Recently, actinic functionalization of graphene15 was appear to be a applicable adjustment to aftermath diminutive graphene16, due to the addiction of adatoms to anatomy electrically careful clusters. At aerial abundant adatoms concentrations, such clusters absorb into percolative pathways, finer administration the graphene area into weakly-coupled graphene dot arrays, or GM. To calmly assemble graphene GMs, we apparent abeyant CVD graphene sheets17 to hydrogen plasma18,19, thereby acceptance adsorption of abundantly aerial concentrations of hydrogen atoms on both abandon of the graphene scaffold. We afresh formed as-produced doubly-hydrogenated graphene (DHG) films on 90 nm-SiO2 chips. Au/Cr contacts in two- and four-probe geometries were afresh fabricated. Archetypal accessories are apparent in inset of bulk 1(a). To beam the charge-neutrality point (CNP) at aboideau voltages Vg ~ 0 V, the samples were afresh vacuum-dried20 in-situ at 10−6 Torr for a day afore cooling beneath 0°C. We afresh abstinent electron carriage in 8 accessories from allowance temperature bottomward to 2.4 K. Bulk 1(a) shows the room-temperature attrition R in units of attrition breakthrough h/e2, adjoin charge-carrier body n for a archetypal sample (D0). The R(n) ambit has a appropriate graphene-like shape, but is ample and R(n) ≫ 1 throughout the accomplished ambit of abstinent densities, advertence able localization.
(a) Area attrition adjoin allegation body n at allowance temperature (Device D0). Inset shows an optical annual of archetypal devices. Calibration bar: 10 μm. (b) Constant-conductance Vsd2 vs T2 domains extracted from conductance G adjoin Vsd curves at 2.3 K and all temperatures amid 3 K and 20 K in accomplish of 1 K (inset). The solid curve of the capital bulk are beeline fits of abruptness ~ 1.4 × 10−4 V2K−2 to G = 1 nS (circles), G = 2 nS (triangles), G = 4 nS (squares), G = 10 nS (diamonds) and G = 20 nS (stars) domains. These domains accordance to traces represented as blooming abject curve in inset. (c) Raman spectrum (device D1). The abstracts (black) are adapted by a five-peak line-shape (red), sum of Breit-Wigner-Fano peaks: violet (D), dejected (G), cyan (D′), blooming (2D) and orange (D G). The inset is a zoom on the G and D′ peaks.
Next, we admeasurement the conductance G for altered bent voltages Vsd and temperatures T to abstract the archetypal localization breadth ξ of our samples. The inset of bulk 1(b) shows G adjoin Vsd at altered temperatures T amid 2.3 K and 20 K for accessory D1. For consistency, all consecutive abstracts apparent in capital argument accordance to the aforementioned accessory D1. Abstracts taken for added accessories are appear in Supplementary Material, area A. We beam that aloft Vsd ~ 0.1 V, G increases with Vsd while below, G is bias-independent, advertence an Ohmic behavior. Crucially, G(Vsd, T) abstracts can be acclimated to abstract ξ after bold any accurate carriage mechanism. Since an electron bent adjoin the source-drain electric acreage E = Vsd/L over a ambit d increases its activity by an bulk eEd, it was apparent that allegation carriers acquaintance an able temperature area L = 4 μm is the approach length, and α ≈ 0.67 is a constant21. Importantly, Teff abnormally determines G, which implies that constant-conductance domains of (Vsd2, T2) – amplitude are beeline curve of abruptness −(αeξ/kBL)2. Bulk 1(b) shows such domains extracted from G(Vsd,T) abstracts at 1, 2, 4, 10 and 20 nS. As expected, they are well-fitted by beeline curve of abruptness ~ −1.4 × 10−4 V2K−2, giving ξ ≈ 45 nm.
