Magnetostrictive, magnetic, and transport properties of correlated electron systems

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Magnetostrictive, magnetic, and transport properties of correlated electron systems

Dissertation

zur Erlangung des akademischen Grades des Doktors der Naturwissenschaften

an der Universität Konstanz Fachbereich Physik

vorgelegt von

Roland Schleser

geboren in Uccle (Belgien)

Tag der mündlichen Prüfung: 10. Januar 2002

Referent: Prof. Dr. P. Wyder (Grenoble)

Referent: Prof. Dr. G. Schatz

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List of Figures

2.1 Structure of a single vortex and of the Abrikosov vortex lattice . . . 6

2.2 Illustration of the critical state model. . . 9

2.3 Magnetostriction and magnetization for different models . . . 10

2.4 Field dependence ofF_p and J_c for different models . . . 11

2.5 Demagnetizing effects for different geometries . . . 12

2.6 Sample deformation due to inhomogenous flux distribution. . . 12

3.1 Magnetostriction: Simple brass cell . . . 15

3.2 Empty cell measurement . . . 16

3.3 Setup for simultaneous measurement of∆L,ρ and M . . . 17

3.4 Magnetostriction: Torquemeter based setup . . . 17

3.5 Block diagram of magnetostriction setup . . . 18

3.6 Two-beam cantilever torquemeter . . . 19

3.7 Compensated four-beam cantilever torquemeter . . . 19

3.8 Block diagram and coil subsystem of susceptibility setup . . . 20

3.9 Optimisation of the coil subsystem through a correction term . . . 20

3.10 Block diagram of extraction method setup . . . 21

3.11 Movement in a field gradient: origin of negativeB˙ . . . 22

3.12 Block diagrams of resistivity setup . . . 23

3.13 Block diagrams of resistivity setup . . . 23

3.14 CuBe cell used for measurements of electrical resistivity under pressure . . 24

4.1 Photograph: Comparison of sample geometries . . . 25

4.2 Magnetostriction on a cubic sample⊥to B . . . 27

4.3 Magnetostriction on a cubic samplekto B . . . 27

4.4 Magnetostriction in different cylindrical sample geometries . . . 29

4.5 Magnetostriction on a cylindrical sample with different suspensions . . . . 30

4.6 Magnetostriction on a spherical sample⊥to B . . . 31

4.7 Magnetostriction on a spherical sample in directionk toB . . . 31

4.8 Magnetostriction∆D/D and ∆L/L on a thin cylinder kto B . . . 32

4.9 Magnetostriction⊥B on a hollow cylinderkto B . . . 34

4.10 Double peak structure due to specific suspension configuration. . . 35

5.1 Torque measurement on a cubic sample . . . 38

5.2 Magnetization measured on a cylindrical sample via an extraction technique 40 5.3 Magnetization multiplied by magnetic field. . . 42

5.4 Comparison of−(M B)to ∆L/L . . . 42

6.1 T dependence of B_c2 determined by resistive transition . . . 44

6.2 Difference betweenBirr andBc2 . . . 45 vii

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6.3 Maximum and minimum estimations forb . . . 47

6.4 Comparison of resistivity measurements . . . 48

7.1 Thermal expansion measurement on a NbSe₂ single crystal . . . 52

7.2 Magnetostriction on NbSe₂ (c axiskBk measurement direction) . . . 53

7.3 Magnetostriction on NbSe₂ (B ⊥c axisk measurement direction) . . . 54

7.4 Magnetostriction on NbSe₂ (Bkc axis ⊥measurement direction) . . . 54

7.5 ‘Reversible’ magnetostriction in NbSe₂ . . . 55

7.6 ‘Irreversible’ magnetostriction in NbSe₂ . . . 55

7.7 Magnetization in NbSe₂: torque signal . . . 56

8.1 Detail of the peak region in magnetostriction on a cylindrical sample . . . 60

8.2 Comparison of magnetostriction with B >˙ 0and B <˙ 0. . . 61

8.3 T dependence of the magnetostrictive irreversibility (cubic sample) . . . . 62

8.4 T dependence of the magnetostrictive irreversibility (cylindrical sample) . 62 8.5 Comparison of normalizedT dependence of low field and peak amplitudes 63 8.6 Scaling analysis on a cubic NbTi sample in the peak region . . . 65

8.7 Scaling analysis on a cylindrical NbTi sample in the peak region . . . 65

8.8 Scaling analysis on a NbSe₂ sample in the peak region . . . 66

8.9 Scaling plot for a cylindrical NbTi sample . . . 67

8.10 Scaling plot for a NbSe₂ sample . . . 67

8.11 Scaling on intermediate B maximum on a cubic NbTi sample . . . 68

8.12 Scaling analysis on a cubic NbTi sample for low B . . . 68

8.13 Scaling on intermediate B maximum on a cylindrical NbTi sample . . . . 69

8.14 Scaling analysis on a cylindrical NbTi sample for intermediate B . . . 69

8.15 Scaling on intermediate B maximum on a NbSe₂ sample . . . 70

8.16 Scaling analysis on a NbSe₂ sample for intermediate B . . . 70

8.17 Scaling of∆B_on,off with sample diameter . . . 71

9.1 Extraction magnetization: lowB memory field . . . 74

9.2 History dependence in the peak region forB <˙ 0 . . . 75

9.3 History dependence in the peak region forB >˙ 0 . . . 75

9.4 Selected minor loops at low and intermediateB . . . 76

9.5 Detail: sweep rate independence of history effects . . . 77

9.6 History dependence at intermediateB . . . 78

9.7 Detail: history dependence at lowB . . . 78

10.1 Doniach diagram and Abrikosov-Suhl resonance . . . 82

10.2 YbCo₂Ge₂: resistivity measurements under hydrostatic pressure . . . 83

10.3 YbCo₂Ge₂: changes of resistivity through hydrostatic pressure. . . 84

10.4 YbCo₂Ge₂: measurement of magnetic anisotropy by torque . . . 86

11.1 Peierls transition: mechanism . . . 87

11.2 B-T phase diagram for a spin-Peierls system . . . 88

11.3 Example of torque measurement and its derivative . . . 90

11.4 Fit of TSP(B)using Cross’s model . . . 91

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Chapter 1

Introduction

1.1 Motivation

Since the discovery of superconductivity by H. Kamerlingh Onnes [1] 90 years ago, fundamental research on the properties of the superconducting state has been as much in the focus of attention as the efforts in realizing demanding technical applications, both subjects being revitalized by the discovery of high-temperature superconductivity by Bednorz and Müller in 1986 [2].

A key property in the physics of type II superconductors, as contemplated from the technical as well as from the fundamental points of view, is the critical current density J_c. Hereby we denominate not the depairing current, but the maximal current density below the onset of a voltage across the sample, and therefore, of dissipation.

