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Effect of dipole–dipole interaction on self-control magnetization oscillation in double-domain nanomagnets
Solid State Communications 152 (2012) 561–565
Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Effect of dipole–dipole interaction on self-control magnetization oscillation in double-domain nanomagnets Y.J. Gao a , Y.J. Guo a,b , J.-M. Liu b,c,∗ a
Department of Physics, Jiangsu Institute of Education, Nanjing 210013, China
b
Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China
c
International Center for Materials Physics, Chinese Academy of Sciences, Shenyang 110016, China
article
info
Article history: Received 27 February 2011 Received in revised form 2 December 2011 Accepted 14 December 2011 by S. Miyashita Available online 22 December 2011
abstract A double-domain model with long-range dipole–dipole interaction is proposed to investigate the selfoscillation of magnetization in nano-magnetic systems driven by self-controlled spin-polarized current. The dynamic behavior of magnetization oscillation is calculated by a modified Landau–Lifshitz–Gilbert equation in order to evaluate the effects of the long-range dipole–dipole interaction. While the self-oscillation of magnetization can be maintained substantially, several self-oscillation regions are experienced as the dipole–dipole interaction increases gradually. © 2011 Elsevier Ltd. All rights reserved.
Keywords: A. Magnetic metals B. Spin-polarized current D. Self-oscillation
As a spin-polarized current (SP current) passes across a domain wall in magnetic metals, the conduction electron would exert a spin torque originating from the s–d exchange interaction between the localized spins on the domain wall [1,2]. Such a spin torque causes the wall spins to be canted out of the wall plane, and can even reverse the domain wall moment once the current density exceeds certain critical value. Recently, a great deal of attention has been focused on the magnetization reversal in nanomagnet driven by spin torque from itinerant electrons [3,4]. Such current induced magnetization reversal has shown its promising application potentials in ultrahigh density storage devices, in which the traditional approach requires a magnetic field too high to generate [5–8]. As a new route toward magnetic information storage applications, some vital drawbacks make this current induced route impractical. One of them is the low switching speed, which straightly affects the reading/writing speed. Searching for other scenario with ultrafast magnetization reversal capability is one of the attracting topics in spintronics. In addition, the magnetization reversal via the long-term stabilized magnetization oscillation is of special interest too [9–16]. In earlier works, the oscillation is excited by applying current pulses of high frequency, but may not
∗ Corresponding author at: Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China. Tel.: +86 25 83596595; fax: +86 25 83595535. E-mail address: [email protected] (J.-M. Liu). 0038-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2011.12.026
stabilize for a long time due to the damping effects [17]. To solve this problem, Chen et al. proposed a self-control SP current scheme for the self-control oscillation in a single-domain nanomagnet [18]. In this scheme, a single-domain model is investigated under certain effective magnetic field, while the SP current density is proportional to the magnitude of the z-component of magnetization M. As a consequence of the negative feedback effect of the spin torque, a steady oscillation of the magnetization can be obtained by a proper combination of SP current and effective field. This steady oscillation in a nanomagnet makes the difference from other approaches, but the single domain model may not provide adequate basis for practical applications where usually the systems under consideration consist of more than one domain. Therefore, the self-control SP current scheme must be modified to include more effective field terms associated with the multidomain nanomagnet, because the inter-domain interactions must be taken into account. Consequently, in this work, we address a nanomagnet consisting of two interactive domains, and intend to propose a double-domain model in which the interaction between the two domains will be taken into consideration. To analyze the stability associated with the magnetization oscillation in this double-domain model, a modified Landau– Lifshitz–Gilbert (LLG) equation including the SP current term is usually necessary [17–20]. The effective field in the LLG equation is attributed to the on-site and inter-domain interactions. The former one takes the regular scheme as the single domain model in Ref. [18]. For the latter one, two domains are considered and the correlation between them may cause a different dynamic
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Y.J. Gao et al. / Solid State Communications 152 (2012) 561–565
where the first term is from the on-site interactions, and the second one from the dipole–dipole interaction between the two domains. The effective field from the on-site interaction is suggested as: hIon = hx , hy − hk mIy , hz − hde mIz ,
(3)
where hx , hy , and hz represent the three components of external magnetic field hex , hk is the magnetic anisotropy field along the y-axis, hde is the demagnetization field along the z-axis. For the double-domain model, the difference in Ms between domain A and domain B are relatively small. We can propose that the interdomain interaction for all the domains is almost the same, and the onsite interactions, such as hex , hk and hde , are equal for the two domains. Similar to the configuration set in Ref. [18] and referring typically to Permalloy nanomagnets, hk = 500 Oe is taken and hde = 5000 Oe is much larger than hk . hex is applied along the x-axis with a magnitude of 2500 Oe [9,17,18]. While the nanomagnet is assumed to have negligible domainwall energy, only the dipole–dipole interaction is considered, which can be expressed as:
hAdi
=
MsB
3 mB · r r
|r |5 3 mA · r r A
− MsB
hBdi Fig. 1. (Color online) A schematic drawing of the double domain model with the spin torques.
