Impact of Eddy Currents and Crowding Effects on High-Frequency Losses in Planar Schottky Diodes
ABSTRACT In this paper, we present the influence of eddy currents, skin and proximity effects on high-frequency losses in planar terahertz Schottky diodes. The high-frequency losses, particularly losses due to the spreading resistance, are analyzed as a function of the ohmic-contact mesa geometry for frequencies up to 600 GHz. A combination of 3-D electromagnetic (EM) simulations and parameter extraction based on lumped equivalent circuit is used for the analysis. The extracted low-frequency spreading resistance shows a good agreement with the results from electrostatic simulations and experimental data. By taking into consideration the EM field couplings, the analysis shows that the optimum ohmic-contact mesa thickness is approximately one-skin depth at the operating frequency. It is also shown that, for a typical diode, the onset of eddy current loss starts at ~ 200 GHz, and the onset of a mixture of skin and proximity effects occurs around ~ 400 GHz.
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Article: Terahertz technology[show abstract] [hide abstract]
ABSTRACT: Terahertz technology applications, sensors, and sources are briefly reviewed. Emphasis is placed on the less familiar components, instruments, or subsystems. Science drivers, some historical background, and future trends are also discussedIEEE Transactions on Microwave Theory and Techniques 04/2002; · 2.23 Impact Factor
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ABSTRACT: The terahertz region of the electromagnetic spectrum, spanning from 100 GHz through 10 THz, is of increasing importance for a wide range of scientific, military and commercial applications. This interest is spurred by the unique properties of this spectral band and the very recent development of convenient terahertz sources and detectors. However, the terahertz band is also extremely challenging, in large part because it spans the transition from traditional electronics to photonics. This paper reviews the importance of this frequency band and summarizes the efforts of scientists and engineers to span the "terahertz technology gap." The emphasis is on solid-state circuits that use nonlinear diodes to translate the functionality of microwave technology to much higher frequencies.IEEE Journal of Solid-State Circuits 11/2005; · 3.06 Impact Factor
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ABSTRACT: Theory and tools for analysis and design of millimeter- and submillimeter-wave multipliers are discussed. Experimental work is reviewed. The Schottky diode model at submillimeter frequencies, use of Schottky multiplier chains versus direct higher-order multipliers, and the effect of cooling on Schottky diode multipliers are discussed. Alternative diodes such as the high electron mobility varactor (HEMV), the barrier-intrinsic-n<sup>+</sup> diode (BIN), the barrier-n-n<sup>+ </sup> diode (BNN), the quantum well diode (QWD), and the single barrier varactor (SBV) are discussed, with attention given to their potential submillimeter frequency multipliersProceedings of the IEEE 12/1992; · 6.91 Impact Factor
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(Article begins on next page)
Abstract—In this work, we present the influence of eddy
currents, skin and proximity effects on high frequency losses in
planar terahertz Schottky diodes. The high frequency losses,
particularly losses due to the spreading resistance, are analyzed
as a function of the ohmic-contact mesa geometry for frequencies
up to 600 GHz. A combination of 3-D EM simulations and
lumped equivalent circuit based parameter extraction is used for
the analysis. The extracted low frequency spreading resistance
shows a good agreement with the results from electrostatic
simulations and experimental data. By taking into consideration
the EM field couplings, the analysis shows that the optimum
ohmic-contact mesa thickness is approximately one-skin depth at
the operating frequency. It is also shown that, for a typical diode,
the onset of eddy current loss starts at ~200 GHz; and the onset
of a mixture of skin effect and proximity effect occurs around
Index Terms—Current distribution, electromagnetic coupling,
eddy current, geometric modeling, resistance, parameter
extraction, proximity effect, Schottky diodes, skin effect,
submillimeter wave devices, submillimeter wave integrated
HE increasing interest in terahertz (THz) applications 
has generated a technological pressure on searching for
reliable, compact, room-temperature operational and feasible
for circuit integration type of THz devices . To date, GaAs-
based planar Schottky diodes have been successfully
demonstrated as one of the competitive and promising types of
THz devices [3-7] operating as mixers and frequency
In view of this, there is a continuous demand for optimizing
the Schottky diode circuit performance. For operation up to
THz frequencies, the diode geometry-dependent parasitic
couplings and series resistance (Rs) of the diodes have been
identified as the limiting factors for the diode circuit
Manuscript received November 22, 2010. The research leading to these
results has received funding from the European Community's Seventh
Framework Programme ([FP7/2007-2013]) under grant agreement no. 242424
and the European Space Agency
21867/08/NL/GLC. This research has also been carried out in the GigaHertz
Centre in a joint research project financed by the Swedish Governmental
Agency of Innovation Systems (VINNOVA), Chalmers University of
Technology, Omnisys Instruments AB, Wasa Millimeter Wave AB, and SP
Technical Research Institute of Sweden.
