NL2009014C2 - Method for determining a spring constant of a cantilever of an atomic force microscope or scanning probe microscope, and a calibration device for a cantilever. - Google Patents

Method for determining a spring constant of a cantilever of an atomic force microscope or scanning probe microscope, and a calibration device for a cantilever. Download PDF

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Publication number
NL2009014C2
NL2009014C2 NL2009014A NL2009014A NL2009014C2 NL 2009014 C2 NL2009014 C2 NL 2009014C2 NL 2009014 A NL2009014 A NL 2009014A NL 2009014 A NL2009014 A NL 2009014A NL 2009014 C2 NL2009014 C2 NL 2009014C2
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Netherlands
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cantilever
voltage
electrode
pull
spring constant
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NL2009014A
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Dutch (nl)
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Fred Keulen
Marnani Hamed Sadeghian
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Univ Delft Tech
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01Q—SCANNING-PROBE TECHNIQUES OR APPARATUS; APPLICATIONS OF SCANNING-PROBE TECHNIQUES, e.g. SCANNING PROBE MICROSCOPY [SPM]
    • G01Q40/00—Calibration, e.g. of probes

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  • Physics & Mathematics (AREA)
  • Health & Medical Sciences (AREA)
  • General Health & Medical Sciences (AREA)
  • General Physics & Mathematics (AREA)
  • Nuclear Medicine, Radiotherapy & Molecular Imaging (AREA)
  • Radiology & Medical Imaging (AREA)
  • Measurement Of Length, Angles, Or The Like Using Electric Or Magnetic Means (AREA)