behavior of the magnetization oscillation. It is assumed that the magnetic moment would concentrate in a relatively small region of the domain, and then an effective spin moment can be used to replace the domain. In this situation, the correlation between the two domains can be simplified as a dipole–dipole interaction between the two effective spin moments. And also, the effect from the motion of domain wall, an important phenomenon of spin dynamics, can be removed from our calculations. Our results indicate that the dipole–dipole interaction would destroy the steady oscillation as its intensity exceeds some critical value. To consider the dipole–dipole interaction between the two domains, we proposal the spatial configuration of the two domains, as shown in Fig. 1 where the two domains (domain A and domain B) are aligned with a separation vector r with domain A set at the zero-point O (0, 0, 0), and the spin transfer torques onto the two domains are also depicted. The dynamics of magnetization precession for each domain is described by the LLG equation, and the SP current as an additional spin torque term is introduced, as done earlier [18,21–29]. Accordingly, the spin motion in each domain can be described as: 1 + α 2 dM I
γ
dt
= −M I × hIeff − −
1 MsI
α MsI
M I × M I × hIeff
MI × MI × SI ,
(1)
where α is the damping coefficient, γ the gyromagnetic ratio, M the magnetization vector and Ms is the saturation magnetization, t the time, heff the effective fields, S = Sep the SP current with magnitude S along the direction of ep . The superscript I = A, B, represents domain A or B, respectively. We set the dimensionless moment vector m = M /Ms with its three components (mx , my , mz ). In this scheme, we include a self-controlled SP current, with the magnitude S varying according to S = S0 · mz . The effective field heff contains two components: hIeff = hIon + hIdi ,
(2)
= Ms
|r |5
−
MsA
mB
|r |3 mA
|r |3
(4)
.
For simplicity, we first consider a specific case with r = (0, 0, r0 ) and then extend the relevant conclusion to more general situations. Here, the effect of r0 and Ms as the crucial parameters on the dynamics of magnetization oscillation will be investigated in details. For r = (0, 0, r0 ), the effective field can be expressed as: hAeff
=
hx −
+ hBeff
hx −
mA MsA 3x r0
+
ex +
hy −
hz − hde mAz + MsB
=
mB MsB 3x r0
hz −
−
MsB
−
MsA
ex +
+
ey
r03
ez
r03 hy −
2mA MsA 3 z r0
mBy
hde mBz
2mBz
hk mAy
hk mBy
mAy r03
(5)
ey
ez .
Clearly, here r0 characterizes the strength of the dipole–dipole interaction. According to previous work, [18] the spin torque induced by effective field can be denoted as Γ1 , Γ2 , Γ3 , representing the contributions from hex , hk , hde , respectively. Compared with Γ2 , Γ3 (hk /hde = 0.1) is significant and makes the spin move mainly on the x–y plane rather than the others, while external magnetic field along the x-axis induces an oscillation of m perpendicular to it. Such a special combination of Γ1 , Γ2 , Γ3 together with the spin torque from the SP current allows a delicate balance among those forces on the moment, giving rise to the steady self-oscillation of m(t ), where t is time. Similar to the on-site interaction, the dipole–dipole interaction also imposes a spin torque on the moment motion through the effective field in the LLG equation. Subsequently, we investigate the dynamics of the magnetization oscillation in the double domain model by two approaches. In the first approach, the same initial state is imposed for each of the two domains, and it is easy to conclude that they have the synchronous motion. Therefore, the two domains as a whole act as a single one. In the latter approach, the two domains with different initial states (MsA ̸= MsB or mA (t = 0) ̸= mB (t = 0)) are investigated, and the effects from different spatial configurations of the two domains are also considered.
Y.J. Gao et al. / Solid State Communications 152 (2012) 561–565
a
563
a
b
b c
Fig. 2. (Color online) (a) Time (t) dependent oscillation of magnetization with its three components mx , my , and mz at Ms = 1.0 µB and r0 = 48 nm. (b) The spatial trajectory of the spin precession in (a).
For the first approach, we have MsA = MsB and mA (0) = m (0). The two domains would exhibit the identical magnetization motion because they have the same motion equation with the equal initial state. Such a result indicates that the dipole–dipole interaction weakens the Γ3 while enhancing Γ2 , which would destroy the delicate balance for the steady synchronic selfoscillation of the two domains. To illustrate this effect, the LLG equation is solved numerically through the fourth-order Runge–Kutta algorithm with parameters α = 0.01, γ = 1.7 × 107 Oe−1 s−1 , S0 = 120 Oe and ep along the z-axis, setting Ms and r0 as input parameters. Given a set of values for Ms and r0 , perfectly periodic oscillations of mA (t ) (or mB (t )) can be retained even after a long enough time of relaxation, implying the steady oscillation. What should be noted is that the stability of domain A and domain B is always the same, and the oscillation mA (t ) and mB (t ) would be stable or unstable synchronously. Thus, only one of them (here we choose mA (t )) is presented to define the regions of oscillation, noted as m(t ). As an example, at Ms = 1 µB and r0 = 48 nm, the calculated m(t ) is shown in Fig. 2(a) with its three components (mx , my , mz ), while in Fig. 2(b) the spatial trajectory of the magnetization oscillation is plotted to illustrate the non-attenuated property. Hereafter, we mainly address the dynamic behavior of mx (t ) in response to various parameters, while the other two components are similar in terms of the oscillation behavior. To illustrate the stability of the magnetization oscillation, we first exclude the effect from the dipole–dipole interaction B
Fig. 3. (Color online) Three parameters (a) ⟨mx ⟩, (b) Ax , (c) ωx of the oscillation of component mx as a function of r0 , respectively, at Ms = 1.0 µB .
