Journal of Superconductivity and Novel Magnetism https://doi.org/10.1007/s10948-018-4748-y
ORIGINAL PAPER
Effect of Chemical Composition on Volume and Surface Magnon Creation in Multilayer Cox Pt1−x /Pt Ahmed Qachaou1 · Mohamed Mehdioui1 · Nadia Ait Labyad1 · Atika Fahmi1 · Mohamed Lharch1 Received: 3 April 2018 / Accepted: 22 May 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract The existence of two types of magnon populations created in volume and in the surfaces of the Cox P t1−x /P t multilayer was demonstrated. A band structure of these magnons was evidenced by the shape of the calculated excitation spectra. Two bands of energy allowed for a creation of these magnetic excitations are obtained. “Creation gaps” and associated lifetimes are also calculated. The comparison of the calculated relaxation times with experimental results is more than satisfactory. Keywords Heisenberg spin Hamiltonian · Creation of magnons in surface and volume · Lifetimes and gaps of the creation of magnons · Energy bands allowed for magnons creation
1 Introduction The multilayer alloys Cox P t1−x /P t are characterized by important properties such as a good chemical stability [1], a high coercivity [2], an important Kerr rotation [3], and a strong perpendicular anisotropy [4–6] which are factors of choice in the field of magnetic storage and more generally in the spintronics devices [7–10]. The fundamental advantage of using magnetic elementary excitations (magnons) as information carriers [11–13] is that when a magnon propagates in a sample no Joule effect takes place because no charge transport is involved. In low-dimensional systems, the break of symmetry perpendicular to the surface and the reduction of the number of coordination in this surface lead to a differentiation between magnons created
Ahmed Qachaou
[email protected] Mohamed Mehdioui
[email protected] Nadia Ait Labyad
[email protected] Atika Fahmi
[email protected] Mohamed Lharch
[email protected] 1
LPMC, Department of Physics, Faculty of Sciences-UIT, Kenitra, Morocco
within the surface from those of bulk. This difference depends on the relative values of the exchange integrals at the surface and the volume. Moreover, the lifetime of these modes is different. Therefore, the excitation energy and lifetime are tow important factors in magnon-based devices domain. We report in this article a contribution to a theoretical study of the alloying effect on the evolution of these two factors. Thus, we used the Heisenberg model to calculate the excitation spectrum Elz (k , x) for the two types of magnons. We have also established expressions of the gaps necessary to their creation and of their lifetimes. The results obtained are in line with what is given in previous studies. Indeed, assuming that the effect of the substrate is limited to the interfaces with the magnetic layer, we compared our calculated lifetimes with experimental results available for Co/Cu (001) [10].
2 The Heisenberg Spin Hamiltonian In this work, we represent the studied multilayers, based on Cox P t1−x alloy, by a number c of magnetic monolayers (ML) each containing a number p of planes parallel to Oxy and perpendicular to the quantization axis Oz. The thickness of a ML is estimated assuming an average spacing of 2.5 A˚ between two successive magnetic planes [14, 15]. The corresponding Heisenberg spin Hamiltonian (HSH) containing the terms of exchange (Hex = Hex + dip
