. The results ofthe temperature dependence of lattice constants described aboveindicate the presence of the structure transformation ofκ-(BETS)2FeBr4 in the temperature range of 60-140 K. Thestudies on the crystal structure analyses at low temperaturesare now in progress and the details will be reported elsewhere.

A band dispersion and Fermi surface was calculated on thecrystal structure at room temperature based on the extendedHuckel approximation as shown in Figure 4. Table 1 showsoverlap integrals, mid-gap energies, and the bandwidth of theupper band of theκ-(BETS)2FeBr4 salt together with those ofthe κ-(BETS)2FeCl4 salt.9b The intradimer overlap integral (p,77.33× 10-3) of theκ-(BETS)2FeBr4 salt is much larger thanthe other overlaps (a, -22.41;q, 8.11; andc, 35.14× 10-3)reflecting the dimer structure of theκ-type salts. The banddispersion ofκ-(BETS)2FeBr4 has four energy branches corre-sponding to the existence of four donor molecules in one unitof the donor sheet. There is a mid-gap between the upper twobranches and the lower two branches due to the strongdimerization of the donor molecules as is often observed inκ-type BEDT-TTF salts and the upper band is effectively half-filled consequently. The energy of the mid-gap of theκ-(BETS)2FeBr4 salt is 0.68 eV, which is much larger than thatof the κ-(BETS)2FeCl4 salt (0.48 eV) corresponding to thestrength of the intradimer interaction (p), suggesting the strongerelectron correlation in theκ-(BETS)2FeBr4 salt. The Fermisurface of this salt is a two-dimensional circle closed in theXV boundary as a result of space group symmetry, similar tothose of the isostructuralκ-(BETS)2FeCl4 andκ-(BETS)2GaCl4salts which have been confirmed by the Shubnikov-de Haas

Figure 2. Molecular structures of the BETS radical cation and theFeBr4- anion with the labeling of the atoms at room temperature.

Figure 3. Temperature dependence of the lattice constants ofκ-(BETS)2FeBr4. Filled squares: cooling run. Opened circles: warmingrun.

Figure 4. Band structure and Fermi surface ofκ-(BETS)2FeBr4.

Table 1. Overlap Integrals, Mid-Gap Energies, and Bandwidths ofthe Upper Band

a p q c Emid-gap bandwidth

κ-(BETS)2FeBr4 -22.41 77.33 8.11 35.14 0.68 1.44κ-(BETS)2FeCl4a -16.93 64.37 9.08 32.76 0.48 1.28

a Taken from ref 9b.

308 J. Am. Chem. Soc., Vol. 123, No. 2, 2001 Fujiwara et al.

Figure 2.1: Labeling of the atoms in the BETS and FeBr−4 ; (taken from [9]).

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2 The organic metal κ-(BETS)2FeBr4

interaction betweenπ metal electrons of the donor moleculesand localized magnetic moments of the anions has receivedmuch attention in creating new types of conducting systems.7,8

The largest target among these systems may be an antiferro-magnetic or ferromagnetic organic superconductor. We haveinvestigated BETS salts involving transition metal halides suchas λ-(BETS)2MCl4 [BETS ) bis(ethylenedithio)tetraselena-fulvalene; M) Ga and Fe] with a superconducting (M) Ga)or insulating (M) Fe) ground state.9 Furthermore, we haverecently found an exotic organic conductor,λ-(BETS)2FexGa1-x-BryCl4-y (0 < x < 1, 0< y < 0.5) exhibiting a superconductor-to-insulator or superconductor-to-metal transition. In thesesystems, the strongπ-d coupling between theπ-metal electronsystems of the BETS molecules and the localized Fe3+ magneticmoments of the anions plays an important role in breaking thesuperconducting state of these systems and causes coopera-tively an antiferromagnetic insulating state.10 Contrary to theλ-type (BETS)2MCl4 system, the other main modifications,κ-(BETS)2MX4 (M ) Ga, Fe, and In; X) Br and Cl), retaintheir metallic states down to 2 K.9b,11 A crystal structure andan extended Hu¨ckel tight-binding band calculation suggestκ-(BETS)2FeCl4 to be a two-dimensional metal, which isconsistent with recent low-temperature magnetoresistance ex-periments.12 However, the ground states ofκ-(BETS)2FeBr4 atthe lowest temperature region remained unclear. In this paper,we report structures, magnetic, and electrical properties of anantiferromagnetic organic metal,κ-(BETS)2FeBr4.13 We alsoinvestigated a magnetic field dependence of the electricalresistivities and its anisotropy. Furthermore, we also present astudy on a specific heat measurement to clear the coexistenceof the antiferromagnetic spin ordering and superconductivity.

Results and Discussion

Crystal and Band Structure Determination of K-(BETS)2-FeBr4. Plate-shaped crystals were prepared electrochemicallyfrom a 10% ethanol-chlorobenzene solution containing BETS

and tetraethylammonium iron(III) tetrabromide. A constantcurrent of 0.7 µA was applied for 2-4 weeks at roomtemperature or 40°C. Several of the crystal structures of theλ-andκ-(BETS)2MX4 salts (M) Ga, Fe and In; X) Cl and Br)have been reported previously.9b,11 Figure 1 shows the crystalstructure ofκ-(BETS)2FeBr4 determined at room temperature.One BETS molecule and one-half of the FeBr4

- anion arecrystallographically independent, and the FeBr4

- anion is on amirror plane. The molecular structures of BETS radical cationand FeBr4- anion with the labeling of the atoms at roomtemperature are shown in Figure 2 and the BETS molecule hasa slightly nonplanar molecular structure. It was found that thereis a positional disordering of the carbon C(2) atom of theethylene group, which is indicated by the higher temperaturefactor as well as the length of the C(1)-C(2) bond at roomtemperature. As shown in Figure 1b, the BETS molecules formdimers and adjacent BETS dimers are arranged in a roughlyorthogonal manner characteristic of theκ-type packing motif.3

There is no essential difference between the already-knownmolecular arrangements inκ-(BETS)2FeBr4 and κ-(BETS)2-

(7) (a) Day, P.; Kurmoo, M.J. Mater. Chem.1997, 7, 1291-1295. (b)Graham, A. W.; Kurmoo, M.; Day, P.J. Chem. Soc., Chem. Commun.1995,2061-2062. (c) Kurmoo, M.; Graham, A. W.; Day, P.; Coles, S. J.;Hursthouse, M. B.; Caulfield, J. L.; Singleton, J.; Pratt, F. L.; Hayes, W.;Ducasse, L.; Guionneau, P.J. Am. Chem. Soc.1995, 117, 12209-12217.(d) Martin, L.; Turner, S. S.; Day, P.; Mabbs, F. E.; McInnes, E. J. L.Chem.Commun.1997, 1367-1368.

(8) (a) Clemente-Leo´n, M.; Coronado, E.; Gala´n-Mascaro´s, J. R.; Go´mez-Garcıa, C. J.; Rovira, C.; Laukhin, V. N.Synth. Met.1999, 103, 2339-2342. (b) Ouahab, L.Coord. Chem. ReV. 1998, 178-180, 1501-1531. (c)Coronado, E.; Go´mez-Garcı´a, C. J.Chem. ReV. 1998, 98, 273-296.

(9) (a) Kobayashi, H.; Udagawa, T.; Tomita, H.; Bun, K.; Naito, T.;Kobayashi, A.Chem. Lett. 1993, 1559-1562. (b) Kobayashi, H.; Tomita,H.; Naito, T.; Kobayashi, A.; Sasaki, F.; Watanabe, T.; Cassoux, P.J. Am.Chem. Soc. 1996, 118, 368-377. (c) Akutsu, H.; Arai, E.; Kobayashi, H.;Tanaka, H.; Kobayashi, A.; Cassoux, P.J. Am. Chem. Soc. 1997, 119,12681-12682. (d) Sato, A.; Ojima, E.; Kobayashi, H.; Hosokoshi, Y.; Inoue,K.; Kobayashi, A.; Cassoux, P.AdV. Mater. 1999, 11, 1192-1194. (e)Tanaka, H.; Kobayashi, A.; Sato, A.; Akutsu, H.; Kobayashi, H.J. Am.Chem. Soc. 1999, 121, 760-768.

