IEEE TRANSACTIONS ON COMPUTERAIDED DESIGN
OF
INTEGRATED CIRCUITS AND SYSTEMS,
VOL.
12,
NO.
9,
SEPTEMBER
1993
zyx
327
A
Numerical Method to Compute Isotropic Band Models from Anisotropic Semiconductor Band Structures
Antonio Abramo, Franco Venturi, Enrico Sangiorgi,
zyxwvut
ember,
IEEE,
Jack
M.
Higman, and Bruno Riccb
AbstractA
numerical method for the determination of iso tropic band models has been developed and applied to silicon. The resulting model accurately approximates both density of states and group velocity of the corresponding anisotropic band structure, thus providing an excellent agreement to both the collision and nonhomogeneous terms of the Boltzmann trans port equation. The model, represented through a simple set of energywave vector tables, has been implemented into a Monte Carlo device simulator, but can also be extended to alternative methods for solving Boltzmann equation. Simulation of homo geneous silicon shows a very good agreement with available ex perimental data. Comparison with results obtained using the complete anisotropic band structure, both in homogeneous and nonhomogeneous silicon devices, confirms the validity of the model.
I. INTRODUCTION
HE
Monte Carlo (MC) method is widely recognized
T
s the most accurate tool to simulate submicrometer devices and, for this reason, a continuous effort is devoted to the development of suitable MC microscopic models achieving a good tradeoff between accuracy and com putational efficiency. As for the former aspect, models based on a single band [l] are no longer adequate to de scribe highfield phenomena, in which hot carriers play a significant role; conversely, the use of the complete an isotropic semiconductor band structure (ASBS)
zyxwvut
2] [4]
requires a huge CPU time, due eyentially to the an isotropic dispersion relationship
zyxwvutsrq
(
k).
In this context, a good alternative is provided by multiband isotropic models [5][7] which accurately describe transport phenomena in semiconductors, are significantly less CPUtime consum ing, and can be easily implemented in alternative methods to solve Boltzmann transport equation
[8][lo].
The models of
[5],
[6]
are given in terms
of
analytical electron and hole bands that fit the ASBS density of states
(DOS)
as a function of energy. The
DOS
in fact, con
Manuscript received June 29, 1992; revised March
zyxwvutsrq
1993. This paper was recommended by
D.
Scharfetter.
A.
Abramo and
E.
Sangiorgi are with the Department
of
Electronics, University of Bologna, Italy.
F.
Venturi is with the Department
of
Information Technology, Univer sity of Parma, Italy. J. M. Higman was with the Beckman Institute, University
of
Illinois at UrbanaChampaign. He is now with Advanced Products Research and De velopment Laboratory, Motorola, Inc., Austin. TX 78721.
IEEE
Log
Number 9208765.
tains
integr ted
information over the whole band structure and, in particular, the electronphonon scattering rate at a given energy is proportional to the
DOS
of the final state. Therefore, the fitting of the
DOS
ensures that the simpli fied models feature the same
ver ge
electronphonon scattering rate of the more complex ASBS. The fitting
of
the
DOS
in itself, however, does not ensure that carrier dynamics, which depends on carrier velocity, is properly treated in the simplified model. In add ion, in the ASBS the carrier groupyelocity (1
h)
V;E( k
)
is a function of the wave vector
k,
while for any isotropic bandmodel it depends on the magnitude of the wave vector
11
k
11
only, thus on the energy
E.
This fact leads to the conclusion that a comparison between the carrier velocity of the ASBS and that of isotropic models can be done only through suitable averages. In this framework, this paper advances the state
of
the art, presenting a new
isotropic
band model for bipolar transport in silicon conceived to best fit the behavior in energy of both density of states and a mean group velocity calculated from the ASBS. The
E(k)
rela tionship is given in terms of numerical isotropic bands, one for each
of
the main symmetry points in the conduc tion and valence band. The band model is consistently used in the evaluation of the scattering probabilities, in cluding electronphonon and electronimpurity interac tion. The reduction from the threedimensional momen tum space of the ASBS to the onedimensional energy space lowers the memory and CPU time requirements to the level of the single band models. Extensive comparisons with results obtained using the complete ASBS demonstrate the validity of this approach. This paper is organized in the following way: in Sec tion
I1
the physical background of the problem is sketched; Section I11 discusses the validity of the approach by means of comparison with simulations performed using the
ASBS
model; in Section IV the numerical procedure for obtain ing the isotropic model is described; the simulation results obtained with the new model are given in Section V; and finally, conclusions are drawn in Section VI. 11.
