Céline Baudouin
email: baudouin.geol@gmail.com

Lydéric France

Trace element partitioning between wollastonite and alkaline 2 silicate magmas 3 4 5 6 7 8

Keywords: Wollastonite, Rare earth element, partition coefficient, alkaline magmas 46 47 Wollastonite

The partitioning of trace elements between wollastonite and melt provides a choice tool 29 for understanding differentiation processes and trace element fractionation in alkaline-rich and 30 silica-undersaturated magmatic systems at crustal conditions, but very few data are currently 31 available. Here we provide the first partition coefficients and associated lattice strain 32 parameters between wollastonite and silicate magmas of Oldoinyo Lengai (Tanzania). Trace 33 element partitioning of isovalent cations shows a parabolic dependence between the partition 34 coefficients and ionic radii explained by the lattice strain model with the site radius (r 0 ) 35 decreasing with increasing charge from r 0 1+ = 1.2 Å to r 0 5+ = 0.6 Å. Bivalent cations are 36 moderately incompatible (D Mg = 0.12 and D Sr = 0.5) to compatible (D Mn = 1). High field 37 strength elements such as Zr and Nb are strongly incompatible in wollastonite (D < 0.01), and 38 rare earth element (REE) partition coefficients increase with decreasing ionic radius from D La 39 = 0.19 to D Lu = 2.8. The crystallization of wollastonite could eventually strongly influence 40 REE fractionation (and more specifically the light-heavy REE ratios) during magmatic 41 differentiation of alkali-rich foiditic melts and should therefore be considered to fully 42 understand trace element evolution and partitioning in alkaline and silica-undersaturated 43 magmas.

Introduction

48

is the ratio between tetrahedral (T) and octahedral (O) sites: T:O ratios are 3:2 for wollastonite 75 and 1:1 for diopside (Thompson et al., 2016). In wollastonite, trace element cations of various 76 charges (1+ to 5+) and ionic radii are substituted into the three octahedral Ca sites of similar 77 ionic size (on average 0.99 Å), compared to the two Ca sites of different ionic sizes in 78 clinopyroxene (Ohashi and Finger, 1978;Thompson et al., 2016). Element partitioning can be 79 modelled by the lattice strain model (LSM), defined by the 'strain-free' partition coefficient 80 D 0 , site elasticity (apparent Young's Modulus) E, and site radius r 0 (e.g., Wood and Blundy, 81 1997). Trace element partitioning of 1+ to 4+ cations between wollastonite and melt has only 82 been established at mantle conditions (3 GPa, 1420 °C) for silicate-carbonate melt (Law et al., 83 2000). Law et al. (2000) highlighted relatively high D 0 for 2+ and 3+ cations and a strong 84 dependence between ionic charge and r 0. They suggested that the relatively lower temperature 85 conditions associated with carbonate-rich silicate melts (e.g., at Oldoinyo Lengai) could have been previously analysed and described as alkali-rich nephelinites [START_REF] De Moor | Volatile-rich silicate melts from Oldoinyo Lengai volcano (Tanzania): 432 Implications for carbonatite genesis and eruptive behavior[END_REF] Mitchell and Dawson 2012) (Fig. 1b). Interstitial glasses exhibit high LREE concentrations 159 (e.g., 255-313 ppm La) and concentrations decrease from LREEs to HREEs (0.8-1.1 ppm 160 Ho, 0.3-0.36 ppm Lu, La/Lu = 800-960). Very high large ion lithophile element (LILE) 161 contents are observed, in particular for Ba (2214-2920 ppm) and Sr (2671-3484 ppm) (Table 162 2). Glasses have relatively low concentrations of the transitional metals Sc, Cr, and Ni (<4 163 ppm) and high Zn and V contents (320-452 ppm Zn, 149-207 ppm V) (Table 2). Glasses are 164 also characterized by a fractionation between Nb and Ta (374-549 ppm Nb, Nb/Ta = 69-78), 165 high Zr content (722-1034 ppm), and low pentavalent cation contents (2.6-7.1 ppm Mo, 166 0.81-1.22 ppm Sb) (Table 2). Wollastonite-melt partition coefficients (D i ) for chemical elements i of a given valence 186 are correlated to ionic radii and display parabolic distributions, as observed for other silicate 187 minerals (e.g., [START_REF] Adam | Trace element partitioning between mica-and amphibole-393 bearing garnet lherzolite and hydrous basanitic melt: 1. Experimental results and the 394 investigation of controls on partitioning behaviour[END_REF][START_REF] Dalou | DOUBLE FIT: 428 Optimization procedure applied to lattice strain model[END_REF]Wood and Blundy, 1997). D i can 188 be quantified using the classical LSM equation defined by [START_REF] Blundy | Prediction of crystal-melt partition coefficients from elastic 400 moduli[END_REF] as:

where D i is a function of cation radius (r i in Å), the ideal radius of the crystallographic Ca site 190 occupied by the cations (r 0 ), the partition coefficient of the ideal cation (D 0 ) with radius r 0 , 191 elastic constant E (Young's modulus), Avogadro's constant N, the universal gas constant R, 192 and temperature T (in Kelvin). The best-fit LSM parameters have been extracted using the 193 DOUBLE FIT software [START_REF] Dalou | DOUBLE FIT: 428 Optimization procedure applied to lattice strain model[END_REF].

198

Monovalent Rb and K cations in wollastonite display very low contents that are near the 199 detection limit (Fig. 3). Law et al. (2000) [START_REF] Dawson | Peralkaline nephelinite-natrocarbonatite relationships at Oldoinyo 415 Lengai, Tanzania[END_REF]Purton et al., 2000) and probably accounts for this deviation. Oppositely,

213

D Zn is lower than expected, probably due to the preference of Zn for tetrahedral over 214 octahedral sites (Purton et al., 2000). The iron partition coefficient is not correlated with the can incorporate Fe 3+ in tetrahedral sites (e.g. Thompson et al., 2016). The presence of Fe 3+ in 217 the melt is expected, as the OL samples commonly contains ferric minerals like andradite and clinopyroxene (Kjarsgaard, 1998;Klaudius and Keller 2006). Fe 3+ and Fe 2+ in wollastonite 219 are likely distributed between Si 4+ and Ca 2+ sites, respectively.

220

The trivalent parabola is well predicted by the LSM and is mainly constrained by the 221 REEs (Fig. 4b). D 0 3+ and r 0 3+ values are close to those of lutetium (D Lu = 2-3.6; r Lu = 0.861 222 Å). Among the REEs, Lu displays the closest partitioning value to that of Ca (Fig. 4b). Other has two oxidation states in our sample (i.e., 3+ and 4+) [START_REF] Adam | Trace element partitioning between mica-and amphibole-393 bearing garnet lherzolite and hydrous basanitic melt: 1. Experimental results and the 394 investigation of controls on partitioning behaviour[END_REF].

226

Using the partition coefficients determined for Zr, Hf, Th, and U, we obtained the best-fit 227 tetravalent LSM parameters D 0 4+ = 0.011, E 4+ = 51 GPa, and r 0 4+ = 0.72 Å (Fig. 5). For GPa, and r 0 5+ = 0.598 Å.

232

Our LSM parameters display a strong correlation with valence. The ideal cation radius 233 r 0 decreases significantly with cation charge from r 0 1+ = 1.2 Å to r 0 5+ = 0.6 Å (Fig. 5). The The only previous study providing partition coefficients for wollastonite focused on 242 Ca-rich melt at 1420 °C and 3 GPa (Law et al., 2000), but these experimental conditions and 243 melt composition are not directly relevant to crustal alkaline magmatic systems (Dawson,244 1998; [START_REF] Dawson | Evolution of natrocarbonatite from a 425 wollastonite nephelinite parent: evidence from the June, 1993 eruption of Oldoinyo Lengai, 426 Tanzania[END_REF]Weisenberger et al., 2014). We nevertheless compare our partition and 10 to 100 times higher, respectively (Fig. 3). As bivalent and trivalent cations have ionic 249 potentials near that of Ca 2+ , they have higher partition coefficients than 1+, 4+, and 5+ cations (Fig. 6). REE partition coefficients are 6-7 times higher at the crustal conditions studied 251 herein, and, notably, HREEs behave compatibly. The difference in REE partitioning between 252 the two studies is likely related to the incorporation of minor elements in the Ca site. For 253 example, Al iv or Na contents in clinopyroxene strongly impact REE partitioning (Bennett et 254 al., 2004;Blundy et al., 1998;Hill et al., 2000;Wood and Trigila, 2001). Wollastonite could 255 therefore be affected by a comparable process in the presence of other minor elements such as 256

Fe (Matsueda, 1974;Shimazaki and Yamanaka, 1973), Mn (Brown et al., 1980) Wood and Blundy, 1997) and wollastonite are richer in Ca sites within their crystal lattices.