(10) (a) Kobayashi, H.; Sato, A.; Arai, E.; Akutsu, H.; Kobayashi, A.;Cassoux, P.J. Am. Chem. Soc.1997, 119, 12392-12393. (b) Kobayashi,H.; Sato, A.; Tanaka, H.; Kobayashi, A.; Cassoux, P.Coord. Chem. ReV.1999, 190-192, 921-932. (c) Sato, A.; Ojima, E.; Akutsu, H.; Kobayashi,H.; Kobayashi, A.; Cassoux, P.Chem. Lett. 1998, 673-674. (d) Kobayashi,H.; Akutsu, H.; Ojima, E.; Sato, A.; Tanaka, H.; Kobayashi, A.; Cassoux,P. Synth. Met.1999, 103, 1837-1838.

(11) Kobayashi, A.; Udagawa, T.; Tomita, H.; Naito, T.; Kobayashi, H.Chem. Lett.1993, 2179-2182.

(12) (a) Tajima, H.; Kobayashi, A.; Naito, T.; Kobayashi, H.Solid StateCommun. 1996, 98, 755-759. (b) Kushch, N.; Dyachenko, O.; Gritsenko,V.; Pesotskii, S.; Lyubovskii, R.; Cassoux, P.; Faulmann, C.; Kovalev, A.;Kartsovnik, M.; Brossard, L.; Kobayashi, H.; Kobayashi, A.J. Phys. IFrance 1996, 6, 1997-2009. (c) Harrison, N.; Mielke, C.; Rickel, D.;Montgomery, L.; Gerst, C.; Thompson, J.Phys. ReV. 1998, B57, 8751-8754.

(13) Ojima, E.; Fujiwara, H.; Kato, K.; Kobayashi, H.; Tanaka, H.;Kobayashi, A.; Tokumoto, M.; Cassoux, P.J. Am. Chem. Soc. 1999, 121,5581-5582.

Figure 1. (a) Crystal structure ofκ-(BETS)2FeBr4. (b) Projection alongthe b-axis. The short chalcogen-chalcogen contacts:c1 [Se(1)-Se-(2)] ) 3.796(3),c2 [Se(3)-S(4)]) 3.593(6),c3 [Se(1)-S(4)]) 3.656-(5), c4 [S(1)-S(3)] ) 3.362(8), andc5 [S(1)-S(2)] ) 3.369(8) Å.The short distances:d1 [Br(3)-S(2)] ) 3.693(6),d2 [Br(1)-S(3)] )3.708(6),d3 [Br(1)-Br(3)] ) 4.137(4), andd4 [Br(2)-Br(3)] ) 4.637-(5) Å. The overlap integrals:p 77.33,a -22.41,c 35.14, andq 8.11× 10-3.

A NoVel Antiferromagnetic Organic Superconductor J. Am. Chem. Soc., Vol. 123, No. 2, 2001307

Figure 2.2: (a) Structure of the crystal and (b) orientation of the BETS moleculeswithin the layers in the κ configuration (taken from[9]).

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2.3 Band structure and electronic properties

2.3 Band structure and electronic properties

FeCl4.9b,11 The analyses of the BETS intermolecular distancesshow that there is only one Se-Se type short contact (<3.80Å), 3.796(3) Å [c1: Se(1)-Se(2)], and the shortest Se-S andS-S distances are 3.593(6) [c2: Se(3)-S(4)] and 3.362(8) Å[c4: S(1)-S(3)], respectively. Two-dimensional conductionlayers (theac-plane) composed of the BETS dimers and theinsulating layers of the tetrahedral anions with localizedmagnetic Fe3+ moments are arranged alternately along thebdirection (Figure 1a). In the anion layer, the shortest Fe-Fedistance is 5.921(3) Å along thea-axis and 8.504(5) Å alongthe c-axis. Between the FeBr4

- anion molecules, the shortestBr-Br distance along thea-axis is 4.137(4) Å [d3: Br(1)-Br-(3)], which is much shorter than that along thec-axis, 4.637(5)Å [d4: Br(2)-Br(3)]; however, these distances are still longerthan the sum of the van der Waals radii of bromines (3.7 Å).From these results the direct intermolecular interaction betweenthe FeBr4- anion molecules is considered to be weak butrelatively stronger along thea-axis than along thec-axis. Thereare several short Br-S contacts almost equal to the sum of thevan der Waals radii of bromine and sulfur atoms (3.7 Å) betweenBETS and the anion molecules as indicated in Figure 1a. Theshortest Br-S distance is 3.693 (6) Å [d1: Br(3)-S(2)]. In theλ-(BETS)2FeCl4 salt, the FeCl4

- anion gets into the donorlayer and there is a short Cl-Se contact (3.528 Å) between

BETS and the FeCl4- anion;9b however, no Br-Se contact is

observed in theκ-(BETS)2FeBr4 salt. Therefore, the interac-tion between theπ conduction electrons of the BETS layer andthe Fe3+ 3d localized spins of the anion molecules in theκ-(BETS)2FeBr4 salt is weaker than the case of theλ-(BETS)2FeCl4 salt. The temperature dependence of the latticeparameters ofκ-(BETS)2FeBr4 in the temperature range 270-7K is shown in Figure 3, where changes of the lattice constantsat cooling (filled squares) and warming (opened circles) cyclesare presented. Figure 3 displays the presence of some structuralanomalies in the temperature range of 60-140 K. Around 100K, the b-parameter shows a hysteresis-like behavior, i.e., thereis a decrease of theb-parameters for about 0.1 Å between 110and 90 K at cooling, and practically the same reverse changebetween 120 and 140 K at warming run. On the other hand,thea-parameter of the lattice has a remarkable change at around60 K both for cooling and for warming cycles. The results ofthe temperature dependence of lattice constants described aboveindicate the presence of the structure transformation ofκ-(BETS)2FeBr4 in the temperature range of 60-140 K. Thestudies on the crystal structure analyses at low temperaturesare now in progress and the details will be reported elsewhere.

A band dispersion and Fermi surface was calculated on thecrystal structure at room temperature based on the extendedHuckel approximation as shown in Figure 4. Table 1 showsoverlap integrals, mid-gap energies, and the bandwidth of theupper band of theκ-(BETS)2FeBr4 salt together with those ofthe κ-(BETS)2FeCl4 salt.9b The intradimer overlap integral (p,77.33× 10-3) of theκ-(BETS)2FeBr4 salt is much larger thanthe other overlaps (a, -22.41;q, 8.11; andc, 35.14× 10-3)reflecting the dimer structure of theκ-type salts. The banddispersion ofκ-(BETS)2FeBr4 has four energy branches corre-sponding to the existence of four donor molecules in one unitof the donor sheet. There is a mid-gap between the upper twobranches and the lower two branches due to the strongdimerization of the donor molecules as is often observed inκ-type BEDT-TTF salts and the upper band is effectively half-filled consequently. The energy of the mid-gap of theκ-(BETS)2FeBr4 salt is 0.68 eV, which is much larger than thatof the κ-(BETS)2FeCl4 salt (0.48 eV) corresponding to thestrength of the intradimer interaction (p), suggesting the strongerelectron correlation in theκ-(BETS)2FeBr4 salt. The Fermisurface of this salt is a two-dimensional circle closed in theXV boundary as a result of space group symmetry, similar tothose of the isostructuralκ-(BETS)2FeCl4 andκ-(BETS)2GaCl4salts which have been confirmed by the Shubnikov-de Haas

Figure 2. Molecular structures of the BETS radical cation and theFeBr4- anion with the labeling of the atoms at room temperature.

Figure 3. Temperature dependence of the lattice constants ofκ-(BETS)2FeBr4. Filled squares: cooling run. Opened circles: warmingrun.

Figure 4. Band structure and Fermi surface ofκ-(BETS)2FeBr4.

Table 1. Overlap Integrals, Mid-Gap Energies, and Bandwidths ofthe Upper Band

a p q c Emid-gap bandwidth

κ-(BETS)2FeBr4 -22.41 77.33 8.11 35.14 0.68 1.44κ-(BETS)2FeCl4a -16.93 64.37 9.08 32.76 0.48 1.28

a Taken from ref 9b.

308 J. Am. Chem. Soc., Vol. 123, No. 2, 2001 Fujiwara et al.

Figure 2.3: Band structure of κ-(BETS)2FeBr4 and Fermi surface in the metallic state;(taken from [9]).

The band dispersion and Fermi surface calculated by Fujiwara et al.[9] from thecrystal structure at room temperature, based on the extended Hückel approximation,are shown in Fig. 2.3.

