Effect of the applied magnetic field and the layer thickness on the magnon properties in bilayers Co/Pt and symmetrical trilayer Pt/Co/Pt

Effect of the applied magnetic fi eld and the layer thickness on the magnon properties in bilayers Co/Pt and symmetrical trilayer Pt/Co/Pt

Q3

M. Mehdioui

^a

, A. Fahmi

^a

, H. Lassri

^b

, M. Fahoume

^a

, A. Qachaou

^a,n

Q1

Q11 aLaboratoire de physique de la matière condensée (LPMC), Faculté des Sciences, B.P. 133-14000 Kenitra, Morocco

bLaboratoire de physique des matériaux (LPM), Faculté des Sciences, Ain chock Mâarif, B.P. 5366 Casablanca, Morocco

a r t i c l e i n f o

Article history:

Received 16 August 2013

Keywords:

Q12 Exchange

Heisenberg spin hamiltonian Spin wave theory

Magnon

Excitation spectrum Magnetization per spin

a b s t r a c t

We have studied the elementary magnetic excitations and their dynamics in multilayer Co(tCo)/Pt(tPt) and Pt(tPt)/Co(tCo)/Pt(tPt) under an applied magneticfield. The Heisenberg hamiltonian used takes into account the magneto-crystalline and surface anisotropies, the exchange and dipolar interactions. The calculated excitation spectrum

ε

NðkÞpresents a structure with two sub-bands corresponding to the magnons of surface and volume respectively. The existence of a gap of creating these magnons is also highlighted. The lifetimes deduced from these gaps are in good agreement with the results of previous studies. The thermal evolution of the magnetization mz indicates that the system undergoes a dimensional crossover 3D–2Dwhen the temperature increases. The calculated and measured magne- tizations are compared and they are in good agreement. The exchange integral and critical temperature values deduced from these adjustments are in very good agreement with the results of previous works.

&2013 Elsevier B.V. All rights reserved.

1. Introduction

The Co/Pt multilayers are promising materials for high density magnetic and magneto-optical recording media owing to the much more less volume per bit exigence due to its high interface anisotropy that attains K

u

0

:

6 erg

=

cm

²

and its high coercivity.

The big perpendicular anisotropy suggests that the magnetic properties of Co/Pt are structure and interface dependent in the ultra thin

fi

lms Co/Pt. The interface dependence of the magnetic state is related to the spin orbit coupling at the interfaces. The hybridization of 5d bands of Pt atoms acquires it certain polariza- tion that in

fl

uences the interface proprieties among which is present the interfacial anisotropy that plays an important role in these systems. This type of samples can be crystallized at low temperatures in hcp or more often in fcc structure. The perpendi- cular anisotropy parallel to (111) seems to be optimized in multi- layers Co/Pt [1]. Especially, with the increasing of the number of planes, the shape anisotropy (dipolar interactions) gradually enforced and more and more enters in the competition with the perpendicular anisotropy seeing that shape anisotropy enforces the magnetization in-plane. On the other side the coercivity is governed by the fact that two kinds of clusters with different types of ordering spin polarization are in a competition. When the number of ordered clusters is predominant the ordered magnetic macrostate is stable [2,3].

The con

fi

ned magnons in the surface are promising property in an alternative material dedicated to spintronic devices owing to a weak relaxation time, frequencies (in the GHz

–

THz range) and low barrier magnon creation. The difference in dynamic of surface and bulk magnons investigated in the cases of in plane easy axis and out-plane one con

fi

rm the importance played by the surface anisotropy in governing the magnetic state in the low dimension- ality. At the same time the dipolar interactions are in competition with surface anisotropy and become dominant at the limit of systems in bulk by increasing the thickness.

In this paper we study the magnetic properties of Co(t Co)/

Pt(t Pt) and Pt(t Pt)/Co(t Co)/Pt(t Pt) multilayers. We have estab- lished the corresponding Heisenberg hamiltonian and diagona- lized it by Green's function techniques. It contains the direct exchange, dipolar interaction, magnetocrystalline and surface aniso- tropy terms. We have calculated the excitation spectrum ε

^N

ð k Þ and magnetization per spin for varying t Co, t Pt and H. The calculated

ε

^N

ð k Þ consists of two different sub-bands corresponding to the surface and volume magnons. The calculated and measured magne- tizations are compared and they are in good agreement. The obtained values of exchange integrals and critical temperatures deduced from this comparison are compatible with results generally found for this type of systems containing transition metals such as Co.

