1 Tip-sample control usin quartz tunin forks in near-field scanninoptical microscopes Contributed by Xi Chen and Zhaomin Zhu 1 Introduction In near-field scannin optical microscopy (NSOM), a subwavelenth sized probe of visible liht is used to map the optical properties of a sample with a resolution limited primarily by the tip size and tip to sample separation[1]. NSOM desin specifications require that the tip sample distance control should be achieved with sub-nanometer precision. Several distance control mechanisms have been applied to NSOM, includin electron tunnelin[,3], capacitance[4], photon tunnelin[5], and atomic force[1,6]. The last mechanism detects the atomic force between the tip and sample when they are in very close proximity. Quartz tunin forks have been successfully used as force sensors for tip-sample reulation in NSOM[7,8]. Their very hih 3 5 mechanical quality factor Q( 1 1 ) provides a built-in hih ain and makes them very sensitive to sub-pn forces 4 when used at or near their resonance frequency ( 1 Hz to.5 1 6 Hz). Compared to optical forces measurement schemes [1,6], their advantae is that the measurement of the oscillation amplitude uses the piezoelectric effect native to quartz crystal, yieldin an electric sinal proportional to the applied forces and make them small, robust and simple to operate This review is oranized as follows. In section a model for the quartz tunin fork will be iven. In section 3, issues concernin optimal piezoelectric current detection will be addressed. The calibration of the measured piezoelectric sinal with the actual amplitude of the fork oscillation at the tip level will also be shown in detail. In section 4 the thermodynamic limit to the sinal detection will be discussed as resultin from both the rownian noise of the tunin fork and the Johnson noise associated with the sinal measurement. In section 5, the behavior of the tunin fork in a conventional proportional-interal (PI) closed loop will be presented. It ll be shown, in particular, that the presently common notion accordin to which hih quality factor Q tunin forks are intrinsically slow for scannin probe microscopy application is wron. On the contrary, it will be demonstrated that usin an optimized PI control loop scannin speed can be increased up to two orders of manitude at the cost of an expected increase in noise level. The major conclusion is that one should always seek maximum quality factor Q [8]. The effective mass harmonic oscillator model There are two ways to drive a tunin fork. In this section, we will use the one-dimensional effective mass harmonic oscillator model to describe the dynamics of a tunin fork driven mechanically by dither. In eneral, the resonant dynamics of a tunin fork can be modeled by a harmonic oscillator no matter which drivin method is used. or small amplitudes u() t, each arm of the tunin fork can be thouht of as a sprin with constant k stat. Consequently, u() t is the solution of an effective harmonic oscillator equation of motion driven at frequency ω. The equation of motion for such an oscillator is then simply: u m + exp( ) + kstatu = iωt, (-1) t m is an effective mass to be determined and is the where mechanical drivin force imposed by the dither. is a lossy term proportional to the velocity of the tunin fork. It represents a dra force opposin the movement of the fork arms. We will assume that this dra force is the sum of the forces resultin from the tip sample shear force interaction and from an internal dra force correspondin to the viscous losses occurrin in the tunin fork arm. can be characterized by a phenomenoloical parameter γ as: u = mγ t. (-) The effective mass m is chosen in a way that the resonant frequency f of this harmonic oscillator is compatible with k stat the sprin constant. In the limit of small dra force which corresponds the realistic condition γ << ω, this condition is fulfilled by writin: f k = =. (-3) stat π ω m We can solve the equation of motion for the tunin fork and we et: ( ) exp( ) 1 u ( ) t = iωt ω ω + iγω, m () t = im γωu () t. (-4)