In the figure, SdH oscillations with several frequenciesare clearly seen. The Fourier transform FT spectrum in thePM phase right inset of Fig. 1 shows three fundamentaloscillations, , , and 1 and 2 corresponding to the FScross-sectional areas of 2.4%, 20%, and 100% of the firstBrillouin zone FBZ, respectively. These are consistent withthe previous report;12,13 and are assigned to the closedlens orbit and the magnetic breakdown orbit, respectivelyFig. 2a, while the origin of the is not clear. According tothe band structure calculation based on the crystal structureat room temperature, the energy bands are degenerated at thezone boundary.2 However, the energy bands should begapped at the zone boundary in low temperatures, becauseboth the and oscillations are observed in the PMphase.12,13 The gap at the zone boundary probably arisesfrom some structural phase transition, which changes onlythe symmetry of the crystal.11,15 Because of the presence of alarge internal field, splits into two frequencies 1 and 2corresponding to the FS’s of the up and down spinelectrons.8,16 On the other hand, the FT spectrum in low fieldbelow Hc shows the low frequency oscillation correspond-ing to the 1.5% of FBZ, which has not been ever reportedleft-hand inset of Fig. 1. Because is observed only belowHc, it is very likely that this orbit is formed by the recon-struction of the FS due to the long range magnetic order ofthe Fe spins.

Figure 3a shows the FT spectra of the oscillation be-low Hc at various temperatures. By fitting the data with theLifshitz-Kosevich formula,17 we obtain the effective massmeff=1.1me Fig. 3b, where me is the free electron mass.This value is much smaller than 5.2 and 7.9me for and ,respectively.12,13

Figure 4 shows the oscillation for various values of ,which is the angle between H and the b axis. As shown in theinset of Fig. 4, the frequency follows the cos−1 dependence,suggesting that the FS is cylindrical.

Figure 5 shows the resistance as a function of for vari-ous magnetic fields at 1.8 K. In the figure, characteristic os-cillatory behavior is evident. As the field increases, the peakbecomes more prominent while the peak position does notchange. In addition, the position is periodic in tan as shownin Fig. 6, where two periods denoted as and are ob-

FIG. 1. The interlayer resistance as a function of magnetic fieldperpendicular to the conducting ac layers at 0.3 K. Insets show thesemilog plots of the FT spectra in low and high fields.

FIG. 2. a FS in the PM phase. b Reconstructed FS in theCAF phase. The thick square shows the FBZ in each phase.

FIG. 3. a FT spectra of at various temperatures. b Tem-perature dependence of the FT amplitude divided by temperaturemass plot.

FIG. 4. SdH oscillations at various field directions. Inset showsthe cos−1 dependence of the frequencies.

KONOIKE et al. PHYSICAL REVIEW B 72, 094517 2005

094517-2

Figure 2.4: 2D representation of the Fermi surface of κ-(BETS)2FeBr4 (a) in themetallic state above the antiferromagnetic (AFM) transition and (b) in the AFMstate (taken from[10]).

There is a significant mid-gap energy in κ-(BETS)2FeBr4 of 0.68 eV due to the dimeri-sation and the bandwidth is 1.44 eV. This leads to two effectively half-filled bands re-sulting in a metallic behavior. The Fermi surface in the metallic state shows a closedcircle in the ka-kc plane. Along the b-axis the Fermi surface is a slightly warped cylinder.

5

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2 The organic metal κ-(BETS)2FeBr4

If we also take into account the other Brillouin zones, we get a closed α orbit, and opensheets,shown in Fig. 2.4(a), as the neighboring Fermi surfaces have a little gap betweenthem. Under high fields, the electrons can tunnel through this gap, and we get a neworbit called β. this effect is called magnetic breakdown (MB). The α orbit, has an areaof 20% of the first Brillouin zone, while the bigger β orbit has an area equal to the firstBrillouin zone.An elegant way to study quasi 2D metals is observing Shubnikov-de Haas oscillations[11]. As we see later in the theory chapter the amplitude and the observed frequenciesreveal a lot of information about the structure of the Fermi surface.Yet there are two separated β oscillations observed, differing only in about 1% [12,13].[14, 15] suggested that the two β frequencies are due to an internal magnetic fieldand, therefore, a result of the spin splitting of the energy levels of the Fermi surface,which we discuss later in the theory section. κ-(BETS)2FeBr4 shows an AFM orderingbelow a Néel temperature of 2.5K, that coexists with the superconducting (SC) phase.Fig. 2.5 shows the phase diagram for the field and the temperature

figuration for interlayer resistance measurements. The resis-tance was measured by a conventional four-terminal ac tech-nique. The measurement was carried out by using a dilutionrefrigerator with a 20 T superconducting magnet down to25 mK at NIMS. The samples were aligned in the field by aspecially designed rotator with 0.03° precision.

III. RESULTS

Figure 1(a) shows the field dependence of the resistanceat various temperatures for the magnetic field exactly parallelto thea-axis. As the magnetic field increases, superconduc-tivity is broken at 1.8 T, and then a steplike increase corre-sponding to the metamagnetic transition of the Fe3+ spins, isobserved(inset of Fig. 1).8 At 27 mK, the resistance in-creases almost linearly with field above 2 T, and then showsan abrupt drop by three orders of magnitude at about 8 T. Inthe region between 11 T and 14 T, the resistance is zerowithin the instrumental resolution. This behavior gives us aclear evidence of the FISC in this salt. At higher fields above

17 T, the FISC seems to be completely removed. As thetemperature increases, the FISC becomes unstable and al-most suppressed at 0.81 K. The zero resistance is observedonly in very low temperature region below about 0.3 K. At1 K, a small hump is observed around 12.6 T, the reason ofwhich is not clear.

Figure 1(b) shows the magnetic phase diagram forH iaconstructed from the data in Fig. 1(a). The superconductingcritical field is defined as the middle point of the resistivetransition. The low field superconducting phase is sur-rounded by the AF phase, where the AF phase boundarycorresponds to the metamagnetic transition field. The FISChas the maximumTcs,0.5 Kd at 12.6 T.

The field dependence of the resistance at various tempera-tures and the magnetic phase diagram forH ic are shown inFig. 2. The overall feature is similar to the case forH ia, butthe low field superconducting phase is more stable in thisfield direction. The anomaly in the resistance correspondingto the transition from the canted AF state(CAF) to the para-magnetic state is seen as indicated by the arrow in the insetof Fig. 2(a), but not so clear. The FISC is also observed inthis field direction in almost the same temperature-field re-gion for H ia.

FIG. 1. (a) Field dependence of the interlayer resistance at vari-ous temperatures forH ia. Inset shows the data in low field region.(b) Magnetic phase diagram forH ia. Closed circles, open circles,and closed triangles shows the superconducting, metamagnetic andthe field induced superconducting transitions, respectively. Opensquares shows the metamagnetic transition determined from theSQUID and transport measurements(Ref. 8). Dotted line is theguide to the eyes. Shaded areas show the calculated superconduct-ing phases by using Fisher’s theory(Ref. 14).

FIG. 2. (a) Field dependence of the interlayer resistance at vari-ous temperatures forH ic. Inset shows the data in low field region.(b) Magnetic phase diagram forH ic. Closed circle, open circle, andclosed triangle shows the superconducting, canted antiferromag-netic and the field induced superconducting transitions, respec-tively. Shaded areas show the calculated superconducting phases byusing Fisher’s theory(Ref. 14).

KONOIKE et al. PHYSICAL REVIEW B 70, 094514(2004)

094514-2

Figure 2.5: Phase diagram for magnetic field in the bc plane. Here CAF stands forcanted AFM and FISC is field induced superconductivity (taken from[16]).

In the AFM state we have to take the spins into account. By theory [16] there are twotypes of spin arrangement: the horizontal type that doubles the size of the primitivecell and that is more stable than the AFM type which orders vertically and leaves theprimitive cell unchanged. In the AFM state the crystal lattice unit cell carries spinsshown in Fig. 2.6, where the metallic state unit cell is indicated as solid line. The hor-izontal type of ordering seen in 2.6(c) has opposite spins at each boundary of the unitcell in the c-direction, but the same spins at each end in a-direction. As the unit cellhas a translation symmetry in a and c directions we need to modify the original unitcell for the horizontal ordering type and double it in c direction. Now we have the samespin at each boundary of the unit cell. In the vertical type of ordering Fig.2.6(b) wealready have the same spin at each end of the unit cell. Therefore it stays unchanged.Assuming the horizontal ordering, the primitive cell in the AFM state, consists of

6

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2.3 Band structure and electronic properties

for other Huckel parameters.3,37) These inter-dimer interac-tions work cooperatively to afford J ¼ JB þ JC þ 2Jc ¼622K with frustrating interactions J0 ¼ Jt þ Js ¼ 174K(top two lines of Table V), resulting in the mean-field J0 J ¼ 448K (Appendix D). This is significantly large incomparison with other salts (Table V), indicating largeantiferromagnetic coupling in the -system.

