A. Circuit Analysis of IPBM-SICM

The equivalent circuit of the pipette–sample system in IPBM-SICM is described in our previous work [24] and is illustrated in Fig. 2. The pipette resistance inside the pipette, i.e., Rp, is in series with the resistance of the tip opening region, i.e., Rt, and both of these are in parallel with pipette capacitance Cp. Then, the combined Rp, Rt, and Cp are in series with bath electrolyte resistance Rb. Stray capacitance Cstray is in parallel with the above combination. By simultaneously applying an ac voltage Uac at an angular frequency ω (2πf) coupled with a dc voltage Udc across the two electrodes, the value of the above parameters can be determined by the following measurements, assumptions, and calculations [25].

Fig. 2.

Circuit analysis of the pipette–sample system in IPBM-SICM.

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Fig. 3.

Analysis of experimental circuit parameters for different positions of the nanopipette. (a) Position a: Tip does not touch the solution. (b) Position b: Tip is immersed several millimeters into the solution but far from the dish. (c) Position c: Tip senses the Petri dish.

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First, we calculate capacitance Cstray by assuming that capacitance Cstray remains constant. This assumption is reasonable as the overall external circuit does not change during the experiment. On the basis of this assumption, we can obtain capacitance Cstray in advance before the pipette is immersed into the solution [position a, as shown in Fig. 3(a)], where Rp is infinite and Cp is almost zero. In this position, almost all of the current passes through Cstray. Assuming that the total ac current is measured as I∗ac, Cstray can be calculated by

View Source\begin{equation*}C_{\mathrm{stray}}=I_{\mathrm{ac}}^{\ast}/(U_{\mathrm{ac}}\times\omega).\tag{1}\end{equation*}

Rp+Rt+Rb=Rdc=Udc/Idc.(2)

View Source\begin{equation*}R_{p}+R_{t}+R_{b}=R_{\mathrm{dc}}=U_{\mathrm{dc}}/I_{\mathrm{dc}}.\tag{2}\end{equation*}

Y(jω)=A+jB(3)

View Source\begin{equation*}Y(j\omega)=A+jB\tag{3}\end{equation*}

Y(jω)=b(1+x2a)1+x2a2+j(bx(1−a)1+x2a2+ωCstray)(4)

View Source\begin{equation*}Y(j\omega)=\frac{b(1+x^{2}a)}{1+x^{2}a^{2}}+j\left(\frac{bx(1-a)}{1+x^{2}a^{2}}+\omega C_{\mathrm{stray}}\right)\tag{4}\end{equation*}

Rb=(A+(B′)2A−b)−1Rp+Rt=1b−RbCp=1ω×1Rb×11−bRb×(A−b)B′(5)(6)(7)

View Source\begin{align*}&\,R_{b}=\left(A+\frac{(B^{\prime})^{2}}{A-b}\right)^{-1}\tag{5}\\&\,R_{p}+R_{t}=\frac{1}{b}-R_{b}\tag{6}\\&\,C_{p}=\frac{1}{\omega}\times\frac{1}{R_{b}}\times\frac{1}{1-bR_{b}}\times\frac{(A-b)}{B^{\prime}}\tag{7}\end{align*}

Lastly, when the tip is close to the sample [position c, as shown in Fig. 3(c)], Rt increases as the tip–sample separation decreases. As the known Rp remains constant, the changes of other parameters, including Rt, Rb, and Cp, can be calculated from (5), (6), and (7), respectively. Finally, all of the parameters can be obtained. Table I shows a group of parameters obtained for a specific pipette in one experiment using the aforementioned procedures.

