Magnetic ﬂeld eﬁects in the layered organic superconductor · reported up to now for an organic...

Physik-Department Walther-Meißner-Institut Bayerische Akademie

Lehrstuhl E23 fur Tieftemperaturforschung der Wissenschaften

Magnetic field effectsin the layered

organic superconductor

α-(BEDT-TTF)2KHg(SCN)4

Diplomarbeit

Sebastian Jakob

Themensteller: Prof. Dr. Rudolf Gross

Garching, den 13. Marz 2007

Technische Universitat Munchen

Abstract

Over the last 15 years, organic charge transfer salts were studied intensively, asthey are model systems for low dimensional metals. While some of them exhibitcharge density wave and spin density wave ground states, some others becomesuperconducting. These ground states can be influenced by magnetic fields andpressure.

At the Walther-Meißner-Institut in recent studies onα-(BEDT-TTF)2KHg(SCN)4 under pressure, a superconducting transition at110 mK was discovered. With the available experimental setup the propertiesof this superconducting state could only be mapped out partially. Therefore,in this diploma work a new measurement setup was established. This setupincorporates a dilution refrigerator and the possibility of precise orientation ofthe magnetic field in a spherical angle of 4π.

Thus, it was possible to accurately determine the dependence of the super-conducting state in the highly anisotropic material α-(BEDT-TTF)2KHg(SCN)4on the orientation and strength of a magnetic field at very low temperatures.

α-(BEDT-TTF)2KHg(SCN)4 was found in a broad temperature range downto . 0.2Tc to be a strongly coupled s-wave superconductor.

It turned out, the superconducting state in α-(BEDT-TTF)2KHg(SCN)4 ishighly anisotropic with respect to the orientation of a magnetic field paralleland perpendicular to the conducting layers. In fact, it is the highest anisotropyreported up to now for an organic superconductor. While for field perpendicularto the conducting layers superconductivity is suppressed by orbital effects, forfields parallel to the conducting layers it is paramagnetically limited, but ex-ceeds the Chandrasekhar-Clogston paramagnetic limit. In addition the criticaltemperature in α-(BEDT-TTF)2KHg(SCN)4 shows anisotropic behaviour withrespect to the orientation of a magnetic within the conducting layers.

iii

iv ABSTRACT

In den letzten 15 Jahren wurden die organischen Ladungstransfersalze inten-siv untersucht, da sie Modellsysteme fur niedrigdimensionale Metalle darstellen.Wahrend einige Salze Ladungs- und Spin-Dichtewellen als Grundzustandaufweisen, werden andere supraleitend. Diese Grundzustande konnen durchmagnetische Felder und Druck beeinflußt werden.

Bei kurzlich am Walther-Meißner-Institut durchgefuhrten Experimenten anα-(BEDT-TTF)2KHg(SCN)4 unter Druck wurde ein supraleitender Ubergangbei 110 mK entdeckt. Mit dem verfugbaren Versuchsaufbau konnten die Eigen-schaften dieses Zustandes nur teilweise untersucht werden. Deshalb wurde imRahmen dieser Diplomarbeit ein neuer Versuchsaufbau errichtet. In diesemkommt ein Mischkuhler zum Einsatz und es besteht die Moglichkeit einMagnetfeld prazise innerhalb eines Raumwinkels von 4π auszurichten.

Hierdurch war es moglich, die Abhangigkeit des supraleitenden Zustandesim hoch anisotropen Material α-(BEDT-TTF)2KHg(SCN)4 von Magnetfeldernbei sehr niedrigen Temperaturen zu bestimmen.

Es stellte sich heraus, daß α-(BEDT-TTF)2KHg(SCN)4 in einem weitenTemperaturbereich hinab bis zu . 0.2Tc ein stark gekoppelter s-WellenSupraleiter ist. Des weiteren ist der supraleitende Zustand inα-(BEDT-TTF)2KHg(SCN)4 hoch anisotrop in Bezug auf die Ausrichtung desMagnetfeldes parallel und senkrecht zu den leitenden Schichten des Materi-als. Diese Anisotropie ist die bisher hochste bekannte im Bereich der organ-ischen Supraleiter. Wahrend fur ein senkrecht zu den leitenden Schichten aus-gerichtetes Feld die Supraleitung durch Bahneffekte unterdruckt wird, ist dieSupraleitung fur ein Feld parallel zu diesen Schichten paramagnetisch limitiertund ubertrifft das “Chandrasekhar-Clogston paramagnetic limit”. Außerdemzeigt die kritische Temperatur in α-(BEDT-TTF)2KHg(SCN)4 eine Anisotropiein bezug auf die Ausrichtung eines Magnetfeldes parallel zu den leitendenSchichten.

Contents

Abstract iii

1 Introduction 1

2 Theoretical background 32.1 Superconductivity in general . . . . . . . . . . . . . . . . . . . . 3

2.1.1 The Meißner-Ochsenfeld Effect . . . . . . . . . . . . . . . 32.1.2 BCS Theory . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.3 Ginzburg-Landau Theory . . . . . . . . . . . . . . . . . . 52.1.4 Parameters influencing superconductivity . . . . . . . . . 6

2.2 Superconductivity in layered compounds . . . . . . . . . . . . . . 92.2.1 Highly anisotropic superconductors . . . . . . . . . . . . . 92.2.2 Quasi two-dimensional superconductors . . . . . . . . . . 11

3 The organic metal α-(BEDT-TTF)2KHg(SCN)4 133.1 Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Band structure and electronic anisotropy . . . . . . . . . . . . . . 163.4 T-p phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Experimental setup 214.1 Measurement setup - inside the dewar . . . . . . . . . . . . . . . 21

4.1.1 Dilution refrigerator . . . . . . . . . . . . . . . . . . . . . 224.1.2 Insert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.1.3 Vector magnet . . . . . . . . . . . . . . . . . . . . . . . . 274.1.4 Magnet mounting and rotation of the insert relative to

the magnet . . . . . . . . . . . . . . . . . . . . . . . . . . 294.1.5 Pressure cell . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2 Measurement setup - outside the dewar . . . . . . . . . . . . . . 344.2.1 The technique of four point resistance measurement . . . 344.2.2 Measuring sample resistance . . . . . . . . . . . . . . . . 354.2.3 Measuring pressure . . . . . . . . . . . . . . . . . . . . . . 364.2.4 Thermometry . . . . . . . . . . . . . . . . . . . . . . . . . 374.2.5 Superconducting magnet power supplies . . . . . . . . . . 384.2.6 Grounding and stable line power . . . . . . . . . . . . . . 384.2.7 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2.8 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

v

vi CONTENTS

5 Results and Discussion 495.1 Sample characterisation . . . . . . . . . . . . . . . . . . . . . . . 50

5.1.1 Measurements at ambient pressure . . . . . . . . . . . . . 505.1.2 Measurements under pressure . . . . . . . . . . . . . . . . 54

5.2 Superconducting state at p = 2.8 kbar . . . . . . . . . . . . . . . 575.2.1 Transition curves, definition of critical values . . . . . . . 575.2.2 Magnetic field perpendicular to conducting layers . . . . . 595.2.3 Magnetic field parallel to the conducting layers . . . . . . 635.2.4 ϕ-dependence of critical field . . . . . . . . . . . . . . . . 695.2.5 θ-dependence of critical field . . . . . . . . . . . . . . . . 71

6 Summary 75

7 Appendix 777.1 Cooling power of a dilution refrigerator . . . . . . . . . . . . . . 77

7.1.1 Theoretical cooling power . . . . . . . . . . . . . . . . . . 777.1.2 Cooling power at the sample space . . . . . . . . . . . . . 78

7.2 Inversion curve of 3He . . . . . . . . . . . . . . . . . . . . . . . . 797.3 Gas handling system . . . . . . . . . . . . . . . . . . . . . . . . . 807.4 Technical specifications of vector magnet . . . . . . . . . . . . . . 817.5 Dependence of the sample resistance on the measuring current . 837.6 Earth magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . 83

Bibliography 89

Acknowledgements 91

Chapter 1

Introduction

Since the discovery of superconductivity (1911) there were considerations, if andhow to make use of its properties at room temperature. In 1964 a proposal byW. A. Little [1] gave impetus on the research on conductors based on organicmolecules. He supposed, a pairing mechanism for electrons would be estab-lished, when conducting polymers are embedded in a polarizable medium. Thistheoretical prediction stimulated the research in the field of organic conductorsenormously and in 1980 organic superconductivity was found for the first timein a charge transfer salt of TMTSF (tetramethyltetraselenafulvalene) [2]. Up tonow the highest transition temperature reached in this class is about 14 K [3],but these compounds turned out to be of very high interest also for other reasons.

The basic structural units of the organic charge transfer salts are partiallycharged flat organic molecules, which are packed in stacks or in layers sepa-rated from each other by counterions (in most cases inorganic monovalent ions).The significant overlap between the molecular π-orbitals and a fractional chargetransfer to the counterions lead to the formation of partially filled conductionbands. Due to the chainlike or layered arrangement of the molecular blocks,the conductivity is highly anisotropic. These metals can be considered as quasi-one-dimensional (q1d) or quasi-two-dimensional (q2d) conductors. The reduceddimensionality combined with relatively low concentrations of charge carriersgive rise to strong electron correlations and numerous competing instabilitiesof the normal metallic state. The strength of the instabilities can be tuned byslight chemical modifications or by changing external parameters such as tem-perature, pressure, or magnetic field.

A very interesting and intensively studied compound is thequasi-two-dimensional organic metal α-(BEDT-TTF)2KHg(SCN)4. It is thefirst compound showing low enough transition temperature to a charge densitywave state (T = 8 K) that the important part of the magnetic field-temperaturephase diagram could be studied in detail [4], [5], [6] [7]. By applying hydrostaticpressure this CDW state can be suppressed at p = 2.5 kbar [8], [9], [10]. It isalready known since some time that the title compound shows also an incom-plete transition to a superconducting state at temperatures below 300 mK [11].But it was only recently shown at the Walther-Meißner-Institut that a sharptransition to a superconducting state appears for pressures higher than 2.5 kbar

1

2 CHAPTER 1. INTRODUCTION

at temperatures around 100 mK, when the CDW state is just suppressed [12],[10]. The superconducting state in α-(BEDT-TTF)2KHg(SCN)4 is of high in-terest, as this compound is known to be one of the most anisotropic organicconductors and the superconducting state exists in a close vicinity to the CDWstate. In this diploma work the anisotropy of the upper critical fields paralleland perpendicular to the conducting layers but also the in-plane anisotropy wereinvestigated by resistance measurements under a pressure of 2.8 kbar.

For doing these experiments a dilution refrigerator system with a pressurecell and a vector magnet was established. The new setup allows also rotationof the vector magnet against the dilution refrigerator. Therefore a real two-axisrotation of the magnetic field versus the sample was possible.

Chapter 2

Theoretical background

Superconductivity emerges from a macroscopical occupation of a quantum me-chanical ground state in metals. Vanishing resistance below a certain (“critical”)temperature is the most prominent phenomenon of properties of a material inthe superconducting state. It was also the property that lead to the discoveryof superconductivity (H. Kamerlingh-Onnes, 1911). In the following chapterthe theory used in this diploma thesis shall be sketched. For a more detaileddescription see, for example, references [13] [14] [15].

2.1 Superconductivity in general

2.1.1 The Meißner-Ochsenfeld Effect

The assumption that the only property of a superconductor is its vanishingresistance and, thus, it is an ideal conductor leads to a contradiction from thepoint of view of thermodynamics:

An ideal conductor is cooled below its critical temperature, then an externalmagnetic field is applied (“zero-field cooling”). According to Lenz’s law and theinfinite conductivity of an ideal conductor, a current is induced. This currentcauses a magnetic field directed opposite to the external field. The two fieldscompensate and inside the volume of the ideal conductor there is no field.

In a second run, at first, at a temperature above the critical temperature, theexternal field is applied. As in this temperature region the resistance is finite,the induced currents vanish and the field can penetrate the sample. Now thetemperature is lowered (“field cooling”) and the field stays inside the sample,also in the state of infinite conductivity.

Thus, there are two different states reached, for the same values of mag-netic field and temperature. This contradicts the definition of an equilibriumthermodynamic state.

In 1933, W. Meißner and R. Ochsenfeld performed the experiment, sketchedabove. They obtained the result, that also in the field cooling case the sampleexpelled the magnetic field from its bulk. By this experiment the superconduct-ing state was proven to be a thermodynamical state.

However, the magnetic field is not expelled from the whole volume of thesuperconducting sample. It enters the sample from the surface and its strength

3

4 CHAPTER 2. THEORETICAL BACKGROUND

declines exponentially over a characteristic penetration depth λ according to

H (x) = H (0) e−xλ (2.1)

Here H (0) denotes the field strength at the surface, x is the distance inside thesample measured from the surface.

Above a certain critical field strength Hcth (and below the critical temper-ature) superconductivity is suppressed and the field enters the sample. At thecritical field strength, the increase of the superconductor’s total energy by ex-pelling the field becomes equal to the difference between the energies of thezero-field superconducting state and the normal state. So above the criticalfield strength it is energetically more favourable for the superconductor to letthe field penetrate its interior and go into the normal conducting state.

2.1.2 BCS Theory

The microscopic theory of superconductivity, developed by Bardeen, Cooperand Schrieffer (BCS Theory) in 1957, describes the superconducting state bythe coupling of electrons in pairs (“Cooper pairs”). These pairs consist of twoelectrons with antiparallel wave vectors of the same absolute value and an-tiparallel spins (k ↑,−k ↓). The attractive interaction is explained by mutualscattering of the electrons mediated by virtual phonons (q = 0). The Fermisurface in the lowest energy state at T = 0 is smeared, so that scattering ispossible. The smearing of the Fermi surface is similar (but not equal!) to thesmearing of the Fermi surface in metals for T ≈ Tc.

The width ∆k of the smearing of the Fermi surface is given by

∆k ∼ kF2∆0

EF(2.2)

where EF is the Fermi energy, kF is the Fermi wave number and ∆0 is

∆0 = 2~ωD exp(− 1

N (0)V

)(2.3)

where ωD is the Debye frequency of the superconductor, N (0) is the density ofstates at the Fermi level and V is the attractive potential between the electronsof one pair. ∆0 is the energy gap between the ground state (Cooper pair) andthe lowest state, that can be occupied by a quasi-particle (single electron). Thus,to break a Cooper pair, at least an amount of energy equal to 2∆0 is necessarywithin the BCS theory. ∆0 is related to the critical temperature Tc:

∆0 = 1.76kBTc (2.4)

Near Tc the gap varies with temperature according to

∆ ∝ (Tc − T )12 (2.5)

Cooper pairs can only exist in the region ∆k at the Fermi surface. Makinguse of the uncertainty principle ∆x∆k ∼ 1, the characteristic size of a Cooperpair can be estimated:

∆x ∼ 12∆okF

~2k2F

2m=~vF

4∆0(2.6)

2.1. SUPERCONDUCTIVITY IN GENERAL 5

where vF is the Fermi velocity. This size of a Cooper pair is called coherencelength ξ. Exact calculations for the ground state (T = 0) lead to

ξ0 = 0.18~vF

kBTc(2.7)

2.1.3 Ginzburg-Landau Theory

The Ginzburg-Landau (GL) Theory is based on the theory of second-order phasetransitions developed by L. D. Landau. The order parameter chosen, is the effec-tive “macroscopic” wave function Ψ (r) of the superconducting electrons, thatis finite at T < Tc and zero at T ≥ Tc. It is normalised in such a way that|Ψ (r)|2 = 1

2ns, where ns is the density of superconducting electrons. The GLtheory is only valid near the critical temperature Tc, but in this range it is easierto handle, than the above mentioned BCS theory, from which it may be derived[16]. It is capable of treating variations of the Cooper pair density in space.

The equation for the Gibbs free energy of a superconductor in a magneticfield is minimised using a variational method for the order parameter Ψ (r). Asresult the first Ginzburg-Landau equation

αΨ + βΨ |Ψ|2 +1

4m

(i~∇+

2e

cA

)2

Ψ = 0 (2.8)

and its boundary condition(

i~∇Ψ +2e

cAΨ

)· n = 0 (2.9)

are obtained. Here m is the electron mass, e the charge of an electron, c thevelocity of light, A is the vector potential corresponding to the magnetic field:H = ∇×A, and n is the unit vector normal to the surface of the superconduc-tor. α < 0 and β are phenomenological expansion coefficients, characteristic forthe material.

