Spontaneous time reversal symmetry breaking at individual grain boundaries in graphene
Kimberly Hsieh, Vidya Kochat, Tathagata Biswas, Chandra Sekhar Tiwary, Abhishek Mishra, Gopalakrishnan Ramalingam, Aditya Jayaraman, Kamanio Chattopadhyay, Srinivasan Raghavan, Manish Jain, Arindam Ghosh
SSpontaneous time reversal symmetry breaking at individual grain boundaries ingraphene
Kimberly Hsieh , ∗ , Vidya Kochat , , ∗ , Tathagata Biswas , , Chandra Sekhar Tiwary , ,Abhishek Mishra , Gopalakrishnan Ramalingam , Aditya Jayaraman , KamanioChattopadhyay , Srinivasan Raghavan , , Manish Jain & Arindam Ghosh , Department of Physics, Indian Institute of Science, Bangalore 560 012, India Department of Materials Engineering, Indian Institute of Science, Bangalore 560 012, India Centre for Nano Science and Engineering, Indian Institute of Science, Bangalore 560 012, India and Materials Research Center, Indian Institute of Science, Bangalore 560 012, India ∗ Graphene grain boundaries have attracted interest for their ability to host nearly dispersionlesselectronic bands and magnetic instabilities. Here, we employ quantum transport and universal con-ductance fluctuations (UCF) measurements to experimentally demonstrate a spontaneous breakingof time reversal symmetry (TRS) across individual GBs of chemical vapour deposited graphene.While quantum transport across the GBs indicate spin-scattering-induced dephasing, and henceformation of local magnetic moments, below T . n & × cm − ) and low temperature ( T . Structural disorder in graphene originates from defectsclassifiable into two categories - point defects (vacancies,Stone-Wales defects) and extended defects such as grainboundaries (GBs). Vacancies result in localized statesclose to zero energy leading to magnetic moment forma-tion in graphene, experimentally confirmed by the ob-servation of spin-split resonances in scanning tunnelingmicroscopy (STM) at monovacancies [1], measurementsof spin currents [2] and possibility of Kondo effect [3, 4].GBs lead to local modification of graphene band struc-ture by introducing weakly dispersing, nearly flat elec-tronic bands with enhanced density of states (DOS), ei-ther at zero energy (translational GB(2 , | (2 , , | (3 ,
5 n m5 n m a b c FIG. 1. a , Scanning electron micrograph of a typical pair ofgraphene grains with grain size ≈ µ m forming a GB in be-tween. Scale bar, 10 µ m b , Bright field TEM image of the GBformed between two grains. Selected area electron diffraction(SAED) pattern in inset shows the misorientation angle be-tween the grains ≈ ◦ . Scale bar, 5 nm. c HRTEM imageof the GB region where line and point defects are outlined.Scale bar, 1 nm. ization (WL) compared to single-crystalline grains, indi-cating stronger intervalley carrier scattering due to lat-tice disorder [10, 17]. However, a comprehensive studyof the symmetry-breaking mechanisms at graphene GBsthrough direct measurements of universal conductancefluctuations (UCF) in the inter- and intra-grain regionshas so far been lacking.Magnetic ordering has been predicted at GBs, eitherby localization at non-trivially coordinated C-rings [7] orassisted by strain, e.g. in translational line defects withoctagon-pentagon pairs [5, 6]. In realistic GBs realizedduring chemical vapour deposition (CVD) growth, nucle-ation centers grow independently and fuse in local bond-ing environment. Such GBs comprise multiple defect re-alizations, including vacancies, Stone-Wales defects, in-termittent 1D line defects [12, 20], multi-membered C-rings [7, 21] etc., causing strong increase in charge car-rier scattering and electronic noise [14, 18, 19]. Despiteboth numerical [5–7] and spectroscopic [8, 9, 20, 22] ev-idences of large enhancement in local DOS and spin-splitting, no tangible impact of e-e interaction at theGBs has so far been observed. This work combinesquantum transport and UCF to probe local charge andspin excitations across individual graphene GBs. TheUCF magnitude, determined by the symmetry of the un-derlying Hamiltonian via the Wigner-Dyson parameter β [23], reveals a full spontaneous lifting of the time re-versal symmetry (TRS) in the GB region for T . n & × cm − . The temperature and density-dependence of UCF links the TRS lifting to a frozen mag-netic state arising from the GB defect sites.We measured three devices (D1, D2 and D3)from CVD-synthesized graphene (Supplemental Mate-rial (SM) section S1), optimized to ensure partial fu- a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b ( − ) B (T) GB
SG GB0.1 1 100.11 / K T (K) - 6 - 1 C ( e / h ) T (K) SG n ( cm -2 ) cd eb -8 -6 -4 -2 0 2020406080 R ( k ) n ( cm -2 ) SGGB - 6 - 1 T (K) n ( cm -2 ) GB GBSG cooperon 𝒓 𝒓 𝒓 𝒓 a1 a2 a3 𝑩 𝝓 𝑩𝚫𝝈 𝒄 𝝂 ( 𝑩 ) TRSBroken TRS diffuson 𝒓 𝒓 𝒓 𝒓 FIG. 2. a , Schematic showing a1 , a pair of quantum crossingsrepresenting diffusons, a2 , a pair of quantum crossings repre-senting Cooperons, and a3 , the expected behaviour of ν ( B ) asa function of B for TRS-invariant and TRS-broken systems. b , Sheet resistance ( R (cid:3) ) as a function of gate voltage ( V BG )for the intra-grain (SG) and inter-grain (GB) regions of D1at T = 0 . µ m. c , Magnetoconduc-tance measurements are shown for n = − × cm − at T = 0 . T = 0 .
