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Antivortex gyrotropic motion in a nanocontact
Journal of Magnetism and Magnetic Materials 323 (2011) 499–503
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Antivortex gyrotropic motion in a nanocontact C.E. Zaspel n Department of Environmental Sciences, University of Montana-Western, Dillon, MT 59725 USA
a r t i c l e in f o
abstract
Article history: Received 13 August 2010 Available online 15 October 2010
A vortex–antivortex pair can form in the free layer of a nanocontact device as a result of the Oersted field produced by the current. In a large-scale free layer having uniform magnetization boundary conditions an in-plane external magnetic field will tend to confine the vortex–antivortex pair, which undergoes gyrotropic motion about the nanocontact after an initial displacement from the static equilibrium position. With the vortex pinned to a defect at the nanocontact the antivortex dynamics can be isolated and gyrotropic precession of the antivortex will be the dominant mode. The frequency of antivortex precession increases as the external magnetic field increases, and the frequency decreases as the nanocontact current increases. & 2010 Elsevier B.V. All rights reserved.
Keywords: Magnetic vortex Nanocontact Antivortex
1. Introduction The magnetic vortex in a confined geometry has generated much theoretical and experimental research during the past few years owing to its potential information storage [1] applications. A particular application involves the magnetic vortex, which tends to form in a confined nanoscale system such as a submicron disk, where the vortex is the actual ground state because of the competition between the exchange and magnetostatic energies. For this state over most of the disks magnetization will be in-plane and parallel to the disk edge, which minimizes magnetostatic poles. In-plane magnetization at the vortex center will result in a high energy exchange singularity, which is eliminated by the formation of an out-of-plane vortex core at the expense of magnetostatic pole formation at the disk faces. Typically in permalloy disks these vortex core radii are the order of 5–10 nm with two polarization states perpendicular to the vortex plane. Since the two polarization states of the vortex core have potential for information storage, it is necessary to understand the vortex core dynamic properties. For applications such as vortex core switching, the dynamics of the vortex core is dominated by the so-called gyrotropic mode, which is a sub-GHz precession of the vortex core after an initial displacement from the vortex center by a magnetic field pulse. This particular mode was directly observed [2] using time-resolved Kerr microscopy, and later [3] using time-resolved X-ray imaging. The precession is observed after the vortex is displaced from its static equilibrium position at the disk center by an in-plane external field pulse. After the pulse turns off, the displaced vortex core spirals back toward the disk center.
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For practical applications it is necessary to have an external control over the vortex core polarization either by an external field or electric currents. A recent promising method [4,5] to switch the core magnetization direction uses the vortex gyrotropic mode as well as the formation of vortex–antivortex (VA) pairs excited by a spin-polarized current. The physical mechanism involves the formation of VA pairs of opposite core polarities, and subsequent annihilation of the original vortex and antivortex leaving a vortex of switched polarity, which is a complicated effect to study using analytical theoretical techniques because it involves the interaction of VA pairs. Another application is the spin-torque nanooscillator [6–8], where a vortex is formed at the nanocontact and the gyrotropic mode is excited by a spin-polarized current giving an output in the sub-GHz range. The output features can be improved by synchronization [7] of two nano-oscillators, and recently [8] it was shown that four nanocontacts are synchronized, or phaselocked through the VA interaction. Motivated by the need to better understand VA pair dynamics, in this article a simple thin-film device is proposed allowing the dynamics of an antivortex in the VA pair to be experimentally and theoretically investigated. The dynamics of the simpler single vortex in the nanodisk is now well understood and has been investigated using both numerical [9,10] and analytical [11] techniques. In particular, the gyrotropic mode frequency is simply related [11] to the disk aspect ratio, f 0:2oM L=R, where oM ¼30 GHz is a characteristic frequency for permalloy, and L and R are the disk thickness and radius, respectively. However, much less has been done in relation to the dynamics of either the VA pair or the isolated antivortex. Recently numerical simulations [12] have shown that the single isolated antivortex can form in an asteroid-shaped magnetic particle with the gyrotropic mode excited by the application of a field pulse. Similar simulations [13] have been done for a clover-leaf sample, which also exhibits an
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C.E. Zaspel / Journal of Magnetism and Magnetic Materials 323 (2011) 499–503
antivortex state. In this case the effect of spin-polarized current as well as an external field on the gyrotropic mode amplitude were investigated, indicating that there can be suppression of this mode. The dynamics of the single antivortex are of interest, but because of the fabrication difficulty of the asteroid or clover-leaf shape it is difficult to investigate this isolated structure experimentally. In this article a device is proposed that can be used to investigate the dynamic properties of a single antivortex as a VA pair forms in a nanocontact free layer. Previously the VA pair, referred to as the vortex dipole, has been observed [14] in a permalloy ellipse in a nanopillar device excited by a spin-polarized current. Usually in a confined system such as a thin disk a single vortex is formed, since this structure minimizes the edge magnetostatic charge. However, in the nanopillar the VA pair can also form because of the competing effects from the Oersted field resulting from the electric current and the effective pinning field between the fixed and free layers. The pair tends to form because the free layer magnetization is uniform far from the VA pair, which will minimize the magnetostatic interaction energy between the layers; however, the Oersted energy is minimized by the formation of the vortex dipole symmetric about the ellipse center. Spin-torque then forces the vortex dipole to undergo gyrotropic motion about the center of symmetry. Gyrotropic motion of a single vortex has also been observed [15,16] in a nanocontact where the free layer is very large in relation to the nanocontact. In this case it is expected that the boundary condition (uniform magnetization) far from the nanocontact implies the existence of a VA pair. Indeed, the vortex dipole is a static solution [17,18] of the Landau–Lifshitz equation where the effective field is due to the exchange interaction only. Past numerical simulations [16] for the nanocontact with both the external field and the spin polarization perpendicular to the free layer suggest that a vortex forms as a result of the Oersted field at the nanocontact with eventual orbital motion of the vortex about the nanocontact. However, since the simulation was performed on a confined system of 1000 nm diameter, the formation of a stable antivortex was not observed or it was possibly forced outside of the system. In the following a very large thin film with a single nanocontact in a uniform magnetic field is considered. A large enough nanocontact current will tend to nucleate the vortex core with the formation of an antivortex to give the uniform boundary conditions defined by the magnetic field. The antivortex has a static equilibrium position outside the nanocontact, and the gyrotropic motion of this antivortex is investigated.
