Landau Quantization of a Narrow Doubly-Folded Wrinkle in Monolayer

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Landau Quantization of a Narrow DoublyFolded Wrinkle in Monolayer Graphene Chuanxu Ma, Xia Sun, Hongjian Du, Jufeng Wang, Mingyang Tian, Aidi Zhao, Yasushi Yamauchi, and Bing Wang Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.8b02243 • Publication Date (Web): 24 Oct 2018 Downloaded from http://pubs.acs.org on October 24, 2018

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Landau Quantization of a Narrow Doubly-Folded Wrinkle in Monolayer Graphene Chuanxu Ma§,†, Xia Sun§,‡,†, Hongjian Du§, Jufeng Wang§, Mingyang Tian§, Aidi Zhao§, Yasushi Yamauchi‡, Bing Wang §,* §

Hefei National Laboratory for Physical Sciences at the Microscale and Synergetic

Innovation Center of Quantum Information & Quantum Physics, Key Laboratory of Strongly-Coupled Quantum Matter Physics (CAS), University of Science and Technology of China, 96 Jinzhai Road, Hefei, Anhui 230026, P. R. China ‡

National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan *Email: [email protected]; Phone: +86-551-63602177

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ABSTRACT Folding can be an effective way to tailor the electronic properties of graphene, and has attracted wide study interest in finding its novel properties. Here we present the experimental characterizations of the structural and electronic properties of a narrow graphene wrinkle on a SiO2/Si substrate using scanning tunneling microscopy/spectroscopy. Pronounced and nearly equally separated conductance peaks are observed in the dI/dV spectra of the wrinkle. We attribute these peaks to pseudo-Landau levels (PLLs) that are caused by a gradient-strain-induced pseudo-magnetic field up to about 42 T in the narrow wrinkle. The introduction of the gradient strain and thus the pseudo-magnetic field can be ascribed to the lattice deformation. A doubly-folded structure of the wrinkle is suggested. Our density functional theory calculations show that the band structure of the doubly-folded graphene wrinkle has a parabolic dispersion, which can well explain the equally-separated PLLs. The effective mass of carriers is obtained to be about 0.02me (me: the rest mass of electron), and interestingly, it is revealed that there exists valley polarization in the wrinkle. Such properties of the strained doubly-folded wrinkle may provide a platform to explore some exciting phenomena in graphene, like zero-field quantum valley Hall effect.

Keywords:

graphene,

doubly-folded

graphene

wrinkle,

scanning

tunneling

microscopy/spectroscopy, pseudo-Landau levels, parabolic band structure, valley polarization


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Folding is a routine manner to achieve diverse forms and functions of materials,1-3 from the paper origami to the coiling of proteins.4 Graphene, a two-dimensional material formed by sp2-carbon with a hexagon honeycomb structure,5 although possessing in-plane stiffness,6 can easily buckle into different shapes, just like a sheet of paper. Folding graphene leads to a structure of multilayered graphene with curved folding edges.7-15 The curvature and stacking in folded graphene, changing the nearest-neighbor hopping and introducing interlayer hopping, will modify the low-energy linear dispersion.5 The existence of curved edges causes lattice deformation and thus could induce strain in the deformed area. Many theoretical works have predicted that a gradient strain can induce gauge field,16-19 which acts like an effective magnetic field and leads to the formation of pseudo-Landau levels (PLLs).20-23 The PLLs have been experimentally observed in some strained graphene structures, like nanobubbles,24, 25 standing wrinkles,26-28 and so-called “molecular graphene”, i.e., assembled CO lattice on Cu(111) with strained graphene-like structure.29 Folding graphene can be an ideal way to alter and tailor the electronic properties of graphene by introducing curved edges.30, 31 Transport measurements show that the resistance of graphene along the folding direction can be modified.7 Recently, Kim et al. reported a controllable way to fabricate relatively wide doubly and multiply folded graphene.32 However, it is not known how the electronic properties of such kinds of wrinkle can be affected by the possibly existed strain. In the present work, we report our experimental measurements of the local electronic properties of a narrow wrinkle in monolayer graphene on a SiO2/Si substrate using scanning tunneling microscopy/spectroscopy (STM/STS). We observe differential conductance peaks in the dI/dV spectra obtained from the wrinkle. The peaks can be most likely attributed to


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PLLs because of the strain-induced pseudo-magnetic field in the wrinkle. The peaks are almost equally separated, quite different from the behavior of massless Dirac fermions in monolayer graphene. According to our observations, we propose a structural model of a doubly-folded graphene wrinkle (DFGW), in some extent similar to a trilayer graphene by considering its nearly parabolic band dispersion. The geometric and electronic structures of the DFGW are calculated using density functional theory (DFT) method. A possible deformation in the DFGW is suggested. In addition, nonequivalent sublattices are observed, which could be attributed to valley polarization in the wrinkle.

