This is the accepted version of the article: Nikolaos Mavredakis, Wei Wei, Emiliano Pallecchi, Dominique Vignaud, Henri Happy, Ramon Garcia Cortadella, Andrea Bonaccini Calia, Jose A. Garrido, and David Jiménez. Velocity Saturation Effect on Low Frequency Noise in Short Channel Single Layer Graphene Field Effect Transistors. ACS Applied Electronic Materials, (2019). 1. : 2626 - . 10.1021/acsaelm.9b00604. Available at: https://dx.doi.org/10.1021/acsaelm.9b00604 Please do not adjust margins Velocity Saturation effect on Low Frequency Noise in short channel Single Layer Graphene FETs Nikolaos Mavredakis*a, Wei Weib, Emiliano Pallecchib, Dominique Vignaudb, Henri Happyb, Ramon Garcia Cortadellac, Andrea Bonaccini Caliac, Jose A. Garridoc, d and David Jiméneza Graphene devices for analog and RF applications are prone to Low Frequency Noise (LFN) due to its upconversion to undesired phase noise at higher frequencies. Such applications demand the use of short channel graphene transistors that operate at high electric fields in order to ensure a high speed. Electric field is inversely proportional to device length and proportional to channel potential so it gets maximized as the drain voltage increases and the transistor’s length shrinks. Under these conditions though, short channel effects like Velocity Saturation (VS) should be taken into account. Carrier number and mobility fluctuations have been proved to be the main sources that generate LFN in graphene devices. While their contribution to the bias dependence of LFN in long channels has been thoroughly investigated, the way in which VS phenomenon affects LFN in short channel devices under high drain voltage conditions has not been well understood. At low electric field operation, VS effect is negligible since carriers’ velocity is far away from being saturated. Under these conditions, LFN can be precicely predicted by a recently established physics-based analytical model. The present paper goes a step furher and proposes a new model which deals with the contribution of VS effect on LFN under high electric field conditions. The implemented model is validated with novel experimental data, published for the first time, from CVD grown back-gated single layer graphene transistors operating at gigahertz frequencies. The model accurately captures the reduction of LFN especially near charge neutrality point because of the effect of VS mechanism. Moreover, an analytical expression for the effect of contact resistance on LFN is derived. This contact resistance contribution is experimentally shown to be dominant at higher gate voltages and is accurately described by the proposed model. other applications of graphene are chemical-biological sensors10-13 Introduction and optoelectronic devices14. The performance of all the above Extensive research has taken place the last decade after the devices and circuits can be degraded by the effect of Low Frequency discovery of graphene1-2 due to its exceptional properties. Carrier Noise (LFN) which can be up-converted to undesired phase noise in mobilities up to 2.105 cm2/V.s and saturation velocities of 4.107 cm/s high frequency circuits and it can also affect the sensitivity of led the scientific community to accept the challenge and take sensors5-14. In addition, LFN analysis can provide significant advantage of graphene in electronic applications by fabricating conclusions regarding the quality and reliability of graphene graphene transistors (GFETs)3-4. Despite the fact that graphene’s zero devices15. bandgap is a deterrent for digital operation, the developments of There are three main mechanisms that generate LFN in GFETs for analog and RF applications is ongoing with very promising semiconductor devices: a) carrier number fluctuation (ΔΝ), b) results. Frequency multipliers5, voltage controlled oscillators6 and mobility fluctuation (Δμ) and c) contact resistance (Rc) contribution THz detectors7-9 are some examples of electronic applications, while (ΔR). ΔΝ model16 is based on trapping/detrapping mechanism where carriers can be captured and then emitted at border traps near the dielectric interface of a semiconductor17. A Random Telegraph Signal a. Departament (RTS) in time domain which results in a Lorentzian spectrum is d’Enginyeria Electrònica, Escola d’Enginyeria, Universitat Autònoma de Barcelona, Bellaterra 08193, Spain. generated by each such trap. Supposing that these traps are b. Institute of electronics, Microelectronics and Nanotechnology, CNRS UMR8520, uniformly distributed, then the superposition of these Lorentzians 59652 Villeneuve d’Ascq, France. c. Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC, Barcelona can cause an inversely proportional trend of the Power Spectral Institute of Science and Technology, Campus UAB, Bellaterra, Barcelona, Spain. Density (PSD) of noise with frequency. For this reason the LFN is also d. ICREA, Pg. Lluis Companys 23, 08010 Barcelona, Spain * Email: nikolaos.mavredakis@uab.es known as 1/f (flicker) noise. In transistors with very small dimensions, Electronic Supplementary Information (ESI) available: At first, a thorough analysis of the limited number of traps can lead to Lorentzian-shape PSDs, but the effect of Velocity Saturation phenomenon on Drift-Diffusion current equation is provided in Section A. Then, an analytical solution for effective length is presented this has not been yet observed in GFETs. Δμ model18 is expressed by in Section B. In Section C, Fig. S1 illustrates the behaviour of chemical potential and empirical Hooge formula and is considered to be caused by longitudinal field locally in the channel at different bias conditions. After that, a more detailed description of the theoretical derivation of both carrier number and fluctuations of carrier mobility. Finally, Rc can also influence LFN and mobility fluctuation models is provided in Section D. There, Fig. S2 presents the this contribution can be very significant in GFETs at short channels behaviour of all the local noise expressions related to the above effects, along the channel at different operating points. The process that integrals are calculated for where Rc contribution cannot be neglected. There are many LFN different cases of chemical potential values is presented in detail in Section E. Fig. models available in bibliography describing the above