2. Device and system model To isolate the antivortex and observe its dynamics it is necessary to form the vortex on a pinning center [19,20] in the nanocontact so it will not undergo gyrotropic motion. Also it is necessary to confine the VA pair so that the antivortex after the displacement by an external field pulse will spiral back to equilibrium thereby exhibiting gyrotropic motion. This will obviously not be an isolated antivortex as in the asteroid or clover-leaf, but the motion of the antivortex can be isolated if the vortex is pinned. A strong pinning center is a hole [20] in a thin film at the center of the nanocontact. This eliminates the exchange singularity at the vortex center without the corresponding increase in magnetostatic energy and it has been shown to inhibit gyrotropic motion of the vortex. Also the VA pair can be confined by the application of an inplane magnetic field where an increase in the field will decrease the VA separation. Therefore, this system will have as a static ground state the stable VA pair with the vortex pinned to the hole at the nanocontact center and the antivortex at some other point in the film. As will be shown later, the antivortex position is controlled by
the strength of the external field as well as the nanocontact current magnitude. The dynamics of the system will be restricted then to gyrotropic motion of the antivortex about this equilibrium point. In this article the dependence of the gyrotropic frequency on both the field and current is investigated using analytical techniques. The nanocontact device is placed in an external in-plane magnetic field defining the x-direction and the current is in the z-direction perpendicular to the film plane. A very simplified version of the device is illustrated in Fig. 1, where the magnetization is in the xy-plane (except for the small vortex core). Here the vortex is confined to the origin and the antivortex is at its equilibrium position in the y-axis. The current flows perpendicular to the layer through a contact of radius, r0 indicated by the shaded gray area, which can be a nucleation site for VA formation as a result of the Oersted field. The hole at the nano-ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi contact center is the order of the exchange length (le ¼ 2A=m0 Ms2 , where A¼1.3 10 11 J/m is the non-homogeneous exchange constant and Ms ¼ 8 105 A/m the saturation magnetization for permalloy giving le E5 nm) and is a pinning site for the vortex. Moreover, the film thickness is the order of a few nm so that the magnetization can be assumed to be uniform in the z-direction. The film magnetization is given by Ms sin y cos j, sin y sin j, cos yÞ in terms of polar and azimuthal angles, where Ms is the saturation magnetization. Rather than using the Landau–Lifshitz– Gilbert equation for the y and j dependent variables to describe the dynamics of the VA pair, we will use the Thiele equation [21], for the collective variables specifying the vortex and antivortex ! ! positions R1 ¼ 0 and R0 relative to the nanocontact center. Here only the antivortex position is time-dependent and it is determined by the solution of the corresponding Thiele equation ! ! ! d R0 d R0 @W aZ ¼ ! G dt dt @ R0
ð1Þ
where vortex dynamics will not be considered owing to the pinning ! and suppression of the vortex gyrotropic mode. Here G ¼ m0 Ms =g R sin y rj ry dV is the gyrovector, and the viscosity from i Rh Gilbert damping is Z ¼ m0 Ms =g ðryÞ2 þsin2 y ðrjÞ2 dV, where
m0Ms ffi1 T, g ¼2.2 105 m/As, and a ¼0.01 for permalloy. The right hand side of Eq. (1) is the force arising from all other sources of energy. In particular, we will consider W¼WEx + WOe +WZ + WMS for the exchange, Oersted, Zeeman and magnetostatic contributions, respectively.
y Antivortex
B
x
Fig. 1. Schematic of the nanocontact-hole structure with the vortex–antivortex static equilibrium position. The nanocontact is the gray circular region with a smaller hole at the origin of the x–y coordinate system. The vortex core is illustrated by anticlockwise circulation about the hole and the antivortex is illustrated by its core magnetic structure on the y-axis. The external field is parallel to the x-axis.
C.E. Zaspel / Journal of Magnetism and Magnetic Materials 323 (2011) 499–503
501
Fig. 2. Vortex–antivortex structure with the vortex at the center of the nanocontact and the antivortex on the y-axis.