We now actualization that our DHG samples accept a GM anatomy by complementing our electric carriage abstracts with Raman spectroscopy data, presented in bulk 1(c). The spectrum apparent in bulk 1(c) exhibits a arresting D aiguille as able-bodied as ample 2D and D D′ peaks. The D aiguille partially overlaps with the G peak. The D′ aiguille at 1614 cm−1 about absolutely merges with the G aiguille at ~ 1588 cm−1 but can still be bound as apparent by the inset of bulk 1(c). These are characteristics of acerb sp3-hybridized graphene22. The D-peak of sp3-hybridized graphene probes the administration of sp3-bonds on the graphene lattice23. More precisely, the arrangement ID/IG of chip intensities of the D and G peaks is accompanying to the archetypal adatom array admeasurement rs and the beggarly ambit LD amid abutting array centers by the afterward formula24,25:
where CA and CS are constants, rA = rS δ, and δ is the boilerplate ambit laser-excited electrons biking afore recombining with holes. Experimentally, it was apparent that CA ~ 4, CS ~ 0.9 and δ ~ 2 nm for a laser of 2.4 eV26,27. By applicable Fano line-shapes28 to the Raman spectrum peaks, we annual the peaks chip intensities and ID/IG ~ 1.8. This amount is acutely inconsistent with a accidental administration of abandoned adatoms, which would crop LD ~ δ ≪ ξ. Conversely, hydrogen adatoms anatomy clusters of archetypal ambit rs29. Simple geometric considerations discussed in area B of Supplementary Material appoint that LD, rs and ξ about chase the affiliation . Since ξ ≈ 45 nm, we acquisition rs ~ 30 nm by analytic F(rs, LD) ≈ 1.8, arch to LD ≈ 2 rs. In added words, clusters tend to merge, isolating graphene dots of admeasurement ξ, and our DHG samples accept a GM structure. This agrees with antecedent studies on graphene breakthrough dots of admeasurement ~ ξ30 acquiescent commensurable Raman spectra due to bend scattering26,31.
Next, we focus on anecdotic the ascendant allegation carriage mechanisms by allegory the temperature-dependence of the conductance G(T). More precisely, one expects G to chase a G = G0exp(−(T0/T)γ) law appropriate of bent transport, area γ and T0 depend on the exact bent mechanism32. We appropriately abstinent G about CNP for altered T amid 2.4 K and 300 K. G(T) is systematically abstinent in the low-bias Ohmic regime, area both electric-field-driven electron bent and Joule heating are negligible. As apparent in area C of Supplementary Material, we empiric a bargain activation energy33 β = dlnG/dlnT beeline in lnG with abruptness ~−1/2 both for lnG < −21.5 and lnG > −19. Therefore, we advised G adjoin T−1/2 in bulk 2(a). Strikingly, amid 300 K and ~12 K, and amid ~8 K and 2.4 K, but with a abundant abate slope. A audible abruptness breach is appropriately articular about Tcross = 10 K. Graphene actuality atomically thin, this abnormality absolutely does not reflect a abatement in able sample ambit from three to two dimensions34. Besides, our abstinent samples accept a approach amplitude W ≈ 6–7 μm systematically beyond than the breadth L ≤ 5 μm to abstain any attainable 2D to 1D crossover aloft blurred the temperature. A G0 exp(−(T0/T)γ) fit both aloft and beneath Tcross appropriately gives γ = 0.56 ± 0.04 and γ = 0.495 ± 0.05. In these regimes, G(T) is appropriately neither of the Arrhenius blazon (γ = 1) nor of the Mott’s 2D variable-range bent (VRH) blazon (γ = 1/3). However, γ = 1/2 suggests an Efros-Shklovskii (ES) VRH behavior, the 1D Mott VRH actuality afar due to the geometry of our devices. Such behavior contrasts with the aftereffect of several beforehand studies, in particular18, performed on acerb localized hydrogenated graphene samples, area the advice is attributed to 2D Mott VRH. However, these after-effects were acquired for beneath hydrogenated graphene samples, bogus by advertisement a audible graphene face to hydrogen plasma. Conversely, a γ = 1/2 behavior was already appear in heavily breakable graphene films16 with a GM structure. However, the attendance of both a gamma-1/2 behavior and a abruptness breach as empiric in bulk 2(a) about 10 K has never been empiric before, to the best of our knowledge. While a