- Email: [email protected]
Nucleation of planar magnetization in ultra-thin magnetic films
~ ELSEVIER
Journal of Magnetism and Magnetic Materials 168 (1997) 9 14
Jeunaalof magatl¢ materials
Nucleation of planar magnetization in ultra-thin magnetic films S.T. C h u i * Bartol Research Institute, Universi~ of Delaware, Newark, DE 19716, USA
Received 5 March 1996;revised 19 August 1996
Abstract
We study theoretically the magnetization reversal of Heisenberg spins in a 2D plane under an external reversing field at finite temperatures. Nucleation is observed in a Monte Carlo simulation for an array of spins interacting with exchange, anisotropic and dipolar interactions. The nucleus is elliptical in shape. The eccentricity is smaller the larger the dipolar interaction is with respect to the exchange. The coercive field seems to depend linearly on the temperature. PACS: 75.60.Ch; 75.10.Hk; 75.70.-i; 64.60.Qb Keywords: Nucleation; Magnetization; Thin film; Simulation
There has been much interest recently in understanding the fundamental physics of magnetic films from 1 to 100 layers [1, 2]. A question of importance is how the spins reverse in the presence of an external field. This issue has been most studied at zero temperature in 'thick' films [3], but very little work has been done at finite temperature in ultrathin films where nucleation of domains through thermal fluctuation becomes possible. Depending on the material parameters, the magnetization can lie parallel or perpendicular to the plane [4, 5]. We have previously investigated the physics of spin reversal when the spins are perpendicular to the film [6]. In this paper, we study the magnetization reversal when the spins lie in direc-
*Corresponding author. Tel.: + 1 (302) 831 8115; fax: + 1 (302) 831 1842;e-mail: [email protected].
tions parallel to the film. Our model consists of Heisenberg spins in a 2D plane interacting with long-range dipolar, short-range exchange and uniaxial anisotropy (in the film plane) interactions at finite temperatures. In real systems, impurity [7] and edge effects can also be important. To elucidate the physics of this complex phenomena it is important to understand one effect at a time. For this reason impurity and edge effects are not considered in this paper. We observe nucleation of domains in Monte Carlo simulations. At low temperatures, the shape of the nucleus is elliptical, the deviation from a circular shape becomes larger, the larger the dipolar interactions is compared with the exchange. Relaxation calculations at zero temperature suggests that dipolar interaction favors a nucleus with small eccentricity, whereas the exchange interaction favors a circular nucleus. The final shape is determined as a compromise between the two.
0304-8853/97/$17.00 @ 1997 ElsevierScienceB.V. All rights reserved PII S0304- 88 53(96)00690- 7
t0
X T Chui / Journal o/Magnetism and Magnetic' Materials 168 (1997) 9-- 14
Analytic arguments suggest an eccentricity of the order of x/-J-A/M2ao, where the effective spin wave 'gap' A = (Hr - H)/2 is half the difference between the coherent rotation field Hr = 2K and the external magnetic field H, J is the effective exchange, M the magnetization and a0 the lattice spacing. At zero temperature, switching occurs as the coherent rotation limit is approached and A approaches zero; the nucleus becomes strip-like with the eccentricity equal to zero, consistent with previous study of thick films at zero temperature [8, 9]. This result seems to be consistent also with the experimental studies near the reorientation transition when the effective gap is exceedingly small [10]. The coercive field is found to depend linearly on the temperature over a substantial range. This is in contrast with the squared root temperature dependence for small particles. An explanation is proposed. We now describe our results in detail. We choose our coordinates so that the film is in the xy plane. For transition metals, the exchange interaction per spin is much larger than the dipolar coupling and the crystalline anisotropy interaction. We thus assume all spins in the direction perpendicular to the film to be parallel to each other and approximate it by an effective block spin. These block spins will then interact with each other with renormalized interactions. For a film of n layers and block spins with 12 columns in the xy directions, the renormalized long-range dipolar interaction is approximately proportional to nZl, whereas the anisotropy and the exchange interactions are increased by a factor of nl 2 and ln, respectively [3, 5]. In addition, there will be short range multipolar interactions. If the block-spin size is so large that the multipolar interactions become comparable to the exchange interaction, then their effect need to be included for an accurate calculation. A simple estimate suggests that the leading multipolar term is of the order of (block spin size~block spin separation) 2 ~ 0.25 times the block dipolar interaction g. In the present work the exchange is about two times g. Thus, we have neglected the multipolar term. In any case, we expect the physics to remain unchanged even when the multipolar correction is incorporated. The use of block spins with their approximate renormalized interaction makes it possible to simulate domain nucleation in