ex ), dipolar interactions (Hdip = H H⊥in
dip
+ H⊥ ), surface
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anisotropy (Hanis ), and applied magnetic field (Hf ield ) is such as: dip
dip
ex ex H = Hex + H⊥in + H⊥out + Hanis + Hf ield + H + H⊥ y y Si Si = − Jij ij Six Sjx + Si Sj + Siz Sjz − Jii⊥in Jii⊥out Si Si −
−α
(Siz )2
−h
i
Siz
+
i
(gμB )2
2rij3
(gμB )2 3 3 Si Sj − 2 Si rij Sj rij Si Si − 2 Si rii Si rii + rij rii 2rii3
where < ij > are pairs of nearest neighbors (NN) in the same plane, while < ii > are NN pairs belonging to two different planes, with i, j, i ∈ (Co, P t). Each layer is characterized by exchange interactions between its inner planes (Jij and Jij⊥in ) and with its neighboring layers (Jij⊥out ). ij and α are, respectively, the in-plane and outof-plane anisotropy and H is the applied field (h = gμB H ). Furthermore, it is well known that the hybridization between the 3d-states of cobalt (Co) and 5d-states of platinum (Pt) leads the atoms Pt neighbors of the Co atoms to acquire a small magnetic moment and also to the existence of the exchange integrals (JP tP t and JP tCo = JCoP t ) [16–18]. The contribution of the CoCo pairs is certainly much greater
Alz m (k, x) =
2 4SCo xco JCoCo 1 − CoCo (k )
than that of the PtPt pairs, even for a low concentration of cobalt x. Similarly, the small size of the spin value SP t is even more pronounced in environments where PtPt pairs are predominant. Therefore, the corresponding term in H (1) will be practically omitted. Then, taking into account the different configurations of these NN pairs CoCo, CoPt, and PtPt with respective concentrations xCo = x and xP t = 1 − x and successively carrying out the Holstein-Primakoff and Fourier transformations, the HSH (1) is expressed in terms of the boson (magnons) operators of creation and annihilation as: H=
N
1 + + + Alz m (k)akl (2) a + (k) a a + a a B km l m kl −km z z klz −km z 2 lz ,m k
with:
SP t SCo CoP t (k )
⊥in ⊥out 2 + SCo ) 8JP tCo + JP tCo + D 4SCo xco (k ) − 2
+ 8xCo xP t JP tCo
2 ⊥in ⊥out 4JCoCo + xCo xP t (SP t +SCo xCo + JCoCo
(1)
(SP t + SCo ) −
2 z 3r⊥ +4xCo xP t 2 SP t SCo (k ) − (SP t + SCo ) + 16 − 2 xCo (xP t SP t − xCo SCo ) 2 r⊥ 1 ⊥out 1 ⊥out 2 ⊥in + SCo xCo + JCoCo + 2JCoCo + xCo xP t (SP t +SCo ) 4JP⊥in J tCo 2 2 P tCo 2 z
3r⊥ +16D − 2 xCo (xP t SP t − xCo SCo ) 2 − δlz , 1 − δlz , N 2 r⊥
kz a 2 + ⊥in ⊥out +α (xP t SP t + xCo SCo ) δlz , 1 + δlz , N + h δlz , m SCo xCo 4 (k )cos JCoCo + JCoCo coskz a 2 z2 3r⊥ kz a kz a ⊥in + ⊥out + + JP tCo coskz a − ]2D 1 − 2 +xco xP t SP t SCo 8JP tCo (k )cos (k )cos 2 2 r⊥ 2 + 2xCo xP t SP t SCo δlz , m±1 (3) × SCo xco ⎧ ⎨
⎫ ⎬ 1 2 Blz m (k, x) = − 3D ⎝ 2 − 1⎠ (k ) + 4xCo xP t SP t SCo δl , m SCo xco ⎩ ⎭ z 2 r ! " z2 r⊥ kz a − 2 − 3D 1 − 2 (k )cos SCo xco + 2xCo xP t SP t SCo δlz , m±1 2 r⊥ ⎛
2rx
2
⎞
(4)
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Alm (k, x) and Blm (k, x) are elements of the dynamic matrix M(k , x). The functions and ±are expressed in k a
the case of a cfc structure by: (k ) = cos kx2a cos y2 k a and ± (k ) = cos kx2a ± cos y2 . The excitation spectra E(k ,# x) are then obtained by# resolving the secular equations: #M(k , x) − E(k , x)I# = 0.