three effects S3 in Section F presents the comparison of IV data and model for the rest of the available devices which are not included in the main manuscript while Fig. S4 in for Metal-Oxide-Semiconductor FETs (MOSFETs)19-22. The same three Section G shows the Low Frequency Noise data of these transistors vs. the proposed mechanisms have also been found to be responsible for LFN in model. Please do not adjust margins Please do not adjust margins a) b) c) Fig. 1 a) Energy dispersion relation of GFET (top) and its capacitive network (bottom) are shown with Cq: quantum capacitance, Ctop, Cback: top and back oxide capacitances, Vc(x): chemical potential, V(x): quasi-Fermi channel potential, VG(B)S-VG(B)S0: top and back gate source voltage overdrives. (Top gate is not present in devices under test of (b) but is included in the capacitive network of (a) to support the generalizability of the model). b) GFET structure. Top left: Optical image of GFET. The channel width W is 12 μm. The inset image shows the RF coplanar wave guide access. The scale bar is 60 μm. Top right: SEM image of our GFET with two-figures bottom gate structure. The scale bar is 1 μm. Bottom: Cross section schematic of our bottom gate structure. c) Raman spectra at 473 nm laser excitation of graphene after device fabrication process. GFETs23-35. Δμ effect is known to dominate in metals18 and ΔΝ in away from CNP, it is not valid near the specific point where the semiconductors17 where trapping/detrapping prevails, and since chemical potential Vc tends to 0 and thus usat becomes very high, graphene can be considered a metal as well as a semiconductor, even higher than Fermi velocity uf(~106 m/s) which is the both of the above effects can contribute to its LFN. In fact, LFN maximum velocity of carriers in graphene. Thus, a two branch nature in GFETs is strongly related to the number of layers since model has been proposed40-41 where a constant usat value is ΔΝ mechanism becomes more significant as this number is considered for a quite low graphene net channel charge so that decreased while in multilayer GFETs Δμ is more important24. As it the GFET operates near CNP, while for higher values of charge, a was mentioned before, ΔR can also play an important role more complicated energy dependent expression is used40-41. But because of the Rc values in GFETs. if the aforementioned complicated model is used in LFN modeling The most significant experimental characteristic of gate bias for short channel GFETs, the equations become so complex, that dependence of LFN in GFETs is the M-shape trend with a it is generally impossible to find an analytical solution. That is why, minimum close to charge neutrality point (CNP)25-28, 34-35. In a an inversely proportional relation between Vc and usat is previous work35, an analytical physics-based bias dependent considered away from CNP. model for long channel single-layer (SL) GFETs was proposed and The fundamental scope of this work is the extension of the model successfully validated with experimental data. Both ΔΝ and Δμ proposed in ref. 35 in order to include the VS effect on LFN. models were shown to contribute in total LFN, especially near Furthermore, a simple analytical expression for ΔR contribution CNP, while the Rc effect on LFN was observed at higher gate taken from Si devices42 is proposed. As described thoroughly in voltage regions but not analytically modeled. In addition, a strong ref. 35, the LFN model is implemented based on the assumption relation between gate bias dependence of LFN and residual that the GFETs’ channel is divided into infinitesimal slices, each of charge near CNP was shown; ΔΝ effect is responsible for M-shape which corresponds to a local noise source19, 22, 36, 42. All these local behavior but as residual charge decreases, a Λ-shape trend can noise sources can be considered uncorrelated and thus, the sum be observed35. Moreover, even a minor increase of drain voltage, of their PSDs results in the total LFN42. In simpler words, by was shown to slightly affect the homogeneity of the channel integrating all the local noise contributors along the device especially near CNP and this results in a small rise of LFN there. channel, the PSD of each LFN mechanism can be calculated. As it Those experiments were conducted at very small drain voltage will be shown, these integrals can be solved analytically, similarly values (VDS=20, 40, 60 mV) and thus, important phenomena that to ref. 35, based on a chemical potential based compact model43- are significant at quite high electric fields such as Velocity 45 and thus, the LFN model can be easily implemented in Verilog- Saturation (VS) could not be studied. A and integrated in circuit simulators. The equivalent capacitive Analytical modeling of LFN in short channel GFETs and the circuit of this model43-45 is shown in Fig. 1a. Graphene charge Qgr contribution of VS effect on it remain largely uninvestigated. In Si is stored in the quantum capacitance (Cq); the chemical potential devices, VS effect causes a reduction of LFN at high electric Vc(x) represents the voltage drop across Cq at position x. Vc(x) is fields36 and this is also the case in GFETs as it will be shown in this defined as the difference between the potential at quasi-Fermi work for the first time. VS effect is generated by optical phonon level and the potential at the CNP, as shown in the energy scattering mechanism and particularly for GFETs, saturation dispersion relation scheme of graphene in Fig. 1a where Vc(0)=Vcs velocity usat is usually approximated inversely proportional to at the source end (x=0) and Vc(L)=Vcd at the drain end (x=L). chemical potential Vc37-39. While this relationship is acceptable Please do not adjust margins Please do not adjust margins a) b) c) Fig. 2 Drift velocity udrift vs. chemical potential Vc for W/L=12 μm/100 nm GFET with a) VGEFF=0 V (CNP) and b) VGEFF=0.12 V at VDS=60 mV (left subplot), 0.3 V (right subplot). usat is also shown with dashed lines. c) Effective mobility μeff vs back gate voltage overdrive VGEFF for W/L=12 μm/100 nm GFET at low and high VDS of 60 mV and 0.3 V. Constant long channel mobility (model parameter μ) is also shown with dashed line. VGS-VGS0, VBS-VBS0 are the top and back gate source voltage depends on the ratio of longitudinal electrical field Ex and the overdrives