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Magnetism in layered ruthenates [Elektronische Ressource] = Magnetismus in geschichteten Ruthenaten / vorgelegt von Paul C. Steffens

193 pages
Magnetism in layered RuthenatesMagnetismus in geschichteten RuthenatenInaugural-Dissertationzur Erlangung des Doktorgradesder Mathematisch-Naturwissenschaftlichen Fakult¨atder Universitat zu Koln¨ ¨vorgelegt vonPaul C. Steffensaus Koln¨Koln, 2008¨Berichterstatter: Prof. Dr. M. BradenProf. Dr. A. RoschVorsitzenderder Prufungsk¨ ommission: Prof. Dr. L. Bohaty´Tag der letzten mundlichen¨ Prufung:¨ 23.10.2007Contents1 Introduction 72 Experimental and theoretical tools 92.1 Inelastic neutron scattering experiments . . . . . . . . . . . . . . . . . . 92.1.1 Magnetic neutron scattering . . . . . . . . . . . . . . . . . . . . . 102.1.2 Using polarized neutrons . . . . . . . . . . . . . . . . . . . . . . 102.1.3 Neutrons and the susceptibility . . . . . . . . . . . . . . . . . . . 122.2 Spin fluctuations in metals . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.1 The susceptibility in the metallic state . . . . . . . . . . . . . . . 15The generalized susceptibility . . . . . . . . . . . . . . . . . . . . 17The exchange interaction . . . . . . . . . . . . . . . . . . . . . . 192.2.2 Calculating the . . . . . . . . . . . . . . . . . . . . 20Relation to the neutron scattering cross section . . . . . . . . . . 23002.2.3 Approximations ofχ (Q,ω) near magnetic instabilities . . . . . . 23Nearly ferromagnetic metal. . . . . . . . . . . . . . . . . . . . . . 23Nearly antif metal. . . . . . . . . . . . . . . . . . . . 242.2.
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Magnetism in layered Ruthenates
Magnetismus in geschichteten Ruthenaten
Inaugural-Dissertation
zur Erlangung des Doktorgrades
der Mathematisch-Naturwissenschaftlichen Fakult¨at
der Universitat zu Koln¨ ¨
vorgelegt von
Paul C. Steffens
aus Koln¨
Koln, 2008¨Berichterstatter: Prof. Dr. M. Braden
Prof. Dr. A. Rosch
Vorsitzender
der Prufungsk¨ ommission: Prof. Dr. L. Bohaty´
Tag der letzten mundlichen¨ Prufung:¨ 23.10.2007Contents
1 Introduction 7
2 Experimental and theoretical tools 9
2.1 Inelastic neutron scattering experiments . . . . . . . . . . . . . . . . . . 9
2.1.1 Magnetic neutron scattering . . . . . . . . . . . . . . . . . . . . . 10
2.1.2 Using polarized neutrons . . . . . . . . . . . . . . . . . . . . . . 10
2.1.3 Neutrons and the susceptibility . . . . . . . . . . . . . . . . . . . 12
2.2 Spin fluctuations in metals . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 The susceptibility in the metallic state . . . . . . . . . . . . . . . 15
The generalized susceptibility . . . . . . . . . . . . . . . . . . . . 17
The exchange interaction . . . . . . . . . . . . . . . . . . . . . . 19
2.2.2 Calculating the . . . . . . . . . . . . . . . . . . . . 20
Relation to the neutron scattering cross section . . . . . . . . . . 23
002.2.3 Approximations ofχ (Q,ω) near magnetic instabilities . . . . . . 23
Nearly ferromagnetic metal. . . . . . . . . . . . . . . . . . . . . . 23
Nearly antif metal. . . . . . . . . . . . . . . . . . . . 24
2.2.4 Contribution of spin fluctuations to the specific heat. . . . . . . . 27
2.3 Spin densities and polarized neutron diffraction . . . . . . . . . . . . . . 29
2.3.1 The measurement of flipping ratios . . . . . . . . . . . . . . . . . 29
2.3.2 Constructing the spin density from flipping ratio data . . . . . . . 31
Maximum Entropy Method . . . . . . . . . . . . . . . . . . . . . . 32
Computation of the maximum entropy map . . . . . . . . . . . . 33
3 Ca Sr RuO and the metamagnetic transition 372-x x 4
3.1 Magnetism in Ca Sr RuO . . . . . . . . . . . . . . . . . . . . . . . . 372 x x 4
3.1.1 Magnetic properties in the metallic state . . . . . . . . . . . . . . 37
3.1.2 The metamagnetic transition . . . . . . . . . . . . . . . . . . . . 43
Metamagnetic transitions, crossovers, and quantum criticality . . 43 transition in the bilayer Ruthenate . . . . . . . . . 45 tr in the single layer Ruthenate – is it
quantum critical? . . . . . . . . . . . . . . . . . . . . . 46
3.1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2 Experimental aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3 Magnetic neutron scattering on Ca Sr RuO . . . . . . . . . . . . . . . 532 x x 4
3Contents
3.3.1 Magnetic origin of the signal and magnetic form factor . . . . . . 53
3.3.2 Spin density in Ca Sr RuO . . . . . . . . . . . . . . . . . . . 551.8 0.2 4
