Abstract:
An atomic force microscope based apparatus for examining a sample includes a cantilever having a cantilever arm and a probe tip where the probe tip is offset laterally from a longitudinal axis of torsion of the cantilever arm, an oscillator that drives the cantilever into oscillation in a flexural mode to cause the probe tip to repeatedly interact with the sample where the tip-sample interaction of the laterally offset probe tip excites torsional motion of the cantilever, and a detection system that detects torsional motion of the cantilever in response to the tip-sample interaction.

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     This application is a continuation of application Ser. No. 11/456,810, filed on Jul. 11, 2006, entitled “Torsional Harmonic Cantilevers For Detection Of High Frequency Force Components In Atomic Force Microscopy,” now U.S. Pat. No. 7,302,833, which is a divisional of application Ser. No. 10/887,608, filed on Jul. 8, 2004, entitled “Torsional Harmonic Cantilevers For Detection Of High Frequency Force Components In Atomic Force Microscopy,” now U.S. Pat. No. 7,089,787, issued on Aug. 15, 2006, of the same inventors hereof, which application and patent are incorporated herein by reference in their entireties. 
    
    
     FIELD OF THE INVENTION 
     The invention relates to cantilevers and imaging methods using the cantilevers and, in particular, to cantilevers for detecting high frequency force components and imaging methods using the same. 
     DESCRIPTION OF THE RELATED ART 
     Scanning Probe Microscopy (SPM) refers to the class of instruments and imaging methods where a probe, typically of very small dimensions, is scanned relative to a sample surface for providing a microscopic analysis of the topographical features or material properties of the sample surface. A variety of SPM is Atomic Force Microscopy (AFM), also referred to as Scanning Force Microscopy (SFM), where a flexible cantilever with a tip is used to scan the sample surface. 
     An atomic force microscope (AFM) has proven to be an exceptionally useful tool for mapping the topography of surfaces at the nanoscale. In an AFM, a flexible cantilever with an atomically sharp tip is brought to the vicinity of a sample surface and the deflections of the cantilever as a result of the attractive and repulsive forces between the tip and the sample are monitored while the cantilever is scanned across the surface. The cantilever can be in a contacting mode of operation where the tip is in constant contact with the sample surface. When continuous contact between the tip and the sample is used, frictional forces between the tip and the sample often result in damages to both the tip and the sample. Dynamic imaging techniques that do not require continuous tip-sample contact have also been introduced. In dynamic imaging modes, the tip is maintained at a short distance from the surface of the sample and makes intermittent contact with the sample surface. Among the various dynamic imaging techniques, tapping-mode atomic force microscopy (TM-AFM) has become the most widely used. In TM-AFM, the tip is vibrated at one of its resonance frequency in vicinity of the sample surface. Dynamic atomic force microscopy methods (such as TM-AFM) largely eliminate tip and sample damage and are therefore widely used in atomic force microscopy imaging. 
     In atomic force microscopy, the force interaction between the tip and sample surface depends on the material properties of the tip and the sample. Measurement of the tip-sample forces enables study of material properties and mapping of chemical composition variations across the sample surface. When the cantilever vibrates in the vicinity of a sample, tip-sample forces will have higher harmonic components that generate high frequency vibrations in the cantilever. These high frequency force components carry information about the tip-sample interaction. Imaging with these higher harmonics has been demonstrated and the result has shown good contrast based on material properties. Unfortunately, the vibration amplitudes of traditional AFM cantilevers at higher harmonics are too small for practical AFM imaging. Specifically, the signals at higher harmonics are 20-30 dB lower than the signal at the driving frequency of the cantilever. Thus, the resolution of imaging using the higher harmonics signals is severely limited. 
     Therefore, it is desirable to provide an apparatus or image method which can enable non-destructive measurement of the high frequency tip-sample force components as result of the vibration of a cantilever near a sample surface. 
