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Excitation and damping of spin excitations in ferromagnetic thin films
Journal of Magnetism and Magnetic Materials 241 (2002) 96–109
Excitation and damping of spin excitations in ferromagnetic thin films J. Wu, N.D. Hughes, J.R. Moore, R.J. Hicken* School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, UK Received 8 February 2001; received in revised form 26 July 2001
Abstract We present the results of optical pump-probe measurements made on Ni81Fe19 films of thickness d ¼ 50; 500 and ( We show that the rise time of the pulsed field within the sample may be determined, and that when d is 5000 A. sufficiently small, the value of the Gilbert damping parameter may be obtained from the decay of the magneto-optical signal. The pulsed field was found to rise more slowly when applied perpendicular to the plane of the sample and as the thickness of the sample was increased. By changing the orientation of the pulsed field relative to the static magnetisation we were able to alter the ellipticity of the trajectory of the precessing magnetisation and observe variations in the decay of the precession. We discuss these effects in terms of eddy current shielding of the rising field, eddy current damping of the motion of the magnetisation, and propagation of spin waves from the point at which the sample response is probed. ( we found the value of the Gilbert damping parameter to be strongly field For the thinnest sample studied (d ¼ 50 A) dependent, so that, contrary to our expectations, the calculated Ferromagnetic Resonance (FMR) line width increased as the field was reduced. Under certain circumstances a second mode of higher frequency was observed in the thickest film. We believe this to be a magnetostatic surface mode that has not previously been observed by means of the optical pump-probe technique. r 2002 Elsevier Science B.V. All rights reserved. PACS: 76.50.+g; 78.20.Ls; 78.47.+p Keywords: Ferromagnetic resonance; Damping; Eddy currents; Magnetostatic waves
1. Introduction Nanostructured ferromagnetic materials exhibit spin excitations that may be very different to those of bulk materials. The subject has attracted considerable interest as researchers attempt to explain the form of the modified spin wave *Corresponding author. Tel.: +44-1392-264153; fax: +441392-264111. E-mail address: [email protected] (R.J. Hicken).
spectrum. The measurement of spin wave frequencies can also yield a wealth of information about magnetic moments, anisotropy and exchange interactions in structured materials [1]. However the damping of the same spin wave excitations has so far received less attention. The damping in a thin film is often described by a single phenomenological term in the equations of motion for the magnetisation. This represents the combined effect of a variety of mechanisms that may include dephasing due to the spin-orbit interaction [2],
0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 9 3 0 - 1
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exchange conductivity effects [3], and the generation of additional spin wave modes either by twomagnon scattering at defects [4] or by non-linear processes [5]. Recently, observed variations in Ferromagnetic Resonance (FMR) line widths in exchange biased films [6] have motivated calculations of two-magnon scattering at surface defects in ultrathin films [7], that can be used to understand the dependence of line width upon film thickness [8]. Numerical simulations have shown that dephasing of the large amplitude precession of the magnetisation of nanoscale elements occurs principally by spin wave generation [9]. Magnetic damping is also of considerable technological importance. Damping effects the bandwidth and quality factors of microwave magnetic devices. The next generation of magnetic recording systems will require sub-nanosecond magnetic switching in both the storage medium and the read and write transducers. The size of the angle through which the magnetisation is switched is influenced by both the rise time of the applied field and the damping. There is already evidence that the field rise time within films with thickness as small as 20 nm is affected by eddy currents [10,11]. The time taken for the magnetisation to relax into a new orientation is affected by the damping although ringing can also be suppressed if the pulse shape and duration are matched to the precessional motion [12]. As device sizes continue to be reduced to micron and even nanometre length scales, it is clearly important to study damping in the nanostructured magnetic materials that these devices contain. Damping has previously been studied by absorption of microwaves of fixed frequency as the value of an applied magnetic field is slowly swept. The damping may then be related to the FMR line width. Variable frequency resonant cavities can be used to measure the dependence of line width on frequency, however the microwave technique lacks the spatial resolution that is required to conveniently investigate structures with small lateral dimensions. Optical techniques are favoured since a probing laser spot can be focused to a diameter of less than a micron. Inelastic light scattering is a powerful means by which to study spin wave frequencies, but the
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instrumental resolution is sometimes insufficient for the line width and hence the damping to be reliably determined. Recently a number of authors have shown that optical pump-probe spectroscopy can be used to observe picosecond spin excitations in the time domain [10,11,13–15]. Excitation with pulsed magnetic fields allows a broad band of spin excitations to be observed as the value of a static bias magnetic field is varied. The dependence of the damping parameter on frequency may then be explored. In this paper we present the results of optical pump-probe measurements made on Ni81Fe19 films of thickness d ¼ 50; 500 and ( The experimental technique offers flex5000 A. ibility in the choice of experimental geometry that allows information to be obtained about the rise time of the pulsed field within the sample. Since the pulsed field is spatially inhomogeneous, we discuss the character of the long wavelength magnetostatic modes that are excited. We will determine the conditions in which propagation of these modes may be neglected so that the damping may be deduced from the decay of the magnetooptical signal. We will also discuss how the eddy current damping may depend upon the geometry and how the damping depends upon the film thickness and the frequency of precession. Finally we show how, under certain circumstances, a surface magnetostatic mode may be observed.
2. Experiment Samples were prepared in a magnetron sputtering system equipped with a computer-controlled shutter and substrate platter. The pressure in the chamber was 3 107 Torr prior to growth and sputtering was performed at an Ar pressure of 3.5 mTorr. The Ni81Fe19 was sputtered onto glass ( layer of substrates and then covered with a 60 A Al2O3. Deposition rates were continuously monitored with quartz crystal oscillators that were calibrated by surface profilometer measurements on thick reference films. The films were first characterised by means of magnetometry measurements. These were performed with a longitudinal Magneto Optical Kerr Effect (MOKE) apparatus
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at a wavelength of 633 nm, and with an Alternating Gradient Magnetometer (AGM) apparatus. Our optical pump-probe technique is similar in principle to that introduced by Freeman and coworkers [16]. The apparatus is shown in schematic form in Fig. 1. The light source is a modelocked Ti:sapphire laser producing pulses with width of approximately 100 fs at a repetition rate of 82 MHz. Each pulse is divided into a pump and a probe part so that the average powers of the pump and probe beams are 300 and 7 mW, respectively. A corner cube retro-reflector mounted on a translation stage is used to set the path length of the pump beam before it arrives at the sample. The pump pulse is then used to trigger a magnetic field pulse generated by a device similar to that shown in schematic form in Fig. 2(a). The complete device consists of an interdigitated photoconductive switch attached to a coplanar transmission line that is terminated with surface mount components and connected to a 20 V
Fig. 2. (a) The interdigitated photoconductive switch and transmission line structures are shown in schematic format. (b) The transmission line and sample are shown in cross-section with the pulsed field and static field for geometries A and B (c) A typical oscilloscope trace is shown (d) The value of the inplane (continuous line) and out of plane (dashed line) components of the pulsed field are shown at a height of 5 mm above the transmission line.
Fig. 1. The optical pump probe apparatus is shown in schematic format.