The most straightforward way of determining the critical current is by measuring the dependence of a voltage across a sample on an electrical current through it. Jc is then defined as the current at which a defined voltage threshold is exceeded. However, these measurements cannot be performed on bulk samples. They are restricted to small cross sections and become increasingly difficult in high performance materials where Jc can easily be of the order of10⁹A/m².

An indirect determination of J_c can be performed using a measurement of the magnetic field dependence of a sample’s magnetization, which brings us to the notion of the critical state:

In a type II superconductor, at magnetic fields higher than the lower critical field Hc1, flux starts to penetrate the sample from the outside in units of one flux quantum Φ₀. In a defect free sample, these vortices would move freely inside the sample and would be distributed uniformly in the bulk due to their mutual repulsion. But by defects in the sample’s structure, corresponding to local minima of the superconducting order parameter, vortices are pinned. Through this mechanism, a flux density gradient builds up, which, through Ampère’s law, generates a bulk current inside the sample.

The key ingredient of critical state theory [3] is that, in magnetizing a sample, these currents will always be equal to the critical value, and that this critical current densityJ_c is a single-valued function of magnetic field and temperature,Jc(B, T). Different models have been put forward for the field dependenceJc(B), and one can determine the model’s parameters by fitting magnetization data.

Deducing, by this method, Jc(B) from measurements of magnetization generates results that may depend strongly on the choice of model. For a model-independent interpretation (i.e. the deduction of a numerical model), the use of a single variable (magnetization) might not be sufficient. One is therefore in need for another, complementary

1

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2 1.1. MOTIVATION technique.

On the other hand, the rich physics of the vortex lattice has become a topic on its own, independent of possible applications in terms of high Jc values. Especially the nature of the so-called peak effect [4] has remained a subject of strong interest over a long time. In the framework of the theory of collective pinning [5], J_c is connected with the pinning of a single vortex in a bundle of a certain correlation volumeVc. A softening of the vortex lattice would then allow easier matching of pinning centers (defects) and vortices, connected with a lowering of V_c, and leading to an increase in J_c. The exact nature of the vortex phase in the peak region (disordered solid or liquid) and of the (melting) transition remains a topic under vivid discussion.

Claims have been put forward that for certain classes of clean superconductors, a first order phase transition between an ordered Bragg glass and a flux liquid is observed [6], and that inside the peak region, a metastable supercooled liquid or a superheated solid may coexist with its respective solid or liquid counterpart. This coexistence might explain a variety of phenomena observed in recent years in the peak region, which involve a dependence of the state of the vortex matter on its thermomagnetic prehistory in a wide range of the H-T phase diagram.

Other work on the peak effect in low-T_csuperconductors has concentrated on the high field side of the peak, comparing the irreversibility field (onset of hysteresis)Hirr and the upper critical field Hc2 determined from magnetization measurements [7] and thereby testing general melting ideas and melting criteria, from which it is still not widely known that they apply to low-Tc materials.

Magnetostriction, in general, is the deformation of a sample in a magnetic field and has been used as a technique, e.g., in the field of Heavy Fermion physics, and there especially in investigations on the pseudo-metamagnetic compound CeRu₂Si₂ [8].

For the case of pinning-induced suprastriction, the deformation is directly caused by the pinning force, i.e. by the force that is transmitted to the sample via the interaction of the bulk screening currents (whose absolute value isJcin the framework of the critical state) with the vortex lines, thereby allowing the study ofJ_cand vortex matter in general.

Therefore it is quite surprising how relatively sparsely (compared with magnetization measurements) the technique has been used for this purpose [9,10,11] (see also [12,13,14]

and references therein).

The initial goals of this work were the following:

• Tosystematically evaluatemagnetostriction as a tool and exploit its usefulness, its possibilities and limitations in the investigation of vortex matter

• To use magnetostriction to evaluateJ_c(B)

• To study the effects of sample geometry on magnetostriction

• To make use of the method’s high resolution at high magnetic fields to investigate different forms of vortex matter, in particular the so-called “peak effect”.

• To start from polycristalline samples of NbTi as a low-Tc, isotropic model system.

Then to proceed to the more complicated case of single crystals of 2H-NbSe₂, a layered, highly anisotropic superconductor.

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CHAPTER 1. INTRODUCTION 3 In the following work we will thus present measurements on both NbTi and NbSe₂. The use of magnetostriction as a method will be demonstrated. The influence of sample geometry on the results will be shown. In the peak region as well as for lower fields, the applicability of scaling laws will be tested. The irreversibility line and melting criteria will be investigated. Finally, striking effects of thermomagnetic history dependence for the peak region as well as for lower fields will be presented.

1.2 Overview

The structure of this work is as follows:

Chapter 2 summarizes aspects of type II superconductivity and the principles of magnetostriction in superconductors, addressing Bean’s critical state theory and some geometrical aspects.

Chapter 3 gives an overview over the experimental methods used throughout the different measurements.

Chapter 4 presents measurements of magnetostriction in NbTi, focusing on the del- icate topic of sample geometry and sample suspension in a situation where a complex distribution of force density inside the sample is to be expected.

Chapter 5 contains a presentation of complementary magnetization measurements using different methods.

Chapter 6 features measurements of resistivity, as well as combined measurements of magnetostriction and resistivity, thereby focusing on the irreversibility line and on fluctuations in sample parameters.

Chapter7presents measurements on a 2H-NbSe₂ single crystal, thereby focusing on a comparison of magnetoquantum oscillations in both magnetostriction and magnetization, leading to surprising results for the pressure dependence of the Fermi surface’s cross section.

Chapter 8treats aspects connected with the “peak effect”, as are, among others, it’s temperature dependence and scaling with the upper critical fieldHc2 in the peak region, but also at lower temperatures.

Chapter9specializes on the aspect of history dependence, i.e. the dependence of the sample’s state on its thermomagnetic prehistory.

The two chapters 10 and 11 form a separate part on their own. They result from a cooperation with the Max-Planck Institute for the Chemical Physics of Solids in Dresden:

Chapter 10deals with high-field magnetization measurements and transport studies under pressure on YbCo₂Ge₂, investigating its low temperature properties and searching for signs of pseudo-metamagnetism.

In chapter 11, high field magnetization studies onα’-NaV2O₅ are presented, aiming at a better understanding of the magnetic phase transition in this system.

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4 1.2. OVERVIEW

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Chapter 2

General aspects of suprastriction

This chapter gives a short overview over the phenomenon and the origins of pinning- induced magnetostriction in superconductors. An introduction to the more specialized topics, i.e. the “peak effect” and thermomagnetic history dependence, will be given in the corresponding chapters (8,9).