The authors are with the GigaHertz Centre, Terahertz and Millimetre
Wave Laboratory, Department of Microtechnology and Nanoscience,
Chalmers University of Technology, SE-42196 Göteborg, Sweden. (phone:
+46-31 772 1739; e-mail: firstname.lastname@example.org).
under ESTEC project No.
performance. Similar limiting factors apply to other high
frequency devices, such as HEMTs, HBTs and HBVs -.
For diodes operating in the THz frequency region,
experimental works have indicated that the series resistance
increased dramatically as a function of frequency, and an
empirical factor has to be included to the diode model in order
to fit the experimental results . This strong frequency
dependency of the series resistance is not explainable using
conventional diode series resistance models [11-16], which do
not take into consideration the electro-magnetic (EM) field
interactions within the diode and the diode surrounding
environment. Thus, it is crucial to understand the geometry
dependent EM field interactions in order to optimize the diode
performance and push the diode operating frequency limits.
In this work, we present a systematic method to analyze the
high frequency losses, including the interactions of the EM
field within and surrounding the surface-channel planar type
 diode (see Fig. 1). In particular, the magnetic field
couplings induced by the time-varying current in the air-
bridge finger is investigated, leading to the findings which
links the high frequency losses to the eddy current, and a
mixture of skin effect and proximity effect. The objective of
this work is to optimize the geometry of ohmic-contact mesas
for diode mixers operating in the sub-millimeter wave region
, i.e. to minimize the geometry-dependent parasitic
couplings as well as the buffer-layer spreading resistance
(Rspreading). In this study, we limit the loss-analysis to the
ohmic-contact mesa region, hereinafter referred to as the
effective spreading resistance, which contributes towards
approximately half of the total diode series resistance, Rs, at
DC for a typical diode. Due to the inherent difficulty of
accurately measuring diode series resistance at high
frequencies, based on e.g. Q-value  or S-parameters ,
and the complexity of the planar diode geometry, this type of
investigation can be pursued more efficiently with numerical
Fig. 1 A schematic of the surface-channel planar Schottky diode.
Impact of Eddy Currents and Crowding Effects on
High Frequency Losses in Planar Schottky Diodes
Aik Yean Tang, Student Member, IEEE, and Jan Stake, Senior Member, IEEE
The approach used in this work is a combination of 3-D
full-wave electromagnetic (EM) simulation and lumped
equivalent circuit parameter extraction. The 3-D EM fields
inside the lossy material (i.e. the ohmic-contact mesa) as well
as the EM fields surrounding the diodes are solved using the
finite element method. An Appendix is also included to
elaborate on the electrostatic analysis, also referred as the EM
analysis at DC (EMDC), performed using COMSOL multi-
physics version 3.5  to obtain the series resistance at DC.
The simulated DC series resistances are then verified by
comparing with the experimental results reported in .
II. GEOMETRY OPTIMIZATION METHODOLOGY
This section describes the diode geometry parameters
studied in this work. The diode structures are constructed and
simulated using Ansoft High Frequency Structure Simulator
(HFSS) . Following this, the lumped equivalent circuit
model parameters are extracted using the Agilent Design
System (ADS)  simulator. The diode spreading
impedance (Zspreading) is extracted as a complex quantity but
analyses are only performed on the real part of the spreading
impedance. This is because our focus in this work is to
minimize the power loss and thermal noise, i.e. minimizing
the series resistance.
A. Device Structure and Geometry
The surface-channel planar diode model is built referring to
the material structure of the GaAs monolithic membrane-diode
(MOMED) technology developed by Jet Propulsion
Laboratory (JPL) , , . Fig. 2 shows a schematic of
the surface-channel planar Schottky diode including its
corresponding series resistance components as well as the
corresponding parasitic elements
Fig. 2 Various components of the series resistance and parasitic elements for
a surface-channel planar Schottky diode. Repi denotes the resistance in the top
epi-layer (i.e. junction layer) when the diode is forward biased. NB! The
drawing is not to scale.
In general, the series resistance for a surface-channel planar
Schottky diode is comprised of the air-bridge finger resistance
(Rfinger), the buffer-layer spreading resistance (Rspreading), the
top junction epi-layer resistance (Repi) and the ohmic-contact
resistance (Rcontact), as stated in (1). In this work, an ideal
ohmic-contact is assumed. Only the spreading resistance of
the cathode ohmic-contact mesa is investigated and other
series resistance components are ignored (i.e. treated as
perfect electric conductors). For the parasitic elements, the
geometry-dependent electrical couplings are modeled as
parasitic capacitors (Cpp1, Cpp2, Cfp and Cfb) whereas the
magnetic coupling within the air-bridge finger is modeled as
the parasitic inductor (Lf). The parasitic capacitances, Cpp1 and
Cpp2, model the electrical coupling between the isolated anode
contact mesa and cathode ohmic-contact mesa through the air
and through the semi-insulating membrane substrate,
respectively. The sum of Cpp1 and Cpp2 is also known as the
pad-to-pad capacitance (Cpp). On the other hand, the fringing
field between the air-bridge finger and the buffer mesa is
represented as the finger-to-buffer capacitance (Cfb), whereas
the fringing field between the air-bridge finger and the epi-
layer is modeled as the finger-to-pad capacitance (Cfp).
