When talking about resolution and sensitivity limitations in magnetic force microscopy, it is unavoidable to address the issue of noise. Therefore in this section, some of the most common theories and different modes of noise which are involved in magnetic force microscopy will be discussed. As we shall see in the concluding section of this chapter, the level of noise will provide us with a fundamental limitation in terms of resolution in magnetic force microscopy.
3.2.1 General discussion on noise
Noise is a random fluctuation of a quantity. If a voltage or current is measured to high resolution, it will be found to show variations around the mean value. The variations are random in a sense that a future instantaneous value of the fluctuation cannot be predicted from the present and past values. The maximum time scale over which the fluctuation can predicted is related to the correlation time, or reciprocal of the characteristic frequency of the spectrum of the noise. The noise spectrum is essentially limited to frequencies less than this characteristic frequency. The magnitude of the noise is a measure of the uncertainty in the system and increases if there are more sources of randomness. The magnitude also increases if a given individual source changes its condition so that its effect on the system is less predictable. In the time the noise sources will reveal their presence by a continuously varying signal. This is the reason why the level of noise is a sensitive measure of quality or reliability of the system. If we consider a general power source, the time–dependent power P (t) of the system, which can be related for example to the mechanical, optical or electrical energy of the system, can be written as the sum of two terms
P (t) = P0+ δP (t), (3.20)
where P0= hP i is the average power defined as
hP i = lim
and δP (t) is the noise term. The spectral power density WP(ωd) at a certain drive frequency ωd is defined by
WP(ωd) = h|∆P (ωd)|2i, (3.22)
here h|∆P (ωd)|2i is the mean square of the fluctuations and the mean square of the noise is given by
hδP2i =
The noise, passing through a band–pass filter with a bandwidth β and transmission of unity around the center frequency ωd, is
hδP2i = 2βh|∆P (ωd)|2i = 2βWP(ωd). (3.24) If the average of the signal is hP i and that of the modulation is P′, then the modulation index ı is given by ı = P′/hP i and the S/N becomes
In our magnetic force microscope, as seen in Fig. 1.2, we can distinguish between three different types of noise. The noise due to cantilever’s thermal noise, optical noise and electronic noise. In general, neglecting electronic noise, thermal noise is dominant at the resonant frequency, whereas interferometer noise is dominant at frequencies away from the resonant peak. The different types of noise sources are discussed in the following paragraphs.
3.2.2 Optical noise
Optical noise can be subdivided into zero–point energy arising from quantum fluctuations and interferometer noise. Starting with the least significant of the two, the optical noise induced by the photons of the laser probe bouncing on the lever, its effect can be neglected; the noise force due to a 1 mW beam is of the order of 10−19N, which is much smaller than the long–range attractive forces [43]. Interferometer noise results from laser noise, Johnson noise δVJ and shot noise δVsh [61]. The fundamental limit for the interferometer noise is given by the shot noise contribution. Shot noise is described by the following equation
hδV2i1/2sh = Rd
p2eηphotoP0β, (3.26)
where R is the resistance of the resistor used for current to voltage conversion in the photodetector, e is the electronic charge and ηphoto is the photodiode efficiency. The signal from the interferometer is proportional to the change in power ∆P , contributing to S/N
S/N ∼ ∆P
with V the interferometer voltage, d the cantilever–fiber separation and λ the laser wavelength. As one can see, the minimum measurable displacement decreases with P01/2. Therefore, it would seem as if increasing laser power would increase the sensitivity of the interferometer and thereby improve performance. However, increasing laser power could lead to increased heating in the cantilever and sample. Second, when using particularly low k and/or high Q cantilevers, the laser light can drive or damp the cantilever motion. Finally, increasing the laser power could eventually lead to unrepairable damage of the laser itself. Nonetheless, it is recommended to optimize the laser power and improve the interferometer performance. This can be done for example by the usage of an external laser coupler, e.g. a 50/50 or (even better) a 10/90 coupler, where 10 % of the laser power is send to the cantilever and 90 % of the reflected light is transmitted to the photodetector [56].
Another option for improvement of the interferometer performance is to enhance the reflectivity of the optical fiber by adding a reflective coating onto the end. This would increase the optical power in the interferometer without increasing the power incident on the cantilever. A less obvious, but perhaps effective way to improve interferometer performance, would be the use of cantilevers with lower spring constants (< 1 N/m). These cantilevers would have an increased thermal noise compared to 3.5 N/m cantilevers, which have been used, but would also give larger frequency shifts for an equivalent force gradient, i.e. increased sensitivity. Furthermore, limiting high speed data acquisition (< 1.0 µm/s) and applying low frequency bandpass filters could improve the final sensitivity of the system.
3.2.3 Thermal noise
The atoms within the cantilever vibrate around an equilibrium position and exert small forces on the cantilever.