u t and () t show a resonance peaked at the f. The amplitude of u() t is iven u = γω / ( ω ω ) + γ ω, (-5) u u = /( ikstatγ ) is the tip oscillation amplitude ω ω = ω ). The above equation shows that the oth () frequency by: where at resonance ( oscillation amplitude of the tip is a orenzian function of frequency as shown in i. 3_1. And the phase of u() t is iven by: γω θ = arctan( ) ω ω. (-6) At this point, we define the quality factor ( Q factor) as Q f / (-7) where f is the full width at half maximum ( WHM ) of the orenzian shaped resonance of the tip oscillation amplitude. On resonance the tip amplitude and the dra force experienced by the tunin fork is iven by u 3Q = ikstat kstat = i 3Qu, (-8) So we see that for small losses, the maximum vibration amplitude u is proportional to Q. The local potential on the tunin fork due to the piezoelectric effect is proportional to u() t. Therefore, a tunin fork can be used as sensitive dra force detector. The optical fiber with its tapered tip lued alon the side of one of the arms of a quartz crystal tunin fork. The tip at the end of the optical fiber protrudes out of the arm on which it is attached. The tip is hard enouh that it doesn t vibrate by itself, instead the tip oscillates with the tunin fork. The fiber tip acts as a shear force pick-up. The tunin fork is used as a force sensor. The tunin fork is vibrated in such a way that the tip oscillates parallel to the sample surface. oth arms of the fork are piezo-electrically coupled throuh the metallic contact pads. The eometry of the contacts and the couplin between the two arms insures that only one resonant vibration mode of the fork is excited. ON resonance, the bendin amplitude of the prons is maximum. Tunin fork is a sensitive sensor for distance control. Whenever the distance between the sample and the tip is chaned, the tunin fork s resonance frequency shift is instantaneous; while the amplitude of the tunin fork oscillation shift is lowest due to relaxation time delay but the sensitivity is the best. And phase shift is in between the above two methods. After all, we can use the voltae output from the tunin fork with phase locked loop circuit as feedback to keep the distance between the sample surface and the tunin fork as a constant. If we drive the tunin fork with a resonant voltae, then the induced current is: I = j π f ( cu + c U ), (-9) 1 p excitation where f is the resonance frequency; c 1 is a constant which is related to properties and eometrical shape of the tunin fork; c p is capacitance of the tunin fork. The first term of the above equation contains the shear force and its amplitude is lare only near resonance; the second term is responsible for the capacitance backround sinal. We can t reduce the drive voltae U excitation to reduce the backround sinal because the tunin fork motion amplitude u is proportional to U excitation. However, we can reduce the couplin capacitance c. p 3 Sinal detection and its calibration The key to implementin tunin forks for force detection is to accurately measure the fundamental resonance of the tunin fork as a function of applied force. This can be done either by shakin the fork at its mechanical resonance with a dither and monitorin the induced voltae, or by directly drivin the tunin fork with a resonant voltae and measurin the induced current. oth schemes have advantaes and disadvantaes. We here discuss how to implement the later, since it provides for a system that is mechanically simpler at the expense of only minimal electronics. When the tunin fork is driven with an external oscillatin voltae U, the sinal to be monitored is the correspondin ac current flowin throuh the tunin fork. At frequencies away from the resonant condition, the system response is nothin more than a capacitor C p (typically a few p for commercial 33 khz quartz forks). The resultin current is Icapa = iπ fcpu which increases linearly with the frequency. At resonance, the current peaks to a value I max, which depends on the dra force experienced by the fork. This useful sinal is interferin with the capacitive stray backround sinal I capa. The total current flowin throuh the tunin fork can be written as TW I i f d E x C U = π (3 11 + p ), where 11 d is the lonitudinal piezoelectric modulus, E is the elasticity modulus (i.e. Youn s modulus) of the fork material, T and W are the width and heiht of the cross section of one tunin fork arm, respectively, and is the lenth of one arm of the tunin fork. The first term in the bracket is due to the shear force, its amplitude is lare only around resonant conditions. The second term is the capacitive round sinal. Since x is