Description

Method for determining a spring constant of a cantilever of an atomic force microscope or scanning probe microscope, and a calibration device for a cantilever 5 The invention relates to a method for determining a spring constant of a cantilever of an atomic force microscope or scanning probe microscope, which cantilever is provided with a scanning probe tip and wherein the cantilever is provided adjacent to an electrode, wherein the method comprises 10 the step of applying a voltage between the electrode and the cantilever until a predetermined condition is met.
Such a method is known from the article "Characterizing size dependent effective elastic modulus of silicon nanocantilevers using electrostatic pull in instability" by H.
15 Sadeghian et al, Applied Physics Letters, AIP, American Institute of Physics, Melville, New York, volume 94, nr 22, 2 June 2009, pages 221903-1 - 221903-3. In this known method the predetermined condition is an electrostatic pull in condition of the cantilever with reference to the electrode, and it is 20 taught to derive Young's modulus of the cantilever, which is a measure for the cantilever's spring constant, from at least the pull in voltage that is required for meeting said electrostatic pull in condition.
The beauty of the approach based on the electrostatic 25 pull in condition is that one can measure the spring constant and avoid the pitfalls of other solutions to measure the cantilever's spring constant or the corresponding Young's modulus. The pull in behavior of the cantilever depends on the interaction of the electrostatic load which is generated by 30 the applied voltage, the cantilever's stiffness and its geometry. It is however independent of the cantilever's mass. The measurement based on the electrostatic pull in condition thus avoids the sensitivity of measuring Young's modulus or the spring constant of the cantilever by the also known method, 35 which is discussed in the above mentioned article, to determine the stiffness by measuring the cantilever's resonance frequency, which measurement is corrupted because of inaccuracies due to mass changes caused by surface contamination, native oxide and other adsorbed layers on the cantilever.
2
The problem with measuring Young's modulus or the spring constant of the cantilever based on the electrostatic pull in condition is the possible occurrence of stiction and the need to apply high voltages when the cantilever is a rela-5 tively stiff. It may also be undesirable as such to bring the cantilever in contact with the electrode.
It is therefore an object of the invention to provide a simple, easy, fast, low-cost and accurate method to determine and calibrate the spring constant of the cantilever of an 10 atomic force microscope or scanning probe microscope which forms a viable alternative for the known method to base the measurement of said spring constant on the electrostatic pull in condition of the cantilever with reference to the electrode .
15 According to the invention the method of measuring the cantilever's spring constant is therefore characterized in that a series of different voltages are applied between the electrode and the cantilever, wherein each voltage is selected to remain below the pull in voltage, and that with each such 20 voltage a corresponding resonance frequency of the cantilever is measured, which series of frequency measurements is used to estimate the pull in voltage at which electrostatic pull in of the cantilever with respect to the electrode will occur, and that the spring constant is thereafter derived from at least 25 this estimated pull in voltage. The manner in which the spring constant according to the invention is determined therefore largely corresponds to the known method to base the measurement of the conditions prevailing when the cantilever is in the pull in position, with the essential difference that the 30 pull in conditions are not actually achieved but estimated from a series of measurements that avoid the pull in position of the cantilever.
The method of the invention is very suited to measure the spring constant of a series of cantilevers in a batch pro-35 cess.
Suitably the pull in voltage is estimated by employing a known relation between the cantilever's Eigen frequency and the cantilever's displacement that is determined by a differential equation describing the relation between the canti 3 lever's position and the voltage applied between the cantilever and the electrode. This differential equation is provided on page 221903 -2, left-hand column of the above mentioned article, which is deemed inserted and fully incorporated herein.
5 Ultimately this means that deriving the spring constant is based on said differential equation describing the relation between the cantilever's position and the voltage applied between the cantilever and the electrode. Albeit that this equation may be used for the determination of the spring constant, 10 this does not mean that it is required to measure the cantilever's deflection; it suffices to measure the cantilever's resonance frequency with different voltages applied between the cantilever and the electrode.
It may be advantageous that an initial distance be-15 tween the cantilever and the electrode prior to applying the series of voltages is varied, and that different series of measurements are made starting each time with a different initial distance between the cantilever and the electrode. This may be used to improve the reliability of the measurement and 20 rule out or suppress the influence of measurement noise.
The invention is also embodied in a microscope, such as an atomic force microscope or scanning probe microscope, having a cantilever that is provided with a scanning probe tip, and is also embodied in a calibration device for such a 25 cantilever.
The microscope and the calibration device of the invention are embodied with means for determining a spring constant of the cantilever comprising means to apply a voltage between the electrode and the cantilever, and calculating 30 means to determine the spring constant from at least a pull in voltage that is required for meeting an electrostatic pull in condition, wherein the means to apply a voltage are embodied to apply a series of different voltages between the electrode and the cantilever, wherein each voltage is selected to remain 35 below the pull in voltage, and that measurement means are provided to measure with each such voltage a corresponding resonance frequency of the cantilever, and that estimator means are provided to estimate the pull in voltage at which electrostatic pull in of the cantilever with respect to the electrode 4 will occur based on the measurement means providing said series of frequency measurements, and that the calculating means determine the spring constant from at least this estimated pull in voltage.
