Method And Control Unit For Demodulation

Abstract

A method for demodulation including the following steps: exciting a vibrationally mounted, at least sectionally bar-shaped oscillating element for oscillating in the range of a resonance frequency of the oscillating element, wherein a temporally varying, in particular periodic, excitation signal is used for excitation, and wherein at least the temporal variation of the excitation signal is known or determined; detecting a modulated oscillation of the oscillating element by means of at least one sensor, wherein the sensor supplies a sensor measurement variable that varies versus time as a function of an amplitude and a phase of the modulated oscillation of the oscillating element. According to the present teaching, it is provided that the method includes the following step: generate a first comparison signal by amplitude modulating a known temporally varying, in particular periodic, demodulation signal by means of the temporally varying sensor measurement variable.

Claims

1. A method for demodulation comprising the following steps: exciting a vibrationally mounted, at least sectionally bar-shaped oscillating element for oscillating in the range of a resonance frequency of the oscillating element, wherein a temporally varying excitation signal is used for excitation, and wherein at least the temporal variation of the excitation signal is known or determined; detecting a modulated oscillation of the oscillating element by at least one sensor, wherein the sensor supplies a sensor measurement variable that varies over time as a function of an amplitude and a phase of the modulated oscillation of the oscillating element; wherein the method comprises the following-step: generate a first comparison signal by amplitude modulating a known temporally varying demodulation signal by the temporally varying sensor measurement variable.

2. The method according to claim 1, wherein the first comparison signal is filtered by at least one element acting as a low-pass filter, thus generating a filtered first comparison signal.

3. The method according to claim 1, wherein the oscillating element is a cantilever.

4. The method according to claim 1, wherein the demodulation signal is generated by using the excitation signal.

5. The method according to claim 1, wherein a phase of the demodulation signal is shifted such that
Δϕ≈ϕ.sub.c
applies or
Δϕ≈ϕ.sub.c±π/2 wherein Δϕ is the phase of the demodulation signal and ilk is the phase of the modulated oscillation.

6. The method according to claim 1, wherein the at least one sensor comprises a piezoresistive element and the sensor measurement variable is the electrical resistance of the piezoresistive element.

7. The method according to claim 6, wherein the demodulation signal is an AC voltage signal and the first comparison signal is generated by a voltage divider, wherein the voltage divider is formed from a first electrical resistance and a second electrical resistance, and wherein the second electrical resistance comprises the electrical resistance of the piezoresistive element.

8. The method according to claim 1, wherein a difference signal is generated by a linear combination of the first comparison signal with a known temporally varying second comparison signal.

9. The method according to claim 8, wherein the difference signal is filtered by at least one element acting as a low-pass filter, thus generating a filtered difference signal.

10. The method according to claim 8, wherein the second comparison signal is generated by of using the demodulation signal.

11. The method according to claim 7, wherein the second comparison signal is generated by a further voltage divider, which is formed from a third electrical resistance and a fourth electrical resistance, wherein the voltage divider and the further voltage divider are preferably part of a bridge circuit, and in that the first comparison signal and the second comparison signal are subtracted from one another to generate the difference signal.

12. The method according to claim 3, wherein a difference signal is generated by a linear combination of the first comparison signal with a known temporally varying second comparison signal, wherein the difference signal is filtered by at least one element acting as a low-pass filter, thus generating a filtered difference signal, wherein the at least one sensor comprises a piezoresistive element and the sensor measurement variable is the electrical resistance of the piezoresistive element, wherein the demodulation signal is an AC voltage signal and that the first comparison signal is generated by a voltage divider, wherein the voltage divider is formed from a first electrical resistance and a second electrical resistance, and wherein the second electrical resistance comprises the electrical resistance of the piezoresistive element wherein the second comparison signal is generated by a further voltage divider, which is formed from a third electrical resistance and a fourth electrical resistance, wherein the voltage divider and the further voltage divider are preferably part of a bridge circuit, and wherein the first comparison signal and the second comparison signal are subtracted from one another to generate the difference signal the filtered difference signal is amplified with a constant gain factor and then subtracted from a target value in order to form a control signal, wherein the control signal is fed to a scanner control in order to adjust the distance between a sample and the cantilever.

13. The method according to claim 9, wherein a difference signal is generated by a linear combination of the first comparison signal with a known temporally varying second comparison signal, wherein the difference signal is filtered by at least one element acting as a low-pass filter, thus generating a filtered difference signal, wherein the second comparison signal is generated using the demodulation signal, wherein the modulated oscillation of the oscillating element is additionally detected with a second sensor, including a second piezoresistive element essentially identical to the piezoresistive element, wherein the second sensor also supplies a sensor measurement variable that varies versus time as a function of the amplitude and the phase of the modulated oscillation of the oscillating element and wherein the sensor measurement variable of the second sensor is the electrical resistance of the second piezoresistive element, in that a first phase-shifted comparison signal is generated by amplitude modulating a known temporally varying phase-shifted demodulation signal by the temporally varying sensor measurement variable of the second sensor, wherein the phase-shifted demodulation signal has a defined known phase shift in relation to the demodulation signal, in that the phase-shifted demodulation signal is also an AC voltage signal, wherein the first phase-shifted comparison signal is also generated by of a voltage divider, wherein the voltage divider is formed from a first electrical resistance and a second electrical resistance and wherein the second electrical resistance includes the electrical resistance of the second piezoresistive element, in that a phase-shifted difference signal is generated by a linear combination of the first phase-shifted comparison signal with a known temporally varying second phase-shifted comparison signal, in that the second phase-shifted comparison signal is generated by the phase-shifted demodulation signal, wherein the second comparison signal is generated by a further voltage divider, which is formed from a third electrical resistance and a fourth electrical resistance, wherein the voltage divider and the further voltage divider are part of a bridge circuit, and wherein the first phase-shifted comparison signal and the second phase-shifted comparison signal are subtracted from one another to generate the phase shifted difference signal, in that the phase-shifted difference signal is filtered by at least one element acting as a low-pass filter, thus generating a filtered phase-shifted difference signal.