-Phase has only two interactions, JP and JC [Fig. 6(b)].As shown in the caption of Fig. 6, JC is larger than JP.

49)

This contrasts with the -phase ET salts, where JC is smallerthan JP; for example, -(ET)2[N(CN)2]X gives JP ¼ 73Kand 78K, and JC ¼ 63K and 56K for the X ¼ Cl and Brsalts, respectively.21) This comes from the difference in thedihedral angle of the non-parallel molecules. As shown inTable III, the dihedral angles of the BETS salts are about ¼ 129, whereas those of the ET salts are nearly 90.Similarly to the universal phase diagram of the -phase,21)

the transfer integral tp attains a maximum near ¼ 90, andrapidly decreases as increases. This leads to small tp and JPin the BETS salts.

When JP is larger than JC, antiferromagnetic couplingalong the oblique axis (JP) is most important, and the moststable magnetic structure is that shown in Fig. 6(b), where alldimers on the upper and bottom edge of the unit cell hold upspins, and dimers on the side edges have down spins. Thisstructure has been called ‘‘vertical type’’ in analogy with thestripe of charge order patterns.43) This is stable in the ETsalts.

If JC exceeds JP,43) antiferromagnetic coupling along c is

most important. In this case, two magnetic structures havethe same energy: (1) the same spins align in a zig-zagmanner along the a axis (horizontal type) and (2) the samespins align along the oblique axis (diagonal type). Althoughonly from these interactions we cannot determine which ismore stable, the diagonal type is inconsistent with the directd–d interaction (vide infra). Then we may consider only thehorizontal antiferromagnetism [Fig. 6(c)]. In summary, -systems of -phases of ET and BETS prefer differentmagnetic structures on account of the difference in thedihedral angles.

In the horizontal order, cooperative Jp ¼ 2JC þ 2JP andfrustrating J0p ¼ 2JP give J0 J ¼ 316K and 376K for theCl and Br salts (Table V). These values are smaller than thatof the -phase, indicating weaker antiferromagnetic fluctua-tion. This agrees with smaller q. This happens because the-phase is close to the frustration of the triangular lattice,which is achieved at JC ¼ JP.

4. d- and dd-Interactions

4.1 -(BETS)2FeCl4The –d and d–d overlap integrals in -(BETS)2FeCl4 are

Table IV. 0 and q calculated from the band structures. C is given byeq. (B2).

0 q 2C=q Ref.

states/eV

-(BETS)2FeCl4 0.70 2.42 4800K Table II

-(BETS)2FeBr4 0.71 1.85 6290K Table III

-(BETS)2FeCl4 0.70 1.85 5950K Table III

-(ET)2Cu(NCS)2 0.70 1.65 7030K 21

(TMTSF)2AsF6 0.73 3.60 3220K 46

Fig. 5. Spin susceptibility ðqÞ of (a) -(BETS)2FeCl4 and (b) -(BETS)2FeCl4, calculated according to eq. (3:1) and the band structures

in Fig. 4.

a

oc

(c) Horizontal Type

a

oc

(b) Vertical Type

(a) -(BETS)2FeCl4 -(BETS)2FeX4

JB

JC

JCJC

JC

JB

Jt

Jc

Jc

Js

JtJC

JC

JC

JP

JPJP

JP

Fig. 6. Inter-dimer interactions of the dimer model with (a) the stable

antiferromagnetic structure of -(BETS)2FeCl4, and with (b) vertical and

(c) horizontal spin alignments of -(BETS)2FeCl4. The dimer J’s are (a)JB: 194K, JC: 128K, Jc: 150K, Js: 170K, and Jt: 4K, and (b), (c) JP:

58K and 67K, JC: 158K and 188K, respectively for the Cl and Br salts.

830 J. Phys. Soc. Jpn., Vol. 71, No. 3, March, 2002 T. MORI and M. KATSUHARA

for other Huckel parameters.3,37) These inter-dimer interac-tions work cooperatively to afford J ¼ JB þ JC þ 2Jc ¼622K with frustrating interactions J0 ¼ Jt þ Js ¼ 174K(top two lines of Table V), resulting in the mean-field J0 J ¼ 448K (Appendix D). This is significantly large incomparison with other salts (Table V), indicating largeantiferromagnetic coupling in the -system.

-Phase has only two interactions, JP and JC [Fig. 6(b)].As shown in the caption of Fig. 6, JC is larger than JP.

49)

This contrasts with the -phase ET salts, where JC is smallerthan JP; for example, -(ET)2[N(CN)2]X gives JP ¼ 73Kand 78K, and JC ¼ 63K and 56K for the X ¼ Cl and Brsalts, respectively.21) This comes from the difference in thedihedral angle of the non-parallel molecules. As shown inTable III, the dihedral angles of the BETS salts are about ¼ 129, whereas those of the ET salts are nearly 90.Similarly to the universal phase diagram of the -phase,21)

the transfer integral tp attains a maximum near ¼ 90, andrapidly decreases as increases. This leads to small tp and JPin the BETS salts.

When JP is larger than JC, antiferromagnetic couplingalong the oblique axis (JP) is most important, and the moststable magnetic structure is that shown in Fig. 6(b), where alldimers on the upper and bottom edge of the unit cell hold upspins, and dimers on the side edges have down spins. Thisstructure has been called ‘‘vertical type’’ in analogy with thestripe of charge order patterns.43) This is stable in the ETsalts.

If JC exceeds JP,43) antiferromagnetic coupling along c is

most important. In this case, two magnetic structures havethe same energy: (1) the same spins align in a zig-zagmanner along the a axis (horizontal type) and (2) the samespins align along the oblique axis (diagonal type). Althoughonly from these interactions we cannot determine which ismore stable, the diagonal type is inconsistent with the directd–d interaction (vide infra). Then we may consider only thehorizontal antiferromagnetism [Fig. 6(c)]. In summary, -systems of -phases of ET and BETS prefer differentmagnetic structures on account of the difference in thedihedral angles.

In the horizontal order, cooperative Jp ¼ 2JC þ 2JP andfrustrating J0p ¼ 2JP give J0 J ¼ 316K and 376K for theCl and Br salts (Table V). These values are smaller than thatof the -phase, indicating weaker antiferromagnetic fluctua-tion. This agrees with smaller q. This happens because the-phase is close to the frustration of the triangular lattice,which is achieved at JC ¼ JP.

4. d- and dd-Interactions

4.1 -(BETS)2FeCl4The –d and d–d overlap integrals in -(BETS)2FeCl4 are

Table IV. 0 and q calculated from the band structures. C is given byeq. (B2).

0 q 2C=q Ref.

states/eV

-(BETS)2FeCl4 0.70 2.42 4800K Table II

-(BETS)2FeBr4 0.71 1.85 6290K Table III

-(BETS)2FeCl4 0.70 1.85 5950K Table III

-(ET)2Cu(NCS)2 0.70 1.65 7030K 21

(TMTSF)2AsF6 0.73 3.60 3220K 46

Fig. 5. Spin susceptibility ðqÞ of (a) -(BETS)2FeCl4 and (b) -(BETS)2FeCl4, calculated according to eq. (3:1) and the band structures

in Fig. 4.

a

oc

(c) Horizontal Type

a

oc

(b) Vertical Type

(a) -(BETS)2FeCl4 -(BETS)2FeX4

JB

JC

JCJC

JC

JB

Jt

Jc

Jc

Js

JtJC

JC

JC

JP

JPJP

JP

Fig. 6. Inter-dimer interactions of the dimer model with (a) the stable

antiferromagnetic structure of -(BETS)2FeCl4, and with (b) vertical and

(c) horizontal spin alignments of -(BETS)2FeCl4. The dimer J’s are (a)JB: 194K, JC: 128K, Jc: 150K, Js: 170K, and Jt: 4K, and (b), (c) JP:

58K and 67K, JC: 158K and 188K, respectively for the Cl and Br salts.

