Spin-wave excitations in evaporated Co/Pt multilayers

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DOI 10.1007/s10948-010-0835-4 O R I G I N A L PA P E R

Spin-Wave Excitations in Evaporated Co/Pt Multilayers

H. Salhi·K. Chafai·O. Msieh·H. Lassri· K. Benkirane·M. Abid·L. Bessais·E.K. Hlil

Received: 23 July 2010 / Accepted: 6 August 2010 / Published online: 16 September 2010

Abstract The magnetic properties of evaporated Co/Pt multilayers have been studied by magnetic measurements and ferromagnetic resonance (FMR). The spin-wave reso- nances were observed in some multilayers in FMR exper- iments, which implied that spin waves were sustained by the whole and propagated through Pt layers. The relation of the resonance fieldHreswith the mode numbernobeys the so-calledn²law and the interlayer coupling strengthJIhas been determined. The temperature dependence of the spontaneous magnetization can be well described by Bloch’s law, in all multilayers. The increase of the spin-wave parameter B with decreasing cobalt thickness has been discussed.

A spin-wave theory has been used to explain the tempera-

H. Salhi·K. Chafai·O. Msieh·H. Lassri (⁾·M. Abid Laboratoire de Physique des Matériaux, Micro-électronique, Automatique et Thermique, LPMMAT, Faculté des Sciences Ain Chock, Université Hassan II, B.P. 5366, Mâarif, Route

d’El Jadida, km-8, Casablanca, Maroc e-mail:[email protected]

H. Lassri

e-mail:[email protected] H. Salhi

Laboratoire de Mécanique, Productique et Génie Industriel, LMPG, Ecole supérieure de Technologie, Université Hassan II, B.P. 5366, Mâarif, Route d’El Jadida, km-8, Casablanca, Maroc K. Benkirane

Ecole Royale Navale, Bd. Sour Jdid, Casablanca, Maroc L. Bessais

ICMPE-CMTR, UMR CNRS 7182, 2-8, rue H. Dunant, 94320 Thiais, France

E.K. Hlil

Institut Néel, CNRS—Université J. Fourier, B.P. 166, 38042 Grenoble, France

ture dependence of the magnetization and the approximate values for the bulk exchange interactionJ_band surface exchange interaction JS for various Co/Pt multilayers have been obtained.

Keywords Co/Pt multilayers·Spin-wave excitations· Exchange interactions

1 Introduction

The field of magnetism in ultrathin films is one of grow- ing importance for both fundamental and technical studies. Magnetic multilayers with artificial periodicity have shown abundant novel properties, such as the giant magnetoresistance, interlayer exchange coupling and enhancement of magnetic moment of ferromagnetic atoms [1–3]. Ther- mal spin excitations in confined ferromagnetic structures become increasingly important, e.g., because they reduce tun- nel magnetoresistance in highly integrated magnetic memo- ries and the stability of stored information [4,5]. Since multilayers are inherently metastable materials on a nanometer scale, the introduction of the period, the number of layers and the relative thicknesses of the magnetic layers and nonmagnetic layers in multilayers will result in many interesting properties, which are sensitive to the microstructures [5–8].

The properties of these materials are mostly governed by the surface properties and hence the interface plays an important role. The discovery of coupled magnetic behavior between layer components in various magnetic multilayer systems has led to an increased interest in two-dimensional systems.

The temperature dependence of the spontaneous magnetization M(T ) well bellow the Curie temperature for most ferromagnetic materials is well described by Bloch’s

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law [9],M(T )=M(0K)(1−BT^3/2), resulting from a lin- earized spin-wave theory for bulk ferromagnets. For two- dimensional ferromagnets, however, calculations within the framework of spin-wave theory do not usually predict aT^3/2 law forM(T ). In contrast, empirical data have verified the validity of Bloch’s law for ultrathin ferromagnetic films in many cases [7,10]. A possible reason for this surprising fact was given by Mathon and Ahmed [11] who predicted an “ef- fectiveT^3/2law” to be valid in a certain temperature range for two-dimensional systems.

Therefore, we studied the temperature dependence of magnetization in Co/Pt multilayers prepared by evaporation under ultrahigh vacuum (UHV) conditions. The comparison between the calculated magnetization and the measured one allowed us a satisfactory estimate of the various exchange integral values.