5 The invention will hereinafter be further elucidated with reference to the drawing.
In the drawing: -figure 1 schematically shows apparatus features that may be embodied in a microscope or calibration device for a 10 cantilever according to the invention; and -figure 2 represents a graph showing the relation between the square of the cantilever's Eigen frequency and its tip displacement.
In the drawing of figure 1 reference 1 denotes a can-15 tilever and reference 2 denotes the electrode adjacent to which the cantilever 1 is provided. The cantilever 1 is provided with a scanning probe tip 3.
The cantilever 1 and the electrode 2 are connected to supply means 4 for applying a voltage between the cantilever 1 20 and the electrode 2, and these supply means 4 preferably are controllable so that different voltages can be applied to the cantilever 1 and electrode 2.
Figure 1 further shows that measurement means 5 are provided for measuring the resonance frequency of the cantile-25 ver 1. Both the measured resonance frequency recorded by the measurement means 5 and the corresponding voltages that are applied to the cantilever 1 and electrode 2 are for instance stored in memory 7 of a computer 6. The computer 6 is also provided with estimator means 8 which uses the measured reso-30 nance frequencies and the corresponding voltages that collectively are stored in the memory 7 to estimate the pull in voltage that is required to ensure electrostatic pull in of the cantilever 1 with reference to the electrode 2. In the invention pull in of the cantilever 1 with reference to the 35 electrode 2 is however never achieved, since the applied voltages are always kept at a level below the voltage that is required for meeting the electrostatic pull in condition.
The estimated pull in voltage of the cantilever 1 is subsequently provided by the estimator means 8 to calculating 5 means 9 forming part of the computer 6, and based thereon the calculating means 9 provide at output 10 a calculated spring constant of the cantilever 1 employing the method known from the prior art for calculating the Young's modulus of the can-5 tilever. In this connection it is further remarked that it is also possible to vary the initial distance between the cantilever and the electrode prior to applying the series of voltages and to make different series of measurements starting each time with a different initial distance between the canti-10 lever and the electrode. In this manner the reliability of the measurement is increased and the influence of measurement noise can be suppressed.
Determining the spring constant according to the method known for determining Young's modulus of the cantilever 15 is theoretically based on a differential equation describing the relation between the cantilever's position and the voltage applied between the cantilever and the electrode. More in particular the pull in voltage is estimated by employing a relation as shown in figure 2 between the (sguare of the) cantile-20 ver's Eigen frequency and the cantilever's displacement that is determined -as is known by the person skilled in the art-by said differential equation describing the relation between the cantilever's position and the voltage applied between the cantilever and the electrode. Concerning the differential 25 equation reference is made to the article mentioned in the introduction "Characterizing size dependent effective elastic modulus of silicon nanocantilevers using electrostatic pull in instability" by H. Sadeghian et al, Applied Physics Letters, AIP, American Institute of Physics, Melville, New York, volume 30 94, nr 22, 2 June 2009, pages 221903-1 - 221903-3.
The measurement means 5 for measuring the resonance frequencies of the cantilever 1 can be selected from any known means to record these resonance frequencies. Information on this can be derived from the following citations:
Analytical beam model C. A. Clifford, M. P. Seah, "The determination of atomic force microscope cantilever spring constants via dimensional methods 35 6 for nanomechanical analysis", Nanotechnology 16 (9) (2005) 1666.
Cleveland 5 J. P. Cleveland, S. Manne, D. Bocek, P. K. Hansraa, "A nondestructive method for determining the spring constant of cantilevers for scanning force microscopy", Review of Scientific Instruments 64 (2) (1993) 403-405.
10 Sader J. E. Sader, J. W. M. Chon, P. Mulvaney, "Calibration of rectangular atomic force microscope Cantilevers", Review of Scientific Instruments 70 (10) (1999) 3967-3969.
15 Thermal noise J. L. Hutter, J. Bechhoefer, "Calibration of atomic-force microscope tips", Review of Scientific Instruments 64 (7) (1993) 1868-1873.
20 Added mass (zelfde paper als Cleveland) J. P. Cleveland, S. Manne, D. Bocek, P. K. Hansma, "A nondestructive method for determining the spring constant of cantilevers for scanning force microscopy", Review of Scientific Instruments 64 (2) (1993) 403-405.
25
Reference cantilever R. S. Gates, M. G. Reitsma, "Precise atomic force microscope cantilever spring constant calibration using a reference cantilever array", Review of Scientific Instruments 78 (8) (2007) 30 086101.
and C. A. Clifford, M. P. Seah, "Improved methods and uncertainty analysis in the calibration of the spring constant of an atomic force microscope cantilever using static experimental 35 methods", Measurement Science and Technology 20 (12) (2009) 125501.
Electroactuation S. Rana, P. M. Ortiz, A. J. Harris, J. S. Burdess, C. J.
7
McNeil, "An electrostatically actuated cantilever device capable of accurately calibrating the cantilever on-chip for afm-like applications", Journal of Micromechanics and Microengineering 19 (4) (2009) 045012.
5 Turning now to Figure 2 a series of measurements 21- 26 are shown reflecting measured resonance frequencies. These frequencies are stored in memory 7 together with the corresponding excitation voltages that are applied between the cantilever 1 and the electrode 2. Using the known relation be-10 tween the displacement w of the cantilever 1 as shown on the X. axis and the square of the resonance frequency shown on the Y. axis, the estimator 8 makes an estimation of the pull in voltage that is required to establish electrostatic pull in of the cantilever 1 to the electrode 2. When pull in occurs by 15 definition the resonance frequency becomes zero due to loss of stability. The pull in situation symbolized by arrow 27 happens when applying the pull in voltage results in a critical deflection of the cantilever 1. The thus estimated pull in voltage is subsequently used in known manner by calculating 20 means 9 for determining the cantilever's spring constant.