830 J. Phys. Soc. Jpn., Vol. 71, No. 3, March, 2002 T. MORI and M. KATSUHARA

Figure 2.6: The two possibilities of the AFM ordering in κ-(BETS)2FeX4(takenfrom[16]).

two metallic-state-primitive-cells. Doubling the volume of the primitive cells leads tohalf the volume of the first Brillouin zone, and a corresponding folding of the origi-nal Fermi surface. This leads to a reconstructed multiple-connected Fermi surface asshown in Fig. 2.4(b) Until today, only the little grey circles shown in Fig. 2.4(b), havebeen revealed by SdH oscillations. The question is whether they really come from theproposed reconstructed Fermi surface or simply reveal a violation of the metallic Fermisurface from the predicted calculated one. In the former case one would expect moreSdH frequencies caused by various orbits on the reconstructed Fermi surface. Assumingthe vertical type of ordering, the Fermi surface stays the same as in the paramagneticmetallic state seen in Fig 2.4(a)

7

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8

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3 Theoretical background

3.1 Electrodynamics in a magnetic field

We want to give a brief introduction to the model of electron motion in a magnetic fieldand Shubnikov de-Haas (SdH) oscillations following [11], for a more detailed theorysee e.g. [17]. We consider electrons in the vicinity of the Fermi level εF , which areresponsible for the conducting properties. We further assume that scattering processescan be approximated, simply by introducing a constant relaxation time τ , which we sayis independent of the electron’s momentum and magnetic field. This τ -approximation isnot always justified as we see later, when we consider the scattering probability with thestates an electron can be scattered in. As we will see the number of states an electroncan be scattered in varies with magnetic field.When a magnetic field B is applied to a metal, the conduction electrons are subject tothe Lorentz force

FL =dp

dt= ev ×B, (3.1)

where p is the electrons momentum, v the velocity and e the charge. From Eq. (3.1) itfollows that the Lorentz force affects the momentum perpendicular to the field, whilethe component of the momentum parallel to the field pB is constant. As the force isalways perpendicular to the velocity, the energy does not change. Therefore the motionof the electron in the momentum space is in the intersection of contours of constantenergy and paths of constant pB. For low fields the change of the momentum duringthe scattering time can be neglected, as the the trajectory of the electron is only slightlycurved. The characteristic radius of curvature (Lamour radius), rL = pF /(eB), wherepF is the Fermi momentum, is much larger than the mean free path `. With all this itcan be shown that, for a current perpendicular to the field, the relative change in theresistivity is

∆ρ(B)

ρ(0)∝( `rL

)2∝ B2. (3.2)

Now we want to look at higher fields where rL ≤ `. From eq. (3.2), the momentum ofeach electron will change within the time τ . This results in a varying electron velocityv(p) = ∂ε(p)/∂p. Now the velocity depends on the momentum, and is always perpen-dicular to the Fermi surface. Using this, we solve the kinetic Boltzmann equation inthe presence of both, electric and magnetic field. For our semiclassical τ -approximationthe solution is the conductivity tensor σαβ in the form

σαβ = − 2e2τ

(2π~)3

∫df0

dεvα(p)vβ(p)dp, (3.3)

9

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3 Theoretical background

where α and β stand for the directions in space, df0/dε is the energy derivative ofthe equilibrium Fermi distribution function and vβ(p) is the velocity averaged over thescattering time τ :

vβ(p) =1

τ

∫ 0

−∞vβ(p, t)et/τdt. (3.4)

Eqs. (3.3) and (3.4) show us, that the conductivity is determined by the average velocity,which depends on the magnetic field through eq. (3.1) and v(p) = ∂ε(p)/∂p. Equations(3.1), (3.3) and (3.4) allow us to calculate the conductivity numerically and therefore theresistivity provided the dispersion law ε(p) and the velocity v(p) are known. Lifshitzand co-workers have shown that the asymptotic behavior of the conductivity in highmagnetic fields is qualitatively determined by the topology of electron orbits on theFermi surface. Using this, important information about the Fermi surface geometry canbe derived from the conductivity.The electron motion along a closed orbit in p-space can be characterized by a cyclotronfrequency:

ωc =2πeB

(∂S/∂ε)pB

=eB

mc, (3.5)

where S is the orbit area andmc =

(∂S/∂ε)pB

2π(3.6)

is the cyclotron mass. Looking at the high-field limit ωcτ 1, the electron completesmany turns in its orbit around the Fermi surface before being scattered.The allowed states for electrons in a magnetic field follow the Landau subbands or

Landau levels and their energy becomes quantised in the direction perpendicular to thefield as

ε(n, pB) =(n+

1

2

)~wc +

p2B

2me, (3.7)

where me is the free electron mass, n an integer number of the level and pB the mo-mentum parallel to the field B. The electronorbits in the magnetic field follow theintersections of Fermi surface and the Landau subbands.

3.2 Shubnikov-de Haas oscillations

The theory of Shubnikov de-Haas (SdH) oscillations is very complex, as one should, inprinciple, consider a detailed problem of scattering processes modified by a quantizingmagnetic field. Fortunately, in our case we can obtain a satisfactory description byfollowing Pippard’s idea that the scattering probability, and hence the conductivity isproportional to the density of states around the Fermi level D(ζ), the states an electroncan scatter into. The density of states oscillates in quantizing magnetic field. It ispossible to show (see e.g. [17, 18]) that the D(ζ) is proportional to the field derivativeof the magnetization.

D(ζ) ∝(mcB

Sextr

)2∂M

∂B, (3.8)

10

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3.2 Shubnikov-de Haas oscillations

where Sextr is the Fermi surface with the biggest or smallest area. Using this, theoscillatory part of the conductivity can be expressed in the form

σ

σ0∝∞∑r=1

1

r1/2

mcB1/2

(∂2S/∂p2B)

1/2extr

cos[2πr

(FB− 1

2

)± π

4

]RT (r)RD(r)RS(r), (3.9)

where σ0 is the background conductivity and

F =Sextr~2πe

(3.10)

the fundamental frequency. Eq. (3.9) will be hereafter referred to as the Lifschitz-Kosevich formula. RT(r),RD(r) and RS(r) are damping factors we want to discussbelow.

3.2.1 Temperature damping RT

The first damping factor originates from smearing of the Fermi distribution with risingtemperature. Roughly speaking, the Fermi distribution for finite temperatures is likethe Fermi distribution for T = 0 K, but with a few unoccupied states below the Fermienergy and a few states occupied above the Fermi energy in a range that broadenswith temperature. The possible orbits and, hence, the electron oscillation frequenciesF = S

2πe~ then slightly differ from each other. This leads to a smearing in the phase,and, hence, to a destructive interference, which decreases the amplitude. The exactexpression for RT is

RT(r) =KrµT/B

sinh(KrµT/B), (3.11)

whereµ =

mc

me(3.12)

is the cyclotron mass normalized to the free electron mass and

K =2π2kBme

~e≈ 14.7.

T

K(3.13)

As eq. (3.11) contains the effective cyclotron mass, while all other parameters are known,it can be used to obtain the effective cyclotron mass, by measuring the amplitude ofthe oscillations at different temperatures.

3.2.2 Impurity damping RD

In a perfect crystal, the free path ` is infinite or 1τ = 0. A finite relaxation time

widens these Landau subbands. This broadening is usually described by the Lorentzdistribution function with the half-width Γ = ~

2τ . The effect on the conductivity canbe expressed in the so-called Dingle temperature and the Dingle damping factor for theoscillation amplitude

RD(r) = exp(− πτ

wcτ

)= exp

(− KrµTD

B

)(3.14)

11

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3 Theoretical background

with the Dingle temperature

TD =~

2πkBτ. (3.15)

Knowing the effective mass µ from the temperature fit, we can obtain the relaxationtime τ from the field dependent amplitude. One should note that τ is not necessarilythe same as the usual transport relaxation time, as internal stains and dislocations alsocontribute.

