Muon Spin Rotation / Relaxation

Wir schaffen Wissen – heute für morgen

Introduction to Muon Spin Rotation and Relaxation - Instrumentation and Technique

Hubertus Luetkens

Laboratory for Muon Spin Spectroscopy Paul Scherrer Institut Muon is a Local Magnetic Probe

Muon probes the local magnetism Muon probes the local magnetic from within the unit cell response of a superconductor (Meissner screening or flux line lattice)

µ+

~Å

~100nm

Hubertus Luetkens Outline

1. Muon Properties – Pion decay – Muon decay – Parity violation – Muon spin precession

2. Muon Spin Rotation / Relaxation (µSR) – Facilities around the world – Muon production at PSI – µSR instruments at PSI – µSR principle – Muon thermalization / Muon stopping sites / Muon stopping ranges – Measurement geometries

3. Muon Spin Rotation / Relaxation on Magnetic Materials – Different static depolarization functions and examples – Magnetic phase separation / coexistence of different magnetic phases – Magnetic fluctuations

4. Muon Spin Rotation on Superconducting Materials – Using low energy µSR to study the Meissner state of superconductors – Using bulk µSR to study the Vortex state of superconductors – Superfluid density and the symmetry of the superconducting gap – Magnetic and superconducting phase diagrams of Fe-based materials

5. Summary

Hubertus Luetkens Muon Properties

Hubertus Luetkens What is a Muon? Cosmics

Muon Flux at sea level: ~ 1 Muon/Minute/cm2

Mean Energy: ~ 2 GeV

Hubertus Luetkens Muon Properties

Elementary particle/antiparticle:

mass: ∼ 200 x electron mass

(105.6MeV/c2)

∼ 1/9 x proton mass charge: + e, oder - e spin: 1/2

magnetic moment : 3.18 x µp (8.9 x µ ), g ≈ 2.00 N gyromagnetic ratio: 85.145 kHz/G unstable particle: mean lifetime: 2.2 µs N(t) = N(0)exp(-t/τµ)

Hubertus Luetkens Muon as Result of Pion Decay

π+

+ (ud) µ νµ

S = 0 (26ns)

Two-body decay  muon has always the energy 4.1 MeV in the reference frame of the pion (assuming mν = 0)

Spin pion = 0  Muon has a spin 1/2 and is 100% polarized (as only left-handed neutrinos are produced)

Hubertus Luetkens Muon Decay

νµ + µ e+

2.2 µs νe

Three-body decay  Distribution of positrons energies Weak-decay of muon  Parity-violation leading to positrons emitted anisotropically

Hubertus Luetkens Anisotropic Muon Decay

Angular distribution of positrons from the parity violating muon decay:

W (E,θ ) = 1+ a(E) cos(θ )

The asymmetry parameter a = 1/3 when all positron energies E are sampled with equal probability.

Positrons preferentially emitted along direction of muon spin at decay time

By detecting the spatial positron emission as a function of time  time evolution of muon spin !!!

Hubertus Luetkens Muon Spin Precession - Bloch Equation

Rabi Oscillations B, z B, z S x and S y precess with

the Larmor frequency ωL Sz Ehrenfest Theorem Bloch Equation   B dm  = γ m× B m dt

θ Larmor frequency ωL = γ B

Hubertus Luetkens Summary - Muon Properties

The three key properties making µSR possible:

1. The muon is 100% spin polarized.

2. The decay positron is preferentially emmitted along the muon spin direction.

3. The muon spin precesses in a magnetic field.

Hubertus Luetkens Muon Spin Rotation / Relaxation

Hubertus Luetkens µSR Facilities around the World pulsed (50Hz) continuous muon beams muon beams

continuous J-PARC (2009) muon beams

pulsed (50Hz) muon beams

From “µSR brochure” by J.E Sonier, Simon-Fraser-Univ., Canada, 2002. http://musr.org/intro/musr/muSRBrochure.pdf Hubertus Luetkens Muon Production at PSI

600 MeV Proton Cyclotron at PSI: Muons from Pion Decay:

Muon kinetic energy: 4 MeV

Hubertus Luetkens PSI Experimental Hall

Target M Thin graphite (5 mm)

Target E 4cm graphite

“necessary component to expand proton beam before SINQ”

Hubertus Luetkens SµS – The Swiss Muon Source

LEM High Field Low-energy muon beam and instrument , tunable energy µSR (0.5-30 keV, µ+), thin-film, near-surface and multi-layer Muon energy: studies (1-300 nm) 4.2 MeV (µ+) 0.3T, 2.9K 9.5T, 20mK

