Damping of magnetization dynamics Andrei Kirilyuk Radboud - - PowerPoint PPT Presentation

ESM Cluj-Napoca - August 2015

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Damping of magnetization dynamics

• Radboud University, Institute for Molecules and Materials, 


Nijmegen, The Netherlands

Andrei Kirilyuk

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Landau-Lifshitz equation

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S N

energy gain:

• torque equation:

Landau & Lifshitz, 1935

Heff

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Landau-Lifshitz equation

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S N

energy gain:

• torque equation:

Landau & Lifshitz, 1935

Heff

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Damping: Landau-Lifshitz vs Gilbert

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Damping: Landau-Lifshitz vs Gilbert

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Damping: Landau-Lifshitz vs Gilbert

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Landau-Lifshitz vs Gilbert

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Damping: Landau-Lifshitz vs Gilbert

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Since the second result is in agreement with the fact that a very large damping should produce a very slow motion while the first is not, one may conclude that the Landau-Lifshitz-Gilbert equation is more appropriate to describe magnetization dynamics.

Landau-Lifshitz vs Gilbert

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• A. Einstein & W.J. de Haas, Experimenteller Nachweis der Amperèschen


Molekülströme, Verhandl. Deut. Phys. Ges. 17, 152 (1915) S.J. Barnett, Magnetization by rotation, Phys. Rev. 6, 239 (1915)

To remember: magnetization = angular momentum

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Einstein – de Haas & Barnett effects

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Angular momentum transfer and two ways of reversal

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H !

M !

H !

M !

( )

! " # $ % & × + × − = dt dM M M H M dt dM

eff

α γ

( )

! " # $ % & × + × − = dt dM M M H M dt dM

eff

α γ precessional (fast) usual (practical)

from spins to field

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Angular momentum transfer and two ways of reversal

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H !

M !

H !

M !

( )

! " # $ % & × + × − = dt dM M M H M dt dM

eff

α γ

( )

! " # $ % & × + × − = dt dM M M H M dt dM

eff

α γ precessional (fast) usual (practical)

from spins to lattice from spins to field

ESM Cluj-Napoca - August 2015

Angular momentum transfer and two ways of reversal

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H !

M !

H !

M !

( )

! " # $ % & × + × − = dt dM M M H M dt dM

eff

α γ

( )

! " # $ % & × + × − = dt dM M M H M dt dM

eff

α γ precessional (fast) usual (practical)

from spins to lattice from spins to field

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measuring the damping

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measuring the damping

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Example 1: thin film configuration

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from the condition that the net torque on M is zero:

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FMR resonance

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FMR resonance

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FMR resonance

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FMR versus applied field angle

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isotropic

• ut of plane easy axis

easy plane

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FMR linewidth

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FMR linewidth

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Example 2: optical pump-probe measurement

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Damping in a Bi:YIG garnet film as a function

• f temperature

pump probe

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Example 2: optical pump-probe measurement

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200 400 600 800 1000 1200 1400 370 K 380 K 390 K 395 K 400 K 405 K 410 K 415 K

Faraday rotation Delay time (ps)

420 K

Damping in a Bi:YIG garnet film as a function

• f temperature

pump probe

TC

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Energy flow via spin waves??

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Semi-quantitative analysis

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Semi-quantitative analysis

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radius laser spot ~20 µm;

• bserved τ < 200 ps;

⇒

s km v 100 >

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Semi-quantitative analysis

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k ω

radius laser spot ~20 µm;

• bserved τ < 200 ps;

⇒

s km v 100 >

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Semi-quantitative analysis

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k ω

≈ = dk d vg ω

radius laser spot ~20 µm;

• bserved τ < 200 ps;

⇒

s km v 100 >

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Semi-quantitative analysis

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k ω

≈ = dk d vg ω

Magnetostatic modes; picture from Demokritov & Hillebrands

radius laser spot ~20 µm;

• bserved τ < 200 ps;

⇒

s km v 100 >

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Semi-quantitative analysis

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k ω

≈ = dk d vg ω

Magnetostatic modes; picture from Demokritov & Hillebrands

s km vg 10 ≤

radius laser spot ~20 µm;

• bserved τ < 200 ps;

⇒

s km v 100 >

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µ-magnetic simulations [Eilers et al, PRB 74, 054411 (2006)]

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s km ps m v 6 70 4 . ≈ ≈ µ

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Experiment: propagation of spin waves

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J

H !

0.5 ns, 40 Oe pulse

part with T. Korn & U. Ebels, SPINTEC, Grenoble

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Experiment: propagation of spin waves

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J

H !

0.5 ns, 40 Oe pulse

part with T. Korn & U. Ebels, SPINTEC, Grenoble

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Experiment: propagation of spin waves

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s km ns m v 140 5 . 3 500 ≈ ≈ µ

J

H !

0.5 ns, 40 Oe pulse

part with T. Korn & U. Ebels, SPINTEC, Grenoble

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Conclusion 1

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• not everything what you measure is damping!