For the simulation, a simplified version of the diode
structure, as shown in Fig. 3, is constructed. In this simplified
structure, the top junction epi-layer of the diode is removed
since this is the region where the non-linear electrical
properties (also known as intrinsic properties) of the diode
take place. This is in line with our objective to study the
extrinsic properties of the diode, i.e. geometry-dependent
parasitic couplings as well as the high frequency spreading
resistance in the cathode ohmic-contact mesa. Furthermore,
the top and bottom etch stop layers are excluded since both
layers are mainly used in the device fabrications but having
negligible influences on the diode performance.
Fig. 3 Cross section views of the simplified surface-channel planar Schottky
diode with (a) slanted mesa walls; and (b) straight mesa walls. NB! Drawings
are not to scale.
In all of the simulations, the radius of the Schottky contact
is 0.5 µm (corresponding to an area of 0.8 µm2) and the
spacing between the anode and cathode ohmic mesas is 15
µm. Both slanted and straight mesa side walls are simulated
in order to compare the diode behavior for different surface-
channel formation methods, e.g. resulting from wet or dry
etching, respectively, during fabrication. For the wet etching
case, the diode mesa geometry is constructed with a 55º angle
between the membrane substrate and the slanted side wall
. For each shape of the mesa side walls, simulations are
performed on various buffer-layer thicknesses (bz), i.e. 0.5 µm,
1 µm, 2 µm, 4 µm, 6 µm. In this work, only the losses in the
cathode ohmic-contact mesa are studied, thus the anode
contact mesa is treated as a lossless conductor.
B. 3-D Electromagnetic Simulation
In order to study the diode geometry-dependent high
frequency performance, 3-D EM simulations are performed
for the frequency band of interest. The objective of this work
is to optimize the diode geometry for operation up to 600 GHz
(0.5 mm), yielding a simulated frequency range of a total of 2
octaves, from 150 GHz up to 600 GHz. Since this frequency
range is below the material plasma frequency (~20 THz) ,
, a DC constant bulk conductivity (ζ) is used. The
displacement current is neglected since its effect is not
significant compared to the conduction current in this highly
doped buffer-layer . Moreover, the charge carrier inertia
 effects are not taken into account. The material
properties for the lossy ohmic-contact mesa simulations are
listed in TABLE I.
LIST OF MATERIAL PROPERTIES FOR LOSSY OHMIC-CONTACT MESA
Propertya Symbol Value
5 x 1018 cm-3
Electron mobility for doping
concentration of 5 x 1018 cm-3 
ζ = qµeNd
1.46 x 105 S/m
a Room temperature material properties are used in this analysis.
The 3-D full wave equation, as in (2), is solved in the HFSS
simulator with the setup shown in Fig. 4. In order to reduce
the computation time, the diode geometry is cut through its
two-fold symmetry plane and a magnetic wall is inserted at the
symmetry plane. Thus, only half of the diode structure is
Fig. 4 The setup for high frequency EM simulations.
In this setup, the simulation is performed by suspending the
diode in an air-channel with both ends of the diode extended
towards the two wave-ports. The EM waves are excited
alternately from both anode and cathode wave-ports, creating
quasi-coaxial like excitation modes. The air-channel is
designed in a way that only the quasi-coaxial mode is allowed
to propagate, at least up to a frequency of 600 GHz. In order to
prevent the onset of unwanted higher frequency modes, the
air-channel is designed with a width of 210 μm (~0.5 λ0 at f ≥
600 GHz) and a height of 200 μm.
At the lower end of the simulated frequency band, it is
important to ensure that the evanescent modes decay and
vanish before reaching the inner reference planes. Thus, the
de-embedded lengths (from wave-ports to the inner reference
planes) have to be long enough. In this simulation, we have
set the de-embedding length to 250 μm (> 0.125 λg at f = 150
GHz). All the simulation results are then de-embedded from
the wave-ports towards the inner reference planes in order to
obtain the high frequency response (S-parameters) close to the
diode . The S-parameter convergences are justified by
monitoring the maximum delta S-parameter (∆S) for the two
consecutive simulation passes. For all the simulations, the
maximum ∆S is limited to 0.005 and a minimum of 3
consecutive converged passes are required. Moreover,
additional initial meshes are seeded in the area close to the
diode surface-channel, both in the air and in the membrane
substrate. For the lossy case, the typical mesh size can reach
105 tetrahedra upon meeting the convergence criteria.