3, 0 .
8, and 4 K for theGB region, clearly exhibiting WL. Dashed lines correspond toHLN fits. d , Quantum interference correction to conductivity∆ σ c in units of e /πh plotted for both SG and GB regions asa function of T for two fixed densities n = − × cm − and − × cm − . e , Scattering rate γ normalized to itsvalue γ
40 K at T = 40 K, plotted versus T for the SG andGB regions at n = − × cm − is shown, where the blackdotted line indicates the temperature regime where Nyquistscattering dominates. sion of crystallites (scanning electron micrograph inFig. 1a). High-resolution transmission electron mi-croscopy (HRTEM) (Fig. 1b) performed on a pair ofsimilarly synthesized grains reveal an average width ∼
10 nm of the disordered region, and a misorienta-tion angle ≈ ◦ between the parent crystallites. TheGBs form a highly disordered region consisting of ar-rays of line dislocations and under-coordinated C-atoms(Fig. 1c) similar to that observed in STM and TEM stud-ies [8, 9, 19, 20, 22].Fig. 2a schematically describes the conceptual basisof our experimental approach. The quantum interfer-ence effect, that underpins both quantum correction toconductivity (∆ σ c ) and UCF ( h δG φ i ), depends on cross-ings of time-reversed path pairs as the electron (or hole)diffuses across the sample over τ D = L /D , the Thou-less time, where L and D are the length of the system and carrier diffusivity, respectively. While ∆ σ c is de-termined by the probability of single self-crossing, thecorrelation function in h δG φ i ∼ h G (0) G ( τ ) i τ requirestwo quantum crossing points, thereby defining closedloops encircled either in the same (diffusons) or opposite(Cooperons) senses with identical structure factors (Figs.2a1 and 2a2). This has two important consequences:first, compared to ∆ σ c ( ∼ ln( τ D γ )), the UCF magni-tude h δG φ i ∼ ( τ D γ ) − is exponentially more sensitive toemergent dephasing processes in two dimensions, where γ is the dephasing rate, and thus a more suitable tool whenthe dephasing processes are confined within spatially re-stricted regions such as the GBs. Second, when TRS islifted, usually by a transverse magnetic field B (cid:29) B φ , B φ being the field corresponding to one flux quantumthreading a phase coherent cell, the Cooperon contribu-tion is removed, decreasing h δG φ i exactly by a factor oftwo. The reduction factor is protected by the symmetryof the underlying Hamiltonian, i.e. h δG φ i ∼ ( e /h ) /β ,where β = 1 for time reversal invariant systems (orthog-onal ensemble), and β = 2 when TRS is absent (unitaryensemble). For a time-reversal invariant system [24], acrossover function ν ( B ) defined as ν ( B ) = N ( B ) N φ = 1 + 2 b ∞ X n =0 (cid:0) n + (cid:1) + b ] (1)where b = 8 πB ( l φ ) / ( h/e ) is the dimensionless magneticfield which captures the reduction in UCF as a functionof B (Fig. 2a3). Here, N ( B ) and N φ are the values of h δG φ i at B and at B (cid:29) B φ , respectively. When TRSis spontaneously removed (magnetic systems), ν ( B ) re-mains unaffected at the scale of B φ , as observed in ferro-magnetic films [25].For electrical transport, the graphene grains weretransferred on to Si/SiO substrates, patterned into Hallbars such that measurements across the GB and withinsingle grain (SG) can be carried out simultaneously(Fig. 2b inset). The excess disorder in the GB regionresults in enhancement in the resistivity by a factor of ∼ − n (Fig. 2b)and a consequent suppression of the carrier mobility( µ SG ≈
480 cm V − s − while µ GB ≈
220 cm V − s − ).Magnetotransport measurements down to T = 0 . n ( ≈ − × cm − ) indicate enhancedWL correction at the GB region (Fig. 2c), signifyingstronger intervalley scattering from short range latticedefects [10, 17, 26]. Fitting (dashed lines in Fig. 2c)the modified Hikami-Larkin-Nagaoka (HLN) expressionfor graphene [27] to magnetoconductance yields both thequantum correction to conductivity ∆ σ c and the dephas-ing length l φ (Fig. S2a). The T -dependence of ∆ σ c inFig. 2d shows that the quantum correction behaves dif-ferently between SG (left) and GB (right) regions, espe-cially at high n . In both cases, we find ∆ σ c ∝ ln( T ) at
100 200 3000.00.20.4 n = -0.8 cm -2 -0.8 cm -2
47 mT / ( - ) Time (s) -0.8 ( ) -5.4 -7 D1 n (10 cm -2 )B (mT) -3 -2 -1 SG GB ( ) B (T) D1 -8 -4 0 4 81.01.52.0 SG ( = ) n (10 cm -2 ) GB dc ba FIG. 3. a , Conductivity fluctuations for GB region of D1 at n = − . × cm − for B = 0 mT (pink) and B = 47 mT(purple), clearly indicating a reduction in the fluctuation mag-nitude at B (cid:29) B φ . b , ν ( B ) plotted for three different n at T = 0 . n is increased. Solid lines are fits to Eq. 1. c , ν ( B )for the SG and GB regions plotted at T = 0 . n = − × cm − indicating that spontaneous TRSbreaking occurs only in the presence of a GB. d , Noise re-duction factor ν ( B = 0) for the SG and GB regions of D1(circles) measured at T = 0 . T = 0 . T = 4 . n . low n , as expected for diffusive non-magnetic conductorswhere dephasing takes place via Nyquist scattering from e-e interaction so that γ = D/l φ ∝ T . Direct evalua-tion