307

Carbonatites have been proposed to result from extreme differentiation of alkali-rich 308 and silica-undersaturated melts (Jones et al., 2013;Weidendorfer et al., 2017). Wollastonite is 309 present in alkaline rocks associated with carbonatites (e.g., [START_REF] Dawson | Peralkaline nephelinite-natrocarbonatite relationships at Oldoinyo 415 Lengai, Tanzania[END_REF]Dawson et al., 310 1996;[START_REF] Eckermann | The chemistry and optical properties of some minerals of the Alnö 440 alkaline rocks[END_REF]Weisenberger et al., 2014), and related to the evolved terms of the 311 liquid line of descent (e.g. Kjarsgaard, 1998;Kjaarsgaard et al., 1995)

  partition coefficients 170 We have determined trace element partition coefficients (D) between the interstitial 171 glass and coexisting wollastonite (Table 2), in which trace elements are incorporated into the 172 Ca 2+ site (Fig. 3). LILEs display very low partition coefficients (D Rb = 0.004 ± 0.002, D Na = 173 0.01 ± 0.008, D Ba = 0.003 ± 0.001), except for Sr 2+ (D Sr = 0.5 ± 0.1). Bivalent transition 174 metals have various D values increasing from Zn 2+ (0.03 ± 0.006) to Co 2+ and Mg 2+ (0.11 ± 175 0.03 and 0.12 ± 0.02, respectively) to Mn 2+ (1.01 ± 0.07). LREEs behave incompatibly (D La = 176 0.18 ± 0.04 to D Sm = 0.98 ± 0.15), whereas HREEs (D Eu = 1.19 ± 0.19 to D Lu = 2.67 ± 0.5), Y, 177 and Cr are the only trace elements that behave compatibly with wollastonite (Fig. 3). REE 178 partition coefficients display an important range of values from LREE to HREE and are 179 positively correlated between each others (e.g., D La = 0.19 ± 0.03; D Lu = 2.8 ± 0.4). HREE, Y 180 and Cr are the only trace elements compatible with wollastonite. HFSE partition coefficients 181 are very low for tetravalent (e.g., D Zr = 0.011 ± 0.007, D Th = 0.005 ± 0.002) and pentavalent 182 cations (D Nb = 0.002 ± 0.001, D Ta = 0.006 ± 0.002).

194

  Partition coefficients for several isovalent cations are necessary to suitably constrain 195 the LSM parameters. In this study, 2+ and 3+ partitioning parabolas are well defined, whereas 196 1+, 4+, and 5+ partitioning parabolas are less so due to the sparsity of D values for those 197 valences (Fig.4).

223

  trivalent cations considered herein are Cr, Sc, Al, and V. D Cr is higher than predicted by the 224 LSM, possibly due to the presence of Cr 2+ and Cr 3+ in wollastonite. Similarly, vanadium likely 225

228

  pentavalent cations, although Mo and Sb exist with different valence states, we assumed that 229 they behave as Nb and Ta and mostly occur as pentavalent cations in our sample (Adam and 230 Green, 2006). Our best-fit parameters for pentavalent cations are D 0 5+ = 0.092, E 5+ = 7106 231

  234ideal partition coefficient D 0 is dependent on the charge of the cations: D 0 values are the 235 greatest for 2+ and 3+ cations (D 0 > 1), whereas those for 4+ and 1+ cations are 200-300 236 times smaller. In general, r 0 and D 0 are greater for low P-T alkali melts than those obtained for 237 silicate-carbonate melts at mantle conditions(Law et al., 2000). 238 239 Trace element partitioning and implications for wollastonite-bearing igneous rocks 240 241

  245 coefficients to those of Law et al. (2000) to characterize the impact of wollastonite 246 crystallization on trace element partitioning during magmatic differentiation. Compared to Ca-247 rich melt at 3 GPa, our monovalent and tetravalent partition coefficients are 10 times lower 248

Fig. 2 .Fig. 3 .

 23 Figure captions

Fig. 4 . 370 Fig. 5 .

 43705 Fig. 4. Wollastonite/melt partition coefficients (D wollastonite/melt ) vs. ionic radius for (a) bivalent

Fig. 6 .

 6 Fig. 6. Partition coefficient vs ionic potential (Z/r) for cations of different valence. Only

447

  Figure 1

  Figure3

  estimated the LSM parameters of wollastonite-melt

	200
	201	partitioning in silicate-carbonate melt at mantle conditions (3 GPa, 1420 °C) for monovalent
	202	cations to be D 0	1+ = 0.158, E 1+ = 51 GPa, and r 0	1+ = 1.17 Å. In Oldoinyo Lengai wollastonite,
	203	our values for D 0	1+ (<0.01) and r 0	1+ (~1.15 ± 0.01 Å) are smaller, whereas E 1+ is slightly
	204	higher (68 ± 8 GPa) (Fig. 4b). We note that monovalent partition coefficients increase with
	205	increasing ionic radius (D Na > D K > D Rb ).
	206		We applied the bivalent cation distribution model to the Ba, Sr, Mg, Co, and Mn
	207	partition coefficients. The best-fit model parameters are D 0	2+ = 2.38, E 2+ = 116.2 GPa, and
	208	r 0	2+ = 1.003 Å. Because Ca (with Si) is the main constituent of wollastonite and is not
	209	dependent on melt composition or intensive parameters, Ca 2+ was not used in the bivalent
	210	LSM. Zn 2+ , Fe 2+ , and Ni 2+ deviate from the parabola (Fig. 4a). D Ni is slightly higher than
		predicted by the LSM; the unusual preference of Ni for octahedral sites rather than low-

211

coordination environments in the melt has been described in previous studies (e.g., George 212 and Stebbins

  , or Mg (Jung 257 et al., 2005). Indeed, Oldoinyo Lengai wollastonites are characterized by higher FeO t content 258 (1 ± 0.2 wt.%) and similar MnO and MgO contents (0.4 ± 0.1 and 0.2 ± 0.05 wt.%, 259 respectively) compared to those of Law et al. (2000; ~0.1 wt% FeO t ). The variability of the 260 partition coefficients may result from the different melt composition between an alkali-rich 261 foiditic silicate melt (43.3 wt% SiO 2 , 7.3 wt% CaO, NBO/T= 1.35, Table 1) and a silicate-262 carbonate melt (37.2 wt% SiO 2 , 47.3 wt% CaO, NBO/T= 2, Law et al. 2000). The effect of

	284
	285	between wollastonite and alkali-rich silicate melt (	), and between clinopyroxene and
	286	phonolitic melt (	) are relatively close for LREE although	is slightly higher that
	287	D LREE (Fig. 3). The difference increase for HREE with	almost 10 times
	288	higher than	(Fig. 3). More importantly, the behavior of LREE and HREE are
	289	different in Cpx and wollastonite; all REE are indeed incompatible in Cpx when HREE are
	290	compatible in wollastonite. This implies that both Cpx and wollastonite fractionation would
	291	result in LREE enrichment in the melt, and that Cpx fractionation leads to an enrichment in
	292	HREE when wollastonite fractionation drives the melt HREE content to lower values. This
		difference of partitioning behavior between Cpx and wollastonite therefore results in different
	303
		determined herein for wollastonite, whereas	and	are 100 times lower than in
	274 306
		decrease of the	(e.g. -40% on D Sc ), consistent with the trend that has been
	276	observed herein for	(Fig. 3). Further investigations and experimental studies are
	277	necessary to examine the effect of minor-element incorporation (Mn, Mg), melt composition
	278	(e.g. Ca and Alkali contents) and intensive parameter (P, T) on REE partitioning in
		wollastonites.

263

melt structure (e.g. NBO/T) and composition on trace element partitioning has been identified 264 in previous studies and should not be neglected

(Schmidt et al. 2006, Michely et al. 2017, 265 Mollo et al. 2016)

. According to those authors, an increasing in NBO/T value would imply an 266 important decrease of the partition coefficients down to one order of magnitude (when NBO/T 267 rises from 0.2 to 2). NBO/T differs by less than a factor 2 between the present study and the 268 one of Law et al. (2000), and therefore cannot explain the diversity of the partition 269 coefficients shown in the Fig.3. However, this small melt structure variation may account for 270 the slight modification of partition coefficients for HREE that imply a behaviour modification 271 from moderately incompatible to compatible. One of main difference between the melt 272 compositions is the CaO content (7 to 47 wt% CaO) that could affect the partition 273 coefficients. Di Stefano et al. (2019), proposed that the addition of CaCO 3 (+ 20%) triggers a 279 Wollastonite is the pyroxenoids calcic end-member, and is likely to provide us with to the connection between tetrahedral chains and octahedrally coordinated cations, 282 wollastonite crystal structure is similar to the one of pyroxene, and partitioning of trace elements is likely close for the two mineral species. Trace element partition coefficients 293 LREE/HREE fractionations while crystallizing Cpx and wollastonite. This difference of 294 behavior between LREE and HREE, and more specifically the compatible character of HREE 295 in wollastonite may become a key information for economic geology, and prospective issues 296 as REE deposits are usually associated with igneous endmembers related to protracted 297 differentiation of alkaline melts (e.g., Verplanck et al., 2014). 298 Zr and Hf partition coefficients in wollastonite are very low relative to those in 299 clinopyroxene. In clinopyroxene, HFSEs substitute for Mg or Fe in the M1 site, which has a 300 very similar size to the Zr and Hf ionic radii (Shannon, 1976), whereas the Ca site in 301 wollastonite is too large to efficiently accommodate HFSEs. Monovalent cation (e.g., Rb) and 302 HFSE (Zr, Nb) partition coefficients in orthopyroxene (Opx) are in the same range as those 304

wollastonite

[START_REF] Adam | Trace element partitioning between mica-and amphibole-393 bearing garnet lherzolite and hydrous basanitic melt: 1. Experimental results and the 394 investigation of controls on partitioning behaviour[END_REF] van Kan Parker et al., 2010)

. D REE of the pyroxenes are 305 lower than because REE are incorporated into the Ca site (e.g. Law et al. 2000;

  . The crystallization of in HREEs and enriched in LREEs. It finally implies that on a REE deposit prospects 316 perspective in the alkaline-carbonatitic systems, wollastonite crystallization should be taken 317 into consideration if we are to track new deposits or to elaborate prospection strategies.

