ThesisPDF Available

Structural investigation of La(2-x)Sr(x)CuO(4+y) - Following staging as a function of temperature

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Abstract and Figures

The cuprate La(2-x)Sr(x)CuO(4+y) – a high-temperature superconductor – was discovered almost three decades ago. However the mechanisms behind the superconductivity in the material for different doping values x and y are still not fully understood. A small part of this large puzzle is added to the pile with this thesis, where results on the structure for several different samples are presented. The emphasis in this thesis is on a specific superstructure thought to be connected to the ordering of interstitial oxygen, known from the isostructural compound La(2)NiO(4+y) as staging. Four single crystal samples with different co-doping values are investigated by the use of both X-rays and neutrons. Staging is observed for all four samples at low temperatures with X-ray measurements. The sample with strontium doping x = 0.00 shows several coexisting staging levels with staging numbers between 2 and 8, with the highest contribution from a staging level between 4 and 5. The co-doped samples show increasing staging number with increasing x. It is found that the staging belongs to a structural phase assumed in space group Fmmm, while the unstaged fraction of the samples are in the Bmab space group. These two structural phases are found to have significantly different lengths of the long crystal axis for the two low x samples, in the order of a fraction of a percent, while the two higher x samples had a difference of only a small fraction of a permille. The temperature dependent phase transitions for both the Bmab structure and the staging reflections are investigated between 5 and 300 K. The critical exponents for the Bmab reflections are found to be significantly lower than results from similar materials in literature, although with transition temperatures consistent with literature for comparable sample compositions. It is found that the critical exponents for the staging reflections increase for increasing doping while the transition temperatures decrease, both consistent with results on the isostructural La(2)NiO(4+y). Results from previous neutron measurements are found to be consistent with the X-ray measurements in this work, and measured reciprocal space maps from this work show a large variety of other superstructure reflections which will be interesting to investigate in the future.
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F A C U L T Y O F S C I E N C E
U N I V E R S I T Y O F C O P E N H A G E N
Master’s thesis
Pia Jensen Ray
Structural investigation of La
2-x
Sr
x
CuO
4+y
Following staging as a function of temperature
Academic advisors: Kim Lefmann and Linda Udby
Submitted: 19 November 2015
ii
buffer
University of Copenhagen
A thesis
submitted to the Faculty of Science
in partial fulfilment of the requirements for the
degree of Master of Science in physics
Niels Bohr Institute
Copenhagen, Denmark
November 2015
Date CPR number Pia Jensen Ray
iii
Contents
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
List of abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
Resumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
I Introductions and theory 1
1 Introduction to the project 3
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 The project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Reading guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Superconductivity 5
2.1 Conventional superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 The Meissner effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 BCS theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 High-temperature superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Type-I and type-II superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.2 Cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.3 Other high-temperature superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Crystallography 13
3.1 Atomic lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.1 Crystal planes and Miller indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1.2 Reciprocal space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Symmetries and space groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2.1 Crystal systems and Bravais lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2.2 The two La
2x
Sr
x
CuO
4+y
crystal systems . . . . . . . . . . . . . . . . . . . . . . . . . . 15
iv Contents
4 Scattering 19
4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.1.1 Scattering cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.1.2 Elastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 Diffraction from crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2.1 The unit cell structure factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2.2 The Laue condition and Bragg’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2.3 The structure factor for specific crystal symmetries . . . . . . . . . . . . . . . . . . 23
4.3 Diffraction from disordered crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3.1 Twinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3.2 Modulated structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.4 X-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.4.1 Extinction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.4.2 X-ray production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.5 Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.5.1 Incoherent scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.5.2 Inelastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.5.3 Neutron production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.6 Choosing between X-rays and neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5 Phase transitions 33
5.1 The order parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.2 First order phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.3 Continuous phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.3.1 The critical exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.3.2 Universality classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.3.3 Saturation of the order parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.3.4 The critical scattering region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
6 The La
2-x
Sr
x
CuO
4+y
compound 37
6.1 The mother compound La
2
CuO
4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6.2 Doping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6.2.1 Doping with strontium to La
2x
Sr
x
CuO
4
. . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6.2.2 Doping with oxygen to La
2
CuO
4+y
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.2.3 Co-doping to La
2x
Sr
x
CuO
4+y
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.3 Crystallographic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.3.1 Twinning from HTT to LTO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.3.2 Staging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
II Experiments 45
7 Measuring staging patterns 47
7.1 The samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7.2 The BW5 instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7.2.1 An overview of the instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7.2.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7.3 The RITA-II instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
7.3.1 An overview of the instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.3.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Contents v
8 Mapping out reciprocal space 55
8.1 The DMC instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8.1.1 An overview of the instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8.1.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
