Jan 1, 2001 - Nanostructure Physics, Royal Institute of Technology, 10044 Stockholm, Sweden .... stant the LandauâLifshitz form of the dissipative term is.
JOURNAL OF APPLIED PHYSICS
VOLUME 89, NUMBER 1
1 JANUARY 2001
Impedance of a ferromagnetic sandwich strip A. Sukstanskii, V. Korenivski, and A. Gromov Nanostructure Physics, Royal Institute of Technology, 10044 Stockholm, Sweden
共Received 21 December 1999; accepted for publication 11 October 2000兲 A theoretical approach for calculating the impedance of a three-layer sandwich, consisting of two metallic ferromagnetic layers separated by a non-magnetic conductive layer, is presented. The Maxwell equations for the electromagnetic field coupled with the Landau–Lifshitz equations for the magnetization dynamics are solved, which enables one to describe the system over a wide frequency range, including the ferromagnetic resonance. Two sandwich strip structures are analyzed, both having thickness much less than the width, and the width much less than the length: a ‘‘closed’’ structure with the magnetic film closing at the edges along the width, and an ‘‘open’’ structure without flux closures where all the layers have the same width. The impedance for the two structures is calculated and analyzed as a function of the physical parameters of the device and frequency. The ‘‘closed’’ structure is more efficient magnetically and exhibits a highly inductive response to much higher frequencies than the ‘‘open’’ structure. The analytical results obtained are directly applicable to practical design of GHz inductive components. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1330763兴
Antonov et al.14 have analyzed the surface impedance of a sandwich strip with a transverse magnetic anisotropy 共magnetic easy axis along the stripe width兲. The authors 共see also Ref. 6兲 have reduced the problem to one dimension, where the fields are considered to depend only on the coordinate normal to the film surface. Such an approach is based on the fact that the film length, l, and the width, b, are usually much greater than the thickness, d, and, hence, the film can be treated as infinite in two dimensions with the fields uniform along these directions. This approach is adequate for long stripes with flux closures along the width. In the general case, especially for structures with air gaps,4 the properties of ‘‘long’’ sandwich stripes of finite width can differ significantly from those predicted using a one-dimensional 共1-D兲 model. This is true even in the case where the width is much greater than the thickness, dⰆb⬃2, with ⫽( d 0 d 1 ) 1/2 being the characteristic flux decay length in the structure, first introduced by Patton,5 and d 0 ,d 1 being the thicknesses of the conductor and of the magnetic layer, respectively. Here is much greater than d 0,1 due to a high value of the magnetic permeability in soft magnetic films. Typical parameters for the structures of interest are d 0 ⬃d 1 ⬃0.2 m, b ⬃20 m, ⬃103 . ⬃6 m is comparable to the halfwidth, and therefore the size effects cannot be neglected for structures having no flux closures at the edges 共no ‘‘flanges’’兲. It should also be expected that at high frequencies the flux leakage through the conductor would significantly increase the magnetic loss in an ‘‘open’’ sandwich. There are several analytical approaches for calculating the inductance in two- and three-dimensional structures 共see, e.g., Ref. 15兲. A theory for a rather complex twodimensional geometry of a magnetic recording head has been developed in Ref. 16. In Ref. 17 共see also numerous references therein兲 a theory accounting for domain walls in slabs and cylinders has been worked out. These models, however, are developed in the framework of a static or quasi-static
I. INTRODUCTION