while Ctop and Cback are the top and back gate critical field Ec. Ec is the value of electric field Ex above which the capacitances, respectively. The sum of top and back gate carriers’ velocity saturates. After a more detailed analysis (see capacitances is defined as C=Ctop+Cback. The quasi-Fermi potential eqn (A1-A2) in ESI A), eqn (2) (bottom of the page) is derived V(x) is the voltage drop in the graphene channel at position x, which represents the change of integral variable from x to Vc and which is equal to zero at the source end (x=0) and equal to VDS at it will be proved to be very significant for the derivation of the IV the drain end (x=L). A good agreement between drain current and LFN analytical expressions; VS effect is considered through data and the above model43-45 is crucial for the good performance saturation velocity term usat. Then by integrating eqn (2) along the of LFN model, since IV (current-voltage) quantities are used in LFN device channel from Source (S) to Drain (D): expressions. Vd Vcs Cq  C   WQgr  dV W   Qgr dVc The extracted LFN model is validated with experimental data C Vs Vcd from bottom-gated SL GFETs where graphene grown by CVD on a ID   (3) Vd   Cq  2C  Vcs   Cq  2C  copper foil was used46-49. The optical and SEM images as well as L  dV L    dVc       the schematic of the graphene device are shown in Fig. 1b. Details Vs sat  Cq  C  Vcd sat  C  on the fabrication process can be found in Experimental data The denominator of the eqn (3) expresses an effective channel section. The quality of graphene after transfer process was length Leff which accounts for the degradation of ID because of VS verified by performing Raman characterization. Fig. 1c shows a effect (see see eqn (A6-A9) in ESI B). The sign of VDS determines typical Raman spectra of graphene on the substrate (SiO2/Si) after the sign of the electric field E and consequently, the sign of the the device fabrication process. The 2D peak at 2703 cm-1 is fitted longitudinal electrical field Ex as described thoroughly in ESI A, B. with a single Lorentzian component with a full width at half- Regarding the value of usat, a two-branch model is used, as maximum (FWHM) of 26 cm-1. The G peak locates at 1584 cm-1 mentioned before: and has a FWHM (G) of 12 cm-1.The ratio of the 2D and G peak  2u f   S , Vc  Vccrit integrated intensities stands around 2.5, which indicates single    hu f layer graphene. The intensity ratio of the D and G peak is very low, usat       N , N , Vc  Vccrit (4) ~0.1, which suggests that a very low defect density is present in  Q  2  2 e  gr  k  c V  a / k  Vc  a / k  e   our fabricated devices.  2e The analytical expressions for coefficient k, Fermi velocity uf, Results and Discussion graphene charge Qgr, bias dependent term g(Vc) and residual charge related term α are given in ESI A, hΩ is the phonon energy The drain current model proposed in ref. 43- 45 and on which the and e is the electron charge. Qcrit is the critical value of graphene LFN analysis of the present work is based, assumes a drift- net charge above which usat is considered inversely proportional diffusion carrier transport with a soft VS model37-41. In more to Vc as shown in bottom branch of eqn (4) while it is constant detail: below Qcrit as it is shown in upper branch of eqn (4).  Qcrit=eΩ2/(2πuf2) and from there Vccrit can also be calculated. I D  WQgr eff E , eff  (1) 1  Ex / Ec All plots of Fig. 2 are for a GFET with L=100 nm. In Fig. 2a and 2b, where W is the width of GFET, μ is the low field carrier mobility, E the drift velocity udrift and the saturation velocity usat are shown is the electric field and μeff is the effective carrier mobility which vs. the chemical potential Vc at the CNP and away from it represents the degradation of mobility μ at high electric fields and respectively. In each graph, udrift at low (left subplot) and high Please do not adjust margins Please do not adjust margins dx Qgr 2 Leff  Cq  2C dx Qgr 2 Leff  Cq  C    Cq  2C          if V  0 dV kgvc sat Cq  C dVc kgvc  C    C  ds   sat   dx Qgr 2 Leff  Cq  2C dx Qgr 2 Leff  Cq  C    Cq  2C          if Vds  0 dV kgvc sat Cq  C dVc kgvc  C    C    sat (2) a) b) c) Fig. 3 Drain current ID vs. back gate voltage overdrive VGEFF, for GFETs with W=12 μm and a) L=300 nm (A300), b) L=200 nm (A200) and c) L=100 nm (A100) at low (left subplot) and high (right subplot) available VDS values (VDS=30 mV, 60 mV, 0.1 V, 0.2 V and 0.3 V). markers: measured, solid lines: model. (right subplot) drain voltage are shown (VDS=60 mV and 0.3 V) as order, probably because of leakages or possible break down of a solid line; usat is also shown with dashed lines. At low drain the devices at higher electric fields. In more detail, for A300, A100 voltage, udrift is much smaller than usat and thus, VS effect is and B100 GFETs all five drain voltages were measured, for B200, negligible. This is the case both at and away from the CNP. On the A200 GFETs, VDS=0.3 V is missing while for B300 GFET, VDS=30 mV contrary, at higher drain voltage, udrift is still smaller but is missing. Fig. 3 presents the transfer characteristics of a) A300, comparable to usat which means that VS effect starts to become b) A200 and c) A100 GFETs at all available drain voltages. The significant for every gate voltage regime. This is also shown in compact model reported in ref. 43-45 was used for simulating terms of electric field Ex in ESI C (Fig. S1). In Fig. 2c, effective drain current and the fitting with experimental data is of high mobility μeff is shown vs. effective gate voltage VGEFF (back-gate quality both near and away CNP at p-type region. There is an voltage overdrive) again for both drain voltage values mentioned asymmetry in drain current data at higher gate voltages since before. As it was expected at low drain voltage, μeff is quite close they are lower in n-type than p-type region and this can be to long channel mobility μ, while for higher drain voltage, VS explained either by a different mobility at the two operating effect causes a significant degradation of μeff. In accordance with regimes or by a parasitic p-n (n-n) diode which is formed between the usat model described in eqn (4), effective mobility is shown to the channel and the contact when the device is biased in the p(n)- get maximized at CNP. In more detail and as it is shown in Fig. 2a