R3 Model Documentation
Click on each section heading to expand/unexpand.
Reload the page to unexpand all sections.
r3 is a compact model for
polysilicon (poly) resistors,
3-terminal JFETs, and diffused resistors (which are really just JFETs,
often with a large pinch-off voltage so they are fairly linear).
Nonlinearity in poly resistors comes from self-heating and, to a lesser extent,
from MOS action, where the potential difference between the resistor body and the
semiconductor region underneath it modulates conduction through the body of the resistor.
Nonlinearity in diffused resistors and JFETs also comes from self-heating and depletion region
modulation with bias, although from pn-junction rather than MOS physics, and in addition
from velocity saturation. r3 models all of these causes of nonlinearity, along with their
geometry and temperature dependencies, and includes models for noise, statistical variation,
parasitic capacitance, and parasitic leakage currents
(which are primarily important for JFETs and diffused resistors).
Terminology and Notation:
device |
any semiconductor component that r3 is applicable to (poly resistor, diffused resistor, 3-terminal JFET) |
non-pinch-off |
operation when neither end of a device is pinched off (analogous to MOSFET strong inversion non-saturation) |
drain pinch-off |
operation when one end of a device is pinched off (analogous to MOSFET strong inversion saturation) |
full pinch-off |
operation when both ends of a device are pinched off (analogous to MOSFET weak inversion) |
Vb |
voltage across the intrinsic body portion of the device
V (i2)–V (i1)
|
Ib |
current through the intrinsic body portion of the device (see equivalent circuit below) |
Vc1 |
voltage across the intrinsic terminal 1 parasitics
V (c)–V (i1)
|
Vc2 |
voltage across the intrinsic terminal 2 parasitics
V (c)–V (i2)
|
Parameters and nodes are set in Courier New font.
k
is Boltzmann's constant and q
is the magnitude of the electronic charge.
The lareg-signal equivalent circuit for r3 is shown below:
The power generated by the electrical part of r3 (the left side of the network above),
Ith,
drives the thermal part of r3 (the right side of the network above), and the behavior of the electrical part
depends on the local temperature rise, Temp(dt),
generated by the thermal part.
The electrical and thermal parts are solved self-consistently.
The parasitic capacitances
Cp1 and
Cp2
include both linear components and non-linear pn-junction components,
for polysilicon resistors and diffused resistors or JFETs, respectively.
The parasitic currents
Ip1 and
Ip2
are pn-junction currents, including breakdown, so are only
relevant for diffused resistors and JFETs.
r3 has the same physical basis and capabilities as the r3_cmc model,
but includes the following enhancements:
-
improved temperature variation modeling (velocity saturation, depletion potential, thermal conductance)
-
improved geometry dependence modeling (depletion potential, temperature coefficients, drain pinch-off smoothing)
-
ability to turn off all nonlinearities (separately or collectively),
or make them arbitrary small, without affecting numerical robustness
-
drain-induced barrier lowering (DIBL) and channel length modulation (CLM) models
for significantly improved modeling of JFETs
-
significantly more accurate pinch-off models, selectable via the new switch parameter
sw_accpo=1 or 2
-
conductance in full pinch-off is limited to a minimum value to prevent numerical problems
-
code improvements, bug fixes, and code style changes
r3 is not backwards compatible with r3_cmc, you will get different
results if you simulate with the same parameters:
-
The added temperature dependence of velocity saturation
has one component that is physically linked to the temperature
dependence of the low field mobility (i.e. the sheet resistance),
so for any diffused resistor or JFET that is significantly affected
by both velocity saturation and self-heating this will cause some difference.
-
The current in full pinch-off is different, even for
sw_accpo=0
(it now gives the correct slope for
∂ln(Ib)/∂Vb
whereas previously it was a factor of two too big—
setting the parameter nst to half
its value recovers the previous behavior).
-
The capacitance is different for
JFETs and diffused resistors; the pn-junction depletion
capacitance for each end of a device smoothly goes to zero
as it enters pinch-off. For diffused resistors that
do not approach pinch-off operation the difference is minor,
for JFETs in drain or full pinch-off the difference is significant.
