One of the major causes of the degradation of VLSI devices is the damage created by high-energy ('hot') electrons as they travel from source to drain along the channel: Some electrons gain from the electric field enough kinetic energy to be injected into the silicon dioxide gate-insulator. This energy then becomes - somehow - the source of degradation: Atomic bonds are broken at the Si-SiO₂ interface and/or within the oxide. Ultimately, the electric charge associated with these defects will cause a shift of the threshold voltage of the device, increased Coulomb scattering for electrons in the channel - and so a degradation of the mobility -, or even catastrophic dielectric breakdown of the gate insulator. In any event, bad news.

Some questions about the physics

1. We must be able to describe the 'source function', i.e. the energy and velocity distribution of those electrons which are able to create the damage. This requires understanding electron transport in Si to a very high level of accuracy and confidence and up to very high electron energies. As explained in detail elsewhere, we have barely started to learn how to describe electron transport in Si in this regime.
2. We must understand the atomic structure of the Si-SiO₂ interface and of the SiO₂ near-interfacial region, in order to understand by which mechanisms the energy is transferred from the electrons to the ions and how damage is ultimately generated. This problem is outside the scope of DAMOCLES, but the literature on the subject is impressive, testifying to its difficulty.
3. We must tackle some new problems, specifically related to the Si-SiO₂ interface. Even if we are able to accurately calculate the energy, momentum, and spatial distribution of electrons everywhere in the device, how do we handle the transition of electrons from Si into SiO₂? Specifically, as illustrated in Fig. 1:
   • Is parallel momentum conserved? Since SiO₂ is thermally grown in amorphous form, the translational symmetry on the interface plane is broken. Therefore, we should not expect necessarily that the component of the electron crystal(?)-momentum, k_||, lying on the interfacial plane should be conserved. Indeed, Weinberg[1] has argued that conservation of parallel momentum is violated as electrons tunnel across the triangular SiO₂ barrier in Fowler-Nordheim tunnel-injection.
   • Is the image-charge a valid concept? The classical, static concept of 'image charge' describes the electrostatic potential caused by a test charge near the interface between two media with different dielectric constants. But can we still apply this concept when quantum mechanical effects (such as tunneling) take place, and when the test-charge (the tunneling or transmitted electron) moves very fast? Again, Weinberg and Hartstein[2] have argued that the image charge arises only when the electron tunnels. Thus, one should rescale the classical image force by the tunneling probability. Similarly, it has been argued[3] that tunneling processes occur too fast for the image charge (microscopically: the polarization charge induced by the test charge at the interface) to form. Dynamic correction should also suppress, in a complicated way, the classical expression for the image potential.
   • What about transport in SiO₂? If the image charge is indeed a valid concept in this context, how should one handle the singularity of the potential right at the interface? And how should one treat transport of electrons in the image-force-lowered barrier region in SiO₂?
   • Is the WKB approximation valid? Usually, tunneling is treated using the standard textbook Wenzel-Kramers-Brillouin (WKB) approximation. But the literature abounds with discussions about its range of validity and there is confusion about the 'prefactor' to be used in front of the 'usual' exponential term.

There are even more problems concerning, for example, the three-dimensional atomic structure of the interface and how one could treat tunneling in a more realistic fashion than what's usually implied by 'abrupt, square' potential barriers; the role of inelastic tunneling processes; the effect that the electron gas in the Si channels has on the dielectric properties of the interfacial region and on electron tunneling and injection, etc. etc. But those few problems mentioned above are sufficient to give an idea of the complexity of the problem.

