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IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 46, NO. 8, AUGUST 1999
A New Look at Impact Ionization—Part II:
Gain and Noise in Short Avalanche Photodiodes
P. Yuan, K. A. Anselm, C. Hu, H. Nie, C. Lenox, A. L. Holmes, B. G. Streetman, J. C. Campbell, and R. J. McIntyre
Abstract— In Part I, a new theory for impact ionization that
utilizes history-dependent ionization coefficients to account for
the nonlocal nature of the ionization process has been described.
In this paper, we will review this theory and extend it with
the assumptions that are implicitly used in both the local-field
theory in which the ionization coefficients are functions only
of the local electric field and the new one. A systematic study
of the noise characteristics of GaAs homojunction avalanche
photodiodes with different multiplication layer thicknesses is also
presented. It is demonstrated that there is a definite “size effect”
for thin multiplication regions that is not well characterized
by the local-field model. The new theory, on the other hand,
provides very good fits to the measured gain and noise. The new
ionization coefficient model has also been validated by Monte
Carlo simulations.
I. INTRODUCTION
T
HE first successful noise theory [1]–[3] for avalanche
diodes assumed that impact ionization is a continuous,
local process, and that the ionization probability is only a
function of the local electric field irrespective of a carrier’s
ionization history. This theory has been widely accepted and
has proved to be an extremely useful model for a wide
variety of avalanche devices. Recently, however, it has been
observed that this theory does not provide a good fit to the gain
curves and that it overestimates the multiplication noise and
underestimates the gain-bandwidth product for devices with
thin multiplication regions [4]–[13]. These discrepancies are
thought to be due to the assumption of locality. It is well
known that after an ionization event, a carrier needs to travel
a certain distance, which is frequently referred to as the “dead
length,” before it can gain sufficient energy from the electric
field to have a nonnegligible ionization probability [14]. This
dead length can be ignored if it is small compared to the
thickness of the multiplication region. On the other hand,
when the thickness of the multiplication region is reduced
to the point that it becomes comparable to a “few” dead
lengths, the assumption of locality fails and it is no longer
valid to assume a continuous ionization process. Numerous
analytical [5]–[9], [24] and numerical [15]–[23] techniques
Manuscript received October 12, 1998; revised January 25, 1999. This work
has been supported by DARPA through the Heterogeneous Optoelectronic
Technology Center and the National Science Foundation. The review of this
paper was arranged by P. K. Bhattacharya.
P. Yuan, K. A. Anselm, C. Hu, H. Nie, C. Lenox, A. L. Holmes, B. G.
Streetman, and J. C. Campbell are with the Department of Electrical and
Computer Engineering, Microelectronics Research Center, The University of
Texas at Austin, Austin, TX 78712 USA (e-mail: jcc@mail.utexas.edu).
R. J. McIntyre is with McIntyre Photon Detection Consultants, Pointe
Claire, P.Q., H9R 2R3, Canada.
Publisher Item Identifier S 0018-9383(99)06012-8.
(a)
(b)
Fig. 1. Schematic of the spatial notation for (a) holes and (b) electrons in
the history-dependent theory. Carriers are created at x0 and ionization occurs
at x: The filled circles are the electrons, and the circles are holes.
have been proposed to address the nonlocal nature of impact
ionization. The numerical models that utilize the Monte Carlo
technique have the advantage of being formally exact, but
their accuracy is frequently limited by the completeness of the
bandstructure and the scattering models used in the simulation.
As a consequence, adequate fits to the experimental data
require several adjustable parameters, thus obviating one of
their advantages relative to analytical models. In addition,
they are very computation intensive. It is in this context
that the analytical model discussed in the previous paper was
developed. In this paper, we will briefly review the historydependent ionization theory that has been described in Part I
[29], extract the implicitly applied assumptions in this theory,
present extended equations for arbitrary injection profiles, and
show that this theory provides excellent fits to the gain and
noise measurements of GaAs APD’s.
II. THEORY REVIEW
This section reviews the gain and noise calculation of the
local-field theory and the history-dependent theory discussed
in Part I. The local-field theory assumes impact ionization
is a continuous and local process, whereas the new theory
recognizes that the ionizing probability of a carrier depends
on its history. Throughout the calculations of this paper, we
use a one-dimensional (1-D) model and assume the n, i, and
p regions are arrayed from left to right, as shown in Fig. 1.