DOLLY General Purpose Surface Muon Instrument GPS Muon energy: 4.2 MeV (µ+) General Purpose Surface Muon Instrument Muon energy: 4.2 MeV (µ+) 0.5T, 300mK 0.6T, 1.8K

Shared Beam Surface Muon Facility (Muon On REquest) GPD Experimental Hall General Purpose Decay Channel Instrument LTF Muon energy: 5 - 60 MeV + - Low Temperature Facility (µ or µ ) Muon energy: 4.2 MeV (µ+) 0.5T, 300mK 3T, 20mK- 4K 2.8GPa

Hubertus Luetkens Principle of a µSR Experiment

Implantation of muons into the sample

Muon

Mass: mµ ≈ 207 me ≈ 1/9 mp Magnetic moment: µµ ≈ 3 µp Charge: +e Lifetime: tµ ≈ 2.2 µs Polarisation: 100 %

Sample

Hubertus Luetkens Principle of a µSR Experiment

InteractionAnisotropicDetection of ofmuon the the decay muondecay positronspin in the with direction the magnetic of the muonenvironment spin

Positron

Positron- Detector

Sample µSR 021 µs Static field distribution Fluctuation rates -3 -4 5 9 Sensitivity: 10 – 10 µB (10 – 10 Hz)

Hubertus Luetkens μSR – Muons Stopping in Matter

Muon Surface Muons Production Energy: 4.1 MeV v ≈ 0.27 c

105 – 106 excess e– Ionization of atoms 2-3 keV

Sample v ≈ 0.007 c • Total stopping time in matter < 0.1 ns Muonium formation (μ+e–) charge cycling 100 eV • Only electrostatic processes v ≈ 0.0013 c → no loss of polarization!

mainly elastic collisions • Penetration ≈ 0.1 mm (dissociation, μ+e– → μ+) stop at interstitial site

thermalized μ+ stopping at an interstitial site

Hubertus Luetkens Interstitial Muon Stopping Sites Electrostatic potential map: CeOFeAs structure:

Positive muon likes to stop: • In the potential minimum • High symmetry sites • Near negative ions ( e.g. O2-, As3-) (muon hydrogen bond like in OH with ~1 A bond length) • Large spaces in the crystal structure

Hubertus Luetkens Muon Implantation Depth

104 Cu Cosmic muons Bulk µSR: 102 Decay beam  “Normal” samples (sub-mm) Bulk µSR 100  Bulky samples + samples in containers or Surface beam pressure cells

-2 10 LE-µSR: Range (mm) Range 10-4  Depth-selective investigations (1–200 nm) LE µSR 10-6 10-4 10-2 100 102 104 Energy (MeV)

Hubertus Luetkens µSR

Bµ

Bµ internal or external field

Hubertus Luetkens μSR Spectra Muon Spin Muon Spin Polarization )~ t  ( Frequency Value of field x x at muon site aP (ω = γ B ) L m m

Damping  Field distribution and/or dynamics

Hubertus Luetkens Field Direction -- Field Distribution

θ detector

+ e

θ angle between magnetic field and muon polarization at t = 0

Hubertus Luetkens Different Measurement Geometries

ZF and LF: Zero field and Longitudinal Field geometry

Typically used for the study of: • Static magnetism • Temperature dependence of the magnetic order parameter • Determination of the magnetic transition temperature • Homogeneity of the sample • Dynamic magnetism • Determination of magnetic fluctuation rates • Slowing down of fluctuations near phases transitions

Hubertus Luetkens Different Measurement Geometries

TF: Transverse Field geometry

Typically used for the study of: • Magnetism • Determination of the magnetic transition temperature • Homogeneity of the sample • Knight shift (local susceptibillity) • Superconductivity • Absolute value and temperature dependence of the London penetration depth • Coherence length, votex structure, vortex dynamics, …

Hubertus Luetkens Muon Spin Rotation / Relaxation on Magnetic Materials

Hubertus Luetkens Magnetism

Hubertus Luetkens Magnetism

Paramagnetism Ferromagnetism

S S S S S S S

N N N N N N N

S S S S S S S

N N N N N N N

fluctuating static T>TC T

How do you measure this? SQUID, PPMS, … Macroscopic techniques (average over the hole sample):

Hubertus Luetkens Magnetism

Paramagnetism Ferromagnetism Antiferromagnetism

S S S S S S S S S S

S N N N

N N N N N N N N N N

N S S S

S S S S S S S S S

S N N N N

N N N N N N N N N

N S S S S

fluctuating static static T>TC T

How do you measure this? Macroscopic techniques (average over the hole sample):

