structure in the mixed phase gautam i. menon imsc, chennai, india
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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 findinga “pancake” vortex a specified distanceaway from another
one
Correlation Functions
Defines average density at r:
A correlation function
The two-point correlation function in a fluid depends only on the relative distance between two points, by rotational and translational invariance.
Sum over all particles
Related to the probability of finding a particle at r1, given a distinct particle at r2
Correlation Functions II
Defines a structure factor
In terms of Fourier components of the density
From the previous definition of (r)
Brackets denote a thermodynamic average
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) = Nq,G
Implies
Correlation Functions III
Inserting the definition
In terms of n2
Correlations IVDefines 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?
Precise consequence for small angle neutron scattering experiments: S(q) decay about (quasi-) Bragg spots
The Bragg Glass proposal
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, short-ranged translational order.
• Topological defects: transitions driven by dislocation and disclination unbinding
Orientational Correlations
Hexatic
Hexatic vs Fluid Structure
Muon-Spin Rotation
Positively charged muons from an accelerator
Muons polarized transverse to applied magnetic field. Implanted within the sample
The -SR Method I
What the muons see
Muons precess in magnetic field due to vortex lines
Muons are unstable particles. Decay into positrons, anti-neutrinos and gamma rays
Muon Spin Rotation II
Muon lifetime » 10-6 s. Muon decay ! positron emitted preferentially with respect to muon polarization. Emitted positron polarization recorded
Muon Spin Rotation III
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 ofmuons 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
Muon-Spin Rotation
Density of vortex lines
Field at point r
In Fourier space. A is the area of the system
Muon Spin Rotation II
Flux quantum
Muon Spin Rotation VI
The sum is over reciprocal lattice vectors of a triangular lattice
Assuming a perfect lattice
Muon-Spin Rotation Spectra
Sonier, Brewer and Kiefl, Rev. Mod. Phys. 72, 769 (2000).
<ΔB>1
λ2
_
This experiment:•no spontaneous fields present greater than ~0.03G above 2.5K
0.1G0.05G
The rate of muon depolarisation in zero-field µSR (ZF-µSR) is a sensitive probe for spontaneous internal magnetic fields.
MgCNi3
•Tc=7K
• Functional form implies s-wave gap
Results:
nnss/m/m**-2-2
MgCNi3
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 glassU.K. Divakar et al. PRL (2004)
Phase Behavior from SR
Probing the glassy stateand its localcorrelations
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
Essmann and Trauble (1968)
Evaporate magnetic material (fine ferromagnetic grains) onto the surface of the sample
Image
Decoration Data
MgB2YBCO
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.
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