Next it is necessary to assume a form for the magnetization ! specifying the variables y and j in terms of R0 . As was done [11] for analytical study of vortex dynamics, we begin with the static solution of the Landau–Lifshitz equation with exchange only and use this as an ansatz to obtain other contributions to the energy. The ansatz is a known solution [17,18] that will minimize the exchange and magnetostatic energies yY1 yY0 jðx,yÞ ¼ tan1 tan1 ð2Þ xX1 xX0 cos yðx,yÞ ¼ cos y1 ðxX1 ,yY1 Þ þ cos y0 ðxX0 ,yY0 Þ
ð3Þ
where (X1,Y1) and (X0,Y0) are the vortex and antivortex coordinates, respectively, for the magnetization independent of the z coordinate. The actual form of the polar angle is only known for a few special cases when exchange is the only contribution to the energy. Otherwise it is necessary to obtain the polar angle from numerical solution of a second order differential equation. Moreover, for application of this ansatz in the infinite thin film the only contribution to the magnetostatic energy is from the effective surface magnetostatic charge at the vortex and antivortex cores, where y a p=2. Since the vortex or antivortex core has an appreciable out-of-plane component only in the core area of radius the order of the exchange length, compared to other length scales in the nanocontact system, it is sufficient to assume that cos yðx,yÞ can be represented by d-functions centered at the appropriate cores simplifying calculation of the parameters in Eq. (1). Based on this ansatz the magnetic structure of the VA pair is illustrated in Fig. 2, where the nanocontact center is located at the core of the vortex. On this scale, the size of the core region is negligible.
3. Antivortex dynamics Using the above ansatz the parameters on the left hand side of Eq. (1) as well as the energies are obtained analytically in the polar coordinate system shown in Fig. 3, where the antivortex core center is specified by (R0,f0) and X0 ¼ R0 sin f0 and Y0 ¼ R0 cos f0 with f0 measured in an anticlockwise direction from the y-axis. These parameters on the left hand side of Eq. (1) as well as all contributions to the energy are calculated with the vortex pinned at the origin (X1 ¼0, Y1 ¼0). The parameters except the Oersted energy do not depend on f0 so they can be evaluated for the special case when the antivortex is on the y-axis (X0 ¼0, Y0 ¼R0). In the small core approximation the gyrovector is the only quantity that depends on the core structure and using Eqs. (2) and (3)
Fig. 3. Polar coordinates (R0,f0) used to describe motion of the vortex core in the (x, y) plane; a half cycle is also indicated.
with the d-function approximation for cos y, the gyrovector is ! G ¼ 2pm0 Ms Lp=g, where p ¼ 71 depending on the direction of the z component of the core magnetization. Using y ¼ p=2 throughout the region of integration with a lower radial cut-off at the exchange length, the damping coefficient is Z ¼ m0 Ms pLln R0 =le =g. Analytic techniques will be used to calculate the energy terms in Eq. (1) assuming that sin y ¼1 throughout the range of integration. First, the exchange energy is given by the integral Z 2 i 2 AL h 2 d r ðryÞ þ sin2 y rj ð4Þ WEx ¼ 2 An analytic expression is obtained for the exchange energy for the case when cos y ffi 0 outside the vortex core resulting in WEx ¼ ALpln
R0 le
ð5Þ
where le is again the lower limit of the radial integration. Oersted and Zeeman energies are best calculated in a polar (r, w) coordinate system with the nanocontact center at r ¼0. For the Oersted energy one must evaluate the integral Z 2 WOe ¼ L Mw Bw d r ð6Þ
where the Oersted field in the w^ direction is Bw ¼ m0 Ir=2pr02 for r rr0, Bw ¼ m0 I=2pr for r 4r0, and the w component of the magnetization is Mw ¼ Ms ðsin jcos wcos jsin wÞ. Finally the Zeeman energy is given by Z 2 WZ ¼ BMs L sin ycos j d r ð7Þ where B is the magnetic field in the x-direction. First the integral in Eq. (6) is evaluated using the time-dependent polar coordinates (R0(t), f0(t)) to specify the antivortex core position as illustrated in Fig. 3. Then using Eq. (2) the magnetization component is expressed in the simple form R0 cos f0 rsin w Mw ¼ Ms qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 r þ R20 2rR0 sin wf0
ð8Þ
C.E. Zaspel / Journal of Magnetism and Magnetic Materials 323 (2011) 499–503
Notice that this depends on the position of the antivortex reflecting the distortion of the structure as the antivortex moves off the y-axis. Since the main contributions to the Oersted and Zeeman integrals are from the regions outside the vortex and antivortex cores, it is again assumed that sin y ¼ 1 over most of the plane and the integral over w can be done exactly resulting in an expression involving elliptic integrals. Next asymptotic techniques are used to evaluate the radial integral. For this step the elliptic integrals are expanded in powers of r for the region, r oY0 and powers of Y0 for the region, r 4Y0 and the radial integrals are performed. By expansion in arbitrary powers, it is possible to find the power where there is no appreciable change in the value of the integral as the power is increased. Using this method, the dominant term in the Oersted energy is determined to be (for the condition, R0 ble) m Ms IL R R0 ln WOe ¼ 0 ð9Þ cos f0 2 R0 where R is the upper limit of the radial integration and time dependence is contained in the variables specifying the vortex core position. For higher order approximations the radial, R0, dependence will be modified, but the cos f0 factor remains owing to the system symmetry. In a similar manner the Zeeman energy is calculated according to the integral in Eq. (7), but the external field is uniform with ! B ¼ Bx^ and the x-component of the magnetization is Mx ¼ Ms cos j where rR0 sin wf0 cos j ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð10Þ ffi r 2 þ R20 2rR0 sin wf0 resulting in the dominant contribution to the energy, R WZ ¼ 2pMs LBR20 ln R0
dR0 df0 m Ms I R ln aZR0 ¼ 0 sin f0 2 R0 dt dt
150
100
I = 0.01A
50 I = 0.005A 2
4
6
8
10
B (mT) Fig. 4. Static equilibrium position of the antivortex versus external field.
