Uncooled detector challenges: Millimeter-wave and

Uncooled detector challenges: Millimeter-wave and terahertz long channel field effect transistor and Schottky barrier diode detectors M. Sakhno, A. Golenkov, and F. Sizov Citation: J. Appl. Phys. 114, 164503 (2013); doi: 10.1063/1.4826364 View online: http://dx.doi.org/10.1063/1.4826364 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v114/i16 Published by the AIP Publishing LLC.

Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

JOURNAL OF APPLIED PHYSICS 114, 164503 (2013)

Uncooled detector challenges: Millimeter-wave and terahertz long channel field effect transistor and Schottky barrier diode detectors M. Sakhno,a) A. Golenkov, and F. Sizov V. Lashkaryov Institute of Semiconductor Physics of the National Academy of Sciences of Ukraine, 41, Nauki Av., Kiev 03028, Ukraine

(Received 4 April 2013; accepted 6 October 2013; published online 23 October 2013) The model of long channel unbiased field effect transistor (FET) as mm-wave/THz detector is developed with account of some parasitic effects. The model offered is compared with the other known FET detector models and experimental data. The obtained responsivity (R) and noise equivalent power (NEP) estimations were compared with those for Schottky barrier diode (SBD) detectors. Within the framework of the model, R and NEP values for Si FETs can be determined in all inversion regions. Limits for performance of these detectors have been estimated. It has been shown that with advanced FET technology, the performance of FET mm-wave/THz detectors can be made similar to that of SBD ones or in high frequency range can surpass it. Influence of parasitic effects and detector-antenna matching on detector parameters is discussed. It has been ascertained that FETs can be preferable in some applications due to smaller parasitic effects. C 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4826364] V

I. INTRODUCTION

Detectors are among the most critical components of the mm-wave/THz imaging systems. To make these systems cost-effective, it is desirable to use uncooled detectors and arrays. To apply them, it is important to estimate an ultimate possible NEP of the detector or array that defines their use in active or passive imaging systems. The primary advantage of uncooled SBD and FET direct detection detectors and arrays (today NEP  1011…1012 W/Hz1/2 and NEP  1010…1011 W/Hz1/2, respectively) over the very sensitive (NEP  1016…1019 W/Hz1/2) but deeply cooled detectors is their cost effectiveness. Compared with other uncooled detectors FET and SBD ones are relatively sensitive, have much wider dynamic range and can be assembled into arrays by integrated technologies. Available now NEPs allow using them in active direct detection systems. Room-temperature operation, relatively low noise (when zero biased), high measurement rates, and availability of technologies make FET and SBD detectors and arrays favorable for mm/THz wave detection in future direct detection active vision systems. Signal rectification in uncooled mm-wave/THz detectors is an advantageous technique for fast direct detection of radiation allowing the possibility of low-cost fabrication of sensors with a sufficient sensitivity. Also, it seems reasonable to combine these detectors into focal plane arrays for applications in real time active or passive imaging. From this point of view, direct detection FETs under zero-bias sourcedrain conditions and zero-bias SBDs are favorable for uncooled vision systems. Both of these detectors are fast, and their operation rate in vision mm-wave/THz systems is limited only by read-out electronics. At low radiation powers, they are square-law detectors in which the response a)

E-mail: [email protected]

0021-8979/2013/114(16)/164503/18/$30.00

is proportional to the incoming power, and both of them can be assembled into arrays. Still up to now, in spite of progress in FET and SBD direct detection detectors realization, and importance of arrays for the known vision systems (see, e.g., Refs. 1–5), the results on array formation have not demonstrated widely the image array function. To our knowledge, 16-pixel 280 and 860 GHz SBD arrays with poly-silicon gate separation for obtaining images (without any optical system and with the distance between the source and array of 2 cm) were demonstrated.6 In Ref. 7, coherent imaging at  ¼ 591.4 GHz by using the scanning small number of CMOS FET detectors was presented. Recently8 32  32 CMOS THz camera chip with 80 lm pitch was demonstrated for active illumination at radiation frequencies 0.65 and 0.9 THz but the dynamic range seems may be sufficient only for limited applications. For successful implementation of mm-wave/THz focal plane arrays (FPAs), each sensitive element in the array having antenna should have low side lobe level and pronounced main lobe. These requirements cannot be easily satisfied for antennas deposited on such high dielectric permittivity e and relatively thick (d > 200 lm)psubstrates as Si or III-V semiffiffi conductors, and in the case of e  d > 0:1  k, there can be a strong detector signal dependence on the antenna position on a substrate, because the latter one can act as a dielectric resonator. It is worth to emphasize that exact evaluation of detector NEP from experimental data of signal levels in mm or THz spectral regions is a rather challenging procedure. This problem is related with the presence of an antenna almost in all FET or SBD THz detectors. Even if the antenna is not initially designed, some parts of detector structure (e.g., wires and biases) can work as antenna9,10 with some unknown parameters. One of the most important parameters to characterize the detector is NEP. Its determination is usually based on

114, 164503-1

C 2013 AIP Publishing LLC V

164503-2

Sakhno, Golenkov, and Sizov

J. Appl. Phys. 114, 164503 (2013)

measurements of the responsivity. However, measured in experiments is the total performance of a system that consists of detector, antenna, and optics. Therefore, to obtain the responsivity of FET itself, the antenna parameters and impedance matching of the detector with antenna should be determined. However, it is difficult to measure directly the antenna impedance and antenna pattern in THz range. Usually, some simulations are used. Thus, responsivity measurements are not fully experimental, because the antenna parameters are obtained from simulation (which can be very erroneous because of substrate dimensions influence, presence of contact wires, buses, etc., contributing to antenna properties). Thus, for example, one should discriminate the detector NEP itself and NEP of the system “detector þ antenna.” The principal factor that limits performance of SBD or FET detectors is the presence of parasitic effects. Zero-bias SBD direct detection detectors based on III-V ternary semiconductor compounds are well developed and in the frequency range of   150…300 GHz seem to reach their limit NEP  1011…1012 W/Hz1/2 restricted by technology reasons. It seems that in this frequency region they can be preferable as single detectors over the FET ones in spectroscopic or preamplified direct detection systems. Mm-wave/THz silicon FET direct detection detectors are now approximately an order worse NEP  1010…1011 W/Hz1/2. It seems is mainly due to non-optimized antenna matching and usage of FETs not optimized for high-frequency radiation operation earlier designed, as a rule, for other aims, e.g., for elements of digital circuits and not for the mm-wave/THz detectors. Some parameters of uncooled FET and SBD mm-wave/THz detectors, together with the parameters of several other uncooled detectors are presented in Table I. In this work, we try to develop FET THz detector theory similar to known of SBD one.23 As there are some confusion in determining the responsivity R and noise equivalent power NEP of detectors with antennas, we tried to specify the application of these parameters to certain cases. We will start

from the internal part of FET. Obtained expression for rectification from I-V dependency, it will be shown that this is consistent with the previous models results. Then, we consider the parasitic effect influence. Then, the matching with antenna and loading will be considered. Comparison of MOSFET and SBD detectors based on models known and used for data analysis (at least for SBDs, see, e.g., Refs. 23–26) is carried out. The known experimental data are also compared. The purpose of this investigation is to consider the upper possible limits of sensitivity, electrical NEPel and optical NEPopt of FET and SBD detectors for mm-wave/THz spectral regions. Another objective of the research is to relate these data to measured quantities with the aim of their applicability as arrays in uncooled direct detection active or passive vision systems at radiation frequencies   100 GHz (but not the preamplified direct detection vision systems having a lot of additional elements such as Dicke switches, LNAs, demodulators (see, e.g., Refs. 27–32) and operating as a rule at  < 100 GHz), taking into account the parasitic effects and antenna-detector mismatch. The parameters pointed out are the key detector parameters as they define the image quality and acquisition time of imaging systems. The paper discusses the ways of improving FET parameters and influence of parasitics in obtaining the limiting performance of FET and SBD detectors. It is shown   2 if the wide aperture antenthat the responsivity Rmeas V nas are used (or, for example, in experiments lenses are   4 if antenna gain is independent on used), and Rmeas V radiation frequency . In experiments, these responsivity dependences can be between  4 and  2. II. GENERAL CONSIDERATIONS AND FET DETECTOR MODEL A. FET detection mechanism

Simplified field effect transistor representation is shown in Fig. 1. Considered here will be the “long-channel” FETs,

TABLE I. Parameters of uncooled direct FET and SBD mm-wave/THz detectors, together with parameters of some other uncooled detectors. Detector type Golay cell Piezoelectric Nb microbolometer SiN membrane VOx microbolometers Zero bias SBDs Zero bias SBDs, ErAs/InAlGaAs/InP Zero bias SBDs, InGaAs/InP Zero bias SBDs, InGaAs Zero bias SBDs, AlAs/InGaAs/InAs SBD in 0.13 digital CMOS, CoSi2/n-Si, each diode of 8 0.4  0.4 lm shunted cells Si FET 65 nm SiGe CMOS and BiCMOS Si FET Si n-MOS FET Si CMOS FET MCT hot electron bolometer

Modulation frequency, Hz 20 102 … 200 102 Up to 1010

3  104

106

Radiation frequency, GHz 3

NEP, W/Hz1/2 9

10

Refs.

30  10 30  103 30  103 (1.6…4.3)  103 4.3  103 150, 300, 400 104 (300…700) 3  1010 (5…20)  1012 1.2  1012 5  1010 3  1011 (estimated) 4.5  1012, 8  1012 3.2  1011

Commercial Commercial 11 12 13 14 15 16 17 18 6

295 1027 650 320 595; 2.91 THz 75–150

1011 6.6  1011 3  1010 3.2  1010 4.2  1011; 4.87  1010 (1–3)  1010

19 1 2 20 21 22

164503-3

Sakhno, Golenkov, and Sizov

FIG. 1. Simplified FET representation. VGS is a gate-source voltage (e.g., from antenna and external voltage), dVDS is a d.c. drain voltage appearing under radiation.

in which the channel length L is larger as compared to the “detection effective channel length” Leff that in Si FETs at room temperature is short (Leff < 100 nm). Leff is the length of spacing, where a.c. signal rectification occurs. The reason of the mm-wave/THz radiation detection in long-channel FET is a rectification of the high frequency  signal at a short distance near the source (if the antenna receiving a signal is connected to the gate-source). This rectification was experimentally observed in Si FET detectors up to radiation frequencies of   4.3 THz21 and  ¼ 9 THz,33 though these FETs cannot operate, e.g., as signal amplifiers in this frequency range (unit gain frequency is a few GHz). Simple explanation why they operate as a square-law detector lies in the fact that channel parasitic capacitances do not shunt the rectified signal at the remaining channel length, and thus the rectified signal can be reliably registered. The rectification phenomenon is related with the fact that the alternating THz signal amplitude decreases exponentially along the FET channel, and is self-rectified at this length Leff due to interaction of electrons in the channel with the gate, thus building up the photoresponse only at this distance and leading to the condition that the signal is dependent on the channel length L only at L < 80 nm34,35 at  > 100 GHz. If L > 100 nm, the a.c. current vanishes, as it was estimated and proved experimentally.34,36,37 For characterization of FET as mm-wave/THz detector, there can be used two points of view: (i) “Transmission line” approach, in which channel and gate form a transmission line for the a.c. voltage (with the impedance per unit length Z ¼ RLCH , and the admittance per unit length Y ¼ jxC 38)) with typical parameters of L (see, e.g., Ref. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (Refs. 38 and 39) propagation constant k ¼ RLCH  jxC L

