Nanoscale Advances View Article Online PAPER View Journal | View Issue Bias dependent variability of low-frequency noise This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. in single-layer graphene FETs† Open Access Article. Published on 26 October 2020. Downloaded on 1/22/2021 12:04:48 PM. Cite this: Nanoscale Adv., 2020, 2, 5450 Nikolaos Mavredakis, *a Ramon Garcia Cortadella,b Xavi Illa, cd Nathan Schaefer,b Andrea Bonaccini Calia,b Anton-Guimerà-Brunet,cd Jose A. Garridobe and David Jiméneza Low-frequency noise (LFN) variability in graphene transistors (GFETs) is for the first time researched in this work under both experimental and theoretical aspects. LFN from an adequate statistical sample of long- channel solution-gated single-layer GFETs is measured in a wide range of operating conditions while a physics-based analytical model is derived that accounts for the bias dependence of LFN variance with remarkable performance. LFN deviations in GFETs stem from the variations of the parameters of the physical mechanisms that generate LFN, which are the number of traps (Ntr) for the carrier number fluctuation effect (DN) due to trapping/detrapping process and the Hooge parameter (aH) for the mobility fluctuations effect (Dm). DN accounts for an M-shape of normalized LFN variance versus gate bias with a minimum at the charge neutrality point (CNP) as it was the case for normalized LFN mean value while Dm contributes only near the CNP for both variance and mean value. Trap statistical nature of the devices under test is experimentally shown to differ from classical Poisson distribution noticed at Received 31st July 2020 Accepted 26th October 2020 silicon-oxide devices, and this might be caused both by the electrolyte interface in GFETs under study and by the premature stage of the GFET technology development which could permit external factors to DOI: 10.1039/d0na00632g influence the performance. This not fully advanced GFET process growth might also cause pivotal rsc.li/nanoscale-advances inconsistencies affecting the scaling laws in GFETs of the same process. extraordinary characteristics such as carrier mobilities up to 2  Introduction 105 cm2 (V s)1 and saturation velocities of 4  107 cm s1 have Nowadays, the limitations on advanced CMOS technologies and placed graphene transistors (GFET) as a quintessential prospective the predictions for deceleration of Moore's law, have led both the for future applications.3 Despite the fact that the lack of bandgap scientic community and semiconductor industry to turn their in graphene due to its semimetal nature makes single-layer (SL) attention at the development of emerging technologies based on GFETs unsuitable for digital electronics, there has been an enor- 2-Dimensional (2D) materials such as graphene.1,2 Graphene's mous increase of analog and RF circuits designed with GFET technology such as: frequency multipliers,4,5 voltage control6,7 and a ring oscillators8 as well as terahertz detectors.9–11 Moreover, GFETs Departament d'Enginyeria Electrònica, Escola d'Enginyeria, Universitat Autònoma de Barcelona, Bellaterra 08193, Spain. E-mail: nikolaos.mavredakis@uab.es are also widely used and tremendously improve the performance b Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC, Barcelona of chemical–biological sensors and optoelectronic devices.12–17 Institute of Science and Technology, Campus UAB, Bellaterra, Barcelona, Spain It is not enough for recently developed GFET technologies just c Instituto de Microelectronica de Barcelona, IMB-CNM (CSIC), Esfera UAB, Bellatera, to exhibit optimal performance, but they should also be reliable Spain and consistent with CMOS ones since the majority of industry d Centro de Investigacion Biomedica en Red en Bioingenieria, Biomateriales y aims to develop new GFET processes based on their pre-existing Nanomedicina (CIBER-BBN), Madrid, Spain e infrastructures designed for silicon devices.18,19 Variability issues ICREA, Pg. Lluis Companys 23, 08010 Barcelona, Spain † Electronic supplementary information (ESI) available: At rst, fundamental are of outmost importance in advanced semiconductor technol- denitions and equations for the IV model are presented in ESI A. Then, Fig. S2 ogies. Regarding GFETs, the thorough study of such effects is in ESI B illustrates some basic statistical characteristics of the variance of W/L crucial for the transition from immature technologies and pro- ¼ 20 mm/20 mm, 50 mm/50 mm GFETs. Aerwards, the complete LFN variance totyped devices mainly fabricated in research labs to large-scale model including velocity saturation effect is demonstrated in ESI C and Fig. S3. wafer production which will lead to a boost of graphene-based Aer that, the analytical proof of eqn (6) of the main manuscript is derived in ESI D. Then, the simplied (gm/ID)4 LFN variance model is introduced in ESI applications and products. There are two sources of variabilities E. Fig. S4 in ESI F demonstrates the behavior of all the local noise variance in graphene: (a) environmental effect variabilities such as inter- expressions, along the channel at different operating points. Finally, Fig. S5 and face traps and (b) material imperfection variabilities such as edge Table S1 in ESI G describe in detail the extraction of LFN mean value and disorders.18 In this work we focus on low-frequency noise (LFN) variance model parameters. See DOI: 10.1039/d0na00632g 5450 | Nanoscale Adv., 2020, 2, 5450–5460 This journal is © The Royal Society of Chemistry 2020 View Article Online Paper Nanoscale Advances variability for SL GFETs mainly derived from interface trap LFN near the CNP always with a L-shape trend even at SL GFETs statistics which, according to our knowledge, is for the rst time while DR has been observed at higher current regime where Rc is investigated. LFN's contribution to the aforementioned circuits is important.53–55 LFN in GFETs can be reduced aer electron-beam very crucial as it can be upconverted to deleterious phase noise in