(r0 → ∞). This situation degenerates into the single-domain model, which was well investigated [18]. For a better presentation, the spatial trajectory of the magnetization is projected into the (θ , ϕ ) state space with the restriction of |m| = 1 with Ms = 1.0 µB . According to the Poincare–Bendixson theorem, the motion in the (θ , ϕ ) space would either approach a fixed point (i.e. the oscillation is unstable) or takes the precession within a finite-size region (limit cycle) [18,30]. Subsequently, we reduce r0 gradually from infinite to model the gradually increasing dipole–dipole interaction. It is seen that the oscillation behavior becomes suppressed when the dipole–dipole interaction surpasses a critical value. For more concrete illustration, we use three parameters: the mean value, the amplitude and the precession frequency of mx , denoted as ⟨mx ⟩, Ax and ωx , respectively, as shown in Fig. 3. Given the different behaviors of these parameters as a function of r0 , especially Ax (r0 ), the oscillation motion of the magnetization can be divided into three regions, marked as I, II and III, respectively. In regions III and II, with increasing dipole–dipole interaction via decreasing r0 , the oscillation behavior gradually deviates from the steady state available in the single domain model. When r0 decreases down to a threshold value, the motion trajectory of the magnetization is no longer be a limit circle but approaches a fixed point. Therefore, region II and region III are called the steady oscillation regions, while no steady oscillation in region I is possible. Now, we discuss the difference between region II and region III. It is shown in Fig. 3 that in regions III and region II, parameters ⟨mx ⟩ and ωx would change gradually as r0 reduces, both in an accelerating manner. However, parameter Ax would approach a minimal close to zero at r0 ∼ 40 nm, which distinguishes region
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Y.J. Gao et al. / Solid State Communications 152 (2012) 561–565
II from region III. At the minimum point, although Ax falls down to zero, components my and mz oscillate as usual, representing a quasi-two-dimensional oscillation on the yz-plane. To explain the minimal of Ax as a function of r0 at the boundary between region II and region III, one refers to the effective field in Eq. (2), where the effective field originates from the on-site interaction and the dipole–dipole interaction. The magnetization oscillation can also be argued to originate from the collective effects of these two parts. Consulting to the LLG equation, which shows a nonlinear behavior in the damping term and spin torque term, the result cannot be seen as the simple combination of two oscillations generated respectively from hon and hdi . Therefore, it is easy to argue that the dipole–dipole interaction plays an opposite role against the oscillation behavior of the two domains as a self-control single domain (in region III), and it leads to another type of oscillation as it reaches a proper magnitude (in region II). More evidences are needed. To further illustrate the oscillation behavior in region II, where Ax shows different r0 -dependence from that in region III, we include some disturbances to parameters Ms and r0 . In the above calculation, the identical initial state is supposed for domain A and domain B, which suggests that domains A and B have the same size (saturation magnetization MSA = MSB ) and the same moment orientation (mA (0) = mB (0)). However, for realistic multi-domain systems, the two domains may be different in the size and orientation. Hereafter, a disturbing parameter 1s is introduced to present the difference in size for each domain. As a simple scheme, we re-formulate the saturation magnetization of domain A as MsA ∼ MsA + 1s MsA and that of domain B as MsB ∼ MsB + 1′s MsB , while keeping the intensity of the dipole–dipole interaction unchanged, which requires MsA · MsB = (MsA + 1s MsA )(MsB + 1′s MsB ). For the difference in the moment orientation between the two domains, similar scheme is also proposed, with component mx reexpressed as mx + 1d mx , while the dimensionless magnetization m is normalized by term [(mx + 1d mx )2 + (my )2 + (mz )2 ]1/2 . By choosing Ms = 1.0 µB again, we perform the relevant calculations including the disturbances. Fig. 4(a) and (b) present the critical distance (rc ) between the two domains as a function of disturbing parameters 1s and 1d . At very small 1s or 1d , an abrupt change of rc from ∼33 to ∼40 nm is identified. Further increasing of 1s (or 1d ) does not impose significant impact on rc , and the calculated rc presents a symmetric variation as a function of 1s and −1s (also for 1d and −1d ). It should be mentioned that rc ∼ 33 nm is just on the boundary between region I and region II, while rc ∼ 40 nm divides region II and region III. The steady oscillation of the two domains under these disturbances is further investigated, and parameters ⟨mx ⟩, Ax , and ωx remain quite robust at r0 > rc ∼ 40 nm. Therefore, a conclusion can be made here: the steady oscillation of the two domains would only exist in region III, which is gradually destroyed as Ax falls down very close to zero. In case of such disturbances, no steady oscillation in region II is available unless the two domains have the identical initial state (i.e. 1s = 1d = 0), which is of no reality. In the other words, if both domains have the same initial state without any disturbance, steady oscillation of the magnetization is available in both region II and region III. However, in case of a slight difference in the initial state between the two domains, region II no longer accommodates any steady oscillation of the magnetization for each domain. These results allow us to distinct the single domain model from our double domain model. Now we extend our discussion to more complicated situations where the two domains are aligned by an arbitrary separation vector r rather than the simple case along the (0, 0, r0 ) direction. For a general sense, the two domains are configured by separation vector r = (r0 sin θ sin ϕ, r0 sin θ cos ϕ, r0 cos θ ). Then the above calculation over the whole (θ , ϕ ) space is carefully performed to
a
b
Fig. 4. (Color online) Critical distance (rc ) between domain A and domain B as a function of disturbing parameters (a) 1s and (b) 1d for a steady oscillation of the magnetization at Ms = 1.0 µB .