The origin is at the interface of the n and i regions, and the
Thus, electrons are swept to
thickness of the i region is
the left and holes to the right. Without losing generality, we
is greater than
assume the electron ionization coefficient
that of hole in this section.
0018–9383/99$10.00  1999 IEEE
YUAN et al.: A NEW LOOK AT IMPACT IONIZATION—PART II
1633
A. The Local-Field Theory
In the local-field theory [1], the position-dependent gain is
given by the expression
pair is equal to that of the holes. The current gain is defined as
the ratio of the number of final - pairs to that of the injected
- pairs. So, the gain due to the initial pair injected at
is
(4)
(1)
for an electron-hole pair that is created at in the multiplication region. The multiplication noise is described in terms
of the current noise power spectral density with units of watts
per Hertz. According to the local-field theory, the noise power
of a pair injection at (injection of
spectral density
electron-hole pairs at continuously in time), is
Since the noise power spectral density is proportional to
, the calculation of noise starts
the ensemble average
and
for an electron-hole pair injected at
from
Considering all the ionization probabilities for the initial pair
and assuming that the carriers are uncorrelated, i.e.,
and
can be expressed as in [29, (16)
and
are solved in the gain
and (17)]. Once
and
can be calculated similarly. Then,
calculation,
is given by the
the excess noise factor of a injection at
expression
(5)
(2)
is the impedance of the
where is the injected current,
is the excess
device and the measurement circuit, and
noise factor which is a function of the injection position
because
itself is a function of
With the assumption that
and only electrons are injected at
, a common
boundary condition, (2) can be written as
(3)
B. The History-Dependent Theory
It is assumed that the carrier that initiates an impact ionization loses essentially all of its energy relative to the band
edge after each impact ionization event. Each event can be
considered as annihilation of a “hot” carrier and the generation
of three new “cold” carriers: two of one type and one of
the other. The newly generated “cold” carriers will then be
accelerated in the electric field and become “hot.” In order to
accurately represent this process, history-dependent ionization
and
are defined to represent the
coefficients
local ionization probability density at for a carrier generated
If an electron generated at can survive until it gets to
at
without ionizing, then the probability for it to ionize in the
is
The ionization probability
distance element
in
thus depends on the electron
of this electron
All these probabilities are defined
survival rate
in (1a) and (1b) of Part I. The survival rate and ionization
probability of holes are defined similarly.
To calculate both the gain and noise, we utilize Method 2
in Part I. With [29, (9)–(13)], we can calculate the ensemble
and
, the average numbers of the
averages
carriers in the two chains generated by the initial electron and
hole injected at separately. Since in each ionization event
the extra electron and hole are always generated in pairs, the
final number of the electrons in the chain started by the initial
C. General Injection Profile
In the local-field theory and the history-dependent theory,
the following assumptions are implicit.
1) There is no interaction between any of the carriers in the
multiplication region except at the moment of impact
ionization.
2) There is no correlation between any carriers, and they
contribute to noise independently. Thus, their noise
spectral density can be added linearly.
These assumptions are quite reasonable for low-level injection, which is the common operating condition for most
APD’s. Assumption 1 also insures a linear response to infor an
jection. According to this assumption, the gain
can be written as a weighted average
arbitrary injection
of the gain of each injection, which yields
(6)
is the total injected current, and
is defined in
where
(1) for the local-field theory and in (4) for the new theory.
Similarly, according to Assumption 2, the current noise
is
spectral density for an arbitrary injection profile
(7)
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IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 46, NO. 8, AUGUST 1999
where
is defined in (2) for the local-field theory and in
(5) for the history-dependent theory. , which is a weighted
, is the measured excess noise factor according
average of
to either theory.
