**UFBA**

# Microscopia de força magnética (MFM)

(Parte **1** de 4)

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Annu. Rev. Mater. Sci. 1999. 29:53–87 Copyright c° 1999 by Annual Reviews. All rights reserved

U. Hartmann Institute of Experimental Physics, University of Saarbrucken, P.O. Box 151150, D-66041 Saarbrucken, Germany; e-mail: u.hartmann@rz.uni-sb.de

KEY WORDS: magnetic domains, magnetic domain walls, magnetic materials, magnetic recording components

This review on magnetic force microscopy does not provide an exhaustive overview of the past accomplishments of the method but rather discusses the present state of the art. Magnetic force microscopy is a special mode of noncontact operation of the scanning force microscope. This mode is realized by employing suitable probes and utilizing their specific dynamic properties. The particular material composition of the probes and the dynamic mode of their operation are discussed in detail. The interpretation of images acquired by magnetic force microscopy requires some basic knowledge about the specific near-field magnetostatic interaction between probe and sample. The general magnetostatics as well as convenient simplifications of the general theory, which often can be used in practice, are summarized. Applications of magnetic force microscopy in the magnetic recording industry and in the fundamental research on magnetic materials are discussed in terms of representative examples. An important aspect for any kind of microscopy is the ultimately achievable spatial resolution and inherent restrictions in the application of the method. Both aspects are considered, and resulting prospects for future methodical improvements are given.

Magnetic force microscopy (MFM) is a straightforward special mode of operation of the noncontact scanning force microscope. Shortly after the invention of the atomic force microscope it was recognized that detection of magnetostatic interactions at a local scale was possible by equipping the force microscope with a ferromagnetic probe, which then could be raster-scanned across any ferromagnetic sample. The near-field magnetostatic interaction for a typical probe-sampleconfigurationturnsouttobefairlystrongandlargelyindependent

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54 HARTMANN of surface contamination. Consequently, MFM is quite easy to perform and applicable under various environmental conditions, in most cases even without any special sample preparation.

MFM is an important analytical tool whenever the near-surface stray-field variation of a magnetic sample is of interest. This is certainly the case for the development and application of magnetic recording components. It is thus not surprising that MFM was first demonstrated on a microfabricated magnetic recording head (1). A big breakthrough occurred when it was demonstrated that even individual interdomain boundaries (2) and in some cases part of their internal fine structure (3) could be analyzed at high-spatial resolution. This resolution is comparable to that so far only obtained by electron microscopebased instrumentation.

At the beginning of the 1990s MFM started to become a widely used method inmagneticmaterialsresearchandinthedevelopmentofmagneticdevices. The recording industry became an important field of industrial application (4), and the number of reported MFM results in basic research, especially on magnetic thin film arrangements, soon became vast (5). The era of method development in the field of MFM changed more and more toward an era of various dedicated applications of a standard scanning probe method. However, this does not imply that there have been no method breakthroughs during the past few years. One recent breakthrough is certainly the demonstration that MFM can be used to image flux lines in low- and high-Tc superconductors (6). MFMs have even extended local detection of magnetic interactions to eddy currents (7) and magnetic dissipation phenomena (8).

Reviews on past accomplishments can be found in References 5 and 9 and in thevariousproceedingsofconferencesdedicatedtoscanningprobemicroscopy or to magnetic materials research. It is not the main goal of this work to add another review on past achievements by presenting an exhaustive list of references to the vast original literature. Rather, the purpose of the present work is to analyze the field of MFM by emphasizing the state of the art, the main applications in basic research and in the magnetic recording industry, and by looking into the future from a general viewpoint, which is based on more than ten years of experience with a powerful magnetic imaging method.

The simplest mode of operation of a noncontact scanning force microscope consists in lifting the cantilever probe up to a certain distance from the sample surface to measure a long-range interaction in terms of a static force exerted on

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MAGNETICFORCEMICROSCOPY 5 theprobe. Thisapproachis, however, notpreferredinreality, becauseafarmore sensitive detection can be realized by utilizing the dynamic properties of the probe. An obvious characteristic describing part of these dynamic properties is the resonant frequency of the cantilever given by

with its spring constant c and its effective mass m. In order to vibrate the probe, the cantilever may be attached to a bimorph piezoelectric plate. Alternatively, a piezoelectric actuator can be used to excite the sample. In some applications it is possible to externally modulate the long-range probe-sample interaction, which also results in cantilever oscillation. The latter possibility is particularly relevant if magnetic interactions are caused by electrical driving currents. The noncontact mode of operation involving sinusoidal excitation is frequently called the dynamic or ac mode.