J. Appl. Phys. 114, 164503 (2013)

with the width W to the length L channel ratio W/L ¼ 20/2 (lm). The value of VGS is taken for the case of maximum experimentally observed signals for FETs investigated. In long FET channels, LRek 1. As the gate resistance is much lower than the channel one, the potential across the gate can be taken constant. The potential difference between the channel and gate exponentially decreases. Therefore, the a.c. voltage across the channel is present only on a short part of it, and the d.c. voltage appears at a short distance near the source. The effective channel length is defined from the condition at which THz voltage signal is decreased by e times: Re(Leff k) ¼ 1. (ii) “Transit time” approach means that considered is the part of the channel, where electrons have time to reach its end. The transit time sp  Leff2 is several ps (if the value for Leff is used from “transmission line” approach), so the short part of channel near the source can operate as a very short-channel transistor. For long channel transistors (if Leff L, practically full rectification occurs at the distance 4Leff), these approaches lead to the same results as in this case of long channel transistors, and the final expressions do not include Leff. Thus, we can consider the FET channel (Fig. 2) as FET series connected with the length Lef f and resistor (other part of the channel with the length L–Leff). Let us assume that the current in some part of the channel is proportional to L1 (no ballistic or quasi-ballistic transport). The rectified signal occurs at Leff is IDS1, the resistance of this part of the channel L with the length Leff is RCH1 ¼ Lef f RCH , and the resistance of LL the rest of the channel is RCH2 ¼ L ef f RCH . This circuit is equivalent to the current source Ieqv with the internal impedance Reqv due to the Norton theorem, where Reqv ¼ RCH1 L kRCH2 þ RCH2 ¼ RCH , Ieqv ¼ RCH1RCH2 ¼ Lef f Ids1 . Therefore, the resulting signal will be the same, if rectification will be over the length of channel (formally, the formulas will be the same). That is why in all the following formulas, L is used instead of Leff. This result will not change, if 4Leff is used instead of Leff because 4Leff < L. It follows from the known recent publications that mmwave/THz radiation detection by using the FET transistors could be described within the classical models (see, e.g., Refs. 2, 38, and 40) even using formulas for a static case, which is in accord with the stated above. In these publications, the equations developed are valid for the strong inversion region, though the maximal mm-wave/THz response is observed in moderate inversion region. Some confusion exists in describing FET operation for THz wave detection. One can note some difference between the classical FET models (e.g., BSIM ones, concerning the technology of ordinary transistors41,42) and the theory

(k  >105 cm1). Here, C is the capacitance between the gate and channel. It is dependent on the gate-source voltage VGS (in the region of strong inversion C ¼ Cox). For estimations, it is accepted C  Cox, x ¼ 2p is the radiation circular frequency,   100 GHz. The transmission line is formed by distributed capacitance of the under-gate dielectric (Cox  (1…5)  1014 F) and channel resistance RCH  1 MX at the gate-source voltage VGS  0.6 V for 1 lm design rules FET

FIG. 2. FET channel d.c. equivalent circuit. Here, eqv means equivalent.

164503-4

Sakhno, Golenkov, and Sizov

developed of THz detection, e.g., in hydrodynamic approximation (see, e.g., Refs. 40, 43, and 44) and used assumptions of strong inversion (e.g., Qi ¼ Cox ðVGS  VT Þ). The Dyakonov-Shur44 theory was originally developed for HEMT. Physics for Si-FET and HEMT are slightly different. For example, Si-FET operates in inversion region but HEMT operates in accumulation region. Dyakonov-Shur theory is based on a simplified usual equation (drift current, stronginversion) but with proper boundary conditions and in the frequency range, where the distributed nature of channel could take place. So, it might be expected to use the traditional models, but taking into account that only a small part of the channel is operational to describe rectification. B. I-V all-region static dependence

In literature, there exist several different titles for transistor operation regions (e.g., weak-inversion, moderateinversion, strong-inversion, sub-threshold, above threshold). To make things clear, in this paper, we use those from Ref. 41, and they are mentioned below. The FET channel inversion region, that is controlled, e.g., by the gate-source voltage VGS, can be separated into three regions:41 The weak inversion region, where the channel (drainsource) current IDS of diffusion character can be described by the following simple expression valid at VGS VT:   VGS VT VDS (1) IDS ¼ I0  1  e ut  e nut : Equation (1) is only a simplified limiting equation (at VGS VT) of more complicated equation for diffusion current41 but not the universal model equation that describes the drain-source current in all operation regimes of FET. The factor I0 includes the width W to length L of the channel ratio W/L. The current in this region is basically of the diffusion type, boundaries for this region are VFB þ /F  VG  VM pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ VFB þ 2/F þ c 2/F þ VSB . The lower boundary is approximately defined at the point, where the inversion begins or where the leakage current becomes dominant. The upper one is treated as the point, where the charge of inversion layer is comparable with that of ionized acceptors in the depletion region. Here, I0 is a constant specific to technology, VDS is the drain-source voltage, ut ¼ kBT/q is the thermal potential, kB is the Boltzmann constant, n  1.5 is the sub-threshold turn-on swing factor,41 which is a function of the gate voltage VGS and other FET parameters, VFB is the flat-band voltage, /F is the Fermi level relatively to the middle of the band gap, VM is the voltage of moderate inversion, c is the transistor body effect factor, VSB is the voltage between the source and bulk. In the moderate inversion region, the simple analytical expressions do not exist. Diffusion and drift components can be comparable in their values. Several approximate formulas exist.41 The strong inversion region, where the channel current is of the drift character   W 1 0 2 IDS ¼ ln Cox  ðVGS  VT Þ  VDS  ð1 þ dÞ  VDS : (2) L 2

J. Appl. Phys. 114, 164503 (2013)

This equation valid at VGS VT is only a simplified limiting equation of more complicated equation for drift current41 but not the universal model equation describing the drain-source current in all operation regimes of FET. In this region, the current is basically the drift one, and VGS  VH ¼ VM þ VZ . Here, VZ  0.5…0.6 V is some constant that is dependent on technology and accuracy of calculations, C’ox is the specific capacitance of the under-gate insulator, VDS is the drain-source voltage, d is the bulk-charge factor accounting for bulk effects, VH is the strong inversion voltage, VM is the moderate inversion voltage. Shown in Fig. 3 are typical different components of the drain-source current in FET channel at various VGS for one of the transistors used in experiments, which demonstrate the importance to consider different regions. As the signal responses have maximum at VGS  0.6 V for transistors investigated (see also Refs. 2, 3, 5, 19, 34, 37, 38, 40, and 45) and from Fig. 3, it is clear that the static current is mainly a diffusion current at this VGS, and thus the signal response process is mainly governed by the diffusion current component. In all these three regions, FET detectors can detect the mm or THz wave signals. However, practically for most of the models for FET sensitivity in the cited papers pointed out, there are expressions that are valid only for the strong inversion region and these expressions are extended for all VGS voltages. But these models have singularity (see below) when VGS ! VT, which is of unphysical behavior. Here, VT is the threshold voltage. Also, as a rule, the parasitic capacitances and/or resistances were not taken into account (only the responsivity for the FET internal part was determined). But both of these factors can lead to the overestimations of the responsivity R and NEP values in FET detectors. As the maximal responsivity RV and minimal NEP in most cases are experimentally (e.g., for transistors investigated here and for those of Refs. 2 and 40) observed in the range of VGS voltages corresponding to the moderate inversion ones, then the expressions that are more general should be used. There exist the analytical models that describe FET current-voltage dependences for all VGS, however, they

FIG. 3. Comparison of calculated drift and diffusion currents in different operation regions of FETs that were experimentally investigated (W/L ¼ 20/2 (lm)) at VGS ¼ 0.6 V (at signal maximum), VDS ¼ 0.05 V. IDS,diffusion/IDS,drif  20 according to the universal model.41 VT ¼ 0.758 V. To calculate the curves, according41 the next parameters were used: /F ¼ 0.4 V, c ¼ 1.1 V1/2, VFB ¼ 1.15 V.

164503-5

Sakhno, Golenkov, and Sizov

J. Appl. Phys. 114, 164503 (2013)

require numerical computation41 or numerical integration (e.g., Pao-Sah integral46), or are rather complicated (e.g., BSIM, EKV, and PSP). Therefore, we choose the “balanced” model that satisfies the demands of moderate complexity, accuracy and lack of numerical computations (e.g., numerical integration or root finding). Here, the

IDS ðVDS ; VGS Þ ¼

ln 2 0    2  Cox ð2nÞ/t VGS þ VT VGS þ VT 1 þ ua þ ub Tox Tox "   2   2 # VGS  VT VGS  VT  nVDS :  ln 1 þ exp  ln 1 þ exp 2n/t 2n/t W L

In Eq. (3), ln is the electron mobility in the channel (given by the parameter u0 in the BSIM file) and ua, ub are the parameters of the BSIM model.42 Basically, the term with mobility degradation does not play an important role when VGS < 1 V, and if it is not used, the result will not change noticeably (for the region of maximal sensitivity). Further considerations for obtaining, e.g., FET responsivity and NEP values for different gate-bias regimes use this expression. The resembling approach to consider the MOSFET responsivity and NEP for different gate-bias regimes was also developed in recent papers21,48 using the Dyakonov-Shur model43,44 based on hydrodynamic approximation and empirical expression49 for the electron density in the inversion layer. When comparing Eq. (3) with experimental data, the parameters W and L were taken without any corrections. The transistor sub-threshold turn-on swing factor n was chosen from the region of weak inversion and VT was chosen from the region of strong inversion for the better fitting with experimental data for static current-voltage dependences. For some unknown reason, the threshold voltage has changed during the period of a few years, so it is impossible now to directly extract it from the BSIM3 model file. In Fig. 3, the experimental and calculated I-V characteristics for one of the investigated Si MOSFETs are shown. C. FET symmetry and second derivatives

FET or SBD responses as mm-wave/THz detectors are conditioned by signal rectification because of I-V nonlinearities in devices. At series expansion of the FET current through the internal part of transistor (the channel) in Eq. (3) near VGS, VDS ¼ 0 points, it follows: IDS ðVGS þ DVGS ; VDS þ DVDS Þ @IDS @IDS ¼ IDS ðVGS ; VDS Þ þ DVGS þ DVDS @VGS @VDS 1 @ 2 IDS @ 2 IDS 2 þ DV þ DVGS DVDS GS 2 @VGS @VDS 2 @VGS þ

1 @ 2 IDS 2 DVDS þ ::::: 2 2 @VDS

semi-empirical expression that is applicable to all regions (see, e.g., Refs. 41 and 47) of drain-source current IDS was used for experimental data description and NEP evaluations. In it, the electron mobility dependence41,42 on VGS was taken into account (used was the effective mobility from the BSIM3 model)

(4)

(3)

In some models, in which the threshold voltage is counted from the source (source referenced model) (see, e.g., Refs. 2, @ 2 ID 35, 38, 40, and 50), the derivative @V 2 does not exist at the DS

VDS ¼ 0 point, as it has different signs astride this point. If, for example, one takes a model (see, e.g., Refs. 2, 38, and 40) 8   > > > A ðVGS  VT ÞVDS  1 V 2 ; VDS  0 < 2 DS   ID ¼ 1 > 2 > > : A ðVGS  VT ÞðVDS Þ  2 ðVDS Þ ; VDS < 0; (5) that is applicable to the strong inversion region, then @ 2 IDS ¼ A; 2 @VDS VDS !0þ

@ 2 IDS ¼ A; 2 @VDS VDS !0

where A is a certain constant. It is clear from this that

(6)