Figure 1. (a) Topographic STM image of a wrinkle in the transferred graphene on SiO2/Si (+1.0 V, 40 pA; 35.7 × 13.5 nm2). The arrows indicate defect sites in the wrinkle. (b) High resolution image of the marked region in (a) (+1.0 V, 40 pA; 11.1 × 8.3 nm2), and (c) magnified image of the wrinkle within the marked rectangle in (b) (2.6 × 5.2 nm2). The line profile across the wrinkle is superimposed in (b), showing its apparent height and width. (d) SEM image (1.5 × 1.5 µm2), indicating the existence of folded wrinkles with various widths.


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Figure 1a shows a topographic STM image of a structure in the graphene sample. This structure has an apparent height of ~6 Å and a width of ~3.0 nm (Figure 1b). The profile displays a slightly asymmetric feature, which is different from the symmetric profiles of the ripples20, 33 (Supporting Information Figure S1). It is particularly interesting since we find it has electronic properties quite different from other hierarchical ripples27,

33

(Supporting

Information Figure S1), as we will discuss below. The high resolution image (Figure 1c) shows that the lattice in the middle region of the structure displays non-uniform hexagonal structures, distorting from the honeycomb structure of monolayer graphene, similar to the nanobubbles.25 We here tentatively assume the structure as a wrinkle in the graphene layer. Such a distortion may be caused both by multiple-layer graphene stacking28 and the possibly introduced strain.25, 34 The existence of many defects in the wrinkle may be caused by fairly strong strain in the wrinkle. The folding edge tilts from the armchair direction by about 7° (Figure 1c). It is noticed that the wrinkle shows a curved structure, which is different from the flat feature in the much wider graphene wrinkles.7, 32 We further checked the graphene sample using scanning electron microscopy (SEM, Helios NanoLab650, acceleration voltage 1 kV, secondary electron mode) after the characterizations by STM. As shown in Figure 1d, the dark lines in the SEM image may correspond to folded wrinkles, similar to the assignment of previous observations in the transferred graphene on SiO2/Si.32 The weak and narrow lines with widths less than 10 nm thus do show the existence of narrow folded wrinkles in the sample.


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Figure 2. (a) dI/dV spectra recorded on the wrinkle (in red) and the graphene sheet (in black) (+1.0 V and 100 pA). The spectrum for the wrinkle is obtained by averaging 24 spectra recorded at different sites within the dashed rectangle in Figure 1b, and the spectrum for the graphene sheet is obtained by averaging 20 spectra from different sites away from the wrinkle at both sides by 2 nm. (b) Normalized (dI/dV)/(I/V) spectra for the wrinkle (in red) and the graphene sheet (in black), respectively. The spectra are vertically shifted for clarity. The PLL indexes N are indicated by assuming the neutrality point of the wrinkle close to the ED of the graphene sheet near the wrinkle. (c) Plot of EN for the PLLs as a function of sgn(N)|N|1/2, by considering the model of monolayer graphene. (d) Plot of EN for the PLLs as a function of sgn(N)[|N|(|N|+1)(|N|+2)]1/2, by considering the model of ABC-stacked trilayer graphene. The linear fit of the data is shown by the dashed red line.

Figure 2a shows the dI/dV spectra recorded at the wrinkle and the graphene sheet away from the wrinkle by about 2 nm. The dI/dV spectrum of the wrinkle is an averaged one over several tens of spectra recorded at the wrinkle sites within the dashed rectangle in Figure 1b. The gap-like feature near the Fermi energy (EF) in the spectrum of graphene sheet is similar to


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our previous observations,35 and can be similarly attributed to the mechanism of the phonon-mediated inelastic tunneling process.36 This feature is a signature of the monolayer graphene. It can be seen that there are oscillations in the spectrum obtained from the wrinkle, while the dI/dV spectrum from the graphene sheet shows an ordinary V-shaped feature with the Dirac point (ED) locating at +0.23 eV, as indicated by the arrow. The nearly equally separated oscillations in the dI/dV spectrum of the wrinkle can be seen more clearly after normalization by dividing its corresponding I/V spectrum, (dI/dV)/(I/V), as shown in Figure 2b and Supporting Information Figure S2. One may note that the STM images of the structure are quite similar to a carbon nanotube (CNT).37, 38 It has been shown that the CNTs can also exhibit equally separated peaks in dI/dV spectra, which has been well explained and assigned to the Coulomb staircases by modeling the suspended CNT as a quantum dot in double barrier tunneling junctions39 or the van Hove singularities because of the quasi-one-dimensional nature of the CNTs.40 However, in our experiment, the sample preparation conditions should not produce CNTs. To minimize the possibility of the structure from a CNT, we also tried to manipulate the structure, using manipulation conditions similar to those reported before.41 In the case of a CNT, it is much easily moved or bent. Our results, however, showed that the structure tends to be unmovable (Supporting Information Figure S3). Similar manipulations have also been performed for other ordinary wrinkles in the sample (Supporting Information Figure S4). It showed that the wrinkle at the end tends to have a deformation change, just as redistributions of ripples in graphene after manipulation. On the other hand, in the measurements of the suspended CNT, the gap between the STM tip and the CNT and the gap between the