irradiation49 while the same can be achieved with the usage of such high frequency applications.4–11 In addition, it can affect the substrates such as boron nitride.50,51 A number of models have sensitivity of sensors12–17 while LFN deviations can also be proved been proposed to describe the behavior of LFN mean value on useful for such sensing applications.18 GFETs but most of them43,45,46 are based on a simple approxi- Moreover, LFN examination can provide crucial information mation for DN model taken from CMOS devices26,29,31 which as far as the quality of the devices and their interfaces is introduces an (gm/ID)2 trend of SIDf/ID2 LFN. The latter approach This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. concerned.20,21 can only be functional under uniform channel conditions.54,55 Open Access Article. Published on 26 October 2020. Downloaded on 1/22/2021 12:04:48 PM. Three main effects are considered responsible for the Recently, a complete physics-based LFN model was proposed53–55 generation of LFN in semiconductor devices and consequently which is valid in all operating regions since it accounts for all GFETs; rstly, the carrier number uctuation mechanism (DN), non-homogeneities of the device. secondly the mobility uctuation mechanism (Dm) and nally While there is a signicant number of works regarding LFN the series resistance (Rc) contribution (DR). DN model stems mean value modeling in GFETs, no studies are available that from the trapping/detrapping process at semiconductor deal with LFN variability in these devices even though it is devices.22,23 In more detail, a free carrier can be captured by an equally signicant to its mean value. The large deviations of active trap near the dielectric interface and within a few kT from LFN observed in GFETs, makes it urgent to develop statistical the Fermi level and then emitted back at the conduction path, LFN models to investigate the physics behind this variability. In and as a result a Random Telegraph Signal (RTS) is generated the present work, such an analysis that combines both the which corresponds to a Lorentzian Power Spectral Density experimental characterization of LFN variance data and the (PSD). For transistors with channels larger than a few hundred derivation of a new physics-based statistical compact model, is nanometers, the number of active traps is high and conse- proposed for the rst time. The proposed model reveals the quently the superposition of the generated Lorentzian spectra relation of LFN variability with the deviation of the parameters can result in LFN PSDs inversely proportional to frequency of the fundamental physical generators of LFN in GFETs (DN, under the condition that the traps are uniformly distributed. Dm). As it is known from CMOS technology, LFN variance is This is also known as 1/f noise and was rst proposed by connected with operating conditions in larger devices,39,40 McWhorter.24 Minimization of device dimensions in advanced which is the case for the GFETs under study in the present CMOS technology nodes has led both the LFN mean value and analysis. The proposed model is based on the recently estab- variance to be dominated by RTS25–27 but this is not yet the case lished chemical-potential based one regarding LFN mean in GFETs. Dm model occurs due to uctuations of the bulk value53 as well as CV–IV behavior.56–58 For more details on the mobility and is described by the empirical Hooge expression28 CV–IV–LFN model see ESI A (Fig. S1).† The principal idea for the while DR one is caused by Rc contribution especially at high LFN mean value model calculation was to divide the device current regimes. Several physics-based models, simpler or more channel into microscopic uncorrelated local noise sources. The analytical ones, are available in literature describing both LFN local noise PSD that originates from each LFN mechanism was mean value29–33 and variance34–40 in CMOS transistors. Most of then calculated and integrated from source to drain in order to the LFN variance models focus on its area dependence for short analytically evaluate the contributions from DN, Dm and DR channel CMOS devices where RTS prevail34–37 while bias LFN; by adding these contributions the total LFN PSD is ob- dependence is also analyzed in some of them.38–40 Character- tained.33 Similarly, in the present work, the variance is calcu- ization and modeling of the standard deviation of the natural lated locally for each LFN mechanism by applying fundamental logarithm of LFN is also very common in bibliography since laws of statistics.39,40 Then, the integration along the channel LFN deviations follow a log-normal distribution.25,27,35–37,39,40 leads to an analytical compact solution with the help of the A lot of research has also been conducted regarding LFN in chemical-potential based model mentioned above. The derived GFETs41–55 and the ndings agree that the same mechanisms model describes very accurately the experimental data and as in (DN, Dm, DR) are responsible for the generation of LFN. In fact, it CMOS devices,39,40 DN and Dm variance present a bias depen- has been shown that the nature of LFN in GFETs strongly relies dence similar to their means. Thus, DN variance is responsible upon the number of layers of the device.47 In transistors with for an M-shape of total LFN variance centered at the CNP, while many layers, volume noise (Dm) prevails while the fewer the Dm variance is more signicant near the CNP. The proposed layers the more signicant the surface LFN (DN) becomes. In this LFN variance compact model can be easily implemented in work, SL GFETs are governed by trapping/detrapping mecha- Verilog-A and annexed at the chemical-potential based model nism which causes an M-shape gate-bias dependence of output mentioned above 53–58 and then included in circuit simulators. LFN divided by squared drain current (SIDf/ID2), referred to 1 Hz, Trapping/detrapping mechanism that generate LFN is inu- with a minimum at the charge neutrality point (CNP). Residual enced by some additional factors at the solution-gated (SG)-SL charge, which dominates at the CNP, can proportionally increase GFETs under study59 resulting in a trap distribution different the LFN minimum. Similarly, non-homogeneous charge density than Poisson which is the case in silicon-oxide devices.27,35–38 More at the channel, caused by a relative high drain voltage, can also particularly, trapping/detrapping can occur either near the surface increase the LFN minimum.53 Dm model can also contribute to of the polyimide substrate or at the graphene–electrolyte interface This journal is © The Royal Society of Chemistry 2020 Nanoscale Adv., 2020, 2, 5450–5460 | 5451 View Article Online Nanoscale Advances Paper where processes of association/dissociation with charged moieties can take place.60–63 In addition, GFET technologies are still in a non-optimal stage and thus, extrinsic traps might be generated during the fabrication process affecting the trap statistics behavior. Results and discussion On wafer LFN and IV measurements were conducted at top SG- SL GFETs59 at three different device geometries; (a) W ¼ 100 mm/ This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. L ¼ 100 mm, (b) W ¼ 50 mm/L ¼ 50 mm and (c) W ¼ 20 mm/L ¼ 20 Open Access Article. Published on 26 October 2020. Downloaded on 1/22/2021 12:04:48 PM. mm where W, L are the width and length of the device respec- tively. Top gate voltage was swept from strong p-type to strong n- type region including the CNP with a step of 5 mV while drain voltage was constant, VDS ¼ 50 mV. Arrays of SG-SL GFETs were fabricated (see Experimental data section for more details on fabrication and measurements procedures) and thus, a signi- cant number of samples were measured for each available geometry in order to have adequate data to characterize LFN statistics. In more detail, the analysis took place at around 48– 50 samples from geometry (a), 25–28 samples from geometry (b) and 23–25 samples from geometry (c) aer the exclusion of outliers. The reason why the number of samples is not constant for each geometry is that some measurements might behave as outliers in some specic operating conditions but not in the whole range. While ID and SID were measured for 100 top gate voltage values, only 11 of them were chosen for the LFN vari- Fig. 2 (a) Drain current ID vs. top gate voltage overdrive VGEFF, for ability analysis in order to speed up the process. These values GFETs with W/L ¼ 20 mm/20 mm, 50 mm/50 mm and 100 mm/100 mm. were extended from high-to low-current regime both at p- and n- Markers: measured data, solid lines: model. (b) Cumulative distribution type regions in order to permit the thorough study of the LFN function (CDF) of natural logarithm of normalized LFN ln(WLSIDf/ID2), referred to 1 Hz, for GFET with W/L ¼ 100 mm/100 mm, shows a log- variance at all the operating conditions. The schematic of the normal distribution. Markers: extracted CDF, solid lines: theoretical device under test is shown in Fig. 1a where the graphene CDF of normal distribution of ln(WLSIDf/ID2). channel, the metal contacts, the SU8 passivation, the electrolyte gate and the reference electrode are shown. Fig. 1b illustrates the average Raman spectrum from 400 points in the W ¼ 20 mm/ response. In Fig. 2a, IV model is validated56–58 for the three GFET L ¼ 20 mm GFET area aer the transfer of graphene (see geometries under test. The transfer characteristics (drain-source Experimental data section). Fig. 1c shows a histogram repre- current ID vs. effective gate voltage VGEFF) are shown for all senting the D/G ratio for each of the measured spectra, indi- regimes of operation, near and away the CNP, with VGEFF calcu- cating the rather low density of defects in the graphene lattice. lated as the gate voltage (VG) minus the voltage at the CNP (VCNP). Fig. 2a also presents the tting from the chemical-potential based IV model validation model,56–58 showing a close match with the experimental data. The The rst step towards the modeling of LFN mean and variance is fundamental parameters of the IV model such as mobility m, top the extraction of physical parameters related to their stationary gate capacitance Ctop, at band voltage VGSO, residual charge r0 Fig. 1 (a) Schematic of a SG-SL GFET. (b) Average Raman spectrum over the whole graphene channel and (c) histogram of D/G ratio repre- sentative of the defects density distribution. 5452 | Nanoscale Adv., 2020, 2, 5450–5460 This journal is © The Royal Society of Chemistry 2020 View Article Online Paper Nanoscale Advances Table 1 IV–LFN (mean value–variance) model parameters Parameter Units W/L ¼ 20 mm/20 mm W/L ¼ 50 mm/50 mm W/L ¼ 100 mm/100 mm m cm2 (V s)1 3500 3800 7000 Ctop mF cm2 2 2 2 VGSO V 0.278 0277 0.33 r0 cm2 3.29  1011 3.29  1011 3.43  1011 Rc U 400 320 380 NT eV1 cm3 2.2  1019 4.93  1019 1.74  1020 — 5.6  104 1.5  103 1.12  102 This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. aH SDR2 U2 Hz1 3  104 4  105 1.8  105 Open Access Article. Published on 26 October 2020. Downloaded on 1/22/2021 12:04:48 PM. Ntcoeff — 2.7  103 2.5  105 3.4  106 NaH — 1.19  1013 3.5  1011 3.6  1010 normalization is that the variance of this term presents a 1/(WL) dependence.35,36,39,40 In Fig. 2b, this form of depiction of LFN data is shown to follow a log-normal distribution (s[ln(LFN)] ¼ Oln(1 + Var(LFN)/E2(LFN)), where s is the standard deviation and E the mean value37,40) for the rst time in GFETs. Cumulative distribu- tion function (CDF) of the natural logarithm of WLSIDf/ID2 for the 100 mm/100 mm GFET is presented at three different VGEFF; markers present the extracted CDF of the random variable X ¼ ln [WLSIDf/ID2] while the solid lines the theoretical CDF under the consideration that X is normally distributed. The precise tting among markers and lines conrms that WLSIDf/ID2 data indeed follow a log-normal distribution. Fig. 3a and b indicate the correlation of the WLSIDf/ID2 variability with VGEFF (or VCNP equiv- alently) and