Imaging performance of trolling mode atomic force microscopy: investigation of effective parameters

Abstract

In this study, we investigate the limitations and influence of various factors on the performance of trolling mode atomic force microscopy (TR-AFM). For this purpose, at first, based on the governing equations of motion and using a conventional control method, a simulation tool capable of correctly simulating the imaging procedure in TR-AFM is developed. Then based on the developed simulation tool, imaging of different surfaces is performed, and the effect of different factors on the image quality is analyzed. The flexibility of nanoneedle in TR-AFM has unpredictable effects on dynamics of system as well as imaging performance. One problem in imaging is due to coexistence of two stable responses (bistability) which can reduce the accuracy of the images. A qualitative investigation of the nonlinear behavior of the TR-AFM reveals that owing to the nonlinear characteristics of the tip–sample interactions, there often exist two stable responses for a given set of parameters. Hence, the possibility of multiple stable responses and their effect on the imaging performance of various surfaces have been thoroughly investigated. Moreover, the influence of horizontal displacement of nanoneedle tip on image quality at different scanning speeds in the both presence and absence of measurement noise are examined. Finally, the scanning operation for a 3D sample using a 3D resonator model considering nanoneedle tip out-of-plane displacement in a real-time operating system is simulated, and the effect of tip out-of-plane displacement and cantilever scanning direction on the image quality are investigated.

Appendix 1: Coefficients of the derived mathematical model

The coefficients $b_{i}$ (i = 1,2,…,16) are as follows:

$$\begin{aligned} b_{1} & = - a_{2} c_{{\alpha \beta }} \mathop \smallint \limits_{0}^{{L_{b} }} f_{b} \left( x \right)g\left( x \right)\,{\text{d}}x + a_{5} c_{\alpha } \mathop \smallint \limits_{0}^{{L_{b} }} f_{b} \left( x \right)\,{\text{d}}x \\ &\quad + \left( { - g\left( {L_{b} } \right)c_{{\alpha \beta }} a_{{11}} - c_{{\alpha \beta }} a_{{20}} \left( {\frac{{\text{d}}}{{{\text{d}}L_{b} }}\,g\left( {L_{b} } \right)} \right) + c_{\alpha } a_{{21}} } \right)\frac{{\text{d}}}{{{\text{d}}L_{b} }}\,f_{b} \left( {L_{b} } \right) \\ &\quad + f_{b} \left( {L_{b} } \right)\left( { - g\left( {L_{b} } \right)c_{{\alpha \beta }} a_{{10}} - c_{{\alpha \beta }} a_{{11}} \left( {\frac{{\text{d}}}{{{\text{d}}L_{b} }}\,g\left( {L_{b} } \right)} \right) + c_{\alpha } a_{{12}} } \right), \\ b_{2} & = - a_{3} c_{{\alpha \beta }} \mathop \smallint \limits_{0}^{{L_{b} }} f_{b} \left( x \right)g^{\prime}\left( x \right)\,{\text{d}}x - a_{4} c_{{\alpha \beta }} \mathop \smallint \limits_{0}^{{L_{b} }} f_{b} \left( x \right)g\left( x \right)\,{\text{d}}x + a_{6} c_{\alpha } \mathop \smallint \limits_{0}^{{L_{b} }} f_{b} \left( x \right)\,{\text{d}}x\\ &\quad + \left( { - g\left( {L_{b} } \right)c_{{\alpha \beta }} a_{9} - c_{{\alpha \beta }} a_{{19}} \frac{{\text{d}}}{{{\text{d}}L_{b} }}\,g\left( {L_{b} } \right) + a_{{22}} c_{\alpha } } \right)\frac{{\text{d}}}{{{\text{d}}L_{b} }}\,f_{b} \left( {L_{b} } \right)\\ &\quad + f_{b} \left( {L_{b} } \right)\left( { - g\left( {L_{b} } \right)c_{{\alpha \beta }} a_{8} - c_{{\alpha \beta }} a_{9} \frac{{\text{d}}}{{{\text{d}}L_{b} }}\,g\left( {L_{b} } \right) + c_{\alpha } a_{{13}} } \right), \\ b_{3} & = - a_{1} \frac{{\text{d}}}{{{\text{d}}L_{b} }}\,f_{b} \left( {L_{b} } \right)\frac{{{\text{d}}^{2} }}{{{\text{d}}L_{b} ^{2} }}\,g\left( {L_{b} } \right)c_{{\alpha \beta }} + \frac{{\text{d}}}{{{\text{d}}L_{b} }}\,f_{b} \left( {L_{b} } \right)a_{{18}} g\left( {L_{b} } \right)c_{{\alpha \beta }} \\&\quad - f_{b} \left( {L_{b} } \right)a_{7} g\left( {L_{b} } \right)c_{{\alpha \beta }} + a_{1} f_{b} \left( {L_{b} } \right)\frac{{{\text{d}}^{3} }}{{{\text{d}}L_{b} ^{3} }}\,g\left( {L_{b} } \right)c_{{\alpha \beta }}\\ &\quad + c_{\alpha } \frac{{\text{d}}}{{{\text{d}}L_{b} }}\,f_{b} \left( {L_{b} } \right)a_{{23}} + c_{\alpha } f_{b} \left( {L_{b} } \right)a_{{14}} - a_{1} c_{{\alpha \beta }} \mathop \smallint \limits_{0}^{{L_{b} }} f_{b} \left( x \right)g^{{\left( 4 \right)}} \left( x \right)\,{\text{d}}x, \\ b_{4} & = f_{b} \left( {L_{b} } \right)a_{{10}} + \frac{{a_{2} \mathop \smallint \nolimits_{0}^{{L_{b} }} f_{b} \left( x \right)^{2} \,{\text{d}}x}}{{f_{b} \left( {L_{b} } \right)}} + \frac{{\left( {\frac{{\text{d}}}{{{\text{d}}L_{b} }}\,f_{b} \left( {L_{b} } \right)} \right)^{2} a_{{20}} }}{{f_{b} \left( {L_{b} } \right)}} + 2a_{{11}} \left( {\frac{{\text{d}}}{{{\text{d}}L_{b} }}\,f_{b} \left( {L_{b} } \right)} \right), \\ b_{5} & = \frac{1}{{f_{b} \left( {L_{b} } \right)}}\left( {\left( {\frac{{\text{d}}}{{{\text{d}}L_{b} }}\,f_{b} \left( {L_{b} } \right)} \right)^{2} a_{{19}} + 2a_{9} \left( {\frac{{\text{d}}}{{{\text{d}}L_{b} }}\,f_{b} \left( {L_{b} } \right)} \right)f_{b} \left( {L_{b} } \right)}\right. \\ &\quad \left. { + f_{b} \left( {L_{b} } \right)^{2} a_{8} + a_{4} \mathop \smallint \limits_{0}^{{L_{b} }} f_{b} \left( x \right)^{2} \,{\text{d}}x + a_{3} \mathop \smallint \limits_{0}^{{L_{b} }} f_{b} \left( x \right)f_{b} {\text{'}}\left( x \right)\,{\text{d}}x} \right), \\ b_{6} & = f_{b} \left( {L_{b} } \right)a_{7} + \frac{{a_{1} \mathop \smallint \nolimits_{0}^{{L_{b} }} f_{b} ^{{\prime \prime }} \left( x \right)^{2} \,{\text{d}}x}}{{f_{b} \left( {L_{b} } \right)}} - a_{{18}} \frac{{\text{d}}}{{{\text{d}}L_{b} }}\,f_{b} \left( {L_{b} } \right), \\ b_{7} & = \frac{{c_{\alpha } }}{{f_{n} \left( {L_{n} } \right)}}\left( {\left( {a_{{26}} \frac{{\text{d}}}{{{\text{d}}L_{b} }}\,f_{b} \left( {L_{b} } \right) + a_{{15}} f_{b} \left( {L_{b} } \right)} \right)\mathop \smallint \limits_{0}^{{L_{n} }} f_{n} \left( z \right)\,{\text{d}}z + a_{{27}} \frac{{\text{d}}}{{{\text{d}}L_{b} }}\,f_{b} \left( {L_{b} } \right)\mathop \smallint \limits_{0}^{{L_{n} }} f_{n} \left( z \right)z\,{\text{d}}z} \right), \\ b_{8} & = \frac{{c_{\alpha } }}{{f_{n} \left( {L_{n} } \right)}}\left( {\left( {a_{{24}} \frac{{\text{d}}}{{{\text{d}}L_{b} }}f_{b} \left( {L_{b} } \right) + a_{{16}} f_{b} \left( {L_{b} } \right)} \right)\mathop \smallint \limits_{{L_{n} - L_{0} }}^{{L_{n} }} f_{n} \left( z \right)\,{\text{d}}z + a_{{25}} \frac{{\text{d}}}{{{\text{d}}L_{b} }}f_{b} \left( {L_{b} } \right)\mathop \smallint \limits_{{L_{n} - L_{0} }}^{{L_{n} }} f_{n} \left( z \right)z\,{\text{d}}z} \right), \\ b_{9} & = c_{\alpha } \left( {\left( {\frac{{\text{d}}}{{{\text{d}}L_{b} }}\,f_{b} \left( {L_{b} } \right)} \right)a_{{28}} - f_{b} \left( {L_{b} } \right)a_{{17}} } \right), \\ b_{{10}} & = - a_{{27}} c_{{\alpha \beta }} \frac{{\text{d}}}{{{\text{d}}L_{b} }}\,g\left( {L_{b} } \right)\mathop \smallint \limits_{0}^{{L_{n} }} f_{n} \left( z \right)z\,{\text{d}}z - \mathop \smallint \limits_{0}^{{L_{n} }} f_{n} \left( z \right)\,{\text{d}}z\left( {a_{{30}} c_{{\alpha \beta }} c_{\alpha } \frac{{\text{d}}}{{{\text{d}}L_{b} }}\,g\left( {L_{b} } \right) + a_{{15}} c_{{\alpha \beta }} g\left( {L_{b} } \right) - a_{{31}} c_{\alpha } } \right), \\ b_{{11}} & = - a_{{25}} c_{{\alpha \beta }} \frac{{\text{d}}}{{{\text{d}}L_{b} }}\,g\left( {L_{b} } \right)\mathop \smallint \limits_{0}^{{L_{n} }} f_{n} \left( z \right)\hat{u}\left( {z - L_{n} + L_{0} } \right)z\,{\text{d}}z \\ & \quad - \mathop \smallint \limits_{0}^{{L_{n} }} f_{n} \left( z \right)\hat{u}\left( {z - L_{n} + L_{0} } \right)\,{\text{d}}z\left( {g\left( {L_{b} } \right)a_{{16}} c_{{\alpha \beta }} + a_{{24}} c_{{\alpha \beta }} \frac{{\text{d}}}{{{\text{d}}L_{b} }}\,g\left( {L_{b} } \right) - a_{{32}} c_{\alpha } } \right), \\ b_{{12}} & = \frac{{c_{\alpha } a_{{30}} \frac{{\text{d}}}{{{\text{d}}L_{b} }}\,f_{b} \left( {L_{b} } \right)\mathop \smallint \nolimits_{0}^{{L_{n} }} f_{n} \left( z \right)\,{\text{d}}z}}{{f_{b} \left( {L_{b} } \right)}} + \frac{{a_{{27}} \frac{{\text{d}}}{{{\text{d}}L_{b} }}\,f_{b} \left( {L_{b} } \right)\mathop \smallint \nolimits_{0}^{{L_{n} }} f_{n} \left( z \right)z\,{\text{d}}z}}{{f_{b} \left( {L_{b} } \right)}} + a_{{15}} \mathop \smallint \limits_{0}^{{L_{n} }} f_{n} \left( z \right)\,{\text{d}}z, \\ b_{{13}}& = \frac{{a_{{24}} \frac{{\text{d}}}{{{\text{d}}L_{b} }}\,f_{b} \left( {L_{b} } \right)\mathop \smallint \nolimits_{0}^{{L_{n} }} f_{n} \left( z \right)\hat{u}\left( {z - L_{n} + L_{0} } \right)\,{\text{d}}z}}{{f_{b} \left( {L_{b} } \right)}} + a_{{16}} \mathop \smallint \limits_{0}^{{L_{n} }} f_{n} \left( z \right)\hat{u}\left( {z - L_{n} + L_{0} } \right)\,{\text{d}}z \\ &\quad + \frac{{a_{{25}} \frac{{\text{d}}}{{{\text{d}}L_{b} }}\,f_{b} \left( {L_{b} } \right)\mathop \smallint \nolimits_{0}^{{L_{n} }} f_{n} \left( z \right)\hat{u}\left( {z - L_{n} + L_{0} } \right)z\,{\text{d}}z}}{{f_{b} \left( {L_{b} } \right)}}, \\ b_{{14}} & = \frac{{a_{{27}} c_{\alpha } \mathop \smallint \nolimits_{0}^{{L_{n} }} f_{n} \left( z \right)^{2} \,{\text{d}}z}}{{f_{n} \left( {L_{n} } \right)}}, \\ b_{{15}} & = \frac{{a_{{25}} c_{\alpha } \mathop \smallint \nolimits_{0}^{{L_{n} }} \hat{u}\left( {z - L_{n} + L_{0} } \right)f_{n} \left( z \right)^{2} \,{\text{d}}z}}{{f_{n} \left( {L_{n} } \right)}}, \\ b_{{16}} &= \frac{{c_{\alpha } a_{{29}} \mathop \smallint \nolimits_{0}^{{L_{n} }} \left( {\frac{{{\text{d}}^{2} }}{{{\text{d}}z^{2} }}\,f_{n} \left( z \right)} \right)^{2} \,{\text{d}}z}}{{f_{n} \left( {L_{n} } \right)}}, \\ \end{aligned}$$

