REVIEW OF SCIENTIFIC INSTRUMENTS 81, 103303 共2010兲
Development and application of setup for ac magnetic field in neutron scattering experiments Sergey Klimko,1,a兲 Kirill Zhernenkov,2,b兲 Boris P. Toperverg,2,c兲 and Hartmut Zabel2 1
Laboratoire Léon Brillouin, 91191 Gif-sur-Yvette, France Institut für Festkörperphys IV, Ruhr Universität Bochum, D-44780 Bochum, Germany
2
共Received 1 February 2010; accepted 9 September 2010; published online 25 October 2010兲 We report on a new setup developed for neutron scattering experiments in periodically alternating magnetic fields at the sample position. The assembly consisting of rf generator, amplifier, wide band transformer, and resonance circuit. It allows to generate homogeneous ac magnetic fields over a volume of a few cm3 and variable within a wide range of amplitudes and frequencies. The applicability of the device is exemplified by ac polarized neutron reflectometry 共PNR兲: a new method established to probe remagnetization kinetics in soft ferromagnetic films. Test experiments with iron films demonstrate that the ac field within the accessible range of frequencies and amplitudes produces a dramatic effect on the PNR signal. This shows that the relevant ac field parameters generated by the device match well with the scales involved in the remagnetization processes. Other possible applications of the rf unit are briefly discussed. © 2010 American Institute of Physics. 关doi:10.1063/1.3495965兴
I. INTRODUCTION
Various types of devices generating periodic magnetic fields in the radio frequency 共rf兲 range are nowadays routinely used on a number of neutron scattering instruments in which ac field is employed to manipulate the neutron spin polarization. Among them are well known rf resonance spin flippers,1,2 which allow to effectively alter the direction of the neutron polarization and serve as key elements in neutron resonance spin echo 共NRSE兲3,4 spectrometers5–9 and reflectometers.10–12 In all those examples, the rf field is applied to the neutron spin, aiming to improve time and/or space resolution of the corresponding neutron methods. On the other hand, there exists a broad range of physical problems, which can be addressed with neutron experiments that require an ac field applied to the sample with and without simultaneously affecting the neutron spin. Among those are remagnetization processes in magnetic films13 and nanostructures,14 as well as kinetics in magnetic colloids, field induced excitations, etc. All of them require to record a response of the sample on an external ac field, providing information within the range of time and/or space 共direct or reciprocal兲 accessed. Here, we describe in detail a device, which produces a homogeneous magnetic ac field with an amplitude of up to 60 Oe and a frequency range 0.02ⱕ f ⱕ 2 MHz over a volume typical for neutron scattering experiments. A set of test measurements with the ac field option installed on the ADAM reflectometer15 共ILL, Grenoble, France兲 was carried out to probe the ac response of thin iron films. It is demonstrated that, indeed, the specular reflection and the off-specular scattering signals strongly depend on the a兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected]. b兲 Also at: Joint Institute for Nuclear Research, 141980 Dubna, Russia. c兲 Also at: Petersburg Nuclear Physics Institute, 188300 Gatchina, Russia. 0034-6748/2010/81共10兲/103303/8/$30.00
ac field amplitude and the frequency within the range covered by our rf unit. This justifies a perspective of our device to be actively used in forthcoming PNR studies16,17 of domain kinetics18 in thin films,19 heterostructures,20 and nanopatterns.21,22 II. BASIC COMPONENTS OF RF DEVICE A. Principle
The main idea of the installation is to use a serial LCR-circuit, where L and R are inductance and resistance of the rf-coils, respectively, and C is the capacity of a combination of one variable capacitor and a set of additional fixed capacitors 共Fig. 1兲. The standard function generator HP 33120A 共Ref. 23兲 with broadband frequency range of 0.1 Hz–15 MHz is used as a signal generator. The output signal of the generator with adjustable amplitude 共50 mVpp–10 Vpp兲 feeds the input of an rf-amplifier AG 1012.24 The ferrite transformer between the amplifier and the LCR circuit is used to match the output resistance of the amplifier 共50 ⍀兲 to the resistance of the LCR-circuit. The decoupling of the resonance circuit and the amplifier by a ferrite transformer and its shielding via an aluminum box significantly reduces reflected power from the resonance circuit to the amplifier and allows to avoid interference induction to the environment with sensitive electronics, such as detectors, encoders, etc. The reactive power at the peak current I = Imax in the rf-coil of inductance L can readily be found from the equation 2 PreactivL = 2 fLImax sin IU ,
共1兲
where IU is the phase shift between current and ac voltage V for a certain frequency f. At the resonance condition, IU is equal to / 2. Then, for a set of typical values, e.g., of inductance L = 50 H, current I = 8 A, and resonance fre-
81, 103303-1
© 2010 American Institute of Physics
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Pick-up coil
Signal generator L,R
Cvar Amplifier
Step motor
Ferrite transformer Rotor plates with teflon and kapton insulators
Oscilloscope
Cadd Gear
Transformer Additional fixed capacitors FIG. 1. The principal scheme of the radio frequency circuit. Variable capacitor Cvar, additional fixed capacitors Cadd, and ferrite transformer are shielded by an aluminum box.
