I
IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN
OF
INTEGRATED CIRCUITS AND SYSTEMS,
VOL.
12,
NO. 3, MARCH 1993
425
A Computationally Efficient Unified Approach to the
Numerical Analysis
of
the Sensitivity and Noise
of
Semiconductor Devices
Giovanni Ghione,
Member,
IEEE,
and Fabio Filicori
Abstract-This paper presents a new computationally effi-
cient unified approach to the numerical simulation of sensitivity
and noise in majority-carrier semiconductor devices, based on
the extension to device simulation
of
the adjoint method, well
known in the sensitivity and noise analysis of electrical net-
works. Sensitivity and device noise analysis based on physical
models are shown to have a common background, since they
amount to evaluating the small-signal device response to an im-
pressed, distributed current source. This problem is addressed
by means of a Green’s function technique akin to Shockley’s
impedance field method. To enable the efficient numerical eval-
uation
of
the Green’s function within the framework of a dis-
cretized physical model, interreciprocity concepts, based on the
introduction of an adjoint device, are exploited. Examples of
implementation are discussed, relevant to the sensitivity and
noise analysis of GaAs MESFET’s.
I.
INTRODUCTION
HE IMPRESSIVE development of CAD tools for
T
semiconductor device design based on physical
models which has taken place over the last ten years only
marginally has touched two important areas of device per-
formance evaluation:
sensitivity
and
noise
simulation.
Although the concept of
sensitivity analysis
as a tool
for device optimization had already been suggested in
[
11,
the only method for device sensitivity analysis proposed
so
far to our knowledge is the technique for evaluating
static small-change device sensitivities with respect to
uniform doping and geometry variations presented in
[2]
and implemented in the HFIELDS simulator
[3].
No
at-
tempt has been made to address other topics, such as ac
sensitivity (i.e., the sensitivity of small-signal parame-
ters) or the statistical sensitivity of the device with respect
to random parameter variations
[4].
Concerning
noise analysis,
classical analysis tech-
niques, such as the
impedancejield
method described by
Shockley, Copeland, and James
[5]
and its extensions and
generalizations have been available for many years (for a
review, see, e.g.,
[6],
[7]).
Although such techniques are
Manuscript received March
11,
1991; revised March 10, 1992. This pa-
G. Ghione is with the Dipartimento di Elettronica, Politecnico di Torino,
F. Filicori is with the Facolta di Ingegneria, Universith di Ferrara, Fer-
IEEE Log Number 9201
145.
per was recommended by Associate Editor D. Scharfetter.
Turin, Italy.
rara,
Italy.
well suited for numerical implementation in two- or three-
dimensional geometries, only applications to one-dimen-
sional structures
[8]
or to quasi-physical models of GaAs
FET’s
have been presented
so
far (see
[9]
and references
therein).
Despite their different purposes, small-change para-
metric sensitivity and the noise analysis of semiconductor
devices have a common background: both analyses re-
quire obtaining the small-signal response of the device to
an equivalent distributed current source impressed either
into the volume of the device or into its surface. Since the
small-signal device model is linear, the response to an
arbitrary impressed harmonic current density can be ex-
pressed in terms
of
the response to a spatially impulsive
dipole source, also called the Green’s function
of
the
problem
[
101
and a superposition integral over the device
volume. Moreover, linearity allows the device response
to be uniquely expressed, for instance, in terms
of
the
open-circuit voltages induced on the device ports. This
choice justifies the name of
vector impedancejield
2
orig-
inally assigned in
[5]
to the open-circuit voltage response
to a spatially impulsive dipole source. It is worth stressing
that the response to a dipole source, which is primary in
noise and sensitivity device analysis, can, however, be
derived from the response to a spatially impulsive scalar
current source (i.e., the
scalar impedanceJield
Z
[5]).
In sensitivity analysis, the source term arises from lin-
ear perturbation of the physical model as a dc (ac) current
density for the static (small-signal) case,
so
that the small-
signal response is directly related to the device sensitiv-
ity. In noise analysis, the distributed current source
models microscopic random fluctuations of the current
density with respect to its average value, and the purpose
of the analysis
is
to relate the power and correlation spec-
trum of the microscopic fluctuations (see, e.g.,
[ll], [5],
[12]
and references therein) to the power and correlation
spectra of the voltage or current fluctuations induced into
the external circuit connected to the device.