@ 2 IDS 2 @VDS

does not exist at VDS ¼ 0. This unphysical behavior is related with referencing the threshold voltage VT from the source.41,50 Still, this model is frequently used2,38,40 when describing the response of FET detectors, taking only the upper part of (5) at VDS > 0 and also extending its applicability to VDS < 0, which is not very admissible as VDS varies symmetrically around the VDS ¼ 0 point. This leads, e.g., to arising non-zero hIDS i current, when excitation is related only to source-drain, which is an unphysical behavior. This problem is usually associated with the problem of the Gummel symmetry test that involves MOSFETs with varying voltages that pass41 through VDS ¼ 0. This is of a little concern for modeling the digital CMOS circuits, e.g., in BSIM3 or BSIM4 models, but is of crucial importance for the high frequency analog CMOS circuits,41,51 in which FETs are used, for example, as mm-wave/THz detectors. Taking VGS ¼ VG0 þ DVGS , VDS ¼ DVDS , DVGS ¼ DVG0 cosðxtÞ, DVDS ¼ DVD0 cosðxt þ D/Þ, where D/ is the phase shift, x ¼ 2p,  is the radiation frequency, and

164503-6

Sakhno, Golenkov, and Sizov

using time-averaging (taking into account that

J. Appl. Phys. 114, 164503 (2013) @ 2 IDS 2 @VDS

can be

discontinuous due to model imperfections) one can obtain 1 @ 2 IDS 2 DVG0 hIDS i ¼ 2 4 @VGS VGS ;VDS 1 @ 2 IDS þ cosðD/ÞDVG0 DVD0 2 @VGS @VDS VGS ;VDS ! 1 @ 2 IDS @ 2 IDS 2 þ þ : (7) DVD0 2 2 8 @VGS @V VGS ;VDS 0 GS VGS ;VDS þ0 Practically, the operation mode VDS ¼ 0 is used in experiments the most frequently, as it provides the lowest noise 2 level. Then, IDS ðVGS ; VDS ¼ 0Þ ¼ 0 ) @@VI2DS ¼ 0 as FET is a GS symmetrical device IDS ðVGS ; VDS Þ ¼ IDS ðVGS ; VDS Þ. Thus, all even VDS derivatives of IDS at VDS ¼ 0 are equal to zero. This concerns only the internal part of FET. If one takes the known models, then in them the component with average of derivatives astride the point VDS ¼ 0 should be canceled as they are odd on VDS. Thus, the average drain current from Eq. (7) 1 @ 2 IDS cosðD/ÞDVG0 DVD0 : (8) hIDS i ¼ 2 @VGS @VDS VGS ;VDS ¼0 Response appears only from source and drain signals mixing. If one apply mm-wave/THz radiation signal at source-drain of the symmetrical FET when connections and parasitic effects are the same at both ends of the channel than no signal should appear. In the experiment, the response can appear, as at high signal frequencies there exist various parasitic capacitances, through which the signal penetrates to the gate (or which in the most cases are connection wires and buses that are asymmetrical). D. Relations with known models

Because of the gate resistance is much lower the channel one, the potential across the gate can be taken as a constant. The potential difference between the channel and gate exponentially decreases. Therefore, the a.c. voltage will be available only at the short part of the channel. The d.c. voltage will appear at a short distance near the source. Then, DVG0 ¼ DVD0 ¼ DV0, D/ ¼ 0, and 1 @ 2 IDS DV 2 : (9) hIDS i ¼ 2 @VGS @VDS VGS ;VDS ¼0 0 Then, the FET response for the internal part of transistor is  1 @IDS Vdet;int ¼ @V hIDS i and if one takes into account that, DS VGS

at VDS ! 0, the differential resistance is equal to the ordinary one @IDS IDS  ¼ rCH : (10) @VDS VDS ¼0 VDS VDS !0 Then, the following expression is valid:

1 1 drCH DV02 : 2 rCH dVGS

Vdet ¼

(11)

The similar expression was obtained earlier3,37,40,52,53 but with the coefficient “1/4” that is because of the derivative here is taken as a mixed derivative with respect to VGS and VDS, and in the papers mentioned, the derivative actually is taken with respect to VDS or distributed channel is considered. The presence of “1/2” instead of “1/4” means that FETs should have better responsivity over SBDs (see below, paragraph 4), which was mentioned long time ago.54 In the papers cited,2,19,38,40 the analysis was considered for the case of strong inversion only, and this analysis was applied for all VG values. It should be denoted that DV02 is frequently treated40 as a fitting parameter, so exact value of this coefficient cannot be extracted from experimental data. Equation (11) using Equation (3) can be written in more explicit form giving the possibility to compare FET detector signal with that for SBD detector (see below). Using Eq. (3), the channel conductivity at VDS ¼ 0 can be presented as rCH ¼ rCH0 fr ðxÞ;

(12) x=2

x=2

Þe where rCH0 ¼ WL ln C0ox nut , fr ðxÞ ¼ 2 lnð1þe ð1þex=2 Þ

(this is the

first term in the Taylor expansion of Eq. (3)), to make things clearer the mobility reduction term was not accounted in this expression), and x ¼ VGSnuVT . Then, t

Vdet ¼

DV02 1 1 @fr : 2 n/t fr @x

(13)

Equation (13) is different from that for SBD detectors by the factor “1/2” (see Eq. (41)), but the FET voltage response (at internal part) also depends on the channel conductivity and its derivative, which are the functions of the gate-source bias, etc., contrary to equations describing the SBD, for which the internal current responsivity is independent of bias at low input powers. III. FET RESPONSIVITY TAKING INTO ACCOUNT PARASITIC COMPONENTS A. FET internal and external parts

To characterize FET as mm-wave/THz detector, one should find the relationship between the output signal and absorbed radiation power, thus, the voltage or current responsivity. To estimate them, one should know the FET circuit. The high-frequency FET equivalent circuit is rather complicated (see, e.g., Ref. 41 and 55). But we are interested in its part between the source and gate. This FET simplified circuit with the connected antenna can be represented by that shown in Fig. 4(a) (see, e.g., Ref. 41 and 55). Here, it is also shown the simplified schematics for SBD frequently used for its (Fig. 4(b)) properties description.23–26 These schematics are presented as similar to each other to let to compare in the same manner the characteristics of FETs and SBDs as mm-wave/THz detectors. The transistor can be separated into two parts: (i) internal part (where rectification occurs) ZGS,int and (ii) external

164503-7

Sakhno, Golenkov, and Sizov

J. Appl. Phys. 114, 164503 (2013)

FIG. 4. FET (a) and SBD (b) simplified schematic representations taking into account the basic parasitic components. ZA is the antenna impedance; VA is the radiation voltage amplitude developed in the antenna; RS ¼ RG þ Rsource in FET is the active series (parasitic) resistance of FET, where RG is the gate active resistance; RS in SBD is the series parasitic active resistance; RD is SBD differential active resistance; CP is the parasitic reactance (usually capacitive), RS is the active serial resistance; ZGS,int is the internal source-gate impedance.

product RV,intg that defines REl V will not change (for frequencies higher than the cut-off frequency). Here, g is the coefficient, which defines the ratio of the power absorbed in the FET internal part to all power absorbed over the whole transistor. It is important to find the voltage RV (or current RI) responsivity and noise equivalent power of FET mm-wave/THz detector. We use the approach similar to that used for SBDs.23 First, we will find the responsivity for the internal part of transistor RV,int. Second, we will find the ratio g of the power absorbed in the FET internal part to all power absorbed over the whole transistor. Then, for FET Rel V ¼ g  RV;int . Let’s find the power that is absorbed in the internal transistor part absorbing the mm-wave/THz radiation. It is ðT 1 1 ReZGS;int IGS;int ðtÞVGS;int ðtÞdt ¼ jDVGS;int j2 ; (16) Pin;int ¼ T 2 jZGS;int j2 0

one (where parasitic components influence of signal characteristic, which can be substantial) CP, RS. In literature, the internal part is called intrinsic and the external one is called extrinsic. It should be noted that frequently in literature RS means the series source resistance41,55 in opposite to the total (source series þ gate) resistance in this paper. Here, it was accepted, against common practice, to facilitate similarities between the circuits and formulas for FET and SBD. B. Internal responsivity

The input impedance of the internal transistor part ZGS,int can be defined by the expression (as input impedance of a transmission line with the length d ¼ L, propagation constant k, characteristic impedance Z0, and loaded by impedance ZL)39 ZðdÞ ¼ Z0 

ZL þ Z0 tanhðkdÞ ; Z0 þ ZL tanhðkdÞ

(14)

and as (LRek) 1, then Z ¼ Z0 due to tanhðLkÞ  1. Thus, the internal source-gate impedance is equal to the transmission line characteristic impedance (for the case of a long channel transistor) sffiffiffiffiffiffiffiffiffi RCH að1  jÞ ¼ pffiffiffiffi ; (15) ZGS;int ¼ Z0 ¼ jxC x 1/2

where a ¼ (RCH/2C) . Here, C is a function of VGS bias. In strong inversion region, C  Cox. In the moderate inversion region, C can have its minimum value C  0.4Cox for the transistor (1-lm design rules technology, W/L ¼ 20/2 (lm), n ¼ 1.625) investigated at frequencies that do not exceed several GHz.41 What it will be for the higher frequency range  > 100 GHz, it is difficult to say. Thus, for estimations below it is accepted C  Cox. For example, with C  Cox for   100 GHz the internal impedance ZGS,int  5.3  (1 – j)  103 X, j ¼ (–1)1/2. Taking C  0.4Cox will increase this value by 1.6 times but this ZGS,int increase will not change the electrical responsivity value REl V (see below) as the

where DVGS,int is the high frequency voltage between the source and drain in the internal part of the transistor, and the coefficient 1=2 is due to using the amplitude value. Then, from (9) and (16) the current responsivity RI,int in the internal part of FET @ 2 IDS @VDS @VGS VDS ¼0 hIDS i ¼ : (17) RI;int ¼ ReZGS;int hPin;int i jZGS;int j2 It can be also written using Eqs. (10), (12), and (15) as sffiffiffiffiffiffiffiffiffiffiffiffi 2rCH0 1 @fr 1 pffiffiffiffi : RFET I;int ¼ n/t xCCH fr @x

(18)

And, in contrast to Eq. (40) for SBD detectors (see below), the internal current FET responsivity RFET I,int (and the voltage response, too) is frequency dependent and is dependent on VGS bias too. In the case of low input power levels, the current and voltage responsivities are related by the equality RV;int ¼ RI;int  RCH :

(19)

The voltage responsivity RV,int can be very large as the channel resistance can have large values in the weak inversion region. This voltage responsivity was calculated38 with account of the parasitic resistances but without of the parasitic capacitances, which, however, does not change the result sufficiently. Finally, as it follows from Eqs. (15), (17), and (19) a @ 2 IDS ; RV;int ¼ 2 pffiffiffiffi RCH @VDS @VGS VDS ¼0 x

(20)

which also can be written using Eqs. (10), (12), and (15) in the form

164503-8

Sakhno, Golenkov, and Sizov

RV;int

1 ¼ n/t

J. Appl. Phys. 114, 164503 (2013)

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 @fr pffiffiffiffi : rCH0 xCCH fr fr @x

(21)

C. Parasitic components influence

One should find the ratio of power that is absorbed in the internal part of the transistor Pin,int to power Pin that is absorbed in transistor as a whole (in its internal and external parts). That is (for the circuit in Fig. 4(a)) Pin;int Z1 2 ReZGS;int =jZGS;int j2 g¼ ¼ ; (22) Pin ZGS ReZGS =jZGS j2 1 . We call where Z1 ¼ ZGS;int kXP , ZGS ¼ Z1 þ RS , XP ¼ jxC P g as the power transmission coefficient, because it is a ratio of two powers. Thus, the electrical responsivity can be written as

Rel V ¼ g  RV;int :

(23)