quency f = 750 kHz, the power circulating in the rf-circuit is about PreactivL ⬇ 15 kW. The maximum voltage drop in the circuit is of the order of 5 kV. Thus, appropriate HF-cables, wires, and insulations are used. The power consumption is mainly due to heating of the rf-coil, cables, and dielectric losses in the capacitors. At the conditions mentioned above, the losses are about 100–120 W. The current amplitude stability in the circuit is defined by the stability of the signal generator 共⫾1.5%兲 and the amplifier 共⫾1%兲. Frequency stability is determined by a signal generator, and it is about of 20 ppm from the specified frequency. B. The variable rotary capacitor
In order to permit fine tuning of the resonance circuit, a high power variable rotary capacitor was used as a key element of the device: with fixed inductance, the frequency adjustment can be done by varying the capacity C only. The basic requirements to the capacitor follow from the following elementary consideration. As it is well known, the imaginary part of the complex impedance Z of a serial LCR circuit and the phase shift between current I = V / Z and applied voltage V = V0exp共i2 ft兲 turn at the resonance frequency f = f res to zero. This condition immediately determines the resonance frequency f res. Using the following set of standard equations: 1 , 共2兲 Z = R + i2 fL + i2 fC I=
冑 冉
V0
1 R2 + 2 fL − 2 fC
= tan−1
冢
2 fL − R
1 2 fC
冣
冊
,
2
ei共2 ft−兲 ,
共3兲
共4兲
yielding: f res =
1 2
冑
1 . LC
共5兲
Aluminum stators plates
FIG. 2. Top view of the variable rotary capacitor with step motor from the right side. The aluminum shielding cover is opened to show the additional electronics inside from the left side.
One should admit that at resonance, the voltage drop on the capacitor VC = IZC depends on the current passing through the circuit and the ratio 冑L / C, VC =
V0 R
冑
L i共2 f t−/2兲 res e . C
共6兲
Since only C is varied for frequency changes, the voltage drop over the capacitor increases with frequency. This puts quite a hard constraint 共high voltage兲 on the insulators of the rotary capacitor at elevated frequencies. The bandwidth of the LCR-circuit does not depend on the capacity ⌬f = R / L. This means that at low frequencies, we have a very broad resonance, while at high frequencies, it becomes very sharp. The capacitor was originally developed for the NRSE option ZETA 共Ref. 7兲 at the ILL. Changes in the frequency band can be achieved by adding additional fixed capacitors. A detailed description of development and improvements of the rotary capacitors can be found in Ref. 25, while here, we briefly review its main abilities for covering the necessary parameters of the LCR circuit and, finally, the output ac field. The rotary capacitor consists of 100 stator and 100 rotor plates 共Fig. 2兲. The stator plates are fixed on the frame of the capacitor, whereas the rotor plates are mounted on a common axis. The capacity depends on the overlap surface of the rotor and stator plates and on the dielectric material in between. Polished aluminum was used as material for rotor and stator plates. The Teflon and Kapton disks were introduced in between as insulator. All rotor plates are fixed on a brass axis, which is connected through an isolated coupler to the step motor block. The step motor is equipped with a transmission gear with a ratio of 1:200. This opens the possibility of fine capacitor tuning and hence of the resonance frequency. For higher currents and frequencies, it has been proposed by K. Thillosen26 to fill in the capacitor by a dielectric liquid—a sort of liquid Teflon 共fluorinate liquid兲, replacing the air gap between the Kapton and Teflon insulators. The tests have shown that it is easily possible to increase current and voltage through the capacitor up to 20 A and 10 kV, respectively, at a frequency of 1 MHz. Due to the special form of the rotor plates, the capacity has a linear dependence