a reasonable size system. The total energy of the system is assumed to be Ev = Eo + H ~ i Sxl, where the 'internal' interaction energy between the block spins is Eo 0.5Eij=xyz,RR' Vis(R - R')S~(R)Sj(R'). The sum is over the 2D positions R , R ' of the block spins in the plane. The potential V = Va + Ve + Va is the sum of the dipolar energy V o i j ( R ) = gDo(R), Do(R) = 6 i j / R 3 - 3RiRfRS; the exchange energy Ve = - Jb(R = R' + d)b~s; and the anisotropy energy Va = -- 2K6(R = R')f~xbj~. Here d denotes the nearest neighbours between the block spins. The simulation reported here follows our earlier work on the equilibrium finite temperature studies of the phase diagram in ultra-thin magnetic films [4, 5] where H = 0. It is carried out for a triangular lattice of 60 × 60 block spins in a rectangular region with aspect ratio xf3/2 under periodic boundary conditions. Under the assumption of periodic boundary conditions, each spin will interact with the other spins in the rectangular region as well as with other periodic images outside the rectangular region. The image contribution to the long-range dipolar potential is summed with the Ewald sum technique. In the present study, we start with the magnetization along the positive x direction and study the system in the presence of a reversing magnetic field. The MC simulation corresponds to solving the master equation for the probability distribution P(s,t) of the spin configuration labelled by s as a function of time t: =
dp dt = ~ [ - w(s ~ s')P(s, t) + w(s' ~ s)P(s', t)]. s"
Here the transition probabilities w are picked so that the condition of detailed balance is satisfied; i.e. w(s ~ s')/w(s' ~ s) = e x p ( - AE/kT) [11], where AE is the energy difference between spin configurations s and s'. This relaxational dynamics explores all paths from the initial to possible final configurations with rates determined by the free-energy barriers. If the free-energy barrier is high for a path, this path will be blocked in a simulation carried out over a reasonable but finite time. This path will become available when the free-energy barrier has decreased to a value comparable to the temperature. We thus expect this dynamics to produce the
S.T.
Chui / Journal of Magnetism and Magnetic Materials 168 (1997) 9-14
correct free energy of nucleation and the correct switching field but a prefactor in the nucleation rate that may be different from that of a stohastic Landau-Gilbert equation. In Fig. la-Fig, lc we show intermediate Monte Carlo configurations for fixed J(0.2), K(0.1) and increasing 9 (0.05, 0.1, 0.15). This clearly demonstrates the spin reversal through domain nucleation. The nucleus is not circular in shape; the larger the 9, the more stripe-like it looks. In Fig. la, the nucleus at the lower right-hand corner is still quite circular. In Fig. lc, the nucleus becomes very long. A similar question was discussed in the case of thick films at zero temperature, where a nucleus in the shape of a strip is predicted [8,9]. We discuss
. . . .
~ ~.--~_. . . .
=~--~ . . . . . . . . .
(a)
~
11
below how this effect can be understood at finite temperature in ultra-thin films. To describe the nucleus, we need a simple expression for the energy including the dipolar interaction. When the external field is equal to H, = 2K, the spins can be rotated together as a whole with no third-order increase in energy. This is the coherent rotation limit. The energy barrier becomes smaller, the closer is H to 2K. Inside the nucleus, the spins are not completely reversed but are only rotated by a small amount of the order of ~b~t = cos- I(H/2K) [12, 8, 9]. Even though it is difficult to write down a simple analytic expression for the dipolar interaction for a general spin configurations, because the deviation from the uniform alignment is small here,
,:':.~ ;-.,'i_
(b)
(c) Fig. 1. The o r i e n t a t i o n of the spins in the xy p l a n e at a t e m p e r a t u r e T = 0.1 for J = 0.2, K = 0.1 a n d g = 0.05, 0.1, 0.15 at i n t e r m e d i a t e times.
S.T Chui / Journal of Magnetism and Magnetic Materials 168 (1997) 9-14
12
the energy can be described analytically in terms of the Fourier transform of the spin deviation
Si(q) = ~I~Si(R) exp(iqR)/,4/N by the spin wave energy as E = Eo + ~ [A + 0.59Drr(q) + Je(q)]lSr(q)] 2 q + [A + Ha -
0.5gDxx(q)
-- 0.5gDyy(q) + Je(q)]lS~(q)l 2.
