new p. vavassori - magnetism.eumagnetism.eu/esm/2018/slides/vavassori-slides2.pdf · 2018. 9....

P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

FT-2: Magneto-optics and Magneto-plasmonics Part 1

P. Vavassori

-Ikerbasque, Basque Fundation for Science and CIC nanoGUNE Consolider, San Sebastian, Spain.


Outline

Magneto-optics

Brief overview of the Magneto-optical Kerr effects (MOKE)

Advanced MOKE: vector magnetometry

Magnetic nanostructures

micro-MOKE

Approach 1: focused beam

Approach 2: microscopy

MOKE from diffracted beams:

simple theory of diffracted MOKE

in conjunction with micromagnetics and MFM

from magnetometry to magnetic imaging


John Kerr1824 - 1907

Michael Faraday1791 - 1867

s

p

s

p

s

p

Reflected Light

z

θ

x

Transmitted Light

Polarization Plane

Sample

The MagnetoOptical Effect

x

z

y

p

s

ps

θ θ

x

z

y

p

s

ps

θ θ

But, what happens if we applied a magnetic field??

→ →

→ →

=

s s p s

s p p p

r rR

r r


The Magneto-Optical Effect

→ →

→ →

=

s s p s

s p p p

r rR

r r

xx xy xz

yx yy yz

zx zy zz

x

z

yHy

p

s

ps

θ θ

x

z

y

Hx

p

s

ps

θ θ

Hz

Polar and Longitudinal Configuration Transverse Configuration

Reflectivity matrixDielectric Tensor


MOKE

The magneto-optic Kerr effect (MOKE) is widely

used in studying technologically relevant

magnetic materials.

It relies on small, magnetization induced changes

in the optical properties which modify the

polarization or the intensity of the reflected light.

Macroscopically, magneto-optic effects arise from

the antisymmetric, off-diagonal elements in the

dielectric tensor.

sssp

pspp

rr

rrSample

rpp = r0

pp+ rppM my

rps - mx - mzrsp mx-mz

Fresnell reflection coefficients

iTM

rTMpp

E

Er =

iTE

rTMps

E

Er =

iTM

rTEsp

E

Er =

iTE

rTEss

E

Er =

−

−

−

=

0

0

0

ˆ

xy

xz

yz

ii

ii

ii

=

0

0

0

00

00

00

ˆ

x = 0 Q mx; y = 0 Q my; z = 0 Q mz;

• Non-destructive;

• High sensitivity;

• Finite penetration depth (~ 10 nm);

• Fast (time resolved measurements);

• Laterally resolved (microscopy);

• Can be easily used in vacuum and

cryogenic systems;

J. Kerr, Philosophical Magazine 3 321 (1877)Z. Q. Qui and S. D. Bader, Rev. Sci. Instrum. 71, 1243 (2000) P. Vavassori, APL 77 1605 (2000)


MOKE origin: classical picture Lorentz force


Here, x (r) is the spin-orbit parameter or coupling constant, which depends on the gradient of the electrostatic potential of the nuclear charges.

Its values are of the order of 10-100meV and, thus, the spin-orbit interaction is much weaker than the exchange interaction (≈ 1eV).

Electron theory of Magneto-Optics

Microscopically, the coupling between the electric field of the propagating light and the

electron spin in a magnetic medium occurs through the spin-orbit interaction splitting of

optical absorption lines (Zeeman effect).


Microscopic origin of Magneto-Optics

s+ and s- Selection rules

Optical transitions between the d-orbitals and the p-orbitals

Only transition from m = 1 to m = 0 are considered for simplicity


Microscopic origin of Magneto-Optics



• Magnetization→Splitting of spin-states (Exchange)

– No direct cause of difference of optical response between LCP and RCP

• Spin-orbit interaction→Splitting of orbital states

– Absorption of circular polarization→Induction of circular motion of electrons

• Condition for large magneto-optical response

– Presence of strong (allowed) transitions

– Involving elements with large spin-orbit interaction

– Not directly related with Magnetization

MOKE results from a lifting of orbital degeneracy due to spin-orbit interaction (SOI) in the presence of spontaneous spin polarization.