Attention is increasingly being given to the potential use of soft ferromagnetic film for inductors in GHz integrated circuits. Such inductors are expected to reduce size and increase efficiency of these circuits. Several planar magnetic inductor designs have been proposed: solenoids, spirals, and meanders 共see, e.g., Refs. 1–3 and references therein兲. An analysis of different types of planar inductors4 showed that a simple structure of a magnetic sandwich formed into a stripe is one of the most promising for GHz applications. Soft magnetic metallic alloy films rather than ferrites are used in integrated designs, since they can be readily incorporated in multilayer structures compatible with the Si-based fabrication processes. Due to the electrical conductivity of these films, the effects of eddy currents in both the conductor and the magnetic films have to be taken into account. In addition, the dynamics of the magnetization in the ferromagnetic films will affect the stripe inductive response. Efficiency of magnetic sandwich inductors has been analyzed using a static magnetic model.4,5 A high frequency model for a magnetic sandwich,6 allowing eddy currents in all the layers, has also been developed. However, in Ref. 6 the permeability, , was taken to be independent of frequency, , i.e., the model was restricted to frequencies small compared with the ferromagnetic resonance 共FMR兲 frequency of the magnetic films. To correctly predict the behavior of the system near the FMR, where the maximum device efficiency is expected, Maxwell’s equations coupled with the Landau–Lifshitz equations for the magnetic moment have to be solved for the structure under study. This approach is used, for example, in calculating the surface impedance tensor of magnetic wires 共see, e.g., Refs. 7–10兲 and of multilayer film with crossed anisotropy,11–13 exhibiting the so-called giant magneto-impedance 共GMI兲 effect. 0021-8979/2001/89(1)/775/8/$18.00
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© 2001 American Institute of Physics
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J. Appl. Phys., Vol. 89, No. 1, 1 January 2001
Sukstanskii, Korenivski, and Gromov
used19兲, a ⫽gH a , 0 ⫽gM 0 , g is the gyromagnetic ratio and H a is the anisotropy field. H is the magnetic field subject to the Maxwell equations: 4 j, c
rot H⫽
rot E⫽⫺
1 B , c t
div B⫽0,
共2兲
where B⫽H⫹4 M, j is the electric current density, jÄ E, E is the electric field induced by a time-varying magnetic field, is the conductivity of the magnets, and c is the light velocity. In these equations we neglected the displacement current term in the first equation since in the GHz frequency range the retardation effects due to this term are negligible. Assuming the time dependence of the fields and magnetization to be proportional to exp(⫺it), we obtain from Eqs. 共1兲 the relations between the magnetization and the field, m ˆ H, where ˆ is the permeability tensor, ⫽ ˆ H, or B⫽
FIG. 1. 共a兲 Magnetic sandwich without flanges–‘‘open’’ structure. 共b兲 Magnetic sandwich with flanges–‘‘closed’’ structure.
approximation, which is inadequate in the FMR regime. The objective of the present article is to develop a twodimensional model for the impedance of a three-layer structure consisting of a nonmagnetic conductor sandwiched between two magnetic conductive layers. The theory is based on solving Maxwell’s equations for the electromagnetic fields and the Landau–Lifshitz equations for the magnetization vector with appropriate boundary conditions at the interfaces between the layers and outer surfaces. The impedance is calculated for two forms of the edges along the width: with and without magnetic closures 共with and without flanges兲. The respective models are termed ‘‘closed’’ and ‘‘open.’’ II. THE MODEL
Let us consider a three-layer structure shown in Fig. 1, consisting of two magnetic layers of thickness d 1 and conductivity 1 , separated by a nonmagnetic layer of thickness 2d 0 and conductivity 0 . The film length and width are l and b, respectively. The magnetization distribution in the magnetic layers is described by M. The magnets are taken to be of the easy-axis type, the easy axis being parallel to the Cartesian Z-axis in the film plane. Neglecting the exchange contribution,18 the linearized Landau–Lifshitz equations for the magnetization vectors in the magnetic layers can be written in the form
mx ⫽ 0 H y ⫺ a m y ⫹ r 共 0 H x ⫺ a m x 兲 , t my ⫽⫺ 0 H x ⫹ a m x ⫹ r 共 0 H y ⫺ a m y 兲 , t
冉
ˆ ⫽ i a 0 ⫽1⫹ a⫽
where m⫽M⫺M0 , M0 is the equilibrium value of the magnetization vector in the magnetic layers, M0 ⫽M 0 ez , M 0 ⫽ 兩 M兩 is the saturation magnetization, ez is the unit vector along the Z-axis, r is the phenomenological relaxation constant 共the Landau–Lifshitz form of the dissipative term is
0
冊
0 , 1
0
4 0 共 a ⫺i r 兲
2a ⫺ 共 ⫹i r a 兲 2
,
4 0 . 2 a ⫺ 共 ⫹i r a 兲 2
共3兲
Introducing the scalar and vector potentials, and A, BÄcurl A, E⫽⫺(1/c) ( A/ t) ⫺ⵜ , Maxwell’s equations 共2兲 take the form 共Coulomb gauge, div AÄ0, is used兲 ˆ ⫺1 curl A兲 ⫽⫺ curl共
冉
冊
1 A ⫹ⵜ , c t
ⵜ 2 ⫽0.