To determine A (r), the Gibbs free energy is minimised with respect to A.From this the second Ginzburg-Landau equation is given:

js = − i~e2m

(Ψ∗∇Ψ−Ψ∇Ψ∗)− 2e2

mc|Ψ|2 A (2.10)

js is the current density in the superconductor.

The Ginzburg-Landau equations can be reduced to the form

ξ2

(i∇+

2π

Φ0A

)2

ψ − ψ + ψ |ψ|2 = 0 (2.11)

∇×∇×A = −iΦ0

4πλ2(ψ∗∇ψ − ψ∇ψ∗)− |ψ|2

λ2A (2.12)

with

ξ2 =~2

4m |α| (2.13)


λ2 =mc2

4πnse2=

mc2β

8πe2 |α| (2.14)

the flux quantum Φ0 =hc

2e

and the dimensionless wave function

ψ (r) =Ψ (r)Ψ0

(2.15)

where Ψ0 is the equilibrium (zero field) order parameter:

Ψ20 =

|α|β

(2.16)

From (2.11) with H = 0 it can be derived, that ξ is the characteristic lengthover that the order parameter can vary ∝ exp

(√2xξ

). ξ is the coherence length.

This quantity was already introduced in the previous section on BCS theory asthe size of a Cooper pair.

λ characterises the penetration depth, over that the magnetic field decaysexponentially inside the superconductor, see equation (2.1).

From λ and ξ another quantity, the GL parameter, can be defined:

κ =λ

ξ(2.17)

From this parameter the influence of a magnetic field on the order parameter(density of Cooper pairs) can be evaluated (see next section).

2.1.4 Parameters influencing superconductivity

Temperature

Superconductivity occurs only below a certain critical temperature Tc. Thiscan be explained with the temperature dependence of the energy gap ∆, thatwas already mentioned in Section 2.1.2. A pair can be broken by supplyingthe energy kBT ∼ ∆. The electrons of a broken pair occupy two states k1 andk2 in reciprocal space. These two states can now not be used any longer byscattering electrons of an intact Cooper pair. Thus as scattering becomes lesseffective, the lowering of the energy per pair becomes less. This again means asmaller energy gap ∆. So, by increasing temperature, more and more Cooperpairs become broken, and breaking them becomes more easy with increasingtemperature until, at Tc, all of the pairs are broken and no superconductivityis present any longer.

Magnetic field

Applying a magnetic field causes the superconductor to expel this field from itsinterior. This is accomplished by supercurrents producing a magnetic field ofsame strength directed opposite to the external field. The expulsion of the fieldleads to an increase of the energy of the superconducting sample. When this

2.1. SUPERCONDUCTIVITY IN GENERAL 7

energy becomes larger than the energy of the sample in the normal state, thesuperconductivity is suppressed.

Depending on the Ginzburg-Landau parameter κ = λξ (see Section 2.1.3)

two kinds of behaviour are known:

For κ < 1√2

there exist Type I superconductors.For κ > 1√

2there exist Type II superconductors.

They are different in the way, how the magnetic field penetrates into them.

For Type I superconductors, the magnetic field penetrates only within thewidth of λ into the surface of the superconductor. Up to a critical field strengthof Hc = Hcth the remaining interior volume of the superconductor is free ofa magnetic field. When exceeding Hc, the superconductor becomes normalconducting and the field penetrates the whole sample, having the same strengthinside and outside.

For Type II superconductors, the magnetic field is expelled up to a valuecalled lower critical field Hc1 in the same way as the Type I shows up to Hc.Exceeding Hc1 the magnetic field enters the volume of the sample in the shapeof line-like vortices. These vortices consist of a normal conducting core flownaround by supercurrents that shield the remaining volume of the sample fromthe magnetic field through the vortex. On further increasing the magnetic fieldmore and more of those vortices enter the sample, arranging in a triangular pat-tern. Reaching the upper critical field Hc2, the packing of the vortices becomesso tight, that the normal cores of the vortices overlap and no superconductingareas can exist any longer. Thus the superconductivity has vanished and thefield strength has the same value inside and outside the superconductor.

To explain this different behaviour of Type I and Type II superconductorsthe boundary region between the field penetrated part and the part free offield are examined. The density of Cooper pairs can vary only over the lengthξ. This means, within this ξ the maximum density is not reached and thusthe superconductor’s energy is increased compared to a region with maximumCooper pair density. Over the length λ the magnetic field can penetrate thesuperconductor and the energy of the superconductor is lowered, as in thisregion no field is expelled. These two effects balance, to minimise the totalenergy of the superconductor. As a result, for Type I superconductors (large ξ,small λ), it is more convenient to expel the field from their interior, while forType II superconductors (small ξ, large λ) a certain amount of magnetic fieldinside them is energetically more favourable.


Hc1 and Hc2 are related to Hcth by κ (for κ À 1):

Hc1 =1√2κ

Hcth ln (κ + 0.5) (2.18)

Hc2 =√

2κHcth (2.19)

For large values of κ the magnetic field penetrates the sample at a low Hc1 andsuperconductivity is present up to a high Hc2. If κ is small, the field is expelledup to a high Hc1, then enters the sample and superconductivity is suppressedat a low Hc2.

For example, the hard Type II superconductor niobium-44 wt.% titaniumhas κ = 24 and Hcth = 2.8 · 105 A/m. Calculating Hc1 and Hc2 from equations(2.18) and (2.19 results in Hc1 = 0.263·105 A/m and Hc2 = 95·105 A/m. Thesevalues fit the experimental ones of Hc1 = 0.112·105 A/m and Hc2 = 96·105 A/m.For Nb κ = 0.781 and Hcth = 1.59·105 A/m. The resulting Hc1 = 0.35·105 A/mand Hc2 = 1.75 · 105 A/m are in the range of the experimental values ofHc1 = 1.39 · 105 A/m and Hc2 = 3.23 · 105 A/m [17], [18], the discrepancyin the case of Nb can be attributed to κNb < 1.

So far we have described, how the magnetic field suppresses superconduc-tivity by orbital effects. This means the suppression originated from the orbitalmovement of charge carriers. Pair breaking can also happen due to Pauli para-magnetism of charge carriers.

In a magnetic field, the electrons states with opposite spin orientations areno longer degenerate due to Zeeman splitting. The energy of an electron withspin parallel (antiparallel) to the field decreases (increases) by µBµ0H.

Cooper pairs consist of a spin up and a spin down electron. In an externalmagnetic field, this configuration stays intact, until it is energetically more con-venient for the material, to allow the individual electron to align parallel to themagnetic field, thus to break Cooper pairs and cause the superconductivity tovanish.

The magnetic field strength necessary to break a Cooper pair and the criticaltemperature of a superconductor are related. This was shown by B. S. Chan-drasekhar [19] and A. M. Clogston [20], involving the free energy of the super-conductor, its energy gap from BCS theory and the Zeeman splitting of energylevels of an electron in an applied magnetic field. The result obtained is therelation

µ0Hp =∆0

µB

√2

which, after inserting ∆0 = 1.76kBTc, becomes

µ0Hp = 1.85Tc (2.20)

This relation is known as the “Chandrasekhar-Clogston paramagnetic limit”.Tc has to enter equation (2.20) in kelvin, then µ0Hp is obtained in tesla.

This paramagnetic limit as a pair breaking effect can only be seen, if su-perconductivity has not already been destroyed by orbital effects caused bythe applied magnetic field exceeding the critical field. Thus, it can be observedmainly in low dimensional superconductors, where orbital effects are suppressedor weakened.

2.2. SUPERCONDUCTIVITY IN LAYERED COMPOUNDS 9

In the case of paramagnetically limited layered superconductors at low tem-peratures Bulaevskii [21] presents an expression for the temperature dependenceof the upper critical field for the magnetic field aligned parallel to the conductinglayers:

Hc2 (‖) =2πTc

√τ√

7ζ (3)γ, τ =

Tc − T

Tc, γ =

12gµB (2.21)

Thus the dependenceHc2 (‖) ∝

√Tc (Tc − T ) (2.22)

holds.

Current

Currents in superconductors are carried by the Cooper pairs. These, so-calledsupercurrents only flow on the outer border of a superconducting region withina width of the penetration depth λ.

In a Type I superconductor a current density js leads to the destruction ofsuperconductivity, when the kinetic energy of the paired electrons reaches thecondensation energy

12nsmv2

s =2π

c2λ2j2

s =H2

c

8π(2.23)

leading to critical current density of

jc =cHc

4πλ(2.24)

Here equation (2.14) was used. This result is an estimate, that does not considerthe decrease of ns on increasing current. A more exact expression from the GLtheory, valid near Tc, is

jc =cHc (T )

3√

6πλ (T )∝

(1− T

Tc

) 32

(2.25)

In Type II superconductors above Hc1 an additional mechanism can destroysuperconductivity: flux motion. A supercurrent flowing through a sample in amagnetic field causes the Lorentz force to act on the vortices inside which themagnetic field penetrates the sample. These vortices are pinned to defects insidethe sample. If the Lorentz force exceeds the pinning force, the vortices movethrough the superconductor, dissipating energy and causing a finite resistanceof the sample. An ideal Type II superconductor without pinning centers wouldshow this dissipative behaviour for arbitrarily small currents. Thus, an idealType II superconductor would not be suitable for constructing superconductingmagnets or wires.

2.2 Superconductivity in layered compounds

2.2.1 Highly anisotropic superconductors

Members of the group of highly anisotropic superconductors are, for example,high temperature superconductors, organic superconductors, and intercalated


compounds. All of those mentioned here, can be described by highly conductinglayers, separated by layers with low conductivity (isolating).

In some of these materials the coherence length ξ (Cooper pair size) is largerthan the interlayer distance d.

ξ > d

So there is still a “connection” between the different conducting layers.The layered character reduces the superconducting currents perpendicular

to these layers, compared to the current flow possible inside the conducting lay-ers. If a magnetic field is applied parallel to the conducting layers, the inducedsuperconducting currents are always lower than in the case of the field per-pendicular to the layers. This means depending on the field orientation, fieldsof different strength are necessary to create a current density that causes pairbreaking. Thus, in highly anisotropic superconductors the critical fields for afield parallel to the conducting layers is larger than the critical field for a fieldapplied perpendicular to the conducting layers.

W. E. Lawrence and S. Doniach have shown, that such superconductorscan still be described by the GL theory for “classical” three-dimensional (3d)superconductivity by introducing an anisotropic mass [22]. There they givefor the upper critical field perpendicular Hc2 (⊥) and parallel Hc2 (‖) to theconducting layers the relations

Hc2 (⊥) =Φ0

2πξ2‖

(2.26)

Hc2 (‖) =Φ0

2πξ‖ξ⊥(2.27)

These can be combined to the well known ratio

Hc2 (‖)Hc2 (⊥)

=ξ‖ξ⊥

(2.28)

ξ‖ denotes the coherence within a layer, ξ⊥ the coherence perpendicular to thelayers.

The angular behavior of the upper critical field Hc2 in the case of an highlyanisotropic superconductor can be described [23] by

Hc2 (θ, T ) =Φ0

2πξ2‖ (T )

√sin2 θ + α2 cos2 θ

(2.29)

with λ⊥ =λ‖α

, ξ⊥ = ξ‖α and α =(

m‖m⊥

) 12

where λ⊥,‖ (T ) = λ⊥,‖

(1− T

Tc

)− 12

and ξ⊥,‖ = ξ⊥,‖

(1− T

Tc

)− 12

θ is the angle measured from the conducting layers of the superconductor.

2.2. SUPERCONDUCTIVITY IN LAYERED COMPOUNDS 11

2.2.2 Quasi two-dimensional superconductors

When the anisotropy of the superconductor increases further, the conductinglayers are not “connected” any longer via Cooper pairs. In this regime holdsthe relation

ξ <d√2

Superconductivity only establishes within the conducting layers of the material.Perpendicular to these superconducting layers a three-dimensional ordering isrealised only by Josephson coupling (superconducting layer/ insulating layer/superconducting layer). Thus the current perpendicular to the conducting layersis limited to Josephson currents. For a field parallel to the conducting layersthis is why the paramagnetic limit becomes more important in a quasi two-dimensional superconductor than in a three-dimensional superconductor. Asthe supercurrents perpendicular to the layers are strongly reduced, the orbitaleffects are suppressed and so the paramagnetic limit becomes the main pairbreaking mechanism.

For the quasi two-dimensional (q2d) regime (ξ < d√2) there exists a model

for dirty superconductors by R. A. Klemm, A. Luther and M. R. Beasley (KLB)[24]. In dirty superconductors, the mean free path of a Cooper pair l is muchsmaller than the coherence length ξ0 (l ¿ ξ0). Bulaevskii presents a model forthe clean limit (l À ξ0) yielding the following expression for the upper criticalfield [21] [25]:

H2c2 (θ)

H2c2 (⊥)

=1 + 2x2 − (

1 + 4x2) 1

2

2x4 sin2 θ(2.30)

where x =Hc2 (⊥)

Hc2 (‖) sin θand Hc2 (‖) =

2πTc

(1− T

Tc

) 12

µ0

√7ζ (3)

Here Hc2 (⊥) and Hc2 (‖) denote a orientation of the magnetic field perpendic-ular and parallel to the conducting layers, respectively.

In purely two-dimensional (2d) superconductors, no coupling between theadjacent conducting layers is given any longer. The superconductivity is limitedto the conducting layers. This is expressed by the relation between coherencelength ξ and layer distance d.

ξ ¿ d

If the layer thickness s is much smaller than the coherence length s ¿ ξ, |Ψ|can be assumed to be constant within the layer. The superconductor can bedescribed as a thin film. For this case, M. Tinkham gives a relation for thedependence of the critical field on the orientation of an applied field to the film[15]: ∣∣∣∣

Hc (θ) sin θ

Hc⊥

∣∣∣∣ +(

Hc (θ) cos θ

Hc‖

)2

= 1 (2.31)

Here Hc⊥ and Hc‖ denote the critical field perpendicular and parallel to theconducting layers, respectively.


For the Hc2 parallel to the conducting layers M. Tinkham [15] gives

Hc2,‖ =2√

6Hcthλ

d(2.32)

where d is the thickness of the film. Using equations (2.14) and (2.13) this canbe rewritten in the form

Hc2,‖ =√

3Φ0

πdξ‖ (0)

√1− T

Tc(2.33)

In [26] this was applied to fit data obtained from measurements on q2d layeredcomposites.

Chapter 3

The organic metalα-(BEDT-TTF)2KHg(SCN)4

The probed material α-(BEDT-TTF)2KHg(SCN)4 is a member of the group ofcharge-transfer-salts following the scheme (BEDT-TTF)aXb. “(BEDT-TTF)”is the abbreviation for bis(ethylenedithio)-tetrathiafulvalene, “X” represents amonovalent anion. The Greek letter denotes the type of arrangement of themolecules.

3.1 Synthesis

α-(BEDT-TTF)2KHg(SCN)4-crystals are grown by electrochemical methods.In a mixture of (1,1,2)trichlorethane and methanol the salts KSCN, Hg(SCN)2and BEDT-TTF are solved. Putting two Pt-electrodes into the solution andapplying a constant current leads to an electrochemical oxidation of BEDT-TTF with the solved salts acting as electrolytes. At a constant temperature of20 C and a very low current density of 1.2 µA

cm2 the crystals grow and can beharvested from the anode after a few weeks. The crystals reach dimensions ofup to 1× 1× 0.2 mm3, typical values are 0.7× 0.2× 0.1 mm3.

Figure 3.1: Schematic of the molecule BEDT-TTF, the radical cation in thecharge transfer salt α-(BEDT-TTF)2KHg(SCN)4 (taken from [27])

13

14 CHAPTER 3. THE ORGANIC METAL α-(BEDT-TTF)2KHG(SCN)4

3.2 Crystal structure

A schematic of the molecule BEDT-TTF is shown in Figure 3.1. The crystalstructure of the charge transfer salt α-(BEDT-TTF)2KHg(SCN)4 is depictedin figures 3.2 and 3.3. In Figure 3.2 the view is aligned with the c-axis onto

Figure 3.2: Structure of α-(BEDT-TTF)2KHg(SCN)4 projected along the c-direction (from [28]).

the ab-plane. Standing almost upright, the organic molecules form layers in theac-plane which are separated by layers consisting of anions. The arrangement ofthe BEDT-TTF part in the ac-plane is shown in Figure 3.3. The “herringbone/fish bone” pattern exhibited by the cation radicals, is denoted by the Greekletter “α”.

The BEDT-TTF part donates electrons to the polymerised anion part(“charge transfer salts”). By this mechanism the two parts are bound together.Within the sheets of the BEDT-TTF part the binding is established by themolecular π-orbitals. Within the sheets of the KHg(SCN)4 part bridges betweenK+ and Hg2+ cations are responsible for the binding. The overall arrangementof the ac-layers alternating along the b-axis is sketched in Figure 3.4.

