The spherical indentation analysis of germanium [7] [8] is reported here for comparison with the further examples in Table 1. Figure 1 provides the basic data from the F_N vs πh^3/2(R/h − 1/3) plot and by using Equations (1) and (2). It shows the penetration resistances k₁ and k₂ as the slopes for the two phases up to 50 mN load.

The phase-transition onset of Figure 1 is at 10.703 mN. The energetic data give the indentation work W_indent = 0.4756 mNµm and the transition energy W_transition = 4.1157 mNµm, as calculated up to 50 mN load. The transformation pressure is also calculated to give the good correspondence of 2.3 GPa with the anvil pressurizing phase-transition at 2.5 GPa. A nevertheless published trial plot in [7] with F_N vs h^3/2 (Equation (3)) for excluding h^3/2 as prescribed by ISO standards and false Johnston’s formula for spherical indentations gave a convex plot

instead of linearity. A further endothermic phase-transition is already indicated at the end of the second straight line.

The spherical indentation onto ZnO with wurtzite structure follows Equation (1) and it reveals two phase-transitions. The analysis had to be performed after repair [10] of the published pop-ins in [12]. The plotted data are in Figure 2, where the included regression data are the basis for the calculations.

The spherical indentation onto ZnO exhibits two phase-transitions, distinguishing 3 polymorphs in the force range up to 100 mN load, as revealed with the plot by application of Equation (1) to the published original load-depth data. The inserted linear regression equations are the basis for the calculation of the energetic terms and the pressure data in Table 1.

The spherical indentation onto a GaN epilayer (R = 4.2 µm) was reported in 2002 [13] and the analysis (after pop-in repair [10] ) with Equation (1) gives the linear plot with two phase-transitions (three linear branches) as shown in Figure 3. The inserted regression formulas allow for the calculation of the onset forces, the energetic terms, and onset pressures in Table 1.

The more recent spherical indentation onto a single crystal of GaN [14] cannot be compared. Unfortunately, the analysis of these data with Equations (1) and (3) using the techniques for the various further materials in Section 4 shows that these published depth data are fitted according to the ISO14577 standard to concur with the disproved Johnson’s formula. These data and conclusions are totally at variance. It would be nice to see the original untreated data.

The results with Ge, ZnO, and GaN are compared in Table 1.

These materials cover maximal loads that are 50 mN for Ge, 100 mN for ZnO, and 250 mN for GaN. More phase-transition onsets are to be expected at higher loads. The force for the first phase-transition onset describes the sensitivity of the materials for their stability with respect to mechanical interactions. It is equally reflected by the sequence of the penetration resistance values k₁ (physical hardness) in Figures 1-3. The penetration depth values are not in the same sequence and neither so the first indentation work that are required for reaching the transition onset. But the transition work values are for the first and second transition of ZnO and GaN in the same sequence as the F_Nkink values. The data reflect the situation of spherical indentations with the same radius covering one or two phase-transitions per sample. The depths are always very low. The results are calculated from 0 to kink, from kink to kink, and from kink to the maximal force. The transition-energy and the onset pressure values can also be calculated for every force of interest, but they cannot be normalized per force as in the pyramidal case. The energy law requires multiplication of the phase-transition F_Nkink values with W_indent/W_applied = F_Nindent/F_Napplied [7] [15] when only the penetration is addressed, for obeying the energy law. The advantage of experimental spherical indentations is the reliable pressure calculations at the transition onsets. The close similarity of flat surface and cap surface for the low depths is very favorable. It allows the comparison with hydrostatic pressurizing and new insights are therewith achieved.