312

wollastonite would thus strongly fractionate LREEs (D LREE < 1) from HREEs (D HREE > 1). depleted

  mineral/melt ) obtained for Oldoinyo Lengai

	358
	359	wollastonite/alkali-rich foiditic silicate melt (NBO/T=1.35) in this study are shown by the
	360	thick black line and open circles; 1 errors are shown in gray. Data from previous studies are
	361	experimental wollastonite/silicate-carbonate melt (37.2 wt% SiO 2 and 47.3 wt% CaO,
	362	NBO/T=2, Law et al., 2000), clinopyroxene/phonolitic melt (NBO/T=0.4, Mollo et al. 2016),
	363	and orthopyroxene/silicate melt (CMAS composition, NBO/T=0.79, van Kan Parker et al.,
		2010). Elements are ranked according to cation charge and increasing ionic radius.

Table 379 Table 1 :

 3791 Major element composition of wollastonites and interstitial melts (wt%)

	Wollastonite	Melt
	n=23	1σ	n=16	1σ

Table 2 :

 2 Trace element concentrations (ppm) in wollastonite and interstitial alkali-rich silicate melt of Oldoinyo Lengai ijolite xenolith 10TL01 and their wollastonite/melt partition coefficients (D). George, A. M., & Stebbins, J. F. (1998). Structure and dynamics of magnesium in silicate 446 melts: A high-temperature 25Mg NMR study. American Mineralogist, 83(9-10), 1022-1029.

	Element	Wollastonite	1σ	Melt	1σ	D	1σ
	Li	4.37	0.31	51.7	2.23	0.085	0.005
	Be	0.13	0.04	19.2	1.98	0.007	0.001
	B	4.44	0.48	28.6	1.92	0.16	0.01
	Na	890	296	101041	400	0.009	0.002
	Mg	1086	100	8383	301	0.13	0.008
	Al	50.9	0.94	18746	716	0.003	8E-05
	K	11.5	0.91	30499	1298	0.000	6E-05
	Sc	0.30	0.08	1.69	0.08	0.18	0.03
	Ti	355	25.3	5967	521	0.06	0.005
	V	1.53	0.07	192	6.12	0.008	0.0003
	Cr	5.83	1.41	2.49	0.69	2.35	0.8
	Fe	7616	1165	105968	1200	0.07	0.006
	Mn	2604	143	2344	120	1.11	0.06
	Co	1.47	0.12	12.2	0.44	0.12	0.007
	Ni	1.45	0.36	2.21	0.24	0.66	0.12
	Zn	8.9	0.92	400	21.8	0.02	0.002
	Rb	0.11	0.03	84.9	3.46	0.0012	0.0003
	Sr	1570	40.5	3242	108	0.48	0.01
	Y	65.9	2.06	32.0	1.15	2.06	0.07
	Zr	10.2	0.11	930	31.7	0.01	0.0003
	Nb	0.38	0.008	498	17.1	0.001	4E-05
	Mo	0.37	0.04	5.05	0.16	0.07	0.005
	Sb	0.07	0.03	1.05	0.05	0.07	0.02

  Nb Ta La Ce Pr Sr Nd Zr Hf Sm Eu Gd Tb Dy Y Ho Er Tm Yb Lu sample / primitive mantle This study, Wo/alkali-rich foiditic silicate melt Law et al. 2000, Wo/silicate-carbonate melt Figure 3 van KanParker et al. 2010, OPX/silicate melt Mollo et al. 2016, CPX/phonolitic melt

	Figure 1 Figure 2	10		Figure 2
	0.01 0.1 1 10 100 1000 10000 0.0001 0.001 0.1 1 0.01 D mineral/melt	melt -Oldoinyo Lengai melt -Law et al. 2000 Wollastonite -Komorowski et al. 2007 Wollastonite -Law et al. 2000 3+ 4+ Rb Ba Th U Wollastonite -Oldoinyo Lengai 1+ 2+ increasing radius	5+
			Na K Rb MgMn Sr Ba	Cr Lu Yb Tm Er Y Ho Dy Tb Gd Eu Sm Nd Pr Ce La	Hf Zr U Th	Ta Nb

other bivalent cations (Fig.4a). This may be attributed to the presence of Fe 3+ , as wollastonite

Mineralogy and Petrology, 166(6), 1545-1561.

Acknowledgments 337

We thank the Tanzania commission for science and technology (COSTECH) for the field 338 permits. The authors acknowledge the help of Robert Dennen for English editing. This

key information on the partitioning of trace elements in the Ca site of silicate phases. Also due Wollastonite crystallization in alkaline-carbonatitic volcanic systems may eventually strongly (1980). Phase relations inferred from field data 411 for Mn pyroxenes and pyroxenoids. Contributions to Mineralogy and Petrology, 74(4), 417standard, J. Anal. At . Spectrom., 23, 1529-1537, doi:10.1039/b801807n.


Shashank Shekhar

Hyungwoo Lee

Duckhyung Cho

Myungjae Yang

Minju Lee

Seunghun Hong
email: seunghun@snu.ac.kr

Nanoscale mapping of noise-source-controlled hopping and tunneling conduction in domains of reduced graphene oxide

Keywords: reduced graphene oxide, electrical noise, charge transport, scanning noise microscopy, domain

We report a nanoscale mapping of noise-source-controlled transport characteristics in the domains of reduced graphene oxide by utilizing noise-source imaging strategies. In this method, current and noise images were measured simultaneously using a scanning noise microscopy and analyzed to map sheet-resistances (R ) and noise-source densities (n eff ). The maps showed the formation of conducting and insulating domains, where the insulating domains exhibited up to three-four orders of higher R and n eff than those of conducting domains. Interestingly, the sheet-conductance (Σ ) and n eff followed rather opposite power-law behaviors like Σ ∝ n eff -0.5 and Σ ∝ n eff 0.5 in conducting and insulating domains, respectively, which could be attributed to the difference in mesoscopic charge transport mechanisms controlled by n eff in domains. Notably, high biases resulted in the increased conductance (∆Σ ) and decreased noise-source density (∆n eff ) following a relationship like ∆Σ ∝-∆n eff 0.5 for both conducting and insulting domains, which could be explained by the passivation of noise-sources at high biases. Furthermore, ∆Σ versus ∆n eff plot on the annealing also followed a power-law dependence (∆Σ ∝-∆n eff 0.5 ) in conducting domains, which could be attributed to carrier generation on the annealing. Our results about mesoscopic charge transports could be significant advancements in fundamental researches and applications.

Introduction

Reduced graphene oxide (RGO) is one of the most promising materials in nanoelectronics, and has been widely studied in materials science, electronics, chemistries, and biologies [START_REF] Georgakilas | Noncovalent functiolization of graphene and graphene oxide for energy materials, biosensing, catalytic, and biomedical applications[END_REF][START_REF] Yeh | On the origin of the stability of graphene oxide membranes in water[END_REF][START_REF] Chen | Graphene oxide: Preparation, functionalization, and electrochemical applications[END_REF][START_REF] Ma | Investigating the energy storage mechanism of SnS 2 -rGO composite anode for advanced Na-Ion batteries[END_REF][START_REF] Sahu | Ultrahigh performance supercapacitor from lacey reduced graphene oxide nanoribbons[END_REF][START_REF] Myung | Ambipolar memory devices based on graphene oxide and nanoparticles[END_REF][START_REF] Eda | Chemically derived graphene oxide: Towards large-area thin-film electronics and optoelectronics[END_REF][START_REF] Zhu | Graphene and graphene oxide: synthesis, properties, and applications[END_REF][START_REF] Mativetsky | Local current mapping and patterning of reduced graphene oxide[END_REF][START_REF] Joung | High yeild fabrication of chemically reduced graphene oxide field effect transistors by dielectrophoresis[END_REF][START_REF] Becerril | Evaluation of solution-processed reduced graphene oxide films as transparent conductors[END_REF][START_REF] Mattevi | Evolution of electrical, chemical, and structural properties of transparent and conducting chemically derived graphene thin films[END_REF][START_REF] Wang | Nanoscale charge distribution and energy band modification in defect-patterned graphene[END_REF][START_REF] Shaw | Nanoscale current spreading analysis in solution-processed graphene oxide/silver nanowire transparent electrodes via conductive atomic force microscopy[END_REF][START_REF] Robinson | Reduced graphene oxide molecular sensors[END_REF][START_REF] Zhu | Schottky diode characteristics and 1/f noise of high sensitivity reduced graphene oxide/Si heterojunction photodetector[END_REF].

