360
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VOL.
8. NO.
4
APRIL
1989
A
General Purpose Device Simulator Coupling Poisson and Monte Carlo Transport with Applications to Deep Submicron
MOSFET’s
FRANCO VENTURI, R. KENT SMITH, ENRICO C. SANGIORGI,
MEMBER,
IEEE,
MARK
R. PINTO,
MEMBER,
IEEE,
AND
BRUNO
RICCO,
MEMBER,
IEEE
AbstractAn efficient, selfconsistent device simulator coupling Poisson equation and Monte Carlo (MC) transport has been developed, suitable for general silicon devices, including those with regions of high dopingkhrrier densities. Key features include an srcinal iteration scheme between MC and Poisson equation and an almost complete vec torization
of
the program. The simulator has been used to characterize nonequilibrium effects in deep submicron nMOSFET’s. Substantial overshoot effects are noticeable at gate lengths of
0.25
pm at room tem peratures.
I. INTRODUCTION
EVICE simulation tools used in semiconductor tech
D
ology development have to date been based primar ily on the driftdiffusion (DD) model for carrier transport, a simplification of the more general Boltzmann transport equation
zyxwvutsrqp
11. Recently reported experimental work on sil icon MOSFET’s [2],
[3]
however, indicates that non equilibrium transport effects, not directly treatable using DD
,
become important in determining intrinsic device characteristics at small geometries.
A
simulator which could treat the Boltzmann equation directly would be able to correctly model these effects, the extent of which still remains unclear. Furthermore, the analysis
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f
hot carrier induced phenomena could be addressed more rigorously The numerical method generally proposed to solve the Boltzmann equation
is
the Monte Carlo (MC) technique, first applied to the charge transport in semiconductors by Kurosawa
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5].
The MC method consists of following in space and time the history of each individual particle and does not rely on the relaxation time approximation. The major drawback of the MC approach comes from the intensive computational requirements that become
[41.
Manuscript received July 27, 1988; revised October 17, 1988 and No vember
4,
1988. The review
of
this paper was arranged by Guest Editor J. Fossum. F. Venturi and B. Ricco are with the Department
of
Electronics, Uni versity
of
Bologna, Bologna, Italy. R.
K.
Smith and M. R. Smith are with AT&T Bell Laboratories, Murray Hill, NJ 07974.
E.
C. Sangiorgi is with the Department
of
Physics, University
of
Udine, Udine, Italy.
IEEE
Log Number 8825906.
particularly severe if the carrier distributions obtained from the MC calculation must be computed selfcon sistently with the electrostatic potential, given by Poisson equation.
As
a result, in the past the MC method has been felt to be impractical for use in studying devices such as MOSFET’s with very heavily doped source and drain re gions
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6].
The purpose of this paper is to describe a new, com putationally efficient, selfconsistent MC simulator which can analyze devices with completely general geometries and impurity profiles. As an example of the applications and usefulness of such a tool, an analysis of nonequilib rium effects in deep submicron MOSFET’s will be pre sented showing that at gate mask lengths of
0.25
pm these can account for an additional
20
percent increase in per formance at room temperature.
11.
THE
MONTE
CARLO
METHOD
The MC method applied to charge transport in semi conductors [7] relies on the simulation of a large number of carriers subject to external forces (electric field) and given scattering mechanisms. A carrier trajectory consists of a sequence of ballistic flights (free flights), terminated by collisions (scattering events). The duration of each free flight and the type of terminating scattering event are se lected stochastically, according to given probabilities de scribing the microscopic processes. Because each particle is followed individually along its entire trajectory, a com plete set of informations, including carrier position, mo mentum and energy can be collected; these data can be suitably averaged to obtain local carrier densities and dis tribution functions in a form similar to that provided by macroscopic models (e.g., DD). Due to its intrinsic nature, the MC method provides a microscopic description of the physical transport phenom ena; in addition it is structured in such a way that new and/or more sophisticated scattering models can be easily added to the simulator. Most importantly, this approach avoids the assumption of the relaxation time approxima tion which may not be valid in some conditions of inter est. Furthermore, energy threshold phenomena like im
02780070/89/04000360 01
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SIMULATOR COUPLING
POISSON
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MC TRANSPORT
36
1
pact ionization, gate current, and hotcarrier degradation can be most successfully treated [4], [SI because the en tire energy distribution function is calculated, while other methods rely
on
average quantities of the distribution.