     SUMMARY OF THE INVENTION 
     According to one embodiment of the present invention, an atomic force microscope based apparatus for examining a sample includes a cantilever having a cantilever arm and a probe tip where the probe tip is offset laterally from a longitudinal axis of torsion of the cantilever arm, an oscillator that drives the cantilever into oscillation in a flexural mode to cause the probe tip to repeatedly interact with the sample where the tip-sample interaction of the laterally offset probe tip excites torsional motion of the cantilever, and a detection system that detects torsional motion of the cantilever in response to the tip-sample interaction. 
     According to another aspect of the present invention, an atomic force microscope based apparatus for examining a sample includes a cantilever having a cantilever arm and a probe tip wherein the probe tip is offset laterally from a longitudinal axis of torsion of the cantilever arm, means for oscillating the cantilever in a flexural mode to cause the probe tip to repeatedly interact with the sample such that the tip-sample interaction of the laterally offset probe tip excites torsional motion of the cantilever, and means for detecting torsional motion of the cantilever in response to the tip-sample interaction. 
     According to yet another aspect of the present invention, a cantilever for atomic force microscopy includes a cantilever arm having a first planar surface adapted for reflecting an optical beam of an optical detection system and a second planar surface opposite the first planar surface, and a probe tip positioned on and projected from the second planar surface, the probe tip being offset laterally by at least 2 microns from a longitudinal axis of torsion of the cantilever arm 
     The present invention is better understood upon consideration of the detailed description below and the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a generalized schematic diagram of a tapping-mode atomic force microscope. 
         FIG. 2A  is a perspective view of a cantilever according to one embodiment of the present invention. 
         FIG. 2B  is a top view of the cantilever of  FIG. 2A . 
         FIG. 3  illustrates the mode shape of the torsional mode of the cantilever of  FIGS. 2A and 2B . 
         FIG. 4  illustrates a split photodetector and a deflection measurement technique using the photodetector for detecting flexural and torsional movements. 
         FIG. 5  is a top view of a cantilever according to a first alternate embodiment of the present invention. 
         FIG. 6  is the flexural vibration spectrum of the cantilever of  FIG. 5  while the cantilever is being tapped on a sample surface at the fundamental resonance frequency. 
         FIG. 7  is the torsional vibration spectrum of the cantilever of  FIG. 5  while the cantilever is being tapped on a sample surface at the fundamental resonance frequency. 
         FIG. 8  is a top view of a cantilever according to a second alternate embodiment of the present invention. 
         FIG. 9  is a top view of a cantilever according to a third alternate embodiment of the present invention. 
         FIG. 10  is a top view of a cantilever according to a fourth alternate embodiment of the present invention. 
         FIG. 11  is a top view of a torsional harmonic cantilever according to one embodiment of the present invention. 
         FIG. 12  is the torsional vibration spectrum of the torsional harmonic cantilever of  FIG. 11  while the cantilever is being tapped on a surface at the fundamental resonance frequency. 
         FIG. 13  illustrates the mode shape of the torsional vibration of the torsional harmonic cantilever of  FIG. 11 . 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     In accordance with the principles of the present invention, a cantilever including a tip at the free end of the cantilever is configured to incorporate a geometric feature that amplifies signals associated with high frequency components of the tip-sample interaction force when the cantilever is vibrated near the surface of a sample. When the cantilever is applied in a scanning probe microscope, the cantilever forms a high bandwidth and highly sensitive mechanical system that enables the detection and measurement of signals associated with the high frequency force components of the non-linear tip-sample interaction. In one embodiment, the cantilever includes a cantilever arm and a tip at the free end of the cantilever arm that is positioned at an offset location from the axis of torsion of the cantilever arm. More specifically, the tip is offset laterally from a longitudinal axis of torsion of the cantilever arm as shown in  FIGS. 2A and 2B . Placing the tip at the offset location from the axis of torsion enhances the torsional motions of the cantilever. As a result, signals related to a significant number of the high frequency force components of the tip-sample interaction force can be amplified to allow new imaging modalities or study of the material properties of the sample under test. 