Previous studies showed that RGO has versatile exotic chemical and physical properties such as a high charge storage capacity, a tunability in electrical properties, a high chemical reactivity, an optical transparency and a mechanical flexibility [START_REF] Georgakilas | Noncovalent functiolization of graphene and graphene oxide for energy materials, biosensing, catalytic, and biomedical applications[END_REF][START_REF] Yeh | On the origin of the stability of graphene oxide membranes in water[END_REF][START_REF] Chen | Graphene oxide: Preparation, functionalization, and electrochemical applications[END_REF][START_REF] Ma | Investigating the energy storage mechanism of SnS 2 -rGO composite anode for advanced Na-Ion batteries[END_REF][START_REF] Sahu | Ultrahigh performance supercapacitor from lacey reduced graphene oxide nanoribbons[END_REF][START_REF] Myung | Ambipolar memory devices based on graphene oxide and nanoparticles[END_REF][START_REF] Eda | Chemically derived graphene oxide: Towards large-area thin-film electronics and optoelectronics[END_REF][START_REF] Zhu | Graphene and graphene oxide: synthesis, properties, and applications[END_REF][START_REF] Mativetsky | Local current mapping and patterning of reduced graphene oxide[END_REF][START_REF] Joung | High yeild fabrication of chemically reduced graphene oxide field effect transistors by dielectrophoresis[END_REF][START_REF] Becerril | Evaluation of solution-processed reduced graphene oxide films as transparent conductors[END_REF][START_REF] Mattevi | Evolution of electrical, chemical, and structural properties of transparent and conducting chemically derived graphene thin films[END_REF][START_REF] Wang | Nanoscale charge distribution and energy band modification in defect-patterned graphene[END_REF][START_REF] Shaw | Nanoscale current spreading analysis in solution-processed graphene oxide/silver nanowire transparent electrodes via conductive atomic force microscopy[END_REF][START_REF] Robinson | Reduced graphene oxide molecular sensors[END_REF][START_REF] Zhu | Schottky diode characteristics and 1/f noise of high sensitivity reduced graphene oxide/Si heterojunction photodetector[END_REF]. In addition, the large scale fabrication of RGO devices is possible in a short span of time under ambient conditions via solution processing methods [START_REF] Myung | Ambipolar memory devices based on graphene oxide and nanoparticles[END_REF][START_REF] Eda | Chemically derived graphene oxide: Towards large-area thin-film electronics and optoelectronics[END_REF][START_REF] Zhu | Graphene and graphene oxide: synthesis, properties, and applications[END_REF][START_REF] Mativetsky | Local current mapping and patterning of reduced graphene oxide[END_REF][START_REF] Joung | High yeild fabrication of chemically reduced graphene oxide field effect transistors by dielectrophoresis[END_REF][START_REF] Becerril | Evaluation of solution-processed reduced graphene oxide films as transparent conductors[END_REF][START_REF] Mattevi | Evolution of electrical, chemical, and structural properties of transparent and conducting chemically derived graphene thin films[END_REF]. Owing to various remarkable properties, RGO has been used in versatile applications such as high-performance supercapacitors, memory devices, highly-sensitive biological and chemical sensors, and high-performance solar cells [START_REF] Georgakilas | Noncovalent functiolization of graphene and graphene oxide for energy materials, biosensing, catalytic, and biomedical applications[END_REF][START_REF] Ma | Investigating the energy storage mechanism of SnS 2 -rGO composite anode for advanced Na-Ion batteries[END_REF][START_REF] Sahu | Ultrahigh performance supercapacitor from lacey reduced graphene oxide nanoribbons[END_REF][START_REF] Myung | Ambipolar memory devices based on graphene oxide and nanoparticles[END_REF][START_REF] Eda | Chemically derived graphene oxide: Towards large-area thin-film electronics and optoelectronics[END_REF][START_REF] Zhu | Graphene and graphene oxide: synthesis, properties, and applications[END_REF].

RGO films have rich microstructures due to the presence of the various kinds of functionalities and the ordered sp 2 and disordered sp 3 hybridization of carbon bondings [START_REF] Ma | Investigating the energy storage mechanism of SnS 2 -rGO composite anode for advanced Na-Ion batteries[END_REF][START_REF] Sahu | Ultrahigh performance supercapacitor from lacey reduced graphene oxide nanoribbons[END_REF][START_REF] Myung | Ambipolar memory devices based on graphene oxide and nanoparticles[END_REF][START_REF] Eda | Chemically derived graphene oxide: Towards large-area thin-film electronics and optoelectronics[END_REF][START_REF] Zhu | Graphene and graphene oxide: synthesis, properties, and applications[END_REF]. Recent works have shown significant local variations in currents in a RGO thin film [START_REF] Mativetsky | Local current mapping and patterning of reduced graphene oxide[END_REF]. Therefore, the nanoscale investigation of the transport properties of RGO should be essential for the further understanding and improvement of RGO-based devices. However, still a little is known about the transport properties of RGO at a nanoscale domain-structure level.

On the other hand, electrical noises, the temporal fluctuations, in the currents provide useful information about materials and devices, and those noises are helpful in understanding charge traps or noise-source densities, charge inhomogeneities, carrier types and densities in devices [START_REF] Xu | Effect of spatial charge inhomogeneity on 1/f noise behavior in graphene[END_REF][START_REF] Sung | Scanning noise microscopy on graphene devices[END_REF][START_REF] Shekhar | Nanoscale direct mapping of localized and induced noise sources on conducting polymer films[END_REF][START_REF] Lee | Nanoscale direct mapping of noise source activities on graphene domains[END_REF]. In a RGO film, various localized noise-sources such as structural defects, electronic traps, functionalities, and impurities are distributed in different domains, generating electrical

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4 noises [START_REF] Lee | Nanoscale direct mapping of noise source activities on graphene domains[END_REF][START_REF] Carbone | Space-charge-limited current fluctuations in organic semiconductors[END_REF][START_REF] Carbone | Trapping-detrapping fluctuations in organic space-charge layers[END_REF][START_REF] Liu | Origin of 1/f noise in graphene multilayers: Surface vs[END_REF][START_REF] Balandin | Low-frequency1/f noise in graphene devices[END_REF][START_REF] Pal | Microscopic mechanism of 1/f noise of in graphene: Role of energy band dispersion[END_REF][START_REF] Jung | Space charge limited conduction with exponential trap distribution in reduced graphene oxide sheets[END_REF][START_REF] Shin | Nanoscale reduction of resistivity and charge trap activities induced by carbon nanotubes embedded in metal thin films[END_REF]. Previously, there have been some studies about the noises in RGO devices, and it has been observed that the noise behaviors depend on the presence of functional groups and disorders in the RGO [START_REF] Robinson | Reduced graphene oxide molecular sensors[END_REF][START_REF] Zhu | Schottky diode characteristics and 1/f noise of high sensitivity reduced graphene oxide/Si heterojunction photodetector[END_REF][START_REF] Xu | Effect of spatial charge inhomogeneity on 1/f noise behavior in graphene[END_REF]. However, it has been very difficult to identify and map the activities of such localized noise-sources in a RGO film at a nanoscale resolution. Recently, we have developed a scanning noise imaging method [START_REF] Sung | Scanning noise microscopy on graphene devices[END_REF][START_REF] Shekhar | Nanoscale direct mapping of localized and induced noise sources on conducting polymer films[END_REF][START_REF] Lee | Nanoscale direct mapping of noise source activities on graphene domains[END_REF][START_REF] Shin | Nanoscale reduction of resistivity and charge trap activities induced by carbon nanotubes embedded in metal thin films[END_REF], allowing us to map the localized noise-sources in domains and boundaries of graphene [START_REF] Lee | Nanoscale direct mapping of noise source activities on graphene domains[END_REF] and polymers [START_REF] Shekhar | Nanoscale direct mapping of localized and induced noise sources on conducting polymer films[END_REF]. Noise-source mappings further helped us in understanding transport mechanisms in conducting channels.

However, the nanoscale effects of noise-sources on mesoscopic transport behaviors in the various domains of RGO have not been explored yet.

Herein, we report a nanoscale direct mapping of the noise-source controlled charge transports in domains of the RGO film, showing hopping and tunneling conduction mechanisms in conducting and insulating domains, respectively. In our strategy, firstly, the images of electrical currents and noises on RGO films were measured simultaneously using a scanning noise microscopy (SNM) [START_REF] Sung | Scanning noise microscopy on graphene devices[END_REF][START_REF] Shekhar | Nanoscale direct mapping of localized and induced noise sources on conducting polymer films[END_REF][START_REF] Lee | Nanoscale direct mapping of noise source activities on graphene domains[END_REF] and analyzed to map the sheet-resistance (R ) and noise-source density (n eff ). The measured maps showed the conducting and the insulating domains, where the insulating domains exhibited up to three orders of magnitude higher R and n eff than those of the conducting domains. Notably, we observed opposite power law relationships like Σ ∝ n eff -0.5 and Σ ∝ n eff 0.5 in conducting and insulating domains, respectively. The results were attributed to the variations in mesoscopic charge transports controlled by the n eff in the RGO domains. provide valuable insights about the effect of noise-sources on mesoscopic transports and should have significant impacts on fundamental researches and applications.

Methods

Device fabrication

We purchased graphene oxide (GO) dispersed in water from Graphene Supermarket (Calverton, NY, USA) for fabricating devices. The concentration of purchased GO solution was 6 mg/ml.

The solution was diluted in DI water to 2 mg/ml via sonication for 10 minutes. Then, GO solution of concentration 2 mg/ml in DI water was dropped on a clean Si/SiO 2 substrate (oxide thickness ~100 nm). The dropped solution on the substrate was kept idle for 30 seconds. Then, the GO was spun coated on the substrate with 2000 rpm for 60 seconds. To obtain RGO, the spin coated GO was reduced by an annealing inside the quartz furnace tube at 450 °C for 60 minutes under the flow of argon (Ar) gas (10 ml/s). The furnace tube was rapidly heated up to a target temperature of 450°C at the rate of 32.5 °C/min under a constant Ar flow (10 ml/s). A metal electrode (50 nm Au on 3 nm thick Ti) was deposited by thermal evaporation through a shadow mask. It is to be mentioned here, that we did not use any lithographic process to keep the RGO surface clean for electrical measurements. preamplifier, and the RMS power of the collected signals was obtained by a homemade RMS-todc converter built using an AD737 chip (purchased from Analog Devices). Note that the obtained noise power was the integrated value of noise (power spectral density) PSDs over the frequency (f) range of the pass band of the band-pass filter. By dividing the square of the measured RMS noise power with the bandwidth of the band pass filter, we obtained the noise PSD at the central frequency of the band. Finally, the measured current and noise PSD maps were analyzed using a network model to obtain the R and n eff maps of the domains of RGO.