The pressure data of germanium has already been compared with anvil pressure as combined with X-ray diffraction date. The low anvil pressure of 2.5 GPa required enormous effort for being detected and was long disregarded, as outlined and discussed in [7]. It corresponds with the onset pressure of 2.3 GPa in Table 1. The present analysis of the spherical indentation is by far easier than anvil experiments but we profit from the X-ray proof for the low-pressure GeI to GeIII phase-transition in a highly rewarding manner. The pressure values for ZnO in Table 1 up to 100 mN load are not large enough for the B4 to B1 (NaCl-type) transition, because the hydrostatic pressure for that phase-transition are reported to start at 9 GPa [16] or 9.8 GPa [17]. There are numerous reports on theoretically calculated caged ZnO polymorphs [18] that are not better space-filling than the wurtzite structure (B4) of these materials and need not to be considered here. The theoretical calculations of 7 bulk polymorph structures of ZnO with respect to expected optical properties is also not helpful, because these lack density predictions [19] and none of these possible structures have been found by anvil pressurizations. The difficulties with resolving low pressure transitions under anvil are already discussed above with germanium. There remain the zinc-blende (=sphalerite) B3 phase and formations of twinned polymorphs. The more highly pressurized of these (onset at 4 GPa) is most likely the (B3) phase of ZnO that could already be epitactically grown on (001) of GaAs on a ZnS substrate at 500˚C. The growth of the film was by metalorganic molecular-beam epitaxy with diethylzinc + O₂, “using electron cyclotron resonance plasma source to excite high density oxygen plasma with low-ion energy of 10 - 20 eV”. The lattice constant of 4.463 ± 0.015 Å was obtained from the RHEED (high-energy electron diffraction) pattern [20] [21]. The fact that the growing of the film succeeded indicate that the pressure for its formation must be rather low. We can thus confidently claim having detected the first synthesis technique for the bulk ZnO B3 polymorph under the diamond calotte cap for further investigation. It is provisionally attributed to the 4 GPa onset pressure. The spacious wurtzite tetrapod with its complicated twinned shape including legs, as obtainable from thermal “vapor deposition on a polymer decorated silicon substrate” [22], must be excluded. The high tendency of ZnO to form twin structures is known [23]. A ZnO twin had already been synthesized by pulsed laser deposition [23]. It can also be obtained by electro-deposition on indium doped tin oxide (ITO) [24]. Again, spherical indentation appears to be the easiest synthesis of this species. These attributions must be confirmed by on-site X-ray diffraction at a synchrotron, or spectroscopically. This will include their further characterization. The spherical indentation results of ZnO reveal that the hydrostatic anvil pressurizing experiments did not resolve these lower pressure phase-transitions. The almost uniform pressure distribution in the low depth spherical indentations is particularly suitable for the suggested investigations.

The GaN B4 phase (wurtzite) transforms upon hydrostatic pressurizing at 47 GPa into the GaN B1 phase (rock-salt) [25]. Another report found this transition pressure at 37 GPa [26]. These publications do not report on transitions at lower pressures that are again not remarked under anvil pressurizing. Thus, the B4 to B3 (zinc-blende = sphalerite) transition pressure was not known. The ab-initio calculations of [27] predict a pressure of 11.45 GPa for the B4 into B3 transition. This value is close to our 12 GPa value in Table 1 and we assign accordingly. Very interesting is our 10 GPa value in Table 1. When compared to ZnO this phase-transition occurs at 0.65 times the depth at 1.75 times the force and at 2.75 times the pressure. This may at first glance indicate twinning of GaN, but why should that require so much force and pressure? Another attribution comes to mind: it could be the formation of the rhombohedral phase (R3m) of GaN that has recently be found in the B3 (zinc-blende, F4—3m) phase of GaN epilayers, as grown on sapphire by molecular beam epitaxy [28]. This would prevent the twinning by pressurizing upon spherical indentation. Unfortunately, no energetics is available for the rhombohedral phase. The rhombohedral GaN (R3m) forms also via migration enhanced encapsulation growth by encapsulation between silicon and graphene [29]. It is thus easily formed. This would also support our attribution of the first phase-transition onset pressure of GaN to the rhombohedral polymorph with space group R3m. Clearly, experimental spherical indentation extends the knowledge from hydrostatic anvil pressurizing. Both assignments should be checked by Synchrotron X-ray diffraction together with the now far easier further analyses under the diamond calotte of these elusive and now easily available polymorphs.

The spherical indentation (R = 10 µm) of GaAs in [32] does not follow the equation for spherical indentations (1) and the Authors cited paper [4]. The analysis of the published loading curve—after repair of the “pop-in” in their Figure 1(a)—with Equation (1) for spheres does not give a straight line with Equation (1), but the concave curve of Figure 4. This indicates the belief of the authors in Johnson’s formula from 1985 and the application of a fitting procedure to the original loading data to concur with such formula.

Conversely, the application of Equation (3) for cones and pyramids as falsely claimed by Johnson to the data of [32] leads, after a short initial effect, to two straight lines, the first steeper as the second by forming a kink. This is shown in Figure 5. The disproved Johnson’s formula requires, of course, linearity with the

exponent 3/2 on h for fitted spherical indentations. However, the putative exothermic event does not represent the endothermic phase-transitions of GaAs, and also the slopes are simulation and iteration artefacts without any value.