(1)
Here H a = 9 ~ R 0 . 7 5 / R 3, A = K - H ~ 2 , e(q) = 0.5~,3[1 - cos(q" d)]. Eo is the energy of the state with all spins aligned along the x-axis. Do{q)= ~R[COS(q" R) -- 1]Do(R ) is identical in form to the dynamical matrix of the 2D electron crystal [5, 13]. In the following, we shall ignore the deviation of the spins out of the xy plane. Because of the extra 'shape anisotropy field' Ha in the energy which flips the spins out of the plane, this process is much less likely. To investigate the nucleus, we are interested in the long wavelength limit where the spin configuration is slowly varying. We thus expand the energy as a power series in q and retain the lowest order term. We finally obtain E -- ~ [A + 0.25gc]~/ci + 1.5Jq2]lSr(q)l 2,
(2)
q
where q is the smallest of q + G for all reciprocal lattice vectors G. This reflects the discrete nature of the lattice. The long wavelength limit of the dipolar term is obtained from previous work on electron crystals [13]. Note that it is anisotropic and only of the first power in q. Previous studies on thick rims uses a three-dimensional expression for D (see Eq. 23 in [9]) which remains finite as the wave vector k approaches zero. The physics of the difference between the two and the three dimensional case is the same as that for plasmon dispersion in two and three dimensions. We can rewrite the expression for E in real space by rewriting iqj as the derivative 8j. We approximate Sy by the angle q5 of the spin with respect to the x-axis. The shape of the nucleus can then be determined from the Euler-Lagrange equation from extremizing the energy E. There are two length scales that determine the rate of change of qS. They are lj = x/J/~/A and lg -- g/A. If Ij >>lo then the energy change of the dipolar term 9/1j is always smaller than the energy change due to the exchange
term J/l 2. The dipolar term is not important and the nucleus will be circular. As the coherent rotation limit is approached, A approaches zero and eventually l0 becomes larger than ls. In that limit the rate of change in the x direction will be controlled by J, whereas that in the y direction will be controlled by g. We thus expect a nucleus ellipsoidal in shape with x dimension of the order of lj and y dimension of the order of 1o. As the temperature approaches zero, A ~ 0, the nucleus becomes a strip, consistent with previous results at zero temperature [8]. This picture is also consistent with the Monte Carlo result that the largest deviation from the circular shape comes from the largest 9. In the simulation, the nucleus is oriented at an angle of 120 ° with respect to the x-axis. This is consistent with Eq. (2) that the form of the dipolar energy is invariant under a 120 ° rotation. Presumably, in that case, one can gain maximum advantage from the external magnetic field. To further verify that this picture is correct, we have calculated numerically the energies of ellipsoidal nuclei of different eccentricities with the same area in a finite size lattice. This is carried out as follows. We focus on a lattice of 20 x 20 spins under periodic boundary conditions [14]. The major axis of the ellipse is set to 5 when the eccentricity is 0.2. At the beginning of the calculation, the spins inside the nucleus are rotated by a finite angle 0. The spins outside the nucleus are aligned along the x-axis. The system is then relaxed so that the local minima of the energy is sought with the quasiNewtom algorithm [15]. We avoid reaching the minima where all spins are aligned along the x-axis by fixing the orientation of the spins that are halfway inside the nucleus. In Fig. 2a, we show the energy of the nuclei of different eccentricities as a function of the rotation angle 0 when the exchange is zero. As expected, for a given eccentricity, a maximum energy barrier is reached at a finite rotation angle. Furthermore, this energy barrier is smaller, the smaller the eccentricity. In Fig. 2b we show the result of a similar calculation but now with the exchange set at a finite value (0.05). The smallest barrier now occurs at an intermediate eccentricity. An estimate of the coercive field can be determined from the threshold at which the nucleation
S. T. Chui / Journal of Magnetism and Magnetic Materials 168 (1997) 9-14
1.0 0.8 ~0.6
,e- . . . . . . . NO
EXCHANGE /
,."
~
0.20 ~, 0.18 b_ 0.16
_ "
~3
,.'",'2..7.:?.2.:~ .-
©
© >
0.4
•~
13
1! . . . .
H v~sg ( T / J = O . 5 ) ~
o.14
(D
0.2 0.0 0.0
~ 0.12
0.2
0.4 0.6 angle
(a)
0.8
.0
@.10 0.0
0.5
! .0
T/J or g/J
1.5
Fig. 3. The coercive field as a function of temperature for J = 0.2, K = 0.1. >-, c~ © c
1 0 1 ~ 5 0
.
0.0 (b)
.
.
0.5
~,r." t'//Z .
~.~-~
,
. . . .