2 Experimental

Co/Pt multilayers were grown by evaporation in UHV under controlled conditions, and the pressure during the film deposition was maintained in the range 2–5×10⁻⁹torr. The rate of deposition (about 0.3 Å/s) and the final thickness were monitored by precalibrated quartz oscillators. The magnetic layer thicknesst_Cowas varied from 4 to 18 Å and that oft_Pt was kept fixed at 12 Å. All the samples were deposited on glass substrates at 423 K on a Pt buffer layer 100 Å thick.

The top layer in all the samples was Pt 20 Å thick. The growth parameters will be designated as (tCo/t_Pt)_q, where q indicates the number of Co layers. Low- and high-angle X-ray diffraction studies were made to verify the periodic structure and to calculate the layer thickness. The high-angle results exhibit the satellite peaks and show that the samples have [111] texture from the presence of only the allowed (111) reflection [7]. Fort_Co≤10 Å, the pattern is in agree- ment with fcc Pt and Co–Pt alloy, but the mean composition of this alloy cannot be determined because the position of the lines is only a little dependent on the composition of the alloy. The magnetization and the magnetic anisotropy were measured with a vibrating sample magnetometer and a torque magnetometer, in the temperature range 5–300 K under a maximum field of 1.7T. TheM–H loops with the external field applied perpendicular to the film plane become perfectly rectangular whentCo≤10 Å, indicating the presence of the out-of-plane easy axis. The ferromagnetic resonance measurements were performed using a spectrometer with X-band microwave frequency of 9.8 GHz.

3 Results and discussion

The magnetization expressed in terms of total volume of Co first increases with decreasingtCoand then starts decreasing

Fig. 1 The n²-dependence of the resonance fields H_res of the spin-wave modes for (Co_{18 Å}/Pt_{12 Å})40multilayer at 300 K

after showing a peak [12]. This indicates clearly that there is a contribution from the Pt atoms present in the layer adjacent to Co atoms, as has been mentioned early and also observed by others [13,14].

The study of the effective anisotropy of these multilayers shows a positive contribution to the interface anisotropy, with K_S =0.6 erg/cm² at 300 K, which reflects a perpendicular magnetic anisotropy. In general, we assume that K_S could be treated as originating from several effects that change the surface spins at the interfaces such as misfit anisotropy, surface roughness and Néel’s anisotropy. When there is some interdiffusion between the magnetic and nonmagnetic layers, polarization effects may modify the magnetic surface anisotropy.

In FMR measurement, we observed spin-wave modes suggesting the interlayer coupling between Co layers. The multilayer becomes a single coupled system, the spin waves may propagate through the nonmagnetic layers and the standing spin-wave modes are sustained by the whole film.

The observed spin-wave field positions for the sample are plotted againstn² in Fig.1. The presence of even and odd spin-wave resonance (SWR) modes implies an inhomoge- neous distribution of magnetization perpendicular to the film plane, and an asymmetrical spin pinning at the two surfaces and interfaces of the Co layer [15].

A model for spin waves in ferromagnetic/weak ferromagnetic multilayer proposed by van Stapele et al. [16] was extended to the case of ferromagnetic/nonmagnetic multilayers by Wang et al. [17]. In perpendicular geometry, for a single magnetic layer in multilayers, the spin-wave dispersion relation can be expressed by

ω

γ =H_res^⊥ −4π Meff+2Ak²

M_S , (1)

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whereH_res is the resonance magnetic field, 4π Meff is the effective magnetization, A is the exchange coupling constant in the magnetic layer andk is the spin-wave number (k=nπ/L). L is the total thickness of the magnetic film sustaining the spin waves and the integernis the spin-wave mode number. When the magnetic layers couple to each other by interlayer exchange interactions, a collective spin- wave mode may appear with overall wave vectorK.Kand kare related by the dispersion relation [16]

cos(ktCo)=cos(ktCo)+ A

tCoAg

ktCosin(ktCo), (2) wheretCois the thickness of a single magnetic layer andAg

is the interlayer exchange coupling constant (per area). In the approximation for smallktCoandKtCo,

K=k

1+ 2A tCoA_g

. (3)

Then the spin-wave dispersion relation of the multilayer film can be expressed by

ω

γ =H_res^⊥ −4π Meff+2A M

1 1+_t_Co^2A_A_g

K². (4)

Kdepends on the boundary conditions. For an ideal pinning boundary condition and for an ideal free boundary, N Kl=mπ,mis also an integer. Thus the spin-wave spectra should satisfy ann²law.

Then we can estimate the interlayer couplingA_g by an- alyzing the experimental results shown in Fig.1using (4).