The crystal structure is triclinic with the dimensionsa = 10.082 A, b = 20.565 A, c = 9.973 A and the angles α = 103.7, β = 90.91,γ = 93.06, resulting in a total volume for the cell of 1997 A3 [28].

3.2. CRYSTAL STRUCTURE 15

Figure 3.3: Perspective view onto a layer of α-(BEDT-TTF)2KHg(SCN)4 alongthe b-axis. Transfer integrals along the stack direction and in between thedifferent stacks are denoted by ci and pi, respectively. [28]

Figure 3.4: Schematic of the layered structure of α-(BEDT-TTF)2KHg(SCN)4resulting in the anisotropic conductivity


3.3 Band structure and electronic anisotropy

The band structure for α-(BEDT-TTF)2KHg(SCN)4 at 100 K, calculated usinga molecular orbital method called extended Huckel tight-binding theory [29], isdepicted in Figure 3.5.

Figure 3.5: Band structure obtained from extended Huckel tight-binding theoryfor α-(BEDT-TTF)2KHg(SCN)4 at T = 100 K (from [29]).

The charge transfer from the BEDT-TTF layer to the anion layer amountsto one electron e− per two BEDT-TTF molecules. As there are four BEDT-TTF molecules in one unit cell there also exist two holes h+ in one cell. Thetwo (energetically) uppermost bands overlap each other and cross the Fermienergy, resulting in a metallic character of α-(BEDT-TTF)2KHg(SCN)4. Thefinite conductivity perpendicular to the layers is caused by a small overlap ofthe molecular orbitals of neighbouring conducting layers, allowing an electronexchange between these conducting layers.

While α-(BEDT-TTF)2KHg(SCN)4 is a chemically quite complex system,its conducting system and, thus, its Fermi surface present themselves to berather simple. The in plane Fermi surface of α-(BEDT-TTF)2KHg(SCN)4 isshown in Figure 3.6. It is derived from the band structure shown in Figure 3.5.Along the b-axis the Fermi surface is slightly warped as a result of the finitedispersion between the layers perpendicular to the ac-plane. In the ac-plane theFermi surface consists of open sheets and closed cylindrical parts, from which(quasi-)one-dimensional and (quasi-)two-dimensional behavior of the electronicsystems evolves, respectively. The closed (quasi-)two-dimensional cylindricalparts lead to quantum oscillation effects in a magnetic field.

From quantum Shubnikov-de Haas and de Haas-van Alphen oscillations thearea of the cylindrical parts can be evaluated to ∼ 15.5% of the cross section ofthe Brillouin zone and a Fermi temperature of TF ≈ 300 K can be derived [27].

3.4. T-P PHASE DIAGRAM 17

Figure 3.6: 2d view of the Fermi surface calculated for α-(BEDT-TTF)2KHg(SCN)4 using the crystal parameters at T = 100 K (from [29]).

For the (quasi-)one-dimensional part of the Fermi surface, conducting chainscaused by the transfer integrals pi (see Figure 3.3) are responsible.

Due to the layered character of its crystal structure, the electronic propertiesof α-(BEDT-TTF)2KHg(SCN)4 are extremely anisotropic. At room tempera-ture the resistivity ratio

ρ⊥ρ‖

> 104 (3.1)

The ratio of the effective transfer integrals within and across the layers amountsto approximately 500 [30], which is one of the highest among the known or-ganic conductors. Nevertheless, the temperature dependence of the resistivityin both directions shows a metallic character upon cooling. The extremely highanisotropy leads to the fact, that even a low temperatures the interlayer trans-port can be coherent or incoherent, depending on the sample quality. This inparticular manifests in distinct features in the high field magnetoresistance asfound recently [30].

3.4 T-p phase diagram

In the family of organic metals α-(BEDT-TTF)2MHg(SCN)4 (M= NH4, K, Tl,Rb) two kinds of ground states appear. While the NH4-salt exhibits a super-conducting state below T = 1 K [27], the ground state of the other salts is ofdensity wave nature [31]. The transition temperatures are TK = 8 K, TTl = 9 K[32] and TRb = 12 K. After extensive investigations over the last fifteen, therenow is a general agreement that below this transition a charge density waveestablishes [7].

The charge density wave (CDW) is an instability of the electronic systemagainst nesting of the quasi-one-dimensional part of the Fermi surface (Figure3.6). Nesting of a Fermi surface is given if parts of the Fermi surface can be


translated by a single wave vector qc onto another part. In that case the elec-tronic system becomes unstable and leads to a redistribution of the charges. Thisphenomenon was first described theoretically by Peierls for a one-dimensional(1d) conductor. It can be observed in several 1d organic metals. The CDWstate in organic metals competes also with the spin-density-wave state, wherethe spin-density becomes modulated. The transition temperature to the CDWstate in α-(BEDT-TTF)2KHg(SCN)4 is the lowest known up to now and al-lowed for the first time to study the magnetic field-temperature phase diagramof a CDW state experimentally (see for instance [4], [9], [5]).

For the present compound the charge density wave state can be influencedby applying pressure as shown by D. Andres et al. [12]. The pressure changesthe overlap integrals of the electron orbitals and thus the Fermi surface, lead-ing to a change in the nesting conditions. Above a certain “critical pressure”Pc ≈ 2.5 kbar the charge density wave state is suppressed and the sample staysin the normal metallic state. In the pressure regime near the charge densitywave state the normal metallic state becomes unstable in magnetic field andtranforms in stepwise transition back to the CDW state.

At very low temperatures, D. Andres et al. discovered the superconductingtransition of α-(BEDT-TTF)2KHg(SCN)4.

Figure 3.7: Pressure-temperature phase diagram of α-(BEDT-TTF)2KHg(SCN)4 as investigated in [8], [10], [33] (taken from [33])

In Figure 3.7 the pressure-temperature phase diagram ofα-(BEDT-TTF)2KHg(SCN)4 is shown as investigated by D. Andres et al. [8],[10], [33]. Traces of superconductivity exist already at ambient pressure belowabout 0.3 K, but there is no bulk superconductivity observed. Starting fromp = 2.5 kbar a sharp transition into a superconducting state is observed with amaximum Tc slightly above 100 mK.On increasing pressure, Tc is reduced by −30 mK

kbar .With their setup [12], [33] D. Andres et al. could measure only the critical

fields perpendicular to the layers. The results for different pressures are given

3.4. T-P PHASE DIAGRAM 19

0 20 40 60 80 100 1200

2

4

6

8

10 HP, 2 kbar TP, 2 kbar HP, 2.5 kbar HP, 3 kbar HP, 3.5 kbar HP, 4 kbar

0H (m

T)

T (mK)

Figure 3.8: Dependence of Hc2 on temperature at different pressures (takenfrom [12]).

in Figure 3.8.

Existing in the vicinity of the charge density wave, the properties of thissuperconducting state are of special interest. In this region superconductivitywas predicted to be either of d-wave type [34], [35] or to be of triplet p-wavetype, where no paramagnetic limit exists [36], [37]. To check either of thesepossibilities, a study of the influence of the strength and orientation of a mag-netic field is expected to be very useful. On the other hand accordingly, it isinteresting to study how the normal state is reflected in the superconductingproperties. Probing this superconducting state is done in this diploma thesis:with the new setup, we studied the complete anisotropy of the superconductingstate for a pressure of p = 2.8 kbar very near to the CDW state.

Chapter 4

Experimental setup

This chapter describes the experimental setup of the experiment. During thisdiploma work a new experimental facility was built up, allowing two-axis rota-tion of a magnetic field with respect to the sample being at millikelvin temper-atures and under pressure. For cooling the samples a dilution refrigerator builtat the Walther-Meißner-Instut was used. The following parts already existedat the beginning of the work: pressure cell, low temperature part of a dilutionunit together with the gas handling system, support system of the dilution unit,superinsulated dewar and vector magnet.

The experimental task of this work consisted in:

• the construction of a separate mounting system of the vector magnet al-lowing mechanical rotation of the magnet system against the dilution unit

• assembly of the parts of the dilution refrigerator and leak testing

• the complete wiring of the setup including filter boxes

• writing a measuring program allowing the control of the vector magnetsystem and the temperature and data acquisition of all necessary param-eters.

4.1 Measurement setup - inside the dewar

In this section details on the insert incorporating a dilution refrigerator unitare presented. After that the vector magnet and its suspension device, allowingrotation of the magnet against the insert with respect to the insert axis, areintroduced. By that, arbitrary orientation of the samples with respect to themagnetic field is possible. Finally, attention is turned towards the pressure cellcontaining the samples and manganin pressure sensor and the arrangement ofthe measurement leads.

21

22 CHAPTER 4. EXPERIMENTAL SETUP

Figure 4.1: Phase diagram for mixtures of liquid 3He and 4He. X denotes theconcentration of 3He in 4He. I, II and I+II specify the regions of normal phase,superfluid phase, and phase separation, respectively (taken from [38]).

4.1.1 Dilution refrigerator

Basic principle

A dilution refrigerator makes use of the solvability of 3He (fermion) in 4He(boson), that is still finite at T=0 K (see Figure 4.1) [39], [38], [40]. It hasbeen experimentally shown, that the heat capacitance C (at constant volume)of 3He atoms in an environment mainly consisting of 3He (concentrated phase)is smaller than that of 3He atoms in an environment mainly consisting of 4Heatoms (diluted phase) [41].

Ccp < Cdp (4.1)

This can be explained by the dependence of C on the Fermi temperature TF

[42], [43]:

C ∝ 1TF

(4.2)

The Fermi temperatures of 3He in concentrated phase and diluted phase are[44]:

TF,cp ≈ 4 K

TF,dp ≈ 0.38 K

The intrinsic energy U of He can be written

U = Q =

T∫

0

Cx dT = γxT 2 (4.3)

4.1. MEASUREMENT SETUP - INSIDE THE DEWAR 23

At the transition from the diluted to the concentrated phase the following ex-pression holds:

U ≈ F = constant (4.4)

F represents the free energy.From equations (4.1) to (4.4) follows a decrease of temperature due to the

dilution process. In a dilution refrigerator this process happens in the mixingchamber leading to a temperature in the millikelvin range.

For a continuous operation of the dilution refrigerator a continuous transi-tion of 3He into the diluted phase has to be maintained. Because of the finiteamount of 3He solvable in 4He, already solved 3He has to be removed fromthe He-mixture. This is done at the distillation chamber (“still”, T=0.7 K) bypumping on the mixture. Due to its higher vapor pressure compared to 4He,essentially only 3He evaporates. The pumped 3He is being “recycled” and sentback into the dilution refrigerator, being precooled by several heat exchangers.

To start a dilution refrigerator, there are two possibilities.The “classical” one makes use of a 1 K pot (pumped 4He bath) to precool the3He. This 1 K pot is installed before the expansion nozzle inside the cryostat.By this precooling, the 3He-4He-mixture is liquified.

The second method uses a compressor installed outside the cryostat andprecooling of the 3He-4He-mixture to 4.2 K. The pressure increase makes the3He-4He-mixture cross the inversion curve on the p-T -phase diagram and thusliquefaction by the Joule-Thomson process is possible. For continuous operationthis method needs a further heat exchanger, precooling the incoming gas in thepumping line just above the still.

The two methods of precooling have also an influence on the still temper-ature. In the classical setup, the 1 K pot intercepts a part of the heat loadbefore it reaches the still. For increasing the evaporation rate at the still andthus the cooling power of the dilution fridge heating of the still is necessary.In the compressor setup, the heat load on the still is higher, so that less addi-tional heating on the still is necessary. In the setup used in this diploma workadditional heating of the still was not necessary.

The dilution refrigerator installed in this setup

For our experiment an existing dilution refrigerator unit of K. Neumaier(Walther-Meißner-Institut) was used. Due to leak problems several parts hadto be renewed before putting it into operation. This dilution unit makes useof the “compressor method” for starting the dilution process. Figures 4.2 and4.3 show a sketch and a photograph of the refrigerator. In the text, labels inbrackets refer to the labels in Figure 4.2.

From the gas handling system the 3He-4He-mixture runs through a tubeand a spiral-shaped heat exchanger (1) embedded in the liquid 4He. In this way(pre-)cooling to 4.2 K is being achieved. The tube for the incoming gas passesthe vacuum pot, to the main pumping line, where the so called Joule-Thomsonheat exchanger (2) is placed. This heat exchanger is surrounded by a radiationshield, to protect it against the 4 K radiation from the Helium bath. TheJoule-Thomson expansion is being accomplished at a flow impedance (nozzle)


Figure 4.2: Crossectionshowing dilution unit andsamplespace.

Figure 4.3: Photograph showing di-lution unit and samplespace. Radia-tion shield and vacuum pot are not in-stalled.


(A) inserted in the capillary connecting the heat exchanger (2) with the heatexchanger (3) that is thermally coupled to the distillation chamber. A secondflow impedance (B) maintaining the pressure difference necessary for the Joule-Thomson process sits inside the capillary, that connects the heat exchanger (3)to the counter flow heat exchanger (4). A schematic of this heat exchanger (4)is depicted in Figure 4.4.

Figure 4.4: Schematic of the construction of the continuous heat exchangerconnecting distillation chamber and mixing chamber. The outer tube containsthe 3He-4He mixture with 3He (cold) moving towards the distillation chamber,in the inner tube condensed 3He (warm) flows to the mixing chamber.

Condensed (concentrated) 3He flows through the inner spiral of the counterflow heat exchanger (4) to the mixing chamber, while diluted 3He makes its wayfrom the mixing chamber to the distillation chamber inside the tube surroundingthis spiral. The two chambers are mechanically connected via Degussit1 tubes(C) with low heat conductance, to avoid a thermal short circuit. In the mixingchamber the 3He can dilute into the 4He and thus cools the chamber.

To improve the thermal coupling between the mixture and the mixing cham-ber, a labyrinth of sintered copper (D) to enlarge the contact surface is installedto the bottom of the mixing chamber. As 3He has a lower density than 4He,the concentrated 3He phase is floating on top of the 3He-4He mixture. Diluted3He moves via the counterflow heat exchanger (4) to the still, because of thelower concentration of 3He there. This gradient in concentration is caused bypumping on the mixture, resulting in evaporation of the 3He from the mixture.The evaporated 3He passes through the pumping line back to the reservoir.

The mixing chamber is made out of copper, the still and suspension partsfor the dilution refrigerator are made of brass. The connections between thedifferent parts of the assembly are either tightened by a sealing of indium wire(screw joints) or soft soldered.

During the experiment the temperature was measured at the importantpoints: still, mixing chamber, pressure cell. The base temperature of the refrig-erator was slightly below 19 mK, the base point of the thermometer calibration.

During the first experiments no heater was attached to the mixing chamber.The cooling power of the unit therefore could not be determined. Heating ofthe pressure cell to 100 mK could be accomplished by 0.3 µW. During that pro-cedure, the temperature of the mixing chamber stayed below 50 mK. Accordingto this, the mounting of the pressure cell shows a considerable heat resistance.

1Degussit: sintered Al2O3 from Friatec AG, Mannheim


4.1.2 Insert

The core of the insert

The insert used in this experimental setup is depicted in Figure 4.5.

Figure 4.5: Insert used in thisexperimental setup. The vac-uum pot is not installed, sothe sample space is visible

The top flange is made of brass and has adiameter of 150 mm. It carries inlets and out-lets for the measuring leads, the vacuum pot’sevacuation line, the helium exhaust line, thecondensing and pumping line for driving the di-lution unit and orifices for refilling Helium andmeasuring the Helium level.

Below the flange, there are four baffles madeof copper, intended to shield radiation heatcoming from the top down to the vacuum pot.

Three sets of twelve measurement leads eachrun through the helium bath down to the lowerflange (diameter 90 mm), where they enter thevacuum space via feed-throughs tightened byStycast 2850FT2. The wires are made of bronze(diameter 0.12 mm) and arranged in twistedpairs. Bronze was chosen because of its lowheat conductivity at a moderate specific electri-cal resistance. Each set of wires is designated toone type of measurement: one set for measur-ing the samples, one set for temperature mea-surements and control at the sample space, andone set for controlling the dilution unit (mea-suring temperature, still heater). The separatearrangement was chosen to avoid an interfer-ence between the measuring and the controllingsignals.

