Analysis of contact stiffness in Ultrasound Atomic Force Microscopy: Three-dimensional time-dependent ultrasound modeling

Ultrasound Atomic Force Microscopy (US-AFM) has been used for subsurface imaging of nanostructures. The contact stiffness variations have been suggested as the origin of the image contrast. Therefore, to analyze the image contrast, the local changes in the contact stiffness due to the presence of subsurface features should be calculated. So far, only static simulations have been conducted to analyze the local changes in the contact stiffness and, consequently, the contrast in US-AFM. Such a static approach does not fully represent the real US-AFM experiment, where an ultrasound wave is launched either into the sample or at the tip, which modulates the contact stiffness. This is a time-dependent nonlinear dynamic problem rather than a static and stationary one. This letter presents dynamic 3D ultrasound analysis of contact stiffness in US-AFM (in contrast to static analysis) to realistically predict the changes in contact stiffness and thus the changes in the subsurface image contrast. The modulation frequency also influences the contact stiffness variations and, thus, the image contrast. The three-dimensional time-dependent ultrasound analysis will greatly aid in the contrast optimization of subsurface nanoimaging with US-AFM.

In AFM, a vibrating cantilever with a sharp tip scans a sample surface, which in turn, influences the deflection of the same cantilever. The motion of the cantilever is measured with the use of the optical beam deflection method [3,4] and gives a high resolution image of the surface [5]. In dynamic mode AFM, the tip-sample interaction influences the vibration mode of the cantilever [6]. The response of the cantilever specifically depends on the tipsample interaction stiffness, i.e., the so-called contact stiffness. The local variations in contact stiffness change the contact resonance frequency of the cantilever and its vibration mode [7].
The further combination with ultrasonic excitation resulted in a variety of methods like UFM [8], AFAM [9,10], UAFM [6], HFM [11]. We simply refer to Ultrasound-AFM (US-AFM) to indicate the common ultrasonic excitation of the different AFM schemes mentioned above. In US-AFM either the tip or the sample is excited with an ultrasonic wave. At ultrasound frequencies (tens of MHz), the cantilever is effectively stiffened, and its stiffness can be tuned to match the stiffness of the contact, improving the image contrast [7,9,12]. With US-AFM, the possibility of imaging objects below the surface of a sample has been shown [9][10][11]13]. Such subsurface imaging capabilities are of great interest in several fields, such as semiconductors [14], life sciences [15], and measurements of local mechanical properties [16].
To analyze and enhance the image contrast in US-AFM or to extract quantitative material properties, the local changes in contact stiffness should be determined [10,17]. Static and stationary simulations have been performed [7] to predict the changes in the contact stiffness. However, static simulations do not fully represent the nonlinear dynamic situation of US-AFM since the effect of ultrasound waves on the contact stiffness is a time-dependent, dynamic problem. The dynamic behavior of the motion of the cantilever is influenced by the tip-sample interaction, specifically the contact stiffness, and exhibits a different resonance frequency when the tip is probing on top or far away from a subsurface feature [11].
This mechanism is responsible for the subsurface imaging contrast and can be evaluated by measuring the changes in contact resonance.
To accurately analyze the contact stiffness in US-AFM, we performed three-dimensional time-dependent ultrasound calculations using the Finite Element Method (FEM), which better represents the actual experimental conditions. Before addressing the time-dependent ultrasound calculations, we first recall the basics of contact theory and describe the FEM simulations for the static stationary case. This later allows a comparison with the timedependent ultrasound results.
An implementation of a three-dimensional tip-sample contact problem using FEM has been described in [17] to estimate the effect of a scanning tip on the contact stiffness in static condition primarily for cavity-like structures (voids). This approach has been reported to be extremely time consuming due to the required fine discretization at the contact area. We followed the same contact approach, and to reduce the computation time and the required memory, we also implemented a semi-analytical approach based on Hertzian contact theory [18,19].
For a spherical tip (radius R) pressing with a constant force F on a semi-infinite and homogeneous medium in the absence of dissipative effects, the contact radius and the local stress In The indentation is used to make a new estimate [17,20] of the reduced Young's modulus of the first layer M 1 st layer = (3F/2δ) 3 6F R . Finally, the contact stiffness k * can be calculated as is the reciprocal of the effective Young's modulus.
Before implementing the ultrasound (time-dependent) wave excitation, the semianalytical approach has been verified in the static stationary case. We show the verification for one case of a rigid inclusion in a rigid matrix (Table I, case 2, depth incl 100 nm, force 0.5 µN). The feature is moved from right below the contact in steps of 10 nm, and the contact stiffness at the tip-sample contact location is extracted at each scan step. Fig. 2(a) shows that there is very good agreement between the outlined semi-analytical procedure (calculated in COMSOL) and the 3D full contact model (calculated in both COMSOL The semi-analytical model has been used for all the cases listed in Table I. Fig 2(b) shows the baseline contact stiffness (K), which is the contact stiffness when the buried feature is far from the tip. Fig. 2(c) shows the contact stiffness variation normalized to the baseline contact stiffness (∆K/K), which is an estimation of the static contrast.
This approach allows for fast evaluations of the behaviors of certain material and load condition combinations. Fig 2(b) shows that the baseline contact stiffness is largely dependent on the size of the scanned tip and the applied force. Fig. 2(c) shows that the contrast decreases when using a smaller tip (compare solid cyan and dashed green lines) and decreases further if the depth of the feature is increased (compare solid cyan and solid red lines).
The static approach gives a reference for the contact stiffness values reached in the stationary state. However, in current US-AFM techniques, the sample is subjected to an ultrasound wave excitation. The wave excitation is based on modulating a carrier frequency, f c , with a modulation frequency, f m , which is equal or close to the contact resonance frequency of the cantilever. Therefore, the acoustic problem must be addressed with a time-dependent approach. The semi-analytical procedure discussed above has been introduced precisely for the purpose of making the dynamic approach feasible in terms of simulation time and available memory. In this section, we refer to case 1 in Table I The distance between the deformed free-surface and the non-deformed line, ε, is the relative displacement between the PMMA surface with respect to the fixed indentation of the tip. turing at TNO.