ID variabilities, respectively for the same device. It is evident that this correlation is very weak, and this observation proves that LFN variance is not related to the variability of the IV characteristics but is mostly due to trap statistics as well as Hooge parameter variations as it will be shown later. See ESI B (Fig. S2)† for the equivalent plots as Fig. 2b and 3a for the rest of the GFET areas. Therefore, for the derivation of the LFN variance model, it is crucial to rst determine the parameters of the LFN mean value model that are sensitive to variations. These are the number of traps Ntr from DN effect and Hooge parameter aH from Dm one. Fluctuations of the two aforementioned parameters from sample to sample lead to different LFN magnitudes and consequently cause LFN variance. Thus, the only way to explicitly model this Fig. 3(a) Variability of WLSIDf/ID2, referred to 1 Hz, is much higher and variance is to examine how the deviation of these parameters uncorrelated with variability of (a) VGEFF and (b) ID for GFET with W/L ¼ 100 mm/100 mm. affect the statistical behavior of the two mechanisms that generate LFN. The WLSIDf/ID2 PSD locally in the device's channel for a slice Dx, for both mechanisms, is given by:53 and contact resistance Rc are extracted and presented in Table 1.  !2 Derived parameters from every GFET are very close, apart from m SdID  e Cq Ntr 2 WLf ¼ (1) and VGSO which are quite heightened for the larger device, which ID DN Qgr Cq þ C W Dx is indicative of the elevated ID data observed there. Also, Rc for the  medium sized device is a little decreased. SdID  e WLaH WLf ¼ (2) ID 2 Dm Qgr W Dx LFN mean value model respectively, where Ntr ¼ WLNt ¼ KTlNT31,32 is the number of As described before, DN, Dm, DR are the main generators of LFN in active traps, Nt is the trap density in (cm2) and NT is the GFETs. Regarding LFN variance, the rst two are going to be volumetric trap density in (eV1 cm3) which is used as a LFN investigated in this work. LFN normalized by the area over mean value model parameter.53 K is the Boltzman constant, T is squared drain current (WLSIDf/ID2), at 1 Hz, is widely used in the absolute temperature, l z 0.1 nm is the tunneling attenu- literature for the study of LFN variance. The reason for using this ation distance, e is the electron charge, Qgr is the graphene This journal is © The Royal Society of Chemistry 2020 Nanoscale Adv., 2020, 2, 5450–5460 | 5453 View Article Online Nanoscale Advances Paper This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. Open Access Article. Published on 26 October 2020. Downloaded on 1/22/2021 12:04:48 PM. Fig. 4 Normalized LFN WLSIDf/ID2, referred to 1 Hz, vs. top gate voltage overdrive VGEFF, for GFETs with (a) W/L ¼ 20 mm/20 mm, (b) 50 mm/50 mm and (c) 100 mm/100 mm. Measured noise from all available samples: star markers, measured ln-mean noise and its 2-sigma deviation: open circle markers, mean and 2-sigma deviation model: lines (mean data and model: black, +2-sigma deviation data and model: purple, 2-sigma deviation data and model: green). charge stored in the quantum capacitance Cq, C ¼ Ctop + Cback is parameters is not well understood but GFET technologies are the sum of top and back interface capacitances, aH is the not as mature as CMOS for example, and this might cause such unitless Hooge parameter which is also used as a LFN mean deviations even for parameters of the same technology. The value model parameter53 and f is the frequency. See ESI A ensuing LFN statistical analysis will conrm that DN and Dm (Fig. S1)† for more details on the denition of different quan- models dene LFN variance similarly as its mean value. tities. To obtain the total LFN PSDs, integration of the local PSDs derived above with an integral variable change from LFN variance model length x to chemical potential Vc should take place.53–55 The SG- Variations in the number of active traps (Ntr) can denitely SL GFETs which are examined in this study are long-channel induce variations in the DN contribution to LFN. Similarly, devices operating at low electric eld region due to small variations in the aH parameter can produce variability in the drain voltage and thus, Velocity Saturation (VS) effect54 is not effect of Dm on LFN. In this section, an analytical LFN variance expected to contribute to LFN variance. Consequently, VS effect model for each of the two LFN generation mechanisms will be is ignored in the derivation of the model that follows in order to derived. First, LFN variance will be calculated locally in the keep the expressions simple. The complete model including VS channel and since the local noise sources are considered contribution to LFN variance is presented in ESI C (see eqn (S6)– uncorrelated,33,53 integration from source to drain can provide (S12), Fig. S3).† It is conrmed that VS is negligible under low the total LFN variance by adding all the local contributions. electric eld conditions while it can increase variance at high Estimating variance locally ensures that each DID/ID deviation electric elds. LFN mean value model is validated with experi- caused by a uctuation (such as a specic trap) at any inni- mental data, averaged in a bandwidth 1–30 Hz, in Fig. 4 for all tesimal area will contribute independently.25,37,39,40 Local vari- the GFETs under test. WLSIDf/ID2 ln-mean data with black ance represents the relative variation induced by each local circular markers and LFN mean value model53 with solid black noise source. Oppositely, only under uniform channel condi- lines are shown for the W/L ¼ 20 mm/20 mm GFET in Fig. 4a, for tions where local variance would be equal throughout the whole the W/L ¼ 50 mm/50 mm GFET in Fig. 4b and for the W/L ¼ 100 channel, it could be taken out of the integral. But this is not the mm/100 mm GFET in Fig. 4c vs. VGEFF with very consistent results case in this work since all non-homogeneities are taken into for all the devices and for all regions of operation. The loga- account in order to implement