of γ (Figs. 2e and S2b) from HLN fits confirm thisproportionality with T . The quantum correction in theGB region at high n ( & × cm − ) however deviatesfrom this behaviour, where we find both ∆ σ c (Fig. 2d,right panel) and γ (Fig. 2e) saturate below T ∼ γ in metals is oftenattributed to inelastic processes from spin-flip scatter-ing [28–30]. Neglecting electron-phonon scattering atsuch temperatures [31, 32], we can write γ = γ ee + γ s ,where γ ee is the e-e induced scattering rate and γ s isthe spin-flip scattering rate due to dilute magnetic im-purities. The observed saturation in γ at temperatures T . γ ee ∝ T ) countering the reduction in γ s abovethe Kondo temperature T K [33, 34]. Such an anoma-lous T -dependence of γ in the inter-grain region hintsat the formation of magnetic moments that can interactat lower temperatures leading to frozen magnetic order-ing [35]. However, the competing effects of localizationand anti-localization due to graphene’s chiral charge car-riers makes it ambiguous to detect or claim such possi-bilities using WL alone.To complement quantum transport, we carried outUCF measurements in two different ways: (1) Fromslow time-dependent fluctuations in the conductance re-lating directly to the ensemble fluctuations of disorder configuration via ergodic hypothesis [24, 36–38] (Fig. 3,SM section S4), and (2) by analyzing the reproducibleand aperiodic fluctuations in G by tuning the Fermienergy (Fig. 4b, SM section S5). The time-dependentconductance fluctuations across the GB of D1 at n = − . × cm − is plotted in Fig. 3a, clearly display-ing a reduction in the relative magnitude of fluctuationsat B = 0 T and B = 47 mT ( (cid:29) B φ ). Fig. 3b showsthe B -dependence of ν ( B ), defined in Eq. 1, from h δG φ i evaluated from time-dependent conductance fluctuationsin device D1 for three different n at the GB region. Atlow n ( ≈ . × cm − ), ν ( B ) shows a clear factor-of-two reduction as B increases beyond ∼
30 mT, whichcorresponds to B φ (Fig. 3b, uppermost panel). This sug-gests TRS to be preserved in GB regions at low n , similarto 2D systems such as exfoliated graphene [26], topolog-ical insulators [39], doped Si/Ge systems [40] and non-magnetic films [36]. However, with increasing n , a pro-gressive reduction in ν ( B ) at B = 0 was observed acrossthe GB approaching unity, and thus B -independent ν , for n & − × cm − (Fig. 3b, bottom panel). The insen-sitivity of ν ( B ) to transverse field at the scale B ∼ B φ is aunique characteristic of systems with spontaneously bro-ken TRS, as observed before in ferromagnetic films [25]and lightly-doped semiconductors in strongly interact-ing regime [38]. A similar trend was observed for D2(Figs. 3d, S5 and S6) where the reduction in ν ( B ) wasobserved in both doping regimes. Remarkably, the spon-taneous breaking of TRS was observed only in the inter-grain region, while the intra-grain region continues toshow a factor-of-two reduction in UCF magnitude with B at similar high densities (Fig. 3c, higher T in Fig. S9).The near B -independence of ν at high n was foundto be ubiquitous to quantum transport across GBs inCVD graphene as shown for D1 in Fig. 3b and D2 inFig. S5. The solid lines in Figs. 3b and c correspond tofits of ν ( B ) according to Eq. 1 with l φ as the only fit-ting parameter. Fig. 3d shows the noise reduction factor ν ( B = 0) = N ( B = 0) /N φ for D1 and D2 measured at T = 0 . . n . At T = 4 . ν ( B ) ( ≈ .
5) at highest experimental n indicate only partial removal of TRS. The factor of tworeduction of ν ( B ) across the SG region was maintainedthroughout the entire density range, implying that TRSis lifted solely in the presence of the GB.To understand the origin of TRS breaking in the GBregion, we then measured the n -dependence of the zero- B magnitude of UCF, which can distinguish between TRSbreaking from external B field and that from an emer-gent frozen magnetic state [41]. For this, h δG φ i wascalculated from reproducible fluctuations in G withinsmall windows of V BG i.e. from E F (SM section S5).The SG region exhibits a factor of ≈ h δG φ i (Fig. 4b, left panel) due to valley symmetry liftingthereby suppressing the UCF from valley triplet chan-nels, a behaviour observed in exfoliated graphene [26]. cm -2 ) G G SG GB-6 -4 -2 0 2 4 6n ( cm -2 ) Within grain Across grain ba D i ff u s o n C oo p e r o n valleyspin ۧ|𝑺 ۧ|𝑺 ۧ|𝑻 ۧ|𝑻 ۧ|𝑻 ۧ|𝑻 ۧ|𝑻 ۧ|𝑻 ۧ|𝑻 ۧ|𝑻 ۧ|𝑻 𝑩 = 𝟎
Valley hybridization TRS breaking Frozen magnetism ۧ|𝑺
FIG. 4. a , Schematic describing the contribution of diffu-son (blue) and cooperon (red) singlet ( | S i ) and triplet ( | T i , | T i and | T i ) states to the UCF magnitude h δG φ i in differ-ent symmetry classes. Valley hybridization leads to a factorof four reduction in h δG φ i while magnetic impurities reduces h δG φ i by a further factor of eight due to gapping of spin diffu-son triplets and all cooperons. b , The variance in conductance h δG φ i within a phase-coherent box of area l φ normalized to itsvalue at the Dirac point δ h G i as a function of n at T = 0 . ≈ ≈
30 reduction in the GB region.
