Download Gautam Menon

Structure in the mixed phase
Gautam I. Menon
IMSc, Chennai, India
The Problem
• Describe structure in a
compact manner
• Correlation functions
• Distinguish ordered and
disordered states. Also
unusual orderings: hexatic
Information: Flux-line coordinates as functions of time
Vortex Structures
• Lines/tilted lines
• Pancake vortices in
layered systems in
fields applied normal
to the layers
• Josephson vortices in
layered systems for
fields applied parallel
to the planes
• Vortex chains and
crossing lattices for
layered systems in
general tilted fields
Address via correlation functions
Probability of finding
a “pancake” vortex
a specified distance
away from another
one
Correlation Functions
Defines average density at r:
Sum over all particles
A correlation function
Related to the probability of finding a particle at r1, given a distinct particle at r2
The two-point correlation function in a fluid depends only on the relative
distance between two points, by rotational and translational invariance.
Correlation Functions II
Brackets denote a thermodynamic average
Defines a structure factor
From the previous definition of (r)
In terms of Fourier
components of the density
Correlation Functions in a Solid
This sum is over lattice sites. It is non-zero only if
q=G (a reciprocal lattice vector), in which case it
has value N, i.e. (q) = Nq,G
Implies
Correlation Functions III
Inserting the definition
In terms of n2
Correlations IV
Defines g(r)
From g(r), S(q)
Just removes an uninteresting q=0 delta-function
Why are correlation functions
interesting?
Experiments measure them!
Theorists like them ……
The generic scattering experiment measures
precisely a correlation function
and from there g(r)
Physical Picture of g(r)
Area under first peak measures number of neighbours
in first coordination shell
Scattering
Intensities as functions of q
Melting from Neutron Scattering
Bragg spots go to rings:
Evidence for a melting
transition
Ling and collaborators
The Disordered Superconductor
• Larkin/Imry/Ma: No
translational long-range
order in a crystal with a
quenched disordered
background.
• Natterman/Giamarchi/Le
Doussal: This doesn’t
preclude a more exotic
order, power-law
translational correlations
The Bragg Glass
Different types of Ordering
What does long range order mean?
What does quasi-long range order mean?
What does short-range order mean?
The Bragg Glass proposal
Precise
consequence
for small angle
neutron
scattering
experiments:
S(q) decay
about (quasi-)
Bragg spots
More exotic forms of ordering
Hexatics
• In 2-d systems,
thermal fluctuations
destroy crystalline
LRO except at T=0.
Positional order
decays as a power
law at low T
• But, orientational
long-range order can
exist at finite but low
temperatures
Hexatics
• In the liquid, short range
order in positional and
orientational correlations
• How do power-law
translational order and the
orientational long-range
order go away as T is
increased?
• Must be a transition – one
or more?
Hexatics: Nelson/Halperin
• Two transitions out of
the low T phase
• Intermediate hexatic
phase, power-law
decay of orientational
correlations, shortranged translational
order.
• Topological defects:
transitions driven by
dislocation and
disclination unbinding
Orientational Correlations
Hexatic
Hexatic vs Fluid Structure
Muon-Spin Rotation
The -SR Method I
Positively charged muons from an accelerator
Muons polarized transverse to applied
magnetic field. Implanted within the sample
What the muons see
Muon Spin Rotation II
Muons precess in magnetic field due to vortex
lines
Muons are unstable particles. Decay into
positrons, anti-neutrinos and gamma rays
Muon Spin Rotation III
Muon lifetime » 10-6 s. Muon decay ! positron
emitted preferentially with respect to muon
polarization. Emitted positron polarization recorded
Muon Spin Rotation IV
The Principle: Can reconstruct the local magnetic
field from knowledge of the polarization state
of the muon when it decays
Need to average over a large number of
muons for good statistics
Muons are local probes
Muon Spin Rotation V
The magnetic field distribution function
Moments of the field distribution function
Moments contain important information, obtain l
Muon-Spin Rotation
Field at point r
Density of vortex
lines
In Fourier space. A is the area of the system
Muon Spin Rotation II
Flux quantum
Muon Spin Rotation VI
Assuming a perfect lattice
The sum is over reciprocal lattice vectors of a
triangular lattice
Muon-Spin Rotation Spectra
_
<ΔB>1
λ2
Sonier, Brewer and Kiefl, Rev. Mod. Phys. 72, 769 (2000).
