Magnons in ultrahigh vacuum deposited Fe/Ag multilayers

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Magnons in ultrahigh vacuum deposited Fe/Ag multilayers

I. El Kiadi

^a,^∗

, H. Lassri

^a

, K. Benkirane

^b

, B. Bensassi

^a

aLaboratoire de Physique des Matériaux, de Micro-électronique, Automatique et Thermique. Université Hassan II, Faculté des Sciences Ain Chock, B.P. 5366 Mâarif, Route d’El Jadida, km-8, Casablanca, Morocco

bEcole Royale Navale, Bd Sour Jdid, Casablanca, Morocco Received 14 July 2006; accepted 6 November 2006

Abstract

We have grown Fe/Ag multilayers with Ag buffer layer, by evaporation under UHV conditions on glass substrates. The magnetic properties of Fe/Ag multilayers are examined as a function of Fe layer thicknesstFe. The temperature dependence of the spontaneous magnetizationM(T) is well described by aT^3/2law in all multilayers. A spin-wave theory has been used to explain the temperature dependence of the magnetization and the approximate values for the bulk exchange interactionJ_band surface exchange interactionJ_sfor various Fe layer thicknesses have been obtained.

Keywords: Fe/Ag multilayers; Magnetization; Anisotropy; Spin-wave excitations; Exchange interaction

1. Introduction

A great deal of interest has been focused in recent years on the properties of artificial magnetic multilayers and ultra- thin magnetic films. Both experimental and theoretical studies have uncovered a number of unusual magnetic properties due to reduced dimensions and the increasing dominance of surfaces and interfaces. The most interesting results are the enhanced or suppressed surface moments[1], the emergence and thickness dependence of magnetic surface anisotropy [2], and the thickness dependence of the Curie temperature [3]. In prac- tice, experimental studies are carried out on single-layer films epitaxially grown on suitable substrates, sandwiched films, or superlattices. The magnetic properties of multilayers are strongly dependent on their detailed structure and composi- tion, which are determined by the growth conditions used during fabrication [4,5]. For example, the degree of mixing between adjacent layers determines the amount of Fe able to contribute to the magnetic properties of the film, and the degree of crystallographic texture within the layers, combined with any surface anisotropy present determines the overall anisotropy of the multilayers. In this paper we calculate the thermal variation of the spontaneous magnetization as a function of the Fe layer thickness and compare it, qualitatively and quantitatively,

∗Corresponding author.

E-mail address:i [email protected](I. El Kiadi).

with experiment results. The system based on Ag is interesting because Ag does not form solid solutions with the transition met- als and therefore the alloying at the interface could be avoided.

2. Experimental details

Fe/Ag multilayers were grown by evaporation in ultrahigh vacuum under controlled conditions, and the pressure during the film deposition was maintained below 5×10⁻⁹Torr. The rate of deposition (about 0.5 ˚A/s) and the final thickness were monitored by precalibrated quartz oscillators. The Fe layer thickness tFe was varied from 7 to 40 ˚A and that of tAg was kept fixed at 50 ˚A. The number of bilayers was varied from 10 to 30. Samples were deposited on glass substrates at 300 K on a Ag buffer layer 50 ˚A thick. The top layer in all the samples was Ag 50 ˚A thick. The growth parameters will be designated as (tFe/tAg)q, whereqindicates the number of bilayers. X-ray diffraction (XRD) profiles, taken in reflection geometry at both low (2θ< 10^◦) and high (35^◦< 2θ< 60^◦) scattering angle, con- firmed the modulated structure and showed a (1 1 0) texture for bcc Fe. The fcc Ag buffer layer is seen to have a (1 1 1) texture. The magnetization was measured using a vibrating sample magnetometer with the external field applied both in the film plane and normal to it in the temperature range from 5 to 300 K under a maximum field of 2 T. A torque magnetometer was employed to measure the anisotropy under a maximum field of 1.5 T at 5 K.

doi:10.1016/j.mseb.2006.11.021

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Fig. 1. The temperature dependence of the normalized magnetization of Fe/Ag multilayers with varying Fe thicknesses, the solid lines were calculated from the spin-wave theory.

3. Results and discussion

Fig. 1 shows the temperature dependence of spontaneous magnetization for different thicknesses of Fe layers. It can be noticed that the magnetization decreases with a decrease in Fe layer thickness (Table 1). An effect that is typically encountered in the multilayers is a change of magnetic moment in the magnetic layers or an induced magnetic moment in the nonmagnetic layers, either due to hybridization of the electronic bands or due to the reduced coordination at the interface (or both).