a complete model. Fundamental rithmic (ln)-mean values are used for better accuracy due to log- statistics theory gives: normal distribution of LFN. The LFN data for the total of the  2 measured samples are also shown with smaller red markers. Dm vf Varðf ðyÞÞ ¼ VarðyÞ (3) model is signicant near the CNP and from there the aH vy parameter can be extracted, DN LFN is responsible for the M- where in this study y ¼ Ntr, aH for DN, Dm models respectively shape of WLSIDf/ID2 and thus, NT parameter can be extracted and f(Ntr, aH) ¼ L(x) (Ntr, aH). from tting the DN model of LFN generation. Finally, the contribution from DR can be identied at higher gate voltage values, where SDR2 can be extracted.55 The LFN mean value DN LFN variance model parameters are presented in Table 1. NT and aH increase The total WLSIDf/ID2 DN is calculated by integrating eqn (1) along with the area of the GFETs, with the greatest increment the channel:33,53 observed in the largest devices (W/L ¼ 100 mm/100 mm), where  !2 ðL they double with respect to the medium sized devices (W/L ¼ 50 SID  1 e Cq Ntr WLf ¼ dx (4) mm/50 mm). Oppositely, SDR2 reduces with the GFET channel ID 2 DN 0 L 2 Qgr Cq þ C W area. The reason for this deviation of LFN mean value 5454 | Nanoscale Adv., 2020, 2, 5450–5460 This journal is © The Royal Society of Chemistry 2020 View Article Online Paper Nanoscale Advances To calculate the variance of eqn (4): and if eqn (8) is solved analytically, eqn (9) is derived. "  # 2 !2 3 gvc is a normalized drain current term,53–58 a ¼ 2r0e is a residual ðL SID  1 e C N charge related expression55–58 and k is a coefficient.53–58 For more WLf ¼ Var4 dx5 q tr Var (5) ID 2 DN 0 L 2 Qgr Cq þ C W details on denitions see ESI A.† eqn (9) predicts an inversely proportional relation of the WLSIDf/ID2 DN variance model to the area of the device 1/(WL). It can be easily proved that since the local noise sources are uncorrelated and thus independent, the following expression is valid: This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.  ðL  ðL Dm LFN variance Open Access Article. Published on 26 October 2020. Downloaded on 1/22/2021 12:04:48 PM. Var f ðyÞdx ¼ L Var½f ðyÞdx (6) Following a procedure similar as in DN case, the total WLSIDf/ID2 0 0 Dm variance is calculated by integrating eqn (2) along the channel:33,53 See ESI D† for the mathematical proof of eqn (6). Because of  ðL eqn (3) and (6), variance of eqn (5) can enter the integral as: SID  aH e WLf ¼ dx (10) 0 ID 2 Dm 0 LQgr "  # ðL !2 12 SID  1 e C WLf ¼ Var L@ A ½Ntr dx q Var ID 2 DN 0 WL2 Qgr Cq þ C Variance of eqn (10) is calculated aer taking into consid- (7) eration eqn (3) and (6) as in DN case since local noise sources are uncorrelated: where the quantity inside the integral corresponds to the local "  # ð variance. In silicon-oxide devices number of traps are known to SID  L e2 Var 2 WLf ¼ Var½aH dx (11) follow Poisson distribution27,35–38 which means that its variance ID Dm 0 LQgr 2 equals to its mean value, Var[Ntr] ¼ Ntr ¼ WLKTlNT. It will be shown later that this is not the case in SG-SL GFETs of this work but it rather is Var[Ntr] ¼ NtcoeffWLKTlNT, where Ntcoeff is used as No information is available on literature regarding Var[aH] a tting LFN variance model parameter. According to the latter and thus, in order to achieve a similar scaling with DN WLSIDf/ID and if the integral variable is changed from x to Vc:53–58 LFN variance model 1/(WL), the following is assumed: Var[aH] "  # ð ¼ aH(NaHWL)1 where NaH is a specic density used as a tting SID  L kTlNtcoeff NT e4 Cq 4 LFN variance model parameter. According to the latter and if Var WLf ¼  4 dx ID 2 DN 0 WL2 Cq þ C Qgr 4 the integral variable is changed from x to Vc:53–58 ð Vcs "  # ð Vcs 16kTlNtcoeff NT e4 Vc 4 SID  4e2 aH k|Vc | þ C ¼  2 3 dVc (8) Var WLf ¼ dVc (12) Vcd WLgvc C Vc þ a=k ðk|Vc | þ CÞ3 ID 2 Dm WLNaH gvc Ck2 Vcd Vc 2 þ a=k and if eqn (12) is solved analytically: "  # SID  Var WLf ¼ ID 2 DN 2 pffiffiffi 2 pffiffiffi 3Vcs 4C 4 kðC 2 þ akÞ 16C 3 kðC 2  2akÞðC 2 þ akÞ 6 sgnðVc Þ  2  sgnðVc Þ 7 6 C þ kVc 2 C þ sgnðVc ÞkVc 7 6 7 6 pffiffiffi 2 7 6 2 2 7 6 2a kðC þ akÞ ða k  sgnðVc ÞC Vc  3aCðC  sgnðVc ÞkVc ÞÞ 3 7 6 sgnðVc Þ   7 6 a þ kV 2 2 7 6 c 7 6 7 pffiffiffi 6 pffiffiffi 2 7 16kTlNtcoeff NT e4 k 6 kðC þ akÞð5C Vc þ sgnðVc Þ2aC ð12C  sgnðVc Þ17kVc Þ þ 3aC kð sgnðVc Þ8C þ 3kVc ÞÞ 7 2 3 2 6 7 WLgvc C 8ðC 2 þ akÞ5 6 6 a þ kVc 2 7 7 6 r ffiffiffi ! 7 6 3C   k 7 6 7 6 þ pffiffiffi C  25aC k þ 35a C k  3a k arctan 6 4 2 2 2 3 3 Vc 7 6 a a 7 6 7 6 p ffiffiffi  6  7 6 7 6 þsgnðV c Þ24 k C  5aC 4 k þ 2a C k 2 2 2 logðC þ sgnðV c ÞkV c Þ 7 4 pffiffiffi 6 5    sgnðVc Þ12 k C  5aC k þ 2a C k log a þ kVc 4 2 2 2 2 V cd (9) This journal is © The Royal Society of Chemistry 2020 Nanoscale Adv., 2020, 2, 5450–5460 | 5455 View Article Online Nanoscale Advances Paper " # SID  Experimental validation of the LFN variance model Var WLf ¼ ID 2 Dm The qualitative performance of the new derived LFN variance rk ffiffiffi model is veried with the data from SG-SL GFETs under test, as 2 3Vcs C arctan V it will be illustrated in the rest of this section. Initially, as it was 4e2 aH 6 c  7 4 pffiffiffiffiffiffi a þ 0:5 sgnðVc Þ a þ kVc 2 5 mentioned before, ln-mean of WLSID f/ID2 data is calculated for WLNaH gvc Ck ak every device under test from all the available samples and used Vcd in the verication of the LFN mean value model as in Fig. 4. (13) Aerwards, standard deviation s(ln[WLSID f/ID2]) data can be This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. easily extracted from the natural logarithms of all the WLSID f/ID2 samples again for each available GFET. This process is followed Open Access Article. Published on 26 October 2020. Downloaded on 1/22/2021 12:04:48 PM. for the derivation of s(ln[WLSID f/ID2]) instead of eqn (15) since Total LFN variance noise measurements are very sensitive and thus the calculation of normalized variance ratio Var[WLSID f/ID2]/E2[WLSID f/ID2] The total WLSIDf/ID2 variance can be calculated as: "  # "  # contained in eqn (15) is not very consistent because of the small   SI SI  SI  numbers both in numerator and denominator. Then, WLSIDf/ID2 Var D2 WLf ¼ Var D2  WLf þ Var D2  WLf (14) variance can be estimated through eqn (15) since s(ln[WLSIDf/ ID ID DN ID Dm ID2]) is already known, by using the ln-mean in the denominator under the approximation that DN and Dm models are uncorre- of normalized variance. In Fig. 4, 2s standard deviation of lated. Even though this is not completely accurate, there are WLSIDf/ID2 (s ¼ OVar) is also shown both for the model and distinguished regions where each of these effects is dominant experimental data with purple (+2s) and green (2s) solid lines (Dm at the CNP and DN at the peaks of M-shape dependence) and markers respectively. The model captures accurately the and thus the aforementioned independency can be assumed dispersion of the data and its bias dependence, conrming the without signicant error. Eqn (9), (13) and (14) formulate the consistency between LFN mean value and variance models for new compact statistical LFN model. As shown before, WLSIDf/ID2 all the three GFETs examined. follows a log-normal distribution as it can be observed by the WLSIDf/ID2 variance and s(ln[WLSIDf/ID2]) are depicted in CDFs illustrated in Fig. 2b and in ESI B (Fig. S2a and b).† the Fig. 5 and 6 respectively for the W/L ¼ 20 mm/20 mm GFET in fundamental expression for this distribution is:37,39 Fig. 5a and 6a, for the W/L ¼ 50 mm/50 mm GFET in Fig. 5b and vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi 6b and for the W/L ¼ 100 mm/100 mm GFET in Fig. 5c and 6c vs. u u 0 SID 1   u Var 2 WLf VGEFF. Experimental data are represented with markers while SID u I s ln 2 WLf ¼u uln@1 þ  D A (15) the total model with solid lines. Dashed and dotted lines in ID t SID Fig. 5 stand for the DN and Dm variance contributions, respec- E 2 WLf ID 2 tively. This representation conrms that the model precisely captures the experimental data for both WLSIDf/ID2 variance and where s(ln[WLSIDf/ID2]) is the standard deviation of the natural s(ln[WLSIDf/ID2]) in the whole range of operation for every GFET logarithm of WLSIDf/ID2 widely used in bibliography,25,27,35–37,39,40 under investigation. E denotes the LFN mean value model and the ratio of variance The DN and Dm models shown in Fig. 5 prove that these with squared mean WLSIDf/ID2 is dened as normalized vari- effects act similarly as in the LFN mean value model, DN effect is ance.25,27,35–37,39,40 The model in eqn (15) follows the scaling responsible for the M-shape of WLSIDf/ID2 variance as it was for dependence of ref. 37, which is Oln[1 + K/(WL)] and for larger the WLSIDf/ID2 mean value while Dm contributes near the CNP as devices turns to O1/(WL) where K/(WL) is the normalized it did for the LFN mean value model while it retains its L shape variance established before.37 Fig. 5 Variance of normalized LFN Var[WLSIDf/ID2], referred to 1 Hz, vs. top gate voltage overdrive VGEFF, for GFETs with (a) W/L ¼ 20 mm/20 mm, (b) 50 mm/50 mm and (c) 100 mm/100 mm. Markers: measured data, solid lines: total model, dashed lines: individual contributions (DN, Dm). Simplified LFN variance model (gm/ID)4 which considers a homogeneous channel is shown with red dashed lines for W/L ¼ 20 mm/20 mm GFET (a) for comparison reasons. 5456 | Nanoscale Adv., 2020, 2, 5450–5460 This journal is © The Royal Society of Chemistry 2020 View Article Online Paper Nanoscale Advances This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. Open Access Article. Published on 26 October 2020. Downloaded on 1/22/2021 12:04:48 PM. Fig. 6 Standard deviation of natural logarithm of normalized LFN s[ln(WLSIDf/ID2)], referred to 1 Hz, vs. top gate voltage overdrive VGEFF, for GFETs with (a) W/L ¼ 20 mm/20 mm, (b) 50 mm/50 mm and (c) 100 mm/100 mm. Markers: measured data, solid lines: total model. trend. The new LFN variance model shows a deviation from the mean value increases as the area gets larger and as a result NT, experimental data at high n-type conduction regime probably aH get also higher as shown in Table 1. Table S1† clearly due to DR contribution which is not included in this study. For demonstrates that variances of Ntr, aH respectively, increase comparison reasons, a LFN variance model based on (gm/ID)2 more strongly than their mean values as the devices' area approximation which considers a uniform channel,26,29,31 is increases. The latter can justify the boost of Ntcoeff parameter shown in Fig. 5a with red dashed line. This approach estimates and the reduction of NaH as the dimensions get larger. The an (gm/ID)4 dependence of LFN variance. For the analysis of reason for these variations of the statistical LFN model the derivation of this expression see ESI E.† As it was expected, it parameters which occur as a physical consequence of the vari- gives acceptable results away from the CNP but it is incapable of ations of the LFN mean value model ones, could be the critical capturing LFN variance near the CNP where the non- inhomogeneities of GFET technologies, as it has already been homogeneity of the device is more intense even for small VDS stated before. values.53 These non-homogeneities can be easily detected from Standard deviation of natural logarithm s(ln[WLSIDf/ID2]) of the illustration of the DN LFN contribution to WLSIDf/ID2 vari- normalized LFN is shown in Fig. 6 and some remarkable ance throughout the channel where, for VGEFF values away from conclusions can be extracted. The above quantity can be used as the CNP, local variance is constant along the channel. Oppo- a gure of merit for LFN variability comparisons between the sitely, near the CNP a steep dip is noticed in the middle of the GFET technology in this work and CMOS ones35–37,40 It can be channel in which the exact CNP is located under low VDS. For concluded that the range of values between 0.1–1 for all the the above observations see ESI F (Fig. S4).