Theoretical Comparison between the Flicker Noise Behavior of Graphene and of Ordinary Semiconductors

Graphene is a material of particular interest for the implementation of sensors, and the ultimate performance of devices based on such a material is often determined by its flicker noise properties. Indeed, graphene exhibits, with respect to the vast majority of ordinary semiconductors, a peculiar behavior of the flicker noise power spectral density as a function of the charge carrier density. While in most materials flicker noise obeys the empirical Hooge law, with a power spectral density inversely proportional to the number of free charge carriers, in bilayer, and sometimes monolayer, graphene a counterintuitive behavior, with a minimum of flicker noise at the charge neutrality point, has been observed. We present an explanation for this stark difference, exploiting a model in which we enforce both the mass action law and the neutrality condition on the charge fluctuations deriving from trapping/detrapping phenomena. Here, in particular, we focus on the comparison between graphene and other semiconducting materials, concluding that a minimum of flicker noise at the charge neutrality point can appear only in the presence of a symmetric electron-hole behavior, a condition characteristic of graphene, but which is not found in the other commonly used semiconductors.


Introduction
In the actual operation of electronic devices, random fluctuations (the "noise") are always superimposed to the deterministic electrical quantity (the "signal") which conveys the desired information. Such fluctuations derive from underlying microscopic phenomena and in sensors they may limit the achievable sensitivity or, as in the case of fluctuationenhanced sensing [1], contribute to the improvement of sensor performance, in particular in terms of selectivity. Several forms of random electrical fluctuations may exist in electronic devices: shot noise, thermal noise, generationrecombination noise, burst noise, and 1/f (flicker) noise, just to cite the main ones [2][3][4][5][6][7][8][9]. They differ for their physical origin and for the dependence of their power spectral density on the physical parameters characterizing the device operation, such as temperature, frequency, bias current, charge density, and material resistivity.
In particular, understanding the physical mechanisms that govern the intrinsic noise of electronic devices, and in particular of sensors, can significantly help the designer in the choice of the best material for their fabrication and of the optimal bias point for their operation.
Here, we focus on 1/f (flicker) noise, so called from the behavior of its power spectral density, which is inversely proportional to the frequency. This type of noise mainly originates from trapping and detrapping of charge carriers, due to impurities (traps) located inside or near the channel where the device current flows. In general, its amplitude is therefore proportional to the trap concentration.
The power spectral density of the 1/f current noise is often approximated with the empirical Hooge formula [10]: where α H is Hooge's constant, I is the mean bias current, N is the number of carriers in the device, and f is the frequency. According to this formula, 1/f noise should be greater when the number N of carriers in the device is lower. This is actually what is generally observed in ordinary semiconductors [11,12], such as silicon or gallium arsenide. However, a different behavior has been observed in measurements performed on graphene samples. Monolayer graphene is a recently isolated material [13] made up by a planar hexagonal lattice of carbon atoms. It is a semiconductor with a zero energy gap [14][15][16][17][18][19], even though an energy gap can be introduced, for example, by lateral confinement [20][21][22], strain [23,24], doping [25][26][27], functionalization [28,29], or introducing a lattice of antidots in the material layer [30,31]. Its dispersion relations around the degeneration points between conduction and valence bands (the so-called Dirac points, i.e., the charge neutrality points) are linear and thus in graphene charge carriers present a zero effective mass. Since its isolation from graphite, it has been the focus of a large research effort (which has more recently extended to a wider family of two-dimensional materials [32][33][34]), because it possesses very attractive electrical, thermal, optical, and mechanical properties [35][36][37][38][39][40][41][42][43][44][45][46][47]. Moreover, graphene exhibits very uncommon physical phenomena [48][49][50], typical of relativistic mechanics, because its effective mass transport equation coincides with the Dirac equation [18,51,52] (the wave equation which describes relativistic spin-1/2 particles). Bilayer graphene is instead made up of two coupled graphene layers [14,15,[53][54][55][56]. Bilayer graphene with Bernal stacking has nonlinear dispersion relations with a zero gap; however, an energy gap can be easily induced applying an orthogonal electric field, which introduces a shift between the electrochemical potentials of the two layers.
Graphene is of particular interest for the implementation of sensors because of its quasi-two-dimensional nature, which leads to a very large surface-to-volume ratio: since the interaction of a sensor with an analyte mainly occurs as a result of surface adsorption, while conduction is a bulk property, a large surface-to-volume ratio will lead to a large relative variation of resistance when the sensor interacts with the analyte.
