SPM methods combined with the corresponding spectroscopy add-ons give information about morphology of the studied objects and the electronic structure down to the atomic scale. In STM experiments the sharp conductive tip is moved above the sample surface in the \,x\, and \,y\, directions. In constant-current STM measurements the tunnelling current \,I_T, which exponentially depends on the distance \,z\, between the tip and sample \,\left[I_T\propto exp(-kz)\right]\, , is kept constant by a feedback loop adjusting the actual \,z\, position, which is recorded to obtain a three-dimensional topography map. The obtained constant current image represents the integral local density of states (LDOS) of the sample surface, \,I_T(x,y,z,U_T)\propto\int_{E_F}^{E_F+eU_T}\rho(x,y,z,E)dE\,, where integration is performed in the energy range between the Fermi levels (E_F) of tip and sample, when they differ by the value \,eU_T, where \,U_T is the bias voltage. One can perform the differentiation of the \,I(x, y, z, U_T)\, signal with respect to the bias voltage (scanning tunnelling spectroscopy, STS). This can be carried out with lock-in technique and gives direct information about the local density of states. If such measurements are performed in the scanning mode at different bias voltages, then it allows one to observe the so-called electron density standing waves of different periodicities that can be used to obtain the electron dispersion relation, \,E(k), of the surface electronic states.
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Further evolution of the scanning tunnelling microscopy led to the development of the method of atomic force microscopy and spectroscopy. Here the interaction force between sample and the scanning tip, \,F(z), is considered and used as a signal for the feedback loop. Here the deflection of the sensor is registered and as a result the image of the constant force is recorded in the 2D space, \,F(x,y). There are several methods for the detection of the interaction of the sensor and the sample, e.g. back-side STM, capacitive, optical, etc. In the modern AFM instruments the oscillating sensors are widely used. In this case the scanning tip oscillates at very high resonance frequency, \,f_0. When such tip is placed very close to the sample surface, the interaction in the junction slightly changes the resonance frequency and this frequency shift, \,\Delta f\,, can be used as a feedback signal to adjust position of a scanning tip. The interaction force dependence on the distance \,z\, between tip and sample and the respective frequency shift can be expressed from the interaction energy via the respective formulas: \,F_z(z)=-\partial E(z)/\partial z\,, \,\Delta f(z)=-f_0/2k_0\cdot\partial F_z(z)/\partial z, where \,k_0\, is the spring constant of the sensor. The interaction energy \,E(z)\, dependence in general case can be fitted either by the power Lennard-Jones potential or by the exponential-based Morse potential.
In the presented distance dependence of \,E(z)\, several regions can be identified: A and B at distance of more than 5 Å the interaction is defined by the long-range van der Waals forces; C and D around minima of the interaction curve, the imaging contrast is defined by short-range attractive and repulsive forces, respectively. This short-range interaction is used for the atomically-resolved operation of AFM.
Both STM and AFM instruments can be also used in the so-called constant height mode. In this case the feedback loop is completely switched off and scanning tip moves at the fixed \,z=z_0\, position. In this case the 2D maps of the tunnelling current, \,I(x,y,z_0), or frequency shift (or interaction force), \,\Delta f(x,y,z_0), \,F(x,y,z_0), are recorded. This mode, e. g. can be used for the more sophisticated measurements allowing to get 3D data sets of the current or force distribution above the sample surface.
For the simulation of the STM images and comparison of them with the experimental data we use method proposed by Tersoff and Hamann. In this method the tunneling current is proportional to the local density of states at the position of the STM tip, i. e. the scanning tip is modelled as an infinitely small object: {\displaystyle I_T(x,y,z)=\sum^{E_n>E_F-eU_T}_{{E_n}\lt{E_F}}\vert \psi(x,y,z,E_n)\vert^2=n(x,y,z,U_T)}.