2.1 Type I vs. type II superconductivity

This section gives a short draft of the phenomenon of superconductivity. For an introduction to superconductivity, the reader is referred, e.g., to the excellent text book by M. Tinkham [15].

The two primary notions connected with the term superconductivity are the absence of electrical resistivity and the perfect shielding of the bulk interior of a superconducting sample from magnetic fields, at low enough field strength and temperature.

Independent of whether a sample is brought into or cooled down in a magnetic field, lossless shielding currents (Meissner supercurrents) will form inside a thin layer at the surface of thicknessλ (the London penetration depth) and perfectly screen the interior from magnetic flux.

In type I superconductors (e.g. Pb), this state will last up to a critical magnetic field H_c, at which it breaks down and magnetic flux uniformly fills the sample again.¹

In the phenomenological theory of Ginzburg and Landau [16]², the surface energy of an inclusion of magnetic flux in the sample depends on the Ginzburg-Landau parameter κ=λ/ξ, whereξ is the correlation length of the macroscopic superconducting wavefunc- tion (also called the superconducting order parameter) ψ. For κ > 1/√

2, the surface energy is negative.

1 In this introduction, we restrict ourselves to the ideal geometry where the sample is infinite in the direction of the external applied magnetic field. One hereby avoids a phenomenon called the intermediate state, which occurs in samples of type I with finite dimensions: In fact, the change of the flux density B outside the sample due to the described expulsion of flux from the sample (flux compression) can lead to a local decrease of the apparent critical field. This in turn may lead to a domain structure in, e.g., a thin disk sample. It has nothing to do with the penetration of flux inside the sample in type II superconductors, since the sign of the surface energy is different in this case.

2The approach of Ginzburg and Landau (GL) was in fact put forward a long time before the birth of the microscopic BCS theory [17] in 1957. However, in 1959, Gor’kov could show [18,19] that GL theory was indeed a limiting case of the BCS description for temperatures not too far from Tc and for small spatial variations of the order parameterψand the magnetic field’s vector potentialA. This simplifies the description of many situations where only a macroscopic or mesoscopic description of superconducting behaviour is wanted.

5

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6 2.2. ORIGINS OF SUPRASTRICTION In a type II superconductor, at the lower critical field H_c1, this leads to the free energy of flux penetrating the sample becoming negative.

The constraint of flux quantisation — stemming from a phase argument inψ— leads to the formation of vortices containing each one single magnetic flux quantum Φ₀ = _2e^¯^h and being shielded by lossless supercurrents [20]. In a perfect crystal, the vortices can move freely and, by their mutual repulsion, form a lattice, the Abrikosov vortex lattice or flux line lattice (FLL).

Figure 2.1: (a) Spatial variation of the superconducting order parameter and the magnetic flux density in a single vortex. (b) Hexagonal Abrikosov vortex lattice (schematically).

Vertical arrows represent one flux quantum each, circular arrows represent screening currents.

2.2 Origins of suprastriction

Suprastriction is the deformation of a sample due to superconductivity. One distinguishes between thermodynamical (reversible) and irreversible (flux pinning induced) suprastriction.

2.2.1 Thermodynamical suprastriction

Thermodynamical suprastriction occurs due to the change in the free energy connected to the magnetization of the sample, yielding a volume change (see, e.g., [12])

1 V

∂V

∂H

P,T

=−µ₀ ∂M

∂P

H,T

(2.1) (M: magnetization;P: pressure; H: magnetic field;V: volume).

Typically, the maximum values (∆L/L)max of the relative magnetostriction (where L is a linear dimension of the sample) are quite small ((∆L/L)_max ≈ 10⁻⁸) and in the present work we will not be concerned about it.

2.2.2 Pinning-induced suprastriction

In the following, we will discuss magnetostriction whose origin lies in the interaction between pinned vortex lines and bulk currents connected with gradients in the magnetic flux densityB. The question about the magnitude and origin of the bulk currents leads us to the concept of the critical state, presented in the next section.

The vortices in the Shubnikov phase present minima of the amplitude of the superconducting order parameter. Since defects in the sample’s crystal structure also reduce

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CHAPTER 2. GENERAL ASPECTS OF SUPRASTRICTION 7 the amplitude of the order parameter, it becomes energetically favorable for the crystal to match defects and vortices, thus hindering vortex motion and provoquing a deformation of the FLL.

Through Ampère’s law, the resulting inhomogenuous field distribution leads to the formation of bulk currents inside the sample³, to a hysteretic magnetization and, by virtue of the forces acting upon the sample through interaction of the currents with the vortices (and thus with the defect centres), to a deformation of the sample itself. In an equilibrium situation, the internal local stress σ then conforms to the equation

∇σ(x)−n(x)f~p = 0, (2.2) which in a one-dimensional treatment reduces to

∂σ

∂x −n(x)f_p = 0, (2.3)

wheref_p is the pinning force per vortex line per unit length andn(x) is the density of vortex lines. The former can be expressed as

f_p = Φ0

µ₀

∂B

∂x , (2.4)

Φ₀ being a flux quantum,

B(x) = Φ0n(x) (2.5)

the local magnetic flux density, andn(x) the density of flux quanta. Together this yields σ(x) =−B_ext² −B²(x)

2µ₀ , (2.6)

withB_ext =µ₀H_ext, whereH_ext is the applied external field.

Considering now the case of an elastically isotropic slab, infinite in two dimensions and with thickness2d, the magnetic fieldH_ext applied parallel to the surface of the slab, we can integrate the above equations to get the sample’s deformation

∆d(Bext)

d =− 1

2Cµ₀d Z d

−d

B²_ext−B²(x)

dx , (2.7)

whereB(x) can be determined by integrating J(x) =µ⁻¹₀ ∂B(x)

∂x = f_p Φ0

= F_p

B , (2.8)

J(x) being the bulk current density connected with the flux density gradient, whose magnitude is to be discussed below. C is an elastical constant assumed isotropical,

Fp(x) =B(x)J(x) =n(x)fp (2.9) defines a macroscopical pinning force density.

In this same situation, the sample’s magnetization becomes M =− 1

µ0d Z d

−d

[B_ext−B(x)]dx . (2.10)

3as opposed to the surface screening currents in the Meissner phase

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8 2.3. THE CRITICAL STATE Therefore, in magnetostriction, one notices an additional multiplicative factor

B_ext² −B²(x)

B_ext−B(x) =B_ext+B(x), (2.11)

in the integrand, which leads to a qualitatively different functional dependence∆d(B_ext), as compared toM(B_ext). Only in situations where the variation ofB over the sample’s volume is negligible, one can extract this factor from the integral and obtains

∆d

d ≈ M B

2C ≈ Fpd

2C . (2.12)

For practical purposes and in the context of measurements presented later in this work, it is also enlightening to consider the case of an infinitely long circular cylinder of diameter D, calculated by Johanson [13], where Dvaries as

∆D

D =−1−ν Cµ₀

Z 1 0

ρ

B_ext² −B²(ρ)

dρ , (2.13)

while the relative change of the lengthL is obtained from

∆L

L =− 2ν 1−ν

∆D

D , (2.14)

where ν is the Poisson ratio of the material. Note the additional factor ρ in the integral, whose origin lies in the larger weight of the outer shells of the cylinder in the integration process.

