robust sub-harmonic mixer at 340 ghz using intrinsic...

Robust Sub-harmonic Mixer at 340 GHz Using IntrinsicResonances of Hammer-Head Filter and ImprovedDiode Model

Cheng Wang1& Yue He1 & Bin Lu1

& Jun Jiang1 &

Li Miao1 & Xian-Jin Deng1 & Yong-zhong Xiong1&

Jian Zhang1

Received: 24 February 2017 /Accepted: 29 August 2017 /Published online: 6 September 2017# Springer Science+Business Media, LLC 2017

Abstract This paper presents a sub-harmonic mixer at 340 GHz based on anti-parallelSchottky diodes (SBDs). Intrinsic resonances in low-pass hammer-head filter have beenadopted to enhance the isolation for different harmonic components, while greatly minimizingthe transmission loss. The application of new DC grounding structure, impedance matchingstructure, and suspended micro-strip mitigates the negative influences of fabrication errorsfrom metal cavity, quartz substrate, and micro-assembly. An improved lumped elementequivalent circuit model of SBDs guarantees the accuracy of simulation, which takescurrent-voltage (I/V) behavior, capacitance-voltage (C/V) behavior, carrier velocity saturation,DC series resistor, plasma resonance, skin effect, and four kinds of noise generation mecha-nisms into consideration thoroughly. The measurement indicates that with local oscillatingsignal of 2 mW, the lowest double sideband conversion loss is 5.5 dB at 339 GHz; thecorresponding DSB noise temperature is 757 K. The 3 dB bandwidth of conversion loss is50 GHz from 317 to 367 GHz.

Keywords Terahertz . Sub-harmonicmixer . Hammer-head filter . Schottky diode . Devicemodeling

1 Introduction

Coherent detection of terahertz and sub-millimeter wave based on GaAs Schottky diodes sub-harmonic mixer (SHM) have many advantages in room temperature, such as low noise

J Infrared Milli Terahz Waves (2017) 38:1397–1415DOI 10.1007/s10762-017-0436-4

* Yue [email protected]

1 Microsystem &Terahertz Research Center, Institute of Electronic Engineering, China Academy ofEngineering Physics, Mianyang, Sichuan 621900, China

http://orcid.org/0000-0002-2380-7224

http://crossmark.crossref.org/dialog/?doi=10.1007/s10762-017-0436-4&domain=pdf

mailto:[email protected]

temperature, low conversion loss, better accessibility of LO power, and wideband operation.However, its conversion loss cannot be ignored by receiver yet; thus, the noise introduced bycascaded intermediate frequency (IF) module is significant. In recent years, although mono-lithic integrated low-noise amplifiers/mixers based on high cutoff frequency transistors aredeveloping rapidly [1, 2], their noise performance has not exceeded the Schottky receiver.Therefore, SHMs are still critical terahertz and sub-millimeter wave detectors for scientificresearch, communication, imaging, and instrument.

Nowadays, SHMs below 500 GHz still adopt hybrid integration technique based on quartzsubstrate and metal cavity. It brings with the advantages including low circuit loss, low cost,and high flexibility. Fundamental mixers based on quasi-optical technique smooth the diffi-culty on fabrication [3]. Nonetheless, they suffer from low coupling efficiency, which leads tohigher loss. Fundamental mixers based on whisker-contacted diodes and mechanical tuner [4]hold excellent performance both at room and low temperature, but they are inconvenient inpractical application. The SHM proposed by B. Thomas [5] has also achieved impressiveresult. After being integrated with IF amplifier [6], LO multiplier [7], or another SHM [8],various receiving modules have been realized. Integrated MMIC membrane SHM suffers fromlossy substrate [9, 10]. Its performance is lower than hybrid integrated mixer.

The design from [5] indeed has many advantages. However, its substrate has large length/width ratio (> 20). Therefore, this architecture is more vulnerable which has many uncer-tainties on micro-assembly. Actually, due to these issues, our previous experiments on it arenot successful. For instance, the circuit is likely to be Blifted up^ during upside-downmounting. It is because of the discrepancy of expansion coefficient of quartz glass andaluminum. The situation would become more serious with a larger aspect ratio.

In this paper, we propose a robust 340-GHz SHM. It utilizes intrinsic resonances ofhammer-head filter, new DC grounding structure, impedance-matching structure, andsuspended micro-strip. The proposed design presents high tolerance towards fabrication errors.It also presents excellent measured performance, including an average DSB conversion loss of5.9 dB, and an average minimum DSB noise temperature of 838 K from 330 to 350 GHz. The3 dB bandwidth is 50 GHz.