Mounting for the pressure cell

The mounting for the pressure cell is depictedin figures 4.2 and 4.3. It is made out of cop-per. Besides mechanically connecting the pres-sure cell to the mixing chamber, it is alsothe only heat conducting link between bothof them. The top plate fixed to the mix-ing chamber and the receptacle for the pres-sure cell are connected via three rods of length65 mm and diameter 5 mm. Into radial boresin these rods, bronze wires surrounded by De-gussit tubes were glued with Stycast 1266. Herethe wires from the dilution unit were connectedto those from the pressure cell by soldering.

2Stycast: epoxy based encapsulant produced by Emerson & Cuming


Over the mounted cell a radiation shield made out of copper (thickness ofwalls 0.5 mm) is placed, to protect the sample space from 4.2 K radiation emittedby the vacuum pot in contact with the liquid Helium bath.

Path of the measurement leads inside the cryostat

To reduce the heat load applied to the mixing chamber, all leads coming fromroom temperature were fixed at parts of the dilution unit so achieving lowertemperatures before connecting them to the pressure cell:

The measurement leads go from the boxes containing the filters (300 K)directly through the liquid Helium bath (4.2 K) to the flange of the vacuumpot. Inside the pot the leads are in thermal contact with the pumping line ofthe dilution unit and the still (0.7 K). The leads are wound several times aroundthe heat exchanger connecting the mixing chamber with the still ((4) in Figure4.2) and fixed at the bottom of the mixing chamber. From here the leads goto the bronze wires sticking out of the mounting’s copper rods. Afterwards thewires are wound around the rods of the mounting and then enter the pressurecell.

For further reduction of the heat load on the mixing chamber, the diameterof the measurement leads made of bronze (and thus their heat conductance)was reduced: down to the feed-throughs into vacuum space the diameter was0.12 mm. Inside vacuum pot, the diameter was reduced to 0.08 mm and fromthe distillation chamber the diameter of the leads was 0.04 mm. The leads werealways arranged in twisted pairs, to avoid corruption of the measured signal andto minimise the pickup of radio frequency noise.

Vacuum pot

Over the whole assembly below the lower flange the vacuum pot is put (seeFigure 4.2). This pot has a thicker upper part (outer diameter 80 mm, length200 mm) and a thinner lower part (outer diameter 38 mm, length 120 mm) thatfits into the magnet’s bore. The pot is made out of 1 mm thick brass. Also,here an indium seal in combination with a tongue and groove joint is used tokeep the inner part of the the pot sealed tight against the Helium.

4.1.3 Vector magnet

The superconducting vector magnet 3 used in this experimental setup consistsof a main coil having one additional correction coil at each end and a split coil,arranged perpendicular to each other. The main coil (vertical coil) has a boreof 38.5 mm and can give a field up to 1.5 T at 63 A (field constant 23.7 mT

A .The second coil (horizontal coil) is capable of fields ≤ 0.33 T (field constant5.625 mT

A ).The field strengths and their homogeneity were probed at room temper-

ature using “Wuntronic” Hall sensors. These Hall sensors were tested on ahomemade Helmholtz configuration, designed and built by R. Doll (Walther-Meißner-Institut). Figure 4.6 shows the field profile along the axis of the main

3Built at the Walther-Meißner-Institut by G. Eska


coil. In a volume of 5× 5× 5 mm3 at the intersection of the two field compo-nents the field of the vertical coil is determined to be constant within 1% of thefield value in the magnet center, while the field of the horizontal coil is constantwithin 3% of the maximum field value.

-10 0 10 20 30 40 50 60 700,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

Ver

tical

coi

l

Depth inside magnet [mm]

Fiel

d st

reng

th [m

T] fr

om

-0,2

0,0

0,2

0,4

0,6

0,8

1,0

1,2-10 0 10 20 30 40 50 60 70

Hor

izon

tal c

oil

Figure 4.6: Homogeneity of the field from the two coils of the vector magnet.The field strength is normalised to the value of the fields at the magnet’s center(dashed line at 60 mm)

The orientation of the applied magnetic field can be changed within theplane spanned by the two field directions from the coils of the vector magnet byaltering the field strength from the individual coils:

θ = arctan(

Hvertical

Hhorizontal

)(4.5)

The angular resolution, that can be reached in this setup is higher than 0.001

(see Section(4.2.5)).The total field strength at an angle θ is given by

Htotal =√

H2vertical + H2

horizontal (4.6)

A problem arising from the use of superconducting magnets are remanentcurrents induced in the superconducting wire of the coils. These currents leadto a remaining field from the magnet, even if no current is passed into the coilsfrom the power supplies. In a setup with a vector magnet, remanent currents notonly influence the effective field strength applied, they also shift the orientation


of the field. During field sweeps these currents may change in strength, so noconstant “offset” current for compensation can be applied to the coils. Thus,it was necessary to compensate a remanent field before a measurement, andafterwards check again, if the remanent field had changed. By this procedurethe change in remanent field can be estimated and thus a possible resultingerror. The remanent fields during the measurements in this diploma work wereup to 0.4 mT in the main coil and not taking them into account would haveseverely influenced some of the measurements. The remanent field in the smallercoil could not be resolved.

No full 360 degree sweeps at low temperatures could be done in this work, asone of the power supplies was only capable of a unipolar current output and thesecond bipolar power supply was lacking necessary current output to do fieldsweeps at a reasonable field strength (>20 mT).

4.1.4 Magnet mounting and rotation of the insert relativeto the magnet

The vector magnet is placed relatively to the insert using a mounting that isdepicted in Figure 4.7. Via this mounting the magnet is fixed to the dewar.

Current is supplied to the coils via two pairs of brass threaded rods of 5 mmdiameter, that have superconducting wires 4 (NbTi) soldered to them (basis forthe design see [45]).

The mounting is carried by three stainless steel threaded rods (diameter5 mm). To the lowest ring (1) in Figure 4.7 the magnet is screwed, the otherrings (2, 3) serve as stabilisation against twisting of the assembly and fix thecurrent leads in their place. Radiation shields (4) were installed at appropriatelevels, so that they close the gap between the outer edge of the baffles (5) fromthe insert and the cryostat.

The flange of the mounting of the vector magnet represents the basis for therotation of the insert around its longitudinal axis (see schematic in Figure 4.8):As presented in figures 4.9 and 4.10, the insert (b) is not fixed to the flange(a) by screws, but onto the insert’s flange two additional semicircle plates (c)are attached. These plates (c) are screwed (1) to the flange of the magnetmounting (b), forming a sandwich structure (a)-(b)-(c). Unless the screws (2)are tightened, the insert (b) can be rotated with respect to the cryostat (andthe magnet). When the desired position is reached, the screws (2) are tightenedand via the shells (3) the insert is pressed onto flange (a) and thereby fixed.An arrow mounted on the insert makes it possible to read the angle of rotationfrom a scale attached to flange (c). The angle can be resolved to an accuracyof <0.2.

4Vacryflux, Typ F-60-14 (0,6) TL; Manufacturer: Vacuumschmelze GmbH, Hanau (now:European Advanced Superconductors GmbH & Co KG, Hanau


Figure 4.7: Mounting for the vector magnet. (1) - (3) rings for stabilisationand carrying the magnet (4) radiation shields (5) radiation shields of the insert(“baffles”)


Figure 4.8: By rotating the insert around its longitudinal axis and the orienta-tion of the magnetic field, a spherical angle of 4π can be scanned.


Figure 4.9: Detailed view of the arrangement of the flanges of magnet mounting(a), insert (b) and plates (c).

Figure 4.10: Cross section showing the arrangement of the flanges.


4.1.5 Pressure cell

For applying pressure to the samples, a pressure cell was used. A cross sectionthrough the pressure cell is depicted in Figure 4.11 (See also [33]).

All metal parts are made of hardened Copper-Beryllium, except for thepiston consisting of non-magnetic tungsten carbide.

Figure 4.11: Schematic of the pressure cell used in this experimental setup(taken from [33])

Into the coaxial bore (“channel”) of the cylindrical cell (diameter 22 mm,length 48 mm) from one side the sample holder with a wiring feed-through isinserted, from the other side the piston for applying pressure. Onto each sideof the assembly a nut is put and tightened.

The sample holder contains a central bore, through that twelve copper wires(diameter 0.1 mm) are threaded and encapsulated by Stycast 2850FT to tightenthe feed-through. These wires are prolonged by soldering platinum wires ofdiameter 20 µm on them. The platinum wires serve as current and voltageleads for the samples and the manganin pressure gauge and are contacted tothose by carbon paste and solder, respectively.

Over the tip of the sample holder a teflon cup filled with silicon oil5 actingas pressure medium is put. The volume inside the teflon cup is about 0.1 cm3.Two washers at each end of the teflon cup prevent a floating of the cup, whenpressure is applied via the piston.

In order to put the samples under pressure, a hydraulic press is used. The“mushroom” is inserted into the pressure cell, that has been placed in a holder(not depicted), ensuring a firm stand. The force applied to the “mushroom” ismonitored by a commercial sensor (sensor “KAM” connected to “Anzeigeeinheit

5Silicon oil: “GKZh94”


AE 702” from A.S.T.6), the pressure at the samples is read from the resistancechange of the manganin coil (details see Section 4.2.3). When the desired pres-sure is reached, the nut behind the piston is tightened and the hydraulic pressreleased.

In this way a room temperature pressure of 5.7 kbar was applied in themeasurements during this diploma work, in earlier measurements up to 15 kbarwere generated in this cell.

4.2 Measurement setup - outside the dewar

This part of the chapter is about the instrumentation for probing the samplesand delivering information about the parameters influencing the samples’ state:magnetic fields, temperature and pressure (see schematic in Figure 4.12). Be-sides, it describes what has to be taken into particular consideration when doingexperiments at low temperatures.

A sketch of the overall arrangement is given in Figure 4.12.

Figure 4.12: Schematic of the arrangement used for the experiments in thisthesis

4.2.1 The technique of four point resistance measurement

To apply this technique, two pairs of measurement leads have to be contactedto the sample. Via two of these leads a current is applied, the other pair of leadsis used for measuring the voltage drop over the sample’s resistance. The voltagedrop over the lead resistance and the contact resistance are not measured. Thisis so, because the voltage leads in the four point measurement carry (nearly) nocurrent and for that reason act as equipotential connections from the sample tothe measurement device. In our case the lead resistance (∼ 70Ω) and the con-tact resistance at the sample (∼ 10− 100Ω) would falsify the measured sample

6A.S.T. Angewandte SYSTEM-TECHNIK GmbH Dresden

4.2. MEASUREMENT SETUP - OUTSIDE THE DEWAR 35

resistance, which is ≤ 20Ω in the normal state, in a two point measurement. Incontrast, for the RuOx thermometers the two point setup could be used, as thelead resistance could be neglected compared to a resistance of several 10 kΩ.One has to keep in mind that when doing experiments at low temperatures,every link between the sample space and the environment around leads to anadditional heat load and is an additional source of noise.

Figure 4.13: Schematic diagram of the measurement arrangement

The in- and outputs of the different devices were brought together in a sin-gle box made out of a copper plate. This box was connected to the cryostatusing cables consisting of six individually shielded twisted pairs surrounded byan additional shielding.

4.2.2 Measuring sample resistance

The resistances of the samples were measured using two lock-in amplifiers, theone being a “Stanford Research Systems DSP Lock-in amplifier model SR830(DSP = Digital Signal Processor)”, the other one a “Princeton Applied ResearchLock-in amplifier model 124A”. The oscillator frequencies were 333 Hz and27 Hz, respectively. Different frequencies were chosen, to avoid interferencewith each other. Time constants of the lock-in amplifiers were set to 3 seconds.The samples were measured using the described above mentioned four pointmethod.

Figure 4.2.2 shows the setup for the four point resistance measurement onone of the samples. For generation of the current through the sample the internaloscillator of the lock-in amplifier was utilized. In series with the sample therewere two resistors placed: one 10 Ω resistor for calibration purposes (Rcal) and


Figure 4.14: Schematic diagram of the four point measurement used for mea-suring the resistance of the sample

another resistor in the megaohm range (Rcurrent). The current through thesample is then determined by

I =Voscillator

Rsample + Rcal + Rcurrent(4.7)

Since,Rcal ∼ Rsample ¿ Rcurrent (4.8)

the current amplitude is mainly determined by Rcurrent, being largely insensitiveto changes of Rsample:

I ≈ Voscillator

Rcurrent(4.9)

The error due to this approximation can be estimated to be below one perthousand for ambient condition resistance of the samples (1 kΩ) and is an ad-ditional order of magnitude smaller for the samples at millikelvin temperatures(10-20 Ω).

The above mentioned calibration resistor was used to set the current ampli-tude and the phase of the detected voltage signal, that was measured using thedifferential input of the lock-in amplifiers. After the calibration the input wasswitched from the resistor to the sample.

Rcal/sample =Vdifferentialinput

I(4.10)

The described circuits were installed in aluminum diecast boxes to shield themfrom radiation.

4.2.3 Measuring pressure

For measuring the pressure at the samples, a manganin coil (Rm ≈ 6Ω is used.The resistance of the manganin coil changes linearly on increasing pressure:0.273%/kbar. Pressure and temperature dependencies of this coil were studiedby D. Andres in a gas pressure setup [33]. The currently applied pressure canbe obtained from

p [kbar] =R (p)−R (0)

R (0) · 2.73 · 10−3


Here R (0) is the resistance at ambient pressure and Rp the resistance at cur-rently applied pressure. By this procedure, the pressure at a few kilobar can beobtained to an accuracy of ±100 bar.

The measurement of the manganin coil’s resistance is also performed usingthe four probe technique. The DC-current source (I = 1 mA) was a “KnickDC-Strom-Calibrator J 152”, the voltage was measured by a “Keithley 2000”Multimeter. It was averaged over two measurements with opposite current toeliminate errors due to an offset or thermoelectrical power. The measurementfor controlling the pressure was done at 15 K. During the measurements on thesamples at dilution refrigerator temperatures, the manganin pressure gauge wasnot measured, to avoid a heating at the sample space (Pm = RI2 = 6µW).

4.2.4 Thermometry

Measurement of temperature

Temperatures below 1 K were measured by means of RuOx resistors. For readingthe resistance AC bridges of model “AVS 45” and “AVS 46” built by “Picow-att (RV-Elektroniikka Oy)” were used. The devices having only an analogueoutput were connected to multimeters “Keithley 195A” to equip them witha GPIB-interface. The room temperature resistances were 800Ω for the ther-mometers used for measuring the temperature at the sample space (pressurecell) and of 2000Ω for the one sitting at the mixing chamber of the dilutionrefrigerator.

For translating the resistance into temperature values, calibrations made atthe Walther-Meißner-Institut were used. These calibrations cover the tempera-ture range from 19 mK up to 2 K. At the base temperature the accuracy of thecalibration is about 5%, above 50 mK the calibration is accurate to ±1 mK.

A Cernox resistor (CX1030-SD) was used for determining temperaturesabove 1 K. It was read by a temperature controller “LakeShore model 340”,that internally calculated the resistance from a calibration curve.

The reference potential of the AVS resistance bridges there was connectedto common ground.

Temperature control at the sample space

For some of the measurements performed, the temperature of the sample spacehad to be stabilized, for others it had to be swept over a certain temperaturerange at a pre-determined rate. The dilution refrigerator provides permanentcooling of the sample space, so for stabilizing a temperature, additional preciselycontrolled heating had to be provided. This was done using a “Dale 100Ω”resistor that was installed next to the sample space. Over the range where theheater was used, its resistance can be assumed to be constant. A “Keithley 220current source” provided the heating current (P (I) = RI2).

The temperature regulation process is based on a computer control (seeSection 4.2.8), that besides stabilizing the temperature also makes temperaturesweeps at a given rate possible. It reads the resistance from the AVS 45 bridgeand from that calculates the current temperature. The rates of the temperaturesweeps were at maximum 2 mK

min to ensure, the temperature sensor (outside the


pressure cell) and the samples had the same temperature or there was at leastnot too high difference between them.

4.2.5 Superconducting magnet power supplies

For this setup a “Heinzinger TNSU 6-60” with a maximum output current of60 A at up to 6 V and an “Oxford IPS 120-10” delivering ±120 A at up to ±10 Vwere available. Lacking a computer controllable interface, the Heinzinger powersupply was controlled utilizing a “Hewlett Packard 3245A Universal source”with GPIB interface. The Oxford power supply was already equipped withthat kind of interface. At the lowest temperatures the use of the Oxford powersupply led to an increase of the base temperature by several mK. This additionalnoise could not be overcome by proper grounding. For that reason, for low-fieldmeasurements a power supply manufactured earlier at the Walther-Meißner-Institut replaced the Oxford one, as it did not disturb the measurements in thisway. Controlling was done via a second HP source on its analogue output. Themaximum output current of this device is ±1 A at up to ±10 V.

