J O U R N A L O F M A T E R I A L S S C I E N C E : M A T E R I A L S I N E L E C T R O N I C S 1 6 (2 0 0 5 ) 323 – 327
Magnetic studies of spin wave excitations in Fe/Pt multilayers
R. ELKABIL 1 , K. BENKIRANE 1 , M. BENRABH 1 , H. LASSRI 2 , A. HAMDOUN 1 , M.
ABID 2 , M. LASSRI 2 , R. KRISHNAN 3
1 Laboratoire de traitement d’information, Facult ´e des Sciences Ben’Msik Sidi-Othmane, B.P. 7955 - Sidi-Othmane, Casablanca, Morocco
2 Laboratoire de Physique des Mat ´eriaux et de Micro- ´electronique, Facult ´e des Sciences Ain Chock, Universit ´e Hassan II, B.P. 5366 M ˆaarif, Route d’El Jadida, km-8, Casablanca, Morocco
3 Laboratoire de Magn ´etisme et d’Optique, URA 1531, 45 Avenue des Etats Unis, 78035 Versailles Cedex, France
The magnetization of Fe/Pt multilayers has been measured as a function of temperature.
For Fe/Pt multilayers with fixed iron layer thickness of 2.5 ˚ A, the magnetization decreases faster with temperature as the platinum layers are made thicker. A simple theoretical model has been used to explain the temperature dependence of the magnetization. From the model the approximate values for the bulk exchange interaction J b and the interlayer coupling strength J I for various Fe/Pt multilayers have been obtained.
C2005 Springer Science + Business Media, Inc.
1. Introduction
The magnetic behavior of two-dimensional systems has been studied both theoretically and experimentally by many authors because it may help to understand funda- mental properties of magnetically ordered systems. One common approach is to study the spontaneous mag- netization, its temperature dependence and the Curie temperature, T C , of ferromagnetic thin films. Experi- mentally, a drastic change of the magnetic behavior is found for films thinner than a critical thickness which varies from author to author [1–5]. The basic problem is whether these observations describe intrinsic properties of 2-dimensional ferromagnets or are mainly produced by structural imperfections (contamination, diffusion, island growth).
One of the most important consequences of the finite size of the systems is the reduction of the Curie tem- perature [6]. In addition, the lowered dimensionality of thin films in relation to the bulk leads to important changes in the shape of the magnetization as a function of temperature and in the critical exponents. Basically, the magnetization curves reported in the literature can be divided into two classes: (i) a behavior similar to bulk ferromagnets with a weak decrease or even a con- stant behavior of magnetization at low temperatures and a sharp drop of the magnetization as the critical tem- perature is approached [7] and (ii) a linear temperature dependence of the magnetization [1, 8, 9]. In general the linear functionality was attributed to 2D spin-wave excitations [4, 10], or to finite-size and superparamag- netic behavior due to island formation [3, 11].
From the effect of the thickness of the nonmagnetic spacer layers on the temperature behavior one may ob- tain information about a possible interlayer coupling.
The M(T ) behavior and the Curie temperature will of course not only be determined by an interlayer coupling but by the interplay between anisotropy and size effects [8, 12]. Furthermore, interdiffusion causing graded in- terfaces and disorder resulting in a distribution of ex- change interactions, can play an important role. Es- pecially when the magnetic layer thickness is in the monolayer regime the latter two factors might dom- inate the temperature dependence. It has been argued [13, 14] that interlayer couplings of the magnetic layers via nonmagnetic layers should be present and could be modeled in magnitude by variation of the spacer layer.
In this paper we describe our studies on Fe/Pt multi- layers prepared by the RF sputtering method.
2. Experimental
The multilayers were deposited onto water-cooled glass substrates by RF diode sputtering. The chamber was first evacuated to a pressure of 1 − 2 × 10 − 7 Torr us- ing a turbomolecular pump. Argon of 5N purity was used as the sputter gas and its pressure was kept con- stant at 6 × 10 − 3 Torr. The purity of starting materials is 99.99%. The RF power density was 2.1 W/cm 2 . The thickness was measured in situ using a pre-calibrated quartz monitor. All the samples were grown on Pt buffer layers 100 ˚ A thick. In one series, the magnetic layer thickness t Fe was varied in the range 2.5–57 ˚ A and that of the Pt layer t Pt was fixed at 18 ˚ A. In the second se- ries, t Fe was fixed at 2.5 ˚ A and t Pt was varied in the range 10–40 ˚ A. The second series was to check if there was any modification in the nature of coupling between magnetic layers when t Pt was changed. X-ray diffrac- tion profiles, taken in reflection geometry at both low
(2 θ < 10 ◦ ) and high (35 ◦ < 2 θ < 60 ◦ ) scattering angle, confirmed the modulated structure and shows a [111] texture for the multilayers with t Fe < 8 ˚ A and [110] texture for t Fe ≥ 8 ˚ A. The magnetisation M was measured using a vibrating sample magnetometer (VSM). The effective anisotropy was measured at 5 K using a torque magnetometer.