This data check proved the already complained data fitting. It appeared therefore necessary to compare the values in the caption of Figure 5 with the ones from the correct analysis of Berkovich indention loading curves onto GaAs. These are taken from Figure 1 in [33] at low penetration (up to 4.5 mN) and from Figure 3(a) of [34] at high penetration (up to 600 mN). These genuine experimental curves were analyzed to yield kink points for endothermic phase-transitions at 2.517 for loads up to 5 mN and 218 mN for loads up to 600 mN. The first of these with onset at 2.517 mN is not resolved at this loading range. The second of these loads in Figure 6 starts pretty close to the one in Figure 5, but the calculated mechanical data are principally different. The second branch is steeper than the first one, indicating the endothermic phase-transition with the onset (kink position) at 218 mN load and 1.40 µm depth. The normalized Berkovich indentation transition-energy with onset at 218 mN is endothermic at +0.161 mNµm/mN [this work], as deduced with Equation (3) for cones and pyramids and its integration in [7] [11]. Thus, the spherical indentation with R = 10 µm onto GaAs is a misleading fitting artifact showing again the data treatment in [32]. The importance of the argument deserves the printing of the Berkovich indentation analysis of the experimental result from [34] in Figure 6.

Figure 6 where Equation (3) is correct for the pyramidal indentations and the Kaupp-plot [11] proves undoubtedly that the phase-transitions of GaAs under load are endothermic.

As we present strong arguments, we must check whether further fitted spherical indentation exhibit the corresponding behavior.

The analysis of the spherical indentation onto aluminum [35] reveals a similar outcome as with GaAs. The sphero-conical indenters are precisely described with a cone half angle of 45˚ and nominal radii of 10 µm and 5 µm. Scanning electron microscopic images indicated “well formed” radii of 8.5 and 5 µm down to depths of 2.5 and 1.5 µm, respectively. The nominal radius of 10 µm corresponds to h_cone = 2.93 µm. This is considerably larger than the maximal depth of <1 µm) for the indentation data of pure Al. Again, the analysis with Equation (1) for spheres does not give a straight line but the concave curve of Figure 7, which clearly indicates a data fitting for concurring with Johnson’s formula and unfortunately ISO standard.

The trial plot with Equation (3) to the spherical indentation in Figure 8, using the same fitted data as for Figure 7 is linear with after the extended initial effect two straight lines. These simulate an exothermic event, because the second branch has the lower slope. This is the false and dangerous result of the fitting error as above with GaAs (Figure 5): the phase transition of aluminum under load is endothermic, as was already shown with Berkovich indentations onto aluminum in [31].

Again, the calculations of elastic moduli according to false Johnson’s formula and its discussion are useless and misleading. The data fitting is again safely confirmed with Figure 7 and Figure 8. The previous branch is again steeper than the following one, simulating a false exothermic event. The known phase-transition of aluminium upon Berkovich indention has the endothermic phase-transition onset at close to 40 mN load (two different sources) [31]. The similarity with the above GaAs case repeats the falsifications by the fittings. Again, the falsified data for an h^3/2 loading parabola retain only the information that there must be a phase-transition at a similar force, as characterized with a Berkovich indentation.

Data fitting destroys the value of spherical indentation and excludes any use of them. “Fitted” data points must not be told or suggested as being “experimental” ones.

Spherical indentations (R = 8.5 µm) of silicon were published in [36]. Photos of a sphere and a description of their indenter setup as “All of the force-displacement measurements were made on a UMIS-2000 instrument”, and also the copious description how “The indentations were carried out” are indicating that they might have been without data-fitting. Only the cone angle of the conoidal indenter was not disclosed. However, the authors relied in their text on the incorrect Johnson’s formula with the impossible F_N vs h^3/2 relation for spherical indentations. This must again be severely challenged. Our analysis of the published “experimental” data pair crosses from their Figure 4(a) (similarly in the further images) give again no straight line when the Equation (1) for spheres is applied to the published data. The concave curve in Figure 9 is obtained instead.