2.3 The critical state

2.3.1 Basic ideas

In 1961⁴ [3], Bean proposed a phenomenological treatment of the magnetization of hard superconductors, which was later to be called the critical state model: It assumes that, in all regions of the sample containing magnetic flux, the bulk current density J(x) connected with the magnetic flux density’s gradient∂B/∂xin the sample has the critical (depinning) valueJc, and that the magnitude of the critical current only depends on the local flux density (at fixed temperature):

|J(x)|=J_c(B(x)). (2.15) Bean’s phenomenological treatment thus created a possibility to determine J_c, a property of high technical impact, from measurements of the magnetization M. This link betweenM andJ_cis of special importance when one considers superconductors with critical current densities J_c>10⁹ A/m² = 1000 A/mm², where direct measurements of Jc become virtually impossible.

Figure 2.2illustrates the distribution of the magnetic flux density in a slab of thickness 2d in the framework of the critical state model, for increasing ((a), left side) and decreasing ((b), right side) external magnetic field. The shaded areas represent magnetization, negative (shaded area above the curve) for increasing and positive (shaded area below the curve) for decreasing magnetic field. Horizontal arrows show the direction of the force on the sample, compressive for increasing, tensile for decreasingB. Bp is the field⁵ of full penetration, i.e. the value of Bext on increasing at which magnetic flux completely starts to fill the sample, starting from an unmagnetized sample.

4This was indeedbeforethe idea of the Abrikosov vortex lattice was widely accepted.

5In the following, we will sometimes somewhat inaccurately (though consistently with recent publi-

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CHAPTER 2. GENERAL ASPECTS OF SUPRASTRICTION 9

Figure 2.2: Illustration of the critical state model for an infinite slab of finite thickness for (a) increasing (B >˙ 0) and (b) decreasing (B <˙ 0) external magnetic field. The curves represent the distribution of magnetic flux in the sample, shaded areas under the curves represent magnetization. For a J_c(B) monotonically decreasing with increasing B, the field of full penetrationB_p is at the same time the field of maximum magnetization (for increasing field). After reducing the magnetic field to0, trapped flux remains inside the bulk of the sample.

2.3.2 Bean’s model

In his calculations, Bean simplified his model even more by assuming an (unrealistic) field independentJ_c.

2.3.3 More realistic models

Soon after Bean’s work, Kim et al. presented a more realistic critical state model [21], where the critical current density depends on the flux density as

J_c=α/(B₀+B), (2.16)

where α and B0 are (temperature dependent) parameters. Anderson was able to in- terprete this functional dependence and the measured temperature dependence α(T) in terms of thermally activated flux creep [22]. A review of the Kim-Anderson flux creep theory is given in ref. [23].

For high-Tc superconductors, an exponential dependence J_c(B) =J_c0exp

|B|

B₀

(2.17) was first proposed by Senoussi et al. [24].

Quite recently, Ikuta et al. compared measurements of magnetostriction on Bi₂Sr₂CaCu₂O₈ [10] and (La1−xSr_x)2CuO₄ [25] to calculated magnetostriction curves [26] for the three models mentioned above and found good quantitative agreement for the case of the exponential model. Figure2.3shows calculated example curves for both magnetization and magnetostriction for the three models mentioned.

cations) use the term “(external) magnetic field” forBext=µ0Hext, the vacuum flux density, instead of Hext.

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10 2.3. THE CRITICAL STATE

Figure 2.3: Magnetostriction (left column) and magnetization (right column) for (a) Bean model, (b) Kim model and (c) exponential model. Graphs taken from [26]

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Figure 2.4: Dependence of the macroscopic pinning force F_p and the critical current densityJc on magnetic flux density for different critical state models

2.4 Geometrical aspects

The present section deals with the implications of different geometries of the sample as well as of its suspension. The consequences of an inhomogeneous flux and force distribution are compared to those being described by a demagnetizing factor.

2.4.1 Demagnetizing factor

For the case of a sample with infinite extension in one direction and a homogenous magnetic field applied in this direction, the magnetic field near its surface will be the same as far from it.

For any real finite sample, however, the sample’s magnetization will give rise to a magnetic dipole field which will be superimposed on the external field and will cause a change in the flux density near its surface. For a diamagnetic sphere, e.g. (see fig.

2.5(b)), demagnetizing effects will cause a compression of flux lines at the equatorial plane. For some geometries, the demagnetizing factor N can be calculated exactly and magnetization measurements can be renormalized correspondingly. For the simple case of a sample of field independent magnetic susceptibilityχ, the magnetization (magnetic dipole moment per unit volume)M becomes

M = χH_ext

1 +N χ , (2.18)

whereHext is the external magnetic field.

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12 2.4. GEOMETRICAL ASPECTS Since the actual self-field depends on χ, demagnetizing effects play a role only for large |χ|(as e.g. for a sample in the Meissner state withχ=−1).

Figure 2.5: Magnetic flux lines near the surface of (a) a sample with very low demagnetizing factor (b) a diamagnetic sphere (demagnetizing factor1/3,χ =−1) (c) a thin slab (demagnetizing factor1,χ=−1)

In superconductors, the demagnetizing factor plays an important role for fields H <

Hc1, where demagnetizing effects can lead to a reduction of the apparentHc1 of a sample.

In type I superconductors, this leads to the formation of the so-called intermediate state (see footnote on page5).

Note that taking into account demagnetizing phenomena in magnetization investigations by including a demagnetizing factor is quantitatively correct, even for simple geometries as a sphere, only for samples with a field independent magnetic susceptibility χ, since the magnetic flux density varies throughout the sample!

2.4.2 Inhomogeneous flux distributions

In our discussion of the critical state and flux pinning phenomena in general, we are dealing with an inhomogeneous distribution not only of vortices and thus magnetic flux, but also of the resulting force densities. This may lead to convex and concave deformation of the sample’s surface and to differential effects, as will demonstrated vividly in chapter 4.

Pioneering work in the calculation of the internal stress distribution due to pinning has been performed by T. Johansen [27,28,13], who made calculations for various sample geometries, including samples infinite in the magnetic field’s direction with quadratic and circular cross section. In his work, he showed how the sample’s external shape is modified as a result of an inhomogenous flux and force distribution (see fig.2.6as an illustration).

