Investigations into unconventional superconductors through the use of small angle neutron scattering

194 Pages·2014·9.18 MB·English

by  A. Cameron

INVESTIGATIONS INTO UNCONVENTIONAL
SUPERCONDUCTORS THROUGH THE USE OF
SMALL ANGLE NEUTRON SCATTERING
ALISTAIR CAMERON
A thesis submitted to
The University of Birmingham
for the degree of
DOCTOR OF PHILOSOPHY
School of Physics and Astronomy
University of Birmingham
September 2013
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Abstract
HerewepresentaninvestigationintothevortexlatticeofKFe As (KFA),BaFe (As P )
2 2 2 1−x x 2
(BFAP) an YBa Cu O (YBCO) by small angle neutron scattering (SANS). KFA and
2 3 7
BFAP are members of the pnictide group of superconductors which, since the discovery
of an iron-based superconductor in 2006 (Kamihara et al., 2006), have become perhaps
the most investigated class of superconductor in recent years. As a relatively new class
of superconductor there is a great deal of interest in clarifying the fundamental mech-
anism in these materials which leads to superconductivity. Here, through the study of
the magnetic vortex lattice, we search for information about the pairing state of two
members of the 122 group of the pnictide superconductors, finding strong evidence for a
nodal gap structure in both materials. In contrast, YBCO is a member of the cuprate
family of superconductors, discovered in 1987 (Wu et al., 1987) shortly after the discovery
of high temperature superconductivity a year earlier (Bednorz and Muller, 1986). The
cuprate superconductors, so named for the copper-oxide layers which form their common
building block, have been under continuous investigation since their discovery, and here
we present the first microscopic study of vortex matter in fields of up to 16.7 T. We
observe the continuation of a field dependent vortex lattice structure which was observed
in measurements at lower fields (White et al., 2009), and strong indications of a static
Debye-Waller effect arising fromdisorder in the vortex lattice. At high field, vortex lattice
melting is observed at a temperature and field which corresponds to a 1st order melting
transition observed in heat capacity measurements (Roulin et al., 1998), and above the
melting transition no SANS signal from the vortex liquid is observed on the time-scale of
our measurements.
i
For Hilary and Andrew Cameron
ii
Acknowledgements
Firstly I would like to thank my supervisor, Ted Forgan, for giving me the opportunity to
undertake a PhD in the Condensed Matter group. Ted is a exceptional physicist to work
with, possessing a deep understanding of the subject at hand and a far reaching interest
and enjoyment of physics. I have thoroughly enjoyed my time as Ted’s student, and am
exceptionally grateful for all the opportunities given to me to engage with a worldwide
community of people in researching a most fascinating subject.
IwouldliketothanktheCondensedMatterGroupattheUniversity ofBirminghamfor
theenjoyable timeIhave hadduring my PhD. Particularly Iwould like tothankElizabeth
Blackburn, with whom I have enjoyed many an experiment and who is always on hand to
giveinsightfuladvise. AspecialthanksgoestoAlexHolmesforexplaining, overthephone,
what to do with a quenched magnet that has sprung a leak in the middle of the night.
From the devices subgroup, I acknowledge Chris Muirhead and Mark Colclough who also
played memorable roles in my undergraduate education here at Birmingham. From my
office, I thank Josh, George, Bindu, Louis and Lingjia for the interesting conversations,
helpful advice and an enjoyable time as a PhD student. A particular thanks goes out to
John White, for showing me the ropes when I got here and teaching me, among other
things, the basics of neutron experiments.
I would particularly like to thank Hazuki Kawano-Furukawa, who has been a fantas-
tic collaborator throughout our investigation into the pnictide superconductors. Aside
from bringing a multitude of samples with some exceptionally skilled alignment of crystal
mosaics, Hazuki has dedicated a great deal of time and effort into helping us all under-
stand what is going on in a complicated set of superconductors. I look forward to future
collaborations.
I am also grateful for the many interactions I have had with physicists from around
the world during my time at the various neutron scattering institutions where the mea-
surements in this thesis. From the ILL I particularly acknowledge Charles Dewhurst and
Bob Cubitt, who have always provided excellent help and advice during their time as our
iii
local contacts, even at untimely hours of the day, night and weekends. From PSI I thank
Jorge Gavilano for his fantastic assistance during our time on SANS-I.
