# 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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- Aik Yean Tang, Erich Schlecht, Robert Lin, Goutam Chattopadhyay, Choonsup Lee, John Gill, Imran Mehdi, Jan Stake[Show abstract] [Hide abstract]

**ABSTRACT:**We present a self-consistent electro-thermal model for multi-anode Schottky diode multiplier circuits. The thermal model is developed for an $n$-anode multiplier via a thermal resistance matrix approach. The nonlinear temperature responses of the material are taken into consideration by using a linear temperature-dependent approximation for the thermal resistance. The electro-thermal model is capable of predicting the hot spot temperature, providing useful information for circuit reliability study as well as high power circuit design and optimization. Examples of the circuit analysis incorporating the electro-thermal model for a substrateless- and a membrane-based multiplier circuits, operating up to 200 GHz, are demonstrated. Compared to simulations without thermal model, the simulations with electro-thermal model agree better with the measurement results. For the substrateless multiplier, the error between the simulated and measured peak output power is reduced from ${\sim} {\hbox{13}}\hbox{\%}$ to ${\sim} {\hbox{4}}\hbox{\%}$ by including the thermal effect.IEEE Transactions on Terahertz Science and Technology - TTHZ. 01/2012; 2(3):290-298. -
##### Conference Paper: Geometry optimization of THz sub-harmonic Schottky mixer diodes

[Show abstract] [Hide abstract]

**ABSTRACT:**We present a systematic analysis of geometry-dependent parasitics for THz sub-harmonic Schottky mixer diodes. In particular, the diode RF power coupling bandwidth has been studied, showing a trade-off between the air-bridge finger length and finger separation for operating frequencies beyond 1 THz.Infrared, Millimeter, and Terahertz Waves (IRMMW-THz), 2012 37th International Conference on; 01/2012 - SourceAvailable from: Jan Stake[Show abstract] [Hide abstract]

**ABSTRACT:**We present an analytic method to extract Schottky diode parasitic model parameters. All the ten unknown model parameters are extracted via a straightforward step-by-step procedure. The challenges for a proper finger inductance and series resistance extraction are discussed and solutions are recommended. The proposed method is evaluated using three sets of S-parameter data for GaAs-based planar Schottky diodes, i.e., data from measurement up to 110 GHz and 3-D electromagnetic full-wave simulations up to 600 GHz. The extracted models agree well with the measured and simulated data.IEEE Transactions on Microwave Theory and Techniques 05/2013; 61(5):1870-1878. · 2.23 Impact Factor

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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

~400 GHz.

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

circuits.

I. INTRODUCTION

HE increasing interest in terahertz (THz) applications [1]

has generated a technological pressure on searching for

reliable, compact, room-temperature operational and feasible

for circuit integration type of THz devices [2]. 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

multipliers.

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: aik-yean.tang@chalmers.se).

under ESTEC project No.

performance. Similar limiting factors apply to other high

frequency devices, such as HEMTs, HBTs and HBVs [8]-[9].

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 [10]. 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

[7] 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

[17], 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 [18] or S-parameters [19],

and the complexity of the planar diode geometry, this type of

investigation can be pursued more efficiently with numerical

simulation techniques.

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

T

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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 [20] to obtain the series resistance at DC.

The simulated DC series resistances are then verified by

comparing with the experimental results reported in [12].

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) [21]. Following this, the lumped equivalent circuit

model parameters are extracted using the Agilent Design

System (ADS) [22] 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) [17], [23], [24]. 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

inductance).

(capacitances and

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).

contact

R

spreading

R

epifingers

RRR

(1)

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.

(a) (b)

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

[25]. 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

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3

octaves, from 150 GHz up to 600 GHz. Since this frequency

range is below the material plasma frequency (~20 THz) [14],

[16], 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 [11]. Moreover, the charge carrier inertia

[26] effects are not taken into account. The material

properties for the lossy ohmic-contact mesa simulations are

listed in TABLE I.

TABLE I

LIST OF MATERIAL PROPERTIES FOR LOSSY OHMIC-CONTACT MESA

Propertya Symbol Value

Dielectric permittivity

Doping concentration

εr

Nd

12.9

5 x 1018 cm-3

Electron mobility for doping

concentration of 5 x 1018 cm-3 [27]

µe

1830

sV

cm

2

Bulk conductivity

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

simulated.

E

j

Ej

1

(2)

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 [28]. 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.

(a)

(b)

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 [29], [30] 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.

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4

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.

aaa

LLL

21

(3)

ccc

LLL

21

(4)

(5)

fp fbf

CCC

(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.

spreadingspreading spreading

jXRZ

(6)

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

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5

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.

1

imagCCC

fpptot

12

Y

(7)

12

1

22

Y imag

LLLL

caftot

(8)

12 11

1

YY imagCa

1

(9)

22 21

YY imagCc

(10)

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

(11).

f

S

ij

ijij

ij

SS

abs err

'

(11)

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,

respectively.

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

S’

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%.

III. RESULT

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.

(a) (b)

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.

2

f

s

2

(12)

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

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6

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

550 GHz.

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.

(a) (b)

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 [31]

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

600 GHz).

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).

)()(

RIfPfP

n

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).

1

crit

f

In this equation, the f

spreading resistances is related to the eddy current loss [32].

)(

2

f

spreadingn loss

(13)

2 and f

4 frequency

f

4

2

2

1

crit

DCspreading

f

f

k

f

kRR

;

10

1

k

(14)

2 dependency in the effective

Page 8

7

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 [33]. 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 [34]. 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

separate methods.

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

Page 9

8

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.

(a)

(b)

Fig. 17 A contour plot of the effective spreading resistance and the

corresponding skin depth for (a) slanted mesa wall; (b) straight mesa wall.

IV. CONCLUSION

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 [31] 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

frequency-dependent conductivity.

APPENDIX

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.

(a)

(b)

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.

Page 10

9

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.

0

J

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 [12]. In these cases,

comparisons between the spreading resistance calculated from

simulations and experimental data from [12] could be

performed.

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 [12]. With

this, the ohmic-contact resistance is estimated to be 1.1 Ω. In

this calculation, the finger resistance (Rfinger) is neglected.

TABLE II

COMPARISON OF DC RS FOR TWO DIODES

0

V

(15)

dAJI

normtot

2

(16)

SC 2R4 SC 2T1

Measurement [12]

REMDC_sim + Rcontact

5.0 – 7.0 Ω

5.6 Ω

10.0 – 13.0 Ω

10.2 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

and reasonable.

ACKNOWLEDGMENT

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.

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