Bias dependent variability of low-frequency noise in single-layer graphene FETs

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 (ΔN) due to trapping/detrapping process and the Hooge parameter (αH) for the mobility fluctuations effect (Δμ). ΔN 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 Δμ 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 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 influence the performance. This not fully advanced GFET process growth might also cause pivotal inconsistencies affecting the scaling laws in GFETs of the same process.

Fig. S1a depicts the equivalent capacitive circuit of the CV-IV chemical potential based model [56][57][58] where quantum capacitance (C q ) is the derivative of graphene charge Q gr and chemical potential V c (x) corresponds to the voltage drop across C q at channel position x. A linear relationship is considered between C q and V c . (C q =k| V c |) where k=2e 3 /(πh 2 u 2 f) [56][57][58] where u f is the Fermi velocity (=10 6 m/s), h the reduced Planck constant (=1,05·10 -34 J.s). V c (x) equals to the potential difference between the quasi-Fermi level and the potential at the CNP, as illustrated in the energy dispersion relation scheme of graphene in the top drawing of Electronic Supplementary Material (ESI) for Nanoscale Advances. This journal is © The Royal Society of Chemistry 2020 Fermi potential at position x, which equals to zero at the Source and V DS at the Drain end respectively. Bias dependent term g vc defined in the main manuscript is calculated as 56 while the drain current expression is 56-58 : Graphene charge is given by 56-58 : and chemical potential at Source and Drain as 56-58 : As thoroughly explained in the main manuscript, the integration of local LFN noise sources across the channel leads to the calculation of the total LFN PSD and its variance. In order to obtain an analytical compact solution based on the recently established chemical potential based IV model 56-58 , a change of integral variable occurs from length x to chemical potential V c : The fluctuations producing LFN are always slight and as a result, the analysis of the propagation of the noise sources to the voltages or currents at the contact terminals reduces to linear analysis. Therefore, the principle of superposition can be used for adding the effects of the local noise sources along the channel 33 . These local fluctuations can be modeled by adding a random local current noise source δI n with a PSD SδI 2 n as shown in the equivalent noise subcircuit 53-54 in Fig.   S1b. The local fluctuations propagate to the terminals resulting in fluctuations of the voltages and currents around the DC operating point.   Fig. S2c and S2d for the same devices and it is apparent that these two quantities are not correlated which means that WLS ID f/I D 2 variance is mostly related with the mechanisms that produce LFN, as proved in the main manuscript.

C. Supplementary Information
contributors, respectively, correspond to the long channel terms named as Var [ΔΝA], Var [ΔμA] in this section. Taking into consideration the effective length L eff which accounts for the degradation of I D because of VS effect, the following expressions are derived: where N= hΩu f /e and hΩ is phonon energy used as an IV model parameter 54-55 which is considerable only at high electric field region. Thus, eqns (S8-S9) become: which can analytically be solved as:

Var[Total]= Var[ΔΝ]+Var[Δμ]
(S12) Eqn (S12) is equivalent to eqn (14) of the main manuscript. Despite the fact that in the LFN mean value model, the VS related terms ΔΝB, ΔμB are subtracted from ΔΝA, ΔΝB respectively 54-55 , variances are always summed since the variance of the difference of two random variables equals to the sum of the variances of these variables.
The graphical representation of LFN variance contributors derived above (eqns S6-S12) is (S13) Variance can get into the sum since local noise sources are considered independent while squared integral variable leads to a double integral notation. Since Λ 2 (x) is function only of one variable (x), the double integral turns into a single integral multiplied with L * . Eqn (S13) proves the validity of eqn (6) of the main manuscript. The same procedure can be applied for Δμ effect.

E. Supplementary Information: ~(g m /I D ) 4 LFN Variance Model
In Fig. 5a    In Fig. S4 G. Supplementary Information: Extraction of N tr , α H , N tcoeff and N Table S1 while their variance and ln-mean value are also estimated.

F. Supplementary Information: Analysis of LFN variance locally in the channel
As it was indicated in the main manuscript, N tcoeff =Var [N tr