Structure and electronic transport in graphene wrinkles

Transcription

1 Supplementary Information: Structure and electronic transport in graphene wrinkles Wenjuan Zhu*, Tony Low, Vasili Pereeinos, Ageeth A. Bol, Yu Zhu, Hugen Yan, Jerry Tersoff and Phaedon Avouris* IBM Thomas J. Watson Research Center, Yorktown Heights, NY 1598, USA 1. Maximum height estimate of the standing collapsed wrinkle. To estimate the maximum height, we assume that the wrinkle has a fixed amount of material (i.e. no sliding of graphene along the surface), and that it adopts the minimum-energy morphology. We therefore need to compare energies of the different structures for a fixed amount of excess graphene in the wrinkle. First, we estimate energy of the folded wrinkle. As illustrated in Fig. S1, the structure of folded wrinkle consists of the right and left ul-shaped curves with similar radiuses and a flat trilayer region. The right ul we approximate y a pair of arcs, concave and convex. The two left uls are approximated y the arcs of the same angles, with radiuses eing different y the van der Waals distance h separating graphene layers. The trilayer region has length λ, and the ilayer has length λ+ξ. The ase is approximated y arcs of angle π/2 and radius R. The energy of the folded wrinkle in this model is given y: E f κ π θ1 θ1 θ2 θ2 θ3 θ 4 = R R1 R1+ h R2 R2 + h R3 R4 h h β θ1 R1+ + θ2 R λ+ ξ β + su ( 2R h) (1) where κ is graphene ending stiffness and β is van der Waals adhesion energy. The first term reflects the ending energy, the second term reflects adhesion energies of the ilayer and trilayer regions, and the last term reflects the adhesion energy cost to peel off

2 graphene from the sustrate. We will use β su =β. The excess length is defined as the length of the graphene fold minus the length of the flat sustrate and it is given y: ( 2 ) ( 2 ) 2 ( 2 ) L= πr + θ R + h + θ R + h + λ+ ξ + θ R + θ R R + h (2) where relationships etween the angles and radiuses are determined y the geometric constraints, see Fig. S1: ξ = R4sinθ4 R3sinθ3, θ1 θ2 = π 2, ( )( θ ) R1sinθ1+ R = h+ R2 + h 1 cos 2, θ3 θ4 π =, R ( 1 cosθ ) h R ( 1 cosθ ) = Minimization of energy in Eq. (1) with respect to the five variational parameters R, R 1, θ 1, R 3, θ 3 for a fixed excess length L from Eq. (2) gives the energy of the fold as a function of L. Similarly, we can estimate the energy of the standing collapsed wrinkle geometry, shown in Fig. S2: κ π 2θ1 2θ 2 Esc = + + βλ + βsu R + h 2 R R1 R2 ( 2 ) (3) The excess length here is given y: L= π R + 2λ+ 2θ R + 2θ R 2R h (4) where θ1 θ2 π 2 = +, R sinθ h 2 R ( 1 cosθ ) = + are found from the geometrical constraints. Therefore, there are three variational parameters: R, R 1, θ 1 which minimize the energy in Eq. (3). Commonly used values of κ=1.4 ev 1 and β corresponding to 4 mev adhesion energy per caron atom 2,3 suggest an intrinsic length scale R = κ 2β 6.8Å. Numerical energy minimization from Eq. (1) and (3) using parameters R =6.8 Å and