3.1 Calculation of the Excitation Spectrum E (k , x ). In a first approximation and for multilayers based on transition metals (TM) and alloys, the dipolar interactions and anisotropy can be overlooked in front of the exchange interaction (D = α = 0), the HSH (2) becomes: H=
N
+ Alm (k, x)akl akm
(5)
l,m k
3 The Analytical Resolution
where:
2 Alz m (k, x) = 4SCo xco JCoCo 1 − CoCo (k ) + 8xCo xP t JP tCo (SP t + SCo ) − SP t SCo CoP t (k ) 1 ⊥out 2 ⊥in ⊥out ⊥out 2 ⊥in 4JCoCo + xCo xP t (SP t + SCo ) 8JP⊥in + S + JCoCo + J x + 2J J +SCo xCo Co Co tCo P tCo CoCo 2 CoCo
1 ⊥out +xCo xP t (SP t + SCo ) 4JP⊥in 2 − δlz , 1 − δlz , N δlz , m tCo + JP tCo 2 kz a 2 ⊥in + 2SCo xCo − JCoCo + 4xco xP t SP t SCo JP⊥in (k )cos tCo 2 2 ⊥out ⊥out + SCo xCo JCoCo + xco xP t JP tCo SP t SCo coskz a δlz , m±1 (6)
exchange is dominant in the present case (J ⊥in J ⊥out ). The diagonalization is then performed first in the matrix M(k , x)= C(k , x) ⊗ P(k , x) P(k , x) resulting in p solutions Ei for a given layer [20]. The matrix C(k , x) represents c layers, whereas P(k , x) For c layers, we obtain then c×p different values of energies j corresponds to p planes of a given layer. These matrices are Ei (1 ≤ i ≤ p and 1 ≤ j ≤ c). For example, for multilayers coupled by an inter-layer exchange. However, intra-layer with p = 3 and c = 5, we obtained: ⎞ ⎛ Din (k , x) 0 A1z 1 (k, x) + w out (x) ⎠ Din (k , x) P(k , x) = ⎝ A1z 1 (k, x) + w in (x) Din (k , x) out 0 0 A1z 1 (k, x) + w (x) Otherwise, the dynamic matrix M(k , x) can be expressed as:
2 J ⊥in + 8x x (S with: w in (x) = 4SCo xCo Co P t P t + SCo ) CoCo ⊥in 2 J ⊥out + x x (S out JP tCo ; w (x) = SCo xCo Co P t P t + SCo ) CoCo (x) + (k ); (x) = JP⊥out ; D (k , x) = −w w in tCo √ ⊥in ⊥in 2 4SCo xCo JCoCo + 8xco xP t JP tCo SP t SCo ; Dout (x) =
√ 2 J ⊥out + x x J ⊥out S S − 12 SCo xCo co P t P tCo P t Co ). The three CoCo
corresponding solutions are as follows: A1z 1 (k, x)+ wout (x) + θi (k , x) where: θi (k , x) =
Ei (k , x) =
$ 1 in 2 (k , x); (w − w out ) − (w in − w out )2 + 8Din 2 $ 1 in 2 (k , x) (w − w out ) + (w in − w out )2 + 8Din 0; 2
By injecting these three values Ei (k , x) for each of the five layers, the matrix C(k , x) is ⎛ ⎞ out Ei (k , x) + w (x) Dout (x) 0 0 0 ⎜ ⎟ Ei (k , x) + wout (x) Dout (x) 0 0 Dout (x) ⎜ ⎟ out (x) ⎜ ⎟ (x) E (k , x) + w D (x) 0 0 D out i out ⎜ ⎟ ⎝ ⎠ Ei (k , x) + wout (x) Dout (x) 0 0 Dout (x) out Ei (k , x) + w (x) 0 0 0 Dout (x) for every value of i = 1, 2, 3. The diagonalization of C(k , x) x) + gives the five energy values: Ej (k ,√x) = Ei (k , √ w out (x) + εj Dout (x) with: εj = − 3; −1; 0; 1; 3 . Therefore, the excitation spectra contain 15 modes: j
Ei (k, x) = A1z 1 (k, x) + 2wout (x) + εj Dout (x) + θi (k , x)
(7)
Figure 1 shows the spectra EN=15 (k , x) calculated for different values of x, on the symmetry axis = L where = πa (0, 0, 0) and L = πa (1, 1, 1) are respectively the center and the edge of the Brillouin zone (BZ). These spectra contain, for any value of x, three groups of modes (for 3 ML). Each group consists of five modes (for 5 plans).