Current and noise mapping

Results and discussion

Nanoscale mapping of sheet-resistances and noise-source activities in domains of RGO

Fig. 2a shows an atomic force microscopy (AFM) topography image of a thin layer of the RGO film. The image shows a clean surface with its roughness of ~1 nm which is a typical value for an individual RGO layer [START_REF] Gomez-Navarro | Electronic transport properties individual chemically reduced graphene oxide sheets[END_REF]. The scanning electron micrograph (supplementary information Fig. S1) also shows clean surfaces without any cracks or residues, indicating the high quality of our RGO samples.

Fig. 2b shows the corresponding current map of the RGO film. For the current mapping, the bias voltage of 0.4 V was applied to the metal electrode and the current output was measured using the conducting (Pt) AFM tip. The current map showed two distinctive regions with low (of the order of nA) and high currents (of the order of µA) separated by boundaries. The high current carrying areas (bright yellow regions) with average width ~500 nm could be the ordered conducting domains of the RGO, whereas low currents areas (dark regions) could belong to disordered insulating domains. The origin of high current regions could be the formation of sp 2 rich carbon atoms in the RGO, as predicted previously [START_REF] Eda | Chemically derived graphene oxide: Towards large-area thin-film electronics and optoelectronics[END_REF][START_REF] Mattevi | Evolution of electrical, chemical, and structural properties of transparent and conducting chemically derived graphene thin films[END_REF]. The sp 2 hybridization of the orbitals is responsible for high conductions in carbon allotropes. Previous works using c-AFM, Raman spectroscopy, near field scanning optical microscopy and electron microscopies show that RGO consists of conducting (sp 2 -rich) and insulating (sp 3 -rich) domains of similar sizes voltages, I increased as V 4 (slope of logI versus logV is ∼4, i.e. I ∝ V 4 ). It is to be noted that at high voltages, the current of insulating domains rapidly approached to the value of that of the conducting domains. These results implicate that the non-linear I-V characteristics of the RGO device arose due to the presence of insulating domains. The sudden rise in the current at higher voltages could be attributed to excess injected carriers in the insulating domains as reported previously [START_REF] Jung | Space charge limited conduction with exponential trap distribution in reduced graphene oxide sheets[END_REF].

In our previous work, we developed a computational iterative method to calculate a sheet resistance R map from a current map on a two-dimensional conducting channel [START_REF] Lee | Nanoscale direct mapping of noise source activities on graphene domains[END_REF]. Taking advantage of the method, we could obtain R maps from the current maps, showing the spatial distribution of R in RGO. The details of the calculation method are described in the supplementary information and supplementary Fig. S2, S3 and S4.

Fig. 2d (i) and (ii) shows R maps on the RGO film, obtained from the current maps Fig. 2b and supplementary Fig. S5, respectively. In the maps, the conducting and insulating domains could be clearly distinguished. The averaged sheet resistance value in the conducting domains was ∼10 7 Ω/ , which is consistent with the previously-reported sheet resistance of RGO on a SiO 2 substrate [START_REF] Eda | Chemically derived graphene oxide: Towards large-area thin-film electronics and optoelectronics[END_REF][START_REF] Zhu | Graphene and graphene oxide: synthesis, properties, and applications[END_REF]. On the other hand, the sheet resistance value in the insulating domains was estimated as ∼10 8 Ω/ and it was about an order of magnitude higher than that inside the conducting domains. This implies that insulating domains in a RGO film can obstruct the

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11 conduction of charge carriers, increasing the total electrical resistance of the RGO film.

Presumably, the oxygen functional groups and other structural disorders in an insulating domain can act as carrier scattering centers obstructing charge conductions, resulting in the high R in a domain.

Fig. 2e shows a map of S I /I 2 (at ∼17.2 Hz) obtained from the noise PSD map measured simultaneously with the current map. Fig. 2b. Here, we first measured the map of noise PSD (S I ) at ∼17.2 Hz via the noise microscopy, and divided the measured S I map by the square of the current map to obtain the S I /I 2 map. During the measurement, the bias voltage of a 0.4 V was applied to the electrodes. In the conducting domains, the S I /I 2 values were ∼10 -6 Hz -1 which is comparable to the previously-reported S I /I 2 values on a pristine graphene [START_REF] Lee | Nanoscale direct mapping of noise source activities on graphene domains[END_REF][START_REF] Liu | Origin of 1/f noise in graphene multilayers: Surface vs[END_REF][START_REF] Balandin | Low-frequency1/f noise in graphene devices[END_REF][START_REF] Pal | Microscopic mechanism of 1/f noise of in graphene: Role of energy band dispersion[END_REF]. On the other hand, the insulating domains in the RGO exhibited the S I /I 2 values of ∼10 -2 Hz -1 which is much higher than the S I /I 2 in the conducting domains. This implies the higher activities of electrical noise-sources in insulating domains than in the conducting domains of RGO. Any electrical potential barriers or structural defects, which can produce charge scattering with a relatively long trapping time, can produce fluctuation in electrical currents, thus generating electrical noises. For example, it is reported that the electrical and structural inhomogeneities and defects produce localized electronic states which can trap charge carriers for certain time periods [START_REF] Xu | Effect of spatial charge inhomogeneity on 1/f noise behavior in graphene[END_REF][START_REF] Pal | Microscopic mechanism of 1/f noise of in graphene: Role of energy band dispersion[END_REF]. Such localized states produced by electrical and structural defects or scattering are collectively termed as noise-sources. Previous works suggested that an insulating domain in RGO has abundance of lattice disorders, dangling bonds, and chemical functional groups [START_REF] Eda | Chemically derived graphene oxide: Towards large-area thin-film electronics and optoelectronics[END_REF][START_REF] Zhu | Graphene and graphene oxide: synthesis, properties, and applications[END_REF][START_REF] Mativetsky | Local current mapping and patterning of reduced graphene oxide[END_REF][START_REF] Joung | High yeild fabrication of chemically reduced graphene oxide field effect transistors by dielectrophoresis[END_REF][START_REF] Becerril | Evaluation of solution-processed reduced graphene oxide films as transparent conductors[END_REF][START_REF] Mattevi | Evolution of electrical, chemical, and structural properties of transparent and conducting chemically derived graphene thin films[END_REF].

Such nanostructured defects in the insulating domain can provide localized electronic states and act as charge traps which randomly trap and release charge carriers in the domain [START_REF] Joung | High yeild fabrication of chemically reduced graphene oxide field effect transistors by dielectrophoresis[END_REF][START_REF] Zhu | Schottky diode characteristics and 1/f noise of high sensitivity reduced graphene oxide/Si heterojunction photodetector[END_REF] shown in the supplementary information. In brief, we modeled the RGO film as a two-dimensional network on a x-y plane comprised of small RGO segments of a square shape with its area of (∆l) 2 . A RGO segment at a (x, y) was assumed to have a sheet resistance R (x, y)

and an effective noise-source density n eff (x, y). Each node at a (x, y) can be considered as a point where a conducting AFM tip was placed to measure I(x, y) and S I (f, x, y) in our experiments.

Each node at (x, y) was connected with four neighboring nodes through small RGO channels with its area of (∆l) cm -2 eV -1 . At the domain boundaries, the n eff was about ∼10 15 cm -2 eV -1 . In a quantitative manner, the results directly show that the density of electronic traps is much higher in the insulating domain in RGO than in the conducting domain, which has been only theoretically estimated before. The lattice disorders, dangling bonds, and chemical functional groups, which are abundant in the insulating domains, could act as electronic traps and result in the high n eff in the domains, as suggested in previous works [START_REF] Joung | High yeild fabrication of chemically reduced graphene oxide field effect transistors by dielectrophoresis[END_REF][START_REF] Becerril | Evaluation of solution-processed reduced graphene oxide films as transparent conductors[END_REF][START_REF] Mattevi | Evolution of electrical, chemical, and structural properties of transparent and conducting chemically derived graphene thin films[END_REF][START_REF] Wang | Nanoscale charge distribution and energy band modification in defect-patterned graphene[END_REF][START_REF] Shaw | Nanoscale current spreading analysis in solution-processed graphene oxide/silver nanowire transparent electrodes via conductive atomic force microscopy[END_REF][START_REF] Robinson | Reduced graphene oxide molecular sensors[END_REF][START_REF] Zhu | Schottky diode characteristics and 1/f noise of high sensitivity reduced graphene oxide/Si heterojunction photodetector[END_REF]. In RGO, the dominant charge transport mechanism can be hopping [START_REF] Mattevi | Evolution of electrical, chemical, and structural properties of transparent and conducting chemically derived graphene thin films[END_REF]. In a conducting domain of RGO, a large number of hopping states exist. Thus, the charge carriers mainly flow through the hopping states and the trap-to-trap transition of carriers is limited [START_REF] Kajen | Charge transport in lightly reduced graphene oxide: A transport energy perspective[END_REF][START_REF] Arkhipov | Effective transport energy versus the energy of most probable jumps in disordered hopping systems[END_REF][START_REF] Fishchuk | Nondispersive charge-carrier transport in disordered organic materials containing traps[END_REF][START_REF] Shekhar | Mapping nanoscale effects of localized noisesources activities on photoconductive charge transport in polymer-blend films[END_REF][START_REF] Fishchuk | Transition from trapcontrolled to trap-to-trap hopping transport in disordered organic semiconductors[END_REF]. In this case, the charge traps obstruct the charge conduction by reducing the carrier mobility [START_REF] Fishchuk | Nondispersive charge-carrier transport in disordered organic materials containing traps[END_REF][START_REF] Fishchuk | Transition from trapcontrolled to trap-to-trap hopping transport in disordered organic semiconductors[END_REF].

