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.