3.2.3 Spin damping RS

As we need it very often later, we will have a deeper look at the spin damping factor,as it is described in [18]. In magnetic fields B, an electron with spin parallel to the fieldhas a lower energy, and and electron with a spin antiparallel to the field has a higherenergy compared to the case without field. Each energy level without field is split intotwo new energy levels

ε± 1

2∆ε, (3.16)

where∆ε =

1

2gβ0B (3.17)

andβ0 =

e~me

(3.18)

β0 is twice the Bohr magneton and g the Landé factor. As this splitting of the Landaulevel, can be seen as two Landau levels, close to each other, we can formulate the energydifference as a phase difference

φ =2rπ∆ε

βB(3.19)

between the oscillations coming from spin-up and spin-down electrons. Here β = µβ0

and we multiply the phase difference with r for the r-th harmonic. Two successiveLandau levels that are βB = ~ωc apart, pass trough a Fermi surface having a phasedifference of 2π. The phase difference between two cylinders coming from the spinsplitting is the fraction ∆ε

βB of the full phase difference between two Landau tubes. Ina case where the phase difference is π we speak of spin zeros. The superposition of thespin-up and spin-down oscillations is, therefore, an interference that leads to a dampingfactor in the original amplitude

RS(r) = cos(1

2rφ)

= cos(rπ∆ε

βB

)= cos

(1

2rπgµ

). (3.20)

We consider κ-(BETS)2FeBr4 in this thesis. As our material contains magnetic Feions, we have to take into account interaction of conducting electrons with magneticmoments, which is spin-dependent and the new field dependent φ can be written as

φ = 2π[gµ

2

(1− Bex

B

)]. (3.21)

12

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3.2 Shubnikov-de Haas oscillations

In the quasi 2D (q2D) materials, the area of the Fermi surface cross section perpen-dicular to the field B, rises with higher angles as 1/ cos(θ), and by definition leads toan increasing cyclotron mass µ(θ) = µ(0)/ cos(θ). Inserting this and our new phasedifference into eq.(3.20) we get a modified spin damping factor

RS(r) = cos[ rπgµ

2 cos θ

(1− Bex

B

)]. (3.22)

Later on, we will focus on the first harmonic r = 1, for all damping factors, since thesecond and higher harmonics are hard to observe, only for the strong α oscillations wecould see a tiny second harmonic.

13

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14

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4 Experimental setup

4.1 Sample preparation and sample holder

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 Measurement techniques4.2.1 Resistance measurementsIn 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

Figure 4.1: Circuit diagram for the four-point measurement, (taken from: [19])

The samples where contacted with annealed 10µm platin wires that were glued ontothe sample with graphite paste, in an ac four-point measurement arrangement, shownin Fig. (4.1). With this arrangement we can measure the sample resistance, withoutthe wire and contact resistances. We wanted to measure the SdH oscillations in theAFM state. The transition from the AFM to the normal metallic state, is the lowestfor magnetic field parallel a. Therefore in this experiment the samples were rotated inthe bc-plane. For that orientation the transition field was as high as possible, whichwas important to see as many SdH oscillations in the AFM state as possible. Sampleresistanc at low temperatures was below 0.6Ohm, and the contact resistances were 50Ωso we needed to take currents lower or equal to 100µA, as we did not want to overheat

15

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4 Experimental setup

the sample.We measured with a lock-in amplifier, with an ac current with a frequencybetween 50Hz and 250Hz. The lock-in measurement filters all frequencies but themeasurement frequency, all other frequencies vanish, effectively eliminating most noise.

4.2 3He-system

To measure the SdH oscillations at low field with reasonable amplitudes, it is necessaryto cool the sample below 1K, as we have very high effective cyclotron masses that dampour oscillation amplitude strongly with rising temperatures. For that a 3He setup wasused. 3He was condensed an then pumped, to use the cooling power of the vaporizingliquid 3He. By decreasing the pressure in the 3He space, due to the vapor pressurecurve, the temperature of the 3He and the temperature of the sample then decreased.The lowest T achieved by pumping 3He was 0.4K.

4.3 Superconducting magnet

To apply magnetic fields to the sample, we used a superconducting magnet from Cryo-genics, that could produce magnetic fields up to B = 14 T at a current of 101A whencooled to T = 4.2 K and up to 16T at 115A when cooled to 2.2K. The magnet consistsof an outer coil, made from NbTi, and an inner coil made of Nb3Sn. The magnet isequipped with a shunt connected to a heater. The shunt is in parallel with the coils andforms a closed SC electric circuit. If heated, the shunt has a finite ohmic resistance,and the closed SC circuit is opened, so the magnetic field in the coil can be changedand an electric current can be applied. If the heater is switched off and the shunt be-comes superconducting the current stays in the coil and the magnetic field in the coilis conserved. This is used to keep a constant magnetic field.

16

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5 Results and discussion

5.1 Temperature and field dependent resistance

5.1.1 Superconducting transition

0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 50 . 0 0

0 . 0 5

0 . 1 0

0 . 1 5

0 . 2 0

R [Oh

m]

T [ K ]

A F M t r a n s i t i o n

S C t r a n s i t i o n

S C

Figure 5.1: The temperature-dependent resistance below 3.0K

In Fig. 5.1 we see the temperature dependent resistance. We see the AFM transitionat 2.45K, and the SC transition starting from 1.4K. In our salt only the BETS layersbecome superconducting, so the electrons have to tunnel the insulating FeBr4 layers, aswe measure the resistance perpendicular to the planes.

5.1.2 Magnetoresistance

In Fig. 5.2 we see the magnetoresistance from 0 to 14T. At low angles the SC transitionis at low fields, starting almost at zero field and the SC state is fully broken at 0.25T. Thelower boundary, where the SC state is broken, is moving to higher fields for increasingangles up to 2.5T for field parallel to the c-axis[20]. Above the SC transition we see theAFM state and SdH oscillations. In the AFM state there is a clear hysteresis betweenup and down sweep, seen in the inset of Fig. 5.2, between up and down sweep. The AFMstate is broken at 5.5T, indicated by the end of the hysteresis. Above the AFM statethe paramagnetic state begins, showing also SdH oscillations we are going to discusslater.

17

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5 Results and discussion

0 2 4 6 8 1 0 1 2 1 40 . 0 00 . 0 20 . 0 40 . 0 60 . 0 80 . 1 00 . 1 20 . 1 40 . 1 60 . 1 80 . 2 00 . 2 2

0 2 4 60 . 1 4

0 . 1 6R [Oh

m]

B [ T ]

p a r a m a g n e t i c s t a t ea n t i f e r r o m a g n e t i c s t a t e

s u p e r c o n d u c t i n g s t a t e

T = 0 . 4 3 K

R [Oh

m]

B [ T ]

a n t i f e r r o m a g n e t i c s t a t e

Figure 5.2: Magnetoresitance from 0 to 14T, for 0.43K and θ = 2.5 with an insetfrom 0 to 6T

5.2 The antiferromagnetic state: Shubnikov-de Haasoscillations

0 1 0 0 2 0 0 3 0 0 4 0 0

FFT A

mplitu

de [a

rb. un

its]

F r e q u e n c y [ T ]

δ

η

T = 0 . 4 2 K3 T < B < 5 T

θ = 0 °

Figure 5.3: Fast Fourier transformation (FFT) spectrum of the SdH oscillations in theAFM state in the field window 3-5T. The δ-oscillation has a frequency of 62T whilethe new frequency Fη = 178T .

In the AFM state, we see two frequencies in the SdH oscillations, one the δ-frequency

18

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5.2 The antiferromagnetic state: Shubnikov-de Haas oscillations

with 62 T that has already been reported by Konoike[10], and a new frequency with178 T, which hast not been reported yet.

5.2.1 δ frequency orbit

We want to further analyze the δ oscillations, which is the dominating frequency in theSdH spectrum in the AFM state.

Cyclotron mass

0 . 4 0 0 . 4 5 0 . 5 0 0 . 5 5 0 . 6 0 0 . 6 5 0 . 7 0 0 . 7 52 0

3 0

4 0

5 0

6 0| A o s c | = a m 2

c T / s i n h ( K m c T / B m )

FFT A

mplitu

de δ

[arb.

units]

T e m p e r a t u r e [ K ]

µδ = 1 . 3 B m = 3 . 7 8 T

Figure 5.4: T -dependence of the FFT amplitude of the δ-oscillations and the effectivemass fit.

As we need the cyclotron mass of the δ-oscillations later for analyzing their angulardependence, we measured the amplitude of the oscillations at different temperaturesand θ = 0 and fitted the data with the temperature-dependent part of the Lifschitz-Kosevich formula

|A| = aT

sinh(KµT

B

) , (5.1)

where a is a fitting factor, µ the normalized cyclotron mass and B the inverse average

of the used field window B =(

( 1Bstart

+ 1Bend

)/2)−1

. We get µ = 1.3. This differs fromthe value of other publications([10, 19] mc = 1.1me ). A reason for this might be thatwe analyzed only a small temperature window, as we got the data, as a byproduct frommeasuring the cyclotron mass of the η oscillations.

19

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5 Results and discussion

Angle dependence

We know the frequency of the oscillations is proportional to the area of the Fermisurface projection perpendicular to the field. As our Fermi surfaces are cylinders, weget a simple dependence of the frequency on the angle

F (θ) =F0

cos(θ). (5.2)

This formula we fit to our measured frequencies shown in Fig. 5.5 on the left. Weget a Frequency F0 = 62.1± 0.2 T at θ = 0. The frequency is consistent with otherpublications [10]. Initially we used the angel dependent F(θ) to determine the exactsample orientation.

