Bimodal atomic force microscopy

In AFM, feedback loops control the operation of the microscope by keeping a fixed value a parameter of the tip's oscillation.[11] If the main feedback loop operates with the amplitude, the AFM mode is called amplitude modulation (AM). If it operates with the frequency shift, the AFM mode is called frequency modulation (FM). Bimodal AFM might be operated with several feedback loops. This gives rise to a variety of bimodal configurations.[12] The configurations are termed AM-open loop, AM-FM, FM-FM.[13] For example, bimodal AM-FM means that the first mode is operated with an amplitude modulation loop while the 2nd mode is operated with a frequency modulation loop. The configurations might not be equivalent in terms of sensitivity, signal-to-noise ratio or complexity.

Let's consider the AM-FM configuration. The first mode is excited to reach free amplitude (no interaction) and the changes of its amplitude and phase shift are tracked by a lock-in amplifier. The main feedback loop keeps constant the amplitude, at a certain set-point ${\displaystyle A_{1}}$ by modifying the tip vertical position (AM). In a nanomechanical mapping experiment, ${\displaystyle \phi _{1}}$ must be kept below 90°, i.e., the AFM is operated in the repulsive regime. At the same time, an FM loop acts on the second eigenmode. A phase-lock-loop regulates the excitation frequency ${\displaystyle f_{2}}$ by keeping the phase shift of the second mode at 90°. An additional feedback loop might be used to maintain the amplitude ${\displaystyle A_{2}}$ constant.

The theory of bimodal AFM operation encompasses several aspects. Among them, the approximations to express the Euler-Bernoulli equation of a continuous cantilever beam in terms of the equations of the excited modes,[2][8][10][14] the type of interaction forces acting on the tip,[15][16][17] the theory of demodulation methods[18] or the introduction of finite-size effects.[19]

In a nutshell, the tip displacement in AFM is approximated by a point-mass model,[13]

${\displaystyle {\frac {k_{i}}{4\pi ^{2}f_{i}^{2}}}{\ddot {z_{i}}}+{\frac {k_{i}}{2\pi f_{0i}Q}}{\dot {z_{i}}}+k_{i}z_{i}=F_{i}\cos(2\pi f_{i}t)+F_{ts}(t)\,}$

where ${\displaystyle f_{i}}$, ${\displaystyle f_{0}i}$, ${\displaystyle Q_{i}}$, ${\displaystyle k_{i}}$, ${\displaystyle F_{i}}$, and ${\displaystyle F_{ts}}$ are, respectively, the driving frequency, the free resonant frequency, the quality factor, the stiffness, the driving force of the i-th mode, and the tip–sample interaction force. In bimodal AFM, the vertical motion of the tip (deflection) has two components, one for each mode,

${\displaystyle z(t)=z_{0}+z_{1}(t)+z_{2}(t)\approx A_{1}cos\left(2\pi f_{1}t-\phi _{1}\right)+A_{2}cos\left(2\pi f_{2}t-{\frac {\pi }{2}}\right)\,}$

with ${\displaystyle z_{0}}$, ${\displaystyle z_{1}}$, ${\displaystyle z_{2}}$, as the static, the first, and the second mode deflections; ${\displaystyle A_{i}}$, ${\displaystyle f_{i}}$ and ${\displaystyle \phi _{i}}$ are, respectively, the amplitude, frequency and phase shift of mode i.

The theory that transforms bimodal AFM observables into material properties is based on applying the virial ${\displaystyle V_{i}}$ and energy dissipation ${\displaystyle E_{diss}}$ theorems to the equations of motion of the excited modes. The following equations were derived[20]

${\displaystyle V_{1}={\frac {1}{T}}\int _{0}^{T}F_{ts}(t)z_{1}(t)dt=-{\frac {k_{1}A_{1}A_{01}}{2Q_{1}}}\cos {\phi _{1}}\,}$

${\displaystyle V_{2}={\frac {1}{T}}\int _{0}^{T}F_{ts}(t)z_{2}(t)dt\approx -{\frac {k_{2}A_{2}^{2}\Delta f_{2}}{f_{02}}}\,}$

${\displaystyle E_{diss1}=\int _{0}^{T}F_{ts}(t){\dot {z}}_{1}(t)dt=-{\frac {\pi k_{1}A_{1}}{Q_{1}}}(A_{1}-A_{01}\sin(\phi _{1}))\,}$

where ${\displaystyle T=T_{1}T_{2}}$ is a time where the oscillation of both modes are periodic; ${\displaystyle Q_{i}}$ the quality factor of mode i. Bimodal AFM operation might be involve any pair of eigenmodes. However, experiments are commonly performed by exciting the first two eigenmodes.

The theory of bimodal AFM provides analytical expressions to link material properties with microscope observables. For example, for a paraboloid probe (radius ${\displaystyle R}$) and a tip-sample force given by the linear viscoelastic Kelvin-Voigt model, the effective elastic modulus ${\displaystyle E_{eff}}$ of the sample, viscous coefficient of compressibility ${\displaystyle \eta _{com}}$, loss tangent ${\displaystyle \tan \rho }$ or retardation time ${\displaystyle \tau }$ are expressed by[21]

${\displaystyle E_{eff}=4{\sqrt {2}}{\frac {Q_{1}}{\sqrt {R}}}{\frac {k_{2}^{2}}{k_{1}}}{\frac {\Delta f_{2}^{2}}{f_{02}^{2}}}{\frac {A_{1}^{3/2}}{A_{01}^{2}-A_{1}^{2}}}\,}$

${\displaystyle \eta _{com}={\frac {E_{eff}}{\omega _{1}}}\left[{\frac {A_{01}\sin {\phi _{1}}-A_{1}}{A_{01}\cos {\phi _{1}}}}\right]\,}$

${\displaystyle \tan \rho =2\pi \omega _{1}{\frac {\eta _{com}}{E_{eff}}}=2\pi \omega _{1}\tau \,}$

For an elastic material, the second term of equation to calculate ${\displaystyle \eta }$ disappears because ${\displaystyle A_{1}=A_{01}\sin {\phi _{1}}}$ which gives ${\displaystyle \eta =0}$. The elastic modulus is obtained from the equation above. Other analytical expressions were proposed for the determination of the Hamaker constant[15] and the magnetic parameters of a ferromagnetic sample.[16]

Atomic-scale imaging of graphene,[22] semiconductor surfaces[23] and adsorbed organic molecules[24] were obtained in ultra high-vacuum. Angstrom-resolution images of hydration layers formed on proteins and Young's modulus map of a metal-organic frame work, purple membrane and a lipid bilayer were reported in aqueous solutions.[20][25][26]

Bimodal AFM is widely used to provide high-spatial resolution maps of material properties, in particular, mechanical properties. Elastic and/or viscoelastic property maps of polymers,[27][28][29][30][31] DNA,[32] proteins, protein fibers,[33] lipids[34][19] or 2D materials[35][36] were generated. Non-mechanical properties and interactions including crystal magnetic garnets, electrostatic strain, superparamagnetic particles and high-density disks were also mapped.[37][38] Quantitative property mapping requires the calibration of the force constants of the excited modes.[39]