It should be pointed out that as at the input transistor impedance ZGS,int increase with VGS decrease in the weak inversion region, the internal voltage sensitivity RV,int increases, due to the decrease of power that is absorbed in the internal part 2 DS j remains Pin,int, and the detected voltage RCH @V@DSI@V GS VDS ¼0 the same (for VGS < VT). It can be easily seen from Eq. (21), 1 ffiffiffi @fr p1ffiffi x=2 , as fr ðxÞ  2ex ; for x  5 and thus p @x  2 e fr

fr

which exponentially increases with decreasing VGS (and x). Thus, at VDS ! 0 the voltage sensitivity RV,int can reach large-scale values (see Fig. 5), as was pointed out in Ref. 38. But as the power transmission coefficient g is going down with the transistor internal input impedance ZGS,int increase (ZGS,int  (RCH)1/2) from (15)) in the same proportion, the resulting electrical responsivity Rel V ¼ gRV;int of FET remains constant (see Fig. 6). This clearly indicates that taking into account, the external part is necessary. Shown in Fig. 5 is the dependence of the internal responsivity Rv,int on VGS. In Fig. 6, the electrical

FIG. 6. Dependence of calculated electrical responsivity RVel and electrical NEPel for MOSFET investigated with 1-lm design rules technology W/L ¼ 20/2 (lm).

responsivity Rel V ¼ RV;int g dependence is shown, which takes into account the power transmission coefficient g dependence, and thus characterizes the transistor as a whole. In Figs. 5 and 6 also are presented the calculated NEP-dependences (NEP ¼ N/RV) for FET internal part taking into account that the noise level N is due only to the lowest possible thermal noise (Johnson-Nyquist noise). It should be noted that RV,int cannot be achieved in any real device due to the presence of gate RG and source Rsource resistances and parasitic capacitance CP, which play the role of parasitic effects (in other words, g < 1 always). These factors are important in evaluation of FET THz detector responsivity at VGS ! 0. When the channel resistance grows up considerably, the voltage responsivity saturates (first shown experimentally in Ref. 40). To see the detected signal dependence on the radiation frequency, let us present g in the following form. From Eqs. (15) and (22) after some algebraic manipulations, it follows that: g¼



x 1þ xc1

1=2

1

 3=2 ; x x þ þ xc2 xc3

where the coefficients xc1 ¼ qffiffiffiffiffiffiffi RCH xc3 ¼ ð2C2P RS aÞ2=3 , a ¼ 2C . ox

FIG. 5. Dependence of the calculated internal voltage responsivity RV,int from Eq. (20) and internal NEPint. For FET based on 1-lm technology W/L ¼ 20/2 (lm), VT ¼ 0.758 V, n ¼ 1.625. The radiation frequency  ¼ 77 GHz. Note that very large values at low VGS are caused by large channel resistance.

 2 RS a

(24)

, xc2 ¼ ð2CP RS Þ1 ,

Here, x is the circular radiation frequency, and xci are certain coefficients. The estimated voltage responsivity RVel for different devices is shown in Fig. 7. At  ¼ 77 GHz, the FET detector responsivity estimated for 0.35-lm design rules technology with the channel width-to-length ratio W/L ¼ 1/1 (lm) should have the electrical responsivity approximately 20 times larger than for FET detector responsivity manufactured by 1-lm design rules technology with W/L ¼ 20/2 (lm). With increase of the radiation frequency , this difference increases. The following devices were compared: (1) Experimentally investigated FET manufactured by 1-lm technology, W/L ¼ 20/2 (lm), CP ¼ 4 fF, RS  200 X,

164503-9

Sakhno, Golenkov, and Sizov

J. Appl. Phys. 114, 164503 (2013)

For FET investigated (1-lm technology, W/L ¼ 20/2 2 DS (lm)),  c ¼ x2pc ¼ 2:7  109 Hz, @V@DSI@V j ¼ 2  105 VA2 , GS VDS ¼0 7 V RCH ¼ 1:1  106 X, Rel V0 ¼ 2  10 W . Formally, the “zero” 2

FIG. 7. Comparison of calculated electrical responsivities for different devices. 6

VGS  0.6 V, RCH  1.1  10 X (at VGS  0.6 V), Cox  35  1015 F, a  4.0  109 X/s1/2,  c1 ¼ xc1/2p  6.0  1013 Hz,  c2 ¼ xc2/2p  9.8  1010 Hz,  c3 ¼ xc3/2p  1.8  1010 Hz. (2) FET detector based on 1-lm technology with low parasitic resistances: W/L ¼ 4/2 (lm), CP ¼ 1 fF, RS  15 X, RCH  5.5  106 X (at VGS  0.6 V), Cox  7  1015 F, a  2  1010 X/s1/2,  c1  2.8  1017 Hz,  c2  5.4  1012 Hz,  c3  2.25  1011 Hz. (3) Transistors based on 0.35-lm technology: W/L ¼ 1/1 (lm), CP ¼ 0.2 fF, RS  426 X, RCH  1.26  106 X (at VGS  0.5 V), Cox  4.6  1015 F, a  1.2  1010 X/s1/2,  c1  1.2  1014 Hz,  c2  9.3  1011 Hz,  c3  2.93  1011 Hz. The other value VGS ¼ 0.5 V instead of 0.6 V was chosen due to different threshold voltage. The higher Cox value is due to thinner oxide. Parasitic values per unit width are almost the same. (4) SBD (see paragraph 4): RS ¼ 10 X, RD ¼ 3 kX, CP ¼ 5 fF, n ¼ 1.2. Parasitic parameters for FET were taken from the BSIM3 model file provided by the manufacturer. Parameters for the gate resistance was chosen from the technology datasheet (gate resistance is not accounted in the BSIM3 model). As usually,  c1 >  c2 >  c3 let us consider the FET operation at a sufficiently large frequency  >  c3 . Then, the term in Eq. (24) will dominate and the power transmission

 coefficient can be written as g  xxc3 3=2 , and thus @ 2 IDS 3=2 ¼ 2aR x x2 Rel CH V @VDS @VGS VDS ¼0 c3 1 @ 2 IDS ¼ 2 RCH x2 (25) @VDS @VGS VDS ¼0 Cp RS does not depend on the parameter a ¼

qffiffiffiffiffiffiffi RCH 2Cox

(from the physi-

cal view point, the parasitic capacitance CP shunts the internal part ZGS,int, and thus the power is mainly absorbed in RS). This formula can be formally rewritten in the form very similar to SBD (see below)  2 xc el  R ; (26) Rel V V0 x 2

@ IDS 2 1 pffiffiffiffiffiffiffiffiffi ffi where Rel V0 ¼ RCH @VDS @VGS jVDS ¼0 , xc ¼ C R R . p

S CH

@ IDS current responsivity Rel I0 ¼ RCH @VDS @VGS jVDS ¼0 for one of the investigated FET Rel I0  21A=W, which is basically larger than the theoretically possible upper value for RI of SBD detector (at T ¼ 300 K, RI ¼ 19.3 A/W). This is due to the lack of the factor “1/2” (FET responsivity is based on mixed derivatives, see Eq. (9) or Eq. (20), and thus can principally be twice higher as compared to SBD detectors, though one should take into account the other existing factors that level this advantage, see below). Also, the equation for Rel V can be rewritten in a form

Rel V 

1 C2p RS

2 Rel I0 x :

(27)

This formula is useful for estimations as Rel I0 is within the range 10…40 A/W for almost every transistor. For transisA tors studied here, Rel I0  21 W . There, the following expressions can be useful for some estimation: pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4kTRCH pffiffiffiffiffiffiffiffi RCH C2p RS 2 el  4kT x; (28) NEP ¼ Rel Rel V I0 where only the thermal noise, which seems to be the most important at VDS ¼ 034 for FET detectors NEPel is included. Using Cp ¼ Cx W; RS ¼ qW1 þ q2 WL ; RCH ¼ WL RCH0 (that is valid for W > L, the common model used, e.g., in BSIM3) the following expression can be written: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4kTRCH0 2 2 pffiffiffiffiffiffiffi x ðCx q1 LW þ C2x q2 W 5=2 L1=2 Þ: (29) NEPel ¼ Rel I0 Here, RCH0 (X), Cx (F/m), q1 (Xm), q2 (X/ⵧ) are technological parameters for given design rules. For example, for 1-lm and 0.35-lm design rules, the parameters RCH0 ¼ 1.1  107 X (VGS ¼ 0.6 V) and RCH0 ¼ 1.3  106 X (VGS ¼ 0.5 V), respectively. The parameters Cx and q2 are the same (Cx ¼ 2  1010 F/m, q2 ¼ 16 X/ⵧ for 1-lm technology, and q2 ¼ 12 X/ⵧ for 0.35-lm technology), and q1 ¼ 9  104 and 4.1  104 Xm, respectively. The value NEPel (see Eq. (29)) decreases with W, thereof for a given L value, the transistor with less W value will have lower NEP. Optimizing this expression with technological constraints (1 lm  L  5 lm, 1 lm  W  20 lm), the optimal FET dimensions can be evaluated (and they are L ¼ 1 lm and W ¼ 1 lm). For example, for 1.0-lm CMOS technology, one can evaluate the optimal “zero” current responsivity el 13 W/Hz1/2 at Rel I0  21A=W and electrical NEP  2.8  10   77 GHz. For 0.35-lm CMOS technology, these values el 13 W/Hz1/2. NEP are Rel I0  21A=W and NEP  2.5  10 for 0.35-lm technology is better, e.g., due to thinner oxide that leads to a lower channel resistance. So, NEPel is smaller for FET with lower W, which is in accord with the previous results (for FET operation in the normal mode).56 However,

164503-10

Sakhno, Golenkov, and Sizov

J. Appl. Phys. 114, 164503 (2013)

estimations within this model should be used only as the rough ones. D. Matching with antenna and loading

When using FET as a detector with antenna, it is important to find the part of power that is absorbed in transistor to maximal possible power that can be taken off from antenna. We consider the case when transistor is connected to antenna (with input impedance Zant ) directly (transistor input impedance is ZGS , assuming antenna connection to gate-source). In the case, when a transmission line is used, Zant means the impedance at the end of the transmission line. The maximal power Pant,max that can be supplied to FET channel from antenna at plane wave illumination57,58 is Pant;max ¼ G

k2 I0 : 4p

(30)

It is for the case of ideal impedance matching between antenna and loading when ZGS ¼ Z*ant, k is the wavelength in vacuum (air), and I0 is the radiation intensity that is falling down on antenna from air. In the case of impedance mismatch power that supplied to FET is Pant ¼ ga Pant;max :

(31)

For the transfer coefficient ga, one can write57 ga ¼

4ReZant ReZGS jZant þ ZGS j2

:

(32)

Thus, the optical responsivity (as a ratio of the output signal to the power collected to the limit from antenna) can be defined as el Ropt V ¼ RV  ga :

(33)

el Obviously, Ropt V  RV due to possible mismatch of antenna with the detector impedances. Comparison of optical responsivity for different devices is shown in Fig. 8 assuming the constant (independent of frequency) antenna impedance Zant ¼ (100 – j100) X.