Hubertus Luetkens Magnetism

Scattering techniques: Local probes: (neutrons, X-rays) (µSR, NMR, …)

Strength of muon spin rotation / relaxation: Investigation of magnetically inhomogeneous materials: • Chemical inhomogeneity (“dirty samples”) • Competing interactions, coexistence of different magnetic orders, short range order, magnetic frustration • Magnetism and superconductivity (competition and coexistence)

Hubertus Luetkens Magnetically Inhomogeneous Materials

Homogen: 1.0 0.8 0.6 Mhom 0.4 0.2 0.0 -0.2 -0.4 -0.6 Muon Spin PolarisationMuon -0.8 -1.0 0 1 2 3 4 5 6 7 8 9 10 Time (µs)

Inhomo- 1.0 0.8 geneous: 0.6 0.4 0.2 Minhom = Mhom 0.0 -0.2 -0.4

Muon Spin PolarisationMuon -0.6 -0.8 -1.0 0 1 2 3 4 5 6 7 8 9 10 Time (µs) Amplitude = Magnetic volume fraction Frequency = Size of the magnetic moments (order parameter) Damping = Inhomogeneity within the magnetic areas Hubertus Luetkens Sensitivity of the Technique

m

µSR time window: 10-20 µsec  static moments as low as 0.001 μB can be detected by μSR  Frequencies down to 50 kHz detectable Fields of few Gauss (10-4 T)

Hubertus Luetkens Different Depolarization Functions

dedicated P(t) for each problem

Hubertus Luetkens Magnetism of Single Crystals

Hubertus Luetkens Simple Magnetic Sample – Single Crystal

Single Crystal with θ = π / 2

Single Crystal with θ ≠ π / 2

 in a single crystal the amplitude of the oscillatory component indicates the direction of the internal field

Hubertus Luetkens Example:

Single Crystalline UGe2

Hubertus Luetkens Single Crystalline UGe2

Cavendish Lab.

• Two magnetically inequivalent muon stopping sites

• No oscillations for P(0) pointing along a-axis  Internal fields parallel to a-axis (θ = 0) • Oscillations around zero  because single crystal

S. Sakarya et al., Phys. Rev. B 81, 024429 (2010)

Hubertus Luetkens Magnetism of Polycrystals (Powders)

Hubertus Luetkens Simple Magnetic Sample – Polycrystal

Polycrystal (powder)

Hubertus Luetkens Simple Magnetic Sample – Polycrystal

Polycrystal: Ideal Case

Hubertus Luetkens Simple Magnetic Sample – Polycrystal

Polycrystal: Ideal Case Polycrystal: Real Case

►

►

►

Hubertus Luetkens Example: Magnetism of Polycrystalline LaOFeAs

Hubertus Luetkens Polycrystalline LaOFeAs

• Zero Field Muon Spin Rotation • Static commensurate magnetic order T-dependence of the Fe magnetization with high precision, TN = 138 K 100% magnetic volume fraction

Mössbauer spectroscopy

Hyperfine field Bhf (0) = 4.86(5) T => Saturation moment µ = 0.25 - 0.32 µB

Neutron scattering

Saturation moment µ = 0.36(5) µB

Hubertus Luetkens Polycrystalline LaOFeAs

• Zero Field Muon Spin Rotation • Static commensurate magnetic order • T-dependence of the Fe magnetization with high precision, TN = 138 K • 100% magnetic volume fraction

Mössbauer spectroscopy

Hyperfine field Bhf (0) = 4.86(5) T => Saturation moment µ = 0.25 - 0.32 µB

Neutron scattering

Saturation moment µ = 0.36(5) µB

Hubertus Luetkens Polycrystalline LaOFeAs

• Zero Field Muon Spin Rotation • Static commensurate magnetic order • T-dependence of the Fe magnetization with high precision, TN = 138 K • 100% magnetic volume fraction

O La Mössbauer spectroscopy µ Hyperfine field Bhf (0) = 4.86(5) T => Fe Saturation moment µ = 0.25 - 0.32 µB As Neutron scattering

