IEEE ELECTRON DEVICE LETTERS, VOL. 18, NO. 7, JULY 1997 361 Elementary Scattering Theory of the Si MOSFET Mark Lundstrom Abstract— A simple one-flux scattering theory of the silicon ) characteristics MOSFET is introduced. Current–voltage ( are expressed in terms of scattering parameters rather than a mobility. For long-channel transistors, the results reduce to conventional drift-diffusion theory, but they also apply to devices in which the channel length is comparable to or even shorter than the mean-free-path. The results indicate that for very short channels the transconductance is limited by carrier injection from the source. The theory also indicates that evaluation of the drain current in short-channel MOSFET’s is a near-equilibrium transport problem, even though the channel electric field is large in magnitude and varies rapidly in space. C HANNEL lengths and supply voltages of MOSFET’s are rapidly decreasing, and a key issue is to maximize the saturated drain current for short-channel, low-voltage devices. Identifying the limiting value of as approach zero and understanding the role of velocity overshoot are important issues that have been examined experimentally and theoretically –. This letter presents a simple scattering model for which should be useful for interpreting detailed simulations and for guiding device design. Electrons are injected from the source into the channel across a potential barrier whose height is modulated by the gate voltage. Carriers drift across the channel and are collected by the drain. The existence of the source to channel barrier is well known and routinely considered when treating drain-induced barrier lowering or weak inversion operation. The barrier also exists above threshold, but the channel charge screens the gate voltage, so has less influence on the surface potential and the transconductance drops below its bipolar limit. This analogy between the MOSFET and the bipolar transistor (and the particular importance of the source to channel barrier) is well known ,  but is frequently ignored in contemporary MOSFET analysis. This practice has been justified because transport across the channel has been the limiting factor, but transport across the source-channel barrier will increase in importance and will ultimately limit as Scattering theory relates the steady-state current to transmission and reflection coefficients –. Under saturated conditions, the two-section model displayed in Fig. 1 applies. The source is treated as a reservoir of thermal carriers which injects a flux to the source-channel barrier. A fraction of the source flux transmits across the source-channel barrier and enters the channel. A fraction of the flux injected into Manuscript received February 5, 1997. This work was supported by the Semiconductor Research Corporation under Contract 96-SJ-089. The author is with the Department of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907-1285 USA. Publisher Item Identifier S 0741-3106(97)05092-1. the channel transmits across and exits the drain, and a fraction backscatters from the channel and reenters the source. From Fig. 1, we can write the steady-state flux entering the drain as (1) where we have assumed that the flux entering the source from the channel does not backscatter. At the entrance to the channel , we can deduce the electron density from the positively and negatively directed fluxes as (2) is the velocity of the positively and negativelywhere directed fluxes (assumed equal) and is the depth into the channel. Solving (2) for and inserting the result in (1) we find (3) By integrating (3) over and using (4) we find (5) (The approximation assumed in (4) allows us to express the final result in familiar form, but is one of the reasons for the word “elementary” in the title of this letter.) Equation (5), the key result of this letter, provides a simple expression for without invoking the use of a mobility, a quantity of unclear significance in nanoscale MOSFET’s. In the ballistic limit and the maximum current is controlled by the injection velocity at the thermal source. More generally, the key issue is to evaluate the channel backscattering coefficient, The most accurate approach would involve Monte Carlo simulation, but simpler, approximate approaches would also be useful. Monte Carlo simulations show strong velocity overshoot and extreme off-equilibrium transport in short-channel MOSFET’s, but simple estimates of turn out to be reasonably accurate. If there were no electric field in the channel, the 0741–3106/97$10.00 © 1997 IEEE 362 IEEE ELECTRON DEVICE LETTERS, VOL. 18, NO. 7, JULY 1997 is just as given by (6). An expression which works more generally results by multiplying (6) by (7) to obtain (8b) Fig. 1. Two-section model of the MOSFET under saturated conditions. The first section describes the transmission of carriers from the source into the channel and the second transmission across the channel. A third section would generally be required, but under saturated conditions, carriers that enter the drain end of the channel cannot transmit to the source. Fig. 2. Backscattering coefficient for electrons injected into a Si slab with a constant electric field. Two cases, slabs 0.1 and 0.5 m long, are considered. The points are the results of one-dimensional Monte Carlo simulations, and as given by (8). the lines are analytical estimates for