Phase Coherent Effect of UHV Dynamic Force Microscopy with Phase Locked Oscillator B. I. Kim, and S. S. Perry Department of Chemistry University of Houston Revised ( 09 14 99 ) Abstract Phase locked oscillator(plo) with phase locked loop for ultra high vacuum(uhv) dynamic force microscopy has been developed to detect tip-sample interaction and force microscopy. The measured the bandwidth of PLO using electrostatic force modulation between tip and sample is over 340Hz, which is practical scanning speed to get an image. By measuring frequency shift versus distance( f-d ) curve and imaging on atomically clean sample surface, we demonstrate that this technique improved the signal to noise ratio with unconditional stability over all force regime owing to phase coherent effect. PLO method is an extremely effective way to excite the cantilever coherently with constant excitation voltage, compared with that of conventional FM technique without phase coherence. 1. Introduction Dynamic force microscopy(dfm) under UHV has been developed to have a true atomic resolution and several groups have succeeded in getting an atomic resolution image such as atomic defect structure[1-4]. UHV DFM is applicable to insulators as well as conductors because it makes use of interactive forces between tip and sample. In this imaging mode, a novel technique known as frequency modulation has been employed to
detect the interaction between tip and sample[5]. Spontaneous vibration of cantilever induced by thermal excitation keeps tracking its resonance frequency. When the oscillating cantilever is brought to near the sample surface, the interaction between tip and sample shifts the resonance frequency by f, which is used as the probing signal. However its usefulness is restricted to the situations in which the sensor provides sufficient selectivity of amplitude response at the resonance[6]. Recently Dürig et al. used phase locked oscillator(plo), which enables cantilever to keep the phase difference constant during the oscillation[6,7]. They applied it to the adhesion and the dissipation studies successfully. More recently Balmerin et. al., using PLO combined with digital electronics, obtained the atomic resolution of Si(111) and RbBr(001) with stable imaging condition. However, the difference between direct cantilever resonance oscillator(dro), conventional FM technique without phase coherence, and PLO has been rarely verified so far. In this paper we will describe development of PLO and its bandwidth measurement using electrostatic force modulation technique. Through direct comparison PLO with DRO in the measurement of f-distance curve, we will demonstrate phase coherent effect of PLO is extremely advantageous in tip-sample interaction study and stable force microscopy. 3. Experiment We used omicron UHV STM/AFM head with optical beam deflection detection system. For dynamic mode measurement, the cantilever can be oscillated by a thin piezo attached underneath the cantilever stage. Data acquisition control of microscope head and data analysis were carried out using RHK electronics and software. Figure 1(a) shows
schematic diagram of PLO for coherently exciting cantilever beam oscillations, which is similar to the one described in [5, 7]. It consists of detector, Phase shifter and PLL which is absent in DRO. The vertical component of the cantilever vibration was detected with quad-cell position sensitive photodiode (PSD) in the optical beam deflection technique and used as the input signal of phase locked loop(pll). The VCO output of PLL was used as an excitation signal of cantilever after phase shifter, which was used for proper tracking of resonance frequency. The frequency change f of PLO output was used as the feedback signal in the dynamic mode operation. A n+ doped Si cantilever with resistivity 0.01-0.02Ωcm( Nanosensor ) was employed for measurement. The silicon oxide on the surface of tip was removed for 3 min by 1keV Ar sputtering to get a clean Si surface and checked with STM measurement. Stiff silicon cantilevers with resonance frequency about f 0 =270kHz and spring constant k = 21-78N/m were used as the force sensor to prevent cantilever from jumping into contact with the surface. Oscillation peak to peak amplitude of the cantilever was about 30nm under UHV environment with base pressure ~ 1 10-10 torr. The VC(100) single crystal was cleaned with argon ion sputter/ anneal cycles in ultrahigh vacuum. After sputtering sample with the accelerating potential of Ar ion beam 1 kev for 15min at room temperature, the sample was annealed by electron beam heating to 1200ºC for 5min. Fig. 1(b) shows the schematic block diagram of the electronic circuit in the PLO. The response signal from detector is clipped to a certain value by limiter to reduce of the spur noise due to the change of input amplitude. The analog multiplier as phase detector generates the phase difference between the excitation and response with the frequencydoubled component, which is eliminated by loop filter. If the excitation signal is given by
