what is the difference in magnetism of size...

Freie Universität Berlin International Graduate School HUB 1. June 20071

Freie Universität Berlin

Klaus BaberschkeKlaus Baberschke

Institut für ExperimentalphysikInstitut für ExperimentalphysikFreie Universität BerlinFreie Universität Berlin

Arnimallee 14 DArnimallee 14 D--14195 Berlin14195 Berlin--Dahlem GermanyDahlem Germany

WHAT IS THE DIFFERENCE IN MAGNETISM WHAT IS THE DIFFERENCE IN MAGNETISM

OF SIZE CONTROLLED AND OF BULK MATERIAL? OF SIZE CONTROLLED AND OF BULK MATERIAL?

Today it is well known fact that the reduction of dimensionality of solid materials imposes extraordinary new features. Discovery and understanding of the properties of nanostructures, quantum dots, nanowires and other low-dimensional interfaces, ….. have lead to numerous technological applications. Prominent exam ples are applications in information processing and information -storage technologies, new light sources, lasers etc. Spintronics has emerged as a new field of semiconductor electronics which uses both the charge and spin for unique functionalities.

wwwwww..physikphysik..fufu--berlinberlin.de/~.de/~babbab


J inte

r

TCNi

TCCo

IEC

Ni

CoCu

Part I: Fundamentals

• Curie temperature TC ,

• Orbital- and spin- magnetic moments,

• Magnetic Anisotropy Energy (MAE)

Part II: Spin dynamics in nano-magnets:

• Element specific magnetizations and TC’s in trilayers.

• Interlayer exchange coupling and its T-dependence.

• Gilbert damping versus magnon-magnon scattering.






UHV – ac susceptibility

film prepared and measured in-situ


For thin films the Curie temperature can be manipulated

thickness (ML)0 5 10

0

200

400

600

Ferromagn.

Paramagn.finite sizeNi(001)/Cu(001)

M

M

T=300Kd=7.6

P. Poulopoulos and K. B.J. Phys.: Condens. Matter 11, 9495 (1999)

Th.v. Waldkirch, K.A. Müller, W. Berlinger, PRB (1973)

Fe3+-VO /SrTiO3

ν/1

)()()( −=

∞−∞

cdT

dTT

C

CC

Note that some figures in theweb-version are missing due tofile-size.


Crossover of MCo(T) and MNi(T)

Two order parameter of TCNi and TC

Co

A further reduction in symmetry happens at TClow

760 800 840 880-60

-40

-20

0

20

x2

Nor

m.X

MCD

Diff

eren

ce(a

rb.u

nits)

Photon Energy (eV)

Cu(001)

2.1ML Cu

4ML Ni

Cu(001)

2.1ML Cu

4ML Ni

1.3ML Co

-edgesL3,2Co Ni -edgesL3,2

45K

0 40 80 120 160 200 2400

100

200

300

500

750

1000

Mag

netiz

atio

nM

(Gau

ss)

Temperature (K)

MNieasy

MCoeasy

H ,ext k

MNiAFM

[110]

[100]

Msat

Msat

2√

[110

]


−≡−−≡ −− 2}22{)2( 2/1

2ψ

Orbital magnetism in second order perturbation theory

The orbital moment is quenched in cubic symmetry

⟨2- LZ 2-⟩ = 0,

but not for tetragonal symmetry

L± • S

LZ • SZ

+λL •S +_

d1

W.D. Brewer, A. Scherz, C. Sorg, H. Wende, K. Baberschke, P. Bencok, and S. Frota-PessoaDirect observation of orbital magnetism in cubic solidsPhys. Rev. Lett. 93, 077205 (2004)andW.D. Brewer et al. ESRF – Highlights, p. 96 (2004)

H '= µBH•L + λL•S


Enhancement of Orbital Magnetism at Surfaces: Co on Cu(100)

λ

λλ

/10

/)2(3

/)1(

expnDd

n

nDdn

dD

S

L

eCeBAe

MM

−−=

−−=

−−

Σ+Σ+

=

M. Tischer et al., Phys. Rev. Lett. 75, 1602 (1995)


Giant Magn. Anisotropy of SingleCo Atoms and Nanoparticles

P. Gambardella et al., Science 300, 1130 (2003)

n = number of atoms

Induced magnetism in molecules

T. Yokoyama et al., PRB 62, 14191 (2000)

XMCD

XMCD

CO

~531 eV e-

∼10ML NiCu

µ

µL


Orbital and spin magnetic moments deduced from XMCD

H. Ebert Rep. Prog. Phys. 59, 1665 (1996)

860 880 900

L2 edgeL

µLµS

3 e dge

Ninorm

.XM

CD

(arb

.uni

ts)

Photon Energy (eV)

µLµS

dELL )2( 23 µµ ∆−∆∫

dELL )( 23 µµ ∆+∆∫

?