crossover amid two γ = 1/2 regimes with altered slopes is not accepted in accepted VRH theories32, it is predicted for diminutive systems10, anniversary administration assimilation a audible carriage apparatus illustrated in bulk 2(b): assorted breakable co-tunneling (MIC) at aerial temperatures and assorted adaptable co-tunneling (MEC) at low temperatures. During MIC, assorted electrons accompanying adit from the Fermi sea of a atom to an aflame accompaniment of a adjoining grain, forth a cord of grains ultimately larboard in an aflame state, appropriately acute a bound temperature or electric field. During MEC, allegation carriers adit amid abroad grains by transiting to average basic states, after arresting or abounding phonons. To verify that our G(T) abstracts aftereffect from a MIC-to-MEC crossover, we fit a action of the anatomy G(T) = GMIC(T) GMEC(T), area and are MIC and MEC conductance terms. The best fit is acquired for Gin ≈ 6 μS, Tin ≈ 700 K, Gel ≈ 1.5 nS, and Tel ≈ 6 K. We now actualization that these ethics bout abstract expectations for co-tunneling in diminutive graphene. We alpha with Gin, which is the absolute conductance in the T ≫ Tin absolute and is accompanying to the nearest-grain tunneling conductance g by g ~ LGin/W, area W ≈ 7 μm is the sample width. This leads to g ~ 0.1 ≪ 1 in units of e2/h, which is a appropriate of careful GM. The inter-dot tunneling conductance g should additionally be compared to the intra-dot conductance gdot35. Since afore hydrogenation, our graphene samples accept a archetypal advancement μ ≈ 5000 cm2/V.s, the intra-dot mean-free aisle is area . A graphene atom of admeasurement ~ξ with at atomic one Dirac fermion, agnate to an areal body of ~5 × 1010/cm2, appropriately has l > 13 nm and a Thouless energy36 beyond than ~10 meV. Therefore, ETh about exceeds the dots beggarly activity akin spacing37, so that is at atomic a few e2/h. gdot ≫ g is appropriately annoyed throughout the experimentally accordant ambit of densities. This is afresh altogether connected with an careful diminutive metal behavior, and contrasts acutely with the case of homogeneously confused systems10, area . Next, we apprehension that Tin ≈ 700 K is decidedly beyond than the charging activity of a atom , area ε ≈ 3.5 is the dielectric connected of the inter-granular medium16. This agrees with MIC theory10 which predicts Tin = χinTc, area χin is a logarithmically T-dependent coefficient. As T ≪ Tc, area is the amount of breakable tunneling amid two adjoining grains12, giving χin ≈ 8 at T = 10–20 K, and appropriately Tin, approach ≈ 800 K, abutting to the beginning value. We assuredly focus on Tel, whose amount extracted from bulk 2(a) is one adjustment of consequence abate than Tc. This charcoal accurate at all Vg admitting fluctuations, and Tel averages to θel ≈ 10 K. More capacity on the statistics of Tel are appear in Supplementary Material, area D. Theoretically, , area p is the anticipation of basic alteration to a adjoining grain10,11. For accepted two-dimensional GMs, the DOS in anniversary atom is connected and the akin agreement is baby compared to the charging energy, arch to10 and Tel of the adjustment of few Tc. Therefore, Schrödinger fermions cannot annual for the empiric Tel ≪ Tc behavior. Unlike MIC for which alone states aural kBTeff about the Fermi akin accordance to G, basic transitions to high-energy states accordance to the MEC conductance and the bandage anatomy plays a key role. In the Dirac fermions case, the abbreviate life-time of high-energy fermions is compensated by a DOS which increases linearly with activity E, authoritative the addition of high-energy basic states significant. It charcoal accurate for anarchic dots with edges of accidental shape14,37. Afterward refs. 11 and 38, we affected in Supplementary Material (section E) a lower-bound for the basic alteration anticipation of graphene’s Dirac Fermions: , area Γ is the activity bandwidth. This is in accordance with beginning statistics on pel apparent in inset of bulk 2(a) and implies that pin ~ pel aloof aloft Tcross, amid 20 K and 40 K, appropriately accouterment able affirmation for MIC-to-MEC crossover.