In contrast, the UCF magnitude in the GB region ex-hibits a drastic reduction (Fig. 4b, right panel) by afactor of ≈
30 as n is increased. This unique and un-precedented reduction can be quantitatively understoodfrom a combination of valley hybridization, TRS break-ing and suppression of spin triplet channels in the pres-ence of static (measurement time short compared to Ko-rringa relaxation time) spin-dependent scattering, as de-picted schematically in Fig. 4a. The static spin tex-ture, or ‘frozen magnetic state’, at large n may hap-pen when the defect-bound magnetic impurities interactvia RKKY (Ruderman-Kittel-Kasuya-Yosida) exchange,forming long or short ( e.g. a spin glass) range spin-ordered states [35, 42]. Thus, the UCF measurement ingraphene containing a GB suggests a rather unexpectedeffect of doping, which manifests in both the valley and(static) spin polarization when carrier density is madesufficiently large.To estimate the energy scale for local moment inter-action at the GB, we study the effect of temperature on ν ( B ). The normalized magnetonoise ν ( B ) for the GBregion in device D2 at n = 6 . × cm − with vary-ing temperature is shown in Fig. 5a (data at higher T inFig. S7a). Evidently, the spontaneous TRS breaking oc-curs only at temperatures . ν ( B ) approaches ∼ T is increased (data for D2 at T = 4 . T = 8 K in Fig. S8). To estimatethe exchange interaction between moments, we first es-timate the Kondo temperature T K ’
20 K from the T -dependence of the sheet resistance R (cid:3) at finite B whereWL corrections are suppressed (Fig. S14). This T K is -8 -7 D1 N ( = ) T (K) GB D1 ( h / e ) T (K)
SG GB -9 -8 -7 N ( = ) T (K) ( ) D2 B (mT) a b c FIG. 5. a , ν ( B ) plotted for D2 at high electron density n =6 . × cm − showing spontaneous TRS breaking at zerofield only at temperatures T . b , T -dependence of thereduced sheet resistance ρ (cid:3) (in units of h/e ) averaged fromthe resistance fluctuations measurements at B = 0 T for SG(blue) and GB (red) regions at n = − × cm − for D1. c , Normalized variance N = S σ /σ at B = 0 as a functionof temperature is plotted for GB region of D1 at n = − . × cm − clearly indicating a sharp increase in N ( B = 0) atlower temperatures. compatible with studies on irradiated graphene [3, 4].The RKKY interaction between moments can be esti-mated as [43] k B T RKKY ∼ j a / πv F ~ R ∼ . a ≈ .
246 nm and Fermi velocity ingraphene v F ≈ m s − ), where the Kondo exchange j ≈ . T K and the DOS in the GB region, D ( E F ) ∼ .
05 eV − (Fig. S12) [5–7]. Such a large j value agrees with previ-ous theoretical calculations [44–46]. The average defectdistance R ≈ T RKKY agrees reasonably wellwith the T dependence of ν ( B = 0) (Fig. S7b), show-ing continual increase in ν ( B = 0) up to T ≈
10 K,after which the decrease in ν ( B = 0) can be attributedto the loss of phase coherence through thermal averag-ing (Fig. S7a). The estimated values of T K and T RKKY signal a competition between Kondo singlet formationand a frozen magnetic state [47]. To gain further insightinto the nature of this magnetic state, we have measuredthe time-averaged resistivity ρ (cid:3) at n = − × cm − for the GB and SG regions of D1 simultaneously. A dis-tinctive feature of the T -dependence of ρ (cid:3) in the GB is anoticeable downturn at T . e-e interaction corrections. Such resistivity down-turn at low- T is strongly indicative of spin-glass freez-ing resulting from reduced spin-flip scattering [48, 49].Additionally, the normalized variance N ( B = 0) of theGB region increases rapidly by nearly an order of mag-itude on cooling from ∼ . . ∼ ∼ ∗ These authors contributed equally to this work.; Present address: Materials Science Centre, IndianInstitute of Technology Kharagpur, Kharagpur 721302,India; Present address: Department of Physics, Arizona StateUniversity, Tempe, AZ 85287, USA; Present address: Department of Metallurgical andMaterial Engineering, Indian Institute of TechnologyKharagpur, Kharagpur 721302, India[1] M. M. Ugeda, I. Brihuega, F. Guinea, and J. M. G´omez-Rodr´ıguez, Phys. Rev. Lett. , 096804 (2010).[2] K. M. McCreary, A. G. Swartz, W. Han, J. Fabian, andR. K. Kawakami, Phys. Rev. Lett. , 186604 (2012).[3] J.-H. Chen, L. Li, W. G. Cullen, E. D. Williams, andM. S. Fuhrer, Nat. Phys. , 535 (2011).[4] Y. Jiang, P.-W. Lo, D. May, G. 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We are grateful to H. R. Krish-namurthy, S. Banerjee and S. Dutta for useful discus-sions. K.H., V.K. and A.G acknowledge the Departmentof Science and Technology (DST) for a funded project.K.H. and A.G. also thank the National NanofabricationCenter, CeNSE, IISc (NNfC) for providing clean roomfacilities. pontaneous time reversal symmetry breaking at individual grainboundaries in graphene: (Supplemental Material)
Kimberly Hsieh , a , b , Vidya Kochat , , a , Tathagata Biswas , , Chandra SekharTiwary , , Abhishek Mishra , Gopalakrishnan Ramalingam , Aditya Jayaraman ,Kamanio Chattopadhyay , Srinivasan Raghavan , , Manish Jain & Arindam Ghosh , Department of Physics, Indian Institute of Science, Bangalore 560 012, India Department of Materials Engineering,Indian Institute of Science, Bangalore 560 012, India Centre for Nano Science and Engineering,Indian Institute of Science, Bangalore 560 012, India Materials Research Center, Indian Institute of Science, Bangalore 560 012, India Present address: Materials Science Centre,Indian Institute of Technology Kharagpur, Kharagpur 721302, India Present address: Department of Physics,Arizona State University, Tempe, AZ 85287, USA and Present address: Department of Metallurgical and Material Engineering,Indian Institute of Technology Kharagpur, Kharagpur 721302, India a equal contributions b email: [email protected] a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b