In this study, three cases are simulated for each diode
geometry variation. In the first two cases, the diode is
simulated without taking into account any conductive losses.
All the conductors, i.e. air-bridge finger, ohmic-contact mesas
and ohmic pads, are treated as perfect electric conductors.
These two cases are the open-circuited and short-circuited
diode structures as shown in Fig. 5. For the third case, a short-
circuited diode structure similar to Fig. 5 (b) is simulated.
However, in this case, the simulation includes the conductive
loss for the cathode ohmic-contact mesa and the Maxwell
equations are solved inside the mesa. In these simulations, the
physical perception of the field distribution in the lossy mesa
could be visualized and understood. The scattering responses
(S-parameters) from the simulations for various diode
geometries are used to estimate the geometry dependent
parasitic elements and effective spreading resistances, as
elaborated in Section II-C.
Fig. 5 Simulated cases for each of diode geometry (a) open-circuited diode
for the lossless case; and (b) short-circuited diode for both the lossless and
lossy ohmic-contact mesa cases.
C. Lumped Equivalent Circuit Model
In order to study the spreading resistance in the buffer-
layer, we have extended the conventional lumped equivalent
circuit ,  to model losses due to EM couplings. For
simplicity, the high frequency losses due to magnetic
couplings between the air-bridge finger and the ohmic-contact
mesa are represented as the effective spreading impedance
(Zspreading), as shown in Fig. 6, instead of model with mutual
inductance and resistance elements.
Fig. 6 The lumped equivalent circuit used to study the high frequency losses.
A combination of direct extraction and least square error
fitting is used for the parameter extractions. For the least
square error fitting procedure, high degrees of freedom in the
fitting process, without proper boundary conditions, would
lead to inconsistent results converging towards improper local
minimums. Thus, a simplified lumped equivalent circuit as
shown in Fig. 7 is used. For the lossless cases, the pad
inductors are simplified as stated in (3) and (4), and the finger-
to-pad (Cfp) and finger-to-buffer capacitor (Cfb) are expressed
as (5). This simplification reduces the number of unknowns
from 10 to 7 for the lossless cases.
(a) open-circuited diode (lossless case)
(b) short-circuited diode (lossless case)
(c) short-circuited diode (lossy buffer-layer case)
Fig. 7 The simplified version of lumped equivalent circuits for (a) an open-
circuited diode in a lossless case; (b) a short-circuited diode in a lossless case;
(c) short circuited diode in a lossy cathode ohmic-contact mesa case.
In the case of the lossy ohmic-contact mesa, the effective
spreading impedance (Zspreading) is extracted using the lumped-
equivalent circuit, as shown in Fig. 7(c). As in (6), this
spreading impedance consists of a real part, related to the
power dissipation, and an imaginary part, related to the energy
storage. In this work, the ohmic-contact mesa geometry
optimization is performed by analyzing only the real part of
the impedance (Rspreading). An example of a plot of the real
and imaginary parts of the effective spreading impedance is
shown in Fig. 8.
Fig. 8 An example plot of the effective spreading impedance for a diode with
slanted mesa wall (bz = 6 µm).
D. Parameter Extraction Procedure
For parameter extractions, the parasitic capacitors and
inductors are extracted using the lumped equivalent circuit for
the two lossless cases, respectively (shown in Fig. 7 (a) and
(b)). These parasitic capacitors and inductors are then used in
the lumped equivalent circuit for the lossy ohmic-contact mesa
case (Fig. 7(c)) to estimate the frequency-dependent spreading
resistance. An overview of the parameter extraction procedure
is shown in Fig. 9.
Fig. 9 A flow chart showing the procedure of parameter extraction.
Using the „Π‟ network configuration at the lower simulated
frequency range, the total parasitic capacitances (Ctot) are
estimated using (7) in the open-circuited case, whereas the
total parasitic inductances (Ltot) are estimated using (8) in the
short-circuited case. The anode and cathode pad capacitances
are estimated using (9) and (10) from the short-circuited case.
These values are extracted at low frequencies in order to
reduce the effect of parasitic inductance on the extraction of
parasitic capacitance for the open-circuited diode case, and
vice versa for the short-circuited diode case.
In order to guide the least square error fitting, the total
parasitic capacitances and total inductances estimated from (7)
and (8) are used as the initial guess for the pad-to-pad
capacitance (Cpp) and the finger inductance (Lf), respectively.