In addition, we propose an improved lumped element equivalent circuit (LEC) model forplanar Schottky junction, which is much more completed than the published models. It takescurrent-voltage (I/V) behavior, capacitance-voltage (C/V) behavior, carrier velocity saturation,DC series resistor, plasma resonance, skin effect, and several noise generation mechanismsinto consideration thoroughly. The simulation coincides with the measurement within atolerance around 1 dB.

2 Design of Sub-harmonic Mixer

2.1 Circuit Architecture

The architecture of SHM is shown in Fig. 1. The following strategies are adopted in the design:(1) To minimize the transmission loss in circuit—firstly, the anti-parallel Schottky barrierdiodes (SBDs) are embedded at the edge of RF input rectangular waveguide; secondly, theintrinsic resonances of hammer-head filter are introduced to enhance signal choke. Onehammer-head filter is inserted between RF input port and IF output port and another isbetween LO input port and SBDs. (2) To realize impedance matching with the simplest

1398 J Infrared Milli Terahz Waves (2017) 38:1397–1415

DC GNDSilver epoxy

SBDs

RF waveguide

LO waveguide IF Transformer

RF and LO choke filter

RF choke filter

(a)

(b)

0 1 2 3 4 5 6

-40

-20

0

)B

d(

re

te

ma

ra

pS

Frequency (GHz)

S(1,1,)

S(2,1)

(c)Fig. 1 a Architecture of 340-GHz sub-harmonic mixer. b The IF impedance transformer-simulated structure. cThe simulated S parameter

J Infrared Milli Terahz Waves (2017) 38:1397–1415 1399

Resonator 2 f0 ≈ 331GHz

Resonator 1 f0 ≈ 245GHzResonator 3 f0 ≈ 337GHz

Cg2

Cg1

Cr1

Cr2 Lr3

Lr1

Lr2 L1

(a) (b)

(c)

L1

C1 C2

L1 L12L11

C1 C2C11 C12 C21 C22

L12L11

C1 C20.5*C110.5*C12

0.5*C220.5*C21

0.4mm

0.28mm 0.2mm

65 m

175 m

50 m

(d)

(e)


structure, the RF and LO coupling probe provides conjugate impedance matching directly fordiodes. (3) To reduce the LO pumping power, the IF port is designed to have a characterimpendence of 100 Ohm. An IF transformer converts the IF impedance from 100 to 50 Ohmfurther which is a five-step-quarter-wavelength uniform response filter. The IF output band isfrom 500 MHz to 6 GHz, as shown in Fig. 1b, c. (4) To reduce the errors of micro-assembly,the quartz substrate with thickness of 50 μm is suspended in the cavity for removing thebackside grounding. It uses silver epoxy to connect the Au layer on the end of the circuit andthe mental cavity directly, which provides DC grounding, as shown in Fig. 1. This structurelocates far from the RF port and has low impedance. Therefore, it further improves thetolerance of SHMs.

2.2 Intrinsic Resonance in Hammer-Head Filter

The introduction of intrinsic resonances in hammer-head filter is a significant improvement inour design. It should be acknowledged that hammer-head filter has been adopted in SHMs fora long history [11, 12]. Figure 2a shows an equivalent circuit of the simplest 3-order stepped-impedance filter. The low-impedance transmission line is equivalent as capacitors (C1, C2)and high-impedance line is recognized as inductor (L1). In contrast, a schematic of 3-orderhammer-head filter is shown in Fig. 2b. It utilizes the additional stripline branches, which bringthe additional reactance including L11, L12, C11, C12, C21, and C22. As a result, thereactance of C1, C2, and L1 in original 3-order stepped-impedance filter is enhanced. Afterthat, the hammer-head filter can achieve better rejection with equal or shorter circuit length.However, since the RF and LO choke filter in current design needs to cover a wide frequencyrange from 150 to 380 GHz, the conventional solution is to increase the order of hammer-headfilter to more than 7 [7]. It definitely introduces more loss and circuit length. In this article, wecan also achieve sufficient rejection with 3-order filter by using resonant modes in hammer-head filter.

Figure 2c presents three resonant modes in the RF and LO choke filter. Resonator 1 at245 GHz locates in the main stop-band of hammer-head filter. Edge coupling betweenhammer-head branches brings with coupling capacitors (Cg1, Cg2). They work with inductorsof high-impedance line (L11, L1, L12) to establish resonator 1. It significantly enhances thesignal rejection by introducing a pole in the main stop-band. The edge coupling capacitors(Cr1, Cr2) between hammer-head branches and main stripline cooperate with stripline induc-tors (Lr1, Lr2, Lr3) to form resonator 2 and 3 as shown in Fig. 2c. They introduce additionalpoles to the first parasitic stop-band of hammer-head filter at 331 and 337 GHz, respectively.The resonating frequencies of resonators 2 and 3 differ from each other slightly, due to thedifference in inductor Lr2 and impedance transition of low-impedance stripline. Figure 3shows the simulated S parameter of the RF and LO choke filter. The simulated rejection ismore than 14.5 dB during 150–190 GHz, and more than 27.3 dB during 300–380 GHz.