supply by means of a coaxial cable. The device was placed between the pole pieces of an electromagnet designed to allow wide field optical access. The coplanar strips were always perpendicular to the plane of incidence of the probe beam but the static field was applied either perpendicular to, or in the plane of incidence as shown in Fig. 2(b). We refer to these as geometries A and B, respectively. A photolithographic lift-off technique was used to define the Au tracks on an intrinsic GaAs(1 0 0) substrate. The transmission line structure consisted of coplanar strips with width and separation
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( and resistance per of 30 mm, thickness of 600 A unit length r of about 30 O/mm. When the pump pulse is directed onto the interdigitated region the GaAs between the fingers conducts and a current pulse propagates along the transmission line. The use of the interdigitated structure allows the pump spot to be defocused so that the amplitude of the current pulse is insensitive to any small motion of the pump spot as the translation stage is moved. The magnetic field associated with the current interacts with the sample that is placed face down on the transmission line. The probe beam is intensity stabilised and expanded by a factor of 10 before being focused through the glass substrate to a spot size of approximately 20 mm on the surface of the Ni81Fe19 film. In this study measurements were made with a p-polarised probe beam of wavelength 730 nm incident at an angle of 431. The pump beam was chopped at a frequency close to 200 Hz and the optical rotation of the probe beam measured with a polarisation bridge in a phase sensitive manner. It is important to establish the magnitude and temporal profile of the current in the transmission line so that the form of the pulsed field can be determined. It is useful to consider a simple model in which a parallel wire transmission line is described by the Telegrapher’s equations [17]. Initially the transmission line is biased with voltage V and the load impedance ZL of the photoconductive switch is effectively infinite. The voltage reflection coefficient at the open circuit switch is unity. The voltage V on the transmission line can then be thought to arise from two counterpropagating pulses of voltage amplitude p V =2 and ffiffiffiffiffiffiffiffiffi ffi current amplitude 7V =2Z0 ; where Z0 ¼ L=C is the characteristic impedance of the transmission line, and L and C are the inductance and capacitance per unit length. When the switch is gated, the load impedance becomes finite and the voltage reflection coefficient becomes equal to ðZL Z0 Þ=ðZL þ Z0 Þ: A current pulse of amplitude V =ðZL þ Z0 Þ then propagates along the transmission line. We note that if the pump pulse is sufficiently intense that ZL 5Z0 ; the magnitude of the field pulse is limited by the value of Z0 : The value of Z0 depends upon the ratio of width w to separation s of the coplanar strips, and the
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dielectric constant er of the substrate [18]. In our case w=s ¼ 1; er ¼ 13; and we estimate that C ¼ 100 pF/m, L ¼ 740 nH/m and Z0 ¼ 86 O. The transmission line is also lossy and dispersive due to the finite dc resistance of the tracks and, for sufficiently high frequencies, due to the skin effect. The temporal profile of the current is complicated by reflections of the current pulse that occur at impedance mismatches between different parts of the device. Surface mount resistors of value 100 O were used to terminate the transmission line and a 500 MHz oscilloscope1 was attached across one of the resistors in order to monitor the shape of the pulse, as shown in Fig. 2(c). The values of the surface mount resistors must be chosen carefully if the value of the current in the transmission line is to be determined. At GHz frequencies small value resistors have considerable stray inductance while large values have considerable stray capacitance.2 If the component has resistance RSM and total impedance Z then jRSM =Zj is found to be closest to unity when RSM is of the order of 100 O. Small changes in gradient in the trace shown in Fig. 2(c) suggest the presence of multiple reflections. These become more pronounced if smaller resistor values are used since the current pulses are attenuated less by the resistors and instead they are reflected from the power supply with a round trip time of the order of a few nanoseconds. The available surface mount capacitance values generally decrease as the frequency of operation is increased and our experience is that, in practice, these components have little effect on the operation of the device shown in Fig. 2(a). The pulse in Fig. 2(c) has a finite rise time that reflects the bandwidth of the oscilloscope rather than that of the pulse itself. We expect that the current in the semiconductor substrate rises to its full value within 1 ps, the time taken for the photo-excited electrons to thermalise. Then the geometry of the interdigitated structure and the dispersion of the transmission line cause the rise time to be broadened to a value of around 10 ps at the 1
LeCroy LT342 Digital Sampling Oscilloscope. See for example Philips Components Application Note, ‘‘Space-saving 4-resistor arrays in 0804 package’’. 2
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sample. The decay of the pulse can be fit with an exponential that yields a relaxation time of 1.5 ns. By extrapolating back to zero time delay we estimate a peak pulse height of 17 mA. The current pulse propagates at a velocity of about 108 m/s on the transmission line. The full length of the transmission line is filled with current after about 30 ps, after which the current is essentially uniform along the length of the line as the pulse slowly decays. Assuming that the current is uniformly distributed through the cross-sectional area of the Au tracks we are able to calculate the magnetic field generated by the transmission line by numerical integration of the Biot–Savart law. The calculated field values change only slowly as the surface of the track is approached and the calculated magnetic field at a height of 5 mm above the tracks is shown in Fig. 2(d). We use these values of the pulsed field in the calculations presented below.
3. Theory The response of the magnetisation M of a ferromagnetic material to a pulsed field may be described by the Landau–Lifshitz–Gilbert equation qM a qM ¼ jgjM Heff þ M ð1Þ qt M qt in which a is the Gilbert damping constant, Heff is the total effective field acting upon the magnetisation and g ¼ g p ð2:80Þ MHz/Oe where g is the spectroscopic splitting factor. Measurements are normally made in the presence of a static applied field H that causes the sample to occupy a single domain state. Heff then includes H; the pulsed field hðtÞ within the film, and effective fields associated with the anisotropy, demagnetising and exchange energies. In this paper we simulate the trajectory of the uniform mode. The exchange field vanishes and the thin film is assumed to have the usual 4p demagnetising factor. We do not attempt to calculate the associated eddy current distributions. This is a complicated non-local problem that requires the simultaneous solution of Maxwell’s equations and lies beyond the scope of the present
study. When the orientation of M is specified by polar coordinates Eq. (1) reduces to two coupled first order linear differential equations [19] that may be solved with standard numerical techniques. Once the time-dependent vector magnetisation has been calculated, the magneto-optical response may be calculated from generalised Fresnel coefficients that describe the linear MOKE effect [11]. For small amplitude oscillations of the magnetisation, an algebraic expression may be obtained for o; the circular frequency of the uniform mode. In general the films possess a small in-plane uniaxial anisotropy. When the static field H is applied in the sample plane at an angle f relative to the easy axis, the anisotropy energy has the form Ku cos2 ðf yÞ in which Ku is the uniaxial anisotropy constant, and y is the angle between M and H: The value of y is given by MH sin y Ku sinð2ðf yÞÞ ¼ 0;
ð2Þ
while the uniform mode frequency is given by 2 o 2Ku cosð2ðf yÞÞ ¼ H cos y þ g M 2Ku H cos y þ cos2 ðf yÞ þ 4pM : M ð3Þ It is often observed that, when the film thickness is ( there is an additional less than about 100 A, uniaxial anisotropy present that has its axis perpendicular to the film plane. In this case Eq. (3) must be modified by introducing an effective demagnetising field, ð4pMÞeff ¼ 4pM H> ; in which H> is the perpendicular anisotropy field. The latter field may be associated with discontinuities in the film that reduce the shape anisotropy, or with surface anisotropy or magnetoelastic effects. The pulsed field shown in Fig. 2(d) is spatially inhomogeneous and the Fourier power spectra of both field components have maxima at in-plane wave vector values of k8 E460 rad=cm; that correspond to a wavelength of about 140 mm. The induced dynamical magnetisation therefore consists of a superposition of long wavelength modes. If we write the in-plane wave vector component as k8 ; the effective field associated
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with the exchange interaction may be written as ð2A=MÞk82 ; where A is the exchange constant. Assuming values of A ¼ 1:22 106 erg/cm and M ¼ 860 emu/cm3 for bulk Ni81Fe19 we obtain an exchange field value of 6 104 Oe for k8 ¼ 460 rad=cm that can be safely neglected. However the finite in-plane wave vector also gives rise to dipolar fields and then Eq. (1) must be solved simultaneously with the magnetostatic Maxwell equations as described by Damon and Eshbach [20]. Let k> be the wave vector component perpendicular to the film plane. For an isotropic film, when the exchange interaction is neglected, the general solution includes a surface mode (k> E0) and a manifold of volume modes with a series of allowed k> values. Let us consider the two geometries used in our study. In geometry A, the in-plane wave vector is orthogonal to M and the volume modes are degenerate with frequency o0 given by Eq. (3) with Ku set equal to zero. The surface mode lies above the degenerate volume modes. In the limit jk8 dj51; which will be satisfied in all of our experiments, the frequency of the surface modes is given by 2 2 o o0 k8 dð4pMÞ2 ; ð4Þ ¼ þ g g 2 to first order in k8 d: In geometry B, the in-plane wave vector is parallel to M: The degeneracy of the volume modes is lifted and the surface mode is replaced by a volume mode for which k> E0 at the bottom of the volume spin wave manifold. The frequency of this mode is given by 2 2 o o0 Hð4pMÞk8 d ; ð5Þ ¼ g 2 g to first order in k8 d: Eqs. (4) and (5) allow us to determine when the finite wave vector of the excited modes leads to a significant departure from the uniform mode frequency. They also allow us to evaluate the group velocity, vg ¼ qo=qk8 ; so that we may decide whether significant mode propagation is likely during our measurement. Finally, we must consider the effect of exchange upon the magnetostatic mode spectrum. Eqs. (4) and (5) will not be significantly affected since they describe modes for which k> E0: However the frequencies of the remaining volume modes will be
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shifted upwards. We can obtain an estimate of the size of this effect in the limit that k8 ¼ 0: The frequency of the modes in an isotropic film is given by 2 o 2A 2 2A 2 k ¼ Hþ H þ 4pM þ k> ; ð6Þ g M > M where the allowed values of k> form a series in which k> ¼ pp=d and p ¼ n d; where n is an integer and the value of d lies between 0 and 1 depend upon the pinning at the film interfaces [21]. The lowest frequency modes correspond to k> E0; which we have already discussed, and then k> Bp=d: This latter mode may lie above or below that described by Eq. (4) depending upon the film thickness.
4. Results Before making pump-probe measurements the in-plane anisotropy of the films was first characterised by Longitudinal Magneto Optical Kerr Effect (MOKE) and Alternating Gradient Magnetometer (AGM) measurements. MOKE hysteresis loops for the three films are shown in Fig. 3. ( films exhibit a simple We see that the 50 and 500 A in-plane uniaxial anisotropy with hard axis satura( tion fields of 10 and 11 Oe respectively. The 5000 A film was found to be isotropic within the film plane with a larger saturation field of 130 Oe. In the AGM measurements a single turn current loop was placed around the sample to provide a known magnetic moment so that the sample moment could be determined. The magnetisations of the ( films were found to be close to the 500 and 5000 A bulk value of 861 emu/cm3, while the magnetisa( sample was 296726 emu/cm3 at tion of the 50 A room temperature. For the pump-probe measurements the samples were cut into rectangular strips with width of about 2 mm before being overlaid on the transmission line. The long edge of the strip was perpendicular to the length of the transmission ( line and the in-plane easy axes of the 50 and 500 A samples were oriented at 531 and 401, respectively to the plane of incidence. The probe spot was first scanned across the coplanar strips in geometry A.
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Fig. 3. Normalised MOKE loops are shown for Ni81Fe19 films ( (b) 500 A ( (c) 5000 A. ( The continuous and of thickness (a) 50 A dotted curves were taken with the field applied parallel to the easy and hard axes respectively.