A excite sin( 0 excite ω t + θ ) and the cantilever response Α responsecos ( ω 0 t + θ response), their product is A sin( t + θ ) Α cos( ω t + θ ). The response signal has excite ω0 excite response 0 response been written as a cosine and the voltage controlled oscillator (VCO) output as a sine. This product is time-averaged with loop filter over one period 2π to have a sinusoidal output ω 0 characteristic. ω 0 2π 2π ω0 A excitesin( ω0 t + θ excite ) Αresponse cos( ω0 t + θ response) dt 0 1 = A excitea responsesin( θ excite -θ response) (1) 2 It is a null when the tip-sample distance causes the cantilever oscillation frequency to be exactly equal to that of the reference signal and simultaneously 90º out of phase(sine and cosine have 90 phase difference each other). Since this time averaged dc output is used as the input of VCO, it is an accurate measure of its frequency. Therefore the phase locked loop circuit serves as frequency detector which is known to have generally reduced noise characteristics- 6dB at maximum compared to conventional demodulator which uses frequency dependent phase discriminator such as a dual LC filter[9]. Since the frequency of VCO output is exactly the same with that of PLL input in the locked state, the VCO output is used as an excitation signal to drive cantilever. PLL plays the role of waveform converter effectively with constant excitation voltage regardless of the amplitude value of response signal[2]. A phase shifter was adjusted precisely so that phase relationship between the response of cantilever vibration and excitation signal from VCO insures maximum positive feedback on resonance. RMS to DC converter
circuit(ad536 Analog Device) was constructed to measure the root mean square amplitude, which can be deduced from amplitude vs. distance curve. Figure 2(a) shows a loop filter in second-order composed of output resistance R 1, R 2 and capacitance C, which determine the bandwidth of PLL. The loop response function of PLL on modulation frequency ω between θ excite and θ response can be written as [9] 2 jωωn (2ζ ω n / KoKd ) + ωn H( jω ) = (2) 2 ω + 2 jζω ω + ω n 2 n where natural frequency and damping constant are given by K K ω o d n = (3) R C R C 1 + 2 1 KoKd 1 ζ = ( R2C + ) (4) 2 R C + R C K K 1 2 o d for given sensitivity of phase detector K d and that of VCO K o. Since the shifts of resonance frequency f due to the tip-sample interaction in the force regime of interest is less than 1kHz, the maximum frequency deviation f m between a carrier s center frequency f c and its upper or lower was chosen by ±800Hz, ±0.30% of f c (=270kHz). The natural frequency ω n =2π 360 rad/sec and damping constant ζ=1.0 were selected so that the phase error, θ excite -θ response, due to f m sin(ω n t) in the loop, can be less than 1 radian or 57, leaving a 33 margin for noise. We used a simple integrated PLL circuit LM 565(National Semiconductor) with K d =0.68V/rad, K o = 4.1 (rad/sec) for V supply voltage V, and maximum operating frequency 500 khz[10]. In LM565, the center frequency of VCO was adjusted with variable resistor R vco and fixed capacitor C vco in f c
Fig. 2(b) by the relation f c = R vco 0.3 C vco under unlocked state. In addition, R vco can be used to make the offset of f output zero under locked state. 33.6 With K o K d = V f c /sec, f c =270 khz and V=14V, we can find time constants R 1 C = 1.96 10-2 sec and R 2 C = 4 10-4 sec from Eq. (3) and Eq. (4). For a given output resistance R 1 = 3.6kΩ, we find C 1 =5.44µF and R 2 =73.5Ω. Fig. 2(b) shows the actual circuitry with LM565. The limiter is composed of diodes and amplifier to clip the response signal to ±0.6 V. Since with a deviation of 10%, the output of demodulator of LM565 will produce approximately 300 mv peak to peak output, we can expect output sensitivity of PLL at carrier frequency f c to be 3000/ f c (mv/hz). The resultant sensitivity of f output at f c =270kHz was designed to be 5mV/Hz using amplifier with gain 455. The low pass amplifier with cut-off frequency 1kHz followed after PLL to eliminate high frequency ripple. In addition, the offset voltage between pins 6 and 7 is 100 mv and output common mode voltage is 4.5V. It is desirable to amplify and level shift(offset) this signal to ground so that plus and minus output voltages can be obtained for frequency shifts above and below resonance frequency. R so was used to set the output at zero volts with no input signal. Since VCO has thermal drift of f with 2 10-4 f c Hz/ C, the circuits had been turned on for several hours before measurement of the frequency to get a thermal stabilization. Since frequency drift with supply voltage is 0.002 f c Hz/V, regulators 7805 and 7905 were used to get stable powers +7V and 7V, respectively. A 0.001 µf capacitor is needed between pins 7 and 8 to stop parasitic oscillations.