When measuring the voltage at the transistor gate-source by a voltmeter, the FET channel and voltmeter input circuit form the voltage divider. The voltage part at the voltmeter (the voltage transfer coefficient from transistor to voltmeter) is40,59 1 ; gl ¼ (34) R 1 þ CH Zload where Zload ¼ Rload k j2pfmod1 Cload is the impedance of the measuring device (for example, for one of the investigated FETs, the voltmeter parallel connected resistor Rload ¼ 10 MX, capacitor Cload ¼ 90 pF, and modulation frequency fmod ¼ 172 Hz. Thus, Zload  (5.0–j5.1)  106 X). The resulting voltage measured by the voltmeter at FET as a detector is as follows: Vdet ¼ Pant;max RV;int gga gl : The measured responsivity is ¼ Rmeas V

Vdet ; Pmeas

(36)

where Pmeas is some power. To obtain its value, the data of some area S absorbing radiation are needed. Frequently, the pitch size is used19,21 though other areas, e.g., diffraction limited area or effective area (Seff ¼ (k2/4p)G, where G is an antenna gain) can be applied. In high dielectric permittivity and thick (d > 100 lm) substrates, the different wave modes can arise strongly modifying an antenna gain factor G57,58 in Eq. (30) and unpredictably changing the detector responsivity and NEP. The resulting responsivity measured for one of the FETs investigated as a detector in linear and semi-log coordinates is shown in Fig. 9. One can see that in logarithmic coordinates observed here is the exponential signal decay at VGS < 0:4 V, which is related with increase (RCH  exp(–VGS/n/t)) of the channel resistance and thus reducing the matching coefficient gL with a constant voltmeter input resistance. The voltage responsivity RV ¼ V/P was calculated with signals V measured with lock-in amplifier (Stanford SR-830, intrinsic noise level  25 nV/Hz1/2). The power P from BWO sources at the detector placement was estimated with the help of thermal detector (pyroelectric type). The experimental dependences of RV are smooth as the signal-to-noise ratio (SNR), as a rule, is high (e.g., at responsivity maximum SNR > 500, the signal level V from 1 to 30 mV, the “instrumental” noise level < 2 lV). Using Eq. (35), it was determined the level of measured FET voltage signal. From Eq. (35), one can estimate the frequency dependence of detected signal on RV that looks like Vdet ¼ Pant;max RV;int gga gl :

FIG. 8. Comparison of calculated RVopt for SBD, FET 1-lm design rules technology, 0.35-lm FET technology and “optimized” MOSFET based on 1-lm technology. Zant ¼ (100–j100) X.

(35)

(37)

When comparing with experimental data, the method of experimental data obtaining should be taken into account. In experiments, usually the detected voltage Vdet and irradiation intensity I0 are known values. The latter one can be obtained, e.g., using thermal detectors. However, to obtain

164503-11

Sakhno, Golenkov, and Sizov

J. Appl. Phys. 114, 164503 (2013)

FIG. 9. Comparison of experimental (points) and calculated (solid lines) voltage responsivities vs. the gate voltage VGS for FET with W/L ¼ 20/2 (lm). Radiation frequency   77 GHz, modulation frequency f ¼ 172 Hz. (a) RVmeas is presented in linear coordinates and (b) RVmeas is presented in semi-log coordinates. To fit the calculations and experimental data, the latter ones were multiplied by certain coefficient as the power introduced to FET channel is uncertain, because of an unknown antenna gain G and antenna input impedance. (Normalization is equivalent to usage the following antenna parameters: G ¼ 0.4, Zant ¼ (10–j30) X).

the responsivity of the detector RVmeas, the radiation power Pmeas supplied to detector should be known, and that one is related with the irradiation intensity I0 and some effective area Smeas absorbing radiation. This area is usually uncertain: it can be the diffraction limited area, pixel area, antenna area, etc. That is why it is difficult to compare the responsivities cited in various papers. In other worlds, it is impossible to obtain the performance of detector itself Rel V from the performance of a system Rmeas without knowing the antenna V parameters. Here, as the electrical responsivity Rel, one holds the parameter that characterizes the detector itself (defined by the ratio of detector voltage to power absorbed in detector). As for the optical responsivity Ropt, taken here is the parameter that characterizes the detector þ antenna circuit (defined by the ratio of detector voltage to the maximal power that might be given by the antenna at a given irradiation intensity), and for the measured responsivity Rmeas is apprehended the detector þ antenna þ (method of measurements) composition. It is clear that the electrical responsivity should not depend on the antenna or optics type. One can see that the influence of coefficients ggagL product that determines the power absorbed by FET detector is crucial for determination of registered detector signals and thus, for responsivity R and NEP evaluations. When taking the values of these coefficients estimations g  0.1 from Eq. (24) for the radiation frequency   77 GHz, ga  0.2 at Zant  (100–j100) X from Eq. (32), and the FET impedance ZFET  (200–j500) X, from Eq. (34), gL  1 at voltmeter Rinput  10 MX, and antenna gain G  1, one has with RV,int at VGS  0.6 V (see Fig. 4) the value of Rmeas  4  103 V/W. V In Ref. 19, the voltage sensitivity observed in CMOS detec 6  103 V/W at   300 GHz and tors obtained Rmeas V perhaps are connected with larger gain G > 1 due to the resonances in thick substrates used and other FET parameters that were not pointed out. From Eq. (24) in high frequency approximation (when all other terms in denominator of Eq. (24) can be neglected), it follows g   3/2 (e.g., for 1-lm technology transistor with

W/L ¼ 20/2 (lm) studied here  > 20 GHz). From Eq. (32), ga < 1 is some coefficient dependent on the antenna input impedance that is depending on the antenna type and radiation frequency. When antenna dimensions are close to the wavelength, in general, simple dependences of ga on radiation frequency do not exist. For some particular cases of broad-band antennas, e.g., spiral ones, in vacuum or deposited on very electrically thin substrates, the antenna impedance can be weakly dependent on the radiation frequency and, thus, ga  const in this frequency range. From Eq. (34), gL is independent on  and always gL  1 but it is dependent on the parameters of the voltage divider. From Eq. (20), RV,int is proportional to  1/2. From Eq. (30), Pant,max   2 if G is not dependent sufficiently on the radiation frequency (e.g., wire or planar antennas). However, for the case of wide aperture antennas (e.g., lens or horn) the effective area remains almost unchanged with frequency, which is equivalent to increase in the antenna gain G quadratically with frequency.57,58 Thus,   2 , if the wide aperture antennas are used (or, for Rmeas V example, in experiments lenses are used), and Rmeas   4 in V other cases. So, the responsivity and NEP of FET detectors in the radiation high frequency range, in general, is strongly dependent on . To mitigate this situation, the special antenna types with high G (thus high directivity), which are sensitive in some frequency band with a narrow field of view, should be applied. Thus, in the limiting case in the radiation high frequency  Vdet   4, which is similar to range one can observe Rmeas V the frequency dependence of SBD point contact detectors60 and also follows from considerations examined below (see Sec. V). In experiment, one can observe the dependences between  4 and  2 depending on the methods of measurements. IV. GENERAL CONSIDERATIONS OF SBD DETECTORS

SBD uncooled detectors have long been used since 1940 s for microwave detection and mixing because of their

164503-12

Sakhno, Golenkov, and Sizov

J. Appl. Phys. 114, 164503 (2013)

TABLE II. Parameters of some SBD detectors. RS, X; composition 3, ErAs/InAlAs

RD, X

 c, GHz

RV,int, V/W 1

5 … …

 c  157,  c,series ¼ (RSCP)  2.6 THz  c  89,  c,series ¼ 708 GHz … …

7  105

10

 c  6.7,  c,series  2 THz

6400

5.5

…

860

2900 45, 1.1  1.1 lm2, AlAs/InGaAs/InAs 19 2600 19, zero-bias 1000–3000 8 (for 16 pixel cell structure), SBD in 130-nm logic CMOS, polysilicon gate separation ErAs/InAlGaAs

CP, fF 20

high sensitivity and ability to operate at ambient or cryogenic temperatures. Together with recently introduced FET detectors, SBDs have advantages in their fast response time compared with other room temperature detectors (see Table I), such as Golay cells, pyroelectric detectors, or different kind of uncooled bolometers. SBDs have advantage over p-n-junction diodes for high-frequency application (up to the radiation frequency   10 THz60) due to avoidance of minority-carriers storage effects. Here, we will consider quasi-optical SBDs that can be assembled into arrays of direct detection detectors6,22 like quasi-optical matrix FETs.1–5,8 A responsivity of 300 to 1000 V/W was measured for a quasi-optical SBD detector over a frequency range from 150 to 400 GHz.14,16,61 Even much higher voltage responsivities were observed in such kind of detectors (see Table II). The question is, what can be the difference in upperlimit NEP itself estimations (electrical NEPel) and measured NEPmeas with account of parasitic effects and measuring technique of such quasi-optical detectors and possible ways of their realization, and how to account for the possible drawbacks of their operability and applications as uncooled detectors? The devices compared here as direct detection and quasi-optical low-level mm-wave/THz detectors for their detection properties are based on nonlinear current-voltage characteristics. The available literature describing parameters of SBDs are extremely vast (mostly as mixing and heterodyne devices). The primary advantage of SBD and FET detectors and arrays over very sensitive (NEP  1016…1019 W/Hz1/2)64 but deeply cooled detectors is their cost effectiveness though they are not so sensitive (see Tables I and II). But when compared to other uncooled detectors they are relatively sensitive and have much wider dynamic range in contrast, e.g., to that of thermal detectors. Room-temperature operation, possible relatively low noise (when zero biased), high measurement rates, small dimensions, and availability of technologies make these detectors, among others, favorable for mm/THz wave detection. The typical current-voltage characteristic for SBDs is shown in Fig. 10. Here, simplified analysis for SBDs is considered to compare and estimate their ultimate performance characteristics as detectors with those of silicon FETs. This consideration is undertaken to examine possible principal

12

6800 (7.9 A/W), NEP ¼ 1.410

Refs 1/2

W/Hz

 ¼ 200…400 GHz,25 000, NEP  3…81011 W/Hz1/2 6500; measured 4000 V/W at 100 GHz, 400 V/W at 900 GHz Measured 300–1000 V/W at  ¼ 150…440 GHz, estimated NEP ¼ 5…201012 W/Hz1/2 107 (estimations without bias),  ¼ 280 GHz, measured 5100 with amplifier, NEP ¼ 2.91011 W/Hz1/2,  ¼ 860 GHz, measured 355, NEP ¼ 3.21011 W/Hz1/2 4500,  ¼ 94 GHz

15 18 62 14 26

63

differences in the responsivity and NEP of these devices. Shown in Fig. 4(b) is the SBD a.c. standard simplified equivalent schematic representation that is frequently used in estimations of SBD parameters.23–26 The influence on SBD characteristics of the skin effect, dielectric relaxation, finite transit time, etc., will not be discussed here (they usually are the second order effects and being incorporated principally do not change the values of RS, Cp60,65). The dominant current transport mechanisms in SBDs are the emission of electrons over the barrier and the quantum mechanical tunneling of electrons through the barrier. A widely recognized model that takes into account both of these effects is the thermionic-field emission model.66 Generalized I-V characteristics of an ideal junction in either forward or reverse bias can be written in the form     V IðVÞ ¼ Is exp 1 ; (38) n/t where I(V) is the current through the SBD, IS is the reverse bias saturation current, n is the ideality factor (n is approximately 1 to 2, depending on the fabrication process), the thermal voltage /t ¼ kBTjun/q  25.85 mV at T ¼ 300 K, where kB is the Boltzmann constant, and q is the electron charge.

FIG. 10. Typical SBD I-V characteristic. The rise of I for large negative V is due to the breakdown.

