In the previous section, basically WKB approximation was used for the tunneling probabilities through vacuum and insulator barriers. Let us examine this assumption. For tunneling, the criterion for applicability of WKB approximation can be written as¹⁰:

where d is the barrier thickness, H - its height, E - energy of the particle, and m - its mass.

Fig. 8 Energy diagram for the tip-sample tunneling through vacuum and insulator layers, with applied bias.

Schematic representation of the energies involved in the tunneling junction considered in our problem is presented in Fig. 8. For the vacuum barrier, we have the average height H = f + ½V and particle has energy E. (Note that in Fig. 8 E=0 is taken at E_F of the sample, rather than the tip, so some formulas appear differently from the ones used in Sections II.2 - 4). For typical values    (H-E) > f - ½V ~ 2 eV   and   d ~ 11 Å    the above WKB criterion becomes:    11*(0.51*2)^½ = 11.1 >> 1

As we can see, for the vacuum barrier the use of WKB-type formula may be justified. One can also note that similar approximations have been successfully used for modeling of "cold emission". Therefore, at the desired level of accuracy, we can leave this simplification intact.

From Fig. 8 it is clear however, that for bias values V ~ E_CBM the WKB criterion is certainly not satisfied. As a matter of fact, the energy of the particle can be below, equal or above the barrier height (E_CBM in this case). Therefore, even with the square barrier assumption, the complete formula for the tunneling probability should be used. Fortunately, for this case it has been derived in most textbooks on quantum mechanics (e.g. [10]):

H - barrier height x - energy of the electron, m - electron mass deq - "equivalent" barrier width given by deq = dins(meff)½ - combination of the insulator thickness dins and effective mass meff.

Fig. 9 Comparison of the tunneling probabilities in case of WKB-type approximation (solid line) and exact formula for tunneling through a square barrier (dashed line). Barrier height is 10 in both cases.

For comparison of the tunneling probabilities resulting from the WKB-type formula used in Sections II.3-4 and the above expression see Fig. 9. The barrier height was set to 10 in both calculations. Note, that the former formula (solid line in Fig. 9) gives a sharp change at the energy equal to the barrier height, whereas the latter expression (dashed line in Fig. 9) exhibits a smoother rise, and never flattens out at the higher energies. Since the dI/dV extrema basically correspond to the extrema of the tunneling probability, the position of the first peak will not be exactly at the barrier height, and there will be more peaks at higher energies, albeit not quite as well pronounced. We can expect the position of the peak then to be related to the barrier height (E_CBM) and the width of the peak to depend on d_eq.

Fig. 10 Resonance peak dependence on ECBM and deq. Note that the peaks are significantly shifted from respective ECBM's.

The parameters of the resonance peak do depend significantly on both ECBM and deq, as shown in Fig. 10 (compare to Fig. 5). The peaks themselves became broader and more asymmetric compared to Fig. 5. Only for values of deq > 5 Å the peaks are actually easy to discern (in the simpler model peaks, albeit small ones, are visible for all values of dins). In addition, we do indeed observe extra oscillations at higher energy ends of the curves.

Dependence of the peaks' parameters on the model's variables can be summarized as follows (Fig. 11). Peak position is almost invariably different from E_CBM, and for each value of E_CBM the patterns of deviations are very similar (Fig. 11 left). For a fixed value of E_CBM peak height depends on d_eq as a * d_eq^b where a and b are constants (Fig. 11 center). a depends strongly on E_CBM, and b = 2.3 in all cases. The width of the resonance depends strongly on d_eq, but only slightly on E_CBM (Fig. 11 right). Actually, it is hard to define a width for such asymmetric peaks (width at half maximum was used in Fig. 11 right), so E_CBM dependence may not be real.

	Fig. 11 Resonance peak parameters dependence on ECBM and deq: position (left), height (bottom left), width (bottom right). All three exhibit strong dependence on deq , but only the position and the height depend strongly on ECBM.

And finally, an attempt was made to reproduce the experimental data (Fig. 2) with this improved model. The resulting curves are presented in Fig. 12, they are still qualitatively similar to the ones from Fig. 2. The values of E_CBM were taken as 2.1 eV for CaF₁ and 3.5 eV for CaF₂ (arrows in Fig. 12), with the above mentioned shifts accounting for the correct peak positions (at 2.3 and 3.7 eV respectively - dashed lines in Fig. 12). Vacuum barrier widths (d_tip) were taken the same as in the previous model (10 and 11 Å respectively). "Equivalent" thickness d_eq was chosen to produce peak heights close to the observed ones, the values were 9 and 10 Å respectively. If we recall that d_eq = d_ins(m_eff)^½ and the values of the actual insulator thickness d_ins used before (2.7 and 3.93 Å), we would need m_eff ~ 10.

Fig. 12 Calculated tunneling spectra obtained from a minimal model for tunneling through an insulator film. Spectra are shown for CaF2 and CaF1, analogous to the data in Fig. 2.