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Vibration isolation analysis for a scanning tunneling microscope A. I. Oliva, Victor Sosa, R. de Coss, Raquel Sosa, N. L6pez Salazar, and J. L. Peila Centro de Investigacicin y Estudios Avanzados de1 IPN, Unidad Mbrida, Departamento de Fisica Aplicada, Apdo. Postal 73-Cordemex.. 973 IO, Mhrida, Yucatcin, Mhico (Received 7 October 1991; accepted for publication 11 February 1992) We analyze the efficiency of a vibration isolation system (VIS) for a scanning tunneling microscope as a function of the different parameters involved. The VIS consists of a stack of several metallic plates, separated by rubber elements with known properties. We show three-dimensional graphs obtained for different values of parameters such as rigidity (spring) constant (K), damping constant (C), mass (M), and the number of stages (n). Analyzing the K dependence of the position of the main peaks, we find a parabolic behavior when the damping constant is small, with a slight deviation for larger values. I. INTRODUCTION The scanning tunneling microscope (STM) was invented by Binnig and Rohrer.’ For this work they were awarded the Nobel Prize for Physics in 1986.2 Since then, the STM has been used to obtain surface topography with atomic resolution and electronic properties of metals and other materials. It can be used in vacuum or under atmospheric pressure and actually it has become a necessary tool in research laboratories because it is possible to perform nondestructive surface analysis. The principle of operation of a STM is based on maintaining a tunneling current in the gap formed between a sample and a sharpened metallic tip, when they are very closely spaced (several angstroms apart). If this current is controlled by means of an electronic feedback circuit, and the sample surface is scanned by means of piezoelectric elements, one can obtain the topography of the surface and/or some electronic properties. To achieve atomic resolution, the STM must be provided with a system capable of reducing the external perturbations in order to obtain good stability in the tunnel junction. Some authors suggest l-pm (10 - l2 m) resolution needed to detect the corrugation of the atomic surface in metals.3’4 If the resolution is higher, it is possible to detect, at low temperatures, molecular vibrational motion and phonons both in the tip an,d the sample.5*6 Different STM designs have used different methods for vibration isolation. These methods have been modified with time to obtain better results. Examples are the first STM of Binnig and Rohrer’ using superconducting magnetic levitation at liquid-helium temperature; the use of helical springs in two or three stages, damped with eddy currents; and recently, the use of viton elements between metallic plates to form a stack. The tunnel junction is fabricated over the stack. The use of the last method has made it possible to design a more simple and compact STM with equal or higher performance. Several STM8lo with atomic resolution have been constructed with this kind of vibration isolation method. Okano et a1.,3 and recently Hammiche er al.,” presented a theoretical model to describe the vibration isolation system (VIS) for a STM for a simple geometry based on a number of stages separated by rubber elements with 3326 Rev. Sci. Instrum. 63 (6), June 1992 known properties, with one-dimensional motion. However, they did not analyze the behavior for different values of the relevant parameters, to obtain better results in the VIS. In this work we present a study, based on Okano’s model, of the influence of the different parameters involved in the VIS, namely, the mass (M), the number of stacks (n), the rigidity constant (K), and the damping constant ( C), in order to recommend the best conditions to achieve good results. Some authors”3*‘2 recommend that the ratio between the natural frequency of the system and the frequency of the external perturbation be greater than at least two orders of magnitude in order to avoid resonance effects. II. THEORY Okano’s model is based on the effect caused by the external vibrations on the tunnel junction located on the last stage of a one-dimensional system. To define some concepts, we first assume that the VIS consists of a simple mass-spring system of two stages. In this case, the equations of motion for the two coupled oscillators are: mlBl + KIXl -I- K2(X1 - X2) = KIXb sin wt, .. m2X,+K2W2--X,) =o, (1) (2) where w is the angular frequency of the external perturbation and Mi and Ki are the mass and the rigidity constants of every stage, respectively, and Xb the amplitude of the external perturbation. The global transfer function of the complete system is dB = 20 log(xZ - xl)/xb (3) We assume that the VIS consists of n stages with only one degree of freedom (Fig. 1) and Ci and Ki have known values for every stage. Drawing the free-body diagram for every mass, applying Newton’s second law, and assuming an external perturbation Y. ejwc and oscillations Xi ej”‘, we obtain the following equations. For mass one, - w2M,X, +jwC,Xl + K,Xl + f&(X' -X2) +jwC*M 0034-6746/92/063326-04!