The cantilever oscillation amplitude

*A*(

*t*), while scanning a step of height Δ

*z*, is expressed as [

1]

$A\left(t\right)={A}_{\mathsf{\text{sp}}}+\Delta z\cdot \left(1-\mathsf{\text{exp}}\left(-{\omega}_{\mathsf{\text{0}}}t/2Q\right)\right),$

(1)

where *ω*_{0} is the cantilever resonant frequency, *Q* is the cantilever quality factor, and *A*_{sp} is the set point amplitude. Thus, the cantilever transfer function *C*(*s*) takes the form $C\left(s\right)=\frac{1}{\left(1+s{\tau}_{c}\right)}$, where *τ*_{
c
} is the time constant of the cantilever and is equal to ${\tau}_{c}=\frac{2Q}{{\omega}_{\mathsf{\text{0}}}}$. The frequency response of the actuator *G*(*s*) and the cantilever deflection signal detector *K*(*s*) has a constant gain equal to DC gain and don't add extra phase lag (it can be assumed that *G*(*s*)·*K*(*s*)· =*G*_{0}·*K*_{0} ≈ 1) in the bandwidth of interest. Indeed, the pole frequency of the detector transfer function [*ω*_{det}] should be at least ten times less than the cantilever resonant frequency ${\omega}_{\mathsf{\text{det}}}={\omega}_{\mathsf{\text{0}}}\phantom{\rule{0.3em}{0ex}}/\phantom{\rule{0.3em}{0ex}}10$. The pole frequency of the transfer function *C*(*s*) is equal to ${{\tau}_{c}}^{-1}={\omega}_{\mathsf{\text{0}}}\phantom{\rule{0.3em}{0ex}}/\phantom{\rule{0.3em}{0ex}}2Q\ll {\omega}_{\mathsf{\text{det}}}$ (if $Q~100$).

Suppose the feedback controller is an integral controller with time constant

*τ*_{
i
} whose transfer function

*R*(

*s*) is

$R\left(s\right)=-\frac{1}{s{\tau}_{i}}$. Then, the frequency-dependent open-loop gain becomes

$\left(-\frac{1}{s{\tau}_{i}}\cdot {G}_{\mathsf{\text{0}}}\cdot {K}_{\mathsf{\text{0}}}\cdot \frac{1}{\left(1+s{\tau}_{c}\right)}\right)$. Thus, the characteristic polynomial of the loop control's frequency response

*D*(

*s*) can be written as

$D\left(s\right)={s}^{2}+\frac{1}{{\tau}_{c}}s+\frac{{G}_{\mathsf{\text{0}}}{K}_{\mathsf{\text{0}}}}{{\tau}_{c}{\tau}_{i}}\approx \left(s+\frac{1}{{\tau}_{c}}\right)\left(s+\frac{{G}_{\mathsf{\text{0}}}{K}_{\mathsf{\text{0}}}}{{\tau}_{i}}\right).$

(2)

For stability of the loop control, we need to have significantly different frequencies for the real poles of the transfer function:

$\frac{{G}_{\mathsf{\text{0}}}{K}_{\mathsf{\text{0}}}}{{\tau}_{i}}\ll \frac{1}{{\tau}_{c}}.$

(3)

In the case of such characteristic polynomials, the transient response is described by two exponential function, the fast function having time constant *τ*_{
c
} and the slow function, $\frac{{\tau}_{i}}{{G}_{\mathsf{\text{0}}}{K}_{\mathsf{\text{0}}}}$. As a result, the speed of a closed-loop control system (that is, without loss of surface) is determined by the time constant $\frac{{\tau}_{i}}{{G}_{\mathsf{\text{0}}}{K}_{\mathsf{\text{0}}}}$. Feedback speed, the speed of the actuator, is limited in tapping mode by the stability condition of the loop control (Equation 3). Thus, the feedback speed is limited by the cantilever time constant *τ*_{
c
}.

Increasing scan speed leads to a loss of surface when a 'downward step' is scanned or a parachuting effect. If an 'upward step' is scanned, it leads to instability of the loop control [1, 2].

Let us find the maximum scan speed without loss of surface. The transient response of the loop control to a capacitive displacement sensor output (if the high-frequency pole (frequency

${{\tau}_{c}}^{-1}$, Equation

2) is ignored) can be written as

$\frac{\Delta Y\left(s\right)}{\Delta Z\left(s\right)}=\frac{1}{\left(1+s{\tau}_{i}\phantom{\rule{0.3em}{0ex}}/\phantom{\rule{0.3em}{0ex}}{G}_{\mathsf{\text{0}}}{K}_{\mathsf{\text{0}}}\right)}.$

(4)

Then, the transient response of the loop control for a downward step of height Δ

*z* takes the form

$\Delta y\left(t\right)=\Delta z\cdot \left(1-{e}^{-{G}^{\mathsf{\text{0}}}{K}^{\mathsf{\text{0}}}t\phantom{\rule{0.3em}{0ex}}/\phantom{\rule{0.3em}{0ex}}{\tau}_{i}}\right).$

(5)