1.0 1.5 angle
,
2.0
2.5
Fig. 2. The energy costs to create nuclei of eccentricities from 0.2 to 1 in increments of 0.2 as a function of the angle of rotation in radians; H = 0.39, K = 0.2, g = 0.2: (a) The exchange is set to zero, lines for increasing energy are for increasing eccentricities. (The dotted-dashed, short dashed, dotted, solid and long dashed lines are for eccentricities 0.2, 0.4, 0.6, 0.8 and 1, respectively.)(b) The exchange was set to 0.05. The line symbols are the same as in (a). The lowest energy barrier corresponds to an eccentricity of 0.6.
occurs during the finite simulation time (4000 M C steps/spin) [16]. The coercive field as a function of temperature and as a function of g is shown in Fig. 3. Over a substantial temperature range, the temperature dependence seems to be approximately linear. This linear temperature dependence can be understood as follows. At high temperatures, the external field at which nucleation occurs is smaller than that for coherent rotation. We thus expect A to be not close to zero and that the nucleus to be not as elongated. F o r simplicity we thus neglect the dipolar contribution. Following s t a n d a r d practice in the study of nucleation and instantons we employ the saddle point m e t h o d and a p p r o x i m a t e the 'pinning energy' E~ = - K cos2(~b) + H cos(~b) by expanding it
a b o u t its local m a x i m u m ~bM and obtain Ee(~b) Ee(~aM) -- fl(q~ -- (aM)z/2, where ]~ = ( n 2 - 4KE)/2K. We assume the angle q5 to be only a function of the radial distance and obtain the E u l e r - L a g r a n g e equation (0rq~)/r + 02q~ _ fl(~b _ ~bM) = 0. The solution can be expressed in terms of Bessel functions. With the b o u n d a r y conditions that the solution is well-behaved at the origin and that the angle is zero at the b o u n d a r y R of the nucles, we get the solution qS(r) = ~bM[1 Jo(~-flr)/Jo(xfflR)]. the nucleus is also a variational tremizing the energy with respect the condition [17] O6,E~b'lr=nstituing in the expression for E and -
-
The radius R of parameter. Exto it, we obtain EIr=R = 0. SubqS, we obtain the
result that x/fiR = c, where J'o(C)/Jo(c)= 1. The energy of the barrier EM can be obtained by direct substitution and we obtain EM = qSMf, 2 where the f a c t o r f i s given by the integral
f = f l u[(J'°(u)/J°(u))2 + 0"5(J°(u)/J°(c))Z]du"
W h e n H is close to 2K, 4)2 is proportional to 2 K - H. We finally obtain the result that EM is p r o p o r t i o n a l to 2K - H . The coercive field Hc is determined by the condition that the Boltzman factor exp(-Ev,/kBT) is substantial. Substituing everything back, we obtain a coercive field given by Hc = 2K - C1T ln(C2/t), where C1,2 are constants and t is the simulation time. As the temperature approaces zero, the nucleus becomes strip-like. We
14
S.Z Chui / Journal of Magnetism and Magnetic Materials 168 (1997) 9-14
expect a deviation from this linear temperature dependence. To summarize, because of the competition between the dipolar and the exchange interaction, nuclei formed during in-plane magnetization reversal are ellipsoidal in shape. These domains are more stripe-like, the smaller the effective in-plane anisotropy is. For multilayers with strong interplane coupling, the effective dipolar coupling can be modified. We hope to investigate this in more detail in future studies.
Acknowledgements This work is supported by the Office of Naval Research under contract N00 014-94-1-0213.
References [1] G. Prinz, Science 1092 (1990) 250. I-2] B. Heinrich and J.F. Cochran, Adv. Phys. 42 (1993) 524. 1-3] See, for example, H.N. Bertram and J.G. Zhu, in: Solid State Physics, Vol. 46, Eds. H. Ehrenreich and D. Turnbull (Academic Press, New York, 1992), p. 271.
[4] For a recent summary from a theoretical perspective, see S.T. Chui, Phys. Rev. B 50 (1994) 12559. [5] S.T. Chui, Phys. Rev. Lett. 74 (1995) 3896. [6] S.T. Chui, J. Appl. Phys. 79 (1996) 4951. [7] S.T. Chui, Phys. Rev. B 51 (1995) 250. [8] R.M. Gotdstein and M.W. Muller, Phys. Rev. B 2 (1970) 4585. [9] M.-H. Yang and M.W. Muller, IEEE Trans. Magn. 7 (1970) 705. [10] R. Allenspach and A. Bischof, Phys. Rev. Lett. 69 (1992) 3385. [11] K. Binder and D.W. Heermann, Monte Carlo Simulation in Statistical Physics, 2nd edn. (Springer, New York, 1992). [12] S.T. Chui, J. Appl. Phys. 78 (1995) 3965. [13] S.T. Chui (Ed.), Physics of the 2D electron solid (International Press, Boston, 1995). [14] The difference in the Monte Carlo results for the 20 x 20 and the 60 x 60 systems is small. The relaxation calculation is quite demanding in that it requires the inversion of large (N × N by N x N) matrices. For this reason we have picked the 20 x 20 system. [15] D. Kahaner et al., subroutine UNCMND, in: Numerical Methods and Software (Prentice-Hall, Englewood Cliffs, NJ, 1989) Ch. 9. El6] The dependence on the simulation time is expected to be only logarithmic. [17] J. Matthews and R.L. Walker, in: Mathematical Methods of Physics (Benjamin, New York, 1965) p. 312.