region giving rise to different contact resistances 50 .The model is and 2b, usat becomes maximum at CNP and consequently Ec, symmetric so it can provide identical behaviour at p- and n-type which is proportional to usat, is also maximized (see eqn (A1) in ESI regimes away from CNP. The best fit was achieved at the whole A) while the ratio Ex/Ec gets minimum (see Fig. S1 in ESI C). As a p-type region, near CNP and up to a value of VGEFF≈0.15 V in n- result from eqn (1), μeff is maximum at CNP. type regime. We focused our attention on this region which IV and LFN data were measured for six different GFETs with W=12 presents the highest transconductance, a crucial figure of merit μm and for three available channel lengths; L=300 nm for A300, for RF applications. Table 1 presents the parameter set of the IV B300 devices, L=200 nm for A200, B200 devices and L=100 nm for model which includes the long channel carrier mobility (μ), the A100, B100 devices (see Experimental Data section for more back gate capacitance (Cback), the flat band back gate voltage details on fabrication and measurements). The back gate voltage (VBSO), the contact resistance (Rc), the inhomogeneity of the was swept from VG=0 to 1.4 V with a step of 50 mV for transfer electrostatic potential (Δ) which is related to the residual charge characteristics while for LFN spectra the sweep was from VG=0.6 density ρ043-45 and the phonon energy (hΩ) which is related to VS to 1.3 V with a step of 50 mV, covering regions both away and effect. One parameter set is used for all bias conditions at each near CNP. The measured frequency range was from 1 Hz to 1 kHz. GFET; all the above parameters except hΩ are extracted for low Moreover, five different drain voltage values were recorded for drain voltages, while hΩ is extracted for the higher ones. Fig. 4a both IV and LFN setups (VDS=30m, 60m, 0.1, 0.2 and 0.3 V) in order shows the equivalent noise subcircuit35, 42 where a random local to cover the low and high electric field region which is crucial for current noise source δIn with a PSD SδI2n is used to model the local studying VS effect. In a few cases, some drain voltages were fluctuations. This noise subcircuit and its operating principles is omitted since the IV as well as LFN data were completely out of analysed thoroughly in ref. 35. Fig. 4b and 4c illustrate the LFN Please do not adjust margins a) b) c) Fig. 4 a) Equivalent noise subcircuit with a local current noise source. Relative power spectral density of drain current noise SID for b) W/L=12 μm/300 nm A300 GFET and c) W/L=12 μm/100 nm A100 GFET with back gate voltage overdrive (VGEFF = −0.3, −0.2, −0.1, 0, 0.1 V) at low VDS= 30 mV (left subplot) and high VDS= 0.3 V (right subplot). The solid line corresponds to a 1/f slope. V   cs NA  A1  k   2  k C arctan    c  2 2  V   k ln   kVc  2C ln  C  kVc    2k  k  C 2 gvc      Vcd (5)        V  2C 3k C 2  ak k C 2  ak 3aC 2  a 2k  2C 3V 2C 3 k C 2  3 k  cs  c  k     arctan  Vc   C  kV   kVc2     B1  c   NB V V  3         c ccrit (6a) 2 k  C2 S  4 2  a  kVc2     k 3C  aC k ln   C  kV       c  2  Vcd V     cs        C  kVc   C  2C  ak  ln  2 2    2    k Vc2    a2 k  C  kVc   C 3Vc  2C  3kVc   aC 2  4C  3kVc   2   CVc  C   k Vc   B1  k     k   NB V V     c ccrit N 2 5/2  C 2  ak  C  kV    kV 2   C 2  ak     c  (6b)   c                  Vcd V V B 2k  2C  cs A2  kVc2    cs  k   A  CVc    B V V   arctan  Vc   0.5ln a  kVc2  (7)   (8a) gvc  2  S  k     c ccrit  Vcd  Vcd Vcs B2       B V V   k Vc2   2C ln Vc  Vc2    N  NA  NB,    A   B N   (8b) (9) k  k  V c ccrit  cd 2 g ms 2  g md Wk Cback 2 R  SR 2 , g ms, d  Vcs, d (10) 2 2 Leff C   1  2  g ms  g md   Rc   * , : Top sign refers to Vc>0 and bottom sign to Vc<0. * Eqns (6a, 6b, 8a, 8b) are defined in cases where limits of integration have the same sign and the absolute value of both of them is below (eqns (6a, 8a)) or above (eqns (6b, 8b)) Vccrit. See main text below and ESI E for better understanding. * Eqns (6a, 8a) correspond to the integral: “LFN Near CNP” while eqns (6b, 8b) correspond to the integral “LFN Away CNP” which are defined in ESI E. Please do not adjust margins Please do not adjust margins spectra of A300 and A100 GFETs respectively with a slope close (ΔΝB, ΔμB) is subtracted by the long channel LFN (ΔΝA, ΔμA) for to 1/f, at five different effective gate potentials (VGEFF=-0.3, -0.2, - both ΔΝ and Δμ noise contributions which agrees with the 0.1, VCNP, 0.1 V). Low and high drain voltages are shown (VDS= 30 findings in MOSFETs36. Even for a negative drain voltage the VS mV, 0.3 V) at left and right subplots respectively. The 1/f trend of effect reduces LFN. More specifically, ΔΝA, ΔμA terms do not LFN is aparent in all cases. change sign with negative drain voltage since they are derived The derivation of ΔΝ and Δμ models is thoroughly analysed in ref. from the first term of the right hand of eqn (2) where its sign is 35 by considering a linear dependence of the quantum not affected by drain voltage polarity. This can be proved since capacitance Cq and the chemical potential Vc (Cq=k∙|Vc|) 35, 43. NT, the integration of ΔΝA, ΔμA in eqns (5, 7) from Vcs to Vcd gives which is the dielectric volumetric trap density per unit energy (in negative results but gvc is also negative and thus, ΔΝA, ΔμA LFN eV-1cm-3), is used as a first model parameter related to ΔΝ effect35 terms are positive as in the case of positive drain voltage. On the while αH, which is the unitless Hooge parameter, is used as a other hand, ΔΝB, ΔμB terms are derived from the second term of second model parameter related to Δμ effect35. For both ΔΝ and the right hand of eqn (2) and thus, they change their signs for Δμ cases, the total PSD of normalized drain current noise divided negative drain voltage. In this case, eqns (6, 8) remain identical by squared drain current at 1 Hz is calculated by considering the with just calculating the integrals from Vcd to Vcs, which results in integral from S to D (see eqn (A15, A22) for ΔΝ and Δμ, positive solution. Thus, eqn (9) always results in reduction of ΔΝ, respectively in ESI D). Then by applying eqn (2), the integral Δμ terms regardless of the polarity of drain voltage. It is very variable of LFN changes from x to Vc (see eqn (A16, A23) for ΔΝ crucial to notice that during the integration process (see eqns and Δμ, respectively in ESI D). The