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on the rotation angle and can smoothly be changed from 0.14 to 34 nF. Optionally, the capacity range can be extended up to range from 4 pF to 250 nF. The effective resistance of the LCR-circuit 共including the resistance of the rf-coil, capacitors, and wires兲 depends on the resonance frequency. In our applications for neutron reflectometry work, it varies from 0.6 to 3.5 ⍀ depending on frequency. For an efficient usage of the power amplifier, it is necessary to adjust the output resistance of the amplifier 共50 ⍀兲 with the input resistance 共which changes with frequency兲 of the LCR-circuit in a broad 50 kHz–2 MHz frequency range, providing a power transfer from 0 to 1000 W. The degree of the adjustment can be estimated by the value power standing wave relation 共PSWR兲 PF + PR , PF − PR
PSWR =
共7兲
where PF is the power forwarded to, while PR is the power reflected from the resonance circuit. If the power amplifier and the resonance circuit are not matched well, then part of the energy cannot be transferred from the amplifier to the circuit and it is reflected back toward the amplifier. Reflected power has the following implications in radio frequency circuits: 共1兲 increase of the energy losses and 共2兲 that a damage of the amplifier can occur. The best matching corresponds to PSWR= 1 where no reflected power occurs. In this case, the whole energy from the amplifier is transferred to the resonance circuit. To accomplish matching between the amplifier and resonance circuit, we fabricated a power impedance matching transformer 共Fig. 1兲 with “Amidon” powdered-iron toroidal cores.27 In order to improve the efficiency of the transformer, two toroids were glued together. This decreases the heating of the core and avoids saturation of the ferrites. The primary and secondary reelings on the toroid are connected to the amplifier and to the LCR-circuit, respectively. For the proper matching of the output and input impedances, the correct number of turns in both primary and secondary reelings has to be chosen. The simplest method for determining the correct number of turns is the following: the inductance of the primary winding L1 can be found from the ratio between the amplifier output impedance Rout and the working frequency f l, L1 = Rout/2 f l ,
共8兲
and the number of the turns can be calculated by the following equation:27
冑
N1 = 100
L1 , AL
共9兲
where L1 should be taken in H and the AL 共core constant from iron powder specification兲 in H / 100 turns. The number of the turns for the secondary winding is defined as the square root of the ratio of LCR-circuit input impedance Rin and the amplifier output impedance Rout, N2 = N1
冑
Rin . Rout
共10兲
The value of the input impedance Rin can be calculated via measuring a quality factor Q of the resonance circuit,
Rin =
1 Q
冑
L . C
共11兲
The advantage of this transformer is that the LCR-circuit is floating 共decoupled from ground connection兲; thus the induced interference with neighboring electronics such as amplifiers for the neutron detectors and monitor detectors is decreased significantly. C. Radiofrequency and probe coils
In our experiments, we used a solenoid type radiofrequency coil. For PNR experiments, the coil should provide a gap required for neutrons passing through the inner coil space without scattering on the coil material. As a material of the coil body, we have chosen fiber glass because of its good mechanical and thermal properties and electric insulation. The Ohmic resistance of the coil is about 0.2 ⍀. The inductance is 150 H. The coil was tested at high frequencies and high currents. The maximum tested frequency and current for the coil were 1.8 MHz and 30 A. At higher current values, the coil requires air cooling. The amplitude and frequency of the oscillating magnetic field in the coil are monitored by a probe coil placed inside. The voltage from the probe coil is read by the oscilloscope 共HP-54624A兲.23 From the probe voltage, one can estimate the amplitude of the magnetic field at the position of the probe coil. The magnetic flux through the surface S p of the probe coil is ⌽=
冕
BdS,
共12兲
Sp
where B is the magnetic field induction. Assuming that the magnetic field is homogeneous over the surface area S p, the electromotive force on the n turns of the probe coil E pc can be written as E pc = −
d⌽ dB =− nS p , dt dt
共13兲
and for oscillating field B共t兲 = B0 cos共2 ft − 兲, the amplitude of the magnetic field is as follows: B0 =
E pc0 , 2 f · n · S p