- 8 0 - 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0 8 06 08 0

1 0 01 2 01 4 01 6 01 8 0

Frequ

ency

[T]

θ [ D e g ]

F ( θ ) = F 0 / c o s ( θ)

Figure 5.5: Left: The frequency of the oscillations plotted against the polar angle.

Oscillations amplitude

Our main aim when measuring the angle dependence of δ-oscillations was measuringspin zeros. The reason was a theory [8] that predicts the absence of spin zeros. Byfitting the Lifshitz-Kosevich Formula to the data

|Aosc| = a ·Kµ2T

[1− exp

(− BMB

B⊥

)]2exp

(− KµTD

B⊥

)B1/2 cos2 θ sinh

(KµTB⊥

) ·∣∣∣ cos

( πµg

2 cos θ

)∣∣∣ (5.3)

where BMB the magnetic breakdown field, B⊥ = B cos θ the field component per-pendicular to the ac-plane, TD the Dingle temperature, a a prefactor and θ the angle.

20

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5.2 The antiferromagnetic state: Shubnikov-de Haas oscillations

- 7 5 - 6 0 - 4 5 - 3 0 - 1 5 0 1 5 3 0 4 5 6 0 7 50 . 0 1

0 . 1

1

1 0

A m p l i t u d e g = 0 g = 1 . 4 g = 0 . 6

FFT A

mplitu

de δ

[arb.

units]

T h e t a [ D e g ]Figure 5.6: Amplitude against the angle. As we can show there are no spin zeros inthe range −70 ≤ θ ≤ 70.

We take TD and a as fitting parameters and set BMB = 20T, µ = 1.3, K = 14.7 T/K

and an averaged B⊥ =(

( 1Bstart

+ 1Bend

)/2)−1

cos θ, with Bstart = 3T and Bend = 5T,as well as g = 0 (solid line), g = 0.6 (blue line) and g = 1.4 (broken line) in Fig.5.6.The measured data points fit well to the fit curves for g = 0. To ensure that there is nog factor other than g≤ 0.25 possible, we fitted the data with all reasonable g factors.Where the fits for g 6= 0 differ from the fit for g = 0, we have two or more points closeto the g = 0 fit, and far away from the g 6= 0 fit. Only for g factors smaller than 0.25,the spin zeros would be at such high angles that we have no data for those angles. Aspin splitting factor g ≤ 0.25 is unrealistic. Therefore, we can say that there are no spinzeros.

5.2.2 η frequency orbit

We found new oscillations and a new peak in the FFT of the SdH oscillations. Wename the frequency Fη = 178T. The third harmonic of δ would have a frequency ofF3δ = 186T. This is close to our new frequency, so one could think that we found F3δ

instead. But we can determine the oscillation frequency to 1.5T accuracy, so we clearlyfound a new frequency.

21

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5 Results and discussion

0 . 3 5 0 . 4 0 0 . 4 5 0 . 5 0 0 . 5 5 0 . 6 0 0 . 6 5 0 . 7 00

1

2

3

4

FFT A

mplitu

de η

[arb.u

nits]

T e m p e r a t u r e [ K ]

µη = 2 . 7 5

| A o s c | = a m 2c T / s i n h ( K m c T / B m )

B m = 3 . 7 8 T

Figure 5.7: Fit of the effectiv mass of the η oscillations.

Effective Mass

To determine the cyclotron mass we proceeded as for the δ oscillations. By fitting wedetermined the cyclotron mass to be mc = 2.75 me as shown in 5.7.

Comparison with Fermi surface reconstruction model

Looking at the Fermi surface Fig.5.8 there are three different pockets: the little δ circles,almost rectangular grey-dyed "bone" in the middle of the Brillouin zone and the star-like areas around the δ-circle. We neglect the gap in the Brillouin zone. By defining theδ area as the intersections of two α ellipses, we can calculate the area of the bone. Todo so, we split up the areas in the Fermi surface, as the upper area Aup, consisting ofthe α area, the δ areas and the star areas, the middle area Amid containing the bone,and the lower area Adown = Aup, being the same as the upper area. The upper area inthe first Brillouin zone of the normal metallic state, is:

Aup =α

4+α

2− δ

2+α

4− δ

2= α− δ = Adown (5.4)

Here we denote Aδ by δ and the same for the others. The area in the middle is

Amid = b+ 2[β

2− 2(α

2+ (

s

4− δ

4

))], (5.5)

where b is the bone area and s is the complete star area. Amid , Adown and Aup

together are equal to the area of the metallic state first Brillouin zone and the latter isequal to β.

22

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5.3 The normal metallic state

b

s

Adown

Amid

Figure 5.8: Fermi surface in the AFM state, with horizontal ordering (proposed in[10]).

β = Amid +Adown +Aup = 2(α− δ

)+ b+ 2

[β2− 2(α

2+ (

s

4− δ

4

))]= 2α− 2δ + β − 2α− s+ δ + b

0 = −δ − s+ b

b = δ + s

(5.6)

We see that the area of the bone is equal to the area of the star plus the δ area. This isconsistent with a theorem of Luttinger [21], as the bone is hole like, and the star and theδ are electron like. As the bone has a bigger surface and is therefore harder to observewe assume that the new found frequency is the star shaped surface. Thus observationof the new SdH with corresponding cyclotron mass appears to be consistent with modelof the Fermi surface reconstruction, thus supports the prediction of the horizontal AFMorder [16]. For an unambiguous assignment as well as to verify the Fermi surface moreexperimental data is necessary.

5.3 The normal metallic state

5.3.1 Field induced superconductivity

We measured the magnetoresistance for T = 0.42K and θ = 90 for B parallel tothe c-axis in the full field window, as shown in 5.9 . At roughly 1.8T we see the SCtransition, to normal conductivity. We see a feature between 11 and 14T. This featuresis due to FISC. We subtract the monotonic magnetoresistance, assuming it to be linearin B, with the result shown in Fig. 5.9(b) Bex is the middle of the feature, we get after

23

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5 Results and discussion

- 2 0 2 4 6 8 1 0 1 2 1 4 1 60 . 00 . 10 . 20 . 30 . 40 . 50 . 60 . 7

R[Ohm

]

B [ T ]

( a )

1 0 1 1 1 2 1 3 1 4 1 5

Resis

tance

[Ohm

]

M a g n e t i c f i e l d [ T ]

1 2 . 6 T

( b )

Figure 5.9: Field induced superconductivity (FISC) at around 12.6 T, for field perfectlyparallel to conducting layers. (a) Resistance against magnetic field, (b) The samedata in the field window 10 to 15T, after subtracting the monotonic background. Theresistance minimum reveals the effective field of the π-d exchange interaction.

subtracting the monotonic magnetoresistance. We get Bex = 12.6T, which is consistentwith earlier measurements [20].There are two mechanisms that break SC, in a magnetic field. One is the Zeeman effectthat breaks apart Cooper pairs, if they are in a spin singlet. This happens when theenergy gain from arranging the spin parallel to the field is higher than the energy gainfrom forming a Cooper pair. The other effect is the orbital effect, where the SC currentnecessary to shield the vortices in the SC layer becomes over critical. For q2D SC theorbital effect is suppressed, if the magnetic field is applied parallel to the SC layers. Inthis case only the Zeeman effect destroys SC.In our materials we have a magnetic ordering of the iron 3d spins in the metallic state,that creates an internal effective field acting on the conduction π-electrons in BETSlayers. This internal field then breaks the cooper pairs by the Zeeman effect. Thishappens in the paramagnetic state above 4.4T in Fig. 5.9 If this internal field iscompensated by an external field, the paramagnetic pair breaking effect does not anylonger destroy SC and we see FISC. The exchange field we measure is 12.6T, consistentwith other measurements [20]. This effect we measure is also called Jaccarino-Petercompensation effect[14].Uji et al. found in 2001 FISC in λ-(BETS)2FeCl4 [4], while FISC was proposed forκ-(BETS)2FeBr4 by Cépas et al,[14], and found experimentally by Fujiwara et al. [22]and Konoike et al.[20]. As the magnetic field we have to compensate is around 12.6 T,and therefore in our field range and the suppression of the orbital pair breaking effectis very sensitive to orientation of the sample relative to the field, we used it to measurethe real zero degree angle position of the sample on the rotator.