check the stability of the magnetization oscillation. It is found that for the spatial lines (0, ϕ ) and (π , ϕ ), which represent that the two domains are separated along the ±z-axis respectively, the dynamic behaviors are the same with each other. For other configurations, the dynamic behaviors are different. As the vector r deviates from the ±z-axis, the stability of the magnetization oscillation will be reduced. For r = (0, 0, r0 ) and equal initial states for two domains (MSA = MSB and mA (0) = mB (0)) with Ms = 1.0 µB , the threshold value is rc ∼ 33 nm. This threshold value increases up to ∼40 nm if the initial states are different (MSA ̸= MSB or mA (0) ̸= mB (0)) by inducing disturbances 1s (or 1d ). For the other cases with different r, this threshold reaches up to >54 nm. This suggests that the steady oscillation of magnetization can only be maintained at a weaker dipole–dipole interaction. Thus in this case, region III has to be divided into region IIIa (40 nm < r < 54 nm) and IIIb (r > 54 nm), and the steady oscillation can only exist in region IIIb if the dipole–dipole interaction does not align along the ±z-axis. Finally, we come to investigate the role of Ms in modulating the stability of the magnetization oscillation and this modulation can be expected looking at Eq. (1), noting that Ms = 1.0 µB in all above calculations. Allowing Ms to vary from 1.0 to 100 µB , the Ms –r0 phase diagram is evaluated and the calculated results are shown in Fig. 5(a). It is clearly shown that for a larger Ms , the steady oscillation of the magnetization cannot be maintained unless at a larger distance between the two domains. Based on this phase diagram, we repeat the dynamic behaviors of the magnetization
Y.J. Gao et al. / Solid State Communications 152 (2012) 561–565
565
a
II but stable in region III. This allows a distinguishing of region II from region III. The phase diagram in Fig. 5(a) can be linearly scaled by redefining the inter-region boundaries in the Ms − r03 space. Such a linear relation suggests that the dynamic behaviors would not change in response to the variation of Ms if a constant ratio Ms /r03 is reserved. This scaling relation may be satisfied by a proper balancing the on-site interaction and the SP current. In the other hand, the total saturation magnetization (in a unit volume) taking the form of N · M s /V for a multi-domain system is proportional to ratio Ms /r03 . Therefore, it can be suggested that for a slightly inhomogeneous nanomagnet, the oscillation state is only determined by the total saturated magnetization, no matter whether the size and inter-separation of those domains are slightly different from one and another.
b
Acknowledgments This work was supported by the Natural Science Foundation of China (11074113, 50832002), the National 973 Projects of China (2011CB922101, 2009CB623303), the Natural Science Foundation of Jiangsu Province, China (BK2008024), and the Priority Academic Program Development of Jiangsu Higher Education Institutions, China. References [1] [2] [3] [4] [5]
Fig. 5. (Color online) (a) The Ms − r0 diagram for the oscillation dynamics of the magnetization. (b) The linear scaling relation between Ms and r03 for the boundaries between regions I, II, and III.