An interesting theory [71] exploited the electrostatic screening of the trapped carriers and the peculiar properties of the graphene band structure to explain the observed fea-tures. We have developed a different approach [72] that leads to analogous results, exploiting a model based on the conservation of charge neutrality and on the mass action law (which has to be satisfied if the main fluctuations in flicker noise are slow compared to the generation-recombination times of carriers).
Here, we extend this model to the case of generic semiconductors, and we use it to explain the origin of this discrepancy between the behavior observed in graphene samples and that typical of common semiconductors. In particular, we show that a minimum of the 1/f noise at the charge neutrality point is expected in materials (such as graphene) where electrons and holes have an identical mobility, in the presence of a quite low potential disorder. This is actually the case of suspended graphene or of bilayer graphene (where the electrostatic effect of randomly located charged impurities is strongly screened). In common semiconductors, where the electron and hole behavior is in general different [50,73], this local minimum in the flicker noise power spectral density is much less apparent and is totally suppressed by the inevitable presence of potential disorder.

Simulation Model
In order to relate the (microscopic) phenomena, consisting in the motion of the charge carriers and their capture and reemission by the traps to the (macroscopic) currents at the terminals of the device, we can use the Ramo-Shockley theorem [74,75] (then generalized by Pellegrini with the electrokinematics theorem [71,76,77]). We assume the scattering phenomena to take place on timescales much faster than the considered trapping/detrapping events, in such a way as to be able to actually define a drift current. The current at the terminals is given by where q is the modulus of the elementary charge; v ! n and v ! p are the drift velocities of electrons and holes, respectively; F ! ðx, yÞ is the electric field that would be produced in the point with coordinates ðx, yÞ by a unit potential applied to the electrode for which we want to compute the current (while the other electrode is grounded and in the absence of mobile charges) [74]; E ! ðx, yÞ is the electric field actually present in the device; μ n (μ p ) and n (p) are the mobilities and surface densities of electrons (holes); and A is the area of the device, equal to the product of the width W and the length L. Since (and thus along E ! ðx, yÞ) represents the weight with which the carrier motion in each point of the device contributes to the total current, thus being analogous to the sensitivity coefficient introduced by Vandamme et al. [78][79][80]. Since for the purpose of the present study the relative weight of the contributions from the different regions of the device is not essential, we will assume a constant F ! ðx, yÞ (equal to 1/L and parallel to E ! ðx, yÞ), i.e., a uniform sensitivity coefficient.
In this approximation, the instantaneous current at the contacts is given by [72] In the case of three-dimensional devices, this integral has to be replaced with an integral over the volume V, and n and p become volumetric densities (instead of surface densities). The average current can be written as If, as it is usually the case [71], the effect of trapping and detrapping events on carrier densities is by far prevalent with respect to that on the mobilities and on the electric field, we can express the current fluctuations due to carriers being trapped and detrapped as where Δn and Δp are the fluctuations of electron and hole concentrations, respectively. Therefore, the ratio of the current fluctuation to the average current is equal to where the total number of electrons and holes in the area A have been defined as N = Ð A ndxdy and P = Ð A pdxdy, respectively, while their variations as an effect of the trapping-detrapping phenomena have been indicated with ΔN and ΔP.
In order to compute ΔN and ΔP, we use the following approach [72] (which differs from that used in Ref. [71] but leads to similar results). Let us consider a single trap, and let us define χ the number (0 or 1) of electrons in the energy level of the impurity affected by the trapping events. In the following, we will indicate with Δ the variations with respect to the values with no trapped electron (i.e., when no electron is trapped all the variations Δ are zero). When an electron or a hole is trapped, the number of the corresponding charge carriers varies instantaneously by one, but, over time intervals exceeding the thermal generation-recombination time, the mass action law and electroneutrality must be satisfied. Therefore, for a generic variation Δχ of the number χ of electrons in the trap (Δχ = χ, since Δχ = 0 if no electron is trapped, while Δχ = 1 if one electron is trapped), the variations ΔN and ΔP of carriers in the device have to satisfy The first relation of Equation (7) can be obtained differentiating the relation PN = c (i.e., the mass action law with both members multiplied by the square of the area A). (Equivalently, it can be obtained noting that the mass action law has to be valid both before and after the examined trapping event. Therefore, both ðΔP + PÞðΔN + NÞ = c and PN = c have to be satisfied; subtracting the two equations and neglecting the second-order term ΔPΔN with respect to the other ones, the first relation of Equation (7) is obtained.) The quantity c is a constant which generally depends on the type of semiconductor and on the temperature, and thus, it does not change as a consequence of trapping events. Even in the case of graphene, in which c depends also on the position of the Fermi level [81], we can assume its variation due to a trapping event to be negligible.
