Hot electrons and low bias

Hot-electron problems affecting the reliability of devices in the present VLSI technology consist mainly in damage at the Si-SiO₂ interface caused by the presence of either secondary carriers created by impact-ionization[1], or of hot carriers which can be directly injected into the SiO₂ gate insulator. While the details of the damaging process are still being debated in the scientific and engineering communities, it is clear that both types of processes are threshold processes, that is, they depend on the presence of carriers which have kinetic energies exceeding the energy-gap (1.1 eV in Si at 300K), or the energy of the Si-SiO₂ interface barrier, about 3.2 eV.

As devices get smaller and the applied bias is lowered, one may wonder whether these hot-electron problems are going to disappear as the supply bias is reduced below these thresholds. As a general rule in nature, sharp threshold are always 'smeared out' by various sources of broadening. The case of interest here is no exception. As also shown by the group at the University of Birmingham[2] [3], DAMOCLES[4] has indicated that the Coulomb interactions among carriers are responsible for a strong 'thermalization' of the charge carriers, resulting in the presence of high-energy tails above the expected 'cut-off' at the supplied bias.

The 'kink' effect in Si SOIs at low bias

Here we consider the example of impact ionization in a 0.25 µm gate-length (0.18 µm effective channel length) Si SOI FET with a 7nm-thick gate oxide on a 80 nm uniformly-doped (10¹⁷ cm^-3) Si layer. According to naive expectations, as the voltage applied between the source and the drain of the device is lowered below the energy of the Si band-gap, one would not predict the occurrence of any impact ionization events: The applied bias is simply not sufficiently large to cause electrons to acquire enough kinetic energy to impact ionize. Yet, experimentally, this is not what is observed: SOI's are devices in which holes created by impact ionization events occurring manily at the channel/drain junction drift towards the substrate and remain 'trapped' for long times (in the typical time scale of electron transport). Before eventually recombining, they contribute to a charging of the 'body', which is seen as a shift of the threshold of the device. Since holes 'pile up' in the substrate til a dynamic equilibrium between recombination (at the bottom Si-SiO₂ interface) and generation (via impact ionization) is established, the device acts as a 'integrator' of the generated charge, allowing the detection of ionization events even at levels so low that would go undetected in conventional 'bulk' nMOSFETs. Figure 1 (kindly provided by Dr. G. Shahidi) shows clearly the so-called 'kink effect': at a source-to-drain bias of only 0.8 V, the threshold of the device is seen to shift to progressively smaller values, as indicated by the drain current becoming larger. Eventually, at much larger values of the applied bias, conventional avalanche breakdown is observed. Similar observations on a variety of devices [5] [6] [7] have confirmed that indeed impact ionization is responsible for the presence of the 'kink', as well as of the harder-to-detect substrate current in n-channel MOSFETs[8]. Even more interesting, gate-currents have been observed in devices biased at voltages well below the expected 3.2 V-threshold[6] [9].

So: Where are the electron finding the energy to overcome the thresholds of these processes?

Dynamically-screened e-e scattering

Figure 2 provides the answer. There, the energy distribution of the electrons entering the heavily-doped drain region are shown, as calculated with DAMOCLES runs including (red line) or suppressing altogether (blue line) the long and short-range Coulomb interactions. The blue line shows three distinct regions: At very low energies are the cold electrons in the drain. Intermediate-energies (0.3 to 1.0 eV) are populated by 'channel electrons' accelerated by the longitudinal field in the channel. Finally, above the maximum energy a ballistic electron can gain (1.0 eV, given by the applied voltage of 0.8 V plus the Fermi energy in the source contact), one sees a high-energy tail dropping exponentially with slope given by the lattice temperature. These 'above-supply-voltage' tails have been studied in the past[10] [11] and have been shown to be caused by 'superlucky' electrons which have absorbed more phonons than they have emitted[12].

While the presence of these 'superlucky' electrons will indeed cause a small amount of ionization, many more high-energy electrons are seen in the red curve. This is the anticipated result of the electron-electron interaction.