24

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5.3 The normal metallic state

6 8 1 0 1 2 1 4 1 60 . 0 60 . 0 70 . 0 80 . 0 90 . 1 00 . 1 10 . 1 20 . 1 30 . 1 40 . 1 50 . 1 60 . 1 70 . 1 80 . 1 9

4 7 °4 4 °4 1 °3 8 °3 3 . 5 °

2 4 . 5 °2 9 °

0 . 5 °

9 °- 2 0 °

2 0 °1 5 . 5 °1 4 °1 2 . 5 °1 1 °8 °9 . 5 °6 . 5 °3 . 5 °

Amplit

ude i

nvers

e FFT

[arb.

units]

B [ T ]

- 2 . 5 °

Figure 5.10: The inverse FFT of the oscillation for the β frequency, conserving theamplitude and arranging them by the angles. Different curves for different angles arevertically shifted, different measurements for the same angle colored differently, butshifted to the same position.

25

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5 Results and discussion

5.3.2 β Shubnikov-de Haas oscillations

In 5.10 we see the inverse FFT of the β oscillations. The inverse FFT for the angleswere shifted to different positions to compare the inverse FFT of different angles. Forsome angles we measured the oscillations several times, so all the measurements wereput on the same position, but colored differently. We see that the inverse FFT are zeroor show a minimum for some B. These points we call nodes or beats.From theory we would predict that the nodes are shifting towards a constant Bex withrising angle by the formula∣∣∣Aosc∣∣∣ =

∣∣∣A cos[ πµg

2 cos θ

(BexB− 1)]∣∣∣ (5.7)

Where A is the angle- and field- dependent amplitude, and the cos function the mod-ulation due to the spin up and downs. µ is the cyclotron mass of β. We fit, as shownin Fig. 5.11. We fit only the nodes with Eq.(5.7) as other factors of the oscillation am-plitude also depend on the magnetic field, and only the nodes depend solely on ourequation (5.7).

0 . 0 7 0 . 0 8 0 . 0 9 0 . 1 0 0 . 1 1

- 0 . 5

0 . 0

0 . 5

1 . 0

Amplit

ude

1 / B [ 1 / T ]

θ = 9 °B _ e x = 1 2 . 2 5 T

g = 2 . 1 2

Figure 5.11: Fit of the nodes, with (5.7) using Bex and g as fitting parameters, shownas an example for θ=9.

From FISC we know the Bex = 12.6T. From Fig. 5.12 we see that the exchange fieldis only for θ = 0 equal to the exchange field we know from field FISC. In betweenθ = 0 and 20, Bex and g differ from their θ = 0 values and show a strange behavioras seen in Fig. 5.12 and Fig. 5.13. Above 20, the behavior of Bex and g stabilizes atBex = 12.95T and g = 2.05. For high angles around 45 we see some instabilities in Bexand g. This might come from errors in the 1

cos θ term, as this term gets very sensitive toerrors in the angle for high angles. It seems as in the range θ = 0 to 20 that anothermechanism than spin-splitting is involved.

26

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5.3 The normal metallic state

- 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 4 0 5 01 2 , 21 2 , 41 2 , 61 2 , 81 3 , 01 3 , 2

B ex [T

]

θ [ D e g ]

( a )

Figure 5.12: The exchange field as obtained by the fit. Clearly visible, the Bex is notconstant. The error bars show the errors estimated by the fitting algorithm.

- 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 4 0 5 01 , 6

1 , 8

2 , 0

2 , 2

2 , 4

g

A n g l e t h e t a

( b )

Figure 5.13: The g factor obtained from the fit. The error bars show the errorsestimated by the fitting algorithm.

27

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5 Results and discussion

5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7

- 0 . 1

0 . 0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

0 . 6

0 . 7

4 7 °5 3 °5 6 °

4 4 °4 1 °3 8 °3 6 . 5 °3 3 . 5 °2 9 °2 4 . 5 °2 0 °1 5 . 5 °1 4 °1 2 . 5 °9 . 5 °8 °6 . 5 °

3 . 5 °0 . 5 °

5 °

- 2 . 5 °- 1 0 °

B [ T ]

- 2 0 °

6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7- 0 . 1

0 . 0

0 . 1

4 7 °5 3 °5 6 °

4 4 °4 1 °

3 8 °3 6 . 5 °

3 3 . 5 °

2 9 °

2 4 . 5 °

2 0 °

1 5 . 5 °

B [ T ]

Figure 5.14: Left: SdH oscillations of the α frequency, results of the inverse FFT forthe α frequency, arranged in the same way as for β. Right enlarged window startingfrom θ = 15.5 showing the only visible beats.

5.3.3 α Shubnikov-de Haas oscillations

In Fig.5.14 we see the evolution of the SdH oscillations corresponding to the α-orbit.Below 20 there are no beats visible. Above 20, we see beats appearing, becomingmore pronounced at increasing angles. The angle range above 20 is the same anglerange where the beats in β more or less behave as predicted by theory, showing almostconstant Bex and g. As there are not enough visible beats to fit in α, to obtain Bexand g, we try to compare the beats we would expect, using the Bex and g from the βoscillations, to the beats we find in the α oscillations, shown in Fig. 5.15.We see that below the exchange field, the theory nodes are where we expect them.

But we also see that from theory we would expect also nodes where in the measurementwe see a clear maximum. Above the exchange field, the nodes in theory do not fit atall with the nodes we measure, as we see nodes in theory where we do not see nodesin the measurement and nodes in the measurement where we do not see nodes in thetheory. Further investigation is necessary, to find if the absence of nodes below 20 inthe oscillations corresponding to the α orbit, and the strange behavior in the oscillationscorresponding to the β orbit are related. The absence of nodes in α-oscillations was alsoobserved in the sister compound κ-(BETS)2FeCl4 [23]. It is further to investigate whythe nodes we expect from theory and the nodes we measure differ for the α-frequency.

28

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5.3 The normal metallic state

5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7- 0 . 1 0- 0 . 0 8- 0 . 0 6- 0 . 0 4- 0 . 0 20 . 0 00 . 0 20 . 0 40 . 0 60 . 0 80 . 1 00 . 1 20 . 1 40 . 1 60 . 1 80 . 2 0

Amplit

ude i

nvers

e FFT

[arb.

units]

4 7 °5 3 °

5 6 °

4 4 °4 1 °

3 8 °3 6 . 5 °3 3 . 5 °

2 9 °

2 4 . 5 °

2 0 °

1 5 . 5 °

B [ T ]

- 2 0 °

Figure 5.15: The enlarged window starting from θ = 15. The crosses and filledsquares mark the positions where we would expect nodes from theory.

29

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30

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6 Summary

In this thesis we measured the low-temperature magnetoresistance in κ-(BETS)2FeBr4down to temperatures of 0.4K and magnetic fields up to 16T. We analyzed Shubnikov-de Haas (SdH) oscillations in the paramagnetic as well as in the antiferromagnetic(AFM) state.In the low-temperature AFM state we have confirmed the δ frequency Fδ = 62T re-ported by Konoike in [10]. Moreover, we found another frequency Fη = 178T in theAFM state. This frequency supports a theory by Mori et al. [16] predicting that theAFM ordering imposes a superstructure in the Fermi surface. The new frequency is tobe further investigated to certify the area in the AFM Fermi surface which is responsiblefor it.A theory [7, 8] predicted the absence of Zeeman splitting effect on the SdH oscillations.This is reflected in the absence of "spin-zeros" in the angular dependence of the oscil-lations amplitude. We have shown that there are, indeed, no spin zeros up to high tiltangles. Hence, we can provide experimental evidence for this theory.In the normal metallic paramagnetic state we observed Fα = 800T and fast β os-cillations with Fβ = 4300T in agreement with earlier measurements [10, 19]. Bothfrequencies show a field-dependent spin splitting effect due to an internal magnetic fieldthat allows us to evaluate the internal field Bex and the spin splitting factor g. Varyingthe angle between the magnetic field and the normal to the conducting layers, we foundbelow 20, Bex to vary from 12.3T to 13T and the g factor to vary from 1.8 to 2.25,for the β oscillations; for the α oscillations we found no beats. Above 20 Bex wasapproximately constant, ≈13T and g ≈ 2.05 for the β oscillations. We also saw nodesin the α oscillations, but were not able to fit for more than three angles. For the βoscillations below 20 and for the α oscillations in general further studies are requiredto understand their behavior.

31

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32

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Bibliography

[1] R. Gross & A. Marx. Festkörperphysik (Oldenbourg Verlag München, (2012)), 1stedn.

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Acknowledgments

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 thesis

• 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 small samples.

• H.Fujiwara for providing the samples

• the master student of this group, Sergej Fust, and all the other students andmembers of the WMI for the nice working atmosphere.

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