14. The method according to claim 13, wherein the defined known phase shift of the phase-shifted demodulation signal in relation to the demodulation signal is 90°.

15. The method according to claim 13, wherein the phase-shifted demodulation signal is identical to the demodulation signal except for the defined known phase shift.

16. The method according to claim 13, wherein the filtered difference signal and the filtered phase-shifted difference signal are processed by a signal processing unit in order to calculate an output amplitude and/or an output phase.

17. The method according to claim 14, wherein the phase-shifted demodulation signal is identical to the demodulation signal except for the defined known phase shift, wherein the filtered difference signal and the filtered phase-shifted difference signal are processed by a signal processing unit in order to calculate an output amplitude and/or an output phase, wherein the output amplitude (A.sub.out) is calculated according to
A.sub.out=((DIFFS_f).sup.2+(DIFFS_Q_f).sup.2).sup.1/2 and/or the output phase (ϕ.sub.out) is calculated according to
ϕ.sub.out=atan(DIFFS_Q_f/DIFFS_f), wherein A.sub.out is the output amplitude, DIFFS_f is the filtered difference signal, DIFFS_Q_f is the filtered phase-shifted difference signal, bout is the output phase, and atan is the arc tangent.

18. A control unit for demodulation, wherein the control unit is connectable to at least one excitation means for exciting a vibrationally mounted, at least sectionally bar-shaped oscillating element, wherein the control unit can be connected to at least one sensor for detecting a modulated oscillation of the oscillating element, wherein the control unit is formed to execute a method according to claim 1.

19. A device comprising a control unit according to claim 18, the device further comprising the at least one excitation means, wherein the control unit is connected to the at least one excitation means, the device further comprising the at least one sensor, wherein the control unit is connected to the at least one sensor, and the device including the oscillating element.

20. The device according to claim 19, wherein the at least one excitation means comprises a piezo actuator.

21. The device according to claim 19, wherein at least one low-pass filter is provided.

22. The device according to claim 19, wherein the oscillating element is a cantilever.

23. The device according to claim 19, wherein the at least one sensor comprises a piezoresistive element and the sensor measurement variable is the electrical resistance of the piezoresistive element.

24. The device according to claim 23, wherein a voltage divider is provided to generate the first comparison signal, wherein the voltage divider is formed from a first electrical resistance and a second electrical resistance and wherein the second electrical resistance includes the electrical resistance of the piezoresistive element.

25. The device according to claim 24, wherein the device comprises a further voltage divider, which is formed from a third electrical resistance and a fourth electrical resistance, in order to generate a second comparison signal by the further voltage divider, wherein the voltage divider and the further voltage divider are part of a bridge circuit.

26. The device according to claim 22, wherein a scanner control is provided to adjust the distance between a sample and the cantilever.

27. The device according to claim 25, wherein a second sensor, comprising a second piezoresistive element essentially identical to the piezoresistive element, is provided whose sensor measurement variable is the electrical resistance of the second piezoresistive element, in that a voltage divider is provided to generate the first phase-shifted comparison signal, wherein the voltage divider is formed from a first electrical resistance and a second electrical resistance and wherein the second electrical resistance includes the electrical resistance of the second piezoresistive element, in that a further voltage divider is provided, which is formed from a third electrical resistance and a fourth electrical resistance to generate a second phase-shifted comparison signal by the further voltage divider, wherein the voltage divider and the further voltage divider are part of a bridge circuit.

28. The device according to claim 27, wherein an AC voltage source is provided to generate the demodulation signal and an AC voltage source is provided to generate a phase-shifted demodulation signal.

29. The device according to claim 27, wherein a module for generating the difference signal and a phase-shifted difference signal is provided, wherein a low-pass filter is provided to generate a filtered difference signal by filtering the difference signal, and a low-pass filter to generate a filtered phase-shifted difference signal by filtering the phase-shifted difference signal, and a signal processing unit is provided to process the filtered difference signal and the filtered phase-shifted difference signal and to calculate an output amplitude and/or an output phase.

30. The method according to claim 3, wherein the oscillating element is a cantilever of an atomic force microscope.

31. The method according to claim 4, wherein the demodulation signal is generated as a linear function of the excitation signal.

32. The method according to claim 10, wherein the second comparison signal is generated as a linear function of the demodulation signal

33. The device according to claim 22, wherein the oscillating element is a cantilever of an atomic force microscope.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0093] The present teaching is now explained in more detail with reference to exemplary embodiments. The drawings are exemplary and while they are designed to illustrate the idea of the present teaching, they are in no way intended to narrow or even conclusively reproduce the idea of the present teaching.

[0094] FIG. 1 illustrates a block diagram of a known method for amplitude-modulated atomic force microscopy (AM-AFM)

[0095] FIG. 2 illustrates a block diagram of the use of two possible embodiments of a method according to the present teaching for AM-AFM

[0096] FIG. 3 illustrates a block diagram of a simplified design for executing a further embodiment of the method according to the present teaching

[0097] FIG. 4 illustrates a block diagram of a further simplified design for executing a further embodiment of the method according to the present teaching

[0098] FIG. 5 illustrates a block diagram of a minimalistic design for executing a further embodiment of the method according to the present teaching

[0099] FIG. 6 illustrates a block diagram of a design for executing a further embodiment of the method according to the present teaching

DETAILED DESCRIPTION

[0100] FIG. 1 shows a block diagram of an amplitude-modulated atomic force microscopy (AM-AFM), wherein a method according to the prior art is used for the demodulation of a modulated oscillation of an oscillating element in the form of a cantilever 1. The cantilever 1 comprises a cantilever tip 13 in order to be able to determine the topography of a sample 10 with high resolution. For this purpose, the sample 10 is placed on a scanner 11, which can move the sample 10 parallel to three mutually normal spatial directions x, y, z, wherein the two double arrows indicate shifts in the drawing plane of FIG. 1, i.e., parallel to the x and z direction.