共14兲
where E pc0 is the amplitude of the signal measured by the oscilloscope in volts and f is the frequency of the magnetic field in hertz. III. AC-POLARIZED NEUTRON REFLECTOMETRY A. Specular reflection and off-specular scattering under ac field
Polarized neutron reflectivity under external ac field via the device described above is a new method, which we term ac polarized neutron reflectometry 共ac PNR兲. This method promises to provide information on the remagnetization rate in thin magnetic films, multilayers, and micro- to nanopatterns. Such information is highly required in view of current and forthcoming spintronic applications.28–30 Since more than a decade, PNR is well recognized as a tool for detailed
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a)
y
Coherence volume
H = H b − H 0 cos(2πft )
x
b)
αf M
αi
ki
kf Q
z
Domain wall
FIG. 3. 共Color online兲 The layout of the experimental setup. 共a兲 Sketch of the ADAM neutron reflectometer 共ILL, Grenoble兲: 共1兲 monochromator, 共2兲 Be filter, 共3兲 polarizer, 共4兲 spin flippers, 共5兲 sample, sample holder with radio frequency coil, 共6兲 static Helmholtz coils, 共7兲 beam stop, 共8兲 analyzer, and 共9兲 detector. 共b兲 Scheme of the experiment geometry: incident neutrons with the wave vector kជ i impinge onto the sample surface at the angle ␣i and ជ = kជ f − kជ i. dc field reflected under the angle ␣ f with the wave vector transfer Q Hb and ac field H共t兲 = H0 cos共2 ft兲 are set collinearly with the incident ជ of domains. They polarization 共blue arrows兲 and magnetization vectors M are separated by nearly 180° domain walls 共vertical red lines兲 moving against each other with velocities vជ during a remagnetization process.
investigations of spintronic materials in an applied dc field.16,17 Here, we describe the first test experiments showing that ac PNR methods can substantially extend the capabilities of PNR by not only probing the magnetization distribution in space but also following its evolution in time. The ac field device was set up as an add-on option of the ADAM reflectometer15 at the Institute Laue-Langevin, Grenoble, France. The device generates at the sample position an ac field of amplitudes and frequencies as described above. Additionally a dc-field with a maximum magnitude of 35 Oe generated by Helmholtz coils can be applied to the sample. The sample is placed inside the rf-coil together with a pick-up coil for probing and controlling the amplitude of the field at the sample. The sample was mounted on a sample holder made of boron nitride. Nearly perfect collinearity between the incident neutron polarization vector and dc and ac field directions was achieved in order to guarantee a no detectable beam depolarization above the level of 1.5% due to the 98.5% efficiency of the polarizing/analyzing system. All other details are like in a usual PNR experiment with four specular reflectivities: two non-spin-flip 共NSF兲, R++ and R−−, and two spin-flip 共SF兲, R+− and R−+, reflectivities. The diffuse scattering is measured additionally with a position sensitive detector. In this case, only the incident polarization of the neutron beam can be altered with spin flipper, but no spin analysis of the scattered beam is employed. The layout of the experimental setup realized on the reflectometer ADAM and the scheme of the scattering geometry are sketched in Fig. 3. ac PNR test measurements were carried out to probe the frequency and amplitude range of the ac field affecting the neutron reflectivity from a single crystalline iron film of about 100 nm thickness. An ac magnetic
FIG. 4. 共Color online兲 NSF reflectivity curves R++ 共top兲 and R−− 共middle兲 measured from Fe film in dc field Hb = 25 Oe set collinear with an ac field with the amplitude H0 = 55 Oe at frequencies f = 0.3, f = 0.45, and f = 1.5 MHz. In the latter case, the curves are indistinguishable from those measured in saturating dc field Hb = 25 Oe with no ac field on the sample. Small SF reflection seen 共bottom panel兲 at saturation is mostly attributed to the imperfect efficiency of the polarizing/analyzing system. Some enhancement of SF signal at intermediate frequency may be due to a contamination of SF off-specular scattering depicted in Fig. 5.