The amplitude of the Kerr signal exhibits two maxima that allow the positions of the coplanar strips to be identified. The signal from the three samples is shown in Fig. 4 at positions above and in between the coplanar strips. While a qualitatively similar behaviour is observed for the 50 and ( samples, the data for the 5000 A ( sample is 500 A clearly different being reminiscent of two modes beating together. Fast Fourier transforms were taken of this last set of data and the resulting power spectra confirm the presence of two modes of different frequency. Measurements were also ( samples in geometry made on the 500 and 5000 A B as shown in Fig. 5. This time the power spectra ( sample show just a single mode. for the 5000 A We have simulated the response of the two thinner samples, and one of the curves for the ( sample that exhibits single mode beha5000 A viour, using the uniform mode solution of Eq. (1). It is necessary to first specify the form of the pulsed
Fig. 4. The Kerr rotation measured above one track, in between the tracks, and above the other track of the ( transmission line in geometry A is shown for (a) d ¼ 50 A, ( H ¼ 210 Oe (c) d ¼ 5000 A, ( H ¼ 216 Oe (b) d ¼ 500 A, H ¼ 216 Oe. The normalised power spectra of the curves in (c) are shown in (d). The smooth simulated curves assume the parameter values shown in Table 1.
field hðtÞ: By this we mean the field within the sample since this may lag the applied field due to shielding by an induced eddy current distribution. Our experience is that 8 eh0 > > > < ðe 1Þ expðtrise =tÞð1 expðt=trise ÞÞ; tptrise ; h¼ trise otptrefl ; h0 expðt=tÞ; > > > : t > trefl ; h0 ½expðt=tÞ R expððt trefl Þ=tÞ ;
ð7Þ
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Fig. 6. The magnitude of the pulsed field specified by Eq. (6) has been sketched for the case that jh0 j ¼ 1:
Fig. 5. The Kerr rotation measured above one track, in between the tracks, and above the other track of the ( transmission line in geometry B is shown for (a) d ¼ 500 A, ( H ¼ 216 Oe. The normalised H ¼ 210 Oe (b) d ¼ 5000 A, power spectra of the curves in (b) are shown in (c). The smooth simulated curves assume the parameter values shown in Table 1.
is the simplest form for hðtÞ that adequately reproduces the main features of the experimental data. The form of the pulse has been sketched in Fig. 6. The peak field is close to h0 ; while the rise time of the pulse is controlled by the parameter trise : Other functional forms were tried for the rise of the pulse but the exponential form was found to give the best results. It also seems the most likely form if the rise time is largely determined by the decay of eddy currents. The decay of the pulse is assumed to follow that of the current shown in Fig. 2(c) with the same time constant t: We find that, in order to correctly reproduce the relative phase of successive oscillations of the Kerr rotation, it is essential to include a reflected pulse that arrives at the sample after time trefl : The value of trefl is of order 100 ps and so the reflection cannot be seen in Fig. 2(c) due to the limited
bandwidth of the oscilloscope. The values of trefl and R used for each sample are shown in Table 1. Since the three samples were placed and probed at different positions above the transmission line some variation in their values is not surprising. Once the position of the magnetisation has been calculated the instantaneous Kerr rotation is evaluated from Eqs. (5), (7) and (8) of Ref. [11]. These equations apply to the interface between two semi-infinite media, while in fact the probe beam passes from air, and through the glass substrate, before impinging upon a Ni81Fe19 film of finite thickness. Since the reflection coefficient at the air-glass interface is small, we ignore the weak reflection from this surface but take account of the modified angle of incidence, 271, in the denser (n ¼ 1:55) optical medium. Since the optical skin ( at the depth of the Ni81Fe19 is just 270 A wavelength of 730 nm, the assumption that the Ni81Fe19 is effectively semi-infinite is reasonable for the two thicker films. Since the optical constants of Ni81Fe19 are not well known at 730 nm, we instead use values determined for pure Ni. These are 2.35+4.24i and 0.00740.0046i for the refractive index [22] and magneto-optic con( sample the stant [23] respectively. For the 50 A value of the magneto-optic constant was scaled in proportion to the reduced room temperature magnetisation. Given the uncertainties in the values of the optical constants and the exact ( magnitude of the pulsed field, and since the 50 A film is thin compared to the optical skin depth, we do not attempt to reproduce the exact magnitude
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Table 1 Parameter values used in modelling the time resolved MOKE data are shown. The letters A and B refer to the two measurement geometries described within the text ( d (A)
50
500
5000
M (emu/cm3) g Hsat (Oe) H> (kOe) trise (ps)
296 2.15 10 2.12 10–40 (A)
860 2.11 130 0 90 (B)
trefl (ps) R a
205 0.2 0.121 (A)
800 2.00 11 0 20–70 (A) 70 (B) 95 0.5 0.0083 (A) 0.0076 (B)
165 0.2 0.0129 (B)
of the measured Kerr rotation, although experiment and simulation rarely disagree by a factor of more than two. By manually adjusting the values of trise ; trefl ; R and the damping constant a; our aim has been to reproduce the initial rise of the Kerr signal, and then the phase and decay of the ensuing oscillations. Since the orientation of the pumping field varies continuously above the strips and the probe spot has a finite diameter, the simulations are an average of 3 points that are each 10 mm apart. The good agreement between data and simulation in Figs. 4 and 5 attest to the success of this approach and the values of parameters used in the simulation are listed in Table 1. In practice the biggest problem faced in performing the simulations is that the frequency of oscillation is slightly different between data and simulation. Wide field optical access to the sample was obtained at the expense of the homogeneity of the field generated by the electromagnet and, while the field is certainly uniform over the area of the sample, additional uncertainty is introduced into the measurement of the field value. The frequency discrepancy causes successive oscillations of the simulation to accumulate an increasing phase error relative to the experimental data. Clearly the smallest phase error occurs for the first peak in the curve, and it is this that we try to reproduce by varying the trise parameter. We also wished to explore the effect of the magnitude of the static field H ¼ jHj upon the
response of the magnetisation. Measurements were performed in geometry A with the probe spot above one of the coplanar strips. The response obtained for different values of H has been plotted ( sample in Figs. 7–9. Power spectra for the 5000 A again confirmed the presence of two modes while ( samples showed only those for the 50 and 500 A single peaks. We have only simulated the Kerr ( samples for which response for the 50 and 500 A single mode behaviour was observed. The simulated curves are again found to be in reasonable agreement with the experimental data. However there is a striking difference between the 50 and ( samples. For the latter the entire series of 500 A
Fig. 7. In (a) the measured and simulated Kerr rotation are ( sample when the probe spot is above one of shown of the 50 A the tracks of the transmission line. The smooth simulated curves assume the parameter values shown in Table 1 and the values of the damping constant a shown in (b). The effective FMR linewidth DH ¼ 2ao=g is plotted in (c).
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Fig. 10. Mode frequencies are plotted as a function of static ( (+, lower continuous curve), 500 A ( (W, field H for d ¼ 50 A ( (K, long dashed curve and ’, short dashed curve), 5000 A upper continuous curve). The curves assume the parameter values in Table 1 and are discussed in the text. Fig. 8. The measured and simulated Kerr rotation are shown ( sample when the probe spot is above one of the for the 500 A tracks of the transmission line. The smooth simulated curves assume the parameter values shown in Table 1.
( Fig. 9. The measured Kerr rotation is shown for the 5000 A sample when the probe spot is above one of the tracks of the transmission line.
curves is well described by the single damping ( factor shown in Table 1. However for the 50 A sample the damping of the oscillations appears to
increase as the value of H is decreased. The values of a used in the simulations have been plotted in Fig. 7(b). The frequencies of all the modes observed in all three samples are plotted in Fig. 10. The measured frequencies are much lower ( sample, suggesting a smaller demagfor the 50 A netising field. Since the value of M is fixed by the AGM measurement, a value of H> ¼ 2:12 kOe was introduced to reproduce the measured fre( sample quencies in Figs. 7 and 10. For the 5000 A we have assumed that the higher frequency mode is the Damon Eshbach surface mode. We have therefore used Eq. (4) to generate a theoretical curve for which the best fit value of k8 corresponds to a wavelength of 215 mm. The other modes are treated as the uniform modes of the respective samples, and Eqs. (2) and (3) have been used to generate the theoretical curves in each case.