Ning's experiment and DAMOCLES simulation

The typical 'testing ground' for models attempting to model electron injection into SiO₂ is the simulation of the devices Ning and co-workers[4] employed to measure the injection probability. These are long-channel MOSFETs on p-type substrates. By connecting the source and drain contacts together, negatively biasing the substrate, positively biasing the gate, and shining light on the device, one creates electron-hole pairs deep in the substrate. While the holes drift away to the substrate, electrons are accelerated towards the interface, 'falling down' the large substrate-to-channel 'cliff'. Most of the electrons 'bump' against the Si-SiO₂ barrier, thermalize, and are eventually collected by the source or the drain contacts and contribute to the 'channel current', I_channel. But a few electrons (typically, one in a million or billion!) will be lucky enough to acquire so much energy as to be injected by tunneling at high energy into the oxide, or even jump over the top of the barrier. These electrons will be measured as 'gate current', I_gate. By varying the substrate bias, V_sub, and the gate bias, V_gate, from the ratio I_channel/I_gate one extracts the injection probability as a function of the accelerating potential and field in the oxide. Figure 2 illustrates the situation for a substrate bias of 7.5 V.

Some answers about the physics

DAMOCLES has been used to mimic the experimental scenario and confirm the following answers to the questions raised above (See Refs. [5] and [6] for ample discussion of these issues):

1. Conservation of parallel momentum. At a microscopic level, there are theoretical[7] and experimental[8] indications that even amorphous SiO₂ exhibits some order after all: The Si-O network organizes itself in six-membered rings of O-Si-O elements in a fashion very similar to the structure of the most common form of crystalline SiO₂ found in nature, alpha-quartz. The average size of these rings, about 3nm, is larger than the wavelength of a high-energy electron hitting the interface. Therefore, the electron cannot possibly 'feel' the disorder over larger distances. Rather, the interface will appear to be an interface between crystalline Si and crystalline SiO₂. Thus, we should expect that parallel momentum is conserved in the Si-to-SiO₂ transition. This is confirmed by the results of the simulation (left frame of Fig. 3): Simulations done ignoring momentum conservation, but, instead, allowing all electrons with total energy larger than the Si-SiO₂ barrier height (about 3.2 eV ignoring image-force effects), yield much larger injection probability than experimentally observed.
2. Static, classical image force. As argued by Puri et al.[9] and by Binning and co-workers[10] in the context of scanning-tunneling-microscopy, the time-scale involved in tunneling (or transmission over the barrier, we may add) is relatively slow. Here 'relatively' means relative to the time required by the surface charge to re-polarize itself as the electron tunnels. This is so because the frequency of the surface plasmons is very high, so that there's plenty of electrons in the valence bands of Si to induce a very fast and complete screening of the field associated with the injected electron. The singularity at the interface associated to the image-charge is removed by many-body effects[11]. Once the image force concept is accepted, electrons must be allowed to scatter in the SiO₂. The Monte Carlo approach employed in the past[12], with minor corrections[13] seems to be adequate, as shown in the right frame of Fig. 3.
   
   

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   Figure 3. Calculated probability of electron injection into SiO₂ at a fixed substrate bias as a function of the square-root of the oxide field. At left, the experimental data from Ref. [4] (solid black line) are compared to DAMOCLES results accounting for (red circles) or ignoring (solid green dots) conservation of parallel momentum. At right, comparison is made with simulation results obtained by ignoring image-force corrections (blue open triangles), or accounting for it, but ignoring (solid green dots) or accounting for (red circles) scattering in SiO₂. (Adapted from Ref. [5]).

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3. Transfer-matrix for tunneling. Finally, DAMOCLES is quite non-chalant and cavalier about tunneling issues. Indeed, tunneling does not seem to be the dominant effect in Ning's experiments. Nevertheless, enough arguments have been brought up by Zhakharova and colleagues[14] to indicate that at least a transfer-matrix approach, together with an empirical k·p-model for the SiO₂ band structure, is required in order not to excessively brutalize the physics.

