Magnetic Resonance(s) European School on Magnetism, Brno 2019 - - PowerPoint PPT Presentation

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Magnetic Resonance(s) European School on Magnetism, Brno 2019 - - PowerPoint PPT Presentation

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Magnetic Resonance(s) European School on Magnetism, Brno 2019 Laurent Ranno laurent.ranno@neel.cnrs.fr Institut N eel - Universit e Grenoble-Alpes 18 septembre 2019 laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 1 / 58

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Magnetic Resonance(s) European School on Magnetism, Brno 2019

Laurent Ranno laurent.ranno@neel.cnrs.fr

Institut N´ eel - Universit´ e Grenoble-Alpes

18 septembre 2019

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 1 / 58

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Error Bars

This lecture will last 90 minutes ±180. Should I rush ?

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 2 / 58

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Outline

Dynamics of one (electronic) spin

Electron Spin Resonance (ESR)

Ferromagnetic dynamics

Ferromagnetic Resonance (FMR) Uniform, Non uniform Modes Magnetic Objects : Domain Wall, Vortex

Mossbauer, NMR

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 3 / 58

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SLIDE 4

Resonance vs ”usual” Magnetometry

Classical Magnetometry = Magnetic Moment m(H,T,angle ...) An alternative is to detect a Magnetic Resonance to determine : resonance field (or resonance frequency) amplitude of the resonance width of the resonance line as a function of external parameters (Applied Magnetic Field, Geometry, Temperature ...) and hopefully to extract interesting magnetic infos

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 4 / 58

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Gyromagnetic factor

A magnetic atom is characterised by its quantum number J (could be J=S only) The atom angular momentum is L = J. The atom magnetic moment is m = gµBJ. The gyromagnetic factor is the ratio magnetic moment / angular momentum i.e. γ = −gµB

  • = − ge

2me γ is negative (today), both moments are antiparallel. UNLIKE the usual choice in spintronics where electrons magnetised up are labelled spin up (and in reality are spin down).

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 5 / 58

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Larmor precession

When a magnetic field H is applied, a Zeeman energy appears : Ez = −µ0 m. H The torque Γ which is applied to the magnetic moment is :

  • Γ =

m ∧ µ0 H The torque corresponds to a change of angular momentum :

  • Γ = d

L dt Finally : d

m dt = γ

m ∧ µ0 H The equation corresponds to a precession around the applied field : Larmor Precession Angle ( m, H) = constant (constant energy, no relaxation).

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 6 / 58

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Electron Spin Resonance (ESR)

In an applied magnetic field H, a spin 1/2 can absorb a photon = electromagnetic wave hν = gµBµ0H To allow the transition, the ac field from the wave must be perpendicular to the applied field (Sz is not anymore an eigen state)

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 7 / 58

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What can we learn from Larmor precession ?

d m dt = γ m ∧ µ0 H hν = gµBµ0H if we know ν and g : field sensor, calibration if we know ν and H : g sensor, orbital/spin contributions, chemical infos

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 8 / 58

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Numbers

Applied Field µ0Hz, J=S= 1

2

∆E = gµBµ0Hz = hν When g=2 ν=10 GHz gives resonance at 0.357 Tesla ν=100 MHz gives resonance at 3.57 mTesla MHz electronics = small field resonance (mT) GHz microwave = Tesla field µ0Hres (electromagnets and superconducitng coils)

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 9 / 58

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Measurement

Two types of setups/measurements to get the resonance At fixed field, scan the frequency, detect absorption At fixed frequency, scan the field, detect absorption The absorption line or its derivative is measured Extract resonance field, amplitude of absorption, width of the line (peak-to-peak)

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 10 / 58

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Table ESR

10-100 MHz ESR for practicals Helmholz Coils for the applied field

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 11 / 58

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What to learn from ESR ?

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 12 / 58

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What to learn from ESR ?

Amplitude of the resonance signal : number of resonating species Position of the resonance line : shift from free electron g-factor environment effects on the orbitals (orbital moment is impacted) interaction with nucleus moment (hyperfine splitting of the energy lines) Line width : intrinsic (narrow) distribution of g-factors (wider line) inhomogeneous field, sample

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 13 / 58

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DPPH calibration

DPPH : 2,2-Diphenyl-1-picrylhydrazyl (free radical) Chemical Formula C18H12N5O6

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 14 / 58

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DPPH for calibration

For a free electron g = 2.0023 (2 + correction QED) For DPPH (very light atoms, small SO), g = 2.0035 γ = 1.76 1011Hz/T(28.041 GHz/T) The resonance field at 9.750 GHz for DPPH is 0.347703 T (g=2.0035) For DPPH, g factor varies from 2.003 to 2.0045 depending on preparation and environment (solvant in particular).

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 15 / 58

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Hyperfine Structure

The electronic spin feels the field of the nuclear spin. It is then a (S=1/2,I=1/2) system : singlet state + triplet state Singlet-triplet degeneracy is lifted when the field is applied

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 16 / 58

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ESR and magnetometry

NV center magnetometry Balasubramanian et al. NatMat2009

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 17 / 58

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NV center magnetometry

Ta/CoFeB (1.5 nm)/MgO, perpendicular M.