Saturation moment µ = 0.36(5) µB

Hubertus Luetkens Polycrystalline LaOFeAs

• Zero Field Muon Spin Rotation • Static commensurate magnetic order • T-dependence of the Fe magnetization with high precision, TN = 138 K • 100% magnetic volume fraction

contact field

• Mössbauer spectroscopy

• Hyperfine field Bhf (0) = 4.86(5) T => Saturation moment µ = 0.25 - 0.32 µB

Neutron scattering

Saturation moment µ = 0.36(5) µB

Hubertus Luetkens Polycrystalline LaOFeAs

• Zero Field Muon Spin Rotation • Static commensurate magnetic order • T-dependence of the Fe magnetization with high precision, TN = 138 K • 100% magnetic volume fraction

contact field

• Mössbauer spectroscopy

• Hyperfine field Bhf (0) = 4.86(5) T => Saturation moment µ = 0.25 - 0.32 µB

• Neutron scattering

• Saturation moment µ = 0.36(5) µB

C. de la Cruz et al., Nature 453, 899 (2008) Hubertus Luetkens Randomly Oriented Magnetic Moments - Short Range Magnetism - Magnetic Disorder

Hubertus Luetkens Randomly Oriented Moments m

Hubertus Luetkens Randomly Oriented Moments

Kubo-Toyabe function

Hubertus Luetkens Example: Kubo-Toyabe depolarization due to nuclear moments

InN and MnSi

Hubertus Luetkens InN & MnSi InN MnSi Semiconductor system lacking inversion symmetry Study of the hydrogen-related defect chemistry itinerant-electron magnet MnSi

Y.G. Celebi et al., Physica B 340-342, 385 (2003) R.S. Hayano et al., Phys. Rev. B 20, 850 (1979) Polarization

In the paramagnetic state, a KT function is very often observed reflecting the small field distribution created by the nuclear moments Hubertus Luetkens Detection of Magnetic Phase Separation -

Coexistence of Different Magnetic Phases

Hubertus Luetkens Example:

Magnetic Phases of Polycrystalline YBaCo2O5.5

Hubertus Luetkens Magnetic Structures in YBaCo2O5.5 ZF-µSR on YBaCo O : 2 5.49 Different magnetic phase – different µSR signal • PM – non-magnetic • FM – Ferrimagnetic (simple structure) • AFM1 – antiferromagnetic (complicated) • AFM2 – antiferromagnetic (complicated)

Multiple µSR frequencies • Complicated magnetic structure with increased magnetic unit cell • Phase separation into various magnetic structures

Temperature dependence f(T) Second order transition (smooth change) First order transition (abrupt change)

Phase separation and complicated magnetic structure

Hubertus Luetkens Magnetic Structures in YBaCo2O5.5 Fourier analysis of YBaCo O : 2 5.49 Different magnetic phase – different µSR AFM2 AFM1 F PM signal • PM – non-magnetic • FM – Ferrimagnetic (simple structure) • AFM1 – antiferromagnetic (complicated) • AFM2 – antiferromagnetic (complicated)

Multiple µSR frequencies • Complicated magnetic structure with increased magnetic unit cell • Phase separation into various magnetic structures

Temperature dependence f(T) • Second order transition (smooth change) • First order transition (abrupt change)

Phase separation and complicated magnetic structure

Hubertus Luetkens Magnetic Structures in RBaCo2O5.5

Different magnetic phase – different µSR signal • PM – non-magnetic • FM – Ferrimagnetic (simple structure) • AFM1 – antiferromagnetic (complicated) • AFM2 – antiferromagnetic (complicated)

Multiple µSR frequencies • Complicated magnetic structure with increased magnetic unit cell • Phase separation into various magnetic structures

Temperature dependence f(T) • Second order transition (smooth change) • First order transition (abrupt change)

Phase separation and complicated magnetic structure

H. Luetkens et al., Phys. Rev. Lett. 101, 017601 (2008)

Hubertus Luetkens Example:

Magnetic Phase Separation in URu2Si2

Hubertus Luetkens URu2Si2 Neutron scattering: Muon spin relaxation: F. Bourdarot et al., condmat/0312206 A. Amato et al., J. Phys.: Condens. Matter 16 (2004) S4403

V 2 m

Phase separation into magnetic and non-magnetic regions.

Only the combination of neutron scattering and muon spin relaxation allowed the correct interpretation of the data.

Hubertus Luetkens Example: Microscopic Coexistence of Superconductivity

and Magnetism in RuSr2GdCu2O8

Hubertus Luetkens RuSr2GdCu2O8 Resistivity: Magnetization: (superconductivity) (ferromagnetism)

µSR:

~100%

Magnetic volume Structure: T. Nachtrab et al., Phys. Rev. Lett. 92 (2004) 117001 Mikroscopic coexistence of superconductivity and magnetism

C. Bernhard et al., Phys. Rev. B 59 (1999) 14099

Hubertus Luetkens Static Magnetism Probed by ZF µSR Single Crystals – Magnet

• Frequency: Size of the magnetic moments • Damping: Inhomogeneity • Amplitude: Polycrystals – Magnet Magnetic volume fraction

