phys. stat. sol. (b) 243, No. 1, 165–168 (2006) / DOI 10.1002/pssb.200562408

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Original

Paper

Spin fluctuations in coupled two-dimensional magnetic trilayers

H. Wende*, 1, C. Sorg1, M. Bernien1, A. Scherz2, P. J. Jensen1, N. Ponpandian1,

and K. Baberschke1

1 Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany 2 SSRL, Stanford Linear Accelerator Center, 2575 Sand Hill Road, Menlo Park, California 94025, USA

Received 16 May 2005, revised 7 July 2005, accepted 25 July 2005

Published online 29 November 2005

PACS 75.10.Dg, 75.70.Cn, 78.70.Dm

Magnetic trilayer systems consisting of two ultrathin ferromagnetic films of Co and Ni separated by a

non-magnetic spacer of Cu are investigated in order to analyze the effect of spin fluctuations in the ultra-

thin film limit (2D). The element specificity of the X-ray magnetic circular dichroism technique is applied

to study the temperature dependence of the Co and Ni magnetization separately. The effect of (i) the inter-

layer exchange coupling and (ii) the enhanced 2D spin fluctuations on the shift of the critical temperature

of the Ni films is determined by varying both the Cu and the Ni thicknesses. For a detailed understanding

of these effects the experimental results are compared to a microscopic many-body Green’s function the-

ory. Both experiment and theory provide clear indications that for nanostructured magnets a static mean

field description is insufficient.

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

The investigation of layered magnetic structures is a very active field nowadays. On the one hand, this

interest origins from the new magnetic properties of these nanostructures like the ‘giant’ magnetoresis-

tance (GMR) [1, 2] which depends on the alignment of the individual ferromagnetic layers coupled

through non-magnetic spacers. Various technical devices which are already established (e.g. magnetic

read-heads) or under development (e.g. magnetic random access memories (MRAM)) make use of these

new properties [3]. On the other hand, it turns out that the fundamental understanding of these new ef-

fects is not yet complete especially in the ultrathin film limit as discussed e.g. in [4, 5]. Here, we will

study the effect of spin fluctuations in ultrathin coupled trilayer systems. These samples shown sche-

matically in Fig. 1a) are prototypes for the investigation of ferromagnets (FM1, FM2) coupled via a non-

magnetic spacer (NM) and thereby represent the fundamental building block of magnetic multilayers.

The important two competing effects are visualized in Fig. 1a): When decreasing the thickness of FM1

the spin fluctuations for this film are enhanced. Therefore, the magnetization at a fixed temperature is

reduced since the Curie temperature C

T of the films is reduced according to the finite size scaling (see

e.g. [6]). However, the interlayer exchange coupling (IEC) inter

J suppresses these fluctuations. With de-

creasing thickness of the spacer layer the strength of the coupling increases and therefore the magnetiza-

tion is enhanced. The effects of varying the thicknesses of the FM1 and NM layers were analyzed sepa-

rately up to now: The oscillatory behavior of inter

J versus the thickness of the spacer layer (NM) is stud-

ied in detail in theory [4] as well as in experiment by ferromagnetic resonance (FMR) [8] and X-ray

magnetic circular dichroism (XMCD) [9]. It was shown by Jensen et al. that the effects of the enhanced

spin fluctuations which become prominent when entering the 2D-limit by reducing the thickness of FM1

* Corresponding author: e-mail: wende@physik.fu-berlin.de, Phone: +49 30838-56147, Fax: +49 30838-53646

166 H. Wende et al.: Spin fluctuations in coupled two-dimensional magnetic trilayers

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com

IEC ~ 1dNM

2

dNMdFM1

M (T)FM1

Cu (100)

FM2 (Co)

NM (Cu)FM1 (Ni)

spin fluctuations

b)a)

temperature (K)

mag

netiz

atio

n(k

A/m

)

0 5 0 100 150 200 250 300 3500

100

200

300

400Ni 3ML Co

2.1 ML3.1 ML4.2 ML

∆ T =70 KC,Ni ∆T =67 KC,Ni ∆T =77 KC,Ni

4.2 ML Cu

NiCu(100)

Fig. 1 a) Possible behavior of the sublayer magnetization FM1

( )M T in a coupled magnetic trilayer sys-

tem. FM1

( )M T decreases strongly with a decreasing film thickness FM1

d due to the increasing effect of spin

fluctuations and reduced coordination number. In contrast, a pronounced increase of FM1

( )M T is caused

by the interlayer exchange coupling (IEC) in particular for small thicknesses NM

d of the spacer layer.