3.4 Magnetic correlations in Ca Sr RuO . . . . . . . . . . . . . . . . . . 581.8 0.2 4
3.4.1 Below the metamagnetic transition . . . . . . . . . . . . . . . . . 58
Detailed structure of the signal. . . . . . . . . . . . . . . . . . . . 60
Energy dependence. . . . . . . . . . . . . . . . . . . . . . . . . . 60
Antiferromagnetic nature of the signal and possible ferromag
netic contribution. . . . . . . . . . . . . . . . . . . . . . 62
Relation to the band structure . . . . . . . . . . . . . . . . . . . . 64
Nesting of theα andβ bands . . . . . . . . . . . . . . . . . . . . 68
Relation to macroscopic measurement methods . . . . . . . . . 68
3.4.2 Temperature dependence . . . . . . . . . . . . . . . . . . . . . . 69
Quasi ferromagnetic signal as function of Temperature . . . . . . 70
Overall evolution of the signal at higher temperatures . . . . . . 72
3.4.3 The dependence on the magnetic field . . . . . . . . . . . . . . . 74
Response at different fields . . . . . . . . . . . . . . . 74
Enhancement of fluctuations at the transition . . . . . . . . . . . 77
3.4.4 Above the metamagnetic transition . . . . . . . . . . . . . . . . . 80
Excitations in the high field state . . . . . . . . . . . . . . . . . . 80
Modelling the spin wave . . . . . . . . . . . . . . . . . . . . . . . 82
Is it a magnon? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.5 Magnetic correlations in Ca Sr RuO . . . . . . . . . . . . . . . . . 881.38 0.62 4
3.5.1 Magnetic response close to the ferromagnetic instability . . . . . 89
3.5.2 A universal description of the magnetic response . . . . . . . . . 91
3.5.3 Temperature dependence . . . . . . . . . . . . . . . . . . . . . . 96
Qualitative description . . . . . . . . . . . . . . . . . . . . . . . . 97
A quantitative model . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.6 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 100
4 Strontium Ruthenate 107
4.1 Sr RuO and spin triplet superconductivity . . . . . . . . . . . . . . . . . 1072 4
4.2 Basic properties of Sr RuO and magnetic fluctuations . . . . . . . . . . 1092 4
4.3 Measurement of magnetic excitations in Sr RuO by inelastic neutron2 4
scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.3.1 Neutron scattering experiments on Sr RuO . . . . . . . . . . . . 1122 4
4.3.2 Polarization analysis . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.3.3 Quantitative of the susceptibility . . . . . . . . . . . . . 118
4.3.4 Comparison with NMR data . . . . . . . . . . . . . . . . . . . . . 121
4.3.5 Further possible implications of the results . . . . . . . . . . . . 123
4.3.6 Magnetic fluctuations in the superconducting state . . . . . . . . 126
4.4 Sr RuO and Ti doping . . . . . . . . . . . . . . . . . . . . . . . . . . . 1282 4
4.4.1 In the ordered state: 9% Ti . . . . . . . . . . . . . . . . . . . . . 129
4Contents
4.4.2 Near the critical concentration: 2.5% Ti . . . . . . . . . . . . . . 132
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5 Bilayer Ruthenates 137
5.1 Spin density in Ca Ru O . . . . . . . . . . . . . . . . . . . . . . . . . . 1373 2 7
5.1.1 The bilayer Ruthenate Ca Ru O and its metamagnetic transition1373 2 7
5.1.2 Measurement of the spin density in Ca Ru O . . . . . . . . . . 1403 2 7
Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.2 Magnetism in Ti doped Sr Ru O . . . . . . . . . . . . . . . . . . . . . . 1473 2 7
5.2.1 Ti doping and magnetic order in Sr Ru O . . . . . . . . . . . . 1473 2 7
5.2.2 Magnetic order probed by elastic neutron scattering . . . . . . . 149
5.2.3 excitations in Sr (Ru Ti ) O . . . . . . . . . . . . . 1543 0.9 0.1 2 7
5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6 Summary, conclusions and outlook 161
A Appendix 165
A.1 The calibration of magnetic scattering intensity . . . . . . . . . . . . . . 165
A.2 Some remarks about the maximum entropy algorithm . . . . . . . . . . 167
Kurzfassung (Deutsch) 169
Abstract (English) 171
Danksagung 173
List of Publications 175
Bibliography 177
Index 188
Offizielle Erklarung 193¨
5Contents
61 Introduction
BydiscussingthemagnetismofthelayeredRuthenates, thisthesis addressesa vari
ety of intriguing phenomena in solid state physics. The restriction on this single class
of materials does by no means limit the number of interesting topics to be discussed:
they include metamagnetic transitions, quantum critical behavior, magnetic order in
general, magnetic fluctuations in highly correlated materials, and unconventional su
perconductivity.
