Pati S.K., Enoki T., Rao C.N. R. (eds.) Graphene and Its Fascinating Attributes

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December 8, 2010 19:26 World Scientiﬁc Review Volume - 9in x 6in Chap13

204 A. N. Pal and A. Ghosh

where

V 1

(WL)Q



(x, E)τ

(x)

1+ω

(x)

(E)(1 − f

(E))dxdE (6)

V 2

(WL)Q



2µ

avg

Σ(x)

(x, E)τ

(x)

1+ω

(x)

(E)(1 − f

(E))dxdE

(7)

V 3

(WL)



avg

(x)

(x, E)τ

(x)

1+ω

(x)

(E)(1 − f

(E))dxdE (8)

The ﬁrst term arises entirely due to number ﬂuctuation, the second term

describes the joint eﬀect of both number and mobility ﬂuctuation, and the

ﬁnal term represents the mobility ﬂuctuation alone.

Number ﬂuctuation: For a uniform trap distribution, the integral over

E of n

(E)(1−f

(E)) is approximately k

). The limits x1and

x2 are such that all the traps are taken into account, in our experimental

frequency range. The ﬁrst term can be calculated by integrating over τ as

V 1

(WL)Q

)

(

)(9)

Using typical values of n

), we get S

V 1

∼ 10

−30

/Hz,whichisfar

lower compared to the experimental values obtained in recent experiments.

Moreover, S

V 1

∝ 1/n

, irrespective of the number of graphene layers in

the device. However, in recent noise experiments in graphene it has been

observed that the conductivity noise magnitude in single and multilayer

device behaves oppositely with gate voltage. In case of SLG devices the

noise magnitude goes down with increasing carrier density, while it increases

in case of multilayer devices. These inconsistencies indicate that the number

ﬂuctuations to be only of minor importance in graphene.

Mobility ﬂuctuation: Calculating the integral exactly in (8) is not

straightforward. From (8), we ﬁnd the mobility ﬂuctuation term mainly

depends on average mobility µ

avg

and Σ. Hence,

V 3

∼ Σ

avg

(10)

December 8, 2010 19:26 World Scientiﬁc Review Volume - 9in x 6in Chap13

Noise in Graphene Transistors 205

To get the density dependence of S

V 3

we should calculate the density de-

pendence of Σ and µ

avg

. Within the semiclassical model of conductivity

Σ(x) can be expressed as

Σ(x)=

(



)

imp

(11)

where τ

as the scattering time arising due to the Coulomb scattering only

andwehavetakenδQ ∼ n

imp

, the density of charge impurities trapped

at the SiO

interface. Using the expressions for the scattering rates, cal-

culated for both single

52,53

and bilayer/multilayer graphene

for Coulomb

scattering, it can be shown from (9) that Σ(x) is independent of density for

both single and multilayer graphene.

Hence, the density dependence of

noise only arises due to that of µ

avg

only. The important fact is that the

average mobility of graphene transistors comes from all scattering mech-

anisms, whereas only Coulomb scattering is responsible for the mobility

ﬂuctuations. So one can extract the dependence of µ

avg

from the den-

sity dependence of the Drude conductivity σ = neµ

avg

.Wefocusonthe

most relevant experimental situation, where q

/2k

> 1. Recently,

52–54

the density dependence of σ has been calculated within Thomas-Fermi ap-

proximation for both single and bilayer/multilayer graphene for all possible

scattering mechanisms. In Table 1, a comparison of density dependence of

σ for various scattering mechanisms has been summarized.

Table 1. Dependence of σ for various scattering

mechanisms in experimentally relevant situation

where q

/2k

> 1.

SLG BLG/MLG

Bare Coulomb Scattering σ ∼ nσ∼ n

Screened Coulomb σ ∼ nσ∼ n

Short-range scattering σ ∼ n

σ ∼ n

Using Table 1, one can write µ

avg

∼ n

,where−1 <β<0forSLG

and 0 <β<1 for BLG and MLG. Hence, the contribution from mobility

ﬂuctuations S

V 3

∼ n

2β

, which predicts that in case of SLG the noise

magnitude always decreases with density, whereas, for BLG, the noise power

spectral density increases with increasing density. This opposite behavior is

a direct consequence of the band structures. In case of multilayer graphene

we expect noise to behave in the same way as BLG because of their similar

parabolic band structure. Noise can form an excellent probe to detect SLG

from other forms of graphene.

December 8, 2010 19:26 World Scientiﬁc Review Volume - 9in x 6in Chap13

206 A. N. Pal and A. Ghosh

3. Experimental Section

3.1. Sample preparation and characterization

(a)

(b)

Fig. 2. (a) Raman spectra for SLG, BLG, FLG, and MLG showing the characteristic

G and 2D peaks (b) Gate voltage characteristics of graphene devices: for comparison,

the ratio of resistivity (ρ) and the resistivity at the CNP (ρ

) are plotted as a function

of (V

− V

) at T = 100 K. The inset shows the optical micrograph and outline of a

t ypical graphene device. The scale bar is 5µm.