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suspended CNT and the substrate formed the double barrier tunneling junctions.39 If we assume that the structure is a CNT, in our experimental setup, the presence of the CNT on a conducting graphene substrate is hard to form double barrier tunneling junctions to produce Coulomb staircases. Taking into consideration of our sample preparation conditions and the manipulation results, we believe that the structure is most likely a certain type of wrinkle in the graphene sheet rather than a CNT. It is noted that the possibility of the peaks due to the van Hove singularities of the quasi-one-dimensional electronic system, similar to the CNTs,40 could not be fully excluded. However, as we can see that despite their similar feature in the STM images the ordinary wrinkles do not show peaks in dI/dV spectra (Supporting Information Figure S1), we tend to believe that the wrinkle may not be a good one-dimensional electronic system to present van Hove singularities. Therefore, the observed oscillations in the dI/dV should have an origination different from the CNT. Considering that the oscillations appear together with the visible distortion and defects around the wrinkle, we attribute the oscillations to the PLLs caused by the strain-induced pseudo-magnetic field24-28, 34 in the wrinkle. Since the wrinkle is formed in the monolayer graphene, we first consider the possibility of massless Dirac fermions in the wrinkle. Figure 2c shows that the plot of the PLLs as a function of |N|1/2, where N is the PLL index. The PLLs obviously do not obey the linear relation of Dirac fermions.24-27, 34 Instead, as shown in Figure 2d, one can see that the PLLs meet the following relation, which is used to describe the massive fermions in ABC-stacked trilayer graphene,42

EN − E± = sgn( N )[(2ehvF2 )3/2 / γ 12 ]B3/2

N ( N + 1)( N + 2)


(1)

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where γ1 is the interlayer hopping energy in trilayer graphene, υF is the Fermi velocity of graphene, N (= 0, ±1, ±2, …) is the PLL index, signs “+” and “−” are applied to the electron-like and the hole-like carriers, respectively, e is the electron charge, and ћ is the reduced Plank constant. The pseudo-magnetic field strength (Bps) is about 42 T derived from the linear fitting. In the data analysis, we assume that the neutrality point is around the ED of the graphene sheet near the wrinkle, and the energy positions of the PLLs were chosen according to the local maxima in the dI/dV spectra. In the fitting, we adopt the parameters γ1 ≈ 0.4 eV and υF ≈ 1 × 106 m/s. However, as we will discuss below, due to the existence of curvature in the wrinkle, the interlayer coupling could be much smaller than the value in trilayer graphene,43, 44 and the Fermi velocity may also be slow down by a certain small percentage.45 The value of the pseudo-magnetic field strength in the wrinkle may be somewhat overestimated, but it may not affect the main topic of this study.

Figure 3. (a) Normalized (dI/dV)/(I/V) spectra, (b) STM image (10.0 × 9.3 nm2) showing the


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correspondingly labeled sites for acquisition of the spectra (+1.0 V and 100 pA). The spectra are vertically shifted for clarity. Spectra i-iii are from the graphene sheet for comparison. (c) Plot of PLL peak energies of every spectrum as a function of sgn(N)[|N|(|N|+1)(|N|+2)]1/2. The dashed line is guide for eyes only. (d) Plot of pseudo-magnetic field strengths, Bps, against the relative site positions for the square and circle marked sites, correspondingly as labeled in (c). Sites 1, 2, 4, 6 and 7 are acquired across the wrinkle, and other sites are along the wrinkle. The values of Bps are obtained from the linear fitting slopes of each set of (EN − ED) ~ sgn(N)[|N|(|N|+1)(|N|+2)]1/2 data in (c).

Due to the existence of defects around the wrinkle, we further examine the effect of these defects on the distribution of the pseudo-magnetic field, as shown in Figure 3. It is seen that the normalized (dI/dV)/(I/V) spectra taken at the labeled sites 1-17 show nearly equidistant peaks, although the peak positions vary in a certain degree in individual spectra (Figure 3a,b). In comparison, there is no such feature in the spectra from the graphene sheet at sites i-iii. The spectra recorded at defect sites (13 and 16) also show some peaks, but the features are quite different from the spectra obtained at the wrinkle sites in both sides away from the defects. We attribute the peaks in spectra 13 and 16 to the localized defect states since these states just appear at the defect sites and do not obviously affect the electronic properties of the wrinkle, judging from the spectra obtained at sites away from the defects by about 1 nm. The observed range affected by the defects is consistent with the reported ranges for some well-defined defects in graphene sheets,35, 46, 47 where the states of defects tend to be localized. Hence, the contributions of the existing defects to the states for the sites away from the defects over about 1 nm should be negligible. Similar to the analysis shown in Figure 2, the PLL peak