J Supercond Nov Magn Fig. 1 EN =15 (k , x) calculated on the axis of BZ, for: a x = 0.1, b x = 0.25, c x = 0.50, and d x = 0.7
Fig. 2 Ep (k , x) calculated on the axis, for x = 0.25 and e p = 5, f p = 7, g p = 15, and h p = 30
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Figure 2 shows, however, that for a magnetic monolayer with a small number of planes (N = p ≤ 15) and a fixed x = 0.25, Ep (k , x) contains the same number p of modes. Whereas for thicker layer (p > 15), as the number p increases, the number of modes in the spectrum differs from p and decreases, in accordance with previous results [15, 19–21]. It also appears that each spectrum is formed by a structure with two different sub-bands. The sub-band of surface magnons (Sbs ) corresponds to the lowest energies; it is formed, as seen below in section 3.2.2, by the two modes Es,1 (k , x) and Es,2 (k , x). The volume sub-band (Sbb ) is located at the highest energies; it consists of the remaining modes (p-2). For p increasing beyond 15, the Sbb covers the Sbs in the center of the Brillouin zone (BZ).
3.2 Analyze and Discussion: Magnons Created in Volume and in Surface The CoP t/P t multilayers (as generally for low dimensionality systems) are characterized by (i) symmetry breaking along the axis perpendicular to the plane of the magnetic layers, (ii) the difference in the number of first neighbors of a cobalt atom located in the volume or inside the surfaces or interfaces. This would lead to a clear differentiation of the exchange intensity responsible for the creation of magnons. One would expect, therefore, a creation of at least three types of magnons in volume, surface, and interface. The magnons created in the interfaces also depend on another important factor which is the overlap of the conduction electron states of the substrate with those of the magnetic layer. However, in the present calculation, we limit ourselves to an exchange configuration where the contribution of the conduction electrons of the substrate is assumed to be negligible (J ⊥in J ⊥out ). 3.2.1 Magnons Created in Volume Each plane lz of a magnetic layer is supposed to interact only with its two neighboring planes lz ± 1. Therefore, the exchange configuration is reduced to J ⊥out = 0 and J ⊥in = J ⊥ . The spectrum expression of the modes created in a plane lz , for all x, is deduced from (7) above as: kz a Elz (k, x) = α(k , x) − γ (k , x)cos (8) 2 where
α(k, x) kz a 2
=
e1 (x) − e2 (0, x)(k ) + := γ (0, x) (k )cos kz2a
and γ (k , x)cos such that: e1 (x) = 4x SCo xJCoCo + 2(1 − x) (SP t + SCo ) JP tCo + ' ( ⊥ + 2x(1 − x) (SP t + SCo ) JP⊥tCo , e2 (0, x) 4x SCo xJCoCo √ =4x SCo xJCoCo CoCo +2(1 − x)JP tCo SP t SCo CoP t
√ ⊥ + 2(1−x) SP t SCo JP⊥tCo . and γ (0, x) = 2x xSCo JCoCo
To discuss the expression (8), we assume that the 1 individual state | ϕn >= (2S)− 2 Sn− |0 of each created magnon (spin deviation) is located on the site n carrying the spin Sn undergoing the deviation. ) cr The potential ) energy V (r − Rn ) = Vncr (r ) of this creation V cr (r ) = n n ) is generated by the exchange field J (ρ)S(ρ) created ρ
by neighbors of this site located at a distance ρ. Rn is a translation vector in the direct crystal lattice. The propagation of the created magnon from a given site to its nearest neighbors corresponds to a state of spin deviation constructed as ) a Bloch sum of localized individual states: | ψk >= 11 ek Rn |ϕn . Therefore, the interpretation of N2 n