Claims (7)

1. Werkwijze voor het bepalen van een veerconstante van een cantilever (1) van een atomic force microscoop of scanning probe microscoop, welke cantilever (1) voorzien is van een scanning probe tip (3) en waarin de cantilever (1) ge- 5 plaatst is naast een elektrode (2), waarin de werkwijze de stap van het aanbrengen van een spanning tussen de elektrode (2) en de cantilever (1) omvat, en het afleiden van de veerconstante van tenminste een pull-in spanning welke vereist is voor het realiseren van een elektrostatische pull-in conditie 10 van de cantilever (1) ten opzichte van de elektrode (2), met het kenmerk, dat een serie verschillende spanningen toegepast worden tussen de elektrode (2) en de cantilever (1), waarin iedere spanning ingesteld wordt op een waarde die onder de pull-in spanning blijft, en dat met elke dergelijke spanning 15 een corresponderende resonantiefrequentie van de cantilever (1) gemeten wordt, welke serie frequentiemetingen gebruikt wordt voor het schatten van de pull-in spanning bij welke elektrostatische pull-in van de cantilever (1) ten opzichte van de elektrode (2) zal plaatsvinden, en dat de veerconstante 20 afgeleid wordt van tenminste deze geschatte pull-in spanning.Method for determining a spring constant of a cantilever (1) of an atomic force microscope or scanning probe microscope, which cantilever (1) is provided with a scanning probe tip (3) and in which the cantilever (1) is mounted is located next to an electrode (2), wherein the method includes the step of applying a voltage between the electrode (2) and the cantilever (1), and deriving the spring constant from at least one pull-in voltage that is required for realizing an electrostatic pull-in condition 10 of the cantilever (1) with respect to the electrode (2), characterized in that a series of different voltages are applied between the electrode (2) and the cantilever (1), wherein each voltage is set to a value that remains below the pull-in voltage, and that with each such voltage a corresponding resonance frequency of the cantilever (1) is measured, which series of frequency measurements is used for estimating the pu 11-in voltage at which electrostatic pull-in of the cantilever (1) relative to the electrode (2) will take place, and that the spring constant 20 is derived from at least this estimated pull-in voltage. 2. Werkwijze volgens conclusie 1, met het kenmerk, dat de pull-in spanning geschat wordt onder gebruikmaking van een relatie tussen de Eigen-frequentie van de cantilever en de verplaatsing van de cantilever die bepaald wordt door een dif- 25 ferentiaal-vergelijking die de relatie beschrijft tussen tenminste de positie van de cantilever en de spanning die toegepast wordt tussen de cantilever (1) en de elektrode (2) .2. Method as claimed in claim 1, characterized in that the pull-in voltage is estimated using a relationship between the Eigen frequency of the cantilever and the displacement of the cantilever determined by a differential equation that describes the relationship between at least the position of the cantilever and the voltage applied between the cantilever (1) and the electrode (2). 3. Werkwijze volgens conclusie 1 of 2, met het kenmerk, dat een initiële afstand tussen de cantilever (1) en de 30 elektrode (2) voorafgaand aan het toepassen van de serie spanningen gevarieerd wordt, en verschillende series metingen uitgevoerd worden die telkens aanvangen met een verschillende initiële afstand tussen de cantilever (1) en de elektrode (2).3. Method as claimed in claim 1 or 2, characterized in that an initial distance between the cantilever (1) and the electrode (2) is varied prior to the application of the series of voltages, and different series of measurements are carried out which each start with a different initial distance between the cantilever (1) and the electrode (2). 4. Werkwijze volgens één der conclusies 1-3, met het 35 kenmerk, dat het afleiden van de veerconstante gebaseerd is op een differentiaal-vergelijking die de relatie beschrijft tus- sen tenminste de positie van de cantilever en de spanning die toegepast wordt tussen de cantilever en de elektrode.4. Method as claimed in any of the claims 1-3, characterized in that the derivation of the spring constant is based on a differential equation describing the relationship between at least the position of the cantilever and the voltage applied between the cantilever and the electrode. 5. Werkwijze volgens één der voorgaande conclusies, met het kenmerk, dat de veerconstante van een serie cantile- 5 vers wordt bepaald in een batch proces.5. Method as claimed in any of the foregoing claims, characterized in that the spring constant of a series of cantilevers is determined in a batch process. 