The second relation of Equation (7) enforces the electroneutrality of the device: assuming that the overall device, including the bias electrodes, was neutral before the examined event, the total change of charge deriving from the trapping event has to be zero: qΔP − qðΔN + ΔχÞ = 0 (see Figure 1), where we have included the variations in the number of free holes, of free electrons, and of electrons captured by the trap, which all contribute to the total charge.

Journal of Sensors
The system of Equation (7) has the following solutions: In these solutions, the quantities N and P appear only through their ratio; therefore, the exact choice of the area where N and P are evaluated is irrelevant.
As can be seen from Equations (7) and (8), the necessity to satisfy both the mass action law and electroneutrality uniquely determines the values of the variations ΔN and ΔP deriving from a trapping event: these electron and hole fluctuations are therefore fully correlated.
Substituting these expressions into Equation (6), we obtain Δi Therefore, for a single trap, we have (S I is the flicker noise power spectral density): where, neglecting the contribution of the average value, S χ has a Lorentzian dependence on frequency, with a characteristic relaxation time (since Δχ is a random telegraph signal) [82]. If we suppose the many traps present in the device to be reciprocally independent, the overall noise spectrum can be obtained summing up their spectra. The combination of the effects of traps with properly distributed time constants leads then to 1/f noise [83,84]: with η being a coefficient which depends on the concentration, distribution, and properties of the traps, while γ is a number close to 1.
We will report the flicker noise power spectral density as a function of n − p (i.e., the charge density divided by −q), which in actual experiments is the quantity that can be adjusted by tuning the bias voltage of a gate capacitively coupled to the device.
From Equation (8), we can observe that if the electron concentration is much larger than that of holes, i.e., when N ≫ P, we have ΔN≈−Δχ while ΔP ≈ 0, i.e., the variation in the trap charge is completely screened by electrons. In a similar way, when the hole concentration strongly dominates (i.e., P ≫ N), ΔP ≈ Δχ and ΔN ≈ 0, which means that the trap charge variation is completely screened by holes. In intermediate conditions, the variation in the trap charge is screened by variations of both electron and hole concentrations. In particular, at the neutrality point (when N = P), the trap charge is screened for a half by a variation of the hole number and for the other half by an opposite variation of the electron number: ΔN = −Δχ/2 and ΔP = Δχ/2.
From Equation (11), we observe that the flicker noise power spectral density S I vanishes when μ p P − μ n N = 0. The behavior of S I is symmetrical around this point if μ n and μ p coincide; otherwise, such a symmetry is absent. If μ n = μ p , the point in which S I vanishes and around which S I is symmetrical is the charge neutrality point (where N − P = 0); this condition is obtained for a Fermi energy corresponding to the middle of the gap between the conduction and valence bands if such bands are symmetric.
The quantities N and P depend, through the energy dispersion relations, on E F , i.e., on the relative position of the Fermi energy with respect to the local value of the potential energy. However, the nonuniform distribution of charged dopants and impurities (including the randomly located charged traps themselves) introduces a potential disorder (i.e., a random spatial variation of the potential energy) which can substantially alter this result. In order to introduce the effect of this random spatial energy variation in our calculations, for each value E F of the Fermi energy we average our results over a Gaussian distribution of energies around E F : Journal of Sensors represents a normalized Gaussian distribution with null average and standard deviation σ.
The standard deviation σ of this Gaussian represents an estimation of the effect on the potential profile, and thus on the relative position E F of the Fermi energy with respect to the potential energy, of the random charged impurities which represent the sources of the potential disorder. Therefore, the value of σ does not depend only on the "strength" of the disorder sources but also on the electrostatic screening efficiency of the material, which can be estimated through the derivative ∂ðn − pÞ/∂E F (or from the quantum capacitance, which is proportional to it [45]). If this derivative is larger, the same charged impurities induce a smaller variation of E F (and thus a smaller σ), since this smaller variation of E F is sufficient to screen the electrostatic effect of impurities through an opposite variation of the mobile charge density −qðn − pÞ.