1. Methods
1. Graphene growth and device fabrication
Graphene grains are grown by low pressure chemical vapour deposition (CVD) on AlfaAesar (25 µ m) Cu foils. The Cu foil is wrapped to form an enclosure, and placed inside a1 inch quartz tube reaction chamber. Initially, 100 SCCM of H gas was passed through thereaction chamber for flushing out impurities from the Cu surface and subsequently, the Cufoil was annealed at 1000 ◦ C for 4 hours while the flow rate was reduced to 50 SCCM of H .A mixture of CH , H and N gases in the ratio of 5:500:1000 was passed for ∼
30 secondsfollowing which, the furnace was cooled to room temperature under a constant flow rate of50 SCCM of H . The Cu foil with graphene was then spin-coated with polymethyl methacry-late (MicroChem PMMA 950K) at 2000 RPM and allowed to dry at room temperature. Thepolymer-graphene-Cu foil stack was then suspended in 0.1 M ammonium persulphate solu-tion to etch the Cu foil. The floating PMMA-graphene stack was rinsed thoroughly withDI water and transferred onto clean Si/SiO (285 nm) substrates by dissolving the PMMAusing hot acetone. Suitable pairs of graphene grains were chosen and patterned by electronbeam lithography followed by metallization of the contacts. a cb Figure S1.
Optical microscope images of the device at various stages of fabrication. a
Selected pairof CVD graphene grains etched into Hall bar geometry using reactive ion etching with O plasma. b Electrical contacts defined by e-beam lithography. c Final image of the device after metallization.Scale bar, 10 µ m. . Transport measurements Magnetotransport measurements were either carried out in a Janis He-3-SSV refriger-ator (for 0 . ≤ T ≤
40 K) or a Leiden MNK-126 He/ He dilution refrigerator (for0 .
03 K ≤ T ≤ ∼
800 realizations). For time-dependent conductance fluctuations, an AC four-probe Wheatstone bridge configuration was used to measure the fluctuations in resistanceat a fixed n . The time series data was digitized and decimated in multiple stages to obtainthe power spectral density (PSD), which was then integrated over the experimental band-width to obtain the normalized variance N = R S G G df = h G i G (Fig. S3a). Care was taken tominimize heating of the electrons by ensuring that the source-drain bias V SD . k B T /e .
3. DFT calculations
All first principles calculations have been performed within the framework of SIESTA [2]code. We use norm-conserving pseudopotentials [3] and Generalized Gradient approxima-tion [4] (PBE) for exchange correlation functional. A double- ζ plus polarization (DZP) basisset and a mesh cut-off of 300 Ry have been chosen for Brillouin zone integrations. For thecalculations of both GB(2 , | (2 ,
0) and GB(5 , | (3 , × ×
50 k-grid have been em-ployed. All the atomic positions are relaxed using conjugate-gradient algorithm until theforces were less than 0.04 eV / ˚A. 3
2. Weak localization measurements for D1 at T = 0 . K GB SG n = -6 cm -2 ( p s - ) T (K)-8 -6 -4 -2 0100200300400500 SG l ( n m ) n (10 cm -2 )GBD1, 0.3 K a b Figure S2. a , The l φ values at 0.3 K obtained from the magnetoresistance fitting for graphene isplotted for the SG and GB regions as a function of n . b , The scattering rates γ (obtained fromthe HLN fits to the magnetoconductance data) plotted as a function of temperature T for the SGand GB regions at n = − × cm − . S3. Noise for D1 at B = 0 T -10 -9 -8 -7 -6 -5 -4 -3 -2.2-0.6 S / ( H z - ) f (Hz) n (10 cm -2 ) / ( - time (s) a -6 -4 -2 0 210 -18 -17 -16 A S / n (10 cm -2 ) GB SG D1, 4.2 K b Figure S3. /f noise for D1 at zero magnetic field. a , The normalized power spectrum for D1at T = 4 . n showing clear 1 /f behaviour. The inset shows theconductivity fluctuations for these n clearly showing that the fluctuation magnitude is increasedat low n . b , The area-normalized power spectral density ( A × S σ /σ ) is shown for the SG and GBregions of D1 as a function of n at T = 4 . /f behaviour in the frequency domain as shown inFig. S3a. The area-normalized power spectrum ( A × S σ /σ ) as a function of n at 4.2 K isshown in Fig. S3b for the SG and GB regions. The noise in the GB region is larger thanthe SG region indicating more dynamic scattering arising from the localized states at theGBs [5]. S4. UCF from time-dependent conductance fluctuations
For time-dependent conductance fluctuations, an AC four-probe Wheatstone bridge con-figuration was used to measure the fluctuations in resistance at a fixed n and a fixed B .Two resistors, R R R ∼ R B at a fixed V BG ,the bridge is balanced by varying R f s (1000 samplesper second) using a 16-bit ADCs (analog-to-digital converters) NI USB-6210 from NationalInstruments with a maximum sampling rate of 250 Kilosamples per second, 4095 samplesof on-board memory and an input impedance of ∼
10 GΩ, which can be interfaced withLabVIEW programs. The data is stored in the on-board memory temporarily and thentransferred to the hard disk of the computer in segments for further processing.The time series data was then decimated in multiple stages to obtain the power spectraldensity (PSD) using the fast Fourier transform (FFT) technique, which was then integratedover the experimental bandwidth to obtain the normalized variance N = R S G G df = h G i G ateach B for a fixed V BG . Care was taken to minimize heating of the electrons by ensuringthat the source-drain bias V SD . k B T /e . More details of the experimental technique canbe found in Ref. [6]. The evaluation of the variance h δG φ i was obtained from statistically5eaningful windows of time ( ≈ ≈ G ). S5. UCF from gate voltage-dependent conductance fluctuations