Randomly oriented static moments

Hubertus Luetkens Remember: Different Depolarization Functions

dedicated P(t) for each problem

Hubertus Luetkens Magnetic Dynamics / Fluctuating Fields Slowing down of magnetic fluctuations near a magnetic phase transition: For Gaussian distributed fluctuating fields (with width < >) produced by fluctuating spins with𝟐𝟐 a single 𝝁𝝁 fluctuation time the∆𝑩𝑩 muon spin relaxation rate is given by: 𝝉𝝉𝒄𝒄 = < > + 𝟐𝟐 𝟐𝟐 𝝉𝝉𝒄𝒄 𝝁𝝁 𝝁𝝁 𝟐𝟐 𝝀𝝀 𝟐𝟐 𝜸𝜸 ∆𝑩𝑩 𝑳𝑳 𝒄𝒄 For the fast fluctuation limit𝟏𝟏 the𝝎𝝎 𝝉𝝉 relaxation function is:

= −𝝀𝝀𝝀𝝀 with 𝑷𝑷 𝒕𝒕 𝑷𝑷 𝟎𝟎 𝒆𝒆 = < > 𝟐𝟐 𝟐𝟐 𝝀𝝀 𝜸𝜸𝝁𝝁 ∆𝑩𝑩𝝁𝝁 𝝉𝝉𝒄𝒄 P.C.M. Gubbens et al., Hyperfine Interactions 85, 245 (1994)

Hubertus Luetkens

Magnetic Dynamics / Fluctuating Fields

Persistent low frequency dynamics in highly frustrated magnets down to For the very slow fluctuation limit the lowest temperatures longitudinal relaxation rate of the 1/3-tail is given by: 𝑳𝑳 𝝀𝝀 / =

𝝀𝝀𝑳𝑳 ∝ 𝟏𝟏 𝝉𝝉𝒄𝒄 𝝂𝝂𝝁𝝁

( fluctuation rate = ) 1 𝜇𝜇 𝜈𝜈 𝜏𝜏𝑐𝑐 J.A. Hodges et al.,PRL 88, 077204 (2002)

Hubertus Luetkens Muon Spin Rotation on Superconducting Materials

Hubertus Luetkens Superconductivity

Below the superconducting temperature Below the transition temperature a super- the material conducts the electric current conductor expels a magnetic field from without any resistance. its inner core.

Heike Kamerlingh Onnes Walther Meissner

Technical applications:

Resistivity Elektronics, magnets, power industry, …

Quelle: http://staff.ee.sun.ac.za/wjperold/Research/research.htm

Hubertus Luetkens MRI Communication - Satellites Magnetic Screening in the Meissner State

Meissner Ochsenfeld Effekt Field penetration at the surface of a superconductor

B(z) Superconductor

λ

0 z λ is in the nm scale λ = London penetration depth  Low energy µSR Hubertus Luetkens Using Low Energy Muon Spin Rotation to Study Superconducting Materials

Hubertus Luetkens Low Energy µSR

Variable muon energy: 0 – 30 keV 104 Cu Cosmic muons  102 Decay beam Depth-sensitive local magnetic Bulk µSR 100 spin probe on the nm length scale Surface beam

10-2 • Thin films

Range (mm) Range • Near-surface regions 10-4 LE µSR • Multilayers 10-6 • Buried interfaces 10-4 10-2 100 102 104 Energy (MeV)

Hubertus Luetkens Low Energy µSR Apparatus

electrostatic mirror

“surface” µ+ beam, ~4 MeV

Spin MCP detector UltraHighVacuum Spin-rotator Einzel lens moderator (E x B) -10 (LN2 cooled) ~ 10 mbar

Einzel lens

(LN2 cooled)

Start detector (10 nm C-foil)

Conical lens

Sample cryostat e+ detectors

Hubertus Luetkens Low Energy µSR Apparatus

electrostatic mirror

“surface” µ+ beam, ~4 MeV

Spin MCP detector Moderator Spin-rotator Einzel lens moderator (E x B) (LN2 cooled) • Source of low energy muons

Einzel lens

(LN2 cooled)

Start detector (10 nm C-foil)

Conical lens

Sample cryostat e+ detectors

Hubertus Luetkens Low Energy µSR Apparatus

electrostatic mirror

“surface” µ+ beam, ~4 MeV

Spin MCP detector Moderator Spin-rotator Einzel lens moderator (E x B) (LN2 cooled) 15 eV µ+ 4 MeV µ+

Einzel lens (LN cooled) 2

Start detector (10 nm C-foil)