Details of the theory that underlies r3 can be found in
[1] and [2].
Details of robust parameter extraction algorthms for r3
can be found in [2] and [3].
|
Instance Parameters:
Name |
Default |
Min. |
Max. |
Unit |
Description |
m |
1 |
1 |
inf |
|
multiplicity factor (number in parallel) |
w |
1.0e-6 |
0.0 |
inf |
m |
design width of resistor/JFET body |
l |
1.0e-6 |
0.0 |
inf |
m |
design length of resistor/JFET body |
wd |
0.0 |
0.0 |
inf |
m |
dogbone width (total; not per side) |
a1 |
0.0 |
0.0 |
inf |
m2 |
area of node n1 partition |
p1 |
0.0 |
0.0 |
inf |
m |
perimeter of node n1 partition |
c1 |
0 |
0 |
inf |
|
# contacts at node n1 terminal |
a2 |
0.0 |
0.0 |
inf |
m2 |
area of node n2 partition |
p2 |
0.0 |
0.0 |
inf |
m |
perimeter of node n2 partition |
c2 |
0 |
0 |
inf |
|
# contacts at node n2 terminal |
trise |
0.0 |
|
|
°C |
local temperature delta to ambient (before self-heating) |
nsmm_rsh |
0.0 |
|
|
|
number of standard deviations of local variation for rsh |
nsmm_w |
0.0 |
|
|
|
number of standard deviations of local variation for w |
nsmm_l |
0.0 |
|
|
|
number of standard deviations of local variation for l |
sw_noise |
1 |
|
|
|
switch to include noise: 0=no and 1=yes |
sw_et |
1 |
|
|
|
switch to include self-heating: 0=no and 1=yes |
sw_lin |
0 |
|
|
|
switch to force linearity: 0=no and 1=yes |
sw_mman |
0 |
|
|
|
switch to enable mismatch analysis: 0=no and 1=yes |
The switch parameters can also be specified as model parameters,
and a value specified on an instance line
overrides a value specified on a model card.
The following diagram shows how the instance parameters should
be determined for a typical resistor layout
(note that the end region dogbone may be asymmetric between the two sides).
The total width of an end region is
where the contact width, contact spacing, and contact-to-edge distances,
wc,
wc2c,
and wc2e, respectively,
are shown in the layout.
If c1 is zero
(which happens for non-end sections of a multi-section model)
then a1
and p1
should be calculated as half of the area and perimeter, respectively, of
the body of the resistor (i.e. 0.5wl
and l, respectivey).
If c1 is greater than zero
then the area and (non-body adjacent) perimeter of the left end region should
be added to these values. Similarly for a2
and p2.
|
Special Parameters:
Name |
Default |
Min. |
Max. |
Unit |
Description |
version |
2 |
|
|
|
model version |
subversion |
0 |
|
|
|
model subversion |
revision |
0 |
|
|
|
model revision |
tmin |
-100.0 |
-250.0 |
27.0 |
°C |
minimum ambient temperature |
tmax |
500.0 |
27.0 |
1000.0 |
°C |
maximum ambient temperature |
gmin |
1e-12 |
0.0 |
inf |
S |
minimum parasitic conductance |
imax |
1.0 |
0.0 |
inf |
A |
current at which to linearize diode currents |
scale |
1.0 |
0.0 |
1.0 |
|
scale factor for instance geometries |
shrink |
0.0 |
0.0 |
100.0 |
% |
shrink percentage for instance geometries |
rthresh |
1.0e-3 |
0.0 |
inf |
Ω |
threshold to switch end resistance to V=I*R form |
These model parameters do not affect the core calculations of the r3 model,
but define quantities related to simulator variables, such as
gmin and
imax,
if there is any optical shrink, and if there is a unit difference between
netlisted instance parameter geometries and meters.