In order to formulate the problem into a more familiar
form, it will be helpful to recall that the equations
of
the
discretized
device model can be interpreted in terms of an
equivalent electrical network (cf. e.g.,
[13]
and refer-
ences therein). Therefore, the analysis of device sensitiv-
ity and noise can be more easily understood by consider-
0278-0070/93$03.00
@
1993 IEEE
I
426
IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN
OF
INTEGRATED CIRCUITS AND SYSTEMS,
VOL.
12,
NO.
3.
MARCH
1993
ing the well-known problem
of
evaluating the sensitivity
and noise of an
electrical network
[14],
[15].
Also in the
latter case, sensitivity and noise can be evaluated from the
linear response of the network to current sources related,
respectively, to parameter variations or to the noise be-
havior of the individual components. The network anal-
ogy also offers a straightforward interpretation of the sca-
lar impedance field. In fact, if the device model is
discretized on a N-node mesh through a suitable numeri-
cal scheme, the discretized scalar impedance field
Z,
is
just the
transimpedance
between an internal node
j
and
the electrode
i.
Thus the potential induced by a distribu-
tion of current generators can be simply evaluated by
su-
perposition, which is a discrete form of the spatial super-
position integral appearing in a Green’s function
formulation.
From a computational standpoint, the evaluation of the
discretized scalar impedance field within the framework
of a frequency -domain small-signal numerical device
simulator
[
161,
is a straightforward but computationally
intensive task. In fact, it is necessary to repeatedly per-
form a device analysis while placing a single current
source in each of the
N
internal nodes, with a computa-
tional cost amounting to the solution of
N
(N
=
lo3
-
lo4)
linear systems of dimension
=N,
or, at least, to one
LU
factorization and
N
backsubstitutions for each fre-
quency. While a Green’s function formulation cannot be
avoided in noise analysis, where the total available noise
power at the device terminals is obtained by power sum-
mation of spatially uncorrelated internal noise sources,
’
sensitivity analysis does not strictly require a Green’s
function approach. This happens because the small-change
device sensitivity with respect to a parameter can be de-
scribed as the response to a well-defined current source
distribution proportional to the parameter variation, and
can, therefore, be obtained by globally solving a single
current-driven problem. Nevertheless, in the Green’s
function approach the device sensitivity with respect to a
parameter is expressed as a volume integral (or, in dis-
cretized form, as a node summation), whose integrand can
be interpreted as a
distributed
parametric sensitivity. The
knowledge of the spatial behavior of the sensitivity clearly
allows deeper insight into which regions of the device dis-
play high or low sensitivity with respect to some param-
eter and is, therefore, a valuable design tool.
A
computationally efficient technique for the noise and
sensitivity analysis
of
semiconductor devices, which
drastically reduces the computational cost of evaluating
the Green’s function, can be derived from the principle of
the
adjoint system approach
[14], [15], [17]
proposed
during the late
1960’s
for the noise and sensitivity anal-
ysis of large electrical networks. The adjoint approach
avoids the repeated evaluation of the network response to
a current generator placed in turn in each of the internal
nodes by exploiting the fundamental property of direct-
’
The
formulation can
be
extended
so
as
to
deal with spatially correlated
noise sources;
see
Section
11-C.
adjoint linear system pairs, i.e., their being mutually re-
ciprocal, or interreciprocal. The idea is trivial in a recip-
rocal network where the direct and adjoint system coin-
cide; thus the transimpedance
ZG
where
i
is
any of the
internal nodes and
j
is an external port, is equal to
qi.
and
can, therefore, be evaluated from the solution of one lin-
ear problem, i.e., by connecting a current generator to
portj and computing the resulting internal potentials. For
any nonreciprocal linear network simple rules ultimately
derived from the application of Tellegen’s theorem enable
the definition of an
adjoint network
such that the direct
and adjoint pair are interreciprocal
[
141.