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 18 / 58

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Ferromagnetic Resonance (FMR)

In a ferromagnetic material : The volume torque acting on magnetisation is : Γ = µ0 M ∧ Heff . The variation of volume angular momentum is d

L dt =

Γ

d M dt = γ

Γ, γ < 0 So, d

M dt = γ

µ0 M ∧ Heff Similar to Larmor

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 19 / 58

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LLG equation

Experimentally Larmor precession does not last forever and the magnetisation aligns with the field Need relaxation Landau-Lifshitz (LL) equation (1935) :

d M dt = µ0γ

M ∧ H + α

  • M

Ms ∧ (

M ∧ H) Landau-Lifshitz-Gilbert (LLG) equation (1955)

d M dt = −µ0γ

M ∧ H + αLLG

  • M

Ms ∧ d M dt

αLLG has no unit Typically 0.001 (low damping) to 0.1 (fast relaxation) Mathematically, one can transform LL into LLG : same dynamics.

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 20 / 58

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Relaxation of the magnetic moment

The field is applied along z. In NMR, Bloch equations (1946) are used and take into account two relaxations The relaxation of mz : longitudinal relaxation Characteristic time T1 d mz dt = γ m ∧ µ0 H − mz − msat T1 Energy needs to be transferred out, to the lattice (spin-lattice relaxation time) The relaxation of mx and my : transverse relaxation, in Bloch equations : Characteristic time T2 d mx dt = γ m ∧ µ0 H − mx T2 Energy stays in the spin system (spin-spin relaxation time)

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 21 / 58

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Alpha

In ferromagnets, LL and LLG introduced a phenomelogical constant (isotropic) relaxation, characterised by damping constant α. Its value is measured from resonance experiments or relaxation experiments Its minimum theoretical value is not zero (intrinsic damping) can be evaluated from band structure calculation.

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 22 / 58

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Low alpha material

Co2FeAl, effect of annealing. Low damping when well ordered.

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 23 / 58

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Damping Contributions

intrinsic inhomogeneous sample eddy currents Spin Transfer Torque Spin Pumping

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 24 / 58

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Inhomogeneous Samples

FMR : allows to compare series of samples (with thickness or growth conditions) Easier to estimate homogeneity than Magnetometry (think of a M(H) loop)

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 25 / 58

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Eddy Currents

skin depth (conductivity dependent) Typically 100 nm at 10 GHz negligible for ultrathin films (the field penetrates) eddy current : losses, heating (microwave oven). Water-based large samples (bio environment) ⇒ need to go to lower frequency for ESR

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 26 / 58

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Spin Transfer Torque

Spin Transfer Torque : acts as an extra contribution to LLG d M dt = µ0γ M ∧ H + (αLLG + αSTT)

  • M

Ms ∧ d M dt αSTT ∝ j.P = electron flow. spin polarisation Increase damping (faster relaxation, less ringing) Decrease (Cancel) damping (spin pumping, STT oscillators)

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 27 / 58

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Effective field

Landau-Lifschitz-Gilbert equation d▼ dt = γ▼ × ❍eff + α▼ × (▼ × ❍❡✛ ) (1) ❍❡✛ the effective field : ❍❡✛ = − 1 µ0 ∂E ∂▼ The effective field includes contributions from the applied field (Zeeman energy), the demagnetising field (shape anisotropy), magnetocrystalline and exchange energies.

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 28 / 58

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What to do with this equation ?

Once the external parameters are known (geometry, field). Write the Total energy : EZ+Ed+Emc+EA Look for M equilibrium position Determine the shape of the energy minimum (curvature) Determine the natural frequency of the oscillations around the equilibrium position

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 29 / 58

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Resonance Fields

Linear development around the equilibrium position of M Free energy F (spherical coordinates), M = Ms ur : dF = −µ0d M. Heff = d

  • M. ∂F

∂ M Heffθ = − 1 µ0Ms ∂F ∂θ , Heffφ = − 1 µ0Mssinθ ∂F ∂φ (all steps in Baselgia et al. PRB 1988) (ω2 γ2 ) = 1 M2

s

(Fθθ( Fφφ sin2θ + cosθ sinθ Fθ) − ( 1 sinθFθφ − cosθ sin2θFφ)2) Line-width FWHM : ∆H = α dω/dH γ Ms (Fθθ + 1 sin2θFφφ)

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 30 / 58

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Example : Soft Film

Uniform film, not patterned Only Zeeman + Shape anisotropy : Field in-plane : Bres =

  • (µ0M)2+4∗ ω2(1+α2)

γ2

−µ0M 2

At 10 GHz : Bres = √

(µ0M)2+4∗0.342−µ0M 2

Field out-of-plane : Bres = ω(1+α2)

γ

+ µ0M At 10 GHz : Bres = 0.34T + µ0M

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 31 / 58

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Example : Soft Film

FMR gives M (magnetisation) magnetometry (VSM-SQUID) gives m (magnetic moment) Useful to compare