−=

xx

xxxy

xyxx

00

0

0~ xy = i0 Q mz;

( )( )

( ) 0~

2

2

2=

+ E

tcErotrot

0

00

0

0

2

2

=

−

−

−−

z

y

x

zz

xxxy

xyxx

E

E

E

n

n

xyxx in =2

Maxwell Equation

Eigenequation

Eigenvalue Eigenmodes：LCP and RCP

Without off-diagonal terms：No difference between LCP & RCP

( ) ( )tznkieEE −= 00 ( ) 0ˆ2 =−− EEnnE n

A simple case: M || z

Different modes :different speed and attenuation

Therefore, incidentlight becomeselliptically polarizedafter propagation in a MO activematerial


Phenomenology of MO effect

Linearly polarized light can be decomposed to LCP and RCP

Difference in phase causes rotation ofthe direction of Linear polarization

Difference in amplitudes makes Elliptically polarized light

In general, elliptically polarized lightWith the principal axis rotated

Different speed(phase lag)

Different attenuation


1

220

212

21

20

12

1

sinsin1sin1cos

n

nn

n

n

−=−=−=

rpp = r0

pp+ rppM my

rps - mx - mzrsp mx-mz

xy = i1Q mz; xz = -i1Q my; yz = i1 Q mx; xy = -yx; zx = -xz; zy = -yz;

sssp

pspp

rr

rr

General case: Oblique incidence and arbitrary direction of M


Polarization conversion

Summary of phenomenology

Ei Er Ei Er

M

M = 0

pxpx

py( )0

−=

xx

yx

yppx

py


( )

+=

=

= sincos

11~

ir

re

r

rr

rE

pp

spi

pp

sp

sp

pp

r

K K

pp

sp

r

rRe

ss

ps

r

rRe

pp

sp

r

rIm

ss

ps

r

rIm

rsp


x

y

x’

y’

l/4plate

h

E0

E0sinh

E0cosh

E E i i j= +0(cos sin )h h

Opticaxis

( )

E E i i e j

E i j

E i

i

' (cos sin )

cos sin

'

= +

= +

=

−

02

0

0

h h

h h

x

y

h

E’

E

Measurement of ellipticity

E0 Er

E’rM


l/4

l/2

45°

22.5°

p

sp

s p

s

p

s

wollastone

45°p

s

wollastone

Measurement of ellipticity & rotation: high sensitivity

+M

-M

+M

-M

=0

+M

-M


Measurement of ellipticity & rotation: high sensitivity


Measuring K and K

Modulation polarization technique for recording the longitudinal and polar Kerr effects, which are proportional to the magnetization components mx mz..

POLARIZER Glan-Thompson

PHOTOELASTIC MODULATOR (50kHz)

ORIGINAL ELLIPTICTY

MODULATED ELLIPTICTYPREAMPLIFIED PHOTODIODE

POLARIZER

HeNe LASER

p-polarized beams-p polarized reflected

beam (elliptical polarization)

y

x

Lock-in

Electromagnet

E

More details in: P. Vavassori, Appl. Phys. Lett. 77, 1605 (2000)

K Kspol - mx – mzppol mx- mz


Io

Transverse Kerr effect

Laser

Polarizer

Detector

p-polarized light (TM)M

Er = rpp Eo

rpp = ro

pp + rm

ppmy

Ir = Er (Er)*

Ir = Io+ DIm DIm/Io a my

The reflected beam is p-polarized.Variation of intensity and phase.

y

Y. Souche et al.JMMM 226-230, 1686 (2001);JMMM 242-245, 964 (2002).

M fm

Dfm a my

E

E

Polarizer

l/4


Supplementary information

Examples of application of MOKE for magnetic characterization of materials and nanosctructures


Vector analysis of reversal

H

MMyMx

x

y

My Mx-1.0

-0.5

0.0

0.5

1.0

-1.0

-0.5

0.0

0.5

1.0

-10000 -5000 0 5000 10000

-1.0

-0.5

0.0

0.5

1.0

my

mx

Fig. 2

mz

Field (Oe)

MgO

Fe(300 nm)

NiO(1.4 nm)Fe(7 nm) 180-nm-thick CoNiO

-10000 -5000 0 5000 10000

0

30

60

90

120

150

180

210

240

10000 5000 0 -5000 -10000

180

210

240

270

300

330

360

390

420

-10000 0 10000

0.5

1.0

10000 0 -10000

0.5

1.0

Branch up

out

in

Ro

tatio

n a

ng

le (

de

g.)

out

in

Branch down

Ro

tatio

n a

ng

le (

de

g.)

Field (Oe)

m

Field (Oe)

m

Field (Oe)

- F. Carace, P. Vavassori, G. Gubbiotti, S. Tacchi, M. Madami, G. Carlotti, and T. Okuno, Thin Solid Films 515/2, 727 (2006).- A. Brambilla, P. Biagioni, M. Portalupi, P. Vavassori, M. Zani, M. Finazzi, R. Bertacco, L. Duò, and F. Ciccacci, Phys. Rev. B. 72, 174402 (2005).- P. Vavassori, Appl. Phys. Lett. 77, 1605 (2000) .