For safety reasons, in parallel to the magnet coils, resistors were installed toshunt them in case of a quench. To be sure, that there is no mismatch betweenthe set output current on the power supplies and the real current through thevector magnet coils two 1 mΩ resistors capable of high currents were inserted inseries with the current leads to the coils. The voltage drop over these resistorswas read by Keithley 195A multimeters and from that the actual current couldbe calculated. No mismatch to an accuracy of 10−4A (≤ 2µT )7 could be found.

The angular resolution of the vector magnet is given by the minimum stepsize by that the current through the magnet coils can be changed. With theavailable power supplies, the angular resolution of the vector magnet was limitedby the digital resolution of the Oxford power supply ∆I = 0.1 mA. For the ana-logue controlled ones no such resolution limit could be realised (∆I ≤ 10−6 A).The angular resolution can be obtained from equation (4.5): A horizontal fieldof Hhorizontal = 10 mT applied by the analogue supply can be changed at aminimum step of

∆θ = arctan23.7 mT

A · 0.0001 A

10 mT≈ 0.014

At 200 mT the angular resolution is ∆θ < 0.001.To have a proper tool to do field sweeps at arbitrary orientation and field

orientation sweeps at arbitrary field strength, a controlling software had to bedeveloped (see Section 4.2.8).

4.2.6 Grounding and stable line power

The cooling power of a dilution refrigerator is very low (several µW at T =100 mK). Therefore, unlike in other types of experiments, here only small ex-citations and signals can be used. Moreover, to prevent additional unintendedheating, special efforts have to be made to reduce any possible sources of noise.

7Earth magnetic field at laboratory site ≈ 48µT


To do so, in this setup a ground potential independent from the line groundwas chosen and a separate power line was used (schematic see Figure (4.2.6)).The former consists of a steel rod driven into the soil of the institute’s courtyardwith direct connection to the laboratory via a 25 mm2 copper wire (from hereon referred to as “common ground”). The separate power line was achieved asfollows: An electric motor fed from the normal power line drives a generatorthat provides a “clean” power line for several experimental setups. To excludeinterference between those setups an isolating transformer is installed in betweenthem and the generator’s output.

Due to the in line transformer and the parallel experimental setups only alimited amount of power can be drawn from the line. Because of this, the lesscrucial, that is less sensitive to noise or line fluctuations, devices were operatedfrom the normal line, powered via isolating transformers.

The grounds of the gas handling system, running on a normal power line,and the cryostat were separated by connecting the tubes of the condensing andpumping line by using isolating o-ring connectors and clamps made of plastic.

Figure 4.15: Schematic diagram of the grounding arrangement

4.2.7 Filtering

The following section will be concerned with another aid for gaining stableand reliable measurements at low temperatures, besides a good connection to asuitable ground.

Aside from the load coming from the noise emitted by the measurementdevices and other electrical devices in the laboratory, another source of heatingof the sample is radiation collected by the measuring leads from outside of thecryostat which is then guided directly to the sample space. The source of thisradio frequency noise can be transmitters for radio and television broadcasts,cell phones and other wireless means of communication.


One of the main sources for this radiation is the transmitter “Großsendean-lage Bayerischer Rundfunk” which is working at a frequency of 801 kHz withan output power of 100 kW and is located around Ismaning at a distance ofabout 6 km from the Walther-Meißner-Institut. The density of energy at themeasuring site due to this transmitter’s signal is about

w =Pemitted

4πd2=

100 kW4π(6000 m)2

= 220µWm2

So an antenna area of 1 cm2 gives 0.022 µW which is in the range of the coolingpower of the dilution refrigerator at base temperatures. Antenna area can be,for example, at soldered joint of the wires, as there the twisted pair wires areuntwisted.

To lower this radiation based heat load to the sample, low-pass filters wereinstalled on the system. There now follows a short overview on the workingprinciples of L-C elements which are the basis of the low-pass filters used in theexperimental setup.

Low pass filters

The frequency dependent impedances of an inductor L and a capacitor C are:

XL = iωL (4.11)

XC =−i

(ωC)(4.12)

Here ω = 2πf is the angular frequency of the signal and i denotes a π2 shift of

the phase of the current with respect to the voltage.

The L-C element shown in Figure 4.16 represents a frequency dependentpotential divider. For high frequencies most of the voltage drop is over the highimpedance inductor while that over the low impedance capacitor only accountsfor a small part. The output of the L-C element is in parallel to the capacitor,so only for low frequencies does the full amount of applied AC voltage “reach”the output, for high frequencies this amount decreases.

Figure 4.16: Schematic of a lowpassfilter

The voltage measured at the output divided by the input voltage gives theattenuation ratio a, usually expressed in decibels:

a[dB] = 20 · log(

Voutput

Vinput

)(4.13)


Using the equations (4.11) and (4.12) and applying Kirchhoff’s circuit laws andOhm’s law to the circuit of the low pass filter, the above mentioned attenuationratio can be expressed:

VC

V= 1− VL

V= 1− XL

VXtotal

V= 1− XL

XL + XC; (4.14)

Substituting this into equation (4.13) gives

a (f) [dB] = 20 log(

XL

XL + XC

)= 20 log

(1− 1

1− XC

XL

)=

= 20 log

(1− 1

1− 1(2πf)2LC

)(4.15)

Designing the low pass filters used in this experimental setup

When designing the low pass filters, the goal was to have as high attenuationas possible and at the same time not too high resistance at the frequency rangeat which the measurements were taken. Thus the sample (in parallel to thecapacitor, see Figure 4.19) was still reached by a high enough voltage to givereliable measured values. To obtain an attenuation of a = −40 dB at a frequencyof f = 300 kHz, a capacitor C = 1 nF, and an inductor L = 1000 µH were used.

The filters were tested using a spectrum analyser model “Rohde und SchwarzFSP7”. Several frequency spectra from zero up to 1 GHz were recorded todetermine, whether the low pass filters produce suitable results or not. As canbe seen from Figure 4.17, up to a frequency of 2 MHz the attenuation measuredusing the spectrum analyser coincides more or less with the filters’ calculatedbehavior.

For higher frequencies, as shown in Figure 4.18, there is a mismatch betweenthe calculation and the measured data. While in the curve from the simulationthe attenuation constantly becomes higher, the real attenuation drops, afterreaching its highest value of approximately -80 db and oscillates; the oscillationsare most probably caused by antenna effects of the filters. This behavior mayalso be due to the properties of the real electrical components used in the filterswhich were not taken into consideration in the calculations. The discrepancybetween real and ideal filter elements leads to a huge difference in the attenuationof both at higher frequencies.

Final setup of the low pass filters

The filters used in this experiment are connected in series with the measuringleads, which are twisted in pairs. So for every twisted pair of leads there aretwo filters. From the measurement device to the sample a signal passes thefirst filter, next the sample, then the second filter and reaches the measurementdevice again. Figure 4.19 schematically shows the total circuit for one of thepairs of measurement leads.

The filters were grouped by twelve and each set was placed on a separate holematrix board (Figure 4.20). To have the ability of (ex)changing or adjustingthe filters depending on the experiment and the equipment used, these matrixboards are fitted on connectors of Type ”Harting Gds A-D”.


0,00 0,25 0,50 0,75 1,00 1,25 1,50 1,75 2,00-120

-110

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

-120

-110

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Sim

ulated attenuation [db] of low pass filterM

easu

red

atte

nuat

ion

[db]

of l

ow p

ass

filte

r

Frequency [MHz]

Figure 4.17: Measured attenuation of input signal due to filtering in a frequencyrange from 0 to 2MHz

-400

-350

-300

-250

-200

-150

-100

-50

0

0 100 200 300 400 500 600 700 800 900 1000 1100-80

-70

-60

-50

-40

-30

-20

-10

0

Mea

sure

d at

tenu

atio

n [d

b] o

f low

pas

s fil

ter

Frequency [MHz]

Sim

ulated attenuation [db] of low pass filter

Figure 4.18: Measured attenuation of input signal due to filtering in a frequencyrange from 0 to 1GHz


Figure 4.19: Block diagram of the arrangement of filters and sample

Figure 4.20: Front and back view of the card carrying twelve low pass filters


One pair of the resulting pluggable cards finds its counterpart in a boardfitted with two female Harting connectors on which the wiring of the socket forthe connection to the measurement equipment ends and from which the wiresdown into the cryostat start. On Figure 4.21 of this board also the black doublerows of sockets (2) can be seen, where the in- and out-going pairs of wires can beindividually permuted, permitting low effort customisation of the wiring. Alsothe possibility exists to bypass the filters for individually selected pairs of wiresif needed. To optimise electrical contact at these linking points, sockets andcorresponding plugs are chosen to have gold plated surfaces.

The whole board is installed in an aluminum diecast box (see Figure 4.21)that is attached via a helium tight flange to the insert.

Figure 4.21: View into the assembled filter box, the filter cards (figure 4.20) arenot yet installed onto their receptacles (3), on the left hand side the feed-throughfor the wires to the helium bath can be seen (1). To the black connectors (2)the measurement leads are attached.

4.2.8 Software

The new established experimental setup meant the available means of acquiringdata could no longer be used. A new package of measurement software wascreated using the graphical programming language “LabVIEW 8.0” from Na-tional Instruments. The package includes three applications: one for controllingthe superconducting magnet power supplies, one for gathering data from themeasurement devices and one for controlling temperature.

Control for the superconducting magnet power supplies

This application was written to control the orientation of the applied magneticfield and its strength. The utilized power supplies are addressed simultaneously


(within a small time interval) by the computer. To maintain a given rate ofsweeping the field strength or field orientation, this has to happen at a constanttime interval.

The application supports the afore mentioned power supplies and can beused with any power supply that can be controlled via an analogue input bya voltage proportional to the output current. It can also be easily fed withthe field constants for different magnets, so it is not restricted to only onesetup. For each setup, maximum allowed field strength and sweep rates can bespecified, to protect the magnet from damage. The application comes also withan option to set “offset-”fields to compensate remanent currents in the magnet’scoils, separate from the other functions. Furthermore, the program features anautomatic routine, that processes a given list of operations one after anotherfor unattended operation of the vector magnet. This routine can also influencethe measurement program presented in the next section to automatically switchto the appropriate recording device and create a new file to store data. Valuesthat can be changed individually using the program are: total field strength,field orientation, vertical and horizontal field. Another option of the programis an automatical adjustment of a program cycles’ length to adapt the programto the computer’s speed.

The program does not measure the actual current applied to the magnet,but assumes, that all the commands it sends are executed. For example a bro-ken connection between the Hewlett Packard universal source and an analogueinput of a power supply would not be detected by the program as the universalsource would not return any error message, as it works properly. The programwould display a field, but no current would be passing through the magnet.

LabVIEW does not produce source code in a classical manner, but a kindof block diagram. This diagram is too large to be printed in this format at areasonable size. Therefore, the basic structure of the program is presented withthe following sequence:

• Get the current value from the system timer

• Check, whether user inputs are in conflict with the specified limits of themagnet, if so inform the user

• Calculate the current field components from angle and total field (not in“single component” mode)

• Calculate the field components to reach from angle and total field to reach

• Compare current values and values to reach

• Depending on the result decide:

– Desired values not reached: decrease/ increase values

– Desired values reached: do not change the values

• Translate the (new) values into the corresponding output currents

• Send the commands to the power supplies and wait for receipt from thesupplies.


• Get the current value of the system timer, subtract the earlier receivedvalue from it and subtract this value from the time interval set for oneprogram cycle. Depending on the difference decide

– Difference is negative. Increase the time interval of one programcycle* and start at the beginning of this sequence.

– Difference equal to or above zero. Decrease the time interval of oneprogram cycle*, wait for the difference in time and start at the be-ginning of this sequence

* (only if automatic correction of cycle length is set enabled)

Gathering measurement data

This application was written for recording data from the instruments. It fea-tures 11 input channels, of which one is the independent “X” channel while theremainder of the channels act as dependent “Y” channels.

One measurement device can be assigned to each channel where the commu-nication between device and computer is done using GPIB/ IEEE488 protocoland interface. There is also the possibility built in to receive data directly fromthe before mentioned magnet controlling application by internal data exchangewithin the LabVIEW runtime environment. The read values from the devicescan be processed by means of rescaling and incorporating an offset.

The data acquired can be stored in an ASCII file, following the schemeX-Value [Tabulator] Y1-Value [Tabulator] . . . Y10-Value. This file also containsinformation about the devices used and the labels assigned to the individualchannels by the user. Optionally a copy of the file without this “header”-information can be stored separately. The character of the decimal point sep-arator to be stored can be selected to be “comma” or “point” to maintaincompatibility with different software products for data evaluation.

Additionally up to two Y-channels can be monitored simultaneously on thefly in two graph windows on the graphical user interface. These graphs allowzooming the data and other basic analysing tools like a crosshair for probingpoints on the graph. While the storing to the file is triggered when the changeof the X-value exceeds a user-preset value, the data in the graphs is updatedevery program cycle. As in the previously described program, also here a rou-tine is implemented, to control the program’s cycle length and adapt it to thecomputer’s calculation power. For each channel there is a configuration option,that allows the current device’s behavior to be adjusted without having to pauseor stop the main measuring program.

The program is held in a modular structure, providing program internal in-terfaces for easily adding support for further devices in the future. Also furtherdata formatting procedures can be implemented following the same scheme.One example for the already integrated formatting procedures is the use of thefitting routine for a calibrated Ruthenium Oxide Thermometer that was takenover from D. Andres. From a given resistance it calculates the correspondingtemperature based on a calibration dataset.

The following sequence gives a simplified illustration of the actual program:

• Get current value from the system timer


• Check, whether the user has changed settings of the channels. Dependingon the result decide:

– No channel changed: No further actions, proceed with the program

– Changes occurred: reinitialise internal values of the affected chan-nel(s) to comply with new settings, then proceed with the program

• scan for active channels and for each of them:

– Read data from measurement device

– Format read data according to the scaling factor and the offset oraccording to fitting data applied

– Depending on device setup:

∗ Check for device specific parameters (for example, current range)and send command(s) to the device (for example, new measuringrange) if necessary

∗ Do calculations for timed processes (for example changing polar-ity of Knick DC-Strom-Calibrator) and execute them if requiredconditions are satisfied

– open the device configuration program (running in parallel to appli-cation) if initiated by the user.

• Check for the set interval for X-channel value being exceeded and storedata to file, force operating system to write this file to the disk.

• Add data points of chosen channels to graphs

• Process actions set by the user on the graph

• Get the current value of the system timer, subtract the earlier receivedvalue from it and subtract this value from the time interval set for oneprogram cycle. Depending on that difference, decide

– Difference is negative. Increase the time interval of one programcycle* and start at the beginning of this sequence.

– Difference equal to or above zero. Decrease the time interval of oneprogram cycle*, wait for the difference in time and start at the be-ginning of this sequence

* (only if automatic correction of cycle length is set enabled)

Controlling Temperature

To improve the quality of the data from temperature sweeps, the manuallyexecuted “floating setpoint” technique was replaced by a computer based ad-justment of heating power.

The resulting program gets the temperature values measured from one ofthe channels (user selectable) of the above mentioned application for gatheringdata. Based on this current temperature the two averaged temperatures arecalculated where the averaging interval is user determined. Additionally, tworates of temperature change are determined from the current temperature value


and the one stored from the program’s last cycle. For these rates the integrationtimes can be determined by the user. For keeping a certain temperature sweeprate, the two rates are compared to the stored rate, that was set by the user.If the calculated rates deviate from the stored rate, the power source of theheater is addressed to change its output to match the desired value. As soonas the given end temperature is reached, this temperature is stabilised. Thisstabilisation uses the same principles as the sweeping mode, but the deviationof the actual temperature from the set point temperature is used instead of thedeviation of the temperature’s changing rate. The programm works both withcurrent and voltage sources and can be extended to supply additional sources,as it is designed in a modular manner. Depending on the kind of source selected,the corresponding laws for calculation of the output power are applied. Thismeans the user only has to provide the kind of source, the heater’s resistanceand the maximum power to be applied to it.

During the experiment the temperature could be stabilised at ±50 µK, thesweep rate within a few hundredth of mK per minute.

Again the structure of the program is illustrated with the following sequence.

• Get temperature from the other application

• Determine the difference between the current temperature and the tem-perature to reach and determine the difference between the temperaturemeasured during the last program cycle and this one

• Determine time interval for every program cycle

• Based on the time interval for every program cycle and the input timeintervals for each averaging/ integrating process create a separate array.