$02.00 -X,) = Fl, @II1992 American (4) Institute of Physics 3326 Downloaded 26 Oct 2006 to 148.247.195.130. Redistribution subject to AIP license or copyright, see http://rsi.aip.org/rsi/copyright.jsp FIG. 1. Diagram of the vibration isolation system (VIS) of n stages analyzed in this work. The system consists of a stack formed by plates and rubbers with a one-dimensional motion. Frequency (Hz) FIG. 2. VIS response for four, six, and eight stages. M = 0.03 kg, K = 80,OCHlN/m, and C = 5 Ns/m for each stage. where F, = (K, + @C, ) Ye. For mass r, - w2MJ, + jwC,(X, - x, - , ) + K,(X, - x,- ,) Our principal aim was to find optimal values of the parameters K, C, M, and n in a VIS to give the best con- ditions for reaching the 10 - 12-m vibration amplitude necessary in the tunnel junction. If we try to solve the problem analytically and thus to obtain the optimum values, it is very complicated when the number of stacks is higher than two. This problem has widely been analyzed and demonstrated in previous work’3+‘4 for two stages. In that theoretical model we found an expression for a transfer function with two parts: a real part and an imaginary part. The real part is strongly influenced by the rigidity constant K and depends inversely on the square of frequency. In the real part of the behavior of the resonance peaks is similar to a spring-mass system. The imaginary part depends strongly on the damping constant and the variation with frequency is more smooth. The imaginary part determines the amplitudes of the resonance peaks. However, our interest is a combination of both parts, because in a real problem, both are joined. Also, the mass M, involved in both parts, is another important parameter and its value depends on the necessities and the available space. The current tendency is to design compact STMs. To solve the equations [(4)-(7)] presented before, we developed a FORTRAN program to calculate the effect of every parameter on the last stage of the STM. We make graphs for a wide range of the VIS parameters. With these graphs we studied the efficiency and the behavior of the VIS. In the next paragraphs we present a discussion of the results obtained from the analysis. One must choose the number of stages needed for the VIS, the required vibration isolation, and the expected external perturbations. We assume, for all calculations, a 10e6-m external perturbation and a lo-I*-m vibration amplitude desirable in the last stage. This implies that we need a transfer function which reduces the amplitude by 10e6 times or 120 db. In Fig. 2 we show the transfer function for a VIS with four, six, and eight stages. We use M = 0.03 kg, K = 8.1 X lo4 N/m, and C = 5 Ns/m for every stage. The curve for 4 stages has the main peak at a frequency of 100 Hz and it is possible to reduce the external perturbation by 3327 STM vibration +JG+,(~,-X,+1) +K+l(X,--X,+,) =o (5) where r = 2,3 ,... n - 1, and for the nth mass, -u~2~,~,,+jwcn(x,--x,-,) +K,(X,-xX,-1) =o. (6) The transfer function between the last stage and the external perturbation for this system is given by dB = 20 log(X,,/YO) or, dB = 20 log{[Re(X,,)2 + Im(X,)2]“2/Yo}. (7) We solved the problem varying the values of different parameters (K and C) for the rubber elements used in the VI.5 These values can be determined using experimental methods. There exist different kinds of rubber materials used between each stage, but viton is the most widely used in vacuum conditions. To analyze the behavior of the VIS we need to know properties like K and C. Both values depend on the size, the geometry, and the material type. We measured K and C for a toroidal viton rubber. The rigidity constant K was determined using the equation K = F/6 (Hooke’s law), where F is an applied force and S the corresponding deformation. To measure the damping constant C, we used the equation obtained from a critical damping analysis, given by C = ( - 2M/t) ln(X/X,), where M is the mass, X the deformation during time t, and X0 the initial deformation. Average values obtained for toroidal vitons (3.6-mm external diameter, and 1.3-mm-diam cross sectional) are K = 8.12X lo4 N/m and C = 3.5 Ns/m. All the experiments were done with 12 N as the maximum applied load. Ill. DISCUSSION Rev. Sci. Instrum., Vol. 63, No. 6, June 1992 isolation 3327 Downloaded 26 Oct 2006 to 148.247.195.130. Redistribution subject to AIP license or copyright, see http://rsi.aip.org/rsi/copyright.jsp 60 86 II 4 Frequency (Hz) FIG. 3. Same as Fig. 2, but using A4 = 0.3 kg in every stage. only four orders of magnitude ( -- 80 dB) approximately at about 1 X lo3 Hz. With these characteristics it is not possible to reach our desired value ( - 120 dB). But if we increase the number of stages, we can reach this goal. In the same Fig. 2, with eight stages we obtained - 120 dB at 700 Hz. From this analysis we c.an determine an adequate number of stages, depending on the space, cost, and perturbation level. Mass is another important parameter for VIS in a STM. By experience, we know that if we increase the mass, the VIS will work better. Figure 3 is similar to Fig. 2 but it was obtained with a mass ten tilmes higher (0.3 kg). The other parameters remain the same. In Fig. 3 we can observe, for the same number of stages, the lower values in frequency reached for higher mass. Some authors’2*15 mention the use of high mass values in the VIS for the STM with excellent results. In order to