If there is a simulator parameter that is defined for any of the above
model parameters the simulator value is used in preference to the
default listed above.
|
Model Parameters:
Name |
Default |
Min. |
Max. |
Unit |
Description |
type |
-1 |
|
|
|
resistor/JFET type: -1=n-body and +1=p-body |
tnom |
27.0 |
-250.0 |
1000.0 |
°C |
nominal (reference) temperature |
lmin |
0.0 |
0.0 |
inf |
μm |
minimum allowed drawn length |
lmax |
9.9e09 |
lmin |
inf |
μm |
maximum allowed drawn length |
wmin |
0.0 |
0.0 |
inf |
μm |
minimum allowed drawn width |
wmax |
9.9e09 |
wmin |
inf |
μm |
maximum allowed drawn width |
jmax |
100.0 |
0.0 |
inf |
A/μm |
maximum current density |
vmax |
9.9e09 |
0.0 |
inf |
V |
maximum voltage w.r.t. control node nc |
tminclip |
-100.0 |
-250.0 |
27.0 |
°C |
clip minimum temperature |
tmaxclip |
1000.0 |
27.0 |
1000.0 |
°C |
clip maximum temperature |
sw_accpo |
0 |
|
|
|
switch for accurate pinch-off model: 0=no and 1/2=yes |
grpo |
1e-12 |
0.0 |
0.1 |
|
minimum body conductance in pinch-off (ratio w.r.t. Vc=0) |
rsh |
100.0 |
0.0 |
inf |
Ω/□ |
sheet resistance |
xw |
0.0 |
|
|
μm |
width offset (total) |
nwxw |
0.0 |
|
|
μm2 |
narrow width width offset correction coefficient |
wexw |
0.0 |
|
|
μm |
webbing effect width offset correction coefficient (for dogboned devices) |
fdrw |
1.0 |
0.0 |
inf |
μm |
finite doping width offset reference width |
fdxwinf |
0.0 |
|
|
μm |
finite doping width offset width value for wide devices |
xl |
0.0 |
|
|
μm |
length offset (total) |
xlw |
0.0 |
|
|
|
width dependence of length offset |
dxlsat |
0.0 |
|
|
μm |
additional length offset for velocity saturation calculation |
nst |
1.0 |
0.1 |
5.0 |
|
subthreshold slope parameter |
atsinf |
0.0 |
0.0 |
inf |
V |
saturation smoothing parameter for wide/long devices |
atsl |
0.0 |
0.0 |
inf |
V·μm |
saturation smoothing parameter 1/l coefficient |
dfinf |
0.01 |
0.0 |
inf |
/V0.5 |
depletion factor for wide/long devices |
dfw |
0.0 |
|
|
μm/V0.5 |
depletion factor 1/w coefficient |
dfl |
0.0 |
|
|
μm/V0.5 |
depletion factor 1/l coefficient |
dfwl |
0.0 |
|
|
μm2/V0.5 |
depletion factor 1/(w*l) coefficient |
sw_dfgeo |
1 |
|
|
|
switch for depletion factor geometry dependence: 0=drawn and 1=effective |
dpinf |
2.0 |
0.1 |
inf |
V |
depletion potential for wide/long devices |
dpw |
0.0 |
|
|
V·μmdpwe |
depletion potential w dependence coefficient |
dpwe |
0.5 |
|
|
|
depletion potential w dependence exponent |
dpl |
0.0 |
|
|
V·mudple |
depletion potential l dependence coefficient |
dple |
2.0 |
|
|
|
depletion potential l dependence exponent |
dpwl |
0.0 |
|
|
V·μmdpwe+dple |
depletion potential wl dependence coefficient |
ecrit |
4.0 |
0.0 |
1000.0 |
V/μm |
velocity saturation critical field |
ecorn |
0.4 |
0.0 |
ecrit |
V/μm |
velocity saturation corner field |
du |
0.02 |
0.0 |
1000.0 |
|
mobility reduction at ecorn |
dibl1l |
0.0 |
0.0 |
inf |
μmdibl1le |
dibl linear component l dependence coefficient |
dibl1le |
1.0 |
0.0 |
inf |
|
dibl linear component l dependence exponent |
dibl2l |
0.0 |
0.0 |
inf |
V1-dibl2e·μmdibl2le |