Reconstruction of the magnetization vector during the reversal


Element sensitivity (and layer sensitivity)

XMCD x-ray magnetic circular dichroism (FeL3 Py hysteresis loop and CoL3 thresholds), → chemical and magnetic sensitivity

PyCuCo

230 nm

10 nm

-1000 -500 0 500 1000

0.1725

0.1730

0.1735

0.1740

0.1745

0.1750

Rotation

MO

KE

sig

nal (a

rb. units)

Field (Oe)

-1000 -500 0 500 10000.065

0.066

0.067

0.068

0.069

0.070

0.071

0.072

0.073

0.074

0.075Ellipticity

MO

KE

sig

nal (a

rb. units)

Field (Oe)

MOKE XMCD

-1000 -500 0 500 1000-5.4

-5.2

-5.0

-4.8

-4.6

-4.4

-4.2

-4.0

Fe

Sig

na

l

Field (Oe)

-1000 -500 0 500 1000-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Co

Sig

na

l

Field (Oe)

Simulations

-1000 -500 0 500 1000-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

Permalloy

MO

KE

sig

nal (a

rb. units)

Field (Oe)

-1000 -500 0 500 1000-0.004

-0.003

-0.002

-0.001

0.000

0.001

0.002

0.003

0.004

Cobalt

MO

KE

sig

na

l (a

rb.

un

its)

Field (Oe)

JOURNAL OF PHYSICS D-APPLIED PHYSICS 41, 134014 (2008)


0 20 40 60 80 100

-0,010

-0,005

0,000

0,005

0,010

0,015 Py S Rotation P Rotation

s Ellipticity

P Ellipticity

Radia

nts

Incidence Angle (degrees)

0 20 40 60 80 100

-0,04

-0,02

0,00

0,02

0,04

0,06 Co S Rotation P Rotation

s Ellipticity

P Ellipticity

deg

rees

Incidence Angle (degrees)

0 1 2 3 4 5 6

-0,015

-0,010

-0,005

0,000

0,005

0,010

0,015

0,020

Py S Rotation

P Rotation

s Ellipticity

P Ellipticity

deg

rees

Photon energy (eV)

0 1 2 3 4 5 6

-0,030

-0,025

-0,020

-0,015

-0,010

-0,005

0,000

0,005

0,010

0,015

0,020

0,025

0,030

0,035

0,040

0,045

Co S Rotation

P Rotation

s Ellipticity

P Ellipticity

de

gre

es

Photon energy (eV)

Element sensitivity (thin layer regime)


Layer sensitivity


Ht = Ht0 Sin(2ft)

H

Lock-in 1: Ref.

freq. 50-100kHz

Lock-in 2:Ref

freq. f

mx0 mx = mx0 Sin (2ft)

MOKE transverse susceptibility: anisotropy

The quantity measured with the Lock-in2 is proportional to the transverse

suceptibility ct = D0 / Ht0.

It can be shown that : 1/ct = (Eo''(eq)/ eq) where Eo(eq) is the total free energy and eq is the

average magnetization, which makes an angle eq with the EA.

H = 700 OeHt = 35 Oef = 156 Hz

0.0

0.4

0.8

1.2

0

30

6090

120

150

180

210

240270

300

330

0.0

0.4

0.8

1.2

1/c

M

t (O

e

Vo

lt-1)

ea

ea ea

ea

1/c

Mt (

arb

. u

nit

s)


Configurational anisotropy

Flower state: higher energy

Leaf state: lower energy

Even small perturbations from uniform magnetization which must exist in anynonellipsoidal magnet give rise to a (strong) angular dependence of magnetostaticdipolar energy in symmetric squared particles, which should be magnetically isotropicin the approximation of M uniform. This anisotropy is called configurational anisotropy

Py squares 150 x 150 x 15 nm3

H

fourfold symmetry, at first order, eightfold symmetry at second order

R. P. Cowburn et al. Phys. Rev. Lett. 81, 5414 (1998)

H = 0

P. Vavassori, et al., Phys. Rev. B 72, 054405 (2005)


Kerr microscopy: focused laser beam

Laser

Detector

CCD

PHYSICAL REVIEW 72, 224413 (2005)