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positions of each spectrum are plotted as a function of sgn(N)[|N|(|N|+1)(|N|+2)]1/2 in Figure 3c. By fitting the slopes of each set of data, we obtain the site-dependent Bps and plot the values against the relative site positions in Figure 3d (Supporting Information Figure S5). At some sites, the peaks only obviously present at positive voltages, and we just used these peak positions for fitting. The fields both along and across the wrinkle fluctuate in a certain range, from about 37 to 47 T, showing a site dependence on the relative distance away from the defects in both sides. Along the wrinkle, in the middle region and at the sites (e.g., 11, 12, and 17) close to the defects, the pseudo-magnetic fields are smaller than that in the transition regions (e.g., sites 5 and 10). This could be due to the smaller gradients in the middle region and the sites very close to the defects, as the strength of the pseudo-magnetic field is proportional to the gradient of the strain. In the middle region, relatively small gradients can be expected because of the large distances away from the defects, while in the sites very close to the defects, the small gradients could be caused by releasing the strain near the defects.48 For the pseudo-magnetic fields across the wrinkle, the fluctuation of the fields is relatively small. The information from the measurements of the site-dependent pseudo-magnetic fields strongly support that the peaks could be reasonably ascribed to the PLLs.


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Figure 4. (a) Side view and (b) top view of the structural model of the DFGW used in our DFT calculations. Only the z coordinates of the atoms are fixed in the monolayer graphene regions on both sides, as labeled by double arrows in (a). The supercell is marked by the red rectangle in (b). (c) Left panel: Simulated STM image of the DFGW by integrating the states from 0 to 1.0 eV (4.3 × 4.3 nm2). Right panel: Experimental STM image of the wrinkle (+1.0 V, 40 pA, 4.3 × 4.3 nm2). Note that a different wrinkle orientation is used in the simulations. (d) The LDOS of carbon atoms at the center of the topmost layer of the DFGW (in red) and the monolayer graphene sheet (2 nm away from the folding edge, in black). (e) Calculated band structures of the DFGW. (f) Typical charge density distributions of the up-dispersed band (upper panel) and the down-dispersed band (lower panel) of the DFGW, with the isosurfaces of 0.0006 and 0.0002 e/Å3, respectively.

On the basis of the results and discussion above, especially the good accordance of the


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peak positions with those in trilayer graphene, a doubly-folded graphene wrinkle, which has a structure of curved triple layers, as shown in Figure 4a, could be a reasonable model to describe the wrinkle. We carried out DFT calculations to verify the DFGW structure using a plane-wave basis set. Due to the computational limitation in the calculations, the folding edge is modeled along the armchair direction (Figure 4b), somewhat different from the experimentally observed tilting angle of 7° from the armchair direction. The asymmetric feature of the wrinkle is simulated by different curvature radiuses at left and right folding edges. In the structural optimization, only the z coordinates of partial monolayer graphene (among regions labeled with arrows in Figure 4a) between neighboring wrinkles are fixed, and all other atoms are relaxed until the force on each atom is less than 0.01 eV/Å. The width of the DFGW is about 3 nm, which is consistent with the experiment measurement in Figure 1b. The apparent height is about 12 Å, which is larger than the height of 6 Å from above STM image, but consistent with the height of ∼10 Å for DFGWs from atomic force microscopy measurements.7, 32 In an STM image, the apparent height of a structure is mainly caused by the difference in density of states (DOS), and may deviate from its actual height. Figure 4c (left panel) shows the simulated STM image of the DFGW at +1.0 V under constant current mode plotted with the charge density of 0.5 e/Å3. It can be seen that the simulated image is in good agreement with the measured STM image (Figure 4c, right panel). In this comparison, one may see that in both images the contrast at the left folded edge changes slowly, but the contrast at the right folded edge changes much sharply. Such an agreement between the experimental and the calculated results can be another signature to support our proposed model.


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To examine the reason of the oscillations in the dI/dV spectra of the wrinkle, the local DOSs (LDOSs) of the DFGW are calculated, as shown in Figure 4d. No obvious peak feature can be seen in the LDOSs, and there is no obvious difference in the LDOSs between the carbon atoms in the topmost layer of the DFGW and in the graphene sheet (2 nm away from the folding edge). Considering the fact that in our DFT calculations the geometric structure of the narrow DFGW is relaxed, we may exclude that the oscillations in the dI/dV spectra of the wrinkle are resulted from the electronic structure of trilayer graphene. This is consistent with the dI/dV spectra in Figure S1c, in which no PLLs are detected for two wrinkles with more homogeneous honeycomb structure. Therefore, there should exist a gradient strain that causes the observed PLLs in the DFGW. The observed defects in the DFGW may reflect the existence of the strain. The calculated band structure shows a parabolic dispersion in the relaxed DFGW (Figure 4e), which is similar to the parabolic dispersion in ABC-stacked trilayer graphene,42 but completely different from the linear dispersion in monolayer graphene. The charge density distributions of the up-dispersed and the down-dispersed parabolic bands are shown in Figure 4f. It is clearly seen that the states mainly distribute within the wrinkled trilayer region. The nearly equally separated PLLs observed in our experiment can thus be explained by the parabolic dispersion of the bands in the DFGW. In our DFT calculations, an infinite length of the DFGW with perfect carbon lattice is considered. Since the observed defect states are mainly localized within the sites away from the defects by about 1 nm, the DFGW model with the infinite length should be used to describe the middle region of the wrinkle observed in our experiment.