The purpose of the present paper is twofold. First, a
unified treatment of the sensitivity and noise analysis of
semiconductor devices is provided in which, for the first
time to our knowledge, sensitivity analysis is reduced to
evaluating the device response to a set of equivalent, dis-
tributed
perturbation
current density sources. Second, the
idea of exploiting the interreciprocity properties
of
an ad-
joint system is extended to the numerical analysis of sen-
sitivity and noise of semiconductor devices.
A drift-diffusion majority-carrier model will be as-
sumed as the basis
of
the discussion, and application of
the adjoint method to GaAs MESFET’s will be consid-
ered as a case study. Nevertheless, the adjoint approach
can in principle be extended to two-carrier or nonstation-
ary (hydrodynamic) models. In order to derive the discre-
tized adjoint device from the direct one by formal appli-
cation of network rules
[
141,
the small-signal majority
carrier model is expressed in an admittance-like form, al-
ready proposed by the authors for computational expe-
diency
[
181,
[
191.
The extension
of
the adjoint approach
to two-carrier models which is currently under develop-
ment requires on the other hand the use of more general
forms of the interreciprocity theorem, in which the simple
formal network analogy is no longer helpful.
The paper is structured as follows: Section
I1
presents
a unified discussion
of
sensitivity and noise analysis in
terms of the small-signal response of the device to a har-
monic, distributed current source. Section
I11
introduces
the adjoint technique, while its numerical implementation
is described in Section IV. Finally, Section V is devoted
to the discussion
of
some results concerning examples of
sensitivity and noise analysis of GaAs MESFET’s.
11.
A
UNIFIED
APPROACH
TO
SENSITIVITY
AND
NOISE
ANALYSIS
OF
SEMICONDUCTOR
DEVICES
From a physics-based standpoint, parametric sensiti
v-
ity and noise analysis apparently have very little in com-
mon. In sensitivity analysis, the effect of parameter vari-
ations on the device response is investigated; in the
particularly important case where the parameter change is
small, the device behaves as a linear system
(small-change
sensitivity analysis). In noise analysis, a deterministic de-
vice model is considered involving
average
values of mi-
croscopic quantities, such as the charge density and elec-
tric potential. Then microscopic fluctuations are
I
421
GHIONE AND FILICORI: UNIFIED APPROACH TO NUMERICAL ANALYSIS OF SEMICONDUCTOR DEVICES
superimposed, which can be modeled as random, distrib-
uted, harmonic current sources, and the response to such
sources is investigated
[5].
Since current fluctuations have
small amplitude, the device again behaves as a linear sys-
tem. Therefore, both small-change sensitivity and noise
analysis involve the evaluation of the small-signal re-
sponse
of
the device to a distributed source term related
either to parameter variations or
to
microscopic current
or
charge fluctuations. However, while noise in the external
circuit connected to the device naturally appears as the
response to microscopic current fluctuations occurring
with the device, the interpretation of the static and small-
signal sensitivity as the response to distributed dc or ac
current sources is less straightforward. The present sec-
tion is aimed at giving a formal background to the above
ideas
so
as to establish a common framework for small-
change sensitivity and noise device analysis.
A. ne Physical Model
Although most of the concepts introduced in the fol-
lowing sections can be readily applied to other, more
complex, physical models, a majority-carrier drift-diffu-
sion model will be assumed as the basis for the discus-
sion. In large-signal operating conditions the model equa-
tions read
where
J
=
q[-npV4
+
DVn]
is the drift-diffusion cur-
rent density and
J,
is an impressed distributed current
source, which is zero under normal operating conditions
and
is
introduced here for formal expediency, to be ex-
ploited later. In the above equations
n
is the electron den-
sity,
4
the electric potential,
p
and
D
the (field-depen-
dent) electron mobility and diffusivity,
q
the electron
charge,
E
the semiconductor permittivity, while
NA
is
the
net equivalent ionized donor density; for the sake of sim-
plicity, the ionization is assumed to be constant. The
boundary conditions associated to the drift-diffusion sys-
tem are well known, see,
[20].
From the Poisson-conti-
nuity system, an equation can be derived in the potential
only, which is useful to establish a direct analogy with
the nodal formulation of an electrical network. On sub-
stituting in
(1)
the charge density derived from
(2),
the
following expression is obtained:
-€V2$
+
V
[-qNDfpV+
-
€pV24V4
+
qDVNDf
+
EDVV~~]
E
3(+,
4)
=
-V
J,
(3)
where
4
=
a$/&.