TABLE I Calculated Parameters for Different Positions in Fig. 3 Based on the Model in Fig. 2

B. Simplified Circuit

In our experiment, Rb is almost negligible as its magnitude is about three orders less than that of Rp, as shown in Table I, which was obtained using a tip with an inner radius of approximately 75 nm. The same conclusion can be also made through experimental results of Unwin group [23]. Therefore, we can neglect Rb to obtain a simplified circuit, as shown in Fig. 4. In the simplified circuit, the total capacitance Ctotal, which includes capacitance Cstray and capacitance Cp, is in parallel with the total resistance, which is called the solution resistance Rsol. The current through the solution resistance path, i.e., Isol, is in phase with the applied ac voltage and can be obtained by finding the product of Iac and cosθ, as shown in Fig. 4(b). The in-phase current is inversely related to the solution resistance Rsol and is thus sensitive to the tip–sample separation. As a result, it can be used as a feedback signal to regulate the tip–sample separation. Using the simplified circuit model, dc voltage Udc is not needed for the calibration of all parameters. The unknown parameters Rsol and Ctotal in the simplified circuit model can be calculated by

View Source\begin{align*}R_{\mathrm{sol}}=&\,U_{\mathrm{ac}}/(I_{\mathrm{ac}}\times\cos\theta)\tag{8}\\C_{\mathrm{total}}=&\, (I_{\mathrm{ac}}\times\sin\theta)/(U_{\mathrm{ac}}\times\omega).\tag{9}\end{align*}

Fig. 4.

Simplified circuit of the pipette–sample system in (a) IPBM-SICM and (b) current analysis.

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From (8) and (9), we can determine the solution resistance Rsol and capacitance Ctotal from our experimental measurements and hence obtain their dependence on the tip–sample separation, as shown in Fig. 5. Fig. 5(a) indicates that the solution resistance Rsol increases but the capacitance Ctotal remains relatively unchanged when the tip–sample separation is reduced. In addition, the zoomed-in data in Fig. 5(b) show that Ctotal slightly increases when the tip is very close to the sample surface, which may be the reason why Iac slightly increases with the decrease in the tip–sample separation, as shown in Fig. 6. This current increase may be also attributed to additional capacitance paths that were formed in the vertical direction of the tip region that accompanied the increase in resistance Rt.

Fig. 6 shows the experimental approach curves that were obtained when the pipette tip approached the Petri dish. These curves show the relationships between different parameters with respect to the tip–sample separation. The total ac current Iac remains relatively constant because it flows mainly through the dominant capacitance path, and it will remain almost constant with relatively constant capacitance. However, the inset in Fig. 6 shows that current Iac slightly increases because of the slightly increased capacitance, as previously discussed. The approach curves of the dc current Idc (see the black line in Fig. 6) and the in-phase ac current Isol (see the red line in Fig. 6) reflect the sensitivity of the control feedback to the tip–sample separation in the dc mode and IPBM mode, respectively. Fig. 6 shows that, as the tip approaches the sample, the sensitivity of the IPBM mode is initially the same as that of the dc mode; however, it becomes less than that of the dc mode as the tip moves closer to the sample surface. This loss in sensitivity may be attributed to the increasing sample capacitance at smaller tip–sample separation [26]. In our experiment, we chose the control setpoint to be 94%–98% of the reference current Isol (recorded far from the sample), and it is located in the overlapping region of the current Isol and current Idc approach curves. Therefore, the selected setpoint value will provide control performance that is similar to that in the dc mode.

Fig. 5.

(a) Normalized and (b) absolute change of calculated solution resistance and capacitance with tip–sample separation in IPBM mode. The modulation frequency is 15 kHz.

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Fig. 6.

Approach curves for the relationship between different parameters versus the tip–sample distance in the IPBM mode. As the tip–sample separation reduces, the amplitude of (blue line) the experimental ac current remains almost constant, but the ac current through (red line) the solution resistance path reduces and the change matches well with the right part of (black line) the dc current versus the tip–sample separation. The modulation frequency is 15 kHz.

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A. System Setup

The setup of the IPBM-SICM system with CC is essentially the same as the homemade SICM system previously reported [24] (as shown in Fig. 9). The system consists of a scan head, a closed-loop feedback control module based on a control computer running a real-time Linux operation system, and a human–computer interaction module that is based on a host computer running the Windows XP operating system. The scan head is equipped using an XY scanner with a maximum XY scan range of 100 μm×100 μm (Physik Instrumente, P517.3CD), an independent z-axis piezo with a range of 38 μm (Physik Instrumente, P753.31C), and a three-axis motorized stage (9062-XYZ-PPP, New Focus Corporation, USA) for the large movement of the tip with submicrometer resolution. The control computer, which is equipped with a data acquisition card (PCI-6251, National Instruments, USA), samples the feedback input Isol and the actual displacement of the z-axis piezo, runs the control module, and outputs the control value to regulate the XYZ movement of the tip or sample. The host computer communicates with the control computer via the Internet and provides the interface through which the user can control the scanning procedure or display the topography of the sample. Other key and peripheral devices of the system include an optical microscope, a function generator (Tektronix, AFG3022B), a patch clamp amplifier (Heka, EPC800USB), a lock-in amplifier (Stanford Research Systems, SR830), a piezocontroller, and a motorized stage controller. In addition, the CC circuit is the built-in part of the patch clamp amplifier.