3. Results and discussion
Fig. 1 shows the temperature dependence of M for sev- eral values of t Fe thicknesses. It can be noticed that T C decreases when t Fe decreases. The low-temperature magnetization was studied in detail for a few samples.
For three-dimensional magnetic films, the magnetiza- tion has a T 3 / 2 dependence due to classical spin-wave excitations. In such cases, according to spinwave the- ory, the temperature dependence should follow the re- lation:
M(5 K) − M(T )
M(5 K) = BT 3 / 2 . (1) In all cases this behaviour is observed for tempera- tures as high as T C / 3. The spin-wave constant B de- creases from 48 × 10 − 6 K − 3 / 2 for t Fe = 8 . 4 ˚ A to 7 . 3 × 10 − 6 K − 3 / 2 for t Fe = 57 ˚ A. These values are much larger than the value of 5 × 10 − 6 K − 3 / 2 found for bulk Fe. The B versus 1 / t Fe is plotted for the samples with 8 . 4 ≤ t Fe ≤ 57 ˚ A in Fig. 2. It is seen that the exper- imental points align well in a straight line. The values extrapolated to 1 / t Fe = 0 are in good agreement with those found for the bulk Fe. It was observed that the parameters B in Equation 1 depend on t Fe according to:
B(t Fe ) = B ∞ + B S / t Fe , (2) where B ∞ is the bulk spin-wave parameter of Fe and B S surface B value. The interface anisotropy strongly
Figure 1 The temperature dependence of the magnetization for Fe thick- nesses. All data were fitted using Equation 13 as indicated by solid lines.
Figure 2 The t
Fe−1dependence of the B.
affects the thickness dependence of the magnetization.
The linear relation between the spin-wave parameter B and the reciprocal of the magnetic film thickness was reported also by Gradmann and co-workers [15, 16] for Fe(110) films on W (110).
The M(T ) plots of the Fe(2.5 ˚ A)/Pt(t Pt ) multilayer films for t Pt = 10 , 20 and 40 ˚ A are shown in Fig. 3.
It can be clearly seen that, as t Pt is decreased, M(T ) changes from quasi-linear behaviour to more bulk-like behaviour. The interlayer coupling, whose strength is expected to be weak and depends on the Pt layer thick- ness, effectively alters the magnetic properties of these multilayer films consisting of the 2.5 ˚ A Fe layers which individually behave as 2D ferromagnets without the coupling.
To understand better how the interlayer coupling between neighboring Fe layers affects the magnetic
Figure 3 Calculated (continuous line) and measured (symbols) temper-
ature dependence of the normalized magnetization of Fe/Pt multilayers
with varying Pt thicknesses.
behavior of these films, we have used a simple model to describe these multilayer films. We suppose that the multilayer (X n /Y m ) q is formed by an alternate deposi- tion of a magnetic layer (X ) and non magnetic one (Y ).
The multilayer is characterized by the number (q) of pairs (X / Y ), the number (n) of atomic planes in the magnetic layer a and the number (m) of atomic planes in the non magnetic layer. We chose the lattice unit vec- tors (e x , e y , e z ) so that e z is perpendicular to the atomic planes. We note by S i αµ 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 z . (3) H e describes the exchange interactions in the same magnetic layer (bulk and surface) as well as the ex- change interactions between adjacent magnetic layers:
H e = − J b b
< i αµ, j αµ>
S i αµ S j αµ +
< i αµ, j α
µ>
S i αµ S j α µ
− J S S
< i αµ, j αµ> S i αµ S j αµ
− J I I
< i αµ, j α
µ
>
S i αµ S j α µ
, (4)
where J b and J S are the bulk and surface exchange in- teractions. J I is the interlayer coupling strength which depends on the number m of atomic planes in the non magnetic layer.
The contribution of the surface anisotropy is esti- mated by:
H a = − D ⊥ s
i αµ
S i Z αµ
2+ D
//s i αµ
S i X αµ
2− S i Y αµ
2, (5)
where D ⊥ and D
//are the surface anisotropy parameters for the uniaxial out of plane and in plane components.
