Modeling and design aspects of millimeter-wave and submillimeter-wave Schottky diode varactor frequency multipliers

Abstract

Design and optimization of Schottky varactor diode frequency
multipliers for millimeter and submillimeter wavelengths are generally
performed using harmonic balance techniques together with
equivalent-circuit models. Using this approach, it is difficult to
design and optimize the device and multiplier circuit simultaneously.
The work presented in this paper avoids the need of equivalent circuits
by integrating a numerical simulator for Schottky diodes into a circuit
simulator. The good agreement between the calculated and published
experimental data for the output power and conversion efficiency
originates from the accurate physical model. The limiting effects of
multiplier performance such as breakdown, forward conduction, or
saturation velocity are discussed in view of the optimum circuit
conditions for multiplier operation including bias point, input power,
and loads at different harmonics. It is shown that the onset of forward
or reverse current flow is responsible for the limitation in the
conversion efficiency

Full text

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700IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 48, NO. 4, APRIL 2000
Modeling and Design Aspects of Millimeter-Wave
and Submillimeter-Wave Schottky Diode
Varactor Frequency Multipliers
Jesús Grajal, Viktor Krozer, Member, IEEE, Emilio González, Francisco Maldonado, and Javier Gismero
Abstract—Design and optimization of Schottky varactor
diode frequency multipliers for millimeter and submillimeter
wavelengths are generally performed using harmonic balance
techniques together with equivalent-circuit models. Using this
approach, it is difficult to design and optimize the device and mul-
tiplier circuit simultaneously. The work presented in this paper
avoids the need of equivalent circuits by integrating a numerical
simulator for Schottky diodes into a circuit simulator. The good
agreement between the calculated and published experimental
data for the output power and conversion efficiency originates
from the accurate physical model. The limiting effects of multi-
plier performance such as breakdown, forward conduction, or
saturation velocity are discussed in view of the optimum circuit
conditions for multiplier operation including bias point, input
power, and loads at different harmonics. It is shown that the onset
of forward or reverse current flow is responsible for the limitation
in the conversion efficiency.
Index
technique, numerical modeling, Schottky diode modeling, semi-
conductor simulation, submillimeter-wave multipliers.
Terms—Frequencymultipliers,harmonicbalance
I. INTRODUCTION
V
quencies. The key points in the progress of the performance of
Schottkyvaractorfrequencymultipliershavebeentheenhanced
physical insight into and optimization of submillimeter-wave
Schottky diode operation [1], the improvement in frequency
multiplier analysis methods since the original work of Siegel
and Kerr [2]–[5], and in physical analytical Schottky diode
models [5]–[9], as well as numerical physical device models
[4], [10]–[12]. The performance of active devices is defined not
only bytheir inherentcharacteristics, butalso bythe embedding
circuit. This coupling can be taken into account by including
a numerical physical model into a circuit simulator [10], [11],
[13]. The main problem in frequency varactor circuit design
today is the inability to reproduce the experimental results
ARACTOR frequency multipliers play a vital role in
developing all-solid-state power sources at terahertz fre-
Manuscript received March 4, 1999. This work was supported in part by the
EuropeanSpaceAgency.TheworkofJ.GrajalwassupportedbytheComunidad
AutónomadeMadrid.TheworkofV.KrozerwassupportedbytheMinisteriode
EducaciónyCulturaMadrid,Spain,undertheProgramaNacionaldeFormación
de Personal Investigador.
J. Grajal, E. González, F. Maldonado, and J. Gismero are with the ETSIT,
Technical University of Madrid, 28040 Madrid, Spain.
V. Krozer is with the Technical University of Chemnitz, D-09126 Chemnitz,
Germany.
Publisher Item Identifier S 0018-9480(00)02529-1.
without additional empirical parameters. Furthermore, the
limitations of multiplier operation at high powers and/or high
frequencies is not well understood. Finally, a predictive design
and circuit analysis tool is still missing, which is essential for
the design of integrated frequency multipliers.
The scope of this paper is to present a circuit analysis tool,
which works without empirical parameters and is able to pre-
dict the required device and circuit parameters of a frequency
multiplier. We focus on the circuit design and operation aspects
of frequency multipliers for millimeter and submillimeter
bands. As a design tool, we employ the harmonic balance
method (HBM) together with a physics-based drift-diffusion
(DD) numerical device simulator. Our simulator incorporates
accurate boundary and interface conditions for high forward as
well as reverse bias, including impact ionization, nonconstant
recombination velocity, self-consistent incorporation of the
tunnelling, and image-force effects [11]. The validation of the
numerical simulator has been performed by comparison of
simulated device and circuit characteristics with experimental
results obtained for submillimeter-wave Schottky diodes fabri-
cated at the Technical University Darmstadt (TUD), Darmstadt,
Germany, and the University of Virginia (UVa), Charlottesville,
and for a number of multiplier circuits published in the litera-
ture [4], [5].
The integration of numerical simulators for active devices
into circuit simulators avoids the need of an equivalent-circuit
model extraction. This new philosophy accounts for the de-
vice–circuit interaction and provides another degree of freedom
to improve the performance of circuits because they can be
designed from both a device and circuit point-of-view.
The simulation tool utilized in this paper and its implementa-
tion are presented in Section II. The validation of the tool is out-
lined in Section III, including a comparison of measured dc and
RF performance characteristics with simulated values. In Sec-
tion IV, an analysis of the performance of the frequency multi-
pliers is presented together with an identification of the limiting
mechanisms. Simulated results and a detailed discussion for the
limitingmechanismsareprovidedinSectionV,includingbreak-
down and velocity saturation effects. Based on these results, the
synthesis of an optimum doubler and tripler circuit is dealt with
in Section VI. The optimization of the device parameters such
as doping concentration and profile, layer thicknesses, etc. are
omitted in this paper, but will be dealt with in a future paper.
This paper concludes with a summary of the results.
0018–9480/00$10.00 © 2000 IEEE