     One useful application of the cantilever of the present invention is in a tapping-mode AFM where the cantilever can be used to study material properties of the sample. However, the cantilever of the present invention can be applied in many other applications, both within scanning probe microscopy or beyond. In sum, the cantilever of the present invention can be applied when the high frequency tip-sample interaction force components are of interest. 
       FIG. 1  is a generalized schematic diagram of a tapping-mode atomic force microscope. In the TM-AFM of  FIG. 1 , the cantilever is vibrated at a frequency close to one of its flexural resonances, typically the fundamental resonance frequency, in the vicinity of the sample surface so that the tip makes intermittent contacts (tapping) with the surface once in every oscillation period. During the scan across the surface, the amplitude of vibration is maintained at a constant value through a feedback loop that adjusts the height of the cantilever base. Specifically, a light beam and a photodetector are used to measure the motion of the cantilever at the driving frequency. The feedback signal therefore reflects the topography of the surface. In  FIG. 1 , the feedback loop includes an optical lever detection system and a feedback controller. 
     The cantilever of the present invention incorporates a geometric feature to realize amplification of high frequency force components of the interaction between a sample and the tip of the cantilever. The geometric feature can be implemented in various forms to accomplish the signal amplification. According to one embodiment of the present invention, a cantilever includes a tip at the free end that is positioned at an offset location from the axis of torsion of the cantilever arm. The offset tip placement enhances the torsional motion of the cantilever. Specifically, when the offset tip hits the sample surface, the interaction forces excite the torsional modes of the cantilever. In the present description, a cantilever with an offset tip placement for enhancing torsional motion will be referred to as a “torsional cantilever.” However, it is important to note that a “torsional cantilever” of the present invention may, when driven at a given frequency, vibrate in a flexural mode, a torsional mode or other vibrational modes. The use of the term “torsional” is not intended to limit the vibrational modes of the cantilever of the present invention to torsional motion only. 
     Furthermore, in the present description, the term “axis of torsion” refers to an axis of the cantilever where the cantilever is not displaced by torsional vibration when the cantilever is vibrated. That is, the axis of torsion is where the cantilever does not move in the torsional mode. More specifically, the axis of torsion generally extends in the longitudinal direction of the cantilever arm. For a symmetrical cantilever, such as a rectangular cantilever, the axis of torsion is the center line of the cantilever arm perpendicular to the base of the cantilever. However, for cantilevers in other geometries, the axis of torsion is not necessarily the center line of the cantilever arm and moreover, is not necessarily a straight line. The axis of torsion for a cantilever of any given geometry can be determined, such as by use of a simulation software. In conventional cantilevers, the probe tip is always positioned on the axis of torsion of the cantilever arm. 
       FIG. 2A  is a perspective view of a cantilever according to one embodiment of the present invention.  FIG. 2B  is a top view of the cantilever of  FIG. 2A . Referring to  FIGS. 2A and 2B , a torsional cantilever  100  is a rectangular cantilever and includes a cantilever arm  102  projecting from a base. A probe tip  106  is positioned near the free end of cantilever arm  102 . For torsional cantilever  100 , the axis of torsion is the center line of cantilever arm  102  along the longitudinal axis of the cantilever arm, indicated by a dash-dot line  104  in  FIGS. 2A and 2B . Probe tip  106  is an offset position relative to the axis of torsion (line  104 ). That is, probe tip  106  is offset laterally from the longitudinal axis of torsion of the cantilever arm. Specifically, in the present embodiment, probe tip  106  is positioned a distance “d” from the axis of torsion. By placing probe tip  106  away from the axis of torsion, the probe tip will be displaced when the cantilever is vibrated in the torsional mode. The torsional motion of the cantilever is then enhanced, resulting in the amplification of high frequency harmonics of the tip-sample interaction force. 