3 h=3.4 Å leads to the values of the variational radiuses of the left and right uls in the folded wrinkle: R Å, R Å (see Fig. S1 caption) to e in very good agreement with the values found from the DFT optimized geometry of 5 6 Å 4. The minimum energy of the standing collapsed wrinkle from Eq. (3) as a function of L is given as Esc L 14.78, while minimum energy of the folded wrinkle from Eq. (1) is given β R 2 R y E f L The equal energy condition E sc = E f is satisfied for Lm 24.7R, β R R which defines a transition height from standing collapsed wrinkle to folded wrinkle as R + λ + R 1 cosθ + R sinθ, where λ 7.9R is found from Eq. (4). The height of the ( ) standing wrinkle at the transition (i.e. the maximum height) is aout 12.4R 8.4 nm, very close to L m /2. 2. Electrostatic modeling of the trilayered folds regions We model the electrostatics of the graphene fold as a tri-layer graphene system, assuming that the graphene layers are electrically decoupled from one another. Through the Poisson equation, the Dirac point potential in each layer with respect to Fermi energy can e computed as follows, d V = ε n + V ( ) ( ) d V = ε n + n + V V = n + n + n + V g 1 C g (5) where V 3 is given a priori. d = 3.4Å is the graphene interlayer separation, ε is the free space permittivity, C g is the ack gate capacitance and V g is the applied gate ias. In Fig. 4a of the main manuscript, the calculated carrier densities assumed a finite electron-hole puddle densities n = cm -2 estimated from Hall measurements. The fractional carrier population in the graphene layer closest to the gate, i.e. n 1 /n where n = n 1 +n 2 +n 3, is closer to unity at larger V g. On the other hand, the layer densities are more equally

4 distriuted when V g is iased near the Dirac point. This carrier redistriution within the trilayered graphene system is a consequence of nonlinear screening 4, and is crucial to explaining our experimental oservations. 3. Diffusive transport modeling along/across a graphene fold We discuss first electronic transport along a graphene fold. The effective electrical conductivity σ eff in the diffusive limit can e written as, W W W = ( + + ) + (6) W W σ f σ σ σ f σ eff where σ j refers to the electrical conductivity in the j th layer and σ is the electrical conductivity in monolayer graphene i.e. control devices. W f is the width of the graphene fold, estimated from SEM to e.14µm, and W is the device width. In addition, the electrical conductivity σ as a function of the carrier density n can e determined through Hall measurements. The carrier mean-free-path, λ MFP (n), can simply e derived from σ = π λ 5. If each graphene layer in the fold also follows the same λ MFP (n) 2 4e π h n MFP functional relationship, then the respective σ j are also known. In this case, the calculated σ eff is shown in Fig. 4c of the main manuscript, yielding good agreement. Electronic transport across a graphene fold can e modeled in similar fashion, with σ eff written as, 1 Lf L Lf 1 = σ eff L σ1 σ2 σ3 L σ (7) where L f is the length of the graphene fold, estimated from SEM to e.14µm, and L is the device length. 4. Quantum transport modeling of standing collapsed wrinkle

5 Here we elaorate on the electronic transport calculation of the standing collapsed graphene wrinkle in the main manuscript. We assume that the transport direction is along the armchair direction, as illustrated in Fig. 5a. The Hamiltonian H is descried y a nearest neighor couplings, p z tight-inding model 6 including oth in-plane and out-of-plane (8) H= Vaa+ taa + saa i i i ij i j ij i j i < ij> ij where Vi denote the on-site energy, t ij the in-plane coupling and s ij the out-of-plane coupling. Explicitly, they are expressed as, t s ij ij ( vij pi ) ( vij pj ) vij pi vij pj = r r v p v p ij ij ij i ij j r r p v p v = αγ exp δ rij rij ij p i ij j ij ε π pp (9) where p i refers to the local out-of-plane vector, v ij is the ond vector and r ij = v. ij Parameters r.34nm refers to the equilirium graphene interlayer separation, p π γ.119ε pp is the out-of-plane coupling energy, δ.185 3r where r =.142nmis the caron-caron ond-length and α 1.4is a fitting parameter 6. Electronic transport across the structure is calculated using the non-equilirium green function method 5 within the Landauer formalism, assuming periodic oundary condition along the transverse width direction. The transmission function ( k, y E) Τ can then e calculated. The finite temperature device conductance can e calculated using, G = f E Τ k E de (1) 2 2e 1 E µ exp 2 ( ) ( y, ) h kbt kbt ky