If the edge coupling capacitors (Cg1, Cg2) are increased by reducing the distance betweenstriplines, the resonator 1 at 245 GHz in Fig. 2c further moves into 150–190 GHz band. The

�Fig. 2 a Schematic of traditional 3-order stepped-impedance low-pass filter. b Schematic of 3-order hammer-head low-pass filter. cHigh-frequency schematic of 3-order hammer-head low-pass filter for LO and RF choke in340 GHz SHM. Three resonating modes have been indicated in 245, 331 and 337 GHz. d The dimension ofhammer-head filter. e Cross-sectional view


signal rejection will be better. Considering the balance between circuit length and performance,we have not taken that strategy in RF and LO choke filter. However, this strategy has beenadopted in RF choke filter, as shown in Fig. 1. If only signal isolation is required, one resonatoris sufficient, whose maximum rejection is better than 30 dB. Nevertheless, to expand the pass-band of LO signal, two resonators (R1 and R2) are employed. The simulation result shows thatS21 < − 36.7 dB in 300–380 GHz, S11 < − 20.8 dB in 150–190 GHz.

In addition, since these resonators are independent from the capacitance between striplineand cavity, the fabrication errors have limited influences on the performance of filters.

2.3 Impedance Matching and Fabrication Errors

A similar RF choke hammer-head filter is designed to block the leak of RF signal in LOwaveguide illustrated in Fig. 4. Meanwhile, to minimize the loss of impedance-matchingstructure, directly coupling from the standard rectangular waveguide is performed for RF andLO signals. As shown in Fig. 5, one pad of anti-parallel SBDs is mounted at the edge of RFchoke filter. It realizes wideband RF virtual grounding. The height of WR.8 rectangularwaveguide is 356 μm. The width of quartz substrate and stripline of RF probe is 250 and200 μm, respectively. Therefore, the physical dimension of RF probe approaches 1/4 λRF of

0 50 100 150 200 250 300 350 400-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Frequency (GHz)

)B

d(rete

marap

SS11S21

LO stopband150-190GHz

RF stopband300-380GHz

IF passbandDC-15GHz

Resonator 1f0 ≈ 245GHzResonator 2f0 ≈ 331GHzResonator 3f0 ≈ 337GHz

Fig. 3 Simulated S parameter ofRF and LO choke hammer-headfilter

Cg2

Cg1

L1

Resonator R1

C1 C2 C3

Lp1 Lp2

Lp3 Lp4

Resonator R2Fig. 4 Simulated structure of RFchoke hammer-head filter


input signal. With the purpose of enhancing robustness, we have not reduced these dimen-sions. However, this method is more likely to introduce spurs in the curve of conversional lossdue to horizontal resonant mode of RF probe. We eliminate them by tuning the probe carefully.

On one hand, we utilize the RF probe and a zigzag high-impedance transmission line torealize 1/2 λLO transition, since the RF and LO choke filter provides LO virtual grounding onlynearby its input strip. Thus, a LO virtual grounding at the edge of RF probe is achieved.Otherwise, 1/2 λLO transition will be extremely long. On the other hand, this design may setlimitation on bandwidth, but it maintains a reasonable balance for conversion loss on bothsides of center frequency.

Figure 6 presents the simulated single-sideband (SSB) conversion loss versus differentfabrication errors, corresponding to Fig. 5. The simulated results show a 0.3 dB variation oflowest conversion loss under +/− 20 μm displacements on physical dimensions and +/− 15°ro-tation of diode mounting. These errors hold greater but still acceptable influence on bandwidth.The displacements and rotations used in simulation is the worst case for realization. Therefore,the final design is pretty robust.

