Magnetic studies of spin wave excitations in Fe/Mn multilayers

Author’s Accepted Manuscript

Magnetic studies of spin wave excitations in Fe/Mn multilayers

H. Salhi, R. Moubah, A. El Bahoui, H. Lassri

PII: S0304-8853(16)30810-1

DOI: http://dx.doi.org/10.1016/j.jmmm.2016.12.068 Reference: MAGMA62276

To appear in: Journal of Magnetism and Magnetic Materials Received date: 24 May 2016

Revised date: 14 November 2016 Accepted date: 17 December 2016

Cite this article as: H. Salhi, R. Moubah, A. El Bahoui and H. Lassri, Magnetic studies of spin wave excitations in Fe/Mn multilayers, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2016.12.068

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1

Magnetic studies of spin wave excitations in Fe/Mn multilayers

H. Salhi

, R. Moubah

, A. El Bahoui

, H. Lassri

a

LPMMAT, Faculté des Sciences Ain Chock, Université Hassan II de Casablanca, B.P. 5366 Mâarif, Casablanca, Morocco

b

LMPG, Ecole supérieure de technologie, Université Hassan de Casablanca, Casablanca, Morocco.

Abstract

The structural and magnetic properties of Fe/Mn multilayers grown by thermal evaporation technique were investigated by transmission electron microscopy, vibrating sample magnetometer and spin wave theory. Transmission electron microscopy shows that the Fe and Mn layers are continuous with a significant interfacial roughness. The magnetic properties of Fe/Mn multilayers were studied for various Fe thicknesses (t

). The change of magnetization as a function of temperature is well depicted by a T

law. The Fe spin-wave constant was extracted and found to be larger than that reported for bulk Fe, which we attribute to the fluctuation of magnetic moments at the interface, due to the interfacial roughness. The experimental M (T) data were satisfactory fitted for multilayers with different Fe thicknesses;

and several exchange interactions were extracted.

Keywords: Fe/Mn multilayer; Magnetization; Spin wave excitations; Exchange interactions

1. Introduction

Investigation of magnetic multilayers is a subject of importance on both the fundamental and

technological levels. These heterostructures are widely used in many potential applications

such as read heads, or magnetic memory-storage cells [1,2,3]. At the same time, the

mechanisms implicated in their interlayer exchange coupling are not trivial [4] and several

models based on total energy calculations, Ruderman-Kittel-Kasuya-Yosida (RKKY) or

Green function method have been developed to study their properties [5,6].The properties of

these heterostructures are greatly affected by their microstructure [7,8,9]. In order to exploit

2

these systems in technological applications, one is requested to carefully study their structural and magnetic properties. In this study, we investigate experimentally and theoretically the structural and magnetic properties of Fe/Mn multilayers. Transmission electron microscopy and vibrating sample magnetometer measurements were performed to study experimentally the structural and magnetic properties, respectively. The spin wave theory based on the Holstein-Primakoff calculations at the lowest order approximation is used to calculate the exchange interactions. Up to now, the spin wave theory has never been used to investigate the magnetic properties in the Fe/Mn multilayers [10,11]. The spin wave theory is one of the milestones in magnetism and it is of a fundamental importance [12,13,14]. We model the thermal variation of the spontaneous magnetization as a function of the Fe thickness and compare it, qualitatively and quantitatively, with experimental results, which allow to extract various exchange integral values.

2. Experimental

Fe/Mn multilayers were grown by thermal evaporation in ultrahigh vacuum chamber on oriented silicon substrates. The pressure during growth was in the range 3-5 × 10

Torr. The deposition rate was about 0.3 Å/s and the layer thicknesses were controlled by precalibrated quartz oscillators. For the magnetic measurements, the Fe thickness (t

) was varied from 20 to 100 Å, while that of Mn (t

) was fixed at 50 Å. X-ray reflectivity (XRR) patterns of the resulting multilayers show peaks typical of periodic structures and the x-ray diffraction (XRD) in the high angle range indicates that Fe crystallizes in a bcc phase with the (110) family plane orientation parallel to the substrate. Transmission electron microscopy (TEM) observations were carried out using a JEOL 2000FXII [15]. For sample TEM preparation, the Nviosion40 FIB was used to extract the cross-sectional specimens, from the initial wafer, using lift-out method. Magnetization measurements were performed using a vibrating sample magnetometer (VSM).

25 nm

3

Fig. 1. Bright field cross-section TEM image of a Fe/Mn multilayer grown on a Si substrate.