D. A. Allwood, et al., J. Phys. D: Appl. Phys. 36, 2175 (2003)

Py wires 175 nm wideMicro-MOKE

Py wire 200 nm wideSingle loop

Py wire 100 nm wide1000 loops


Polar

Longitudinal and transverse

Kerr microscopy: imaging

M

M


Magnetometry of ultra-small nanostructure

EBIDelectron beam induced deposition

Scanning electron microscopy image of the set of EBID cobalt structures


APPLIED PHYSICS LETTERS 100, 142401 (2012)

t = 20 nm

single sweep measurement sensitivity of approximately 1 x10-15 Am2

sensitivity of 10-12 to 10-13 Am2 for the latest generation of SQUID magnetometer

t = 5 nm

9 averagesSingle sweep

Magnetometry of ultra-small nanostructure


Diffraction of light by an array

"Diffracted-MOKE: What does it tell you?",

M. Grimsditch and P. Vavassori J. Phys.: Condensed Matter 16, R275 - R294 (2004).

As is well known for optical gratings, when a beam of light is incident upon a sample that has a structure comparable to the wavelength of radiation, the beam is not only reflected but is also diffracted. If the material is magnetic, one may ask whether the diffracted beams also carry information about the magnetic structure.


Examples of D-MOKE loops

Peculiar structures due to- Collective properties- Interference effects

P. Vavassori, et al., J Appl. Phys. 99, 053902 (2006)

M. Grimsditch, P. Vavassori, et al., Phys. Rev. B 65, 172419 (2002)


P. Vavassori, et al., J. Appl. Phys. 101, 023902 (2007)

P. Vavassori, et al.,Phys. Rev. B 59 6337 (1999)



-2000 -1500 -1000 -500 0 500 1000 1500 2000

-1,0

-0,5

0,0

0,5

1,0

1,5

Rflected

No

rm.

Sig

na

l-2000 -1500 -1000 -500 0 500 1000 1500 2000

-1,0

-0,5

0,0

0,5

1,0

1,5

1st order

No

rm.

Sig

na

l

-2000 -1500 -1000 -500 0 500 1000 1500 2000-3,0

-2,5

-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

2,5

3,0

Field (Oe) Field (Oe)

Field (Oe)

2nd

order

No

rm.

Sig

na

l

-2000 -1500 -1000 -500 0 500 1000 1500 2000

-2,5

-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

2,5

3,0

No

rm.

Sig

na

l

3rd order

-2000 -1500 -1000 -500 0 500 1000 1500 2000-1,5

-1,0

-0,5

0,0

0,5

1,0

No

rm.

Sig

na

l

4th order

-600 -400 -200 0 200 400 600

-0.274

-0.273

-0.272

-0.271

-0.270

-0.269

-0.268

Reflected

D-M

OK

E I

nte

nsi

ty (

arb

. u

nits

)

Field (Oe)

H

-2000 -1500 -1000 -500 0 500 1000 1500 2000

-1,0

-0,5

0,0

0,5

1,0

1,5

Rflected

No

rm.

Sig

na

l

-2000 -1500 -1000 -500 0 500 1000 1500 2000

-1,0

-0,5

0,0

0,5

1,0

1,5

1st order

No

rm.

Sig

na

l

-2000 -1500 -1000 -500 0 500 1000 1500 2000-3,0

-2,5

-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

2,5

3,0

Field (Oe) Field (Oe)

Field (Oe)

2nd

order

No

rm.

Sig

na

l

-2000 -1500 -1000 -500 0 500 1000 1500 2000

-2,5

-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

2,5

3,0

No

rm.

Sig

na

l

3rd order

-2000 -1500 -1000 -500 0 500 1000 1500 2000-1,5

-1,0

-0,5

0,0

0,5

1,0

No

rm.

Sig

na

l

4th order

Inte

nsity (

norm

. sig

nal)

-600 -400 -200 0 200 400 6000.838

0.840

0.842

0.844

0.846

0.848

0.850

0.852

0.854

0.856

Field (Oe)

D-M

OK

E In

tensi

ty (

arb

. u

nits

)

2nd

order

H

Reflected

Reflected

2nd order

2nd order

-600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 Field (Oe) Field (Oe)

3.0

2.0

1.0

0.0

-1.0

-2.0

-3.0

1.0

0.5

0.0

-0.5

-1.0

1.0

0.5

0.0

-0.5

-1.0

1.0

0.5

0.0

-0.5

-1.0

-2000 -1000 0 1000 2000

Inte

nsity (

norm

. sig

nal)

-2000 -1000 0 1000 2000

IncidencePlane (xz)

H

Laser532 nm50 mW

E

Diffraction

pattern

coil

coil

Photo

dete

cto

r

IncidencePlane (xz)

H

Laser532 nm50 mW

E

IncidencePlane (xz)

H

Laser532 nm50 mW

IncidencePlane (xz)

H

Laser532 nm50 mW

E

Diffraction

pattern

coil

coil

Photo

dete

cto

r


Intuitive explanation of D-MOKE loops

Peculiar structures due to- Interference effects- Collective propertiesO. Geoffroy et al., J. Magn. Magn. Mat. 121 (1993) 516

Y. Souche et al., J. Magn. Magn. Mat., 140-144 (1995) 2179

H

rpp = ro

pp + rm

ppmy(x,y)

M M


Physical-optics approximation provides a very simple and physically transparent

description.