Graphene ﬂakes were prepared on 300 nm SiO

on n

doped silicon

substrate (the backgate) by micromechanical exfoliation of highly oriented

pyrolytic graphite (HOPG). All ﬂakes were characterized by Raman spec-

troscopy,

and subsequent atomic force microscopy indicated the FLG and

MLG devices in the present case to consist of 3 −4and∼ 14 layers, respec-

tively. 40 nm gold (99.99%) contacts were deﬁned using standard electron

beam lithography technique. To keep the disorder level comparable, we

have cleaned the Si/SiO

substrates in standard RCA solution (1:1:5 so-

lution of NH

OH (ammonium hydroxide) + H

(hydrogen peroxide) +

O (water) at 75 or 80

C) followed by acetone and isopropyl alcohol

prior to the graphene deposition. Figure 2(a) shows the characteristic Ra-

man spectra for diﬀerent graphene ﬂakes where the intensity ratio of the G

peak to the 2D peak can be seen to increase with increasing layer number.

Figure 2(b) shows the gate voltage characteristics of the devices. In all the

cases, charge neutrality point (CNP) was shifted to a ﬁnite gate voltage

due to the intrinsic doping (see Fig. 3 also), primarily arising from the im-

purities inside the SiO

. Hence in order to compare the inﬂuence of gating

we have plotted the ratio of resistivity to that at the CNP as a function of

December 8, 2010 19:26 World Scientiﬁc Review Volume - 9in x 6in Chap13

Noise in Graphene Transistors 207

− V

), where V

is the backgate voltage at CNP. Figure 2(b) clearly

demonstrates the ambipolar transistor action in all devices, although the

eﬀect of gating decreases with increasing layer numbers. Mobility of SLG,

BLG, FLG, and MLG devices was calculated to be 1100, 1160, 2450, and

1200 cm

−1

respectively.

(b)

(a)

(c)

(d)

Fig. 3. The resistance and the normalized noise power spectral density (N

) as func-

tions of bac k gate voltages are shown for: (a) SLG (b) BLG (c) FLG and (d) MLG

devices at T = 100 K. The thick solid lines are guide to the eye. The insets in eac h ﬁg-

ure correspond to the bandstructure at particular voltages for corresponding graphene

ﬂake .

3.2. Noise characteristics in graphene based FET devices

Noise in the graphene devices were measured in low frequency ac four-probe

method.

In all devices, the normalized resistance noise PSD behaved

as S

(f)/R

∼ 1/f

,withα ranging from 0.8 − 1.2.

Here, instead

of focusing on noise magnitude at a speciﬁc frequency, we compute and

December 8, 2010 19:26 World Scientiﬁc Review Volume - 9in x 6in Chap13

208 A. N. Pal and A. Ghosh

analyze the total variance of resistance ﬂuctuations (N

)integratedoverthe

experimental bandwidth. Figures 3(a)–(d) shows the variation of N

and

the corresponding average resistance as a function of backgate voltage (V

)

for SLG, BLG, FLG, and MLG devices, respectively. The most striking

observation of Fig. 3 is the unique gate voltage dependence of noise in case

of SLG,

44,47

noise decreases with increasing n on either side of the CNP (the

Dirac point). But in all other cases, noise magnitude behaves oppositely.

The observed characteristics can be qualitatively explained by our model

described in Sec. 2, which shows that the bandstructure plays an important

role in determining the behavior of noise in graphene. To compare the noise

performance from diﬀerent devices, people often calculate the dimensionless

parameter γ

from the empirical relation given by Hooge,

(1/(A

),whereA

is the area of graphene channel between two

voltage probes and n is the density of charge carriers inside graphene.

Fig. 4. (Color Online) Temperature dependence of Hooge parameter for SLG, FLG and

MLG devices, far from the CNP (n =2.4 × 10

/cm

Figure 4 shows the variation of Hooge parameter γ

in SLG, FLG and

MLG, at a particular density n (2 × 10

/cm

) over a wide temperature

(T ) range. The exponential increase in γ

in all devices with increasing

T could be readily understood in the framework of charging/discharging of

the trap states at SiO

and graphene interface which are known to be ac-

tivated processes. Our analysis indicates the activation energies (17 ±1.5),

(17 ±0.6), and (22 ±2) meV in SLG, FLG, and MLG respectively,

being

December 8, 2010 19:26 World Scientiﬁc Review Volume - 9in x 6in Chap13

Noise in Graphene Transistors 209

expectedly similar since the devices were fabricated on the same Si/SiO

wafer. However, the most intriguing aspect of Fig. 4 is nearly two orders of

magnitude lower noise in FLG and MLG in comparison to SLG that was

consistently observed in other similar devices as well. A simple understand-

ing of this can be obtained by considering eﬃcient screening of underlying

trap potential ﬂuctuations for FLG and MLG due to their nearly three di-

mensional character, and existence of parabolic bands that provide ﬁnite

DOS at low energies. The three dimensional nature increases with increas-

ing number of layers making most of FLG/MLG structure insensitive to

extraneous ﬂuctuations. Indeed, with considerable ﬁeld eﬀect behavior,

and with γ

∼ 10

−5

−10

−6

FLG forms a promising material for low noise

nanoelectronic applications, outperforming competing nanomaterials such

as carbon nanotubes in this respect.