Imust alsothankmysuperfriendsandbrillianthousemates, whoseclosefriendshiphas
givenmeagreatdealofsupportovertheyearsandledtomorefunandexciting adventures
thanI dare to mention. They deserve a great deal of commendation in displaying patience
during the numerous times when I have brought the conversation into a realm of physics
somewhat removed from everyday life. A special mention goes out to the J-Tones, with
whom I have enjoyed many hours pretending to be a Saxophonist. To them, I promise
that one day I will remember what key the next piece is in.
I save the final thanks for my parents, for all the love and support they have given
me. If I had to put down a moment where physics became something I found interesting
it would have to be sometime around the age of seven, when my father sat me down
and explained a very basic model of the atom. Whilst the idea of wavefunctions and
condensates were still a far distant concept at that point, something about it grabbed my
attention and has yet to let go.
iv
CONTENTS
1 Theory 2
1.1 Thermodynamic description of Superconductivity . . . . . . . . . . . . . . 3
1.2 London Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 London Theory and the Flux Line Lattice . . . . . . . . . . . . . . . . . . 9
1.4 Anisotropic London Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Non-Local corrections to London Theory . . . . . . . . . . . . . . . . . . . 15
1.6 BCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.7 Unconventional Superconductivity . . . . . . . . . . . . . . . . . . . . . . . 22
1.8 Ginzburg-Landau Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.9 The Abrikosov Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.10 Small angle neutron scattering . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.11 Magnetic scattering of neutrons from a periodic potential . . . . . . . . . . 31
1.12 The Ewald Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.13 Instrumental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.14 Resolution of the instrument . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.15 Cryostats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.16 ‘Grasp’ Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2 The Pnictides 47
2.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2 KFe As . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2 2
2.2.1 KFe As sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2 2
2.2.2 Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.2.3 Structure of the vortex lattice . . . . . . . . . . . . . . . . . . . . . 57
2.2.4 Field dependence of the vortex lattice form factor . . . . . . . . . . 63
2.2.5 Temperature dependence of the vortex lattice form factor . . . . . . 68
2.3 Study of the anisotropic vortex lattice where the field is no longer applied
in the basal plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.3.1 Structure of the vortex lattice with B tilted from the c axis . . . . . 74
2.3.2 Angle dependence of the vortex lattice form factor . . . . . . . . . . 78
2.3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.3.4 Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.4 BaFe (As P ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2 1−x x 2
2.4.1 Literature review for BaFe (As P ) . . . . . . . . . . . . . . . . 83
2 1−x x 2
2.4.2 Sample and Experimental Considerations . . . . . . . . . . . . . . . 85
2.4.3 Field dependence of the VL structure . . . . . . . . . . . . . . . . . 87
2.4.4 Field dependence of the vortex lattice form factor . . . . . . . . . . 91
v
2.4.5 Temperature dependence of the vortex lattice form factor . . . . . . 93
2.4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
2.4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
2.4.8 Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3 YBCO 104
3.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.1.1 Crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.1.2 Electronic Structure and Magnetic correlations . . . . . . . . . . . . 108
3.1.3 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.1.4 The question of the pseudogap . . . . . . . . . . . . . . . . . . . . . 113
3.1.5 The Vortex Lattice in the cuprates . . . . . . . . . . . . . . . . . . 114
3.1.6 Previous investigations into de-twinned YBCO . . . . . . . . . . . . 116
3.2 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.3 Investigation with B parallel to the c axis . . . . . . . . . . . . . . . . . . 121
3.3.1 Structure of the Vortex lattice . . . . . . . . . . . . . . . . . . . . . 121
3.3.2 Irreversibility in the Lattice . . . . . . . . . . . . . . . . . . . . . . 128
3.3.3 Investigation into the vortex lattice Form Factor . . . . . . . . . . . 136
3.3.4 Investigation into the vortex lattice Form Factor at higher temper-
atures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
3.3.5 Vortex lattice melting . . . . . . . . . . . . . . . . . . . . . . . . . 149
3.3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
3.3.7 Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
3.4 Other work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
