The resolution that can be achieved by MFM is determined by a combination of many factors, of which tip geometry, tip–sample distance and instrument sensitivity are the most important. To determine the ultimate resolution that can be achieved by MFM, one has to address the question what is actually meant by the term
‘ultimate’. For this purpose we consider an array of equally closely spaced objects, as shown in Fig. 3.4. From this array, the definition of resolution can be transferred to the spatial frequency domain as the minimum spatial wavelength that can still be observed. The upper limit of the spatial frequency is defined as frequency above which the signal becomes non–detectable. The upper limit of the spatial frequency is usually determined by the background noise level, i.e. the signal–to–noise ratio (S/N) should exceed a certain value. In principle the S/N can be made arbitrarily high by increasing the measurement time, because that decreases the measurement bandwidth. However, this is only valid in the case where the background noise has a flat frequency spectrum.
In practice, the noise strongly increases when the frequency becomes very low and is commonly referred to as the 1/f noise. In general the bandwidth β is related to the measurement time between the pixels ∆t and the total number of pixels ℵ, the combination of which determines the measurement time of the total image,
β = 1
From Eq. 3.33 we can see that for large ℵ the bandwidth is inversely proportional to the measurement time.
This means that decreasing the background noise by increasing the S/N is limited for large measurement times, where the 1/f will start to dominate.
Let us now examine the tip transfer function as defined by Eq. 3.15. If we assume a minimum bandwidth of 200 Hz, which acceptable under normal conditions, we can link the noise in the frequency domain to the noise in the spatial domain through the scan speed. From hereon we can determine the resolution by calculating the wavelength where the signal drops below the noise level multiplied with the desired S/N. Because the minimum S/N is quite low and usually the tip transfer function is very steep around this point, we set the S/N
Figure 3.4: Intuitive examination of the relationship between resolution, sensitivity and noise levels. From top to bottom, an array of equally spaced rectangular objects as observed by a SPM probe for degrading object size (black) and constant background noise level (red). In the upper picture the background noise level is negligible in comparison to the detection signal and maximum resolution and sensitivity is obtained.
For smaller object size, first the resolution of the observed object degenerates. For even smaller object size, the background noise level will start to influence the detection signal up until a point were the noise level is indistinguishable from the detection signal and all information is lost (bottom).
ratio to unity. We call this wavelength the critical wavelength λc [62]. Using the critical wavelength and the background noise level we can predict the resolution of MFM. Of course, as said before, noise stems from many sources such as thermal excitation of the cantilever, electronic crosstalk, mechanical vibrations, etc. Many of these sources however, can be eliminated by proper design of the detection system and operation in either UHV and/or low–temperature. Therefore, in an ideal situation, we are only left with the thermal noise in the cantilever itself. This provides us with a fundamental limit to MFM resolution. In order to derive an analytical expression for this limit, let us assume that the tip’s height h is much larger than the film thickness t, and that the film thickness loss factor e−2kkz can be ignored for wavelengths close to the critical wavelength. In other words, Eq. 3.15 transforms into σtip∗ (kk) = b2qM,tip, which is the point–like representation of the surface charge × the typical probe dimensions b = b(x, y). For the force function, as defined in Eq. 3.13, we adopt the sinc function sinc(x), also called the sampling function¶, with this Eq. 3.13 transforms into
F(kk) = −µ0Mkb2sinc
kxb 2
sinc
kyb 2
σtip∗ (kk), (3.35)
in which Mkis the transverse component of the surface magnetization. To calculate the critical wavelength, we assume a perpendicular medium with a harmonic magnetization distribution Mz= |Mz| sin(2πx/λ)sin(2πy/λ), with identical wavelengths in both x– and y–directions. The force amplitude as function of wavelength in the z–direction is then given by
∆ |Fz(λ)| = µ0 Mk
|Mz| b2sinc2
πb λ
. (3.36)
For a fixed wavelength, the force has a maximum amplitude at b = λ/2, so when the tip diameter is exactly half of the wavelength of the stray field. This is logical, because the for smaller wavelengths, charges with opposite polarity will be beneath the tip and reduce the force. So, from eliminating the tip dimension b from Eq. 3.36, we are left with the maximum force amplitude
¶This is a function that arises frequently in signal processing and the theory of Fourier transforms. There are two definitions in common use. The one used here defines
sinc(x) =
1 for x = 0
sin x
x otherwise , with normalization
+∞
´
−∞
sinc (x) dx = π. (3.34)
(∆ |Fz| (λ))max= µ0
Mk |Mz|
λ π
2
. (3.37)
This force should be larger than the base noise level as defined in Eq. 3.31 and therefore the critical wave-length λc yields
λc= π
s √4kBT γβ µ0
Mk
|Mz| . (3.38)
This is fundamental limit of MFM resolution that theoretically is possible. To get an idea of the value of this limit; Abelmann et al. [62] calculated the critical wavelength for dynamic mode operation in MFM for k = 3 N/m, ω = 2π75 kHz, Q = 300 and β = 200 Hz as a function of tip–sample distance for magnetization values of Mk and Mz equal to 1 MAm−1. They found that for tip–sample separation of ∼ 6 nm, the critical wavelength λc ∼ 10 nm. For higher Q values (Q ∼ 40, 000) they found that the theoretically achievable resolution can be improved by a factor of 2. With respect to this theoretical analysis of λc however, it should be stated that although it is useful in a sense, its meaning for practical MFM is limited. Usually, much smaller signals can be detected. One of the reasons might be that the magnetic charge density in general will not be on the uttermost front of the tip’s surface nor at the top surface of the medium. It is much more likely that charges will be distributed over a volume with a thickness of at least the exchange length, which can be several nanometers. On the other hand, contributions of absorbed molecules on the sample surface can result in an effective tip–sample distance, which is larger than the physical tip–sample separation. To conclude, in practice there will also be other noise sources other than the thermal noise contributing to the overall noise level, such as the 1/f noise, laser power fluctuations, electronic crosstalk etcetera. Even in the thermal noise level, characterized by the Q factor, can be an uncertainty as a result of damping effects, meniscus formation, and eddy currents in the sample caused by time varying magnetic stray field of the tip. All these effects will reduce the Q factor if the tip approaches the sample. It is therefore wise to measure the Q factor of resonance at the same tip–sample distance used during the measurements.
Experimental setup
4.1 Introduction
The basis of the setup we employ stems from the attoAFM–I from by attocube, a compact atomic force microscope particularly designed for applications at low and ultra low temperatures. The principle of operation of this SPM consists of scanning the sample of interest below a fixed cantilever and measuring the cantilever’s deflection with a fiber based optical interferometer as shown in Fig. 1.2. The cantilever, sample, piezoelectric actuators and fiber ensemble is mounted onto a microscope stick. The microscope stick is inserted into a vacuum tube that is placed in a cryostat equipped with an active damping system and a soundproof isolation box. The magnetic force microscope is operated by the SPM1000 Rev.8.5, a universal SPM controller, and the PLLPro2, a phase–locked–loop amplifier, both provided by RHK Technology. In this chapter we will outline the most important elements, which are subdivided into mechanical, optical and electrical components. Furthermore, we will discuss alterations and improvements that have been made to this setup.