3 Improved LEC Model of Schottky Diodes

The SHM in this article uses anti-parallel SBDs measured by Terahertz Research Center,CAEP. The measured parameters of single diode are saturated current Isat = 9.1×10−15 A, ideal

Y

X

Z`

LO Virtual GND

WGBS

RF Virtual GND

LO in

RF inIF out

QSDP

SBDR

QSLF

Fig. 5 Detail of RF impedance-matching structure

310 320 330 340 350 360 3706789

1011121314151617181920

Frequency (GHz)

SSB

Con

vers

ion

Loss

(dB

)

WGBS +20umWGBS -20umQSLF +20umQSLF -20umQSDP +20umQSDP -20umSBDR +15 degreeSBDR -15 degreeOriginal Design

Fig. 6 Simulated single-sideband(SSB) conversion losses versusvarious fabrication errors. WGBS:horizontal length of waveguidebackshort to RF probe; QSLF:vertical height of suspended quartzsubstrate to mental channel;QSDP: horizontal displacement ofsubstrate in channel; SBDR: therotation of SBDs compared with Xdirection


factor n = 1.12, barrier height Vbar = 0.69 V, DC Series resistor RS = 21 Ohm, total capacitanceCtotal = 10 fF. The junction capacitance is estimated as Cj0 = 1.8 fF. Diode’s passive structure ismodeled by electromagnetic (EM) simulator, including anode pad, cathode pad, anode finger,mesa, and GaAs substrate. A coaxial probe, as shown in Fig. 7a, is inserted to replace thenonlinear Schottky junction [13]. Therefore, EM model predicts the parasitic elements ofdiodes exactly. Then, our model concentrates on active junction.

There are already several modeling methods for Schottky junction, such as lumped elementequivalent circuit (LEC), drift-diffusion (DD), hydrodynamic (HD), and Monte Carlo (MC)[14]. Although the last three models depend more on simulation of practical physical behaviorof individual electron, which are supposed to be more accurate, LEC model based onexperimental data and physical principles has been proved to be an effective method. It iswidely accepted below 1 THz.

(a)

(b) (c)

(d)

Ij(Vj)

Cj(Vj)

Repi Li

Cd

Rsp

Rresidual

Vdiode

Idiode

SchottkyJunction

Undepletedepi Layer

GaAs N+Substrate &

Ohmic contact

Vj

Ij(Vj)

Cj(Vj)

Repi Li

Cd

Zsp

Ij(Vj)

Cj(Vj)

Rs

Gleakage

Ishot+IflickerVthermal1+Vhot_electron

Vthermal2

Fig. 7 a EM model of anti-parallel Schottky Diodes. The coaxial probe removes Schottky junction. b BasicSchottky diode model. c Enhanced Schottky diode model which takes velocity saturation, plasma resonance, andskin effect into consideration. d Proposed model in this article


Figure 7b is the basic diode model in [15]. It uses classical thermal emission and diffusiontheory to predict the current-voltage (I/V) relationship of Schottky diode. Its capacitance-voltage (C/V) relationship adopts abrupt doping approximation. A series resistor RS accountsfor total DC resistance. Figure 7c introduces an enhanced LEC model [16]. It describes carrierinertia and the displacement current by using an inductor Li and a capacitor Cd. The carriervelocity saturation is solved by limiting the maximum current through junction. The compleximpedance from skin effect in high frequency is included in Zsp.

The improved model of us is shown in Fig. 7d. Our model is established based on data fromprobe test. It does not only cover the above issues with better equations, but also incorporatenoise model. These are all listed as follows.

3.1 Forward and Backward Current-Voltage (I/V) Behavior

Actually, thermal emission and diffusion theory describes I/V behavior of diode in positive biaspretty well [15]. However, deep forward bias, backward avalanche breakdown, and leakagecurrent cannot be ignored, too. Under deep forward bias, the junction is linearized. Thus, Imax isset to limit the maximum current of diode. The maximum voltage Vmax is defined as (1).

Vmax ¼ nVT lnImaxIS

þ 1

� �ð1Þ

where IS is saturation current, VT =KBT/q is thermal voltage, n is ideal factor of quantumtunneling, KB is Boltzman’s constant, T is temperature, and q is charge of single electron.

When Vj > Vmax, junction current Ij is defined as (2).

I j V j� � ¼ Imax þ V j−Vmax

� � Imax þ ISð ÞnVT

ð2Þ

When Vmax ≥ Vj ≥ − 10nVT, Ij is defined as (3).

I j V j� � ¼ IS exp

V j

nVT

� �−1

� �ð3Þ

When − 10nVT > Vj ≥ − Vbmax − Vbv, the backward current with avalanche breakdown isdescribed by (4).

I j V j� � ¼ IS

e−10

nVTV j þ 11nVT� �

−1� �

−Ibvexp −V j þ Vbv

nbvVT

� �ð4Þ

where Vbv is breakdown voltage corresponding to breakdown current Ibv. Generally, Vbv ischosen as Ibv = 1 μA. nbv is an ideal factor for breakdown. Vbmax is defined by Imax and limitsthe maximum backward current as in (5).

Vbmax ¼ nbv⋅VT ⋅lnImax

Ibv

� �ð5Þ


Finally, when Vj < −Vbmax −Vbv, (6) is used to describe deep avalanche breakdown in largenegative bias.