164503-13

Sakhno, Golenkov, and Sizov

J. Appl. Phys. 114, 164503 (2013)

The reverse bias temperature dependence of the saturation current can be expressed as 

IS ¼ S  A 

2 Tjun

  /b ð0Þ ;  exp  n  /t

(39)

where ub(0) is the equilibrium Schottky potential barrier, S is the junction area, Tjun is the temperature of the junction, the Richardson constant A* ¼ 4pqkB2m0/h3 ¼ 120 A/(cm2R2), where m0 is the free electron mass. In semiconductors m0 should be changed to effective electron mass m* that will decrease the Richardson constant by m*/m0 times (that means, which thermo-ionic current decreases by the same factor). The modified Richardson constant in dependence of reverse bias67 can change from A* ¼ 30 A/(cm2R2) in moderate electric fields to A*  7 A/(cm2R2) at biases U ¼ 1 V. In GaAs, as a rule, it is accepted A*  8.2 A/(cm2R2). Expression (38) describes the current-voltage characteristics of metal-semiconductor junction but does not do that in the SBD as included under the exponent is the voltage applied to the junction, but not the bias voltage, that is applied to SBD, and it does not take into account the series resistance RS and junction capacitance CP. With ub(0)  0.7 V at T ¼ 300 K (ub(0) changes within 0.6 to 1.0 V for more than 40 metals in GaAs SBDs68) at reverse bias. From IS-Eq. (39), it follows IS  1.5  1014 A for the junction area S  1 lm2, and its resistance at zero bias R0  1.8  1012 X from (38) (A* ¼ 8 A/(cm2R2),

dI 1 q/b ð0Þ n/t t jV¼0 ¼ Anu n ¼ 1). Thus, R0 ¼ dV  T 2  expðnk T Þ ¼ I , B jun S

SBD (with ub(0)  0.7 V) operating as sub-THz detector should be forward biased, which leads to additional noises up to several times and even orders in dependence on modulation frequency primary decreasing the 1/f noise component.23–25,62,69 So, it is important to lower the barrier height to obtain SBD operation at zero bias to exclude the 1/f noise. By lowering the effective barrier of SBD junction, the differential resistance can be reduced significantly. With ub(0)  0.2 V (see, e.g., Al/GaAs d-doped SBDs,70 InGaAs/InP62,71) IS  (1.5…10)  106 A for the similar junction area and zero bias resistance R0  8  103 X, and thus such kind of SBDs as detectors having ub(0)  (0.2…0.3) V can be used at zero bias. One of the disadvantages of zero-bias III-V SBDs based mainly on ternary alloys is their difficulties for monolithic integration into arrays with read-out electronics (which are usually based on silicon). Another limitation of SBDs is that their performance exhibits strong temperature dependence, as the current flow through the barrier is the thermionic one (IS changes approximately by a factor of 2 every 20 K). SBD can be separated,23 like a FET, into two parts: the internal (where rectification occurs) and the external ones (where parasitic resistances and capacitances are present). We consider the junction capacitance as the external part. Let us obtain the responsivity expression for the internal part RV;int . It is the ratio of the averaged current and power dissipated in the internal part of SBD.

Contrary to FET detector, the responsivity for SBD can be found in an analytical form in the case of no bias (without usage of the Taylor expansion). However, the equation derivation using the Taylor expansion of current is simpler23 @ 2 I @V 2 V¼0 1 RI;int ¼ ¼ : (40) @I 2n  /t 2 @V V¼0 The detected voltage from Eq. (40) is VdSBD

 1 2 DV 2 dI d I DV 2 1   ¼  2¼ 4 4 n  /t dV dV

(41)

that is different from the similar expressions for FET detector (Eqs. (11) and (13)) by the factor “1/2,” but FET voltage response depends also on the gate-source bias, etc. At T ¼ 300 K and n ¼ 1, RI,int ¼ 19.3 A/W is a maximal possible internal current responsivity of SBD. And for the voltage responsivity RV measured at low frequencies (when SBD is connected to a high impedance load such as, e.g., oscilloscope) RV,int  RI,int  RD, where RD is the zero-bias differential resistance. It should be noted that, in contrast to FET (similar formulas for FET are (11) and (13)), but Eq. (41) for SBD has “1/2” in denominator. It is caused by the fact that FET being a 3-terminal device, and the detected signal is related to mixed derivatives (which have no 1=2 coefficient with them in series expansion). The low-level (square-law region at the amplitude of the wave DV < n/t), the SBD current responsivity RI (defined by the ratio of current through the SBD to power absorbed in SBD) under the power with radiation frequency  for the circuit shown in Fig. 4(b) can be written23 Rel I ¼ RI;int  g;

(42)

where g is the power transmission coefficient (the ratio of dissipated power in the internal part to the total power) g¼

1 ð1 þ RS =RD Þ  ð1 þ ð= c Þ2 Þ

;

(43)

where the cut-off frequency  c c ¼

ð1 þ RS =RD Þ1=2 2p  CP  ðRS  RD Þ1=2

:

(44)

For instance, with “good” values of different parameters (see Table II) RS  10 X, RD  3  103 X and typical zero-bias capacitance for small area SBDs CP  5 fF,  c  184 GHz. At radiation frequency   100 GHz RI  15 A/W with n ¼ 1. For electrical NEPel ¼ N/RVel, at   350 GHz and only for thermal noise NI ¼ (4kBTDf/R)1/2  2.4  1012 A, it will give NEPel  6.8  1013 W/Hz1/2, and input impedance Zin ¼ (12.8–j 91) X. For   350 GHz, the NEP values of Virginia Diodes VDI Model WR2.2ZBD will be NEPopt  5  1011 W/冑Hz. Taking into account the 1/f

164503-14

Sakhno, Golenkov, and Sizov

noise or generation-recombination (G-R) noise at VB 6¼ 0, it will increase (worsen) the estimated NEPel values. For high-frequency SBDs, several characteristic frequencies are mentioned in literature,15 and they all are named as the “cut-off” frequency. The shunt cutoff frequency  shunt is the frequency at which the real part of the intrinsic diode impedance drops below the magnitude of the imaginary part. When RD RS, as it is true for most of zero-bias rectifiers,  shunt  (2pRDCP)1, which is equal to  shunt  11 GHz for the above pointed parameters. The series-cutoff frequency, well known in Schottky diode mixers and multiplier technology, is the frequency at which the real part of the junction impedance equals the series resistance and is given by  series  (2pRSCP)1  3.2 THz. And the third cut-off frequency given by Eq. (24) is  c 184 GHz. As a rule, in papers (where the cut-off frequency exceeds 200 GHz) frequently pointed out for cut-off frequency is the upper frequency  series, though for estimations of the frequency dependence for sensitivity, the lower frequency  c should be used. It is worth to emphasize that the cut-off frequencies are the functions of SBD detector resistance, and as such quantities they will depend on the bias. At RD  6.5 kX and n ¼ 1.6 (see, e.g., Ref. 63), the zero bias SBD internal responsivity can be estimated as RI,int 11.7 A/W giving for the internal voltage responsivity RV,int  RI,intRD  75  103 V/W. For comparison, in one of the investigated FETs (Si MOSFET, 1-lm design rules) the internal estimated responsivity RV,int  243  103 V/W and ZGS,int  5.7  (1–j)  103 X at VGS ¼ 0:6 V. But the real frequency dependent SBD characteristics are limited by undesirable effects caused by parasitic capacitances and parasitic resistances. Here, they are taken as RS and CP. Some of those known values are summarized in Table II. For typical zero bias, SBD parameters pointed above (RS  10 X, RD  3  103 X, and CP  5 fF),  c  184 GHz. Comparing with typical values for Si FET used as detectors for 1-lm design rule technology with polysilicon gate (W  20 lm, L  2 lm), Rsource  50 X, RG  160 X, CP  4 fF, a  4  109 X/s1/2, which gives  c3  20 GHz. These values for, e.g., FETs with the metallic gate (or multifinger design) and W  4 lm, Rsource  10 X, RG  5 X, CP  1 fF, a  2  1010 X/s1/2. For these figures, the cut-off frequency is  c3  225 GHz. V. NEP ESTIMATIONS

Noise equivalent power NEP ¼ N/RV (W/Hz1/2), where N is the noise level and RV is the responsivity, usually is the most important quantity of a detector. If the responsivity is small and the noise level is small too, a low noise amplifier can be used to get appropriate signal level. But if the detector noise level is appreciable, even in the case of relatively high responsivity, the NEP can be poor and not much information can be extracted from the signal. One should discriminate the electrical, measured, and optical responsivities, which, in turn, define the electrical (NEPel), that is defined by the current through the detector; the measured (NEPmeas) and the

J. Appl. Phys. 114, 164503 (2013)

optical (NEPopt) that are calculated from the noise N and responsivity levels. Here, for obtaining the NEP values facility with BWO sources and the routine with taking into account the beam profile were used.22 It is important to find the possible upper limit NEP at a given frequency and its radiation frequency dependence for both FET and SBD detectors, and try to make a conclusion which is better, e.g., for vision system applications. The electrical responsivity Rel V characterizes the detector characterizes the detector itself, optical responsivity Ropt V þ antenna circuit, and measured responsivity Rmeas characV terizes the detector þ antenna þ (method of measurement and, e.g., lenses used) arrangements. From these parameters, various NEP values can be estimated. The quantity Rel V is the maximal responsivity that can be obtained, if an ideally matched detector with antenna and optical system with antenna occurs. This is an upper possible limit (the same is for NEPel) for detectors. Thus, it is important to differ between the electrical NEP of the detector itself (NEPel) without antennas attached, and optical NEPopt with an antenna connected to the detector. A. FET detectors

Accounting for the case of no voltage applied to FET source-drain contacts, the noise is only the thermal one.34 For FET electrical NEPFET,el, one has from Eqs. (20) and (24) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4kTRCH : (45) NEPFET;el ¼ RFET;el V In high radiation frequency  limit (where  >  c3), it will be RFET;el  12 , (that follows from Eqs. (20) and (24)). In the V case of weak frequency  dependence for one has from Eq. (45) NEPFET;el  2 : V

@ 2 IDFET @VGS @VDS

; RS ; Zant ,

(46)

Taking into account that Pant,max   2, when antenna gain G is not dependent sufficiently on radiation frequency (e.g., when only wire or printed antennas without lenses or focusing optics are used), the optical NEPFET,meas should have the following dependence on the radiation frequency  NEPFET;meas  4: V

(47)

For above mentioned Si FET parameters (1-lm technology, W/L ¼ 20/2 (lm)), the estimations made above at gain G  1 with RV,int ¼ 2.4  105 V/W at VGS  0.6 V (see Fig. 5) give RVmeas  570 V/W. Thus, for only thermal noise Nth observed in Si FETs34 at VDS ¼ 0 and   77 GHz NEPFET,meas will be NEPFET,meas  2.4  1010 W/Hz1/2, which corresponds to experimental NEP estimations (at RCH  1.1 MX, VGS  0.6 V, and T  300 K, Nth ¼ (4kBTRCHDf)1/2  1.3  107 V/Hz1/2, Df ¼ 1 Hz) and NEPFET,el  5.8  1012 W/Hz1/2. In the case, when Zant ¼ (100–j100) X the responsivity RVmeas ¼ 4.1  103 V/W, and thus NEPFET,meas  3.3  1011 W/Hz1/2.