These competing effects may lead to dramatic effects of the critical behavior in the limit indicated by the

arrows, i.e., FM1

0d Æ and NM

0d Æ . b) Temperature dependence of the Ni magnetization for a staircase

trilayer sample with constant Cu (4.2 ML) and Co (3 ML) thickness. The solid lines represent scaled stan-

dard magnetization curves (see text). In addition the magnetization curves for the uncoupled case are indi-

cated by the dashed lines.

cannot be explained in a mean field theory (MFT) any more [10]. It turns out that higher order spin-spin

correlations have to be included in the theory to describe the experimental data. In the present work we

study these two effects in a combined way by analyzing the relative shift of the critical temperature

C CT TD / of Ni in Co/Cu/Ni/Cu(100) trilayers versus the Ni (FM1) and the Cu (NM) thickness. The

method of choice is the X-ray absorption spectroscopy which allows for an element specific probing of

the magnetic properties with XMCD [7]. The element specific Ni magnetization is shown in Fig. 1b) for

a staircase sample with fixed Co and Cu, and variable Ni thickness. A clear shift of the Ni ordering tem-

perature towards higher temperatures (solid points) can be determined for the coupled films in compari-

son to the uncoupled case (dashed lines). The figure will be discussed in detail later.

The trilayer samples were prepared and measured in situ at the UE56/2-PGM1 beamline at BESSY

under ultrahigh vacuum conditions (base pressure p 102 10

-

< ¥ mbar). The use of the third generation

synchrotron radiation facility is necessary for a reliable determination of the magnetization close to the

quasi-ordering temperature C,Ni*T of the coupled Ni film, e.g. in order to analyze the magnetization tail in

detail. Details on the film preparation and the measurement procedure can be found elsewhere [11]. The

measurements were carried out in an applied field of 3H ª kA/m. This ensured that no magnetic do-

mains are formed close to the ordering temperature. However, care is taken that the field is small enough

that no sizable tail is induced in the magnetization curve.

The calculations performed here make use of a Heisenberg Hamiltonian which considers the isotropic

exchange, the Zeeman term, and the dipole interaction:

20

0 5

1 13( ) ( )

2 2 i j

i j

ij i j i i j ij ij i ij j

i j i ij

H J rr

µµ µ µ µ µ µ

,

π

· , Ò

= - - + - .È ˘Î ˚  ÂS S H r r (1)

Collective magnetic excitations are taken into account by using a many-body Green’s function ap-

proach [12]. Higher order spin–spin correlations have to be included in order to model the experimental

results. This is performed by approximating higher-order Green’s functions in the equation of motion by

the Tyablikov-decoupling [13].

At first, we turn to the experimental results. In order to study the effect of the spin fluctuations in 2D

films as a function of the Ni thickness, staircase trilayers were prepared with a shutter to create macro-

scopic steps of about 2 mm width. On these Ni steps the Cu and Co films are grown each with the same

phys. stat. sol. (b) 243, No. 1 (2006) 167

www.pss-b.com © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Original

Paper

thickness. In general, the interlayer exchange coupling also depends on the Ni film thickness in an oscil-

latory manner [4]. We will show here that the assumption of a constant inter

J is justified for Ni film thick-

nesses ≥3 layers, since the strong increase of C,Ni C,Ni/T TD with a decreasing Ni film thickness can be

explained solely by the action of enhanced spin fluctuations when approaching the 2D limit. On the other

hand, for the thinnest Ni film investigated the observed C,Ni C,Ni/T TD cannot be explained with the trans-

versal spin fluctuations alone. Rather one has to take into account in addition e.g. an enlarged inter

J for

Nid = 2 ML with respect to thicker Ni films. The temperature dependence of the Ni magnetization for one

of these staircase trilayer samples is presented in Fig. 1b). The magnetization curves of the coupled sys-

tem (solid symbols) are obviously shifted in temperature in comparison to the uncoupled system (dashed

lines). The magnetization curves of the uncoupled Ni films were determined before for individual Ni

films in accordance with the finite size scaling describing the change of the Ni Curie temperature

C, Ni Ni( )T d versus the Ni thickness. The data points are not given for these samples for clearer representa-

tion. In order to determine the shift of the curve in temperature a standard magnetization curve is scaled

to the data points. This yields a ‘quasi-critical temperature’ C, Ni*T which is assigned to the resonance-like

maximum of the susceptibility [14, 15]. Interestingly, with this procedure a shift of C,NiTD of about 70 K

is determined for all the Ni thicknesses (2.1 ML, 3.2 ML and 4.2 ML). However, it was shown earlier