Fig. 9.

Configuration of homebuilt IPBM-SICM.

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All nanopipettes were fabricated from 1.0-mm-outer-diameter 0.5-mm-inner-diameter borosilicate capillaries (1B100F-4, World Precision Instruments, Shanghai Trading Company Ltd., China) using a CO ₂-laser-based pipette puller (P2000, Sutter Instrument, USA), providing tip openings with inner radii of approximately 75 nm and a half cone angle of 4 ∘–5 ∘ (confirmed by SEM data, data not shown). In our experiment, the nanopipette was backfilled with a phosphate buffer solution (PBS) and immersed in the PBS solution, and dc resistance Rdc was found to be approximately 40–50 MΩ.

In this paper, the ac current Iac was amplified by the I– V converter in the patch clamp amplifier at a gain of 0.1 mV/pA, and cutoff filter frequency is set at 100 kHz. The lock-in amplifier detects the amplitude, phase, and X-out (see Appendix II) of the ac current and outputs the RMS values after they are amplified by a factor of ten times. The values of Uac are set at the maximally allowable value for which Isol does not saturate the lock-in amplifier input.

B. Sample Preparation

A microgrid made from PDMS was prepared to test the performance of the CC method in IPBM-SICM. The microgrid was fabricated using a soft lithographic approach [27] as follows. First, a 10 : 1 mixture of PDMS and curing agent (Sylgard 184, Dow Corning) was poured onto a master mold, which is a silicon calibration grating (Digital Instruments, P/N 498-000-026) with a 10-μm pitch and a 200-nm step depth), and it was then baked on a hotplate (PC-600, Corning Inc., USA) at 70 ∘C for 4 h. Then, PDMS was peeled from the master mold after PDMS had hardened. Finally, a replica of the microgrid in PDMS was obtained with which the surface contacting the master mold was imprinted with the microgrid structure.

C. Applying CC With the Same Tip–Sample Separation as in the Pure IPBM Mode

To investigate how the CC improves the SNR, we first compare the scan results obtained with and without CC by maintaining the same tip–sample separation during scanning. As shown in Fig. 10, when applying CC or not, the height images seem similar; however, the error images seem different. In error images, fine structures pointed by the arrows can be only distinguished from the background noise with CC, which is due to the improved SNR. Without CC, Uac is limited to 22 mV and the error signal of fine structures is covered by the noise, as shown in Fig. 10(b). On the contrary, the expanding signal will stand out from the same noise level after CC when Uac is set to 60 mV, as shown in Fig. 10(d). Therefore, the quality of the error image in the IPBM mode is improved with the CC method because of the improved SNR.

Fig. 10.

Comparison of SICM scan images with and without the CC method at the same tip–sample separation (the setpoints are set to 95%). (a) Height image and (b) Isol error image without CC; Uac is set to 22 mV. (c) Height image and (d) Isol error image with 3.5-pF CC; Uac is set to 60 mV. All of the scan images are obtained with a modulation frequency of 15 kHz, Lock in amplifier setting of status III and a scan rate of 0.5 Hz. Bar: 10 μm.