6. Microscoop zoals een atomic force mircoscoop of scanning probe microscoop, met een cantilever (1) die voorzien is van een scanning probe tip (3), welke cantilever (1) geplaatst wordt naast een elektrode (2), en uitgevoerd is met 10 middelen (4, 5, 6) voor het bepalen van een veerconstante van de cantilever (1), omvattende middelen (4) voor het aanbrengen van een spanning tussen de elektrode (2) en de cantilever (1), en rekenmiddelen (9) voor het bepalen van de veerconstante uit tenminste een pull-in spanning die vereist is voor het reali-15 seren van een elektrostatische pull-in conditie van de cantilever (1) ten opzichte van de elektrode (2), met het kenmerk, dat de middelen (4) voor het aanbrengen van de spanning uitgevoerd zijn voor het toepassen van een serie verschillende spanningen tussen de elektrode (2) en de cantilever (1), waar-20 bij iedere spanning geselecteerd wordt op een waarde onder de pull-in spanning, en dat meetmiddelen (5) voorzien zijn voor het meten bij iedere dergelijke spanning van een corresponderende resonantiefrequentie van de cantilever (1), en dat schattingsmiddelen (8) voorzien zijn voor het schatten van de 25 pull-in spanning bij welke elektrostatische pull-in van de cantilever (1) ten opzichte van elektrode (2) zal plaatsvinden, gebaseerd op de meetmiddelen (5) die voorzien in genoemde serie frequentiemetingen, en dat de rekenmiddelen (9) de veerconstante bepalen uit tenminste deze geschatte pull-in span-30 ning.6. Microscope such as an atomic force circuit or scanning probe microscope, with a cantilever (1) provided with a scanning probe tip (3), which cantilever (1) is placed next to an electrode (2), and is equipped with 10 means (4, 5, 6) for determining a spring constant of the cantilever (1), comprising means (4) for applying a voltage between the electrode (2) and the cantilever (1), and computing means (9) for determining the spring constant from at least one pull-in voltage required for realizing an electrostatic pull-in condition of the cantilever (1) relative to the electrode (2), characterized in that the means (4) for applying the voltage are designed for applying a series of different voltages between the electrode (2) and the cantilever (1), with each voltage being selected for a value below the pull-in voltage and that measuring means (5) are provided for measuring at each such voltage of a corresponding resonance frequency of the cantilever (1), and that estimation means (8) are provided for estimating the pull-in voltage at which electrostatic pull-in of the cantilever (1) with respect to electrode (2) will take place based on the measuring means (5) which provide for said series of frequency measurements, and that the calculating means (9) determine the spring constant from at least this estimated pull-in voltage. 7. Calibratieinrichting voor een cantilever (1) die voorzien is van een scanning probe tip (3), welke cantilever (1) voorzien is naast een elektrode (2) en uitgevoerd met middelen (4, 5, 6) voor het bepalen van een veerconstante van de 35 cantilever (1), omvattende middelen voor het aanbrengen van een spanning tussen de elektrode (2) en de cantilever (1) en rekenmiddelen (9) voor het bepalen van de veerconstante uit tenminste een pull-in spanning die vereist is voor het realiseren van een elektrostatische pull-in conditie van de canti- lever (1) ten opzichte van de elektrode (2), met het kenmerk, dat de middelen (4) voor het aanbrengen van een spanning uitgevoerd zijn voor toepassing van een serie verschillende spanningen tussen de elektrode (2) en de cantilever (1), waarbij 5 iedere spanning geselecteerd wordt op een waarde onder de pull-in spanning, en dat middelen (5) voorzien zijn voor het meten bij iedere dergelijke spanning van een corresponderende resonantiefrequentie van de cantilever (1), en dat schattings-middelen (8) voorzien zijn voor het schatten van de pull-in 10 spanning bij welke elektrostatische pull-in van de cantilever (1) ten opzichte van de elektrode (2) zal plaatsvinden, gebaseerd op de meetmiddelen (5) die voorzien in genoemde serie frequentiemetingen, en dat de rekenmiddelen (9) de veercon-stante bepalen uit tenminste deze geschatte pull-in spanning.A calibration device for a cantilever (1) provided with a scanning probe tip (3), which cantilever (1) is provided next to an electrode (2) and provided with means (4, 5, 6) for determining a spring constant of the cantilever (1), comprising means for applying a voltage between the electrode (2) and the cantilever (1) and calculating means (9) for determining the spring constant from at least one pull-in voltage required for realizing an electrostatic pull-in condition of the cantilever (1) relative to the electrode (2), characterized in that the means (4) for applying a voltage are designed for applying a series different voltages between the electrode (2) and the cantilever (1), wherein each voltage is selected for a value below the pull-in voltage, and that means (5) are provided for measuring at each such voltage a corresponding resonance frequency of the c antelever (1), and that estimation means (8) are provided for estimating the pull-in voltage at which electrostatic pull-in of the cantilever (1) relative to the electrode (2) will occur based on the measuring means (5) which provide for said series of frequency measurements, and that the calculating means (9) determine the spring constant from at least this estimated pull-in voltage.
NL2009014A 2012-06-15 2012-06-15 Method for determining a spring constant of a cantilever of an atomic force microscope or scanning probe microscope, and a calibration device for a cantilever. NL2009014C2 (en)