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Figure 5. (a) Comparison of the geometric structures before (in brown) and after (in cyan) introducing a deformation, which leads to a gradient strain and yields a pseudo-magnetic field of 42 T. Only the atoms in the topmost layer of the DFGW are shown. (b) Plot of the displacements of ux and uy along x and y directions, respectively, between the carbon atoms (in brown and cyan before and after deformation) in the upper end of the DFGW.

There are different ways to introduce a pseudo-magnetic field in graphene.21, 49, 50 We similarly suggest that the pseudo-magnetic field is caused by a gradient strain in the DFGW. The observed defects can be a signature of the existing strain, where the formation of the defects is to release the strain in the DFGW. The gauge field or vector potential A in graphene can be written as21, 22

β   Ax = ± a (ε xx − ε yy )   A = m2 β ε xy  y a

(2)

where the x-axis is chosen along the zigzag direction of the graphene lattice, and a ≈ 1.4 Å is the C-C bond length in graphene, β = −∂ ln t / ∂ ln a ≈ 2 , while t ≈ 3.0 eV is the electron hopping energy between pz orbitals located at nearest neighbor atoms. The strain tensor (εxx,

εyy, εxy) associates to the deformation of the graphene layer and can be given by 15 Environment ACS Paragon Plus

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 ∂u ε xx ≈ x ∂x   ∂u y ε yy ≈ ∂y   1  ∂u ∂u  ε xy ≈  x + y  2  ∂y ∂x  

(3)

where ux and uy are the displacements along x and y directions. The strain-induced pseudo-magnetic field (given in unit h/e ≡ 1 ) perpendicular to the graphene plane is

Bps =

∂Ay ∂x

−

∂Ax β  ∂ε xy ∂ε xx ∂ε yy = − 2 + − a  ∂x ∂y ∂y ∂y

2 2  β  ∂ 2u x ∂ u y ∂ u y = − 2 + −   a  ∂x∂y ∂x 2 ∂y 2 

  

(4)

The pseudo-magnetic field is determined by the gradient of strain. Here we simulate a possible deformation with the displacements along x direction (µx) and y direction (µy) described by xy   µ x ( x, y ) = R  2 2  µ ( x, y ) = − 4 x + y y  2R

(5)

The corresponding strain-induced pseudo-magnetic field is

Bps =

β aR

(6)

The parameter R determines the gradient of the strain and the pseudo-magnetic field Bps. Figure 5a compares the simulated geometric structures of the topmost layer in the DFGW before and after the deformation at a yielding pseudo-magnetic field of 42 T, corresponding to our experimental observations. We here consider a model of 26 carbon atoms in x direction and 30 zigzag rows in y direction for the DFGW in the topmost layer, where the displacement


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is defined as zero between the two carbon atoms (at the lower-left corner in Figure 5a) before and after deformation. Obvious displacements can be observed in the upper part of the wrinkle, while the honeycomb structure is still kept after the deformation. Figure 5b shows the displacements along x and y directions for the carbon atoms in the uppermost zigzag row. For the carbon atom of 26 (at the upper-right corner), its displacements reach to ux = 0.67 Å and uy = −1.39 Å. At such a large distortion, the C-C bond in a graphene may be broken, which may explain self-consistently the observed defects in the DFGW. 21, 51


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Figure 6. (a) (1×10) unit cell for pristine (left) and strained (right) trilayer graphene. The nominal strain ε eff = ∆L / L , where L is the length of the unit cell along y direction, and ∆L is the length change after introduction of the strain. (b) DFT calculated spectra of the trilayer graphene under various strains. Each spectrum is averaged over 4 atoms in the sixth carbon rings in the topmost layer, as marked by the red atoms in the dashed square in (a). For all of the calculated spectra, a Gaussian broadening of 25 meV is applied. (c) Enlarged plot of the


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calculated spectrum for ε eff = 2% . Note that ε eff = 1% corresponds to a pseudo-magnetic field of about 44 T, and ε eff = 2% to 88 T. (d) Peak positions (with respect to ED) against sgn(N)[|N|(|N|+1)(|N|+2)]1/2, extracted from the spectra in (b). For a more direct confirmation, we performed DFT calculations by including the strains in the DFGW, unlike the calculations for pseudo-magnetic field in graphene using cluster models on the basis of tight-binding (TB) approach.21, 51 We here consider the displacement of atoms only along the armchair direction in the wrinkle region, that is, u x ( x, y ) ≡ 0 , and

u y ( x, y ) =

y2 , so that we still have the relation for the pseudo-magnetic field as given in eq 6. 2R