The Hamiltonian H z is considered in the case where a magnetic field H app is applied to the system in the z direction:
H z = − g µ B H app
i αµ
S i z αµ . (6)
Further we denote by
the summation on the sites of the bulk layer planes ( = b). Surface layer planes ( = s) or the surfaces planes coupled via the non magnetic layer ( = I ). The symbol <> denotes the pairs of nearest-neighbors atoms or adjacent magnetic planes.
In the Holstein-Primakoff formulation [17], the cre- ation and annihilation operators (a + i αµ and a i αµ ) for each atomic spin are related to the spin operators by:
S i X αµ + i S i Y αµ = (2 S) 1 / 2 f i αµ (2 S)a i αµ and (7) S i X αµ − i S i Y αµ = (2 S) 1 / 2 a i + αµ f i αµ (2 S) .
In the frame work of non interacting spin wave the- ory, the linear approximation of the Holstein-Primakoff method is sufficient to describe the main magnetic be- havior and the correction terms are quite-small at low temperatures (T < T C / 3) [18, 19]. So, the value of
f i αµ (2 S) is fixed to 1.
We pass from the atomic variables (a i + αµ and a i αµ ) to the magnon variables (b + k αµ and b k αµ ) after a two- dimensional Fourier transformation. We show that:
H = H 0 + A s k ,αµ
b k αµ b − k αµ + b + k αµ b − + k αµ
+ s k ,αµ
B k b + k αµ b k αµ + b k ,αµ
C k b k + αµ b k αµ
+
k ,<αµ,α
µ
>
D k b + k αµ b k α
µ
+
I k ,<αµ,α
µ
>
E k b + k αµ b k α
µ
, (8)
where
A = 2 D
//S ,
B k = ( J S (2 n
//− ( λ + k + λ − k )) + 2 J b n ⊥ + 2D ⊥ ) S + 2 J I n S + g µ B H app ,
C k = (4 n ⊥ + 2n
//− ( λ + k + λ − k )) J b S + g µ B H app , D k = − J b S λ k ,
E k = − J I S λ k . (9)
H 0 is a constant term, the coefficients λ + k , λ − k , λ k and λ k depend on the crystallographic structure of the magnetic layer. n
//represent the number of nearest- neighbors sites in the same atomic plane, while n ⊥ (n ) is the number of nearest-neighbors in the adja- cent plane in the same (adjacent) magnetic layer. For bcc(110) (n
//= 4 and n ⊥ = 2) with the lattice constant a and in the case where the non magnetic layer don’t dis- turb the succession order of the magnetic atomic planes (n = 2):
λ + k = λ − k = 4 cos(ak x √
2 / 2) cos(ak y / 2) , λ k = λ k = 4 cos(ak y / 2) , (10)
while for fcc(111), n
//= 6 , n ⊥ = 3 and n = 3, the coefficients ( λ + k , λ − k , λ k and λ k ) become:
λ + k = λ − k = 4 cos(ak x √
6 / 4) cos(ak y √ 2 / 4) + 2 cos(ak y √
2 / 2) , λ k = λ k = 4 cos(ak x √ (11)
6 / 12) cos(ak y √ 2 / 4) + 2 cos(ak x √
6 / 6) .
The spin system is characterized by 2 nq × 2 nq equa-
tions, then the resulting secular equation is given by a
(2 nq × 2 nq) matrix:
W (2nq × 2nq) =
U (nq × nq) V (nq × nq)
− V (nq × nq) − U (nq × nq)
,
where U (nq × nq)
=
U 1 (n × n) U 2 (n × n) U 3 (n × n) U 1 (n × n) U 2 (n × n)
· · ·
· · ·
· · ·
U 3 (n × n) U 1 (n × n) U 2 (n × n) U 3 (n × n) U 1 (n × n)
,
V (nq × nq) =
V (n × n)
· ·
· V (n × n)
,
U 1 (n × n) =
B k D k
D k C k D k
· · ·
· · ·
· · · D k C k D k
D k B k
,
U 2 (n × n) =
0 · · · 0
· · ·
· · ·
· · ·
0 · ·
E k 0 · · · 0
, U 3 (n × n) =
0 · · · 0E k
· · 0
· · ·
· · ·
· · · 0 · · · 0
,
V (n × n) =
2 A
0
·
·
· 0
2A
. (12)
Among the 2 (n × q) eigenvalues of the matrix W (2nq × 2nq ), we consider the n × q positive ones which correspond to the n × q magnon excitation branches ω r k (r = 1 , 2 , . . . n × q). The reduced magnetization versus temperature is computed numerically from:
m(T ) = 1 − 1 N k nq S
k , r
1 exp ω
rk