Further, previous works based on an effective medium approximation theory [START_REF] Fishchuk | Transition from trapcontrolled to trap-to-trap hopping transport in disordered organic semiconductors[END_REF] show that the mobility (µ) can be inversely proportional to the square root of the charge trap density as

µ ∝ n eff -0.5 , (1)
when the trap states have a Gaussian distribution in the energy domain with a rather large distribution width as reported for the RGO [START_REF] Li | Enhancing efficiency of perovskite solar cells via surface passivation with graphene oxide[END_REF]. Note that the conductivityΣ is proportional to neµ, and the conducting domains have nearly uniform carrier density (n) as carbon atoms have mostly sp 2 bondings [START_REF] Mativetsky | Local current mapping and patterning of reduced graphene oxide[END_REF]. Thus, this result is consistent with our observation of Σ ∝ n eff -0.5 in the conducting domains. On the other hand, the insulating domains have only few numbers of hopping states and have a large number of trapping states. In this case, the charge conduction through the hopping states is limited, and charge carriers are mainly flown by tunneling from a trapping state to another trapping state [START_REF] Li | Enhancing efficiency of perovskite solar cells via surface passivation with graphene oxide[END_REF][START_REF] Arkhipov | Analytic model of carrier mobility in doped disordered organic semiconductors[END_REF][START_REF] Arkhipov | Effect of doping on the density of states distribution and carrier hopping in disordered organic semiconductors[END_REF][START_REF] Eda | Insulator to semimetal transistions in Graphene oxide[END_REF]. In such trap-to-trap conduction, the carrier mobility can have a direct correlation with the trap density as traps assist in conduction [START_REF] Arkhipov | Analytic model of carrier mobility in doped disordered organic semiconductors[END_REF][START_REF] Arkhipov | Effect of doping on the density of states distribution and carrier hopping in disordered organic semiconductors[END_REF][START_REF] Eda | Insulator to semimetal transistions in Graphene oxide[END_REF][START_REF] Vianelli | Observation of different transport regimes and large magnetoresistance in graphene oxide layers[END_REF].

Particularly, when the trap density is rather high as in the insulating domains of RGO, the mobility µ can be proportional to the square root of the charge trap density [START_REF] Fishchuk | Transition from trapcontrolled to trap-to-trap hopping transport in disordered organic semiconductors[END_REF]. Then, the relationship Σ ∝ n eff 0.5 is consistent with our result. It is also worth mentioning that our results are consistent with previous studies about the electrical characteristics of RGO with varying temperature measurements [START_REF] Pingel | Comprehensive picture of p-yype doping of P3HT with the molecular acceptor F 4 TCNQ[END_REF][START_REF] Zuo | Impact of doping on the density of states and the mobility in organic semiconductors[END_REF]. For example, previous works show that the electrical

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15 transport of the RGO at different temperatures strongly depends on trap or noise-source densities which are determined by the reduction status of RGO. At a low reduction status when noisesource density is high, an activated conduction is observed in the RGO at a room temperature, which was attributed to the trap-to-trap transition of carriers. At a low temperature or in a highlyreduced GO film, there are not sufficient energy states for a direct transition among the traps, and hopping mechanism dominates. In this case, conducting domains and hopping of carriers can determine the conduction mechanism. On the other hand, at a room temperature, the insulating domains and trap-to-trap transition could contribute dominantly to the electrical conduction, which is consistent with our works [START_REF] Pingel | Comprehensive picture of p-yype doping of P3HT with the molecular acceptor F 4 TCNQ[END_REF][START_REF] Zuo | Impact of doping on the density of states and the mobility in organic semiconductors[END_REF]. These results provide important insights about the role of electronic traps as dominant noise-sources on the charge transports in different domains of RGO. Considering that the noise-sources can be a critical factor determining the characteristics of RGO-based devices, our work should significantly contribute to the basic researches and applications of RGO. In order to estimate the effect of electric fields on the R , we mapped the R values at the bias voltages of 0.2 V and 2 V, as shown in the Fig. 3a and 3b, respectively. The maps were calculated from corresponding current maps measured by the method described in supplementary information. The insulating domains showed R of the order of10 9 Ω/ at the low bias (0.2 V). At the high bias of 2 V, we observed over two orders of magnitude decrease in the R (<10 7 Ω/ ) in the insulating domains, as can be seen from the comparison of Fig. 3a and3b. On the other hand, we did not observe any significant changes in R for conducting domains due to the increase in the voltage. Note that the R of the insulating domains drastically decreased and nearly approached to the values of those of the conducting domains at the high bias. The results show that the high bias affected the electronic transport of the RGO, mostly in insulating domains. The distinctive behavior of R in the insulating domain of RGO could be attributed to the presence of the high density of defects, functionalities and disorders creating a high potential barrier [START_REF] Wang | Nanoscale charge distribution and energy band modification in defect-patterned graphene[END_REF]. A high bias could have provided sufficient energies to charge carriers to overcome the barrier Those injected carriers could outnumber intrinsic carriers in insulating domains resulting in rapid decrease in R [START_REF] Shekhar | Nanoscale direct mapping of localized and induced noise sources on conducting polymer films[END_REF][START_REF] Carbone | Space-charge-limited current fluctuations in organic semiconductors[END_REF][START_REF] Carbone | Trapping-detrapping fluctuations in organic space-charge layers[END_REF].

Effect of bias voltages on the noise-source activities in domains of RGO

M A N U S C R I P T A C C E P T E D

To observe the effects of a high bias on a noise-source density, we mapped and compared the n eff values at the bias voltages of 0.2 V and 2 V, as shown in the Fig. 3c and3d, respectively. The maps were calculated from the S I /I 2 maps (supplementary Fig. S7) obtained from the same area of the RGO device from which R map was measured in Fig. 3a. The maps show an overall decrease in n eff of the RGO at a high bias. The insulating domains exhibited the high n eff ∼10 15 cm -2 eV -1 at the low bias voltage of 0.2 V, whereas the n eff decreased by nearly three orders of magnitude at the high bias of 2 V. On the other hand, the n eff values in the conducting domains of the RGO decreased slightly at 2 V as compared to 0.2 V. The results indicate that the high bias voltage significantly affected the noise-source activities of the RGO. Presumably, noise-sources would have been passivated by excess injected carriers, leading to a large decrease in the n eff at the high bias voltage [START_REF] Carbone | Trapping-detrapping fluctuations in organic space-charge layers[END_REF][START_REF] Tietze | Fermi level shift and doping efficiency in pdoped small molecule organic semiconductors: A photoelectron spectroscopy and theoretical study[END_REF][START_REF] Tietze | Doped organic semiconductors: Trap-filling, impurity saturation, and reserve regimes[END_REF][START_REF] Shang | Trade-off between trap filling, trap creation, and charge recombination results in performance increase at ultralow doping levels in bulk heterojunctions solar cells[END_REF]. Since insulating domains have rather many noise-sources compared to conducting domains, we can expect the drastic decrease of the n eff in the insulating domains. It should be mentioned that such decrease in noise-sources at a high bias can make RGO devices advantageous for a high voltage operation because the most of organic materials-based devices exhibit increased noise-sources at a high bias. ) between ∆Σ and ∆n eff was observed in both conducting and insulating domains. The least square fittings (solid lines) of the data indicate that exponent t of the fitting curves on a log-log scale were ∼0.5±0.1 for both conducting and insulating domains.

A high bias generates a large number of carriers in RGO, which could shift a Fermi level towards a conduction (valance) band when electrons (hole) are majority carriers . Due to the shift in a Fermi level, traps and noise-sources whose energy levels are below the Fermi level would be passivated by charge carriers [START_REF] Tietze | Fermi level shift and doping efficiency in pdoped small molecule organic semiconductors: A photoelectron spectroscopy and theoretical study[END_REF][START_REF] Tietze | Doped organic semiconductors: Trap-filling, impurity saturation, and reserve regimes[END_REF][START_REF] Shang | Trade-off between trap filling, trap creation, and charge recombination results in performance increase at ultralow doping levels in bulk heterojunctions solar cells[END_REF]. When a high bias is applied, there exist excess electrical carriers in a system. Hence, some fraction of the noise-sources can be effectively filled with the carriers and do not contribute to the noise generation. The passivation of noise-sources refers to such screening of the noise-sources by the excess carriers. Since a Fermi level in RGO is a slowly varying function of n [START_REF] Zhu | Graphene and graphene oxide: synthesis, properties, and applications[END_REF], the Fermi level shift ∆E F is expected to be small. Then, ∆n eff and ∆n can be approximated as ∆n eff =-A∆E F , ∆n = B∆E F and ∆n=-E∆n eff where A, B, and E are proportionality constants. A change in a conductivity by a bias voltage change can be written as