Journal of Magnetism and Magnetic Materials 168 (1997) 9 14
Jeunaalof magatl¢ materials
Nucleation of planar magnetization in ultra-thin magnetic films S.T. C h u i * Bartol Research Institute, Universi~ of Delaware, Newark, DE 19716, USA
Received 5 March 1996;revised 19 August 1996
Abstract
We study theoretically the magnetization reversal of Heisenberg spins in a 2D plane under an external reversing field at finite temperatures. Nucleation is observed in a Monte Carlo simulation for an array of spins interacting with exchange, anisotropic and dipolar interactions. The nucleus is elliptical in shape. The eccentricity is smaller the larger the dipolar interaction is with respect to the exchange. The coercive field seems to depend linearly on the temperature. PACS: 75.60.Ch; 75.10.Hk; 75.70.-i; 64.60.Qb Keywords: Nucleation; Magnetization; Thin film; Simulation
There has been much interest recently in understanding the fundamental physics of magnetic films from 1 to 100 layers [1, 2]. A question of importance is how the spins reverse in the presence of an external field. This issue has been most studied at zero temperature in 'thick' films [3], but very little work has been done at finite temperature in ultrathin films where nucleation of domains through thermal fluctuation becomes possible. Depending on the material parameters, the magnetization can lie parallel or perpendicular to the plane [4, 5]. We have previously investigated the physics of spin reversal when the spins are perpendicular to the film [6]. In this paper, we study the magnetization reversal when the spins lie in direc-
*Corresponding author. Tel.: + 1 (302) 831 8115; fax: + 1 (302) 831 1842;e-mail: [email protected].
tions parallel to the film. Our model consists of Heisenberg spins in a 2D plane interacting with long-range dipolar, short-range exchange and uniaxial anisotropy (in the film plane) interactions at finite temperatures. In real systems, impurity [7] and edge effects can also be important. To elucidate the physics of this complex phenomena it is important to understand one effect at a time. For this reason impurity and edge effects are not considered in this paper. We observe nucleation of domains in Monte Carlo simulations. At low temperatures, the shape of the nucleus is elliptical, the deviation from a circular shape becomes larger, the larger the dipolar interactions is compared with the exchange. Relaxation calculations at zero temperature suggests that dipolar interaction favors a nucleus with small eccentricity, whereas the exchange interaction favors a circular nucleus. The final shape is determined as a compromise between the two.
0304-8853/97/$17.00 @ 1997 ElsevierScienceB.V. All rights reserved PII S0304- 88 53(96)00690- 7
t0
X T Chui / Journal o/Magnetism and Magnetic' Materials 168 (1997) 9-- 14
Analytic arguments suggest an eccentricity of the order of x/-J-A/M2ao, where the effective spin wave 'gap' A = (Hr - H)/2 is half the difference between the coherent rotation field Hr = 2K and the external magnetic field H, J is the effective exchange, M the magnetization and a0 the lattice spacing. At zero temperature, switching occurs as the coherent rotation limit is approached and A approaches zero; the nucleus becomes strip-like with the eccentricity equal to zero, consistent with previous study of thick films at zero temperature [8, 9]. This result seems to be consistent also with the experimental studies near the reorientation transition when the effective gap is exceedingly small [10]. The coercive field is found to depend linearly on the temperature over a substantial range. This is in contrast with the squared root temperature dependence for small particles. An explanation is proposed. We now describe our results in detail. We choose our coordinates so that the film is in the xy plane. For transition metals, the exchange interaction per spin is much larger than the dipolar coupling and the crystalline anisotropy interaction. We thus assume all spins in the direction perpendicular to the film to be parallel to each other and approximate it by an effective block spin. These block spins will then interact with each other with renormalized interactions. For a film of n layers and block spins with 12 columns in the xy directions, the renormalized long-range dipolar interaction is approximately proportional to nZl, whereas the anisotropy and the exchange interactions are increased by a factor of nl 2 and ln, respectively [3, 5]. In addition, there will be short range multipolar interactions. If the block-spin size is so large that the multipolar interactions become comparable to the exchange interaction, then their effect need to be included for an accurate calculation. A simple estimate suggests that the leading multipolar term is of the order of (block spin size~block spin separation) 2 ~ 0.25 times the block dipolar interaction g. In the present work the exchange is about two times g. Thus, we have neglected the multipolar term. In any case, we expect the physics to remain unchanged even when the multipolar correction is incorporated. The use of block spins with their approximate renormalized interaction makes it possible to simulate domain nucleation in