Fig. 2. Change of spontaneous magnetization as a function of Fe thicknesses t

recorded at different temperatures 300 on 5 K. The Mn thickness (t

) was fixed at 50 Å.

3. Results and discussion

The structural quality of samples was studied using TEM technique. A low- magnification cross-sectional TEM image of a Fe/Mn with 10 repetitions is shown in figure 1. As seen in the figure, all layers are continuous with a uniform thickness. However, the interfaces between Fe and Mn are significantly rough. The Fe and Mn thicknesses were found to be 50 and 200 Å, respectively.

T

he spontaneous magnetization of the Fe/Mn multilayers were measured at 5 and 300 K for various Fe thicknesses ranging from 20 to 100 Å, the Mn thickness was kept at 50 Å (Fig. 2).

At 300 K, the magnetization decreases quickly with the decrease of Fe thickness and falls sharply for Fe thicknesses smaller than 50 Å. However at 5 K, the magnetization does not

20 40 60 80 100

1400 1500 1600 1700

t_Mn= 50 Å T = 5K



M(emu/cm3)

t_Fe(Å)

4

change significantly and remains almost constant. This behavior can be attributed to the finite size effect, which results in a low Curie temperature (T

) for multilayers with small Fe thicknesses [16]. One can notice that the magnetization at 5 K is similar to that reported in the bulk of Fe (1700 emu/cm

) [17]. We note that for other systems such as Fe/W or Fe/Au, an exaltation of Fe moment was reported [18,19,20,21], which can be understood by the increase of the Fe–Fe distance close to the interface. Our results show that the Fe/Mn interface does not play a significant role in affecting the Fe moments. Figure 3 displays the change of the spontaneous magnetization as a function of temperature for Fe/Mn multilayers with various Fe thicknesses. The slopes of the M (T) curves increase with the increase of Fe thickness, highlighting a reduction of T

for multilayers with small Fe thicknesses, which is in agreement with the deduced results from figure 2. Due to spin-wave excitations, the magnetization has a T

change and the temperature dependence can be expressed using the following formula [22]:

(5 ) ( )

(5 )

M K M T

M K B T

  (1)

Where B is the spin-wave constant, and M (5 K) is the magnetization at 5 K. It should be noted that this equation is only applicable for temperatures smaller than T

/3. The deduced B parameter decreases from 35.5×10

to 10.2×10

K

with increasing t

from 20 to 100 Å, respectively. It is important to point out that the B constant is much larger than the value found for bulk Fe (5×10

K

). The difference in B can be understood by taking into account the interfacial effects. In order to determine the interfacial contribution, the B versus 1/t

is plotted in figure 4. As expected, the experimental points fit well in a straight line, which is a signature of an interfacial behavior.

Fig. 3. Measured (symbols) and calculated (continuous line) temperature dependence of the normalized spontaneous magnetization of Fe/Mn multilayers for different Fe thicknesses

Using the generic interfacial model, the B parameter can be expressed using this equation:

5

Fe Int Bulk

Fe t

B B t

B( ) 

(2)

Where B

is the bulk spin-wave parameter of Fe and B

is the interfacial spin-wave parameter value.

The extrapolated B value for 1/t

= 0 is in line with that reported for bulk Fe 5×10

K

. B

is found to be 6.25×10

K

Å. This value supports that the increase in B, is mainly associated with the interfacial contribution. Indeed, several experimental and theoretical reports have demonstrated that if the fluctuation of the surface magnetic moments is larger than the interior moments, B

should be greater than the B bulk [23]. This scenario is in line with the TEM analysis showing a large interfacial roughness, which should induce an important fluctuation of interfacial magnetic moment. The interfacial anisotropy significantly influences the magnetization with decreasing the magnetic layer thickness. To investigate how the Fe neighboring affects the exchange coupling, we extend the spin wave model developed by Pinnettes and Lacroix [24] for ferromagnetic thin films to the Fe/Mn multilayers case. The multilayer (X

/Y

)

is assumed to be composed by an alternate deposition of a magnetic film (X) and a non-magnetic one (Y). We consider that the Mn layer does not contribute to the magnetization, since its magnetic moments are antiparallely coupled. q is the number of repetitions, n and m are the atomic plane number of Fe and Mn, respectively. The lattice unit vectors (

,

,

) are chosen of such kind that the

eZ

axis is perpendicular to the atomic planes. S

is the spin operator of the atom i (i=1, 2…, N) in the plane  (=1, 2…, n) of the Fe film µ (µ=1, 2…, q).