The electric field in the nth order diffracted beam, due to the periodic modulation of

the “effective” reflectivity r’pp is:

End = Eo fnfn = S r’pp exp{i n G•r} dS where n integer, G reciprocal lattice vector

and S is the unit cell.

r’pp = ropp + rmpp my(x,y,H)

End = Eo (r

opp fn

nm + rmpp fnm ) with ropp(i, n, dots, subst), rmpp(i, n, dots, Q)

fnnm = S exp{i n G•r} dxdy non-magnetic form factor

fnm (H)= Dot my(x,y,H) exp{i n G•r}dxdy magnetic form factor

Ind = En

d (End)* DIn

m (my) = AnRe[fnm]+Bn Im[fnm]

Simple theory of diffracted-MOKE


DInm a An Re[fn

m] + Bn Im[fnm]

Re[fnm]= Dot my cos(n Gx x) dS

Im[fnm]= Dot my sin(n Gx x) dS

Im[f1m] = 0

Saturated state Re[f1

m] > 0

What are the differences due to?

y

x

period L

Unit cell

Gx = 2/L reciprocal lattice vector

Diffracted spots in the scattering planey

x

y

x

1st order

Im[f1m] < 0

Re[f1m] = 0

Im[f1m] > 0

Re[f1m] = 0

2nd order

Im[f2m] < 0 large

Re[f2m] = 0

y

x

y

x

3rd order

Im[f2m] < 0 very large

Re[f2m] = 0

Two-domain state: asymmetric M distribution


y

x

Tuning the sensitivity to selected portions of the dot!

Immaginary contribution:highlight any asymmetric (y-mirror symmetry breaking) magnetic (my) behaviour

Real contribution:y-mirror symmetry my

y

x

period L


Asymmetry to induce the desired vortex rotation

Narrow channel pins

the magnetization2 m

-600 -400 -200 0 200 400 600

-0.274

-0.273

-0.272

-0.271

-0.270

-0.269

-0.268

Reflected

D-M

OK

E Inte

nsity (

arb

. units)

Field (Oe)

3

1

2

-600 -400 -200 0 200 400 6000.838

0.840

0.842

0.844

0.846

0.848

0.850

0.852

0.854

0.856

Field (Oe)

D-M

OK

E In

tensity (

arb

. u

nits)

2nd

order

-600 -400 -200 0 200 400 600-0.46

-0.44

-0.42

-0.40

-0.38

-0.36

-0.34

1st order

D-M

OK

E Inte

nsity (

arb

. units)

-600 -400 -200 0 200 400 600-0.285

-0.280

-0.275

-0.270

-0.265

-0.260

-0.255

1st order

-600 -400 -200 0 200 400 600-0.231

-0.230

-0.229

-0.228

-0.227

-0.226

-0.225

-0.224

Field (Oe)

2nd

order

H

H

H

H

H

-600 -400 -200 0 200 400 600

-3

-2

-1

0

1

2

3

Field (Oe)

2nd

order (Re[fd

m] + 0.65 * Im[f

d

m])

-600 -400 -200 0 200 400 600-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

No

rma

lize

d D

-MO

KE

sig

na

l

Reflected

Field (Oe)

H

H H

H H

-600 -400 -200 0 200 400 600-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

1st order (Re[f

d

m] + 0.8 * Im[f

d

m])

No

rma

lize

d D

-MO

KE

sig

na

l

-600 -400 -200 0 200 400 600-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

1st order (Re[f

d

m] + 0.8 * Im[f

d

m])

-600 -400 -200 0 200 400 600-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

No

rma

lize

d D

-MO

KE

sig

na

l

Field (Oe)

2nd

order (Re[fd

m] + 0.65 * Im[f

d

m])

P. Vavassori, R. Bovolenta, V. Metlushko, and B. Ilic, J Appl. Phys. 99, 053902 (2006)

Measured Calculated


Measured D-MOKE loops from square rings

-2000 -1000 0 1000 2000

-1.0

-0.5

0.0

0.5

1.0

Norm

alize

d Kerr

sign

al

0th order

-2000 -1000 0 1000 2000

-1.0

-0.5

0.0

0.5

1.00

th order

-2000 -1000 0 1000 2000

-15

-10

-5

0

5

10

15

1st order

Field (Oe)

-2000 -1000 0 1000 2000

-3

-2

-1

0

1

2

3

1st order

Norm

alize

d Kerr

sign

al

Field (Oe)

(a) (b)

Reflected

1st order 1st order

Reflected

H H

Square lattice (4.1x4.1 m2) of

Permalloy square rings (2.1 m

side). Nominal width 250 nm.

