Scanning Tunneling Spectroscopy (STS) measurements for insulating systems are notoriously difficult and often yield ambiguous results because of exponentially vanishing tunneling currents for biases within the forbidden energy gap of insulators. This inherent limitation can be circumvented by measurements in films only few atomic layers thick. In recent experiments with CaF₂ films on Si(111) an unexpected peak in STS spectra has been found at energy close to that of conduction band minimum of CaF₂. This result is interpreted using simple models of tunneling through thin insulating films.

Scanning Tunneling Spectroscopy (STS) over the last decade has proven to be a very valuable tool in studies of clean metal and semiconductor surfaces and various adsorbates on these surfaces^1,2. Application of this technique to insulators however faces two major obstacles.

First of all even Scanning Tunneling Microscopy (STM) imaging seems to be hardly possible for insulating surfaces, since the tunneling current will be very efficiently attenuated inside the insulating film. The only way to avoid this exponential dampening of the tunneling current is to inject electrons into the conduction band of the insulator, where they should be able to propagate. A typical insulator has an optical forbidden energy gap of about 10 eV and such extreme bias voltages are not practical for STM. Fortunately, for an insulator film grown on a semiconductor substrate, the energy difference between the conduction band minimum (CBM) and the Fermi energy (E_F) can be smaller than the gap because of the band alignment. Fig. 1 (left) illustrates this phenomenon for the case of CaF₂ film on Si(111) substrate. The Fermi level of the system is pinned near the top of the Si valence band. The top of the valence band of bulk-like CaF₂ is placed at 8.5 eV below E_F. Therefore, even though the band gap of CaF₂ is 12 eV, the CBM should be only about 3.5 eV above E_F. So STM imaging is in fact possible for positive sample biases of 3.5 eV and above^4,5. Attempts to stabilize the tunneling current at a bias voltage below the CBM causes the tip to approach the underlying Si and to pick up insulating CaF₂ debris or crash-land on the sample.

Fig. 1 Schematic of chemically-selective tunneling into insulator films, using CaF₂ on Si(111) as example:
Left: Stabilizing the tip at a sample bias that allows tunneling of electrons from the tip into the conduction band minimum (CBM) of the insulator.
Right: Acquiring a current image at a bias where electrons from the tip enter the gap of the insulator.
Bands with empty states are outlined with red, filled states - blue and shaded gray. Dashed line marks E_F.

In STS mode the feedback loop of the STM is turned-off, the bias voltage is ramped between the specified values and the tunneling current is recorded. Thus an I(V) curve is obtained in one point above the surface. The procedure can be repeated for multiple points (usually a square array). Note, that during the motion between these points the feedback loop has to be on, which means that a specified sample bias - tunneling current combination is maintained. Again, for insulators the bias needs to be rather high ( e.g. 3.5 eV for CaF₂, as explained above), which effectively limits the bias range available for STS². Basically the tunneling current drops exponentially for bias voltages smaller than the stabilization value, and since a typical setup only has sensitivity that spans a few decades, the smallest possible bias voltage is limited by the smallest detectable tunneling current. That problem can be somewhat alleviated either by the correct choice of stabilization parameters, or by employing other STS techniques².

The second obstacle of performing STS with insulators is related to the interpretation of the STS data. The I(V) curve is dominated by the exponential dependence of the tunneling current I on the bias voltage V. One way to obtain sample specific information is to calculate (dI/dV) / (I/V) - a quantity called normalized conductance, which has been shown to approximate the density of states of the sample^2,6. The problem is that for bias voltages below the CBM the tunneling current I = 0 for bulk insulators and semiconductors, while the conductance dI/dV remains finite. The normalized conductance (dI/dV) / (I/V) thus has a singularity near the CBM, which has been experimentally observed for bulk semiconductors⁷. The appearance of this singularity can be avoided by using alternative forms of the normalized conductance, but one has to be careful not to introduce any artifacts into the resulting spectra.