The signal square wave output pin 4 (6.3 V pp ) of VCO, which is 90 degree out of phase with respect to response signal, was used as the constant excitation voltage of cantilever in the PLO. Its amplitude and phase were controlled with variable gain amplifier and phase shifter, respectively. By sweeping center frequency of VCO, f c, with R vco in Fig. 2(b) until f c equals to cantilever resonance f 0, the acquisition of lock-in state between θ excite and θ response at f 0 can be done without requiring a spontaneous excitation. This technique could allow operation over wide span of cantilever resonance frequency without more technical complications such as heterodyne technique. Fig. 3(a) shows the setup for the measurement of the bandwidth of PLO and f-d curve. Ac voltage, sin( Ω t), was applied between tip and sample directly using VCO of a V ac function generator (TEMA:2MHz function generator) to induce sinusoidal electrostatic force. A well-known square law dependence of the capacitive force F c on the probing signal induces an attractive electrostatic force modulation. 1 C( z) 2 Fc ( z) = Vac (1 cos(2ω t)) (5) 4 z where C(z) is the capacitance between tip and sample and a function of the instaneous tip-sample distance z. The shift of resonance frequency f under this force field is also sinusoidal function with modulation angular frequency 2Ω whose amplitude can be controlled with V ac and z for a give vibration amplitude a. A tip-sample distance z was adjusted with z-offset piezo under constant peak to peak amplitude 30nm, V ac =10V, and Ω =1Hz so that sinusoidal signal was modulated at resonance f0 with deviation f 2π 50Hz and frequency 2Ω. The signals near arrows in Fig. 3(a) represent such an effect schematically. After the sinusoidal f signal after PLL is filtered in a low pass filter in
order to eliminate the dc component, root mean square amplitude was measured with RMS-to-DC converter as the input frequency 2Ω changes to obtain the frequency response of PLO. The input voltage of VCO of function generator was swept to increase the input frequency. The measured output of RMS-to-DC goes to computer via ADC. RHK STM 100 provides DAC and ADC for voltage sweep and data acquisition. When this sweep voltage is switched to the input of z-piezo, frequency shift vs. distance( f d) curve can be acquired. Fig. 3(b) shows the measured closed-loop transfer function between θ excite and θ response of PLO. Due to mixing effect of Eq. (5), the effective input frequency is twice of the output frequency Ω of VCO in Fig. 3(a). The bandwidth of PLO is about 340Hz, at which the frequency response has declined to a factor of 0.707 times (f 3dB ) from its low frequency value. This value facilitates practical scanning speed to get an image. The inset of Fig. 3(b) shows the transfer function of PLL measured independently using frequency shift keying(fsk) output of function generator DS335 (Stanford Research) with frequency deviation f at f c =270kHz. The solid line is the calculated transfer function based on a theoretical modeling of PLL response Eq. (2) with the natural frequency ω n = 2π 360 rad/sec and damping constant ζ=1.0. The bandwidth of PLL shows nearly the same value with that of PLO. This fact implies that bandwidth of PLO is limited by that of PLL. In other words, the frequency response of tip-sample system in Fig. 3(a) is flat at least to the bandwidth of PLL. The time to change the resonance frequency from f 0 to f 0 + f due to tip-sample interaction is proportional to 2l/v ~ 1/f 0, the time that acoustic wave propagates along the cantilever with length l with velocity v until new condition of normal mode is established. Therefore the bandwidth of tip-sample