latter is essential since the IV (A19, A26, and A29-A32) of ESI D) that results in the analytical model is a chemical potential based model35, 43-45. Since eqn (2) expressions of eqns (5-8), different cases should be considered has two terms on the right hand side, this results in two ΔΝ depending both on the signs of Vcs, Vcd and if their absolute value related terms, ΔΝA and ΔΝB, and two Δμ related terms, ΔμA and is lower or higher than Vccrit since different equations are valid in ΔμB, which are derived in the format of integrals before being each such region. In more detail, there are four separate regions solved analytically. (see eqns (A19-A20, A26-A27) of ESI D). VS where Vc might be; a) lower than negative Vccrit, b) higher than phenomenon contributes to LFN both through Leff, which is negative Vccrit but lower than Vc=0 (CNP), c) higher than Vc=0 (CNP) contained in constants A1, B1, A2, B2 which are defined in ESI D, but lower than positive Vccrit and d) higher than positive Vccrit. If and through second right hand term of eqn (2). ΔΝA and ΔμA, both Vcs, Vcd belong to the same region then the eqns (5-8) are which come from the first right hand term of eqn (2), do not solved from Vcs to Vcd.. On the other hand, if Vcs, Vcd belong to include usat and in fact are the same with the ΔΝ-Δμ LFN terms different regions then the integrals that result in eqns (5-8) should extracted in ref. 35 with the only difference of presenting Leff be split into as many sub-integrals as required, corresponding to instead of L. ΔΝB and ΔμB on the other hand, include usat since the four regions mentioned above, to ensure that the limits of they come from the second right hand term of eqn (2) (see eqns integration of each one of the new sub-integrals have the same (A29 and A31) for ΔΝB and eqns (A30 and A32) for ΔμB of ESI D) polarity and their absolute values are both higher or lower than and they represent the main correction to LFN that is proposed in Vccrit. This happens since eqns (5-8) are valid and can be solved this work along with the contribution of Leff. The behaviour of all only in each such region. This process is described in detail in eqns the contributing terms to LFN locally in the channel is shown in (A33- A34) of ESI E where the sub-integrals are added when is Fig. S2 of ESI D. The integrals of ΔΝA, ΔΝB, ΔμA and ΔμB terms are then solved analytically in eqns (5-8) for the case of a positive drain voltage. Effective length Leff can be solved analytically (see eqns (A10-A11 in ESI B). Eqn (9) shows that the VS induced LFN a) b) c) Fig. 5 Output current noise divided by squared drain current SID/ID2, referred to 1 Hz, vs. back gate voltage overdrive VGEFF, for a) ΔΝ effect, b) Δμ effect and c) all noise contributions for W/L=12 μm/100 nm GFET. Dashed lines represent ΔΝΑ, ΔΝΒ contributors of ΔΝ effect in (a), ΔμA, ΔμB contributors of Δμ effect in (b) and total ΔΝ and Δμ noise mechanisms in (c). ΔR contributor is also shown with dotted lines in (c). Please do not adjust margins a) b) c) d) e) f) Fig. 6 Normalized drain current noise divided by squared drain current referred to 1 Hz, SIDf/ID2, vs. back gate voltage overdrive VGEFF, for GFETs with W=12 μm and a, d) L=300 nm (A300), b, e) L=200 nm (A200) and c, f) L=100 nm (A100): Upper plots show the highest and lowest available VDS value depending on the GFET. (A300: VDS=30 mV, 0.3 V, A200: VDS=30 mV, 0.2 V, A100: VDS=30 mV, 0.3 V) while down plots show the rest of available VDS values (A300: VDS=60 mV, 0.1 V, 0.2 V, A200: VDS=60 mV, 0.1 V, A100: VDS=60 mV, 0.1 V, 0.2 V) markers: measured, solid lines: model, dashed lines: long channel model from ref. 35. Table 1. IV and LFN Noise model Parameters Parameter Units A300 B300 A200 B200 A100 B100 (L=300 nm) (L=300 nm) (L=200 nm) (L=200 nm) (L=100 nm) (L=100 nm) μ cm2/(V∙s) 300 300 670 670 630 300 Cback μF/cm2 1,35 1,35 1,35 1,35 1,35 1,35 VBSO V 0,88 0,92 0,89 0,89 1,03 0,9 Δ eV 0.105 0.116 0.082 0.082 0.097 0.105 hΩ eV 0.016 0.016 0.017 0.017 0.018 0.017 Rc/2=RS,D Ω 260 198 219 223 131 176 NT eV-1cm-3 8·1019 7·1019 2.1·1019 2.3·1019 5.5·1018 1.5·1019 αH - 5·10-3 3·10-3 1.1·10-3 1.5·10-3 2.5·10-4 9.5·10-4 SΔR2 Ω2/Hz 2·10-2 7·10-3 3.4·10-3 9·10-3 5·10-4 4.5·10-3 needed in order to take the total solution. SI SI SI SI In the present work, a simple model for ΔR contribution is also D  D  D  D (11) 2 ID 2 2 2 ID ID ID derived taken from Si devices42 since the effect of Rc on LFN is N  R significant especially at higher-gate voltages as it will be shown The behaviour of all compact expressions of LFN related terms of later. ΔR model is described in eqn (10) where SΔR2 expressed in eqns (5-10) is analysed in Fig. 5. Normalized drain current LFN Ω2/Hz, is the third parameter of the proposed LFN model and gms, divided by squared drain current and referred to 1 Hz, SIDf/ID2 is gmd are the source and drain transconductances45. In order to shown vs. effective gate voltage VGEFF for a low and a high drain calculate the total LFN, the three different contributions have to voltage value (VDS=60 mV, 0.3 V). In Fig. 5a ΔΝA, ΔΝB and ΔΝ be added as: terms of carrier number fluctuation mechanism are presented. ΔΝ effect is responsible for M-shape of LFN35 and this is also confirmed in Fig. 5a. Apart from ΔΝA long channel term, ΔΝB also follows an M-shape trend. For low drain voltage, ΔΝB is negligible Please do not adjust margins Please do not adjust margins and ΔΝA coincides with ΔΝ. This is totally acceptable since at this is tested for the rest of drain voltages and it can be observed that region, VS effect does not contribute at all to drain current as well its behavior remains precise. For drain voltages up to 0.1 V the as to LFN. On the other hand, for higher drain voltage, the difference between the short and long channel model is negligible contribution of VS effect is apparent. The carrier number but above this value, VS effect significantly reduces LFN PSD and fluctuation term ΔΝ which is calculated by the subtraction of ΔΝA this is accurately captured by the proposed model. ΔR and ΔΝB as it is shown in eqn (9), is significantly lower than ΔΝA contribution is significant away from CNP especially for higher which in fact represents the long channel case. Similar drain voltage levels where LFN PSD is almost constant and ΔR conclusions can be extracted in Fig. 5b where the