Similarly, the boundary conditions can
be entirely expressed in terms
of
the unknown
4
and its
spatial derivatives; for the sake of brevity, these will be
denoted as
x(4,
s)
=
0,
where
x
is a partial differential
operator acting on the contour of the device and
s
is a set
of
external sources applied at the device terminals.
B.
Sensitivity Analysis
To carry out
a
formal sensitivity analysis, let us set the
impressed term
J,
to zero and make
3
and
x
explicitly
dependent on a
parameter
set
p,
which can be either a
single parameter, a discrete set of parameters, or a con-
tinuous function (e.g., the doping profile). Since from the
standpoint of a numerically discretized problem the pa-
rameter set always has finite dimension, we shall unre-
strictively consider
p
to
be a vector
of
dimension
Np.
We
have, therefore,
3(+,
4,
p)
=
0
with associated boundary
conditions
x
(4,
s,
p)
=
0.
Let us suppose we apply
to
the
device a dc bias
so
superimposed on small-amplitude time-
varying
ac
generators
s^,
i.e., a small-signal excitation; at
the same time, let the parameters undergo a small vari-
ation
Ap
with respect to a reference value
po:
s
=
so
+
s^(t)
P
=
Po
+
AP.
(4)
(5)
(6)
The resulting potential distribution will be written as
4
=
40
+
A40
+
d(0
+
Ad@)
where
<bo
is
the dc response with nominal parameters,
A+o
t>e dc perturbation caused by the parameter variation,
4(t)
the small-signal potential due to the time-varying part
of the impress!d generators
s^(t),
i.e., the small-signal re-
sponse and
A4(t)
the variation of the latter due to the
parameter variation.
I,n
the linear approzimation, one
clearly has
A$J~
a
Ap, 4(t)
a
s^(t),
while
A+(t)
a
s^(t)Ap.
If the expressions of
(4),
(9,
and
(6)
are introduced into
(3)
and the operator and boundary conditions are ex-
panded in Taylor series around the point
(so,
do,
po),
the
zeroth-order term yields the nominal (i.e., unperturbed)
dc problem:
T+O,
0,
Po)
=
0
x(40,
$0,
Po)
=
0
(7)
(8)
while first-order terms in
Ap
yield the perturbed dc prob-
lem:
3#A&
=
-3,Ap
(9)
X#NO
=
-xpAP
(10)
where
3a
stands for the gradient of
3
with respect to
CY,
and all gradients are evaluated at the operating point
(40,
so,
po)
and in dc steady-state conditions
(4
=
0).
Simi-
larly, by taking the terms of first order in
3,
one has the
ac small-signal problem, which can be written in the fre-
quency domain:
Y(46(4
=
0
(11)
X#d(W>
=
-x,s^(w)
(12)
where
$(U)
and
$(U)
are the Fourier transforms of the
small-signal potential and impressed sources, respec-
tively, and
y(w)
=
3+
+
j~34
is
the small-signal admit-
tance operator arising from the linearization of the drift-
diffusion model. Notice that in
(9)
3$
=
y(0).
Finally,
,
428
IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN
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INTEGRATED CIRCUITS AND SYSTEMS,
VOL.
12.
NO.
3.
MARCH
1993
the second-order bilinear terms of the order
8Ap
yield the
perturbation of the ac small-signal problem, which reads
in the frequency domain:
(13)
(14)
Y
(4A6
(U)
=
-
[
Y
g
(w)A+o
+
Y,(+Pl6
(U)
xgA6(4
=
-[xQQA+o
+
XgpAP16(4
where
ya
is the gradient of
with respect to
a.
On in-
troducing the inverse operator of 9,
2,
the above equa-
tions can be rewritten in the following more explicit form:
Y
(W6
(0)
=
-
[
Yg
(d53(0)3,
+
Yp
(w)l3
b)AP
(15)
(16)
Inspection of the above systems suggests the following
interpretations. It is clear from
(9)
that evaluating the
static device sensitivity with respect to parameter vari-
ations reduces to computing the small-signal response to
a dc distributed source term
A J,
(r)
such as:
(17)
with boundary conditions given by
(10).