backscattering coefficient could be estimated from (6) where is the mean-free-path [9, Eq. (2.2.2)] or [10, Eq. (33b) with ]. When a channel electric field is present, we can use Price’s observation that carriers which travel only a short distance down a potential drop are unlikely to reemerge—even if they backscatter . Recognizing this as the Bethe condition for thermionic emission (in reverse) we take the critical distance to be the distance over which the potential drops by to write (7) Finally, we note that is deterwhere mined by the electric field profile very near the source where the electrons have been heated by no more than about , so can be estimated from the low field mobility in [10, (A9) with ], to write (8a) which is [12 (33)]. Equation (8a), however, approaches 1 as the electric field is reduced toward zero. In a short-length sample (such as the short channels we have in mind), some electrons will transmit across the field free channel, so Its value Fig. 2 shows the results of a Monte Carlo evaluation of compared with the simple analytical expression (8b). A model “channel” with or m and a constant electric field are assumed. The agreement is not exact, but the simple expression agrees remarkably well with the Monte Carlo simulations. Equation (8) raises a subtle point. The thermal velocity in (8) is the two–dimensional (2-D) thermal velocity since it refers to transport in the inversion layer, but in (5) it is the three–dimensional (3-D) thermal velocity because it refers to carriers injected from the source. (The appropriate thermal velocity is the velocity directed at a plane, for a nondegenerate 2-D or 3-D gas.) Note that presented here can be done more the simple “derivation” of formally with essentially identical numerical result (but with more complicated expressions) . To establish a connection between the scattering approach and conventional models, consider a device with a channel , then by inserting (8a) in (5), length long enough that we find an expression for (9) which is identical to the conventional result except for the Conventional theory implicitly assumes appearance of an infinite supply of carriers at the source end of the channel. is maximized by increasing the Equation (9) shows that mobility or electric field at the source, which reduces backscattering. Even though strong off-equilibrium transport occurs, is determined by carriers near the source which have not been greatly heated. In nanoscale MOSFET’s, therefore, the low field mobility continues to have physical significance as a measure of the mean-free-path in the critical region. Finally, note that this scattering viewpoint provides an explanation for the inflection point in the channel velocity versus position characteristics observed in Monte Carlo simulations (e.g. , ). This point marks the transition from a thermally injected flux to an off-equilibrium flux in the channel. The drain current is largest when this thermal flux is maximized and when channel backscattering is minimized. A high field gradient causing the at the source end of the channel will reduce velocity to approach, but not exceed, the thermal injection velocity. Finally, what is the role of velocity saturation and velocity overshoot? Both effects should influence the drain to source saturation voltage because both influence the carrier density in the channel and, therefore, the self-consistent field at the Some workers see a difference source, which determines in the potential barrier at the source when velocity overshoot is treated , but others find little difference . This issue deserves further attention. It is also clear from (5), however, LUNDSTROM: ELEMENTARY SCATTERING THEORY OF THE Si MOSFET that an “effective velocity” deduced from is a measure of the average velocity at the source end of the channel and not of the average velocity in the channel. This is well known, but scattering theory gives the upper limit as the thermal injection velocity. Of course, the injection velocity of the degenerate Fermi gas may exceed cm/s, which may explain reports of velocity overshoot in MOSFET’s , . Generally, source injection, channel backscattering at the source end, and velocity saturation or overshoot in the channel should all be considered in a current balancing approach , . Finally, note that our attention has been focused on the steady-state current. Clearly, velocity overshoot will reduce the transit time, and, therefore, increase As expressed in (9), the key result is remarkably similar to conventional velocity saturation models which use a piecewise linear velocity-field model —except that is replaced by In silicon, these two velocities happen to be similar, which may explain why the velocity saturation model works remarkably well, even for devices with very short channels for which velocity overshoot is not expected to occur. The models may, however, predict different temperature dependencies because decreases with temperature while increases. The well-known increase in as the temperature is reduced  is presumably a result of the fact that although in (5) decreases, this is offset by reduced scattering at low temperatures. Careful studies of the temperature-dependent transconductance would be a useful test of the scattering model. This letter presented a very simple scattering theory of the MOSFET. 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