The initial guesses for the finger capacitance (Cf) is 1 fF and
for the pad inductances (La, Lc) are 1 pH. In the least square
error fitting procedure, the error functions are expressed as
Sij = S-parameters from HFSS simulations
ij = S-parameters from lumped equivalent circuit simulations
ij = 11, 12, 21, 22
For the lossless cases, the fitting procedure is performed
across the frequency range of 150 GHz to 600 GHz. The
parasitic capacitors and inductors are estimated through a
sequence of fine-tuning using open-circuited and short-
circuited cases, as shown in Fig. 9. The convergence criterion
is defined as a less than 10% difference between the extracted
Lf, Cpp and Cf of current iteration with previous iteration,
In order to extract the effective spreading resistance, the
parasitic elements extracted from the lossless cases are used.
The least square error fitting procedure is then performed in
which the effective spreading impedance is fine-tuned for a
partition of 10 GHz bands within the frequency range of 150
GHz to 600 GHz. In general, all the errors calculated using
(11) are less than 8%.
In addition to errors introduced from the S-parameter fitting
procedure, other potential error sources affecting the spreading
resistances result are the accuracy of other parasitic elements
extracted from the lossless cases. Thus, a simple error
analysis is performed for the case of slanted mesa wall diodes
with a 6 μm thick buffer-layer. In this analysis, the sensitivity
of the extracted spreading resistance, upon a 10% error in each
parasitic element (i.e. Lf, La, Lc, Cf, Cpp, Ca and Cc), is
evaluated by assuming a zero error correlation between these
parasitic elements. The analysis shows that the spreading
resistances in this model are most sensitive to the deviation of
Ca. The S-parameter fitting errors for this error analysis,
calculated using (11), are less than 10%. By assuming a
symmetrical behavior of the error, the extracted spreading
resistance is deduced to be within an error margin of +/- 6%.
A. EM Field Distribution and Eddy Currents
By examining the EM fields of the diodes, as shown in Fig.
10 and Fig. 12, the strong frequency-dependent losses are
attributed to eddy currents effects. The generation of eddy
currents in the ohmic-contact mesa could be explained by the
Faraday‟s law, where the electric field in the mesa is
developed due to the time-varying magnetic field induced by
the current through the air-bridge finger. Fig. 10 shows the
low frequency (i.e. 150 GHz) normalized magnetic fields and
the corresponding current density vectors for a thin and a thick
ohmic-contact mesa. Comparatively, the plots indicate a more
uniform magnetic field penetration in area under the air-bridge
finger for the thin than the thick ohmic-contact mesa, showing
the importance of current crowding phenomena.
Fig. 10 The current density vectors and the magnetic field (normalized to the
corresponding maximum field) for diodes with ohmic-contact mesa of a
thickness of (a) 1µm ; and (b) 6µm at 150 GHz.
In conventional diode resistance models, the skin effect is
the only frequency-dependent loss mechanism considered.
The frequency-dependent skin depth (δs) is calculated using
(12). For the diode geometry analyzed, the skin depth is
approximately 1.8 μm at 600 GHz, indicating that the 0.5, 1
and 2 μm buffer-layer thicknesses are less than or
approximately one skin-depth. Nevertheless, the frequency-
dependent losses are consistently observed for these buffer-
layers. Thus, it is obvious that the skin effect is not a
dominating loss mechanism in this case.
As the frequency increases, the loss mechanisms become
more complicated, with a mixture of skin effect and proximity
effect. In order to confirm the existence of both current
crowding effects, an additional simulation is performed to
solve the EM fields inside the air-bridge finger and the ohmic-
contact mesa for the case of a 6 μm thick slanted mesa walls
buffer-layer. The rectangular cross-section of the air-bridge
finger is inherently a simpler way to visualize these effects.
The current distributions in the air-bridge finger and the mesa
are shown in Fig. 11. For the air-bridge finger which is
located in the air-channel, the skin effect causes a symmetrical
current crowding at the outer air-bridge finger. On the other
hand, for the air-bridge finger which is located close to the
ohmic-contact mesa, the proximity effect causes an
asymmetrical current crowding phenomenon. In this case,
more current are pushed towards the outer finger facing the
mesa compared to the outer finger facing away from the mesa.
As a result of both effects, the total current distribution within
the air-bridge finger, in close proximity to the ohmic-contact
mesa, becomes asymmetrical.
Fig. 11 Normalized current distribution within the air-bridge finger and the
ohmic-contact mesa showing a mixture of skin effect and proximity effect at
Fig. 12 shows the normalized magnetic field and current
density vector of a thick ohmic-contact mesa at the frequency
of 550 GHz for a slanted mesa walls case. Comparing to Fig.
10, it is clear that the circulation of current within the mesa
under the air-bridge finger, due to Faraday‟s law, is more
pronounced at a higher frequency. Thus, the loss mechanisms
involved for the geometry of planar Schottky diodes become
more complex as the frequency is increased. The similar loss
mechanism occurs for the straight mesa walls buffer-layer.