The data-fitting is again additionally secured with the trial Kaupp-plot (F_N vs h^3/2) in Figure 10, using the so called “observed” but in reality, fitted loading curve data pairs (the data crosses next to the simulated curve in [36] ). This second proof is evident by the linearity with exothermic unsteadiness at about 30 mN and very pronouncedly at 80 mN load.

As shown with GaAs and aluminium, the residual information of exothermic unsteadiness tells only that phase-transitions will be found by Berkovich indentation. The steepness of the preceding lines in Figure 10 is again higher than

that of the following ones. This seems to be indeed typical for the undue fitting procedure. The genuine silicon phase-transitions are all endothermic: the Berkovich indentation onsets of Si (100) at 4, 15 and 25 (data taken from [37] ) or 29 and 81 mN (data taken from [38] ) were analyzed in [39] ). They reveal endothermic phase-transitions.

Unfortunately, the more recent spherical indentation with R = 5 µm of [40] also used Johnson’s formula and the Authors do not disclose their cone angle. We therefore do not discuss it here.

Datye et al. in [41] published spherical indentation data of silicon carbide (SiC). They used their SiC-N brand “that is similar to (0001) of the single crystal”. With their sphere radii of 25 µm and 7.5 µm they reached fully elastic or elastic and plastic indentations, respectively. The data were “fitted” to “Hertz spherical contact solution”, which means: the disproved Johnson’s equation was again the fitting target. These published load-depth curves with fitted depths do again not follow the Equation (1) for spherical indentations. They give the concave curve when tested with Equation (1) and they analyze linear with the trial Kaupp-plot (that is only valid for pyramids and cones) according to Equation (3) without unsteadiness up to 475 mN load. After short initial effect, the simulated slopes of 1030 (R = 7.5 µm) mN/µm^3/2 or 3742 (R = 25 µm) mN/µm^3/2 are obtained with correlation coefficients of 0.9997 or 0.9999, respectively. These do however not describe any materials’ property, but only reflect the fitting efficiency. We nevertheless determined these slopes despite the data-fitting, for provisionally checking the influence of the tip radius influence. Interestingly, despite the fitting treatment of the depths, the ratio of these slopes (3.7) is similar to the ratio of the radii (3.3). It should be further studied whether such a relation holds also for spherical indentations with untreated experimental depth data. The mayor errors of the simulating procedure appear to be the modulus E* as calculated with the false Johnson formula.

The phase-transition pressures of SiC have been calculated to 102 and105 GPa (hexagonal 6H to 1B) or (cubic 3C to 1B = sodium chloride) phase, respectively, and the experimental shock data have them at about 100 GPa [42]. This is far from being reached with the maximal force of 500 mN in [41].

There seems to be the same data fitting techniques in all of the here analyzed cases, but we still need further analyses with a crystalline oxide.

The authors of [43] used a polished and rinsed (001) surface of MgO for a “spherical indenter in diamond” with an iterated radius of 9.5 µm. Their published Figure 1(c) in [43] is completely reversible up to 300 mN load and 0.4 µm depths. The data fitting is clearly revealed by their reference to [4] and again by the concave plot that results by application of Equation (1) to the published force-depth curve. We need thus not deal any further with this report.

For rounding up our knowledge of the falsifying effects of data-fitting spherical indentations we also need the analysis of a technical multi-component material. A spherical indentation onto a standard microhardness steel block (500 HV30; H/E = 0.04), using a sphero-conical tip with radius of 7.2 µm [44] and cone half angle of 45˚ [35] (we calculate h_cone = 2.11 µm) appeared appropriate. Actually, the nominal radius of 5 µm ± 6.6 nm (we calculate h_cone = 1.46 µm) was increased by a “well-fitting simulation” to 7.2 µm. And the “nominal values of 210 GPa and 0.3 were assumed for Young’s modulus and Poisson’s ratio in all simulations”. The load-depth data were taken from Figure 5(a) in [44]. The application of Equation (1) for spherical indentation is valid for every tip radius as long as the h_cone value is not surpassed. But the depth values for a certain force are strongly dependent on the tip radius (cf. Section 4.4.). Again, concave curves (not shown here) but not straight lines are obtained both for nominal 5 µm or iterated 7.2 µm radius. These analyses tell that our already multiply complained data-fitting was again performed in [44]. Figure 11 ensued, when the Kaupp plot according to Equation (3) was applied to the clearly fitted loading data. It exhibits three linear branches that simulate two unsteadiness points both with misleading exothermic behavior. Neither the slopes nor the exothermicities are usable. The known phase-transitions of steels and iron are endothermic. A numerical comparison with their onset forces would require a Berkovich indentation of this particular multi-component steel. It should be stressed that the radius R must be measured but never be simulated. It plays an important role for the outcome of experimental spherical indentations.