However, these calculations relate only to the cross section of an infinite sample, whereas in discussing geometrical aspects we will mainly be concerned by the deformation of the sample’s end faces.

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Figure 2.6: Illustration of the (strongly exaggerated) deformation of a sample’s cross section at various stages in a magnetization cycle (after T.H. Johansen [13]): (a) unmagnetized, (b) partially magnetized, (c) fully penetrated by magnetic flux, (d) after reversal of sweep direction

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14 2.4. GEOMETRICAL ASPECTS

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Chapter 3

Experimental techniques

This chapter describes measurement techniques used in this work. Apparati that have been newly built or changed significantly in the frame of the present work are described in somewhat more detail.

More information on a concrete measurement may be found in the corresponding chapter/section.

3.1 Magnetostriction

Two different setups have been used to measure sample length changes, both based on the capacitance technique.

3.1.1 High precision setup

This setup (see [8,29]) consists of a closed brass cell (figure 3.1) with Stycast^r as glue and as electrical insulation between the metallic parts. The sample, via a sample holder, deforms a circular metallic diaphragm which forms one electrode of a capacitor, the other one being rigid and fixed. The force on the sample, being of the order ofF ≈50 N, makes the setup tolerant against vibrations, resulting e.g. from the cooling water flow in the coil of a resistive high field magnet. Absolute resolutions of the order ofδl≈0.05Å can be obtained this way.

Figure 3.1: Simple brass cell (drawing after [30]) 15

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16 3.1. MAGNETOSTRICTION To get an idea about the background signal resulting from magnetostriction of the cell’s components themselves, measurements on the empty cell have been performed at different temperatures. Fig. 3.2shows the resulting magnetic field dependence converted to absolute length units (note that all magnetostriction measurements in later chapters are normalized to the sample’s dimension).

∆

Figure 3.2: Magnetostriction measurement on the empty measuring cell for two different temperatures

The curve has a broad minimum atB ≈5 Tand a subsequent upturn. The zero shift in the T = 4.2 K backsweep curve is most certainly due to long-time relaxation of the insulating layers of the cell (Stycast^r glue). In most later measurements, a longer waiting period was applied between cooling down of the setup and the start of the measurements, in order to avoid thermal drift.

Since the vast majority of our measurements using this cell only focuses on the irreversible (hysteretic) part of the data, and since this background is still quite small compared with the signal in most cases, it was in general not subtracted from the measured data. The amplitude of the very small hysteresis of unknown origin visible in the high-field upturn has an absolute amplitude of only δ(∆L) ≈ 0.5Å and therefore was of no relevance either. In fact, it was also strongly reduced after a sufficient relaxation time.

The calibration of the setup was performed by deforming the membrane of the pressure cell via the sample holder, using a MICOS PM-90 linear translation unit, driven by a 2-phase stepping motor which in turn is controlled by a microstepping controller (MICOS SMC-compact), yielding a repeatability of∆x≈0.4µm. Since this calibration was done at ambient temperature, one has to account for thermal expansion of the thin space between membrane and counter electrode. This generates an error in the absolute magnetostriction values whose order of magnitude should be clearlyδ(∆L/L)<10%.

Simultaneous measurement of magnetostriction and electrical resistivity In a specific series of measurements (see section6.2), the (cubic) sample was electrically isolated from the cell and contacted to perform a four-point resistivity measurement. All

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CHAPTER 3. EXPERIMENTAL TECHNIQUES 17 four contacts where aligned on one surface of the sample, perpendicular to the magnetic field, while magnetostriction was measured in a direction perpendicular toB. An additional Hall probe was fixed (electrically insulated) on the side of the sample opposite to the contacts (see fig. 3.3for an illustration).

Figure 3.3: Setup for simultaneous measurement of ∆L,ρ and M

3.1.2 Low force setup

The primary requirement for this second setup was to reduce the force exerted by the cell on the sample, to be able to measure samples being brittle and/or having a small cross section, as it proved to be the case for the measurements on 2H-NbSe₂ perpendicular to the crystal’s c axis (see section7.2.4).

The sample presses down a four-beam, two-plate parallelogram torquemeter (fig.

3.4, see also section 3.2.1). The parallelogram configuration, realised using two CuBe torquemeters separated by two spacers of equal thickness, avoids tilting of the lower torquemeter plate, forming the moving electrode of a capacitor, versus the counter electrode. If necessary, a tip is glued to the bottom of the sample to ensure one well-defined contact point.

The force on the sample when clamped between sample holder and torquemeter is as low as F ≈ 0.5 N, but the resolution one can obtain in this setup only approaches δl≈1Å.

Figure 3.4: Torquemeter based setup. The sample holder may vary depending on sample geometry and is shown only schematically.

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18 3.2. MAGNETIC PROPERTIES 3.1.3 Description of the apparatus

Both magnetostriction setups detect length changes by measuring capacitance. This is realised through a fully-balanced computer controllable capacitance bridge (AH2500A).

Temperature at zero magnetic field is measured using a Germanium or Cernox (TM) resistive thermometer through a LR-700 resistance bridge. Whenever temperature control (stabilizing or sweeping) is needed, it is realized through a LR-130 temperature controller with an input signal from a SR830 lock-in amplifier, connected through a GR1616 passive to a capacitive (magnetic field independent) thermometer (fig. 3.5), or using a membrane pressure controller (in the range1.5 K< T <4.2 K). For lower temperatures (T <1.4 K), measurements are performed in a³He system with the setup immersed in liquid³He where temperature control is realized by controlling³He gas pressure.

Figure 3.5: Block diagram of magnetostriction setup

3.2 Magnetic properties

To measure magnetic properties, two different setups have been used: A capacitive torquemeter allowing to determine the magnetic anisotropy of a sample, and a differential (moving sample) AC susceptibility apparatus, which can also be used as an extraction magnetometer.

3.2.1 Magnetization: Torque method

The torque on a sample in a magnetic field is proportional to the vector cross product of the magnetic field and the magnetization of the sample, while the force on the sample is proportional to the scalar product of the magnetization and the magnetic field gradient.

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CHAPTER 3. EXPERIMENTAL TECHNIQUES 19 Measuring the capacitance change due to the deformation of a cantilever torquemeter (fig. 3.6) on which a sample is mounted is one of the easiest ways to investigate its magnetic properties. It is widely used in fermiology and whereever absolute values of magnetization are not required.