共4兲
Let us consider the case in which a time-dependent external voltage, U⬃exp(⫺it), is applied to the stripe at z ⫽0 and z⫽l. Since the length of the stripe, l, is much greater than its transverse dimension 共but much smaller than the wave length of the signal, lⰆc/ ), the field distribution can be considered to be independent of z. Therefore, the scalar potential inside the sandwich can be sought in the form ⫽⫺E e z, where E e ⫽U/l has the meaning of a ‘‘driving’’ electric field. Noting that in our geometry only transverse x and y components of the magnetic field are nonzero, it follows that A x ⫽A y ⫽0, and the only nonzero z component of the vector potential in the magnetic layers satisfies the equation
2 A z(1,2) 共1兲
⫺i a
x2
⫹
2 A z(1,2) y2
⫹
2i
␦
A (1,2) ⫽⫺ 2 z
U 4 e , c l
共5兲
where ␦ ⫽c/(2 1 e ) 1/2 is the skin depth and e is the effective transverse permeability, e ⫽( 2 ⫺ 2a )/ . Hereafter the upper index j⫽1,2 corresponds to the upper (d 0 ⬍x⬍d⫽d 0 ⫹d 1 ) and lower (⫺d⬍x⬍⫺d 0 ) magnetic layer, j⫽0 to the conductor (⫺d 0 ⬍x⬍d 0 ), and j⫽3 and 4 to the space above (x⬎d) and below (x⬍⫺d) the film, respectively.
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J. Appl. Phys., Vol. 89, No. 1, 1 January 2001
Sukstanskii, Korenivski, and Gromov
In soft ferromagnetic alloys the anisotropy field is small compared to the saturation magnetization, the typical ratio H a /4 M 0 ⬃10⫺3 . Therefore in a broad frequency range 关up to the FMR frequency, ⫽( a b ) 1/2, b ⫽ a ⫹4 0 ] * the effective permeability is large, 兩 e 兩 Ⰷ1. The skin depth is much smaller than in a nonmagnetic media with the same conductivity, since ␦ ⬃ ⫺1/2 . In the vicinity of the FMR e frequency e ( ) exhibits a typical resonant behavior:20 at ⯝ its real part goes through 0, whereas the imaginary * part has a maximum. Similar to 共5兲 the equation for the vector potential in the conductor has the form
2
A z(0)
x2
⫹
2
A z(0)
y2
4 U ⫹ 2 A z(0) ⫽⫺ , c 0 l ␦0 2i
共6兲
where ␦ 0 ⫽c/(2 0 ) 1/2 is the classical skin depth. The current density in the conductor, j z(0) , and in the magnetic layers, j z(1,2) , is j z(0) ⫽ 0
冉
冊
U i (0) ⫹ Az , l c
j z(1,2) ⫽ 1
冉
冊
U i (1,2) . ⫹ Az l c
共7兲
At the interfaces between the layers (x⫽⫾d 0 ) and at the film surfaces (x⫽⫾d) the standard boundary conditions are used. Namely, the continuity of the vector potential and the tangential component of the magnetic field, H y 关the continuity of the normal component of the magnetic induction 共in our case, B x ⫽ A z / y) is fulfilled automatically, if the two conditions above are satisfied兴. The latter can be readily expressed in terms of the vector potential space derivatives: H y⫽
冉
冊 冉
冊
a Az a Az 1 1 ⫹i B y ⫹i Bx ⫽ ⫺ . e e x y
共8兲
In the conductor and in the external region H y ⫽B y ⫽⫺ A z / x. Expressed in terms of the vector potential and its space derivatives, the boundary conditions become A z(0) 共 ⫾d 0 兲 ⫽A z(1,2) 共 d 0 兲 ,
冉 冊 冉 冊 A z(0) x
A z(3,4) x
⫽
x⫽⫾d 0
冉 冉
A z(1,2) 共 ⫾d 兲 ⫽A z(3,4) 共 ⫾d 兲 ,
a A z(1,2) 1 A z(1,2) ⫺i e x y
1 A z(1,2) a A z(1,2) ⫽ ⫺i e x y x⫽⫾d
冊 冊
,
共9兲
x⫽⫾d 0
dx
2
d 2 ¯A z(0) 共 x 兲 dx 2
⫹
⫹
2i
␦
2
¯A 共z1,2 兲 共 x 兲 ⫽⫺
2i
¯A 共z0 兲 共 x 兲 ⫽⫺ 2 ␦0
U 4 e , c l
4 U , c 0 l
共11兲
as well as the boundary conditions analogous to 共9兲. The second term in 共10兲 depends on both coordinates and satisfies homogeneous equations, with the right-hand side equal to zero:
2 ˜A z(1,2) x
2
2 ˜A z(0) x2
⫹
⫹
2 ˜A z(1,2) 2
⫹
2 ˜A z(0)
2i
y
y2
⫹
2i
␦2
˜A z(1,2) ⫽0, 共12兲
˜A z(0) ⫽0. 2 ␦0
The boundary conditions for ˜A z (x,y) at the interfaces between the layers (x⫽⫾d 0 ) are similar to 共9兲; at the outer surface, however, ˜A z (⫾d,y)⫽0. Our boundary conditions for functions ¯A z (x) and ˜A z (x,y) are approximations. They reflect the fact that without a voltage applied to the stripe the magnetic flux is concentrated within the sandwich. When a current is flowing through the stripe, the external flux must be accounted for. Since dⰆb, the component of the magnetic induction normal to the film surface, ¯B x ⫽ ¯A z / y, is negligibly small and the vector potential ¯A can be considered independent of y. The combination of these two parts in the vector potential enables us to develop a 2-D model for the fields intrinsic to the sandwich as well as account for the external flux. This approach is found to accurately reproduce the ‘‘air-core’’ inductance of the stripe in the absence of magnetization. Boundary conditions at the edges, y⫽⫾b/2, apply to the total vector potential. These conditions are determined by the physical layout of the edges and will be discussed in the next section. We begin by solving Eqs. 共11兲 for ¯A z (x). The solutions can be written in the form ¯A z(0) ⫽C 0 cosh共 ␣ 0 x 兲 ⫹D 0 sinh共 ␣ 0 x 兲 ⫺ p, ¯A z(1) ⫽C 1 cosh关 ␣ 1 共 x⫺d 0 兲兴 ⫹D 1
.