The normalization problem described in the previous section can be averted if we utilize thin films, where the tunneling current I remains finite inside the band gap. The states of the silicon substrate decay exponentially through the CaF₂ film, leaving a finite density of states at the CaF₂ surface available for tunneling.

Fig. 2 Tunneling spectra of CaF1/Si(111) and CaF2/CaF1/Si(111). Two sharp onsets in the (dI/dV) spectra characterize the respective conduction band minima and provide chemical selectivity. The normalized (dI/dV)/(I/V) spectra exhibit resonances at the CBM. Different line types represent different tip-sample separations.

The experimental⁵ (dI/dV)/(I/V) tunneling spectra obtained from CaF₂/Si(111) and CaF₁/Si(111) interfaces are presented in Fig. 2. The conduction band edges correspond to onsets in the (dI/dV) curves. In the normalized spectra well-defined peaks are observed at the CBM. The peak is three times as high as the continuum above it for CaF₂ and five times as high for CaF₁. In search of an explanation for this phenomenon several options can be considered⁵. Here a model for tunneling through a thin insulator film is developed, based on established approaches for planar tunneling⁶. The outcome produces the observed resonances and even their absolute height, suggesting that the most important physical effects are indeed captured in the model with a simple barrier structure. Some possible extensions of this approach for more realistic barrier potentials will be discussed as well.

Since the appearance of the STM most tunneling theories are based on Bardeen's tunneling current formalism⁸, which has been adapted for the STM by Tersoff and Hamann⁹:

where f(E) is the Fermi function, V - the bias voltage, M_mn - the tunneling matrix element between the states y_m of the tip and y_n of the surface. In special cases M_mn and the whole above expression can be simplified. In particular case of interest to STS of semiconductors and insulators we are dealing with the applied bias of the order of 2 eV, which is not small in comparison to kT (0.026 eV at room temperature). For tip-sample bias smaller than a typical work function ( 4-5 eV ) the tunneling current at a fixed location on the surface can be approximated⁶ as:

where r_T(E+eV) is the density of states of the tip, r_S(E) is the density of states of the sample and T(E,V) is the transmission probability of the electron of energy E through the tunneling barrier. Even in this simplified model, a number of assumptions about all the three functions in the integrand can be made to construct more or less realistic models. Several such models will be considered in the following sections.

The first approximation is to assume that both the tip and the sample are essentially metallic and have the same work function f and constant density of states. The assumption of a constant tip density of states is often made, lacking detailed knowledge of the tip states. The Si density of states has little structure at the energies under consideration, i.e., about 4 eV above the valence band maximum.

Assume that the tunneling probability through the vacuum is given by:

T_vac(E,V) = , where

which corresponds to WKB approximation for tunneling through a barrier of the average height.

Assume that the total tunneling probability T(E,V) = T_vac(E,V)T_ins(E,V).

The following are the values of the "fixed" tunneling barrier and environment parameters:

f = 4 eV - work function for the tip and the sample

kT = 0.026 eV - assume room temperature

d_tip = 11 Å - tip-sample separation (typical value for sample bias of 3-4 V)

d_ins - thickness of the insulator film in Å

These are the values used in all the following models, unless specified otherwise.

In the simplest approximation one can neglect the effect of a finite temperature and consider sharp step functions instead of the Fermi function f(E) for metallic density of states. The tunneling probability through the insulator T_ins is assumed to be:

That is electrons tunneling into the gap see a constant height barrier, and electrons tunneling into the conduction band of the insulator propagate freely.

Fig. 3 Normalized conductance calculated including the contribution from the tunneling probability only. Adjustable parameters: dins = 3 Å ECBM = 3 eV

Amazingly enough, even this simplest model produces the resonant peak at CBM in normalized conductance (Fig. 3), and the absolute values that are not too far from the observed ones (compare to Fig. 2). Typical values are assumed for the insulator thickness d_ins and E_CBM.

The next logical step is to include the effects of the thermal broadening that will be present to some extent at room temperature. It is done by assuming Fermi distribution for the occupied states in both the tip and the sample. That is assume:

Fig. 4 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.