III. EXPERIMENT
In order to study the multiplication characteristics of APD’s
with thin multiplication regions, a series of PIN homojunction
devices were grown in a Varian GEN II molecular beam
epitaxy reactor on n GaAs substrates. All the doping levels
used in each layer were determined with Hall effect measurements, and the error was conservatively estimated to be within
50% due to the run-to-run fluctuation. Reflection high-energy
electron diffraction (RHEED) was employed to calibrate the
thickness of each layer. The i regions were 100, 200, 400,
500, 800, and 1600 nm thick. The i layers were sandwiched
between a 1500-nm p layer on the top and a 200-nm n layer
below. The i region was nominally undoped with a background
10 cm ,
n-type doping concentration approximately 2
while the designed doping in the p and n layer varied from
10 cm to 4
10 cm . A layer of 30-nm-thick,
1
heavily doped ( 10 cm ) p GaAs was grown on the
top to insure a good p contact. Mesas with a diameter of
100 m were formed by etching in a H SO : H O : H O
(1 : 1 : 10) solution. The metal contacts were patterned with
standard photolithography and lift-off processes. Au-Cr served
as the p contact and Ni-AuGe-Au was used as the n contact.
The n contact covered the whole surface of the wafer except
the mesas. The contacts were annealed in nitrogen ambient at
450 C for 30 s.
The photocurrent and dark current curves were measured
with an HP 4145B semiconductor parameter analyzer. For the
photocurrent and noise measurements, a He-Ne laser beam
633 nm) was carefully focused on the top of
(wavelength
the mesa. The possible contamination of mixed injection is
always a concern for this type of measurement. Consequently,
we have designed the devices and the measurement system
to limit the influence of mixed injection. The diameter of the
focused optical signal was kept small ( 10 m) relative to that
of the mesas (100 m) in order to minimize the amount of light
incident on the mesa edges. Based on published absorption
data [25], we estimate that the photon “leakage” through the
1.5- m-thick top p layer is 0.15%, and the reflection of the
semiconductor-air interface is 30%. Therefore, only about
0.1% of the incident photons enter the semiconductor and
pass to the high field i layer before being absorbed. Since the
hole injection efficiency of these survival photons is far from
unity and the total external quantum efficiency of the device
is approximately 5%, we conservatively estimate that the ratio
of hole injection to the total is less than 2%, which results in
an essentially pure electron injection into the multiplication
region. In addition, the ionization rate of injected holes is
lower than that of the electrons in GaAs. Thus, according to
(6), the error in the gain due to mixed injection is less than
, which is the ratio of the gain of pure
2% times
hole injection to that of pure electron injection and is less
than one, while that of the noise is the product of 2% with the
Fig. 2. Measured (dashed lines) and fitted (symbols) gain curves for GaAs
homojunction APD’s. The curves are fitted with the history-dependent theory
and the corresponding model for and :
Fig. 3. Relationship of junction capacitance and area of the GaAs homojunction APD’s with multiplication layer thicknesses of 100–1600 nm. The
symbols represent the experiment results, whereas the dashed lines represent
the fitted lines.
ratio of
, which is not necessarily
and
in
greater than one due to
GaAs. At low voltage, the photocurrent response is relatively
flat, independent of bias which facilitates the identification of
unity gain. All of the devices measured were able to achieve a
gain of at least 30. The dark currents were less than 100 nA at
90% of breakdown. It was thus possible to measure the noise
at high gain. The measured gain, using the photocurrent at low
bias as a unity-gain reference, is shown in Fig. 2.
The i-layer thickness of each wafer was checked by mea) characteristics at low
suring the capacitance–voltage (
bias of a series of devices having different diameters. The
capacitance-versus-area curves are shown in Fig. 3. The almost perfect linearity permitted the thickness of the i region
to be determined with a high degree of accuracy. It was found
YUAN et al.: A NEW LOOK AT IMPACT IONIZATION—PART II
1635
Fig. 4. Schematic of the noise measurement apparatus.
that the thickness deviation from the initial design was more
significant on thin devices because of Be diffusion during
the MBE growth. The electric field profiles of devices with
different doping levels in the p and n regions were calculated
with a commercial simulator, Medici. As pointed out by Hu
et al. [10], the extent of the depletion region into the p and n
layers cannot be neglected, especially for thin i-region devices.
In these experiments, the thicknesses of the n and p depletion
regions, as estimated with the depletion approximation, were
subtracted from the total depletion thickness that was determeasurements. The depletion approximation
mined by
is not very accurate near the edges of the depletion region;
it slightly overestimates the total depletion region thickness.
However, this appears to be a small effect, especially when
the doping levels on the two sides are high. In order to reduce
the extent of the field into the p- and n-type layers adjacent to
the i region and thus reduce the uncertainty in the thickness of
the multiplication thickness, the doping in the p and n layers
was kept high.