factors α and γ can be made by employing a reasoning analogous to that obtained under the usual TBA method for the electronic structure. So, e1 (x) = e1 (x) + e1⊥ (x) is an integral with a single center representing the distortion energy (parallel e1 (x) and perpendicular e1⊥ (x) to the plane lz ) of | ϕn >, located at a given site n, by the exchange field created by its neighbors n = n. The other two terms represent the transfer integrals (i) within the same plane lz of the magnetic layer (two-center integral): e2 (k , x) = e2 (0, x)(k ) and (ii) between this plane lz and its two adjacent lz ± 1 (three-center integral): planes γ (0, x) + (k )cos kz2a . Furthermore, since c ≤ cos kz2a ≤ 1, where the limits 1 and c are obtained at the center and at the edges of the BZ respectively, the expression (8) of Elz (k, x) is bounded between two values, Emin (k , x) = Elz (k , kz = 0, x) = α(k , x) − γ (0, x) + (k ) and Emax (k , x) = Elz (k , kz , x) = α(k , x) − c × γ (0, x) + (k ) for k in BZ. Thus, the determination of these bounds shows that for the symmetry axes = X, = L and = K, the lower bound Emin (k , x) has the same expression and it corresponds to a component of k parallel to the surface: kz = 0. Whereas the expression of the upper bound Emax (k , x) is different and it corresponds to a component π of k parallel to the surface: kz = 2π a for Emax , kz = a and k = 3π for E respectively. The existence for Emax z max 2a of these two boundaries of Elz (k, x) leads to define a permitted energy band to create the magnons in volume, characterized by a bandwidth: Wb (k , x) =| Emax (k , x) − Emin (k , x) |
(9)
We show in Fig. 3 the behavior of x) = γ (0, x) | + (k ) |. The dependence of bandwidth as a function of the alloying effect and wave vector provides information on the creation and dynamics of magnetic excitations. Thus, the difference between the bandwidths calculated on the different axes above reflects the importance of the effect of the system symmetry on this creation. The axis is the most Wb (k ,
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favorable, while the axis would be the most unfavorable to this creation. The situation of the axis is intermediate. Moreover, whatever the symmetry axis, Wb (k , x) admits a maximum value Wb ( , x) showing that in the center of the BZ, this creation is clearly more favored. It then decreases gradually with k increasing and reaches a value almost zero at the edges of the BZ. Otherwise, the increasing
of the cobalt content x favors clearly the magnon creation in the center and inside, whereas at the edges of the BZ, the variation of x remains without effect. To further explore this creation, we have tried to follow the two parameters that control it: the gap Egb and the lifetime τb of these created magnons. Thus, the existence of the lower bound Emin makes it possible to define Egb as Egb (k , x) = Emin (k , x):
kz a 2 2 ⊥ + Egb (k , x) = 4SCo x JCoCo 1 − CoCo (k ) + 2SCo x JCoCo 2 − (k )cos 2 +8x(1 − x) JP tCo SP t + SCo − SP t SCo CoP t (k ) kz a (10) +4x(1 − x)JP⊥tCo 2 (SP t + SCo ) − SP t SCo + (k )cos 2 giving at the center of the BZ: Egb (x) = 4SCo x 2 JCoCo (1 − CoCo ) + 8x(1 − x) JP tCo (SP t + SCo
( √ √ − SP t SCo CoP t + JP⊥tCo SP t + SCo − SP t SCo .
Figure 4 shows a phase diagram
JP tCo
= f ()
JCoCo
describing the stability of the magnons creation state. This diagram is obtained by plotting the equilibrium curve: Egb (x) = 0 reached when there is total compensation of the distortion energy of the spin wave functions in the plane lz by the transfer energy between this plane and its two adjacent planes lz ± 1. We thus find that for a fixed ratio
of the exchange integrals
JP tCo
JCoCo
, the increasing x widens
the stability domain of the magnon creation. Moreover, when the x is fixed, the enhanced exchange in the layers plane between the Co-Pt pairs reduces this stability. While strengthening the exchange between Co-Co pairs increases the domain extent of the created magnons stability. The hyperbolic behavior of the diagram also clearly shows the relevance of magnetocrystalline anisotropy effect on this stability. The most favorable situation is obtained at the intersection between the curve f () and the line =
JCoCo
.