⫻sinh关 ␣ 1 共 x⫺d 0 兲兴 ⫺ p,
x⫽⫾d
In the structures of interest here the width is much larger than the thickness, dⰆb, and there is a small parameter in the problem, d/bⰆ1, which is used to facilitate the analysis below. Equations 共5兲 and 共6兲 as well as the boundary conditions are linear. Therefore, we can use the superposition principle, seeking a solution to Eqs. 共5兲 and 共6兲 as a sum of two parts: ¯ 共 x 兲 ⫹A ˜ 共 x,y 兲 . A共 x,y 兲 ⫽A
d 2 ¯A z(1,2) 共 x 兲
777
共10兲
The first term in 共10兲 does not depend on the y coordinate and satisfies the inhomogeneous one-dimensional equations with a nonzero right-hand side,
共13兲
¯A z(2) ⫽C 2 cosh关 ␣ 1 共 x⫹d 0 兲兴 ⫹D 2 sinh关 ␣ 1 共 x⫹d 0 兲兴 ⫺p, where
␣ 20 ⫽⫺
2i
␦ 20
,
␣ 21 ⫽⫺
2i
␦
2
,
p⫽⫺
icU , l
共14兲
and the coefficients C j , D j are determined from the boundary conditions. It is easy to see that the system under con¯ (⫺x), from which folsideration has a symmetry, ¯A (x)⫽A lows that D 0 ⫽0. For the external regions ‘‘3’’ and ‘‘4’’, where ⫽0, the solution is less straightforward. It is well known 共see, e.g.,
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J. Appl. Phys., Vol. 89, No. 1, 1 January 2001
Sukstanskii, Korenivski, and Gromov
Ref. 21兲 that the vector potential calculated for any structure with at least one infinite dimension diverges logarithmically at infinity. The standard approach to avoid this divergence is to use a cut-off distance of the order of the characteristic size of the structure, in our case the length of the film, l. We shall, however, use an approach,22 in which the external vector potential is taken to be proportional to the three-dimensional vector potential of a current carrying sheet, since in our case dⰆb:
¯A z(3) ⫽A ¯ z(4) ⫽C 3
再 冋
¯A z(3,4) 共 d 兲 ⬇C 3 1⫹ln
冊
2l , b
¯B (3,4) 共 ⫾d 兲 ⬇⫺C 3 y
. 共17兲 b
共18兲
p sinh ␣ 0 d 0 , ⌬
⫺d
dx ⬘
b/2
⫺b/2
dy ⬘
冕
l/2
⫺l/2
dz ⬘
␦共 x⬘兲 关共 x⫺x ⬘ 兲 ⫹ 共 y⫺y ⬘ 兲 2 ⫹ 共 z⫺z ⬘ 兲 2 兴 1/2 2
共15兲
at y⫽0, z⫽0. Integrating in 共15兲, one obtains for the magnetic potential outside the film22
冋
册冎
共16兲
.