??

/µ

µL

S

Fe40 Fenm Fe 4 /V4 Fe2/V54/ V2

bulk Fe

0

0.06

0.12/FMR: (Fe+V)

XMCD: (Fe)µ µL / S

XMCD(V)µ µL / S

µSµL

510 520 530 700 720 740-1

0

1

2

FeV

x5

prop. µL

prop. µS

Photon Energy (eV)

g<2 g>2

µS µL µS µ L

L3 L2

XM

CD

Inte

gral

(ar

b. u

nits

)( ) ( )( ) d

Zdh

2L3L

dZ

dZd

h2L3L

L2N

NdE

T7S23N

NdE2

∫

∫

=+

+=⋅−

? µ? µ

? µ? µ


1. Magnetic anisotropy energy = f(T)

2. Anisotropic magnetic moment ≠ f(T)

≈ 1µeV/atom is very small compared to≈ 10 eV/atom total energy but all important

B (G)

100

100 20000

500

300

300

M (G

)

[111]

[100]

Ni[100]

[111]

0.1 G∆µL ~~

Characteristic energies of metallic ferromagnets

binding energy 1 - 10 eV/atom

exchange energy 10 - 103 meV/atom

cubic MAE (Ni) 0.2 µeV/atom

uniaxial MAE (Co) 70 µeV/atom

K. Baberschke, Lecture Notes in Physics, Springer 580, 27 (2001)

anisotropic µL ↔ MAE

ξLSMAE ∝ ∆µL4µBBruno (‘89)

g|| - g⊥ = geλ(Λ⊥-Λ||)

D= ∆gλge

MAE = ?M·dB ˜ ½ ?M·?B ˜ ½ 200 · 200 G2

MAE ˜ 2·104 erg / cm3 ˜ 0.2 µeV / atom

Magnetic Anisotropy Energy (MAE) and anisotropic µL


Growth of artificial nanostructuresbcc, fcc → tetragonal, trigonal

fcc Cu(001) substrate (ap=2.55Å)

Bain patha + ε1

c + ε2

There are only 2 origins for MAE: 1) dipol-dipol interaction ∼ (µ1• r)(µ2• r) and2) spin-orbit coupling ? L S (intrinsic K or ∆Eband)



“volume”, “surface” and “interface” MAE

R. Hammerling et al., PRB 68, 092406 (2003)

full trilayer grows in fct structuret=T/TC(d)



O. Hjortstam, K. B. et al. PRB 55, 15026 (’97)

-0.005 0.000 0.005 0.010

-100

0

100

200

α =1α =0.05

Ni

c/a=1.08

c/a=0.69c/a=0.79

K (

µeV

)V

∆ Orbital moment (µ )B

K

(µeV/atom)

V

Orbital moment

B

( /atom)

µ

0.75 0.85 0.95 1.05

0.05

0.06

0.07

SO (001]SO+OP [110]SO+OP [001]

SO[110]

bcc fcc

-100

0

100

200

300

400

500

exp.

c/a

Magnetic Anisotropy Energy MAE and anisotropic µL


The surface and interface MAE are certainlylarge (L. Néel, 1954) but count only for onelayer each. The inner part (volume) of a nano-structure will overcome this , because theycount for n-2 layers.

C. Uiberacker et al.,PRL 82, 1289 (1999)

SP-KKR calculation for rigit fcc and relaxed fct structures

R. Hammerling et al.,PRB 68, 092406 (2003)

layer resolved ∆Eb=ΣKi at T=0


( )AsGa 1-x In x

Virtual Crystal Approximation (VCA) X-ray diffraction of pseudobinary

COMPOSITION (x in Ga1-xInxAs)

0 0.2 0.4 0.6 0.8 1.0

2.45

2.50

2.55

2.60

VIRTUALCRYSTAL

( Vegard's law)

Ga - As

In - As

X - RAY

GaAs InAs

Mikkelsen & Boyce PRL , 1412 (82)49

EXAFS a local probe


Determination of orbital- and spin- magnetic moments

Which technique measures what?