For three dimensional magnetic films, the magnetization has aT^3/2dependence due to spin-wave excitations, and the temperature dependence should follow the relation:

[M(5 K)−M(T)]

M(5 K) =BT^3/2. (1)

In all cases this behavior is observed for temperatures as high as TC/3[6,7](where TC is the Curie temperature). The spin- wave constantBdecreases from 27×10⁻⁶K⁻^3/2fortFe= 7 ˚A to 8.4×10⁻⁶K⁻^3/2fortFe= 40 ˚A. These values are much larger than the value of 5×10⁻⁶K⁻^3/2found for bulk Fe[8].

The B versus 1/tFe is plotted for the samples with 7≤tFe≤40 ˚A inFig. 2. It is seen that the experimental points align well in a straight line. The values extrapolated to 1/tFe= 0 are in good agreement with those found for the bulk Fe. It was observed that the parametersBin Eq.(1)depend ontFeaccording

Fig. 2. Thet_Fe⁻¹dependence of theB.

to:

B(t_Fe)=B_∞+Bs

tFe

, (2)

whereB_∞ is the bulk spin-wave parameter of Fe and Bs the surface B value. The surface anisotropy strongly affects the thickness dependence of the magnetization. The linear relation between the spin-wave parameterBand the reciprocal of the magnetic film thickness was reported for Co/V multilayers[9]

and Fe(1 1 0) films on W(1 1 0)[8,10].

The apparent universality of Bloch’sT^3/2—law for the temperature dependence of the spontaneous magnetization, and of generalization thereof, is considered by Kray[11]. It is argued that in the derivation one should not only consider the exchange interaction between the spin, but also the other interactions between them, leading to elliptical spin precession and devia- tions from the parabolic dispersion of magnons. Also interaction effects are important to explain the apparent universality of gen- eralized Bloch law exponentsn, defined byM(T) =M(0)−const.

Tⁿ, valid in wide temperature rangeT1<T<T2, and for dimen- sionalityD= 1, 2, and 3.

Let us discuss the results on the anisotropy. The effective anisotropyKeffof the multilayers can be expressed on the basis of phenomenological model, as follows:

K_eff =K_b+2Ks

tFe

, (3)

whereKbandKsare the bulk and surface anisotropies (the lat- ter is the contribution of the surface atoms), respectively. The bulk anisotropyKb=−2πM²+Kcr+Kme, where the first term is demagnetization energy arising from the film geometry, the

Table 1

The magnetization and fitting results from Eq.(14)for Fe(tFe)/Ag(tAg= 50 ˚A)

tFe( ˚A) M(emu/cm³)±50 emu/cm³at 300 K Jb/kB(K)±10 K Js/kB(K)±10 K

7 1100 135 71

10.5 1340 130 61

20 1710 134 70

40 1720 140 71

Jbis the bulk exchange interaction between neighbouring Fe atom andJsis the surface exchange interaction.

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Fig. 3. ThetFedependence of the productKeff·tFeat 5 K.

second term is the crystalline anisotropy, and the last term is the anisotropy arising from magneto-elastic interactions.Fig. 3 shows the linear variation of the productKeff·tFeas a function of tFein Fe/Ag at 5 K. A fit of the data at 5 K yieldsKs= 0.3 erg/cm² and from the slope inFig. 3we obtainKb=−1.2×10⁷erg/cm³. To understand better how the exchange interactions between neighbouring Fe atoms affects the magnetic behavior of these films, we attempted to use a simple model to describe these multilayer films. We suppose that the (Xn/Ym)q multilayer is formed by an alternate deposition of a magnetic layer (X) and non-magnetic one (Y). The multilayer is characterized by the number (q) of bilayers (X/Y), the number (n) of atomic planes in the magnetic layerμ(μ= 1, 2,. . .,q) and the number (m) of atomic planes in the non-magnetic layer. We chose the lattice unit vectors (e_x,e_y,e_z) so thate_z is perpendicular to the atomic planes. We note bySiαμ the spin operator of the atomi (i= 1, 2,. . .,N) in the planeα(α= 1, 2,. . .,n) of the magnetic layerμ.

The system Hamiltonian is given by:

H=He+H_a^s+H_a^b+Hz (4) Hedescribes the exchange interactions in the same magnetic layer (bulk and surface) as well as the exchange interactions between adjacent magnetic layers:

H_e= −J_bb

<iαμ,jαμ>S_iαμS_jαμ+b

iαμ,jαμS_iαμS_jαμ

−J_ss

iαμ,jαμS_iαμS_jαμ−J_II

iαμ,jαμS_iαμS_jαμ, (5) whereJbandJsare the bulk and surface exchange interactions.JI

is the interlayer coupling strength and the parameterJIdepends on the numbermof atomic planes in the non-magnetic layer.