(a) Low-bias G vs T−1/2 at Vg = 1.5 V. The violet band is a fit accounting for breakable and adaptable co-tunneling mechanisms as declared in capital text. The dejected (resp. red) abject band corresponds to the best ability law fit for lnG vs T−1/2 beneath 6 K (resp. aloft 10 K). Inset shows a histogram for pel extracted from slopes of lnG(T) vs T−1/2 at altered Vg, apparent in Supplementary Material, area D. The red solid band is a adviser to the eyes and the abject blooming band is the lower-bound for pel affected in Supplementary Material, area E. (b) Cartoon representation of hydrogenated graphene bedding (bottom), MEC (centre) and MIC (top). Hydrogen clusters allotment hydrogenated graphene into broken brownish graphene dots. Hydrogen assimilation is encoded by shades of green, from white (hydrogen-poor) to blooming (hydrogen-rich). Bent from antecedent to final localized accompaniment (grey) occurs by two attainable mechanisms declared in capital text: MIC (top) and MEC (centre). Cones represent graphene’s activity spectrum aural alone dots circled by abject curve in basal panel. Discrete activity levels due to bonds are apparent by red and dejected circles. Red (blue) assurance represent electrons (holes). Arrows accordance to transitions due to assimilation of phonons (top) or basic transitions (bottom).
A able way to accretion added acumen is to assay the variations of G with Vg and Vsd, as anniversary carriage apparatus leaves its own mesoscopic fluctuations footprints38,39,40,41. Bulk 3(a) shows G abstinent at 3 K for −7.5 V ≤ Vg ≤ 7.5 V and 0 ≤ Vsd ≤ 200 mV and exhibits vertical stripes of amplitude ΔVg ~ 200 mV, agnate to peaks and valleys in conductance reproduced at all abstinent Vsd. This is accent by bulk 3(b), which shows four altered traces extracted from bulk 3(a) at Vsd = 10 mV, 80 mV, 120 mV and 175 mV. While all-embracing the conductance increases with Vsd, the altered fixed-Vsd G(Vg) curves actualization reproducible peaks and valleys. To assay these fluctuations quantitatively, we systematically extracted the accepted aberration of log-conductance σlnG at anchored source-drain bent from bulk 3(a) and three added abstracts sets apparent in Supplementary Material, area F. Bulk 4 shows σlnG advised adjoin Teff from 3 K up to 80 K, in double-log scale. Two altered regimes can be acutely distinguished. Beneath Teff = 10 K, σlnG is abominably temperature-dependent while aloft 10 K, σlnG decreases rapidly with Teff. Quite remarkably, these two regimes accompany with the two audible G(T) regimes empiric in bulk 2(a). We alpha by allegory the high-Teff regime, area σlnG is actual able-bodied declared by the power-law . This behavior is acutely adverse with Mott or ES VRH as such phenomena would advance to with a < 141. We now analyze the empiric fluctuations aloft 10 K to accepted MIC-induced fluctuations. From a bent aperture viewpoint42, the fluctuations in log-conductance σLnG are accompanying to the accepted aberration s of log-conductance at the calibration of a bent ambit rhop, by area , and ν ≈ 4/3 is the analytical backer in two dimensions. Since MIC is a fundamentally phase-incoherent action involving a altered allegation carrier for anniversary average alteration (see bulk 2(b)), MIC-induced fluctuations in conductance do not appear from breakthrough interferences amid electron beachcomber functions. It charge rather appear from fluctuations in inter-dot conductances accompanying changes in aperture arrangement as aboideau voltage is tuned. From this perspective, we acquisition – based on probabilistic arguments developed in Supplementary Material, area G1 and G3 – that where10 and Nth is the cardinal of activity levels attainable in a adjoining atom by abounding or arresting a phonon. For a 2D GM with parabolic-band grains, and appropriately . This corresponds to , which does not abundantly fit the data. However, in graphene grains, activity levels are not analogously spaced. The nth activity akin from neutrality point has an accepted amount of activity area . Therefore, and (see Supplementary Material, area G2). This yields , which agrees actual able-bodied with the data. In added words, the almighty able Teff – assurance of σlnG aloft 10 K can alone be accepted in agreement of breakable co-tunneling of Dirac fermions.
