Muons in condensed matter research Tom Lancaster Durham University, UK

Condensed matter physics in a nutshell Lev Landau ( )

T>T C T<T C

T>T C T<T C

What can we do with muons? Mu vs. H atom Chemistry: - gases, liquids & solids - best test of reaction rate theories - Study “unobservable” H atom rxns. - Discover new radical species Mu vs. H in Semiconductors: - Until recently, μ + SR → only data on metastable H states in semiconductors! Probing Magnetism: unequalled sensitivity - Local fields: electronic structure; ordering - Dynamics: electronic, nuclear spins Probing Superconductivity: - Coexistence of SC & Magnetism - Magnetic Penetration Depth λ - Coherence Length ξ Quantum Diffusion: μ + in metals (compare H + ) Mu in nonmetals (compare H) Muonium (Mu= μ + e - ) as light hydrogen The muon μ + as a probe

Muon spectrometers at ISIS muSR HiFi

quadrupole magnet photomultiplier tubes cryostat muons Helmholtz magnet

Example: Muons as a probe of magnetism in condensed matter

or Uniformly weakly magnetic Non-magnetic, with strongly magnetic impurities Susceptibility gives average information and therefore can give the same response for the situations sketched above  SR gives local information and therefore can distinguish between these two situations.

Cu(NO 3 ) 2 (pyz) S=1/2 Cu 2+ chains with J=10.3 K Muons reveal order at 107 mK Phys Rev B (2006) We estimate J’=0.046 K

Superconductivity The physics of vortices

Vortices are not stable on their own They cost an infinite amount of energy Due to swirliness at infinity Vortices: topological excitations in condensed matter

One solution: stabilize a vortex with a (magnetic) field This is how a vortex lattice in a superconductor is formed The only way to do this involves quantized flux

Muons measure the superfluid stiffness (or penetration length)

The Uemura plot

What can be done for vortices can be done for other topological excitations Example: the skyrmion lattice

What can be done for vortices can be done for other topological excitations Cu 2 OSeO 3 arXiv:

Semiconductors: muonium as light hydrogen Energy levels of muonium in an applied field Scale set by hyperfine interactions Mu = μ + e - bound state

Semiconductors in brief Hydrogen is present in all semiconductors Muonium is a similar defect Two muonium sites: 1) Tetrahedral 2) Bond centred A rare example of being able to study metastable states in matter

Transport and diffusion Provides fundamental information on quantum diffusion and, more recently, on charge transport in energy materials.

Conclusions Muons are a powerful tool for studying condensed matter systems Recent successes include insights into pnictide superconductivity, low-dimensional magnetism, battery materials… Bare muons are (mainstream) probes of magnetism and superconductivity Muonium spectroscopy remains uniquely powerful in condensed matter and chemistry

Relaxation rate allows us to determine the spin transport mechanism Diffusion: f(ω) ~ ω -1/2 Ballistic: f(ω) ~ ln(J/ω) Rate determined by the autocorrelation function f(ω) F.L. Pratt et al., PRL (2006)

Evidence for spin diffusion In Cu(pyz)(NO 3 ) 2 Power law fits for T N <T<J give n≈0.5 Power law fit gives unphysical value when T>J and when used for 2D compound. F. Xiao et al. in preparation (2014).

The magnet is a toy universe Using molecules we can build magnets with a variety ground states Each ground state is a vacuum for a toy universe

Dimensionality determines the behaviour of the magnet (or universe) In two dimensions: Flatland

Dimensionality determines the behaviour of the magnet or universe In one dimension: Lineland

No magnetic order for Heisenberg chains and planes above T=0 Sidney Coleman ( )

Example: specific heat