5. Discussion We begin by discussing the qualitative variation of the data seen in Figs. 4 and 5. As the probe spot is scanned across the transmission line the orientation of the pulsed field changes as shown in Figs. 2(b) and (d). The effect upon the magnetisation depends upon the initial orientation
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Fig. 11. The transmission line and sample are shown in crosssection. In (a) and (b) the trajectory of M is shown in schematic form for geometry B and A respectively. In (c) the expected form of the eddy current distribution and its associated magnetic field is shown.
of M; as shown schematically in Fig. 11. In geometry A, both in-plane and out of plane pulsed fields exert a torque upon the magnetisation. The ellipticity of the orbit of the magnetisation is different in each case and so the interference of the longitudinal and polar Kerr effects is rather different, leading to different shaped curves above and in between the tracks of the transmission line [15]. The deviation of the magnetisation is generally larger for an in-plane pulsed field and so a larger Kerr response is observed above the tracks where the pulsed field lies in the plane of the film. In geometry B the in-plane component of the pulsed field exerts no torque on the magnetisation. The Kerr response is driven entirely by the out of plane component of the pulsed field and so the shape of the curves does not change at different positions above the transmission line. Rather the
amplitude of the response decreases as the spot is moved away from the centre of the line. Larger values of trise are required to simulate the curves in Fig. 4 (geometry A) when the pulsed field is perpendicular to the plane of the film (see Table 1), that is, when the probe spot is between rather than above the strips. However in Fig. 5 (geometry B) a single value of trise is sufficient for all positions of the probe spot. Similar variations of trise were reported previously in Ref. [11]. This behaviour can be understood from consideration of the likely distribution of eddy currents within the sample as shown in Fig. 11(c). The eddy currents, which are essentially a negative mirror image of those in the transmission line, oppose the rise of the component of pulsed field perpendicular to the film plane, but enhance the component of field parallel to the film plane. The difference in the values of trise above and between the tracks in geometry A is therefore a measure of the time required for the eddy currents to be dissipated within the film. Let us assume that the current path in the sample has the same values of inductance per unit length, L; and resistance per unit length, r; as the transmission line. The expected time constant for the decay of the eddy currents is then L=r ¼ 25 ps which is indeed of similar magnitude to the observed values of trise : The values of trise increase with film thickness as r becomes smaller. We now discuss the nature of the excited modes in more detail. In section III we stated that the Fourier spectrum of the pulsed field in Fig. 2(d) has maximum amplitude when k8 ¼ 460 rad=cm: This yields values of k8 d ¼ 0:00023; 0.0023 and ( respectively. For 0.023 for d ¼ 50; 500 and 5000 A, geometry B, when H ¼ 210 Oe, g=2.11 and M ¼ 860 emu/cm [3] Eq. (5) gives a mode frequency that is shifted from that of the uniform mode by less than 0.05 GHz for all three thicknesses. However, for geometry A Eq. (4) predicts that the frequencies will be increased by 0.01, 0.13 and 1.17 GHz in order of increasing film thickness. We conclude that the only frequency to be significantly shifted from that of the uniform ( sample in geometry mode is that of the 5000 A A. Since we have no evidence for strong pinning of spins at the interface, we expect that only the k> E0 mode will be excited when the film
J. Wu et al. / Journal of Magnetism and Magnetic Materials 241 (2002) 96–109
thickness is small compared to skin depth of the exciting microwave field. At a frequency of 5 GHz the skin depth in Ni81Fe19 is just a few microns, and so it seems conceivable that modes with finite ( k> values could only be excited in the 5000 A sample. Let us consider the modes observed in Figs. 4(d) and 5(c). From Eq. (6), we would expect the series p ¼ 0; 1; 2; 3 to give frequencies of 4.56, 4.67, 5.02, 5.55yGHz. No such series of modes is observed. However the higher order modes couple less strongly to a nearly uniform exciting field, and it is possible that the first few modes might be excited without being clearly resolved. We would expect the single mode observed in geometry B, in Fig. 5(c), to have a frequency essentially equal to that of the uniform mode. For geometry A, we suggest that the higher frequency mode in Fig. 4(d) is the surface mode while the lower mode is the lowest lying volume mode with k> Bp=d: The higher frequency mode cannot be a volume mode since it is not observed in Fig. 5(c). The fit to the higher mode frequency in Fig. 10 yielded a wavelength of 215 mm for the surface mode which is close to the value of 140 mm suggested by the Fourier transform of Fig 2(d). Finally we consider the values of a shown in Table 1 that were used to reproduce the decay of the oscillatory Kerr signal. Since we excite a spatially non-uniform dynamical magnetisation, we must decide whether the apparent decay arises from damping of the magnetisation precession, or propagation of spin waves away from the fixed point of measurement. Eqs. (4) and (5) allow us to calculate the group velocity and hence the distance travelled by a group of waves within the 2 ns time ( we window of our experiment. For d ¼ 50 A calculate distances of 3.6 and 0.07 mm for geometries A and B, respectively. These distances are small compared to the probe spot diameter of 20 mm and the width of the perturbed magnetisation profile. Propagation effects do not therefore contribute to the observed decay of the ( sample. The propagation Kerr signal for the 50 A distance increases in proportion to the film thickness and so we conclude that propagation effects can be neglected for all three films in ( samples geometry B, but not for the 500 or 5000 A in geometry A.
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( sample we see that larger values For the 500 A of the constant a are required for the simulations in geometry A than those in geometry B. This is consistent with the idea that propagation enhances the decay of the signal in geometry A, although no significant variation of a was observed for the scan across the transmission line in geometry A. The ( fact that the value of a is larger for d ¼ 5000 A ( than for d ¼ 500 A in geometry B suggests that eddy current damping may be a significant factor. We remark that eddy current damping of the magnetisation precession is a non-local effect that depends upon the spatial distribution of the dynamical magnetisation through the entire sample. We would expect a single damping constant associated with each geometry, but, given the very different forms of the dynamical magnetisation shown in Fig. 11, the eddy current damping would generally be different for geometries A and B. The ( largest value of a was obtained for the 50 A sample. This may be due to the presence of structural disorder that also causes the room temperature magnetisation to be reduced. The value of a is essentially independent of the applied ( sample in geometry B, field strength for the 5000 A suggesting that the damping is largely intrinsic in origin. However a strong field dependence is seen ( sample in Fig. 7(b). Most authors for the 50 A discuss the effects of damping in terms of DH; the expected FMR line width. It is difficult to obtain closed algebraic forms for DH; particularly for the elliptical precession that occurs when the magnetisation lies in the plane of a thin film. However numerical calculations suggest a general trend in which DH increases monotonically with frequency [24]. In some simple cases, such as when the magnetisation lies perpendicular to the film plane, DH is given by the simple form o o DHðoÞ ¼ DHð0Þ þ 2a0 ¼ 2a : ð8Þ g g We make a distinction between our effective damping constant a; and a0 ; a frequency independent damping constant that derives from the intrinsic damping. The term DHð0Þ represents the effect of frequency independent extrinsic damping mechanisms such as two magnon scattering. In Fig. 7(c) we have plotted the values of 2ao=g
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obtained from our experiments. Eq. (8) suggests that DH should be proportional to the frequency of oscillation and hence, for small H values, pffiffiffiffi ffi proportional to H : Instead we see that DH decreases with increasing field. This suggests that, for this sample a different process such as a higher order spin wave process contributes to the damping as the field is reduced and the amplitude of precession increases. In conclusion, we have used optical pump probe spectroscopy to study the dependence of the rise and damping of picosecond spin excitations in ferromagnetic thin films upon the film thickness. The pulsed field within the sample was found to rise more slowly when the pulsed field was applied perpendicular to the plane and as the thickness of the sample was increased. We attribute these effects to eddy current shielding of the rising field. By changing the orientation of the applied field relative to the transmission line structure we were able to alter the ellipticity of the trajectory of the precessing magnetisation and observe a significant variation in the decay of the magneto-optical signal. However in the present study this variation is largely due to the propagation of magnetostatic surface waves away from the point of measure( ment. For the thinnest sample studied (d ¼ 50 A) we found the damping to be strongly field dependent, and, contrary to our expectation, the calculated FMR line width increased as the field was reduced. The damping mechanism responsible for this behaviour has not yet been identified. To our knowledge this is the first time that a magnetostatic surface mode has been unambiguously identified in an optical pump-probe experiment. The coherence between the surface mode and a volume mode is particularly interesting and suggests that the interaction between two such modes might be explored. However we have also seen that the group velocity of the surface mode becomes significant as the film thickness is increased. A better model including propagation effects is required if the damping of these modes is to be reliably determined. We suggest that spin waves of any desired wavelength could be excited if the existing transmission line structure were replaced by suitable arrays of lithographically defined wires. We expect that the optical pump
probe technique will allow us to investigate a wide variety of spin excitations in future, and that the information provided about rise times and damping will be of assistance in achieving picosecond magnetic switching.
Acknowledgements The authors gratefully acknowledge the financial support of the UK Engineering and Physical Sciences Research Council (EPSRC).
References [1] J.A.C. Bland, B. Heinrich (Ed.), Ultrathin Magnetic Structures II, Springer, Berlin, 1995 (Chapter 3). [2] V. Kambersky, Czech J. Phys. B 26 (1976) 1366. [3] W.S. Ament, G.T. Rado, Phys. Rev. 97 (1955) 1558. [4] M. Sparks, R. Loudon, C. Kittel, Phys. Rev. 122 (1961) 791. [5] H. Suhl, J. Phys. Chem. Solids 1 (1957) 209. [6] R.D. McMichael, M.D. Stiles, P.J. Chen, W.F. Egelhoff Jr., J. Appl. Phys. 83 (1998) 7037. [7] R. Arias, D.L. Mills, Phys. Rev. B 60 (2000) 7395; R. Arias, D.L. Mills, J. Appl. Phys. 87 (2000) 5455. [8] A. Azevedo, A.B. Oliveira, F.M. de Aguiar, S.M. Rezende, Phys. Rev. B 62 (2000) 5331. [9] E.D. Boerner, H.N. Bertram, H. Suhl, J. Appl. Phys. 87 (2000) 5389. [10] Y. Acreman, C.H. Back, M. Buess, O. Portmann, A. Vaterlaus, D. Pescia, H. Melchior, Science 290 (2000) 492. [11] J. Wu, J.R. Moore, R.J. Hicken, J. Magn. Magn. Mater. 222 (2000) 189. [12] M. Bauer, R. Lopusnik, J. Fassbender, B. Hillebrands, Appl. Phys. Lett. 76 (2000) 2758; T.M. Crawford, P. Kabos, T.J. Silva, Appl. Phys. Lett. 76 (2000) 2113. [13] W.K. Hiebert, A. Stankiewicz, M.R. Freeman, Phys. Rev. Lett. 79 (1997) 1134. [14] T.M. Crawford, T.J. Silva, C.W. Teplin, C.T. Rogers, Appl. Phys. Lett. 74 (1999) 3386. [15] R.J. Hicken, J. Wu, J. Appl. Phys. 85 (1999) 4580. [16] M.R. Freeman, J. Appl. Phys. 75 (1994) 6194. [17] B.I. Bleaney, B. Bleaney, Electricity and Magnetism, 3rd edition, Oxford University Press, Oxford, 1976 (Sections 9.3–9.5). [18] K.C. Gupta, R. Garg, I.J. Bahl, Microstrip Lines and Slotlines, Artech House, 1979. [19] P.R. Gillete, K. Oshima, J. Appl. Phys. 29 (1958) 529. [20] R.W. Damon, J.R. Eshbach, J. Phys. Chem. Solids 19 (1961) 308.