oscillation in these regions I, II, and III. In region I, the dipole–dipole interaction is too strong for the system to maintain any steady oscillation, because of either a too large Ms for the domains or too small r0 between them. In region II, the steady oscillation can be maintained only with the identical initial state, and also the separation vector r = (0, 0, r0 ), representing the case of single domain. In region IIIa, the steady oscillation can exist as the dipole–dipole interaction is along the z-axis with the SP current along the z-axis too, while the initial states for the two domains are different. Such a result indicates that the oscillation may only appear in the magnetic wire system (with the SP current along the wire direction). In region IIIb, the steady oscillation of the magnetization for both domains can be reached even if the two domains align in an arbitrary orientation. It is suggested that for a nanomagnet consisting of randomly aligned domains, a steady oscillation of the magnetizations for all these domains is possible in region IIIb. Therefore, whether the oscillation is stable or not should be the only criterion for the region division. For example, in region I, the oscillation is always unstable, while in region II a steady oscillation can only be maintained with the identical initial state for the two domains. So region I and region II can be distinguished from the stability under the condition of ‘identical initial state’. When the two domains have different initial states, and the separation vector r is aligned along the ±z-axis, the oscillation is unstable in region
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Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Effect of dipole–dipole interaction on self-control magnetization oscillation in double-domain nanomagnets Y.J. Gao a , Y.J. Guo a,b , J.-M. Liu b,c,∗ a
Department of Physics, Jiangsu Institute of Education, Nanjing 210013, China
b
Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China
c
International Center for Materials Physics, Chinese Academy of Sciences, Shenyang 110016, China
article
info
Article history: Received 27 February 2011 Received in revised form 2 December 2011 Accepted 14 December 2011 by S. Miyashita Available online 22 December 2011
abstract A double-domain model with long-range dipole–dipole interaction is proposed to investigate the selfoscillation of magnetization in nano-magnetic systems driven by self-controlled spin-polarized current. The dynamic behavior of magnetization oscillation is calculated by a modified Landau–Lifshitz–Gilbert equation in order to evaluate the effects of the long-range dipole–dipole interaction. While the self-oscillation of magnetization can be maintained substantially, several self-oscillation regions are experienced as the dipole–dipole interaction increases gradually. © 2011 Elsevier Ltd. All rights reserved.
Keywords: A. Magnetic metals B. Spin-polarized current D. Self-oscillation
As a spin-polarized current (SP current) passes across a domain wall in magnetic metals, the conduction electron would exert a spin torque originating from the s–d exchange interaction between the localized spins on the domain wall [1,2]. Such a spin torque causes the wall spins to be canted out of the wall plane, and can even reverse the domain wall moment once the current density exceeds certain critical value. Recently, a great deal of attention has been focused on the magnetization reversal in nanomagnet driven by spin torque from itinerant electrons [3,4]. Such current induced magnetization reversal has shown its promising application potentials in ultrahigh density storage devices, in which the traditional approach requires a magnetic field too high to generate [5–8]. As a new route toward magnetic information storage applications, some vital drawbacks make this current induced route impractical. One of them is the low switching speed, which straightly affects the reading/writing speed. Searching for other scenario with ultrafast magnetization reversal capability is one of the attracting topics in spintronics. In addition, the magnetization reversal via the long-term stabilized magnetization oscillation is of special interest too [9–16]. In earlier works, the oscillation is excited by applying current pulses of high frequency, but may not
∗ Corresponding author at: Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China. Tel.: +86 25 83596595; fax: +86 25 83595535. E-mail address: [email protected] (J.-M. Liu). 0038-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2011.12.026
stabilize for a long time due to the damping effects [17]. To solve this problem, Chen et al. proposed a self-control SP current scheme for the self-control oscillation in a single-domain nanomagnet [18]. In this scheme, a single-domain model is investigated under certain effective magnetic field, while the SP current density is proportional to the magnitude of the z-component of magnetization M. As a consequence of the negative feedback effect of the spin torque, a steady oscillation of the magnetization can be obtained by a proper combination of SP current and effective field. This steady oscillation in a nanomagnet makes the difference from other approaches, but the single domain model may not provide adequate basis for practical applications where usually the systems under consideration consist of more than one domain. Therefore, the self-control SP current scheme must be modified to include more effective field terms associated with the multidomain nanomagnet, because the inter-domain interactions must be taken into account. Consequently, in this work, we address a nanomagnet consisting of two interactive domains, and intend to propose a double-domain model in which the interaction between the two domains will be taken into consideration. To analyze the stability associated with the magnetization oscillation in this double-domain model, a modified Landau– Lifshitz–Gilbert (LLG) equation including the SP current term is usually necessary [17–20]. The effective field in the LLG equation is attributed to the on-site and inter-domain interactions. The former one takes the regular scheme as the single domain model in Ref. [18]. For the latter one, two domains are considered and the correlation between them may cause a different dynamic
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Y.J. Gao et al. / Solid State Communications 152 (2012) 561–565
where the first term is from the on-site interactions, and the second one from the dipole–dipole interaction between the two domains. The effective field from the on-site interaction is suggested as: hIon = hx , hy − hk mIy , hz − hde mIz ,
(3)
where hx , hy , and hz represent the three components of external magnetic field hex , hk is the magnetic anisotropy field along the y-axis, hde is the demagnetization field along the z-axis. For the double-domain model, the difference in Ms between domain A and domain B are relatively small. We can propose that the interdomain interaction for all the domains is almost the same, and the onsite interactions, such as hex , hk and hde , are equal for the two domains. Similar to the configuration set in Ref. [18] and referring typically to Permalloy nanomagnets, hk = 500 Oe is taken and hde = 5000 Oe is much larger than hk . hex is applied along the x-axis with a magnitude of 2500 Oe [9,17,18]. While the nanomagnet is assumed to have negligible domainwall energy, only the dipole–dipole interaction is considered, which can be expressed as:
hAdi
=
MsB
3 mB · r r
|r |5 3 mA · r r A
− MsB
hBdi Fig. 1. (Color online) A schematic drawing of the double domain model with the spin torques.