2. Experiments

Co/Pt multilayers were grown by evaporation in UHV under controlled conditions, and the pressure during the

fi

lm deposition 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 Contents lists available at ScienceDirect

journal homepage:www.elsevier.com/locate/jmmm

Journal of Magnetism and Magnetic Materials

0304-8853/$ - see front matter&2013 Elsevier B.V. All rights reserved.

http://dx.doi.org/10.1016/j.jmmm.2013.09.062

nCorresponding author. Tel.:þ212 670776064.

Q4

E-mail address:ahqachaou@yahoo.fr (A. Qachaou).

was maintained in the range of 2

–

5 10

⁹

Torr. The rate of deposi- tion (about 0.3 Å/s) and the

fi

nal thickness were monitored by precalibrated quartz oscillators. The magnetic layer thickness t Co was varied from 4 to 18 Å and that t Pt was kept

fi

xed 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 (t Co/Pt)

_q

, where q indicates the number of Co layers. Low and high angle X-ray diffraction studies were made to verify the periodic struc- ture and to calculate the layer thickness. The high-angle results exhibit the satellite peaks and show that the sample has [111]

texture from the presence of only the allowed (111) re

fl

ection [4].

For tCo

r

10 Å, the pattern is in agreement with fcc Pt and CoPt alloy, but the mean composition of this alloy cannot be deter- mined because the position of the line is only less dependent on the composition of the alloy. The magnetization and the magnetic anisotropy are measured with a vibrating sample magnetometer and a torque magnetometer, in the temperature range 5

–

300 K under a maximum

fi

eld of 1.7 T. The M

–

H loops with the external

fi

eld applied perpendicular to the

fi

lm plane become perfectly rectangular when tCo

r

10 Å, indicating the presence of the out- of-plane easy axis. The ferromagnetic resonance measurements were performed using a spectrometer with a X-band frequency of 9.8 GHz. The experimental results that we used in the case of Pt/

Co/Pt are taken from the literature [17].

3. Spin hamiltonian

For a system with N ferromagnetic planes parallel to (OX,OY), the Heisenberg hamiltonian expressed with a magnetization easy axis orthogonal to the plane of the layers is

Q5

H

¼

∑

〈ij〉

J

_ij^J

ð Δ

¹J

S

^x_i

S

^x_j

þ Δ

²J

S

^y_i

S

^y_j

þ Δ

?

S

^z_i

S

^z_j

Þ

∑

〈ii′〉

J

_ii^?_′

ð Δ

1J

S

^x_i

S

^x_i′

þ Δ

2J

S

^y_i

S

^y_i′

þ Δ

?

S

^z_i

S

^z_i′

Þ

þ 1 2

∑

〈ij〉

ð g μ

B

Þ

²

r

³_ij

! S

i

! S

j

3 ð ! S

_i

! r

ij

Þð ! S

_j

! r

ij

Þ r

²_ij 0

@

1 A

þ 1 2

∑

〈ii′〉

ð g μ

B

Þ

²

r

³_ii′

! S

i

! S

i′

3 ð ! S

_i

! r

ii′

Þð ! S

_j

! r

ii′

Þ r

²_ii′ 0

@

1 A

þ 1

2 ð α

¹

þ α

²

Þ∑

i

0

S

^z_i

2 þ g μ

B

H

∑

i

S

^z_i

ð 1 Þ

where

∑′

means that the sum is on the surface sites J

_ij^J

is the exchange integral between nearest neighbors in the same plane, J

_ii′^?

the exchange integral between nearest neighbors belonging to different planes of indices k and k

′

. α

1

, α

2

are respectively the surface anisotropies of the two interfaces of a given plane and

Δ ¼ ð Δ

¹J;

Δ

²J;

Δ

¹?