ð11Þ
This is the increase in energy above the uniform state as a result of separation of the VA pair; moreover, it does not depend on the orientation, f0 of the VA pair in the xy-plane. If the antivortex displacement from the y-axis is small, then cos j0 4 0 and the Oersted field will tend to lower the energy as the separation is increased, but the Zeeman energy will increase as the separation increases; therefore, the external field tends to confine the VA pair. Finally, Eq. (9) clearly indicates that the antivortex must be on the y-axis at static equilibrium. To determine the static equilibrium position and investigate the dynamics of the antivortex, the radial and azimuthal components of the Thiele equation are solved using the calculated Oersted and Zeeman energies df0 dR0 m0 Ms I R A GR0 cos f0 4pMs BR0 þ aZ ¼ 1 ln 2 R0 dt dt pR0 ð12Þ G
200
R0 (nm)
502
ð13Þ
To numerically solve Eqs. (12) and (13) it is necessary to estimate an effective system size R, but because of the logarithmic dependence this estimate is not critical. Therefore, one can consider R to be the lateral extent of the free layer. In the following the upper limit, R¼104 nm is used. First, the static structure is illustrated in Fig. 4 for j0 ¼0 by solution of Eq. (12) giving the vortex position on the y-axis versus the static Zeeman field for fixed current. Notice that this deviates slightly from linearity for small values of magnetic field. For fixed values of Zeeman field, the core position exhibits a simple linear increase with current.
Fig. 5. Gyrotropic motion of the vortex core for I ¼0.01 A and B0 ¼5 mT.
Gyrotropic motion is excited by application of a magnetic field pulse along the static field direction, moving the antivortex in or out from R0 depending on whether the pulse is in the positive or negative y direction, respectively. In this case the total field can be expressed as B ¼B0 +B(t), where the first term is the static field and the second term is a step-function pulse that is switched on and off. In the following the initial displacement of the antivortex is related to the magnitude of B(t) and frequencies are calculated for various values of B0. After the pulse is turned off the antivortex will spiral back to the equilibrium position on the y-axis. In Fig. 5 this type of motion is schematically illustrated by numerical solution of Eqs. (12) and (13) for I¼0.01 A and B0 ¼5 mT with a pulse starting the antivortex core at 100 nm. The relatively large amplitude gyrotropic motion of the core is clearly seen as it spirals toward the static equilibrium at X0 ¼0 and Y0 ¼86 nm. To investigate the effects of Zeeman field and nanocontact current on the gyrotropic frequency, Eqs. (12) and (13) are solved numerically for an initial condition given by the antivortex core on the y-axis, but slightly displaced from the static equilibrium position. The effect of magnetic field on the frequency for two different fixed currents is illustrated in Fig. 6. Here is noticed a significant frequency increase as the magnetic field is increased with an approximately linear relation above about 5 mT. Moreover, the current dependence appears to be rather small. This current dependence is better shown in Fig. 7, which was obtained
C.E. Zaspel / Journal of Magnetism and Magnetic Materials 323 (2011) 499–503
5
4
Freq. (GHz)
I = 0.005 A 3 I = 0.01 A
2
1
2
4
6
8
10
12
B0 (mT) Fig. 6. Frequency versus external field for two different values of the current.
2.0
Freq. (GHz)
1.5
503
There is also a critical current related to the initial formation of the VA pair. This can be estimated from the Oersted energy required to overcome the exchange barrier as the pair forms. Since the VA pair is small at its initial formation, and the main contribution to W is from exchange, it is best to use the Belavin–Polyakov [17] energy, WBP ¼4pA independent of the VA separation. Equating WOe and WBP, using parameters for permalloy, the current necessary for VA formation depends on the separation. If one chooses the separation to be the order of the exchange length, then this current is the order of 4–6 mA. For smaller separations it will be necessary to evaluate energies on a discrete lattice, but this method should be sufficient to give an order of magnitude estimate. If the current is large enough to form a VA pair and the separation is roughly greater than 2le, then any value of the current can support gyrotropic antivortex motion since the VA pair is a metastable state. Moreover, the existence of the metastable VA state implies that hysteresis effect might be observed as the current and magnetic field are varied. In conclusion, both an Oersted field and a uniform external magnetic field will result in a stable static VA structure in a thin film nanocontact device. Excitation by a field pulse gives gyrotropic motion of the antivortex exhibiting a frequency that increases as the magnetic field increases and decreases with increasing current from the confining effect of the external field. This is similar to the confining effect seen for the vortex in the thin disk where increased confinement leads to higher frequency [11]. This device provides a method to isolate antivortex motion, and observe the effects of Oersted and Zeeman fields on VA formation and gyrotropic frequency. Indeed, current and field-dependent hysteresis effects have been observed [15] for the case of in-plane spin polarization, and the VA metastable state could provide a possible explanation.
1.0 References
B = 5 mT
0.5
0.005
0.015
0.010
0.020
I (A) Fig. 7. Frequency versus current for B0 ¼5 mT.
for a fixed field of 5 mT. There are other lower amplitude higher frequency modes [11] that were not considered here. However, it is possible to investigate these modes using the same device and theoretical techniques through a higher order Thiele equation [22] that contains an effective antivortex mass as well as third order time derivatives.
4. Additional considerations and conclusion These data were calculated for small displacements from the static equilibrium position, but the non-linear frequency shift is very small for larger displacements. For example if the initial displacement is approximately the equilibrium, Y0 starting precession at y¼2Y0, then the antivortex will complete half of an orbit to the origin where it can annihilate the vortex at the origin. For this large amplitude the non-linear frequency shift is only a few percent.
