Probing Exfoliated Graphene Layers and Their Lithiation with Microfocused X-rays

X-ray diffraction is measured on individual bilayer and multilayer graphene single-crystals and combined with electrochemically induced lithium intercalation. In-plane Bragg peaks are observed by grazing incidence diffraction. Focusing the incident beam down to an area of about 10 μm × 10 μm, individual flakes are probed by specular X-ray reflectivity. By deploying a recursive Parratt algorithm to model the experimental data, we gain access to characteristic crystallographic parameters of the samples. Notably, it is possible to directly extract the bi/multilayer graphene c-axis lattice parameter. The latter is found to increase upon lithiation, which we control using an on-chip peripheral electrochemical cell layout. These experiments demonstrate the feasibility of in situ X-ray diffraction on individual, micron-sized single crystallites of few- and bilayer two-dimensional materials.

: Schematic multilayer representation of a flat sample probed by specular X-ray reflectivity as described within Parratt's recursion formalism. Red arrows represent X-rays incident onto the sample at an angle θ with respect to the surface. of a flat sample under an angle θ, is determined by the sample's composition at and near its surface (within a certain depth z into the bulk). To calculate this intensity, Parratt S2 introduced a recursive formalism approximating a flat sample as a slab consisting of N where and Here, a m is an amplitude factor and f m is a Fresnel reflection coefficient. σ m denotes the root mean square (rms) density variation at the interface between layers m and m − 1 (see S1). XRR cannot distinguish between a rough interface at z = z m between two media with refractive indices n m and n m−1 and a profile of the refractive index n m (z) that varies smoothly across the interface. As long as the thicknesses d m and d m−1 exceed σ m , the Névot-Croce factor s m may thus be derived using S4 For a given angle θ, Eq. 5 is calculated in recursive manner starting with m = N and ending with m = 1. Since the sample is penetrated only little by X-rays in the relevant case of small θ, the substrate may be considered infinitely thick and hence r N = 0. The intensity of X-rays specularly reflected from the sample is obtained as where I 0 is the intensity of the incident X-ray beam.
Due to the finite lateral extent of both the graphene flake and its supporting Si substrate, depending on θ the footprint of the incident X-ray beam may exceed either the former or even both (see Figs. 3a-c and S2a-c). We approximate the profile of the incident X-ray beam by an ellipse of width 2w and height 2h (w and h are the semi axes parallel to y and z · cos θ, respectively) as shown in Fig. S2d. The width of the footprint on the sample is identical to 2w irrespective of θ. This is not the case for its length 2h . Instead, h = h/ sin θ. The area of the elliptical X-ray beam footprint is thus A bfp has to be compared to both the lateral dimensions of a given graphene flake under study as well as of its supporting substrate. Especially at low angles θ, the X-ray beam footprint falls only partially on the graphene flake (probed area A G ). Another part of area A SiO 2 Figure S2: Schematic of the X-ray beam footprint (red ellipse) on a sample consisting of a graphene flake (dark grey) supported on a SiO 2 -terminated Si substrate (light grey), similar to Figs. 3a-c in the main text. (a-c) Illustration of the repartition of the X-ray beam footprint area for increasing incidence angle θ. Schematic top views are included in the bottom row.
(d) Elliptical profiles of the incident X-ray beam (upper part) and of its footprint on the sample (lower part). (e) Exemplary θ-dependence of the weighting factors A and B required for Eq. 12. The lower row shows snapshots of the substrate, graphene flake and footprint areas at three different values θ from which the weighting factors were computed. This example corresponds to the case of the bilayer graphene sample of Fig. 3d in the main text.
falls on the graphene-uncovered substrate, see Figs. S2a-c. Taking into account the lateral dimensions of both a given graphene flake and its SiO 2 -terminated Si substrate, A G (θ) and A SiO 2 (θ) are determined in straight forward manner by geometrical computation (Boolean S-5 operations on surfaces). Consequently, the intensity of specularly reflected X-rays (Eq. 10) comprises two weighted contributions and may be rewritten as where r 0,G and r 0,SiO 2 are the computed recursions (Eq. 5) for the graphene flake-covered and -uncovered parts of the sample, respectively. The weighting factors are In the upper panel of Fig. S2e, we plot A(θ) and B(θ) for the case of the graphene bilayer shown in Fig. 3d (main text). As is to be expected, B dominates at small θ and A dominates at larger θ. The initial steep increase of B(θ) up to θ S = 0.08 • reflects the