J. Wu et al. / Journal of Magnetism and Magnetic Materials 241 (2002) 96–109 [21] A.H. Morrish, The Physical Principles of Magnetism, Krieger, Florida, 1983 (Section 10.10). [22] D.R. Lide (Ed.), Handbook of Chemistry and Physics, 75th edition, Chemical Rubber, Boca Raton, 1994.
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[23] G.S. Krinchik, V.A. Artem’ev, Sov. Phys. JETP 26 (1968) 1080. [24] J.F. Cochran, R.W. Qiao, B. Heinrich, Phys. Rev. B 39 (1989) 4399.
Excitation and damping of spin excitations in ferromagnetic thin films J. Wu, N.D. Hughes, J.R. Moore, R.J. Hicken* School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, UK Received 8 February 2001; received in revised form 26 July 2001
Abstract We present the results of optical pump-probe measurements made on Ni81Fe19 films of thickness d ¼ 50; 500 and ( We show that the rise time of the pulsed field within the sample may be determined, and that when d is 5000 A. sufficiently small, the value of the Gilbert damping parameter may be obtained from the decay of the magneto-optical signal. The pulsed field was found to rise more slowly when applied perpendicular to the plane of the sample and as the thickness of the sample was increased. By changing the orientation of the pulsed field relative to the static magnetisation we were able to alter the ellipticity of the trajectory of the precessing magnetisation and observe variations in the decay of the precession. We discuss these effects in terms of eddy current shielding of the rising field, eddy current damping of the motion of the magnetisation, and propagation of spin waves from the point at which the sample response is probed. ( we found the value of the Gilbert damping parameter to be strongly field For the thinnest sample studied (d ¼ 50 A) dependent, so that, contrary to our expectations, the calculated Ferromagnetic Resonance (FMR) line width increased as the field was reduced. Under certain circumstances a second mode of higher frequency was observed in the thickest film. We believe this to be a magnetostatic surface mode that has not previously been observed by means of the optical pump-probe technique. r 2002 Elsevier Science B.V. All rights reserved. PACS: 76.50.+g; 78.20.Ls; 78.47.+p Keywords: Ferromagnetic resonance; Damping; Eddy currents; Magnetostatic waves
1. Introduction Nanostructured ferromagnetic materials exhibit spin excitations that may be very different to those of bulk materials. The subject has attracted considerable interest as researchers attempt to explain the form of the modified spin wave *Corresponding author. Tel.: +44-1392-264153; fax: +441392-264111. E-mail address: [email protected] (R.J. Hicken).
spectrum. The measurement of spin wave frequencies can also yield a wealth of information about magnetic moments, anisotropy and exchange interactions in structured materials [1]. However the damping of the same spin wave excitations has so far received less attention. The damping in a thin film is often described by a single phenomenological term in the equations of motion for the magnetisation. This represents the combined effect of a variety of mechanisms that may include dephasing due to the spin-orbit interaction [2],
0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 9 3 0 - 1
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exchange conductivity effects [3], and the generation of additional spin wave modes either by twomagnon scattering at defects [4] or by non-linear processes [5]. Recently, observed variations in Ferromagnetic Resonance (FMR) line widths in exchange biased films [6] have motivated calculations of two-magnon scattering at surface defects in ultrathin films [7], that can be used to understand the dependence of line width upon film thickness [8]. Numerical simulations have shown that dephasing of the large amplitude precession of the magnetisation of nanoscale elements occurs principally by spin wave generation [9]. Magnetic damping is also of considerable technological importance. Damping effects the bandwidth and quality factors of microwave magnetic devices. The next generation of magnetic recording systems will require sub-nanosecond magnetic switching in both the storage medium and the read and write transducers. The size of the angle through which the magnetisation is switched is influenced by both the rise time of the applied field and the damping. There is already evidence that the field rise time within films with thickness as small as 20 nm is affected by eddy currents [10,11]. The time taken for the magnetisation to relax into a new orientation is affected by the damping although ringing can also be suppressed if the pulse shape and duration are matched to the precessional motion [12]. As device sizes continue to be reduced to micron and even nanometre length scales, it is clearly important to study damping in the nanostructured magnetic materials that these devices contain. Damping has previously been studied by absorption of microwaves of fixed frequency as the value of an applied magnetic field is slowly swept. The damping may then be related to the FMR line width. Variable frequency resonant cavities can be used to measure the dependence of line width on frequency, however the microwave technique lacks the spatial resolution that is required to conveniently investigate structures with small lateral dimensions. Optical techniques are favoured since a probing laser spot can be focused to a diameter of less than a micron. Inelastic light scattering is a powerful means by which to study spin wave frequencies, but the
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instrumental resolution is sometimes insufficient for the line width and hence the damping to be reliably determined. Recently a number of authors have shown that optical pump-probe spectroscopy can be used to observe picosecond spin excitations in the time domain [10,11,13–15]. Excitation with pulsed magnetic fields allows a broad band of spin excitations to be observed as the value of a static bias magnetic field is varied. The dependence of the damping parameter on frequency may then be explored. In this paper we present the results of optical pump-probe measurements made on Ni81Fe19 films of thickness d ¼ 50; 500 and ( The experimental technique offers flex5000 A. ibility in the choice of experimental geometry that allows information to be obtained about the rise time of the pulsed field within the sample. Since the pulsed field is spatially inhomogeneous, we discuss the character of the long wavelength magnetostatic modes that are excited. We will determine the conditions in which propagation of these modes may be neglected so that the damping may be deduced from the decay of the magnetooptical signal. We will also discuss how the eddy current damping may depend upon the geometry and how the damping depends upon the film thickness and the frequency of precession. Finally we show how, under certain circumstances, a surface magnetostatic mode may be observed.
2. Experiment Samples were prepared in a magnetron sputtering system equipped with a computer-controlled shutter and substrate platter. The pressure in the chamber was 3 107 Torr prior to growth and sputtering was performed at an Ar pressure of 3.5 mTorr. The Ni81Fe19 was sputtered onto glass ( layer of substrates and then covered with a 60 A Al2O3. Deposition rates were continuously monitored with quartz crystal oscillators that were calibrated by surface profilometer measurements on thick reference films. The films were first characterised by means of magnetometry measurements. These were performed with a longitudinal Magneto Optical Kerr Effect (MOKE) apparatus
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at a wavelength of 633 nm, and with an Alternating Gradient Magnetometer (AGM) apparatus. Our optical pump-probe technique is similar in principle to that introduced by Freeman and coworkers [16]. The apparatus is shown in schematic form in Fig. 1. The light source is a modelocked Ti:sapphire laser producing pulses with width of approximately 100 fs at a repetition rate of 82 MHz. Each pulse is divided into a pump and a probe part so that the average powers of the pump and probe beams are 300 and 7 mW, respectively. A corner cube retro-reflector mounted on a translation stage is used to set the path length of the pump beam before it arrives at the sample. The pump pulse is then used to trigger a magnetic field pulse generated by a device similar to that shown in schematic form in Fig. 2(a). The complete device consists of an interdigitated photoconductive switch attached to a coplanar transmission line that is terminated with surface mount components and connected to a 20 V
Fig. 2. (a) The interdigitated photoconductive switch and transmission line structures are shown in schematic format. (b) The transmission line and sample are shown in cross-section with the pulsed field and static field for geometries A and B (c) A typical oscilloscope trace is shown (d) The value of the inplane (continuous line) and out of plane (dashed line) components of the pulsed field are shown at a height of 5 mm above the transmission line.