ΔμA, ΔμB and model successfully predicts this behavior. The IV and LFN models’ Δμ terms of mobility fluctuation effect are shown. Again, for low validation for the remaining devices B300, B200 and B100 is drain voltage, ΔμB is much lower than ΔμA and as a consequence presented in ESI F and G (Fig. S3 and S4 respectively). ΔμA dominates. For high drain voltage though, the difference The three extracted LFN parameters, NT, αH and SΔR2 are shown in ΔμA- ΔμB, which results in Δμ according to eqn (9) is significant. Table 1. The value of NT ranges from ~5.5·1018 -8·1019 eV-1cm-3 VS effect is responsible for the reduction of both ΔN and Δμ LFN depending on the device and is a little lower than values extracted mechanisms and this is the case at higher electric field regime. in other works33-35 related to GFETs while is also closer to typical Even the long channel terms ΔΝA and ΔμA are lower in VDS=0.3 V values of MOSFETs21-22, 42. Regarding αH, is between ~2.5·10-4 - than VDS=60 mV and this can be explained by the presence of Leff 5·10-3 which is similar to ref. 35 and higher than MOSFETs21-22, 42. in denominators of terms A1, A2 of eqns (5, 7) respectively (See Finally, SΔR2 parameter ranges from ~3.4·10-3 -2·10-2 Ω2/Hz which ESI D, eqns (A19, A26)). Finally at Fig. 5c all three contributions of is much than MOSFETs21, 42 but this is reasonable since the contact normalized LFN (ΔN, Δμ and ΔR) are shown. ΔN effect is resistance Rc of the measured GFETs is more intense than Si responsible for the M-shape which becomes less deeper at higher devices. drain voltage due to non-homogeneous channel condition35. Δμ effect contributes near CNP35 for both high and low drain voltages while ΔR is negligible near CNP but has a strong impact at higher Conclusions gate voltages where Rc is also dominant. Away from CNP, ΔR In conclusion, a comprehensive physics-based analytical LFN contribution is similar for both drain voltage levels while at CNP, model for short channel SL GFETs is proposed in the present study where ΔR is anyway negligible, its level is much lower at low drain which is proved to be very consistent. This model extends the voltage value. The later can by justified in terms of eqn (10) since model derived in ref. 35 in order to deal with effects that are Vcs,d tend to 0 at low drain voltage and near CNP operating important when gate is scaled down, i.e. in high frequency conditions. devices. In particular, VS can contribute strongly to LFN at higher The proposed analytical LFN model is validated with experimental electric fields and short gate lengths. The analysis of VS effect and data from devices under test46-49. In Fig. 6 we show the normalized the way it affects drain current and consequently LFN, is total LFN SIDf/ID2 vs. effective gate voltage VGEFF for the devices thoroughly examined. Carrier number and mobility fluctuation whose transfer characteristics are presented in Fig. 3. More mechanisms are the main contributors to LFN and VS particularly, the LFN for the A300 GFET with L=300 nm is shown phenomenon affects both of them. At low drain voltage regime, in Fig. 6a and 6d, the LFN for the A200 GFET with L=200 nm is VS effect contribution is negligible since longitudinal electric field shown in Fig. 6b and 6e and the LFN for the A100 GFET with L=100 is much lower than critical field but for higher drain voltages, VS nm is shown in Fig. 6c and 6f. Upper plots correspond to the LFN effect becomes significant and as a result, leads to a reduction of results from the highest and lowest drain voltage while lower LFN especially at CNP. The proposed model is validated with novel plots to the rest of the drain voltages available. LFN data is experimental data from CVD grown GFETs46-49 at three different represented by round markers while model is shown with solid short channel lengths (L=300, 200, 100 nm) and in every case it lines. For comparison, long channel model proposed in ref. 35 is accurately captures the reduction of LFN at higher drain voltages, also shown in dashed lines. In fact, this long channel model equals something presented for the first time. At lower drain voltages, it to the sum of ΔΝA, ΔμA and ΔR terms of eqns (5,7, and 10), coincides with the long channel model proposed in ref. 35. ΔN respectively. In the upper plots, it is clear that the proposed effect models the M-shape of the LFN data near CNP while Δμ model coincides with the long channel model at low electric field effect contributes mainly near CNP. Moreover, a compact model regime where the VS effect is not significant. The agreement with for the effect of contact resistance on LFN is also derived with very the data is consistent apart from some regions away from CNP at consistent results at higher gate voltage regimes where contact n-type regime where the IV model was also not consistent (See resistance is dominant for GFETs. All the equations are solved in a Fig. 3) due to asymmetries of the data. At higher drain voltage compact way which makes the resulting model suitable for circuit region, the proposed short channel model is very accurate. It simulators. In general, the extension of the well-established predicts a reduction of LFN in comparison with long channel model of ref. 35 in order to cover the VS effect contribution to model and this fully agrees with experimental data especially at LFN is of high importance. Graphene is used in RF applications CNP where ΔR is not significant. This phenomenon is analyzed and where the demand for high maximum oscillation frequencies modeled for the first time. In the down plots of Fig. 6, the model makes essential the use of short channel devices and high drain Please do not adjust margins voltage levels. The latter are the device dimensions and operating supported by the Severo Ochoa Centres of Excellence conditions where VS effect becomes important. The VS effect is programme, funded by the Spanish Research Agency (AEI, grant found to reduce significantly the LFN. no. SEV-2017-0706). Experimental Data References Devices fabrication (1) K. S. Novoselov, A. K. Geim, S. V. Morozov, Y. Zhang, S. V. Dubonos, I. V. Grigorieva and A. A. Firsov, Science, The main features of our GFETs structure are the bottom gates 2004, 306, 666. with native oxide. We designed different gate length of 100 nm, (2) A. K. Geim and K. S. Novoselov, Nature Materials, 2007, 200 nm and 300 nm. The channel width is 2x12 μm. 