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Citations (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
WO2012064193A1 (en) * 2010-11-12 2012-05-18 Technische Universiteit Delft Method for determining a spring constant for a deformable scanning probe microscope element, and scanning probe microscope and calibration device arranged for determining a spring constant for a probe element

Patent Citations (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
WO2012064193A1 (en) * 2010-11-12 2012-05-18 Technische Universiteit Delft Method for determining a spring constant for a deformable scanning probe microscope element, and scanning probe microscope and calibration device arranged for determining a spring constant for a probe element

Non-Patent Citations (2)

* Cited by examiner, † Cited by third party
Title
J. P. CLEVELAND ET AL: "A nondestructive method for determining the spring constant of cantilevers for scanning force microscopy", REVIEW OF SCIENTIFIC INSTRUMENTS, vol. 64, no. 2, 1 January 1993 (1993-01-01), pages 403 - 405, XP055024743, ISSN: 0034-6748, DOI: 10.1063/1.1144209 *
SUNIL RANA ET AL: "An electrostatically actuated cantilever device capable of accurately calibrating the cantilever on-chip for AFM-like applications; An electrostatically actuated cantilever", JOURNAL OF MICROMECHANICS & MICROENGINEERING, INSTITUTE OF PHYSICS PUBLISHING, BRISTOL, GB, vol. 19, no. 4, 1 April 2009 (2009-04-01), pages 45012, XP020153379, ISSN: 0960-1317, DOI: 10.1088/0960-1317/19/4/045012 *

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