[0101] The cantilever 1 is fastened to a piezo actuator 14, by means of which the cantilever 1 can be caused to oscillate with a frequency fin the range of a resonance frequency of the cantilever 1, i.e., the cantilever 1 is vibrationally mounted on the piezo actuator 14. For excitation, a known temporally varying excitation signal u.sub.act is used in the form of a periodic AC voltage with an amplitude A.sub.act:

u.sub.act=A.sub.act*sin(2*π*f*t),

wherein t is the time and the excitation signal u.sub.act is generated by means of an AC voltage source 18. This means that the piezo actuator 14 is operated with this excitation signal u.sub.act. The oscillation of the cantilever 1 is modulated by the interaction with the sample 10, in particular by Van der Waals forces between the cantilever 1 or the cantilever tip 13 and the surface of the sample 10, wherein the modulation can relate to the amplitude and the phase. This means that the modulated oscillation of the cantilever 1 has an amplitude A.sub.c and a phase ϕ.sub.c, which are initially unknown.

[0102] In order to detect the modulated oscillation of the cantilever 1, a piezoresistive element 2 is provided in the exemplary embodiment of the prior art, which is integrated in or on the cantilever 1 and whose electrical resistance ΔR changes according to the modulated oscillations of the cantilever 1 (due to the deformation of the cantilever 1 associated with the oscillation). In other words, the electrical resistance ΔR of the piezoresistive element 2 represents a sensor measurement variable SMG, which varies versus time as a function of the amplitude A.sub.c and the phase ϕ.sub.c.

[0103] The piezoresistive element 2 is analyzed by means of a bridge circuit 6, wherein the bridge circuit 6 is supplied with a DC voltage U.sub.DC. The bridge circuit 6 comprises a voltage divider 4 and a further voltage divider 5 (each indicated by a dashed line), wherein the voltage divider 4 comprises a first electrical resistance R1 and a second electrical resistance R2 and wherein the further voltage divider 5 comprises a third electrical resistance R3 and a fourth electrical resistance R4. In the exemplary embodiment shown, the second electrical resistance R2 is formed by the sum of a further electrical resistance R′ and the electrical resistance ΔR of the piezoresistive element 2, i.e.

R2=R′+ΔR.

[0104] Preferably, the first electrical resistance R1, the further electrical resistance R′, the third electrical resistance R3 and the fourth electrical resistance R4 are selected, or in the case of adjustable resistances are set, to equal the value R, i.e.

R1=R′=R3=R4=R.

[0105] This is assumed in the following for the exemplary embodiment shown in FIG. 1.

[0106] The modulated oscillation of the cantilever 1 leads to a temporally varying voltage differential u.sub.d on the bridge, which is output by an operational amplifier 8 in the exemplary embodiment shown and reads as follows:

u.sub.d=U.sub.DC*(R+ΔR)/(2*R+ΔR)−U.sub.DC/2=U.sub.DC*(ΔR)/(4*R+2*ΔR).

[0107] Using the approximation normally valid in practice

ΔR<<R

results in:

u.sub.d=U.sub.DC/(4*R)*ΔR=U.sub.DC/(4*R)*A.sub.c*sin(2*π*f*t+ϕ.sub.c).

[0108] In order to perform a demodulation and obtain A.sub.c or ϕ.sub.c, an elaborate and expensive lock-in amplifier 15 is provided according to the prior art, which is supplied with u.sub.d. The excitation signal u.sub.act is also supplied to the lock-in amplifier 15 as a reference signal, wherein a phase ϕ.sub.ref of the excitation signal u.sub.act is shifted by means of a phase shifter 16 of the lock-in amplifier 15. In particular, an adaptation of ϕ.sub.ref to ϕ.sub.c can be achieved by the phase shifter 16, i.e.

ϕ.sub.ref≈ϕ.sub.c.

[0109] This adjustment can be made in a known and automated manner by exciting the cantilever 1 to oscillate, initially without a sample 10, and by varying the phase ϕ.sub.ref by means of the phase shifter 16 until an output signal u.sub.lia at the output of the lock-in amplifier 15 is maximized.

[0110] The output signal ma of the lock-in amplifier 15 is determined as follows: The phase-shifted reference signal is multiplied by a multiplier 17 of the lock-in amplifier 15 with the voltage differential u.sub.d, whereby this multiplication provides a multiplied signal u.sub.m. The multiplied signal u.sub.m is then filtered by means of an electronic low-pass filter 3 of the lock-in amplifier 15 and is finally amplified by means of an amplifier 7 of the lock amplifier 15, which ultimately results in the output signal u.sub.lia.

[0111] In the exemplary embodiment of FIG. 1, the multiplied signal u.sub.m is basically (based on the above approximation ΔR<<R, which is always assumed here and hereinafter)

u.sub.m=U.sub.DC/(4*R)*A.sub.act*A.sub.c*sin(2*π*f*t+ϕ.sub.c)*sin(2*π*f*t+ϕ.sub.ref)

and for the stated case of the adjusted phase of the reference signal

u.sub.m=U.sub.DC/(4*R)*A.sub.act*A.sub.c*sin.sup.2(2*π*f*t+ϕ.sub.c)

or by converting using the trigonometric identities

u.sub.m=U.sub.DC/(8*R)*A.sub.act*A.sub.c*(1−cos(4*π*f*t+2*ϕ.sub.c)).

[0112] With the exception of the known factors U.sub.DC, R and A.sub.act, A.sub.c thus results in a constant component of u.sub.m. The high-frequency portion of u.sub.m containing ϕ.sub.c is cut off by using the electronic low-pass filter 3. Since the amplification of the amplifier 7 is known, it is therefore possible to in principle directly infer A.sub.c from the output signal ma, or the output signal ma represents the amplitude A.sub.c of the modulated oscillation of the cantilever 1 except for the aforementioned known factors.