dibl nonlinear component l dependence coefficient |
dibl2le |
1.0 |
0.0 |
inf |
|
dibl nonlinear component l dependence exponent |
dibl2v |
0.1 |
0.01 |
inf |
V |
dibl nonlinear component voltage offset |
dibl2e |
0.5 |
0.01 |
1.0 |
|
dibl nonlinear component voltage exponent |
clm1l |
0.0 |
0.0 |
inf |
μmclm1le |
clm linear component l dependence coefficient |
clm1le |
1.0 |
0.0 |
inf |
|
clm linear component l dependence exponent |
clm1c |
0.0 |
0.0 |
inf |
/V |
clm linear component V(nc) dependence coefficient |
clm2l |
0.0 |
0.0 |
inf |
V1-clm2e·μmclm2le |
clm nonlinear component l dependence coefficient |
clm2le |
1.0 |
0.0 |
inf |
|
clm nonlinear component l dependence exponent |
clm2v |
0.1 |
0.01 |
inf |
V |
clm nonlinear component voltage offset |
clm2e |
0.5 |
0.01 |
1.0 |
|
clm nonlinear component voltage exponent |
rc |
0.0 |
0.0 |
inf |
Ω |
resistance per contact |
rcw |
0.0 |
0.0 |
inf |
Ω·μm |
width adjustment for contact resistance |
fc |
0.9 |
0.0 |
0.99 |
|
depletion capacitance linearization factor |
isa |
0.0 |
0.0 |
inf |
A/μm2 |
diode saturation current per unit area |
na |
1.0 |
0.0 |
inf |
|
ideality factor for isa |
ca |
0.0 |
0.0 |
inf |
F/μm2 |
fixed capacitance per unit area |
cja |
0.0 |
0.0 |
inf |
F/μm2 |
depletion capacitance per unit area |
pa |
0.75 |
0.0 |
inf |
V |
built-in potential for cja |
ma |
0.33 |
0.0 |
1.0 |
|
grading coefficient for cja |
aja |
-0.5 |
|
|
V |
smoothing parameter for cja |
isp |
0.0 |
0.0 |
inf |
A/μm |
diode saturation current per unit perimeter |
np |
1.0 |
0.0 |
inf |
|
ideality factor for isp |
cp |
0.0 |
0.0 |
inf |
F/μm |
fixed capacitance per unit perimeter |
cjp |
0.0 |
0.0 |
inf |
F/μm |
depletion capacitance per unit perimeter |
pp |
0.75 |
0.0 |
inf |
V |
built-in potential for cjp |
mp |
0.33 |
0.0 |
1.0 |
|
grading coefficient for cjp |
ajp |
-0.5 |
|
|
V |
smoothing parameter for cjp |
vbv |
0.0 |
0.0 |
inf |
V |
breakdown voltage |
ibv |
1.0e-6 |
0.0 |
inf |
A |
current at breakown |
nbv |
1.0 |
0.0 |
inf |
|
ideality factor for breakdown current |
kfn |
0.0 |
0.0 |
inf |
μm2 |
flicker noise coefficient - see (*) note below on the unit |
afn |
2.0 |
0.0 |
inf |
|
flicker noise current exponent |
bfn |
1.0 |
0.0 |
inf |
|
flicker noise 1/f exponent |
sw_fngeo |
0 |
|
|
|
switch for flicker noise geometry calculation: 0=drawn and 1=effective |
ea |
1.12 |
|
|
V |
activation voltage for diode temperature dependence |
xis |
3.0 |
|
|
|
exponent for diode temperature dependence |
xvsat |
0.0 |
|
|
|
exponent for saturation velocity temperature dependence |
tc1 |
0.0 |
|
|
/K |
resistance linear TC |
tc2 |
0.0 |
|
|
/K2 |
resistance quadratic TC |
tc1w |
0.0 |
|
|
μm/K |
resistance linear TC 1/w coefficient |
tc2w |
0.0 |
|
|
μm/K2 |
resistance quadratic TC 1/w coefficient |
tc1l |
0.0 |
|
|
μm/K |
resistance linear TC 1/l coefficient |
tc2l |
0.0 |
|
|
μm/K2 |
resistance quadratic TC 1/l coefficient |
tc1wl |
0.0 |
|
|
μm2/K |
resistance linear TC 1/(w*l) coefficient |
tc2wl |
0.0 |
|
|
μm2/K2 |
resistance quadratic TC 1/(w*l) coefficient |
tc1rc |
0.0 |