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Antivortex gyrotropic motion in a nanocontact C.E. Zaspel n Department of Environmental Sciences, University of Montana-Western, Dillon, MT 59725 USA
a r t i c l e in f o
abstract
Article history: Received 13 August 2010 Available online 15 October 2010
A vortex–antivortex pair can form in the free layer of a nanocontact device as a result of the Oersted field produced by the current. In a large-scale free layer having uniform magnetization boundary conditions an in-plane external magnetic field will tend to confine the vortex–antivortex pair, which undergoes gyrotropic motion about the nanocontact after an initial displacement from the static equilibrium position. With the vortex pinned to a defect at the nanocontact the antivortex dynamics can be isolated and gyrotropic precession of the antivortex will be the dominant mode. The frequency of antivortex precession increases as the external magnetic field increases, and the frequency decreases as the nanocontact current increases. & 2010 Elsevier B.V. All rights reserved.
Keywords: Magnetic vortex Nanocontact Antivortex
1. Introduction The magnetic vortex in a confined geometry has generated much theoretical and experimental research during the past few years owing to its potential information storage [1] applications. A particular application involves the magnetic vortex, which tends to form in a confined nanoscale system such as a submicron disk, where the vortex is the actual ground state because of the competition between the exchange and magnetostatic energies. For this state over most of the disks magnetization will be in-plane and parallel to the disk edge, which minimizes magnetostatic poles. In-plane magnetization at the vortex center will result in a high energy exchange singularity, which is eliminated by the formation of an out-of-plane vortex core at the expense of magnetostatic pole formation at the disk faces. Typically in permalloy disks these vortex core radii are the order of 5–10 nm with two polarization states perpendicular to the vortex plane. Since the two polarization states of the vortex core have potential for information storage, it is necessary to understand the vortex core dynamic properties. For applications such as vortex core switching, the dynamics of the vortex core is dominated by the so-called gyrotropic mode, which is a sub-GHz precession of the vortex core after an initial displacement from the vortex center by a magnetic field pulse. This particular mode was directly observed [2] using time-resolved Kerr microscopy, and later [3] using time-resolved X-ray imaging. The precession is observed after the vortex is displaced from its static equilibrium position at the disk center by an in-plane external field pulse. After the pulse turns off, the displaced vortex core spirals back toward the disk center.
n
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For practical applications it is necessary to have an external control over the vortex core polarization either by an external field or electric currents. A recent promising method [4,5] to switch the core magnetization direction uses the vortex gyrotropic mode as well as the formation of vortex–antivortex (VA) pairs excited by a spin-polarized current. The physical mechanism involves the formation of VA pairs of opposite core polarities, and subsequent annihilation of the original vortex and antivortex leaving a vortex of switched polarity, which is a complicated effect to study using analytical theoretical techniques because it involves the interaction of VA pairs. Another application is the spin-torque nanooscillator [6–8], where a vortex is formed at the nanocontact and the gyrotropic mode is excited by a spin-polarized current giving an output in the sub-GHz range. The output features can be improved by synchronization [7] of two nano-oscillators, and recently [8] it was shown that four nanocontacts are synchronized, or phaselocked through the VA interaction. Motivated by the need to better understand VA pair dynamics, in this article a simple thin-film device is proposed allowing the dynamics of an antivortex in the VA pair to be experimentally and theoretically investigated. The dynamics of the simpler single vortex in the nanodisk is now well understood and has been investigated using both numerical [9,10] and analytical [11] techniques. In particular, the gyrotropic mode frequency is simply related [11] to the disk aspect ratio, f 0:2oM L=R, where oM ¼30 GHz is a characteristic frequency for permalloy, and L and R are the disk thickness and radius, respectively. However, much less has been done in relation to the dynamics of either the VA pair or the isolated antivortex. Recently numerical simulations [12] have shown that the single isolated antivortex can form in an asteroid-shaped magnetic particle with the gyrotropic mode excited by the application of a field pulse. Similar simulations [13] have been done for a clover-leaf sample, which also exhibits an
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antivortex state. In this case the effect of spin-polarized current as well as an external field on the gyrotropic mode amplitude were investigated, indicating that there can be suppression of this mode. The dynamics of the single antivortex are of interest, but because of the fabrication difficulty of the asteroid or clover-leaf shape it is difficult to investigate this isolated structure experimentally. In this article a device is proposed that can be used to investigate the dynamic properties of a single antivortex as a VA pair forms in a nanocontact free layer. Previously the VA pair, referred to as the vortex dipole, has been observed [14] in a permalloy ellipse in a nanopillar device excited by a spin-polarized current. Usually in a confined system such as a thin disk a single vortex is formed, since this structure minimizes the edge magnetostatic charge. However, in the nanopillar the VA pair can also form because of the competing effects from the Oersted field resulting from the electric current and the effective pinning field between the fixed and free layers. The pair tends to form because the free layer magnetization is uniform far from the VA pair, which will minimize the magnetostatic interaction energy between the layers; however, the Oersted energy is minimized by the formation of the vortex dipole symmetric about the ellipse center. Spin-torque then forces the vortex dipole to undergo gyrotropic motion about the center of symmetry. Gyrotropic motion of a single vortex has also been observed [15,16] in a nanocontact where the free layer is very large in relation to the nanocontact. In this case it is expected that the boundary condition (uniform magnetization) far from the nanocontact implies the existence of a VA pair. Indeed, the vortex dipole is a static solution [17,18] of the Landau–Lifshitz equation where the effective field is due to the exchange interaction only. Past numerical simulations [16] for the nanocontact with both the external field and the spin polarization perpendicular to the free layer suggest that a vortex forms as a result of the Oersted field at the nanocontact with eventual orbital motion of the vortex about the nanocontact. However, since the simulation was performed on a confined system of 1000 nm diameter, the formation of a stable antivortex was not observed or it was possibly forced outside of the system. In the following a very large thin film with a single nanocontact in a uniform magnetic field is considered. A large enough nanocontact current will tend to nucleate the vortex core with the formation of an antivortex to give the uniform boundary conditions defined by the magnetic field. The antivortex has a static equilibrium position outside the nanocontact, and the gyrotropic motion of this antivortex is investigated.