[0113] For the AM-AFM, the numerical value of A.sub.c is not typically evaluated per se, but ma is compared to a target value SP by means of an operational amplifier 8′, wherein the target value SP is determined or defined in a known and preferably automated manner. For this purpose, the amplitude A.sub.c or ma is monitored as the cantilever 2 approaches the sample 10. If the value of the amplitude A.sub.c and/or u.sub.lia decreases by a given percentage within a given time, the then available value of u.sub.lia is used as the target value SP.

[0114] The operational amplifier 8′ thus generates a control signal CS that is fed to a scanner control 9, which adjusts or regulates the distance (measured in the z-direction in FIG. 1) between the sample 10 and the cantilever 1 or the cantilever tip 13 by means of the control signal CS. A corresponding output signal OS of the scanner control unit 9, on the basis of which the scanner 11 performs the movement of the sample 10—basically parallel to all three spatial directions x, y, z—, therefore corresponds to the topography to be determined of the sample 10.

[0115] In a block diagram analogous to FIG. 1, FIG. 2 shows the application of two embodiments of a method according to the present teaching for amplitude-modulated atomic force microscopy, whereby the latter can be operated much more easily and cost-effectively, in particular without a complex and expensive lock-in amplifier 15. The cantilever 1 including the cantilever tip 13 is again excited by means of the piezo actuator 14 to oscillate with the frequency fin the range of a resonance frequency of the cantilever 1. For this purpose, the excitation signal u.sub.act is generated by means of the voltage source 18, as described above for the known exemplary embodiment of FIG. 1.

[0116] The piezoresistive element 2 in turn comprises the temporally varying electrical resistance ΔR as a sensor measurement variable SMG, which together with the further electrical resistance R′ forms the second electrical resistance R2 of the voltage divider 4 of the bridge circuit 6, also as described above.

[0117] However, the bridge circuit 6 is not operated with the DC voltage U.sub.DC, but with a known temporally varying demodulation signal DMS in the form of an AC voltage. FIG. 2 shows two variants (marked with (a) and (b) in FIG. 2) for generating the demodulation signal DMS, between which can be toggled by means of a switch 19.

[0118] In variant (a), the excitation signal u.sub.act of the AC voltage source 18 is used as the demodulation signal DMS, so that in this case the demodulation signal DMS is a linear function of the excitation signal u.sub.act. A phase Δϕ of the demodulation signal DMS can be adjusted by means of a phase shifter 16 upstream of the switch 19.

[0119] In variant (b), a separate AC voltage source 18′ is provided for generating the demodulation signal DMS. In particular, the phase of the Δϕ demodulation signal DMS can be set with the AC voltage source 18′. Furthermore, it is of course also possible to generate the demodulation signals DMS with the AC voltage source 18′ in such a way that it corresponds to the excitation signal u.sub.act, excluding the phase Δϕ.

[0120] Due to the temporal changes of ΔR or R2, the demodulation signal DMS is amplitude-modulated in the voltage divider 4, generating a first comparison signal VS1. A second comparison signal VS2 is generated in the exemplary embodiments of FIG. 2 in the further voltage divider 5 by means of the demodulation signal DMS, wherein the second comparison signal VS2 is proportional to the demodulation signal DMS and thus is a linear function of the demodulation signal DMS—and in the case of the variant (a) also of the excitation signal u.sub.act.

[0121] The two comparison signals VS1, VS2 are then subtracted from each other using the bridge circuit 6 in order to generate a difference signal DIFFS, wherein the difference signal DIFFS is to a certain extent automatically generated in the bridge branch of the bridge circuit 6 and is present in the illustrated design examples as an output signal of the operational amplifier 8, which is arranged or switched in the bridge branch of the bridge circuit 6.

[0122] The phase Δϕ of the demodulation signal DMS can in particular be set such that

Δϕ≈ϕ.sub.c

applies, wherein this occurs analogously to what is described above in connection with FIG. 1. For this purpose, it is sufficient to vary the phase Δϕ of the demodulation signal DMS without sample 10, preferably automatically, until the constant component of the difference signal DIFFS is maximized.

[0123] In this case,

R1=R′=R3=R4=R

results in the following difference signal DIFFS completely analogous to the above statements regarding the multiplied signal u.sub.m:

DIFFS=1/(4*R)*A.sub.act*A.sub.c*sin.sup.2(2*π*f*t+ϕ.sub.c)=1/(8*R)*A.sub.act*A.sub.c*(1−cos(4*π*f*t+2*ϕ.sub.c)).

[0124] In other words, the amplitude A.sub.c of the modulated oscillation of the cantilever 1 results directly from the constant component of the difference signal DIFFS, since R and A.sub.act are known. In the exemplary embodiments shown in FIG. 2, the temporally varying portion of DIFFS is cut off by using the electronic low-pass filter 3 and the filtered difference signal DIFFS_f is obtained, which can be amplified with the amplifier 7 for reasons related to more practical handling.

[0125] It should be noted that an electronic low-pass filter 3 is not necessary by many practical applications since downstream elements often have a low-pass characteristic, so that the same effect is essentially achieved as with the electronic low-pass filter 3, i.e., the filtered difference signal DIFFS_f is generated.

[0126] In the exemplary embodiments of FIG. 2, the filtered difference signal DIFFS_f is compared after its amplification with the target value SP by means of the operational amplifier 8′, wherein the target value SP is determined or defined in a per se known and preferably automated manner, as already explained in detail in FIG. 1. The operational amplifier 8′ thus generates a control signal CS that is fed to a scanner control 9, which adjusts or regulates the distance (measured in the z-direction in FIG. 1) between the sample 10 and the cantilever 1 or the cantilever tip 13 by means of the control signal CS. The corresponding output signal OS of the scanner control unit 9, on the basis of which the scanner 11 performs the movement of the sample 10—basically parallel to all three spatial directions x, y, z—therefore again corresponds to the topography to be determined of the sample 10.