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Bae and E. Pop, Applied Physics Letters, 2010, 97(8), 082112. (40) S. Thiele and F. Schwierz, Journal of Applied Physics, 2011, 110(3), 034506. Please do not adjust margins Supplementary Information for: Velocity Saturation effect on Low Frequency Noise in short channel Single Layer Graphene FETs Nikolaos Mavredakis*a, Wei Weib, Emiliano Pallecchib, Dominique Vignaudb, Henri Happyb, Ramon Garcia Cortadellac, An- drea Bonaccini Caliac, Jose A. Garridoc, d and David Jiméneza a Departament d’Enginyeria Electrònica, Escola d’Enginyeria, Universitat Autònoma de Barcelona, Bellaterra 08193, Spain b Institute of electronics, Microelectronics and Nanotechnology, CNRS UMR8520, 59652 Villeneuve d’Ascq, France. c Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC, Barcelona Institute of Science and Technology, Campus UAB, Bellaterra, Barcelona, Spain d ICREA, Pg. Lluis Companys 23, 08010 Barcelona, Spain * nikolaos.mavredakis@uab.es A. Supplementary Information: Drift-Diffusion current equation when Velocity Saturation effect is in- cluded. How the integral variable changes from dx to dVc. Eqn (1) of the main manuscript calculates the drain current ID of the device when Velocity Saturation (VS) effect is taken into consideration. It is known that43-45: dV d dV  dVc dV Cq  C u (Eq.A1) E , Ex    ,  , Ec  sat dx dx dx dVc C  where all the above quantities are defined in the main manuscript apart from ψ which is the electrostatic potential. If eqn (A1) is replaced into the effective mobility term of eqn (1) of the main manuscript and if this is then inserted into drain current term of eqn (1) of the main manuscript then the drain current ID is given as:  dV  dV  dV I D  WQgr  WQgr  WQgr (Eq.A2)  dV  dVc dx  dVc  dx  dV  C  2C  dx 1 dV 1   1  q  sat dx   dV  sat dx  Cq  C  1 sat dx Then from eqns (A1, A2) we can end up with eqn (2) of the main manuscript with k=2·e3/(π·h2·u2f) 43-44 where uf is the Fermi velocity (=106 m/s), h the reduced Planck constant (=1,05·10-34 J·s). Bias dependent term g(Vc) which expresses the normalized drain current ID is calculated as43: VDS 3 Vcs  Vcd 3 V Wk gVc   g Vc    k sgn Vcd  Vcd4  sgn Vcs  Vcs4   DS , I D  Vcd   gVc (Eq.A3) Vcs k 3 4C k 2 Leff while graphene charge is given by43-45: Qgr  k 2 Vc2   / k  (Eq.A4) and chemical potential at source and drain as43: C  C 2  2k Ctop VGtop  VGtop 0  Vs ,d   Cback VGback  VGback 0  Vs ,d  Vcs ,d  (Eq.A5) k where α=2.ρ0.e is a residual charge (ρ0) related term, VGtop-VGtop0 and VGback-VGback0 are the top- and back- gate voltage overdrives, respectively. The discrimination between positive and negative VDS defines the sign of electrical field E and consequently the signs of second terms of the right hand of eqn (2) of the main manuscript. In more detail, if VDS>0 then dV<0 and E=-dV/dx>0 (top branch of eqn (2)) while if VDS<0 then dV>0 and E=-dV/dx<0 (bottom branch of eqn (2)). This relation between dx and dVc is very crucial for the calculations of LFN as it will be shown later35. B. Supplementary Information: Leff calculation In the denominator of eqn (3) of the main manuscript, Leff is defined which represents an effective length to take into account VS effect. The thorough procedure of its extraction will be presented below. From eqn (A2) we can take: I D  dV  Cq  2C  dV I   Cq  2C  ID     WQgr   I D dx  D dV    WQgr  dV (Eq.A6) sat dx  Cq  C  dx sat  Cq  C      If we integrate each term of eqn (A6) from S (Source) to D (Drain): Vd I   C  2C  Vd ID L    dV    WQgr  dV q D  (Eq.A7)   Vs sat  Cq  C  Vs Again the sign of VDS defines the sign of the second left hand term in eqn (A7). If VDS>0 then dV<0 and - dV>0 and if eqn (A1) is also taken into account, eqn (A7) becomes: Vd Vcs Cq  C   WQgr  dV W   Qgr dVc Vs Vcd C (Eq.A8) ID   Vd   Cq  2C  Vcs   Cq  2C  L    dV L    dVc  Vs sat  qC  C   Vcd sat  C  On the other hand, if VDS<0 then dV>0 and -dV<0 and similarly to before eqn (A7) becomes: Vd Vcs Cq  C   WQgr  dV W   Qgr dVc Vs Vcd C ID   (Eq.A9) Vd   Cq  2C  V cd   Cq  2C  L    dV L     dVc Vs  sat  Cq  C  Vcs  sat  C  2 Eqn (A8, A9) are identical to eqn (3) of the main manuscript. It is very significant to solve the denominator integrals of the above equations analytically in order to use them in LFN expressions where Leff is present. The positive drain voltage case will be shown while the negative one can be solved similarly. To proceed with the calculation, two cases should be discriminated regarding usat value as described in eqn (4) of the main manuscript; one for Vc0 and bottom sign to Vc<0. C. Supplementary Information: Detailed examination of graphene chemical potential and longitudinal electric field locally in the channel. The behavior of significant quantities such as chemical potential or longitudinal electric field along the channel at different bias conditions regarding gate and drain voltage, can contribute to the understand- ing of the operation of the device and clarify the way that short channel effects such as VS can influence this operation. In Fig. S1, the chemical potential Vc(x) and the longitudinal electric field Ex(x) are pre- sented vs. channel position x at charge neutrality point – CNP (a,d), away from CNP in p-type region (b,e) and away from CNP in n-type region (c, f) for a channel length of L=100 nm for both low and high drain voltage values (VDS=60 mV, 0.3 V). Regarding Vc(x), it can be easily observed that it is almost identical at every position at low drain voltage which indicates a uniform channel while as the drain voltage gets higher this homogeneity is not valid anymore since the fluctuation of Vc(x) is quite large. As far as Ex(x) is concerned, it is much lower than critical field Ec(x) at low drain voltage and this is the reason why VS effect is negligible there while at high drain voltage Ex(x) becomes comparable to Ec(x) and this affects the operation of the device due to the degradation of effective mobility as it is illustrated in Fig. 2c of the main manuscript. Moreover, LFN is also affected as it will be shown in the next section. 