µL , µS in UHV-XMCD

µL + µS in UHV-SQUID

µL / µS in UHV-FMR

per definition:

1) spin moments are isotropic

2) also exchange coupling J S1·S2 is isotropic

3) so called anisotropic exchange is a (hidden)projection of the orbital momentum intospin space

For FMR see: J. Lindner and K. BaberschkeIn situ Ferromagnetic Resonance:An ultimate tool to investigate the coupling in ultrathin magnetic filmsJ. Phys.: Condens. Matter 15, R193 (2003)






Part I: Fundamentals

• Curie temperature TC ,

• Orbital- and spin- magnetic moments,

• Magnetic Anisotropy Energy (MAE)


A whole variety of experiments on nanoscale magnets are available nowadays. Unfortunately many of the data are analyzed using theoretical static mean field (MF) model, e. g. by assuming only magnetostatic interactions of multilaye rs, static exchange interaction, or static interlayer exchange coupling (IEC), etc. We will show that such a mean field ansatz is insufficient for nanoscale magnetism, 3 cases will be discussed to demonstrate the importance of higher order spin-spin correlations in low dimensional magnets.

Physica B 384, 147 (2006)

⟨⟨ ⟩⟩S Si j + −∂

∂t →Spin-Spin correlation function

S i S S S S S S S j i i j i j i≈ − ⟨ ⟩ − ⟨ ⟩ +⟨ ⟩S S i j+ + + + ++− −z z

RPA…


A trilayer is a prototype to study magnetic coupling in multilayers.

What about element specific Curie-temperatures ?

Two trivial limits: (i) dCu = 0 ⇒ direct coupling like a Ni-Co alloy(ii) dCu = large ⇒ no coupling, like a mixed Ni/Co powder

BUT dCu ≈ 2 ML ⇒ ?

1. Element specific magnetizations and TC’s in trilayers.


Ferromagnetic trilayers

U. Bovensiepen et al.,PRL 81, 2368 (1998)

760 780 800 820 840 860 880 900

-40

-20

0

336K

336K

290K

290K

x 2

Ni L3,2

Co L3,2

XM

CD

(arb

.uun

its)

Photon energy (eV)

Cu (001)

2.0ML Co

2.8ML Cu

4.3ML Ni

MCo

MNi

290 300 310 320 3300.0

0.1

0.4

0.8

TC

Co = 340 K

T*C

Ni = 308 K

T (K)M

(arb

. uni

ts)

0

200

400 Ni L3,2Co L3,2

x 1.72

no

rm.

ab

sorp

tion

(a

.u)

760 800 840 880-100

-80

-60

-40

-20

0

20

T = 140K

2.2 ML Co3.4 ML Cu3.6 ML NiCu(001)

Cu(001)

XM

CD

(arb

.uun

its)

h (eV)ν


J.H. Wu et al. J. Phys.: Condens. Matter 12 (2000) 2847

-0.02

0.00

0.02

EFM2

ENM

EFM1

ETOT

0 1 2 3 4 5 6 7d NM

60

30

0

30

60 T1/ max

FM coupled

AFM coupled

1/m

ax (

arbi

trar

y un

it)

-0.010

-0.005

0.000

0.005

T=0oK

T=250oK

(c)

(a)

(b)

χ

χ

T (

o K)

∆

∆

∆∆∆∆

E=E

AFM

EFM

(eV

/ML)

∆-

+

+

Single band Hubbard model: Simple Hartree-Fock (Stoner) ansatz is insufficientHigher order correlations are needed to explain T C-shift

Enhanced spin fluctuations in 2D (theory)∆

T /

TC

Ni

3 K

1 K

1 2

Co/Cu/Nitrilayer

d (ML)Ni

J =3 Kinter

MF

0 3 4 5 60.0

0.3

0.6

0.9 Tyablikov (or RPA)decoupling

, mean field ansatz (Stoner model) is insufficientto describe spin dynamics at interfaces of nanostructures S S ⟨ ⟩i j

+z

⟨⟨ ⟩⟩S Si j + −∂

∂t →Spin-Spin correlation function

S i S S S S S S S j i i j i j i≈ − ⟨ ⟩ − ⟨ ⟩ +⟨ ⟩S S i j+ + + + ++− −z z

RPA…

P. Jensen et al. PRB 60, R14994 (1999)


FM1 (Ni)