1/2 where Q 0 ⫽(k 2 ⫺2i ␦ ⫺2 and Q 1 ⫽(k 2 ⫺2i ␦ ⫺2 ) 1/2. Substi0 ) tuting 共20兲 in the boundary conditions ˜A z (⫾d,y)⫽0, we obtain a homogeneous set of linear equations for coefficients a j and b j :
a 0 cosh共 Q 0 d 0 兲 ⫹b 0 sinh共 Q 0 d 0 兲 ⫽a 1 , a 0 cosh共 Q 0 d 0 兲 ⫺b 0 sinh共 Q 0 d 0 兲 ⫽a 2 ,
e Q 0 关 a 0 sinh共 Q 0 d 0 兲 ⫹b 0 cosh共 Q 0 d 0 兲兴 ⫽Q 1 b 1 ⫹
ak a , 1 共21兲
e Q 0 关 ⫺a 0 sinh共 Q 0 d 0 兲 ⫹b 0 cosh共 Q 0 d 0 兲兴 ak a , 2
a 1 cosh共 Q 1 d 1 兲 ⫹b 1 sinh共 Q 1 d 1 兲 ⫽0, a 2 cosh共 Q 1 d 1 兲 ⫺b 2 sinh共 Q 1 d 1 兲 ⫽0. Setting the determinant of 共21兲 equal to 0, the following equation for k is obtained:
where  ⫽ ␣ 0 e / ␣ 1 ⫽( 0 e / 1 ) , 1/2
2e Q 20 ⫹Q 21 coth2 共 Q 1 d 1 兲
⌬⫽cosh ␣ 0 d 0 cosh ␣ 1 d 1 ⫹  sinh ␣ 0 d 0 sinh ␣ 1 d 1
冉
d
⫽Q 1 b 2 ⫹
p C 1 ⫽C 2 ⫽ cosh ␣ 0 d 0 , ⌬
D 1 ⫽⫺D 2 ⫽
冕
册
Substituting 共11兲 and 共17兲 in 共9兲 and solving the resulting system of linear equations for the coefficients C j and D j , we obtain p C 0⫽ , ⌬
⫻
冕
l bl 2x 共共 l 2 ⫹b 2 ⫹4x 2 兲 1/2⫹b 兲共共 l 2 ⫹b 2 ⫹4x 2 兲 1/2⫹l 兲 ln arctan ⫺ 2 2 2 1/2 2 2 1/2 2 2b b 2x 共 l ⫹b 2 ⫹4x 2 兲 1/2 共共 l ⫹b ⫹4x 兲 ⫺b 兲共 b ⫹4x 兲
This expression correctly reproduces the magnetic field at infinity and results in a nondiverging vector potential for a finite-length stripe. The numerical error introduced by approximating the external flux in this way is typically within 10%.23 We need the values of the potentials ¯A z(3) , ¯A z(4) and their derivatives, ¯B (3,4) ⫽ ¯A z(3,4) / x, at x⫽⫾d. Since dⰆb y Ⰶl, we can use the following approximations:
冉
¯A (3,4) 共 x,y,z 兲˜
冊
␣1 2l ⫹ b ln ⫹1 共 cosh ␣ 0 d 0 sinh ␣ 1 d 1 e b ⫹  sinh ␣ 0 d 0 cosh ␣ 1 d 1 兲 .
⫹2 e Q 0 Q 1 coth共 Q 1 d 1 兲 coth共 2Q 0 d 0 兲 ⫺ 共19兲
Let us now proceed to solving Eqs. 共12兲 for ˜A z (x,y). Taking the solution in all the layers to be proportional to exp(iky), we seek ˜A z (x,y) in the form ˜A z(0) ⫽ 兵 a 0 cosh共 Q 0 x 兲 ⫹b 0 sinh共 Q 0 x 兲 其 :e iky ,
˜A z(2) ⫽ 兵 a 2 cosh关 Q 1 共 x⫹d 0 兲兴 ⫹b 2 sinh关 Q 1 共 x⫹d 0 兲兴 其 :e iky ,
共20兲
a 2 2 k ⫽0. 共22兲
Equation 共22兲 is a dispersion relation implicitly defined k as a function of frequency. In the general case, this transcendental equation cannot be solved analytically, since Q 0,1 are functions of k and . However, a numerical solution of 共22兲 for typical sandwich parameters shows that 兩 kd 1 兩 Ⰶ1. Using this condition, 共22兲 can be simplified to obtain the following expression for k:
˜A z(1) ⫽ 兵 a 1 cosh关 Q 1 共 x⫺d 0 兲兴 ⫹b 1 sinh关 Q 1 共 x⫺d 0 兲兴 其 :e iky ,
冉 冊
k⫽⫾
再
2i 2e d 0 d 21 ⫺ ␦ 20 共 e d 1 ⫹d 0 兲 d 1␦ 0 d 0 共 2e 2 ⫺ 2a 兲
冎
1/2
.
共23兲
It should be noted that in the low-frequency limit, →0, 共23兲 reduces to a purely imaginary value, k ⯝⫾i 关 (0)d 0 d 1 兴 1/2.