Fig. 1. The optical pump probe apparatus is shown in schematic format.
supply by means of a coaxial cable. The device was placed between the pole pieces of an electromagnet designed to allow wide field optical access. The coplanar strips were always perpendicular to the plane of incidence of the probe beam but the static field was applied either perpendicular to, or in the plane of incidence as shown in Fig. 2(b). We refer to these as geometries A and B, respectively. A photolithographic lift-off technique was used to define the Au tracks on an intrinsic GaAs(1 0 0) substrate. The transmission line structure consisted of coplanar strips with width and separation
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( and resistance per of 30 mm, thickness of 600 A unit length r of about 30 O/mm. When the pump pulse is directed onto the interdigitated region the GaAs between the fingers conducts and a current pulse propagates along the transmission line. The use of the interdigitated structure allows the pump spot to be defocused so that the amplitude of the current pulse is insensitive to any small motion of the pump spot as the translation stage is moved. The magnetic field associated with the current interacts with the sample that is placed face down on the transmission line. The probe beam is intensity stabilised and expanded by a factor of 10 before being focused through the glass substrate to a spot size of approximately 20 mm on the surface of the Ni81Fe19 film. In this study measurements were made with a p-polarised probe beam of wavelength 730 nm incident at an angle of 431. The pump beam was chopped at a frequency close to 200 Hz and the optical rotation of the probe beam measured with a polarisation bridge in a phase sensitive manner. It is important to establish the magnitude and temporal profile of the current in the transmission line so that the form of the pulsed field can be determined. It is useful to consider a simple model in which a parallel wire transmission line is described by the Telegrapher’s equations [17]. Initially the transmission line is biased with voltage V and the load impedance ZL of the photoconductive switch is effectively infinite. The voltage reflection coefficient at the open circuit switch is unity. The voltage V on the transmission line can then be thought to arise from two counterpropagating pulses of voltage amplitude p V =2 and ffiffiffiffiffiffiffiffiffi ffi current amplitude 7V =2Z0 ; where Z0 ¼ L=C is the characteristic impedance of the transmission line, and L and C are the inductance and capacitance per unit length. When the switch is gated, the load impedance becomes finite and the voltage reflection coefficient becomes equal to ðZL Z0 Þ=ðZL þ Z0 Þ: A current pulse of amplitude V =ðZL þ Z0 Þ then propagates along the transmission line. We note that if the pump pulse is sufficiently intense that ZL 5Z0 ; the magnitude of the field pulse is limited by the value of Z0 : The value of Z0 depends upon the ratio of width w to separation s of the coplanar strips, and the
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dielectric constant er of the substrate [18]. In our case w=s ¼ 1; er ¼ 13; and we estimate that C ¼ 100 pF/m, L ¼ 740 nH/m and Z0 ¼ 86 O. The transmission line is also lossy and dispersive due to the finite dc resistance of the tracks and, for sufficiently high frequencies, due to the skin effect. The temporal profile of the current is complicated by reflections of the current pulse that occur at impedance mismatches between different parts of the device. Surface mount resistors of value 100 O were used to terminate the transmission line and a 500 MHz oscilloscope1 was attached across one of the resistors in order to monitor the shape of the pulse, as shown in Fig. 2(c). The values of the surface mount resistors must be chosen carefully if the value of the current in the transmission line is to be determined. At GHz frequencies small value resistors have considerable stray inductance while large values have considerable stray capacitance.2 If the component has resistance RSM and total impedance Z then jRSM =Zj is found to be closest to unity when RSM is of the order of 100 O. Small changes in gradient in the trace shown in Fig. 2(c) suggest the presence of multiple reflections. These become more pronounced if smaller resistor values are used since the current pulses are attenuated less by the resistors and instead they are reflected from the power supply with a round trip time of the order of a few nanoseconds. The available surface mount capacitance values generally decrease as the frequency of operation is increased and our experience is that, in practice, these components have little effect on the operation of the device shown in Fig. 2(a). The pulse in Fig. 2(c) has a finite rise time that reflects the bandwidth of the oscilloscope rather than that of the pulse itself. We expect that the current in the semiconductor substrate rises to its full value within 1 ps, the time taken for the photo-excited electrons to thermalise. Then the geometry of the interdigitated structure and the dispersion of the transmission line cause the rise time to be broadened to a value of around 10 ps at the 1
LeCroy LT342 Digital Sampling Oscilloscope. See for example Philips Components Application Note, ‘‘Space-saving 4-resistor arrays in 0804 package’’. 2
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sample. The decay of the pulse can be fit with an exponential that yields a relaxation time of 1.5 ns. By extrapolating back to zero time delay we estimate a peak pulse height of 17 mA. The current pulse propagates at a velocity of about 108 m/s on the transmission line. The full length of the transmission line is filled with current after about 30 ps, after which the current is essentially uniform along the length of the line as the pulse slowly decays. Assuming that the current is uniformly distributed through the cross-sectional area of the Au tracks we are able to calculate the magnetic field generated by the transmission line by numerical integration of the Biot–Savart law. The calculated field values change only slowly as the surface of the track is approached and the calculated magnetic field at a height of 5 mm above the tracks is shown in Fig. 2(d). We use these values of the pulsed field in the calculations presented below.
3. Theory The response of the magnetisation M of a ferromagnetic material to a pulsed field may be described by the Landau–Lifshitz–Gilbert equation qM a qM ¼ jgjM Heff þ M ð1Þ qt M qt in which a is the Gilbert damping constant, Heff is the total effective field acting upon the magnetisation and g ¼ g p ð2:80Þ MHz/Oe where g is the spectroscopic splitting factor. Measurements are normally made in the presence of a static applied field H that causes the sample to occupy a single domain state. Heff then includes H; the pulsed field hðtÞ within the film, and effective fields associated with the anisotropy, demagnetising and exchange energies. In this paper we simulate the trajectory of the uniform mode. The exchange field vanishes and the thin film is assumed to have the usual 4p demagnetising factor. We do not attempt to calculate the associated eddy current distributions. This is a complicated non-local problem that requires the simultaneous solution of Maxwell’s equations and lies beyond the scope of the present
study. When the orientation of M is specified by polar coordinates Eq. (1) reduces to two coupled first order linear differential equations [19] that may be solved with standard numerical techniques. Once the time-dependent vector magnetisation has been calculated, the magneto-optical response may be calculated from generalised Fresnel coefficients that describe the linear MOKE effect [11]. For small amplitude oscillations of the magnetisation, an algebraic expression may be obtained for o; the circular frequency of the uniform mode. In general the films possess a small in-plane uniaxial anisotropy. When the static field H is applied in the sample plane at an angle f relative to the easy axis, the anisotropy energy has the form Ku cos2 ðf yÞ in which Ku is the uniaxial anisotropy constant, and y is the angle between M and H: The value of y is given by MH sin y Ku sinð2ðf yÞÞ ¼ 0;
ð2Þ
while the uniform mode frequency is given by 2 o 2Ku cosð2ðf yÞÞ ¼ H cos y þ g M 2Ku H cos y þ cos2 ðf yÞ þ 4pM : M ð3Þ It is often observed that, when the film thickness is ( there is an additional less than about 100 A, uniaxial anisotropy present that has its axis perpendicular to the film plane. In this case Eq. (3) must be modified by introducing an effective demagnetising field, ð4pMÞeff ¼ 4pM H> ; in which H> is the perpendicular anisotropy field. The latter field may be associated with discontinuities in the film that reduce the shape anisotropy, or with surface anisotropy or magnetoelastic effects. The pulsed field shown in Fig. 2(d) is spatially inhomogeneous and the Fourier power spectra of both field components have maxima at in-plane wave vector values of k8 E460 rad=cm; that correspond to a wavelength of about 140 mm. The induced dynamical magnetisation therefore consists of a superposition of long wavelength modes. If we write the in-plane wave vector component as k8 ; the effective field associated
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with the exchange interaction may be written as ð2A=MÞk82 ; where A is the exchange constant. Assuming values of A ¼ 1:22 106 erg/cm and M ¼ 860 emu/cm3 for bulk Ni81Fe19 we obtain an exchange field value of 6 104 Oe for k8 ¼ 460 rad=cm that can be safely neglected. However the finite in-plane wave vector also gives rise to dipolar fields and then Eq. (1) must be solved simultaneously with the magnetostatic Maxwell equations as described by Damon and Eshbach [20]. Let k> be the wave vector component perpendicular to the film plane. For an isotropic film, when the exchange interaction is neglected, the general solution includes a surface mode (k> E0) and a manifold of volume modes with a series of allowed k> values. Let us consider the two geometries used in our study. In geometry A, the in-plane wave vector is orthogonal to M and the volume modes are degenerate with frequency o0 given by Eq. (3) with Ku set equal to zero. The surface mode lies above the degenerate volume modes. In the limit jk8 dj51; which will be satisfied in all of our experiments, the frequency of the surface modes is given by 2 2 o o0 k8 dð4pMÞ2 ; ð4Þ ¼ þ g g 2 to first order in k8 d: In geometry B, the in-plane wave vector is parallel to M: The degeneracy of the volume modes is lifted and the surface mode is replaced by a volume mode for which k> E0 at the bottom of the volume spin wave manifold. The frequency of this mode is given by 2 2 o o0 Hð4pMÞk8 d ; ð5Þ ¼ g 2 g to first order in k8 d: Eqs. (4) and (5) allow us to determine when the finite wave vector of the excited modes leads to a significant departure from the uniform mode frequency. They also allow us to evaluate the group velocity, vg ¼ qo=qk8 ; so that we may decide whether significant mode propagation is likely during our measurement. Finally, we must consider the effect of exchange upon the magnetostatic mode spectrum. Eqs. (4) and (5) will not be significantly affected since they describe modes for which k> E0: However the frequencies of the remaining volume modes will be
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shifted upwards. We can obtain an estimate of the size of this effect in the limit that k8 ¼ 0: The frequency of the modes in an isotropic film is given by 2 o 2A 2 2A 2 k ¼ Hþ H þ 4pM þ k> ; ð6Þ g M > M where the allowed values of k> form a series in which k> ¼ pp=d and p ¼ n d; where n is an integer and the value of d lies between 0 and 1 depend upon the pinning at the film interfaces [21]. The lowest frequency modes correspond to k> E0; which we have already discussed, and then k> Bp=d: This latter mode may lie above or below that described by Eq. (4) depending upon the film thickness.