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Damping channels: intrinsic vs extrinsic

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damping via magnetoelastic interactions breathing Fermi-surface in metals extrinsic: two-magnon scattering

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damping via magnetoelastic interactions breathing Fermi-surface in metals extrinsic: two-magnon scattering

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Phenomenology based on magneto-elasticity

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‘Dissipative’ part of magnetic field

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‘Dissipative’ part of magnetic field

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so that the total effective field is

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Heating rate

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Heating rate

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Heating rate

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Magnetostriction

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the adiabatic magnetostriction coefficients are defined as

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Magnetostriction

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the adiabatic magnetostriction coefficients are defined as the time-varying magnetostrictive strain is then

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Finally: the Gilbert damping tensor

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thus, a changing M produce a changing strain; the crystal viscosity tensor determines the heating rate per unit volume

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Finally: the Gilbert damping tensor

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thus, a changing M produce a changing strain; the crystal viscosity tensor determines the heating rate per unit volume

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Finally: the Gilbert damping tensor

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thus, a changing M produce a changing strain; the crystal viscosity tensor determines the heating rate per unit volume from this, the Gilbert damping tensor is rigorously given by

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Experiments vs theory

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Experiments vs theory

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the theoretical prediction is that

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Theoretical vs measured damping parameters

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Theoretical vs measured damping parameters

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damping via magnetoelastic interactions breathing Fermi-surface in metals extrinsic: two-magnon scattering

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damping via magnetoelastic interactions breathing Fermi-surface in metals extrinsic: two-magnon scattering

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Ferromagnetism of metals

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‘breathing’ Fermi-surface

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following Steiauf and Fähnle, PRB 72, 0064450 (2005); see Kambersky, Can J. Phys. 48, 2906 (1970); Kunes and Kambersky, PRB 65, 212411 (2002)

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• 1. Adiabatic regime

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we confine the treatment to the adiabatic regime: several ps to nanoseconds (single-electron spin fluctuations can be integrated out):

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• 2. Dissipative free-energy functional

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the existence of such functional is postulated:

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• 2. Dissipative free-energy functional

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the existence of such functional is postulated:

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• 3. Translate this to the electronic level

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as outputted from the density functional theory

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• 3. Translate this to the electronic level

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as outputted from the density functional theory

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• 3. Translate this to the electronic level

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as outputted from the density functional theory as the total number of states is conserved

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• 3. Translate this to the electronic level

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as outputted from the density functional theory as the total number of states is conserved

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• 3a. Spin-orbit coupling

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• 3a. Spin-orbit coupling

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for a lattice of simple cubic symmetry this gives

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• 3a. Spin-orbit coupling

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for a lattice of simple cubic symmetry this gives

N.B.: this is a difficult point, usually not much discussed!

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• 4. Semiempirical extension of DFT

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Redistribution of the occupation numbers provided by scattering processes

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• 4. Semiempirical extension of DFT

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Redistribution of the occupation numbers provided by scattering processes Approximated by

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• 5. Consider homogeneous situation

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• 5. Consider homogeneous situation

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Anisotropy and ‘damping’ fields:

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where the damping matrix:

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• 7. Same relaxation times around the Fermi surface

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Finally: equation-of-motion

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scalar damping parameter is

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sidenote: damping vs anisotropy

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In many discussion you find the direct relation between damping and magnetocrystalline anisotropy - equations show that this is not entirely correct:

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Results: Fe, Co, Ni

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bcc Fe hcp Co fcc Ni

two eigenvalues of the damping matrix vs direction of M

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Anisotropic FMR linewidth

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Anisotropic FMR linewidth

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Temperature dependence of damping

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the higher T, the shorter => less damping?? is this reasonable??

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Temperature dependence of damping

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the higher T, the shorter => less damping?? is this reasonable??

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Temperature dependence of damping - 2

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Bhagat, Lubitz, PRB 10, 179 (1974)

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Interband transitions at higher temperature

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note that this also includes the ‘breathing Fermi surface’ part for transitions inside the same band

Gilmore et al, PRL 99, 027204 (2007)

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Interband transitions at higher temperature

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note that this also includes the ‘breathing Fermi surface’ part for transitions inside the same band

Gilmore et al, PRL 99, 027204 (2007)

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damping via magnetoelastic interactions breathing Fermi-surface in metals extrinsic: two-magnon scattering

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damping via magnetoelastic interactions breathing Fermi-surface in metals extrinsic: two-magnon scattering

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Two-magnon scattering

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FMR is the lowest frequency, isn’t it??

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Spin waves in thin films

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Dispersion relations

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Dispersion relations

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States available for scattering

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Angular dependence of 2-magnon damping

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Different types of defects

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Experiments??

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films with different anisotropy, roughly corresponding to the initially defined ones.

Srivastava et al, J. Appl. Phys. 85, 7838 (1999);

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Summary:

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• not obvious experimentally, lots of artefacts

intrinsic versus extrinsic mechanisms phenomenology versus ab-initio

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