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J. Appl. Phys., Vol. 89, No. 1, 1 January 2001
Sukstanskii, Korenivski, and Gromov
779
Since the dispersion relation 共22兲 has two solutions, k and ⫺k, a general expression for the function ˜A z (x,y) must contain not only the term proportional to exp(iky) but also a term proportional to exp(⫺iky). For example, ˜A z (x,y) in the conductor should have the form ⫹ iky ˜A z(0) ⫽ 兵 a ⫹ 0 cosh共 Q 0 x 兲 ⫹b 0 sinh共 Q 0 x 兲 其 e ⫺ ⫺iky ⫹兵a⫺ . 0 cosh共 Q 0 x 兲 ⫹b 0 sinh共 Q 0 x 兲 其 e
共24兲
According to 共23兲, the real and imaginary parts of k have the same sign. Without a loss of generality, hereafter we consider Re(k)⬎0, Im(k)⬎0. ⫺ Coefficients a ⫺ 0 and b 0 in 共24兲 共and corresponding co⫺ ⫺ efficients a 1,2 and b 1,2 in the magnetic layers兲 are generally ⫹ different from a ⫹ 0 , b 0 . Solving these equations and taking into account the central symmetry of our structure, ˜A z (⫺x, ⫾ ˜ z (x,y), all coefficients a ⫾ ⫺y)⫽A j , b j can be expressed in ⫹ terms of one, for example, a 0 : ⫹ b⫾ 0 ⫽⫾a 0
冋
1 Q cosh共 Q 0 d 0 兲 cosh共 Q 1 d 1 兲 ⌬⫾ 1
⫹ e Q 0 sinh共 Q 0 d 0 兲 sinh共 Q 1 d 1 兲
册
ka ⫾ cosh共 Q 0 d 0 兲 sinh共 Q 1 d 1 兲 , ⫺ ⫹ a⫹ 1 ⫽a 2 ⫽a 0
e Q 0 sinh共 Q 1 d 1 兲 , ⌬⫺
⫹ ⫹ a⫺ 1 ⫽a 2 ⫽a 0
e Q 0 sinh共 Q 1 d 1 兲 , ⌬⫹
⫺ ⫹ b⫹ 1 ⫽⫺b 2 ⫽⫺a 0
e Q 0 cosh共 Q 1 d 1 兲 , ⌬⫺
⫹ ⫹ b⫺ 1 ⫽⫺b 2 ⫽⫺a 0
e Q 0 cosh共 Q 1 d 1 兲 , ⌬⫹
共25兲
ternal air-path reluctance is much larger than the leakage reluctance through the conductor, so the fringing magnetic flux is negligible. In our case this means that the average value of the component of the magnetic induction normal to the film edges is zero:
冕
冉 冊 冕
d
0
dxB y x,⫾
b ⫽ 2
0
⫺d
冉 冊
dxB y x,⫾
b ⫽0. 2
共27兲
Such boundary conditions are similar to the approximation known as the ‘‘effective magnetic wall,’’ first proposed in Ref. 24 for modeling the magnetostatic spin waves in films of a finite cross section 共see also Ref. 25兲. Since B y ⫽⫺ A z / x, we obtain from 共27兲 the following relations for the vector potential at y⫽⫾b/2: 共26兲
⌬ ⫾ ⫽Q 1 sinh共 Q 0 d 0 兲 cosh共 Q 1 d 1 兲 ⫹ e Q 1 cosh共 Q 0 d 0 兲 sinh共 Q 1 d 1 兲 ⫾
FIG. 2. Flux decay length as a function of frequency.
ka sinh共 Q 0 d 0 兲 sinh共 Q 1 d 1 兲 .
Thus, we can write down an expression for ˜A z (x,y) in the layers, containing only one undefined coefficient, a ⫹ 0 . Since all the coefficients in ¯A z (x) in 共13兲 are completely defined in 共18兲 and 共19兲, the expressions for the total vector ¯ z (x)⫹A ˜ z (x,y) contain only one undepotential A z (x,y)⫽A fined coefficient, which is to be determined from boundary conditions at the edges.