behavior of the magnetization oscillation. It is assumed that the magnetic moment would concentrate in a relatively small region of the domain, and then an effective spin moment can be used to replace the domain. In this situation, the correlation between the two domains can be simplified as a dipole–dipole interaction between the two effective spin moments. And also, the effect from the motion of domain wall, an important phenomenon of spin dynamics, can be removed from our calculations. Our results indicate that the dipole–dipole interaction would destroy the steady oscillation as its intensity exceeds some critical value. To consider the dipole–dipole interaction between the two domains, we proposal the spatial configuration of the two domains, as shown in Fig. 1 where the two domains (domain A and domain B) are aligned with a separation vector r with domain A set at the zero-point O (0, 0, 0), and the spin transfer torques onto the two domains are also depicted. The dynamics of magnetization precession for each domain is described by the LLG equation, and the SP current as an additional spin torque term is introduced, as done earlier [18,21–29]. Accordingly, the spin motion in each domain can be described as: 1 + α 2 dM I
γ
dt
= −M I × hIeff − −
1 MsI
α MsI
M I × M I × hIeff
MI × MI × SI ,
(1)
where α is the damping coefficient, γ the gyromagnetic ratio, M the magnetization vector and Ms is the saturation magnetization, t the time, heff the effective fields, S = Sep the SP current with magnitude S along the direction of ep . The superscript I = A, B, represents domain A or B, respectively. We set the dimensionless moment vector m = M /Ms with its three components (mx , my , mz ). In this scheme, we include a self-controlled SP current, with the magnitude S varying according to S = S0 · mz . The effective field heff contains two components: hIeff = hIon + hIdi ,
(2)
= Ms
|r |5
−
MsA
mB
|r |3 mA
|r |3
(4)
.
For simplicity, we first consider a specific case with r = (0, 0, r0 ) and then extend the relevant conclusion to more general situations. Here, the effect of r0 and Ms as the crucial parameters on the dynamics of magnetization oscillation will be investigated in details. For r = (0, 0, r0 ), the effective field can be expressed as: hAeff
=
hx −
+ hBeff
hx −
mA MsA 3x r0
+
ex +
hy −
hz − hde mAz + MsB
=
mB MsB 3x r0
hz −
−
MsB
−
MsA
ex +
+
ey
r03
ez
r03 hy −
2mA MsA 3 z r0
mBy
hde mBz
2mBz
hk mAy
hk mBy
mAy r03
(5)
ey
ez .
Clearly, here r0 characterizes the strength of the dipole–dipole interaction. According to previous work, [18] the spin torque induced by effective field can be denoted as Γ1 , Γ2 , Γ3 , representing the contributions from hex , hk , hde , respectively. Compared with Γ2 , Γ3 (hk /hde = 0.1) is significant and makes the spin move mainly on the x–y plane rather than the others, while external magnetic field along the x-axis induces an oscillation of m perpendicular to it. Such a special combination of Γ1 , Γ2 , Γ3 together with the spin torque from the SP current allows a delicate balance among those forces on the moment, giving rise to the steady self-oscillation of m(t ), where t is time. Similar to the on-site interaction, the dipole–dipole interaction also imposes a spin torque on the moment motion through the effective field in the LLG equation. Subsequently, we investigate the dynamics of the magnetization oscillation in the double domain model by two approaches. In the first approach, the same initial state is imposed for each of the two domains, and it is easy to conclude that they have the synchronous motion. Therefore, the two domains as a whole act as a single one. In the latter approach, the two domains with different initial states (MsA ̸= MsB or mA (t = 0) ̸= mB (t = 0)) are investigated, and the effects from different spatial configurations of the two domains are also considered.
Y.J. Gao et al. / Solid State Communications 152 (2012) 561–565
a
563
a
b
b c
Fig. 2. (Color online) (a) Time (t) dependent oscillation of magnetization with its three components mx , my , and mz at Ms = 1.0 µB and r0 = 48 nm. (b) The spatial trajectory of the spin precession in (a).
For the first approach, we have MsA = MsB and mA (0) = m (0). The two domains would exhibit the identical magnetization motion because they have the same motion equation with the equal initial state. Such a result indicates that the dipole–dipole interaction weakens the Γ3 while enhancing Γ2 , which would destroy the delicate balance for the steady synchronic selfoscillation of the two domains. To illustrate this effect, the LLG equation is solved numerically through the fourth-order Runge–Kutta algorithm with parameters α = 0.01, γ = 1.7 × 107 Oe−1 s−1 , S0 = 120 Oe and ep along the z-axis, setting Ms and r0 as input parameters. Given a set of values for Ms and r0 , perfectly periodic oscillations of mA (t ) (or mB (t )) can be retained even after a long enough time of relaxation, implying the steady oscillation. What should be noted is that the stability of domain A and domain B is always the same, and the oscillation mA (t ) and mB (t ) would be stable or unstable synchronously. Thus, only one of them (here we choose mA (t )) is presented to define the regions of oscillation, noted as m(t ). As an example, at Ms = 1 µB and r0 = 48 nm, the calculated m(t ) is shown in Fig. 2(a) with its three components (mx , my , mz ), while in Fig. 2(b) the spatial trajectory of the magnetization oscillation is plotted to illustrate the non-attenuated property. Hereafter, we mainly address the dynamic behavior of mx (t ) in response to various parameters, while the other two components are similar in terms of the oscillation behavior. To illustrate the stability of the magnetization oscillation, we first exclude the effect from the dipole–dipole interaction B
Fig. 3. (Color online) Three parameters (a) ⟨mx ⟩, (b) Ax , (c) ωx of the oscillation of component mx as a function of r0 , respectively, at Ms = 1.0 µB .