UCF measurements were performed in D2 using the two-probe resistance configuration [1].The phase coherence length ( l φ ) was extracted from the WL MC curves using the modifiedHikami-Larkin-Nagaoka expression [7]. Successive gate voltage windows of 4 V interval werechosen such that the conductance does not change appreciably but significant fluctuationsare nevertheless present for a statistically meaningful analysis ( ∼
800 realizations). This
18 19 20 21 22-0.04-0.020.000.020.04 R ( k ) V BG (V) Residuals of the fit
18 19 20 21 221.31.41.5 R ( k ) V BG (V) Experimental data 4 th order polynomial fit -12 -8 -4 0 4 8 12 16 20 24 28 32024 R ( k ) V BG (V) -12 -8 -4 0 4 8 12 16 20 24 28 320.20.40.60.8 G ( e / h ) V BG (V) a bd c Figure S4. a , A typical plot of two-probe resistance as a function of back gate voltage for graphene,split into successive 4 V interval windows. b , The resistance fluctuations within a 4 V window rangeplotted along with its fourth order polynomial fit. c The residuals obtained from the fourth orderpolynomial fit of the curve in b . d The calculated values of h δG i corresponding to the differentgate voltage windows of the representative curve in a .
6s demonstrated in Fig. S4a for a typical transfer characteristic curve of graphene. Thetwo-probe resistance in each V BG window was then fitted with a fourth-order polynomialas shown in Fig. S4b, and then the variance of the fluctuations was calculated from theresiduals of the fit as shown in Fig. S4c. The fitting with a fourth-order polynomial wasdone to subtract a smooth background for calculating the fluctuations and to suppress slowlyvarying changes in the conductance. Fitting with polynomials of other orders do not giveany qualitative difference in the analysis [8, 9]. Finally, the magnitude of conductancefluctuations is obtained from the resistance fluctuations as h δG i = h δR ih R i (1)where h R i is the average four-terminal resistance corresponding to that particular gate volt-age window. However, this measured h δG i arises from the entire sample and in order toobtain the magnitude of conductance fluctuations within a phase-coherent box of area l φ × l φ ,the principle of superposition is invoked which gives h δG ih G i = 1 N h δG φ ih G φ i (2)where N = LW/l φ is the number of phase coherent boxes within the channel of length L and width W . Since h δG φ i = σ and h δG i = σW/L , h δG φ i = L W h δG i l φ (3)Both the phase breaking length ( l φ ) and the variance of conductance fluctuations ( h δG i )are dependent on carrier density and have to be evaluated experimentally to extract thedensity dependence of h δG φ i . 7
6. Magneto-noise data for devices D2 and D3
1. D2 at T = 0 . K D2, 0.3 K -0.5 ( B ) -4.1 n (10 cm -2 ) B (mT) -6.1
Figure S5. ν ( B ) for D2 at T = 0 . n in units of 10 cm − . Solidlines are fits to the crossover function. . D2 at T = 4 . K -0.04 -0.02 0.00 0.02 0.04 -0.9 B (T) -0.04 -0.02 0.00 0.02 0.04 B (T) -1.8 -0.04 -0.02 0.00 0.02 0.04 B (T) -3.0 -40 -20 0 20 4012
B (mT) -4.8 𝜈 ( B ) n (10 cm -2 ) D2, 4.5 K -0.04 -0.02 0.00 0.02 0.04 B (T) -0.04 -0.02 0.00 0.02 0.04 B (T) -0.04 -0.02 0.00 0.02 0.04 B (T) -40 -20 0 20 4012 𝜈 ( B ) n (10 cm -2 ) D2, 4.5 K
Figure S6. ν ( B ) for D2 at T = 4 . n in units of 10 cm − . Solidlines are fits to the crossover function. . D2 at higher T -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06
12 B (T)
D2, n = 6.1 x 10 cm -2 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06
12 B (T)
10 K -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06
12 B (T)
14 K 𝜈 ( B ) -60 -40 -20 0 20 40 6012 B (mT)
19 K ( = ) T (K) a b
Figure S7. a , ν ( B ) for GB of D2 at T = 7, 10, 14 and 19 K for n = 6 . × cm − . Solidlines are fits to the crossover function. b , The reduction factor ν ( B = 0) of the time-dependentUCF in the intergrain region plotted as a function of temperature T at high electron density n = 6 . × cm − . From Fig. S7, we observe that the spontaneous TRS breaking in graphene GBs happensonly at ultra-low temperatures below ∼ − ν ( B = 0) continues to increase as T increases up to ≈
10 K, after which ν ( B = 0) starts to decrease as quantum interferenceeffects begin to disappear due to loss of phase coherence. At higher temperatures, the 1 /f noise can no longer be attributed entirely to UCF [10].10 . D3 at T = 8 K B (mT) ( B ) D3, 8 K n (10 cm -2 ) -2.4-3-4.8-6 Figure S8. ν ( B ) for D3 at T = 8 K for different carrier densities n in units of 10 cm − . Solidlines are fits to the crossover function. . Single grain magneto-noise for D2 at T = 4 . K -0.04 -0.02 0.00 0.02 0.04
12 B (T) -0.9 n (10 cm -2 )SGD2, 4.5 K -0.04 -0.02 0.00 0.02 0.04
12 B (T) -2.4 -0.04 -0.02 0.00 0.02 0.04
12 B (T) -3.9 -40 -20 0 20 4012 B (mT) -6.6 𝜈 ( B ) -0.04 -0.02 0.00 0.02 0.04
12 B (T) n (10 cm -2 )SGD2, 4.5 K -0.04 -0.02 0.00 0.02 0.04
12 B (T) -40 -20 0 20 4012 B (mT) 𝜈 ( B ) Figure S9. ν ( B ) for the single grain region (without GB) of D2 at T = 4 . n in units of 10 cm − , showing factor-of-two reduction in the magneto-noise even athigh n unlike the case of the inter-grain region. Solid lines are fits to the crossover function.