4. Results Before making pump-probe measurements the in-plane anisotropy of the films was first characterised by Longitudinal Magneto Optical Kerr Effect (MOKE) and Alternating Gradient Magnetometer (AGM) measurements. MOKE hysteresis loops for the three films are shown in Fig. 3. ( films exhibit a simple We see that the 50 and 500 A in-plane uniaxial anisotropy with hard axis satura( tion fields of 10 and 11 Oe respectively. The 5000 A film was found to be isotropic within the film plane with a larger saturation field of 130 Oe. In the AGM measurements a single turn current loop was placed around the sample to provide a known magnetic moment so that the sample moment could be determined. The magnetisations of the ( films were found to be close to the 500 and 5000 A bulk value of 861 emu/cm3, while the magnetisa( sample was 296726 emu/cm3 at tion of the 50 A room temperature. For the pump-probe measurements the samples were cut into rectangular strips with width of about 2 mm before being overlaid on the transmission line. The long edge of the strip was perpendicular to the length of the transmission ( line and the in-plane easy axes of the 50 and 500 A samples were oriented at 531 and 401, respectively to the plane of incidence. The probe spot was first scanned across the coplanar strips in geometry A.
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Fig. 3. Normalised MOKE loops are shown for Ni81Fe19 films ( (b) 500 A ( (c) 5000 A. ( The continuous and of thickness (a) 50 A dotted curves were taken with the field applied parallel to the easy and hard axes respectively.
The amplitude of the Kerr signal exhibits two maxima that allow the positions of the coplanar strips to be identified. The signal from the three samples is shown in Fig. 4 at positions above and in between the coplanar strips. While a qualitatively similar behaviour is observed for the 50 and ( samples, the data for the 5000 A ( sample is 500 A clearly different being reminiscent of two modes beating together. Fast Fourier transforms were taken of this last set of data and the resulting power spectra confirm the presence of two modes of different frequency. Measurements were also ( samples in geometry made on the 500 and 5000 A B as shown in Fig. 5. This time the power spectra ( sample show just a single mode. for the 5000 A We have simulated the response of the two thinner samples, and one of the curves for the ( sample that exhibits single mode beha5000 A viour, using the uniform mode solution of Eq. (1). It is necessary to first specify the form of the pulsed
Fig. 4. The Kerr rotation measured above one track, in between the tracks, and above the other track of the ( transmission line in geometry A is shown for (a) d ¼ 50 A, ( H ¼ 210 Oe (c) d ¼ 5000 A, ( H ¼ 216 Oe (b) d ¼ 500 A, H ¼ 216 Oe. The normalised power spectra of the curves in (c) are shown in (d). The smooth simulated curves assume the parameter values shown in Table 1.
field hðtÞ: By this we mean the field within the sample since this may lag the applied field due to shielding by an induced eddy current distribution. Our experience is that 8 eh0 > > > < ðe 1Þ expðtrise =tÞð1 expðt=trise ÞÞ; tptrise ; h¼ trise otptrefl ; h0 expðt=tÞ; > > > : t > trefl ; h0 ½expðt=tÞ R expððt trefl Þ=tÞ ;
ð7Þ
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Fig. 6. The magnitude of the pulsed field specified by Eq. (6) has been sketched for the case that jh0 j ¼ 1:
Fig. 5. The Kerr rotation measured above one track, in between the tracks, and above the other track of the ( transmission line in geometry B is shown for (a) d ¼ 500 A, ( H ¼ 216 Oe. The normalised H ¼ 210 Oe (b) d ¼ 5000 A, power spectra of the curves in (b) are shown in (c). The smooth simulated curves assume the parameter values shown in Table 1.
is the simplest form for hðtÞ that adequately reproduces the main features of the experimental data. The form of the pulse has been sketched in Fig. 6. The peak field is close to h0 ; while the rise time of the pulse is controlled by the parameter trise : Other functional forms were tried for the rise of the pulse but the exponential form was found to give the best results. It also seems the most likely form if the rise time is largely determined by the decay of eddy currents. The decay of the pulse is assumed to follow that of the current shown in Fig. 2(c) with the same time constant t: We find that, in order to correctly reproduce the relative phase of successive oscillations of the Kerr rotation, it is essential to include a reflected pulse that arrives at the sample after time trefl : The value of trefl is of order 100 ps and so the reflection cannot be seen in Fig. 2(c) due to the limited
bandwidth of the oscilloscope. The values of trefl and R used for each sample are shown in Table 1. Since the three samples were placed and probed at different positions above the transmission line some variation in their values is not surprising. Once the position of the magnetisation has been calculated the instantaneous Kerr rotation is evaluated from Eqs. (5), (7) and (8) of Ref. [11]. These equations apply to the interface between two semi-infinite media, while in fact the probe beam passes from air, and through the glass substrate, before impinging upon a Ni81Fe19 film of finite thickness. Since the reflection coefficient at the air-glass interface is small, we ignore the weak reflection from this surface but take account of the modified angle of incidence, 271, in the denser (n ¼ 1:55) optical medium. Since the optical skin ( at the depth of the Ni81Fe19 is just 270 A wavelength of 730 nm, the assumption that the Ni81Fe19 is effectively semi-infinite is reasonable for the two thicker films. Since the optical constants of Ni81Fe19 are not well known at 730 nm, we instead use values determined for pure Ni. These are 2.35+4.24i and 0.00740.0046i for the refractive index [22] and magneto-optic con( sample the stant [23] respectively. For the 50 A value of the magneto-optic constant was scaled in proportion to the reduced room temperature magnetisation. Given the uncertainties in the values of the optical constants and the exact ( magnitude of the pulsed field, and since the 50 A film is thin compared to the optical skin depth, we do not attempt to reproduce the exact magnitude
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Table 1 Parameter values used in modelling the time resolved MOKE data are shown. The letters A and B refer to the two measurement geometries described within the text ( d (A)
50
500
5000
M (emu/cm3) g Hsat (Oe) H> (kOe) trise (ps)
296 2.15 10 2.12 10–40 (A)
860 2.11 130 0 90 (B)
trefl (ps) R a
205 0.2 0.121 (A)
800 2.00 11 0 20–70 (A) 70 (B) 95 0.5 0.0083 (A) 0.0076 (B)
165 0.2 0.0129 (B)
of the measured Kerr rotation, although experiment and simulation rarely disagree by a factor of more than two. By manually adjusting the values of trise ; trefl ; R and the damping constant a; our aim has been to reproduce the initial rise of the Kerr signal, and then the phase and decay of the ensuing oscillations. Since the orientation of the pumping field varies continuously above the strips and the probe spot has a finite diameter, the simulations are an average of 3 points that are each 10 mm apart. The good agreement between data and simulation in Figs. 4 and 5 attest to the success of this approach and the values of parameters used in the simulation are listed in Table 1. In practice the biggest problem faced in performing the simulations is that the frequency of oscillation is slightly different between data and simulation. Wide field optical access to the sample was obtained at the expense of the homogeneity of the field generated by the electromagnet and, while the field is certainly uniform over the area of the sample, additional uncertainty is introduced into the measurement of the field value. The frequency discrepancy causes successive oscillations of the simulation to accumulate an increasing phase error relative to the experimental data. Clearly the smallest phase error occurs for the first peak in the curve, and it is this that we try to reproduce by varying the trise parameter. We also wished to explore the effect of the magnitude of the static field H ¼ jHj upon the
response of the magnetisation. Measurements were performed in geometry A with the probe spot above one of the coplanar strips. The response obtained for different values of H has been plotted ( sample in Figs. 7–9. Power spectra for the 5000 A again confirmed the presence of two modes while ( samples showed only those for the 50 and 500 A single peaks. We have only simulated the Kerr ( samples for which response for the 50 and 500 A single mode behaviour was observed. The simulated curves are again found to be in reasonable agreement with the experimental data. However there is a striking difference between the 50 and ( samples. For the latter the entire series of 500 A
Fig. 7. In (a) the measured and simulated Kerr rotation are ( sample when the probe spot is above one of shown of the 50 A the tracks of the transmission line. The smooth simulated curves assume the parameter values shown in Table 1 and the values of the damping constant a shown in (b). The effective FMR linewidth DH ¼ 2ao=g is plotted in (c).
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Fig. 10. Mode frequencies are plotted as a function of static ( (+, lower continuous curve), 500 A ( (W, field H for d ¼ 50 A ( (K, long dashed curve and ’, short dashed curve), 5000 A upper continuous curve). The curves assume the parameter values in Table 1 and are discussed in the text. Fig. 8. The measured and simulated Kerr rotation are shown ( sample when the probe spot is above one of the for the 500 A tracks of the transmission line. The smooth simulated curves assume the parameter values shown in Table 1.
( Fig. 9. The measured Kerr rotation is shown for the 5000 A sample when the probe spot is above one of the tracks of the transmission line.
curves is well described by the single damping ( factor shown in Table 1. However for the 50 A sample the damping of the oscillations appears to
increase as the value of H is decreased. The values of a used in the simulations have been plotted in Fig. 7(b). The frequencies of all the modes observed in all three samples are plotted in Fig. 10. The measured frequencies are much lower ( sample, suggesting a smaller demagfor the 50 A netising field. Since the value of M is fixed by the AGM measurement, a value of H> ¼ 2:12 kOe was introduced to reproduce the measured fre( sample quencies in Figs. 7 and 10. For the 5000 A we have assumed that the higher frequency mode is the Damon Eshbach surface mode. We have therefore used Eq. (4) to generate a theoretical curve for which the best fit value of k8 corresponds to a wavelength of 215 mm. The other modes are treated as the uniform modes of the respective samples, and Eqs. (2) and (3) have been used to generate the theoretical curves in each case.