Fig. 12 Normalized EM field distributions due to a mixture of skin effect and
proximity effect for a 6 µm thick buffer-layer at 550 GHz (a) distribution of
the magnetic field and current density vector of the diode; (b) magnetic field
vectors at the A-A‟ cross section of the diode.
B. Reactive Parasitic Elements
For quantitative studies, the EM field couplings are
modeled as parasitic capacitances and inductances, as
discussed in Section II.A. For the Schottky diode design, it is
essential that the geometry-dependent parasitic capacitances
are minimized, especially the parasitic capacitance across the
Schottky diode port (refer to Fig. 6). A larger parasitic finger
capacitance (Cf) presents a lower impedance (reactance)
current path, shorting the non-linear diode junction. Fig. 13
shows the result of the extracted parasitic capacitances. For
completeness of the model, the extracted finger inductance (Lf)
is plotted as well in the same figure.
Fig. 13 A comparison of the parasitic capacitances and inductance as a
function of buffer thickness.
By extrapolating the extracted Cf values towards zero buffer
thickness, the finger-to-pad capacitances (Cfp) for diodes with
slanted and straight mesa walls are estimated to be
approximately 1.3 fF. For diodes with straight mesa wall, the
coupling between the air-bridge finger to the buffer (Cfb) and
the coupling between anode and cathode mesa (Cpp) is not
significant. Thus, the increase of total parasitic capacitance
for the diodes with slanted mesa walls as a function of buffer-
layer thickness are mainly due to the increase of Cfb and Cpp.
The extracted Cf values are close to the measured value in 
for diodes with similar geometry, confirming the validity of
the extraction method. The result also indicates that the effect
of displacement current on the current distribution in the
ohmic-contact mesa is not significant, due to the inherently
high reactance (e.g. ~130 Ω for a 2 fF parasitic capacitance at
C. Formulation of the Effective Spreading Resistance
In this work, the extracted resistance is treated as an
effective spreading resistance (Rspreading) contributed by
various power loss mechanisms (Pn), such as eddy currents
and current crowding due to skin effect and proximity effect.
A simplified total power loss (Ploss) in the ohmic-contact mesa
is written as (13).
where n = number of high frequency loss mechanisms
By analyzing the slope of the extracted spreading
resistances as a function of frequency in the logarithmic scale,
the spreading resistances show f
dependency within the simulated frequency band (i.e. 150
GHz to 600 GHz). The effective spreading resistance is
formulated as (14).
In this equation, the f
spreading resistances is related to the eddy current loss .
2 and f
2 dependency in the effective
For losses due to a mixture of skin effect and proximity effect,
the frequency dependency of spreading resistance is
mathematically complicated. For a simple case of two parallel
conductors, the analytical solution of the total current involves
cosh and sinh functions . Thus, the f 4 dependency in the
effective spreading resistances is interpreted as the
representation of additional current crowding losses.
A weighted curve-fitting is performed to acquire the
coefficients for further quantitative analysis. The weight
factors are assigned to be 20, 10, 5 and 1 for the frequency
range of 150 GHz to 200 GHz, 210 GHz to 250 GHz, 260
GHz to 300 GHz and 310 GHz to 600 GHz, respectively. By
assigning the coefficient k to 0.1, the critical frequencies, i.e.
fcrit1 and fcrit2, can be calculated from the coefficients fitted to
the extracted resistances. These critical frequencies are
defined as the frequencies when 10% (i.e. k = 0.1) of the
related loss mechanisms begin to dominate the overall loss
mechanisms . Fig. 14 shows a plot of the extracted
spreading resistances and the fitted effective spreading
resistance curves whereas Fig. 15 shows the calculated
geometry-dependent critical frequencies. The result of RDC is
addressed in Section III-D.
The result shows that the critical frequencies for slanted
mesa diodes are generally lower than the straight mesa wall
diodes. A lower critical frequency indicates an „earlier‟ onset
of the corresponding high frequency loss mechanisms.
Therefore, the slanted mesa wall diodes exhibit higher
frequency-dependent losses compared to the straight mesa
wall diodes. For both types of diodes, the fcrit1 decreased as
the buffer-layer thickness is increased from 0.5 μm to 2 μm.
This critical frequency is almost constant for the thickness
larger than 2 μm. On the other hand, an obvious decrease in
the fcrit2 is observed for the slanted mesa wall diodes thicker
than 2 μm. Thus, for a thick buffer-layer, the loss mechanisms
due to a mixture of skin effect and proximity effect are more
severe for slanted mesa wall diodes than straight mesa wall
diodes due to a larger effect of magnetic coupling.
Fig. 14 Extracted high frequency effective spreading resistances for diodes
with various buffer-layer thicknesses.
Fig. 15 Geometry-dependent critical frequencies for high frequency losses.