I j V j� � ¼ − Imax þ − V j þ Vbv

� �−Vbmax

� Imax

nbv⋅VT

� �ð6Þ

Actually, saturated current IS, ideal factor n, series resistor RS, barrier height Vbi, breakdownvoltage Vbv, breakdown current Ibv, and breakdown ideal factor nbv can all be extracted directlyfrom I/V curve. Furthermore, considering the inevitable leakage current, a parallel conductanceGleakage is employed, which improves the accuracy in low current. Figure 8 presents themeasured and simulated I/V behavior of our anti-parallel SBDs. The series resistors are alsoincluded in DC simulation. Consequently, they coincide with each other extremely well.

3.2 Capacitance-Voltage (C/V) Behavior and Carrier Velocity Saturation

Firstly, before further discussion, it is meaningful to pay some attention on the measurement ofdevice’s C/V behavior. Multi-Frequency Capacitance Measurement Unit (MFCMU) ofAglient B1500A semiconductor analyzer is used for our C/V measurement. It extracts theimpedance of device: Z = X − jY through pumping AC voltage. The current through capacitordepends on simple relationship in (7).

IC ¼ C j⋅dV j

.dt ð7Þ

And then, the device’s capacitor is calculated byY = 1/(2πf0C), where f0 is frequency of ACsignal. Nevertheless, for capacitor with strong nonlinearity, such as junction capacitance inSBDs, the output current given by (7) is invalid. The relationship should be alternated by (8).

IC ¼ dQdt

¼ d C jV j� �dt

¼ dC j

dV jV j þ C j

� �dV j

dtð8Þ

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

-15

10-12

10-9

10-6

10-3

100

V diode (V)

)A( e

doi

d I

Measurement 01Measurement 02Measurement 03Proposed Model

-1 -0.5 0 0.5 1-15

-10

-5

0

5

10

15

MeasuredProposed Model

I diode(mA)

Vdiode (V)

Uncertainty ofsmall current

Fig. 8 Simulated and measured current-voltage (I/V) relationship of anti-parallel Schottky barrier diodes


Consequently, there is a difference between measured and authentic values of nonlinearjunction capacitor as in (9).

Cmeas: ¼ ∂C j

∂V jV j þ C j

� �ð9Þ

where Cmeas. refers to measured C/V data acquired directly from MFCMU, Cj refers toauthentic junction capacitor.

The solution for the above transcendental equation is in (10).

C j V j� � ¼ 1

V j∫V j

0 Cmeas: tð Þdt V j≠0

C j 0ð Þ ¼ Cmeas: 0ð Þ V j ¼ 0

8<: ð10Þ

As a result, we obtain the authentic C/V data which indicates the variation of space chargesin depleted layer correctly. It also matches well with the quasi-static capacitance measurementin semiconductor analyzer, which gets capacitor through injecting charges and measuring thechange of voltage. But our approach holds higher resolution when the capacitor is in tens of fF.

Furthermore, abrupt doping approximation is made for SBDs generally [15]. It assumes thatthe doping density in epitaxial layer is uniform and has a sudden change in the interfacebetween epitaxial layer and N+ layer. Nonetheless, the formula in [16] could not match themeasurement due to the non-uniform doping, edge effect of anode and adjustment of barrierheight. In our model, (11) and (12) for junction capacitor are employed, which do not onlyoffer us good agreement under negative bias but also avoid mathematical singularity underpositive bias.

WhenVj ≤ FC ⋅ (ϕbi − VT), junction capacitor relies on (11).

C j V j� � ¼ γ V j

� � AεsWD V j� � ¼ C j0 1−

V j

ϕbi−VT

� �−M

ð11Þ

WhenVj > FC ⋅ (ϕbi − VT), (12) is used for modeling Cj in deep positive bias.

C j V j� � ¼ C j0

1−FCð ÞM 1þ M V j−FC ϕbi−VTð Þ� �ϕbi−VTð Þ 1−FCð Þ

� �ð12Þ

where A is area of anode, εs is dielectric constant of epitaxial layer, Cj0 is capacitor under zero-bias, M is grading coefficient, ϕbi is barrier height, FC is forward-bias depletion capacitancecoefficient, WD(Vj) is width of depleted layer, γ(Vj)is the correction factor for edge effect [17].