Basically the cantilever torquemeter can be used in two ways:

1. By mounting the setup in the field center (zero gradient), one measures the torque on the sample. To get a non-zero torque signalτ, the magnetization must have a non-zero component perpendicular to the field:

~τ =~p_m×B ,~ (3.1)

where~p_m is the sample’s total magnetic moment

2. By mounting the sample at a distance from the field center (non-zero gradient), the force on the sample proportional to the field gradient and to the sample’s magnetization creates an additional torque on the cantilever (Faraday method):

τ_Faraday=l p_mdB

dz cosα , (3.2)

wherel is the distance of the torquemeter’s suspension point to the center of the sample, pm the magnetic moment andα the angle between~pm and B.~

Figure 3.6: Two-beam cantilever torquemeter (original drawing from [31])

Since again capacitance has to be measured, the block diagram describing the torque setup would be identical to the one for the magnetostriction setup (fig. 3.5) for most situations.

For one measurement, where a calibration of magnetization data was required, the sample also being rather large, a compensated four-beam torquemeter was used. It was realized by adding a second CuBe spring separated from the first by a spacer, around which a coil with well-defined geometry (height h = 1.4 mm, diameter D = 4.8 mm, N = 20 windings) was wound (see figure 3.7). By letting a current flow through the coil and comparing the resulting torque signal in a magnetic field to a sample’s signal, magnetization data can be calibrated.

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20 3.2. MAGNETIC PROPERTIES

Figure 3.7: Compensated four-beam cantilever torquemeter (detail) 3.2.2 AC susceptibility

An AC susceptibility apparatus was built in the course of the present work to measure magnetic susceptibility in sample volumes of about 10 mm³ at temperatures 0.3 K <

T <10 K.

The primary goal in its design was to measure magnetic susceptibility of Heavy Fermion and other high susceptibility systems, but it can also be used for measuring magnetization of e.g. superconducting samples using the extraction method (see section 5.2).

The coil system was designed so that a sample with a maximum diameter of D = 2.8 mm and a maximum length l = 5 mm or a sample holder for powder of similar dimensions could be fitted.

The bobbin was made in a way so that the excitation field amplitude shows little spatial variation inside the sample length and that the signal of a point like test particle (calculated as the product of the field generated by the outer coil and the pickup sensitivity of the inner compensated coil pair) does not vary over the sample’s dimensions in either of the two secondary coils, resulting in a reduced sensitivity against sample mispositioning. Figure 3.9 presents an uncompensated profile from coils with simple rectangular cross section, a corrected profile and the correction term, realized through a small volume fraction of rectangular cross section in the secondary coils without windings.

To measure, the sample is moved into one of the two pickup coils using the same linear translation unit described in section3.1.1, and the pickup signal is recorded. Then the same is done for the second coil, which is wound oppositely. The difference of the two signals gives twice the signal due to the sample, while any incomplete compensation of the two pickup coils is compensated.

3.2.3 Magnetization: Extraction method

For the few measurements using this method, no changes have been made to the hardware of the susceptibility apparatus. The primary coil (see fig.3.8) is not used, the secondary (pickup) coil connected to a lock in amplifier. The synchronization signal is generated via a photoelectric barrier by the motor unit, which in turn moves the sample sinusoidallywith a small amplitude (δz <1 mm, see 3.2.4) and a low frequency (f < 1 Hz). The signal picked up by the lock in amplifier is then proportional to the magnetization of the sample.

3.2.4 Comparison of methods:

Special issues concerning superconductors

Some special care has to be taken while measuring magnetization on any sample showing magnetic hysteresis. This is especially true for type II superconductors. Especially in the

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CHAPTER 3. EXPERIMENTAL TECHNIQUES 21

Figure 3.8: Block diagram and coil subsystem of susceptibility setup

Figure 3.9: Optimisation of the coil subsystem. The calculated profile is the position dependence of the signal of a point-like test sample (arbitrary units). The deviation of the secondary coils from a simple rectangular cross-section gives a correction term which results in a “flat” profile, tolerant against small sample positioning errors.

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22 3.2. MAGNETIC PROPERTIES

Figure 3.10: Block diagram of extraction method setup

high field region, whereJ_c is reduced, small oscillations in the external field around the sample can result in large relative changes of the internal magnetization. The oscillating excitation field of an AC susceptibility apparatus or the change of external field due to the movement of the sample in a small field gradient, both of the order of a fewmT, may be enough to alter or completely annihilate the metastable critical state.

One can avoid or reduce these difficulties by taking care that for all external fields the external magnetic field’s time derivative never changes sign. This can be done by correspondingly choosing the excitation amplitude (for AC susceptibility measurements), the sample translation and the sweep rate of the external field. I.e. one has to choose a large enough sweep rate of the external magnetic field which makes sure that no sign change in B˙_ext occurs, where B_ext is the sum of the monotonically swept external field and the oscillatory field component (see fig.3.11for an illustration).

But a reduction in the amplitude of the sample’s movement leads to a decrease in sensitivity. And the possible increase in the sweep rate of the externalB field is limited by the integration time needed, which is at least one period of the sample’s movement.

Measurements in SQUIDs suffer from the same drawback (see [32] for an analysis of the problem), though Ravikumar et al. [33] claim to have found a solution to this issue which they call “half scan technique”, since the sample only traverses half the way of a normal SQUID scan, thereby sensing only a monotonic field change in time in the gradient.

One of the convincing advantage of measurements using the torque or magnetostriction techniques is thus the immobility of the sample, avoiding all problems due to oscillating external fields. Tenya et al. have demonstrated this advantage of the Faraday method in measurements on CeRu₂ [34].

In addition, the dependence on the magnetic field amplitude ~τ = p~_m×B~ causes the torque method to be comparatively sensitive at high magnetic fields. A similar argument holds for magnetostriction: In calculating the sample’s total magnetostriction, one integrates over a term B_ext² −B², while for magnetization, the corresponding term is B_ext−B, where B_ext is the external magnetic field and B is the local magnetic flux density in the sample. In comparison, magnetostriction includes an additional factor

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Figure 3.11: At higher fields, the movement of the sample in the field gradient can lead to a negative time derivative of the external field sensed by the sample. Shown in this graph is the time dependence of the external field around the sample for a sweep up in field withB˙ = 3 mT/s, the sample being moved in a gradient with a total amplitude of δB≈B·10⁻⁴ at a frequency of aboutf ≈0.16 Hz.

B_ext+B (see section 2.2.2 for more details on the calculation of magnetostriction). It is therefore no coincidence that measurements of torque and magnetostriction show a certain similarity in the form of the measured curves.

Note, however, that the torquemeter technique also presents one important disadvantage: Even when used in the Faraday configuration (mounted in the gradient of the external field), an intrinsic torque may still be present due to some anisotropy of the sample, e.g. of purely geometrical origin (i.e. even when the sample is polycristalline). With the inhomogeneous flux distribution present in a type II superconductor, a slight tilting of the sample or a small asymmetry in its shape may cause such an additional unwanted torque

~τ_anisotropy. Then, one measures the sum of two torques: ~τ_total = ~τ_Faraday+~τ_anisotropy, making it difficult to extract the desired (Faraday) signal. This will be illustrated in chapter5. Subtraction of the unwanted background signal, which can be obtained inde- pendently by performing an additional measurement in zero gradient, can help reducing the problem.