0 50 100 150 200 250 300

0.80 0.85 0.90 0.95 1.00

20 Å 50 Å 80 Å 100 Å

M(T)/M(5 K)

T (K)

6

Fig. 4. The spin wave constant B dependence of Fe/Mn multilayers versus inverse of Fe thicknesses.

The full Hamiltonian can be expressed as: H = H

+H

H

depicts the exchange interactions within a same Fe film and also the exchange interactions between adjacent films [25], while H

describes the contribution of the interfacial magnetic anisotropy. H

is given by the following formula:





 i j

j i

b b

e J S S

H











,







 



 '

' ,

j i j i

S











 

 i j

j i

S S S

J









,

int '' ''

'' '' ,











 i j

j i

I

I S S

J









J

and J

are the interfacial and bulk exchange interactions of Fe, respectively. J

is the

0.01 0.02 0.03 0.04 0.05 0.06

0 5 10 15 20 25 30 35

°

B (10-6K-3/2)

1/t_Fe(A^-1)

7

interlayer exchange coupling. Σ

and Σ

denote the summation on sites of bulk and interfacial layer planes. While Σ

is the contribution for the surface planes coupled through Mn layer. <

> describes the pairs of nearest-neighbors atoms or adjacent magnetic planes.

We note that H

can expressed as:





2 ( ) ( )

)

( _i^X _i^Y

s

i Z

i s

i

a D S D S S

H _{ }

 



 



 ^

  ⁽⁵⁾

D

and D

are the out-of-plane and in-plane components of the interfacial anisotropy parameters, respectively. D

= D

+ D

and D

(K) = K

a

/ k

, where k

is the Boltzmann constant and a is the lattice parameter. According to the Holstein-Primakoff formulation [26], the operators of creation (a

) and annihilation (a

) for each atomic spin are linked to the operator spin by the following formula:

1 2

1 2

(2 ) (2 )

(2 ) (2 )

Z X

i i i i

Z X

i i i i

S i S S f S a

S i S S a f S

   

  ^ 

  



  



(6)

Using the non-interacting spin wave theory, the linear approximation of the Holstein- Primakoff approach is appropriated to depict the magnetic behavior. With this assumption, the corrections are small for temperatures smaller than T

/3 [27]. Thus, we fix the value of

to 1.

The atomic variables (a

a

) are substituted by magnon variables (b

b

) after a two-dimensional Fourier transformation. As a result, we obtain the following relationship:

 

8

 

0

( , ) ( , )

S

k k k k

k

S b

k k k k k k

k k

I

k k k k k k

k k

H H A b b b b

B b b C b b

D b b E b b

   



   

 

     

     

 

 

 

 

  

  

   

 





 

 

Where: (7)





AS ^ 

2





2 _int   ^   ^

 SJ n J n J n S D D

B_k _{k b S I} ^





 _{b k V}

k J S n n

C 2 (  )

' b k

k

J S

D   

' I k

k J S

E  ^

(10)

Where H

is a constant term, 

and 

are coefficients associated with the crystalline structure of Fe. n

is the number of nearest-neighbors sites within the same atomic plane, and



n

and n

are the numbers of interfacial and volume nearest-neighbors in the adjacent plane and in the same Fe, respectively. For a specific site in the interfacial plane of the Fe layer, n

is the number of the nearest-neighbor sites in the adjacent layer across Mn layer. For Fe bcc (110), n

and n

are found to be equal to 4 and 2 respectively. In case where the Mn spacer does not influence the succession order of the magnetic atomic planes (n

3):

k= 4 cos (ak_x√2/2) cos (aky/2) (8)

'

k= 4 cos (ak_y/2)

The motion formula can be expressed as follow:

9

 

 











 



 









H t b

b i

H t b

b i

k k

k k

, ,

















(9)

The spin system can be described using 2nq × 2nq equations, which results in secular equation:

 

 

2 0

2 0

k k k k k k k k k

k k k k k k k k k

C B b D b E b Ab

A b D b E b C B b

      

      







   

  

  

  

      



     



(10)

The resulting secular equation from the Heisenberg formula of motion is given by a 2n × 2n matrix:

 

 

 

 

 

2 2

,

nq nq nq nq

nq nq

nq nq nq nq

U V

W

V U

 



 

 

 