field H共t兲 = H0 cos共2 ft − 兲 with amplitudes H0 up to 60 Oe and frequencies f = 300– 1800 kHz was applied along one of the easy axes 共the Y axis兲 within the film plane and perpendicular to the wave vector transfer Q normal to the surface. The additional bias dc field of Hb = 25 Oe was set collinear to the ac field 共and incident polarization兲 in order to overcome the coercive field Hc ⬇ 21 Oe and hence to prepare the initially magnetized state of the sample in zero ac field. On the other hand, an ac field H ⬍ −共Hc + Hb兲 = −46 Oe should ensure remagnetization of the film in a quasistatic limit, although full remagnetization may not take place over the full dynamical range, as we shall see later. In Fig. 4, a set of NSF and SF reflectivity curves collected in Hb = 25 Oe and ac field with the amplitude H0
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FIG. 5. 共Color online兲 Intensity maps of the specular and diffuse scattering R+ and R− recorded at different frequencies. ␣i and ␣ f are the incident and exit angles, respectively. The specular intensity ridge runs along the diagonal ␣i = ␣ f .
= 55 Oe and different frequencies is compared with those measured in saturating field Hb = 25 Oe, but with ac field switched off. In all curves, one can clearly see the critical angles ␣c⫾ of the total reflection for neutron spins aligned along with or opposite to the bias field Hb. In saturation, both NSF reflectivities, R++ and R−−, show plateaus at the wave vector transfers Q ⬍ Qc− = 共4 / 兲sin ␣c+ and, respectively, Q ⬍ Qc+ = 共4 / 兲sin ␣c−, while at 0 ⬍ f ⬍ 1 MHz, each of the reflectivities reveals both plateaus. Above the critical edges, respective reflectivity curves rapidly drop down, showing thickness oscillations at higher values of Q. Most of the small SF specular reflectivity R⫾⫿ in the bottom panel of Fig. 4 can be attributed to the efficiency of about 98.5% of our spin polarizing/analyzing system. This indicates a very good collinearity between the neutron polarization vector and the external dc and ac fields, as well as with the magnetic induction in the film. Some SF signal above the level of 1.5% 共due to imperfect polarization analysis兲 seen at f = 0.3 MHz may be caused by small deviations of domain magnetization from the ac and dc field direction or by a contamination of SF off-specular scattering 共see below兲 from small magnetic inhomogeneities. Such a contamination is unavoidable because of the finite angular resolution of the instrument. Here, we do intend to further speculate on the origin of a small SF signal in the specular channel and rather concentrate on dramatic ac field effects seen in the NSF reflectivity. It is well known16 that the positions of the critical edges, Qc⫾ = 4冑共Nbn ⫾ Nbm兲, are determined by the nuclear, Nbn, and magnetic, Nbm, scattering length densities. N is the nuclear and magnetic number density. The fact that the critical values Qc⫾ remain unchanged in ac fields implies that the absolute value M = Nbm / 共2mnn兲 of the magnetization31 is not affected by the ac field. Moreover, the value of the magnetization M ⬇ 22 kG is close to that in saturation, and the
absence of SF reflected intensity suggests that the NSF specular signal under ac field is just a linear combination of intensities reflected from the alternatively magnetized states, ¯ 兩R+兩2 + 共1 − w ¯ 兲兩R−兩2 , R++ = w
共15兲
¯ 兲兩R+兩2 + w ¯ 兩R−兩2 . R−− = 共1 − w
共16兲
Here, R⫾ are reflection amplitudes for positive or negative ¯ is the probability of the film magnetization directions, and w state with positive magnetization direction averaged over the period T = 1 / f. If no ac field is applied, then due to the posi¯ = 1, R++ = 兩R+兩2 and R−− = 兩R−兩2, tive saturating dc bias field w as seen from the corresponding curve in Fig. 4. Alternating the saturating field direction from positive to negative by a negative ac field H共t兲 ⬍ −共Hc + Hb兲 results in an exchange between R++ and R−−. Figure 4 shows that the ac field mixes up both NSF reflectivities such that each of them reveals both plateaus at the total reflection. The data on off-specular scattering were recorded at f = 0.3, 0.45, 0.6, 0.8, 1.0, and 1.5 MHz and combined into a set of maps displaying the scattering intensity distributions I⫾共␣i , ␣ f 兲 over