6 where f ( E) is the Fermi Dirac distriution. The resistance associated with the standing collapsed wrinkle can then e calculated after sutracting off the quantum contact resistance. In our calculations, we assume that the electrostatic doping of the flat region to e.2ev, and undoped in regions which are raised, namely the collapsed ilayer and the structure sutended from it. Temperature is taken to e 3K as per experiments. 5. Temperature dependence of conductivity The conductivities of the graphene device at 4.2K and 3K are shown in Fig. S3. We oserve that the conductivity is nearly unchanged when the temperature is decreased from 3K to 4.2K. 6. Measurements of Hall Moility We performed standard Hall measurement to otain the resistivity tensor and then the conductivities σ xx and σ xy, from which the carrier moility µ and carrier density n can e extracted. 1 σ xy µ = (11) B σ xx 2 2 σ xx(1 + µ B ) n = (12) µ q where B is the magnetic field. These quantities are plotted in Fig. S4, otained at 3K. We emphasize that the extraction method reaks down when the graphene is iased near the Dirac point, the range highlighted in the plot. The oserved downturn in the moility is unphysical, an artifact of the extraction method which ignores the two carrier nature of transport near the Dirac point 7. Outside this region, the measured moility clearly shows a decreasing moility with increasing doping. 7. Dirac point shifts due to folds

7 The statistical sampling of Dirac voltage for the graphene Hall-ars with and without fold is shown in Fig.S5: (a) across fold vs no fold; () along the fold vs no fold. The statistics of Dirac voltage value indicate that the presence of a fold does not lead to significant changes in the Dirac point shifts, hence of the doping level. This indicates that most of the trapped impurities reside in the sustrate or the SiO 2 -graphene interface.

8 Reference: 1 Chopra, N. G. et al. Fully Collapsed Caron Nanotues. Nature 377, , (1995). 2 Girifalco, L. A. & Lad, R. A. Energy of cohension, compressiility, and the potential energy functions of the graphite system. Journal of Chemical Physics 25, , (1956). 3 Zacharia, R., Ulricht, H. & Hertel, T. Interlayer cohesive energy of graphite from thermal desorption of polyaromatic hydrocarons. Physical Review B 69, (24). 4 Kuroda, M. A., Tersoff, J. & Martyna, G. J. Nonlinear Screening in Multilayer Graphene Systems. Physical Review Letters 16, 11684, (211). 5 Datta, S. Electronic transport in mesoscopic systems. (Camridge University Press, 1997). 6 Uryu, S. & Ando, T. Electronic intertue transfer in doule-wall caron nanotues. Physical Review B 72, 24543, (25). 7. Zhu, W., Pereeinos, V., Freitag, M. & Avouris, P. Carrier scattering, moilities, and electrostatic potential in monolayer, ilayer, and trilayer graphene. Physical Review B 8, 23542, (29).

9 R 2 θ θ 4 2 λ R 4 R π/2 θ 1 R1 h h ξ R 3 θ 3 Fig. S1. Schematics of the folded graphene wrinkle. The minimum energy from Eq. (1) corresponds to the values of the variational parameters R = R, R1.95R, R2 3.25R, θ1.764π, θ2.264π, R3.714R, R R, θ π, θ4.246π, where R =6.8 Å. θ 1 R 1 θ 2 R 2 λ h R π/2 Fig. S2. Schematics of the standing collapsed graphene wrinkle. The minimum energy from Eq. (3) corresponds to the values of the variational parameters R 2π ( 2 π) R R1.967R, R2 3.41R, θ1.685π, θ2.185π, where R =6.8 Å. = +,

10 a c Conductivity σ (ms) No fold 4.2K 3K V BG -V Dirac Across fold 4.2K 3K V BG -V Dirac Along fold 4.2K 3K V BG -V Dirac Fig.S3. Conductivity as a function of V BG -V Dirac at 4.2K and 3K for a graphene device (a) with no fold, () measured across the fold and (c) along the fold. a µ (Vs/cm 2 ) V BG -V Dirac n (x1 12 /cm 2 ) V BG -V Dirac Fig.S4. Extracted (a) Hall moility and () carrier density in graphene at 4K, via the standard Hall measurement procedure.

11 a V Dirac no fold across fold V Dirac no fold along fold Device type Device type Fig. S5. Statistics of Dirac voltage of graphene Hall-ars (a) across fold vs no fold, and () along fold vs no fold. The statistics are ased on the data from 42 devices.