Stiffness calibration of qPlus sensors at low temperature through thermal noise measurements

Non-contact atomic force microscopy (nc-AFM) offers a unique experimental framework for topographical imaging of surfaces with atomic and/or sub-molecular resolution. The technique also permits to perform frequency shift spectroscopy to quantitatively evaluate the tip-sample interaction forces and potentials above individual atoms or molecules. The stiffness of the probe, k, is then required to perform the frequency shift-to-force conversion. However, this quantity is generally known with little precision. An accurate stiffness calibration is therefore mandatory if accurate force measurements are targeted. In nc-AFM, the probe may either be a silicon cantilever, a quartz tuning fork (QTF), or a length extensional resonator (LER). When used in ultrahigh vacuum (UHV) and at low temperature, the technique mostly employs QTFs, based on the so-called qPlus design, which actually covers different types of sensors in terms of size and design of the electrodes. They all have in common a QTF featuring a metallic tip glued at the free end of one of its prongs. In this study, we report the stiffness calibration of a particular type of qPlus sensor in UHV and at 9.8 K by means of thermal noise measurements. The stiffness calibration of such high-k sensors, featuring high quality factors (Q) as well, requires to master both the acquisition parameters and the data post-processing. Our approach relies both on numerical simulations and experimental results. A thorough analysis of the thermal noise power spectral density of the qPlus fluctuations leads to an estimated stiffness of the first flexural eigenmode of ≃2000 N/m, with a maximum uncertainty of 10%, whereas the static stiffness of the sensor without tip is expected to be ≃3300 N/m. The former value must not be considered as being representative of a generic value for any qPlus, as our study stresses the influence of the tip on the estimated stiffness and points towards the need for the individual calibration of these probes. Although the framework focuses on a particular kind of sensor, it may be adapted to any high-k, high-Q nc-AFM probe used under similar conditions, such as silicon cantilevers and LERs.