angles of incidence, ␣i, and scattering, ␣ f . Two pairs of those maps, the one measured at f = 0.3 MHz 共upper row兲 and the other obtained at f = 1.5 MHz 共bottom row兲, are depicted in Fig. 5. In all maps, the specularly reflected intensity is seen as a ridge tracing along the diagonal ␣i = ␣ f , while off-specular scattering is concentrated into spots beside the specular ridge. Two critical edges of the total reflection for two spin states can be distinguished on the specular ridges in the first R+ and R− maps measured at f = 0.3 MHz, but only one critical edge is seen in the respective maps recorded at f = 1.5 MHz. This is in good agreement with the reflectivity curves shown in Fig. 4, confirming the fact that SF processes contribute little to the specular reflectivity. This, however, is not the case for off-specular
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scattering, which is almost entirely spin-flip in origin. Such a conclusion can immediately be made considering the bottom row of maps, which exhibit a strong asymmetry in the positions of the diffuse scattering spots with respect to the specular ridge. From the pronounced asymmetry, we infer16,32,33 that the sample is in a well magnetized state.34 Indeed, the positions of the diffuse scattering spots are determined by the Yoneda enhancement factors, which amplify neutron wave fields refracted into the film if the incident and/or scattered wave vectors are close to the critical edges.16 In the magne⫾ tized film, there are two critical angles: ␣⫾ c , where sin ␣c ⫾ = 共 / 4兲Qc . Therefore, both incoming and scattered wave fields, and hence diffuse scattering, are enhanced if either ⫾ ⫾ ⫿ ␣i = ␣⫾ c and ␣ f = ␣c or ␣i = ␣c but ␣ f = ␣c . In the first case, NSF diffuse scattering experiences enhancement peaked at the critical edges ␣i = ␣ f = ␣⫾ c on the specular ridge. This sort of diffuse scattering is fairly symmetric and may be caused by nuclear inhomogeneities, e.g., interfacial roughness, or by fluctuations of magnetization vector components parallel to the mean induction of the film. Neither of these two possibilities takes place in the present magnetic film under the probing ac field. Instead, in all panels in Fig. 5, the diffuse scattering peaks are centered at either ␣i = ␣+c and ␣ f = ␣−c for positive polarization, or, alternatively, at ␣i = ␣−c while ␣ f = ␣+c . Two peaks in the upper panels have different amplitudes, while in the lower panels, only one of the peaks survives. This all means that during the scattering process, the neutron spin projection onto the mean induction 共Y axis in Fig. 3兲 alters its sign: diffuse scattering is caused by magnetization vector fluctuations with projections onto the X-axis normal to the mean film induction. The mean induction solely contributes to specular reflection and provides a splitting between spin states. Its magnitude is determined by lateral averaging over the neutron coherence length lc.35 The coherence length is estimated16 as lc ⬇ / ␣⌬␣ ⬃ 50 m for the neutron wavelength = 4.4 Å impinging onto the sample at glancing angle ⬃␣ ⬃ 10 mrad with uncertainty ⌬␣ ⬃ 1 mrad. Fluctuations of magnetization, of whatever nature, on a smaller length scale cause diffuse scattering. Usually, it is ascribed to magnetic domains.16 However, in our case, offspecular scattering cannot be suppressed by a dc field, which just exceeds the coercive field Hc, where domains are expected to be erased. Maps taken in a dc field above Hc 共not shown here兲 look very similar to those displayed in the lower panels of Fig. 5. The diffuse scattering can only be suppressed in dc fields as high as 120 Oe, i.e., greatly exceeding Hc = 21 Oe.36 This lets us suggest that the observed diffuse scattering may be due to either thermal magnetization fluctuations or to some sort of imprinted static magnetic inhomogeneities. A quantitative analysis of the maps in the distorted wave born approximation 共DWBA兲 共Ref. 16兲 determines the size of magnetic fluctuations to be of the order of a few micrometers and amplitudes of the magnetization deviations up to a few degrees. We also note that the positions of the critical edges on the specular ridges at f = 1.5 MHz are close to those in saturation fixed by a bias field of 25 Oe.