† The experimental transistors under test as depicted in Fig. 6 are similar with the data were measured at VDS ¼ 50 mV, but it is quite certain that results obtained from CMOS devices with similar dimensions, the inconsistency of the (gm/ID)4 term would be more signi- indicating a decent performance for the GFETs under cant for higher VDS values. study.35–37,40 Another crucial observation is the weak bias Ntcoeff, NaH parameters of DN and Dm effects respectively, are dependence of s(ln[WLSIDf/ID2]) data with a rather smooth extracted and shown in Table 1. NaH is calculated from the CNP uctuation near the CNP and a slight decrease away from the while Ntcoeff is then adjusted to t the M-shape. It is clear that CNP, which is remarkably captured by the proposed model. Ntcoeff values for all GFETs are far from unity which means that Despite the fact that the proposed WLSIDf/ID2 variance model Ntr does not follow a Poisson distribution as in silicon-oxide focuses on the bias dependence, its scaling with the area is also devices. As stated before, this might be both due to the nature crucial. DN and Dm WLSIDf/ID2 variance models follow a 1/(WL) of the traps generated in graphene–electrolyte interface43,60–63 trend as it is clear from eqn (9) and (13) and as a result total LFN and because of process-induced extrinsic traps of the specic variance expression in eqn (14) behaves similarly. SIDf/ID2 mean SG-SL GFETs. In order to prove the validity of the obtained value model presents the same dependency.53–55 In addition, Ntcoeff, NaH values, a thorough analysis was conducted where the s(ln[WLSIDf/ID2]) also follows an O1/(WL) trend due to the large LFN mean value model parameters (NT–Ntr, aH) were extracted device dimensions.25,27,35–37,40 However, due to inhomogeneities for each of the measured samples for all available GFET areas. of the GFET technologies in general, extracted parameters both The variance of these parameters as well as their ln-mean value of the LFN mean value (NT, aH) and variance (Ntcoeff, NaH) were then estimated, allowing to derive the LFN variance model models, differ from device to device. parameters Ntcoeff, NaH, given that Var[Ntr] ¼ NtcoeffWLKTlNT and Var[aH] ¼ aH(NaHWL)1. These values are proven to be identical with the ones extracted from Fig. 5 and shown in Table 1, which Conclusions provide the best possible tting of the model. This result This work investigates thoroughly the bias dependence of LFN conrms that charge traps do not present a Poisson distribution variability in large area SG-SL GFETs. To our knowledge, this is in the GFETs under study. For the complete analysis see ESI G the rst time that such a study that combines both the analysis (Fig. S5 and Table S1).† As it was mentioned before, WLSIDf/ID2 of statistical LFN data for graphene devices and the extraction of This journal is © The Royal Society of Chemistry 2020 Nanoscale Adv., 2020, 2, 5450–5460 | 5457 View Article Online Nanoscale Advances Paper a physics-based LFN variance model is presented. More The derived results contribute to the thorough understanding specically, an analytical compact model based on carrier of the nature of charge traps statistics in solution-gated devices number DN and mobility uctuation Dm effects has been and they also provide the tools to quantify and predict LFN proposed and implemented for circuit simulators. The devel- variability in SL GFETs. This framework is considered critical opment of such an effective tool is critical for the boost of for upscaling the production of graphene electronics from graphene circuit design, where LFN variability should be accu- research labs into larger-scale dedicated fabrication facilities. rately predicted to prevent performance deterioration in certain applications. The latter is one of the most signicant novelties Experimental data of this work. Besides, it is experimentally proven that LFN This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. variability does not count on the variability of IV quantities such Electrical characterization of LFN Open Access Article. Published on 26 October 2020. Downloaded on 1/22/2021 12:04:48 PM. as VCNP, ID but it is directly linked to the number of traps Ntr and To measure the DC transfer curves and the LFN spectra accu- Hooge parameter aH variations regarding DN and Dm mecha- rately, the drain to source current was pre-amplied in a rst nisms respectively. The derived compact model precisely covers amplication stage with a gain of 104. The signal was then high the measured LFN variance over the whole range of operation, pass ltered, thus canceling its low frequency (i.e. DC level) from strong conduction to the CNP at both p- and n-type components. The resulting signal was further amplied and low regimes. DN and Dm LFN variance models exhibit a similar pass (anti-aliasing) ltered in a second stage with a 102 gain. bias dependence with the corresponding DN and Dm mean value The signals were digitalized using a NI DAQ Card in all char- ones. Thus, DN effect accounts for the M-shape of LFN variance acterization procedures. To extract the power spectral density, similarly as it is known to cause an M-shape for its mean value the drain to source current was measured under different gate while Dm provides a L-shape in the bias dependence of LFN bias conditions for 10 s at each point. variance, which contributes signicantly at the CNP, analo- gously to the Dm contribution to LFN mean value. It is also Fabrication of SG GFETs worth mentioning that variances of both DN and Dm mecha- Arrays of SG-SL GFETs were fabricated on a 10 mm thick poly- nisms must be computed locally in the channel in order to imide (PI-2611, HD MicroSystems) lm spin coated on a Si/SiO2 guarantee that each parameter's (Ntr, aH) deviation at any 400 wafer and baked at 350  C. A rst metal layer (10 nm Ti/ position in the channel will have an individual impact on LFN 100 nm Au) was deposited by electron-beam vapour and then variability. A simpler variance model with a (gm/ID)4 shape is structured by a li-off