Comparison of
(10)
with
(12)
clearly shows that these boundary condi-
tions are equivalent to a source term impressed on the de-
vice periphery. According to the type of problem consid-
ered, either the volume or the boundary term can vanish.2
Similarly, the ac sensitivity analysis amounts to evaluat-
ing the small-signal response to an ac distributed source
term
A
Js(r,
U)
bilinearly related to the small-signal po-
tential and to the parameter variation
Ap,
such as
xgA6(4
=
-[xggZ(0>3,
+
xgp16(wP.
V
*
AJ,
=
3,Ap
v
*
A.f,
=
r
Yg
(U)
55
(0)
3,
+
Y,(w)l
6
b)AP
(18)
with boundary conditions given by
(16),
again equivalent
to an ac boundary source.
A final remark is suggested by comparison of
(1
3)
and
(9).
While for the dc sensitivity analysis the equivalent
source term depends only
on
the results of the dc analysis
with reference parameter values, the ac source terms also
contain the variation of the dc potential distribution,
A#o.
Therefore, the ac sensitivity analysis always requires the
solution of Np dc small-change sensitivity problems nec-
essary in order to set up the Np different perturbation dis-
tributed ac current sources to be used in
(13)
and
(14).
Thus the ac sensitivity analysis is intrinsically more com-
putationally intensive than the dc one, independent of the
technique exploited for evaluating the device response to
the source term.
Response to harmonic current sources-Green
s
Func-
tion Approach:
It has already been shown that the dc and
ac sensitivity analyses amount to evaluating the small-sig-
nal response to an equivalent sensitivity current density
volume source (cf.
(9),
(13)),
subject to a set
x
of linear
boundary conditions, which may include a sensitivity sur-
For
instance, the sensitivity to dimension variations not affecting the
distributed parameters
of
the device can
be
formulated as already implied
in
[4],
so
as to lead to a boundary source term only.
face excitation (see
(lo), (14)).
As
already pointed out,
either the volume or the surface excitation can vanish,
according to which sensitivity parameter is considered.
Finally, a surface excitation also appears in evaluating the
nominal small-signal response of the device (see
(12)).
Since in linear operating conditions the superposition
principle holds, the device response to surface and vol-
ume sensitivity sources can be separately evaluated
so
as
to characterize the equivalent generators appearing at the
device ports, while the nominal small-signal response can
be uniquely derived from a linearly independent set of
small-signal parameters. For the sake of simplicity, we
shall concentrate mainly on the treatment of
volume
sources, which are also fundamental in noise analysis.
Let
us
generally indicatt a harmonic impressed volume
current density source as
Js(r,
a),
where
r
is the position
vector, and the corresponding induced small signal poten-
tial as
&(r,
U),
which is the solution to the linear problem:
Y(w)6
=
-v
*
js<4
(19)
where the operator y, already introduced in the last sec-
tion, is, for the sake of brevity implicitly associated to the
boundary conditions defined by the boundary operator
x.
It can be noted that the boundary condition set
x
is now
homogeneous,
that is, it does not include any impressed
surface generators. For the drift-diffusion model defined
by
(3),
the explicit expression of
(19)
can be readily de-
rived as
v
[POV
+
poe~~6
-
(qnoE
-
+
joE)v61
=
-v
*
j,(w)
(
20)
where
po
and
Do
are the dc operating point electron
rno-
bility and diffusivity, the dc electric field,
no
the dc
electron concentration,
+
(U)
the Fourier transform of the
small-signal electric potential and
-
is
the small-signal
equivalent mobility tensor
(21)
where
Z
is the identity tensor and
V,
the voltage equivalent
of temperature.
The potential induced by the volume source
js(r,
U),
$(r),
can be generally expressed in terms of the
vector
Green
's
jimction
2
and
of
a spatial superposition integral
[lo]
as
$(r,
U)
=
s,
Z(r,
r',
0)
j(r',
U)
dr'.
(22)
As shown in
[5]
(see also
[7]),
2
can be derived from a
potential
Z
as
Z(r,
r',
w)
=
VrZ(r,
r',
U).