The contribution of the surface anisotropy is estimated by:

H_a^s= −D^⊥_ss

iαμS^Z_iαμ² +D^//_s s

iαμ(S_iαμ^X² −S_iαμ^Y² ), (6)

whereD^⊥_s andD^//s are the surface anisotropy parameters for the uniaxial out of plane and in plane components.

H_a^b= −D_b^⊥b

iαμS_iαμ^Z² +D^//_b b

iαμ(S^X_iαμ² −S_iαμ^Y² ), (7) whereD^⊥_b andD^//_b are the bulk anisotropy parameters for the uniaxial out of plane and in plane components. Ds=Ksa²/S² and Db=Kba³/S² are the normalized terms associated with the surface and bulk anisotropies (where a is the lattice parameter).

The HamiltonianHzis considered in the case where a magnetic fieldHappis applied to the system in thezdirection:

HZ= −gμBHapp

iαμS^z_iαμ. (8)

Further we denote by^Ξthe summation on the sites of the bulk layer planes (Ξ=b), surface layer planes (Ξ=s) or the surfaces planes coupled via the non-magnetic layer (Ξ=I). The symbol < > denotes the pairs of nearest-neighbours atoms or adjacent magnetic planes.

In the Holstein–Primakoff formulation[12], the creation and annihilation operators (a⁺_iαμandaiαμ) for each atomic spin are related to the spin operators by:

S_iαμ^X +iS_iαμ^Y =(2S)^1/2fiαμ(2S)aiαμ and

S_iαμ^X −iS_iαμ^Y =(2S)^1/2a⁺_iαμf_iαμ(2S) (9) In the frame work of non-interacting spin-wave theory, the linear approximation of the Holstein–Primakoff method is suffi- cient to describe the main magnetic behavior and the correction terms are quite-small at low temperatures (T<TC/3)[6,7]. So, the value offiαμ(2S) is fixed to 1.

We pass from the atomic variables to the magnon variables after a two-dimensional Fourier transformation:

H =H0+As

kαμ(b_kαμb₋_kαμ+b⁺_kαμb⁺₋_kαμ) +s

kαμB_kb⁺_kαμb_kαμ+b

kαμC_kb⁺_kαμb_kαμ +

kαμ,αμ)D_kb_kαμ⁺ b_kαμ+I

kαμ,αμ)E_kb⁺_kαμb_kαμ

+Fb

kαμ(b_kαμb₋_kαμ+b⁺_kαμb₋⁺_kαμ) (10) where

A=2D^//s S;

Bk=2(Js(n^//−λk)+Jbn^⊥+D^⊥_s )S+2JInS+gμBHapp; Ck=2((2n^⊥+n^//−λk)Jb+D^⊥_b)S+gμBHapp;

D_k= −J_bSλ_k; E_k= −J_ISλ_k; F =D^//_bS

(11) H0is a constant term and the coefficientsλk,λ_kandλ_kdepend on the crystallographic structure of the magnetic layer.n^//repre- sent the number of nearest-neighbours sites in the same atomic plane, while n^⊥ (n) is the number of nearest-neighbours in

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the adjacent plane in the same magnetic layer. For example, for bcc(1 1 0) (n^//= 4 andn^⊥= 2) with the lattice constant a and in the case where the non-magnetic layer do not disturb the succession order of the magnetic atomic planes (n= 2):

λk =4 cos(akx

2/2) cos ak_y

2

, λ_k =λ_k =4 cos

aky

2

. (12)

The spin system is characterized by 2nq×2nq equations, then the resulting secular equation is given by a (2nq×2nq) matrix:

W^(2nq^×^2nq)=

U^(nq^×^nq) V^(nq^×^nq)