Þ is the magnetocrystalline anisotropy. H is the applied magnetic

fi

eld, S the spin and D is the dipolar interaction coef

fi

cient, with

D ¼ ð g μ

B

Þ

²

r

³_ij

ð 2 Þ

Applying both Holstein

–

Primakoff [5] and Fourier [6] transforma- tions we have for

Q6

k

_J

¼ ð k

x;

k

z

Þ

Hð

k

_J

Þ ¼

∑

lm

∑

k

A

_lm

ð k

_J

Þ a

_k^þ

J;l

a

_kJ;m

þ

¹2

B

lm

ð k

_J

Þð a

k_J;l

a

k;m

þ a

_k^þ

J;l

a

^þ_k

J;m

Þ ð 3 Þ

a

_k^þ

J;l

and a

_kJ;l

are the creation and annihilation operators of magnons in the state k for the plane l. The matrix elements A

lm

(k)

and B

lm

(k) are A

lm

ð k Þ ¼ S

∑

δJ

("

J

^J

ð 4 Δ

?

ð Δ

¹J

Δ

2J

Þ cos ð k

_J

δ

J

ÞÞ::

þ 1

2 D

_δJ

1 3 4 r

^x2_δ

J

þ r

^y2_δ

J

r

²_δ

J

!!

cos ð k

_J

δ

J

Þ 1

!

þ h þ α ð δ

l;1

þ δ

l;N

Þ

#

þ

∑ δ?

ðð 4 Δ

^z

^J

^?

þ D

_δ?

Þ ð 2 δ

l;1

δ

l;N

ÞÞ

"

þ J

^?

ð Δ

¹J

þ Δ

²J

Þþ 1 þ 3 4

r

^x2_δ?

þ r

^y2_δ

?

r

²_δ

?

!!

cos ð k δ

?

Þ

!

δ

^l;mþ1

#)

δ

^l;m

ð 4 Þ

B

_l;m

ð k Þ ¼ S

∑ δJ

J

^J

ð Δ

²J

Δ

¹J

Þ cos ð k

_J

δ

J

Þ

h

n

þ D

_δJ

3 4 r

^x2_δ

J

þ r

^y2_δ

J

r

²_δ

J

!

þ 3

2 r

^x_δ

J

r

^y_δ

J

r

²_δ

J

!!

cos ð k

_J

δ

J

Þ

#

∑

δ?

J

^J

ð Δ

²J

Δ

¹J

Þ cos ð k

_J

δ

J

Þ

h

þD_δ? 3 2

r^x2_δ?þr^y2_δ? r²_δ

?

!

þ3 r^x_δ? r^y_δ? r²_δ

?

!!

cosðk?

δ

?Þ

#

δ

l;mþ1

)

δ

l;m

ð 5 Þ

δ

J

¼ ð r

^x_δ

J;

r

^y_δ

J;

r

^z_δ

J

Þ is the vector relating two

fi

rst neighbor sites belonging to the same plane, and δ

?

¼ ð r

^x_δ

?;

r

^y_δ

?;

r

^z_δ

?

Þ is the vector connecting two sites belonging to two adjacent planes.

The determined excitation spectra ε ð k

_J

Þ are the eigenvalues of

H

diagonalized by using the techniques of retarded Green's functions de

fi

ned as

G

_l;m

¼

〈〈

a

k_Jl;

a

_k^þ

Jm〉〉

ð 6 Þ

and G

^′_l;m

¼

〈〈

a

^þ_k

Jl;

a

_k^þ

Jm〉〉

ð 7 Þ

Therefore the matrix expression of Eq. (1) is A ε

I

B

B A ε

I

G 0

G

^′

0

¼ 1 2 π

1 0 0 0

ð 8 Þ We have then a set of 2N coupled equations where A ¼ ð A

_lm

Þ and B ¼ ð B

lm

Þ are N order matrices de

fi

ned by previous equations (3)

–

(5). Energies ε ð k

_J

Þ are poles of Green's matrix

_G′^G0₀

. They are also the eigenvalues of the matrix: M ¼

^AB B

A

. 3.1. Results

3.1.1. Excitation spectrum ε

^N

ð k Þ

–

analytical resolution

For samples involving 3D-transition metals (such as Fe, Ni, Co), it is well established that the dipolar interactions and anisotropy are very small compared to the exchange. Therefore, we limit ourselves to consider in this section, the effect of exchange alone ð D ¼ α ¼ 0 Þ . The extension to other effects (dipole and anisotropy) is performed thereafter

H

¼

∑

lm∑

k

A

lm

ð k

_J

Þ a

_k^þ

J;l

a

k_J;m

ð 9 Þ

with A

lm

ðkÞ ¼ S∑

δJ

½2J

_δ^J_J

ð Δ cos ðk

J

δ

J

ÞÞ δ

l;m

þS∑

δ?