Such a selection of the gradient strain allows us to implement the DFT calculations with a large enough k-points grid using a slab model of a (1×10) unit cell, where the ABC-stacked trilayer graphene is considered, as shown in Figure 6a. Our calculations show that the DOSs are very sensitive to the k-points grids. A large enough number of k points is necessary for a highly accurate result, but this necessity limits our selection of a larger unit cell. We first tested the DOSs in single layer graphene by introducing the strains. We find that the method used here can well describe the PLLs in single layer graphene (Supporting Information Figure S6), which gives the results in consistent with those obtained using the TB approach.21, 51 Figure 6b shows the calculated spectra for trilayer graphene with various ε eff . In comparison with the unstrained trilayer graphene, additional peaks occur and become more and more pronounced with the increase of the strain. The spectrum with ε eff = 2% is enlarged in Figure 6c for showing the less pronounced peak features. The peak positions with respect to ED (zero energy) are plotted according to the relation (EN−ED) ~ sgn(N)[|N|(|N|+1)(|N|+2)]1/2


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in Figure 6d, which can approximately describe the linear relation. Therefore, our results based on the DFT calculations may be qualitatively used to describe the PLLs in the ABC-stacked trilayer graphene. However, in comparison with the experimental results, the energy positions (or peak separations) from calculations are much smaller than that from the experimental observations. This can be rationalized by considering the difference in effective mass of electrons between the ABC-stacked graphene and the DFGW. As we will show below (see Figure 7), the measured effective mass in the DFGW is much smaller than that in the ABC-stacked graphene. A smaller effective mass of electrons (or holes) indicates a smaller electron (and hole) pocket, which will result in a larger separation between the Landau levels. The curved structure in the DFGW could be one reason to cause such the difference. A more accurate calculation is demanded to describe the DFGW not only including the strain but also including the curved structure.


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Figure 7. (a) and (b) dI/dV maps, acquired at bias voltages of +0.28 and +0.91 V, respectively, within the same area (10.0 × 9.9 nm2). The inset in (b) shows the topographic image obtained simultaneously during acquisition of the dI/dV map (+0.91 V and 100 pA). (c) Schematic drawing of the wavefunction and probability density of eigenstates with j = 1, 2. (d) The magnified map of the marked rectangle in (a) with superposed sublattices A and B. (e) Histogram of the differential conductance dI/dV of the map in (d). In the histogram, the fitting curves indicate three distinct peaks, labeled as A, B and hol, corresponding to the conductance at sublattices A and B, and the hollow site, respectively.

The presence of defects in the wrinkle also helps to estimate the effective mass of the carriers by measuring the standing wave using a one-dimensional particle-in-a-box model. We measured the dI/dV maps at various bias voltages. Figure 7a,b show the dI/dV maps obtained at +0.28 and +0.91 V, respectively. One dim node (conductance minimum in the spatial


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distribution, as indicated by the line profile along the wrinkle) can be seen in the middle region of the wrinkle in Figure 7b. If we simply treat the region between the defects as a one-dimensional infinite quantum well, we have Ej = (π2ћ2j2)/(2meffl2), where j is the quantum number ( j = 1, 2, 3, …), and l ≈ 9 nm is the distance between the defects. The existing defects mainly perform as barriers to confine electrons within the segment. The pattern with one node can thus be attributed to the eigenstate of j = 2 (Figure 7c). We thus obtain the effective mass of the carriers, meff ~ 0.02me, where me is the rest mass of electron. This effective mass is smaller, actually about half of the value than that in stacked trilayer graphene52 or graphite,53 which may also indicate their distinct difference in electronic properties between the DFGW and stacked trilayer graphene or graphite. Figure 7d shows a magnified map (from the marked rectangle in Figure 7a) with superimposed sublattices A and B (in grey and red dots). It shows that the sublattices A and B give nonequivalent conductance. The histogram of the pattern is plotted in Figure 7e. The fitting peaks of A and B well reflect the distributions of nonequivalent conductance from sublattices A and B, while the lowest peak (“hol”) is the conductance distribution for the hollow sites. Since the map was acquired at 0.28 eV, close to the ED of the graphene sheet, the variation of the conductance intensity in the map may reflect the change of the energy of ED. Therefore, the nonequivalent conductance of sublattices A and B in the map can be ascribed to the lift of the degeneracy of the two sublattices. This result suggests that there exists local valley polarization54, 55 in the wrinkle. The valley polarization in graphene can be induced by the atomic spatial potential variation56 or by the strain in a curved structure.28 Hence, the existence of the valley polarization is consistent with our suggestion about the introduction of


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strain in the wrinkle. Moreover, since the valley polarization spontaneously breaks the time-reversal symmetry in graphene,5, 16 such a narrow DFGW can provide a platform to realize zero-field quantum valley Hall effect.57 In summary, we characterize the structure and its electronic properties of a narrow graphene wrinkle using STM/STS. From both of the topography and the electronic behaviors, we suggest a doubly-folded structure for this narrow wrinkle, which is well supported by our DFT

calculations

and

image

simulations.