proportional to U, one cannot reduce the drive voltae U in order to reduce the backround sinal. To eliminate the effect of the packae capacitance, a bride circuit was used (i 3_1a). The transformer yields two wave forms phase shifted from each other by 18. y appropriately adjustin the variable capacitor, the current throuh the packae capacitance is neated by the current throuh the variable capacitor. A standard operational amplifier was used for current-to-voltae conversion. The current to voltae ain of the circuit has been calibrated for dc to 1 khz and was found to be described as Z R fr C ain = / 1 + ( π ) where f is the frequency, R =9.51M Ω, and parallel with C =.6p is the stray capacitance in R. Havin calibrated the measurement system, the impedance of the tunin fork can be accurately measured. The ratio of the output to input for a bare fork is shown as the closed circles in i 3_b. The response fits well to the orentzian line shape Af f / Q / ( f f ) + ( f f / Q) (3-1) with A=17.14, f =3733.3kHz, and Q=855, shown as the solid line. This result demonstrates that effect of the packae capacitance has been eliminated and that the harmonic oscillator is an excellent model for the response of the tunin fork. Values for the RC circuit mode can be determined by comparin the above-mentioned line shape to the formula for the ain of the amplifier with the RC resonator as the input impedance, yieldin R = Z / A=.494MΩ, RQ/π f = =.5kH, and ain C = 1/(4 π f ) =1.14f. R is lare enouh that it s attributed completely to mechanical dissipation associated with motion of the quartz. been determined by interferometrically measurin the physical amplitude of oscillation of one arm of the tunin fork while simultaneously measurin the output voltae of the system. This interferometric technique is described in detail in Ref 1. To interpret this measurement, it is necessary to define the relationship between the measured output voltae and the oscillation amplitude of one arm of the tunin fork. The output voltae is sensitive only to the antisymmetric mode of the tunin fork, Vout = c( x1 x), where c is a constant and x 1 and x are the amplitude of motion of the two arms of the fork. When drivin the fork with an external voltae, only the antisymmetric mode is excited, yieldin x1 = x. Thus, we define V = cx = x / α. The calibration yields α =59.6 ±.1 out 1 1 pm/mv. Viewin the tunin fork as a current source, it is convenient to write the above-mentioned equation in terms of the current to voltae convertin resistor, α = β / Z ain, yieldin β =5.55 m/a. ecause the chare separation in the tunin fork is amplitude dependent, calibration in terms of current yields an accurate measure of the amplitude of the tunin fork oscillation. 4 Thermodynamic limits to force detection The fundamental measurement limit of the ability to measure the resonance is dictated by the intrinsic system noise. These limits can be determined experimentally by measurin the output noise with the input rounded. The resultin output is shown in i 4_1as the filled circles. A noise analysis of the circuit shows that the two primary noise sources are Johnson noise of the feedback resistor, 4kTR V / 3 Hz, and Johnson noise associated with mechanical dissipation in the fork, as manifested by the R in the series RC equivalent circuit, 4 ktrz ( / R) ain ( ff / Q)/ ( f f ) + ( f f / Q) V / Hz, where k is the oltzmann constant and T is the temperature. All other parameters have already been determined experimentally. These two noise terms add in quadrature. It is shown as a solid line in i 4_1. i 3_1 The measurement system (a) and system response (b)[from Ref 8]. Another important calibration parameter is the amplitude of oscillation of the tunin fork as a function of the output voltae, which was denoted by the parameter α. This has
i 4_1 The noise spectrum of the measurement circuit in i 3_1a [from Ref 8]. The power spectrum of the noise associated with the tunin fork can be interated so as to et the root mean square (rms) voltae noise, Vrms = 4 ktr( Zain / R) ( π f / Q) V µ V which evaluates to = 3.81 at room temperature. rms This can be related to the thermal motion of the arms of the tunin fork by takin the time averae of the previous definition relatin the output voltae to the motion of the arm of the fork V = c( x x ) = c ( x + x x x ) rms 1 1 1 = c x 1 where it is assumed that the arms are only very weakly coupled and that their motion is uncorrelated. The inteferometric measurement allows one to convert this to an rms displacement of the arm, xrms = αvrms.3pm. This is the random motion of one arm of the fork due to thermal fluctuations. With this value for x rms, an effective sprin constant can be calculated via the equipartition theorem, K = kt / xrms 4. kn / m. This value arees well with the theoretically determined sprin constant from 3 3 the formula K = EWT /(4 ), where E is Youn s modulus of the fork material, is the lenth of one arm of the fork, W, T are the width and heiht of the cross section of one arm of the fork, respectively. The thermal enery