• Append the new calculated temperature/ -differences to the array anddelete the oldest entry

• Calculate the averaged temperatures, and the averaged temperature chang-ing rates from the arrays based on the time intervals given by the user

• Compare those averaged values with the currently measured value

• Determine whether the desired temperature has already been reached orif the sweep shall go on.

• Based on this decision weight the deviation of temperature/ temperaturerate from the value to reach by the factor given by the user (proportionalregulation)

• Calculate the increase or decrease to be applied to the output power anddetermine the corresponding voltage/ current value for the heating source

• send the new value to the heating source

• Start at the beginning of this sequence

Chapter 5

Results and Discussion

First this section presents a few measurements done at ambient pressure tocharacterise the samples and test the equipment. The main experimental partof this diploma work is done at a pressure of 2.8 kbar, to suppress the densitywave state in α-(BEDT-TTF)2KHg(SCN)4 and to get an homogeneous super-conducting state with maximal Tc.

Figure 5.1: Definition of the angles used for describing the orientation of themagnetic field ~B. ~a, ~b∗, ~c represent the crystal axes of the sample, where ~b∗

is the component of ~b perpendicular to the ac-plane. ϕ is the azimuthal angle,with ϕ = 0 at ~B ‖ ~c, θ is the polar angle, with θ = 0 defined as ~B ‖ac-plane.The magnetic field is labeled ~B.

Two samples were measured simultaneously in this setup. As the sampleswere aligned by eye, the orientation of the samples with respect to the directionof the magnetic field is not exactly the same. The error in the alignment of thesamples is about ±3.

49

50 CHAPTER 5. RESULTS AND DISCUSSION

Due to its layered structure, α-(BEDT-TTF)2KHg(SCN)4 has a much higherupper critical field for a magnetic field applied parallel to the layers thanfor a field applied perpendicular. Thus, the superconductivity in α-(BEDT-TTF)2KHg(SCN)4 is very sensitive to a misalignment of the magnetic fieldparallel to its layers. Therefore, it is only possible to well align the field parallelto the conducting layers of one sample at a time. For this reason only the dataof one sample will be presented.

In Figure 5.1 the definition of the angles describing the orientation of themagnetic field with respect to the sample is depicted. The vectors ~a, ~b∗ and~c represent the crystal axes, where the component of ~b perpendicular to thehighly conducting ac-plane is labeled ~b∗. The magnetic field is denoted ~B. Theazimuthal angle ϕ gives the orientation of the field within the ac-plane, ϕ = 0

being defined as parallel to the ~c-axis. The polar angle θ gives the orientationof the magnetic field with respect to the ac-plane. For θ = 0 the magnetic fieldis aligned parallel to the ac-plane.

The current during the measurements was applied perpendicular to the con-ducting layers and the interlayer resistance was recorded.

5.1 Sample characterisation

5.1.1 Measurements at ambient pressure

Measurements in zero magnetic field

6 8 10 12 140

100

200

300

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500

600

Temperature [K]

Res

ista

nce

[]

TCDW

0 50 100 150 200 250 3000

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[]

Temperature [K]

ambient pressure

Figure 5.2: Temperature dependence of the sample resistance. The inset showsthe anomaly (hump) at 8.5 K caused by the transition into the charge densitywave state.

In Figure 5.2 the temperature dependence of the sample resistance at zero

5.1. SAMPLE CHARACTERISATION 51

magnetic field is presented. From room temperature down to about 175 Kthe resistance decreases slowly by 3% of its initial value. From 175 K on tolower temperatures the resistance decreases more strongly, leading to a negativecurvature in the plot. At about 8.5 K a hump can be seen. This is the wellknown anomaly at the transition to the charge density wave phase, first reportedby Sasaki et al. [46].

At low temperatures the resistance is mainly due to scattering at impuritiesor defects in the crystal structure. From the ratio of resistance at room tem-perature and low temperature an estimation of the sample’s purity is possible.Using data from the resistance measurement depicted in Figure 5.2 this ratiofor the present sample is

R (300 K)R (1.5 K)

=3000 Ω25 Ω

= 120

Compared to values for this material from literature, the probed sample canbe considered as of high quality.

0 50 100 150 200 250 3000

2

4

6

8

10

12

14

16

18

20

22

Res

ista

nce

[]

Temperature [mK]

H=0 ambient pressure

Figure 5.3: Sample resistance as a function of temperature showing the super-conducting transition in zero field at ambient pressure.

Below 200 mK a broad transition to a superconducting state is observed (seeFigure 5.3), which is not accomplished at the base temperature of 25 mK. Thebroadness of the transition and the finite resistance at 25 mK can be attributedto the inhomogeneous character of superconductivity. Only a part of the samplebecomes superconducting. This behavior occurs in the charge density wave partof the phase diagram and was described in detail in [10], [12].


Measurements in magnetic field

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

Res

ista

nce

[]

Field strength [mT]

perpendicular

parallel

T=55mK

Figure 5.4: Sample resistance as a function of the field strength (low fieldregime). The curves represent data for the field parallel and perpendicularto the conducting layers, respectively.

The behavior of the sample resistance as function of an applied magnetic fieldis shown in Figure 5.4. One curve was recorded with the field perpendicular,the second curve with field parallel to the conducting layers at a temperatureof 55 mK. The transition from the superconducting state to the normal statefor the perpendicular orientation sets in immediately and ends approximatelyat 20 mT. From there on the sample shows a strong magnetoresistance of thenormal state. In the parallel orientation no clear transition is seen up to 300 mT(not depicted). With increasing field the superconductivity gradually becomessuppressed, resulting in the increase of the resistance. This different behaviourfor perpendicular and parallel field orientation reflects the anisotropy of thepresent compound. The spikes in the curve for the field parallel to the layershave their origin in the noise from one of the power supplies (see Section 4.2.5).

Figure 5.5 shows the dependence of the resistance on the orientation of themagnetic field at 300 mT. Already at this relatively small magnetic field clearevidences for angular dependent magnetoresistance oscillations are observed.The strong maximum at θ ∼ 85 (magnetic field almost perpendicular to theconducting layers) is characteristic for the charge density wave state [47]. Thesharp dips near θ = 0 and 180 are due to the superconducting state. Theasymmetry of the curve reflects the skewed shape of the corrugated Fermi cylin-der along its longitudinal axis (triclinic crystal structure, see Section 3).

The interlayer resistance in a magnetic field up to 600 mT applied perpen-dicular to the conducting layers at 70 mK is shown in Figure 5.6. At thattemperature the superconducting transition is finished already at 10 mT. For


0 20 40 60 80 100 120 140 160 18020

25

30

35

40

45

50

55

60

R

esis

tanc

e [

]

Angle [deg]

Figure 5.5: Dependence of the interlayer resistance on the field orientation. Atθ = 90 the field is perpendicular to the conducting layers. The field strengthis 300 mT, T = 60 mK.

0 100 200 300 400 500 6000

10

20

30

40

50

60

70

80

90

100

110

120

Res

ista

nce

[]

Field strength [mT]

Figure 5.6: Interlayer resistance at ambient pressure in a magnetic field perpen-dicular to the conducting layers.


higher fields a very strong parabolic increase of the magnetoresistance in the nor-mal state of α-(BEDT-TTF)2KHg(SCN)4 is observed. At 0.5 T the resistance isalmost a factor five larger than the zero-field value. This behaviour is again char-acteristic of the charge density wave state of α-(BEDT-TTF)2KHg(SCN)4(seefor example [46]).

5.1.2 Measurements under pressure

As mentioned in Section 4.1.5 a pressure of 5.7 kbar was applied at room tem-perature. It is well known that the pressure in these kinds of pressure cellsdecreases considerably on cooling. The pressure at low temperature was deter-mined from the manganin coil as described in Section 4.2.3. This gave a valueof p15 K = 2.8 kbar. So at low temperatures the critical pressure of 2.5 kbar isexceeded.

Measurements without magnetic field

0 50 100 150 200 250 3000

500

1000

1500

2000

2500

3000

3500

0 2 4 6 8 10 12 140

100

200

300

400

Res

ista

cne

[]

Temperature [K]

Temperature [K]

Res

ista

nce

[]

pressurised

Figure 5.7: Temperature dependence of sample resistance. The inset shows thelow temperature part. No anomaly at 8.5 K is present any longer. This indicatesthe suppression of the charge density wave state.

In Figure 5.7 the resistance behavior from room temperature down to lowtemperatures is shown. In comparison to the ambient pressure data (see Figure5.2), the resistance is lower over the whole temperature range. The lower resis-tance at room temperature can be explained from the increased overlap of theelectron π-orbitals perpendicular to the layers of the crystal due to the appliedpressure, resulting in an increase of the effective interlayer transfer integral. Theresistance increases slightly from room temperature down to T = 165 K. Below165 K the resistance decreases, resulting in a negative curvature of R (T ) downto approximately 15 K. Below 15 K the curvature turns positive.


The low temperature part is shown in the inset of Figure 5.7. The humpshowing the onset of the charge density wave state is no longer seen. Thedisappearing of this feature is an indication for the applied pressure being abovecritical. As a consequence the charge density wave is suppressed and the sampleis in the purely metallic state. Using the data from the resistance measurementunder pressure (see Figure 5.7), the resistance ratio is obtained:

R (300 K)R (1.5 K)

=2370 Ω19 Ω

= 125

This is about the same value as in the ambient pressure case, the scatteringprocesses inside the sample do not notably change due to pressure.

0 10 20 30 40 50 60 70 800

2

4

6

8

10

12

14

0 20 40 60 80 100 120 140 1600

2

4

6

8

10

12

14

0.1Rn

Res

ista

nce

[]

Temperature [mK]

pressure p=2.8 kbar

B=0mT

0.9Rn

Tc

Res

ista

nce

[]

Temperature [mK]

B=4mT

Figure 5.8: Resistance as a function of temperature in zero field at an pres-sure of 2.8 kbar. The inset shows the behaviour with a field of 4 mT appliedperpendicular to the conducting layers.

Below about 110 mK a sharp transition into a superconducting state isobserved (Figure 5.8). The transition temperature is in very good agreementwith the earlier data [12]. The transition width ∆Tc defined from the decrease90% to 10% of the normal state resistance (Rn in Figure 5.8) amounts to 7.5 mKand also supports the high quality of the crystal under investigation.

Measurements in magnetic field

The behavior of the resistance as a function of temperature at an applied fieldperpendicular to the conducting planes is depicted in the inset of Figure 5.8.Down to a temperature of about 60 mK the sample is in the normal conductingstate, its resistance shows only a weak (metallic) dependence on temperature.Below 60 mK the transition into the superconducting state sets in. Comparingthe recordings at zero field and 4 mT, it is noticed, that the transition in an


applied field of 4 mT (12.9mK) is slightly broader than the zero field transition(7.5mK).

0 20 40 60 80 100 120 140 160 18020

22

24

26

28

30

32

34

R

esis

tanc

e [

]

Angle [deg]

Figure 5.9: Dependence of the resistance on the field orientation. At θ = 0

the field is parallel to the conducting layers. The field strength is 200 mT,temperature 50 mK, pressure 2.8 kbar.

Figure 5.9 shows the angular dependence of the sample resistance at a pres-sure of 2.8 kbar. As in the ambient pressure case (depicted in Figure 5.5) twosharp dips at a field orientation parallel to layers originate from the supercon-ducting transition. The main difference is the decrease of the resistance arounda field orientation perpendicular to the conducting layers in contrast to the in-crease that is seen in the ambient pressure data. This behavior of the AMROis another indication for the suppression of the charge density wave state [47],[48], [49].

It was examined, to what extend the superconducting transition is influencedby the measuring current through the sample. Two transitions R (B) at atemperature of T = 30 mK were recorded for a current strength of I = 50 nAand I = 25 nA, respectively. The data are presented in Figure 5.10. Thecurves coincide pretty well. It can be recognised, that at the lower currentscattering becomes higher. From those results, it was assumed, that the currentstrength used during the measurements (50 nA) has no definitive influence onthe measurements. Another comparison for currents of 100 nA and 50 nA wasdone at a temperature of 90 mK. There also no clear dependence on the currentstrength could be retrieved.

5.2. SUPERCONDUCTING STATE AT P = 2.8 KBAR 57

0 1 2 3 4 5 6 7 80

3

6

9

12

15

50 A

Sam

ple

resi

stan

ce [

]

Field strength [mT]

25 A

T=30mK

Figure 5.10: Superconducting transition as a function of a magnetic field per-pendicular to the conducting planes at a temperature of 30 mK for a current of25 nA and 50 nA through the sample.

5.2 Superconducting state at p = 2.8 kbar

5.2.1 Transition curves, definition of critical values

The data were obtained from temperature and field strength sweeps. For thetemperature sweeps the magnetic field was kept constant, during the field sweepsthe temperature was stabilised. For temperature sweeps the rates ranged from0.5 to 2 mK

min, field sweeps were done at 0.001-0.1 mTs . The sweep rates were

chosen so as to avoid artificial hysteresis due to the sweep. In addition, thefield sweeps were always slow enough, so that they did not induce a significantoverheating.

Figure 5.11 shows a typical example for a transition curve at constant mag-netic field. To determine the critical temperature the construction shown inFigure 5.11 was applied:The recorded curve was divided in three parts: the range below the transition,the range of the transition and the range above the transition. To each rangea straight line was fitted. The intersection points of the lower and upper linewith the line of the transition range give the lower and upper estimates of thecritical value of Bc2, respectively.

The same procedure was applied for field sweeps at constant temperature(Figure 5.12).

We verified that the field dependence of the critical temperature and thetemperature dependence of the critical field do not depend considerably on theconstruction scheme. We, therefore, present in the following mainly data withTu (B) and Bc2u (T ) and refer to them as critical temperature and critical field,


0,00 0,03 0,06 0,09 0,12 0,15

0

5

10

15

20

25

Tl

Sam

ple

resi

stan

ce [

]

Temperature [K]

Tu

Figure 5.11: Construction for the determination of the upper and lower criticaltemperature, respectively.

0 2 4 6 8 10 12

0

3

6

9

12

15

Res

ista

nce

[]

Magnetic field strength [mT]

Bc2u

Bc2l

Figure 5.12: Construction for the determination of the upper and lower criticalfield, respectively.


respectively. When the “lower” critical temperature Tl (field Bc2l) is used forcomparison, it is explicitly stated.

In [50] simultaneous torque magnetisation and electrical resistivity measure-ments were done on an organic superconductor. The determination of uppercritical field there led to the conclusion, that Bc2u is the appropriate value inthe transition to for describing Bc2.

In this work all transition curves were recorded on increasing and decreas-ing the temperature (field) to minimise influences of non-equilibrium and timeconstants.

5.2.2 Magnetic field perpendicular to conducting layers

To map out the phase diagram of α-(BEDT-TTF)2KHg(SCN)4 for perpendic-ular magnetic fields, two kinds of measurements were done.

The first set of recordings was done in constant magnetic fields directedperpendicular to the conducting planes of the sample. In these fields the tem-perature at the sample space was swept over a suitable interval containing thesuperconducting transitions. The strength of the fields during these tempera-ture sweeps ranged from 0 up to 4.66 mT.

0 10 20 30 40 50 60 70 80 90 100 110 120 1300

2

4

6

8

10

12

14

Res

ista

nce

[]

Temperature [mK]

0mT1.4mT2.8mT

4.6mT

Figure 5.13: Transition curves at constant field perpendicular to the conductinglayers obtained from varying temperature.

Secondly constant temperatures in the range from 21 mK up to 90 mK werestabilised and field sweeps with the field perpendicular to the conducting planeswere made for ranges covering the transition field strength. These field sweepswere performed with positive and negative direction of the magnetic field. Fromthe asymmetry of the resulting curve, a possibly existing remanent field of thesuperconducting magnet could be obtained and included in the evaluation ofthe measured data.


-8 -6 -4 -2 0 2 4 6 80

2

4

6

8

10

12

14

Res

ista

nce

[]

Magnetic field [mT]

30m

K

50m

K

70m

K

90m

K

Figure 5.14: Transition curves at constant temperatures obtained from varyingthe field perpendicular to the conducting layers. The minima of the curves areall shifted to a positive value of the magnetic field. This offset represents theremanent field of the superconducting magnet.

The obtained curves were evaluated using the “three line” construction pre-sented in Section 5.2.1.

Some of the transition curves from that the data were obtained are shown infigures 5.13 and 5.14. For temperature sweeps at constant field the transitionshave the same shape. The curves in Figure 5.14 show a clear offset due to aremanent field of the superconducting magnet. After warming up the cryostatand cooling down again a similar set of measurements was done immediately.These curves had no offset, as expected from a “virgin” superconducting coiland very low magnetic fields.