½2J

_δ^?_?

ð2 δ

l;1

δ

l;N

Þ δ

l;m

δ

l;m

S

∑ δ?

½ 2J

_δ^?

?

cos ð k

_J

δ

?

Þð δ

l;m1

þ δ

l;mþ1

Þ ð 10 Þ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

The resolution of the system (8) leads to the resolution of the following secular equations:

det ½ A ε

I ¼

0 ð 11 Þ

ε

^N

ð k

_J

Þ ¼ A ð k

_J

Þþɛ

N

W ð 12 Þ A ð k

_J

Þ ¼ A

11

ð k

_J

Þ , W ¼ A

12

.

ɛN

are real coef

fi

cients. In Fig. 1 we present the spectra ε

^N

ð k

_J

Þ for different values of the magnetic layer thickness t Co assuming an average space of 2

–

2

:

5 Å between two successive ferromagnetic planes [2,3,7]. For small thicknesses of the magnetic layer (3 Å

r

tCo

r

10 Å Þ deposited on a non- magnetic layer Pt with constant thickness (t Pt ¼ 12 Å), the excita- tion spectra ε

^N

ð k Þ have N modes for N ferromagnetic planes Fig. 1(a)

–

(d). We also note that these spectra consist of two sub- bands of different concavities. One, noted sb

s

, is formed by two convex branches and appears to higher energies ε

^N

ð k Þ ; it

corresponds to modes of optical magnons. The other, noted sb

b

, is concave and is formed of the remaining modes; it corresponds to lower energies and is assigned to magnon of acoustic type. We will see in Section 4 that sb

s

is attributed to the surface modes while sb

b

corresponds to the bulk modes.

Moreover, the existence of a gap E

g

is highlighted on these spectra. This gap represents the potential barrier necessary for the creation of magnons in the multilayer studied. It corresponds to the lowest eigenvalue of the spin Hamiltonian ε

N;min

ð k ¼ 0 Þ ¼ E

gb

for bulk-magnons and ε

N;min

ð k ¼ π

=

a Þ ¼ E

gs

for surface magnons.

Based on Eqs. (4) and (5) we obtain

E

gb

¼ E

1;1

W ¼ S ½ 4 Δ ^J

J

þ 2 ð J

_?

J

_J

Þþ h þ α ð 13 Þ

E

gs

¼ E

N1;N1

W ¼ S ½ 4 Δ ^J

J

þ 2 ð J

_?

þ J

_J

Þþ h þ α ð 14 Þ 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

Fig. 1.

Q10 Excitation spectra calculated forJ_?4J_J and afixedα¼10. (a)tCo¼3 Å; (b)tCo¼4 Å; (c)tCo¼4.8 Å; (d)tCo¼10 Å; (e)tCo¼30 Å.

3.1.2. Magnetization per site

Introducing the solutions ε

^N

ð k Þ of Eq. (11) the magnetization per spin is given as

m

z

ð T Þ ¼

〈

S

z〉

S ¼ 1 s NS ð 2 π Þ

² ∑^N

l¼1

Z

〈

a

_l;k

a

_l^þ_;k〉

dk

x

dk

z

¼ 1 s NS ð 2 π Þ

² ∑^N

l¼1

Z

1

e

^{εlðkJÞ=kBT}

1 dk

x

dk

z

ð 15 Þ s is the elementary Fermi surface. k

B

is the Boltzmann constant and N is the number of planes. We restrict ourselves to a low temperature range corresponding to a non-negligible contribution of magnons (usually T

r

Tc

=

3). Therefore only the low-frequency modes i.e. low wave vectors k are excited and

fi

rst order- development of ε

^l

ð k

_J

) around k

_J

¼ ð 0

;

0 Þ is allowed [2,3]. Thus we obtain

m

z

ð T Þ ¼ 1 g

3=2

e

^{8SJJðΔ1Þ=kBT}

1 8 π

³⁼²

1 2S ð J

_?

Þ

¹⁼²

1

SJ

_J

ð k

B

T Þ

³⁼²

ð 16 Þ for a low temperature range T

≪

8SJ

_J

ð Δ 1 Þ= k

B

. While, for the range of high temperatures 8SJ

_J

ð Δ 1 Þ= k

B≪

T

r

Tc

=

3, we have

m

z

ð T Þ ¼ 1 k

B

T 8 π ^NSJ

J

log

0 BB BB B@

2k

B

T 8SJ

_J

ð Δ 1 Þþ 4SJ

_?:

1

1 þ 1 4SJ

_?