 

  

 

   

 

(11)

The two n x n matrices U and V are given by:

 

 

 

,

n n

nq nq

n n

V

V

V







 

 

  

 

 

 

  

 

 

^{n n}

2A 0

V ,

0 2A



 

 

 

  

 

  

  

 

 

 

 

10

 

   

     

     

   

1 2

3 1 2

3 1 2

3 1

,

n n n n

n n n n n n

nq nq

n n n n n n

n n n n

U U

U U U

U

U U U

U U

 

  



  

 

 

 

 

    

 

 

   

 

  

 

 

 

 

 

 

1

,

k k

k k k

n n

k k k

k k

B D

D C D

U

D C D

D B



 

 

 

    

 

     

    

 

 

 

 

 

2

0 0

0

0 0

n n

k

U and

E



  

 

    

 

    

  

  

 

   

 

    

 

 

3

0 0

0

0 0

k

n n

E

U

    

   

 

    

  

  

 

    

 

    

 

(12)

11

0 1 2 3 4

500 1000 1500 2000 2500



kx

12

Fig. 5. Spin wave excitation spectrum of the spin waves as a function of k_x (k_y=k_x/ 2) for Fe/Mn multilayers with tFe=20Å, tMn=50Å, S=1, JI nt=50K, Jb=100K, JI=0.03K, D^=-0.15K, D^//=0K

The n×q positive equations correspond to the n×q magnon excitation branches



 .

These branches can be sorted into n groups of q quasi-degenerate components in the case

where J

remains enough small with respect to the effective interlayer exchange coupling

(Fig.5). The gap separating the different spectra for the inner atomic planes is attributed to the

magnetic anisotropy. We note that this gap is known to be responsible for stabilizing

magnetization in thin films and increases with anisotropy [24]. For the atomic planes located

in the upper and lower interfaces, their branches present a tiny gap, which indicates that the

anisotropy is too small for the interfacial modes.

13

The reduced magnetization as a function of temperature was computed using the following formula:

r

k k,r k

B

1 1

m (T) 1 -

N nqS ω

exp 1 k T

  

  

 



(13)

N

is a coefficient which depicts the number of k points considered in the first Brillouin region.

Making use of equation (13), we got nice fits for the M (T) curves for all the Fe/Mn multilayer with different Fe thicknesses. The obtained M (T) fits are displayed in figure 1, reasonable matching to the experimental data can be observed. By choosing S = 1, and D

= 0 K, the values of J

and J

were deduced from the fits and are shown in Table 1. The extracted bulk exchange interaction is in agreement with that reported in Fe bulk [28]. The strength of the interlayer coupling depends on t

, which is given by the following formula: J

=M

H

t

/4 where M

is the saturation magnetization of layers and H

is the saturation field [29,30]. It is found that the interlayer coupling is too small with respect to both the bulk and interfacial exchange interaction coupling. This is not surprising given the large Mn thickness (50 Å), which separates the Fe layers. Nonetheless, its influence on the magnetic properties is important. The different Fe films are coupled by the interlayer exchange coupling, and spin waves propagate through the whole multilayer at low temperature. It is interesting to point out that D

which reflects the interfacial perpendicular anisotropy is found to increase with increasing t

. This is surprising, since generally the interfacial anisotropy should be more important at small Fe thickness. This behavior can be understood by taking into account the interfacial roughness of these samples. At small Fe thickness, the influence of the Fe/Mn roughness is dominant, which reduces D

, as the t

increases the interfacial roughness becomes less important thus D

increases.

Table 1: Extracted parameters from Eq. (4) for Fe/Mn multilayers for different Fe

thicknesses.

14 4. Conclusion

The structural and magnetic properties of Fe/Mn multilayer were studied. Transmission electron microscopy has shown that the layers are continuous and uniform, with a significant roughness. The spin-wave constant was found to be much higher than that reported in bulk Fe, due to fluctuation of interfacial magnetic moments. A simple model permitted to deduce the several exchange interactions with different Fe thicknesses. Finally, this study will be useful to understand the magnetic properties of Fe/Mn multilayers.

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Highlights

 The structural and magnetic properties of Fe/Mn multilayers were studied

 Fe and Mn layers are continuous with an important interfacial roughness

 The Fe spin-wave constant is larger than that reported for bulk Fe due to the fluctuation of the interfacial magnetic moments

 Using spin wave theory, many exchange interactions were extracted for various Fe

thicknesses