Fig. 4 is a block diagram of the apparatus that was used
to measure the avalanche noise. A stable 633-nm He-Ne CW
laser was used as the optical source. An iris was placed in front
of a 40 objective to shield all but the central beam and ensure
pure electron injection. The noise power spectral density was
measured at 50 MHz with a bandwidth of 4 MHz with an HP
8970B noise figure meter. The gain of the preamplifier was
carefully chosen to avoid transient saturation. To eliminate
the environmental noise and any signal resonances in the
transmission line, the device was contacted with a shielded microprobe and careful impedance matching steps were adopted.
The noise from all the components following the photodiode
was calibrated and normalized with a commercial noise source.
In order to calibrate the measurement, the shot noise of the
photodiode at unity gain was measured as a function of the
photocurrent. This step is essential since the circuit impedance,
especially the contact resistance to the device, varies from
device to device. The noise power density and the dc current
were measured under illumination and in the dark alternatively
at the same voltage. The gain was defined as
(8)
Fig. 5. Average electric field near breakdown of the devices with various
multiplication layer thicknesses.
where
is the photocurrent,
is the dark current, and
is the photocurrent at unity gain. The avalanche noise power
density was determined as the difference between the noise
power density under illumination and that in the dark. Care
in the growth and device fabrication yielded devices that were
very stable, and their dark current and dark noise power density
were far less than the corresponding photocurrent and noise
power density under illumination until the gain was very high.
is widely used as the noise
The excess noise factor
as a function of dc
figure of merit for an APD.
gain for the GaAs APD’s with different multiplication region
thicknesses is shown in (10). In the local-field noise theory,
an effective is often used as an adjustable parameter to fit
the noise curves. Following this procedure, we have obtained
values of 0.24, 0.31, 0.39, and 0.48 for multiplication region
thicknesses of 100, 200, 500, and 800 nm, respectively.
IV. DISCUSSION
Fig. 5 shows plots of the average electric field in the i
region at the breakdown voltage in the different devices. For
simplicity, the average fields were determined by the ratio
of the breakdown voltage to the i region thickness. A more
accurate calculation of the electric field profiles in the devices
was employed in the fitting. Bulman’s measurements of the
ionization coefficients in GaAs [26] and the rapid increase of
will approach unity as the
electric field predict that
multiplication region gets thinner. On the contrary, if we use
(3) with a modified , the measurement of the excess noise
decreases with increasing
factor shows that the effective
plot of [10], it can be
electric field. In addition, in the
seen that the measured excess noise factors are slightly but
consistently lower in the low-gain regime than the curves that
have been calculated with the local-field theory.
In the history-dependent theory for ionization that is presented in Section VI-C of Part I, the dead length is treated by
and
modifying the electric field with correlation functions
Correlation functions that have been empirically found
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IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 46, NO. 8, AUGUST 1999
to provide good fits to our data are
(9)
and
are the correlation lengths that depend on
where
the scattering mechanisms and the electric field. Using these
correlation functions, the effective electric fields become
(10)
and the ionization coefficients are represented by the conventional parameterization
Fig. 6. Excess noise factor F (M ) of the GaAs devices of different multiplication region thicknesses. The symbols are the measured data, and the lines
are calculated with the history-dependent model.
(11)
and
are constant. Some of the Monte
where
Carlo simulations have shown that the voltage drop across
the dead length remains relatively constant [16], [22]–[24].