3 a) b) c) d) e) f) Fig. S1 Graphene chemical potential Vc(x) (upper plots) and longitudinal electric field Ex(x) (down plots) vs. channel position x, for a, d) VGEFF=0 V (CNP), b, e) VGEFF=-0,12 V and c,f) VGEFF=0,12 V at VDS=60 mV, 0.3 V for W/L=12 μm/100 nm. D. Supplementary Information: Thorough procedure for calculation of local LFN and examination of VS effect on it for both carrier number and mobility fluctuations effect. As described in ref. 35, the LFN methodology applied here considers a noiseless channel apart from an elementary slice between x and x+Δx. This local noise contribution can be represented by a local current noise source with a PSD SδI2n. Without entering into much detail since these are described thoroughly in ref. 35, the PSD of the total noise current fluctuation at the drain side SID due to all different sections along the channel is obtained by summing their elementary contributions SδI2nD assuming that the con- tribution of each slice at different positions along the channel remains uncorrelated 35, 42: L S I 2  , x  L S I 2  , x  1 L S ID   nD dx   GCH 2 R 2 n dx  2  xS I 2  , x  dx (Eq.A12) 0 x 0 x Leff 0 n since with VS effect included we obtain36, 42:   dI 2  V 2   W eff Qgr 2 2 x   x  G R    d   2 2      (Eq.A13) CH   dV   I d    Leff W eff Qgr   Leff    and 4 S I 2 , x   GCH 2 R 2 S I 2 , x  (Eq.A14) nD n Regarding carrier number fluctuation effect, the PSD of the normalized local noise source divided by squared drain current is given in ref. 35 (eqn (4)-pp 10). If this is inserted in eqn (A12) above then the integral below expresses the total LFN PSD normalized with squared drain current at 1 Hz regarding ΔΝ mechanism: 2 SID KT  NT L  e Cq  I D2 f   WL2eff 0  Qgr Cq  C  dx (Eq.A15)   As the LFN model is based on a chemical potential based model43-45, the integral variable should change from x to Vc according to eqn (2) of the main manuscript and this is the point where VS effect enters noise calculations. We will proceed with the case of VDS>0 but the procedure is similar for a negative VDS. After applying eqn (2) of the main manuscript at eqn (A15): SID KT  NT Vcs  e  2  Cq  2   Q 2 L  C  C    C  2C      Qgr      gr eff  q  (Eq.A16) q f     dVc I D2 WL2eff Vcd    Cq  C    kg vc  C  sat  C    Eqn (A16) can be split into two integrals as it is shown below: 2 2 SID KT  NT Vcs  e   Cq  Qgr 2 Leff  Cq  C  I D2 f A  WL2eff V  Qgr   C  C  kg   C dVc  (Eq.A17) cd    q  vc and 2 2 SID KT  NT Vcs  e   Cq    Cq  2C  f B  V  Qgr     dVc (Eq.A18)  Cq  C  sat 2 I D WL2eff cd    C  If Qgr is replaced by eqn (A4) and Cq=k|Vc|35, 43 then we have: 4 KT  NT e2 V SID A1 cs Vc2 gvc Vcd Vc2   / k  C  k Vc  f  dVc with A1  (Eq.A19) I D2 A   CWLeff and SID Vcs Vc2 k Vc  2C A1 f  B1  dVc with B1  (Eq.A20) Total ΔΝ   k V  C 2 B 2 2 I D Vcd usat V   / k c 2 c Leff LFN at 1 Hz is then given: SID SID SID 2 f   2 f A  f B (Eq.A21) I D I D I D2 As far as mobility fluctuation effect is concerned, a similar process is followed where: 5 SID H e L 1 I 2 D f   WL 2 eff 0  Qgr dx (Eq.A22) The integral variable changes from x to Vc according to eqn (2) of the main manuscript and eqn (A22) becomes: H e 1  Qgr 2 Leff  Cq  C    Cq  2C   Vcs SID I 2 D f   WL 2 eff Vcd Q  gr   kgvc  C  sat    C   dVc  (Eq.A23) Similarly as before, eqn (A23) is split into two new integrals: H e 1  Qgr 2 Leff  Cq  C   Vcs SID I 2 D f  A  WL 2 eff Vcd Q  gr   kgvc  C     dVc (Eq.A24) and H e   Cq  2C  Vcs SID 1 I 2 D f  B  WL 2 eff Vcd Q gr  sat  C  c dV (Eq.A25) again If Qgr is replaced by eqn (A4) and Cq=k|Vc|35, 43 then we have: 2 H e V SID I 2 f  A  A2 cs gvc Vcd  C  k Vc dVc with A2  WLeff Ck (Eq.A26) D and SID Vcs k Vc  2C A2 f  B2  dVc with B 2  (Eq.A27) I 2 D  B Vcd  usat V  a / kc 2  Leff Total Δμ LFN at 1 Hz is given: SID SID SID 2 f   2 f  A  f  B (Eq.A28) I D I D I D2 Terms ΔΝB, ΔμB change depending on the above condition. For the first case where usat is constant we take: SID B1 cs V Vc2 k Vc  2C S Vcd V 2   / k f  dVc (Eq.A29)   k V  C 2 B 2 2 I D c c and B 2 cs k Vc  2C V SID S Vcd Vc2   / k f  B  dVc (Eq.A30) I D2 while for the second case where usat is inversely proportional to sqrt(Vc2+a/k) we have: 6 SID V B1 cs Vc2 k Vc  2C f   dVc (Eq.A31)    k Vc  C  2 B 3/2 2 I D N Vcd V 2   / k c and B 2 cs k Vc  2C V SID N Vcd Vc2   / k f  B  dVc (Eq.A32) I D2 a) b) c) d) e) f) Fig. S2 Normalized drain current noise divided by squared drain current referred to 1 Hz, Sδinf/ID2, vs. channel potential x for ΔΝ (upper plots) and Δμ (down plots) noise mechanisms for a, d) VGEFF=0 V (CNP), b, e) VGEFF=-0,12 V and c,f) VGEFF=0,12 V at VDS=60 mV, 0.3 V for W/L=12 μm/100 nm. ΔΝΑ, ΔΝΒ contrib- utors of ΔΝ effect are also shown in upper plots while ΔμA, ΔμB contributors of Δμ effect are also shown in down plots respectively. For simplicity the terms of eqns (A19, A20, A21, A26, A27, A28) from now on will be referred as ΔΝΑ, ΔΝΒ, ΔΝ, ΔμA, ΔμΕ, Δμ respectively. In the case of a negative VDS, the above integrals are transformed depending on eqn (2) of the main manuscript. It is important to mention here that terms ΔΝΑ, ΔμA are identical to those extracted in ref. 35 where the long channel LFN model was presented. Terms ΔΝB, ΔμB deal with the VS effect on LFN and this is clear since usat term is included in eqns (A20, A27). As it was described before, usat is constant for Vc0 where Vcs>Vcd while eqn (A34) refers to the case of VDS<0 where VcdVcs instead of Vcs>Vcd. F. Supplementary Information: IV plots for the rest of the devices similarly to Fig. 3 of the manuscript a) b) c) Fig. S3 Drain current ID vs. back gate voltage overdrive VGEFF, for GFETs with W=12 μm and a) L=300 nm (B300), b) L=200 nm (B200) and c) L=100 nm (B100) at low (left subplot) and high (right subplot) available VDS values (VDS=30 mV, 60 mV, 0.1 V, 0.2 V and 0.3 V). Markers: measured, solid lines: model. 9 G. Supplementary Information: LFN plots for the rest of the devices similarly to Fig. 6 of the manu- script a) b) c) d) e) f) Fig. S4 Normalized drain current noise divided by squared drain current referred to 1 Hz, SIDf/ID2, vs. back gate voltage overdrive VGEFF, for GFETs with W=12 μm and a, d) L=300 nm (B300), b, e) L=200 nm (B200) and c, f) L=100 nm (B100). Upper plots show the highest and lowest available VDS value depending on the GFET. (B300: VDS=60 mV, 0.3 V, B200: VDS=30 mV, 0.2 V, B100: VDS=30 mV, 0.3 V) while down plots show the rest of available VDS values (B300: VDS=0.1 V, 0.2 V, B200: VDS=60 mV, 0.1 V, B100: VDS=60 mV, 0.1 V, 0.2 V). Markers: measured, solid lines: model, dashed lines: long channel model from ref 35. 10