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s
already mentioned, the computational constraints would seem to limit extensive use of this technique in technology development in particular if real, multidimen sional devices are treated.
111.
FEATURES
F
THE
SIMULATOR In order to adequately deal with advanced devices op erating under extreme conditions (for example, with high internal fields) a MC simulator should present the follow ing characteristics: a) the carrier distribution function and the electrostatic potential (obtained via Poisson equation) must be computed selfconsistently b) should be CPU time efficient to allow an extensive use in technology de velopment; c) include accurate physical models for the dominant scattering mechanisms, though saving the com putation efficiency; d) include terminating regions where safe (i.e., equilibrium) boundary conditions can be im posed; e) should be able to deal with “rare events” such as low carrier concentrations compared to high ones and a few hot carriers in the presence of a multitude of colder ones. The first point, as will be shown later, is important in submicron devices where nonequilibrium phenomena cause the electrostatic potential to differ significantly from that obtained with conventional (DD) simulations. The desired consistency can be achieved by means of an iter ative procedure coupling Poisson equation and the MC transport solution and, to this end, the potential obtained with DD calculations represents an obvious choice of ini tial guess. Very serious problems, however, arise from the noise inevitably associated with the MC carrier con centration, and they seriously threaten the convergence of the iterative procedure.
As
for the last point, the general problem
is
that rate events require the simulation of a vast number of common ones to achieve sufficient degree of confidence in their statistical description. This problem can be tackled by means of multiplication techniques expanding the statis tical significance of occurring rare events; however pre vious applications of the algorithm
[12],
[8]
seem to be unsufficiently sophisticated to deal with complex devices as silicon MOSFET’s. In this context the present work advances the state of the art in two aspects. First it features an effective method to couple MC transport and Poisson equation providing a viable solution for the problem of the statistical noise mentioned above. Second it presents a new approach for a welldesigned implementation of a MC code on a vector processor machine. The obtained improvements have made it possible to perform costeffective MC simulations for such complex devices as advanced silicon submicron MOSFET’s.
As
for the other features mentioned above, our simu lator makes use of sophisticated techniques available in the literature and already used elsewhere
[7][9],
[ll], [12]. In some of these cases, however, several improve ments have been introduced to adapt such techniques to the needs of real device simulations. In particular an im proved version of the multiplication method [SI,
[
121 used to deal with scarcely populated regions has been devel oped to suit the multiparticle simulation case. In this case suitable weight factors
W,
are assigned to each geometric element
so
that the density of simulated “virtual” parti cles remains constant across the device and so does the statistical noise.
In
fact, when a particle enters the geo metric element
zyxw
its weight is compared with
Wi.
f the former is greater than the latter, the particle is “split” into a set of smaller subparticles; in the opposite case, the particle is gathered with other “light” ones possibly present in the same cell. The gathering algorithm does not affect the amount of total charge present in the device be cause the new particle weights are chosen to be close but in general not to coincide with the reference one
(
W,
)
associated with the cell. The whole process of splitting and gathering particles keeps constant the amount of mo bile charge present in the device during the simulation. This choice is justified even in a selfconsistent scheme, because the mobile charge is dominated by that in the source and drain regions. In order to keep constant the statistical error in the re gion of the current flow, the weights
Wi
re automatically chosen
so
that the density of “virtual” particles is uni form along the whole interface, including source, chan nel, and drain regions. In this way the density of simu lated particles in the vertical direction of the channel decreases sharply following the behavior of the actual car rier density. For the study of the energy tails, instead, regions within the device can be specified where the multiplication al gorithm is applied to the most energetic electrons (re garded as “interesting” rare events) in a recursive way
so
that any range of the distribution function can be stud ied. The MC package requires an initial guess for the solu tion
of
the electrostatic potential. Solutions obtained from an advanced DD simulator, e.g., [lo], provide remark ably good guesses for most problems, certainly much bet ter than any
zyxw
d
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oc
guess, thus speeding up the MCPois son convergence. Furthermore, the computation time spent in calculating the DD solution is an extremely small fraction of the MC solution time. The MC analysis is generally performed only in a small domain or window within the device as simulated by DD, thus avoiding as much computational effort as possible that would be expended in simply reproducing the DD results. The ideal domain would consist of only those re gion(s) where the DD model is expected to be in error, e.g., the channel of a MOSFET. However, because only equilibrium guarantees a correct choice of boundary con ditions, the MC domain must extend in regions where vir tually no electric field is present, e.g. the MOSFET source and drain.