a reasonable size system. The total energy of the system is assumed to be Ev = Eo + H ~ i Sxl, where the 'internal' interaction energy between the block spins is Eo 0.5Eij=xyz,RR' Vis(R - R')S~(R)Sj(R'). The sum is over the 2D positions R , R ' of the block spins in the plane. The potential V = Va + Ve + Va is the sum of the dipolar energy V o i j ( R ) = gDo(R), Do(R) = 6 i j / R 3 - 3RiRfRS; the exchange energy Ve = - Jb(R = R' + d)b~s; and the anisotropy energy Va = -- 2K6(R = R')f~xbj~. Here d denotes the nearest neighbours between the block spins. The simulation reported here follows our earlier work on the equilibrium finite temperature studies of the phase diagram in ultra-thin magnetic films [4, 5] where H = 0. It is carried out for a triangular lattice of 60 × 60 block spins in a rectangular region with aspect ratio xf3/2 under periodic boundary conditions. Under the assumption of periodic boundary conditions, each spin will interact with the other spins in the rectangular region as well as with other periodic images outside the rectangular region. The image contribution to the long-range dipolar potential is summed with the Ewald sum technique. In the present study, we start with the magnetization along the positive x direction and study the system in the presence of a reversing magnetic field. The MC simulation corresponds to solving the master equation for the probability distribution P(s,t) of the spin configuration labelled by s as a function of time t: =
dp dt = ~ [ - w(s ~ s')P(s, t) + w(s' ~ s)P(s', t)]. s"
Here the transition probabilities w are picked so that the condition of detailed balance is satisfied; i.e. w(s ~ s')/w(s' ~ s) = e x p ( - AE/kT) [11], where AE is the energy difference between spin configurations s and s'. This relaxational dynamics explores all paths from the initial to possible final configurations with rates determined by the free-energy barriers. If the free-energy barrier is high for a path, this path will be blocked in a simulation carried out over a reasonable but finite time. This path will become available when the free-energy barrier has decreased to a value comparable to the temperature. We thus expect this dynamics to produce the
S.T.
Chui / Journal of Magnetism and Magnetic Materials 168 (1997) 9-14
correct free energy of nucleation and the correct switching field but a prefactor in the nucleation rate that may be different from that of a stohastic Landau-Gilbert equation. In Fig. la-Fig, lc we show intermediate Monte Carlo configurations for fixed J(0.2), K(0.1) and increasing 9 (0.05, 0.1, 0.15). This clearly demonstrates the spin reversal through domain nucleation. The nucleus is not circular in shape; the larger the 9, the more stripe-like it looks. In Fig. la, the nucleus at the lower right-hand corner is still quite circular. In Fig. lc, the nucleus becomes very long. A similar question was discussed in the case of thick films at zero temperature, where a nucleus in the shape of a strip is predicted [8,9]. We discuss
. . . .
~ ~.--~_. . . .
=~--~ . . . . . . . . .
(a)
~
11
below how this effect can be understood at finite temperature in ultra-thin films. To describe the nucleus, we need a simple expression for the energy including the dipolar interaction. When the external field is equal to H, = 2K, the spins can be rotated together as a whole with no third-order increase in energy. This is the coherent rotation limit. The energy barrier becomes smaller, the closer is H to 2K. Inside the nucleus, the spins are not completely reversed but are only rotated by a small amount of the order of ~b~t = cos- I(H/2K) [12, 8, 9]. Even though it is difficult to write down a simple analytic expression for the dipolar interaction for a general spin configurations, because the deviation from the uniform alignment is small here,
,:':.~ ;-.,'i_
(b)
(c) Fig. 1. The o r i e n t a t i o n of the spins in the xy p l a n e at a t e m p e r a t u r e T = 0.1 for J = 0.2, K = 0.1 a n d g = 0.05, 0.1, 0.15 at i n t e r m e d i a t e times.
S.T Chui / Journal of Magnetism and Magnetic Materials 168 (1997) 9-14
12
the energy can be described analytically in terms of the Fourier transform of the spin deviation
Si(q) = ~I~Si(R) exp(iqR)/,4/N by the spin wave energy as E = Eo + ~ [A + 0.59Drr(q) + Je(q)]lSr(q)] 2 q + [A + Ha -
0.5gDxx(q)
-- 0.5gDyy(q) + Je(q)]lS~(q)l 2.