[0127] In the exemplary embodiments of FIG. 2, a control unit 12, which is formed to execute the method according to the present teaching, is integrated into the scanner control 9. The control unit 12 is connected to the AC voltage source 18 (indicated by the dotted line in FIG. 2) in order to control the latter, wherein the control unit 12 is of course also indirectly connected to the piezo actuator 14. The control unit 12 is also indirectly connected to the piezoresistive element 2 by the bridge circuit 6 and the downstream elements (operational amplifier 8, electronic low-pass filter 3, amplifier 7, operational amplifier 8′).

[0128] Furthermore, in the variant (a), the electronic control unit 12 is preferably connected to the phase shifter (16) (indicated by the dashed-dotted line in FIG. 2) in order to be able to adjust the phase Δϕ of the demodulation signal DMS.

[0129] Furthermore, in the variant (b), the electronic control unit 12 is preferably connected to the AC voltage source 18′ (indicated by the dashed-dotted line with two dots between the dashes in FIG. 2) in order to be able to adjust the demodulation signal DMS and in particular the phase Δϕ of the demodulation signal DMS.

[0130] It is also conceivable that the operational amplifier 8 and/or the electronic low-pass filter 3 and/or the amplifier 7 and/or the operational amplifier 8′ and/or parts of the bridge circuit 6—e.g. R1, R′, R3 and R4—can be integrated into the control unit 12.

[0131] It should be emphasized that the method according to the present teaching can for example also be used for other dynamic operating modes of an atomic force microscope. In the exemplary embodiments of FIG. 2 described above, the following applies for general phases Δϕ and ϕ.sub.c using the trigonometric identities:

DIFFS=1/(4*R)*A.sub.act*A.sub.c*sin(2*π*f*t+ϕ.sub.c)*sin(2*π*f*t+=1/(8*R)*A.sub.act*A.sub.c*(cos(ϕ.sub.c−Δϕ)−cos(4*t*f*t+ϕ.sub.c+Δϕ).

[0132] By now setting the phase Δϕ of the demodulation signal DMS such that

Δϕ≈ϕ.sub.c±π/2

applies, the first cos-Term, which is constant and thus determines the constant component of DIFFS, disappears. Such an adjustment is in turn possible, preferably automated, by varying Δϕ until the constant component of DIFFS disappears.

[0133] However, this means that the phase Δϕ of the demodulation signal DMS can be used to determine the phase ϕ.sub.c of the modulated oscillation of the cantilever 1, since the described setting ϕ.sub.c is known relative to π±/2.

[0134] Accordingly, phase-modulated atomic force microscopy (PM-AFM) or frequency-modulated atomic force microscopy (FM-AFM) can also be operated using the method according to the present teaching, whose modes require the measurement of ϕ.sub.c.

[0135] FIG. 3 shows an example of a design for executing a simplified embodiment of the method according to the present teaching, which is used according to FIG. 3 for demodulating the oscillations of an oscillating element embodied as a cantilever 1. For reasons of clarity, no excitation means are drawn in and no downstream elements for processing the difference signal DIFFS are shown apart from the electronic low-pass filter 3.

[0136] The embodiment of FIG. 3 does not use a bridge circuit 6, but only the voltage divider 4 for generating the first comparison signal VS1. The voltage divider 4 is operated with the (known) demodulation signal DMS, which is designed as an AC voltage by means of the AC voltage source 18′. The phase of the demodulation signal DMS is also determined or the phase Δϕ of the demodulation signal DMS can be set accordingly by means Δϕ of the AC voltage source 18′. Since the second electrical resistance R2 of the voltage divider 4 is composed of the additional electrical resistance R′ and the electrical resistance ΔR of the piezoresistive element 2, which is integrated in or on the cantilever 1, the amplitude of the demodulation signal DMS is modulated by the sensor measurement variable SMG temporally varying due to the oscillation of the cantilever 1 or by the temporally varying electrical resistance ΔR. The first comparison signal VS1 is tapped as a corresponding partial voltage at the voltage divider 4 and compared by means of the operational amplifier 8 to the (known) second comparison signal VS2.

[0137] In the exemplary embodiment of FIG. 3, the second comparison signal VS2 is generated by means of an AC voltage source 18″, thus ensuring that VS2 is known. The second comparison signal VS2 can be generated as a linear function of the demodulation signal DMS, in particular in the same way as the demodulation signal DMS.

[0138] In the exemplary embodiment of FIG. 3, the control unit 12 is connected to the AC voltage sources 18′, 18″ in order to ensure a suitable generation of the known demodulation signal DMS and the known comparison signal VS2.

[0139] In particular, the phase of the demodulation signal DMS can then be adjusted by means Δϕ of the control unit 12 such that

Δϕ≈ϕ.sub.c

applies. As already described above in FIG. 2, this concrete, preferably automated, method for setting the phase Δϕ of the demodulation signal DMS involves varying Δϕ without sample 10 until the constant component of DIFFS or, if applicable, of DIFFS_f is maximized.

[0140] The amplitude A.sub.c of the modulated oscillation of the cantilever 1 again is proportional to the constant component of the difference signal DIFFS. The high frequency component of the difference signal DIFFS contains the phase ϕ.sub.c of the modulated oscillation of the cantilever 1. By low-pass filtering the difference signal DIFFS, its constant component remains, which does not contain the phase ϕ.sub.c of the modulated oscillation of the cantilever 1. This means that the filtered difference signal DIFFS_f can be used for controlling the amplitude-modulated atomic force microscopy, analogous to the case of the embodiments described above according to FIG. 2.