Thickness 30 nm.

Note intense peaks in the diffracted loopsP. Vavassori, et al., Phys. Rev. B 67, 134429 (2003)


Square ring structures

1 saturated state

2 onion state 4 reversed onion state 5 saturated state

1 saturated state 4 reversed onion state 5 saturated state

2 onion state

-2000 -1000 0 1000 2000

-8

-4

0

4

8

Norm

alize

d sign

al 1st order

Field (Oe)

-2000 -1000 0 1000 2000-2

-1

0

1

2

Field (Oe)

1st order

Norm

alize

d sign

al

3 horseshoe state

3 vortex state

H

H


Quenching structures in intermediate states

and image them with MFM

-1000 -500 0 500 1000-3

-2

-1

0

1

2

3

Norm

alize

d Kerr

sign

al

Field (Oe)

H

-1000 -500 0 500 1000

-15

-10

-5

0

5

10

15

Norm

alize

d Kerr

sign

al

Field (Oe)

H

(c)

(b)

P. Vavassori, M. Grimsditch, V. Novosad, V. Metlushko, B. Ilic, P. Neuzil, and R. Kumar, Phys. Rev. B 67, 134429 (2003)


Y. Suzuki, C. Chappert, P. Bruno, and P. Veillet, J. Magn. Magn. Mater. 165 516 (1997) only for size >> l

“effective” reflectivity

An = Re[ro

pp* rmpp]

Bn = Im[ro

pp* rmpp]

An and Bn (i, n, dots, sub, Q)

r’pp = ropp + rmpp

For inhomogeneous gratings ropp = ropp, dot + ropp, sub

DInm (my) = 2 fn

nm { AnRe[fnm]+Bn Im[fn

m] }

About An and Bn

An interesting characteristic of D-MOKE related to this : the absolute value of (DI/Io)n is increased up to several times the specular value. Effect due to the compensation of the non-magnetic component of the light diffracted by the magnetic dots and the light diffracted by the (non-magnetic) complementary part of the substrate.

Io,n = |ro

pp, dot|2 fnnm + | ropp, sub |2 f’n

nm = ( |ropp, dot|2 - | ropp, sub |2 )fnnm


532 n

m la

ser

Different approach: towards Fourier imaging? 1st step

H(1,0)(-1,0)

(0,1)

(0,-1)(-1,-1) (1,-1)

(1,1)(-1,1)

Normal incidence: the scattering plane is defined by the selected spot

y

x

Sensitivity to (mx , my)

Sensitivity to mx

-2000 -1000 0 1000 2000

-0.0465

-0.0460

-0.0455

-0.0450

-0.0445

-0.0440

-0.0435

-0.0430

-0.04252

nd order m

x

MO

KE

in

ten

sity (

arb

. u

nits)

Field (Oe)

Vectorial D-MOKE

(DInm)norm = An Re[fn

m]- Bn Im[fnm]

(DI-nm)norm = −An Re[fn

m]- Bn Im[fnm]

Sensitivity to my

-2000 -1000 0 1000 2000-16.55

-16.50

-16.45

-16.40

-16.35

-16.30 2nd

order my

MO

KE

in

ten

sity (

arb

. u

nits)

Field (Oe)

-2000 -1000 0 1000 2000

-1.0

-0.5

0.0

0.5

1.0

Norm

alize

d Kerr

sign

al

0th order

-2000 -1000 0 1000 2000

-1.0

-0.5

0.0

0.5

1.00

th order

-2000 -1000 0 1000 2000

-15

-10

-5

0

5

10

15

1st order

Field (Oe)

-2000 -1000 0 1000 2000

-3

-2

-1

0

1

2

3

1st order

Norm

alize

d Kerr

sign

al

Field (Oe)

(a) (b)