3.3 Resistivity

Resistivity measurements have been performed using a four-wire technique, with thin copper or platinum wires (the diameter depending on the sample size) glued to the

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24 3.4. RESISTIVITY UNDER PRESSURE sample using silver epoxy glue H20E (A+B) by Polytech.

Two different setups were used:

1. a LR-700 AC resistance bridge by Linear Research. This bridge works at a frequency of16.6 Hz (fig. 3.12).

2. a combination of a90 Hz AC current source with a lock-in amplifier (fig. 3.13).

Figure 3.12: Block diagrams of resistivity setup using a LR-700 resistance bridge

Figure 3.13: Block diagrams of resistivity setup using a combination of AC current source and lock in amplifier

The latter setup proves to be less susceptible to noise (and spikes) generated by the resistive magnets at extremely low resistances (µΩrange), the former allows for a higher degree of measurement automation.

3.4 Resistivity under pressure

Some of the resistivity measurements on YbCo₂Ge₂ have been done under hydrostatic pressure, using an apparatus built by the author during his diploma thesis [35]. These measurements have been performed at Darmstadt University of Technology. Figure3.14 shows a drawing of the pressure cell used.

To measure the pressure inside the cell, a small (V ≈0.25 mm³) piece of lead inside a minute coil (100 windings, linear dimensions about(1 mm)³) is put into the sample space.

Together with an external coil around the inner cell, the lead’s magnetic susceptibility is

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Figure 3.14: CuBe cell used for measurements of electrical resistivity under pressure.

The elements marked with an asterisk (*) exist in two versions, one made from hardened CuBe, the other from a special steel alloy (Thyrodur^r by Thyssen), the latter one sup- porting higher tensile stress (allowing pressures p_max ≥ 3 GPa), but having the minor disadvantage of being slightly ferromagnetic.

measured to determine the superconducting transition temperature T_c, whose pressure dependenceTc(p)is well known.

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26 3.4. RESISTIVITY UNDER PRESSURE

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Chapter 4

Magnetostriction in NbTi

The following chapter presents measurements relating to more general aspects of magnetostriction in NbTi. Special attention is given to issues related to sample geometry and to the heterogeneous force density distribution inside the samples. Topics such as a more refined analysis of the peak region, the irreversibility line, thermomagnetic history dependence effects and scaling are deferred to later chapters.

4.1 Introduction and motivation

Samples of polycristalline NbTi have been chosen for the present work by virtue of their relatively highT_c= 9.5 KandH_c2(T = 0) = 14 Tand to avoid anisotropy related effects.

Those would have unnecessarily complicated the interpretation of our measurements, which were also intended as a systematic evaluation of magnetostriction as a method.

A cylindrical rod of Nb₅₆Ti₄₄ alloy has been supplied by Goodfellow SARL, from which all the samples of different shape described below have been machined and cut (see fig. 4.1).

Figure 4.1: Photograph of all sample geometries measured by magnetostriction, together with sample number.

In addition, a small rod, a few mm long, with a cross section of aboutA ≈ 1 mm² was cut to perform resistivity measurements (sample]2b). From these measurements, a residual resistivity ratio ofR(300 K)/R(12 K)≈1.33was determined. See chapter 6for more details on resistitivity measurements.

All measurements have either been performed in a14 Tsuperconducting warm bore magnet or in a 20.3 T resistive magnet, both located at the Grenoble High Magnetic

27

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28 4.2. MEASUREMENTS ON DIFFERENT SAMPLE GEOMETRIES Field Laboratory.

Measurements at T = 4.2 K were done in a liquid Helium bath; T = 1.5 K was reached by pumping the bath, higher and intermediate temperatures were obtained using an electrical heater, which was controlled using a capacitive (magnetic field independent) temperature sensor, or using a membrane pressure controller. Measurements at T = 0.4 K were performed in a³He system where the measuring cell was immersed in liquid

3He.

4.2 Measurements on different sample geometries

Measurements on samples of substantially different geometry are presented, stressing the importance of the form and suspension of the samples. A detailed discussion will follow in section4.3.

4.2.1 Measurements on a cubic sample

To be able to compare magnetostriction measured perpendicular and parallel to the external magnetic fieldB, without changing the sample’s orientation with respect to B, a sample of cubic shape (d= 6.2 mm) and, later on, a spherical sample (see section4.2.2) have been cut.

Figs.4.2and4.3show magnetostriction measurements on the cubic sample]2, where the measurements have been performed perpendicular and parallel to the magnetic field, respectively.

The deformation in the direction of the magnetic field is opposite to the one perpendicular to B, and its value is larger. At first sight, this is what simple considerations taking into account only mechanical properties of the sample would suggest: The compression of the sample perpendicular toB due to the interaction between bulk currents and vortices leads to an elongation in the direction ofB.

An estimation of the apparent Poisson ratio from the relation [13]

∆L

L = −2ν 1−ν

∆D

D (4.1)

yields a slightly magnetic field dependentνin the range(0.37±0.03)< ν <(0.44±0.03) (where the errors stem from uncertainties in the calibration of the cell), which seems reasonable. Equation4.1holds rigorously only in the case of an infinite circular cylinder, which may explain the dependence onB. Especially one must consider that the geometry of the cross section is not conserved (see figure2.6) and that due to the finite length of the sample in the direction of the magnetic field, the stress distribution is far from being 2-dimensional.

At fields slightly below Bc2, one observes a sharp maximum, known for a long time from magnetization, transport and magnetostriction measurements as the “peak effect”, which will be treated in detail in chapter8.

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CHAPTER 4. MAGNETOSTRICTION IN NbTi 29

∆

Figure 4.2: Measurements on a cubic sample with the measurement direction perpendicular to the applied magnetic field. The arrow through the curves points in the direction of increasing temperature.

∆

Figure 4.3: Measurements on a cubic sample with the measurement direction parallel to the applied magnetic field. The arrow through the curves points in the direction of increasing temperature.

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30 4.2. MEASUREMENTS ON DIFFERENT SAMPLE GEOMETRIES 4.2.2 Measurements on cylindrical samples

To be able to estimate the influence of different sample geometries on magnetostriction measurements, several cylindrical samples were cut, having ratios of diameter over length ranging fromD/L≈0.25 up toD/L≈10.