(1)
Here H a = 9 ~ R 0 . 7 5 / R 3, A = K - H ~ 2 , e(q) = 0.5~,3[1 - cos(q" d)]. Eo is the energy of the state with all spins aligned along the x-axis. Do{q)= ~R[COS(q" R) -- 1]Do(R ) is identical in form to the dynamical matrix of the 2D electron crystal [5, 13]. In the following, we shall ignore the deviation of the spins out of the xy plane. Because of the extra 'shape anisotropy field' Ha in the energy which flips the spins out of the plane, this process is much less likely. To investigate the nucleus, we are interested in the long wavelength limit where the spin configuration is slowly varying. We thus expand the energy as a power series in q and retain the lowest order term. We finally obtain E -- ~ [A + 0.25gc]~/ci + 1.5Jq2]lSr(q)l 2,
(2)
q
where q is the smallest of q + G for all reciprocal lattice vectors G. This reflects the discrete nature of the lattice. The long wavelength limit of the dipolar term is obtained from previous work on electron crystals [13]. Note that it is anisotropic and only of the first power in q. Previous studies on thick rims uses a three-dimensional expression for D (see Eq. 23 in [9]) which remains finite as the wave vector k approaches zero. The physics of the difference between the two and the three dimensional case is the same as that for plasmon dispersion in two and three dimensions. We can rewrite the expression for E in real space by rewriting iqj as the derivative 8j. We approximate Sy by the angle q5 of the spin with respect to the x-axis. The shape of the nucleus can then be determined from the Euler-Lagrange equation from extremizing the energy E. There are two length scales that determine the rate of change of qS. They are lj = x/J/~/A and lg -- g/A. If Ij >>lo then the energy change of the dipolar term 9/1j is always smaller than the energy change due to the exchange
term J/l 2. The dipolar term is not important and the nucleus will be circular. As the coherent rotation limit is approached, A approaches zero and eventually l0 becomes larger than ls. In that limit the rate of change in the x direction will be controlled by J, whereas that in the y direction will be controlled by g. We thus expect a nucleus ellipsoidal in shape with x dimension of the order of lj and y dimension of the order of 1o. As the temperature approaches zero, A ~ 0, the nucleus becomes a strip, consistent with previous results at zero temperature [8]. This picture is also consistent with the Monte Carlo result that the largest deviation from the circular shape comes from the largest 9. In the simulation, the nucleus is oriented at an angle of 120 ° with respect to the x-axis. This is consistent with Eq. (2) that the form of the dipolar energy is invariant under a 120 ° rotation. Presumably, in that case, one can gain maximum advantage from the external magnetic field. To further verify that this picture is correct, we have calculated numerically the energies of ellipsoidal nuclei of different eccentricities with the same area in a finite size lattice. This is carried out as follows. We focus on a lattice of 20 x 20 spins under periodic boundary conditions [14]. The major axis of the ellipse is set to 5 when the eccentricity is 0.2. At the beginning of the calculation, the spins inside the nucleus are rotated by a finite angle 0. The spins outside the nucleus are aligned along the x-axis. The system is then relaxed so that the local minima of the energy is sought with the quasiNewtom algorithm [15]. We avoid reaching the minima where all spins are aligned along the x-axis by fixing the orientation of the spins that are halfway inside the nucleus. In Fig. 2a, we show the energy of the nuclei of different eccentricities as a function of the rotation angle 0 when the exchange is zero. As expected, for a given eccentricity, a maximum energy barrier is reached at a finite rotation angle. Furthermore, this energy barrier is smaller, the smaller the eccentricity. In Fig. 2b we show the result of a similar calculation but now with the exchange set at a finite value (0.05). The smallest barrier now occurs at an intermediate eccentricity. An estimate of the coercive field can be determined from the threshold at which the nucleation
S. T. Chui / Journal of Magnetism and Magnetic Materials 168 (1997) 9-14
1.0 0.8 ~0.6
,e- . . . . . . . NO
EXCHANGE /
,."
~
0.20 ~, 0.18 b_ 0.16
_ "
~3
,.'",'2..7.:?.2.:~ .-
©
© >
0.4
•~
13
1! . . . .
H v~sg ( T / J = O . 5 ) ~
o.14
(D
0.2 0.0 0.0
~ 0.12
0.2
0.4 0.6 angle
(a)
0.8
.0
@.10 0.0
0.5
! .0
T/J or g/J
1.5
Fig. 3. The coercive field as a function of temperature for J = 0.2, K = 0.1. >-, c~ © c
1 0 1 ~ 5 0
.
0.0 (b)
.
.
0.5
~,r." t'//Z .
~.~-~
,
. . . .
1.0 1.5 angle
,
2.0
2.5
Fig. 2. The energy costs to create nuclei of eccentricities from 0.2 to 1 in increments of 0.2 as a function of the angle of rotation in radians; H = 0.39, K = 0.2, g = 0.2: (a) The exchange is set to zero, lines for increasing energy are for increasing eccentricities. (The dotted-dashed, short dashed, dotted, solid and long dashed lines are for eccentricities 0.2, 0.4, 0.6, 0.8 and 1, respectively.)(b) The exchange was set to 0.05. The line symbols are the same as in (a). The lowest energy barrier corresponds to an eccentricity of 0.6.