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D. Applying CC With Larger Tip–Sample Separation

With the CC method, a larger tip–sample separation (higher setpoint) can be applied during scanning. Without compensation, the setpoint should be set to lower than 98% because of the low SNR (see the green line in Fig. 8). When values of 3.5 or 5 pF are compensated, the setpoint can be set to be 98% with an increased value of Uac. Moreover, the experimental results show that images obtained with enough CC, which allow a large tip–sample separation [see Fig. 11(f)], have similar quality to those obtained without CC, which requires a small tip–sample separation [see Fig. 11(a)]. However, as shown in the red circle in Fig. 11(h), at larger tip–sample separation with smaller CC, i.e., 3.5 pF, parachuting (image blurring in the downhill region of grating) issue occurs sometimes due to the relatively low SNR (see the red line in Fig. 8). In addition, a larger tip–sample separation reduces the sample distortion during scanning. During practical operation, the electrolyte filling level is always different from the equilibrium value that is determined by the capillary tension, which leads to a hydrostatic force that is inversely proportional to the tip–sample separation [28]. In this case, the sample suffers from a larger hydrostatic force as the tip–sample separation is reduced, thus causing a larger sample distortion.

Fig. 11.

Comparison of SICM scan images with and without the CC method at different tip–sample separations. (a) Height image without CC when setpoint is 95% and Uac is set to 22 mV. (b) We failed to obtain the image without CC when setpoint is 95% and Uac is set to 22 mV. (c) and (d) Height images with 3.5-pF CC and 60-mV Uac, whereas (c) and (d) are obtained at 95% and 98% tip–sample separation, respectively. (e) and (f) Height images with 5-pF CC and 160-mV Uac, whereas (c) and (d) are obtained at 95% and 98% tip–sample separation, respectively. Lower panel (g) shows line profiles from the red line in (a), the blue line in (c), and the black line in e. The inset is a zoom-in of the height profiles in (g). (h) Contrast of line profiles from the red line in (a), the blue line in (d), and the black line in (f). The inset is a zoom-in of the height profiles in (h). All of the scan images are obtained with a modulation frequency of 15 kHz, lock-in amplifier setting of status III, and a scan rate of 0.5 Hz. Bar: 10 μm.

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E. Applying CC at Larger Modulation Frequency

With the CC method, a larger modulation frequency can be applied during scanning. When the modulation frequency is set to 15 kHz, without CC Uac is limited to 20 mV, we can obtain the image at 94% setpoint, although the SNR [see the red line in Fig. 12(f)] is very low. If f increases to 25 kHz, Iac will increase and Uac has to be reduced to a lower value (limited to 9.5 mV). This will further reduce Isol, and thus, the noise to reference Isol becomes larger [see the green line in Fig. 12(f)]. Such a low SNR is unacceptable for imaging. However, after applying CC with a larger Uac, the SNR can be significantly improved [see the blue line in Fig. 12(f)], and it can be also improved at a larger modulation frequency of 25 kHz [see the black line in Fig. 12(f)]. Therefore, with the improved SNR after CC, SICM can function at a larger modulation frequency, i.e., 25 kHz. Moreover, the line profiles show that the images' quality with CC [see Fig. 12(c) and (d)] seems similar with the one obtained without CC [see Fig. 12(a)]. However, the low-SNR problem can be also inferred from the error images, and the one without CC (data not shown) is more blurring.

Fig. 12.

Comparison of SICM results with and without the CC method at different modulation frequencies. (a) Height image without CC when the modulation frequency is 15 kHz and Uac is set to 20 mV. (b) We failed to obtain the image without CC when the modulation frequency is 25 kHz and Uac is maximized to be 9.5 mV. (c) Height image with 5.9-pF CC when the modulation frequency is 15 kHz and Uac is set to 180 mV. (d) Height image with 5.9-pF CC when the modulation frequency is 25 kHz and Uac is set to 100 mV. (e) Contrast of the line profiles from the red line in (a), the blue line in (c), and the black line in (d). The inset is a zoom-in of the height profiles in (e). (f) Experimental approach curves with and without the CC method at different modulation frequencies. All of the scan images are obtained when the lock-in amplifier is set at status II, the scan rate is 0.5 Hz, and the setpoints are set to 94%. Bar: 10 μm.