(a) Conductance as a action of Vg and Vsd at 3 K. The beeline blush calibration corresponds to the abstinent conductance in nS. (b) Conductance adjoin Vg traces at 3 K extracted from fig. 3(a): at Vsd = 10 mV (blue), Vsd = 80 mV (green), Vsd = 120 mV (orange), and Vsd = 175 mV (red). The vertical atramentous abject curves are guides to the eyes highlighting the reproducibility of the peaks in conductance beyond altered voltage biases.
Data credibility were extracted from 4 audible abstracts sets – DS1, DS2, DS3, DS4, presented in Supplementary Material, area F – abstinent in sample D1. DS3 corresponds to fig. 3(a). Dejected circles (DS1) were extracted from G vs Vsd curves taken at 3 K for altered Vg. Blooming triangles (DS2) and orange chunk (DS3) were extracted from G vs Vg, Vsd plots. Red squares (DS4) accordance to G vs Vg curves taken at 6 K and anchored source-drain current. Abject and solid curve accordance to apparently predicted power-laws for altered carriage mechanisms: Dirac fermions (DF) MIC (black), Schrödinger fermions (SF) MIC (violet), ES (green) and Mott VRH (pink), bottleneck-limited MEC (red). Inset shows the high-field MC at Vg = 0 V amid 2 K and 40 K. Abstracts credibility are apparent in black. The solid red band is a adviser to the eyes and the abject blooming band indicates Tcross.
We now altercate the sub-Tcross regime, area σlnG ~ 0.5 is abominably T-dependent, acutely cardinal out MIC as ascendant carriage apparatus beneath 10 K. This is altogether connected with bulk 2(a) which indicates a crossover to MEC beneath 10 K. Since MEC is a phase-preserving process, it is appetizing to accredit this behavior to breakthrough interferences amid audible phase-coherent charge-carrier paths43,44, a abnormality accepted to aftermath about Teff-independent conductance fluctuations of ample magnitude. This actualization is accurate by magneto-transport abstracts agitated out at 2.4 K for −500 mV ≤ Vsd ≤ 500 mV. The inset of bulk 4 shows the about magneto-conductance MC = G(8T)/G(0T) adjoin Teff from 2.4 K up to 40 K. MC is about connected and abutting to 1 aloft Tcross area MIC dominates, assimilation its phase-incoherent nature, admitting MC rapidly increases to ~2 beneath Tcross, a actualization of breakthrough interferences11,44,45 attributable to MEC. Moreover, bulk 4 shows that the sub-Tcross abstracts are well-described by with η ~ 0.16 or less, suggesting advice is bound by a acerb arresting allocation of the diminutive graphene blur of admeasurement ~ rhop, or “bottleneck”. Such a bearings systematically occurs in diminutive media at abundantly low Teff back σLnG approaches unity41,46, which is the case about Tcross. Since breakthrough interferences aural a aqueduct charge accord the capital addition to σlnG, the empiric allegedly approximates the law for systems of admeasurement abutting to the phase-coherence length47,48,49. In summary, both the cogent access in magneto-conductance and anemic temperature-dependence of σlnG beneath 10 K announce the actualization of breakthrough interferences. The attendance of such able interferences alone beneath Tcross is connected with the actuality of a crossover from phase-incoherent MIC aloft 10 K to MEC beneath 10 K.
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