III Results and analysis 57
9 Staging measurements 59
9.1 Initial treatment of BW5 staging data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
9.1.1 Correcting for changing lattice parameter . . . . . . . . . . . . . . . . . . . . . . . . . . 59
9.2 Fitting to find integrated intensities for BW5 staging data . . . . . . . . . . . . . . . . . . . 67
9.2.1 Fitting La
1.96
Sr
0.04
CuO
4+y
peaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
9.2.2 Fitting La
1.935
Sr
0.065
CuO
4+y
peaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
9.2.3 Fitting La
1.91
Sr
0.09
CuO
4+y
peaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
9.2.4 Fitting La
2
CuO
4+y
peaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
9.3 Alternative ways to find the transition temperature . . . . . . . . . . . . . . . . . . . . . . . . 80
9.3.1 Taking extinction and twinning at the phase transition into account . . . . . 80
9.3.2 Slope analysis of the integrated intensities . . . . . . . . . . . . . . . . . . . . . . . . . . 83
9.4 Analysing the Bmab and staging phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . 85
9.4.1 Choosing which part of the data to fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
9.4.2 Fit results for the phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
9.5 Results from RITA-II measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
10 Reciprocal space maps 93
10.1 Raw data from DMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
10.1.1 Combining many datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
10.2 Transformed data as reciprocal planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
10.3 Observations in the reciprocal space maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
10.3.1 Staging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
10.3.2 Other superstructures and forbidden reflections . . . . . . . . . . . . . . . . . . . . . 98
IV Discussion of results 99
11 Discussions 101
11.1 Lattice parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
11.1.1 Two different structural phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
11.1.2 Lattice length as a function of doping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
11.2 Comparing measured intensities for X-ray and neutron data . . . . . . . . . . . . . . . . . 104
11.2.1 About absolute scale intensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
11.3 Discussion of phase transition results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
11.3.1 Critical exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
11.3.2 Transition temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
11.4 Further comments on staging in literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
11.4.1 Phase transition behavior for varying staging number . . . . . . . . . . . . . . . . . 109
11.5 Other superstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
11.6 Peak widths and shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
12 Conclusions 111
12.1 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
vi Contents
Appendixes 113
A Calculations 115
A.1 Derivation of the Bragg condition for scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
A.2 Proof that the Bragg and Laue conditions are equivalent . . . . . . . . . . . . . . . . . . . . 116
A.3 Loss of flux due to typical sample environments on BW5 . . . . . . . . . . . . . . . . . . . . 117
A.4 Typical twinning separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
A.5 X-ray attenuation lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
B Additional crystallographic information 119
B.1 Overview of space group investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
B.2 Examples of crystal refinement results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
C Mathematical functions 123
C.1 Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
C.2 Lorentzian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
C.3 Voigt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
C.4 Power law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
D Additional stag ing figures 127
D.1 All raw BW5 staging measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
D.2 Extra BW5 data analysis figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
E Additional reciprocal space figures 149
E.1 All DMC maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Bibliography I
Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I
Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I
Websites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII
Lookup tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII
Reports and theses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII
Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX
Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX
vii
List of Figures
Below is a list of figures in the report, with short descriptions and their page locations.
2.1 Kamerlingh Onnes’ measurements of resistance in mercury . . . . . . . . . . . . . . . . . . . 6
2.2 Magnetic field lines for a superconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Mediation o f a bosonic pair of electrons via phonons . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Timeline for the history of superconducting compounds . . . . . . . . . . . . . . . . . . . . . . 8
2.5 The magnetization of a superconductor as a function of applied field . . . . . . . . . . . 9
2.6 Magnetic field lines for a superconductor in the vortex state . . . . . . . . . . . . . . . . . . . 9
2.7 Examples of typical cuprate unit cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.8 Typical phase diagram for a cuprate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1 The basis vectors and angles i n a crystal lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Simple examples of crystal planes in a crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 The P, I, F, and C lattice centerings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4 Symmetries o f tetragonal crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5 Symmetries o f ortorhombic crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.1 Particle energies versus wavelengths for differ ent scattering probes . . . . . . . . . . . . . 20
4.2 A sim ple look at scattering vectors and angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.3 Requirements for constructive interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.4 Examples of reciprocal space for twinned and modulated crystals . . . . . . . . . . . . . . 26
4.5 Relating di ffere nt energy units for X-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.6 Relating di ffere nt energy units for neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.7 X-ray and neutron scatteri ng lengths compared . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.1 A typic al phase transition with saturated order parameter and critical scattering . . 36
6.1 The Bm ab crystal structure of La
2
CuO
4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.2 Comparison o f the tetragonal and ortorhombic unit cells . . . . . . . . . . . . . . . . . . . . . 38
6.3 Phase diagrams following dopi ng for La
2x
Sr
x
CuO
4
and La
2
CuO
4+y
. . . . . . . . . . . . . 39
6.4 The twinning of La
2x
Sr
x
CuO
4+y
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.5 Angular distance between twinning reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.6 CuO
6
tilt structures and staging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
7.1 Photos of the samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
7.2 Overview o f the BW5 instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
viii List of figures
7.3 Overview of the RITA-II