and
can be expressed in terms
The correlation lengths
and
,
of the voltage drops across the dead lengths
respectively, as shown in (28) of Part I. However, in most
and
are
cases,
good approximations. Both the gain curves and the excess
noise curves for all the measured devices were fit using
V and
V and assuming a pure
electron injection. This was done by solving the recursive
equations (12), (13), (16), and (17) of Part I with successive
iterations. Since several iteration steps were required and the
calculated results of each step are sensitive to the accuracy
of the previous values, especially when the gain is high,
relative error)
a conservative convergence criterion ( 10
and a high-density mesh (200–400 points) were employed
for each iteration to avoid unnecessary errors introduced by
the iterations. Fig. 2 compares the calculated gain curves
(symbols) with the measured gain (dashed lines), and Fig. 6
shows the calculated excess noise curves with the experimental
results (symbols) superimposed. Our fitting parameters are
cm
cm
V/cm
V/cm
(12)
and reported
Although the local-field theory and the
in [26] can give a fair fit to the gain curves of APD’s
with different thicknesses, Figs. 6 and 7 show they predict
when the device thickness
the wrong tendency of
decreases. Recently, [28] reported an effort to solve this
problem in a piecewise manner within the frame of the
local-field theory. Here, we note that not only does the
history-dependent theory with the correlation-length ionization
coefficient model achieve excellent agreement with gain curves
using a single set of parameters for and for all the devices
Fig. 7. Comparison of the excess noise factor F (M ) calculated with the
local-field theory (dashed lines) and the history-dependent theory (solid lines).
The thicknesses of the multiplication of the APD’s are marked on the edge.
measured, but it also provides a good fit to the excess noise
factor including the low gain regime, which is not the case
with the local-field theory even with modified ’s.
When the multiplication region is long, the case for which
the local-field theory is most appropriate, agreement between
the two models would be expected. In both theories, ionization
coefficients are used to describe carriers’ ionization probability densities. However, the ionization coefficients cannot be
measured directly, and all the published data are calculated
results based on a particular APD theory and the measurement
of physical parameters, such as gain or noise. As a result,
the ionization coefficients reflect the characteristics of the
model employed and provide good benchmarks to compare
models. In the local-field theory, the ionization coefficients
and
are only functions of the local electric field,
whereas in the model described in Part I, they depend on a
YUAN et al.: A NEW LOOK AT IMPACT IONIZATION—PART II
1637
Fig. 8. Schematic of the spatial variation of the ionization probability density
(0jx) (the dashed line) of a hole traveling along x without ionization. The
solid line shows the history-independent ionization coefficient in the local-field
theory.
carrier’s history. For convenience of comparison, we will first
consider the history-dependent ionization coefficients
and
in the regime where a carrier has traveled a long
distance, relative to the dead length, in a constant electric field
without ionizing. According to the ionization model described
in (9)–(11), the energy accumulated from the field will be
balanced by energy loss through nonionizing scattering events.
For such an energetic carrier, which we will refer to as a
’s in (10) become equal to
,
saturated hot carrier, the
the local electric field, because the ’s defined in (9) are
normalized. Correspondingly, (11) returns to the conventional
local model form, (13). Fig. 8 illustrates the spatial variation
(the dashed line) in a constant electric field. In the
of
beginning, the hole is “cold” and has little probability to ionize.
If the hole travels far enough without ionizing, it becomes a
“saturated hot” carrier and its ionization probability density
becomes “saturated” or history insensitive. The ionization
coefficient model that is used in the local-field theory is shown
in the solid line in Fig. 8. According to our fitting, in a GaAs
PIN structure with a flat electric field in the i region, the
ionization coefficients of the saturated hot carriers have a
simple relationship with the electric field
V/cm
V/cm
cm
cm
(13)
In Fig. 9, these ionization coefficients are compared with
those reported by Bulman et al. [26]. It is clear that (13)
yields coefficients that are generally parallel to but uniformly
greater than [26], and the difference increases with increasing
Fig. 9. Comparison of the measured (E ) and (E ) in [26] with the
(Ee ;e ) and (Ee ;e ) of saturated hot carriers in the history-dependent
theory.
electric fields. There are several factors that contribute to
these differences. First, (13) applies only to the ionization
coefficients (or ionization probability density) of saturated hot
carriers, whereas the data reported in [26] is an averaged effect.