Conical lens ∼500 nm ∼100 µm Sample cryostat s-Ne, s- Ar, Ag at 6K e+ detectors s-N2 . Efficiency ~ 10-4 – 10-5 Hubertus Luetkens

Low Energy µSR Apparatus

electrostatic mirror

“surface” µ+ beam, ~4 MeV

Spin MCP detector Moderator Spin-rotator Einzel lens moderator (E x B) (LN2 cooled) 7.5 – 20 keV µ+ 4 MeV µ+

Einzel lens (LN cooled) 2

Start detector (10 nm C-foil)

Conical lens ∼500 nm ∼100 µm Sample cryostat s-Ne, s- Ar, Ag at 6K e+ detectors s-N2 . Efficiency ~ 10-4 – 10-5 Hubertus Luetkens

Low Energy µSR Apparatus

electrostatic mirror

“surface” µ+ beam, ~4 MeV

Spin MCP detector

Spin-rotator Einzel lens moderator (E x B) Electrostatic mirror (LN2 cooled) • Deflection of the low energy muons Einzel lens (LN cooled) 2

Start detector (10 nm C-foil)

Conical lens

Sample cryostat e+ detectors

Hubertus Luetkens Low Energy µSR Apparatus

electrostatic mirror

“surface” µ+ beam, ~4 MeV

Spin MCP detector Spin Rotator Spin-rotator Einzel lens moderator (E x B) (LN2 cooled) Rotates muon spin for different experimental conditions

Einzel lens (LN2 cooled)

Start detector (10 nm C-foil)

E-Field

Conical lens

Sample cryostat e+ detectors

B-Field

Hubertus Luetkens Low Energy µSR Apparatus

electrostatic mirror

“surface” µ+ beam, ~4 MeV

Spin MCP detector

Spin-rotator Einzel lens moderator Trigger detector (E x B) (LN cooled) 2 • Start of the LE-µSR measurement

Einzel lens • Time-of-flight measurements (LN2 cooled)

Start detector (10 nm C-foil)

Conical lens

Sample cryostat e+ detectors

Hubertus Luetkens Low Energy µSR Apparatus

electrostatic mirror

“surface” µ+ beam, ~4 MeV

Spin MCP detector

Spin-rotator Einzel lens moderator Sample cryostat (E x B) (LN cooled) 2 • Deceleration and acceleration of the µ+

Einzel lens (LN cooled) 2

Start detector (10 nm C-foil)

Conical lens

Sample cryostat e+ detectors

Hubertus Luetkens Low Energy µSR Apparatus

electrostatic mirror

“surface” µ+ beam, ~4 MeV

Spin MCP detector

Spin-rotator Einzel lens moderator Sample cryostat (E x B) (LN cooled) 2 • Deceleration and acceleration of the µ+

Einzel lens (LN cooled) 2

Start detector (10 nm C-foil)

Conical lens

Sample cryostat e+ detectors

Hubertus Luetkens LE-µSR Apparatus

Hubertus Luetkens LE-µSR Apparatus

Hubertus Luetkens Example: Magnetic field profile in type-I and type-II

superconductors

Hubertus Luetkens Meissner State Measurements

1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 Muon Spin PolarisationMuon -0.8 -1.0 B(z) 0 1 2 3 4 5 6 7 8 9 10 Time (µs) 1.0 0.8 Superconductor 0.6 0.4 in the Meissner State 0.2 0.0 -0.2 -0.4 -0.6 Muon Spin PolarisationMuon -0.8 -1.0 Bext 0 1 2 3 4 5 6 7 8 9 10 Time (µs) 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 Muon Spin PolarisationMuon -0.8 -1.0 λ 0 1 2 3 4 5 6 7 8 9 10 Time (µs) z 0

Hubertus Luetkens Magnetic field profiles in YBa2Cu3O7-δ and Pb

YBa Cu O , T=20K, T =87.5K Lead, Tc=7.0(2) K, hext = 91.5(3)G, 2 3 7-δ c ξ λ h = 91.5(3) G, 0.01 0 = 90(5)nm, 0 = 58(3)nm 0.01 ext ξ λ 0 = 1.5 nm fixed, 0 = 137(10) nm

6.66K B (T) B (T) κ = λ0/ξ0 ~ 90

6.19K h exp(-z/λ(T)) ext κ = λ0/ξ0 ~ 0.6 3.4 keV 8.9 keV 15.9 keV 20.9 keV 29.4 keV 1E-3 1E-3

2.85K 0 50 100 150 z (nm) 0 50 100 150 z (nm) local: exponential Non-local: non-exponential

st 1 experimental determination of ξ0 in a non-local T.J. Jackson et al., Phys. Rev. Lett. 84, 4958 (2000). A. Suter et al., Phys. Rev. Lett. 92, 087001 (2004). superconductor; confirmation of some predictions A. Suter et al., Phys. Rev. B72, 024506 (2005). of BCS (~50 years after theory!!)