In the latter case the initial vertical actuator speed is

${\upsilon}_{v}=\frac{\Delta y\left(\mathsf{\text{0}}\right)}{\Delta t}=\frac{\Delta z\cdot {G}_{\mathsf{\text{0}}}{K}_{\mathsf{\text{0}}}}{{\tau}_{i}}.$

(6)

Assuming that there is no loss of surface by the probe, the horizontal scan speed

*υ*_{
H
} is related to the vertical actuator speed

*υ*_{
v
} by

${\upsilon}_{H}={\upsilon}_{v}\cdot tg\left(a\phantom{\rule{0.3em}{0ex}}/\phantom{\rule{0.3em}{0ex}}2\right)=\frac{\Delta z\cdot {G}_{\mathsf{\text{0}}}{K}_{\mathsf{\text{0}}}\cdot tg\left(a\phantom{\rule{0.3em}{0ex}}/\phantom{\rule{0.3em}{0ex}}2\right)}{{\tau}_{i}},$

(7)

where *a* is the apex angle of the diamond tip.

From Equation

3, it follows

$\frac{{\tau}_{i}}{{G}_{\mathsf{\text{0}}}{K}_{\mathsf{\text{0}}}}\approx 10{\tau}_{c}=\frac{20\cdot Q}{{\omega}_{\mathsf{\text{0}}}}$ yielding

${\upsilon}_{H}=\frac{\Delta z\cdot {\omega}_{\mathsf{\text{0}}}\cdot tg\left(a\phantom{\rule{0.3em}{0ex}}/\phantom{\rule{0.3em}{0ex}}2\right)}{20\cdot Q}$

(8)

An increase in the actuator speed is caused by an increase in the error signal

*e*(

*t*) =

*A*(

*t*) -

*A*_{sp}. For a step of height Δ

*z* <

*A*_{fr}-

*A*_{sp}, where

*A*_{fr} is the free-air amplitude (the amplitude of the cantilever oscillation without touching the surface), the error signal is

*e*(0) = Δ

*z*. That's why the velocity

*υ*_{H} depends on the step height Δ

*z*. For Δ

*z* = (

*A*_{fr}-

*A*_{sp}), the scan speed becomes

${\left({\upsilon}_{H}\right)}_{\mathsf{\text{lim}}}=\frac{\left({A}_{\mathsf{\text{fr}}}-{A}_{\mathsf{\text{sp}}}\right)\cdot {\omega}_{\mathsf{\text{0}}}\cdot tg\left(a\phantom{\rule{0.3em}{0ex}}/\phantom{\rule{0.3em}{0ex}}2\right)}{20\cdot Q}.$

(9)

For higher steps, the initial probe speed doesn't increase as the error signal is saturated at *e*_{max} = *A*_{fr}-*A*_{sp}. For scan speed *υ*_{
H
} > (*υ*_{
H
})_{lim}, the tip doesn't touch the surface and loses sample surface.

For example, let us find the scan speed limit for the SPM NanoScan-3D [8] where the probe is a piezoceramic cantilever with a diamond tip. This device allows you to scan the surface topography and to produce indentation and sclerometry simultaneously. If the set point amplitude is *A*_{sp} = 0.8·*A*_{fr} (where the cantilever free-air amplitude is *A*_{fr} = 100 nm), the cantilever resonance frequency is *f*_{0} = 11.5 kHz, the quality factor is 100, and the apex angle of a diamond tip is 120° [8], then the scan speed limit is approximately (*υ*_{
H
})_{lim} ≈ 12.5 μm/s.

The loop control is a high-pass filter for the error signal which is related to the height step Δ*z* by $e\left(t\right)=\Delta z\cdot {K}_{\mathsf{\text{0}}}\cdot {e}^{-\frac{t{G}^{\mathsf{\text{0}}}{K}_{\mathsf{\text{0}}}}{{\tau}_{i}}}$. In the case of parachuting, the loop control is opened by the loss of sample surface by the probe. The error signal is saturated at *e*_{max} = (*A*_{fr}-*A*_{sp}) ≈ 0.2 *A*_{fr}. To avoid, or at least reduce, the parachuting region, the dynamic controller should increase the error signal *e*_{max} [2] or reduce the integral controller time constant *τ*_{
i
}.

According to the algorithm implemented on FPGA, if the error signal is more than a threshold

*e*_{th}, the integrator time constant is reduced according to

${\tau}_{i}\left(t\right)={\tau}_{i}-g\cdot \left(e\left(t\right)-{e}_{\mathsf{\text{th}}}\right),$

(10)

where *g* is the 'gain' of the dynamic controller.

As the tip scans over an upward step, the probe oscillation amplitude is reduced. It can be reduced to zero for the height step Δ*z* >*A*_{sp} and scan speed *υ*_{
H
} > (*υ*_{
H
})_{lim} (Equation 9). A higher scanning speed can damage both the sample and the tip. A decrease of the time constant *τ*_{
i
} can cause instability of the closed-loop. According to the found algorithm, the scanning speed is reduced for the threshold of the amplitude *A*_{low} <*A*_{sp}. Scanning at the lower speed is continued as long as the error signal is reduced and the oscillation amplitude is restored.