At any point in the multiplication region, there is always
a spectrum of carriers ranging from cold ones to saturated
hot ones. The local-field theory assumes that at any point
all the carriers of the same type have the same ionization
probability density, which is inevitably an average over the
spectrum of the carriers and lower than that of a saturated
hot carrier. The solid line in Fig. 8 shows the average, i.e.,
, used in the local-field theory. Thus, our coefficients
for the saturated hot carriers are generally higher than the
results in [26]. A second factor is the influence of the deadspace, particularly at high electric fields, where the mean
free path between ionizing collisions becomes shorter and
closer to the dead length. In devices with thin multiplication
regions, the effect of the dead pace is more important. One
way to see this is to consider that after traveling a dead
length, the carriers go an additional average distance before
ionization. In thin multiplication regions, the field is high
and
in
and the saturated ionization rate, referred to as
Fig. 2(a) of Part I, is also very high. Consequently, the carriers
ionize relatively quickly after traversing the dead length. This
means that the carriers tend to spend a larger portion of their
lifetime between ionization events acquiring energy from the
field and less time as “hot” carriers. At any time there are
more cold carriers and the statistical weight of the cold carriers
becomes more significant. Therefore, in the thin devices, the
measured ionization coefficients, which are also averages over
the lifetime of the carriers, become even less than the values of
the saturated hot carriers. Third, the local-field theory, which
was used in the work by Bulman et al., overestimates the
contribution of the peak field region in an abrupt p n junction.
While [26] does not contain sufficient details to support a
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IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 46, NO. 8, AUGUST 1999
Fig. 10. Electric field profile in the devices of [27] and the corresponding
effective electric fields Ee ;e of electrons and Ee ;h of holes in the
history-dependent theory.
careful simulation, we still can analyze their measurement
qualitatively. Fig. 10 illustrates the wedge-shaped electric field
profile of the p n junction, where the solid line is
Since the local-field theory assumes that all carriers of the
same type have the same ionization probability, which is a
function only of the local electric field, the ionization would
be expected to occur primarily near the peak electric field,
and the extracted coefficients would be weighted heavily to
that field. However, since the ionization process is nonlocal,
the ionization rates will also be influenced by the slope of
the field. The dotted and dashed lines in Fig. 10 illustrate the
effective electric field of an electron generated at and that of
a hole injected at zero, respectively. We note that the effective
field maxima for both cases are lower than the original peak
The net result is that the local-field theory will tend to
underestimate the ionization coefficients in a p n junction.
One conclusion we can draw here is that any measurement of
an effective ionization coefficient is an averaged result of the
specific electric field profile and valid only for that particular
structure, so the result cannot be applied to other electric field
profiles directly without the risk of significant error.
Since there are no Monte Carlo simulation results on GaAs
that show the spatial behavior of the ionization coefficients,
the model described by (9)–(11) was applied to Higman’s
of Si devices with different electric field
simulations of
profiles [27]. As shown in Fig. 11, the Monte Carlo results can
be almost perfectly fit by these empirical formulas. The dead
are also described very
space and the rising slope of
well.
V. SUMMARY
The history-dependent avalanche theory presented in Part
I has been reviewed and extended for an arbitrary injection
profile. This extension facilitates the estimation of the amount
of error that can be caused by mixed injection. This model
has been shown to agree with Monte Carlo simulations of
the ionization coefficients of Si. It has also been demonstrated
that GaAs homojunction APD’s with thin multiplication layers
Fig. 11. Model of the history-dependent (w jx) compared to Monte Carlo
simulations on Si [27]. The lines are calculated with the model, and the
symbols are the Monte Carlo simulation results. The parameters of (w jx)
are adjusted for Si.
show a significant decrease in multiplication noise compared
to the local-field theory. Further, the noise for fixed gain decreases with decreasing thickness of the multiplication region,
a result that is also counter to the local-field noise theory.
The new theory, on the other hand, provides excellent fits to
the measured avalanche gain and noise with a single set of
parameters for a broad range (100–1600 nm) of multiplication
thicknesses. The parameterized ionization coefficients that
provide good fits to the GaAs APD’s reported here have
been compared to the ionization coefficients reported in [26]
and the differences have been discussed. We have concluded
that measured ionization coefficients are actually a weighted
average, which strongly depends on the structure of the devices
and cannot be used without the possibility of significant error
for other devices with different field profiles.
ACKNOWLEDGMENT
The authors would like to thank J. E. Bowers for discussions
that initiated this collaborative research.
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P. Yuan, photograph and biography not available at the time of publication.
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C. Hu, photograph and biography not available at the time of publication.
H. Nie, photograph and biography not available at the time of publication.
C. Lenox, photograph and biography not available at the time of publication.
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B. G. Streetman, photograph and biography not available at the time of
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J. C. Campbell, photograph and biography not available at the time of
publication.
R. J. McIntyre, photograph and biography not available at the time of
publication.