[10] that the crucial property is the relative shift C,Ni C,Ni/T TD which increases from about 30% for 4.2 ML

Ni up to 230% for 2.1 ML Ni. Furthermore, Jensen et al. demonstrated that the mean field theory can only

account for relative shifts of up to C,Ni C,Ni/ 5%T TD ª [10] which reveals that higher order spin-spin correla-

tions must be included to describe the experimentally observed shift of more than 200% found here. This

is visualized in Fig. 2a) where C,Ni C,Ni/T TD is plotted versus the Ni thickness. The data points correspond-

ing to the results of Fig. 1b) are the upper solid circles connected by the dashed line. The lower lying

points are determined for two trilayer systems with the thicknesses: Ni: 2.8 ML and 3.8 ML, Cu: 3.0 ML

and Co: 2.0 ML. The solid lines are results of the calculation for different values of inter

J corresponding to

the specific Cu thickness.

Obviously, the theoretical model applied here (including collective spin excitations) is able to repro-

duce the trend of the strong increase of C,Ni C,Ni/T TD towards smaller Ni thicknesses. In this representa-

tion the dependence of the relative shift on the Cu thickness can only be included by an array of curves.

For a more detailed analysis the calculated dependence of inter

J and C,Ni C,Ni/T TD on the Cu thickness is

presented in Fig. 2b). We have shown in earlier investigations with the ferromagnetic resonance that the

2 4 6 8 10

Cu thickness (ML)

0.4

0.2

0.0

0.2

0.4

0.6

-400

-200

0

200

400

600

Jinter (

eV/atom

)µ

Jinter

4 ML Ni3 ML Ni

experiment

1 2 3 4 5 6Ni thickness (ML)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

2. 3

∆TT C

, Ni

/

theory (RPA)

C,N

i

∆T

T C,N

i/

C,N

i

( eV/atom):µJinte r

b)

a)

516

310

86

FM

AF

M

Fig. 2 a) Dependence of the relative shift of the critical temperature C,Ni C,Ni/T TD on the Ni thickness. The

experimental results of two staircase trilayer samples (solid points connected with dashed lines) with Cu

spacer thicknesses of 4.2 ML (above) and 3.0 ML (below) are given. In addition, the results of the calcu-

lation including collective spin excitations (spin waves) are presented for different values of interJ (solid

lines) corresponding to the specific Cu thicknesses. b) Calculated oscillatory dependence of interJ and

C,Ni C,Ni/T TD on the Cu spacer thickness. Please note that the shift C,NiTD is always positive.

168 H. Wende et al.: Spin fluctuations in coupled two-dimensional magnetic trilayers

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com

2 3 4 5 6 7 8 9 10 111

2

3

4

5

6

Cu (ML)

Ni(

ML)

-1.5

-0.9

-0.3

0.3

0.9

1.5

2.0

∆ TC,Ni

TC,Ni

functional behavior of the interlayer exchange coupling follows the experimental results [8]. However, in the present work we analyze the relative shift of the critical temperatures versus the Cu thickness. Re-cently, we have shown that C,Ni C,Ni/T TD is not linearly connected to

interJ [16]. The effect of this non-

linearity can be seen in the calculated compressed functional behavior of C,Ni C,Ni/T TD versus Cu

d shown in Fig. 2b). For this calculation the dependence of

interJ on

Cud described in [4] serves as an input parameter.

The action of the enhanced spin fluctuations on the relative shift of the critical temperature can clearly be seen from the curve for 3 ML of Ni in comparison to 4 ML. Here, the dependence on the Ni thickness can also only be visualized by an array of curves. The two figures given in Fig. 2 reveal the necessity of a combined visualization of the dependence on both the Ni and the Cu thickness. This is shown in the contour plot of C,Ni C,Ni/T TD presented in Fig. 3. The steep rise of the relative shift in temperature can be made out definitely for Ni thicknesses smaller than three layers. Also the interference effects of the two oscillatory contributions modeling the depend-ence on the Cu thickness can obviously be identified. In conclusion, this figure demonstrates the crucial interplay of both the Ni and Cu thickness resulting in enhanced/reduced spin fluctuations for

Ni Cu0d d, Æ – an effect that cannot be described in a static mean field theory.

Acknowledgements This work was supported by BMBF (05 KS4 KEB/5) the DFG (Sfb290 TP A2).

References

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Fig. 3 (online colour at: www.pss-b.com) Contour

plot of the calculated C,Ni C,Ni/T TD versus the Ni

and Cu thickness. For a clearer representation

C,Ni C,Ni/T TD is multiplied by –1 in the regime with

antiferromagnetic coupling (inter

0J < ).