The unsteadiness onsets in Figure 11 cannot be termed phase-transition values, as they would falsely claim “exothermic” behavior and the slopes are worthless. They are equally false as in the cases for GaAs, Al, and Si. Despite the obvious errors of data-fitting for obeying ISO 14577 standards, the authors of [44] relied on the false Johnson formula for “Young’s modulus” determinations from a simulated loading curve. These are nothing else than faked results.

Very strange is the publication of Figure 3 in [44]. The tip radius had been shifted from 50 µm ± 1.4 nm to 115 µm, because the data could “only be well fitted” to the simulated Johnson loading curve with R = 115 µm. Such behavior is

absurd. The Authors of [44] did not at all recognize that the course of a spherical indentation strongly depends on the radius R. They tried to justify their shifting with the inconceivable claim that “a slight flattening of the spherical shape over the 4.5 µm radius contact region, arising from as little as 0.11 µm at the center, is sufficient to produce an increase in spherical radius of this amount” (from R = 50 µm ± 1.4 nm into R = 115 µm!). Tip radii are mechanical data that cannot be shifted by iterations “as necessary”. While all of that already proves the multiple data treatment, we nevertheless checked the fitted data with our test methods. The application of Equation (1) give the differently sized concave curves (not imaged here) for the data pairs with R = 115 µm and for the simulated curve with nominal R = 50 µm ± 1.4 nm. This confirms the data fitting, as in all of the other tested cases in Section 4.

Furthermore, both simulated curves for the spherical indentation “of a steel standard hardness block (900 HV30 nominal” with H/E = 0.4 in Figure 3 of [44] ) give linearity when trial plotted according to Equation (3). The simulated R = 50 µm radius curve gives an exothermic unsteadiness at about 100 mN and 0.150 µm (not imaged here) with our calculated pressure of p = 2.11 GPa. Conversely, the simulated R = 115 µm curve in [44] gives the single straight line in Figure 12 again with the maximal force at 200 mN from the first to the last data point without any unsteadiness. This only tells the high fitting precision that is obtained with the published equations in e.g. [6] (as written down in Section 1). Our calculated pressure in Figure 12 at its end with 200 mN load gives p = 1.58 GPa. This clearly misses the unsteadiness at p = 2.11 GPa with R = 50 µm, as such a high pressure is not reached with a more than twice as high radius R. It stresses however the enormous influence of the tip radius. In the present case one must be alerted of not mixing up the fitted spherical indentation (that is totally

worthless) with a pyramidal or conical indentation that would give a totally different slope (yet unknown for steel 900 H/E 0.4). It appears rather strange that the paper [44] could pass the Reviewers and Editors for its publication in the Journal of Materials Research. Fortunately, our mathematically sound analyses are now able to detect the invalid simulations, fittings, and iterations, by using Equation (1).

The plot in Figure 12 again underlines the unique power and necessity of checking publication data from spherical indentations with the correctly deduced Formula (1) in [5] and [7]. It reveals: both of the manipulated data in Figure 3 of [44] (R = 115 µm or of nominal 50 µm) are totally worthless due to various simulations, iterations, and fittings for concurring with the disproved Johnson equation that does not take care of the depth-dependent R/h ratio during the penetration. Even worse, the influence of the tip radius to the applied pressure was not acknowledged when nominal tip radii were changed by iterations (5 into 7.2 or 50 into 115 µm).

A very recent report deals with the spherical indentation of several steels with a ball of radius 250 µm, in e.g. Figure 10 of [45]. The Authors used the techniques of [35] and published several “experimental” indentation curves. We analyzed the one for DC01 steel. The test with Formula (1) for spheres gives the concave curve (not imaged here) corresponding to the ones that we always obtained when the spherical indentation data were fitted. This paper [45] covers high loads up to almost 200 N. The trial test with Equation (3) for conical or pyramidal indentations gives three linear branches at very high load with two unsteadiness points, simulating again worthless exothermic behavior. This F_N vs h^3/2 plot in Figure 13 proves that the simulation, iteration, and fitting techniques from 1993 [44] and 1995 [35] are unfortunately still in active use.