[0141] Likewise analogous to the exemplary embodiments of FIG. 2, the phase of the demodulation signal DMS can be adjusted, preferably automated, by means Δϕ of the control unit 12 such that

Δϕ≈ϕ.sub.c±π/2

applies. This is done by varying Δϕ until the constant component of DIFFS disappears, whereby ϕ.sub.c relative to π±/2 is also determined directly. In other words, the embodiment of the method according to the present teaching according to the simplified design of FIG. 3 is also suitable, among other things, for use in phase-modulated atomic force microscopy or frequency-modulated atomic force microscopy.

[0142] The dashed-dotted line in FIG. 3 indicates that the control unit 12 can be connected to one or more elements downstream of the electronic low-pass filter 3, so that an indirect connection between the control unit 12 and the piezoresistive element 2 is given.

[0143] FIG. 4 shows a block diagram of a further simplified design for executing a further embodiment of the method according to the present teaching. This structure differs from that of FIG. 3 only in that no separate AC voltage source 18″ is used to generate the second comparison signal VS2, but rather the demodulation signal DMS generated by means of the AC voltage source 18′. Specifically, the demodulation signal DMS is tapped at the input of the voltage divider 4 and amplified by means of the amplifier 7 in order to generate the second comparison signal VS2. The control unit 12 is accordingly connected to the AC voltage source 18′ in order to appropriately control the latter and to be able to adjust the demodulation signal DMS, in particular its Δϕ phase, as desired.

[0144] What is stated above regarding FIG. 3 otherwise also applies to FIG. 4. This means that the two comparison signals VS1, VS2 are subtracted from one another by means of the operational amplifier 8 in order to generate the difference signal DIFFS.

[0145] If the phase Δϕ of the demodulation signal DMS is set, preferably automated, such that

ϕϕ≈ϕ.sub.c

applies, the amplitude A.sub.c of the modulated oscillation of the cantilever 1 is proportional to the constant component of the difference signal DIFFS. The high frequency component of the difference signal DIFFS contains the phase ϕ.sub.c of the modulated oscillation of the cantilever 1. By low-pass filtering the difference signal

[0146] DIFFS, its constant component remains, which does not contain the phase ϕ.sub.c of the modulated oscillation of the cantilever 1. This means that the filtered difference signal DIFFS_f can be used for controlling the amplitude-modulated atomic force microscopy, analogous to the case of the embodiments described above according to FIG. 2.

[0147] If the phase Δϕ of the demodulation signal DMS is set, preferably automated, such that

Δϕ≈ϕ.sub.c±π/2

applies, the constant component of DIFFS will disappear and is thus ϕ determined up to π±/2. In other words, the embodiment of the method according to the present teaching according to the simplified design of FIG. 4 is also suitable, among other things, for use in phase-modulated atomic force microscopy or frequency-modulated atomic force microscopy.

[0148] The dashed-dotted line in FIG. 4 indicates that the control unit 12 can be connected to one or more elements downstream of the electronic low-pass filter 3, so that an indirect connection between the control unit 12 and the piezoresistive element 2 is given.

[0149] Regarding the exemplary embodiments of FIG. 3 and FIG. 4, it should be noted that a separate electronic low-pass filter 3 does not necessarily have to be provided for low-pass filtering, since other elements downstream of the difference signal DIFFS can have a low-pass characteristic. In such cases, a low-pass filtering of the difference signal DIFFS occurs to a certain extent automatically. Examples of such a downstream element would be, for example, signal adjustment elements, the scanner controller 9 (also called feedback controller), the scanner 11, anti-aliasing filter, or a power amplifier.

[0150] FIG. 5 shows an example of a minimal design for executing a simplified or “minimal” embodiment of the method according to the present teaching, which is used according to FIG. 5 for demodulating the oscillations of an oscillating element embodied as a cantilever 1.

[0151] The method, which is realized by way of example with the design according to FIG. 5, is essentially limited to the core of the present teaching, according to which the first comparison signal VS1, in particular the constant component of the first comparison signal VS1, is already proportional to the amplitude of the modulated oscillation A.sub.c or directly dependent on the phase ϕ.sub.c of the modulated oscillation. In other words, the first comparison signal VS1 basically already provides the desired information about the amplitude A.sub.c or phase ϕ.sub.c of the modulated oscillation, wherein the information about the ratio (instead of the exact numerical value) of the temporally varying amplitude A.sub.c of the modulated oscillation is sufficient for many applications.

[0152] For reasons of clarity, no excitation means are shown in FIG. 5 either and no downstream elements are shown apart from the electronic low-pass filter 3 (in this case for processing the first comparison signal VS1).

[0153] The embodiment of FIG. 5 only uses the voltage divider 4 for generating the first comparison signal VS1. The voltage divider 4 is operated with the (known) demodulation signal DMS, which is designed as an AC voltage by means of the AC voltage source 18′. The phase of the demodulation signal DMS is also determined or the phase Δϕ of the demodulation signal DMS can be set accordingly by means Δϕ of the AC voltage source 18′. Since the second electrical resistance R2 of the voltage divider 4 is composed of the additional electrical resistance R′ and the electrical resistance ΔR of the piezoresistive element 2, which is integrated in or on the cantilever 1, the amplitude of the demodulation signal DMS is modulated by the sensor measurement variable SMG temporally varying due to the oscillation of the cantilever 1 or by the temporally varying electrical resistance ΔR. The first comparison signal VS1 is tapped at the voltage divider 4 as a corresponding partial voltage.

[0154] The first comparison signal VS1 has a constant component and a temporally varying component, the latter having a relatively high amplitude (half amplitude of the demodulation signal DMS for R1=R′=R). Since there is no linear combination with a second comparison signal VS2, this amplitude of the temporally varying component is also not further reduced.