Fig. 3 Permitted energy bandwidth for creating magnons calculated on the axis, for different concentrations x of cobalt: Wb (k , x) and Ws (k , x) (see section 3.2.2)
1 τbcal (k ,
x)
=
#2 2π ## Egb (k , x)# D(x, EF )
10
9 x=0.1 x=0.15 x=0.25 x=0.35 x=0.5 x=0.7
8 7 6
x=0.1 x=0.15 x=0.25 x=0.35 x=0.5 x=0.7
9 8 7
5 4 3
6 5 4 3
2
2
1
1
0
(11)
We assume that the density of states of the TM-based alloy can be fairly well expressed in an approximation of rigid bands by: D(x, EF ) = xDCo (EF ) + (1 − x)DP t (E) where DCo (EF ) 4.16 (eV −1 /at) and DP t (EF ) 5.65 (eV −1 /at) [22, 23]. Figure 5i shows the lifetime evolution τbcal (k , x) of the created magnons calculated on the axis of the BZ. For any value of x, it decreases rather slowly in a small interval near the center from a maximum bound τb,max = τbcal ( , x), then begins a decay with a faster slope and reaches a minimal value τb,min = τbcal (L, x) near the edge L of the BZ. The values obtained for these lifetimes vary between a few units and some tens of femtoseconds (fs). This scale of τb would correspond to the short-range magnetic interactions. A short length of spin-spin correlation would then be sufficient to stabilize the local ordering on the atomic spins that would be separated by a distance of the order of a few lattice sites. Table 1 also shows that these created magnons last longer in the
Ws (meV)
JP tCo
Wb(meV)
2
The lifetime of the created magnons is also calculated using the same procedure that we employed in [19] by:
Γ
Λ
L
0
Γ
Λ
Γ
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3.2.2 Magnons Created in Surface
0.8 x=0.1 x=0.15
To demonstrate the existence of the magnons created in the surfaces, we followed the analysis suggesting that during a perturbation “d” responsible for generation of a surface in a layer, the exchange within this surface can undergo a softening or a strengthening (J = J ⊥ ∓ dJ ⊥ ) [27]. In the case of a softening of the exchange, as for the studied samples, the spectrum of the modes created in the surface Es (k , x) is below that of the modes created in volume and its expression is:
x=0.25
0.6
x=0.35
|| Δ b= 2 JCoPt || JCoCo
0.4
x=0.5 x=0.7
Δb
0.2
0
Es (k , x) = Emin (k , x) −
2 γ (k , x) 1 − η(k , x) 2η(k , x) (12)
−0.2
−0.4
Stability −0.6
0
0.5
||
1
/
1.5
2
JCoPt J ||
CoCo
Fig. 4 The stability phase diagram Egb (x) = 0 of magnons creation in volume as a function of x
center than at the edge of the BZ. They last almost twice for dilute alloys (x ≤ 0.25) and around three times for the highest cobalt concentrations (x ≥ 0.35). In addition, for any value of the component k , the lifetime of the created magnons decreases as x increases from 0.1 to 1. This can be qualitatively explained by the fact that the barrier of potential Egb , necessary to create these magnons, is recognizable in their contribution to the potential of mixture Vsd between the s and d states of Co atoms [20, 25, 26]. Therefore, τb is governed by the exchange interactions, reinforced by the growth in cobalt concentration leading to a strengthened gap, hence a decrease in τb . This is in very good agreement with the results of previous studies giving values in fs for magnon lifetimes in multilayers with an exchange configuration J ⊥ > J [7–9].