γ V j� � ¼ 1þ b1

WD V j� �ra

þ b2WD V j� �2r2a

ð13Þ

where b1 = 1.5, b2 = 0.3, and ra is the radius of anode.Although, there are also other equations [18, 19] for removing singularity in deep positive

bias, the effectiveness for them is still under validation. SBDs are not a simple circuit with


series capacitor and resistor. There are complex junction and parasitic elements. For instance,we cannot eliminate the nonlinear forward Bturn on^ current. Even inconspicuous parasiticelements can have significant influence on final positive C/V data. Therefore, the measurementin positive bias is questionable. Figure 9 gives the measured and simulated C/V data of anti-parallel SBDs. Although Cj is calculated to be 20–30 fF under 0.8–1.0 V from (12), themeasurement presents a capacitor above 1 pF. The reactance in deep positive bias is notdominated by Cj anymore. So, it is extremely hard to achieve good agreement between modeland measurement. The simulated C/V relationship does not only exceed the model’s Cj

remarkably, but also has difference at 1 and 100 MHz. Fortunately, positive reactance holdslimited influence on overall impendence, since its value is small.

In addition, carrier velocity saturation is caused by the limitation of maximum electronvelocity in epitaxial layer. This phenomenon happens at sufficient high frequency or underlarge pumping voltage. It is also a critical restrictive factor for varactor multiplier. Manyapproaches are proposed to solve this issue [16, 20, 21]. For instance, a current-dependedresistor is introduced to limit the maximum current for varactor [20]. The method used in [16]gives a current limitation, which assumes all of the carriers are moving at maximum velocity.But it suffers from inaccuracy and non-uniform doping density. Since the width of depletedlayer cannot change faster than the movement of Btransition front^ of space charge, wesuppose that setting a maximum speed for movement of depleted layer can be a more advisablestrategy. Based on it, the width of depleted layerWD depending on Cj is deduced through (11)and (13) as shown in (14).

WD V j� � ¼ r2a

2b2

C j V j� �AεS

−b1ra

� �−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC j V j� �AεS

−b1ra

� �2

−4b2r2a

s24

35 ð14Þ

Given by [16], the maximum electron velocity at 5 kV/m electrical field is calculated to bevmax = 2.9×105 m/s.

Whend(WD(Vj))/dt ≤ vmax, current IC introduced by junction capacitor Cj employs (8).

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 11e0

1e1

1e2

1e3

1e4

1e5

1e6

1e7

Vbias (V)

)Ff(j

C

CV (f0=1MHz) Meas.CV (f0=100MHz) Simu.CV (f0=1MHz) Simu.

Fig. 9 Simulated and measuredcapacitance-voltage (C/V) rela-tionship of anti-parallel Schottkybarrier diodes


When d(WD(Vj))/dt > vmax, based on (8), (11), and (14), the saturated current correspondingis expressed by (15).

IC ¼ V j þC j V j� �

C0j V j� �

!dC j V j� �

j

dt

¼ V j þC j V j� �

C0j V j� �

!b2r2a−

C j V j� �

γ V j� �

AεS

!224

35AεS dWD V j

� �dt

¼ V j þC j V j� �

C0j V j� �

!b2r2a−

C j V j� �

γ V j� �

AεS

!224

35AεSvmax

ð15Þ

Eventually, we successfully incorporate the carrier velocity saturation in our model entirelybased on C/V curve, without the details of doping density. Through tuning the value of vmax,good agreement can be acquired.

3.3 DC Series Resistor, Plasma Resonance, and Skin Effect

Theoretically, DC series resistor of SBDs can be calculated exactly based on the resistivity ofepitaxial layer, buffer layer, and Ohmic contact. Practically, this method underestimates theseries resistor f compared with the measurement. In our model, we only calculate the DCresistor Repi in epitaxial layer Repi by (16). The extra DC resistor in measurement is put inRresidual as shown in Fig. 7d.

Repi ¼ tepiq⋅A⋅μepi⋅ND

1þ 4tepiπra

� �−1

ð16Þ

where tepi is thickness of epitaxial layer, σepi is conductivity, μepi is mobility of electron, andND is average doping density, in epitaxial layer respectively.

The plasma resonance refers to the carrier inertia and the displacement current in the un-depleted regions of the SBDs. This phenomenon has been modeled by an inert inductor Li anda displacement capacitor Cd [16, 22]. They are believed to be important above 1 THz.However, there are nearly no published experimental data to verify this theory. In fact, Liand Cd have inconspicuous influence towards the total impendence below 500 GHz. Forcompleteness, inert inductor Li and displacement capacitor Cd is still given by (17) and (18).

Li ¼ Repim*μepi

.q ð17Þ

Cd ¼ εepi.

Repi⋅σepi� � ð18Þ

Where m* is effective mass of individual electron.The next factor to be considered is the skin effect. It is a main factor of performance

deterioration for SBDs. The complex impedance introduced by skin effect has been deduced in[22, 23]. Nonetheless, the frequency-dependent reactance in the complex impedance cannot be


compatible with modeling language: Verilog-A. As a result, we have only incorporated twokinds of skin effect resistors in our model as shown in (19).

