Probes and Cantilevers for SPM

     The most popular materials are monocrystalline silicon and Si₃N₄. Silicon nitride cantilevers offers advantage over silicon ones in the sense that they may be produced thinner and, hence, more flexible (having lower stiffness) but Si₃N₄ does not possess perfect manufacturability for tips machining, therefore Si₃N₄ tips are usually inferior to those made of silicon. Most recently, merits of both materials have been coupled together in hybrid probe construction, which features flexibility of silicon nitride lever and sharpness of silicon tip [1442].
     Also cantilevers made of tungsten, nickel and another materials are in use worldwide [157, 581, 602, 870, 921, 922, 943, 1022, 1065, 1328]. Material through its inherent mechanical properties (elasticity module or Young's module E, module of rigidity G) and density defines stiffness, resonant frequency and Q-factor (see below). Also, material properties of the coating should be taken into consideration if it spreads over the whole cantilever surface.
     Using cantilevers made of low resistance material such as metals or highly doped silicon ensures that no electrostatic charges will collect at the tip apex. Gathering of electrostatic charges results in distortion of the images and is especially crucial in Scanning Tunneling and Electrical Force Microscopy studies.

     A number of cantilever geometries were proposed since the invention of AFM and even earlier when Scanning Tunneling Microscopy was the only SPM technique of comparable principle of operation. The most preferable among researchers are rectangular thin bar and triangular lever forms. Triangular form often used in such contact mode experiments where twisting a tip along the only symmetry axis of the lever is undesirable. Range of geometries of the serial cantilevers manufactured by MikroMasch is rather wide:

MikroMasch Data from publications reviewed Length l, ?µm 90 - 460 > 10 Width w, ?µm 35 - 60 > 3 Thickness s, ?µm 0.7 - 7.5 > 0.1 Force constantk, N/m 0.01 - 91 0.001 - 400 Resonant frequency fo, kHz 7 - 420 3 kHz ~ 10 Mhz

     Stiffness is defined by force constant k measured in N/m (or sometimes nN/nm). Usually employed cantilevers possess force constants in the range of 0.01 - 100 N/m. The "soft" cantilevers which k is below 0.1 N/m are chosen mainly while using contact mode in order to affect the sample in minimal extent. The rigid ones with k>1 N/m are often used in non-contact or dynamic modes since they exhibit high resonant frequencies and small oscillation amplitudes of about several tens Angström. It provides wide dynamic frequency range and substantially raises sensitivity.

Fig. 1.1 Schematic of vertical deflection	Fig. 1.2 Schematic of lateral deflection

     By default term "force constant" stands for that of determining rigidness of vertical deflection of the cantilever (Fig.1.1). Theoretical expression for estimation of this force constant in the case of the simplest rectangular cantilever is given by following formula:

     In a number of SPM techniques especially in Lateral Force Microscopy (LFM) tip is moved in touch with the surface of the sample and twisting of the cantilever apart from vertical deflection also take place as depicted in Fig 1.2. Stiffness of the cantilever suffered from such kind of applied external force is determined by lateral force constant, which can differ from that of "normal" k at great extent. The formula for k_lat has the same structure but additional parameter of h defining tip height appears:

     By additional processing of commercial cantilevers Kageshima et al. [1437] enhanced the lateral force sensitivity of Atomic Force Microscopy for detection of molecular scale interactions. Sensors of two types were fabricated by this group. As a result, the lateral force constants were reduced about 10-fold and 180-fold for both types respectivelly that allowed for improvement of lateral force resolution up to 1 nN.
     Expressions for determination of force constants of triangular levers are more complicated [1308]. See also a useful paper of Newmeister and Ducker [1218].
     It is commonly adopted that triangular lever much better withstands lateral forces in comparison to rectangular one of identical normal stiffness. Rigorous calculations performed by Sader [2622] show that the reality is quite the contrary. This unexpected theoretical result is to be approved experimentally.
     It is nontrivial task however to measure exactly the force constant of a particular cantilever mainly due to the fact that geometrical parameters of every cantilever cannot be exactly controlled and determined individually in mass production. Therefore, theoretically calculated stiffness may substantially differ from the real one. Especially it concerns of thickness control during the etching process of cantilever micromachining. Simple calculation shows that variation of thickness in ?±20% results in variation of k value in the range of about -50% ~ +70%. To cope with this unsertainty a number of empiric and semiempiric techniques of force constant measuring and calibration are developed [E008, 859, 1300, 1205, 1206, 1621, 2626].
     MikroMasch produces a series of cantilevers with calibrated force constant. Calibration procedure is taken from paper of J. Sader et al. [1206].
     Determination of mechanical properties of materials at micro and nanoscale needed for microprocessor design and development of micro-elecrtomechanical systems (MEMS) is a challenge due inadequate testing equipment since it must be capable of applying loads on the order of 10^-6 to 10^-9 Newtons. AFM with well-calibrated silicon cantilevers is quite promising technique for such measurements. Comella and Scanlon [756] report development of cantilever based technique for determination of elasticity of thin films. In their simple and reliable approach microcantilevers of material investigated are formed by ordinary lithographical method, then AFM silicon cantilever with well-calibrated force response is used to measure microbeams deflection under defined loadings. Authors claim that the method developed is applicable to beams of thickness ranging from Angströms to microns. Vice versa, using calibrated loading (for example, with indenter) one can measure unknown force constant of the AFM cantilever [E011].
     Visit also Materials Research section and subsections within for numerous examples of materials characterization.