In contrast, the maps in the upper panels of Fig. 5 are more symmetric, and they can be described as a weighted incoherent sum wIs+ + 共1 − w兲Is− of two maps Is+ and Is− measured above Hc or of those maps displayed in the lower panels. This is similar to our discussion of the specular reflectivities in Fig. 4. From this observation, we infer that at 0.3 MHz, our film spends a fraction of each period in a multidomain state with domain dimensions greater than lc. These large domains cannot be resolved in off-specular scattering. However, magnetic fluctuations inside large domains would cause diffuse scattering. A detailed analysis of the nature of those fluctuations and their role in the remagnetization process is, however, far beyond the scope of the present report. Here, we restrict ourselves to considering only one particular aspect of ac reflectometry, which is the determination of the rate at which the magnetization reversal takes place. The rate is mostly due to kinetics of large domains, which can be analyzed via the specular part of the ac reflectometry.
B. Remagnetization rate
Let us assume that the film is initially magnetized in the positive direction by a dc bias field Hb ⬎ Hc at t = 0. At a given moment of time t ⬎ 0, the film becomes partially demagnetized under field H共t兲 = Hb − H0 sin共2 ft兲 with the ac field amplitude H0. This partially demagnetized state is characterized by the surface fraction w = w共t兲 of the film with positive magnetization. The instantaneous probability w ¯ averaged over the ac period can = w共t兲 and the probability w easily be estimated taking into account that the remagnetization process from a positive saturation may only be initialized by a negative ac field with the amplitude H0 ⬎ Hb + Hc. ¯ =1 Otherwise, the sample remains in saturation and w共t兲 = w at any frequency of the ac field. On the other hand, the condition H0 ⬎ Hc + Hb does not yet guarantee that remagnetization indeed occurs. This is due to the fact that remagnetization does not happen instantly but takes a certain transient time after being triggered at the moment of time t = t1 = x1T, where x1 =
1 Hb + Hc arcsin 2 H0
共17兲
is a fraction of the period T. At t1, the external field H共t1兲 = Hb − H0 sin共2 ft1兲 ⬍ −Hc overcomes the negative coercive field −Hc. If the ac frequency is sufficiently low, then the sample may arrive at negative saturation at the time t1 + ⬍ T / 4. Then, it may remain in this state in a further decreasing and subsequently increasing ac field up to the time t2 = x2T. At this moment, the net external field reaches a positive critical value H共t2兲 = +Hc, and hence, x2 =
1 Hb − Hc 1 arcsin . − 2 2 H0
共18兲
For t ⬎ t2 + , the field finally drives the system into the positively saturated state, which persists till the end of the cycle at t = T. If the reversal cycle is fully accomplished then the prob¯ = w0, where ability w
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w0 = 1 − 共x2 − x1兲
共19兲
is just proportional to the mean fraction of the period during which the sample is magnetized parallel to the positive bias field. This fraction is independent of the remagnetization time, and the ac frequency and varies within the range 1 / 2 ⱕ w0 ⱕ 3 / 4 as a function of two parameters: ac field amplitude H0 ⱖ Hb + Hc and bias field Hb ⱖ Hc. The highest value ¯ w0 = 3 / 4 is attained at H0 = 2Hb = 2Hc, while the lowest, w = 1 / 2, is approached at H0 Ⰷ Hb , Hc. The full remagnetization cycle during an ac period T may, however, only proceed if t1 + ⱕ t1⬘, i.e., if x1 + f ⱕ x1⬘, where t1⬘ is the moment of time when the increasing negative field again reaches the value H共t1⬘兲 = −Hc. Hence, x1⬘ =
1 1 Hb + Hc − arcsin . 2 2 H0
共20兲
Alternatively, the remagnetization process is terminated and/or reversed before negative saturation is attained. This means that there exists a threshold frequency f c = −1共x1⬘ − x1兲,
共21兲
above which the remagnetization cycle remains incomplete, whereas at f ⱕ f c, a total magnetization reversal is possible ¯ is independent of the frequency. At higher and the weight w ¯ starts to vary with f. By frequencies f ⬎ f c, the weight w detecting the onset point of this dependency, one can, in principle, determine the parameter . The threshold frequency f c = f c共H0 , Hb兲 in Eq. 共21兲 depends on the ac field amplitude H0 and bias field Hb and varies within the range 0 ⱕ f c ⱕ 1 / 2. The lower limit f c = 0 is reached at the lowest ac amplitude H0 = Hb + Hc, while the upper limit, f c = 1 / 2 is approached at H0 Ⰷ Hb + Hc. ¯ 共f兲 cannot be deduced within A general dependency w the model-free treatment applied above and remagnetization kinetics may even not be characterized by a transient time , while the latter may also depend on the applied field. It is well known that the domain wall speed depends on the external field amplitude.37 In the simplest scenario, one can neglect this dependence assuming that at H共t兲 ⱕ −Hc, the fraction w共t兲 of positively magnetized volume linearly decreases until it reaches a minimum value. This occurs when after negative external field passing its minimum becomes again equal to −Hc. After that moment of time, the magnetization stays constant until H共t兲 = +Hc. Then, it again linearly increases with further increasing positive field and finally saturates. As a result, within the range f ⱖ f c, ¯ 共f兲 = 1 − 共x2 − x1兲 w
fc . f
共22兲
¯ = 1 in the From this equation and Eq. 共21兲, it follows that w ¯ 共f兲 merges with its limit f c → ⬁, while at f = f c, the function w frequency independent limit w0 given in Eq. 共19兲, as it should.
