Model for the 1/f noise behavior of graphene

Several experimental results have been reported on the behavior of the 1/f noise power spectral density S of graphene as a function of charge density. For monolayer graphene a V, M or ȿ shape have been observed in different conditions, while in the case of bilayer graphene only a V shape (becoming an M shape for high carrier densities) has been measured. Here we present two approaches we have developed [1] to explain these results. The origin of the 1/f noise is attributed to charges moving into and out of traps located near the graphene sheet. These fluctuations modify the concentrations n and p of the electrons and holes in the sample and therefore cause a variation of the current at the terminals through the relation / ( /) / c c A ' ' i I n n dx dy A ³ , where nc =n+p, A is the area of the sample and I is the average current at the terminals. In our first approach, ȴnc is related to the fluctuation of the trap occupancy through the variation of the potential profile U and by enforcing charge neutrality. Taking into account the distribution of the trap relaxation times, we obtain, for the power spectral density: 2 2 / / ( ) ( / ( )) c c S I Af a a n J K , where / a nU c c w w , / a nU w w n , nn=nͲp, the coefficient Ș depends on the characteristics of the traps, and Ȗ is usually 1. In the alternative approach, we express the integral of ȴnc as the sum of the variations ȴN and ȴP, determined by the trapping event, of the number N of electrons and of the number P of holes in the sample. These variations can be analytically determined as a function of N and P enforcing the neutrality of charge and the mass-action law. Including the effect of the distribution of the trap relaxation times, we obtain: 2 2 / '/ ( ) (( ) / )c S I Af N P n J ' ' K . The quantities ac , a, N and P are computed exploiting the dispersion relations of graphene, and the results obtained with the two approaches are very close. The curve of S as a function of the charge density has an M shape both for monolayer and for bilayer graphene (Figs. 1-2), but for bilayer graphene this behavior extends over a larger range of charge density values, as a consequence of the quite different dispersion relations. The curves we have obtained vanish in the Dirac point, where the variation of trap occupancy has an opposite effects on electrons and holes and thus ȴnc is null. Finally, we have taken into account the effect of potential disorder, averaging the value of S/I2 over a Gaussian distribution of the potential U (Figs. 3-4). After averaging, in monolayer graphene the behavior of S for weak potential disorder has a V shape around the Dirac point and an M shape over a larger charge density range, but has a ȿ shape (where the minimum at the Dirac point disappears) when the potential disorder increases. Instead, in bilayer graphene the simultaneous effect of the different dispersion relation around the Dirac point and of the larger screening on the random impurities which generate the potential disorder makes a ȿ shape very unlikely to observe, in agreement with experimental data.