     Resonant frequency f_o varies from several kHz up to several Mhz [163] and analytically is expressed as:

where m and m_tip are masses of a cantilever and a tip respectively. In the simplest case of rectangular cantilever neglecting with the tip mass one can derive:

     It can be easily seen that the decisive factor to increase resonant frequency is to shorten the length of a cantilever. At the same time cantilever can be done rather "soft" by means of width or thickness adjustments compensating rise of force constant due to appearance of l in the denominator of the expression for k. The cantilevers with high resonant frequencies f_o and low k are the better choice for tapping mode investigations since tip most gently tapping the surface oscillating at the same time with several hundred kilohertz [519, 1578].
     Eigenfrequency of the cantilever can significantly vary if it is coated with rather thick coating like, for instance, DLC. Salvadori at al [85] investigated dependence of f_o from the thickness of DLC coating deposited on the cantilevers of 3,5-5 ?µm thicknesses using a specially derived formula for a thin and uniform coating with thickness s. It was found to have almost linear fashion. The coating of about 0,3 ?µm raise eigenfrequency of cantilever by 50%. Although, deposition of thick DLC films results in severe bending of cantilever as shown in photographs presented by Lemoine at al [82]. This bending is originated from the intrinsic stress in DLC films. It can be released substantially by incorporating silicon atoms (~15%) in a-C:H film. But authors note that stress-induced cantilever deflections are too large for direct use of the cantilevers in an AFM microscope. Nevertheless, such a bending gave rise to works on stress measurements in thin films grown on silicon cantilever beams (see, for example, [1649]).
     Fritz et al. [1211] reported on the successful deposition of a thin copper film onto conductive silicon cantilevers of MikroMasch produce by means of galvanic displacement technique. This procedure did not lead to bending of the lever and kept the resonant frequency almost unchanged. To demonstrate the effectiveness of this approach for tribological studies, the coated cantilevers were chemically functionalized with alkanethiol monolayers. The effect of changed surface energy was detected with adhesion measurements in water and ethanol.

     Q-factor shows a part of the whole vibration energy that system looses during a full cycle of oscillation. In other words, it defines minimal external vibration force within given range of frequencies needed to maintain oscillation of the system. Ideally, no external force is needed since it contributes to overall noise reducing sensitivity. In highly dumping ambients such as liquids substantial external forces are needed thus Q-factor is less than in ultrahigh vacuum in 2-3 order of magnitude. In mechanical systems it approximates as:

where b is a damping factor. Thus Q-factor depends upon mechanical characteristics of cantilever and damping properties of ambient. The more is Q, the more is sensitivity. See the recent publication on the theme by Yasumura et al. [1210].

     The choice of the material depends upon the purposes it is intended for. When the highest possible hardness is necessary, for example in nanoscratching experiments or Scanning Spreading Resistance Microscopy (SSRM), the cantilever is furnished with diamond tip [602, 1606, 1609]. Another way is to use conventional tip with diamond-like coating (DLC) [1336].
     Resistance of the tip and the whole cantilever against aggressive ambients especially in the case of fluids is of primary importance as well. This problem is often solved using protective coatings such as Cr-Au, Pt etc.
     Protective coatings at the same time can serve as conductive coatings. Besides, TiN, W₂C, TiO, doped DLC and many others are used. Cantilevers with such tips are suitable for Scanning Tunneling Microscopy (STM), Electric Force Microscopy (EFM) and other conductance sensitive techniques. In principle, silicon tip itself can be used for this purpose if made of highly doped silicon. But it easily damaged in contact mode techniques such as mentioned above SSRM where very high contact forces of 10^-5 N are applied to the specimen [157].
     Another class of coatings includes magnetic ones such as Co, Ni, Fe and a number of special alloys for use in Magnetic Force Microscopy (MFM) and Scanning Tunneling Microscopy investigations [642, 873, 1111, 1117, 1122, 1123, 1126, 1176, 1441]. You can choose those manufactured by MikroMasch here.
     Since the midst 1990's a novel approach of functionalization of usual tip apex with another molecules in order to build specific probes and sensors is developing intensively. Till now a different micro-, macromolecules and functionalization addenda was proposed and investigated as candidates for supertips: carbon nanotubes [30, 78, 818, 1325, 1380, 1381, 1456, 1464, 1610-1620], W₂C nanotubes [1445], proteins [974], adamantane [270], caltrop shaped molecules based on a differentially substituted tetrahedral silicon atom [E009] and many others [451, 1211, 1622-1631]. Besides inherent extrasharpness the functionalized tips possess selectivity with respect to the features on the sample surface having different chemical nature and ideally suit to map their distribution and perform single-molecule force spectroscopy studies. Of course, to measure such tiny forces down to 10^-12 N and gradients of 10^-5 N/m quite sensible and intelligent registration and feedback systems are necessary.
     Functionalization of the tip apex with various agents allows for preferential imaging of different atoms at the sample surface. Ke et al. [13] investigated theoretically the effect of tip morphology on the image contrast for a GaAs(110) surface for the three atomically different tip apexes of a Si tip: (1) Si apex with a half-filled dangling bond; (2) Ga apex with an empty dangling bond; and (3) As apex with a fully filled dangling bond. The study revealed a great impact of the dangling-bond states of different atoms at the tip apex on the image contrast. Calculation revealed that the Ga apex will image the As sublattice, and the As apex will image the Ga sublattice, and in the case of the Si apex, it is possible to image only the As sublattice or both the As and Ga sublattices, depending on the tip-sample separation.