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4
APRIL
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zy
In device simulation a grid is used to discretize Pois son’s equation. Furthermore the grid is also used to dis cretize the electric field during a carrier’s flight as it is essential to account for the
local
electric field on each trajectory, particularly in areas such as that near the drain of a MOSFET where the field is very nonuniform. To this purpose we have used a general twodimensional trian gular grid to discretize the simulation domain. This choice has two major advantages over other grids (e.g. rectan gular). First, as with stateoftheart DD simulators, e.g. [lo], triangular grids permit the treatment of devices with nonplanar geometries which are increasingly prevalent in VLSI technologies. Second, the computation of carrier trajectories becomes vastly simplified with triangular ele ments as the electrostatic potential is assumed linear and hence the field is constant therein. In the calculation of the free flight time duration, it is most convenient to have the total scattering probability per unit time constant and in particular independent on the electron momentum and energy. An artificial scattering mechanism, the socalled selfscattering
zyxwvuts
111,
is, there fore, introduced such that the sum of the true scattering rate and of the selfscattering rate
is
always constant. However, because of the boundary condition constraints the MC domain will always include low field regions where the energy dependent scattering rate is low, to gether with highfield regions with very high scattering rate. This causes the number of selfscatterings to be un acceptably high because in the low field regions the con stant scattering rate is made up mostly by the selfscatter ing rate. To avoid this problem we developed an algorithm which suitably discretizes the total scattering rate in order to minimize the number of selfscatterings while saving the convenience of the method. This scheme, described in details in
[8],
limits the number of selfscatterings to less than 10 percent of the total with a speedup factor of ten, even in the most challenging applications. Finally, the physical models and scattering mechanisms included in this study are those discussed in [8]. In par ticular the acoustic and optical phonons, collision with ionized impurities and impact ionization are described in
[7],
while the surface scattering model adopted comes from
[9].
IV. SELFCONSISTENT OUPLING
CHEMES
When a significant fraction of the electron population inside a device is under nonequilibrium conditions, the carrierdistributions obtained via MC must be computed selfconsistently with the electrostatic potential. This point is demonstrated in Fig. 1 which shows a comparison of surface densities for a MOSFET with a gate mask length
Lgate
=
0.25
pm. Notice first that the DD model predicts a much higher carrier concentration at the surface essen tially because, in contrast with MC calculations, it ne glects field induced electron heating within the channel, thus confining the carrier position in a narrow portion of the potential energy well, close to the surface. Second, the MC concentrations obtained using the DD potential
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0.4
0.5
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.6
0.7
DISTANCE
(pm)
Fig.
1
Electron surface concentration for a
MOSFET
with a gate mask length
Lgate
=
0.25
zyxwv
n
at
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cs
=
1.5
V,
V,,
=
2.0
V
calculated with
DD
(solid), nonselfconsistent MC (dasheddotted) and selfconsistent MC (dashed) approach.
(i.e., non selfconsistent MC) are unacceptably different from the fully selfconsistent solution. Thus nonselfcon sistent approaches like that described in [8], though much simpler and faster than complete solutions, can only be used when most of the electron population is reasonably well described by the DD model. To obtain the selfconsistent solution, our program starts from an initial guess generated by a DD simulator and then iteratively solves the transport problem (by MC) and Poisson equation until convergence is achieved.