Hubertus Luetkens Interlude: Other applications of LEM – Physicist’s Lego

Magnetic properties Electric properties P CMR M I FM MF FE SC AFM AFE Combinations

Fundamental Science Applications - Competition of ground states - Spintronics (Switching of currents with magnetic fields) - Coexistence of ground states - Sensors (Magnetism, pressure, temperature, gases, …) - Coupling of different ground states - Actuators - (Giant) proximity effects - Non-volatile memory (MRAM, …) - Electric field effects (2D-electron gas, ...) - … - …

FM AFM FM AFM P CMR P P SC SC SC SC FM AFM FM AFM P CMR

Hubertus Luetkens Using Bulk Muon Spin Rotation to Study Superconducting Materials

Hubertus Luetkens Magnetic Screening

Meissner Ochsenfeld Effekt Type II Superconductors

Hubertus Luetkens Vortex (Abrikosov) phase

H Flux Line Lattice Hc2 (0)

Hc1 (0) H

Tc T Bitter Decoration Magnetic flux quantum U. Essmann et al., (1967) -15 φ0 = h/2e = 2.067 · 10 Wb Flux line 2ξ

Coherence length ξ

H λ Penetration depth λ

Hubertus Luetkens Field Distribution in a Type II S.C.

Bext

Bext

Since the muon is a local probe, the µSR relaxation function is given by the weighted sum of all oscillations:

Hubertus Luetkens µSR a Type II Superconductor

8K T < Tc 0.1 H

Hc2 0

H -0.1 p (B) c1 0.2 76K Tc 0.1

0 B

Asymmetry -0.1 T > T 0.2 c 120K H

0.1 Hc2

0

-0.1 p (B) Hc1 -0.2 0.2 0.4 0.6 0.8 Tc T Time (µs)

YBa2Cu3O6.97 Pümpin et al., Phys. Rev. B 42, 8019 (1990) Bext Hubertus Luetkens Extract Information from the µSR data

Field distribution depends on the Theory microscopic parameters of

Maisuradze et al., superconductivity ξ and λ (2008) Theoretical models of the flux line p(B) lattice to fit the µSR data 

Structure, symmetry of the flux line lattice

Pd-In alloy Vortex motion

Charalambous et al., Phys. Rev. B 66, Characteristic lengths: magnetic penetration 054506 (2002) depth λ, radius of the vortex core - coherence length ξ

Classification scheme of superconductors

Hubertus Luetkens Extract Information from the µSR data

Pümpin et al., 0.08 YBa Cu O Single crystals Phys. Rev. B 42, 2 3 6.97 8019 (1990) • Anisotropy field distribution 0.06 λ = 130 (10) nm

Polycrystals or sintered sample 0.04

• Disorder of pinning sites 0.02 • Strong smearing of the field distribution 0.00 44 46 48 50 Frequency (MHz)

0.06 Pd-In alloy Sonier et al., PRL 83, 4156 0.05 Charalambous et al., (1999) YBaCuCuOO 0.04 22 3 3 6.956.95 Phys. Rev. B 66, single crystal crystal 054506 (2002) 0.03 λ 0.02 = 150 (4) nm

0.01

0.00

811 812 813 814 815 816 817 Frequency (MHz)

Hubertus Luetkens Extract Information from the µSR data

Pümpin et al., 0.08 YBa Cu O Phys. Rev. B 42, 2 3 6.97 8019 (1990) 0.06 λ = 130 (10) nm

0.04

0.02

Ginzburg-Landau model 0.00 44 46 48 50 Frequency (MHz)

A µSR measurement of the second moment London model of the field distribution allows to determine the London penetration depth λ.

The damping of a TF-µSR spectrum is

proportional to the super fluid density ns (number of Cooper pairs).