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 32 / 58

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Uniform vs Non Uniform modes

Uniform Mode : all spin precess together Non Uniform Mode : FMR can excite non uniform modes magnetostatic modes (surface or volume), (see for example

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 33 / 58

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Coupled Modes

A coupled Bilayer can give rise to acoustic (in phase) and optical (out-of-phase) modes

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 34 / 58

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Experimental set-ups for FMR

Cavity (fixed f, scanning H) vs BroadBand (scanning f) X-band (10 GHz), generator, waveguide, detection scheme (diode) ... Broadband Source (VNA), waveguide, VNA

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 35 / 58

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Experimental set-ups

Fixed frequency, Resonating Cavity FMR

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 36 / 58

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Fixed f spectrometer

Band Frequency (GHz) Wavelength (cm) Bres (T) free electron S 2-4 (3) 10 0.11 X 8-12 (9.5) 3 0.34 K 18-27 1.3 0.8 Q 35 0.86 1.3 W 75-110 (95) 0.31 3.3

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 37 / 58

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Experimental set-ups

9.8 GHz (X-band) resonant cavity (mode TE102)

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 38 / 58

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GHz sources

Older Technology : Klystrons it is a type of vacuum tube. A beam of electrons is shaped into ac current and then coupled to transfer its energy to an electromagnetic wave. Newer Technology : Gunn diode Solid State technology. Negative differential resistance which makes total resistance zero and creates oscillations.

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 39 / 58

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Gunn diode

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 40 / 58

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GHz Waveguides

One frequency : Dedicated Guide Non reciprocal device (circulator)

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 41 / 58

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X-band cavity

rectangular cavity. Applied field is along short direction. Erf is minimum in center, Brf is maximum. ( ! pumping direction)

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 42 / 58

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Detection

RF diode + lock-in integrated analyser (power meter) Electric detection (small sample)

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 43 / 58

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X-band Sensitivity

10 GHz cavity Best sensitivity : 1012 spins (narrow line, not hyperfine splitted).

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 44 / 58

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SLIDE 45

Peak-to-peak line width

If the FMR resonance line has a Lorentzian shape : f (B) = A

(B−Br)2 D2

+ 1 Line amplitude : A. Line center : Br. FWHM=2D df dB = −2A(B − Br) D2( (B−Br)2

D2

+ 1)2 Peak to peak width of the line derivative (FMR measurement) ∆Bpp = 2D √ 3 = FWHM √ 3

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 45 / 58

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BroadBand FMR

Choosing the applied field is often necessary Frequency must be swept to find the resonance Generator, waveguide, sample holder must accept a broadband signal.

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 46 / 58

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VNA-FMR

Vector Network Analyser (wave out, wave in). The phase of the electric wave can be analysed (not only the intensity).

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 47 / 58

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Connecting the Sample

Signal is transmitted using coaxial cables, coax connectors, microbonding to striplines / coplanar waveguide

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 48 / 58

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Connecting the Sample

RF Probes can also be used to mechanically connect the sample.

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 49 / 58

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Raw signal

the signal is an electromagnetic wave. It can be reflected at many interfaces. A cable is a resonating object. Evaluating the electric field at the sample is a complicated task

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 50 / 58

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Electric Detection of FMR

Bolometric response : Heating of the sample when Wave is absorbed.

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 51 / 58

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Domain Wall-FMR

A domain wall in a potential minimum is equivalent to a mass in a potential well

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 52 / 58

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Nuclear Magnetic Resonance

Some nuclei carry a nuclear spin I Since mp

me =2000

for a nucleus γn = − ge

2mn is 2000 times smaller compared to γe

NMR frequencies (MHz) are 2000 smaller compared to EPR frequencies (GHz) Similar interest to go to high frequency to improve resonance line resolution

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 53 / 58

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NMR example

Co/Cu multilayer : where is Co ? NMR in Spin-echo mode. Sensitivity 1016 Co spins Environment (Co-Cu mixing) and strain (line shift) See P. Mendels’presentation ESM2011 for more details about NMR

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 54 / 58

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Mossbauer : Principle

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 55 / 58

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Mossbauer : Fe57

6 allowed transitions when a gamma photon is absorbed (ground state I=1/2, excited state I=3/2)

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 56 / 58

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Mossbauer

FeRh : Are all Fe equivalent ? Need Fe57 (Sn=1/2) in the sample (only 2% in natural Fe). here 150nm thick films Get local field on the nucleus (line splitting) Get local field direction (ratio of lines intensities)

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 57 / 58

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Bibliography

Bland-Heinrich Book series (Ultrathin Magnetic Structures I,II, III) Hillebrands, Ounadjela/Thiaville Book series (Spin Dynamics in Confined Magnetic Structures) Review (70 pages) on metallic films FMR : M. Farle, Rep. Prog.

  • Phys. 61, 755 (1998)

laurent.ranno@neel.cnrs.fr Magnetic Resonances ESM2019Brno 58 / 58