In order to determine the interlayer exchange coupling constant A_g, we assumed that the fcc Co layers in multilayers have the same exchange coupling constant as an fcc Co single layer film (ACo=1.3×10⁻⁶erg/cm) [18,19]. Us- ing this value, we obtained the interlayer coupling constant JI =Ag/q=10⁻³ erg/cm² for tCo=18 Å. This value is much near the interlayer coupling constant in Co/Pt multilayer system [20,21]. A positive sign ofJ_Imeans ferromagnetic coupling and agrees with our expectation.

Figure2shows the temperature dependence ofMfor several values oftCothicknesses. It can be noticed that the Curie temperatureTCdecreases with decreasing Co thickness due to reduced coordination. For three-dimensional magnetic films, the spontaneous magnetization has aT^3/2dependence due to the classical spin-wave excitations. In such cases, according to spin-wave theory, the temperature dependence should follow the relation

M(5K)−M(T )

M(5K) =BT^3/2. (5)

In all cases, this behavior is observed for temperatures as high asTC/3. The spin-wave parameter (B) decreases from

Fig. 2 Calculated (continuous line) and measured (symbols) temperature dependence of the normalized magnetization of Co(tCo)/Pt(tPt=12 Å) multilayers with varying Co thicknesses

Fig. 3 Spin-wave parameter B plotted vs. 1/tCofor Co/Pt multilayers

32.5×10⁻⁶K⁻^3/2 for tCo=4 Å to 7.5×10⁻⁶K⁻^3/2 for t_Co=18 Å (Fig.3). These values are much larger than the value of 1.8×10⁻⁶K⁻^3/2found for bulk Co. The increase of the spin-wave parameter with decreasing magnetic layer thickness has been discussed in a number of papers. In par- ticular, it should be pointed out thatBwas found to increase linearly with the inverse magnetic layer thickness in some cases [10,22,23], but in other systems a nonlinear variation was observed [4,7]. It can be clearly seen that the exchange interactions and the interface anisotropy strongly affect the thickness dependence of the magnetization.

We extended the model for spin waves in ferromagnetic thin films proposed by Pinettes and Lacroix [24] to the case of ferromagnetic/nonmagnetic multilayers. We suppose that the multilayer(X_n/Y_m)_q is formed by an alternate deposition of a magnetic layer (X) and nonmagnetic one (Y).

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The multilayer is characterized by the number (q) of bilay- ers (X/Y), the number (n) of atomic planes in the magnetic layerμand the number (m) of atomic planes in the nonmagnetic layer. We chose the lattice unit vectors(e_X,e_Y,e_Z)so that e_Z is perpendicular to the atomic planes. We denote by S_iαμ the spin operator of the atom i (i=1,2, . . . , N) in the plane α (α=1,2, . . . , n) of the magnetic layer μ (μ=1,2, . . . , q).

The system Hamiltonian is given by:H=H_e+H_a. H_edescribes the exchange interactions in the same magnetic layer (bulk and surface) as well as the exchange interactions between adjacent magnetic layers:

H_e= −J_b _b

iαμ,j αμ

S_iαμS_{j αμ}+

iαμ,j αμ

S_iαμS_{j α}μ

−J_s s iαμ,j αμ

S_iαμS_{j αμ}

−J_I I iαμ,j αμ

S_iαμS_{j α}μ, (6)

whereJbandJSare the bulk and surface exchange interactions.JI is the interlayer coupling strength, which depends on the number m of atomic planes in the nonmagnetic layer.

The contribution of the surface anisotropy is given by H_a= −D^⊥

s iαμ

S_iαμ^Z² +D^//

s iαμ

S^X_iαμ² −S_iαμ^Y² , (7)

whereD^⊥andD^//are the surface anisotropy parameters for the uniaxial out-of-plane and in-plane components, respec- tively, andD_eff² =D²_⊥+D_||².Deff(K)=KSa²/ kB, where a is the lattice constant andkBis the Boltzmann constant. Fur- ther, we denote bythe summation on the sites of the bulk layer planes (=b), surface layer planes (=s) or the sur- face planes coupled via the nonmagnetic layer (=I). The symbol denotes the pairs of nearest-neighbors atoms or adjacent magnetic planes.

In the Holstein–Primakoff formulation [25], the creation and annihilation operators (aiαμ anda_iαμ⁺ ) for each atomic spin are related to the spin operators by

S_iαμ^X +iS_iαμ^Y =(2S)^1/2f_iαμ(2S)aiαμ and S_iαμ^X −iS_iαμ^Y =(2S)^1/2a_iαμ⁺ f_iαμ(2S).