Hubertus Luetkens Classification of SCs: Uemura Plot

• Conventional superconductors have

low TC and high superfluid density

• Unconventional superconductors have relatively low superfluid density (“dilute superfluid”)

R. Khasanov, H. Luetkens, et al., Phys. Rev. B 78, 092506 (2008)

Hubertus Luetkens Symmetry of the Superconducting Gap Temperature dependence of the superfluid density (number of Cooper pairs):

YBa Cu O RbOs2O6 Ba1-xKxFe2As2 2 3 6.95

R. Khasanov et al., PRB 72, 104504 (2005) R. Khasanov et al., PRL 102, 187005 (2009) J.E. Sonier et al, PRL 72, 744 (1994)

s-wave gap Multiple s-wave gaps d-wave gap

V. B. Zabolotnyy et al., Nature 457, 569 (2009)

Hubertus Luetkens Muon Spin Rotation to Investigate Phase Diagrams of Magnetic and Superconducting Materials - (Coexistence, Competition, Cooperation)

Hubertus Luetkens Some Phase Diagrams of Fe-based SC

H. Luetkens et al., Nature Materials 8, 305 (2009)

Ba1-xKxFe2As2

E. Wiesenmayer et al., Phys. Rev. Lett. 107, 237001 (2011) Bendele et al., Phys Rev Lett 104, 087003 (2010)

Hubertus Luetkens Summary

Hubertus Luetkens Muon Spin Rotation / Relaxation (µSR)

Magnetism: Superconductivity:

• Local probe • Field distribution of vortex lattice • Magnetic volume fraction • Penetration depth • Coherence length • µSR frequency • Vortex dynamics -3 -4 • Magnetic order parameter (10 – 10 µB) • Temperature dependence • Absolute determination of penetration depth λ(0) • µSR relaxation rate • Homogeneity of magnetism • Temperature dependency of • Penetration depth, λ−4 ∝ <∆B2>2 ∝ σ2 2 • Magnetic fluctuations • Superfluid density ns/m* ∝ <∆B > ∝ σ • Time window: 10 5 – 10 9 Hz  Symmetry of the SC gap function

Hubertus Luetkens Thanks

Figures, slides, and ideas taken from: A. Amato, A. Suter, T. Prokscha, E. Morenzoni, PSI B. Cywinski, University of Huddersfield A. Yaouanc, CEA Grenoble S. J. Blundell, University of Oxford

Thank you for your attention!

Hubertus Luetkens Literature BOOKS •A. Yaouanc, P. Dalmas de Réotier, MUON SPIN ROTATION, RELAXATION and RESONANCE (Oxford University Press, 2010) •A. Schenck, MUON SPIN ROTATION SPECTROSCOPY, (Adam Hilger, Bristol 1985) •E. Karlsson, SOLID STATE PHENOMENA, As Seen by Muons, Protons, and Excited Nuclei, (Clarendon, Oxford 1995) •S.L. Lee, S.H. Kilcoyne, R. Cywinski eds, MUON SCIENCE: MUONS IN PHYSICS; CHEMISTRY AND MATERIALS, (IOP Publishing, Bristol and Philadelphia, 1999) •INTRODUCTORY ARTICLES •S.J. Blundell, SPIN-POLARIZED MUONS IN CONDENSED MATTER PHYSICS, Contemporary Physics 40, 175 (1999) •P. Bakule, E. Morenzoni, GENERATION AND APPLICATIONN OF SLOW POLARIZED MUONS, Contemporary Physics 45, 203-225 (2004). REVIEW ARTICLES, APPLICATIONS •P. Dalmas de Réotier and A. Yaouanc, MUON SPIN ROTATION AND RELAXATION IN MAGNETIC MATERIALS, J. Phys. Condens. Matter 9 (1997) pp. 9113-9166 •A. Schenck and F.N. Gygax, MAGNETIC MATERIALS STUDIED BY MUON SPIN ROTATION SPECTROSCOPY, In: Handbook of Magnetic Materials, edited by K.H.J. Buschow, Vol. 9 (Elsevier, Amsterdam 1995) pp. 57-302 •B.D. Patterson, MUONIUM STATES IN SEMICONDUCTORS, Rev. Mod. Phys. 60 (1988) pp. 69-159 •A. Amato, HEAVY-FERMION SYSTEMS STUDIED BY µSR TECHNIQUES, Rev. Mod. Phys., 69, 1119 (1997) •V. Storchak, N. Prokovev, QUANTUM DIFFUSION OF MUONS AND MUONIUM ATOMS IN SOLIDS, Rev. Mod. Physics, 70, 929 (1998) •J. Sonier, J. Brewer, R. Kiefl, µSR STUDIES OF VORTEX STATE IN TYPE-II SUPERCONDUCTORS, Rev. Mod. Physics, 72, 769 (2000) •E. Roduner, THE POSITIVE MUON AS A PROBE IN FREE RADICAL CHEMISTRY, Lecture Notes in Chemistry No. 49 (Springer Verlag, Berlin 1988)

Hubertus Luetkens