It appears that at this stage the model captures the essence of the process, so it will be examined in some detail. First we should assign realistic values to d_tip and d_ins. In case of CaF₂ (Fig. 2 right curves) the stabilization bias voltage was about 4 V, which corresponds to about 11 Å tip-sample separation. CaF₂ layer grows on top of CaF₁, so if we take CaF₂ layer + the fluorine plane from CaF₁ layer as the insulator, the thickness should be 1¼*3.14 = 3.93 Å. Similarly, for CaF₁ (Fig. 2 left curves) the tip was stabilized at about 3.5 V (d_tip = 10 Å) and the insulator layer thickness is 2.7 Å. Now, the only adjustable parameter in the model is the position of the CBM. Since in this model the peak position coincides with E_CBM, we can set it to 2.3 eV for CaF₁ and 3.7 eV for CaF₂, to match the observed values from Fig. 2. The dI/dV and (dI/dV)/(I/V) curves obtained from this model are presented in Fig. 4. Note the similarity to Fig. 2, even in the absolute values of the normalized conductance. Such similarity is surprising for modeling results with basically no adjustable parameters.

Fig. 5 Peak height dependence on ECBM and dins.

The model also provides some predictions, such as the dependence of the peak height on ECBM and dins. The resonance peak basically scales with both in bilinear fashion: (dI/dV)/(I/V)max ~ 2.4 (eV Å)-1 dins ECBM (see overview in Fig. 5)

Fig. 6 Apparent singularity in (dI/dV)/(I/V) near ECBM = 4 eV, as insulator thickness d tends towards infinity.

Another very important observation, is that as dins tends towards infinity, so does the peak height. As it was pointed out before, in practice there is always the smallest detectable tunneling current limiting our observations for thick (i.e. bulk-like) films, so only a certain range of the normalized conductance will be observed. If the peak height corresponding to a particular film thickness is sufficiently greater than the observable range, than the normalized conductance will appear to have a singularity as it approaches ECBM (Fig. 6). For ECBM = 4 eV asymptotic dins approaches infinity behavior is reached already for film thickness 15 Monolayers (ML). But finite peaks in the normalized conductance should still be observed for 1-5 ML thick insulator films, with this limiting width decreasing with increasing ECBM.

Fig. 7 Normalized conductance singularity near ECBM. Comparison of the model curve with data for bulk GaAs.

It is interesting to note that this modeling of the singular behavior seems to be quite universal. In Fig. 7 the model's prediction is compared to the actual data for bulk GaAs. Note that in this case the only adjustable parameter in the model was the value of E_CBM = 1.02 eV, which was assigned in the original paper by Martensson and Feenstra⁷.

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.

The simplicity of the two models examined above in some detail implies that this list of possible improvements should be fairly long and by no means exhaustive. A more realistic representation of the vacuum barrier would be the first obvious next step. To start, one could use the actual formula for the tunneling probability through a triangular barrier, rather than the "average" square one used in the above discussion. Next step would be the inclusion of the image charge effects, long recognized6 to be important for tunneling. Generally the introduction of image charge potentials amounts to smoothening of all the "sharp" features of the barrier potential.

Another interesting effect to consider, would be the voltage drop in the insulator. In the two models examined, it was set to zero, but in fact it is probably not. As suggested in Fig. 1, one would expect to have some voltage drop in the insulator, when tunneling through it (i.e. for biases within the gap). That drop is expected to be smaller than the one in the vacuum, because of the large dielectric constants of insulators (e.g. e = 7 for CaF₂). Once electrons start passing through the conduction band of the insulator, the voltage drop will probably become negligible, however one still may ask whether some effect will be left in the first atomic layer.

For an even more realistic consideration one would need to solve the 3-dimensional, rather than planar problem. The framework for performing such calculations has not been established sufficiently well though, so at this point both the modeling and the interpretation of these models are far from being straightforward.

This work would not have been possible without the guidance of my advisor Prof. F. Himpsel and very helpful discussions with Prof. D. Huber. I would like also to thank fellow graduate students R. Boutchko and R. Haslinger for sharing their experiences regarding numerical methods.

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