In the recent years, two members of the family of layered Ruthenates have made
these materials widely recognized and attracted significant attention: firstly, this is
Sr RuO – an unconventional superconductor in which the Cooper pairs are most2 4
likely in a triplet state. Secondly, in Sr Ru O the concept of metamagnetic quantum3 2 7
criticality is currently under active debate.
While there are also layered Ruthenates with a higher number of layers, these two
cases – the layered perovskite with one or two layers – are the only ones
considered here. Starting from these two materials, a variety of other substances
are obtained which have very different properties and are of interest on their own
right. Substituting Strontium by Calcium, one can continuously vary the chemical
composition and arrives finally at Calcium Ruthenate (Ca RuO or Ca Ru O ) which2 4 3 2 7
has, driven by structural distortions, entirely different properties. These substances
are therefore very well suited to study the interplay between the structural, electronic,
orbitalandmagneticdegreesoffreedom. Anotherinterestingvariationisachievedby
substituting Ruthenium with Titanium, which leads to magnetically ordered states.
This thesis contains the results of experimental work. For the investigation of mag
netic properties, neutron scattering is an extremely powerful tool. A number of differ
ent neutron scattering techniques has been applied, thereby addressing very diverse
aspects of magnetism in the layered Ruthenates and yielding a detailed picture of
these materials.
This thesis contains four large chapters, which are organized as follows:
• Chapter 2 does not contain experimental results, but briefly summarizes the
foundations of the experimental techniques, some theoretical background in
cluding a summary of the relevant formulae, and details of some computational
methods used in the discussion of the data. The following chapters refer to this
information at many occasions.
71 Introduction
• The longest part of the thesis is Chapter 3, which is devoted to the single
layer Ruthenates with mixed Ca/Sr content, Ca Sr RuO . It focuses on the2 x x 4
paramagnetic metallic region of the phase diagram, in particular on the Sr
concentrations x=0.2 and 0.62. The most interesting physical phenomenon to
be discussed is the metamagnetic transition, which may have similar properties
as the metamagnetic transition in Sr Ru O . Very strong magnetic fluctuations3 2 7
ofdifferentcharacterarerelatedtointerestingbehaviorandarethoroughlystud
iedasfunctionoftemperatureandmagneticfield. Themoststrikingobservation
is how the nearly antiferromagnetic Ca Sr RuO can be turned into a ferro 1.8 0.2 4
magnet by the magnetic field.
• Continuing with the single layer Ruthenates, Chapter 4 discusses the magnetic
fluctuations in Sr RuO , which are highly important as they may be responsible2 4
for (spin triplet) pairing in this material. By characterizing a weak but essential
part of the fluctuation spectrum that had not been identified so far, it provides
a full and consistent description of the magnetic response in Sr RuO . Briefly,2 4
some results are also presented on Sr Ru Ti O .2 1 x x 4
• Finally, Chapter 5 addresses two – rather different – aspects of the magnetism
in the bilayer Ruthenates. In Ca Ru O , a material which also shows a meta 3 2 7
magnetic transition, though quite different from Ca Sr RuO , a spin density1.8 0.2 4
study has been performed. In Ti doped Sr Ru O it is shown that similar to the3 2 7
single layer Sr Ru Ti O magnetic order – an incommensurate spin density2 1 x x 4
wave – is induced at Ti contents of only a few percent, and this magnetic order
is characterized in detail. Furthermore, the excitations have been studied.