Rsp ¼ 1

4⋅π⋅δbuf ⋅σbufþ 1

2⋅π⋅δbuf ⋅σbuf⋅ln

rocra

� �ð19Þ

δbuf ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπ⋅ f ⋅μo⋅σbuf

p ð20Þ

where δbuf is skin depth, σbuf is conductivity, and μ0 is permeability, in buffer layer respec-tively. f is the operation frequency. roc and ra are the radius of ohmic contact and anode.

The first item in (19) refers to the resistor spreading across the interface between epitaxiallayer and buffer layer. The second item in (19) refers to the resistor spreading from the edge ofanode to the edge of Ohmic contact. Therefore, it can be a reasonable approximation for theskin effect resistance.

3.4 Four Kinds of Noise Generation Mechanisms

Generally, there are several noise-generation mechanisms for SBD, including shot noise,flicker noise, thermal noise, hot electron noise, and noise introduced by scattering betweenenergy valley. The last item is neglected for its complexity. So, there are only four effectsincluded in the model.

The shot noise is a series of random current pulses through the junction, whose averagenumber is in positive proportion to the DC junction current Ij. It is described by a current-typenoise source as in (21).

i2shotΔf

¼ 2qI j V j� � ð21Þ

Flicker noise represents the random fluctuation of carrier density, which is in reverseproportion to the deviation from the center operation frequency. Therefore, it is called B1/f^noise generally and is modeled as a current-type noise source in (22).

i2flickerΔf

¼ Kf ⋅ I j V j� �� Af ð22Þ

where Kf is flicker noise coefficient and Af is flicker noise exponent.Actually, it is a big challenge to take the two parameters Kf and Af through experiment.

Because the phase noise introduced by LO signal is hard to be calibrated. Fortunately, the IFband of receiver locates far away from the region influenced by 1/f noise. Thus, flicker noise isnot significant.

Thermal noise depending on blackbody radiation is a main factor of noise. In the simplestnoise model for SBDs, thermal noise depends on series resistor entirely. In our model, voltage-


type noise sources Vthermal1 and Vthermal2 have been introduced, which refer to the thermalnoises of Repi and Rsp+Rresidual, respectively, as shown in (23).

V2thermal1

Δf¼ 4⋅k⋅T ⋅Repi

V2thermal2

Δf¼ 4⋅k⋅T ⋅ Rsp þ Rresidual

� �8>>><>>>:

ð23Þ

where k is Boltzmann’s constant.The last item is hot electron noise, which is caused by the redistribution of electrons in the

epitaxial layer. Hot electron noise can be significant when the junction current Ij is relativelylarge. It has been noticed by [24]. However, the total series resistor has been taken intocalculation, which overestimates the contribution of hot electron noise. In our case, only theresistor in epitaxial layer is calculated as shown in (24).

V2hot electron

Δf¼ 4⋅k⋅Kh⋅Repi ð24Þ

where Kh is the constant for hot electron noise. When the diameter of junction approximates1 μm, the energy relaxation time is 1 ps, the mobility of electron is 5000 cm2/V, doping densityis around 2×107 cm−3, and Kh is 2.447×10

7 K/A2 [24].At the end, all of the theories and equations proposed in this article are programmed using

analog hardware description language: Verilog-A. A packaged model in design kit for our anti-parallel SBDs has been established, which is compatible with mainstream commercial har-monic balance simulators.

Fig. 10 a Photograph of quartzcircuit in 340 GHz SHM. bPhotograph of entire module of340 GHz SHM


4 Measurements

The photograph of 340 GHz SHM is presented in Fig. 10. Based on the proposed SHMstructure, we successfully reduce the length/width ratio of substrate from more than 20 to 13.The requirements on mechanical milling and micro-assembly also decrease. The maximumfabrication error is +/− 13um. After incorporating these errors in the simulation, no significantperformance degradation is observed. Our design is robust under current fabrication process.

The double-sideband (DSB) conversion loss and DSB noise temperature are measuredthrough modified Y factor method. A 999.99 °C infrared blackbody from Shanghai Institute ofTechnical Physics services as high-temperature radiator. Room temperature environmentprovides low-temperature blackbody radiation. The IF output band is 1 GHz +/− 50MHz. Avariable attenuator switches the noise figure of IF amplifier chain between 0.64 and 2.56 dB.Thus, the DSB conversion loss and DSB noise temperature can be extracted simultaneously,which resembles method used in [5]. It should be indicated that the 999.99 °C infraredblackbody has not been optimized for 340 GHz band. It may slightly lower blackbodyradiation power compared with expectation in theory. The antenna used in measurement is adiagonal horn with 25 dBi gain and 21.4 mm physical length. Its loss is estimated to be 0.8 dB,

310 315 320 325 330 335 340 345 350 355 360 365 3703

4

5

6

7

8

9

10

11

12

13

Frequency (GHz)

)B

d(ss

oL

noisrev

no

CB

SD

Conv. Loss Meas.Conv. Loss Simu.