III. IMPEDANCE OF A ‘‘NARROW’’ SANDWICH
As was mentioned in the Introduction, we will be considering two geometries. Let us begin with the ‘‘open’’ structure 关Fig. 1共a兲兴 having no magnetic film closing the edges 共no flanges兲. We note 共see, e.g., Ref. 16兲 that the ex-
冉 冊 冉 冊
A 0,⫾
b b ⫽A d,⫾ 2 2
¯ 共 d 兲 ⫺A ¯ 共 0 兲. or ˜A 共 0,⫾b/2兲 ⫽A 共28兲
Substituting ¯A z (x) and ˜A z (x,y) obtained in the previous section, we obtain a ⫹ 0 : p a⫹ 0 ⫽ 关 cosh共 ␣ 0 d 0 兲 cosh共 ␣ 1 d 1 兲 ⌬ ⫹  sinh共 ␣ 0 d 0 兲 sinh共 ␣ 1 d 1 兲 ⫺1 兴 sec
冉 冊
kb . 2
共29兲
Parameter k in Eq. 共23兲 is complex, k⫽k ⬘ ⫹ik ⬙ . There⫹ fore, a ⫹ 0 decreases exponentially with b, a 0 ⬃exp(⫺b/2), ⫺1 where the quantity ⫽(k ⬙ ) is the effective decay length of field at the edges. As seen from Fig. 2, where is plotted as a function of frequency, the effective decay length monotonically decreases with increasing frequency. In the lowfrequency limit, →0, the decay length obtained from 共23兲 reaches its maximum value (0)⫽( (0)d 0 d 1 ) 1/2, which coincides with the decay length calculated in Ref. 5 in a dc model. We now have all the coefficients defining through 共13兲 ¯ z (x) and 共20兲 the total vector potential A z (x,y)⫽A ˜ ⫹A z (x,y). They all are proportional to parameter p and, consequently, to the applied voltage, U. Using 共7兲, we can
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readily calculate the total current in the stripe, I tot , which can be written as a sum of two terms related to ¯A z (x) and ˜A z (x,y): I tot⫽¯I ⫹˜I , ¯I ⫽
再
共30兲
冋
再 冉 冊
冉 冊 册冎 ␣ 1d 1 2
,
共31兲
冋
冉 冊冋
2bU kb f l⌬ 2 ⫹
1 2bU 0 sinh共 ␣ 0 d 0 兲 ⫹ cosh共 ␣ 0 d 0 兲 sinh共 ␣ 1 d 1 兲 l⌬ ␣ 0 ␣1 ⫹2  sinh共 ␣ 0 d 0 兲 sinh2
Z open⫽
˜I ⫽
冉 冊册
1 Q 1d 1 cosh共 ␣ 0 d 0 兲 tanh Q1 2
⫻ 关 cosh共 ␣ 0 d 0 兲 cosh共 ␣ 1 d 1 兲 ⫹  sinh共 ␣ 0 d 0 兲 sinh共 ␣ 1 d 1 兲 ⫺1 兴 ,
共32兲
where f (x)⫽tan(x)/x. The impedance of the ‘‘open’’ structure, Z open⫽U/I tot , is then
冉 冊册
1 ␣ 1d 1 l⌬ 0 sinh共 ␣ 0 d 0 兲 ⫹ cosh共 ␣ 0 d 0 兲 sinh共 ␣ 1 d 1 兲 ⫹2  sinh共 ␣ 0 d 0 兲 sinh2 2b ␣ 0 ␣1 2 ⫹f
0 sinh共 Q 0 d 0 兲 Q0
冋
冉 冊 册冎
kb 0 1 Q 1d 1 sinh共 Q 0 d 0 兲 ⫹ cosh共 ␣ 0 d 0 兲 tanh 关 cosh共 ␣ 0 d 0 兲 cosh共 ␣ 1 d 1 兲 ⫹  sinh共 ␣ 0 d 0 兲 sinh共 ␣ 1 d 1 兲 ⫺1 兴 2 Q0 Q1 2
⫺1
. 共33兲
The inductance, L⫽⫺Im(Z)/ , and the quality factor, Q⫽ 兩 Im(Z) 兩 /Re(Z) of the ‘‘open’’ structure are plotted in Figs. 3共a兲 and 3共b兲, respectively, for different widths and for the following physical parameters of the sandwich: d 0 ⫽0.25 m, d 1 ⫽0.2 m, l⫽1 mm, 0 ⫽5⫻1017 s⫺1 , 0 ⫽9⫻1015 s⫺1 , 4 M 0 ⫽2.1 kG, H a ⫽17 Oe, g⫽2⫻107 (s•Oe) ⫺1 , and r ⫽10⫺2 . For these parameters, the FMR frequency is ⯝2 ⫻1.9 GHz. The inductance monotoni* cally decreases with frequency, and its dc value increases with increasing width, which indicates that the edge effects play a dominant role. The maximum value of the quality factor does not exceed 1.2 for a 20 m wide stripe. It is interesting to note that L and Q are monotonic functions at the FMR frequency. The inductance as a function of width is shown in Fig. 4 for several frequency values. At low frequencies 共curve 1 in Fig. 4兲, the inductance has a pronounced maximum at b ⯝20 m (L⬇3.6 nH兲. At higher frequencies the peak in L is less pronounced. These effects are due to the contribution from the edges: for widths small compared to the characteristic flux decay length, , the main part of the screening current flows in the conductor, crowding at the edges, and induces large dissipation. Let us continue with analyzing the ‘‘closed’’ structure shown in Fig. 1共b兲. Due to the low magnetic reluctance of the flanges the magnetic flux is largely contained within the flanges, and leakage and fringing are small. As a result, the magnetic field distribution within the film can be considered as uniform along y. This means that within the film B x ⫽ A z / y⬇0, and we can omit the second term in 共10兲. The total current is then I tot⫽¯I , and the impedance has the form
Z closed⫽
l⌬ ␣ 0  关 cosh共 ␣ 0 d 0 兲 sinh共 ␣ 1 d 1 兲 2b 0 ⫹  sinh共 ␣ 0 d 0 兲 cosh sinh共 ␣ 1 d 1 兲兴 ⫺1 .