2. Device and system model To isolate the antivortex and observe its dynamics it is necessary to form the vortex on a pinning center [19,20] in the nanocontact so it will not undergo gyrotropic motion. Also it is necessary to confine the VA pair so that the antivortex after the displacement by an external field pulse will spiral back to equilibrium thereby exhibiting gyrotropic motion. This will obviously not be an isolated antivortex as in the asteroid or clover-leaf, but the motion of the antivortex can be isolated if the vortex is pinned. A strong pinning center is a hole [20] in a thin film at the center of the nanocontact. This eliminates the exchange singularity at the vortex center without the corresponding increase in magnetostatic energy and it has been shown to inhibit gyrotropic motion of the vortex. Also the VA pair can be confined by the application of an inplane magnetic field where an increase in the field will decrease the VA separation. Therefore, this system will have as a static ground state the stable VA pair with the vortex pinned to the hole at the nanocontact center and the antivortex at some other point in the film. As will be shown later, the antivortex position is controlled by
the strength of the external field as well as the nanocontact current magnitude. The dynamics of the system will be restricted then to gyrotropic motion of the antivortex about this equilibrium point. In this article the dependence of the gyrotropic frequency on both the field and current is investigated using analytical techniques. The nanocontact device is placed in an external in-plane magnetic field defining the x-direction and the current is in the z-direction perpendicular to the film plane. A very simplified version of the device is illustrated in Fig. 1, where the magnetization is in the xy-plane (except for the small vortex core). Here the vortex is confined to the origin and the antivortex is at its equilibrium position in the y-axis. The current flows perpendicular to the layer through a contact of radius, r0 indicated by the shaded gray area, which can be a nucleation site for VA formation as a result of the Oersted field. The hole at the nano-ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi contact center is the order of the exchange length (le ¼ 2A=m0 Ms2 , where A¼1.3 10 11 J/m is the non-homogeneous exchange constant and Ms ¼ 8 105 A/m the saturation magnetization for permalloy giving le E5 nm) and is a pinning site for the vortex. Moreover, the film thickness is the order of a few nm so that the magnetization can be assumed to be uniform in the z-direction. The film magnetization is given by Ms sin y cos j, sin y sin j, cos yÞ in terms of polar and azimuthal angles, where Ms is the saturation magnetization. Rather than using the Landau–Lifshitz– Gilbert equation for the y and j dependent variables to describe the dynamics of the VA pair, we will use the Thiele equation [21], for the collective variables specifying the vortex and antivortex ! ! positions R1 ¼ 0 and R0 relative to the nanocontact center. Here only the antivortex position is time-dependent and it is determined by the solution of the corresponding Thiele equation ! ! ! d R0 d R0 @W aZ ¼ ! G dt dt @ R0
ð1Þ
where vortex dynamics will not be considered owing to the pinning ! and suppression of the vortex gyrotropic mode. Here G ¼ m0 Ms =g R sin y rj ry dV is the gyrovector, and the viscosity from i Rh Gilbert damping is Z ¼ m0 Ms =g ðryÞ2 þsin2 y ðrjÞ2 dV, where
m0Ms ffi1 T, g ¼2.2 105 m/As, and a ¼0.01 for permalloy. The right hand side of Eq. (1) is the force arising from all other sources of energy. In particular, we will consider W¼WEx + WOe +WZ + WMS for the exchange, Oersted, Zeeman and magnetostatic contributions, respectively.
y Antivortex
B
x
Fig. 1. Schematic of the nanocontact-hole structure with the vortex–antivortex static equilibrium position. The nanocontact is the gray circular region with a smaller hole at the origin of the x–y coordinate system. The vortex core is illustrated by anticlockwise circulation about the hole and the antivortex is illustrated by its core magnetic structure on the y-axis. The external field is parallel to the x-axis.
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Fig. 2. Vortex–antivortex structure with the vortex at the center of the nanocontact and the antivortex on the y-axis.