occurs during the finite simulation time (4000 M C steps/spin) [16]. The coercive field as a function of temperature and as a function of g is shown in Fig. 3. Over a substantial temperature range, the temperature dependence seems to be approximately linear. This linear temperature dependence can be understood as follows. At high temperatures, the external field at which nucleation occurs is smaller than that for coherent rotation. We thus expect A to be not close to zero and that the nucleus to be not as elongated. F o r simplicity we thus neglect the dipolar contribution. Following s t a n d a r d practice in the study of nucleation and instantons we employ the saddle point m e t h o d and a p p r o x i m a t e the 'pinning energy' E~ = - K cos2(~b) + H cos(~b) by expanding it
a b o u t its local m a x i m u m ~bM and obtain Ee(~b) Ee(~aM) -- fl(q~ -- (aM)z/2, where ]~ = ( n 2 - 4KE)/2K. We assume the angle q5 to be only a function of the radial distance and obtain the E u l e r - L a g r a n g e equation (0rq~)/r + 02q~ _ fl(~b _ ~bM) = 0. The solution can be expressed in terms of Bessel functions. With the b o u n d a r y conditions that the solution is well-behaved at the origin and that the angle is zero at the b o u n d a r y R of the nucles, we get the solution qS(r) = ~bM[1 Jo(~-flr)/Jo(xfflR)]. the nucleus is also a variational tremizing the energy with respect the condition [17] O6,E~b'lr=nstituing in the expression for E and -
-
The radius R of parameter. Exto it, we obtain EIr=R = 0. SubqS, we obtain the
result that x/fiR = c, where J'o(C)/Jo(c)= 1. The energy of the barrier EM can be obtained by direct substitution and we obtain EM = qSMf, 2 where the f a c t o r f i s given by the integral
f = f l u[(J'°(u)/J°(u))2 + 0"5(J°(u)/J°(c))Z]du"
W h e n H is close to 2K, 4)2 is proportional to 2 K - H. We finally obtain the result that EM is p r o p o r t i o n a l to 2K - H . The coercive field Hc is determined by the condition that the Boltzman factor exp(-Ev,/kBT) is substantial. Substituing everything back, we obtain a coercive field given by Hc = 2K - C1T ln(C2/t), where C1,2 are constants and t is the simulation time. As the temperature approaces zero, the nucleus becomes strip-like. We
14
S.Z Chui / Journal of Magnetism and Magnetic Materials 168 (1997) 9-14
expect a deviation from this linear temperature dependence. To summarize, because of the competition between the dipolar and the exchange interaction, nuclei formed during in-plane magnetization reversal are ellipsoidal in shape. These domains are more stripe-like, the smaller the effective in-plane anisotropy is. For multilayers with strong interplane coupling, the effective dipolar coupling can be modified. We hope to investigate this in more detail in future studies.
Acknowledgements This work is supported by the Office of Naval Research under contract N00 014-94-1-0213.
References [1] G. Prinz, Science 1092 (1990) 250. I-2] B. Heinrich and J.F. Cochran, Adv. Phys. 42 (1993) 524. 1-3] See, for example, H.N. Bertram and J.G. Zhu, in: Solid State Physics, Vol. 46, Eds. H. Ehrenreich and D. Turnbull (Academic Press, New York, 1992), p. 271.
[4] For a recent summary from a theoretical perspective, see S.T. Chui, Phys. Rev. B 50 (1994) 12559. [5] S.T. Chui, Phys. Rev. Lett. 74 (1995) 3896. [6] S.T. Chui, J. Appl. Phys. 79 (1996) 4951. [7] S.T. Chui, Phys. Rev. B 51 (1995) 250. [8] R.M. Gotdstein and M.W. Muller, Phys. Rev. B 2 (1970) 4585. [9] M.-H. Yang and M.W. Muller, IEEE Trans. Magn. 7 (1970) 705. [10] R. Allenspach and A. Bischof, Phys. Rev. Lett. 69 (1992) 3385. [11] K. Binder and D.W. Heermann, Monte Carlo Simulation in Statistical Physics, 2nd edn. (Springer, New York, 1992). [12] S.T. Chui, J. Appl. Phys. 78 (1995) 3965. [13] S.T. Chui (Ed.), Physics of the 2D electron solid (International Press, Boston, 1995). [14] The difference in the Monte Carlo results for the 20 x 20 and the 60 x 60 systems is small. The relaxation calculation is quite demanding in that it requires the inversion of large (N × N by N x N) matrices. For this reason we have picked the 20 x 20 system. [15] D. Kahaner et al., subroutine UNCMND, in: Numerical Methods and Software (Prentice-Hall, Englewood Cliffs, NJ, 1989) Ch. 9. El6] The dependence on the simulation time is expected to be only logarithmic. [17] J. Matthews and R.L. Walker, in: Mathematical Methods of Physics (Benjamin, New York, 1965) p. 312.