When including potential disorder, we will report the flicker noise power spectral density as a function of hn − pi, i.e., n − p averaged over the same Gaussian distribution of energies: The average over the potential disorder (see Equation (12)) decreases the dependence of the noise spectrum on the charge density, with a smoothing effect which increases with the disorder strength.

Graphene
First of all, let us analyze the case of monolayer and bilayer graphene (with Bernal stacking). In this material, the electron and hole bands are approximately symmetric and μ n = μ p . Therefore, Equation (6) becomes Analogous simplifications can be performed in Equations (9)-(12); for example, Equation (11) becomes For graphene, we compute the carrier concentrations n and p by integrating the product of the density of states and of the occupation function over the whole energy range: where DOS is the density of states and f is the Fermi-Dirac occupation function for electrons (therefore, 1 − f is the occupation function for holes). The DOS, in turn, depends on the dispersion relations of monolayer graphene or bilayer graphene, i.e. [14,18,53,54]: where κ is the difference between the wave vector and the Dirac point, γ = ℏv F (with ℏ the reduced Planck constant and v F = 8:73 × 10 5 m/s the Fermi velocity of graphene), γ 1 = 0:39 eV is the graphene interlayer coupling, and Δ (the term which is responsible of the possible band gap opening) is the difference between the on-site energies in the two layers, which is approximately proportional to the carrier concentration n − p (more in detail, Δ ≈ 1:4 × 10 −18 eV m 2 × ðn − pÞ). In Figures 2-5, we report some results obtained for graphene at 300 K, without potential disorder averaging. In Figures 2 and 4, we report the quantities ΔP/Δχ and ΔN/Δχ (evaluated from Equation (8)) as a function of n − p (for monolayer and bilayer graphene, respectively). As in the general case, the charge variation in the traps is mainly screened by holes when the holes are the dominant carriers, while it is mainly screened by electrons when the In Figures 3 and 5, we report the quantity ðS I /I 2 ÞðAf γ /ηÞ (evaluated from Equation (11)) as a function of n − p (for monolayer and bilayer graphene, respectively). Since in graphene μ n = μ p , the behavior of the flicker noise power spectral density is exactly symmetrical with respect to the neutrality point. In particular, in the charge neutrality point (Dirac point), where ΔN = −Δχ/2 and ΔP = Δχ/2, the fluctuation of the current (and therefore the noise power spectral density) completely vanishes (see Equation (16)). Comparing Figures 3 and 5, we notice that the dip around the Dirac point is wider in bilayer graphene than in monolayer graphene, as a consequence of their different dispersion relations.
In Figures 6 and 7, we show analogous results obtained averaging the spectrum of monolayer ( Figure 6) and bilayer (Figure 7) graphene at 300 K over potential disorder. In particular, we report the value of ðhS I i/I 2 ÞðAf γ /ηÞ (i.e., the integral appearing in Equation (12)) as a function of hn − pi (i.e., the charge density divided by the electron charge,    Journal of Sensors averaged according to Equation (14)). Both for monolayer and bilayer graphene, we show the results for the following four different values of σ: 10, 20, 30, and 40 meV. In both cases, we observe that the minimum of the spectrum at the charge neutrality point tends to disappear increasing the strength of the disorder: therefore, the behavior of the flicker noise power spectral density evolves from an "M"-shaped one to a "Λ"-shaped one. However, as we can deduce from the corresponding values of ∂ðn − pÞ/∂E F (which in the charge neutrality point is equal to about 7 × 10 12 cm −2 eV −1 in monolayer graphene and to about 44 × 10 12 cm −2 eV −1 in bilayer graphene), the screening of potential fluctuations is about six times larger in bilayer graphene than in monolayer graphene. This is a consequence of the different dispersion relations of the two materials, with bilayer graphene characterized by flatter bands near the Dirac points [50]. Therefore, for similar impurity distributions, values of σ should be considered for monolayer graphene that are approximately six times larger than those considered for bilayer graphene. As a consequence, for realistic values of disorder, monolayer graphene does not usually exhibit a minimum of the flicker noise power spectral density at the charge neutrality point.