8SJ

_J

ð Δ 1 Þþ 4SJ

_?

2

" #1=2

1 CC CC

CA

ð 17 Þ

A

fi

rst analysis of m

z

(T) clearly shows that the studied multi- layer undergoes a crossover of dimensionality from a behavior of the bulk systems (3D) at low temperatures (Eq. (16)) to a behavior of quasi-two-dimensional systems (2D) at high temperatures (8SJ

_J

ð Δ 1 Þ= k

B≪

T

r

Tc

=

3) (Eq. (17)). Both thermal behaviors of the magnetization are shown in Fig. 2.

4. Discussion

4.1. Excitation spectra

4.1.1. Bulk and surface magnons

To understand the existence and nature of the two sub-bands in the spectra ε

^N

ð k Þ above, we followed the effect of two factors α

(Fig. 3) and t Co (Fig. 1) that are directly or indirectly related respectively to the impacts of surface and volume on properties of studied multilayers.

It thus appears that when the surface anisotropy α ^increases,

the sub-band sb

b

corresponding to the low energy remains practically unchanged at the edge of the Brillouin zone (BZ), while in the center thereof, the width of the band de

fi

ned by W

_b

¼ j ε

^bN;max

ð k ¼ 0 Þ ε

^bN;min

ð k ¼ 0 Þj undergoes a growth rate Δ ^W

^b

ð α Þ= Δα

which is negative having an order of ð W

b

ð α ¼ 0 Þ W

b

ð α ¼ 30 ÞÞ=

ð 0

–

30 Þ ¼ 4

:

73. For cons, the second sub-band sb

_s

undergoes under the effect of α , a net displacement in block of energy for all values of k in BZ. Its bandwidth W

s

ð α Þ ¼ j ε

^sN;max

ð k ¼ π

=

a Þ

ε

^sN;min

ð k ¼ π

=

a Þj undergoes a growth rate Δ ^W

^s

ð α Þ= Δα negative also but more pronounced with a value of ð W

s

ð α ¼ 0 Þ W

s

ð α ¼ 30 ÞÞ=

0

–

30 ¼ 9

:

73 qualitatively giving a double rate compared to the case of bulk magnons Δ ^W

^s

ð α Þ= Δα

C

2 Δ ^W

b

ð α Þ= Δα ^{(see Table 1}

(a)). We also note, as shown in Fig. 3(d), that the two modes within the surface sub-band have different symmetries. The lower mode is insensitive to the variation of the ratio ð α

1=

α

2

Þ of the anisotropy of the two surfaces for the same magnetic layer; it can be regarded as corresponding to a mode compatible with the union of all elements of symmetry of the two surfaces of a magnetic layer.

While, the superior mode undergoes a signi

fi

cant change when the ratio α

¹=

α

²

varies; it is rather only compatible with symmetry elements common to both surfaces.

When the thickness of the magnetic layer t

_Co

increases (N increases) the number of modes in sb

b

increases in the center of the BZ, while at the edge of this zone, these modes are almost totally degenerate. However, the number of modes of the sub- band sb

s

remains

fi

xed to two. Both sub-bands sb

b

and sb

s

practically overlap for thicker layers (beyond N ¼ 20). For small thicknesses t

Co

¼ 3 and 4 Å, the spectrum contains only two modes with different concavities. One corresponds probably to a state in the sub-band sb

b

and the other in sb

s

. We also note that the effect of the variation of the thickness of the magnetic layer t

Co

over the width of the two sub-bands W

_b

and W

_s

is reverse to that of the effect of anisotropy α presented above. This is W

b

which is much more sensitive when t

Co

increases. Indeed, the increase of the width of the sub-band sb

_b

rate is double that of the sub-band sb

s

with t

Co

varying: Δ ^W

^b

ð tCo Þ= Δ ^tCo

C

2 Δ ^W

^s

ð tCo Þ= Δ ^{tCo (see}

Table 1).

In conclusion, the increasing of the surface anisotropy α ^effect

promotes the second sub-band sb

s

, while the

fi

rst sub-band sb

b

is favored for thick magnetic layers. This allows us to suggest that sb

s

is a band formed by two modes of surface magnons, while sb

b

is rather attributed to modes of bulk magnons.