∆σ = eµ h n h -eµ l n l = e(µ l +∆µ)(n l +∆n) -eµ l n l = eµ l ∆n + en l ∆µ +e∆µ∆n, (2)
where subscripts h and l stand for the high and low bias conditions, respectively. A low bias noise-source density n eff_l can be expressed in terms of µ l by equation (1) as, n eff_l = .µ l -2 , where D is a proportionality constant. A high bias can affect a noise-source density by a large extent, and it can also bring some change in a mobility [START_REF] Carbone | Space-charge-limited current fluctuations in organic semiconductors[END_REF][START_REF] Carbone | Trapping-detrapping fluctuations in organic space-charge layers[END_REF][START_REF] Jung | Space charge limited conduction with exponential trap distribution in reduced graphene oxide sheets[END_REF]. Thus, a noise-source density at a high bias n eff_h can be written as n eff_h = .(µ l +∆µ) -2 . Assuming, ∆µ/µ l <<1 and applying Taylor expansion to n eff_h , we can obtain

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 20 ∆ = - = . . ! " -. # $ ≈ -. &''_ ) = -. * &''_ ")/ (3)
Thus, ∆µ dependence on n eff can be approximated as ∆µ=-,. . _# $-/ ∆ . Substituting µ and∆µ values in equation ( 2), we can obtain

∆σ = -.,/ _# $ / ∆ -,. # _# $-/ ∆ + .,/ $-/ ∆ _# ∆ . ( 4
)
Analytically, a Fermi level shift can passivate more noise-sources if a noise-source density is high at that point. If the noise-source density change ∆n eff is originating from the Fermi level shift, then the ∆n eff would be rather large at the places where the n eff_l would be large.

Experimentally, we also observed the similar behavior showing a linear dependence between n eff and ∆n eff (see supplementary Fig. S8). Thus, n eff_l in equation ( 4) can be substituted by C∆n eff ,

where C is a proportionality constant. Then, equation ( 4) can be modified as

∆σ = -.,/1 $ / ∆ / -,.1 $-/ # ∆ $ / + .,/1 $-/ ∆ / . ( 5
)
If ∆n eff is much larger than initial carrier density n l , which holds in our cases due to the passivation of a large number of noise-sources at the high bias, we can neglect 2 nd term in the equation ( 5) and it can be further simplified as

∆σ = -.,/1 $ + .,/1 $ ) ∆ / . ( 6
)
This result indicates a relationship ∆σ ∝ ∆n eff 0.5 , explaining our experimental results. Some data deviate from the approximation, resulting in the broadening of data points in the plots.

Interestingly, equation ( 6) is also valid for the plots of insulating domains, implicating that a similar hopping process could dominate transports in an insulating domain at a high bias. A high

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 21
bias creates a plenty of carriers in insulating domains, and those carriers make hopping more favorable than trap-to-trap transitions [START_REF] Fishchuk | Nondispersive charge-carrier transport in disordered organic materials containing traps[END_REF]. range (1 Hz-10 kHz), the normalized PSD of a conducting domain as well as an insulating domain followed a f -β behavior for all voltage conditions (0.2 to 3V). The exponent β was estimated from the slopes of the spectra, ranging from 0.9 to 1.1. These results show that the conducting and the insulating domains of the RGO device had a typical 1/f noise behavior at low frequencies as previously reported, implying that the noises originated in the multiple noise-sources of varying relaxation times (trapping-de-trapping time) [START_REF] Robinson | Reduced graphene oxide molecular sensors[END_REF][START_REF] Zhu | Schottky diode characteristics and 1/f noise of high sensitivity reduced graphene oxide/Si heterojunction photodetector[END_REF][START_REF] Lee | Nanoscale direct mapping of noise source activities on graphene domains[END_REF][START_REF] Balandin | Low-frequency1/f noise in graphene devices[END_REF].

The noise PSD of both conducting and insulating domains were decreased by nearly four orders of magnitude as a bias voltage was increased from 0.2 to 3 V. Interestingly, a broad peak appeared at ∼3 kHz in the PSD spectra of the insulating domains at low voltages (0.2 to 0.4 V), whereas the peak did not exist in the spectra of the conducting domains. The voltage dependence of noise PSD reduction could be due to the passivation of noise-sources by a high bias [START_REF] Carbone | Trapping-detrapping fluctuations in organic space-charge layers[END_REF]. The broad peak in the insulating domains could be attributed to the distribution of relaxation processes [START_REF] Russell | Direct observation of molecular cooperativity near the glass transition[END_REF]. Usually, the relaxation function (ϕ) for a specific relaxation process decays exponentially and follows a relation like ϕ ∝ e -t/τ where τ is the relaxation time [START_REF] Russell | Direct observation of molecular cooperativity near the glass transition[END_REF][START_REF] Raychaudhury | Measurement of 1/f noise and its application in materials science[END_REF]. The noise PSD of this relaxation process generates a Lorentzian noise of a type 2τ/[1+(2πfτ) 2 ], and numerous such relaxation processes having different τ result in the 1/f kind of noise [START_REF] Eda | Insulator to semimetal transistions in Graphene oxide[END_REF]. On the

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 22
other hand, previous works show that heterogeneities, organic functionalities and disorders in the RGO could produce a different kind of relaxation process which decays non-exponentially (e.g. polynomial etc) [START_REF] Russell | Direct observation of molecular cooperativity near the glass transition[END_REF][START_REF] Raychaudhury | Measurement of 1/f noise and its application in materials science[END_REF]. In an insulating domain of the RGO, the different clusters of atoms with -OH and -COOH groups (2-4 nm of size) could cooperatively produce additional non-exponential relaxations [START_REF] Kaniyankandy | Ultrafast relaxation dynamics in graphene oxide: Evidence of electron trapping[END_REF][START_REF] Islam | Evidence of interfacial charge trapping mechanism in polyaniline/reduced graphene oxide nanocomposites[END_REF][START_REF] Gong | Unusual dielectric-strength of Debye relaxation in monohydroxy alcohols upon mixing[END_REF]. Such relaxation process generates a broadened peak in a noise PSD versus frequency spectrum [START_REF] Russell | Direct observation of molecular cooperativity near the glass transition[END_REF][START_REF] Kaniyankandy | Ultrafast relaxation dynamics in graphene oxide: Evidence of electron trapping[END_REF][START_REF] Gong | Unusual dielectric-strength of Debye relaxation in monohydroxy alcohols upon mixing[END_REF]. Fig. 4a shows the R map of the RGO sample annealed for 15 minutes. Fig. 4b is the R map of the same region after subsequent annealing for 30 minutes. The furnace tube was rapidly heated up to the target temperature of 450 °C at a rate of 32.5 °C/min, and the RGO was annealed at the temperature for different time periods under the flow of argon (Ar) gas (10 ml/s).

Effect of thermal annealing on the sheet-resistance and noise-source activities

We made scratch marks on the device to identify a specific location and used them to come back to the same area for mapping after the annealing process. A bias voltage of 0.4 V was applied during electrical measurements. Most of the part of the RGO (Fig. 4a) showed R values higher It has been previously reported that during the annealing process, the oxidized regions of the GO can be slowly converted to the sp 2 configuration which leads to the delocalization of the carriers [START_REF] Eda | Chemically derived graphene oxide: Towards large-area thin-film electronics and optoelectronics[END_REF][START_REF] Zhu | Graphene and graphene oxide: synthesis, properties, and applications[END_REF][START_REF] Mativetsky | Local current mapping and patterning of reduced graphene oxide[END_REF]. Further, the conducting domains grow in the size and contributed in the electrical conduction [START_REF] Eda | Chemically derived graphene oxide: Towards large-area thin-film electronics and optoelectronics[END_REF][START_REF] Zhu | Graphene and graphene oxide: synthesis, properties, and applications[END_REF][START_REF] Mativetsky | Local current mapping and patterning of reduced graphene oxide[END_REF]. Previously, it was reported that the annealing of the RGO produces large conducting domains which are rich in sp 2 hybridized carbon atoms [START_REF] Mativetsky | Local current mapping and patterning of reduced graphene oxide[END_REF][START_REF] Lee | Friction and conductance imaging of sp 2 -and sp 3 -hybridized subdomains on single layer-graphene oxide[END_REF][START_REF] Ramrez | Synthesis of conducting graphene/Si 3 N 4 composite by spark plasma sintering[END_REF][START_REF] Saha | Investigation of the surface plasmon polariton and electrochemical properties of covalent and non-covalent functionalized reduced graphene oxide[END_REF]. Hence, annealing process increases the concentration of delocalized π electrons due to the formation of large sp 2 conducting domains. In a way, an annealing process is similar to a doping process and increases the carrier concentration, resulting in a small shift in the Fermi level. Note that the R maps in our method allowed us to monitor the microscopic changes in the domains of the RGO on an annealing, providing important insights about the growth mechanism and electrical conduction of the domains.

  Interestingly, the increased bias voltage resulted in the increased conductivity ∆Σ and decreased noise-source density ∆n eff with a relationship like ∆Σ ∝-∆n eff 0.5 in both conducting and insulating domains, which was explained by the passivation of noise-sources at the high bias voltage. The thermal annealing process increased the area of conducting domains. Most of the increased conductivity and decreased noise-source density, while the conducting domains exhibited the relationship like ∆Σ ∝-∆n eff 0.5 . These results show that charge transports and noise-sources are closely related, and the noise-source activities even play a determinant role in the nanoscale charge transport phenomena of RGO films. These results

Fig. 1 .