On the contrary, in sufficiently clean samples of bilayer graphene, an "M"-shaped behavior is often observed, with a minimum at the Dirac point. A similar "M"-shaped behavior has been observed also in the case of suspended monolayer graphene, for which potential disorder is much weaker than for monolayer graphene on a substrate.

Ordinary Semiconductors
Let us now move on to the case of ordinary semiconductors, such as silicon and gallium arsenide, for which a minimum of the flicker noise power spectral density at the charge neutrality point has never been experimentally observed. Also in this case, we can adopt the numerical procedure described in Section 2, substituting (in the case of 3dimensional channels) areas with volumes. In order to simplify the calculations and to reach general results, we neglect the details of the band structure of the materials and, assuming a constant effective mass, we use the semiclassical expressions for the carrier concentrations [85,86]: where E C is the conductance band minimum, E V is the valence band maximum, k B is the Boltzmann constant, and T is the absolute temperature.
In particular, we perform our calculations using the material parameters for silicon and for gallium arsenide.
In Figures 8 and 9, we report the results obtained for silicon at 300 K without potential disorder averaging. More in detail, in Figure 8, we show the ratios ΔP/Δχ and ΔN/Δχ (evaluated according to Equation (8)) as a function of n − p, while in Figure 9, we report the behavior of the quantity ðS I /I 2 ÞðAf γ /ηÞ (evaluated according to Equation (11)) as a function of n − p.
In Figures 10 and 11, we report the analogous results for gallium arsenide at 300 K without potential disorder averaging.
As we see from the reported parameters, contrary to graphene, these semiconductors have quite different mobilities for electrons and holes. This is the reason of the clear asymmetry observed in the results, with a larger noise spectrum in the region where transport is dominated by electrons, i.e., the carriers with higher mobility. We can also observe that the asymmetry is stronger in gallium arsenide, for which the difference between electron and hole mobility is larger. Due to the difference in mobility, the spectrum does not vanish any more at the charge neutrality point (where n − p = 0) but where μ p P − μ n N = 0. Moreover, with respect to graphene (for which, in the absence of disorder, this vanishing point presented a sharp reduction of the spectrum with respect to both the hole-dominated and the electron-dominated conduction regions), here, due to the spectrum asymmetry, the minimum point is much less prominent. Indeed, especially in the case of gallium arsenide, it is nearly indistinguishable from the hole-dominated region, where the spectrum is already very low.
Therefore, when the effect of potential disorder is introduced, by averaging over a Gaussian energy distribution with standard deviation σ according to Equation (12), the local minimum of the spectrum completely disappears and the spectrum exhibits a "Λ" shape, with a maximum centered in the electron-dominated region, where the mobility is larger. The behavior of ðhS I i/I 2 ÞðAf γ /ηÞ (i.e., the integral appearing in Equation (12)) as a function of hn − pi (given by Equation (14)) is reported in Figures 12 and 13, at 300 K, for silicon and for gallium arsenide, respectively, for four different values of σ: 10, 20, 30, and 40 meV. The described behavior is observed for both semiconductors but is even more apparent in the case of gallium arsenide, for which a minimum in the spectrum was already hard to recognize in the absence of disorder.
The fact that in a real three-dimensional material it is difficult, if not impossible, to modulate the position of the Fermi level uniformly in the whole volume, further prevents a perfect balance of electron and hole current fluctuations in the overall device and thus makes it impossible to experimentally observe a flicker noise cancellation analogous to that described for graphene.

Conclusions
Exploiting a model based on charge neutrality and on the mass action law, we have compared the flicker noise behavior of graphene and of more common semiconductors, such as silicon and gallium arsenide. We have concluded that a minimum of the flicker noise power spectral density can be  Journal of Sensors observed at the charge neutrality point only in materials where electrons and holes have a highly symmetric transport behavior, in particular have approximately the same mobility (the case of graphene), and only if the potential disorder is low. When these conditions are satisfied, the opposite variations of electron and hole concentrations induced by trapping phenomena reciprocally cancel, leading to no current fluctuations. Instead, in the presence of electron/hole asymmetry (the case of ordinary semiconductors) or of a significant degree of potential disorder, such a minimum disappears. We believe that this analysis, clarifying the mechanisms that cause or prevent a reduction, at proper bias conditions, of the intrinsic device flicker noise, can be useful for the design of low-noise devices and in particular of highsensitivity sensors.

Data Availability
The numerical data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
The authors declare no conflict of interest.