-2000 -1000 0 1000 2000

-1.0

-0.5

0.0

0.5

1.0

Norm

alize

d Kerr

sign

al

0th order

-2000 -1000 0 1000 2000

-1.0

-0.5

0.0

0.5

1.00

th order

-2000 -1000 0 1000 2000

-15

-10

-5

0

5

10

15

1st order

Field (Oe)

-2000 -1000 0 1000 2000

-3

-2

-1

0

1

2

3

1st order

Norm

alize

d Kerr

sign

al

Field (Oe)

(a) (b)

Im

Re

Mz

x


K. Postava et al. “Null ellipsometer with phase modulation,” Opt. Express 12, 6040 (2004)

compensator

analyzer

photodetector

photoelasticmodulator

polarizer

laserrLock-inI

I2Lock-in Re[Drm/r0] = DIm/I0

Im[Drm/r0] = Dfm/f0

Dfnm/fn

o = Bn Re[fnm]−An Im[fn

m]

DInm/In

o = An Re[fnm]−Bn Im[fn

m] DI−nm/I−n

o = −An Re[fnm] − Bn Im[fn

m]

Df−nm/f−n

o = −Bn Re[fnm] − An Im[fn

m]

Re[fnm]

Im[fnm]

D-MOKE problem fully solved

M fm

Dfm a my

E

E


Samples: arrays of NiFe triangular rings

Triangular rings (2.1 m side).

Nominal width 250 nm.

Nominal thickness 30 nm.

ZEP 520 Resist (0.3 m)

PMGI Resist (0.3 m)

Exposed Areas

Double Layer Resist Spin-coating

EB Patterning Si substrate

Resist Development

EB Deposition

(FeNi target)

Lift-off process nanoelement

Electron beam lithography


+1

-1

-2000 -1000 0 1000 2000

-0,002

-0,001

0,000

0,001

0,002

-2000 -1000 0 1000 2000

-0,003

-0,002

-0,001

0,000

0,001

0,002

0,003

Am

plit

ude(m

V)

Field(Oe)

-1 order Re(Drpp/rpp)

-1 order Im(Drpp/rpp)

Am

plit

ude(m

V)

Field(Oe)

-2000 -1000 0 1000 2000

-0,001

0,000

0,001

-2000 -1000 0 1000 2000

-0,002

-0,001

0,000

0,001

0,002

Am

plit

ude(m

V)

Field(Oe)

+1 order Re(Drpp/rpp)

Am

plit

ude(m

V)

Field(Oe)

+1 order Im(Drpp/rpp)

Normal incidence: extraction of magnetic form factors

Dfnm/fn

o = Bn Re[fnm]−An Im[fn

m]

DInm/In

o = An Re[fnm]−Bn Im[fn

m] DI−nm/I−n

o = −An Re[fnm] − Bn Im[fn

m]

Df−nm/f−n

o = −Bn Re[fnm] −An Im[fn

m]

Re[fnm]

Im[fnm]


Fig. 2

nmf nmf

-0.10

-0.05

0.00

0.05

-0.02

0.00

0.02

-2 -1 0 1 2

-0.12

-0.06

0.00

0.06

-2 -1 0 1 2

-0.08

-0.04

0.00

0.04

-0.05

0.00

0.05

0.10

-0.03

0.00

0.03

2nd

order horizontal

2nd

order vertical

3rd

order horizontal

CalculatedExperimental

1st

order horizontal

Form

fact

orField (kOe)

3rd

order vertical

1st

order vertical

Exp and calculated Re[fm] and Im[fm] – horizontal and vertical plane


-2000 -1000 0 1000 2000-0,14

-0,12

-0,10

-0,08

-0,06

-0,04

-0,02

0,00

0,02

0,04

0,06

0,08

0,10

Re

Im

Am

plit

ud

e

Field (Oe)

-2000 -1000 0 1000 2000-0,06

-0,04

-0,02

0,00

0,02

0,04

0,06

Re

Im

Am

plit

ud

e

Field (Oe)-2000 -1000 0 1000 2000

-0,020

-0,015

-0,010

-0,005

0,000

0,005

0,010

0,015

0,020 Re

Im

Am

plit

ude

Field (Oe)

-2000 -1000 0 1000 2000

-0,10

-0,08

-0,06

-0,04

-0,02

0,00

0,02

0,04

0,06 Re

Im

Am

plit

ud

e

Field (Oe)

-2000 -1000 0 1000 2000-0,08

-0,06

-0,04

-0,02

0,00

0,02

0,04 Re

Im

Am

plit

ud

e

Field (Oe)

-2000 -1000 0 1000 2000

-0,06

-0,04

-0,02

0,00

0,02

0,04

0,06

0,08

0,10

0,12

Re

Im

Am

plit

ude

Field (Oe)