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In the framework of non-interacting spin-wave theory, the linear approximation of the Holstein–Primakoff method is sufficient to describe the main magnetic behavior, and the correction terms are quite small at low temperatures (T < TC/3) [26,27]. So, the value offiαμ(2S)is fixed to 1.

We pass from the atomic variables (aiαμ, a_iαμ⁺ ) to the magnon variables (bkαμ, b⁺_kαμ) after a two-dimensional Fourier transformation; we show that

H=H₀+A s k,αμ

b_kαμb₋_kαμ+b⁺_kαμb⁺_−kαμ

+ s k,αμ

B_kb⁺_kαμb_kαμ+ b k,αμ

C_kb⁺_kαμb_kαμ

+

k,αμ,αμ

Dkb⁺_kαμb_kαμ

+ I k,αμ,αμ

E_kb⁺_kαμb_kαμ, (9)

where A=2D^//S, B_k=2

J_s

n^//−λ_k +J_bn+D S+2JIn S, Ck=2

2n+n^//−λk Jb S, D_k= −J_bSλ_k,

E_k= −J_ISλ_k.

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H0 is a constant term, the coefficients λ_k and λ_k de- pend on the crystallographic structure of the magnetic layer.

n^// represent the number of nearest-neighbors sites in the same atomic plane, whilen(n ) is the number of nearest- neighbors in the adjacent plane in the same (adjacent) magnetic layer. For fcc(111), (n^//=6 andn^⊥=3) with the lattice constant a and in the case where the nonmagnetic layers do not disturb the succession order of the magnetic atomic planes (n =3),

λ_k=4 cos

ak_x

√6 4

cos

ak_y

√2 4

+2 cos

ak_y

√2 2

,

λ_k=4 cos

ak_x

√6 12

cos

ak_y

√2 4

+2 cos

ak_x

√6 6

.

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The equation of motion is given by

⎧⎪

⎪⎨

⎪⎪

⎩ i∂b_kα

∂t = [bkα, H], i∂b₋⁺_kα

∂t = [b_−kα⁺ , H].

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The spin system is characterized by 2nq×2nq equations, with the resulting secular equation:

(Ak+Bk+ωkα)bkα+Ckb_kα+(D+2F)b⁺₋_kα=0, (D+2F )bkα+(A_k+B_k−ω_kα)b₋_kα=0. (13)

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Fig. 4 Spin-wave excitation spectrum vs.k_x(ky=k_x√

2) for fcc(111) ferromagnetic multilayer withq=3;n=5; S=0.78; D^//=0 K;

D^⊥=5 K;J_I=10⁻²K andJ_b=J_s=200 K

We consider then×q positive ones, which correspond to then×q magnon excitation branchesω^r_k (r=1,2, . . . , n×q). These branches can be classified into n groups of q quasi-degenerate components in the usual case where the J_I remain sufficiently small compared to the effective in- tralayer exchange strength (Fig.4).

The reduced magnetization versus temperature is com- puted numerically from:

m(T )=1− 1 N_knqS

k,r

1 exp(_k^ω^r^k

BT)−1

. (14)

The coefficientN_kindicates the number ofkpoints taken in the first Brillouin zone. In (14), the zero-point fluctuations effects have not been taken into account.

Using (14), satisfactory fits were obtained for theM(T ) data for all of the Co/Pt multilayer films. TheM(T )theory curves obtained from the fits for the films are shown in Fig.2, well matching the experimental data points. Taken S =0.78, D^⊥ =5.5K (0.6 erg/cm²) and JI =10⁻²K (10⁻³ erg/cm²), the values of J_b andJ_S are found to be equal to (200±20)K for all the multilayers. The derived exchange interaction constants all consistently fall in the range expected for the exchange interaction in ultrathin fcc Co [28].

4 Conclusion

The spin-wave modes were analyzed with the existing spin- wave resonance theory of multilayers and the interlayer exchange coupling constant was obtained. The temperature dependence of the magnetization of Co/Pt multilayers has been investigated for various Co layer thicknesses. The spin- wave constant B is found to increase with decreasing cobalt

layer thickness, which can be understood as a consequence of the reduced coordination of surface spins. A simple model has allowed us to obtain numerical estimates for the bulk exchange interactions and the surface exchange interactions for various Co/Pt multilayers.

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