 1 Fig. 1. Schematic diagram showing the experimental set-up for the nanoscale mapping of noise-source

Fig. 1

 1 Fig.1shows a schematic diagram of a set-up for the current and the noise imaging on a RGO

[ 7 , 9 ,Fig. 2 .

 792 Fig. 2. Nanoscale mapping of sheet-resistances (R ) and noise-source activities in RGO.(a) Contact

Fig. 2c is

  Fig. 2c is a typical current-voltage (I-V) curve of conducting and insulating domains. To

  Fig.2eand supplementary Fig.S6, respectively. The detailed calculation method for the n eff is

  (i) and (ii)), the effective noise-source density n eff (x, y) at each point could be calculated from the S I (f, x, y) values at the corresponding node at (x, y) (details in the supplementary information).The n eff maps show the inhomogeneous distributions of the trap densities on the RGO film. TheM A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT13 conducting domains exhibited n eff values in the range of ∼10 11 to 10 13 cm -2 eV -1 . On the other hand, the insulating domains exhibited rather high n eff values in the range of ∼10 13 to 1015

Fig. 2g is

  Fig. 2g is a scatter plot showing the relation between the effective noise-source density n eff and the sheet-conductance Σ on a RGO film. Each data point in the plot shows a pair of n eff and Σ values at an individual position on the RGO film. The data from the conducting and

Fig. 3 .

 3 Fig. 3.Effect of bias voltages on the noise-source activities in the conducting and the insulating domains

  domains, resulting in enhanced carrier injections from the electrodes.

Fig. 3e shows

  Fig. 3e shows scatter plots of ∆Σ versus ∆n eff in conducting (black curve) and insulating (red curve) domains. Each data point in the plot represents a pair of ∆n eff and ∆Σ values at the individual positions of the RGO film. The Σ (or n eff ) map at 0.2 V was subtracted from corresponding map at 2 V to obtain ∆Σ (or ∆n eff ) values. A positive correlation following a

Fig.3f and

  Fig.3f and 3g show the frequency dependence of normalized noise PSD (S I /I 2 ) at different

Fig. 4 .

 4 Fig. 4. Effect of thermal annealing on the sheet resistance and noise-source activities in the conducting

than 10 9

 9 Ω/ in the sample annealed for 15 minutes. Only a few tiny patches of conducting domains of average size ∼100 nm exhibited R values below 10 8 Ω/ . Since the boundaries of the conducting domains had large R (~10 8 Ω/ ), the conducting domain could be easily visualized in the Fig.4a and 4b. One of the conducting domains is marked by blue dotted lines in Fig.4a and 4b. When the RGO sample was subsequently annealed for another 30 minutes (Fig.4b), the conducting domains became larger and exhibited rather low R (∼106 Ω/ ) values. The insulating domains also showed a considerable decrease in R and most of the regions of the insulating domains exhibited R values below 10 8 Ω/ after annealing for 30 minutes. The results show that the annealing of the RGO could restore the conductivity in both conducting and insulating

Fig. 4c shows

  Fig.4cshows the n eff map of the RGO annealed for 15 minutes. The n eff map was calculated from S I /I 2 map (supplementary Fig.S9) obtained on the same area where the R map in the

Fig. 4d shows

  Fig.4dshows the n eff map of the RGO after a subsequent annealing for 30 minutes at 450 o C.

This work was supported by BioNano Health-Guard Research Center funded by the Ministry of Science, ICT (MSIT) of Korea as Global Frontier Project (No. 2013M3A6B2078961). S.H. also acknowledges the support from the MIST (Nos. 2017R1A2B2006808 and 2014M3A7B4051591) and the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 682286).

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

26 lower than 10 14 cm -2 eV -1 , indicating that overall n eff decreased by nearly two orders. The conducting areas with a low n eff became larger and their boundaries got extended. The central regions of conducting domains exhibited a very low n eff value of ∼10 10 cm -2 eV -1 which is comparable to that of the graphene [START_REF] Lee | Nanoscale direct mapping of noise source activities on graphene domains[END_REF]. These results clearly show that the prolong annealing can significantly reduce the n eff of the RGO film. Previous studies show that an annealing provides a thermal energy which could reduce lattice disorders, dangling bonds, and chemical functional groups in RGO [START_REF] Zhu | Graphene and graphene oxide: synthesis, properties, and applications[END_REF][START_REF] Mativetsky | Local current mapping and patterning of reduced graphene oxide[END_REF][START_REF] Joung | High yeild fabrication of chemically reduced graphene oxide field effect transistors by dielectrophoresis[END_REF][START_REF] Becerril | Evaluation of solution-processed reduced graphene oxide films as transparent conductors[END_REF][START_REF] Mattevi | Evolution of electrical, chemical, and structural properties of transparent and conducting chemically derived graphene thin films[END_REF]. It was reported that the annealing process could remove the defects like -OH and -COOH functionalities and dangling bonds [START_REF] Gomez-Navarro | Atomic structure of reduced graphene oxide[END_REF][START_REF] Chen | Self healing of defected graphene[END_REF]. The removal of these functionalities creates crystallographic misfits in structures which are further healed to sp 2 hybridization, increasing the structural order-ness. Further, previous results by TEM imaging have shown such a healing process of defected graphene/RGO to sp 2 configuration by an annealing [START_REF] Gomez-Navarro | Atomic structure of reduced graphene oxide[END_REF][START_REF] Chen | Self healing of defected graphene[END_REF]. The annealing also restores graphitic ordered structures, resulting in the reduction of electronic traps and low n eff in the domains. Our methods enabled the microscopic monitoring of annealing effects on the properties of RGO films including the growth of sp 2 rich carbon film by an annealing. Fig. 4e shows the scatter plot of a change in Σ versus a change in n eff on the subsequent annealing. To obtain ∆Σ (or ∆n eff ) values, Σ (or n eff ) at 15 minutes annealed film were subtracted from corresponding values of the same film after subsequent 30 minutes annealed film. The data points of conducting and insulating domains are plotted with black and red colors, respectively. Note that the conducting and the insulating domains exhibited different behaviors in the ∆Σ versus ∆n eff plot. The data points of the conducting domains (black color) showed a power-law dependence following ∆Σ ∝-∆n eff 0.5 as indicated by the red fitting line. However, we could not establish any correlation in insulating domains due to the wide variation of the data.

Presumably, in conducting domains, the subsequent annealing of a RGO sample removes defects and increases carrier concentrations [START_REF] Joung | High yeild fabrication of chemically reduced graphene oxide field effect transistors by dielectrophoresis[END_REF][START_REF] Becerril | Evaluation of solution-processed reduced graphene oxide films as transparent conductors[END_REF][START_REF] Mattevi | Evolution of electrical, chemical, and structural properties of transparent and conducting chemically derived graphene thin films[END_REF]. Previously, it has been observed that the annealing of the RGO can lead to a Fermi level shift in the disordered graphitic sample due to generation of additional carriers [START_REF] Coletti | Charge neutrality and bandgap tuning of epitaxial graphene on SiC by molecular doping[END_REF][START_REF] Negishi | Band-like transport in highly crystalline graphene films from defective graphene oxides[END_REF]. Therefore, we can apply equation ( 6) in this case as well which can explain the ∆Σ ∝-∆n eff 0.5 behavior in the conducting domains of the RGO. The subsequent annealing of insulating domain of the RGO could partially remove functional groups. However, ∆n eff could not be very large due to incomplete reduction, resulting in a wide variation of ∆Σ and ∆n eff data points on the plot. These results show that our method can be a useful tool to quantitatively investigate the effect of annealing on charge transport properties in the domain of a RGO film.

Conclusions

In summary, we report a strategy to measure the nanoscale noise-source-controlled charge transports in the domains of RGO films showing different conduction mechanism in conducting and insulating domains. In this method, we analyzed the maps of the sheet-resistance (R ) and noise-source density (n eff ). The maps showed two distinctive regions with rather high or low R (n eff ) values, which can be identified as conducting and insulating domains, respectively. In fact, the insulating domains exhibited up to 3-4 orders higher values of R and n eff , which could be attributed to the presence of functionalities and defects. Interestingly, the sheet-conductance (Σ ) and noise-source density in the conducting and the insulating domains were found to be correlated by an inverse (Σ ∝ n eff -0.5 ) and a direct (Σ ∝ n eff 0.5 ) power-law dependence, respectively. The different behaviors can be attributed to the difference in conduction

28 mechanisms controlled by n eff in the domains. Notably, a high bias passivated the noise-sources, resulting in a large increase in the conductance and decrease in the noise-source density.

Furthermore, the change in the sheet-conductance ∆Σ versus the change in noise-source ∆n eff due to a high bias also exhibited a power law behavior like ∆Σ ∝-∆n eff 0.5 which could have been associated with noise-source passivation by the high bias voltage. A thermal annealing improved the electrical properties of the RGO, resulting in increased sheet-conductance and decreased noise-sources. Furthermore, the conducting domains evolved to larger in sizes, and the ∆Σ versus ∆n eff plot followed a relationship like ∆Σ ∝-∆n eff 0.5 presumably due to the enhancement of the carrier density on the annealing. Our results of mapping the nanoscale effects of noise-source activities provide valuable insights on mesoscopic transports in RGO domains and can have significant impacts on basic researches and practical applications.

ASSOCIATED CONTENT

Supplementary Data

Calculation method for sheet-resistance and noise-sources by a network model with a computational method based on a finite element method. Current and noise PSD maps.

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. †These authors contributed equally.

Notes

The authors declare no competing financial interests.

ACKNOWLEDGMENT