PHYSICAL
ACKGROUND This section introduces preliminary physical consider ations as a theoretical basis of our work. Section I11 in stead shows the results of some tests that have been per
02780070/93 03.00
@
1993
IEEE
formed in order to validate these theoretical assumptions. Therefore, all results shown in Figs.
16
refer to the ASBS and not to the isotropic model discussed in this paper. As already discussed in
zyxwvutsrq
5],
he
zyxwvutsr
OS(E)
contains
zyxwvu
n
tegrated
information over the ASBS. In addition, the elec tronphonon scattering rate is proportional to the
DOS
(E).
Although the information contained in the
DOS
is neces sary to correctly treat transport in semiconductors, carrier dynamics also plays a fundaqental role. In particular, the carrier group velocity
zyxwvutsrq
g
k)
is defined as the vector (1
/A)ViE(
zyxwvutsrq
nd, therefore, carries the information about the detailecj shape of the band structure. Notice that, in general,
Gg
k)
s not a function of energy only, and thus electrons at the same energy can experience large differences in velocity due to the anisotropicity of the band structure. Conversely, the group velocity in2n isotropic band model is a function of the magnitude of
k
and, there fore, of energy only. Hence, an isotropic band model re lated to the ASBS group velocity should feature, at each energy, a group velocity
u,(E)
corresponding to a suit able
average
of all the group velocities of the ASBS at the same energy. Following this approach, we chose to define
u,(E)
as the magnitude of the group velocity
Jg(k)
averaged over surfaces+
of
constant energy. In particular, we weighted all the
k
states contributing to
u,(E)
by their
DOS,
thus obtaining: where: Therefore, an isotropic model, reproducing both
DOS(E)
and
u,(E),
will feature the same
average
elec tronphonon scattering rate and group velocity that the carriers would have in the ASBS if they were uniformly distributed over each equienergy surface. The leading idea was that the consistent treatment of carrier scattering and dynamics should lead to excellent approximation of both the+ collision and the nonhorflogeneous terms
((df(?,
k,
t)/dt),,,,
and
Gg
V,f(
3,
k,
t
of the Boltz mann transport equation, that depend on the semiconduc tor
DOS
and carrier gFup velocity, respectively. From the ASBS
E(
k)
data computed on a regular grid (spaced 1
/20 of
?r/a,,
a,
being the lattice constant) using the wellknown empirical pseudopotential method
zyxwvut
1
11,
the two functions of interest (namely
DOS(E)
and
zyxwvut
g
E))
were obtained following the method of
[12].
For pur poses peculiar to the derivation of the present model (see Section IV for details), the irreducible wedge of the first Brillouin zone
(1
48
of
the whole zone) was divided into three parts
Si,
=
1,
3),
each centered on a selected
direction
Fig.
1.
ASBS
along principal directions: different line types emphasize the different regions around the following principal points: (a) conduction band:
0.85X
dotted);
r
(dashed);
L
(solid). (b) valence band:
X
(dotted);
r
(dashed);
L
(solid).
ici
nd made up of all the nearest_
’
vectors (usually known as Voronoi polyhedra). The
k,;
vectors were cho sen at the local energy minima (maxima)
of
the conduc tion (valence) band:
zyxw
.85X
’,
and
L
for electrons;
X,
I’’
and
L
for holes (see Fig.
1).