As Bean pointed out [3], pinning induced magnetization (and, using the same ar- guments, magnetostriction) increases with the linear dimension perpendicular to B of the sample (thought infinite in the direction of B). This means that magnetostriction does not only depend on the geometry (as, in our case, the ratio D/L), but also on the absolute size of the sample. However, the intention in our investigation was to get a qualitative idea of the influence of sample geometry on the general magnetic field dependence of magnetostriction. An additional aim was to choose a geometry suited for further measurements opening the possibility of quantitative interpretation as, e.g., the determination ofJc(B, T) in a large field range.

Fig. 4.4 shows example magnetostriction measurements at T = 4.2 K on a range of cylindrical samples with different values for D/L. Some of the curves are not closed.

This is due in part to the difference between the unmagnetized state at the beginning of the measurement and the magnetized (trapped flux) state at the end, when starting from a zero-field cooled sample. But part of it is also due to thermal drift of the measuring cell (due to the number of samples having to be measured in a relatively short time, the time between cooling of the system and the measurement was shortened).

The results are remarkable and unexpected. One would be prepared for marked variations, at low fields, due to differences in the demagnetization factor and due to the different absolute sizes. One might also consider an instability of the Euler type when subjecting a flat sample to compressive stress, leading to a qualitatively different result for this single sample (sample ]5, fig. 4.4(d)). However it is figure4.4(c) that offers the biggest surprise in showing a crossing between theB >˙ 0andB <˙ 0curves or, expressed differently, a change of sign in the irreversible (hysteretic) magnetostriction component.

A second feature to be remarked is the small shoulder between the peak and Bc2. In addition, one should mention that apparent differences in the irreversibility field of the order∆Birr≈0.3 Tappear between different samples with no systematic dependence on the sample’s geometry. This means that certain properties, as areB_c2 and B_irr vary slightly from sample to sample. As a reasong for this, one might imagine inhomogeneities in the rod from which all samples have been manufactured, or an influence of mechanical treatment (machining and/or cutting). This topic will be further discussed in the context of resistivity measurements presented in section6.2.2.

The suspicion soon arose that the way the sample is suspended inside the measurement cell might influence the magnetostriction results. In fact, it is clear that for a sample lying on a flat surface, the points where it is suspended microscopically are not well defined experimentally. We therefore performed a series of measurements, using susbstantially different types of suspension, on a single cylindrical sample (sample]4, see fig.4.4(c).

The results are presented in fig.4.5. They convincingly demonstrate the influence of the choice of the points where the magnetostriction force from the sample acts on the measuring device and vice versa. This proves that the sample’s deformation parallel to the magnetic fieldB (and therefore perpendicular to the pinning forces) is not a simple elongation or compression in the direction ofB, but also contains a deformation of the end faces. Some calculations in the vicinity of this topic have been performed [27,13,36], but they ony treat deformations perpendicular to the field in infinite samples (see fig.

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∆ ∆ ∆ ∆

Figure 4.4: Sample geometry: Measurements on cylinders with different values for the parameter D/L. For all measurements,T = 4.2 K.

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32 4.2. MEASUREMENTS ON DIFFERENT SAMPLE GEOMETRIES

2.6for an illustration).

4.2.3 Measurements on a spherical sample

As a second sample allowing measurements perpendicular and parallel to B, magnetostriction was measured on a sphere. H. Wendel from MPI-FKF Stuttgart managed to cut a nearly spherical sample (sample] 7, see fig. 4.1).

To check for reproducibility issues, two measurements were made in every direction (parallel and perpendicular to the field), removing the sample from the cell in between.

One distinguishes (fig.4.6and 4.7) a general curve form, but the signal is not quantitatively reproduced. For the case of the parallel measurement, the differences are prominent mostly in the peak region, pointing to differential effects (notice the double peak structure in one curve and the sign change in the other) which will be discussed in section4.3. For the case of the measurement being perpendicular toB, the quantitative difference is even more pronounced. Taking into account that the sample inside the cell was rotated by an unknown angle between the measurements (for measurements parallel toB this plays no role), one could presume that a slight anisotropy in the (polycristalline) sample might have caused an additional deviation. Such an anisotropy might result from a mechanical treatment during the manufacturing of the sample or from slight inhomogeneities in the original rod from which the samples were cut (see section6.2.2). To avoid possible future problems from such an anisotropy, most measurements where performed in the direction of the magnetic field.

4.2.4 Measurements on a thin cylinder

From all the samples presented, the long, thin cylinder (sample ]6, D/L = 1.5/6.8 ≈ 0.22 1) measured parallel to the external magnetic field is the one least subject to demagnetizing effects. Our magnetostriction measurements on different cylindrical geometries also show comparatively little qualitative differences between the two cylinders with the largest D/L (fig. 4.4 (a) and (b)), suggesting that one might already be close to the asymptotic case of the infinite cylinder, and promising at least satisfactory reproducibility.

The thinnest cylinder (sample ] 6) was thus the best choice to abstract at least partially from geometry related effects. Further measurements (chapter 8 and 9) were mostly done on this sample, with the cylinder axis parallel toB(see section4.2.2). Figure 4.8(a) shows magnetostriction measurements at several temperatures for this configuration, measuring the length L of the sample, in a comparison to measurements on the diameter D (fig. 4.8(b)). Since D = 1.5 mm 6.8 mm = L, the signal-to-noise ratio is significantly smaller and the background signal from the cell becomes more important for the measurements onD.

A calculation of the apparent Poisson ratio ν from the irreversibility in the above measurement (T = 1.5 K) ranges from ν = 0.47 for B = 4 T up to ν = 0.6 > 0.5 in the peak region, which is unphysical and must be considered as a finite size effect, if one assumes that there is only negligible volume magnetostriction.

4.2.5 Measurements on a hollow cylinder

From the information obtained by geometry related measurents, (see also discussion below) we deduced that we had to combine the following properties to obtain quantitatively relevant data:

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∆ ∆ ∆

Figure 4.5: Influence of a sample’s suspension on magnetostriction measurements. The dark (red) surface presents the area of contact between the sample and the sample holder/measurement cell. For all measurements, the suspension was of the same type on both sides of the sample. See above for an explanation for curves that are not closed.

For all measurements, T = 4.2 K.

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34 4.2. MEASUREMENTS ON DIFFERENT SAMPLE GEOMETRIES

∆

Figure 4.6: Two magnetostriction measurements under the same external conditions on a spherical sample, withB parallel to the measuring direction

∆

Figure 4.7: Two magnetostriction measurements under the same external conditions on a spherical sample, withB perpendicular to the measuring direction

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Figure 4.8: Comparison of magnetostriction measurements on length and diameter of a cylindrical sample oriented parallel to the magnetic field. The arrows through the curves point in the direction of increasing temperature.

Magnetostrictive, magnetic, and transport properties of correlated electron systems