The critical fields of α-(BEDT-TTF)2KHg(SCN)4 for perpendicular mag-netic field are shown in Figure 5.15. The figure contains data from three seriesof measurements: one run of temperature sweeps at constant field, and tworuns of field sweeps at constant temperature. The second run of field sweepswas done after the cryostat was warmed to ∼ 80 K and afterwards cooled downagain. This temperature cycle might have had an influence on the conditionsinside the pressure cell. In particular, this might be a reason for the values ofBc2u obtained from the second run lying slightly below the values of the firstrun. We note that the pressure itself was checked after cooling down again andwas the same as in the first run of measurements.

A slight positive curvature can be seen in Figure 5.15 from Tc to about60 mK, at lower temperatures the curve can be considered to be linear. Sim-ilar results were obtained by D. Andres et al. [12] from measurements on


0 10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

7

M

agne

tic fi

eld

[mT]

Temperature [mK]

Tu

Bc2u 1st run Bc2u 2nd run

Figure 5.15: Phase diagram of the sample in a magnetic field perpendicular tothe conducting layers.

α-(BEDT-TTF)2KHg(SCN)4 under pressure (see Figure 3.8). The origin ofthe positive curvature in the Bc2-dependence can be due to flux flow effects,but as the temperature sweeps exhibit only little broadening, the influence offlux flow is assumed not to severely influence the phase diagram resulting fromthe measurements.

To compare the obtained B (T )-dependence with the earlier report [12] thepresent data Bc2u (T ) for p = 2.8 kbar are shown along with analogous data for2.5 kbar and 3 kbar [12] in Figure 5.16.

The data from this measurement lie in between the data from [12], as onewould expect. The linearity of the B (T )-dependence can then be interpretedin terms of the GL theory with the orbital pair-breaking mechanism. A lin-ear extrapolation of the low temperature part of Bc2 (T ) (Figure 5.15) givesBc2 (0) ≈ 6 ± 1 mT. Using this value in equation (2.29) for the critical fieldperpendicular to the conducting layers (θ = 90), the GL coherence length ξ0,‖inside the layers can be evaluated:

ξ0,‖ =

√Φ0

2πBc2u,⊥= 2400± 200 A (5.1)

This value can be compared to the in-plane mean free path. As no measure-ments for the determination of the Fermi energy of α-(BEDT-TTF)2KHg(SCN)4were done in this work, data obtained from the de Haas-van Alphen effect [47],


0 10 20 30 40 50 60 70 80 90 100 1100

1

2

3

4

5

6

7

8

Fiel

d st

engt

h [m

T]

Temperature [mK]

2.5 kbar

3 kbar

2.8 kbar

Figure 5.16: Critical fields at different pressures. The curves for p = 2.5 kbarand p = 3 kbar are based on data from [12], the curve for p = 2.8 kbar wasobtained in this work.

[27] are used. From those the Fermi velocity can be estimated:

vF =

√2EF

m∗ ∼ 8.4 · 104 ms

(5.2)

with the effective mass m∗ = 2me. This is the same order of magnitudeas the value of 6.5 · 104 m

s reported in [51]. A value of vF = 9 · 104 ms for

α-(BEDT-TTF)2KHg(SCN)4 was evaluated as upper estimation in [52]. Forthe present sample the scattering time was determined in a former experiment[47] to be τ ≈ 15 ps. From this value and vF = 8.3 · 104 m

s the mean free pathcan be determined:

lmfp = τvF = 12500 A (5.3)

vF = 6.5 · 104 ms leads to

lmfp = τvF = 9800 A (5.4)

The mean free path for the present sample is four to five times larger than thein-plane coherence length and, thus, the sample can be considered to be in theclean limit, lmfp À ξ0.


5.2.3 Magnetic field parallel to the conducting layers

The temperature dependence of the critical field parallel to the conducting layerswas studied. This was done for two orientations of the field within the layers,namely ϕ = 130 and ϕ = 67. The data were obtained from field sweepsand temperature sweeps in ranges suitable for exposing the superconductingtransitions. The field sweeps were done at constant temperatures in the rangefrom 27 mK to 75 mK for ϕ = 130 and in the range from 37 mK to 75 mKfor ϕ = 67. The temperature sweeps were done at constant fields covering therange from 0 up to 288 mT in the case of ϕ = 130 and 0 up to 315 mT inthe case of ϕ = 67, respectively. The sample is mounted slightly tilted. Forthis reason and to compensate remanent fields, the magnetic field was newlyaligned parallel to the conducting layers of the sample before each temperatureand field sweep. The angle θ of the magnetic field was changed in a smallrange around the parallel orientation of the magnetic field to the layers. Theupper critical field for orientations of the magnetic field other than parallel tothe conducting layers is smaller than in parallel orientation. Thus, the sampleresistance increased with increasing deviation from the parallel orientation. Theresistance was recorded during changing θ and from the minima in the curves,the parallel position was determined, as shown in Figure 5.17.

2,4 2,6 2,8 3,0

4

6

8

10

12

14

16

18

(2)(1)

(2)

Sam

ple

resi

stan

ce [

]

Angle [deg]

minimum fromincreasing

minimum fromlowering

(1)

orientation parallel to the conducting layers

Figure 5.17: Determination of the parallel orientation: the values of θ inthe minima obtained from angular sweeps up (1) and down (2) are aver-aged and give the position of the magnetic field parallel to the layers ofα-(BEDT-TTF)2KHg(SCN)4.

First, the angle θ was increased (curve 1), then lowered below the initialposition (curve 2). By averaging the θ values at the two minima, the value forθ parallel to the conducting layers of the sample is obtained. The shift of thetwo curves (1) and (2) relative to each other comes from the time constants of


the setup.The recorded curves were evaluated by the “three line” construction pre-

sented in Section 5.2.1.

120 150 180 210 240 270 300 3300

3

6

9

12

15

18

21

24

27

30

Res

ista

nce

[]

Field strength [mT]

Temperature [mK]75 55 45 35

= 130°

Figure 5.18: Field sweeps for the magnetic field aligned parallel to the conduct-ing layers at different temperatures.

In Figures 5.18 and 5.19 data obtained from field sweeps and temperaturesweeps at ϕ = 130 are shown. The increase of the resistance above the tran-sition in Figure 5.18 is attributed to the normal state magnetoresistance. Thecurves recorded at ϕ = 67 show a qualitatively similar behaviour.

Field orientation ϕ = 130

Figure 5.20 shows the B-T diagram for ϕ = 130. In the figure Bc2u andBc2l obtained from temperature sweeps and field sweeps can be seen. Thecorresponding data from field sweeps and temperature sweeps coincide very well.The overall behaviour is a very steep increase at Tc followed by a flattening atlower temperatures, saturating at about 310 mT. Bc2u and Bc2l both show thesame qualitative dependence on T . Therefore, the point of the transition atwhich the critical field is determined does only influence the absolute value ofthe “ upper critical field” not its dependence on T .

At temperatures within ≈ 20 mK below Tc (0) the critical field increaseslinearly with cooling. From the slope of this linear dependence 11.6mK

mT, theGL coherence length perpendicular to the conducting layers can be calculatedusing the value for Bc2u,⊥ and ξ‖ obtained in Section 5.2.2:

ξ0,⊥ =dBc2u,⊥/dT

dBc2u,‖/dTξ‖ = 13± 1 A (5.5)

This value is just smaller than d√2≈ 14 A and therefore lies at the border


0 20 40 60 80 100 120 140 160 18002468

10121416182022242628

85mT

135mT

185mT

235mT

Res

ista

nce

[]

Temperature [mK]

288mT

0mT

=130°

Figure 5.19: Temperature sweeps at different magnetic fields aligned parallel tothe conducting layers.

0 10 20 30 40 50 60 70 80 90 100 110 1200

30

60

90

120

150

180

210

240

270

300

330

360

Fiel

d st

reng

th [m

T]Fi

eld

para

llel t

o th

e co

nduc

ting

laye

rs

Temperature [mK]

= 130°

Bc2u

Bc2l

Tu

Tl

Chandrasekhar - Clogston

11.6 mT/ mK

(Tc-T)1/2

Figure 5.20: Dependence of Bc2u and Bc2l obtained from field sweeps and tem-perature sweeps at ϕ = 130.


between the 3d regime and the q2d regime. In the intermediate temperaturerange, 40 . T . 90 mK, the behaviour of the curve in Figure 5.20 is Bc2u ∝√

Tc − T . This is shown by the solid line in Figure 5.20 For this proportionalitythere can be two mechanisms responsible: either orbital effect (equation (2.33))or paramagnetic effect (equation (2.21)). For the orbital effect to dominate inthis regime, the sample must be in the 2d limit, that is ξ ¿ d. The ξ⊥ derivedabove is just at border to the 2d limit. There is also no “feature” as a kink inthe Bc2u-curve, when the linear part (3d) goes over to the Bc2u ∝

√Tc − T part

as it would be expected from a dimensional crossover to q2d in the regime ofthe orbital effect (see for example [26]). This would favour another mechanism,related to Pauli paramagnetism as being responsible for the behaviour of Bc2u.

As seen in Figure 5.22, the critical field exceed the ccl limit Hp (Equation(2.20) already at T ≈ 75 mK. There are a number of other layered organic con-ductors, in which the Chandrasekhar-Clogston paramagnetic limitis exceededby up to a factor of 2 [53]. This, however does not mean that the Pauli para-magnetism has no effect on superconductivity. In the present case, the Bc2 (T )saturates at ≈ 310 mT at T = 0, which constitutes ≈ 1.7Hp. The reason forthis high value is most likely many-body interactions which enhance the energygap above its BCS value (Equation (2.4)).

Another mechanism for exceeding the Chandrasekhar-Clogston paramag-netic limit might be spin-orbit-coupling, which is however believed not to occurin α-(BEDT-TTF)2KHg(SCN)4, as no heavy atoms are present in this com-pound.

Field orientation ϕ = 67

In Figure 5.21 the Bc2 (T )-diagram for ϕ = 67 is shown. Again the Bc2 (T )dependence is linear near Tc but deviates from linearity below ≈ 100 mK. As inthe case of ϕ = 130, we attribute this deviation to the paramagnetic effect ofsuperconductivity. Note that although the temperature, at which the deviationstarts, is higher than for ϕ = 130, the corresponding field value 150− 160 mTis approximately the same for both orientations. At 70 mK . T . 100 mKthe critical field varies as ∝ √

Tc − T and, at lower temperatures, saturates at∼ 315 mT. The saturation value is close to that obtained for ϕ = 130 Thusthe dependence of Bc2,‖ on the field orientation within the plane of conductinglayers becomes smaller in the region of the nonlinear Bc2,‖ (T ) dependence. Thiscan directly be seen from Figure 5.22 in which the data Bc2u (T ) and Tu (B)obtained at ϕ = 67 and ϕ = 130 are plotted together, for comparison.

Such a behaviour supports the above conclusion about the Pauli paramag-netism as the dominant pair breaking mechanism at low temperatures, since itis expected to be much less sensitive to the field orientation than the orbitaleffect. This result is a strong argument for the superconducting pairing beingsinglet.

Turning back to the temperature range near Tc, we estimate the interlayercoherence length from the initial slope dBc2

dT = 19mTmK:

ξ⊥ = 8± 1 A (5.6)

This value is more than 1.5 times lower than the estimate made for ϕ = 130,indicating a considerable anisotropy of the orbital effect with respect to ϕ. The


0 10 20 30 40 50 60 70 80 90 100 110 1200

30

60

90

120

150

180

210

240

270

300

330

360

(Tc-T)1/2

Chandrasekhar - Clogston

Fi

eld

stre

ngth

[mT]

Fiel

d pa

ralle

l to

the

cond

uctin

g la

yers

Temperature [mK]

= 67°

Bc2u

Bc2l

Tu

Tl

19 mT/mK

Figure 5.21: Dependence of Bc2u and Bc2l obtained from field sweeps and tem-perature sweeps at ϕ = 67.

value (5.6) is also smaller than d√2≈ 14 A. Therefore, one could consider the q2d

regime theory (Section 2.2.2) to be an adequate description at low temperatures,for the present field orientation. Nevertheless, at T & 0.6Tc the coherence lengthexceeds the value d√

2and the material becomes a 3d superconductor.

It should be noted that, despite the 3d character of superconductivity inthe present compound, it is still extremely anisotropic. The ratio of the coher-ence lengths ξ0,‖ (5.1) and ξ0,⊥ (5.5), (5.6) to the layers yield the anisotropyparameter

γ =ξ0,‖ξ0,⊥

(5.7)

which is 300 ± 25 for ϕ = 67 and 185 ± 15 for ϕ = 130. These values byfar exceed the values obtained for other layered organic superconductors fromcritical field experiments and are comparable to those evaluated from otherkinds of experiments (magnetic torque, Josephson plasma resonance) for themost anisotropic organic and metal-oxide superconductors [54], [53]. One cancompare the obtained anisotropy in the superconducting state with that of thenormal state. Within the Lawrence-Doniach model [22], the parameter γ isdetermined by the ratio of the corresponding effective masses:

γ =(

m⊥m‖

) 12

(5.8)

On the other hand, the effective mass ratio can be estimated for the ratio of the


effective transfer integrals t⊥ and t‖:

m⊥m‖

=(

t‖at⊥d

)2

(5.9)

where a and d are the crystal lattice period within and across the layers, re-spectively. Taking the transfer integral ratio under pressure, t‖

t⊥= 400 [47] and

crystallographic parameters a ≈ 10 A and d ≈ 20 A [28], one obtains

m⊥m‖

≈ 4 · 104 (5.10)

andγ ≈ 200 (5.11)

in excellent agreement with the estimates above.

0 10 20 30 40 50 60 70 80 90 100 110 1200

30

60

90

120

150

180

210

240

270

300

330

360

= 67° Bc2u

Tu

= 130° Bc2u

Tu

Fiel

d st

reng

th [m

T]Fi

eld

para

llel t

o th

e co

nduc

ting

laye

rs

Temperature [mK]

Figure 5.22: Dependence of Bc2u and Tu for the orientations ϕ = 130 andϕ = 67. Both curves saturate at the same magnetic field, the paramagneticlimit.


5.2.4 ϕ-dependence of critical field

In the following, data on the dependence of the upper critical field on the ori-entation of the magnetic field inside the plane parallel to the conducting layersare presented. Within an angular range of 0 to 195 the insert was rotatedby steps of ∆ϕ = 5 with respect to the vector magnet. The magnetic fieldwas set to 200 mT. For each angle ϕ the magnetic field was aligned accuratelyto the conducting layers of the sample. The parallel position was determinedby small θ sweeps as described earlier (Section 5.2.3). After alignment, tem-perature sweeps covering the superconducting transition were recorded. Fromthe curves obtained from the temperature sweeps the upper and lower criticaltemperatures were determined, following the “three line” construction (Section5.2.1.

0 10 20 30 40 50 60 70 80 90 100 110 120 130 1400

2

4

6

8

10

12

14

16

18

20

22

Res

ista

nce

[]

Temperature [mK]

-15°5°25°

45°35°

Figure 5.23: Four transition curves, each recorded at a different value of ϕ. Themagnetic field was 200 mT for all of the curves.

In Figure 5.23 five transition curves recorded at different ϕ are depicted. Al-though the shape of the transition slightly varies upon changing ϕ, the obtainedvalues of Tu and Tl exhibit essentially the same dependence on ϕ. Therefore,only the data on Tu will be presented in the following. The different resis-tance levels above the superconducting transition in Figure 5.23 are due to thestrongly ϕ-dependent normal state magneto resistance of the present sample[30].

The behaviour of Tu is shown in Figure 5.24. The values are renormalisedto the lowest value of Tu at ϕ = 140. As only the interval of 0 to 195 of ϕwas measured, the remaining range up to the full 360 range was constructedby shifting the available data by 180. This procedure is valid, as the measuredproperties of the sample do not depend on the “sign” of the magnetic fielddirection. No discrepancy between the measured data and the shifted date


-30 0 30 60 90 120 150 180 210 240 270 300 330 360

a-axis

T u/Tu(

=140

°)

[deg]c-axis

1.20

1.15

1.10

1.05

1.00

Figure 5.24: Dependence of Tu on the orientation of the magnetic field in theconducting plane. ϕ = 0 along the c-axis. The filled symbols represent themeasured data, the open symbols the same data shifted by +180.

could be recongised in the range where they are overlapping.Tu shows an anisotropy on the orientation of the magnetic field. From its

lowest values at ϕ = 150± 15 Tu increases by almost 20% to its highest valuearound ϕ = 65. The periodicity of this anisotropy in ϕ is 180. Because of thistwo-fold anisotropy the pairing in α-(BEDT-TTF)2KHg(SCN)4 is assumed to beof s-wave type. The d-wave pairing proposed for 2d organic metals would presentitself by a four-fold symmetry of the upper critical field [34], [35], [55]. In ourdata any four-fold-symmetry component could not be resolved within the exper-imental accuracy. Y. Shimojo et al. [56] examined the ϕ-dependence of Hc2 inα-(BEDT-TTF)2NH4Hg(SCN)4This material of the same group as α-(BEDT-TTF)2KHg(SCN)4 shows superconductivity at ambient pressure. They did fieldsweeps at constant temperature and determined the critical field H* at the inflec-tion point of the transition. The ϕ-dependence of H* is reported to be two-fold.The lowest value of H* increases by 15 - 20% to the highest measured value ofH*.