The rate of muon depolarisation in zero-field µSR (ZF-µSR)
is a sensitive probe for spontaneous internal magnetic fields.
0.1G
0.05G
MgCNi3
This experiment:
•no spontaneous fields present greater
than ~0.03G above 2.5K
MgCNi3
 ns/m*l-2
Results: •Tc=7K
• Functional form implies s-wave gap
Important information about the superconducting gap
Results from -Spin Rotation
Underdoped LSCO, Divakar et al.
Muon Spin Rotation LSCO
Why do line-widths
increase with field?
Strong disorder
in-plane, almost
rigid rods
The “true” vortex
glass
U.K. Divakar et al. PRL (2004)
Phase Behavior from SR
Probing the
glassy state
and its local
correlations
Lee and collaborators
Lee and collaborators
Lee and collaborators
Menon, Drew, Lee, Forgan, Mesot, Dewhurst ++…..
Three body correlations in the flux-line glass phase
Nontrivial Information about the Nature
of superconductivity: Uemura Plot
NMR and the Mixed Phase
NMR as a Mixed State Probe
Information obtained is virtually identical to that
obtained in Muon-Spin Rotation
But the probe is different
NMR as a Vortex Probe I
• Interaction of nuclear magnetic moment
with local magnetic field splits nuclear
energy levels
• Nuclear magnetic dipole transitions excited
among these levels by applying a RF field
of an appropriate frequency.
• When the frequency of the RF field is such
that the energy is equal to the energy
separation between the quantum states of
the nuclear spin, energy absorbed. The
resulting resonance can be detected.
NMR as a Vortex Probe
• Since the distances between similar
nuclei in a superconductor are small
relative to vortex separation, sample n(B)
by measuring fields at the sites of nuclei.
• Nuclei uniformly distributed, so sampling
is volume-weighted.
NMR as a Vortex probe III:Method
• In “pulsed NMR” observe time-dependent
transverse nuclear polarization or ``free
induction decay'' of nuclear polarization.
• Here an RF pulse is applied to rotate nuclear
spins from the direction of the local magnetic
field . When the RF field is switched off, nuclear
spins perform a free precession around the local
field and relax back to their initial direction
• The frequency of the nuclear spin precession is
a measure of the local field
• In this technique, different precession
frequencies are observed simultaneously.
NMR as a Vortex Probe IV: Limitations
• Several limitations and added difficulties
associated with the NMR technique which are
overcome in a SR experiment.
• Because the skin depth of the RF field probe is
small, NMR only probes the sample surface. Often
the surface has many imperfections, so strong
vortex-line pinning and a disordered vortex lattice
• The penetration depth of the RF field also limits
the range over which the vortex lattice can be
sampled. Plus additional sources of broadening.
Magnetic Decoration
Decoration Experiments
Evaporate magnetic material
(fine ferromagnetic grains)
onto the surface of the
sample
Image
Essmann and Trauble (1968)
Decoration Data
YBCO
MgB2
Magnetic Decoration
• Several issues: Nature of ordering, how
good are the lattice which are formed
• Hexatic phases
• Correlation between top and bottom of the
sample – how do vortex lines thread the
sample?
• Glassy phases, short-range order
• Melting? Flux-line movement across short
times
Delaunay Triangulation
Fasano et al, PRB’02
Domain States?
Problems?
•
•
•
•
•
Confined really to low fields
Not bulk, only surface information
Useless for dynamics – only static pictures
Yet .. some indicator of lattice quality
Orientational order at surfaces .. maybe
the best way of looking at it
Finally ..
• The structural probes I talked about all
complement each other
• Each provides valuable information, yet
misses many other important things
• Probing at this “mesoscopic” scale is
surprisingly difficult, considering that we
can image the structure of complex protein
molecules to a precision of a few
Angstrom ……………… food for thought.