FM2 (Co)

NM (Cu)

IEC ~ 1dNM

2

dNM

dFM1

Jinter

2D sp

in

fluctu

ation

s

∆TC, Ni

Evidence for giant spin fluctuations [A. Scherz et al.PRB, 73 54447 (2005)]

2D

3D0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

T / TC,Ni

2.1 2.6 3.1 4.2 4.0

d (ML)Ni

1 2 3 4 5 6

Theory / Experiment

C,N

iM

C,N

iM

/(T

= 0

)


2. Interlayer exchange coupling and its T-dependence.

a) J. Lindner, K. B., J. Phys. Condens. Matter 15, S465 (2003)b) A. Ney et al., Phys. Rev. B 59, R3938 (1999)c) J. Lindner et al., Phys. Rev. B 63, 094413 (2001)d) P. Bruno, Phys. Rev. B 52, 441 (1995)

Theory d)

2 3 4 5 6 7 8 9-20

-15

-10

-5

0

5

10

15

20

25

J inte

r (µ

eV/a

tom

)

d (ML)Cu

FMR / / / /

/ / /

→

→XMCD / / /→

Cu Cu Cu(001)

Cu Cu(001)Cu Cu(001)

FMR

Ni Ni

NiNiCo

Co

a)b)c)


in-situ FMR in coupled films

J. Lindner, K. B. Topical Rev., J. Phys. Condens. Matter 15, R193-R232 (2003)

in-situUHV-experiment

theory

FMR



Temperature dependence of Jinter ⇔ ∆ free energy

J =J [ ]inter inter,0 1-(T/T )C 3/2

P. Bruno, PRB 52, 411 (1995) N.S. Almeida et al. PRL 75, 733 (1995)

J =J inter inter,0 [ ]T/T0

sinh(T/T )0T = hv / 2 k d0 F Bπ

J. Lindner et al.PRL 88, 167206 (2002)

Ni7Cu9Co2/Cu(001)

T=55K - 332K

(Fe2V5)50

T=15K - 252K, TC=305K

0 2000 4000 6000 80000

10

20

T3/2

T3/2

T5/2

J (µ

eV/a

tom

)in

ter

v=2.8 10cm/sF

8

v =2.8 10cm/sF 7

T=294K

0 0.2 0.4 0.6 0.80

50

100

150

J (

µeV

/ato

m)

inte

r

(T/T )C3/2

(T/T )C3/2

(T/T )C5/2

v=5.3 10cm/sF6


S. Schwieger, W. Nolting, PRB 69, 224413 (2004)

PRB (2007) submitted.

J(T) ≈ 1- ATn

n ≈ 1.5


T-dependence of interlayer exchange coupling

• What causes the temperature dependence of IEC?• band structure effects (smearing out of Fermi edge)?• spin wave excitations?• Experiment measures only one observable (IEC)

• Theory has the opportunity to switch on/off contributions:¨ Spin wave excitations account for

75% of T-dependence¨ band structure effect (25%)

is not negligible!

• S. Schwieger et al., unpublishedRICHTIGNi7/Cu9/Co2/Cu(001)


T1

T2

effdd

Hmm

×−= t

γ Gilbert damping

Landau-Lifshitz-Gilbert equation(1935)

Bloch-Bloembergen Equation (1956)

2

,,eff

,

1eff

)(d

d

)(d

d

T

m

t

m

TMm

t

m

yxyx

yx

Szz

z

−×−=

−−×−=

Hm

Hm

γ

γ

spin-spin relaxation(transverse)

spin-lattice relaxation(longitudinal)

t

ddm

m×+α

αω1

~T

|M|=const.

Mz=const.

