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Roughness effects on spin wave resonance in thin films
Journal of Magnetism and Magnetic Materials 204 (1999) 199}203
Roughness e!ects on spin wave resonance in thin "lms Daniel Mercier, Jean-Claude Serge LeH vy* Laboratoire de Physique The& orique de la Matie% re Condense& e, Case 7020 Universite& Paris 7-Denis Diderot, 75251 Paris Cedex 05, France Received 25 March 1999; received in revised form 31 May 1999
Abstract Two models of surface roughness are introduced in a transfer matrix approach of spin wave resonance. A one-layer surface roughness is shown to lead in the case of a weak antiferromagnetic coupling to surface modes while an extended roughness leads to an e!ective unpinning with nonlinear contributions. Applications to multilayers are deduced. 1999 Elsevier Science B.V. All rights reserved. PACS: 75.40.Mg; 75.70.Cn; 76.50.#g Keywords: Numerical simulation; Spin wave resonance; Surface roughness
1. Introduction Surface roughness and interface roughness of magnetic multilayers are well known to produce strong changes in magnetic properties and in magnetotransport properties such as giant magnetoresistance [1]. So recently, surface roughness and interface roughness of magnetic multilayers have been thoroughly investigated by means of grazing incidence specular and di!use X-ray scattering [2,3] as well as by surface tunnel microscopy (STM) [4,5]. Di!erent results on the e!ective surface roughness have been obtained. On the large scale observed by X-ray scattering, a fractal model is usually required in order to represent experimental results with a distribution of roughness heights and of lateral coherence lengths. On the
* Corresponding author. Tel.: #33-1-44-274377; fax: #331-46-339401.
short scale observed by STM, sometimes a few one-layer terraces are noticed and in this case, the surface roughness is restricted to just one layer. According to the experimental preparation and way of observation, quite di!erent models for surface and interface roughness must be considered. So, in this paper we choose to study the microscopic roughness de"ned by a one-layer incomplete terrace and compare the results deduced for spin wave resonance with such a roughness to those obtained for spin wave resonance with an extended roughness as it occurs for a Stranski}Krastanov growth with islands [6]. Taking advantage of the lateral coherence of the RF signal in spin wave resonance leads to the de"nition of an e!ective one-dimensional problem for the spin wave propagation, where roughness parameters are de"ned from averaging the one-layer incomplete terrace. These natural parameters, exchange coupling aJ between the rough surface and the "rst bulk layer which is compared to the bulk exchange parameter
0304-8853/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 4 6 6 - 7
200
D. Mercier, J.-C. Serge Le& vy / Journal of Magnetism and Magnetic Materials 204 (1999) 199}203
J, and surface anisotropy dJ also expressed in terms of the same unit, are introduced in a transfer matrix approach to derive the spin wave properties. The magnetic model is given in Section 2, while results are reported in Section 3, and "nally comparisons with results obtained for other models of surface and interface roughness are made in Section 4.
Since the e!ective selection rule for spin wave resonance (k "k "0) implies a large-scale integV W ration over the surface, the magnetic parameters must be averaged over the whole surface, up to the lateral coherence length of the RF signal. After averaging, taking advantage of the translational invariance for directions parallel to the layer because of this averaging enables us to derive a onedimensional problem for the spin deviation b in J layer l
2. 1D magnetic model Neglecting bulk anisotropy as usual for soft magnetic materials, the spin Hamiltonian reduces to three parts, the exchange part H , the Zeeman part H and the surface anisotropy H , with 8 H "! J S ' S , DE D E DE H "!k H ' S 8 D D and "!d (SX ), (1) D D where S is the spin vector of the electron located at D site f and H is the e!ective magnetic "eld which in the present work lies in the direction z normal to the "lm. The magnetic single ion anisotropy d is introduced only for the two surface layers. Since in resonance experiments with perpendicular "eld the magnitude h of this "eld is strong enough to saturate the magnetization and make the equilibrium con"guration of the spins collinear with it, we introduce Pauli operators b to deal with spin wave D excitations, with H
1 i SV " (b>#b ), SW " (b>!b ), D 2 D D D 2 D D 1 SX " !b>b . D 2 D D
(2)
From the Hamiltonian, the equation of motion of Pauli operators for a standing wave of time dependence exp(!iut) is derived and reads i
d 1 (b )"ub "(h#d )b # J (b !b ). E E E 2 DE E D dt E D