For each region
6;
n the
k
space, the corresponding
DOSi
(,E)
and
ugi
E)
were cal culated integrating over all the
k
states in the
Si
egion
so
that:
6(E
E(
zy
d
(3)
and
1
6(E
E(;))
dz
6i
It immediately follows that the total
DOS(E)
and
ug
E)
are related to the previous quantities as follows:
c
DOS;(E)
=
DOS(E)
1
c
ugi
E)
DOSi
(E)
The total
DOS(E)
and
u,(E)
are plotted in Figs.
2
and
3.
In addition, the individual contributions to the
DOS
of
the three portions
6;
n which the Brillouin zone has been divided, are also plotted in Fig.
2.
Looking at Fig.
2,
it can be noticed that the peaks of the electron
DOS,
located at
=
1.9
eV and
=
3
eV, are mainly due to states in the
L
region (solid line); in Fig.
1
the
DOS
maximum at
3
eV appears as the highest minimum at the
L
point; corre
ABRAMO
zyxwvutsrqponmlk
t
al.:
ISOTROPIC
BAND MODELS
zyxwvutsrqpon
valence band conduction band
1329
20,
zyxwvut
I
,
,
,
I

4
2
zyxwvutsrqpon
zyxwvutsrqp
rn2
n
0
4.5
3.0
1.5
0.0
1.5
3.0
4.5
energy [ev] Fig.
2.
DOS(E)
rom
the
ASBS
(thick solid line); the remaining lines show the contributions
of
the
zyxwvutsrq
’;
o the total
DOS
(see Fig.
1).
20
?
15
2
4
10
U

P
zyxwvutsrqponml
E5
M
0
valence band conduction band
zyxwvuts
4.5 3.0
1.5
0.0
1.5
3.0
4.5
energy [eV] Fig.
3.
U, ,?)
from
the ASBS, as defined in
I),
for
valence and conduc tion band.
spondingly in Fig. 3, the electron
ug
shows two pro nounced drops, because the states at these energies ex hibit low group velocity, proportional to the first derivative of the bands. In the energy range
1.93
eV, instead,
ug
exhibits a peak corresponding to the drop in
DOS.
As
for holes, the continuous increase of the
DOS
is given by the contribution of bands around the points
r
and
L
(up to 1.2
eV)
andX(up to
3
eV).
111.
VALIDATION
F
THEORETICAL
SSUMPTIONS
To validate our approach, we now investigate if such
zyxwv
g
E)
s representative
of
the real situation of the electron population under the electric field action in the
ASBS.
In other words, if the carrier distribution function heated by strong and nonhomogeneous fields still produces the same
ug
E)
of equilibrium conditions. This investigation has two aims: first to check whether the carrier dynamics can be reduced to a spherically symmetrical problem; second if the particular expression (1) used for the average group velocity is meaningful.
0
1
2
3
4
energy [ev] Fig.
4.
Electron
U, ,?)
from
the
ASBS
(thick dashed line); electron
U,@)
computed with
MC
in homogeneous cases using the
ASBS
[I31
(thin solid lines). The simulations
refer
to field strengths
of
250
and
500
kV/cm and directions
(
11 1
)
and
(
100).
As can be seen, the different
ug
(E)
are very close to each other, regardless
of
the electric field strength
or
lattice
ori
entation. This strongly suggests that,
for
silicon homogeneous cases, the anisotropic effects are negligible.
As
a first step, we performed
MC
electron simulations of uniformly doped silicon slabs under different electric field strengths and directions using a simulator featuring the
ASBS
1131. This is a recent version of the srcinal simulator described in 131. The main differences are the finer kspace grid for the
E(
k
)
dispersion relationship and the use of a different impact ionization model 1141. We compared the
a
priori
definition of
ug
(E)
(1) with the av erage electron velocity at each energy computed during the simulation. The aim was to determine to which extent the group velocity, averaged over the electron distribution heated by the field, depends on the field direction and magnitude because of anisotropic effects. The results, shown in Fig.
4
or field strengths of 250 and
500
kV/cm and for the two directions
(
11
1
)
and
(
loo), prove that, for silicon, the
average
magnitude of electron velocity versus electron energy is marginally af fected by anisotropic effects and resembles the
u,(E)
computed from the
ASBS.