164503-15

Sakhno, Golenkov, and Sizov

In the case of antenna ideal matching the FET impedance with antenna impedance, gant ¼ 1 (ZGS ¼ Z*ant) NEPFET,opt (ideal) ¼ NEPFET,el  5.8  1012 W/Hz1/2. Optimizing g and gain G in some radiation frequency range, one can improve (decrease) NEPFET,opt by several times. But still these uncooled FET detectors and arrays now seems hardly to be used in direct detection passive vision systems (but not preamplified direct detection systems having a lot of additional elements such as Dicke switches, one or several high gain low-noise amplifiers (LNAs), demodulators27–32 that sufficiently raise the imaging system and the overall system cost. These preamplified direct detection systems as a rule are operating at   100 GHz allowing passive vision regime using Dicke switches, one or several LNAs, demodulators, etc. that allows to decrease the uncooled system NEP to NEPopt  1013 W/Hz1/2. But FET detectors researched up to now (NEPFET,opt  1011 W/Hz1/2) have not sufficiently low NEP (mostly due to parasitic influence and noises) needed even for broad-band (D/ > 0.1) mm-wave/THz passive vision systems for diffraction limited beams (NEP  1013 W/Hz1/2) (see, e.g., Ref. 72). Shown in Fig. 8 is the comparison of the voltage sensitivity radiation frequency dependences (assuming the constant antenna impedance Zant ¼ (100–j100) X over the frequency range) for “simple” Si MOSFET with W/L  20/2 (lm) and polysilicon gate, and “optimized” FET with the metal gate (RS  15 X, CP  1 fF, RG ¼ 5 X, W/L ¼ 4/2 (lm), a  2  1010 X/s1/2, RCH  5  106 X). As one can conclude from Fig. 8, the FET responsivity strongly depends on technology, and thus the parameters of FETs used as detectors. Also, included in Fig. 8 is the curve describing the SBD voltage responsivity radiation frequency dependence for SBD with parameters estimated in Sec. V B. B. SBD detectors

In RV and NEP estimations, only thermal noise is taken into account, which defines the upper values of NEP, though, e.g., 1/f noise in a broad frequency range under forward bias can change the spectral density noise level within the amplitude modulation frequency range by several orders (see, e.g.,Refs. 23–25, 62, and 69) and is, as a rule, important for estimations of these characteristics. For example, in Ref. 26 the “good” figure of NEP ¼ 33 pW/Hz1/2 at the radiation frequency  ¼ 280 GHz was obtained for amplitude modulation frequency f ¼ 1 MHz. 1/f-noise can play an important role in the dynamic operation of SBD detectors due to self-biasing. In the presence of 1/f-noise, the NEP not only depends on the bias level but also on the video bandwidth (output measurement). For SBD, the internal voltage responsivity RV,int ¼ RI,intRD  16 A/W3  103 with X  4.8  104 V/W (n  1.2). 1 The coefficients g ¼ ð1þR =R Þð1þð= 2 ¼ 0:85, S D cÞ Þ ReZSBD ReZant P Then, NEPSBD,opt ga ¼ Pmax ¼ 4 jZ þZ j2  0.093. SBD

ant

¼ (4kBTRDDf)1/2/RVopt  1.85  1012 W/Hz1/2. For ideal antenna matching NEPSBD,opt(ideal) ¼ NEPSBD,el ¼ (4kB TRDDf)1/2/RVel ¼ 1.7  1013 W/Hz1/2 at   77 GHz, (RVopt ¼ RV,intgga ¼ 3810 V/W).

J. Appl. Phys. 114, 164503 (2013)

C. FET and SBD comparison

To do meaningful comparison of some parameters in the formulas obtained, the formulas should be similar or taken “identical” in their form for different devices. However, for 1  AFET  pffiffiffi FET it can be written RFET;el 3=2 , and for V SBD

RSBD;el V

SBD

A



1 , ½1þðx=xc Þ2

x½1þðx=xc3 Þ FET SBD

where A

;A

are some

constants independent of frequency. For SBD, the electrical responsivity is almost constant for x < xc and then decreases as x2 . For FET, the electrical responsivity decreases as x1=2 in the frequency region x < xc3 , and for frequencies x > xc3, it decreases as x2 . So, direct comparison of xc and xc3 is meaningless because of they figure in different formulas. In the case, when x > xc3 and x > xc , the formulas can be written in a similar manner Rel;FET V x

3=2

x2

 AFET xc32 and Rel;SBD  ASBD xc2 . However, for 0.35-lm V FET with W/L ¼ 1/1 (lm) dimensions, it will be valid only for  > 600 GHz. Thus, for SBD and FET direct detection 3=2 detectors, the values of AFET xc3 and ASBD x2c can be compared. The formulas pointed out can be rewritten in the following general form: Rel V 

1 C2p RS

RI0 x2 ;

(48) 2

DS j for where RI0 ¼ RI;int for SBD, and RI0 ¼ RCH @V@DSI@V GS VDS ¼0 FET. They have rather similar values, RI0  10…19 A/W for SBD detectors (depending on the ideality factor n), and RI0 10…40 A/W for FET detectors (depending on n, where n is the subthreshold turn-on swing factor), and CP, RS have their conventional meaning. Which detector is better (for frequencies larger than cutoff frequency, for typical values of parameters  > 400 GHz) depends mainly on the factor P ¼ C21RS . For parameters P mentioned for FET detectors investigated P  3.1 26 2 1  10 F X . For improved “optimized” FET, the parameter P  6.7  1028 F2 X1, and for SBD detector used for estimations P  4  1027 F2 X1. So, improved “optimized” FET can have better performance. These estimations were done for detectors parameters used for curves tracing in Fig. 7. Results of calculation NEPel without this simplification are depicted in Fig. 11. Increase in performance with advance of technology is clearly observed (it is due to thinner oxide at better technology, and thus to the lower channel resistance at the same level of parasitic effects). The results for NEPopt, assuming the constant value of Zant ¼ (100–j100) X, are shown in Fig. 12 together with known experimental data. As one can see the model used for SBD detectors describes relatively well the known experimental data for NEPopt but for FET ones, it is seen the sufficient scatter in experimental data though the general radiation frequency dependence can be noted. The reason of date scatter seems is mainly due to non-optimized technologies for FETs as detectors for THz/sub-THz radiation because of, as a rule, the devices designed, e.g., for digital circuits were used as THz/sub-THz detectors. One can also point out that with

164503-16

Sakhno, Golenkov, and Sizov

FIG. 11. Comparison of calculated electrical NEPel for different devices. FET NEP improvement performance when going from 1 lm technology, W/L ¼ 20/2 (lm) to 0.35 lm technology, W/L ¼ 1/1 (lm), is clearly seen. SBD has better performance.

improving technology for MOSFET design and manufacture, the performance of FETs in radiation high frequency range 1 THz should exceed the SBD ones. Comparing Fig. 11 and Fig. 12, one can point out the appreciable difference between NEPel and NEPopt that is due to the influence of parasitics in both types of detectors that suppress the NEP values about an order. For NEP frequency dependence estimations, the following expression can be obtained: pffiffiffiffiffi pffiffiffiffiffiffiffiffi Rd C2p RS 2 el NEP  4kT x : (49) RI0 Thus, NEP performance at the high frequency pffiffiffiffiffi ( > 400 GHz) can be estimated by the parameter P1 ¼ Rd C2P RS . The lower is this parameter, the better NEP is. For FET, the investigated parameter P1  3.2  1024 F2X3/2. For improved “optimized” FET, the parameter P1 ¼ 3.4  1026 F2X3/2 and for SBD used for estimations P1  1.37  1026 F2 X3/2. Analyzing data on Si FET used as mm-wave/THz detectors and also different SBD detectors, it is difficult to make a conclusion about their advantages over each other as

FIG. 12. Comparison of calculated optical NEPopt for a few devices with the antenna impedance Zant ¼ (100–j100) X. Open marks are data for Si FET detectors and filled marks are for SBD detectors. Numbers at experimental data signs mean the numbers of Refs. Data59 for Si FET (W/L  20/3 (lm)) were obtained without antennas use. By () is marked our data obtained for Si FET at   140 GHz with W/L ¼ 0.5/0.35 (lm) and taken antenna gain G  1 (bow-tie antenna was optimized for  ¼ 300 GHz).

J. Appl. Phys. 114, 164503 (2013)

FIG. 13. Comparison of NEP for room-temperature direct detection SBDs and FETs in the range below 1 THz ( < 1THz). Adapted from Ref. 73. (Copyright 2012 by S. Preu. Reprinted by permission of S. Preu.)

detectors of mm-wave/THz radiation due to lack of standardized measurement procedures and interpretation of measured data (which leads to comparison, e.g., of the electrical with optical responsivity). Zero-bias SBD direct detection detectors based on III-V ternary semiconductor compounds are well developed and perhaps reached their limit NEP  1011…1012 W/Hz1/2 within the frequency range  < 1 THz (see Fig. 13, and Tables I and II). They seem to be preferable as single direct detectors over FET ones in spectroscopic or preamplified direct detection systems. Mm-wave/THz silicon FET direct detection detectors now have approximately an order worse NEP  1010…1011 W/Hz1/2 compared to SBD ones. These worse NEP values seems are mainly because of non-optimized FET impedance matching with antennas (ZFET Zant) and not optimized for high frequency operation (e.g., no multifinger design) FETs earlier designed, as a rule, for other aims and not for detectors. If these tasks are solved, then FETs can have better performance and, perhaps, would satisfy the NEP requirements needed for mm-wave/THz direct detection wide dynamic range active or even passive imaging systems. In Ref. 21, it is pointed out that at high frequencies 1/s > , the signals detected can have increased values compared to signals registered at low frequencies. In Ref. 52, it is stated that in resonant case, the responsivity can be very high 105 V/W at 5 THz. However, experimentally such large responsivity was not observed. At present time, the resonant detection is in a low-temperature HEMT, and the non-resonant detection is in a room-temperature HEMT or MOSFET. One of the possible reasons why no such high responsivity for resonant case was observed is that the antenna also has some frequency dependence, usually with a lot of peaks. So, it is much harder to obtain good response because it requires that peak for antenna received power coincides with the peak of detector response.

VI. CONCLUSIONS

The analysis of silicon FETs as mm-wave/THz detectors was considered using the approach that is valid in all

164503-17

Sakhno, Golenkov, and Sizov

inversion regions. FET peak responsivity occurs in the region of moderate inversion. It is shown that the FET detector performance is mainly limited by parasitic effects. Frequency dependence (in high frequency range) of detected signal is similar for FET and SBD and is  2… 4 dependent consider the antenna type, frequency range and the measurement procedure (e.g., with or without use of lenses). It seems that accounting of the diffusion current one can describe the sensitivity and NEP for Si FETs more correctly, as compared with previous considerations valid for strong inversion region. The considerations have shown that, with advance of technology development, the FET detector performance can be improved due to, e.g., lowering the parasitic effects and increase of oxide capacitance. Because of more advanced technologies applied in Si CMOS area, these detectors in arrays will have some benefits due to much quicker development of silicon manufacturing techniques compared to those of III-V ternary chemical compounds. To operate in passive mm-wave/THz imaging systems, FETs should be optimized by improvement of matching with the antenna impedance and their design as detectors. 1