7. Density functional calculations for atomically sharp grain boundaries
Extended defects
GB (2,0)|(2,0)
GB (5,0)|(3,3) a b
Stone – Wales defect
Vacancies
Point defects c Tilt GBs
GB (2,0)|(2,0)GB (5,0)|(3,3)GB (1,2)|(2,1)
DOS × 10 -2 eV -1 DOS × 10 -2 eV -1 Extended GBs
Figure S10. a , Schematic showing two categories of defects in graphene: point and extendeddefects. b,c , Bandstructure and density of states calculations for graphene with GB containing b , pentagon-octagon defects (GB(2 , | (2 , c , pentagon-heptagon defects (GB(5 , | (3 , S8. Density functional calculations for a grain boundary of finite width
GB(6 , | (6 ,
6) can be thought of as a clean translational GB(2 , | (2 ,
0) sandwiched be-tween two GB(3 , | (5 , , | (5 ,
0) consists of only Stone-Wales5-7 defects while GB(2 , | (2 ,
0) contains only 5-8 defect pairs. To match the periodic-ity in the direction parallel to the GB, five unit cells of GB(2 , | (2 ,
0) and two unit cells ofGB(3 , | (5 ,
0) were used resulting in a matching vector (10,0) in region between the two GBsand (6,6) in the region beyond either GB(3 , | (5 , , | (2 , × ×
50 k-grid was employed. However, for the newly proposed GB(6 , | (6 , , | (6 ,
6) was fixed at ∼
10 ˚A.Fig. S11a shows the band structure of GB(2 , | (2 ,
0) for three cases of zero-, hole- andelectron-doping. The ferromagnetic state arises only for the case of electron doping as theFermi level is raised due to Stoner instability. The localized core states in the GB regionsgive rise to flat bands near the Fermi level which split near the Γ-point resulting in a non-zero magnetic moment. On the other hand, the band structure and density of state (DOS)plots for GB(6 , | (6 ,
6) show spin-splitting not only near the Γ-point but throughout the13 bc d e Figure S11. a , Band structure of GB(2 , | (2 ,
0) where the three panels represent zero doping,hole doping of − . × cm − and electron doping of 5 . × cm − respectively. b , Left panelshows the band structure of and spin polarized density of states (DOS) for GB(6 , | (6 , c,d , Magnetic moment per unit lengthof the GB for both GB(2 , | (2 ,
0) and GB(6 , | (6 ,
6) as a function of c uniaxial strain along thearmchair direction and d carrier density n . e Isosurface showing the spin polarization densityalong the GB(6 , | (6 ,
6) at different densities of hole doping. entire Brillouin zone. The origin of the magnetic moments may be attributed to the in-builtlattice strain due to the additional 5-7 defects around the GB region. It is important to note14hat GB(6 , | (6 ,
6) has a lattice dimension five times larger than that of GB(2 , | (2 ,
0) inthe Y-direction, hence it is necessary to take zone folding into account while comparing theband structures of the two GBs. The additional flat bands near the Fermi level may comedirectly from the random distribution of Stone-Wales defects or via internal strain due totheir presence. The spin-split bands are not only spread throughout the Brillouin zone butalso across an energy range. The bands are split in the range of 1 eV above and below theFermi level, which explains the dependence of the magnetic moment per unit length as afunction of changing carrier density (Fig. S11d). The doping response can also be explainedfrom the spin polarized DOS plots for GB(6 , | (6 ,
6) shown in the right panel of Fig. S11b.The core atoms consist of the 5-8 defects which host almost all the magnetic moments asseen in the isosurface plots of Fig. S11e.Consistent with the findings of previous theoretical studies [11–13], 5-8 defects of trans-lational GB(2 , | (2 ,
0) were found to host a ferromagnetic ground state when the system isdoped [11] or strained [12]. Our theoretical calculations show that application of a tensilestrain along the zigzag direction or compressive strain along the armchair direction sig-nificantly enhances the magnetic moment per unit length along the GB (Fig. S11c). Thestrain is believed to arise during the growth and subsequent transfer process of the graphenegrains on to SiO substrate. Fig. S11c also reveals a striking difference between the two GBs.GB(6 , | (6 ,
6) exhibits a significantly large magnetic moment even in the absence of exter-nal strain. The response to an external strain along the armchair direction is much lesserfor GB(6 , | (6 ,
6) as compared to GB(2 , | (2 ,
0) since the additional 5-7 defects around thecore 5-8 defects in GB(6 , | (6 ,
6) are expected to absorb a significant fraction of the strainapplied.The variation of magnetic moments as a function of carrier density n is plotted inFig. S11d. For GB(2 , | (2 , ∼ − . × cm − ,far beyond the range accessible by an SiO back gate. This can be explained from the sharpresonance in the DOS located at about ± . , | (6 , , | (2 , , | (2 ,
0) as explainedearlier from the band structure and DOS shown in Fig. S11a. The isosurface of spin polar-ization density is showing for GB(6 , | (6 ,
6) in Fig. S11e for the undoped system as well astwo different hole doping densities. The spin polarization value was chosen as 1 . × − µ B for plotting these isosurfaces. For the undoped system, the magnetic moments are local-ized on the atoms along the zigzag edges in the middle of the GB, consistent with the caseof GB(2 , | (2 ,
0) [11, 12]. As the number density is tuned, the C-C dimer connecting thetwo zigzag edges begin to host magnetic moments as well. On increasing the hole densityfurther, some of the atoms of the different sublattice of the zigzag edge reverse their spinpolarizations which leads to a net enhancement of the overall magnetic moment of the GB.These moments may explain the spontaneous breaking of time reversal symmetry at higherdensities, if they interact to result in a frozen magnetic order.