On an average, these forces average out, but on small scales these contributions are nonnegligible and need to be taken into account. When we assume that the cantilever has only one degree of freedom for its motion, the thermal energy of the system is 12kBT . This energy is equivalent to the average mechanical energy. The average thermal energy hUthi of the system can than be expressed as an function of the average amplitude of the cantilever vibration end h∆zthi and the spring constant k
hUthi =1
The time average deflection of the cantilever ∆zrms2 has to be converted into the frequency domain to obtain the noise frequency spectrum. This is done by applying Fourier analysis (Appendix A). We assume that the force of the combined actions of all atoms is equally distributed over all frequency components, so ˆF(ω) = ˆFth, with ˆF the Fourier transform. If the cantilever incorporates a high Q, the noise will predominately focus around the resonance frequency and the noise at dc will be low. In the frequency domain, we can now relate the mechanical energy ˆU to the thermal white noise drive Fth. Integration over all spectral components yields
the total thermal energy, which is equal to 12kBT . This gives us the frequency–independent thermal white tip compliance. For static mode operation the average noise in a bandwidth of β = 2π∆ω is given by [62]
hF2thi =
which is the exact analogon of thermal white voltage noise (Johnson noise) in a electrical resistance√
4kBT Rdβ.
In dynamic mode operation, the situation is slightly more complicated. For excessive study of thermal noise in dynamic mode operation, the reader is referred to for example [43] and [44]. The apparent Q factor of the oscillator system in this case, reflected by the compliance, is defined as
Q′
Q =hzosc2 i
hzth2i. (3.32)
The most obvious way to reduce thermal noise is go to lower temperatures, as has been done with LT systems.
Another way is to use cantilevers with higher spring constants, but as seen in the preceding paragraph, this would limit the sensitivity.
3.2.4 Electronic noise
It is a common experience in electronics that there are good circuits and devices and bad ones. Noise is often introduced into the circuit in a very sensitive way and becomes apparent its value starts to be comparable to detection signals measured in the system itself. The source of the noise is often obscure but is due in some way to a substandard component or a bad contact. In a real measurement, averaging a signal to obtain the true reading becomes impossible voltage differences are several orders of magnitude lower than the level of noise on the signal. The issue of electronic noise in has been discussed ever since the introduction of the first electronic systems for high resolution purposes, for instance by Klipec [63] and Jones [64]. In general, there are four types of noise that bother the instrumentation, i.e. (1) static noise, (2) magnetic noise, (3) common mode noise and (4) cross talk.
The first principal type of noise affecting a experimental setup is known as static noise. Electric fields radiated by power lines and other voltage sources are capacitively coupled to the wires in an instrument circuit.
This coupling results in an alternating noise signal being superimposed on any signal that is transmitted to the wires in the instrument circuit. The most effective way to break down this type of noise, is to break the coupling between the external voltage sources and instrument circuit by means of a static shield which is grounded. When a shield is placed around the wires, the external voltage sources will couple to the shield rather than to the wires within the shield.
Another type of noise is magnetic noise. Experimental laboratory are loaded with stray magnetic fields.
Any time a current goes through a conductor, a magnetic field is produced radially around it. As a result, all power lines, motors, generators, relays and so on radiate magnetic fields of erratic and varying strengths. Two methods are commonly known to fight magnetic noise. The first one is to twist the wires of the instrument, thereby forming series of adjacent loops in the instrument circuit, rather than one loop. Any magnetic field
which goes through the instrument pair will tend to be canceled out by the adjacent loops, as the currents induced by the magnetic fields into adjacent loops in each wire are in opposite directions. In our setup for example, this is done with the electrical wires going to the piezoelectric actuators. The second method is to introduce a material into the instrument circuit which either absorbs or diverts the magnetic field from the wires. This can for instance easily be done by introducing a perfect conductor. In our microscope for example, this method is used at the connection of the cantilever to the dither piezo by means of a Cu plate, with Cu being the perfect conductor. Common mode noise is different from the two types of noise previously discussed.
It stems from the fact that different points throughout a laboratory will be at a different ground potential.
This is due to power currents flowing in ground circuits which, due to the resistance between various points in the laboratory, produce voltage drops in the grounds and therefore different ground potentials. Common mode noise is particularly troublesome with thermocouple circuits. Nowadays common mode noise is hardly an issue, for instance computer manufacturers resolve this problem by introducing high impedances between input terminal and ground.
When transmitting both dc signals and ac signals through multipair cable, there is a tendency for the signal to be superimposed on signals being carried in adjacent pairs. This effect is referred to as crosstalk.
There are two techniques which can be used in combating crosstalk within multipair cables. The first is to use what is termed a balanced circuit, with this, each pair of the cable must be fed by transformer at both ends to isolate the pair circuit in the cable from grounds which may be in the terminal equipment. Balanced lines work because the interfering noise from the surrounding environment is induced into both wires equally.
By measuring the difference between the two wires at the receiving end, the original signal is recovered while the noise is canceled. Any inequality in the noise induced in each wire is an imbalance and will result in the noise not being fully canceled. One requirement for balance is that both wires are an equal distance from the noise source. Another requirement is that the impedance to ground (or to whichever reference point is being used by the difference detector) is the same for both conductors at all points along the length of the line. If one wire has a higher impedance to ground it will tend to have a higher noise induced, destroying the balance.
The second method used is to individually shield each pair and ground the shield to provide isolation between the pair circuits. In our microscope, multipair cables are individually shielded and grounded by using Faraday cages (die–cast Aluminum enclosures). This type of shielding also prevents static mode noise from entering the setup (also see chapter 4).