[0155] In order to in this case remove the temporally varying portion with the low-pass filter 3 and thus generate a filtered first comparison signal VS1_f, the low-pass filter 3 must have a relatively low limit frequency, which in turn restricts the speed of the method or the applications of the method. In other words, the illustrated exemplary embodiment of FIG. 5 results in a limitation of the achievable scan speed when the method according to the present teaching is used for atomic force microscopy. This does not necessarily have to be disruptive, in particular depending on the sample 10 to be examined.

[0156] In the exemplary embodiment of FIG. 5, the control unit 12 is connected to the AC voltage source 18′ in order to ensure a suitable generation of the known demodulation signal DMS.

[0157] In particular, the phase of the demodulation signal DMS can then be adjusted by means Δϕ of the control unit 12 such that

Δϕ≈ϕ.sub.c

applies. Analogous to the description above in FIG. 2, this concrete, preferably automated, method for setting the phase Δϕ of the demodulation signal DMS involves varying Δϕ without sample 10 until the constant component of VS1 or, if applicable, of VS1_f is maximized.

[0158] The amplitude A.sub.c of the modulated oscillation of the cantilever 1 is proportional to the constant component of the comparison signal VS1. The temporally varying component of the first comparison signal VS1 contains the phase ϕ.sub.c of the modulated oscillation of the cantilever 1. As already mentioned above, by low-pass filtering the first comparison signal VS1, its constant component remains, which does not contain the phase ϕ.sub.c of the modulated oscillation of the cantilever 1. In other words, the filtered first comparison signal VS1_f can in principle be used for controlling amplitude-modulated atomic force microscopy, analogous to the case of the embodiments described above according to FIG. 2.

[0159] Likewise analogous to the exemplary embodiments of FIG. 2, the phase Δϕ of the demodulation signal DMS can be adjusted, preferably automated, by means of the control unit 12 such that

Δϕ≈ϕ.sub.c±π/2

applies. This is done by varying Δϕ until the constant component of VS1 disappears, whereby ϕ.sub.c relative to π±/2 is also determined directly. In other words, the embodiment of the method according to the present teaching according to the minimal design of FIG. 5 is also suitable, among other things, for use in phase-modulated atomic force microscopy or frequency-modulated atomic force microscopy.

[0160] The dashed-dotted line in FIG. 5 indicates that the control unit 12 can be connected to one or more elements downstream of the electronic low-pass filter 3, so that an indirect connection between the control unit 12 and the piezoresistive element 2 is given.

[0161] Finally, for the exemplary embodiment of FIG. 5, it should be noted that a separate electronic low-pass filter 3 does not necessarily have to be provided for low-pass filtering, since other elements downstream of the first comparison signal VS1 can have a suitable low-pass characteristic. In such cases, a low pass filtering of the first comparison signal VS1 occurs to a certain extent automatically. Examples of such a downstream element would be, for example, signal adjustment elements, the scanner controller 9 (also called feedback controller), the scanner 11, anti-aliasing filter, or a power amplifier.

[0162] For reasons of clarity, FIG. 6 shows a simplified diagram (in particular without cantilever tip 13, piezo actuator 14 and the voltage source for generating the excitation signal u.sub.act) of an embodiment variant that allows the simultaneous determination of the amplitude A.sub.c and the phase ϕ.sub.c of the modulated oscillation. In this case, the piezoresistive element 2 and a second piezoresistive element 21 are provided on the oscillating element or cantilever 1 for the purpose of detecting its oscillation, wherein the two piezoresistive elements 2, 21 are substantially the same.

[0163] As in the exemplary embodiments of FIG. 2, the electrical resistance ΔR of the piezoresistive element 2 is part of the second electrical resistance R2 of the bridge circuit 6, which is operated with the demodulation signal DMS in the form of an AC voltage. The demodulation signal DMS is generated with an AC voltage source 18′, whereby otherwise the above-mentioned applies analogously to FIG. 2. Accordingly, the filtered difference signal DIFFS_f is generated by means of the piezoresistive element 2, the bridge circuit 6, the operational amplifier 8 and the low pass filter 3.

[0164] Like the piezoresistive element 2, the second piezoresistive element 21 comprises the electrical resistance ΔR that varies versus time according to the modulated oscillation as the sensor measurement variable SMG and is connected analogously to the piezoresistive element 2. In other words, the electrical resistance ΔR of the second piezoresistive element 21 together with a further electrical resistance R′ forms a second electrical resistance R2 of a voltage divider 4 of a bridge circuit 6. This voltage divider 4 further comprises a first electrical resistance R1. The bridge circuit 6 comprises a further voltage divider 5, which is formed from a third electrical resistance R3 and a fourth electrical resistance R4 (the voltage dividers 4, 5 are indicated in FIG. 6 by dashed lines). This bridge circuit 6 is operated with a phase-shifted demodulation signal DMS_Q. In the exemplary embodiment shown, the phase-shifted demodulation signal DMS_Q is identical to the demodulation signal DMS except for a phase shift of 90° and is generated by means of the AC voltage source 18″.

[0165] Correspondingly, in the bridge circuit 6 operated with the phase-shifted demodulation signal DMS_Q the phase-shifted demodulation signal DMS_Q is amplitude modulated due to the temporal variation of ΔR of the second piezoresistive element 21, as a result of which a first phase-shifted comparison signal VS1_Q is generated, which is generated by means of the voltage divider 4 and is tapped between the resistors R1 R2. Similarly, a second phase-shifted comparison signal VS2_Q is generated by means of the phase-shifted demodulation signal DMS_Q, which is generated by means of the further voltage divider 5 and is tapped between the resistors R3 and R4. This means that the second phase-shifted comparison signal VS2_Q is a linear function of, and proportional to, the phase-shifted demodulation signal DMS_Q.