82 Experimental and theoretical tools
2.1 Inelastic neutron scattering experiments
Most experimental data in this thesis have been collected in inelastic neutron scatter
ing (INS) experiments. Neutron scattering is in general a very powerful tool for the
investigation of nuclear and magnetic excitations in condensed matter, because the
neutron interacts as well with the nuclei as with the magnetic moments, offers a wide
rangeofenergyandmomentumtransferandcanpenetratedeepinsidethesample. If
suitable(thismeansoften: largeenough)samplesandsufficientbeam timeareavail
able, one can obtain a degree of information on magnetic correlations and magnetic
order that is hardly achievable by any other method.
Themajorityofexperimentsreportedinthefollowinghavebeenperformedontriple
axis spectrometers; techniques like time of flight spectrometers have not been used
for the results presented here. The triple axis that have been used
work either with cold or with thermal neutrons, thereby covering the range of energy
transfers~ω betweenapproximately0.3meVuptoabout10meV,whichhasbeenthe
interesting range for most problems. (An order of magnitude higher energy transfers
are possible with the technique and frequently used for other problems.)
Thereareseveraltextbooksonneutronscattering, whichcoverallrelevantaspects
of the technique and the underlying theory in extensive detail. Classic references are
for instance Marshall Lovesey’s book [1,2], from which most information was taken,
or the one of Squires [3]. A newer one with a focus on magnetic scattering is the one
of Chatterji [4]. Another recent and very useful one which emphasizes the technical
aspects of triple axis spectrometry, is the book of Shirane [5].
Because this thesis includes no technical development in the field, the reader is
referred to these books, and the extensive theory of neutron scattering shall not be
reviewedhere. Inthefollowingsectionsonlysomeofthemostrelevantformulasshall
be briefly summarized.
The measurements mainly focused on the magnetic scattering in paramagnetic
metallic states. The typical magnetic excitations here are fluctuations. They do not
havethecharacterofwell defineddispersivebrancheslikeforinstancemagnons, but
form a continuum of excitations which contains the information on the spin correla
tions. As the excitations of the itinerant electrons, they are closely connected to the
electronic band structure and Fermi surfaces.
92 Experimental and theoretical tools
2.1.1 Magnetic neutron scattering
In magnetic neutron scattering, the neutron is scattered by the interaction between
themomentassociatedwithitsspinandamagneticmomentinthesample.
The inelastic scattering is usually expressed in terms of the imaginary part of the
wave vector and frequency dependent magnetic susceptibility χ(Q,ω). The formula
for the cross section then reads [1,4]
2 2 Xd σ k r 1f 0 2 −2W(Q) 00ˆ ˆ=N F(Q) e · (δ −Q Q )·χ (Q,ω) (2.1)αβ α β~ω αβ2 −dΩdE k 4πμ k Ti B B1−e αβ
2γewith r = = −5.4 fm, which gives a measure of the strength of the interaction0 2m ce
(same order of magnitude as nuclear scattering lengths). Of special importance are
three factors: (i) the magnetic form factor F(Q), which implies that the signal is large
only at small momentum transfers Q, (ii) the Bose factor, which takes account of the
thermalpopulationofthestatesandtheprincipleofdetailedbalance,and(iii)thesum
contains the geometrical effect that determines how the different components of the
tensorχ contribute. In many cases this term can be simplified to
00 00ˆ ˆ(1−Q )χ +(1+Q )χ (2.2)z zzz xx
Here it becomes evident that it is only the component of χ perpendicular to Q which
contributes to the cross section (χ = χ ). The second important remark is that inxx yy
thisstepχhasbeensplitintoalongitudinal(χ )andtransverse(χ )part. Thechoicezz xx
of z as a quantization and eventual field direction is arbitrary at this point, and in a
paramagnetic state without anisotropies or external magnetic field, all components
would of course be identical. (Theoretically, the condition for the above simplification
zˆis thatS is a constant of motion.)tot
2.1.2 Using polarized neutrons
Applying (2.2) one can in principle, by measuring at different equivalent Q vectors,
00 00separate the two components χ and χ . In some cases this is quantitatively veryxx zz
inaccurate, because other factors like the experimental geometry, form factor effects,
resolution etc. introduce additional errors. The use of polarized neutrons can provide
a great amount of additional information.
Insuchacasethespinoftheneutronsintheincidentbeamispolarizedinacertain
direction,andthechangeoftheneutronspinisanalyzedafterthescatteringprocess,
i. e. one measures aspin flip and anon spin flip intensity. It is possible to choose the
polarization of incident and scattered neutrons completely independently in order to
measure all components (including the off diagonal ones) of χ when using a setup
named CRYOPAD. When one is not interested in these components – and in the
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