(3)
2(u!h!d )b E J "(b !b )J #(b !b )J , J J\ J J\ J J> J J>
(4)
where the J are e!ective layer-to-layer paraJ J! meters. This one-dimensional equation (4) is translated in terms of PoincareH 's transfer matrices ¹ with L reference to both spin amplitude and gradient of spin amplitude on each layer:
b b L> L "¹ ' . L b !b b !b L> L L L\
(5)
Such an expression is quite convenient in order to introduce pinning conditions at the boundaries such as perfect pinning or perfect unpinning or even partial pinning. Starting from layer 1 and ending at layer N de"nes the complete transfer matrix t as M a product of similar transfer matrices including local properties. Since the boundary conditions are known as pinning or unpinnning or partial pinning on limiting surfaces, the characteristic equation for spin waves is easily deduced by equalling to zero a linear combination of matrix elements of the complete transfer matrix t . This polynomial equation is solved numerically by varying the external "eld h in a regular way, as usual. This method applied to the characteristic equation is then used to derive the corresponding eigenvectors. Such a process enables us to deal with any one-dimensional representation of the exchange and magnetization pro"les as already developed in previous works [6,7]. As a matter of fact, the normalized spin wave amplitudes as well as the spin wave frequencies are deduced in such an approach which gives exactly the same results as a standard Green
D. Mercier, J.-C. Serge Le& vy / Journal of Magnetism and Magnetic Materials 204 (1999) 199}203
function approach, but deals with only 2;2 matrices involved in a product. The numerical resolution here deals with the simple resolution of the roots of a polynomial of degree 2N while the numerical resolution of the Green function approach deals with the resolution of a rank N matrix. Here lies the basic numerical advantage of the transfer matrix approach over the classical Green function method which introduces all the eigen modes in a more global picture.
201
In this paper the boundary conditions are chosen to be pinning on both external surfaces. Other boundary conditions can be introduced in the com-
putation and are known to lead more easily to surface modes [6]. The standard values always being the same, u"10 for the frequency and J"10 for the exchange parameter, in order to compare the results with the previous results obtained for di!erent models of roughness [6] or coupling [7], the spin wave spectrum is derived for a "lm of 100 layers. A strong singularity for a zero value of the coupling a of the top layer occurs as shown in Fig. 1 when reporting the resonant "elds as a function of the coupling, in the absence of any surface anisotropy. This singularity is odd as a function of the coupling a of the top layer. It means that for a negative and weak coupling a of the top layer, surface modes are observed. But this singularity is restricted to a very weak antiferromagnetic coupling.
Fig. 1. The resonant magnetic "elds H of the "rst modes of a 100-layer "lm as a function of the roughness coupling parameter a.
Fig. 2. The resonant magnetic "elds H of the "rst modes of a 100-layer "lm with surface anisotropy d"!0.1, as a function of the roughness coupling parameter a. Two surface modes (H'10 000 G) appear when !0.06(a(!0.02.
3. Results for a one-layer roughness
202
D. Mercier, J.-C. Serge Le& vy / Journal of Magnetism and Magnetic Materials 204 (1999) 199}203
In the case of both rather weak coupling and nonzero surface anisotropy, the resonant "elds are reported in Fig. 2 as a function of the coupling a of the top layer. A considerable increase in the extension of the singularity is obtained and this
singularity now occurs for rather large values of the parameter a. More precisely, the spatial behavior of the "rst modes is shown in Fig. 3 with evidence for the existence of several surface modes. The spin wave intensity predicted in such a resonance experiment is reported in Fig. 4 in order to check the occurrence of surface modes with h' 10 000 G and to compare the orders of magnitude of the di!erent modes when including interference e!ects. The intensity is deduced from the classical formula for spin wave resonance [6]
bG N N . I "K G ( (bG ) N N
Fig. 3. The spin wave behavior of the "rst "ve modes of a 100layer "lm with surface anisotropy on both external surfaces d"0.3, and roughness parameter a"!0.04. Two surface modes, symmetric and antisymmetric can be noticed at the bottom of the "gure.
(6)
Two close surface modes can be noticed, a symmetric one, observable by spin wave resonance (SWR) and an antisymmetric one, not observable by SWR. One bulk mode is also observed while other bulk modes are either weak or unobserved because of antisymmetry. For very thin "lms only a very few modes are observed.