[0166] The two phase-shifted comparison signals VS1_Q, VS2_Q are then subtracted from each other using the bridge circuit 6 in order to generate a phase-shifted difference signal DIFFS_Q, wherein the phase-shifted difference signal DIFFS_Q is to a certain extent automatically generated in the bridge branch of the bridge circuit 6 and is present in the illustrated design examples as an output signal of the operational amplifier 8, which is arranged or switched in the bridge branch of the bridge circuit 6.

[0167] An electronic low-pass filter 3, which accordingly outputs a filtered phase-shifted difference signal DIFFS_Q_f, is connected downstream of this operational amplifier 8.

[0168] The filtered phase-shifted difference signal DIFFS_Q_f and the filtered difference signal DIFFS_f are fed to a signal processing unit 20, which is integrated into the control unit 12 in the illustrated exemplary embodiment, as is the scanner control 9. The signal processing unit 20 calculates an output amplitude A.sub.out and/or an output phase ϕ.sub.out based on the signals DIFFS_Q_f and DIFFS_f.

[0169] The control unit 12 is also indirectly connected to the piezoresistive element 2 by the bridge circuit 6 and the respective downstream elements (operational amplifier 8 and electronic low-pass filter 3). The dashed dotted line in FIG. 6 indicates the connection of the control unit 12 to the AC voltage source 18′, so that the control unit 12 can control the AC voltage source 18′ in order to adjust the demodulation signal DMS. The dotted line in FIG. 6 indicates the connection of the control unit 12 to the AC voltage source 18″, so that the control unit 12 can control the AC voltage source 18″ in order to adjust the demodulation signal DMS_Q.

[0170] In order to illustrate the ability to simultaneously determine A.sub.c and ϕ.sub.c, the following is a concrete example for executing the method according to the present teaching by means of a design according to FIG. 6. In this case as well, the cantilever 1 is operated with the frequency f by means of a sinusoidal AC voltage (using a piezo actuator not shown in FIG. 6). The oscillation of the cantilever 1 thus results in the following electrical resistance ΔR in both piezoresistive elements 2, 21:

ΔR=A.sub.c*sin(2*π*f*t+ϕ.sub.c).

[0171] The following demodulation signal DMS is generated with the AC voltage source 18′:

DMS=A.sub.DMs*sin(2*π*f*t),

wherein A.sub.DMS is the amplitude of this AC voltage.

[0172] The following phase-shifted demodulation signal DMS_Q is generated with the AC voltage source 18″, which has a phase shift of 90° compared to the demodulation signal DMS:

DMS_Q=A.sub.DMS*cos(2*π*f*t).

[0173] This means that the phase-shifted demodulation signal DMS_Q and the demodulation signal DMS have the same amplitude A.sub.DMS.

[0174] Analogous to the explanations above,

R1=R′=R3=R4=R

results in the following difference signal DIFFS:

DIFFS≈DMS*ΔR/(4*R)=1/(4*R)*A.sub.DMS*A.sub.c*sin(2*t*f*t)*sin(2*t*f*t+ϕ.sub.c)=1/(8*R)*A.sub.DMS*A.sub.c*(cos(ϕ.sub.c)−cos(4*t*f*t+ϕ.sub.c)).

[0175] For the phase-shifted difference signal DIFFS_Q results analogously:

DIFFS_Q≈DMS_Q*ΔR/(4*R)=1/(4*R)*A.sub.DMS*A.sub.c*cos(2*π*f*t)*sin(2*π*f*t+ϕ.sub.c)=1/(8*R)*A.sub.DMS*A.sub.c*(sin(ϕ.sub.c)+sin(4*π*f*t+ϕ.sub.c)).

[0176] The high-frequency components are removed by the low-pass filters 3, so that the filtered difference signal DIFFS_f and the filtered phase-shifted difference signal DIFFS_Q_f are obtained in the form of the following DC voltages:

DIFFS_f=1/(8*R)*A.sub.DMS*A.sub.c*cos(ϕ.sub.c)

and

DIFFS_Q_f=1/(8*R)*A.sub.DMS*A.sub.c*sin(ϕ.sub.c).

[0177] These signals DIFFS_f and DIFFS_Q_f can be processed in the signal processing unit 20 by means of simple mathematical operations in order to determine the output amplitude A.sub.out and the output phase ϕ.sub.out. For example, the following output amplitude A.sub.out can be calculated:

A.sub.out=((DIFFS_f).sup.2+(DIFFS_Q_f).sup.2).sup.1/2=((1/(8*R)*A.sub.DMS*A.sub.c).sup.2*(cos.sup.2(ϕ.sub.c)+sin.sup.2(ϕ.sub.c))).sup.1/2=1/(8*R)*A.sub.DMS*A.sub.c.

[0178] Since R and A.sub.DMS are known, the exact numerical value of the amplitude A.sub.c of the modulated oscillation can be determined immediately from the output amplitude A.sub.out. If the exact numerical value of the amplitude A.sub.c of the modulated oscillation is not important and a value proportional to A.sub.c is sufficient, the output amplitude A.sub.out can continue to be used. For example, the output amplitude A.sub.out can be used for feeding into the scanner control 9, possibly amplified and/or after prior comparison with a target value SP, analogously as stated above in the description of the exemplary embodiments of FIG. 2.

[0179] Furthermore, the following output phase ϕ.sub.out can be calculated, wherein atan refers to the arc tangent and tan refers to the tangent:

ϕ.sub.out=atan(DIFFS_Q_f/DIFFS_f)=atan(sin(ϕ.sub.c)/cos(ϕ.sub.c))=atan(tan(ϕ.sub.c))=ϕ.sub.c.

[0180] This means that this calculation of the output phase ϕ.sub.out directly results in the exact numerical value of the phase ϕ.sub.c of the modulated oscillation.

[0181] Thus, the amplitude A.sub.c of the modulated oscillation and the phase ϕ.sub.c of the modulated oscillation can be determined at the same time without an adjustment of the phase of the demodulation signal DMS being necessary, as would be necessary in the exemplary embodiments of FIG. 2.