To define the factor η, we assume that for a given magnetic plane lz , the state before the perturbation “d” creating the surface, only the plane lz itself exists and then neither the perpendicular distortion energy e1⊥ nor the transfer energy γ (k , x) through this surface exists. The term α(k , x) in (8) is then reduced to α (k , x) = e1 (x) − e2 (k , x). After the creation of the surface, there appears within the plane lz a perpendicular distortion energy e1⊥ (x), a variation dα k , x) = de1 (x) − de2 (k , x), and a transfer energy γ (k , x) outside the plane lz and through the created surface. The factor η(k , x) can then be defined as the ratio of exchange transfer energy, in one of the two neighboring planes lz + 1 or lz − 1 of lz , by the variation caused by “d” at the exchange in the plane lz itself: η(k , x) =
x τb,max (10−15 s) τb,min (10−15 s) τb,max τb,min
0.10 14.68 7.56 1.918
0.25 11.38 4.09 2.782
0.35 8.51 2.31 3.677
0.50 4.48 1.896 2.362
1 2.46 0.705 3.489
(13)
η has to obey the condition: 0 < η(k , x) < +1 [27]. We assume, in a first-order approximation, that the surface generation process in multilayer alloys would be comparable to that in the pure isotropic system (x = 1 and CoCo = 1) and that: η(k , x) η(k ) =
+ (k ) + (k ) ' ( = ' ( 2 1 + 2j (1 − (k )) 2 1 + 2j λ(k ) (14)
where Table 1 The longevity of magnon created in volume as a function of x in the center and at the edge of the BZ
1 γ (k , x) 2 1 ⊥ 2 e1 (x) + de1 (x) − de2 (k , x)
j=
dJCoCo ⊥ JCoCo
and
λ(k ) = 1 − (k ) = sin2 (kx +
ky ) a4 + sin2 (kx − ky ) a4 . The study of the magnon creation within the surface will then be reduced to an analysis of the solutions k in a domain of the BZ. It follows, in accordance with the condition above, that depending on the value of the ratio j > 0, that is again according to the impact of the perturbation “d,” the domain of the BZ corresponding to the creation of magnons in surfaces is more or less important. Moreover, the condition above imposed to η suggests that
J Supercond Nov Magn
there exists > 0 such that: ≤ η ≤ 1 − ⇒ ≤ 12 . Therefore, for any value of < 0.5, there are two values η1 = and η2 = 1 − bounding the variations domain of the factor η involving the existence of two branches Es,1 (k , x) and Es,2 (k , x) limiting the energy band allowed for a creation of magnons in surface with a width: Ws (k , x) = | Es,1 (k , x) − Es,2 (k , x) | # # γ (0, x) | + (k ) | ## (1 − )2 2 ## = |# − # # 2 (1 − ) #
(15)
It follows that at the edge L of the BZ, the band of the surface magnons is also reduced, as for volume magnons, to a single degenerate mode (see Figs. 2 and 3). We also find that the upper line Es,2 (k , x) is close enough to the volume mode band. It is therefore possible that this spectral component may even be covered by the spectral lines of the volume magnons. Moreover, the existence of the lower line Es,1 (k , x) provides the creation gap by Egs (k , x) = Es,1 (k , x):
kz a ⊥in 1 − 1.0625 + (k )cos Egs (k , x) = 4SCo x 2 JCoCo 1 − CoCo (k ) + 4SCo x 2 JCoCo 2 +8x(1 − x) JP tCo SP t + SCo − SP t SCo CoP t (k ) kz a ⊥in + +4x(1 − x)JP tCo 2 (SP t + SCo ) − 1.0625 SP t SCo (k )cos 2 in
⊥out for = 0.25, JCoCo(P t) = JCoCo(P t) , JCoCo(P t) = 0 and
JCoCo(P t) . Then, the lifetime of these magnons
⊥in JCoCo(P t)
< created in surface is #2 2π ## 1 Egs (k , x)# D(x, EF ) = cal τs (k , x)
(17)
We report in Fig. 5ii the function τscal (k , x) which presents a glabal evolution very similar to that obtained above for the volume magnons. However, τscal (k , x) undergoes a faster
decay when k grows beyond the center of the BZ suggesting that surface magnons are strongly more confined in time and space due to the damping effects [10, 24]. The values obtained for these lifetimes range from a few tens to hundreds of fs. This scale of τs values would correspond to exchange interactions that would also be short range, but with a number of nearer Co-Co pairs reduced from 6 in the volume to 4 in a surface. Table 2 also shows that the longevity of magnons created in the surface at the center