In contrast to the detection of quasistatic forces, the response of the cantilever in the dynamic mode is more complex and deserves some discussion. If the cantilever is excited sinusoidally at its clamped end with a frequency

! and an amplitude –0, the probe tip likewise oscillates sinusoidally with a certain amplitude –, exhibiting a phase shift fi with respect to the drive signal applied to the piezoelectric actuator. The deflection sensor of the force microscope monitors the motion of the probe tip provided that its bandwidth is large enough. The latter requirement clearly favors optical deflection sensors. The equation of motion describing the output from the cantilever sensor is given by

where d0 is the probe-sample distance at zero oscillation amplitude and d.t/ the instantaneous probe-sample separation. Q, apart from the intrinsic properties of the cantilever, which are the lumped effective mass and the resonant frequency, is determined by the damping factor °:

with !0 from Equation 1. ° introduces the influence of the environmental medium, which could be ambient air, a liquid, or ultrahigh vacuum (UHV).

Q thus ranges from values below 100 for liquids, air, or other gases at an appropriate pressure, to more than 100,0 which is sometimes obtained in

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56 HARTMANN

UHV. After the usual building-up, Equation 2 leads to the steady-state solution d.t/ D d0 C – cos.!t C fi/ 4. for the forced oscillator. The amplitude of the probe’s oscillation is given by

The phase shift between this oscillation and the excitation signal amounts to

The above simplified formalism is based on the assumption that the oscillation amplitude – is sufficiently small in comparison with the length of the cantilever. Obviously, the results derived so far describe only free cantilever oscillations, e.g. oscillations at the absence of any probe-sample interaction. This means d0 is still so large that no influence of the sample on the probe’s oscillation can be detected. If d0 is now decreased such that a force F affects the motion of the cantilever then a term F=m has to be added to the left-hand side of Equation 2.

In order to consider almost all interactions that could be relevant in MFM, one has to assume

F D Fµ d;

which, apart from the static interaction, also accounts for dynamic forces. An example of the dynamic forces is eddy currents (7). Because F covers probesample interactions of various types, in particular spatially nonlinear ones; the d.t/ curves monitored by the deflection sensor and found according to Equation 2 may represent anharmonic oscillations. If, however, F.d/ can be substituted by a first-order Taylor series approximation for –0 ¿ d0, then the force microscopedetectsthecomplianceorverticalcomponentoftheforcegradient@F=@z.

On the basis of this approximation, the cantilever behaves under the influence of the probe-sample interaction as if it had a modified spring constant

where c is the intrinsic spring constant entering Equation 1. An attractive probe-sample interaction with @F=@z > 0 will effectively soften the cantilever spring, while a repulsive interaction with @F=@z < 0 will make it effectively

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MAGNETICFORCEMICROSCOPY 57 stiffer. According to Equation 1, the change of the apparent spring constant will modify the cantilever’s resonant frequency to

Provided that @F=@z ¿ c, the shift in resonant frequency is given by

According to Equations 5 and 6, a shift in the resonant frequency will result in a change of the probe’s oscillation amplitude – and of the phase shift fi between probe oscillation and driving signal. 1!;–, and fi are experimentally measurable quantities that can be used to map the lateral variation of @F=@z. Phase and amplitude additionally contain information about the damping coefficient °. Thus a local variation of this quantity can be separated from the local variation of the compliance by measuring the frequency shift and the change in amplitude or the phase shift. The simple harmonic solution in Equation 4 evidently shows that the dynamic mode of operation can be based on the employment of lock-in signal detection methods. The additional use of suitable feedback mechanisms opens up different variants of operation.

The most commonly used detection method, generally referred to as slope detection, involves driving the cantilever at a fixed frequency ! slightly off resonance. According to Equation 9, a change in @F=@z gives rise to a shift in the resonant frequency of 1! and, according to Equation 5, to a corresponding shift 1– in the amplitude of the cantilever vibration. 1– is maximum at that point of the amplitude-versus-frequency curve where the slope is maximum. The sensitivity is ultimately determined by thermal noise. Careful analysis (10) shows that the minimum detectable compliance is given byµ @F s 2kTfl

where –rms is the root-mean-square amplitude of the driven cantilever vibration and fl is the measurement bandwidth. High Q values can be obtained by operation in vacuum, reducing air damping (<10¡3 mbar). It might appear advantageous to maximize sensitivity by obtaining the highest possible Q. With slope detection, however, increasing the Q restricts the bandwidth of the system. If @F=@z changes during scanning, the vibration amplitude settles to a new steady-state value after a sufficient length of time given by