3. Gilbert damping versus magnon-magnon scattering.


k||

ω(k )||

ωFMR

ϕ ϕk|| c< ϕ ϕk|| c>

∆ ωH ( ) = Gil Gγ2MS

ω

ω γ π γ µ0 2 S B

2

S

= (2K - 4 M ), =( /h)gK - uniaxial anisotropy constantM - saturation magnetization

⊥

⊥

∆ ωH ( ) = arcsin2Mag Γ[ω ω2 2 1/2+( /2) ] - 0 ω0/2[ω ω2 2 1/2+( /2) ] + 0 ω0/2√

1 = _∂ M ∂ MγMS

G2(M H )× eff ( )M ×∂ t ∂ t+γ

Landau-Lifshitz-Gilbert-Equation

Gilbert-damping ~ω

2-magnon-scattering

FMR Linewidth - Damping

R. Arias, and D.L. Mills, D.L. Mills and S.M. Rezende in‘ ‘, edt. by B. Hillebrands and K. Ounadjela, Springer Verlag

Phys. Rev. B , 7395 (1999);

Spin Dynamics in Confined Magnetic Structures

6 0

viscous damping, energy dissipation


• Gilbert damping contribution: • linear in frequency • two-magnon excitations (thin films):

non-linear frequency dependence

0

100

200

∆H

(Oe

)

∆H inhom

∆H0*

0 20 40 60 80ω /2π (GHz)

∆H

inhomogeneous broadening

∆HGilbert+∆Hinhom

∆H2-magnon

∆H

real relaxation – no inhomogeneous broadening

two-magnon damping dominates Gilbert damping

by two orders of magnitude:

1/T2~109 s-1 vs. 1/T1~107 s-1

R. Arias et al., PRB 60, 7395 (1999)

2/)2/(

2/)2/(arcsin)(

02

02

02

02

magnon2ωωω

ωωωω

++

−+Γ=∆ −H

eff0with Mγω =K. Lenz et al., PRB 73, 144424 (2006)


Γ γ⋅Γ G α ∆H0

(kOe) (108 s-1) (108 s-1) (10-3) (Oe)Fe4V2 ; H||[100] 0.270 50.0 0.26 1.26 0Fe4V4 ; H||[100] 0.139 26.1 0.45 2.59 0Fe4V2 ; H||[110] 0.150 27.9 0.22 1.06 0Fe4V4 ; H||[110] 0.045 8.4 0.77 4.44 0Fe4V4 ; H||[001] 0 0 0.76 4.38 5.8

• recent publications with similar results:– Pd/Fe on GaAs(001) –

network of misfit dislocationsG. Woltersdorf et al. PRB 69, 184417 (2004)

– NiMnSb films on InGaAs/InP B. Heinrich et al. JAP 95, 7462 (2004)

• two-magnon scattering observed in Fe/V superlattices –

• J. Lindner et al., PRB 68, 060102(R) (2003)

HF FMR K. Lenz et al. PRB 73, 144424 (2006)

0 50 100 150 2000

200

400∆H

PP

(Oe)

f (GHz)

0 4 80

40

80

15 Oe !


s-d-exchange between spin wave and s-electron

R.H. Silsbee, A. Janossy, P. Monod, PRB 19, 4382 (1979)

Precession drives spin current into NM

NM-substrate acts as spin-sink ⇒ ISback = 0

⇒ torque is carried away

⇒ Gilbert damping enhanced by spin-pump effect!

t

MG

t S d

dd

d2eff

MMHM

M ×+×−=γ

γ SpumpI

VM S

γ+

−×=

dtA

dtA ir

MMMIS dd

4πh

Landau-Lifshitz equation + extension

precession Gilbert-damping spin-pump current

pump

(5)

Y. Tserkovnyak, A. Brataas, G.E.W. Bauer, PRB 66, 224403 (2002)

“Spin pump” effects,


B. Heinrich et al., PRL 90, 187601 (2003)

K. Lenz et al.,Phys. Rev. B 69, 144422 (2004)

d no spin-accumulation I = 0 Gilbert-damping enhanced by spin-pump effect

NM SF≥λ ⇒ ⇒⇒

s back

compensation, if both films precess simultaneously (H =H ) only Gilbert contribution remains!

res1 res2

⇒

M M MM

acousticalmode

opticalmode

at „point of contact“ compensation of pumped currents decrease the linewidth


ConclusionFor nanoscale ferromagnets :

• use the reduced temperature t = T/TC

• the orbital magnetic moment is NOT quenched

• the MAE may be larger by orders of magnitude

Higher order spin-spin correlations are important to explain the magnetism of nanostructures.

In most cases a mean field model is insufficient.

A phenomenological effective Gilbert damping parameter gives very little insight into the microscopic relaxation mechan ism. It seems to be more instructive to separate scattering mechanisms within the magnetic subsystem from the dissipative scattering into thethermal bath

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what is the difference in magnetism of size...

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