where a is a radius of the circular contact probe

where a is a radius of the circular contact (probe); r is a distance from the center of the contact – probe place to the hemispherical contact; ρ is a resistivity of the area between the contacts. Spreading resistance R is well approximated for a homogeneous semi-infinite doped sample with resistance ρ during measurements with the radius a of probe curvature (it is assumed that the probe does not penetrate into the sample):
From Eq. (2), it is possible to determine whether the conditions are of tunneling mode or spreading resistance mode. In the tunneling mode the current is small, and overestimated values of resistance R are obtained according to I–V characteristics. The front panel is shown in Fig. 2; topographies are combined (obtained at the different values of bias voltage) because the temperature drift always shifts the scan area for a few tens of nanometers. Then, the user selects the point on the image of topography, in which it is necessary to plot the I–V curve. The selected point is the position in each map of the measured current. Thus, I–V characteristics are constructed, based on the voltage values, at which the set of images, and current values of the selected point on the map of the current distribution were obtained. In other words, the set of current maps allows us to construct I–V characteristic at any point of the resulting topographic image, and each scan is used as a single point of I–V curve.
The technique implementation
Experimental data Fig. 4 demonstrates the picture of ZnO thin film obtained using scanning THZ2 microscope. An analysis of this image discloses that the growth of the film proceeds in two stages, as it is likely, the initial temperature (380 °C) decreased to 350 °C. AFM-topography of the ZnO thin film obtained using the scanning probe microscope is presented in Fig. 5. It was measured in the spreading resistance mode. Height difference of the studied area is 0.36 µm where lighter areas correspond to the tops of the surface relief. As it is seen from the topography, the grains constitute elongated formations of 480–550 nm long, which corresponds to an SEM image of the film (see Fig. 4). At the current map lighter areas correspond to higher values of current. Fig. 6 gives a set of maps with the current distribution obtained in the spreading resistance mode. Bias voltage from −5 to +5 V was applied to the same area. These experimental data are necessary to construct the required I–V characteristics. The result of such construction is shown in Fig. 7. Moreover, some local I–V characteristics were also measured in Nova (program which controls the Atomic force microscope) with applying bias voltage from −5 to +5 V (Fig. 8). It should be noted that even in a high-resistance material like zinc oxide, there are areas with unstable I–V characteristics. An exemplified curve is given in Fig. 9.
Discussion of results As noted above, it is easy to distinguish between the scanning tunneling atomic force microscopy and scanning spreading resistance microscopy employing Eqs. (1) and (2). The higher values of R than those obtained by substituting ρ value (specific resistance) of the sample material into Eq. (2) correspond to the approximation of the tunneling AFM. So, the R value obtained for the direct branch is of the order of about 10¹¹ Ω. As it is known from the band theory [21], tunneling of charge carriers is difficult because of the corresponding position of the Fermi level in metal (probe), where the Fermi level in the semiconductor (dielectric) is in the band gap. As follows from I–V curve (see Fig. 7), the width of the energy gap is close to the band gap for bulk ZnO (3.36 eV). Furthermore, it may be concluded that nanolayers exhibit n-type conductivity (the Fermi level coincides essentially with the bottom of the conduction band). The similar results were obtained in Ref. [22] where InAs nanostructures on GaAs substrates were investigated by conductive AFM. Authors of [22] also used probe tips made of Si with diamond coating (DCP11, see Fig. 4). According to their assumption, such behavior of I–V curve is caused by a formation of a Schottky-type junction between the probe tip and the semiconductor surface.