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GRAJAL et al.: MODELING AND DESIGN ASPECTS OF SCHOTTKY DIODE VARACTOR FREQUENCY MULTIPLIERS701
II. SIMULATION TOOL BASED ON PHYSICAL DEVICE MODEL
The HBM is the most common technique for the design of
large-signal nonlinear microwave circuits. The HBM depends
critically on the accuracy of the nonlinear element model
employed in the analysis. This model must be valid for a wide
range of frequencies, drive levels, and embedding impedances.
The electrical and RF performance characteristics of submil-
limeter-wave Schottky diodes and frequency multiplier circuits
investigated here are based on an accurate physical model,
which combines DD current transport with thermionic and
thermionic-field emission currents imposed at the Schottky
contact.
The model for the carrier transport throughout the bulk is
based on an extended one–dimensional (1-D) DD formulation
[14]. 1-D simulations are adequate for the current flow under
the Schottky metal, whereas two–dimensional (2-D) effects like
current spreading can be accounted for by analytical formulas.
The governing equations are Poisson’s equation and continuity
equations for electrons and holes as follows:
(1)
(2)
(3)
(4)
(5)
where
is the electrostatic potential,
are theelectron concentration, holeconcentration, and
ionizedimpuritydonorconcentration,respectively.
are, respectively, the electron conductioncurrent density, there-
combination rate modeled by the Schockley–Read–Hall recom-
bination, and the generation rate currently restricted to impact
ionization[14].Further,
pendent electron mobility, temperature, affinity, and density of
statesintheconductionband,respectively.Similarequationsalso
hold for holes. In (5), the conventional formulation of the posi-
tion-dependent conduction band potential is augmented by the
imageforceterm.Incontrasttopreviouspublications,wedonot
introduce direct barrier lowering of the Schottky barrier height
[15]. The simulation domain is divided into a nonuniform mesh
of approximately 100 mesh points and the simulation time is of
the order of minutes for one frequency point.
is the permittivity, and
,,and
and representthefield-de-
A. Boundary and Interface Conditions
Dirichlet boundary conditions are imposed at all metal
contacts for Poisson’s and carrier continuity [14]. In the
present model, thermionic and thermionic-field transport at
the Schottky barrier is introduced by means of an interface
condition at the maximum of the barrier at the point
[16].
The position of the maximum is no longer at the metallurgical
Schottky contact because of the influence of the bias-dependent
image force lowering. The interface condition for the carrier
transport is based on the assumption that the carrier distribution
can be modeled by a displaced Maxwellian
(6)
Here,
of the barrier and
sity of electrons—the density that would be present at the top of
the barrier if the electrons could be brought into thermal equi-
librium without disturbing the potential distribution.
classical recombination velocity
recombination velocity after assuming a displaced Maxwellian
shifted by a drift velocity
is the electron concentration at the maximum
represents a quasi-equilibrium den-
is the
is the.
(7)
(8)
(9)
In this formulation, it has been assumed that the Schottky con-
tact is located at
. This interface condition prevents the
unphysical effect of carrier accumulation at the interface [17],
[18]. Between the point
and the metallurgical contact, the
potential is assumed to vary linearly. No mesh points are re-
quired here for the calculation of the potential and current den-
sities. Newton’s method, Gummel’s method, as well as a hybrid
Newton/Gummel solution method are used for the solution of
(1)–(5) with the interface conditions given in (6)–(9).
Tunnelling transport through the barrier is important for
Schottky diodes with high doping in the epitaxial layer. In
our model, the time-independent Schrödinger equation is
solved for arbitrary piecewise-linear potential barriers using
the transfer matrix approach [19], [20]. The grid defined for
Poisson’s equation is also used for Schrödinger’s equation
in the volume of the device, but a number of additional grid
points are generated between
in order to accurately resolve the shape of the conduction band
in this region. The new expression for the current density can
be represented by a simple integration [21] if the transmission
coefficient
is independent of the transversal
component of the kinetic energy
function is of Maxwellian type for the transversal component
of the kinetic energy
and the metallurgical contact
and the distribution
(10)
(11)
where
ture (carrier heating is not considered),
is the Richardson constant,is the lattice tempera-
is the trans-