This process is described by a matrix element M_ee which, in a simple form (Born approximation, no spin, distinguishable particles; but see the expression actually used by DAMOCLES) is:

The dielectric response of the free carriers is represented by the screening parameter ß, frequency (i.e., energy transfer) and wavelength (i.e., momentum transfer) dependent. Often these dependencies are ignored and the simple static, long-wavelength Debye-Hückel expression is employed. However, in small FETs this might be a very bad approximation. We show in Fig. 3 the ratio of the screening parameter (calculated in the high-temperature approximation of Fetter and Walecka[13]) to its (nondegenerate) Debye value in Si. It can be seen that screening becomes negligible at large wavevectors, even in the static approximation. Physically, this means that perturbing potentials of very short wavelength cannot be screened. The electron gas supports collective oscillations (plasmons) of long wavelength, but as the wavelength of the plasmons shortens, they decay (Landau damping) and the electron gas stops responding. Similarly, Fig. 3 shows that at high frequencies (above the plasma frequency), the electron gas is unable to respond.

Along most of the channel of a very short Si FET the Debye approximation is valid: The quasi-ballistic nature of transport implies very narrow energy-distributions (i.e., low electron temperatures, but high average energies). Thus, the interaction among electrons involve small energy (and, often, quasi-momentum) transfers. The relatively low density of free carriers and the strong static, long-wavelength Debye screening dictate that a small role is played by Coulomb interactions along most of the channel. Thus, the suggestion that Coulomb interactions could be invoked to use a Maxwellian distribution along the channel of a short FET[6] does not appear to be well founded. On the contrary, as we approach the drain we face the possibility that a `cold' electron in the heavily-doped drain region interacts with a `hot' electron coming from the channel. In this case, classically the most likely event is a collision in which the hot channel-electron exchanges (loses or, less frequently but very importantly, gains) a small fraction of its kinetic energy with a drain-electron. But since electrons are undistinguishable particles, the `exchange' collision is obtained by switching the role of the two carriers. (This is illustrated in Fig. 4). This results in a very large energy transfer. The matrix element corresponding to this event is thus essentially unscreened: In simple words, the fast `channel electron' is moving too fast for the sluggish `drain electrons' to follow its potential and screen it. As a result, the Coulomb interaction is extremely effective in thermalizing the distribution functions, typical thermalization times being in the 100 fs range.

Anomalous temperature dependence

The resulting generation rate as a function of drain/source bias is illustrated in Fig. 5: The red line (relative to room temperature simulations) shows that generation rate persists to biases well below 1 V. Of interest is also the fact that at a lower lattice temperature the ionization rate is enhanced, as expected, at large biases. But as soon as the bias is lowered towards the ionization threshold, the larger band-gap (and so a higher ionization threshold), large screening (and so a weaker effect of Coulomb collisions), and a reduced high-energy tail of the electron energy distribution at the source (the 'source-function effect') result in a lower generation rate. This is consistent with several experimental observations[5] [7] [8] [9] [14]. While other theoretical results[15] [16] [17] have emphasized the role of quasi-ballistic transport in short devices coupled to the source-function effect, Fig. 6 illustrates that a different lattice temperature causes a small difference in the electron energy distribution under quasi-ballistic transport. Rather, the shift of the energy -threshold for impact ionization, albeit small, affects exponentially the generation rate, since it is sensitive to the high-energy exponential tail of the electron energy distribution.

To conclude with a direct answer to our initial question ("are hot-electron effects a thing of the past?"), it should be said the very seldom one does see an `abrupt' threshold in Nature, broadening and thermalizing effects almost invariably smoothing-out a sharp threshold expected on the basis of oversimplified models. Having realized the importance of the dynamically-screened Coulomb interactions among electrons, the presence of very soft thresholds must be expected. By the same token, the `3-volt threshold' expected for the onset of electron injection into the gate oxide is as fictitious as the `1-volt threshold' we have just discussed. The only difference we might expect is the absence of the `crossover effect', as indeed observed experimentally [9]: The reduced role played by the short-range interparticle collisions in the channel of the device (where gate injection is most likely), and the weaker temperature dependence of the threshold (i.e., the Si-SiO₂ barrier) will cause a `normal' behavior, as far as the temperature of the process is concerned. Yet, the basic idea (a smooth, soft threshold induced by the thermalizing effect of Coulomb scattering) controls this process as well. In a few words, hot-electrons effects are going to be important, even as the applied bias is reduced.

Abstract and post-script version of an article[4] on which this page is based are available from the IBM Research Division CyberDigest.

References

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