−V^(nq^×^nq) −U^(nq^×^nq) , where

U^(nq^×^nq)=

⎛

⎜⎜

⎝

U₁⁽ⁿ^×ⁿ⁾ U₂⁽ⁿ^×ⁿ⁾

U₃⁽ⁿ^×ⁿ⁾ U₁⁽ⁿ^×ⁿ⁾ U₂⁽ⁿ^×ⁿ⁾

· · ·

U₃⁽ⁿ^×ⁿ⁾ U₁⁽ⁿ^×ⁿ⁾ U₂⁽ⁿ^×ⁿ⁾ U₃⁽ⁿ^×ⁿ⁾ U₁⁽ⁿ^×ⁿ⁾

⎞

⎟⎟

⎠

, V^(nq^×^nq)=

⎛

⎜⎜

⎝ V⁽ⁿ^×ⁿ⁾

·

V⁽ⁿ^×ⁿ⁾

⎞

⎟⎟

⎠ ,

U₁⁽ⁿ^×ⁿ⁾=

⎛

⎜⎜

⎜⎝

B_k D_k D_k C_k D_k

· · ·

· · · D_k C_k D_k

D_k B_k

⎞

⎟⎟

⎟⎠

, U₂⁽ⁿ^×ⁿ⁾=

⎛

⎜⎜

⎝

0 · · · 0

· · ·

0 · ·

E_k0 · · · 0

⎞

⎟⎟

⎠ ,

U₃⁽ⁿ^×ⁿ⁾=

⎛

⎜⎜

⎝

0 · · · 0 E_k

· · 0

· · ·

0 · · · 0

⎞

⎟⎟

⎠

, V⁽ⁿ^×ⁿ⁾=

⎛

⎜⎜

⎜⎝ 2A

2F

·

· 2F

2A

⎞

⎟⎟

⎟⎠

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Among the 2 (n×q) eigenvalues of the matrix W^(2nq^×^2nq), we consider the n×q positive ones which correspond to the n×qmagnon excitation branchesω_k^r(r= 1, 2,. . .n×q). These branches can be classified intongroups ofqquasi-degenerate components in the usual case whereJIremain sufficiently small compared to the effective intralayer exchange strength. The low degree of valence electron band overlap between Fe and Ag should be responsible for the weakness of the interlayer coupling [13]. As the Ag layer thick enough (tAg50 ˚A) to magnetically isolate the Fe layers, theJIcan be neglected andm(T) for the Ag/Fe/Ag film (single ferromagnetic layer) is the same asm(T)

for the (Fe/Ag)q multilayer as long as the Fe thickness is the same.Figs. 4(a and b) and 5show, respectively, the influence of Jb,Js andtFe on the low-lying branch, which plays a predomi- nant role in stabilizing the magnetization. For small anisotropies (D/J1), the anisotropy gap is almost the same, but the shape of these low-lying modes varies slowly with thickness. The reduced magnetization versus temperature is computed numer- ically from:

m(T)=1− 1 N_knqS

k,r

1

exp(ω^r_k/k_BT)−1 (14) The coefficientNkindicates the number ofkpoints taken in the first Brouillin zone. The zero point fluctuations effects are neglected.

Using Eq.(14), satisfactory fits were obtained for them(T) data for all of the Fe/Ag multilayer films. Them(T) theory curves obtained from the fits for thetFe= 7, 10.5, 20 and 40 ˚A are shown inFig. 1, well matching the experimental data points. The values ofJbandJsobtained from the fits are listed inTable 1for all films (takenS= 1,JI= 0 K,D_s^⊥= −1.8 K,D^//s =0 K,D_b^⊥=0 K and D^//_b =2 K). The derived bulk exchange interaction constants all consistently fall in the range expected for the exchange interaction of bulk Fe[14,15]. For this parameter, our results remain

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Fig. 4. Spin-wave excitation spectrum of the spin-waves vs.kx(ky=kx/√ 2) for Fe(1 1 0) ferromagnetic multilayer with films in the cases: (a)Jb/Js= 1, (b)Jb/Js= 2, by fixing the other parameters as followsn= 3 (1 monolayer Fe(1 1 0) = 2.03 ˚A),S= 1,JI= 0 K,D^⊥_s = −1.8 K,D^//s =0 K,D^⊥_b =0 K,D^//_b = 2 K.

Fig. 5. Thickness dependence of the low-lying excitation branches versuskx

(ky=kx/√

2) for Fe(1 1 0) ferromagnetic films, by fixing the other parameters as follows;S= 1,Js= 73 K,Jb= 140 K,JI= 0 K,D^⊥_s = −1.8 K,D^//s =0 K,D^⊥_b = 0 K,D^//_b =2 K, (1 monolayer Fe(1 1 0) = 2.03 ˚A).

in the same order than that reported in Co/V multilayers [9].

The surface exchange interactionJsis practically the same for various tFe and its effect on the magnetic properties is rather significant.

4. Conclusions

In conclusion, we have prepared Fe/Ag multilayers by evaporation under UHV conditions and studied their magnetic properties. The thermal variation of the magnetization in ferromagnetic multilayer films is calculated using spin-wave theory.

A simple model has allowed us to obtain numerical estimates for the bulk exchange interactionJband the surface exchange interactionJsat various Fe/Ag multilayers.

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