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F. Applying CC at Higher Scan Speed

With the CC method, a larger scan speed can be used during scanning. Without CC, when the modulation frequency is set to 15 kHz and the setpoint is set to 94%, Uac is limited to 20 mV, and hence, the SNR limits the lock-in amplifier to set at status II. Such induced low bandwidth of the lock-in amplifier makes the gain parameters of PID control larger enough to avoid parachuting or to shorten the parachuting period [29]. However, with large gain parameter, tip will be easy to overshoot [see the red line in Fig. 13(e)] at the uphill of grating when scan rate is 0.5 Hz, as shown in Fig. 13(a). It becomes worse when the scan speed increases to 0.75 Hz, as shown in Fig. 13(b). CC method improves the SNR problem and lets the lock-in amplifier work at status IV, expanding the bandwidth of the lock-in amplifier (please refer to Appendix III). With the expanding bandwidth, no overshoot is found in the image [see Fig. 13(c)] even the one with a faster scan rate at 0.75 Hz [see Fig. 13(d)]. Therefore, with the improved SNR after CC, SICM can function at a larger scan rate frequency, i.e., 0.75 Hz. In addition, our current speed is limited by the bandwidth of the piezoactuator. When at the increasing further scan rate, i.e., 1 Hz, oscillating problem will start to occur (data not shown), and it would not help with a higher modulated frequency.

Fig. 13.

Comparison of SICM results with and without the CC method at different speeds. (a) and (b) Height images without CC when Uac is set to 20 mV, and the lock-in amplifier is set at status II. The difference is the scan rate of (a) is 0.5 Hz, and the one of (b) is 0.75 Hz. (c) and (d) are height images with 5.9 pF CC when Uac is set to 180 mV, the lock-in amplifier is set at status IV. And the difference is the scan rate of c is 0.5 Hz and the one of d is 0.75 Hz. (e) Contrast of the line profiles from the red line in (a) and the blue line in (c). (f) Contrast of the line profiles from the green line in b and the black line in d. The entire scan images are obtained under the following conditions: The modulation frequency is 15 kHz, and the setpoints are set to 94%. Bar: 10 μm.

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Similar to the theoretical consideration of the maximum scan speed in the tapping mode AFM proposed by the Ando group [30], [31], in IPBM-SICM, we can derive the relationship between the scanning speed and the modulation frequency f. Assuming that the sample has a sinusoidal shape with periodicity λ, for an N×N-pixel image, the smallest imaging acquisition time T of SICM can be expressed as

View Source\begin{equation*}T > 16npN^{2}/(\lambda f)\tag{12}\end{equation*}

Dual-phase lock-in amplifiers such as SR830 have two phase-sensitive detectors (PSDs): One multiplies the input signal by the reference signal (provided by either the internal oscillator or an external source), whereas the other multiplies the signal with the reference signal shifted by 90 ∘.

Assuming that the input signal and reference signal are Vsigsin(ωsigt+θsig) and Vrefsin(ωreft+θref), respectively, the product of the first PSD is

View Source\begin{align*}{\hskip-7.5pt}V_{\mathrm{psd1}}\!=&\,V_{\mathrm{sig}}\sin(\omega_{\mathrm{sig}}t+\theta_{\mathrm{sig}})\cdot V_{\mathrm{ref}}\sin(\omega_{\mathrm{ref}}t+\theta_{\mathrm{ref}})\nonumber\\=&\,1/2V_{\mathrm{sig}}V_{\mathrm{ref}}\cos\left[(\omega_{\mathrm{sig}}-\omega_{\mathrm{ref}})t+(\theta_{\mathrm{sig}}-\theta_{\mathrm{ref}})\right]\nonumber\\&\!-1/2V_{\mathrm{sig}}V_{\mathrm{ref}}\cos\left[(\omega_{\mathrm{sig}}+\omega_{\mathrm{ref}})t+(\theta_{\mathrm{sig}}+\theta_{\mathrm{ref}})\right]\tag{13}\end{align*}

Vpsd1=1/2VsigVrefcos(θsig−θref).(14)

View Source\begin{equation*}V_{\mathrm{psd1}}=1/2V_{\mathrm{sig}}V_{\mathrm{ref}}\cos(\theta_{\mathrm{sig}}-\theta_{\mathrm{ref}}).\tag{14}\end{equation*}

In SR830, X can be outputted after being amplified tenfold, and the output is called the 10X output. In addition, both X and Y are continuous signals and therefore have a larger bandwidth than amplitude Vsig and phase θ, which are discrete signals.