D. DC Spreading Resistance
In order to verify the result of the fitted DC spreading
resistances, similar diode geometries are simulated in an
EMDC simulator, where the validity of the EMDC simulations
is also verified with experimental results. In the EMDC
simulations, the DC spreading resistances are calculated using
the Ohm‟s law. Further details of the EMDC simulation setup
and procedures can be found in the Appendix.
Fig. 16 shows a comparison of the DC spreading resistances
simulated in the EMDC simulator and estimated by fitting the
extracted high frequency resistances to the effective spreading
resistance using (14). The figure shows a reasonable
agreement between the spreading resistances acquired from
both methods. It also shows that the estimated DC spreading
resistances for diodes with slanted mesa walls are close to
those estimated for straight mesa walls.
Fig. 16 A comparison of the DC spreading resistances obtained from two
E. Optimized Ohmic-Contact Mesa
The optimization of the ohmic-contact mesa is pursued by
comparing the geometrical-dependent parameters, such as DC
spreading resistance, critical frequencies and parasitic
capacitances. The result indicates a clear advantage of the
straight mesa wall diode over the slanted mesa wall at high
frequency. At low frequency (e.g. 150 GHz), the effective
spreading resistance between both diodes are comparable.
However, lower fcrit1 and fcrit2 are observed for the slanted
mesa wall diodes compared to straight mesa wall diodes.
Thus, as the frequency is increased, the effective spreading
resistances of a slanted mesa wall diode are higher than a
straight mesa wall diode. From the displacement loss point of
view, the straight mesa wall diodes are preferable as well since
the finger-to-buffer capacitance is not significant (Fig. 13).
At DC, a thicker buffer-layer is preferred as the effective
spreading resistances decrease (and saturates at certain
thickness) as a function of buffer-layer thickness. On the
contrary, the high frequency loss increased un-intuitively as
the buffer thickness is increased, where this behavior has been
later related to the effect of EM field couplings. Fig. 17 shows
the effective spreading resistance as a function of frequency
and geometry as well as the corresponding skin depth.
Moreover, with an increase of the buffer thickness, the finger-
to-buffer capacitance is increased for diodes with slanted mesa
wall. Thus, the optimum buffer thickness for the diode design
is approximately one skin-depth at the operating frequency.
Fig. 17 A contour plot of the effective spreading resistance and the
corresponding skin depth for (a) slanted mesa wall; (b) straight mesa wall.
In this work, we have presented a systematic method to
optimize the geometry of surface-channel planar Schottky
diode for high frequency applications. This analysis includes
high frequency losses due to eddy current, skin effect,
proximity effect and displacement losses. The validity of the
methodology and model presented in this work are justified
through the verification of the extracted finger capacitance
with experimental results in  and the verification of
extrapolated DC spreading resistances with EMDC simulation.
Error analysis has also been performed to examine the effect
of potential inaccuracy of other parasitic elements to the final
result, indicating the results are within an error margin of +/-
6%. Straight wall mesa with buffer-layer thickness of 2 µm is
shown to be the optimized geometry in this study. For this
geometry, the critical frequency for the onset of eddy current
effect is approximately ~200 GHz; and the onset of a mixture
of skin effect and proximity effect occurs around ~400 GHz.
This optimization work demonstrates that it is essential to
include the parasitic couplings (both magnetic and electric
field couplings) effects in diode geometry design for high
frequency applications. The result also provides a useful
insight explaining the dramatically increased series resistance
which is not explainable using the conventional models.
Although the top junction epi-layer is not included in the
analysis, it is subjected to similar loss mechanisms as the
ohmic-contact mesa. The effect of the parasitic couplings to
the overall inductance of the diode is also not considered in
this work. Thus, a more physical model including mutual
inductances could be developed further based on the
simplified model proposed in this work.
Although the main focus of this work is the optimization of
the ohmic-contact mesa design for single-finger surface-
channel planar Schottky diodes, similar conclusions may be
drawn for anti-parallel finger Schottky diodes and other high
frequency devices with a similar geometry. In addition, the
method presented could easily be extended to allow for
optimization of other geometries (such as dimensions of the
air-bridge finger, the position and shape of the Schottky
contact, etc). The method could also be used to analyze the
device performances at higher frequencies. In cases where the
charge carrier inertia and dielectric relaxation effects are not
negligible, a similar analysis could be performed by
substituting the constant conductivity with a complex
In this work, EMDC simulations are performed in order to
simulate the spreading resistance at zero frequency.
Furthermore, they are used to verify the DC spreading
resistance extrapolated from the high frequency simulations.
For EMDC simulations, the cathode ohmic-contact mesa is
constructed in the COMSOL multi-physics simulator. As
shown in Fig. 18, only half of the mesa is simulated, due to the
two-fold symmetry of the geometry property. The anode
Schottky contact is modeled by embedding a 2-D sheet layer,
with a boundary condition of 1 V voltage potential. On the
other hand, the faces of the object for ohmic-contacts are
grounded. In this simulation, the specific ohmic-contact
resistance is neglected.