study the effect of the constants K and C, we present three-dimensional graphs using six stages, each with M = 0.03 kg. In Fig. 4(a), a three-dimensional graph was calculated for C = I Ns/m. K values ranged from 1 to lo5 N/m. The scale of K in the graph is divided in two parts to show all the range. The frequency was varied from 1 to lo3 Hz. From Fig. 4(a) we can see how the main peak (at the lowest resonant frequency) drifts slightly to higher values as K is increased. Also, at a 103-Hz frequency the transfer function decreased when the K value is increased. This same effect can be seen better in Fig. 4(b) where we used C = 10 Ns/m. Note that the surface behaves more smoothly for small K values. The main peak appears when K has high values and the other peaks practically disappear due to the effect of the damping constant. If we increase the C value to 100 Ns/m, as we can see in Fig. 4(c), the effect is more pronounced. The transfer function increased at 1 x lo3 Hz and the main peak is weak. If we compare the last three graphs for a fixed frequency we observe that when C increases, the transfer function increases also. Regarding the behavior of the position of the main peaks (frequency f,) in Figs. 4(a), 4(b), and 4(c), we can plot log(271-fM) as a function of K. This graph is presented in Fig. 5. Three lines are shown for different condi3328 Rev. Sci. Instrum., Vol. 63, No. 6, June 1992 (a) (b) FIG. 4. Three-dimensional graphs of the transfer function in a six-stage VIS. K ranges from l-l x lo5 N/m and its scale is divided in two parts to represent the whole range. The parameter values used for every stage were the same as in Fig. 2 except the C value. (a) C= 1 Ns/m; (b) C= IO Ns/m, and (c) C= 100 Ns/m tions. Lines I, II, and III correspond to the main peaks for the C= 1 Ns/m, C = 10 Ns/m, and C= 100 Ns/m graphs, respectively. Applying a linear regression in each line we found the following results: STM vibration isolation 3328 Downloaded 26 Oct 2006 to 148.247.195.130. Redistribution subject to AIP license or copyright, see http://rsi.aip.org/rsi/copyright.jsp 1.8~ , .6.--- ..-- 1.4 1.2 1 0.8 0.8 0.4 0.2 J 1 t!+Ol lE+03 lE+02 1 E+04 1 E+05 -2004 G - 8 - 8 n 8 n g 1 5 1 ) G - s ’ 8 3 0 200 400 600 800 1000 K (N/m) Frequency FIG. 5. Main peak position (fv) of the Figs. 4(a)-4(c) as a function of K. The symbols represent the position of every peak and the solid lines result from linear regressions. Line I (C = 1 Ns/m): fM = 0.0350K0.500 Line II (C= fLw = 0.0264K0.529 Line III 10 Ns/m: (C = 100 Ns/m): fAv = 0.0069K0.647, (8) The slope of line I corresponds to a simple mass-spring system. Assuming this approximation, we can calculate the natural frequency by the equation fiM = (1/2a) (K/M) I’*. In this way, we can find an equivalent mass of the complete system. For line I, we find an equivalent mass of 0.52 kg. Equations obtained from lines II and III show a slight deviation in the slope. This deviation is due to the effect of the damping constant C. With real data obtained from our materials, we simulated a VIS using the graphical model presented. Figure 6 shows the curve obtained. For this calculation we used a stack of six stages with 5 equal masses of 0.026 kg, the sixth mass of 0.636 kg, K = 81 200 N/m, and C = 3.5 Ns/ m. The last heavy mass includes the tunnel junction. From the graph, we can see the behavior of the three last stages of the system and the way that the transfer function improves with the high number of stages. From the graph we see that the desired amplitude (lo-‘* m) is reached at 800 Hz in the last stage. Then, the resonant mechanical frequency of the STM needs to be 8 kHz, to assure that the tunnel junction will reduce the external perturbations in order to achieve atomic resolution. In summary, we obtained a graphical method for vibration isolation analysis where we showed the effect of every parameter involved. This method is useful for STM mechanical design. By means of three-dimensional graphs, we have shown the behavior of the transfer function with frequency for a wide range of the rigidity constant K. We 3329 Rev. Sci. Instrum., Vol. 63, No. 6, June 1992 (Hz) FIG. 6. Transfer function obtained with real data for a STM currently under construction. We use 6 stages with five equal masses (0.026 kg) and the sixth mass is different (0.636 kg). K = 81,200 N/m, and C = 3.5 Ns/m were used for every stage. Responses of the fourth and fifth stages are also shown, to clarify the evolution of damping. find a mass-spring behavior for small values of C with a slight variation when the damping constant is increased. We show using real data the frequency needed to reach the required - 120 dB in the tunnel junction of the STM. ACKNOWLEDGMENTS The authors are very grateful for the support given by CONACyT and COSNET-SEP, Mexico. We also thank Dr. Brian Davies for his technical review of the manuscript. ‘G. Binnig and H. Rohrer, Heiv. Phys. Acta 55, 726 ( 1982). *G. Binnig and H. Rohrer, Lex Pris Nobel en 1986 (The Nobel Foundations), p. 85. 3M. Okano, K. Kajimura, S. Wakiyama, F. Sakai, W. Mizutani, and M. Ono, J. Vat. Sci. Technol. A 5, 3313 ( 1987). 4D. W. Pohi, IBM J. Res. Dev. 30, 417 (1986). 5D. P. E. Smith, M. D. Kirk, and C. F. Quate, J. 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