3.3. Noise in dual gated BLG device

In case of BLG, one can break the interlayer symmetry by applying a per-

pendicular electric ﬁeld across the ﬂake, resulting in a gap between the

conduction and valence band.

4,15,16

Recently, the low-frequency ﬂuctua-

tions, or the 1/f noise, in electrical resistance of bilayer graphene has been

shown to be sensitive to its band structure and the increase in noise with

density has been explained by the diminished ability of BLG to screen the

external potential ﬂuctuations in the presence of ﬁnite bandgap (∆

44,45

However, in noise experiments on BLG so far,

∆

has been tuned only by

varying n with a single (back) gate, where a partial screening of the gate po-

tential leads to excess charge in the upper layer, and hence an electric ﬁeld

between the graphene layers.

Here, we have measured the low-frequency

resistance noise in spatially extended double-gated BLG devices. The main

objective is to achieve an independent tunability of ∆

with both n and E,

where E is the transverse electric ﬁeld across the electrodes, to separate the

inﬂuence of band structure and carrier density on screening. A micrograph

of a typical dual gated device is shown in Fig. 5(a). The resistance (R)- V

characteristics of the device at T ≈ 107 K are shown in Fig. 5(b) for several

diﬀerent values of V

spanning between −30 and +30 V . The charge mo-

bility of the device was estimated to be 1160 cm

−1

, which contained

an intrinsic hole doping of −n

≈ 5.82 × 10

−2

.WhenV

and V

are

diﬀerent, a ﬁnite E is established between the electrodes. Existence of the

electric ﬁeld-induced band gap becomes increasingly prominent at higher

with increasing R at charge neutrality point.

December 8, 2010 19:26 World Scientiﬁc Review Volume - 9in x 6in Chap13

210 A. N. Pal and A. Ghosh

Fig. 5. (a) Optical micrograph of the dual gated BLG device used in the experiment.

The scale bar is 10µm.(b) Resistance versus topgate voltage for various back gate volt-

ages, ranging from 30 V to −30 V (left to right) with an interval of 5 V. Insets show

schematics of corresponding band structures. (c-e) Electrical transport and noise char-

acterization of a BLG device. The resistance and the normalized noise po wer spectral

density (N

) as functions of top gate voltages are shown for various back gate voltages:

sets show typical normalized noise power spectra S

, far from the charge neutrality

point for each back gate voltage. Note that the charge neutrality points and the noise

minimum p oints are not necessarily the same. This can b e explained quantitatively

by considering the noise minimum point in graphene to correspond to zero band gap

between the conduction and the valence bands.

Typical power spectra of resistance noise are shown in the insets of

Figs. 5(c)–(e). Figures 5(c)–(e) show the variation of N

and the corre-

sponding average resistance as functions of V

at three diﬀerent values of

. For all V

, N

shows a minimum at a speciﬁc V

, denoted as V

Nmin

and increases monotonically on both sides of V

Nmin

. Similar behavior

was observed for noise in BLG nanoribbons as well,

which conﬁrms this

to be an intrinsic characteristic of BLG, although V

Rmax

and V

Nmin

are

December 8, 2010 19:26 World Scientiﬁc Review Volume - 9in x 6in Chap13

Noise in Graphene Transistors 211

not equal in our measurements, indicating that noise minimum has been

shifted away from charge neutrality in the presence of ﬁnite E.Morede-

tailed analysis shows that multiple processes involving the charge traps are

active in producing the resistance noise which is intimately connected to

the BLG band structure, being minimum at ∆

∼ 0 even if it corresponds

to a nonzero n.

4. Conclusion

In summary, we have done a comparative study of low frequency ﬂuctu-

ations in electrical resistance of graphene based ﬁeld-eﬀect devices with

varying layer thickness. An analytical model of noise has been proposed

based on the number ﬂuctuation between the interfacial oxide traps and

the correlated mobility ﬂuctuation. Gate voltage dependence of noise in

various graphene devices can well be explained using our model, showing

its intimate connection to the bandstructure. Apart from opposite gate

voltage characteristics between multilayer graphene and SLG, a striking

observation in this study is the extremely low magnitude of noise in case

of multilayer graphene, with Hooge parameter as low as 10

−6

at low tem-

peratures. This makes few/multilayer graphene an attractive candidate for

future nanoelectronics. Also, we have measured the low-frequency resis-

tance noise in bilayer graphene ﬂakes as a function of charge density and

inter-electrode electric ﬁeld. The absolute magnitude of noise is intimately

connected with the BLG band structure, and shows a minimum when the

band gap of the system is zero. The experiments also reveal the charge orga-

nization in BLG-based electronic devices, and the microscopic mechanism

of resistance noise.

Acknowledgments

We acknowledge the Department of Science and Technology (DST) for a

funded project, and the Institute Nanoscience Initiative, Indian Institute

of Science, for infrastructural support. ANP thanks CSIR for ﬁnancial

support.

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