The position of the maxima of the anisotropy in α-(BEDT-TTF)2KHg(SCN)4is at ϕ = 65 and in α-(BEDT-TTF)2NH4Hg(SCN)4 at ϕ = 30. So the ex-tremal values do not coincide as one might have expected. In both salts thevalue of the maximum is about 1.2 times the value at the minimum of the an-gular dependence.

The observed anisotropy with respect to ϕ is consistent with the differentvalues of Bc2u (T ) for ϕ = 67 and ϕ = 130 as presented in Section 5.2.3. Notethat the slope of the Bc2 (T )-dependence at ϕ = 67 (maximum Tu (ϕ) in Figure


5.24) is ≈ 1.5 times higher than the slope at ϕ = 130 (near minimum Tu (ϕ)in Figure 5.24). This difference is bigger than the 20% difference in Tu (ϕ)presented in Figure 5.24. To explain this apparent inconsistency, one shouldtake into account that the field B = 200 mT, at which the dependence Tu (ϕ)was studied, is already in the range where the paramagnetic effect becomesimportant. As mentioned in the previous section, this lead to a decrease of theanisotropy with lowering the temperature.

5.2.5 θ-dependence of critical field

From the series of measurements presented in this subsection, the dependenceof the upper critical field on the orientation of the applied magnetic field withrespect to the conducting layers is obtained at ϕ = 40. Field sweeps weredone at different orientations θ of the magnetic field, covering an interval of−90 < θ < +90, where at θ = 0 the field is aligned parallel to the conductinglayers. These field sweeps were done at a constant temperature of 90 mK.Particular care was taken about the remanent field in the vertical coil of thevector magnet. Failure to take it into account would lead to considerable errorsin the critical field at θ around 90 and in determination of the exact orientationat θ near 0.

0 20 40 60 80 100 120 140 160 180 200 2200

2

4

6

8

10

12

14

16

18

Magnetic field [mT]

Res

ista

nce

[]

0°

-0.39°

-1.14°-2.64°

-28.14°

-0,64°

-0.14°

Figure 5.25: A few transition curves, each recorded at a different value of θ.The temperature was 90 mK for all of the curves.

In Figure 5.25 a few transition curves for different θ-orientations of the mag-netic field near parallel are shown. As before, the critical fields were evaluatedby the “three line” construction introduced earlier in Section 5.2.1.

The resulting dependence of Bc2u on the orientation of the applied magneticfield is shown in logarithmic scale in Figure 5.26. In this figure, Bc2u (θ) isrenormalised to the field Bc2u,⊥ perpendicular to the layers. Bcu2 (θ) shows


-90 -75 -60 -45 -30 -15 0 15 30 45 60 75 901

10

100

Bulaevskii:Bc2,|| = 365 x Bc2,

|| = 0.044°B

c2u(

)/Bc2

u,

[deg]0° is parallel to the conducting layers

Effective Mass ModelBc2,|| =278 x B

|| = 0.052°

Figure 5.26: Angular dependence of Bc2u at a temperature of 90 mK. The solidlines represent fits according to different models (see text for details)

a sharp increase around the field orientation parallel to the conducting layers.From the measured data a ratio

Bc2u,‖Bc2u,⊥

= 215 (5.12)

is obtained. This ratio gives an anisotropy factor

γ =ξ‖ξ⊥

= 215 (5.13)

and This value reflects again the huge anisotropy found in the slope of Bc2

near Tc for parallel and perpendicular fields. It is the largest anisotropy of Bc2

reported in organic metals up to now.The data in Figure 5.26 have been fitted, using three different models: the

effective mass model (3d) (2.29), the model proposed by L. N. Bulaevskii (q2d)(2.30), the model proposed by M. Tinkham (2d) (2.31). All three models fit thedata very well, the q2d model and the 2d model lie on top of each other to anaccuracy of 0.001, thus only one curve is shown in Figure 5.26.

Figure 5.27 shows the same date Bc2u (θ) at θ close to 0. From there cleardeviations between theory and experiment are visible. It should be noted thattheta = 0 represents in this figure the minimum in the corresponding angularsweep (Figure 5.17). The experimental data of Bc2u has the maximal value atthis position, but to positive angles a plateau like shoulder is seen up to θ=1.


-3 -2 -1 0 1 2 310

100

Bulaevskii:Bc2,|| = 365 x Bc2,

|| = 0.044°

Bc2

u()/B

c2u,

[deg]0° is parallel to the conducting layers

Effective Mass ModelBc2,|| =278 x B

|| = 0.052°

Figure 5.27: Same data as in Figure 5.26, for the range of −3 < θ < 3. Thesolid lines represent fits according to different models (see text for details)

Due to this behaviour the fitting curves are slightly shifted to positive values(Looking at the data of Figure 5.27 an even better fit would be possible byshifting the data approximately 0.1 further). This behaviour is not understoodat present. A possible explanation could be a mosaicity of the crystal whichprevents seeing the true anisotropy of the sample. Another reason could bethe fact that X-ray investigations of the present sample showed hints for asecond small crystal being misaligned by 0.2. Finally, one could not rule outa complex flux pinning near parallel orientation in the present sample. Theplateau like behaviour suggests that the intrinsic Bc2u could be even higherthan the apparent value measured in the experiment. Therefore, probably, amore correct estimation of the anisotropy can be obtained from the theoreticalfits to Bc2u as presented below.

At about ±0.3 the 3d model deviates from the other two and yields Bc2u,‖Bc2u,⊥

=

278. The 2d fit ends in a sharp peak at Bc2u,‖Bc2u,⊥

= 365. All fits lie above therecorded data in the region θ = ±0.2. As the fits all coincide in the remainingrange, it is not possible to determine which of them is preferable for the presentdata. From the models of M. Tinkham and L. N. Bulaevskii one obtains theratio γ = Bc2,‖

Bc2,⊥:

bBul = 365 = γBul (5.14)

The effective mass model (EMM) gives

γEMM = 278 (5.15)


In the following table all γ obtained from the measurements are collected:

Bulaevskii EMM ϕ = 130 ϕ = 67 ϕ = 40

γ 365 278 185 300 215

The ratios are all of the same order of magnitude. The values at ϕ = 40,ϕ = 67 and ϕ = 130 are consistent with the anisotropy presented in theprevious section (Figure 5.24).

The temperature of 90 mK for the present angular dependence was chosento get the maximal anisotropy, which is realised near Tc. The temperature de-pendence of the critical fields parallel and perpendicular to the layers suggeststhat the anisotropy reduces on lowering the temperature. As discussed in Sec-tion 5.2.3 this is most likely due to the Pauli paramagnetic effect domination atlowest temperatures. Since the balance between the orbital and paramagneticeffects must influence the angular dependence Bc2 (θ), it would be very interest-ing to perform angular studies at different temperatures. This experiment couldnot be accomplished during this diploma work, but is an important project forfuture work.

Chapter 6

Summary

The low Tc superconducting state (Tc ≈ 110 mK) ofα-(BEDT-TTF)2KHg(SCN)4, existing in close vicinity to the charge densitywave instability, at a pressure p > 2.5 kbar, has been investigated down to20 mK in magnetic fields by resistivity measurements.

For this purpose a new experimental setup has been established. It includesa dilution refrigerator and a vector magnet making it possible to arbitrarily ori-ent a magnetic field within a spherical angle of 4π with respect to the sample.

From measurements in magnetic fields orientated perpendicular and paral-lel to the conducting layers of the high quality sample a high anisotropy of theupper critical field of & 200 was found. This result was supported by a measure-ment in that the field was rotated from the in plane position to an orientationperpendicular to the conducting layers. This anisotropy is the highest reportedso far for organic superconductors.

Due to the low critical temperature the orbital effect destroys superconduc-tivity at ∼ 7 mT for fields perpendicular to the conducting layers. Correspond-ingly, the in-plane coherence length is very large, ξ0,‖ ≈ 2400 A. Nevertheless,the sample studied in this work is found to be in the clean limit.

A paramagnetic limit for magnetic fields parallel to the conducting layers of∼ 310 mT was found, whereby the Chandrasekhar-Clogston paramagnetic limitis exceeded by more than a factor of 1.5. This result suggests a strong couplings-wave superconductivity rather than a triplet pairing

A study of the influence of the field orientation revealed a considerable in-plane anisotropy of the upper critical field for T > 0.5Tc. The anisotropy has atwo-fold symmetry. Therefore, within the resolution of the measurements donein this work, d-wave type superconductivity, that would show a four-fold sym-metry, can be ruled out.

Despite the extremely high anisotropy, the compound behaves mostly as a3d superconductor. Only at some field orientations, at the lowest temperaturesit can be considered as a q2d superconductor.

75

76 CHAPTER 6. SUMMARY

Chapter 7

Appendix

7.1 Cooling power of a dilution refrigerator

7.1.1 Theoretical cooling power

The cooling power of a dilution unit will be derived in the following:

Starting from the second law of thermodynamics

dS =dQ

T(7.1)

followsQ = nT [Scp (T, x)− Sdp (T )] (7.2)

where T is the temperature, x the concentration of 3He in the diluted phase, n

is the throughput (in mols ) of 3He and Sdp and Scp are the molar entropies for

the diluted and concentrated phase, respectively. Inserting herein the entropyfor a degenerate fermi liquid

S (T ) =π2kBNA

2T

TF(7.3)

where NA is the Avogadro constant, kB is the Boltzmann constant and TF theFermi temperature, gives

Q = nπ2kBNA

2T 2

[1

TF,dp (x)− 1

TF,cp

]∝ nT 2 (7.4)

TF,cp and TF,cp refer to the Fermi temperature of the concentrated and the di-luted phase, where the latter one is dependent on the 3He concentration x. Ascan be seen from equation (7.4) the cooling power is proportional to the 3Hethroughput and depends quadratically on the temperature.

77

78 CHAPTER 7. APPENDIX

7.1.2 Cooling power at the sample space

0 20 40 60 80 100 120 1400,0

0,2

0,4

0,6

0,8

1,0

C

oolin

g po

wer

[W

]

Temperature [mK]

~ 3.6*10-11 T2

Figure 7.1: Cooling power at the sample space (pressure cell) of the dilutionunit used (single points). The curve in the graph is a fit ∝ T 2.

From equation (7.4) in the previous section the theoretical cooling power ofa dilution refrigerator is known and, can be evaluated to

Q = 84nT 2 ∝ T 2

where n is the throughput of 3He given in mols . This cooling power equals the

power supplied by the heater when a constant temperature is maintained. Thedata recorded during the measurements lead to the curve depicted in 7.1. Thecooling power at the sample space (pressure cell) marked by full circles in thegraph shows a similar behaviour as the theoretical predicted curve in the graphproportional to T2. The values shown do not represent the cooling power at themixing chamber, which is higher, as between the pressure cell and the mixingchamber the mounting of the pressure cell is placed, that gives an additionalimpedance to the heat flow (“phonon stopper”).

7.2. INVERSION CURVE OF 3HE 79

7.2 Inversion curve of 3He

Figure 7.2: Inversion curve of 3He. In the region on the left hand side of theinversion curve, 3He cools on expansion, on the right hand side it warms (takenfrom [57]).

Figure 7.2 shows the inversion curve for 3He. By increasing the pressure,the temperature below which 3He liquifies, shifts to higher temperatures. Thisis used in the “compressor method” described in Section 4.1.1.


7.3 Gas handling system

The handling system maintains the cycling of the 3He through the dilution unitand contains the storing volume (“reservoir”) for the mixture. A schematic ofthe handling system is depicted in Figure 7.3.

Figure 7.3: Schematic of the gas handling system powering the dilution refrig-erator

The connection between the dilution unit and the handling system in estab-lished by two flexible tubes, one with small diameter (8mm) for the condensationline and one of size DN40.

The pump (througput 20m3

h at 1000 mbar) of the system provides the evap-oration at the distillation chamber and transports the 3He gas through the con-densation line to the Joule-Thomson nozzle. The nitrogen cold trap preventsother eventually present gases than Helium entering the dilution unit. Traces ofof air intruding into the system via a micro leak, would close the nozzle at theJoule-Thomson expansion and with no Helium being able to pass through, thedilution unit would stop operation. The compressor accelerates the initiation ofthe Joule-Thomson process.

7.4. TECHNICAL SPECIFICATIONS OF VECTOR MAGNET 81

7.4 Technical specifications of vector magnet

Year of fabrication: 1980Body material: AluminumBore diameter 38.5 mm (1.5 inch)Outer diameter of magnet: 70 mmHeight of magnet: 120 mmCoils made of: NbTiResistance (room temperature) of vertical coil: 102 ΩResistante (room temperature) of horizontal coil: 23.6 ΩInductance of vertical coil: 61.7 µHInductance of horizontal coil: 5.74 µHField constant of vertical coil: 23.7 mT/AField constant of horizontal coil: 5.625 mT/AHomogeneity of fields: see Section (4.1.3)Temperature dependence of vertical coil’s resistance: see figure 7.5

Figure 7.4: The vector magnet used in the experimental setup.


0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

60

70

Res

ista

nce

of v

ertic

al c

oil [

]

Temperature [K]

Figure 7.5: Temperature dependence of the vertical coil’s resistance.

7.5. DEPENDENCE OF THE SAMPLE RESISTANCE ON THE MEASURING CURRENT83

7.5 Dependence of the sample resistance on themeasuring current

0 50 100 150 200 250 3000

500

1000

1500

2000

volta

ge [n

V]

current [nA]

T=60mK, H=0, p=0

Figure 7.6: Voltage measured on the sample as a function of the current throughthe sample. The slope of the curves corresponds to the resistance of the sample

Figure 7.6 shows the voltage drop over the sample as a function of the cur-rent through the sample. The slope of the curve represents the resistance of thesample. In the regime up to 60 nA the slope can be regarded as independent ofthe current. So no effects from overheating the sample or exceeding the criticalcurrent must be considered here. The measurements in this work were thereforeall done at a current of 50 nA (r.m.s.), for with lower current, the signal fromthe sample becomes more noisy (see Figure 5.10). When increasing the currentabove 60 nA, the resistance of the sample exhibits a strong nonlinearity on thecurrent. At 50 nA the amplitude of the noise was <0.3% of the resistance inthe normal state of the sample.

7.6 Earth magnetic field

Detailed data on the earth magnetic field can be found on the website of the“World Data Center for Geomagnetism, Kyoto”(http://swdcwww.kugi.kyoto-u.ac.jp).

On the site http://swdcwww.kugi.kyoto-u.ac.jp/igrf/point/index.html dataon the earth magnetic field strength can be retrieved after entering latitude,longitude and altitude of the desired position.

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Acknowledgements

This work would not have been possible without the help of many people.

I would like to thank...

• Prof. Dr. Rudolf Gross, for giving me the possibility of doing this workat the Walther-Meißner-Institut.

• Dr. Mark Kartsovnik and Dr. Werner Biberacher, for introducing meto the field of organic superconductors and imparting their knowledge onexperimental techniques and theory to me.

• Dr. Karl Neumaier and Dr. Christian Probst, for their help on buildingand operating the dilution refrigerator.

• Robert Muller, Helmut Thies, Georg Nitschke, Christian Reichlmeier, andJulius Klaus, for manufacturing the parts necessary for the experimentalsetup and in doing so incorporating all my special requests.

• Joachim Geismann and Siegfried Wanninger, for their support on buildingthe measurement setup.

• Yvonne Gawlina, Richard Morschl, and Brian Mills, for proofreading thiswork.

• Elisabeth Hoffmann, Susanne Hofmann, Wolfgang Kaiser, Ludwig Klam,and Thomas Niemczyk for the nice working atmosphere.

In particular, I would like to thank my family, for making it possible for me tostudy physics and for their support during my course of study.

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