This approach is the most simple method which does not take into account the structure of the tip, the chemical interaction between tip and sample, etc. However, in most cases this method provides a good qualitative agreement between theory and experiment, although generally not reproducing the observed surface corrugations. Realisation of this method during simulations is straightforward. DFT code is used for the calculation of the density of states for the energy interval from \,E_F\, to \,E_F+eU_T. Simple estimation for the tunneling current allows to select the respective density contour in the calculations and present it as a constant current STM map acquired at the particular bias voltage: \,n(I)\,[Å^{-3}]\approx 2\times10^{-4}\sqrt{I}\,[nA].
For the description of the AFM mode, the scanning tip is usually separated in two parts - the so-called macro-tip and nano-tip (or tip apex). The interaction of the macro-tip with the surface defines the long-distance tail of the interaction force (regions A and B as discussed earlier) and different shapes of the macro-tips were discussed in the literature. For example the van der Waals interaction between conical tip and flat surface can be expressed as \displaystyle F_z(z_0)=\frac{AR^2(1-\sin\gamma)(R\sin\gamma-z_0\sin\gamma-R-z_0)}{6z_0^2(R+z_0-R\sin\gamma)^2} +\frac{-A\tan\gamma[z_0\sin\gamma +R\sin\gamma+R\cos2\gamma]}{6\cos\gamma(z_0+R-R\sin\gamma)^2}, where \,z_0\, is tip-sample separation, \,A=\pi^2C_6\rho_1\rho_2\, is the Hamaker constant, \,C_6\, is the interaction constant as defined by London, and \,\rho_1\, and \,\rho_2\, are the number densities for tip and sample.
The tip-sample interaction at the short distances, where it determines the appearance of the atomic resolution in imaging, describes the interaction of the nano-tip with the surface and can be modelled in different ways. In our works two approaches were used. In the first one, proposed by Chan et al., where the tip-sample force \,F_{ts}\, can be calculated by a multipole expansion of the tip-sample interaction, treating the influence of the sample on the cantilever tip as a perturbation. In this case the tip-sample interaction energy is \,{\displaystyle E_{ts}(r)=\int \vert \phi(r'-r) \vert^2 V_{ts}(r') dr'}, where \,r\, is the tip position, \,V_{ts}\, is the potential on the tip due to the sample, and \,\phi\, is the electronic state of the tip. Then the tip-sample force can be evaluated as \displaystyle \begin{aligned} F_{ts}(r)&=-\nabla E_{ts}(r)\\ &=-\nabla V_{ts}(r)-\nabla\left[\nabla V_{ts}(r)\int|\phi(r'-r)|^2(r'-r)dr'\right]\\ &=-\nabla V_{ts}(r)-\nabla\left[\nabla V_{ts}(r)\cdot \mathbf{p} \right]=-\nabla V_{ts}(r)-\alpha\nabla\left(|\nabla V_{ts}(r)|^2\right) \end{aligned} where \,\mathbf{p} is the tip polarisation and \,\alpha is its polarisability. For the neutral system, the first monopole term in this expression is equal to zero, giving the final expression for the interaction force: \,F_{ts}(r)\propto -\nabla[\vert\nabla V_{ts}(r)\vert^2]. In these calculations, \,V_{ts}\, is the electrostatic part of the self-consistent potential, i.e. the sum of the Hartree potential and the local part of the ionic pseudopotential. Since these calculations do not take into account the atomic structure of the tip, the overall calculations are not very time consuming, as they include only self-consistent calculations for the respective surface slab and the corresponding treatment of the 3D distribution of the electrostatic potential \,V_{ts}.
In the second approach for the description of the short-distance tip-sample interaction the apex of the scanning tip is modelled as a symmetric cluster build from the W atoms. In this case the cluster is build from the (100) planes of atoms modelling the bcc W lattice, that can be considered as a most realistic and stable configuration for the scanning tip. When studying the interaction between W-tip and the graphene/Ru(0001) sample, the interaction energy can be written as \,\Delta E_{\mathrm{int}}(z)=E_{\mathrm{tip+sample}}(z)-[E_{\mathrm{sample}}(z)+E_{\mathrm{tip}}(z)], where \,E_{\mathrm{sample}}(z)\, and \,E_{\mathrm{tip}}(z)\, are the energies of the isolated tip and sample, and \,E_{\mathrm{tip+sample}}(z)\, is the energy of their interacting assembly.