To further clarify the suggested difference in the nature of the modes, we tried to obtain a qualitative study of the creation and dynamics of two magnon populations through an estimate of the two characteristic factors, namely, the gaps of creation E

gs

and E

gb

and their associated lifetimes τ

s

and τ

b

. Using the same reasoning we have proposed in previous studies on analogous systems M/NM (M ¼ Co, Ni, Fe and alloys) and (NM ¼ Cu, Pt) [2,3,8], we estimated these relaxation times τ

s

for surface and τ

b

for bulk magnons such as

τ

s¹;b

¼ 2 π

ℏ

E

²_g

s;b

D

d

ð E

F

Þ ð 18 Þ

The respective values of the two factors Eg and τ are summar- ized in Table 2. These values clearly show that the magnetic excitations last longer in the volume of the multilayer. Whereas, the more the magnetic layer becomes thicker, the relaxation time of bulk magnons decreases the faster. Indeed, a qualitative analysis of the decay rates of the two relaxation times relative to the thickness of the magnetic layer, in the range tCo

Z

4

:

8 Å for which the distinction between surface and volume nature of created magnons 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

Fig. 2.Thermal variation of magnetization per site calculated fortCo¼4 Å and

10 Å.

is net, gives ð τ

^b

ð tCo ¼ 4

:

8 Þ τ

^b

ð tCo ¼ 10 ÞÞ= 4

:

8

–

10 ¼ 17

:

362 while ð τ

^s

ð tCo ¼ 4

:

8 Þ τ

^s

ð tCo ¼ 10 ÞÞ= 4

:

8

–

10 ¼ 3

:

76. Relaxation of bulk magnons is then

fi

ve times faster than the surface magnons.

Previous studies made by the methods of transfer matrix [9]

and

fi

rst principles [10] calculations also predicted the existence of this difference in the nature of the magnons. They obtained a speci

fi

cation of magnon states at surface and in volume and their

respective lifetimes. Surface states and interface are marked by symmetry breaking at the surface and interface. Moreover, the existence of these surface and volume states which are signi

fi

cantly different is revealed by experimental measurements [11

–

14]. The values given in these works for the relaxation times are in good agreement with the values we obtained (see Table 2): τ

b

is in the range of picoseconds, while τ

s

is in the range of femtoseconds.

4.2. Magnetization per spin

4.2.1. Thermal variation of the magnetization per spin

Fig. 4(a) and (b) is

fi

rstly the temperature dependence of the magnetization per spin m

z

(T) for a variable t Co and t Pt ¼ 12 Å, and secondly for a variable t Pt and t Co ¼ 4 Å for the Co/Pt bilayers. In the case of the three-layer Pt(t Pt)/Co(5 Å)/Pt(t Pt) Fig. 5 presents m

z

(T) for variable t Pt and t Co ¼ 5 Å. The results of our calculations of m

z

(T) (continuous curves) are compared with the results of measurements (different symbols). The agreement obtained is 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

Fig. 3.Effect of surface anisotropy on the excitation spectrum fortCo¼10 Å. (a)α¼0, (b)α¼15, (c)α¼20. (d) Surface modes for various reportsα1=α2whereα1(red) andα2

(blue) are the anisotropies of the two surfaces of a given magnetic layer:α1=α2¼1;40;80;120.

Q8 (For interpretation of the references to color in thisfigure caption, the reader

is referred to the web version of this paper.)

Table 1

Varying the bandwidth as a function of the thickness of the magnetic layer:W(tCo).

α 0 10 15 20 30

(a) Varying the bandwidth as a function of the anisotropy:W(α)

Ws(K) 1034 904 844 816 742

Wb(K) 1074 994 975 956 932

tCo (Å) 3 4 4.8 10 15

(b) Varying the bandwidth as a function of the thickness of the magnetic layer:W(tCo)

Ws(K) 233 377 660 904 1136

Wb(K) 424 648 780 994 2322

Table 2

GapsEgand relaxation timesτobtained for the two magnon populations (bulk and surface).