     The amount of offset displacement or the distance “d” can vary from a minimal displacement (near the axis of torsion) to maximum displacement (near the edge of the cantilever arm). The exact amount of offset is selected based on design choice. In most cases, a minimal displacement of more than the normal process variations or alignment errors should be used to ensure that the probe tip is displaced from the axis of torsion. For example, in one embodiment, the probe tip should be at least about 2 μm from the axis of torsion. It is generally understood that the farther away the probe tip is displaced from the axis of torsion, the more torsional motion the probe tip will experience. Thus, the probe tip can be placed near the axis of torsion for minimum torsional motion and placed near the edge of the cantilever arm to obtain maximum torsional motion. 
       FIG. 3  illustrates the mode shape of the torsional mode of cantilever  100 . As can be seen from  FIG. 3 , locations on the cantilever that are away from the torsion axis (center line in this case) exhibit larger displacement. If the tip offset distance is larger, tip-sample forces will create a larger torque and hence result in a larger torsion angle and better excitation of torsional modes. This is advantageous in terms of signal improvement, however, the effect of torsional modes on the overall cantilever dynamics will be more significant. 
     When cantilever  100  of the present invention is applied in a scanning force microscope, such as the tapping-mode AFM of  FIG. 1 , the same deflection measurement technique can be used to detect the flexural movement as well as the torsional movement of the cantilever.  FIG. 4  illustrates a split photodetector and a deflection measurement technique using the photodetector for detecting flexural and torsional movements. The split photodetector used in a SFM is usually divided into four quadrants Q 1  to Q 4 . The light spot resulted from the reflection of the laser light beam off the cantilever arm is shown as a circular spot about the center of the photodetector quadrants. The movement of the tip of the cantilever is detected by summing and subtracting of the light signals detected in each of the quadrants. In the illustration in  FIG. 4 , it is assumed that the flexural motion of the cantilever tip results in an up-down motion of the light spot and the torsional motion results in a left-right motion of the light spot. To measure the flexural deflection of the cantilever tip, the signals in quadrants Q 1  and Q 2  are summed and the signals in quadrants Q 3  and Q 4  are summed. The difference of the two summed signals is then used to indicate flexural motion. To measure the torsional deflection of the cantilever, the signals in quadrants Q 1  and Q 3  are summed and the signals in quadrants Q 2  and Q 4  are summed. The difference of the two summed signals is then used to indicate torsional motion. In case of a misaligned photodiode and cantilever the cross talk between the flexural and torsional signals has to be taken into account. 
     One advantage of the torsional cantilever of the present invention when applied in a scanning force microscope is an increase in imaging speed. When the torsional deflections are used in the feedback loop of the scanning force microscope, the torsional harmonics, being at higher frequencies, provide a faster response time for the imaging signals. Furthermore, the high frequency torsional harmonic signals can be advantageously applied to enable rapid detection of tip-sample engagement and disengagement. 
     In a conventional cantilever, the torsional vibration or torsional modes have higher resonant frequencies than the fundamental flexural modes. However, the torsional modes of the cantilever are not excited in a conventional cantilever. In accordance with the present invention, the torsional cantilever can be used to excite the torsional mode to enable detection of high frequency force components. When the torsional cantilever of the present invention is applied in a dynamic AFM operation, the torsional cantilever is oscillating in a flexural mode while the tip-sample interaction forces will excite torsional modes of the cantilever. The torsional modes are excited because the tip of the torsional cantilever is located offset from the axis of torsion. The offset placement of the probe tip allows tip-sample forces to generate torsion on the cantilever. Because the torsion arm (the width of the cantilever arm) is shorter than the length of the cantilever, a small tip displacement in the torsional mode will result in a relatively large angular deflection. Therefore, a laser beam reflected from the back of the cantilever can easily detect the torsional motion of the cantilever. Because the torsional modes of the cantilever have high bandwidth, high frequency harmonics of the tip-sample forces will efficiently generate torsional vibrations. 