We

observe

pronounced

and

nearly

equally-separated conductance peaks in the dI/dV spectra acquired at the wrinkle sites. Our DFT calculations also show that the dispersion of the DFGW band structure is parabolic, similar to the parabolic dispersion in ABC-stacked trilayer graphene, but different from the linear dispersion in monolayer graphene. We attribute the peaks to the PLLs that are caused by the strain-induced pseudo-magnetic field, where the equally-separated peaks can be explained by the parabolic dispersion of the band structure in the DFGW. By adopting the trilayer graphene model, we thus derive an average value of 42 T for the magnetic field in the wrinkle, which can be induced by the gradient strain that is originating from the lattice deformations. Our analysis also shows that at the induced field of 42 T in the DFGW the distortion of lattice may even produce defect, which is self-consistent with our observation of the defects in the DFGW. The effective mass of the carriers, meff, is estimated to be about 0.02me in the DFGW, where valley polarization is also observed. Our results shed light that such a narrow DFGW can be a promising platform to explore the novel electronic properties of strained graphene.

Methods. Sample preparation. The monolayer graphene was prepared on the 25 µm copper


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polycrystalline foils with the chemical vapor deposition (CVD) method and then transferred onto a 300 nm SiO2/Si substrate.46 The graphene sample was mounted on the STM sample holder with a good electrical contact. The details of the sample preparation were reported elsewhere.35, 58 The sample was annealed at 600 K with a background pressure better than 10−10 mbar for about 30 h. STM/STS characterizations. Our experiments were conducted using an ultrahigh-vacuum low-temperature STM (UNISOKU Co., Ltd.), performed at 80 K under pressure better than 3 × 10−11 mbar. The tungsten tip was chemically etched and well cleaned before use. All the images were taken in a constant-current scanning mode. The current-voltage (I-V) and the differential conductance (dI/dV) spectra were measured by turning off the feedback loop-gain. The dI/dV spectra were obtained using a lock-in amplifier with a typical sinusoidal modulation of 10 meV (rms) at 1000 Hz. The polarity of the applied voltage refers to the sample bias with respect to the tip. DFT calculations. DFT calculations were performed using a plane-wave basis set, as implemented in the Vienna ab initio simulation package.59 Exchange–correlation interactions are described by the Perdew–Burke–Ernzerhof60 generalized gradient approximation. The projector augmented wave method61 is used to represent the electron–ion interactions. The DFGW structure is modeled in an orthorhombic unit cell (a = 7.43 nm, b = 0.426 nm, c = 2.0 nm). The Brillouin-zone integration is calculated using 6 × 18 × 3 k-point grid, through the gama-centered method. The plane wave energy cutoff is set as 400 eV. The STM image is simulated based on Tersoff and Hamann’s formula62 under constant current mode, plotted with the charge density of 0.5 e/Å3.


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In the calculations of the strained single layer and trilayer graphenes, we used a different slab model with unit cell of (1×10). The Brillouin-zone integration is calculated using a large k-point grid of 48 × 48 × 1, through the gama-centered method. The separation between the layers in ABC-stacked trilayer graphene is 3.35 Å.

Supporting Information The Supporting Information is available free of charge on the ACS Publications website. Images and the dI/dV spectra of other two wrinkles; more (dI/dV)/(I/V) curves of the folded wrinkle; STM tip manipulations on graphene wrinkles; details of the linear fitting of the PLLs; DFT calculations for single layer graphene with various strains (PDF).

Corresponding author [email protected] (B.W.)

ORCID Chuanxu Ma: 0000-0001-6478-5917 Xia Sun: 0000-0002-6556-5200 Aidi Zhao: 0000-0002-6546-4610 Bing Wang: 0000-0002-2953-2196

Author Contributions †

Chuanxu Ma and Xia Sun contributed to this work equally.

Notes The authors declare no competing financial interest. Acknowledgements This work was financially supported by the Ministry of Science and Technology of China (grants 2016YFA0200603), and the National Natural Science


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Foundation of China (grants 91421313, 21421063, 11674297), and Anhui Initiative in Quantum Information Technologies (AHY090300).

List of abbreviations PLL: pseudo-Landau level; DFGW: doubly-folded graphene wrinkle; STM/STS: scanning tunneling microscopy/spectroscopy; DFT: density functional theory; SEM: scanning electron microscopy; EF: Fermi energy; ED: Dirac point ; Bps: pseudo-magnetic field strength; DOS: density of states; LDOS: local density of states; µx: displacement along x direction; µy: displacement along y direction; CVD: chemical vapor deposition; I-V: current-voltage

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Figure 1. (a) Topographic STM image of a wrinkle in the transferred graphene on SiO2/Si (+1.0 V, 40 pA; 35.7 × 13.5 nm2). The arrows indicate defect sites in the wrinkle. (b) High resolution image of the marked region in (a) (+1.0 V, 40 pA; 11.1 × 8.3 nm2), and (c) magnified image of the wrinkle within the marked rectangle in (b) (2.6 × 5.2 nm2). The line profile across the wrinkle is superimposed in (b), showing its apparent height and width. (d) SEM image (1.5 × 1.5 µm2), indicating the existence of folded wrinkles with various widths. 72x50mm (300 x 300 DPI)