tCo (Å) 3 4 4.8 10

E_g;bðKÞ 022.98 017.90 005.61 013.94

τbð10¹²sÞ 006.42 010.57 107.73 017.45

Eg;sðKÞ 446.94 665.86 704.77 1051.3

τsð10¹⁵sÞ 016.97 007.64 006.82 003.06

very good. It appears that as if t Co increases (the number N of planes increases) m

z

(T) also increases. This is because most t Co increases over the magnetic coordination number of Co atom increases. This also explains the fact that the values of the exchange integrals deduced from the adjustment of the values measured and calculated magnetization are increasing with increasing t Co (see Table 3(a)). As against the growth of non- magnetic planes number (t Pt increases) leads to a decreasing magnetization and exchange integrals for a

fi

xed magnetic layer thickness, which is probably due to a reduction of the surface anisotropy and/or enhancing magnetocrystalline anisotropy as t Pt increases gradually, see Table 3(b). This effect is even more pronounced in the case of Pt/Co/Pt trilayer (see Table 3(c)). The values of the critical temperatures and the exchange integrals deducted from these adjustments are consistent with the results of previous works [15,16].

4.2.2. Effect of magnetic

fi

eld on the magnetization per spin for

fi

xed temperature

In Fig. 6 is shown the variation of the magnetization per spin in function of the applied magnetic

fi

eld m

z

ð H

;

T ¼ cte Þ at T ¼ 180 K for the trilayers Pt(10 Å)/Co(5 Å)/Pt(10 Å) (Fig. 6(a)). While m

z

ð H

;

T ¼ cte Þ at T ¼ 300 K is presented for the bi-layers Co(6.6 Å)/Pt(14 Å) in Fig. 6(c) and for trilayers Pt(10 Å)/ Co(5 Å)/Pt(10 Å) in Fig. 6(b). The

agreement reached between our calculated results m

^calc_z

ð H

;

T ¼ cte Þ and the results of previous measures m

^exp_z

ð H

;

T ¼ cte Þ [17,18] is very good. The properties deduced from this adjustment of m

zcalc

and m

zexp

such as the exchange integrals J

_?

and J

_J

gathered in Table 4 are in good agreement with the values usually obtained for similar systems.

A qualitative comparison of results for the bi-layer Co(6.6 Å)/Pt(14 Å) and a symmetric trilayer Pt(10 Å)/Co(5 Å)/ Pt(10 Å)at T ¼ 300 K shows that the capping by a Pt layer reduces the exchange that is due to a reduction in the perpendicular anisotropy. Similarly,the dynamics of magnon population represented by a gap creation and relaxation time is most consistent with that of surface magnons ð τ fs Þ more than the thickness of capping is great. The capping with Pt indeed promotes the contribution of surfaces and interfaces. Moreover,the comparison of the evolution m

z

ð H

;

T ¼ cte Þ for Pt/Co/Pt (Fig. 6(a) and (b)) and for Co/Pt(Fig. 6(c)) highlights a difference in behavior in the area of

fi

eld corresponding to the irreversible rotation of the spins. The trilayer Pt/Co/Pt has indeed need only a Δ ^H ¼ 180 Oe to

fi

nish this step, while for the simple two-layer Co/Pt, performing this step requires a

fi

eld variation of Δ ^H ¼ 1500 Oe. This difference in behavior as a function of magnetic

fi

eld indirectly re

fl

ects the importance of the effect that plays the surface properties in such systems of ultra-thin multilayers.

The presence of an orthogonal gradient of the electronic potential 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

Fig. 4.Comparison between calculated and measured magnetization per spinmz(T) for different thicknesses in Co/Pt. (a)tCo variable andtPt¼12 Å. (b)tPt variable and tCo¼4 Å.m^calcz ðTÞ: solid lines.m^expz ðTÞ: symbols.

Fig. 5.Comparison ofm^calc_z ðTÞ(continuous curves) andm^expz ðTÞ(symbols)[17]for PtðtPtÞ=Coð5 ÅÞ=PtðtPtÞ.