A
flowchart illustrating this procedure, which can be likened to the wellknown “Gummel” iteration procedure
[
131,
is shown in Fig.
2.
It is worth noticing that the output of the MC is the updated carrier concentration
n,
but it can also be specified by an auxiliary function such as the elec tron quasifermi level
&.
The standard way to implement the coupling scheme of Fig.
2
is to substitute the updated MC carrier density
nMC
n Poisson equation, e.g.,
*
(eV$)
=
P
=
q(P
ZMC
k
NDA)
(l)
where
NDA
is the net doping concentration and
E
the di electric constant. Assuming that the hole concentration
p
can be neglected throughout the entire simulation domain,
(1)
becomes linear in and can be easily solved. Unfor tunately, we have found that this simple scheme cannot be applied to cases with high carrier concentrations (as a silicon MOSFET) because of stability problems arising from the unavoidable noise of the MC solution. To clarify this statement, an example is illustrated in Fig.
3
where a solution for a n+nn+ device was modified in the n+ re gion by changing the electron concentration in a grid node from
2
X
lo2’
to
2.0005
X
10’’
to estimate the effect of very small noise
(0.025
percent) in the MC solution. The result is an unacceptable variation in the potential that cannot be recovered with successive iterations. One way to overcome this problem is that of solving Poisson equation very often in order to deal at each step only with extremely small changes in the carrier popula
VENTURI
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MONTE CARLO FLOWCHART
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Fig. 2. Flowchart of the Monte Carlo program
3.0 3’5
I
i
I
I
0 0.5
1.0
1.5
2.0
2.5
DISTANCE
(
pm)
Fig.
3.
Effect of a small localized charge variation
(0.025
percent) on the potential
of
a n+nn+ device. Curves
A
and
B
refer to the linear and nonlinear Poisson case. respectively.
tion unable to produce irreversible divergences in the it eration scheme. To this purpose the electron plasma fre quency
(
=
(
1
fs)’ in the case
of
MOSFET’s [14]) can provide a guideline for the time interval between succes sive solutions of Poisson equation in the iteration scheme (i.e.,
At
<
1
fs). Using such a value the calculations, though reasonably safe, implies very large computation time. In this work we use an alternative method that provides satisfactory convergence property and is much more cost effective in terms of computing resources. Such a method, rather than the MC electron concentration, exploits the equivalent concept of the electron quasifermi level
&
de fined as where
TL
is the lattice temperature,
kB
the Boltzmann con stant,
zyxw
,
he intrinsic concentration and
i?
denotes the de pendence on position. In terms of
q n
the electron cencen tration
n
can be written as
n ?)
=
ni
xp By using this expression Poisson equation becomes nonlinear in
zyxw
c
while
&
is kept fixed. This scheme guar antees stability because even large variations in the MC carrier densities lead to very small changes in the poten tial due to their exponential relationship. In fact, curve
B
in Fig.
3
shows that for the same amount
of
charge vari ation used for the linear case, the potential remains fixed when the nonlinear scheme is used. In Fig.
4
the spatially dependent convergence rate of this coupling scheme is examined for two points along the channel of a 0.25pm gate length MOSFET. In particular Fig. 4(a) illustrates the electron energy plot along the MOSFET surface, showing an equilibrium energy point (40 meV)labeled “A”that lies in the source and a very high energy point
(461
meV)labeled “B”in the chan nel. Fig. 4(c) illustrates the convergence behavior
of
point
B;
the
x
and
y
axes represent the potential and the quasi fermi level that define the status of a grid node. Each ver tical line corresponds to a Poisson solutionwhere the potential is changed and the quasifermi level is kept fixedwhile each horizontal one represents a MC solu tion. The staircaselike behavior of the electrical param eters
(
and
+,J
at point
B
requires a large number of iterations to reach convergence because the actual
solu
tion is far from the initial guess (i.e., the
DD
solution). On the other hand at point
A,
where the electrons are cold, the electrical parameters remain very close to the initial guess and convergence is reached in very few steps (Fig. 403)). To this regard it must be noticed that the scale used to plot Fig. 4(b) is expanded greatly compared to Fig. 4(c). The example of Fig. 4 shows that the coupling with nonlinear Poisson equation guarantees stability but fea tures slow convergence rate in the regions where the elec trons are hot. To increase the convergence speed, we have developed a method based on the use of the electron tem perature
T,
instead of the lattice temperature
TL
in
(3).