     The torsional cantilever of the present invention can be configured in many shapes and geometry depending on the desired application. The salient feature of the torsional cantilever of the present invention is the offset placement of the probe tip from the axis of torsion. The probe tip can be displaced on either side of the axis of torsion.  FIG. 5  is a top view of a cantilever according to a first alternate embodiment of the present invention. Referring to  FIG. 5 , a torsional cantilever  120  is configured as a rectangular cantilever with an enlarged free end. A probe tip  126  is positioned at an offset displacement from the axis of torsion  124  of the cantilever arm. In one implementation, cantilever  120  is able to realize a first torsional resonance frequency that is 16.3 times the fundamental flexural resonance frequency. The wider free end of cantilever  120  has the effect of tuning the torsional resonance frequency of the cantilever as will be described in more detail below. 
       FIGS. 6 and 7  are spectrums of the flexural and torsional vibrations, respectively, of cantilever  120  tapping on a sample surface at the fundamental flexural resonance frequency. From a comparison of the vibration spectrum of the flexural mode ( FIG. 6 ) and the vibration spectrum of the torsional mode ( FIG. 7 ), it can be observed that more high harmonic force components are excited in the torsional mode with better signal to noise ratio. Thus, more information regarding the tip-sample interaction can be retrieved from the torsional vibration spectrum. 
       FIGS. 8-10  illustrate three other alternate embodiments of the torsional cantilever of the present invention. One of ordinary skill in the art, upon being apprised of the present description, would appreciate that the torsional cantilever of the present invention can be implemented using various cantilever configurations and geometries. Referring to  FIG. 8 , a torsional cantilever  140  is formed as a rectangular cantilever with a tapered free end. A probe tip  146  is positioned offset from the axis of torsion (line  144 ). Referring to  FIG. 9 , a torsional cantilever  160  is configured as a rectangular cantilever including two arms  162 A and  162 B separated by an opening  168 . The free end of cantilever  160  is also tapered. A probe tip  166  is positioned at an offset location from the axis of torsion (line  164 ). Finally, referring to  FIG. 10 , a torsional cantilever  180  is formed as a triangular cantilever including arms  182 A and  182 B. A probe tip  186  is positioned at an offset location from the axis of torsion (line  184 ). 
     The torsional cantilever of the present invention has many applications in imaging and in material analysis. In one application, the torsional harmonics (both phase and amplitude) of the torsional cantilever are measured for generating images of the sample surface. In another application, the torsional cantilever is applied to generate a large number of vibration harmonics. When most or all of the harmonic force components are measured, the harmonic force components can be added in the time domain to reconstruct the time-resolved waveform of the tip-sample forces. It is well understood that when all the harmonic force components are added together a force diagram of the tip-sample interaction is constructed. More specifically, a weighted sum of the harmonic force components in the time domain is used to reconstruct the time-resolved waveform of the tip-sample forces. The weighted sum of the harmonic force components is calculated by multiplying each harmonic amplitude by a factor before summation where the factor is the frequency response of the cantilever. A weighted sum is used because the same force amplitude at two different frequencies result in different vibration amplitudes. The difference in vibration amplitudes is taken into consideration by the use of the weighted sum. The torsional cantilever of the present invention enables the generation of torsional harmonics with enhanced amplitude so that measurement of the harmonics can be readily made. Thus, the torsional cantilever of the present invention enables the measurement of the force/distance relations of the tip-sample interaction. 
     According to another aspect of the present invention, a torsional cantilever has a torsional resonance frequency that matches an integer multiple of the fundamental flexural resonance frequency of the cantilever when the cantilever is driven at a driving frequency. Such a torsional cantilever, referred herein as a “torsional harmonic cantilever”, has one of its harmonics coincided with the torsional resonance of the cantilever. In this manner, the torsional motion of the torsional harmonic cantilever at that harmonic frequency will be largely enhanced by the corresponding torsional resonance. The ratio of the torsional resonance frequency to the fundamental flexural resonance frequency of the torsional harmonic cantilever can be tuned by selecting appropriate cantilever geometry since the geometry of the cantilever determines the resonance frequencies of the cantilever. The torsional harmonic cantilever of the present invention can provide improvement in imaging by using the higher order harmonics as the imaging signals. While the tuned resonance frequency will have a high signal to noise ratio, all other harmonics of the cantilever vibration will still have good signal to noise ratio due to the torsional motion. 
