A 0.15 µm channel length nMOSFET was simulated with DAMOCLES. All results below are for the device operating with Vgate=2.5, Vdrain=1.9 and Vsub=0 V, with the source grounded. The lattice temperature is 300 K. The electrons flow from source to drain under the influence of gate- and drain-fields; as they do, they gain energy and impact ionize at the drain end of the channel. Impact ionization consists of an energetic electron colliding with a valence band electron with such force that this valence electron moves to the conduction band. This creates an electron-hole pair, consisting of the electron added to the conduction band and the hole left behind in the valence band. The created electrons are swept out the drain contact, but the majority of the holes flow towards the p-type substrate and are responsible for the substrate current.

The plot below shows the positions of the (approximately) 8000 electrons present in the simulation relative to the device cross section.

The electrons are colored according to their instantaneous kinetic energy: blue for "cold" to red for "hot" electrons (from 0 to 1.5 eV as indicated by the color scale in the figure). The pool of blue electrons in the upper left reside in the source, while those in the upper right reside in the drain. The thin red lines show the metallurgical source- and drain-junctions. Electrons are not simulated inside the heavily-doped source and drain, except for a thin "skin" of particles in the neighborhood of the metallurgical junction. This is done to help reduce computer time: in this calculation, there is no merit in simulating the behavior of the thermal electrons deep within the source and drain.

This plot shows a close up of the previous plot, in the vicinity of the drain-channel junction.

This image makes the presence of hot electrons at the drain-end of the channel more evident. Note how the hottest electrons (the reddest particles) are actually present in the drain, where there is a high density of thermal electrons. The interaction between these very hot and very cool electrons must be treated carefully in order to quantitatively model the electron-hole carrier density created via impact ionization. The metallurgical drain-channel junction is seen in this image as a thin red line.

The corresponding potential distribution in the device cross section is shown in this next plot.

The potential values at the bottom of the conduction band are shown, and vary from approximately -2.0 to 1.5 V. The roughly 3.1 eV jump in the potential when moving from Si into SiO2 is readily seen as a color discontinuity across the top of the image.

The preceding images of electron position and potential can be "combined" into one image, showing more clearly the electron distribution in energy and the driving force responsible for pushing the electrons from source to drain. This is shown in the following plot of total (kinetic + potential) energy versus position from source to drain in the MOS channel along the Si-SiO2 interface.

The bottom of the conduction band is shown as a cyan line, and gives the potential energy value from source (on the left) to drain (on the right). The source has a higher potential energy than the drain; electrons lose potential and gain kinetic energy as they move from source to drain. Kinetic energy is explicitly indicated by the distance of the electron above the potential energy line, and also, by the color of the electron. If there were no scattering, the total energy would remain constant during the transit from source to drain, and as a result, electron trajectories would be horizontal lines in this plot. Scattering causes the total energy to change. Energy may be gained or lost in a scattering event, but for hot electrons, energy loss is favored. Thus, as electrons move from the source towards the drain and gain kinetic energy, there is an increased chance of energy loss due to scattering. Scattering accounts for the distribution in energy of the electron flux as it approaches the drain. This information is important in quantifying hot electron effects.

The same information, total energy versus position in the device, can also be portrayed as a three-dimensional image, as shown below.

The bottom of the conduction band is indicated by the gray surface. The source is at the right center and the drain the lower left of the figure. The view is from above the gate (not shown) towards the channel. The electrons are portrayed as spheres, colored by their kinetic energy. The reservoirs of thermal (blue) carriers in the source and drain are apparent, as is the flux of electrons moving from source to drain. The substrate potential confines the electrons from moving too far away from the Si-SiO2 interface (which is located exactly at the front right edge of the image at y=0.90 µm). The 3.1 eV discontinuity in potential occurs along the y=0.90 µm position: electrons are constrained from falling towards the viewer, i.e., electrons may not move into the region y>0.90 µm.

The velocity of the electrons is plotted below versus position from source (left) to drain (right).

Noteworthy is the fact that the velocity exceeds the saturated velocity in Si, which is about 1x10^7 cm/sec at 300 K. This effect is termed "velocity overshoot", and is the direct result of the distance between scattering events approaching the channel length. The result is off-equilibrium transport, where bulk transport properties no longer apply. While velocity overshoot only occurs near the drain in this example, as the lattice temperature is decreased or the channel length is made shorter, one expects velocities near the source to exceed the saturated velocity as well.

The velocity overshoot effect translates into an increase in device transconductance as the channel length is decreased. If the velocity were limited to the saturated velocity, transconductance would saturate as well as channel length shrinks. The plot below indicates this is not the case, either theoretically or experimentally. The experimental data in this figure derive from the pioneering work done here at Watson Research by George Sai-Halasz and coworkers, described in IEEE Electron Device Lett., vol. 9, no. 9, pp. 464-466, 1988.

Transconductance effects device switching speed; therefore, modeling off-equilibrium transport quantitatively is of great interest to device designers.

As electrons travel from source to drain, they gain sufficient energy to impact ionize. The rate at which electron-hole pairs are generated is shown in this image of the nMOSFET cross section.

The rate is colored by a logarithmic scale, as shown in the figure. Even though the majority of impact ionization occurs within the metallurgical drain region, the majority of holes created still flow towards the substrate as dictated by the local potential. In this device, approximately 90% of the generated holes flow towards the substrate and are responsible for the measured substrate current.

This flow of holes away from the drain and towards the substrate is shown in the image below, which shows hole density (on a logarithmic scale) versus position in the device.

The flux of holes moving towards the substrate is readily identified in this image.

While all of the preceding images have depicted physical quantities in the two-dimensional device cross section, it is important to remember DAMOCLES simultaneously tracks electrons in the three-dimensional wavevector space, i.e., in the Brillouin zone. In the image below, the position of electrons in the Brillouin zone is shown.

These electrons all reside at the drain-end of the nMOSFET channel, and are again colored according to kinetic energy. The lowest energy electrons (blue) are seen to reside near the six energy minima of Si (along the positive and negative axes directions, 85% of the way from the zone center to the zone edge). These electrons are sampled from the thermal population of drain electrons. There are other, more energetic electrons, present as well in this image. They are located away from the valley minima, and correspond to electrons which have gained kinetic energy as they transited from source to drain. In every case, electrons in this image reside in energy states consistent with the Si band structure.

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