(23)
2
can, in turn, be interpreted
[5]
as a
scalar Green 'shnc-
tion,
since the response to a harmonic scalar current source
GHIONE AND FILICORI: UNIFIED APPROACH TO NUMERICAL ANALYSIS OF SEMICONDUCTOR DEVICES
=
Z(IdJ0
).hm)
I
429
i(r,
U)
=
-V
-
j3
can be expressed in terms of
Z
as
$(r,
U)
=
s,
Z(r,
rf,
~)f(t’,
U)
dr’.
(24)
The treatment of an impressed surface current source is
similar to the case of volume sources; only the volume
integral of
(22)
is replaced by a surface integral.
For open-circuit boundary conditions,
Z(ri,
r’,
U),
where
ri
denotes the electrode
i,
coincides with the
vector
impedance jield
originally introduced by Shockley
[5]
as
the open-circuit potential
Pi
6
(Ti)
induced on electrode
i
by a unit dipole source
P
impressed in
rf
(or, equiva;
lently, induced by a spatially impulsive current density
J
=
pS(r
-
rf)).
Similarly, in the same conditions
Z(ri,
r‘,
w)
coincides with Shockley
’s
scalar impedance jeld,
de-
fined [5] as the open-circuit voltage induced on the elec-
trode
i
by a unit current source
t
impressed in
rf
(or,
equivalently, indyced by a spatially impulsive scalar cur-
rent density
i
=
ZS(r
-
f)).
The meaning
of
the scalar and vector impedance fields
for a device having
M
contacts (plus one ground terminal)
is shown in Fig. 1. Expressing the vector Green’s func-
tion in terms
of
a scalar Green’s function, i.e.,
of
the de-
vice response to a scalar current source connected be-
tween the ground and a point of the device, clearly leads
to computational advantages.
Green
s
Function Formulation
of
Sensitivity:
On the
basis of the previous discussion, the expression of device
sensitivities as superposition integrals is straightforward.
Device sensitivities include the dc sensitivities of external
variables such as open-circuit voltages or short-circuit
currents, and the ac sensitivities of small-signal parame-
ters. Since the problem is linear, the small-change sensi-
tivity of any set of parameters can be derived from a lin-
early independent set
of
sensitivities. With no loss of
generality, we shall consider here as the relevant electri-
cal variable the electrode potential
ei
and its open-circuit
variations. Let us suppose that
Np
parameters
pI
-
pN,,
are varied; the impressed distributed current source term
will be linearly related to the parameter variations. For
the dc case:
N”
where
S6k(r).
is the sensitivity of
ei
with respect to
Pk,
which can be expressed as the volume integral of a
dis-
tributed
sensitivity
s:
(r).
For the ac case, the sensitivity of the impedance matrix
elements
zii
can be directly expressed as the sensitivity of
’From the continuity equation, the
tern
-Vrs
. j(r‘)
can
be
interpreted
as the equivalent
scalar
current
densiry
i(r’)
=
jwqk
(r‘)
impressed into
r’.
the open-circuit voltage
Pf
=
zk;
induced on electrode
i
by
a unit current generator applied to the electrode
k.
If
the
distributed source corresponding
fp
a set of parameter
variations
Apk
term is denoted as
AJk(r’>
the variation
&ki
of the impedance matrix element
zki
reads:
NO
Both in the dc and the ac cases, the knowledge
of
the
distributed sensitivity, rather than of only the overall sen-
sitivity, can provide the device designer with valuable in-
sight on how to properly optimize the device structure and
technological process, since it allows the contribution of
specific parts of the device to be singled out as critical.
This information is altogether indispensable if one is in-
terested in
random
parameter variations occurring in the
device volume (e.g., random doping fluctuations or ran-
dom geometry irregularities
[4]);
in this case, which is
not discussed in detail here, the treatment is similar to that
for noise analysis.
C.
Noise Analysis
Noise in the circuit connected to a semiconductor de-
vice can be interpreted as the
small-signal response
to mi-
croscopic fluctuations
SJ
of the current density in the
semiconductor. Random current fluctuations are modeled
as a volume source term like the one introduced in
(3)
and
denoted there as
J,;
the device response is linear since the
amplitude
of
fluctuations is small. For further details on
the physical background, the reader can refer to
[5],
[
111
and references therein.