Fig. 4. The spin wave intensities and resonant "elds for one 100-layer "lm with a"!0.02, d"0.3. The intensities are reported in bold lines, while light lines give the positions of resonant "elds.
D. Mercier, J.-C. Serge Le& vy / Journal of Magnetism and Magnetic Materials 204 (1999) 199}203
4. Conclusive remarks The comparison between the present results for one-layer roughness where SWR e!ects occur for weak coupling and are emphasized by surface anisotropy and those obtained in the case of an extended roughness where the parameter is the average roughness height gives large di!erences. In the case of extended roughness, a nonlinear behavior of the spin excitation occurs because of the exchange gradient. In this case, a mode which appears as localized at the surface can occur because of such an exchange variation which creates a different area, weakly coupled with the rest of the bulk. In this case, the increase of the roughness height is responsible for the appearance of more and more localized modes. This behavior is quite di!erent from that obtained for a one-layer roughness. As a consequence of this comparison, it is clear that SWR measurements, when combined
203
with more structural surface and interface observations, can provide realistic information on the magnetic status of surfaces and interfaces, with applications for multilayers.
References [1] B. Heinrich, J.A.C. Bland, Ultrathin Magnetic Structures, Springer, Berlin, 1994. [2] I.H. Laidler, B.J. Hickey, T.P.A. Hase, B.K. Tanner, R. Schadl, J. Magn. Magn. Mater. 156 (1996) 332. [3] I. Pape, T.P.A. Hase, B.K. Tanner, H. Laidler, C. Emmerson, T. Shen, B.J. Hickey, J. Magn. Magn. Mater. 156 (1996) 373. [4] F. Tsui, J. Wellman, C. Uher, R. Clarke, Phys. Rev. Lett. 76 (1996) 3164. [5] A.K. Schmid, J.C. Hamilton, N.C. Bartelt, R.Q. Hwang, Phys. Rev. Lett. 77 (1996) 2977. [6] D. Mercier, J.-C.S. Levy, J. Magn. Magn. Mater. 163 (1996) 207. [7] M. Jackson, D. Mercier, J.C.S. Levy, J.S.S. Whiting, J. Magn. Magn. Mater. 170 (1997) 32.
Roughness e!ects on spin wave resonance in thin "lms Daniel Mercier, Jean-Claude Serge LeH vy* Laboratoire de Physique The& orique de la Matie% re Condense& e, Case 7020 Universite& Paris 7-Denis Diderot, 75251 Paris Cedex 05, France Received 25 March 1999; received in revised form 31 May 1999
Abstract Two models of surface roughness are introduced in a transfer matrix approach of spin wave resonance. A one-layer surface roughness is shown to lead in the case of a weak antiferromagnetic coupling to surface modes while an extended roughness leads to an e!ective unpinning with nonlinear contributions. Applications to multilayers are deduced. 1999 Elsevier Science B.V. All rights reserved. PACS: 75.40.Mg; 75.70.Cn; 76.50.#g Keywords: Numerical simulation; Spin wave resonance; Surface roughness
1. Introduction Surface roughness and interface roughness of magnetic multilayers are well known to produce strong changes in magnetic properties and in magnetotransport properties such as giant magnetoresistance [1]. So recently, surface roughness and interface roughness of magnetic multilayers have been thoroughly investigated by means of grazing incidence specular and di!use X-ray scattering [2,3] as well as by surface tunnel microscopy (STM) [4,5]. Di!erent results on the e!ective surface roughness have been obtained. On the large scale observed by X-ray scattering, a fractal model is usually required in order to represent experimental results with a distribution of roughness heights and of lateral coherence lengths. On the
* Corresponding author. Tel.: #33-1-44-274377; fax: #331-46-339401.
