Supplementary information for Tunneling Spectroscopy of Graphene-Boron Nitride Heterostructures F. Amet, 1 J. R. Williams, 2 A. G. F. Garcia, 2 M. Yankowitz, 2 K.Watanabe, 3 T.Taniguchi, 3 and D. Goldhaber-Gordon 2 1 Department of Applied Physics, Stanford University, Stanford, CA 9435, USA 2 Department of Physics, Stanford University, Stanford, CA 9435, USA 3 Advanced Materials Laboratory, National Institute for Materials Science, 1-1 Namiki, Tsukuba, 35-44, Japan (Dated: November 9, 211) FABRICATION DETAILS Graphene and h-bn flakes are exfoliated with Nitto tape on oxidized silicon wafer pieces, and annealed at 35C in Ar/H2 atmosphere for 4 hours. We then transfer h-bn on top of graphene prior to any lithographic step, in an attempt to minimize organic contamination at the tunneling interface. The h-bn flakes are coated with a PMMA layer (PMMA 95k, 5% in anisole, spun at 2krpm and baked at 16C for 5 minutes) prior to transfer. We use a 2% solution of potassium hydroxide heated up at 1C to etch the sacrificial layer of SiO 2. After lift-off, the PMMA membrane with the h-bn flake attached to it is transferred into successive DI water baths to avoid KOH contamination. The membrane is then transferred on top of a pierced glass slide, so that the h-bn flake is suspended across the hole. It is then baked at 11C for the edges of the membrane to stick to the glass support. The rigid glass slide is then easily aligned with a micromanipulator on top of the target substrate. The stack is then baked at 11C to promote adhesion, before dissolving the PMMA in acetone. The top gate and contacts to the graphene flake are patterned using e-beam lithography. Prior to metallization, we clean the contacts in UV ozone for 2 minutes to remove residual PMMA contamination. DEVICE GEOMETRY The graphene flakes are annealed in Ar/H 2 at 35 o C and boron nitride flakes are transferred on top of them prior to any other processing, which allows for the interface between the two flakes to be clean (See Fig. S1 for an optical image of a completed device). The top-gated part of the graphene flake is several square microns large. However, the tunnel conductance is an exponential function of the barrier thickness, so the effective tunneling area depends strongly on the cleanliness of the interface. In fact, and as speculated in the main article, impurities in between the boron nitride and graphene can alter the nature of tunneling, resulting in the differences observed in the tunnel conductance g of device A and B. The measured capacitance ratio C T /C G 72(1) for device A(B) is smaller than the theoretical value of 15 given by a parallel plates model for this geometry. This difference is not fully understood but can be due to an effective dielectric constant lower than expected for the h-bn layer, or to an imperfect screening of the electric field of the gates by the graphene sheet. CHARGE PUDDLE SIZE Near the charge neutrality point in graphene, the density of carriers breaks up into a series of n-and p-type puddles [S1, 2], which behave as quantum dots [S3]. The typical size of the charge puddles in our devices can be extracted from the Coulomb diamonds observed in Fig. 4 of the main article. If one defines the capacitance ratios α i = C i /C total where i refers to the top gate (i=t) and back gate (i=g) capacitances, the edges of the Coulomb diamonds have slopes given by: V G V T = 1 α T 2α G and 1 α T 2α G (1) From fits to the edges of the diamonds, it is therefore possible to extract the capacitance ratios C T /C G of 72 and 1 respectively for device A and B. The capacitance of the puddle-induced quantum dots to the back gate is given by the periodicity of conductance oscillations at V T = and as a function of the back gate voltage. We substract a smooth background from g(v T =, V G ) and calculate the fast Fourier transform of δg. For example in the case of device A, we find a periodicity V G of 7 V, which corresponds to a capacitance to the back gate of approximately 2 1 2 F. The back gate capacitance per unit area of the silicon oxide layer has been measured to be 12 nf/cm 2 on different devices, which allows for an estimation of the typical dot size: 2 nm 2 in the case of device A, similar to that observed in Ref. [S2]. A similar procedure is used to extract a puddle area of 72 nm 2 for device B. EXTRACTION OF THE FERMI VELOCITY The tunnel current can be expressed as:
2 I(V T ) It follows that g is given by: ev T ρ(e F + ɛ)t (ɛ, ev T )dɛ. (2) g t = di eρ(e F ev T )T ( ev T, ev T ) (3) dv T + ev T d dv T ρ(e F + ɛ)t (ɛ, ev T )dɛ Using the WKB approximation it is possible to estimate the tunnel transmission as a function of the barrier thickness d, the transverse electron mass in boron nitride m, the average barrier height U and the parallel momentum of the tunneling electron k // : T ( ev T, ev T ) = exp ( 2d 2m U + (.k //) 2 2m ev T 2 (4) In our case, the barrier height U is comparable to half of the band-gap in boron nitride which we approximate by 4 ev. Moreover, for electrons tunneling elastically at the K point, the parallel momentum k // is approximately equal to K 1.7 Å which is fairly high. As a consequence the tunneling energy ev t is small compared to the effective barrier height: U + (.k //) 2 2m, and the tunnel transmission T ( ev T, ev T ) varies slowly with V T as long as inelastic tunneling is negligible, as observed on device A [S4]. In that case, as a first approximation, one can neglect the variations of the tunnel transmission and write: g = di dv t ρ(e F ev T ) + ρ(e F ev T ) + de F d(ev T ) ev T ) d dɛ ρ(e F + ɛ)dɛ de F d(ev T ) (ρ(e F ) ρ(e F ev T ))(5) In graphene the Fermi energy is proportional to n de and the derivative F d(ev T ) should diverge close to the charge neutrality point. However, we ve seen that the density of states saturates at a constant value at low carrier density [see Fig. 1(a) of the main article], and this divergence does not occur. As a consequence, the second term of Eq. 5 remains very small compared to ρ(e F ev ) for all applied voltages used in the experiment and the contours of constant tunnel conductance are very well approximated by curves of constant E F ev. When the carrier density in the graphene sheet is large compared to the intrinsic doping n, the Fermi energy is given by: E F = v F πn = vf π e (C T V T + C G V G + en ) (6) Using this expression it is possible to find an analytical expression for the contours of constant E F ev. We find that these are parabolas of constant curvature: 2 V G V 2 T 2e 3 ± πc G ( v f ) 2. (7) Knowing the back gate capacitance, we can estimate the Fermi velocity from fits to these parabolas, and find a value of 9.45 1 5 m/s. SIMULATION OF THE TUNNEL CONDUCTANCE Our goal in the simulation was not to determine the density of states for disordered graphene from ab-initio calculations, but to see what are the contours of constant tunnel conductance for a density of states resembling what was observed in Fig. 2 of the main article. To this end, we estimated the tunnel conductance from the WKB formula with an empirical density of states reproducing the main features in Fig. 2 (main article). We neglected variations in the tunneling transmission, assuming that g ρ G (E F ev t ). The Fermi energy itself is calculated by integration of the density of states. The density of states is approximated by the theoretical expression: ρ G (E) E which is smoothly truncated to ρ G (E) = ρ under a cutoff energy that we take equal to.1 ev. Randomly placed Lorentzian peaks of random widths and heights are added to this expression to simulate the resonant peaks we observed in Fig. 1(a) (main article). An example of the density of states we use is displayed on Fig. S2. We then numerically integrate this density of states to get the Fermi energy as a function of the carrier density, and calculate the tunnel conductance g t as a function of both gate voltages (Fig. S3). We see that resonant peaks in the density of states give rise to two sets of curves: diagonal straight lines corresponding to constant E F lines, and curves of constant E F ev, similar to the observed features of Fig. 2(b) (main article). S1. J. Martin et al., Nat. Phys. 4, 144 (28). S2. Y. Zhang et al., Nat. Phys. 6, 722 (29). S3. S. Jung et al., Nature Phys. 7, 245 (211). S4. This approximation breaks down if inelastic tunneling can t be neglected, as observed on device B. In that case, K out-of-plane phonons have been shown to considerably lower the effective barrier height, and as a consequence, the tunnel transmission varies much faster as a function of V t.
3 Bare SiO2 Boron nitride Top gate Graphene tunnel junction with a 2 nm thick h-bn tunnel barrier. The contacts and top gate are made by standard e-beam lithography (Ti/Au 1nm/5nm). The edges of the graphene flake are overlaid with dashed lines for clarity FIG. S1:
.5 4.4 ρ G (au).3.2.1.5.5 E (ev) FIG. S2: Simulation of the density of states in disordered graphene
5 5 VBG (V) -5 -.1 V T (V).1 FIG. S3: Simulation of G T as a function of the top gate voltage V T and the back gate voltage V bg