The thought here would be that the greater the tempera ture used in (2) and
(3),
the larger is the change in poten tial caused by the Poisson solver. In this way, in fact, we are exploiting additional information, i.e., the average electron energy, from the MC simulator to adapt (2) to the case of strong nonequilibrium conditions. More in details, starting from the average electron en ergy
E,,
given by the MC calculation, the electron tem perature
T,
is computed as Therefore, (2) and
(3)
become
(4)
364
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EEE TRANSACTIONS ON COMPUTERAIDED DESIGN,
zyxwv
OL.
8,
NO.
zyx
.
APRIL
1989
zy
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0
DISTANCE
pm)
(a)
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0.560
Y
0.555 0.550
I I I
0 0.005
0 01
0.015
0.02
QUASIFERMI LEVEL (V)
(b)
68
0
70
0
72
0
74
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0 78
080
0.82
QUASIFERMI LEVEL
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(C) Fig. 4. Spatially dependent convergence rate for the nonlinear Poisson coupling. (a) Average electron energy at the surface
of
MOSFET with Lgate
=
0.25
pm. (b) Convergence rate
of
point
A
in the domain defined by the potential and the quasifermi level. (c) Convergence rate of point
B
in the domain defined by the potential and the quasifermi level.
and Because
T,
=
TL
for the points such as
A
inside the source and drain regions, the stability of the solution in
1.30

>
J
f
1.26
z
W

1.22 1181
1
I I
I
068 070 072 074 076 078
080
08
PUASIFERMI
LEVEL
VI
Fig.
5.
Convergence rate
of
point
B
in Fig. 4(a) when using the lattice (dashed line)
or
the electronic temperature (solid line) in the nonlinear Poisson equation.
the highly doped regions is still described by Fig. 4(b), while the convergence rate is much faster in the other parts of the device. In Fig.
5
the convergence rate at point B for the pure nonlinear coupling scheme (equations
(2)
and
(3))
and that one which makes use of the electronic tem perature (equations
(5)
and
(6))
are compared. Notice that in the new scheme a Poisson solution is represented by a diagonal line because the variable in the
x
axis has been computed using
TL
to allow comparison between the two methods, It is clear from the presented data that the new approach is over four times faster, while
of
course yield ing the same solution. Considering, as an example, the case of
a
quarter mi cron MOSFET, Fig.
6
shows the
L2
norm of the relative error in the electrostatic potential as a function of the number of selfconsistent iterations. The error first de creases rapidly but eventually is limited by the statistical noise of the MC procedure. Within the accuracy of the MC calculations, we find that a reasonable stopping cri teria is about
0.5
mV in this norm. For the examples we have tried, this tolerance is achieved after about ten iter ations. Fig. 7 illustrates the distribution of the potential update
(A*)
during the convergence process in the case of a MOSFET with
Lgate
=
0.15
pm. Each line connects the points with the same
A*
after the Poisson solution while the difference between two contiguous lines
is
5
mV for Fig. 7(a) and (b) and
0.5
mV for Fig. 7(c). The three plots refer to the situation after the first, fourth, and fourteenth iteration, respectively. At the beginning of the conver gence process (Fig. 7(a)) the update
is
large across the whole device with a maximum (over
50
mV) at the end of the channel where electron heating, hence the differ ence between
DD
and MC solutions, is stronger. How ever, because of the fast convergence process the poten tial update rapidly decreases in few iterations (see Fig. 7(b) where the maximum
A*
is between
510
mV) and eventually levels off (Fig. 7(c)) with a random distribu tion across the whole device and a peak below
1
mV.