(r0 → ∞). This situation degenerates into the single-domain model, which was well investigated [18]. For a better presentation, the spatial trajectory of the magnetization is projected into the (θ , ϕ ) state space with the restriction of |m| = 1 with Ms = 1.0 µB . According to the Poincare–Bendixson theorem, the motion in the (θ , ϕ ) space would either approach a fixed point (i.e. the oscillation is unstable) or takes the precession within a finite-size region (limit cycle) [18,30]. Subsequently, we reduce r0 gradually from infinite to model the gradually increasing dipole–dipole interaction. It is seen that the oscillation behavior becomes suppressed when the dipole–dipole interaction surpasses a critical value. For more concrete illustration, we use three parameters: the mean value, the amplitude and the precession frequency of mx , denoted as ⟨mx ⟩, Ax and ωx , respectively, as shown in Fig. 3. Given the different behaviors of these parameters as a function of r0 , especially Ax (r0 ), the oscillation motion of the magnetization can be divided into three regions, marked as I, II and III, respectively. In regions III and II, with increasing dipole–dipole interaction via decreasing r0 , the oscillation behavior gradually deviates from the steady state available in the single domain model. When r0 decreases down to a threshold value, the motion trajectory of the magnetization is no longer be a limit circle but approaches a fixed point. Therefore, region II and region III are called the steady oscillation regions, while no steady oscillation in region I is possible. Now, we discuss the difference between region II and region III. It is shown in Fig. 3 that in regions III and region II, parameters ⟨mx ⟩ and ωx would change gradually as r0 reduces, both in an accelerating manner. However, parameter Ax would approach a minimal close to zero at r0 ∼ 40 nm, which distinguishes region
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II from region III. At the minimum point, although Ax falls down to zero, components my and mz oscillate as usual, representing a quasi-two-dimensional oscillation on the yz-plane. To explain the minimal of Ax as a function of r0 at the boundary between region II and region III, one refers to the effective field in Eq. (2), where the effective field originates from the on-site interaction and the dipole–dipole interaction. The magnetization oscillation can also be argued to originate from the collective effects of these two parts. Consulting to the LLG equation, which shows a nonlinear behavior in the damping term and spin torque term, the result cannot be seen as the simple combination of two oscillations generated respectively from hon and hdi . Therefore, it is easy to argue that the dipole–dipole interaction plays an opposite role against the oscillation behavior of the two domains as a self-control single domain (in region III), and it leads to another type of oscillation as it reaches a proper magnitude (in region II). More evidences are needed. To further illustrate the oscillation behavior in region II, where Ax shows different r0 -dependence from that in region III, we include some disturbances to parameters Ms and r0 . In the above calculation, the identical initial state is supposed for domain A and domain B, which suggests that domains A and B have the same size (saturation magnetization MSA = MSB ) and the same moment orientation (mA (0) = mB (0)). However, for realistic multi-domain systems, the two domains may be different in the size and orientation. Hereafter, a disturbing parameter 1s is introduced to present the difference in size for each domain. As a simple scheme, we re-formulate the saturation magnetization of domain A as MsA ∼ MsA + 1s MsA and that of domain B as MsB ∼ MsB + 1′s MsB , while keeping the intensity of the dipole–dipole interaction unchanged, which requires MsA · MsB = (MsA + 1s MsA )(MsB + 1′s MsB ). For the difference in the moment orientation between the two domains, similar scheme is also proposed, with component mx reexpressed as mx + 1d mx , while the dimensionless magnetization m is normalized by term [(mx + 1d mx )2 + (my )2 + (mz )2 ]1/2 . By choosing Ms = 1.0 µB again, we perform the relevant calculations including the disturbances. Fig. 4(a) and (b) present the critical distance (rc ) between the two domains as a function of disturbing parameters 1s and 1d . At very small 1s or 1d , an abrupt change of rc from ∼33 to ∼40 nm is identified. Further increasing of 1s (or 1d ) does not impose significant impact on rc , and the calculated rc presents a symmetric variation as a function of 1s and −1s (also for 1d and −1d ). It should be mentioned that rc ∼ 33 nm is just on the boundary between region I and region II, while rc ∼ 40 nm divides region II and region III. The steady oscillation of the two domains under these disturbances is further investigated, and parameters ⟨mx ⟩, Ax , and ωx remain quite robust at r0 > rc ∼ 40 nm. Therefore, a conclusion can be made here: the steady oscillation of the two domains would only exist in region III, which is gradually destroyed as Ax falls down very close to zero. In case of such disturbances, no steady oscillation in region II is available unless the two domains have the identical initial state (i.e. 1s = 1d = 0), which is of no reality. In the other words, if both domains have the same initial state without any disturbance, steady oscillation of the magnetization is available in both region II and region III. However, in case of a slight difference in the initial state between the two domains, region II no longer accommodates any steady oscillation of the magnetization for each domain. These results allow us to distinct the single domain model from our double domain model. Now we extend our discussion to more complicated situations where the two domains are aligned by an arbitrary separation vector r rather than the simple case along the (0, 0, r0 ) direction. For a general sense, the two domains are configured by separation vector r = (r0 sin θ sin ϕ, r0 sin θ cos ϕ, r0 cos θ ). Then the above calculation over the whole (θ , ϕ ) space is carefully performed to
a
b
Fig. 4. (Color online) Critical distance (rc ) between domain A and domain B as a function of disturbing parameters (a) 1s and (b) 1d for a steady oscillation of the magnetization at Ms = 1.0 µB .