       FIG. 11  is a top view of a torsional harmonic cantilever according to one embodiment of the present invention. Referring to  FIG. 11 , a torsional harmonic cantilever  200  includes a cantilever arm formed by a first arm portion  202 A and a second arm portion  202 B. The first and second arm portions are separated by an opening  208 . A probe tip  206  is located at the free end of the cantilever arm and is positioned at an offset displacement from the axis of torsion of the cantilever. Cantilever  200  includes a free end that has an asymmetrical shape where one side of the free end is tapered. By incorporating two separated arm portions in cantilever  200 , each arm portion can bend more easily so that torsional resonance frequencies are lowered. One has to choose proper dimensions for the arms to tune the ratio of the torsional resonance frequency and fundamental flexural resonance frequency to an integer. 
     By using an offset tip placement, torsional harmonic cantilever  200  can enhance torsional motion to enable detection of high frequency force components of the tip-sample interaction. In one implementation, cantilever  200  is able to realize a fundamental torsional resonance frequency that is 9 times the fundamental flexural resonance frequency. 
     In cantilever  200 , the tuning of the torsional resonance frequency to match an integer multiple of the fundamental flexural resonance frequency of the cantilever can be carried out by varying the width of arm portions  202 A and  202 B or by varying the size of opening  208 . The shape of the free end can also be configured to tune the resonance frequency.  FIG. 12  is the torsional vibration spectrum of torsional harmonic cantilever  200  while the cantilever is being tapped on a surface at the fundamental resonance frequency. Referring to  FIG. 12 , the first peak in the vibration spectrum is at the fundamental resonance frequency and the peak at 400 kHz is the torsional resonance frequency. The torsional resonance frequency at 400 kHz is at the 9 th  harmonic of the fundamental resonance frequency and the signal to noise is more than 40 dB. The vibration spectrum of  FIG. 12  illustrates how torsional harmonic cantilever  200  can be tuned to have a fundamental torsional resonance frequency that is 9 times the fundamental flexural resonance frequency. 
       FIG. 13  illustrates the mode shape of the torsional vibration of the torsional harmonic cantilever of  FIG. 11 . The mode shape confirms that by placing the probe tip near the edge of the cantilever arm, the tip will experience large displacement in torsional modes, providing enhanced torsional signals. 
     Torsional cantilever  120  of  FIG. 5  represents another configuration for a torsional harmonic cantilever. The torsional cantilever of  FIG. 5  illustrates how the fundamental resonance frequency can be altered to tune the ratio of the torsional resonance frequency to the fundamental resonance frequency. Returning to  FIG. 5 , the free end of cantilever  120  is made wider than the rest of the cantilever arm. Because the free end is a highly displaced part of the fundamental resonance frequency, by making the free end wider, mass is added to the fundamental resonance frequency which has the effect of lowering the resonance frequency of the fundamental. The mass addition does not change the torsional resonance frequency appreciably but the fundamental frequency is lowered so that the ratio of the two is tuned. 
     In the above description, the use of the term “torsional harmonic cantilever” is not intended to limit the vibrational modes of the cantilever of the present invention to torsional motion only. As described above, the “torsional harmonic cantilever” of the present invention may, when driven at a given frequency, vibrate in a flexural mode, a torsional mode or other vibrational modes. 
     Furthermore, in the torsional harmonic cantilever of the present invention, the torsional resonance frequency and the fundamental flexural resonance frequency have an integer ratio or a ratio slightly below or slightly above an integer ratio, such as within 2% of the integer ratio. Thus, the integer ratio includes whole numbers and fractional numbers slightly greater than or less than the nearest whole numbers so that when the cantilever is driven at a driving frequency at or slightly below or slightly above the fundamental resonance frequency, the torsional resonance frequency is an integer multiple of the driving frequency. 
