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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.
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 [1]–[4]. 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 [5], [6] 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
Scattering theory relates the steady-state current to transmission and reflection coefficients [7]–[9]. 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
From Fig. 1, we can write the steady-state flux entering the
drain as
where we have assumed that the flux entering the source
from the channel does not backscatter. At the entrance to the
, we can deduce the electron density from the
positively and negatively directed fluxes as
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
By integrating (3) over
and using
we find
(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
is just
as given by (6). An expression which works more
generally results by multiplying (6) by (7) to obtain
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
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 [11]. 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
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
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
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) [13].
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
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. [3],
[4]). 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
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 [14], but others find little difference [15]. This issue
deserves further attention. It is also clear from (5), however,
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 [1],
[2]. 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 [16],
[17]. 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 [18]—except that
is replaced
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
decreases with temperature while
The well-known increase in
as the temperature is reduced
[19] 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
This letter presented a very simple scattering theory of
the MOSFET. It emphasizes the critical importance of the
source to channel transition region in these small devices and
the need to design efficient carrier injectors at the source. It
demonstrates that there is an upper limit to
which is set by
thermal injection from the source and that velocity overshoot
in the channel does not extend this limit. It shows that the
concept of mobility continues to have relevance to ultrashortchannel MOSFET’s as a measure of channel backscattering.
Finally, these ideas should also prove useful in interpreting
the results of more detailed simulations and in identifying the
performance-limiting factors in devices.
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