system is considered to be f 0, which is independent of the quality value(q) value and consistent with the result of Fig. 3. While PLO settles down form its transient value to its steady state value to within 1% during the reciprocal of bandwidth when tip-sample interaction occurs, the amplitude decay 1 - exp(-2 Q/Bω 0 ) due to dissipation to its steady state value is less than 100ppm with Q 20,000. Since the amplitude of cantilever still rings while scanning the cantilever with practical scanning rate, input of PLL is clipped to remove the noise in f caused by amplitude fluctuation. Figure 4(a) shows f - d curves measured with PLO on the cleaned VC(100), whose noise level is ( f) rms 1Hz far from sample surface. Subsequent two f-d curves were measured with DRO with quadrature demodulator[11] and PLL one on the same position under the same amplitude and cantilever, as shown in Fig. 4(b) and Fig. 4(c) respectively. Even though both of demodulators used exactly the same response signal from DRO, the noise level of PLL demodulator(fig. 4(c)) is comparable with that of PLO and much better than that of quadrature demodulator(fig. 4(b)), ( f) rms ~ 5Hz. These results are consistent with the general advantage of phase locked technique which is known to be the reduction of noise characteristics 6dB to conventional one. However, both curves of Fig. 4(b) and 4(c) jigger in the repulsive regime because DRO itself is unstable due to reduced selectivity of amplitude response. On the contrary, Fig. 4(a) shows a stable and welldefined value even in the repulsive force regime. The coherent effect of PLO makes f keep tracking the interaction as well as reducing noise level enough to get atomic corrugation of about only 2-3Hz[4]. In other words, PLO enables cantilever to track phase as well as resonance frequency while DRO does only resonance frequency without any restriction to phase. Because the roots of denominator of Eq. (2) have only positive
imaginary parts with considering that transfer function of PLO is the same with that of PLL, phase tracking between θ excite and θ response is unconditionally stable for all gain and frequency. This is the reason why atomic image could be obtained under the stable condition[8]. Therefore PLO scheme is superior to the conventional DRO for cases in which the force sensor exhibits only a weak resonance enhancement such as near atomic force regime in the f -d curve. 5. Conclusion Phase locked oscillator(plo) with phase locked loop for ultra high vacuum(uhv) dynamic force microscopy has been developed to detect tip-sample interaction and force microscopy. The measured the bandwidth of PLO using electrostatic force modulation between tip and sample is over 340Hz, which is practical scanning speed to get an image. The PLO bandwidth can be controlled with the change of loop parameters of PLL in a controlled manner. By measuring frequency shift versus distance( f-d ) curve and imaging on atomically clean sample surface, we demonstrate that this technique improved the signal to noise ratio with unconditional stability over all force regime owing to phase coherent effect. PLO method is an extremely effective way to excite the cantilever coherently with constant excitation voltage, compared with that of conventional FM technique without phase coherence. 6. Reference [1]F. J. Giessibl, Science 267, 68 (1995). [2]S. Kitamura and M. Iwatsuki, Jpn. J. Appl. Phys., Part 2 34, L145(1995).
[3] Y. Sugawara, M. Ohta, H. Ueyama, and S. Morita, Science 270, 1646(1995). [4] R. Luthi, E. Meyer, M. Bammerlin, A. Baratoff, T. Lehmann, L. Howald, Ch. Gerber, and H.-J. Guntherodt, Z. Phys. B 100, 165(1996). [5] T. R. Albrecht, P. Grutter, D. Horne, and D. Rugar, J. Appl. Phys. 69, 668(1991) [6] U. Dürig, H. R. Steinauer and N. Blanc, J. Appl. Phys. 82, 3641(1997). [7] U. Dürig, O. Zuger, and A. Stalder, J. Appl. Phys. 72, 1778(1992) [8] Ch. Loppacher, M. Bammerlin, M. Guggisberg, F. Battiston, R. Bennewitz, S. Rast, A. Baratoff, E. Meyer, H. -J. Guntherodt, Appl. Surf. Sci. 140, 287(1999). [9] F. M. Gardner, Phaselock Techniques(John Wiley & Sons, New York, 1966). [10] Data sheet of LM565( National Semiconductor). [11] P. Horowitz and W. Hill, The Art of Electronics(Cambridge University Press, New York, 1989). 7. Figure Caption
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