5. Discussion We begin by discussing the qualitative variation of the data seen in Figs. 4 and 5. As the probe spot is scanned across the transmission line the orientation of the pulsed field changes as shown in Figs. 2(b) and (d). The effect upon the magnetisation depends upon the initial orientation
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Fig. 11. The transmission line and sample are shown in crosssection. In (a) and (b) the trajectory of M is shown in schematic form for geometry B and A respectively. In (c) the expected form of the eddy current distribution and its associated magnetic field is shown.
of M; as shown schematically in Fig. 11. In geometry A, both in-plane and out of plane pulsed fields exert a torque upon the magnetisation. The ellipticity of the orbit of the magnetisation is different in each case and so the interference of the longitudinal and polar Kerr effects is rather different, leading to different shaped curves above and in between the tracks of the transmission line [15]. The deviation of the magnetisation is generally larger for an in-plane pulsed field and so a larger Kerr response is observed above the tracks where the pulsed field lies in the plane of the film. In geometry B the in-plane component of the pulsed field exerts no torque on the magnetisation. The Kerr response is driven entirely by the out of plane component of the pulsed field and so the shape of the curves does not change at different positions above the transmission line. Rather the
amplitude of the response decreases as the spot is moved away from the centre of the line. Larger values of trise are required to simulate the curves in Fig. 4 (geometry A) when the pulsed field is perpendicular to the plane of the film (see Table 1), that is, when the probe spot is between rather than above the strips. However in Fig. 5 (geometry B) a single value of trise is sufficient for all positions of the probe spot. Similar variations of trise were reported previously in Ref. [11]. This behaviour can be understood from consideration of the likely distribution of eddy currents within the sample as shown in Fig. 11(c). The eddy currents, which are essentially a negative mirror image of those in the transmission line, oppose the rise of the component of pulsed field perpendicular to the film plane, but enhance the component of field parallel to the film plane. The difference in the values of trise above and between the tracks in geometry A is therefore a measure of the time required for the eddy currents to be dissipated within the film. Let us assume that the current path in the sample has the same values of inductance per unit length, L; and resistance per unit length, r; as the transmission line. The expected time constant for the decay of the eddy currents is then L=r ¼ 25 ps which is indeed of similar magnitude to the observed values of trise : The values of trise increase with film thickness as r becomes smaller. We now discuss the nature of the excited modes in more detail. In section III we stated that the Fourier spectrum of the pulsed field in Fig. 2(d) has maximum amplitude when k8 ¼ 460 rad=cm: This yields values of k8 d ¼ 0:00023; 0.0023 and ( respectively. For 0.023 for d ¼ 50; 500 and 5000 A, geometry B, when H ¼ 210 Oe, g=2.11 and M ¼ 860 emu/cm [3] Eq. (5) gives a mode frequency that is shifted from that of the uniform mode by less than 0.05 GHz for all three thicknesses. However, for geometry A Eq. (4) predicts that the frequencies will be increased by 0.01, 0.13 and 1.17 GHz in order of increasing film thickness. We conclude that the only frequency to be significantly shifted from that of the uniform ( sample in geometry mode is that of the 5000 A A. Since we have no evidence for strong pinning of spins at the interface, we expect that only the k> E0 mode will be excited when the film
J. Wu et al. / Journal of Magnetism and Magnetic Materials 241 (2002) 96–109
thickness is small compared to skin depth of the exciting microwave field. At a frequency of 5 GHz the skin depth in Ni81Fe19 is just a few microns, and so it seems conceivable that modes with finite ( k> values could only be excited in the 5000 A sample. Let us consider the modes observed in Figs. 4(d) and 5(c). From Eq. (6), we would expect the series p ¼ 0; 1; 2; 3 to give frequencies of 4.56, 4.67, 5.02, 5.55yGHz. No such series of modes is observed. However the higher order modes couple less strongly to a nearly uniform exciting field, and it is possible that the first few modes might be excited without being clearly resolved. We would expect the single mode observed in geometry B, in Fig. 5(c), to have a frequency essentially equal to that of the uniform mode. For geometry A, we suggest that the higher frequency mode in Fig. 4(d) is the surface mode while the lower mode is the lowest lying volume mode with k> Bp=d: The higher frequency mode cannot be a volume mode since it is not observed in Fig. 5(c). The fit to the higher mode frequency in Fig. 10 yielded a wavelength of 215 mm for the surface mode which is close to the value of 140 mm suggested by the Fourier transform of Fig 2(d). Finally we consider the values of a shown in Table 1 that were used to reproduce the decay of the oscillatory Kerr signal. Since we excite a spatially non-uniform dynamical magnetisation, we must decide whether the apparent decay arises from damping of the magnetisation precession, or propagation of spin waves away from the fixed point of measurement. Eqs. (4) and (5) allow us to calculate the group velocity and hence the distance travelled by a group of waves within the 2 ns time ( we window of our experiment. For d ¼ 50 A calculate distances of 3.6 and 0.07 mm for geometries A and B, respectively. These distances are small compared to the probe spot diameter of 20 mm and the width of the perturbed magnetisation profile. Propagation effects do not therefore contribute to the observed decay of the ( sample. The propagation Kerr signal for the 50 A distance increases in proportion to the film thickness and so we conclude that propagation effects can be neglected for all three films in ( samples geometry B, but not for the 500 or 5000 A in geometry A.
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( sample we see that larger values For the 500 A of the constant a are required for the simulations in geometry A than those in geometry B. This is consistent with the idea that propagation enhances the decay of the signal in geometry A, although no significant variation of a was observed for the scan across the transmission line in geometry A. The ( fact that the value of a is larger for d ¼ 5000 A ( than for d ¼ 500 A in geometry B suggests that eddy current damping may be a significant factor. We remark that eddy current damping of the magnetisation precession is a non-local effect that depends upon the spatial distribution of the dynamical magnetisation through the entire sample. We would expect a single damping constant associated with each geometry, but, given the very different forms of the dynamical magnetisation shown in Fig. 11, the eddy current damping would generally be different for geometries A and B. The ( largest value of a was obtained for the 50 A sample. This may be due to the presence of structural disorder that also causes the room temperature magnetisation to be reduced. The value of a is essentially independent of the applied ( sample in geometry B, field strength for the 5000 A suggesting that the damping is largely intrinsic in origin. However a strong field dependence is seen ( sample in Fig. 7(b). Most authors for the 50 A discuss the effects of damping in terms of DH; the expected FMR line width. It is difficult to obtain closed algebraic forms for DH; particularly for the elliptical precession that occurs when the magnetisation lies in the plane of a thin film. However numerical calculations suggest a general trend in which DH increases monotonically with frequency [24]. In some simple cases, such as when the magnetisation lies perpendicular to the film plane, DH is given by the simple form o o DHðoÞ ¼ DHð0Þ þ 2a0 ¼ 2a : ð8Þ g g We make a distinction between our effective damping constant a; and a0 ; a frequency independent damping constant that derives from the intrinsic damping. The term DHð0Þ represents the effect of frequency independent extrinsic damping mechanisms such as two magnon scattering. In Fig. 7(c) we have plotted the values of 2ao=g
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obtained from our experiments. Eq. (8) suggests that DH should be proportional to the frequency of oscillation and hence, for small H values, pffiffiffiffi ffi proportional to H : Instead we see that DH decreases with increasing field. This suggests that, for this sample a different process such as a higher order spin wave process contributes to the damping as the field is reduced and the amplitude of precession increases. In conclusion, we have used optical pump probe spectroscopy to study the dependence of the rise and damping of picosecond spin excitations in ferromagnetic thin films upon the film thickness. The pulsed field within the sample was found to rise more slowly when the pulsed field was applied perpendicular to the plane and as the thickness of the sample was increased. We attribute these effects to eddy current shielding of the rising field. By changing the orientation of the applied field relative to the transmission line structure we were able to alter the ellipticity of the trajectory of the precessing magnetisation and observe a significant variation in the decay of the magneto-optical signal. However in the present study this variation is largely due to the propagation of magnetostatic surface waves away from the point of measure( ment. For the thinnest sample studied (d ¼ 50 A) we found the damping to be strongly field dependent, and, contrary to our expectation, the calculated FMR line width increased as the field was reduced. The damping mechanism responsible for this behaviour has not yet been identified. To our knowledge this is the first time that a magnetostatic surface mode has been unambiguously identified in an optical pump-probe experiment. The coherence between the surface mode and a volume mode is particularly interesting and suggests that the interaction between two such modes might be explored. However we have also seen that the group velocity of the surface mode becomes significant as the film thickness is increased. A better model including propagation effects is required if the damping of these modes is to be reliably determined. We suggest that spin waves of any desired wavelength could be excited if the existing transmission line structure were replaced by suitable arrays of lithographically defined wires. We expect that the optical pump
probe technique will allow us to investigate a wide variety of spin excitations in future, and that the information provided about rise times and damping will be of assistance in achieving picosecond magnetic switching.
Acknowledgements The authors gratefully acknowledge the financial support of the UK Engineering and Physical Sciences Research Council (EPSRC).
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