Next it is necessary to assume a form for the magnetization ! specifying the variables y and j in terms of R0 . As was done [11] for analytical study of vortex dynamics, we begin with the static solution of the Landau–Lifshitz equation with exchange only and use this as an ansatz to obtain other contributions to the energy. The ansatz is a known solution [17,18] that will minimize the exchange and magnetostatic energies yY1 yY0 jðx,yÞ ¼ tan1 tan1 ð2Þ xX1 xX0 cos yðx,yÞ ¼ cos y1 ðxX1 ,yY1 Þ þ cos y0 ðxX0 ,yY0 Þ
ð3Þ
where (X1,Y1) and (X0,Y0) are the vortex and antivortex coordinates, respectively, for the magnetization independent of the z coordinate. The actual form of the polar angle is only known for a few special cases when exchange is the only contribution to the energy. Otherwise it is necessary to obtain the polar angle from numerical solution of a second order differential equation. Moreover, for application of this ansatz in the infinite thin film the only contribution to the magnetostatic energy is from the effective surface magnetostatic charge at the vortex and antivortex cores, where y a p=2. Since the vortex or antivortex core has an appreciable out-of-plane component only in the core area of radius the order of the exchange length, compared to other length scales in the nanocontact system, it is sufficient to assume that cos yðx,yÞ can be represented by d-functions centered at the appropriate cores simplifying calculation of the parameters in Eq. (1). Based on this ansatz the magnetic structure of the VA pair is illustrated in Fig. 2, where the nanocontact center is located at the core of the vortex. On this scale, the size of the core region is negligible.
3. Antivortex dynamics Using the above ansatz the parameters on the left hand side of Eq. (1) as well as the energies are obtained analytically in the polar coordinate system shown in Fig. 3, where the antivortex core center is specified by (R0,f0) and X0 ¼ R0 sin f0 and Y0 ¼ R0 cos f0 with f0 measured in an anticlockwise direction from the y-axis. These parameters on the left hand side of Eq. (1) as well as all contributions to the energy are calculated with the vortex pinned at the origin (X1 ¼0, Y1 ¼0). The parameters except the Oersted energy do not depend on f0 so they can be evaluated for the special case when the antivortex is on the y-axis (X0 ¼0, Y0 ¼R0). In the small core approximation the gyrovector is the only quantity that depends on the core structure and using Eqs. (2) and (3)
Fig. 3. Polar coordinates (R0,f0) used to describe motion of the vortex core in the (x, y) plane; a half cycle is also indicated.
with the d-function approximation for cos y, the gyrovector is ! G ¼ 2pm0 Ms Lp=g, where p ¼ 71 depending on the direction of the z component of the core magnetization. Using y ¼ p=2 throughout the region of integration with a lower radial cut-off at the exchange length, the damping coefficient is Z ¼ m0 Ms pLln R0 =le =g. Analytic techniques will be used to calculate the energy terms in Eq. (1) assuming that sin y ¼1 throughout the range of integration. First, the exchange energy is given by the integral Z 2 i 2 AL h 2 d r ðryÞ þ sin2 y rj ð4Þ WEx ¼ 2 An analytic expression is obtained for the exchange energy for the case when cos y ffi 0 outside the vortex core resulting in WEx ¼ ALpln
R0 le
ð5Þ
where le is again the lower limit of the radial integration. Oersted and Zeeman energies are best calculated in a polar (r, w) coordinate system with the nanocontact center at r ¼0. For the Oersted energy one must evaluate the integral Z 2 WOe ¼ L Mw Bw d r ð6Þ
where the Oersted field in the w^ direction is Bw ¼ m0 Ir=2pr02 for r rr0, Bw ¼ m0 I=2pr for r 4r0, and the w component of the magnetization is Mw ¼ Ms ðsin jcos wcos jsin wÞ. Finally the Zeeman energy is given by Z 2 WZ ¼ BMs L sin ycos j d r ð7Þ where B is the magnetic field in the x-direction. First the integral in Eq. (6) is evaluated using the time-dependent polar coordinates (R0(t), f0(t)) to specify the antivortex core position as illustrated in Fig. 3. Then using Eq. (2) the magnetization component is expressed in the simple form R0 cos f0 rsin w Mw ¼ Ms qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 r þ R20 2rR0 sin wf0
ð8Þ
C.E. Zaspel / Journal of Magnetism and Magnetic Materials 323 (2011) 499–503
Notice that this depends on the position of the antivortex reflecting the distortion of the structure as the antivortex moves off the y-axis. Since the main contributions to the Oersted and Zeeman integrals are from the regions outside the vortex and antivortex cores, it is again assumed that sin y ¼ 1 over most of the plane and the integral over w can be done exactly resulting in an expression involving elliptic integrals. Next asymptotic techniques are used to evaluate the radial integral. For this step the elliptic integrals are expanded in powers of r for the region, r oY0 and powers of Y0 for the region, r 4Y0 and the radial integrals are performed. By expansion in arbitrary powers, it is possible to find the power where there is no appreciable change in the value of the integral as the power is increased. Using this method, the dominant term in the Oersted energy is determined to be (for the condition, R0 ble) m Ms IL R R0 ln WOe ¼ 0 ð9Þ cos f0 2 R0 where R is the upper limit of the radial integration and time dependence is contained in the variables specifying the vortex core position. For higher order approximations the radial, R0, dependence will be modified, but the cos f0 factor remains owing to the system symmetry. In a similar manner the Zeeman energy is calculated according to the integral in Eq. (7), but the external field is uniform with ! B ¼ Bx^ and the x-component of the magnetization is Mx ¼ Ms cos j where rR0 sin wf0 cos j ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð10Þ ffi r 2 þ R20 2rR0 sin wf0 resulting in the dominant contribution to the energy, R WZ ¼ 2pMs LBR20 ln R0
dR0 df0 m Ms I R ln aZR0 ¼ 0 sin f0 2 R0 dt dt
150
100
I = 0.01A
50 I = 0.005A 2
4
6
8
10
B (mT) Fig. 4. Static equilibrium position of the antivortex versus external field.