共34兲
In Figs. 5共a兲 and 5共b兲 we plot the frequency dependence of the inductance, L, and the quality factor, Q, for the ‘‘closed’’ structure for the same parameters as in Figs. 3共a兲 and 3共b兲. The behavior of L and Q for the ‘‘closed’’ structure differs substantially from that for the ’’open’’ structure.
FIG. 3. Inductance 共a兲 and quality factor 共b兲 versus frequency for the ‘‘open’’ structure for different values of the stripe width.
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J. Appl. Phys., Vol. 89, No. 1, 1 January 2001
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781
FIG. 4. Inductance of the ‘‘open’’ structure versus width at fixed frequencies.
Static values of the inductance are much higher and increase with decreasing width. For a fixed value of b the inductance is almost independent of frequency at ⭐0.8 . In the * FMR region the inductance goes through zero at ⯝ , * reproducing the behavior of the effective permeability, e . The quality factor is much higher than that for the structure without flanges. It has a pronounced maximum (Q⬇5 for the parameters used in Fig. 5兲, almost independent of width, which is natural since L⬃b ⫺1 and R⫽Re(Z)⬃b ⫺1 . In order to determine the main contributions to dissipation in the ‘‘closed’’ inductor, we have varied in 共34兲 the
FIG. 6. Inductance 共a兲 and quality factor 共b兲 for the ‘‘closed’’ structure for different values of the conductivity and damping constant of the magnetic material.
conductivity and the intrinsic damping constant of the magnetic material, with the other parameters the same as in Figs. 2–5. The result is shown in Fig. 6. As the conductivity and damping constant are reduced, Q increases and the point at which it is maximum shifts to high frequencies. We find that for typical geometrical and physical parameters, corresponding to known soft magnetic alloys of high magnetization and high resistivity 共Fe–X–N, Fe–B–Si, etc.兲, the dissipation is almost entirely in the magnetic films and, for the parameters chosen in Fig. 6, is a combined effect with approximately equally strong screening currents and intrinsic relaxation. We also find that, for typical geometries and for a given strength of intrinsic magnetic damping 共with, e.g., r ⫽10⫺2 ), there is little benefit with increasing the resistivity of the magnetic material much beyond 100 ⍀cm, since then the device efficiency is largely determined by intrinsic magnetic relaxation. IV. CONCLUSIONS
FIG. 5. Inductance 共a兲 and quality factor 共b兲 versus frequency for the ‘‘closed’’ structure for different values of the stripe width.
By solving the coupled Maxwell and Landau–Lifshitz equations, the magnetic field and current distributions are obtained for a ‘‘narrow’’ ferromagnetic sandwich stripe, from which the stripe impedance is calculated. The highfrequency properties of the sandwich are shown to strongly depend on the structure of the edges. The inductance and the quality factor for the structure with flanges are much higher than those for the structure without flanges.
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In the ‘‘open’’ structure, screening currents in the conductor associated with the leakage flux through the conductor are the main factor limiting the inductive response of the device. This effect is especially pronounced in narrow sandwich stripes, in which the characteristic flux decay length, , is comparable with the stripe width, b. The overall inductive performance of the ‘‘open’’ inductor is low. The open structure is therefore unfavorable for high-frequency applications. In the ‘‘closed’’ structure the magnetic flux in the conductor is small due to the flux closure at the edges. As a result, eddy currents in the conductor are small and the dissipation in the structure is mainly caused by screening and relaxation in the magnetic material. Since the flux is in the plane of the magnetic film, screening currents can be controlled by varying the thickness of the film, and the device efficiency is ultimately limited by intrinsic magnetic relaxation. Reduced demagnetization and screening result in a strong inductive response, superior to that of air-core designs. ACKNOWLEDGMENTS
Financial support from the Royal Swedish Academy of Sciences 共KVA兲 and the Swedish Research Council for Engineering Sciences is gratefully acknowledged. 1
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