Damping of magnetization dynamics - MAGNETISM.eu - EMA

ESM Cluj-Napoca - August 20151

Damping of magnetization dynamics

!Radboud University, Institute for Molecules and Materials,

Nijmegen, The Netherlands

Andrei Kirilyuk



Landau-Lifshitz equation

3

S

Nenergy gain: !torque equation:

Landau & Lifshitz, 1935

Heff


Landau-Lifshitz equation

3

S

Nenergy gain: !torque equation:

Landau & Lifshitz, 1935

Heff


Damping: Landau-Lifshitz vs Gilbert

4



4



4

Landau-Lifshitz vs Gilbert



4

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


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

5

Einstein – de Haas & Barnett effects


Angular momentum transfer and two ways of reversal

6

H!

M!

H!

M!

( ) !"

#$%

&×+×−=dtdMM

MHM

dtdM eff α

γ( ) !"

#$%

&×+×−=dtdMM

MHM

dtdM eff α

γ

precessional (fast)usual (practical)

from spins to field



6

H!

M!

H!

M!

( ) !"

#$%

&×+×−=dtdMM

MHM

dtdM eff α

γ( ) !"

#$%

&×+×−=dtdMM

MHM

dtdM eff α

γ


from spins to latticefrom spins to field



6

H!

M!

H!

M!

( ) !"

#$%

&×+×−=dtdMM

MHM

dtdM eff α

γ( ) !"

#$%

&×+×−=dtdMM

MHM

dtdM eff α

γ


from spins to latticefrom spins to field


measuring the damping

7


measuring the damping

7


Example 1: thin film configuration

8

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


FMR resonance

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FMR resonance

9


FMR resonance

9


FMR versus applied field angle

10

isotropic out of plane easy axis easy plane


FMR linewidth

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FMR linewidth

11


Example 2: optical pump-probe measurement

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Damping in a Bi:YIG garnet film as a function of temperature

pump

probe


Example 2: optical pump-probe measurement

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0 200 400 600 800 1000 1200 1400

370 K

380 K

390 K

395 K

400 K

405 K

410 K

415 K

Fara

day

rota

tion

Delay time (ps)

420 K

Damping in a Bi:YIG garnet film as a function of temperature

pump

probe

TC


Energy flow via spin waves??

13


Semi-quantitative analysis

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14

radius laser spot ~20 µm; observed τ < 200 ps; ⇒ s

kmv 100>



14

k

ω


kmv 100>



14

k

ω0≈=

dkdvgω


kmv 100>



14

k

ω0≈=

dkdvgω

Magnetostatic modes; picture from Demokritov & Hillebrands


kmv 100>



14

k

ω0≈=

dkdvgω

Magnetostatic modes; picture from Demokritov & Hillebrands

skmvg 10≤


kmv 100>


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

15

skm

psmv 6

704.0

≈≈µ


Experiment: propagation of spin waves

16

JH!

0.5 ns, 40 Oe pulse

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



16

JH!

0.5 ns, 40 Oe pulse




16

skm

nsmv 140

5.3500

≈≈µ

JH!

0.5 ns, 40 Oe pulse



Conclusion 1

17

!

not everything what you measure is damping!


Damping channels: intrinsic vs extrinsic

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19

damping via magnetoelastic interactions

breathing Fermi-surface in metals

extrinsic: two-magnon scattering


19





Phenomenology based on magneto-elasticity

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‘Dissipative’ part of magnetic field

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‘Dissipative’ part of magnetic field

21

so that the total effective field is


Heating rate

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Heating rate

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Heating rate

22


Magnetostriction

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the adiabatic magnetostriction coefficients are defined as


Magnetostriction

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the adiabatic magnetostriction coefficients are defined as

the time-varying magnetostrictive strain is then


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



24




24


from this, the Gilbert damping tensor is rigorously given by


Experiments vs theory

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Experiments vs theory

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the theoretical prediction is that


Theoretical vs measured damping parameters

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Theoretical vs measured damping parameters

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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)


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):


2. Dissipative free-energy functional

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the existence of such functional is postulated:


2. Dissipative free-energy functional

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the existence of such functional is postulated:


3. Translate this to the electronic level

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as outputted from the density functional theory



33




33


as the total number of states is conserved



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as the total number of states is conserved


3a. Spin-orbit coupling

34



34

for a lattice of simple cubic symmetry this gives



34

for a lattice of simple cubic symmetry this gives

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


4. Semiempirical extension of DFT

35

Redistribution of the occupation numbers provided by scattering processes


4. Semiempirical extension of DFT

35

Redistribution of the occupation numbers provided by scattering processes

Approximated by


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:


7. Same relaxation times around the Fermi surface

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Finally: equation-of-motion

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scalar damping parameter is


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:


Results: Fe, Co, Ni

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bcc Fe hcp Co fcc Ni

two eigenvalues of the damping matrix vs direction of M


Anisotropic FMR linewidth

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Anisotropic FMR linewidth

42


Temperature dependence of damping

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


Temperature dependence of damping

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


Temperature dependence of damping - 2

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Bhagat, Lubitz, PRB 10, 179 (1974)


Interband transitions at higher temperature

45

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

Gilmore et al, PRL 99, 027204 (2007)


Interband transitions at higher temperature

45

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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Two-magnon scattering

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


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


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);


Summary:

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!

not obvious experimentally, lots of artefacts

intrinsic versus extrinsic mechanisms

phenomenology versus ab-initio


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Damping of magnetization dynamics - MAGNETISM.eu - EMA