This strongly suggests that the electron distribution function is evenly distributed over each equienergy surface. In addition, we performed nonhomogeneous simula tions to check if the assumption previously discussed would hold also in this more realistic case. We simulated two non uniform, one dimensional test structures whose electric field profiles resemble the longitudinal one of a MOSFET biased with high drain voltage (Fig.
5).
In par ticular, the profiles consist of two constant plateaus con nected by an exponentially increasing and a linearly de creasing section. The profile of the shorter device corresponds to a total voltage drop of 2.88
V
and is ob tained from the other one (total voltage drop 5.76 V) by scaling the longitudinal direction by a factor of two. In Fig.
6
the MC group velocities computed at points in space indicated by bullets in Fig. 5 are compared with the
1330
IEEE TRANSACTIONS ON COMPUTERAIDED DESIGN
OF
INTEGRATED CIRCUITS AND SYSTEMS.
zyxwv
OL.
12,
NO.
9,
SEPTEMBER
1993
zy
400

zyxwvutsrqponmlk
00
2
00 00
.Y
zyxwvut
G
U
zyxwvutsrqponml
zyxwvutsrqpon
zyxwvutsrqponml
A
zyxwvutsrqponmlkjihgfedcb
0
0.0
0.2
0.4
0.6
0.8
distance
[km]
Fig.
5
Electric field profiles used for the onedimensional nonhomoge neous simulations. The two profiles show the same maximum field but dif ferent derivatives.
20
0
0
1
2 3
energy
[ev]
Fig.
zyxwvutsrqponmlk
.
Electron
ug
E)
rom the ASBS (dashed line); electron
ug
E)
com puted with
MC
in nonhomogeneous cases using the
ASBS
1131
(thin solid lines). The simulations
refer
to the electric field profiles
of
Fig.
5.
The labels refer to the ones
of
Fig.
5.
Thus,
from
Fig.
5
the electric field regime in which each curve was computed can be found.
a
priori
definition of
u,(E):
again the good agreement confirms the validity of our assumption. IV.
THE
BAND MODEL The isotropic band model was obtained using the fol lowing procedure.
As
already discussed, the irreducible wedge of the first Brillouin zone was divided into the three portions
Pi,
=
1,
3.
This allows us to determine a single isotropic band for _each symmetry point, with the same effective mass at
kci
and multiplicity
(Z)
of the corre sponding symmetry point in the ASBS. In particular we compute a set of
Ei(k)
featuring isotropic
DOSY
and
U:
in good agreement with the corresponding
DOSi
and
ugi
given by
(3)
and
4),
espectively. Since the functions
Ei(k)
are isotropic, we can write the two basic equations of our model as follows:
1
dk
and
+
where
Zi
s the multiplicity at
kci
and
A
a known propor tionality factor. Equation
6
represents a set of two differ ential equations in one unknown function
Ei(k),
and therefore cannot be solved exactly. An optimum solution can be found, which minimizes the weighted sum of the relative errors
A
given by:
DOSy(E) DOSi(E)
I
DOS;(E)
A
=
(1

W
uF(E) u,;(E)
I
u,;(E)
(7)
where
w
is a weighting factor in the range
[0,
13
allowing to choose whether the
DOS
or the
ug
is to be better ap proximated. In particular, a low value of
w
means a smaller error for
DOS(E)
than for
ug E).
We shall now consider the function
k(E)
nstead of
E(k)
for the following reasons: first because
7),
solved for
k
at a given
E,
is a firstorder nonlinear differential equa tion; second because the
k(E)
s a true mathematical func tion, while the
E(k)
is not. Since
(7)
yields two different solutions for the two cases
dEi/dk
>
0
and
dEi/dk
<
0
and needs boundary conditions, the following constraints were imposed:
1)
k(Ey)
=
k(EF)
=
0,
where,
~y~~
=
Fin
E(;);
ke6,
E?