S9. DOS per site for different types of GBs -20 -15 -10 -5 0 5 10 15 200.000.050.10 D O S pe r s i t e ( e V - ) n ( cm -2 ) pristine graphene GB(3,3)|(5,0) GB(2,0)|(2,0) GB(6,6)|(6,6) Figure S12. DOS per site as a function of n for pristine graphene (green), GB(3 , | (5 ,
0) withonly pentagon-heptagon defects (blue), GB(2 , | (2 ,
0) with only pentagon-octagon defects (red)and finite width GB(6 , | (6 ,
6) with multiple types of defects (yellow).
10. Mapping the defect disorder in a grain boundary of finite width a b C oun t s R (nm)
Figure S13. a , Experimental demonstration of defect-induced distortion on the parallel latticearray of a representative HRTEM image of a finite-width GB. b , Histogram obtained from theHRTEM image in a revealing an average defect distance R ≈ S11. Estimation of Kondo temperature T K The Kondo temperature T K can be extracted from the temperature dependence of thereduced sheet resistance ρ (cid:3) ( R (cid:3) in units of h/e ) at B = 0 . T -independentlongitudinal resistivity ρ c , electron-electron scattering-induced correction ρ e − e [14, 15] anda Kondo contribution ρ K [16, 17]: ρ (cid:3) = ρ c + ρ e − e + ρ K = ρ c + ( µ B − ρ c π A ln ~ k B T τ tr + ρ K , " (cid:18) TT K (cid:19) (2 / . − − . (4)where µ is the device mobility, the coefficient A = 1 + c h − ln(1+ F σ ) F σ i is a measure ofthe interaction strength (where F σ is the Fermi-liquid constant and c is the number ofcontributing multiplet channels), τ tr is the transport time derived from the Drude resistivity,and ρ K is the Kondo resistivity at zero temperature. Fig. S14a show the fits to Eq. 4 forthe resistivity versus temperature data of the SG (blue circles) and GB (red circles) regions17
10 1000.0650.0700.0750.080 GBSG
T (K) ( h / e ) n = -6 cm -2 ( h / e ) e-e K GB ( h / e ) T (K)
GB 0.10.2 ( h / e ) a b Figure S14. a , T -dependence of the reduced sheet resistance ρ (cid:3) (in units of h/e ) measured at B = 0 . n = − × cm − .The solid lines show fit to the Eq. 4, which includes electron-electron interaction corrections ρ e − e and the Kondo contribution to resistivity ρ K . b , The extracted values of ρ e − e (green solid line), ρ K (purple solid line) and the experimentally obtained ρ (cid:3) for the GB region at n = − × cm − are plotted as a function of T to show their relative magnitudes. at n = − × cm − , keeping ρ c , ρ K , , A and T K as fitting parameters. It is evident fromFig. S14a that ρ (cid:3) saturates at a slightly higher temperature for the GB region as comparedto the SG region, which manifests in a relatively high Kondo temperature T K ∼
20 K).Fitting the resistivity data for the GB region at n = − × cm − with Eq. 4, we get ρ c ≈ . h/e ), ρ K , ≈ . h/e ), A ≈ .
47 and T K ≈
20 K respectively. Finally, Fig. S14bcompares the relative contribution of the e-e interaction-induced corrections ρ e − e (green solidline) and the Kondo contribution ρ K (purple solid line) to the total experimental value (redcircles) obtained for the GB region at n = − × cm − .18
12. Temperature dependence of noise N = S σ /σ a b -9 -8 -7 D1 SG GB N ( = ) T (K) -6.9 cm -2 Figure S15. Normalized variance N = S σ /σ at B = 0 as a function of temperature is plotted forSG (blue) and GB (red) regions at n = − . × cm − for device D1. The noise N = S σ /σ at B = 0 in both the SG and GB regions of D1 at n = − . × cm − (Fig. S15) show clear non-monotonic temperature dependences. The noise initiallydecreases as the temperature is lowered in an almost linear fashion ( ∝ T ) as expected fromthe noise models based on carrier tunneling between the channel and trap states spread overan energy range ∼ k B T about the Fermi energy. Below T ≈
10 K (which matches withthe temperature scale in Fig. S7a), quantum interference contributions can no longer beneglected, and the increase in SG noise with decreasing T can be explained using the Feng-Lee-Stone (FLS) theory of UCF noise [18]. However, the comparatively sharper increase innoise of the GB region below T ≈ [1] P. A. Lee, A. D. Stone, and H. Fukuyama, Phys. Rev. B , 1039 (1987).[2] J. M. Soler, E. Artacho, J. D. Gale, A. Garc´ıa, J. Junquera, P. Ordej´on, and D. S´anchez-Portal,J. Phys.: Condens. Matter , 2745 (2002).
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