|
|
/K |
contact resistance linear TC |
tc2rc |
0.0 |
|
|
/K2 |
contact resistance quadratic TC |
tc1dp |
0.0 |
|
|
/K |
depletion potential linear TC |
tc2dp |
0.0 |
|
|
/K2 |
depletion potential quadratic TC |
tc1vbv |
0.0 |
|
|
/K |
breakdown voltage linear TC |
tc2vbv |
0.0 |
|
|
/K2 |
breakdown voltage quadratic TC |
tc1nbv |
0.0 |
|
|
/K |
breakdown ideality factor linear TC |
tc1kfn |
0.0 |
|
|
/K |
flicker noise coefficient linear TC |
tegth |
0.0 |
-inf |
0.0 |
|
thermal conductance temperature exponent |
gth0 |
0.0 |
0.0 |
inf |
W/K |
thermal conductance fixed component |
gthp |
0.0 |
0.0 |
inf |
W/K/μm |
thermal conductance perimeter component |
gtha |
0.0 |
0.0 |
inf |
W/K/μm2 |
thermal conductance area component |
gthc |
0.0 |
0.0 |
inf |
W/K |
thermal conductance contact component |
cth0 |
0.0 |
0.0 |
inf |
s·W/K |
thermal capacitance fixed component |
cthp |
0.0 |
0.0 |
inf |
s·W/K/μm |
thermal capacitance perimeter component |
ctha |
0.0 |
0.0 |
inf |
s·W/K/μm2 |
thermal capacitance area component |
cthc |
0.0 |
0.0 |
inf |
s·W/K |
thermal capacitance contact component |
nsig_rsh |
0.0 |
|
|
|
number of standard deviations of global variation for rsh |
nsig_w |
0.0 |
|
|
|
number of standard deviations of global variation for w |
nsig_l |
0.0 |
|
|
|
number of standard deviations of global variation for l |
sig_rsh |
0.0 |
0.0 |
inf |
% |
global variation standard deviation for rsh (relative) |
sig_w |
0.0 |
0.0 |
inf |
μm |
global variation standard deviation for w (absolute) |
sig_l |
0.0 |
0.0 |
inf |
μm |
global variation standard deviation for l (absolute) |
smm_rsh |
0.0 |
0.0 |
inf |
%μm |
local variation standard deviation for rsh (relative) |
smm_w |
0.0 |
0.0 |
inf |
μm1.5 |
local variation standard deviation for w (absolute) |
smm_l |
0.0 |
0.0 |
inf |
μm1.5 |
local variation standard deviation for l (absolute) |
sw_mmgeo |
0 |
|
|
|
switch for mismatch geometry calculation: 0=drawn and 1=effective |
sw_noise |
1 |
|
|
|
switch to include noise: 0=no and 1=yes |
sw_et |
1 |
|
|
|
switch to include self-heating: 0=no and 1=yes |
sw_lin |
0 |
|
|
|
switch to force linearity: 0=no and 1=yes |
sw_mman |
0 |
|
|
|
switch to enable mismatch analysis: 0=no and 1=yes |
The last four switch parameters can also be specified as instance parameters,
and a value specified on an instance line
overrides a value specified on a model card.
(*) The unit of kfn
depends on the values of the parameters
afn and
bfn;
strictly it is
μmafnA2-afnHzbfn-1
and this becomes μm2 for the default values of
afn and
bfn.
|
Accounting for optical shrinking and unit scaling, in units of microns
the drawn width and length of the resistor body and dogbone width are
The perimeter and area of the resistor body are
The perimeters of each partition (half of the body plus, if contacted, the end region) are
and their areas are
The effective electrical width (see [4]) and length
of the resistor body are
where the effective length offset term is reduced by half or eliminated
entirely when one or no ends of the device are contacted, respectively.