finite lateral extent of the substrate l SiO 2 = 8 mm, as θ S = arcsin(2h/l SiO 2 ) = 0.08 • . This causes the well-known sin θ sin θ S -type increase in reflected intensity in the total-external-reflection regime. S4,S5 The XRR of a multi-layered sample is determined by the profile of electron density perpendicular to the sample surface, i.e., in z-direction. Fig. 3d in the main text shows two schematic examples in its side panels. Here, to first approximation, the densities ρ Si of the Si substrate and ρ SiO 2 of the SiO 2 layer are both assumed independent of z. We model a multi-layered graphene flake assuming identical Gaussian density profiles for each atomic layer. The resulting effective mass density profile S4 of a flake consisting of k graphene sheets stacked upon each other such that neighboring sheets are separated by a distance c can then be described as Here, q is related to the full width at half maximum (FWHM) of each Gaussian as FWHM = S-6 2q √ 2 ln 2 and R is determined by requiring where the bulk density of graphite ρ G times the interlayer spacing in graphite c G = 0.335 nm yields the areal density of one graphene sheet. We thus find R = ρ G c G /q √ 2π. In order to use Eq. (14)  Graphene is known to partially conform to the underlying SiO 2 (see Fig. S4a). S6, S7 We account for this in our calculations as follows. We calculate the θ-dependent XRR response for a density profile representing our systems as described above, however, with a flat SiO 2 surface (zero roughness). We repeat this calculation for different SiO 2 thicknesses d SiO 2 . Then we average all calculated XRR curves in weighted fashion. The probability p(d SiO 2 ) for each calculated curve to contribute to the overall XRR response decreases the more d SiO 2 deviates from the real value d SiO 2 . p(d SiO 2 ) is determined by integrating the probability distribution function (PDF), a normalized Gaussian centered at d SiO 2 with standard deviation σ SiO 2 (to reproduce the Névot-Croce result for the SiO 2 surface, Eq. 8), within ±∆z/2 around d SiO 2 (see Fig. S4b). Here, ∆z is the interval in d SiO 2 for which calculations are performed. In the Névot-Croce approach, the distance b of graphene from the SiO 2 surface (centered at d SiO 2 ) is fixed. In contrast, our approach allows to keep b locked to each d SiO 2 , thereby mimicking conformal adhesion. We find the overall XRR response thus calculated to typically reveal S-7 Figure S3: Schematic density profiles for a 3-layer graphene flake supported on a SiO 2terminated Si substrate. The z-dependent density of graphene sheets is approximated by Gaussian distributions according to Eq. (14). For illustration purposes, we show two extreme scenarios with (a) q ≈ 0.127 nm and (b) q ≈ 0.042 nm.
signatures of the graphene system to an extent that agrees better with the experiment. Fig. S4c is an example, where the orange (blue) curve is calculated for identical parameters but without (with) conformation.
S-8 Figure S4: (a) Schematic of conformation of graphene to the roughness of the underlying SiO 2 . (b) Schematic illustrating how the normalized Gaussian-shaped PDF characterized by the standard deviation σ SiO 2 relates to p(d SiO 2 ), the weight of the contribution of the calculated XRR response of a system with d SiO 2 to the overall XRR response. (c) Exemplary calculations of the overall XRR response of a bilayer graphene system on SiO 2 , without (orange) and with (blue) conformal adhesion taken into account as described in the text.

S-9
In Figs. S5-S7 we again show the measured and calculated data as in Figs

Beam-induced contamination
With the polymer electrolyte present, we systematically observe beam-induced contamination on the sample surface after having performed XRR measurements. In scanning electron micrographs, this contamination appears as broad, dark streaks located where the synchrotron beam irradiated the sample. Three such micrographs are displayed in the top row of Fig. S8. Using atomic force microscopy, we extract height profiles of the contamination at two different locations for each sample (lower row). The thickness tends to be larger at areas that received more integrated irradiation. The deposited material is possibly due to hydrocarbons cracked by X-ray photons or photoelectrons. S8-S10 Among different possible sources, hydrocarbons likely originate from our solidified polymer electrolyte that contains species with non-negligible vapor pressure. Figure S8: Scannning electron micrographs of three samples acquired after XRR measurements (top row). The rim of each graphene flake is demarcated by a white dashed line. The polymer electrolyte is colored yellow. A near-horizontal broad, dark streak easily visible in each micrograph is due to beam-induced contamination. Lower row shows height profiles (obtained by atomic force microscopy) across the beam-induced contamination at two locations indicated in the respective top panel.