short scale observed by STM, sometimes a few one-layer terraces are noticed and in this case, the surface roughness is restricted to just one layer. According to the experimental preparation and way of observation, quite di!erent models for surface and interface roughness must be considered. So, in this paper we choose to study the microscopic roughness de"ned by a one-layer incomplete terrace and compare the results deduced for spin wave resonance with such a roughness to those obtained for spin wave resonance with an extended roughness as it occurs for a Stranski}Krastanov growth with islands [6]. Taking advantage of the lateral coherence of the RF signal in spin wave resonance leads to the de"nition of an e!ective one-dimensional problem for the spin wave propagation, where roughness parameters are de"ned from averaging the one-layer incomplete terrace. These natural parameters, exchange coupling aJ between the rough surface and the "rst bulk layer which is compared to the bulk exchange parameter
0304-8853/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 4 6 6 - 7
200
D. Mercier, J.-C. Serge Le& vy / Journal of Magnetism and Magnetic Materials 204 (1999) 199}203
J, and surface anisotropy dJ also expressed in terms of the same unit, are introduced in a transfer matrix approach to derive the spin wave properties. The magnetic model is given in Section 2, while results are reported in Section 3, and "nally comparisons with results obtained for other models of surface and interface roughness are made in Section 4.
Since the e!ective selection rule for spin wave resonance (k "k "0) implies a large-scale integV W ration over the surface, the magnetic parameters must be averaged over the whole surface, up to the lateral coherence length of the RF signal. After averaging, taking advantage of the translational invariance for directions parallel to the layer because of this averaging enables us to derive a onedimensional problem for the spin deviation b in J layer l
2. 1D magnetic model Neglecting bulk anisotropy as usual for soft magnetic materials, the spin Hamiltonian reduces to three parts, the exchange part H , the Zeeman part H and the surface anisotropy H , with 8 H "! J S ' S , DE D E DE H "!k H ' S 8 D D and "!d (SX ), (1) D D where S is the spin vector of the electron located at D site f and H is the e!ective magnetic "eld which in the present work lies in the direction z normal to the "lm. The magnetic single ion anisotropy d is introduced only for the two surface layers. Since in resonance experiments with perpendicular "eld the magnitude h of this "eld is strong enough to saturate the magnetization and make the equilibrium con"guration of the spins collinear with it, we introduce Pauli operators b to deal with spin wave D excitations, with H
1 i SV " (b>#b ), SW " (b>!b ), D 2 D D D 2 D D 1 SX " !b>b . D 2 D D
(2)
From the Hamiltonian, the equation of motion of Pauli operators for a standing wave of time dependence exp(!iut) is derived and reads i
d 1 (b )"ub "(h#d )b # J (b !b ). E E E 2 DE E D dt E D
(3)
2(u!h!d )b E J "(b !b )J #(b !b )J , J J\ J J\ J J> J J>
(4)
where the J are e!ective layer-to-layer paraJ J! meters. This one-dimensional equation (4) is translated in terms of PoincareH 's transfer matrices ¹ with L reference to both spin amplitude and gradient of spin amplitude on each layer:
b b L> L "¹ ' . L b !b b !b L> L L L\
(5)
Such an expression is quite convenient in order to introduce pinning conditions at the boundaries such as perfect pinning or perfect unpinning or even partial pinning. Starting from layer 1 and ending at layer N de"nes the complete transfer matrix t as M a product of similar transfer matrices including local properties. Since the boundary conditions are known as pinning or unpinnning or partial pinning on limiting surfaces, the characteristic equation for spin waves is easily deduced by equalling to zero a linear combination of matrix elements of the complete transfer matrix t . This polynomial equation is solved numerically by varying the external "eld h in a regular way, as usual. This method applied to the characteristic equation is then used to derive the corresponding eigenvectors. Such a process enables us to deal with any one-dimensional representation of the exchange and magnetization pro"les as already developed in previous works [6,7]. As a matter of fact, the normalized spin wave amplitudes as well as the spin wave frequencies are deduced in such an approach which gives exactly the same results as a standard Green
D. Mercier, J.-C. Serge Le& vy / Journal of Magnetism and Magnetic Materials 204 (1999) 199}203
function approach, but deals with only 2;2 matrices involved in a product. The numerical resolution here deals with the simple resolution of the roots of a polynomial of degree 2N while the numerical resolution of the Green function approach deals with the resolution of a rank N matrix. Here lies the basic numerical advantage of the transfer matrix approach over the classical Green function method which introduces all the eigen modes in a more global picture.
201
In this paper the boundary conditions are chosen to be pinning on both external surfaces. Other boundary conditions can be introduced in the com-
putation and are known to lead more easily to surface modes [6]. The standard values always being the same, u"10 for the frequency and J"10 for the exchange parameter, in order to compare the results with the previous results obtained for di!erent models of roughness [6] or coupling [7], the spin wave spectrum is derived for a "lm of 100 layers. A strong singularity for a zero value of the coupling a of the top layer occurs as shown in Fig. 1 when reporting the resonant "elds as a function of the coupling, in the absence of any surface anisotropy. This singularity is odd as a function of the coupling a of the top layer. It means that for a negative and weak coupling a of the top layer, surface modes are observed. But this singularity is restricted to a very weak antiferromagnetic coupling.