ð11Þ
This is the increase in energy above the uniform state as a result of separation of the VA pair; moreover, it does not depend on the orientation, f0 of the VA pair in the xy-plane. If the antivortex displacement from the y-axis is small, then cos j0 4 0 and the Oersted field will tend to lower the energy as the separation is increased, but the Zeeman energy will increase as the separation increases; therefore, the external field tends to confine the VA pair. Finally, Eq. (9) clearly indicates that the antivortex must be on the y-axis at static equilibrium. To determine the static equilibrium position and investigate the dynamics of the antivortex, the radial and azimuthal components of the Thiele equation are solved using the calculated Oersted and Zeeman energies df0 dR0 m0 Ms I R A GR0 cos f0 4pMs BR0 þ aZ ¼ 1 ln 2 R0 dt dt pR0 ð12Þ G
200
R0 (nm)
502
ð13Þ
To numerically solve Eqs. (12) and (13) it is necessary to estimate an effective system size R, but because of the logarithmic dependence this estimate is not critical. Therefore, one can consider R to be the lateral extent of the free layer. In the following the upper limit, R¼104 nm is used. First, the static structure is illustrated in Fig. 4 for j0 ¼0 by solution of Eq. (12) giving the vortex position on the y-axis versus the static Zeeman field for fixed current. Notice that this deviates slightly from linearity for small values of magnetic field. For fixed values of Zeeman field, the core position exhibits a simple linear increase with current.
Fig. 5. Gyrotropic motion of the vortex core for I ¼0.01 A and B0 ¼5 mT.
Gyrotropic motion is excited by application of a magnetic field pulse along the static field direction, moving the antivortex in or out from R0 depending on whether the pulse is in the positive or negative y direction, respectively. In this case the total field can be expressed as B ¼B0 +B(t), where the first term is the static field and the second term is a step-function pulse that is switched on and off. In the following the initial displacement of the antivortex is related to the magnitude of B(t) and frequencies are calculated for various values of B0. After the pulse is turned off the antivortex will spiral back to the equilibrium position on the y-axis. In Fig. 5 this type of motion is schematically illustrated by numerical solution of Eqs. (12) and (13) for I¼0.01 A and B0 ¼5 mT with a pulse starting the antivortex core at 100 nm. The relatively large amplitude gyrotropic motion of the core is clearly seen as it spirals toward the static equilibrium at X0 ¼0 and Y0 ¼86 nm. To investigate the effects of Zeeman field and nanocontact current on the gyrotropic frequency, Eqs. (12) and (13) are solved numerically for an initial condition given by the antivortex core on the y-axis, but slightly displaced from the static equilibrium position. The effect of magnetic field on the frequency for two different fixed currents is illustrated in Fig. 6. Here is noticed a significant frequency increase as the magnetic field is increased with an approximately linear relation above about 5 mT. Moreover, the current dependence appears to be rather small. This current dependence is better shown in Fig. 7, which was obtained
C.E. Zaspel / Journal of Magnetism and Magnetic Materials 323 (2011) 499–503
5
4
Freq. (GHz)
I = 0.005 A 3 I = 0.01 A
2
1
2
4
6
8
10
12
B0 (mT) Fig. 6. Frequency versus external field for two different values of the current.
2.0
Freq. (GHz)
1.5
503
There is also a critical current related to the initial formation of the VA pair. This can be estimated from the Oersted energy required to overcome the exchange barrier as the pair forms. Since the VA pair is small at its initial formation, and the main contribution to W is from exchange, it is best to use the Belavin–Polyakov [17] energy, WBP ¼4pA independent of the VA separation. Equating WOe and WBP, using parameters for permalloy, the current necessary for VA formation depends on the separation. If one chooses the separation to be the order of the exchange length, then this current is the order of 4–6 mA. For smaller separations it will be necessary to evaluate energies on a discrete lattice, but this method should be sufficient to give an order of magnitude estimate. If the current is large enough to form a VA pair and the separation is roughly greater than 2le, then any value of the current can support gyrotropic antivortex motion since the VA pair is a metastable state. Moreover, the existence of the metastable VA state implies that hysteresis effect might be observed as the current and magnetic field are varied. In conclusion, both an Oersted field and a uniform external magnetic field will result in a stable static VA structure in a thin film nanocontact device. Excitation by a field pulse gives gyrotropic motion of the antivortex exhibiting a frequency that increases as the magnetic field increases and decreases with increasing current from the confining effect of the external field. This is similar to the confining effect seen for the vortex in the thin disk where increased confinement leads to higher frequency [11]. This device provides a method to isolate antivortex motion, and observe the effects of Oersted and Zeeman fields on VA formation and gyrotropic frequency. Indeed, current and field-dependent hysteresis effects have been observed [15] for the case of in-plane spin polarization, and the VA metastable state could provide a possible explanation.
1.0 References
B = 5 mT
0.5
0.005
0.015
0.010
0.020
I (A) Fig. 7. Frequency versus current for B0 ¼5 mT.
for a fixed field of 5 mT. There are other lower amplitude higher frequency modes [11] that were not considered here. However, it is possible to investigate these modes using the same device and theoretical techniques through a higher order Thiele equation [22] that contains an effective antivortex mass as well as third order time derivatives.
4. Additional considerations and conclusion These data were calculated for small displacements from the static equilibrium position, but the non-linear frequency shift is very small for larger displacements. For example if the initial displacement is approximately the equilibrium, Y0 starting precession at y¼2Y0, then the antivortex will complete half of an orbit to the origin where it can annihilate the vortex at the origin. For this large amplitude the non-linear frequency shift is only a few percent.
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