5
−14
1.5
(16)
x 10
x=0.1 x=0.25 x=0.35 x=0.5 x=1
x=0.1 x=0.25 x=0.35 x=0.5 x=1
4
1
τb (s)
τs (s)
3
2
0.5 1
0
0
Γ
0.5
Λ
1
1.5
L
0
0
Γ
0.5
Fig. 5 The lifetime of created magnons calculated on the axis of the BZ as a function of x and k : i Section 3.2.2 below
Λ
τbcal (k ,
1
x) and (ii)
1.5
L
τscal (k ,
x) obtained in
J Supercond Nov Magn
x τs,max (10−13 s) τs,min (10−13 s) τs,max τs,min
0.10 4.918 1.602 3.069
0.25 3.612 0.840 4.30
0.35 2.586 0.497 5.203
0.50 2.338 0.381 6.267
1 2.044 0.251 8.143
ranges between three and eight times more than that at the edge L of the BZ, when the cobalt content increases from 0.1 to 1. In Table 3, we presented a direct comparison of the effect of chemical composition variation on the longevities of the two types of magnons created. It shows that when x increases, the surface magnons last in the center , 30 to 80 times longer than the volume magnons. Whereas at the edge L of the BZ, the surface magnons only last about 20 times longer than the those created in volume. Taking into account the magnetic exchange configuration proposed above neglecting the interface magnons probing the effect of substrate, we have tried a comparison between our calculation results τscal (k , x) and the measured results exp τs given in the bibliography [10] for 8 ML of Co/Cu. exp Figure 6 shows this comparison, with τs (open squares), of τscal (k , x) (curve continuous) calculated for x = 1, N = 24 plans (8 ML of 3 plans each) and k varying on the axis of BZ. The curve τscal (k , x) faithfully reproduces the experimental behavior. The values of the ⊥in exchange integrals (JCoCo(P t) and JCoCo(P t) ) and the magnetocrystalline anisotropies (CoCo(P t) ) deduced from this adjustment are in good agreement with the results given ⊥in < J for materials with a configuration JCoCo CoCo .
4 Conclusion The Heisenberg model was used to calculate the excitation spectrum Elz (k , x) of the magnons created in the magnetic multilayers Cox P t1−x /P t. We have demonstrated the existence of two types of these magnetic excitations: magnons of volume and magnons of surface. The permitted energy bands and the gaps of creation and lifetimes are determined for each type of magnons created. The Table 3 The longevity of the magnons created in volume and in the surface according to the content of Co in the center and at the edge of the BZ x τs,max τb,max τs,min τb,min
0.10 33.501
0.25 31.739
0.35 30.387
0.50 53.303
1 83.089
21.182
20.537
21.477
20.094
35.602
−13
2.5
x 10
2
1.5
τs (s)
Table 2 Evolution of the lifetimes of the magnons created in surfaces as a function of x in the center and at the edge of the BZ
1
0.5
0
0
Γ
0.5
Λ
1
1.5
L
Fig. 6 τscal (k , x) calculated for x = 1, N = 8 × 3, and = 0.25 exp (solid line) compared with τs (open squares) [10]. The adjustment ⊥in parameters are JCoCo = 83.5K, JCoCo = 13.9K, CoCo = 0.7, ⊥in JCoP t = 62.7K, JCoP t = 10.45K, and CoP t = 0.83
increasing alloying effect clearly favors the creation of magnons of wave vectors close to the center of the BZ, but their longevity is becoming more and more disadvantaged as x increases. For any x cobalt concentration, the surface magnons last longer than the volume ones. This difference in longevities of the two types of magnons is more accentuated in the center of the BZ. Surface magnons would be more confined in time and space. The agreement obtained between our calculated results and those provided in previous works is very satisfactory. The exchange parameters deduced from this comparison are in line with what is usually provided for multilayers engaging 3d transition metals.
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