monolayer graphene on hbn round and nearly round bubbles triangular with smooth top triangular and trapezoid with sharp ridges R (L ) (n m )

Transcription

1 Count a 3 monolayer graphene on hbn 2 round and nearly round bubbles triangular with smooth top triangular and trapezoid with sharp ridges 1 b R L) nm) 3 m o n o la y e r h B N o n h B N C o u n t 2 1 ro u n d a n d n e a rly ro u n d tria n g u la r w ith s m o o th to p s c R L ) n m ) m o n o la y e r M o S 2 o n M o S 2 ro u n d a n d n e a rly ro u n d C o u n t R n m ) Supplementary Figure 1. Frequency of occurrence for bubbles of dierent shapes. Shown are histograms of bubble sizes corresponding to dierent shapes. a) monolayer graphene on an hbn substrate ; data collected from an area of 12 µm 2. b) monolayer hbn on an hbn substrate, data collected from an area of 12 µm 2. c) monolayer MoS 2 on an MoS 2 substrate, data collected from an area of 4 µm 2.

2 2 a b δ R tip Supplementary Figure 2. AFM tips used in indentation experiments. a) Scanning electron microscope SEM) image of one of the AFM tips. The dashed circle emphasises its spherical shape. b) Three-dimensional image of another AFM tip obtained using the Bruker calibration procedure see Methods). The plane section indicates the maximum range of indentation depths corresponding to applicability of our numerical tting procedure.

3 3 a 5 F n N ) In d e n ta tio n b F n N ) S a p p h ire s u b s tra te firs t ru n s e c o n d ru n a fte r 3 s e c 2 4 d n m ) n m r n m ) h n m ) n m d n m ) Supplementary Figure 3. Reproducibility of force-displacement curves and denition of zero indentation. a) Two force-displacement FDC) curves taken on the same graphene bubble R =317 nm). Black line shows an FDC taken with the same AFM tip on a sapphire substrate. b) Example of an FDC obtained by full-height indentation of a graphene bubble see Methods) ; the inset shows a prole of the same bubble obtained in the scanning mode prior to indentation. Crossing dashed lines in the main panel indicate δ = zero indentation) ; the vertical part of the FDC corresponds to indentation of the substrate i.e., d reaches the height of the bubble, h max). The distance travelled by the AFM tip until it reaches the substrate, d = 34.8 nm is in excellent agreement with the measured bubble's height h max = 34.3 nm.

4 4 SUPPLEMENTARY NOTE 1. ELASTIC ENERGY. The elastic energy of one bubble, dened by a prole h r) is µλ + µ) 1 E el = 2λ + 2µ) 4π 2 d 2 q q 2 f i,j q) i,j 2 S.1) where i, j = x, y and f ij q) = d 2 re i q r i h r) j h r) S.2) For an isotropic bubble, h r) = hr), where r = r, Supplementary Equation S.1) becomes E el = π µλ + µ) 8 λ + 2µ 2 dqq drr [J qr) J 2 qr)] r h) 2 where J u) and J 2 u) are Bessel functions. In the presence of an isotropic strain, ɛ xx = ɛ yy = ɛ/2, the bubble acquires an additional energy, ɛµλ + µ) E str = λ + 2µ lim q q 2 f i,j q) i,j S.3) S.4) where we have used the relation ɛ q) = 4π 2 ɛδ 2 q). For an isotropic height prole, hr), Supplementary Equation S.1) reduces to E str = πɛµλ + µ) 2λ + 2µ) drr [ r hr)] 2 Alternatively, for an isotropic height prole, hr), one can write the elastic energy as E el = 2π λ [ drr r u r + u r 2 r + rh) 2 ] 2 [ + 2πµ drr r u r + rh) 2 ) 2 ] + u2 r 2 2 r 2 The displacement u r can be obtained from λ + 2µ) r 2 u r + ru r u ) r r r 2 = λ + 2µ) r 2 h ) r h) + µ r rh) 2 S.7) S.5) S.6) SUPPLEMENTARY NOTE 2. INFLUENCE OF TENSILE STRAINS. Tensile strains, ɛ >, modify the aspect ratio of the bubbles, as shown in Supplementary Equation S.7) in the main text. The changes signicant when the term proportional to ɛ becomes a signicant fraction of the total elastic energy. By comparing these contributions in Supplementary Equation S.7), we nd a threshold strain, ɛ th ɛ th c 1 c 2 hmax R ) 2 = c 1 πγ ) 1/2 S.8) c 2 5Y Using γ.15 ev, and the Young modulii for graphene and MoS 2 we nd, for graphene, ɛ th 1.4%, and for MoS 2, ɛ th 1.9%. For larger tensile strains,the aspect ratio tends to h max πγ R S.9) 2c 2 ɛy The bubbles become shorter in the presence of a tensile strain. The dependence of the pressure on the height of the bubble, Supplementary Equation S.18) in the main text is not signicantly modied, and we nd P πγ c V h max S.1)

5 5 SUPPLEMENTARY NOTE 3. INFLUENCE OF COMPRESSIVE STRAINS. In the presence of compressive strains, ɛ <, the aspect ratio h max /R, increases. A suciently high compressive strain will overcome the vdw interaction, and the graphene layer will delaminate, forming a bubble even in the absence of trapped material, or can even completely detach from the substrate. If we do not consider the internal energy of the material inside the bubble, a bubble formed spontaneously with radius R and height h max has an energy, Minimizing this expression with respect to h max, we nd E tot = c 1 Y h4 max R 2 c 2 Y ɛ h 2 max + πγr 2 S.11) The total energy of the bubble, as function of R, is h 2 max R 2 = c 2 ɛ 2c 1 S.12) E tot = Y c2 2 ɛ 2 4c 1 R 2 + πγr 2. S.13) The bubble will be stable and grow towards R if the compressive strain is such that Y c 2 2 ɛ 2 The compressive strain required for the bubble to be unstable is 4c 1 πγ ɛ Y c 2 2 4c 1 > πγ S.14) S.15) These strains are 6.2% and 8.7% for graphene and MoS 2, respectively. On the other hand, for suciently small strains, we can expand the solution of Supplementary Equation S.8) in the main text, and we obtain h max R ) 1/4 c4 γ c 2ɛ 5c 1 Y 5c 1 Y 1c 1 πγ, that is, for ɛ < compressive strain) h max /R will be greater than in the unstrained situation. S.16) SUPPLEMENTARY NOTE 4. INTERACTION BETWEEN BUBBLES. Bubbles can only interact when the in plane displacements relax to the height prole. We can extend eqs.s.1) and S.3) to the case of two non overlapping proles, h 1 r) and h 2 r). Leaving out the self energy of each bubble, we obtain µλ + µ) 1 E int = 2λ + 2µ) 4π 2 d 2 qe i q R i,j q 2 fi,j q) 1 i,j q 2 fi,j q) 2 S.17) where R is the vector connecting the centers of the two bubbles, and fi,j k q) is, as before, the Fourier transform of i h k r) j h k r). E int R) = π 8 µλ + µ) λ + 2µ { dqqj qr) drr [J qr) J 2 qr)] r h 1 ) 2 drr [J qr) J 2 qr)] r h 2 ) 2 } S.18)

6 6 where R = R. A long distances R l, we obtain where E int R) απ 8 µλ + µ) λ + 2µ)R 2 drr r h 1 ) 2 drr r h 2 ) 2 α = lim r 2 dqqj qr) q S.19) S.2) The interaction between bubbles is attractive, favouring their merger into bigger bubbles, as observed experimentally.