This is to facilitate implementation of multi-section sub-circuit models.
The depletion potential and depletion factor geometry dependencies are
where W and
L
are the effective width and length if
sw_dfgeo=1
and the design width and length otherwise (in units of micron).
From these, the pinch-off voltage is computed as
The zero-bias resistor body resistance and conductance factor are
and the contact resistances at each end are
The DIBL and CLM parameter geometry dependencies are
and for the saturation smoothing parameter
The thermal conductance and thermal capacitance include fixed, perimeter, area, and contact components
These scale with drawn, rather than effective, geometries because the effective geometries
for the thermal conductance and capacitance can be different from the effective
electrical geometries. It is easily shown that any difference between drawn and effective
dimensions can be subsumed in the fixed and perimeter component coefficients.
The temperature coefficients for the body resistance vary with geometry as
The geometry dependence of flicker noise is detailed in the section on noise modeling.
|
If T is the
device temperature (in °C) define
Unless otherwise noted, self-heating is included in the calculation of
dT and
rT.
The zero-bias resistance varies with temperature as
which also affects the conductance factor gf,
which varies reciprocally as R0.
The contact resistances at each end vary with temperature as
The effect of velocity saturation in r3 is primarily determined at high
field by the ecrit
parameter. In first order theory this is determined as
Ecrit=vsat/μ0 where
vsat
is the saturation velocity and μ0
is the low field mobility.
R0
varies reciprocally as μ0,
therefore in part the temperature dependence
of Ecrit
should be that of R0.
Including an additional temperature dependence for
vsat gives
The depletion potential varies with temperature as
but for this calculation the effect of self-heating is not included in the
computation of dT.
The thermal conductance temperature variation is
and this is applied without the effect of self-heating included in
rT.
The effective thermal conductance is determined by the whole region through which heat flows,
and when self-heating occurs the temperature of this region is not uniform.
Therefore, the effect of self-heating on gTH
needs to account for the “distributed” nature of the heat flow,
and should be based on an effective temperature that is between that
of the device and the ambient environment.
In r3 this accomplished using the technique presented in [5].
The flicker noise coefficient temperature variation is
For the area and perimeter components of the parasitic pn-junction model,
the saturation current temperature variations are
the built-in potential temperature variations are
(with a physically based modification to smoothly limit to zero for high temperatures),
and the zero-bias capacitance per unit area and per unit perimeter vary with temperature as
The breakdown voltage and breakdown current ideality factor vary with temeprature as
and
|
The current in the resistor body is modeled in r3 by
where the effective conductance of the resistor body is
Expressions for the conductance factor
gf, depletion factor
df, and depletion potential
dp
are detailed in the geometry dependence section, and the temperature dependence for the
first and last of these are detailed in the temperature dependence section.
The effect of self-heating is accounted for in gf.
The effective body voltage
Vb,eff
is the body voltage smoothly limited to the saturation voltage
Vsat
(this form is used to preserve symmetry). The value of
ats
is modified slighty for non-zero sw_accpo.
Vsat is
computed based on the asymptote of the velocity saturation model detailed below.
The channel length modulation terms, those with parameters that start with “clm”,
include a simple linear form and a more physically based non-linear term, and
note that both CLM terms are formulated in terms of
Vb
to capture the influence of the body voltage when it is greater than
Vsat.
Velocity saturation is included through the mobility reduction term
where the field across the body of the resistor is
and
This model has the following characteristics, which are required physically:
it has the value zero for E=0;
it has a slope of zero for E=0;
it is smooth and symmetric about E=0;
it asymptotically approaches
1+(E–ecorn)/Ecrit
for large E.
The effective value of Vc1
is calculated as
which includes limiting in pinch-off and both linear and non-linear DIBL components.
