**9. References**

54 Atomic Force Microscopy – Imaging, Measuring and Manipulating Surfaces at the Atomic Scale

the present samples a very weak or at least less intense magnetic signal than in the case of the Mn-containing films. Therefore, the results of Fig. 12 are in agreement with the initial expectations. Unlike observed in samples with Mn it is possible to identify only a slight decrease in the MFM signal with the tip-sample separation, due to the comparatively lower signal intensity. Here it is important to notice that the present procedure adopted in the MFM measurements is unique in the literature. Consequently, similar results obtained from

For the GeCo samples, we observed that the MFM signal intensity increased with increasing Co concentration [see Fig 12(b)]. The thermal treatment for samples with the same [Co], in principle, didn't intensify the magnetic signal. As an example of increasing MFM signal intensity with [Co], at a tip-sample separation of 500 nm, we observed that the Ge film with [Co] ~ 1.7 at.% showed a MFM signal of ~ 2 mV , and the Ge film with [Co] ~ 7.6 at.% exhibited a MFM signal of ~ 8 mV, both annealed at 500 oC. Still, as can be seen in Fig. 12(b), even the as-deposited Ge samples show magnetic signal, probably due to the existence of magnetically active Co atoms randomly distributed in the amorphous network. For the annealed samples, due to the diffusion of Co and the structural rearrangement of the network, it is expected that the number of magnetically active Co atoms increase (Ko et al., 2006). However, its magnetic activity does not exceed that of the amorphous films due to the formation of CoGe2. Finally, the increasing in the magnetic signal with increasing [Co] is

expected since the number of magnetically active Co atoms probably also increases.

For the SiCo films, the situation seems somewhat different, and not systematic. At first, as shown in Fig. 12(a), the sample with [Co] ~ 2.8 at.% without annealing shows a relatively high value of magnetic signal due to the magnetically active Co. After annealing at 900 oC, its value is diminished, probably due to the formation of CoSi2. In contrast, the as-deposited Si film with the highest [Co] (~ 10.3 at.%) presents an extremely low MFM signal, probably due to the large number of magnetically inactive Co atoms, which may be associated with its highly disordered structure. After annealing at 900 oC, its magnetic activity is significantly increased due to the diffusion and consequent magnetic activation of the Co atoms. However, the magnetic activity is now limited by the existence of CoSi2, and, therefore, its magnetism is less intense than the as-deposited film with [Co] ~ 2.8 at.%, that,

In summary, MFM is a relatively new technique for imaging magnetization patterns with high resolution and minimal sample preparation. The technique is an offspring of AFM and employs a sharp magnetic tip attached to a flexible cantilever. The tip is placed close to the sample surface (from some nanometres to a few micrometres) and interacts with the stray field emanating from the sample. The image is formed by scanning the tip laterally with respect to the sample and measuring the force (or force gradient) as a function of position. The interaction strength is then determined by monitoring the motion of the cantilever using a sensor. Although a lot of effort has been done in order to get quantitative information, MFM is still predominantly a qualitative characterization technique. In the present work, MFM proved to be particularly suitable to study the magnetic properties of Si and Ge-based magnetic semiconductors. In this context, the technique is very efficient to detect magnetic activity in the form of vortices in sub-micrometre structures. As well, a combination of the

others, for quantitative comparison purposes, are non-existent.

in principle, doesn't have the silicide phase.

**7. Conclusion** 


Hartmann U. (1999). Magnetic force microscopy, *Annual Review of Materials Research,* Vol. 29, pp. 53-87.

**4** 

Thin-Lin Horng

*Taiwan, R.O.C.* 

**Vibration Responses of Atomic Force** 

*Department of Mechanical Engineering, Kun-Shan University, Tainan* 

In this investigation, the solution of the vibration response of an atomic force microscope cantilever is obtained by using the Timoshenko beam theory and the modal superposition method. In dynamic mode atomic force microscopy (AFM), information about the sample surface is obtained by monitoring the vibration parameters (e.g., amplitude or phase) of an oscillating cantilever which interacts with the sample surface. The atomic force microscope (AFM) cantilever was developed for producing high-resolution images of surface structures of both conductive and insulating samples in both air and liquid environments (Takaharu et al., 2003 ; Kageshima et al., 2002 ; Kobayashi et al., 2002 ; Yaxin & Bharat, 2007). In addition, the AFM cantilever can be applied to nanolithography in micro/nano electromechanical systems (MEMS/NEMS) (Fang & Chang, 2003) and as a nanoindentation tester for evaluating mechanical properties (Miyahara, et al., 1999). Therefore, it is essential to preciously calculate the vibration response of AFM cantilever during the sampling process. In the last few years, there has been growing interest in the dynamic responses of the AFM cantilever. Horng (Horng, 2009) employed the modal superposition method to analyze the vibration responses of AFM cantilevers in tapping mode (TM) operated in a liquid and in air. Lin (Lin, 2005) derived the exact frequency shift of an AFM non-uniform probe with an elastically restrained root, subjected to van der Waals force, and proposed the analytical method to determine the frequency shift of an AFM V-shaped probe scanning the relative inclined surface in noncontact mode (Lin, et al., 2006). Girard et al. (Girard, et al., 2006) studied dynamic atomic force microscopy operation based on high flexure modes vibration of the cantilever. Ilic et al. (Ilic, et al., 2007) explored the dynamic AFM cantilever interaction with high frequency nanomechanical systems and determined the vibration amplitude of the NEMS cantilever at resonance. Chang et al. (Chang & Chu, 2003) found an analytical solution of flexural vibration responses on tapped AFM cantilevers, and obtained the resonance frequency at arbitrary dimensions and tip radii. Wu et al. (Wu, et al., 2004) demonstrated a closed-form expression for the sensitivity of vibration modes using the relationship between the resonant frequency and contact stiffness of the cantilever and the sample. Horng (Horng, 2009) developed an analytical solution to deal with the flexural vibration problem of AFM cantilever during a

nanomachining process by using the modal superposition method.

The above studies considered the AFM cantilever as a Bernoulli-Euler beam model. The effects of transverse shear deformation and rotary inertia were assumed to be negligible in

**1. Introduction** 

**Microscope Cantilevers** 