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702IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 48, NO. 4, APRIL 2000
Fig. 1.Schematic drawing of the partitioning of the physical model and linear circuit for the harmonic balance calculations.
missioncoefficientforthetunnellingtransport,
for the normal component of the carrier velocity,
the electron distributions in the semiconductor and metal, re-
spectively,
stands for the conduction band in the semicon-
ductor, and
is the Fermi level in the metal. We have as-
sumed that
is a displaced Maxwellian distribution and
a Maxwell–Boltzmann distribution. If tunnelling is not consid-
ered, this equation is similar to that proposed in [16].
istheenergy
andare
is
B. Coupling the Physical Device Model with Circuit
Simulation
Active devices are usually modeled as lumped equivalent cir-
cuits in commercial harmonic balance codes. There exist sev-
eral algorithms for the HBM [2], [22], [23], some of which have
been specifically designed to deal with physical device simula-
tors [24]. The selected algorithm for this study is based on the
solution of a system of nonlinear equations using a modified
Newton’s algorithm or Powell’s algorithm, or the optimization
of an error function by using the Levenberg–Marquardt (LM)
algorithm [25]. The LM algorithm is utilized in the current sim-
ulations because of its efficiency.
In our case, the linear circuit is represented by the general
coupling matrix, defined by
the generator impedance
The harmonic balance error equation for each harmonic
then
network parameters, and
, as can be depicted in Fig. 1.
is
for
(12)
where
(13)
(14)
are the phasors of the excitations, and
monic componentsof thecurrentand voltage atthediode termi-
nals.In thecase offrequency multipliers,
dc voltage component and a single sinusoidal signal. The diode
is always matched at the fundamental frequency. Six to twelve
harmonics are considered in these simulations, and the imped-
ances at the higher harmonics are set to 0.001
resistive and reactive components, if not otherwise specified.
andare the har-
is comprised of the
for both the
Fig. 2.
compared to simulated values for the diode D1038 from TUD. The diode
parameters are indicated in Table I.
Measured current voltage characteristic of a varactor Schottky diode
All the parasitic elements of the diode are absorbed in the
embedding network and included in the impedance seen by the
junction at the different frequencies involved in the multiplier
operation.
III. VALIDATION OF THE PHYSICAL MODEL
The physical model has been validated with measurements
for a large number of diodes in the doping range of
cm cm , different epitaxial layer thick-
nessesandvariousdiodediametersrangingfrom1 m
The simulated current voltage characteristic for all diodes agree
excellently with the respective measurements. An example is
given in Fig. 2 for the diode D1038. The parameters of all the
diodes analyzed in this paper are provided in Table I.
The contribution of the individual components to the total
diode current is indicated in Fig. 2. The simulated capacitance
voltage characteristic is also in excellent agreement with the
measured values, as demonstrated in Fig. 3. The simulation is
extended up to the breakdown voltage, where the junction ca-
pacitanceisknowntoincreaserapidlywithreversevoltage[26].
The RF performance of the simulation tool has been dis-
cussed in [12]. We have performed simulations for two fre-
quencymultiplierspublishedintheliterature[4],[5].Inthesim-
ulations,wedonotuseanyfittingparameters.Inallsimulations,
a barrier height for the
Schottky contact of
is assumed.
The output power and efficiency versus the input power have
been plotted in Fig. 4 for a frequency doubler (2
20 m.
eV
100 GHz)