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Thus for a high-Q cantilever in vacuum (Q D 50,0) and a typical resonant frequency of 50 kHz, the maximum available bandwidth would be only 0.5 Hz, which is unusable for most applications. The dynamic range of the system would be similarly restricted. Because of these restrictions it is not useful to try to increase sensitivity by raising the Q to such high values. Moreover, if the experiments have to be performed in vacuum to prevent sample contamination, it may not be possible to obtain low enough Q for an acceptable bandwidth and dynamic range. Therefore, slope detection is unsuitable for most vacuum applications.

An alternative to slope detection is frequency modulation (FM). In the FM detection system, a high-Q cantilever vibrating at resonance serves as the frequency-determining component of an oscillator. Changes in @F=@z cause instantaneous changes in the oscillator frequency, which are detected by an FM demodulator. The cantilever is kept oscillating at its resonant frequency by utilizing positive feedback. The vibration amplitude is likewise maintained at a constant level. A variety of methods, including digital frequency counters and phase-locked loops, can be used to measure the oscillator frequency with very high precision.

In the case of FM detection, a careful analysis (1) shows that the minimum detectable force gradient is given by that of Equation 1 multiplied by p 2.

However, in contrast to slope detection, Q and fl are absolutely independent in FM detection. Q depends on only the damping of the cantilever and fl is set by only the characteristics of the FM demodulator. Therefore, the FM detection method shows the sensitivity to be greatly increased by using a very high Q without sacrificing bandwidth or dynamic range.

If the probe and sample in a scanning force microscope exhibit a magnetostatic coupling, the major requirement to perform MFM is fulfilled. The manifestation of magnetostatic interactions is obvious if a sharp ferromagnetic tip is brought into close proximity with the surface of a ferromagnetic sample. Raster-scanning of the tip across the surface then allows the detection of spatial variations of the probe-sample magnetic interaction. The long-range magnetostatic coupling is not directly determined by the mesoscopic probe geometry, as for other near-field methods, but rather by the internal magnetic structure of the ferromagnetic probe. As shown in the following, this greatly complicates matters and requires a detailed discussion of contrast formation.

For simplicity’s sake, it is easiest to consider the probe as a needle consisting of bulk material. A sharp ferromagnetic needle naturally exhibits considerable

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MAGNETICFORCEMICROSCOPY 59 magnetic shape anisotropy, which forces the magnetization vector field near the probe’s apex to predominantly align with the axis of symmetry of the probe. On the other hand, sufficiently far away from the apex region, where the probe’s cross-sectionalareaisalmostconstant, themoreorlesscomplexnaturaldomain structure obtained in a ferromagnetic wire is established. This domain structure depends on the detailed material properties represented by the exchange, magnetocrystalline anisotropy, and magnetostriction energies. Lattice defects, stresses, and the surface topology exhibit an additional influence on the domain structure. Because of this complicated situation, it is necessary to develop reasonably simplified magnetic models to describe the experimentally observed features of magnetostatic probe-sample interaction as accurately as possible.

Since it is generally hopelessly complicated to derive the actual magnetization vector field of the aforementioned type of probe from first principles, it is reasonable to apply the following model (12). The unknown magnetization vector field near the probe’s apex, with all its surface and volume charges, is modeled by a homogeneously magnetized prolate spheroid of suitable dimension, while the magnetic response of the probe outside this fictitious domain is completely neglected. The second assumption is that the dimensions and the magnitude of the homogeneous magnetization of the ellipsoidal domain are both completely rigid, i.e. independent of external stray fields produced by the sample. In this way the micromagnetic problem is simplified to a magnetostatic one.

The model allows interpretation of almost all experimental results obtained by MFM on the basis of bulk probes. Moreover, the concept of assuming a single prolate spheroidal domain that is magnetically effective for bulk ferromagnetic probes approaches reality surprisingly well (12). Using this pseudodomain model, the problem is now to determine the probe’s magnetic properties and the probe-sample magnetostatic interaction for a given experimental situation.

The magnetostatic potential created by any ferromagnetic sample is given by

where Ms.r0/ is the sample magnetization vector field and s0 an outward normal vector from the sample surface. The first two-dimensional integral covers all surface charges created by magnetization components perpendicular to the bounding surface, whereas the latter three-dimensional integral contains the

(Parte **1** de 4)