Table 3

Exchange integrals and critical temperatures deduced from the adjustments of m^calcz ðTÞ with m^expz ðTÞ for (a) CoðtCoÞ=Ptð12 ÅÞ; ðbÞCoð4 ÅÞ=PtðtPtÞ and ðcÞ PtðtPtÞ=Coð5 ÅÞ=PtðtPtÞ.

tCo (Å) J_? ðKÞ J_JðKÞ Tc(1K) ΔðKÞ αðKÞ

(a)

3.0 92 17 218 2.220 8.0

4.0 210 12 460 1.010 6.0

4.8 176 14 732 1.010 4.0

10.0 185 19.5 1519 1.010 3.0

tPt (Å) J_?ðKÞ J_JðKÞ Tc(1K) ΔðKÞ αðKÞ

(b)

7.0 400 20.0 840 0.800 1.2

12.0 218 12.0 460 1.005 5.0

15.0 200 12.0 424 1.010 5.0

20.0 214 4.0 436 1.010 3.0

tPt (Å) J_?ðKÞ J_JðKÞ Tc(1K) Δ αðKÞ

(c)

50 100 22 444 1.000 0.17

100 80 13 346 1.000 0.2

200 32 10 148 1.000 0.6

500 28 5 122 1.100 1.1

resulting from an existence of a strong spin

–

orbit coupling in the multilayer leads to large orthogonal anisotropy and uni-axial aniso- tropy

fi

elds more enhanced in the case of symmetrical three-layer Pt/Co/Pt. These uni-axial anisotropy

fi

elds are determined as the

fi

eld required to achieve 90% magnetic polarization along the hard axis [19].

5. Conclusion

We studied the magnon contribution to the magnetic proper- ties of the multilayers Co ð tCo Þ= Pt ð tPt Þ and Pt ð tPt Þ= Co ð tCo Þ= Pt ð tPt Þ

for t Co, t Pt and applied magnetic

fi

eld H varying. The anisotropic Heisenberg hamiltonian used is diagonalized by the method of Green's function. The excitation spectra obtained shows a struc- ture with two sub-bands corresponding to two kinds of magnons related to bulk or surface effects respectively. The surface magnons are characterized by a lifetime ( τ

s

) that is shorter than for bulk excitations ( τ

b

). The values we obtained for τ

b

are in the range of picoseconds, while τ

s

is in the range of femtoseconds. This is in good agreement with the values given in the literature. The increase in the surface anisotropy α effect promotes the surface sub-band sb

s

, while the bulk sub-band sb

b

is favored when the magnetic layers become thicker. For thick magnetic layers, the two sub-bands overlap.

The thermal evolution of calculated magnetization per spin m

^calc_z

ð T Þ shows clearly that the studied multilayer undergoes a crossover of dimensionality from a behavior of bulk systems (3D) at low temperatures (m

^calc_z

ð T Þ varies in a T

³⁼²

law) to a behavior of quasi-two-dimensional systems (2D) at the highest temperatures in the range T

r

Tc

=

3 (m

^calc_z

ð T Þ varies in a T log ð T Þ law).

The comparison between calculated m

^calc_z

ð T Þ and measured m

^exp_z

ð T Þ magnetization leads to a very good agreement. Increasing magnetic layer thickness t Co and

fi

xing non-magnetic thickness 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

Table 4

Values deduced from the adjustment ofm^calc_z ðH;T¼cteÞandm^expz ðH;T¼cteÞin the presence of magneticfield.

Q9

T(1K) J_? ð1KÞ J_Jð1KÞ Eg_H¼HcðmeVÞ τH¼H_cðsÞ Δ α(1K) Co (6.6 Å)

/Pt (14Å)

300 17 13 0.015 5.5110¹² 1.0 4.5

Pt (10 Å)/

Co (5 Å)/

Pt (10 Å)

300 45 13 0.47 5.5510¹⁵ 1.06 6.0

180 25 2.0 0.29 14.5710¹⁵ 1.06 6.0

Fig. 6.Comparison ofm^calc_z ðH;T¼cteÞ(continuous curves) andm^exp_z ðH;T¼cteÞ(black squares) [17,18]for Pt(10 Å)/Co(5 Å)/Pt(10 Å) atT¼180 K (a) and T¼300 K (b).

(c) Co(6.6 Å)/Pt(14 Å) at 300 K.

t Pt favors the magnetic order characterized by an increase of the magnetization, exchange integrals and critical temperatures. As against the growth of t Pt probably reduces the surface anisotropy and/or enhances magnetocrystalline anisotropy leading to a decreasing magnetization and exchange integrals for a

fi

xed mag- netic layer thickness. This effect is even more pronounced in the case of Pt/Co/Pt trilayer. The magnetic

fi

eld enhances the effect of the coating of the magnetic layer by a second non-magnetic layer.

The values of the critical temperatures and the exchange integrals deducted from these adjustments are consistent with the results of previous works.

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