=
gax
E(;).
ke6,
dk

o
for
[E:~ ,
Ei];
dk

0
for
[Ei
?]; dEi dE;
therefore,
E;
=
E(k?).
3)
Ei(k)
E
in
[Eyin,
?].
The above constraints impose that each band consists of two branches: one
electronlike
featuring
dE,/dk
>
0
and one
holelike
featuring
dEi/dk
<
0.
Under these conditions,
(7)
was transformed into a
fi
nitedifference problem and then numerically solved through the bisection technique. The total
DOS(E)
and
u,(E)
computed from
(5)(7)
for different values of
w( Us
corresponding to different
ABRAMO
zyxwvutsrqponmlk
t al.:
ISOTROPIC BAND MODELS
1331
zyx
sets of
zyxwvutsr
i k))
re plotted in Figs.
7
and
8.
It is worth no ticing that while
w
zyxwvutsrqp
0
corresponds to zero error on the total
DOS(E)
(i.e., the
DOS(E)
obtained from the set of
Ei k)
oincides with that of the ASBS),
w
=
1
produces zero error on each
ugi E)
nly. The total group velocity
u,(E)
of the isotropic band structure, instead, is affected by the error made over the
DOSi@)
that are used as weighting factors in the determination of
u,(E)
(see
zyxwvu
5)
and Fig.
8).
Therefore, values of
w
close to unity are not of practical interest. In Figs.
9
and
10
the relative errors made over
DOS
(E)
and
u, E)
have been integrated over energy in the range
zyxwv
4
V and shown as a function of the
w
parameter. At
w
=
0.5,
for example, the error is equally distributed over
DOS(E)
and
u, E),
and
is
=
15
for conduction and
=
20
for valence band, respectively. The electron and hole isotropic band structure obtained with
w
=
0.5
is reported in Fig.
11
,
where different lines refer to the symmetry points of Fig.
1
Notice that the isotropic band related to the
L
symmetry point for elec trons (solid line) reaches a much higher energy than the corresponding branch in the ASBS in Fig.
1.
This can be explained considering that each isotropic band has been determined integrating over all states in each
6,
nd not only over the ones along the
so
called
principal directions
shown in Fig.
1.
The development of this model required also the exten sionof th_e formulae to compute the scattering rates
P 11
k
11,
11
k
for acoustic, optical phonons and ionized impurities from the wellknown analytic band cases to the case where
E(k)
is given by means of numerical tables, as i,n o;r model. This extension wasmade integrating the
P
, k
')
expressions
[
zyxwvut
1
over the
k
'
state space gener ated by our
zyxwvutsrqpo
@')
functions. A similar procedure has been applied in a recent work
[7].
Using the same notation of
[l]
for the sake of simplicity, the scattering rates in our case are given by the following. Elastid acoustic phonon scattering: Intervalley optical phonon scattering:
(9)
Ionized impurity scattering (Brooks and Hemng
[211):
4.5
3.0
1.5
0.0
1.5
3.0
4.5
energy [ev]
Fig.
7.
DOS(E)
of
the isotropic model computed
for
different values of the weighting factor
w.
he curve
w
=
0
corresponds to the exact
DOS(E).
The
w
step is
0.2.

zy
E
15
Y
h
::
10
?
E5

I
1
I
0
1
1
4.5 3.0
1.5
0.0
1.5
3.0
4.5
energy [ev]
Fig.
8.
ug E)
f
the isotropic model computed
for
different values of the weighting factor
w.
he dotted line is the exact
U,@
and
does
not
corre spond to the curve
w
=
1
(see text). The
w
step is
0.2.
40
r .
6e
30
Y
h
U
V
4
M
2
20
4
.
lo
0
~
0.0
0.2
0.4
0.6
0.8
1.0
weighting
factor
w
Fig.
9.
Present algorithm percentage error integrated over energy as a function of the
w
parameter
for
conduction band:
DOS@)
(A)
and
u,(E)
zy
0)
rrors.
I
+4=
PS