1st - h 2nd - h 3rd - h

1st - v 2nd - v 3rd - v

Re and Im parts of the magnetic form factors


0 50 100 150 200

0

50

100

150

200

0 50 100 150 200

0

50

100

150

200

Saturated state

0 50 100 150 200

0

50

100

150

200

0 50 100 150 200

0

50

100

150

200

0 50 100 150 200

0

50

100

150

200


“Peak” state

0 50 100 150 200

0

50

100

150

200

0 50 100 150 200

0

50

100

150

200

0 50 100 150 200

0

50

100

150

200

0 50 100 150 200

0

50

100

150

200

0 50 100 150 200

0

50

100

150

200


0 50 100 150 200

0

50

100

150

200

0 50 100 150 200

0

50

100

150

200

0 50 100 150 200

0

50

100

150

200

0 50 100 150 200

0

50

100

150

200

0 50 100 150 200

0

50

100

150

200

After “peak”


0 50 100 150 200

0

50

100

150

200

0 50 100 150 200

0

50

100

150

200

0 50 100 150 200

0

50

100

150

200

0 50 100 150 200

0

50

100

150

200

0 50 100 150 200

0

50

100

150

200

Towards negative saturation


-2000 -1000 0 1000 2000

-0,12

-0,09

-0,06

-0,03

0,00

0,03

0,06

0,09 Re

Im

Am

plit

ud

e

Field (Oe)

-2000 -1000 0 1000 2000-0,06

-0,04

-0,02

0,00

0,02

0,04

0,06

Re

Im

Am

plit

ude

Field (Oe)

-2000 -1000 0 1000 2000

-0,020

-0,015

-0,010

-0,005

0,000

0,005

0,010

0,015

0,020

Re

Im

Am

plit

ude

Field (Oe)

-2000 -1000 0 1000 2000-0,14

-0,12

-0,10

-0,08

-0,06

-0,04

-0,02

0,00

0,02

0,04

0,06

0,08

0,10

Re

Im

Am

plit

ude

Field (Oe)

-2000 -1000 0 1000 2000-0,06

-0,04

-0,02

0,00

0,02

0,04

0,06

Re

Im

Am

plit

ude

Field (Oe)

-2000 -1000 0 1000 2000

-0,020

-0,015

-0,010

-0,005

0,000

0,005

0,010

0,015

0,020 Re

Im

Am

plit

ud

e

Field (Oe)

Exp and calculated Re[fm] and Im[fm] – horizontal plane

1st - h 2nd - h 3rd - h


-2000 -1000 0 1000 2000-0,08

-0,06

-0,04

-0,02

0,00

0,02

0,04

0,06

0,08

0,10

0,12 Re

Im

Am

plit

ude

Field (Oe)

-2000 -1000 0 1000 2000-0,10

-0,08

-0,06

-0,04

-0,02

0,00

0,02

0,04

0,06

0,08

Re

Im

Am

plit

ude

Field (Oe)

-2000 -1000 0 1000 2000

-0,08

-0,06

-0,04

-0,02

0,00

0,02

0,04

0,06 Re

Im

Am

plit

ude

Field (Oe)

-2000 -1000 0 1000 2000

-0,10

-0,08

-0,06

-0,04

-0,02

0,00

0,02

0,04

0,06 Re

Im

Am

plit

ud

e

Field (Oe)

-2000 -1000 0 1000 2000-0,08

-0,06

-0,04

-0,02

0,00

0,02

0,04 Re

Im

Am

plit

ude

Field (Oe)

-2000 -1000 0 1000 2000

-0,06

-0,04

-0,02

0,00

0,02

0,04

0,06

0,08

0,10

0,12

Re

Im

Am

plit

ude

Field (Oe)

1st - v 2nd - v 3rd - v

Exp and calculated Re[fm] and Im[fm] – vertical plane


Micromagneticsimulations

D-MOKE

MFM(quenchedTo 0 field)

Magnetic imaging proved

APPLIED PHYSICS LETTERS 99, 092501 (2011)


MOKE is a powerful technique for studying technologically relevant magneticmaterials.

MOKE magnetometry based on microscopy provides a noninvasive probeof magnetization reversal for individual ultra-small nano-structures.

Concluding remarks

Next lecture:

Interplay between plasma excitations and MO-activity (magnetoplasmonics).

D-MOKE is a powerful technique to investigate the collective behavior of

magnetic nano-arrays.

new p. vavassori - magnetism.eumagnetism.eu/esm/2018/slides/vavassori-slides2.pdf · 2018. 9....

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