As for the CLM terms, note that DIBL is a function of
Vb,
not the limited form Vb,eff,
because it must include the influence of the body voltage on the device behavior
for Vb>Vsat.
Pinch-off behavior is primarily of importance for JFETs, and three models,
selectable via the sw_accpo switch,
are available.
The computationally simplest, and therefore fastest to simulate, limiting
is selected with sw_accpo=0,
however this model only fairly crudely approximates the physically expected
1–exp(–Vb/φt)
variation of Ib
with Vb in full pinch-off.
Setting sw_accpo=1 selects
a model that more accurately approximates that behavior, at the expense of increased computation time,
and sw_accpo=2 selects an even more
accurate, but slightly more computationally expensive, form.
A minimum conductance is enforced in full pinch-off
to avoid numerical computation issues.
|
The thermal noise for the body of the resistor is
where
geff
is the effective conductance of the resistor body, calculated including
depletion pinching, velocity saturation, and self-heating.
The flicker (i.e. 1/f )
noise is current dependent [6]
and has the physical geometry dependence detailed in [7]
where W and
L
are effective geometries if sw_fngeo is 1
or drawn geometries if it is zero.
Note that W and
L
are in units of micron, not meters, so the model parameter
kfn
should be in units of
μm2
(note that for afn≠2
the unit differs from this; it is recommended that you do not change afn
from its default value of 2);
The thermal noises for the contact resistances at each end are
The shot noises for the parasitic diode currents,
which include components for both the ideal diode and breakdown currents, are
where the ideal saturation currents are
The above form for shot noise is more accurate than the conventional
2qI
form and, in the absence of the breakdown current, is consistent
with the result of van der Ziel [8].
|
The r3 model explicitly includes a capability to model statistical variability,
including for both global (correlated between devices in a die) and local
(i.e. mismatch, uncorrelated between devices on a die) variation.
The statistical model is physically based and is built-in, which obviates the need for
using a sub-circuit (which is conventionally how variability is added
on top of an existing compact model). The form of statistical model
is as described in [9]:
each statistical parameter is characterized by a standard deviation of global variation
σglobal,
a number of standard deviations of global variation
nsig,
a standard deviation of local variation
σlocal
(which varies reciprocally with geometry),
and a number of standard deviations of local variation
nsmm.
A process is characterized by determining σglobal
and σlocal
for each statistical parameter. Corner models are defined
by setting specific values of nsig,
and Monte Carlo type simulations are done by generating random values for
nsig
and, for every device selected for mismatch analysis, for
nsmm.
Note that mismatch is not a pair-wise phenomena,
it comes from atomic level variations (e.g. random dopant fluctuations,
line edge roughness) that affect every device differently.
It is commonly measured from the difference in electrical characteristics between of a pair of devices,
but in circuits like resistor ladders there are more than two devices,
so mismatch needs to be modeled and simulated as variations that are uncorrelated between devices.
If sw_mman is set to 1
r3 includes mismatch. The effective width, effective length, and sheet resistance are varied as
where the subscript nom indicates the
nominal value of the parameter and m
is the multiplicity factor (i.e. number of devices in parallel).
The 0.01 factor in the last expression converts percentage variation to relative variation.
Note that physically line edge roughness in width averages over length,
so the mismatch standard deviation of width varies reciprocally as the square root of length,
line wedge roughness in length averages over width,
so the mismatch standard deviation of length varies reciprocally as the square root of width,
and random dopant fluctuation averages over area,
so the mismatch standard deviation of sheet resistance varies reciprocally as the square root of area.
As noted in [9], the variations in the lateral dimensions width and length
are absolute, and the variation in sheet resistance is relative.
In addition, physically if relative variations in a quantity are large the statistical distribution
they follow is better modeled as log-normal than Gaussian.
This can happen for sheet resistance, and is implemented via the exponential mapping in third expression above.
For small variations in sheet resistance
exp(x)≈1+x
so the distribution form reduces to Gaussian.