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Figure 2. (a) dI/dV spectra recorded on the wrinkle (in red) and the graphene sheet (in black) (+1.0 V and 100 pA). The spectrum for the wrinkle is obtained by averaging 24 spectra recorded at different sites within the dashed rectangle in Figure 1b, and the spectrum for the graphene sheet is obtained by averaging 20 spectra from different sites away from the wrinkle at both sides by 2 nm. (b) Normalized (dI/dV)/(I/V) spectra for the wrinkle (in red) and the graphene sheet (in black), respectively. The spectra are vertically shifted for clarity. The PLL indexes N are indicated by assuming the neutrality point of the wrinkle close to the ED of the graphene sheet near the wrinkle. (c) Plot of EN for the PLLs as a function of sgn(N)|N|1/2, by considering the model of monolayer graphene. (d) Plot of EN for the PLLs as a function of sgn(N)[|N|(|N|+1)(|N|+2)]1/2, by considering the model of ABC-stacked trilayer graphene. The linear fit of the data is shown by the dashed red line. 73x63mm (300 x 300 DPI)


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Figure 3. (a) Normalized (dI/dV)/(I/V) spectra, (b) STM image (10.0 × 9.3 nm2) showing the correspondingly labeled sites for acquisition of the spectra (+1.0 V and 100 pA). The spectra are vertically shifted for clarity. Spectra i-iii are from the graphene sheet for comparison. (c) Plot of PLL peak energies of every spectrum as a function of sgn(N)[|N|(|N|+1)(|N|+2)]1/2. The dashed line is guide for eyes only. (d) Plot of pseudo-magnetic field strengths, Bps, against the relative site positions for the square and circle marked sites, correspondingly as labeled in (c). Sites 1, 2, 4, 6 and 7 are acquired across the wrinkle, and other sites are along the wrinkle. The values of Bps are obtained from the linear fitting slopes of each set of EN − sgn(N)[|N|(|N|+1)(|N|+2)]1/2 data in (c). 97x111mm (300 x 300 DPI)


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Figure 4. (a) Side view and (b) top view of the structural model of the DFGW used in our DFT calculations. Only the z coordinates of the atoms are fixed in the monolayer graphene regions on both sides, as labeled by double arrows in (a). The supercell is marked by the red rectangle in (b). (c) Left panel: Simulated STM image of the DFGW by integrating the states from 0 to 1.0 eV (4.3 × 4.3 nm2). Right panel: Experimental STM image of the wrinkle (+1.0 V, 40 pA, 4.3 × 4.3 nm2). Note that a different wrinkle orientation is used in the simulations. (d) The LDOS of carbon atoms at the center of the topmost layer of the DFGW (in red) and the monolayer graphene sheet (2 nm away from the folding edge, in black). (e) Calculated band structures of the DFGW. (f) Typical charge density distributions of the up-dispersed band (upper panel) and the downdispersed band (lower panel) of the DFGW, with the isosurfaces of 0.0006 and 0.0002 e/Å3, respectively. 99x85mm (300 x 300 DPI)


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Figure 5. (a) Comparison of the geometric structures before (in brown) and after (in cyan) introducing a deformation, which leads to a gradient strain and yields a pseudo-magnetic field of 42 T. Only the atoms in the topmost layer of the DFGW are shown. (b) Plot of the displacements of ux and uy along x and y directions, respectively, between the carbon atoms (in brown and cyan before and after deformation) in the upper end of the DFGW. 64x50mm (300 x 300 DPI)


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Figure 6. (a) (1×10) unit cell for pristine (left) and strained (right) trilayer graphene. The nominal strain εeff = ∆L/L, where L is the length of the unit cell along y direction, and ∆L is the length change after introduction of the strain. (b) DFT calculated spectra of the trilayer graphene under various strains. Each spectrum is averaged over 4 atoms in the sixth carbon rings in the topmost layer, as marked by the red atoms in the dashed square in (a). For all of the calculated spectra, a Gaussian broadening of 25 meV is applied. (c) Enlarged plot of the calculated spectrum for εeff = 2%. Note that εeff = 1% corresponds to a pseudomagnetic field of about 44 T, and εeff = 2% to 88 T. (d) Peak positions (with respect to ED) against sgn(N)[|N|(|N|+1)(|N|+2)]1/2, extracted from the spectra in (b). 169x179mm (300 x 300 DPI)


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Figure 7. (a) and (b) dI/dV maps, acquired at bias voltages of +0.28 and +0.91 V, respectively, within the same area (10.0 × 9.9 nm2). The inset in (b) shows the topographic image obtained simultaneously during acquisition of the dI/dV map (+0.91 V and 100 pA). (c) Schematic drawing of the wavefunction and probability density of eigenstates with j = 1, 2. (d) The magnified map of the marked rectangle in (a) with superposed sublattices A and B. (e) Histogram of the differential conductance dI/dV of the map in (d). In the histogram, the fitting curves indicate three distinct peaks, labeled as A, B and hol, corresponding to the conductance at sublattices A and B, and the hollow site, respectively. 101x122mm (300 x 300 DPI)


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For TOC only 35x15mm (600 x 600 DPI)


Landau Quantization of a Narrow Doubly-Folded Wrinkle in Monolayer

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