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where $c_{{\upalpha }} = {\text{cos}}\left( {\upalpha } \right)$, $c_{\alpha \beta } = {\text{cos}}\left( {{\upbeta } - {\upalpha }} \right)$, and $\hat{u}\left( . \right)$ is unit-step function (Heaviside), $g\left( x \right)$ is a fourth-order polynomial the constants of which are obtained by satisfying the following equations:

$$g\left( 0 \right) = 0 , g^{\prime}\left( 0 \right) = 0,\; g\left( {L_{b} } \right) = 1,\;b_{1} = 0 , b_{10} = 0,$$

(15)

moreover, the coefficients $a_{i}$ (i = 1,2,…,36) are as follows:

$$\begin{aligned} a_{1} & = {\text{EI}}_{b} \\ a_{2} & = A_{b} \rho _{b} \\ a_{3} & = C_{s} \\ a_{4} & = C_{a} \\ a_{5} & = A_{b} {\text{cos}}\left( \beta \right)\rho _{b} \\ a_{6} & = C_{a} {\text{cos}}\left( \beta \right) \\ a_{7} & = {\text{cos}}\left( \alpha \right)^{2} \left( {A_{n} {\gamma }_{L} + K_{m} } \right) \\ a_{8} & = \left( {\left( {C_{{d_{2} }} - C_{{d_{1} }} } \right)L_{0} + C_{m} } \right){\text{cos}}\left( \alpha \right)^{2} + L_{0} C_{{d_{1} }} \\ a_{9} & = {\text{sin}}\left( \alpha \right)\left( {\left( {\left( {C_{{d_{1} }} - C_{{d_{2} }} } \right)L_{0} - C_{m} } \right)L_{{{\text{tm}}}} {\text{cos}}\left( \alpha \right) - \frac{1}{2}C_{{d_{1} }} L_{0} \left( {L_{0} - 2L_{n} } \right)} \right) \\ a_{{10}} & = L_{n} \rho _{n} + m_{{{\text{tm}}}} \\ a_{{11}} & = \frac{1}{2}\rho _{n} L_{n} ^{2} {\text{sin}}\left( \alpha \right) \\ a_{{12}} & = {\text{cos}}\left( \beta \right)\left( {L_{n} \rho _{n} + m_{{{\text{tm}}}} } \right) \\ a_{{13}} & = - {\text{cos}}\left( \alpha \right){\text{cos}}\left( { - \beta + \alpha } \right)\left( {\left( {C_{{d_{1} }} - C_{{d_{2} }} } \right)L_{0} - C_{m} } \right) + C_{{d_{1} }} {\text{cos}}\left( \beta \right)L_{0} \\ a_{{14}} & = {\text{cos}}\left( \alpha \right){\text{cos}}\left( { - \beta + \alpha } \right)\left( {A_{n} \gamma _{L} + K_{m} } \right) \\ a_{{15}} & = \rho _{n} {\text{sin}}\left( \alpha \right) \\ a_{{16}} & = C_{{d_{1} }} {\text{sin}}\left( \alpha \right) \\ a_{{17}} & = {\text{cos}}\left( \alpha \right) \\ a_{{18}} & = L_{{{\text{tm}}}} {\text{cos}}\left( \alpha \right){\text{sin}}\left( \alpha \right)\left( {A_{n} \gamma _{L} + K_{m} } \right) \\ a_{{19}} & = - L_{{{\text{tm}}}} ^{2} \left( {\left( {C_{{d_{2} }} - C_{{d_{1} }} } \right)L_{0} + C_{m} } \right){\text{cos}}\left( \alpha \right)^{2} - C_{{d_{1} }} L_{0} L_{{{\text{tm}}}} \left( {L_{0} - 2L_{n} } \right){\text{cos}}\left( \alpha \right) \\ &\quad + \frac{1}{3}C_{{d_{1} }} L_{0} ^{3} - C_{{d_{1} }} L_{0} ^{2} L_{n} + \left( {C_{{d_{1} }} L_{n} ^{2} + C_{{d_{2} }} L_{{{\text{tm}}}} ^{2} } \right)L_{0} + C_{m} L_{{{\text{tm}}}} ^{2} \\ a_{{20}} & = I_{{{\text{tm}}}} + L_{n} \rho _{n} \left( {{\text{cos}}\left( \alpha \right)L_{{{\text{tm}}}} L_{n} + \frac{1}{3}L_{n} ^{2} + L_{{{\text{tm}}}} ^{2} } \right) + H_{{{\text{tm}}}} ^{2} m_{{{\text{tm}}}} \\ a_{{21}} & = \frac{1}{2}\left( { - {\text{cos}}\left( \alpha \right)L_{n} ^{2} \rho _{n} - 2L_{n} L_{{{\text{tm}}}} \rho _{n} - 2H_{{{\text{tm}}}} m_{{{\text{tm}}}} } \right){\text{sin}}\left( \beta \right) + \frac{1}{2}L_{n} ^{2} {\text{cos}}\left( \beta \right){\text{sin}}\left( \alpha \right)\rho _{n} \\ a_{{22}} & = - {\text{sin}}\left( \beta \right)L_{{{\text{tm}}}} \left( {C_{{d_{2} }} L_{0} + C_{m} } \right) - {\text{sin}}\left( { - \beta + \alpha } \right)\left( {{\text{cos}}\left( \alpha \right)L_{{{\text{tm}}}} \left( {\left( {C_{{d_{2} }} - C_{{d_{1} }} } \right)L_{0} + C_{m} } \right) + \frac{1}{2}C_{{d_{1} }} L_{0} \left( {L_{0} - 2L_{n} } \right)} \right) \\ a_{{23}} & = - L_{{{\text{tm}}}} \left( {{\text{sin}}\left( { - \beta + \alpha } \right){\text{cos}}\left( \alpha \right) + {\text{sin}}\left( \beta \right)} \right)\left( {A_{n} {\gamma }_{L} + K_{m} } \right) \\ a_{{24}} & = C_{{d_{1} }} L_{{{\text{tm}}}} {\text{cos}}\left( \alpha \right) \\ a_{{25}} & = C_{{d_{1} }} \\ a_{{26}} & = \rho _{n} L_{{{\text{tm}}}} {\text{cos}}\left( \alpha \right) \\ a_{{27}} & = \rho _{n} \\ a_{{28}} & = L_{{{\text{tm}}}} {\text{sin}}\left( \alpha \right) \\ a_{{29}} & = {\text{EI}}_{n} \\ a_{{30}} & = \rho _{n} L_{{{\text{tm}}}} \\ a_{{31}} & = \rho _{n} {\text{sin}}\left( { - \beta + \alpha } \right) \\ a_{{32}} & = C_{{d_{1} }} {\text{sin}}\left( { - \beta + \alpha } \right) \\ a_{{33}} & = A_{b} L_{b} \rho _{b} + L_{n} \rho _{n} + m_{p} + m_{{{\text{tm}}}} \\ a_{{34}} & = C_{{d_{2} }} L_{0} + C_{m} + C_{a} {\text{cos}}\left( \beta \right)^{2} L + \left( { - \frac{1}{2}{\text{cos}}\left( { - 2\beta + 2\alpha } \right) + \frac{1}{2}} \right)\left( {\left( {C_{{d_{1} }} - C_{{d_{2} }} } \right)L_{0} - C_{m} } \right) \\ a_{{35}} & = 2\left( {A_{n} {\gamma }_{L} + K_{m} } \right)\left( {\frac{1}{4}{\text{cos}}\left( { - 2\beta + 2\alpha } \right) + \frac{1}{4}} \right) \\ a_{{36}} & = {\text{cos}}\left( { - \beta + \alpha } \right) \\ \end{aligned}$$

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