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GRAJAL et al.: MODELING AND DESIGN ASPECTS OF SCHOTTKY DIODE VARACTOR FREQUENCY MULTIPLIERS703
TABLE I
SCHOTTKY DIODE PARAMETERS FOR
DEVICES FROM TUD AND UVa
Fig. 3.
lines) capacitance voltage characteristic for two diodes from the TUD D1038
and D734. The diode parameters are indicated in Table I.
Comparison between the measured (symbols) and calculated (solid
utilizingaUVa6P4Schottkyvaractor.Theresultshavebeentaken
from [4] with additional simulations achieved with our model
(solid line with “ ” symbols). The techniques used in Fig. 4 are
indicated as: 1) MCHB: self-consistent ensemble Monte Carlo
harmonicbalancesimulator;2)DDHB-fdm:drift-diffusionhar-
monic balance simulator with field-dependent mobility; and 3)
DDHB-am: drift-diffusion harmonic balance with constant av-
eragemobility.Allsimulatorsoverestimatetheoutputpower[see
Fig.4(a)]andtheefficiency[seeFig.4(b)]ofthedoublerat
dBm.Resultsfromourmodelagreewellabove
and reproduce the measured power slope. The bias voltage was
V, and the loads at harmonic frequencies were set to
for
quency,thediodewasmatched.Theloadimpedanceattheoutput
wasoptimizedformaximumoutputpowerforeachinput
powerandobeysavalueof
power
dBm. The unrealistic simulated results at low
inputpowersoriginatefromtheperfectmatchofthediodeateach
powerlevelincontrasttotheexperimentalsetup,wheretheloads
have been optimized for high-power levels.
This is emphasized in Fig. 4(b), which shows that the results
from the Monte Carlo code predict efficiencies close to the the-
oretical limit at low input power levels, whereas the measured
efficiency is low and increases with power level. In our simula-
dBm
,whileatthefundamentalfre-
ataninput
(a)
(b)
Fig. 4.
efficiency as a function of the input power for a frequency doubler at ? ? ???
GHz [4]. The solid line with symbols “?” denotes the results from our model,
the solid line shows the experimental results from [4], and the dashed lines
represent calculated results from [4].
(a) Calculated and measured output power and (b) conversion
tions,we haveused thegeometricalarea andit isoutlinedbelow
that our results would still improvewhen an effective diode area
would have been employed.
We have also compared our simulations with the mea-
sured data for a frequency multiplier using a similar diode
at
GHz [5], illustrated in Fig. 5. At the output, the
balanced diode configuration has been simulated as a direct
parallel connection of two identical diodes. The agreement
with measured data is very good. Simulations of the same
arrangement at
GHz also agree well at high-input
power levels. Although the simulations presented in Figs. 4
and 5 have been performed at different frequencies and with
different circuit structures, we achieve very good agreement at
large power levels, which demonstrates the capabilities of the
simulation tool.
IV. ANALYSIS OF VARACTOR-BASED MULTIPLIER CIRCUITS
The results from a successful circuit analysis of frequency
multipliersshouldpredictthefollowingparametersdetermining

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704IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 48, NO. 4, APRIL 2000
(a)
(b)
Fig. 5.
efficiency as a function of the input power for a frequency doubler at
? ? ?? GHz [5]. The symbols “?” “*” denote the results from our model at
?
? ?? GHz and ?
? ??? GHz, respectively, and the solid line shows the
experimental results from [5].
(a) Calculated and measured output power and (b) conversion
the multiplier performance for a given diode structure: output
power
, conversion efficiency
at the fundamental, output, and idler frequencies at the diode
terminals, bias voltage
, and load impedances at the
remaining harmonics. All conventional multiplier design
methods are based on an equivalent circuit of the Schottky
diode presented in Fig. 6(a) [1], [2], [27]. However, this simple
model cannot explain the decrease in conversion efficiency
beyond a certain input power level, which has been frequently
observed in experimental results. Improved intrinsic diode
models have to be utilized in this case, as shown in Fig. 6(b)
and (c), which describe the physical behavior of the device with
increasing accuracy. As has been pointed out by Grajal et al.
[12], at low-input powers equivalent-circuit models, including
electron velocity saturation phenomena [see Fig. 6(b)] are
sufficient to accurately predict the multiplier performance.
However, at high-power levels and near the maximum of the
conversion efficiency, only accurate physical models [see
, embedding impedances
(a)
(b)
(c)
Fig. 6.
Constant series resistance model. (b) Variable series resistance model. (c)
Proposed physical model including all effects.
Equivalent circuits for different Schottky diode varactor models. (a)
Fig. 6(c)] provide reasonable prediction of the multiplier
performance.
In Fig. 6(a) and (b), the Schottky junction is modeled by
a parallel connection of a voltage-dependent nonlinear capac-
itance and resistance. The series resistance due to the unde-
pleted region in the semiconductor becomes a nonlinear func-
tion of the diode current density at increased current densities
[see Fig. 6(b)]. The nonlinear current dependent series resis-
tance has been considered responsible for the decrease in the
conversion efficiency at high-power levels [1], [5], [27]. It has
been argued that the displacement current becomes comparable
to the maximum current in the material at high operating fre-
quencies. An empirical current-dependent series resistance has
beenintroducedinordertofitthemeasuredresults,accordingto
Fig. 6(b). It has to be emphasized that the above analysis did not
include breakdown effects and its impact onthe RF operation of
the frequency multiplier. Furthermore, the series resistance in-
creases when the current density in the diode approaches the
maximum current density
(15)
with
by the available number of electrons yields the maximum ve-
locity with which the edge of the depletion region can move
being the displacement current. Dividing both sides
(16)