Fig. 11 Simulated and measuredDSB conversion loss versusfrequency

310 315 320 325 330 335 340 345 350 355 360 365 3700

500

1000

1500

2000

2500

3000

3500

4000

4500

Frequency (GHz)

)K(

erutare

pme

Tesi

oN

BS

D

System Noise Temp. Meas.SHM Noise Temo. Meas.SHM Noise Temp. Simu.

Fig. 12 Simulated, measured, andsystem DSB noise temperatureversus frequency


since the average waveguide loss is roughly 0.4 dB/cm. The antenna loss has been calibratedfrom the final data.

Figures 11 and 12 present the DSB conversion loss and DSB noise temperature versus RFfrequency, respectively. The lowest measured DSB conversion loss is 5.54 dB at 339 GHz,corresponding DSB noise temperature is 757 K. Three-decibel bandwidth of DSB conversionloss is 50 GHz, from 317 to 367 GHz. The lowest system DSB noise temperature is 1175 Kwith diagonal horn antenna and IF amplifier of 0.64 dB noise figure. When RF is swept from317 to 367 GHz, 1.5–3 mW LO power is sufficient for pumping SHM. The measured data alsoindicates that within 330 to 350 GHz, the average double sideband (DSB) conversion loss is5.9 dB, the average DSB noise temperature is 838 K, and the average system DSB noisetemperature is 1291 K.

Figure 13 shows the measured and simulated DSB conversion loss and DSB noisetemperature under increasing LO power at 341 GHz. The optimal LO power is 1.99 mWmeasured directly by Erickson PM-4 power meter, while the corresponding DSB conversionloss is 5.65 dB, and DSB noise temperature is 782 K. Through comparison between simulatedand measured data, a LO loss of 1.8 dB has been found. Considering the errors in simulation,the authentic transmission loss for LO signal is estimated to be 1.0 dB.

The simulation predicts the measurement pretty well. In Fig. 11, average simulation error of1 dB can be achieved. In contrast, the most frequently used junction model in Fig. 7b, c onlyshows a difference about 3 dB based on our experience. The simulation of noise also presentsreasonable difference towards the measured data. This proves the effective of the proposedmodel in application of mixer design. Table 1 presents the summery of mixers in 300–400 GHz band. It is noticeable that the SHM in this paper holds comparable performance

0.5 1 1.5 2 2.5 3 3.54

5

6

7

8

9

10

11

12

P LO (mW)

)B

d( sso

L n

oisrevn

oC

BS

D

0.5 1 1.5 2 2.5 3 3.50

200

400

600

800

1000

1200

1400

1600

DS

B N

ois

e T

emp

erat

ure

(K

)

Noise Temp Meas.Noise Temp Simu.Conv. Loss Meas.Conv. Loss Simu.

Fig. 13 Simulated and measuredDSB conversion loss and DSBnoise temperature versus LO inputpower at 341 GHz

Table 1 Summary of Recently Published 300–400 GHz Mixer

Freq (GHz) Harm index DSB loss(dB) DSB noise temp. (K) BW (GHz) Ref.

335 1 − 8.5 1750 20 [3]320–360 1 − 6.0 850 > 65 [4]300–360 2 − 6.3 700 > 60 [5]340 2 − 6.0 770 40 [6]380 2 − 8.0 1625 24 [7]366 2 − 6.9 1220 24 [9]340 2 − 5.54 757 50 This work


with the best published results. The difference is inconspicuous, since the error of measure-ment is estimated to be +/− 0.5 dB.

5 Conclusion

A novel 340 GHz sub-harmonic mixer has been proposed. The application of intrinsicresonance of hammer-head filter, direct impedance matching, and DC grounding structureresults in high tolerance of fabrication. An improved LEC junction model has been proposed,which enhances the accuracy of simulation. The measured data show lowest DSB conversionloss of 5.54 dB, lowest DSB noise temperature of 757 K, and 3 dB bandwidth of 50 GHz at340 GHz band.

Acknowledgments The authors would like to thank Cheng-Li Xie, Wei Huang, Hai-Long Hao, Cheng-Wei Li,and Da-Long Zhou in IEE CAEP for their support on module fabrication and testing.

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