FIG. 6. 共Color online兲 Dependence of weight factor for the magnetization states on the frequency of the ac magnetic field and the field amplitude H0. The dashed lines present the fitting curves by the Eq. 共22兲.
C. Data evaluation and discussion
The data on specular reflectivity acquired at two ac amplitudes of H0 = 55 Oe and H0 = 60 Oe were fitted to Eqs. 共15兲 and 共16兲, and the set of parameters, e.g., nuclear and magnetic scattering length densities 共SLDs兲, thicknesses of Fe film and protecting Pd cap layer, roughnesses, etc., were determined along with the weight factors w共f兲 for each of the frequencies of ac field. The structural parameters and SLDs, Nbn = 8.01⫻ 10−6 Å−2 and Nbm = 4.9⫻ 10−6 Å−2, of the Fe layer are found to be close to their nominal values. The frequency dependencies w共f兲 of the weight factor are plotted in Fig. 6 together with the model curve of Eq. 共22兲 obtained with the variation of a single parameter . All other parameters were fixed to their nominal values: x1 = 0.16, x2 = 0.34, and w0 = 0.656 for H0 = 55 Oe and w0 = 0.682, x1 = 0.14, and x2 = 0.36 for H0 = 60 Oe, respectively. The fit result for the transient time is ⬇ 0.2⫻ 10−6 s.
IV. CONCLUSIONS AND OUTLOOK
We have developed a setup providing an ac magnetic field environment for experiments with polarized neutrons. The frequency and amplitude of the magnetic field can be varied within a broad range of their values. The setup can be used both for the manipulation of the neutron beam spin polarization and of the sample magnetization. We have performed first ac polarized neutron reflectometry experiment with full polarization analysis of the reflected beam and registration of the specular reflected beam as well as the diffuse scattering. Applying an ac field, we have studied the kinetics of the remagnetization process of a thin iron film. It was found that up to 0.4 MHz, the magnetization adiabatically follows the variation of the applied field. In contrast, higher frequencies cause only partial remagnetization of the sample, and above 0.8 MHz, the ac field does not alter the sample magnetization, e.g., via nucleation and propagation of large lateral magnetic domains. The method of ac reflectometry can be used for precise determination of the DW velocity as a function of applied magnetic field. Due to its depths sensitivity, it can also be applied to study layer-by-layer
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103303-8
magnetization kinetics in multilayered structures. In this area of research, the method can deliver a bulk of information inaccessible by other existing tools for the investigation of domain wall propagation, such as the magneto optic Kerr effect38 and the giant magnetoresistance effect.39 In our present device, the range of field is restricted just by the power of about 90 W to be dissipated in the rf-coil and in the sample without additional cooling. The field range achieved is well sufficient for a large variety of magnetically soft materials and systems of current interest 共see references at the end of Sec. I and references therein兲. The field amplitude can readily be increased by a factor of 3 共up to 180 Oe兲 just via increasing the current in the coil. Then, an ac power of about 1 kW can be dissipated via relatively simple air cooling system. Remagnetization in hard magnetic materials may also be of some, more academic interest. Then, a further increase in the ac field amplitude by a factor of 3 or 4 can be achieved via inserting a ferrite core into the coil. However, at higher frequencies, this would require quite sophisticated heat release equipment. ACKNOWLEDGMENTS
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