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LIST OF FIGURES
1.1 Figure illustrating the critical field, reduced to have a slope of 1 at t = 1,
−
plotted as a function of reduced temperature and purity, where λ = 0 is
clean and λ = is dirty. Figure taken from Helfand and Werthamer
∞
(Werthamer et al., 1966). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Magnetisation curves of both a Type I and Type II superconductor, with
the critical field H of the type-I superconductor shown alongside the lower
c
critical field H and upper critical field H of the type-II superconductor. 6
c1 c2
1.3 Diagramshowing therotationof thecoordinatesystem describing themag-
netic field components, b away from the Cartesian basis vectors of the
x,y,z
crystal axes by an angle θ. Rotation takes place in the plane described by
the a vector, meaning that a and y remain equivalent directions under this
transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 a) An s-wave superconducting gap, b) a dx2−y2 gap and c) a dxy gap. The
+ and refer to the phase of the order parameter, where the dashed lines
−
represent planes of broken symmetry. . . . . . . . . . . . . . . . . . . . . . 22
1.5 a) With the reciprocal lattice plane perpendicular to the incoming beam,
the Ewald sphere intersects only with the origin of the reciprocal lattice
plane. b) As the reciprocal lattice plane is rotated, the Ewald sphere inter-
sects the reciprocal lattice to satisfy the Bragg condition for the scattering
vector k′. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.6 Schematic of a SANS instrument. Velocity selected neutrons are sent
through a series of collimation apertures to limit divergence, then are
diffracted from the sample before being incident on a detector. . . . . . . . 37
1.7 Figure describing the effect of instrument conditions and vortex lattice
perfection on diffraction. The line AA′ represents a perfect lattice with
the bold line at an angle theta demonstrating diffraction from a perfectly
collimated beam. However, in reality the collimation of the beam may
contribute an angle φ to the diffraction added to an imperfect lattice
1
described by the angle η results in diffraction at an angle φ . . . . . . . . . 39
2
1.8 Figure showing a basic schematic of a cryostat, based on a design from the
AS Scientific Products online catalogue (Scientific). Features common to
most cryostats are labelled, although precise geometry will vary depending
on design. Since this cryostat does not include an internal magnet, an
external one is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
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1.9 Figure showing a basic schematic of a dilution refrigerator. The diagram
shows the area contained within the inner vacuum chamber, around which
will exist the standard cryostat set-up. The inner vacuum chamber is
thermally connected to a low temperature reservoir, such as a 4He bath, to
provide thermal radiation shielding against the warmer background of the
cryostat. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.1 Crystalstructuresoffourgroupsofiron-basedsuperconductorsa)LaFeAsO
“1111”, b)BaFe As “122”,c)LiFeAs“111”andd)FeSe“11”. Figurefrom
2 2
Ishida et al (Ishida et al., 2009). . . . . . . . . . . . . . . . . . . . . . . . . 48
2.2 The temperature - doping phase diagram for pnictide systems, with su-
perconducting (SC) and antiferromagnetic (SDW) states shown. Picture
reprinted from work by Chen et al (Chen, 2009). . . . . . . . . . . . . . . . 49
2.3 Simplified diagram of the proposed Brillouin zone for the pnictides under-
going s pairing. The hole sheet is shone in red at the centre of the zone
±
with the gap +∆ and the electron sheets are shown in blue at the corners
of the zone with the gap ∆. The proposed interaction for s pairing is
±
−
shown by the vector Q, with the spin density wave vector spanning the hole
and electron pockets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.4 Left: Diffraction pattern from the vortex lattice of LiFeAs at 0.25 T, show-
ing a strongly disordered lattice. Right: Rocking curve at the same field,
showing a wide FWHM indicative of vortex pinning leading to deviations
along the length of the vortices. Both images are from data published by
Inosov et al (Inosov et al., 2010). . . . . . . . . . . . . . . . . . . . . . . . 53
2.5 Sample characterisation measurements for KFA crystals. Top: Magneti-
sation data taken in an applied field of 1 mT. Data shows a T of 3.6 K
c
with a 10-90 % transition width of 0.2 K. Bottom: In-plane resistivity mea-
surements, showing a residual resistivity ratio of 400. Inset: enlarged
∼
region of the resistivity-temperature graph showing the area around the
superconducting transition. Figures reproduced from work by Furukawa et
al (Kawano-Furukawa et al., 2011). . . . . . . . . . . . . . . . . . . . . . . 55
2.6 Example of a KFA sample, comprising a series of single crystals co-aligned
and mounted on four aluminium plates. . . . . . . . . . . . . . . . . . . . . 56
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