     The tuning of the ratio of the torsional resonance frequency to the fundamental flexural resonance frequency of the torsional harmonic cantilever to an integer value or near integer value can be accomplished in primarily two ways. First, the tuning can be accomplished by tuning the fundamental flexural resonance frequency to be an integer divisible of the resonance frequency of the selected torsional resonance frequency. Second, the tuning can be accomplished by tuning the resonance frequency of the selected torsional mode to be an integer multiple of the fundamental flexural resonance frequency. Alternately, both tuning methods can be applied at the same time to tune the ratio to the selected torsional resonance frequency to be an integer multiple of the fundamental flexural resonance frequency. 
     Tuning the fundamental frequency of the torsional harmonic cantilever to be an integer divisible of the resonance frequency of the selected torsional resonance frequency can be accomplished by incorporating geometric features in the cantilever that have dominant effects on changing the fundamental flexural resonance frequency as compared to the resonance frequency of the torsional modes. For instance, increasing or decreasing the length of the cantilever by adding or removing mass to the free end of the cantilever will corresponding increase or decrease the frequency ratio. 
     Alternately, when the cantilever has a rectangular shape or close to a rectangular shape with or without a tapered end, the length to width ratio of the cantilever arm can be adjusted to alter the fundamental flexural resonance frequency. The higher the length to width ratio, the higher the ratio of the torsional to fundamental frequencies. Selecting correct values of length and width will tune the frequency ratio to an integer value or near integer value. 
     The fundamental resonance frequency can also be tuned by increasing or decreasing the effective width or thickness at a highly displaced region of the fundamental mode, which has the effect of decreasing or increasing the fundamental resonance frequency. Typically, this mass modification is applied towards the free end of the cantilever. By decreasing or increasing the fundamental resonance frequency, the ratio of the torsional to fundamental frequencies will correspondingly increase or decrease. 
     Finally, the frequency ratio can also be tuned by increasing or decreasing the effective width or thickness at a high mechanical stress region that has the effect of decreasing or increasing the resonance frequency ratio. Typically, this mass modification is applied towards the base of the cantilever. 
     Tuning the resonance frequency of the selected torsional mode to be an integer multiple of the fundamental flexural resonance frequency can be accomplished by incorporating geometric features in the cantilever that have dominant effects on changing the resonance frequency of the torsional modes as compared to the fundamental flexural resonance frequency. In a cantilever, there are many torsional vibrational modes. In general, the first few (2-3) modes are of interest. 
     First, the resonance frequency of a torsional mode can be tuned by forming arm portions with openings in the body of the cantilever while maintaining the effective width of the cantilever without the openings. One exemplary embodiment of such a torsional harmonic cantilever is shown in  FIG. 11 . Referring to  FIG. 11 , the arm portions and the openings can be characterized by the location of the opening(s) along the body of the cantilever, the separation (“a”) of the arms and the extension (“b”) of the opening(s). Proper tuning can be achieved by adjusting the location and the extension of the opening and the separation of the arms, such as by use of a simulation tool to find the optimum values of the separation and extension lengths. In general, providing arm portions with openings in the body of the cantilever will lower the ratio of the resonance frequencies. The amount that the ratio is lowered is proportional to the extension of the opening. 
     Alternately, increasing the separation (“a”) of the arms will reduce the frequency ratio if the arms are located at a high angular displacement region in that particular torsional mode. Also, the width of the individual arms can also be increased to tune the resonance frequency of the torsional mode, mainly by leading to an increase in the effective width of the cantilever. 
     In  FIG. 11 , torsional harmonic cantilever has two arm portions in parallel. This geometric configuration is illustrative only and is not intended to be limiting. A torsional harmonic cantilever of the present invention can be provided with multiple arm portions. Furthermore, the arm portions do not have to be parallel, that is, the arm portions can be anti-parallel. For instance,  FIG. 10  illustrates a V-shaped torsional harmonic cantilever including two arm portions. 
     The above detailed descriptions are provided to illustrate specific embodiments of the present invention and are not intended to be limiting. Numerous modifications and variations within the scope of the present invention are possible. The present invention is defined by the appended claims.