If sw_mman is set to zero,
then mismatch variation is turned off. Because total variance is the sum of
global variance plus local variance, in this case the variations in the statistical
parameters are computed as
|
For poly resistors isa
and isp should be left
at their default value of zero, in which case the leakage currents to the control terminal are zero
For diffused resistors the parasitic pn-junction (i.e. diode)
currents, with area and perimeter components, are modeled as
Junction breakdown is also included as
giving total parasitic currents for each end of
where a standard SPICE minimum conductance term is included, to aid
numerical solution roubustness.
The parasitic capacitances include bias independent components,
intended for poly resistor modeling, and bias dependent components,
for parasitic pn-junction capacitance modeling for diffused
resistors and JFETs. The complete capacitances are
The parasitic pn-junction capacitances are linearized
for forward biases greater then fc
multiplied by the built-in potential for each component,
and smoothly reduce to zero in pinch-off.
|
Operating Point Parameters:
Name |
Unit |
Description |
v |
V |
voltage across resistor/JFET |
ibody |
A |
current through resistor/JFET body |
power |
W |
dissipated power |
leff_um |
μm |
effective electrical length |
weff_um |
μm |
effective electrical width |
r0 |
Ohm |
zero-bias resistance at nominal temperature |
r_dc |
Ohm |
DC resistance (including bias and temperature dependence) |
r_ac |
Ohm |
AC resistance (including bias and temperature dependence) |
rth |
K/W |
thermal resistance at ambient temperature |
cth |
s*W/K |
thermal capacitance |
dt_et |
K |
self-heating temperature rise |
|
[1] |
R. V. H. Booth and C. C. McAndrew,
“A 3-terminal model for diffused and ion-implanted resistors,”
IEEE Trans. Electron Dev.,
vol. 44, no. 5, pp. 809-814, May 1997.
|
[2] |
C. C. McAndrew,
“Integrated resistor modeling,”
in Compact Modeling: Principles, Techniques and Applications,
G. Gildenblat, Ed., Berlin, Germany: Springer, pp. 271-297, 2010.
|
[3] |
C. C. McAndrew and T. Bettinger,
“Robust Parameter Extraction for the R3 Nonlinear Resistor Model for Diffused and Poly Resistors,”
IEEE Trans. Semicond. Manuf.,
vol. 25, no. 4, pp. 555-563, Nov. 2012.
|
[4] |
C. C. McAndrew, S. Sekine, A. Cassagnes, and Z. Wu,
“Physically based effective width modeling of MOSFETs and diffused resistors,”
Proc. IEEE ICMTS,
pp. 169-174, 2000.
|
[5] |
J. C. J. Paasschens, S. Harmsma, and R. van der Toorn,
“Dependence of thermal resistance on ambient and actual temperature,”
Proc. IEEE BCTM,
pp. 96-99, 2004.
|
[6] |
R. Brederlow, W. Weber, C. Dahl, D. Schmitt-Landsiedel, and R. Thewes
“A physcially based model for low-frequency noise of poly-silicon resistors,”
Tech. Digest IEEE IEDM,
pp. 89-92, 1998.
|
[7] |
C. C. McAndrew, G. Coram, A. Blaum, and O. Pilloud
“Correlated noise modeling and simulation,”
Proc. NanoTech WCM,
pp. 40-45, 2005.
|
[8] |
A. van der Zeil,
Noise in Solid State Devices and Circuits,
John Wiley: New York, 1986, p. 94.
|
[9] |
C. C. McAndrew,
“Statistical modeling using backward propagation of variance (BPV),”
in Compact Modeling: Principles, Techniques and Applications,
G. Gildenblat, Ed., Berlin, Germany: Springer, pp. 491-520, 2010.
|
|
This documentation is distributed as is, completely without warranty, liability,
copyright, or service support. Users may use, modify, copy, reformat, and redistribute
the documentation, both within the user's organization and externally, without restriction.
As a courtesy:
- If you use this documentation, please make an appropriate acknowledgement.
- If you correct a typo or make an enhancement, feed the correction or enhancement back,
to help maintain open-source, useful, and high quality documentation for
everyone who uses the r3 model.
|