     These are height, profile, open angle at the tip apex and apex curvature radius.
     The overwhelming majority of former commercial tips were of pyramidal triangular or square shape with opening angle up to 70?° (Fig 2. a). They were produced by lithography method and didn't actually exhibit sufficient sharpness.
     Using (electro)chemical etching technique one can produce sharper cantilevers from monocrystalline silicon. The bulk shape of this tip is still pyramidal and not so sharp (<40?°, Fig 2. b) but the very end of the tip tends to become more and more thinner as approaching to the apex. The open angle near the apex is about 20?°. But pyramidal profile of the cantilever does not still allow imaging of high aspect ratio structures.
     If high aspect ratio samples having steep and deep walls are to be investigated, you have to choose long and thin tip like a bee STING (Fig 2. c). Such tip is grown on the apex of the conventional tip using Electron Beam Deposition (EBD) technique. Another techniques such as Focused Ion Beam (FIB) machining [2594] or oriented growth of whisker-like silicon needles [1281] are also developed.
     The latest advances in technology of carbon nanotubes allow attachment or direct grow of carbon nanotube as an ideally possible whisker at the tip apex of commercial cantilevers (Fig 2. e).
     Resolution of Scanning Probe Microscopy depends substantially on the sharpness of the imaging tip. Curvature radius of the conventional tip is about 10-15 nm, that of the STING tip is about 5-7 nm. But the highest possible resolution is achievable with HI'RES tips specially developed by MikroMasch (Fig 2. d). Curvature radii of these tips are of the order of 1 nm (!), so fine molecular or even atomic structure can be resolved.
     In some studies quite specific probe tips are required, for example tips with flat apex (Fig 2. f). Such tips are used in tribological, indentation and other studies.

a b c

d e f Fig. 2 Tip variety. (a) Probe with piramidal tip. (b) Probe with conventional silicon tip. (c) Probe with EBD grown tip (STING). (d) TEM image of HI'RES tip. (e) Probe with Carbon Nanotube tip. Image taken from [1612] with permission of Dr. R. Schlaf. (f) Probe with flat tip apex.

     Sometimes there is no apparent dependence between the tip characteristics and imaging parameters. For example, Sedin and Rowlen [21] observed two different trends for measured surface roughness as a function of tip size. Root mean squared (RMS) roughness is one of the most commonly reported measures of surface roughness from AFM images. It was found that at small lateral scan sizes (<500 nm) the image root mean square roughness decreased as tip size increased, but at larger scan sizes (e.g. 5000 nm), the roughness increased with increasing tip size. The authors also emphasized that there is a great variation of tip shapes within a single lot of commercial cantilevers. Frost et al. [27] have shown that the tip quality has a strong influence on the surface roughness parameters extracted from the AFM images particularly for surfaces with a low surface roughness (~1 nm) as generally obtained by means of thin film technologies. For evaluation of the influence of the actual AFM tip quality on the measured surface topography they proposed nanometer-sized sputtered InP-structures which sizes are well-controllable with a set of parameters of the sputtering process.
     There were also attempts to estimate the effect of tip apex size on AFM image using various simulation approaches [132, 698]. The material of the next paragraph is just devoted to tip shape determination methods and related problems.
     Ideally tip apex must be round shaped and terminate with the single atom. Deviations from this ideal situation cause so-called tip artifacts in the scanned images. A collection of tip induced artifacts is gathered by Xu and Arnsdorf [1605]. 10 suggestions to minimize artifacts in SPM images are proposed in [1010]. Artifacts also may be caused by either the microscope design and operation mode or external environmental conditions, i.e. by factors not related to tip imperfectness [1376]. Thorough description of commonly observed artifacts in SPM as well as procedures for tip shape characterization are developed and published by The American Society for Testing and Materials (ASTM) [2800, 2801].