Fig. 18 (a) The 3-D COMSOL EMDC simulation setup for a diode; (b) A
plot of the isopotential and streamline from the EMDC simulation.
The solution is obtained by solving the static form of the
continuity equation as in (15), within the domain of the mesa,
which is similar to solving the Poisson‟s equation. The total
current (Itot) through the diode is obtained using (16) where
Jnorm is the current density normal to the surface in question.
The surfaces integrated are the 2-D Schottky anode contact
and the ohmic-contacts for the extraction of currents for the
anode resistance (Ranode) and cathode resistance (Rcathode)
calculations, respectively. Since only half of the structure is
simulated, a multiplication factor of 2 is included in (16) to
extract the total current. The spreading resistances are then
calculated using the Ohm‟s law.
In order to estimate the resistance from the COMSOL
simulation, a sequence of convergence tests is performed. The
number of mesh elements in the structure is varied until the
changes in both the calculated anode and cathode resistance
saturates. Moreover, the difference between calculated anode
and cathode spreading resistance is monitored until both
spreading resistances converge to a similar value. The
confidence level of the simulation results was validated by
simulating two UVA diodes (SC2R4 and SC2T1) with the
geometry parameters provided in . In these cases,
comparisons between the spreading resistance calculated from
simulations and experimental data from  could be
In this simulation, the simulated series resistance (REMDC_sim)
includes both the top junction epi-layer and ohmic-contact
mesa spreading resistance. The ohmic-contact resistance is
calculated using the DC contact resistance model as well as a
specific contact resistivity of 2 x 10-5 Ω∙cm2 as in . With
this, the ohmic-contact resistance is estimated to be 1.1 Ω. In
this calculation, the finger resistance (Rfinger) is neglected.
COMPARISON OF DC RS FOR TWO DIODES
SC 2R4 SC 2T1
REMDC_sim + Rcontact
5.0 – 7.0 Ω
10.0 – 13.0 Ω
As shown in TABLE II, the series resistances calculated
from the EMDC simulations very well match the experimental
data. This shows that the EMDC simulation results are valid
The authors would like to thank Dr. Tapani Närhi (ESA
ESTEC), Prof. Erik Kollberg, Dr. Tomas Bryllert, Dr. Josip
Vukusic (Chalmers University of Technology), and Peter
Sobis (Omnisys Instrument AB) for many fruitful discussions.
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Aik Yean Tang (S‟09) was born in Kedah, Malaysia. She received the degree
of B.Eng in electrical-electronics engineering (honors) from University
Technology Malaysia, Johor, Malaysia in 2002. She then received her degrees
of M.Eng in nanoscience and nanotechnology and M.Sc. in nanoscale science
and technology from K.U. Leuven, Leuven, Belgium and Chalmers University
of Technology, Göteborg, Sweden, respectively, in 2008.
In year 2002, she was a Silicon Validation Engineer at Intel Technology (M)
Sdn. Bhd, Penang, Malaysia. She was then employed as an Analog IC
Designer in Avago Technologies (M) Sdn. Bhd., Penang, Malaysia end of
year 2004. She is currently a PhD student at the department of
Microtechnology and Nanoscience (MC2), Chalmers University of
Technology, Göteborg, Sweden. Her research interests include modeling,
optimization and integration of Schottky diodes for terahertz applications.
Ms. Tang received the IEEE MTT-S Graduate Fellowship Award in 2010.
Jan Stake (S‟95–M‟00–SM‟06) was born in Uddevalla, Sweden in 1971. He
received the degrees of M.Sc. in electrical engineering and Ph.D. in
microwave electronics from Chalmers University of Technology, Göteborg,
Sweden in 1994 and 1999, respectively.
In 1997 he was a research assistant at the University of Virginia,
Charlottesville, USA. From 1999 to 2001, he was a Research Fellow in the
millimetre wave group at the Rutherford Appleton Laboratory, UK, working
on HBV diode multiplier circuits for submillimeter-wave signal generation.
He then joined Saab Combitech Systems AB as a Senior System Consultant,
where he worked as an RF/microwave engineer until 2003. From 2000 to
2006, he held different academic positions at Chalmers and was also Head of
the Nanofabrication Laboratory at MC2 between 2003 and 2006. During the
summer 2007, he was a visiting professor in the Submillimeter Wave
Advanced Technology (SWAT) group at Caltech/JPL, Pasadena, USA. He is
currently Professor and Head of the Terahertz and Millimetre
Wave Laboratory at the department of Microtechnology and Nanoscience
(MC2), Chalmers, Göteborg, Sweden. His research involves sources and
detectors for terahertz frequencies, high frequency semiconductor devices,
graphene electronics, terahertz techniques and applications. He is also co-
founder of Wasa Millimeter Wave AB.