Fig. 1. The resonant magnetic "elds H of the "rst modes of a 100-layer "lm as a function of the roughness coupling parameter a.
Fig. 2. The resonant magnetic "elds H of the "rst modes of a 100-layer "lm with surface anisotropy d"!0.1, as a function of the roughness coupling parameter a. Two surface modes (H'10 000 G) appear when !0.06(a(!0.02.
3. Results for a one-layer roughness
202
D. Mercier, J.-C. Serge Le& vy / Journal of Magnetism and Magnetic Materials 204 (1999) 199}203
In the case of both rather weak coupling and nonzero surface anisotropy, the resonant "elds are reported in Fig. 2 as a function of the coupling a of the top layer. A considerable increase in the extension of the singularity is obtained and this
singularity now occurs for rather large values of the parameter a. More precisely, the spatial behavior of the "rst modes is shown in Fig. 3 with evidence for the existence of several surface modes. The spin wave intensity predicted in such a resonance experiment is reported in Fig. 4 in order to check the occurrence of surface modes with h' 10 000 G and to compare the orders of magnitude of the di!erent modes when including interference e!ects. The intensity is deduced from the classical formula for spin wave resonance [6]
bG N N . I "K G ( (bG ) N N
Fig. 3. The spin wave behavior of the "rst "ve modes of a 100layer "lm with surface anisotropy on both external surfaces d"0.3, and roughness parameter a"!0.04. Two surface modes, symmetric and antisymmetric can be noticed at the bottom of the "gure.
(6)
Two close surface modes can be noticed, a symmetric one, observable by spin wave resonance (SWR) and an antisymmetric one, not observable by SWR. One bulk mode is also observed while other bulk modes are either weak or unobserved because of antisymmetry. For very thin "lms only a very few modes are observed.
Fig. 4. The spin wave intensities and resonant "elds for one 100-layer "lm with a"!0.02, d"0.3. The intensities are reported in bold lines, while light lines give the positions of resonant "elds.
D. Mercier, J.-C. Serge Le& vy / Journal of Magnetism and Magnetic Materials 204 (1999) 199}203
4. Conclusive remarks The comparison between the present results for one-layer roughness where SWR e!ects occur for weak coupling and are emphasized by surface anisotropy and those obtained in the case of an extended roughness where the parameter is the average roughness height gives large di!erences. In the case of extended roughness, a nonlinear behavior of the spin excitation occurs because of the exchange gradient. In this case, a mode which appears as localized at the surface can occur because of such an exchange variation which creates a different area, weakly coupled with the rest of the bulk. In this case, the increase of the roughness height is responsible for the appearance of more and more localized modes. This behavior is quite di!erent from that obtained for a one-layer roughness. As a consequence of this comparison, it is clear that SWR measurements, when combined
203
with more structural surface and interface observations, can provide realistic information on the magnetic status of surfaces and interfaces, with applications for multilayers.
References [1] B. Heinrich, J.A.C. Bland, Ultrathin Magnetic Structures, Springer, Berlin, 1994. [2] I.H. Laidler, B.J. Hickey, T.P.A. Hase, B.K. Tanner, R. Schadl, J. Magn. Magn. Mater. 156 (1996) 332. [3] I. Pape, T.P.A. Hase, B.K. Tanner, H. Laidler, C. Emmerson, T. Shen, B.J. Hickey, J. Magn. Magn. Mater. 156 (1996) 373. [4] F. Tsui, J. Wellman, C. Uher, R. Clarke, Phys. Rev. Lett. 76 (1996) 3164. [5] A.K. Schmid, J.C. Hamilton, N.C. Bartelt, R.Q. Hwang, Phys. Rev. Lett. 77 (1996) 2977. [6] D. Mercier, J.-C.S. Levy, J. Magn. Magn. Mater. 163 (1996) 207. [7] M. Jackson, D. Mercier, J.C.S. Levy, J.S.S. Whiting, J. Magn. Magn. Mater. 170 (1997) 32.