R. A. Hadi, H. Sherry, J. Grzyb, N. Baktash, Y. Zhao, E. Ojefors, A. Kaiser, A. Cathelin, and U. Pfeiffer, in Proceedings of 2011 IEEE MTT-S International Microwave Symposium (IEEE, 2011), pp. 1–4. 2 E. Ojefors, U. R. Pfeiffer, A. Lisauskas, and H. G. Roskos, IEEE J. SolidState Circuits 44, 1968 (2009). 3 A. Lisauskas, D. Glaab, H. G. Roskos, E. Oejefors, and U. R. Pfeiffer, Proc. SPIE 7215, 72150J–11 (2009). 4 H. Sherry, R. A. Hadi, J. Grzyb, E. Ojefors, A. Cathelin, A. Kaiser, and U. R. Pfeiffer, in Proceedings of 2011 IEEE Radio Frequency Integrated Circuits Symposium (IEEE, 2011), pp. 1–4. 5 F. Schuster, H. Videlier, A. Dupret, D. Coquillat, M. Sakowicz, J. Rostaing, M. Tchagaspanian, B. Giffard, and W. Knap, in Proceedings of 2011 IEEE International Solid-State Circuits Conference (IEEE, 2011), pp. 42–43. 6 R. Han, Y. Zhang, Y. Kim, D. Y. Kim, H. Shichijo, E. Afshari, and O. Kenneth, in Proceedings of 2012 IEEE International Solid-State Circuits Conference (IEEE, 2012), pp. 254–256. 7 S. Boppel, A. Lisauskas, A. Max, V. Krozer, and H. G. Roskos, Opt. Lett. 37, 536 (2012). 8 H. Sherry, J. Grzyb, Y. Zhao, R. A. Hadi, A. Cathelin, A. Kaiser, and U. Pfeiffer, in Proceedings of 2012 IEEE International Solid-State Circuits Conference (IEEE, 2012), pp. 252–254. 9 M. Sakowicz, J. Łusakowski, K. Karpierz, M. Grynberg, W. Knap, and W. Gwarek, J. Appl. Phys. 104, 024519 (2008). 10 D. B. But, O. G. Golenkov, N. V. Sakhno, F. F. Sizov, S. V. Korinets, J. V. Gumenjuk-Sichevska, V. P. Reva, and S. G. Bunchuk, Semiconductors 46, 678 (2012). 11 E. N. Grossman and A. J. Miller, Proc. SPIE 5077, 62 (2003). 12 I. Kasalynas, A. J. L. Adam, T. O. Klaassen, J. N. Hovenier, G. Pandraud, V. P. Iordanov, and P. M. Sarro, IEEE J. Sel. Top. Quantum Electron. 14, 363 (2008). 13 A. W. M. Lee, B. S. Wil, S. Kumar, and J. L. Reno, IEEE Photon. Technol. Lett. 18, 1415 (2006). 14 L. Liu, J. L. Hesler, H. Xu, A. W. Lichtenberger, and R. M. Weikle, IEEE Microw. Wirel. Compon. Lett. 20, 504 (2010). 15 E. R. Brown, A. C. Young, J. Zimmerman, H. Kazerni, and A. C. Gossard, IEEE Microw. Mag. 8, 54 (2007). 16 A. Semenov, O. Cojocari, H.-W. H€ ubers, F. Song, A. Klushin, and A.-S. M€uller, IEEE Electron Device Lett. 31, 674 (2010). 17 D. Schoenherr, C. Bleasdale, T. Goebel, C. Sydlo, H. L. Hartnagel, R. Lewis, and P. Meissner, in Proceedings of the 35th International Conference on Infrared, Millimeter, and Terahertz Waves (IEEE, 2010), pp. 1–2. 18 P. Chahal, F. Morris, and G. Frazier, IEEE Electron Device Lett. 26, 894 (2005).

J. Appl. Phys. 114, 164503 (2013) 19

F. Schuster, D. Coquillat, H. Videlier, M. Sakowicz, F. Teppe, L. Dussopt, B. Giffard, T. Skotnicki, and W. Knap, Opt. Express 19, 7827 (2011). 20 A. Pletersek and J. Trontelj, J. Infrared Millim. Terahertz Waves 33, 615 (2012). 21 S. Boppel, A. Lisauskas, M. Mundt, D. Seliuta, L. Minkevicius, I. Kasalynas, G. Valusis, M. Mittendorff, S. Winnerl, V. Krozer, and H. G. Roskos, IEEE Trans. Microwave Theory Tech. 60, 3834 (2012). 22 F. Sizov, V. Petriakov, V. Zabudsky, D. Krasilnikov, M. Smoliy, and S. Dvoretski, Appl. Phys. Lett. 101, 082108 (2012). 23 A. M. Cowley and H. O. Sorensen, IEEE Trans. Microwave Theory Tech. 14, 588 (1966). 24 E. R. Brown, Solid-State Electron. 48, 2051 (2004). 25 C. Yu, C. Wu, S. Kshattry, Y. Yun, C. Cha, H. Shichijo, and K. K. O, IEEE J. Solid-State Circuits 47, 2335 (2012). 26 R. Han, D. Coquillat, J. Hoy, H. Videlier, W. Knap, E. Brown, and K. K. O, in Proceedings of IEEE Custom Integrated Circuits Conference 2010 (IEEE, 2010), pp. 1–4. 27 S. P. Voinigescu, S. T. Nicolson, M. Khanpour, K. K. W. Tang, K. H. K. Yau, N. Seyedfathi, A. Timonov, A. Nachman, G. Eleftheriades, P. Schvan, and M. T. Yang, in Proceedings of 2007 IEEE International Symposium on Circuits and Systems (IEEE, 2007), pp. 1971–1974. 28 A. Tomkins, P. Garcia, and S. P. Voinigescu, IEEE J. Solid-State Circuits 45, 1981 (2010). 29 L. Gilreath, V. Jain, and P. Heydari, IEEE J. Solid-State Circuits 46, 2240 (2011). 30 Q. J. Gu, Z. Xu, H.-Y. Jian, and M.-C. F. Chang, in Proceedings of 2011 IEEE Radio Frequency Integrated Circuits Symposium (IEEE, 2011), pp. 1–4. 31 E. Weissbrodt, I. Kallfass, A. Hulsmann, A. Tessmann, A. Leuther, H. Massler, and O. Ambacher, in Proceedings of 2011 IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2011), pp. 3843–3846. 32 Z. Chen, C.-C. Wang, H.-C. Yao, and P. Heydari, IEEE J. Solid-State Circuits 47, 2355 (2012). 33 A. Lisauskas, S. Boppel, D. Seliuta, L. Minkevicˇius, I. Kasalynas, G. Valusis, B. Khamaisi, V. Krozer, E. Socher, and H. G. Roskos, in Proceedings of Latin America Optics and Photonics Conference (OSA, Washington, D.C., 2012), p. LM4A.1. 34 R. Tauk, F. Teppe, S. Boubanga, D. Coquillat, W. Knap, Y. M. Meziani, C. Gallon, F. Boeuf, T. Skotnicki, C. Fenouillet-Beranger, D. K. Maude, S. Rumyantsev, and M. S. Shur, Appl. Phys. Lett. 89, 253511 (2006). 35 A. Gutin, V. Kachorovskii, A. Muraviev, and M. Shur, J. Appl. Phys. 112, 014508 (2012). 36 W. Knap, H. Videlier, S. Nadar, D. Coquillat, N. Dyakonova, F. Teppe, M. Bialek, M. Grynberg, K. Karpierz, J. Lusakowski, K. Nogajewski, D. Seliuta, I. Kasalynas, and G. Valusis, Opto-Electron. Rev. 18, 225 (2010). 37 € A. Lisauskas, U. Pfeiffer, E. Ojefors, P. H. Bolıvar, D. Glaab, and H. G. Roskos, J. Appl. Phys. 105, 114511 (2009). 38 S. Preu, S. Kim, R. Verma, P. G. Burke, M. S. Sherwin, and A. C. Gossard, J. Appl. Phys. 111, 024502 (2012). 39 D. M. Pozar, Microwave Engineering, 4th ed. (John Wiley & Sons, 2012), p. 752. 40 M. Sakowicz, M. B. Lifshits, O. A. Klimenko, F. Schuster, D. Coquillat, F. Teppe, and W. Knap, J. Appl. Phys. 110, 054512 (2011). 41 Y. Tsividis and C. McAndrew, Operation and Modeling of the MOS Transistor (Oxford University Press, 2011), p. 752. 42 W. Liu, MOSFET Models for SPICE Simulation, Including BSIM3v3 and BSIM4 (John Wiley & Sons, Inc., 2001). 43 M. Dyakonov and M. Shur, Phys. Rev. Lett. 71, 2465 (1993). 44 M. Dyakonov and M. Shur, IEEE Trans. Electron Devices 43, 380 (1996). 45 M. S. Vitiello, L. Viti, L. Romeo, D. Ercolani, G. Scalari, J. Faist, F. Beltram, L. Sorba, and A. Tredicucci, Appl. Phys. Lett. 100, 241101 (2012). 46 H. C. Pao and C. T. Sah, Solid-State Electron. 9, 927 (1966). 47 Y. Tsividis, K. Suyama, and K. Vavelidis, Electron. Lett. 31, 506 (1995). 48 A. Lisauskas, S. Boppel, M. Mundt, V. Krozer, and H. G. Roskos, IEEE Sens. J. 13, 124 (2013). 49 M. Shur, T. A. Fjeldly, T. Ytterdal, and K. Lee, Solid-State Electron. 35, 1795 (1992). 50 G. Gildenblat, Compact Modeling: Principles, Techniques, and Applications (Springer Netherlands, Dordrecht, 2010). 51 P. Bendix, P. Rakers, P. Wagh, L. Lemaitre, W. Grabinski, C. C. McAndrew, X. Gu, and G. Gildenblat, in Proceedings of the IEEE 2004 Custom Integrated Circuits Conference (IEEE Cat. No.04CH37571) (IEEE, 2004), pp. 9–12. 52 M. I. Dyakonov and M. S. Shur, IEEE Trans. Electron Devices 43, 1640 (1996).

164503-18 53

Sakhno, Golenkov, and Sizov

W. Knap, V. Kachorovskii, Y. Deng, S. Rumyantsev, J.-Q. L€ u, R. Gaska, M. S. Shur, G. Simin, X. Hu, M. A. Khan, C. A. Saylor, and L. C. Brunel, J. Appl. Phys. 91, 9346 (2002). 54 M. Ratni, B. Huyart, E. Bergeault, and L. Jallet, in Proceedings of 1998 IEEE MTT-S International Microwave Symposium Digest (Cat. No.98CH36192) (IEEE, 1998), pp. 1139–1142. 55 C. Enz and Y. Cheng, IEEE J. Solid-State Circuits 35, 186 (2000). 56 B. Razavi, R.-H. Yan, and K. F. Lee, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 41, 750 (1994). 57 C. A. Balanis, Antenna Theory Analysis and Design, 3rd ed. (Wiley, New Jersey, 2005). 58 H. L. Volakis, Antenna Engineering Handbook, 4th ed. (McGrawHill, New York, 2007). 59 F. Sizov, A. Golenkov, D. But, M. Sakhno, and V. Reva, Opto-Electron. Rev. 20, 194 (2012). 60 A. J. M. Kreisler, Proc. SPIE 666, 51–63 (1986). 61 J. L. Hesler and R. M. Weikle, in Proceedings of 2008 33rd International Conference on Infrared, Millimeter and Terahertz Waves (IEEE, 2008), pp. 1–2. 62 J. L. Hesler and T. W. Crowe, in Proceedings of the Eighteenth International Symposium on Space Terahertz Technology, edited by A. Karpov (2007), pp. 89–92.

J. Appl. Phys. 114, 164503 (2013) 63

H. Kazemi, G. Nagy, L. Tran, E. Grossman, E. R. Brown, A. C. Gossard, G. D. Boreman, B. Lail, A. C. Young, and J. D. Zimmerman, in Proceedings of 2007 IEEE/MTT-S International Microwave Symposium (IEEE, 2007), pp. 1367–1370. 64 F. Sizov, Prog. Quantum Electron. 34, 278 (2010). 65 V. G. Bozhkov, Radiophys. Quantum Electron. 46, 631 (2003). 66 S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed. (Wiley-Interscience, 2006), p. 832. 67 F. D. Shepherd, Proc. SPIE 1735, 250 (1992). 68 G. Myburg, F. D. Auret, W. E. Meyer, C. W. Louw, and M. J. van Staden, Thin Solid Films 325, 181 (1998). 69 J. J. Lynch, H. P. Moyer, J. H. Schaffner, Y. Royter, M. Sokolich, B. Hughes, Y. J. Yoon, and J. N. Schulman, IEEE Trans. Microwave Theory Tech. 56, 1592 (2008). 70 V. I. Shashkin, A. V. Murel, V. M. Daniltsev, and O. I. Khrykin, Semiconductors 36, 505 (2002). 71 C. Sydlo, O. Cojocari, D. Sch€ onherr, T. Goebel, P. Meissner, and H. L. Hartnagel, Frequenz 62, 107 (2008). 72 F. F. Sizov, V. V. Zabudsky, A. G. Golenkov, and A. Shevchik-Shekera, Opt. Eng. 52, 033203 (2013). 73 See http://www.lap.physik.uni-erlangen.de/lap/index.php?page=research_ terahertz2, (access 15/03/2013) about information on THz FET.