process. Aerwards, the graphene grown also extracted based on a uniform channel approximation and by chemical vapour deposition on Cu was transferred (process shown for comparison reasons. This approach based on the done by Graphenea s.a.). Graphene was then patterned by well-known (gm/ID)2 model of the DN contribution to LFN oxygen plasma (50 sccm, 300 W for 1 min) in a reactive ion mean value fails to accurately predict the LFN variance near the etching (RIE) aer protecting the graphene in the channel CNP. region with HIPR 6512 (FujiFilm) positive photoresist. Aer the NT, (or Ntr consequently) and aH parameters of the LFN graphene etching, a second metal layer was patterned on the model mean value are also used in LFN variance model together contacts following the same procedure as for the rst layer. The with the newly dened Ntcoeff and NaH parameters which are li-off was followed by an annealing in ultra-high vacuum extracted from statistical LFN data. In silicon-oxide transistors, consisting on a temperature ramp from room temperature to Ntr follows a Poisson distribution and thus, Ntcoeff is close to 300  C. Subsequently, the transistors were insulated with a 3 unity but this is not the case in the devices under test. The latter mm-thick photodenable SU-8 epoxy photoresist (SU-8 2005 is experimentally shown by extracting LFN mean value model Microchem), keeping uncovered the active area of the transis- parameters for every measured sample and then by calculating tors channel and the contacting pads. The polyimide substrate their variance and mean value. was structured in a reactive ion etching process using a thick Regarding geometrical scaling, while both SIDf/ID2 mean AZ9260 positive photoresist (Clariant) layer as an etching mask. value and WLSIDf/ID2 variance models predict a 1/(WL) The neural probes were then peeled off from the wafer and behavior, the corresponding data do not follow this trend. In placed in a zero-insertion force connector to be interfaced with fact, the extracted LFN parameters are not identical for the three our custom electronic instrumentation. Finally, the devices GFETs under test despite the fact that they are devices of the were rinsed for 2 minutes in ethanol to eliminate remaining same technology. On the contrary, LFN mean value parameters resist residues on the graphene channel. increase with device area causing also a large deviation in LFN variance parameters. This device to device parameters' devia- tion could be justied due to increased inhomogeneities Raman characterization of graphene aer transfer observed on recently developed GFET technologies. A SLG sample was transfer onto a SiO2 wafer following the same Eqn (15) expresses the standard deviation of natural loga- process as detailed for the fabrication of GFETs. The Raman rithm of WLSIDf/ID2 – s(ln[WLSIDf/ID2]) – which is taken into spectra at 400 equally spaced points were acquired on the gra- consideration and examined due to log-normal distribution of phene sample, within an area of 20 mm  20 mm. A Witec LFN data and it might be a reliable tool in order to compare LFN spectrometer in backscattering conguration, using a 600 gr variance for different types of transistors. This study represents per nm grating was used. A 488 nm wavelength laser (2.5 mW the rst reported efforts to understand LFN variability in GFETs. power) was focused on the sample with a 50 objective. The 5458 | Nanoscale Adv., 2020, 2, 5450–5460 This journal is © The Royal Society of Chemistry 2020 View Article Online Paper Nanoscale Advances peak intensity for the D and G bands was measured aer 11 S. Castilla, B. Terres, M. Autore, L. Viti, J. Li, A. Y. Nikitin, background subtraction. I. Vangelidis, K. Watanabe, T. Taniguchi, F. Lidorikis, M. S. Vitiello, R. Hillenbrand, K. J. Tielrooij and Data availability F. H. L. Koppens, Nano Lett., 2019, 19(5), 2765–2773. 12 F. Schedin, A. K. Geim, S. V. Mozorov, E. W. Hill, M. I. Blake, The data that support the ndings of this study are available M. I. Katsnelson and K. S. Novoselov, Nat. Mater., 2007, 16, from Ramon Garcia Cortadella. Please, address your requests to 652–655. ramon.garcia@icn2.cat. 13 S. Rumyantsev, G. Liu, M. S. Shur, R. A. Potyrailo and A. A. Balandin, Nano Lett., 2012, 12(5), 2294–2298. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. Conflicts of interest 14 L. H. Hess, M. Seifert and J. A. Garrido, Proc. IEEE, 2013, Open Access Article. Published on 26 October 2020. Downloaded on 1/22/2021 12:04:48 PM. 101(7), 1780–1792. There are no conicts to declare. 15 R. Samnakay, C. Jiang, S. Rumyantsev, M. S. Shur and A. A. Balandin, Appl. Phys. Lett., 2015, 106(2), 23115. Acknowledgements 16 C. Herbert, E. Masvidal-Codina, A. Suarez-Perez, A. Bonaccini-Calia, G. Piret, R. Garcia-Cortadella, X. Illa, This work was funded by the European Union's Horizon 2020 D. E. Corro Garcia, J. M. De la Cruz Sanchez, D. V. Casals, research and innovation program under Grant Agreement No. E. Prats-Alfonso, J. Bousquet, P. Godignon, B. Yvert, GrapheneCore2 785219 and No. GrapheneCore3 881603, Marie R. Villa, M. V. Sanchez-Vives, A. Guimera-Brunet and Skłodowska-Curie Grant Agreement No. 665919 and Grant J. A. Garrido, Adv. Funct. Mater., 2017, 28(12), 1703976. Agreement No. 732032 (BrainCom). We also acknowledge 17 F. Bonaccorso, Z. Sun, T. Hasan and A. C. Ferrari, Nat. nancial support by Spanish government under the projects Photonics, 2010, 4, 611–622. TEC2015-67462-C2-1-R, RTI2018-097876-B-C21 (MCIU/AEI/ 18 G. Xu, Y. Zhang, X. Duan, A. A. Ballandin and K. L. Wang, FEDER, UE) and project 001-P-001702-GraphCat: Communitat Proc. IEEE, 2013, 101(7), 1670–1688. Emergent de grafè a Catalunya, co-funded by FEDER within the 19 A. D. Smith, S. Wagner, S. Kataria, B. G. Malm, M. C. Lemme framework of Programa Operatiu FEDER de Catalunya 2014– and M. Ostling, IEEE Trans. Electron Devices, 2017, 64(9), 2020. The ICN2 is also supported by the Severo Ochoa Centres of 3919–3926. 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