can be thouht of in terms of an effective force actin on the tunin fork. This force has a flat power spectrum, S f, in units of N / Hz. One can calculate the power spectrum from the equation: f / K rms = f x S df f f i( ff / Q) Evaluatin the interal and aain usin the equipartition theorem, one can obtain S = / π f Q( k T / x ) =.44 pn / Hz. 1/ f rms The ratio of sinal voltae to noise voltae, S/N, as a function of bandwidth, f, is iven by S Qs /( β K) = N ( ff / Q) 4 kt ( f/ R + Q/ Zr df ) f ( f f ) ( f f / Q) 4 + where all terms are written as currents, R has been written as Zr / Q, Zr = / C is the resonance impedance of the RC resonator. The noise is dominated by the resonance impedance of the tunin fork as lon as the resonance impedance is sinificantly smaller than R. This is true for f f QR / Zr 1. At larer bandwidths, the noise Q associated with R becomes dominant. This can be remedied by usin larer R ; however, it derades the time response of the amplifier due to the stray capacitance. So, the choice of R is a trade-off between fast response and better S/N ratio. 5 Measurement speed The final issue is the speed with which the measurement can follow chanes in force. Scannin imaes of surfaces are made by fixin the heiht of the probe above the surface usin a closed loop control system. Some combination of the amplitude, and/or phase of the tunin fork sinal is used as the set point in the control loop. When usin conventional Si cantilevers it is well known that one must resort to phase sensitive techniques of system control in order to make the system respond faster than of the open loop response time of the resonator, 1/ τ = π f / Q. These techniques have also been successfully implemented with tunin fork systems[11]. However, in the followin it s demonstrated that conventional proportional and interal (PI) feedback control is sufficient for most uses of tunin forks. The experiment is done at room temperature, pressure, and atmosphere. The setup is shown schematically in i 5_1. A tapered fiber probe is mounted on the fork consistent with operation in a near-field scannin optical microscope. The open loop response of the loaded tunin fork corresponds to the line shape in Eq. (3-1) with f /Q =8 Hz and correspondinly τ = ms. The tunin fork is driven at resonance by a voltae source of fixed amplitude and frequency and the resultin current is measured with the circuit of i 3_1a. This output is detected with a lock-in amplifier whose phase is referenced to the phase of the resonantly driven tunin fork. The X output of the lock-in amplifier is fed to conventional PI feedback electronics which then adjusts the position of the probe above the surface.
5 i 5_1 Schematic of the control system [from Ref 8]. The set point of the feedback is adjusted so that the probe is fixed approximately 1 nm above the surface. To test the response of the trackin system, the vertical position of the sample is intentionally dithered at 1 Hz usin a 1 nm square wave. The output of the control electronics, shown in i 5_, clearly follows the 1 Hz sinal. The rise time of the response is of order.5 ms, 4 times faster than the open loop response. It s clear that simple PI control electronics yield a stable trackin system that can operate at speeds much faster than the response of the tunin fork resonator. Some comments: As we can see from i 5_, the sinal is very noisy. In fact, power spectrum of this sinal shows a lare noise at frequency 1/ τ (i.e., the open loop response frequency). This reason has prevented one from realizin fast scannin speed without reducin the Q factor of the tunin fork. i 5_ The response of the feedback loop to a 1 nm, 1 Hz square wave modulation of the tip-sample separation. Riht fiure is the expanded view of the left [from Ref 8]. References [1] E. etzi, P.. inn, and J. S. Weiner, Appl. Phys. ett. 6, 484 (199). [] U. uri,. W. Pohl, and. Rohner, J. Appl. Phys. 59, 3318 (1986). [3] A. Harootunian, E. etzi, M. Isaacson, and A. ewis, Appl. Phys. ett. 49, 674(1986). [4] E. etzi, Ph.. dissertation, Cornell University, 1998. [5]. Courjon, K. Sarayeddine, and M. Spajer, Opt. Commun. 71, 3 (1989). [6] R. Toledo-Crow, P. C. Yan, Y. Chen, and M. Vaez- Iravani, Appl. Phys. ett. 6, 957 (199). [7] K. Karrai and R.. Grober, Appl. Phys. ett. 66, 184 (1995). [8] R.. Grober, J. Acimovic, J. Schuck,. Hessman, P. Kindlemann, J. Hespanha, and A. S. Morse, Rev. Sci. Instrum. 71, 776 (). [9] K. Karrai and R.. Grober, SPIE 535, 69 (1995). [1] C. Schonenberer and S.. Alvarado, Rev. Sci. Instrum. 6, 3131 (1989). [11]. Giessibl, Appl. Phys. ett. 73, 3956 (1998).