check the stability of the magnetization oscillation. It is found that for the spatial lines (0, ϕ ) and (π , ϕ ), which represent that the two domains are separated along the ±z-axis respectively, the dynamic behaviors are the same with each other. For other configurations, the dynamic behaviors are different. As the vector r deviates from the ±z-axis, the stability of the magnetization oscillation will be reduced. For r = (0, 0, r0 ) and equal initial states for two domains (MSA = MSB and mA (0) = mB (0)) with Ms = 1.0 µB , the threshold value is rc ∼ 33 nm. This threshold value increases up to ∼40 nm if the initial states are different (MSA ̸= MSB or mA (0) ̸= mB (0)) by inducing disturbances 1s (or 1d ). For the other cases with different r, this threshold reaches up to >54 nm. This suggests that the steady oscillation of magnetization can only be maintained at a weaker dipole–dipole interaction. Thus in this case, region III has to be divided into region IIIa (40 nm < r < 54 nm) and IIIb (r > 54 nm), and the steady oscillation can only exist in region IIIb if the dipole–dipole interaction does not align along the ±z-axis. Finally, we come to investigate the role of Ms in modulating the stability of the magnetization oscillation and this modulation can be expected looking at Eq. (1), noting that Ms = 1.0 µB in all above calculations. Allowing Ms to vary from 1.0 to 100 µB , the Ms –r0 phase diagram is evaluated and the calculated results are shown in Fig. 5(a). It is clearly shown that for a larger Ms , the steady oscillation of the magnetization cannot be maintained unless at a larger distance between the two domains. Based on this phase diagram, we repeat the dynamic behaviors of the magnetization
Y.J. Gao et al. / Solid State Communications 152 (2012) 561–565
565
a
II but stable in region III. This allows a distinguishing of region II from region III. The phase diagram in Fig. 5(a) can be linearly scaled by redefining the inter-region boundaries in the Ms − r03 space. Such a linear relation suggests that the dynamic behaviors would not change in response to the variation of Ms if a constant ratio Ms /r03 is reserved. This scaling relation may be satisfied by a proper balancing the on-site interaction and the SP current. In the other hand, the total saturation magnetization (in a unit volume) taking the form of N · M s /V for a multi-domain system is proportional to ratio Ms /r03 . Therefore, it can be suggested that for a slightly inhomogeneous nanomagnet, the oscillation state is only determined by the total saturated magnetization, no matter whether the size and inter-separation of those domains are slightly different from one and another.
b
Acknowledgments This work was supported by the Natural Science Foundation of China (11074113, 50832002), the National 973 Projects of China (2011CB922101, 2009CB623303), the Natural Science Foundation of Jiangsu Province, China (BK2008024), and the Priority Academic Program Development of Jiangsu Higher Education Institutions, China. References [1] [2] [3] [4] [5]
Fig. 5. (Color online) (a) The Ms − r0 diagram for the oscillation dynamics of the magnetization. (b) The linear scaling relation between Ms and r03 for the boundaries between regions I, II, and III.
oscillation in these regions I, II, and III. In region I, the dipole–dipole interaction is too strong for the system to maintain any steady oscillation, because of either a too large Ms for the domains or too small r0 between them. In region II, the steady oscillation can be maintained only with the identical initial state, and also the separation vector r = (0, 0, r0 ), representing the case of single domain. In region IIIa, the steady oscillation can exist as the dipole–dipole interaction is along the z-axis with the SP current along the z-axis too, while the initial states for the two domains are different. Such a result indicates that the oscillation may only appear in the magnetic wire system (with the SP current along the wire direction). In region IIIb, the steady oscillation of the magnetization for both domains can be reached even if the two domains align in an arbitrary orientation. It is suggested that for a nanomagnet consisting of randomly aligned domains, a steady oscillation of the magnetizations for all these domains is possible in region IIIb. Therefore, whether the oscillation is stable or not should be the only criterion for the region division. For example, in region I, the oscillation is always unstable, while in region II a steady oscillation can only be maintained with the identical initial state for the two domains. So region I and region II can be distinguished from the stability under the condition of ‘identical initial state’. When the two domains have different initial states, and the separation vector r is aligned along the ±z-axis, the oscillation is unstable in region
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