Some astronomical conclusions

The success of a model in predicting (or, in our case, postdicting) experimental results should not be considered proof that the model is correct. After all, the Copernican model [A] of the solar system, in its original -- if not historically correct-- formulation based on circular orbits, didn't improve at all the ephemerides computed on the basis of the mathematically -- but not 'psychologically'-- equivalent Tycho Brahe's model [B], or even of the centuries-old Ptolemaic system [C]. (The Prutenic Tables, which replaced the Alfonsine Tables based on the ptolemaic system, did indeed improve upon the latter ones, but at the expense of complicating the Copernican system with many unaesthetical circles and epicycles, bringing it almost at the level of the ptolemaic model). This is an example of an 'incorrect' theory (Thyco's) yielding better agreement with the data ("save the phenomena", in the jargon of those times) than a 'correct' (Copernicus') one. What really matters is that Copernicus' model planted the seed of a 'full-band solar-system model' [D], which finally allowed predictions [E] to be made, which is what counts. Yet, the arguments we have listed above, and the success shown in Fig. 3 are at least suggestive that the gross features of the model are not too far off the mark (...but: Is DAMOCLES like Ptolemy or is it like Kepler?)

The major problem of the model implemented in DAMOCLES remains its heavy computational cost. For some, this is a small price to pay for physical correctness. For others, more empirical and faster methods may be preferable.

The abstract of an article[6] on which this page is based is available from the IBM Research Division CyberDigest.

References

1. Z. A. Weinberg, J. Appl. Phys. 53, 5052 (1982).
2. Z. A. Weinberg and A. Hartstein, Solid-State Commun. 20, 179 (1976).
3. R. Reifenberg, D. L. Haavig, and C. Egert, Surf. Sci. 109, 276 (1981).
4. T. H. Ning, C. M. Osburn, and H. N. Yu, J. Appl. Phys. 48, 286 (1977).
5. M. V. Fischetti, S. E. Laux, and E. Crabbé, J. Appl. Phys. 78, 1058 (1995).
6. M. V. Fischetti, S. E. Laux, and E. Crabbé, in "Hot Carriers in Semiconductors", K. Hess, J.-P. Leburton, and U. Ravaioli Eds. (Plenum, New York 1996), p. 475. (Abstract available from the IBM Research Division CyberDigest).
7. K. Hübner, in "The Physics of SiO₂ and its Interfaces", S. T. Pantelides ed. (Pergamon, New York, 1978), p. 111.
8. E. J. Grunthaner, P. J. Grunthaner, M. H. Hecht, and D. Lawson, in "Insulating Films on Semiconductors", J. J. Simonne and J. Buxo eds. (North-Holland, Amsterdam, 1985), p. 1.
9. A. Puri and W. L. Schaich, Phys. Rev. B 28, 1781 (1983).
10. G. Binning, N. Garcia, H. Rohrer, J. M. Soler, and F. Flores, Phys. Rev. B 30, 4816 (1984).
11. P. A. Serena, J. M. Soler, and N. Garcia, Phys. Rev. B 34, 6767 (1986).
12. M. V. Fischetti, D. J. DiMaria, S. D. Brorson, T. N. Theis, and J. R. Kirtley, Phys. Rev. B 31, 8124 (1985).
13. E. Cartier and F. R. McFeely, Phys. Rev. B 44, 10689 (1991).
14. A. Zhakharova, V. Ryshii, and V. Pesotzkii, Semicond. Sci. Technol. 9, 41 (1994).

1. Copernican system: "De Revolutionibus Orbium Coelestium" (Johannes Petreius, Nürnberg, 1543). Additional information and Web source.
2. Tychonic system and orbit of the comet of 1577: "De mundi aetherei recentioribus phoenomenis" (Uraniborg, 1588). Additional information and Web source.
3. Ptolemaic system: George Trebizond, "Commentary to the Almagest of Ptolemy" (ca. 1482). Additional information and Web source.
4. Kepler's elliptical orbit of Mars (dashed line, Sun at n): "Astronomia Nova..." (Ernst Vögelin, Prague, 1609). Additional information and Web source.
5. Newton's calculations for the orbit of the comet of 1680: "Philosophiae Naturalis Principia Mathematica" (Samuel Pepys, London, 1687). Additional information and Web source.

	damoclesNO-SPAM@watson.ibm.com (last updated: January 26, 1999)